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High momentum diffractive processes and hadronic structure
Progress in Particle and Nuclear Physics 56 (2006) 279–339 www.elsevier.com/locate/ppnp
Review
High momentum diffractive processes and hadronic structure Daniel Ashery School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Israel Received 14 April 2005
Abstract High momentum diffractive processes can be used to study the internal structure of hadrons. This structure is described using the light-cone formalism and the concept of light-cone wave functions (LCWFs). These exclusive observables are related to the more inclusive ones, the structure functions and form factors. Experimental methods to measure the LCWFs through measurements of form factors have proven rather insensitive and alternative approaches are necessary. Measurements of LCWFs using diffractive dissociation are differential and determine the momentum distributions of the valence partons in the hadron. They provide wave function information at the amplitude level. The concept of these measurements relies on the factorization of soft and hard processes. Recent measurements of the pion and the photon LCWFs by diffractive dissociation are described. The measurement of the pion LCWF has been extensively and critically discussed by several authors. These discussions are summarized and conclusions are drawn. These experiments allow observation of the transition from perturbative to non-perturbative QCD regimes. This transition is related to the soft components of the LCWF and to chiral symmetry breaking. The phenomenon of color transparency (CT) is an important component of the experimental approach and is closely related to the LCWF description of the hadronic structure. Observation of this effect in diffractive dissociation of pions interacting with nuclear targets is described and compared with other studies of CT. We finally discuss several future directions in this field. c 2006 Published by Elsevier B.V. Keywords: Hadronic structure; Light-cone wave functions
Contents 1.
Introduction......................................................................................................................280 E-mail address: [email protected].
c 2006 Published by Elsevier B.V. 0146-6410/$ - see front matter doi:10.1016/j.ppnp.2005.08.003
280
2.
3.
4.
5.
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1.1. Hadronic structure................................................................................................... 280 1.2. Applications of the LCWF formalism ........................................................................ 282 1.3. Factorization and hard processes ............................................................................... 284 1.4. Diffraction ............................................................................................................. 285 Color transparency ............................................................................................................ 286 2.1. Quasi-elastic scattering ............................................................................................ 287 2.2. Vector meson production ......................................................................................... 291 2.3. Pion diffractive dissociation to dijets ......................................................................... 297 The pion light-cone wave function ...................................................................................... 302 3.1. Theoretical predictions ............................................................................................ 302 3.2. Distribution amplitudes and form factors ................................................................... 303 3.3. Measurement of the pion LCWF by diffractive dissociation to dijets ............................. 307 3.3.1. Concept of the experiment ........................................................................... 307 3.3.2. Longitudinal momentum distribution ............................................................ 309 3.3.3. Coefficients of the Gegenbauer polynomials .................................................. 311 3.3.4. Transverse momentum distribution ............................................................... 312 3.3.5. Has E791 measured the pion distribution amplitude? ...................................... 314 The photon light-cone wave function ................................................................................... 316 4.1. Predictions of the photon LCWF............................................................................... 316 4.2. Cross section of coherent dijet electroproduction ........................................................ 319 4.2.1. Cross section derived from the LCWF........................................................... 321 4.3. Cross section of coherent photoproduction of dimuons derived from the LCWF ............. 325 4.4. Cross section for dipion production ........................................................................... 326 4.5. The Odderon .......................................................................................................... 327 4.6. Measurement of the electromagnetic component of real photon LCWF ......................... 328 4.6.1. Data analysis.............................................................................................. 328 4.6.2. Results and discussion................................................................................. 329 4.7. Measurement of the hadronic component of the photon LCWF by exclusive dipion electroproduction .................................................................................................... 329 4.7.1. Data analysis.............................................................................................. 330 4.7.2. Results ...................................................................................................... 331 4.8. Measurement of the hadronic component of the photon LCWF by exclusive dijet electroproduction .................................................................................................... 333 Summary and future directions ........................................................................................... 334 Acknowledgements ........................................................................................................... 336 References ....................................................................................................................... 336
1. Introduction 1.1. Hadronic structure The most challenging nonperturbative problem in Quantum Chromodynamics is to determine the structure of hadrons in terms of their quark and gluon degrees of freedom. This understanding is a very important infrastructure in the interpretation of many processes. As is the case for atoms and nuclei, the structure should be described through wave functions that are, in turn, solutions of the appropriate Hamiltonian equation. The wave function should provide the probability amplitude to find the parton in a given position in coordinate space or a given momentum in momentum space. The differences between the Hamiltonian of an atomic or nuclear system and that of a hadron stem from three main reasons:
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• The interaction between quarks and gluons is not well understood. • The number of partons is, in principle, unbounded while the number of electrons or nucleons is finite. • It should be fully relativistic. Several approaches have been developed to tackle these problems, such as using equaltime quantization motivated by non-relativistic potentials or applying the Bethe–Salpeter formalism [1]. However, these calculations have proven to be extremely complex as they require understanding of the solutions over all space and time and boosting which is by itself very hard to do. Hence, their applicability is limited. A powerful method for describing the bound state structure of relativistic composite systems in quantum field theory is the light-cone quantization and Fock expansion [2]. The light-cone QCD wave functions are constructed from the QCD light-cone Hamiltonian: HLC = P + P − − P⊥2 where P ± = P 0 ± P z with P µ the momentum operators [3]. The LCWF ψh for a hadron h QCD with mass Mh satisfies the relation: HLC |ψh = Mh2 |ψh . It has been shown [2] that both the Hamiltonian and all amplitudes are boost, rotational invariant and independent of P µ . These features make these wave functions frame independent. Experimental measurements of LCWFs can therefore provide very important information on the fundamental hadronic interactions described by the Hamiltonian. The LCWFs are expanded in terms of a complete basis of Fock states having increasing complexity [3]. For example, the negative pion has the Fock expansion: |ψπ − = n | π − |n n
) = ψd(Λ ¯ u/π ¯ (u i , k ⊥i, λi )|ud (Λ) (u i , k⊥i, λi )|udg ¯ + ··· +ψ d ug/π ¯
(1)
with longitudinal light-cone momentum fractions: ui =
ki+ ki0 + kiz = , p+ p0 + p z
n
u i = 1,
(2)
i=1
relative transverse momenta k⊥i ,
n
k⊥i = 0 ⊥
(3)
i=1
and spin projections λi . The index i runs over the partons contained in the relevant Fock state. The light-cone momentum fractions of the constituents u i and the transverse momenta k⊥i appear as the momentum coordinates of the LCWF. The light-cone wave function for each quark–gluon combination n: ψn/H (u i , k⊥i , λi ) interpolates between the hadron H and its quark and gluon degrees of freedom. The first term in the expansion is referred to as the valence Fock state, as it relates to the hadronic description in the constituent quark model. This state has a minimal gluon content, must have a small size and hence a large mass. This is further discussed in Section 2. The higher terms are related to the sea components of the hadronic structure, are larger and lighter. It has been shown that the valence Fock state may account for about 25% of the total LCWF and once determined, it is possible to build the rest of the light-cone wave function [4,5].
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The essential part of the wave function is the hadronic distribution amplitude φ(u i , Q 2 ). It describes the probability amplitude to find a q q¯ or a qqq system of the respective lowest-order Fock state carrying fractional momenta u i with ni=1 u i = 1. It is given by an integral of the LCWF over the non-perturbative low-mass region up to a scale Q 2 [6]: 3 kt2 2 2 2 ψ(u i , kti )δ k ti Θ Q − (4) φ(u i , Q ) ∼ d2 k ti . u(1 − u) 0 i=1 i An evolution equation for φ(u i , Q 2 ) from one value of Q 2 to another is given in [6]. 1.2. Applications of the LCWF formalism The structure of LCWFs has been predicted under certain conditions for a variety of hadrons. Some of these predictions and their applications will be discussed in Sections 3 and 4. Here we briefly mention several applications as examples of the large variety of processes where this formalism can be applied. We will point the interested reader to relevant references. 1. DLCQ All the applications need some sort of LCWF. The most promising method for calculating the LCWF of hadrons from the LC hamiltonian is the so-called Discretized Light-Cone Quantization (DLCQ). This is a very difficult and challenging project given the unbounded number of constituents. The method was developed [7] in (1+1) dimensions but is being extended to (3+1) dimensions. Basically it involves discretizing the LC momenta u and k⊥ and applying a cut-off on the invariant mass of the partons in the Fock expansion. This leads to diagonalization of finite matrices and may be simplified by using known physical constraints. Example of applications of (3 + 1) DLCQ include LCWFs and spectrum in the Yukawa theory [8] and the electron magnetic moment [9]. 2. Nucleon structure Given model LCWF, various properties of the nucleon have been calculated. These include the magnetic moments of the proton and the neutron and the axial vector coupling constant measured in neutron β decay [2]. Calculations of the electromagnetic form factor of baryons have been performed already in one of the early works [6]. Parton distribution functions can be calculated from LCWFs. Since they represent the probability to find any parton with fractional momentum x B j they can be obtained by integrating the LCWF over all Fock states and all variables keeping a given fractional momentum u = x B j [2]: 2 (Q) δ(x B j − u i ) (5) d[µn ] ψn/ h (u i , k⊥i , λi ) G a/ h (x B j , Q) = n
where
i
du i d2 k⊥i . . . , d[µn ] . . . = λi ∈n
2
du i d k⊥i
= δ 1−
Nn j =1
uj δ
(2)
Nn j =1
with k⊥ j
du 1 . . . du Nn d2 k⊥1 . . . d2 k⊥Nn .
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The parton distribution functions are usually identified with structure functions F2 (x, Q 2 ) measured in Deep Inelastic Scattering (DIS). It has been shown, however [10] that gluon exchange between the fast, outgoing partons and target spectators causes destructive interference of diffractive channels and thus affects the leading twist structure functions. Use of Eq. (5) can be extended by demanding certain flavor or certain helicity. It is then possible to reproduce measurements of the proton spin structure with an LCWF containing qqq and qqqg Fock states. The process of Deeply Virtual Compton Scattering (DVCS) from a proton is very sensitive to the quark structure of the proton. The photon scatters off a quark thus modifying its momentum as well as the momenta of the spectators. The amplitude for the process is then calculated from the overlap of the initial and the modified LCWFs [11]. The process of high momentum transfer real photon Compton scattering on nucleons, γ N → γ N was calculated [12] using distribution amplitudes that were calculated from QCD sum rules. For some distributions the calculations are able to reproduce experimental results for unpolarized photons and make predictions for several polarized photon scatterings. 3. Mesons Many properties of the π meson were calculated in the seminal work of Brodsky and Lepage [6]. Some of these will be discussed in greater detail in Section 3. The LCWF for the lowest (valence) Fock state is normalized using the measured pion decay constant f π ≈ 93 MeV: fπ du d2k⊥ (Λ) (6) ψd u¯ (u, k⊥ ) = √ . 16π 3 2 3 It is then used to calculate the electromagnetic form factor assuming several forms for the LCWF, given by: 1 1 1 Fπ (Q 2 ) = du dy φπ∗ (y, Q) TH (u, y, Q) φπ (u, Q) 1 + O (7) Q 0 0 with TH the scattering amplitude for the form factor but with the pions replaced by collinear q q¯ pairs. The distribution amplitude [6] φπ (u, Q) is the probability amplitude for finding the q q¯ pair in the pion with u q = u and u q¯ = 1 − u (Eq. (4)). The dimensional counting rules for form factors at large momentum transfer and other hard exclusive processes and the near-conformal behavior of the LCWFs at short distances can also be derived using AdS/CFT duality [13]. Heavy meson decay is a large momentum transfer exclusive process and factorization into perturbative and nonperturbative contributions can be applied, Section 1.3. In calculations of the B → Alν decay [14] Fock state expansion of the initial and final state were used. The amplitude for this process has contributions from two sources. One is the n = 0 parton-numberconserving amplitude with quarks changing flavor. The second is a n = −2 contribution in which a quark and an antiquark from the initial hadron high Fock state annihilate to the leptonic current creating lepton pairs. Similarly, exclusive hadronic decays such as B → π + π − were calculated [15] in pQCD with assumed structure of both the B and π LCWFs. Vector meson (VM) production was treated extensively. In [16] the authors calculate the cross section for diffractive leptoproduction of VM. They make full use of the LCWFs of both the virtual photon and the VM. This is further discussed in Section 2.2. 5. Higher Fock states While the lowest Fock state of hadrons, q q¯ or qqq describes the valence configuration the higher Fock states describe the quark and gluon sea. The hadron fluctuates between the various
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states which live times of the order of
P Mh2 −M 2
where P is the momentum, M and Mh are the Fock
state mass and hadronic mass, respectively. The higher Fock states may be probed in a variety of processes. Deep Inelastic Scattering (DIS) of leptons has been a major tool for studies of nucleon structure. The measured structure functions are a reflection of the whole Fock state expansion. The observation of a contribution from strange quarks to the nucleon spin is a result of probing the higher Fock state components. Similarly, measurements of the charm structure functions [17] reveal the charm components in the higher Fock states with lifetimes of the order of ∼ m1cc¯ . These components are also referred to as the “intrinsic strange” and “intrinsic charm” sea. The experimental results were analysed along these lines and were found [18] to be consistent with about 1% probability for intrinsic charm in the proton. Experimental observations show large asymmetry in charm production between the “leading” charm mesons (that contain a quark present in the beam) and non-leading charm mesons [19]. This effect was predicted by various models including one relating it to the intrinsic charm [20]. The decay J/ψ → ρπ has been interpreted as decay of the cc¯ into a high Fock state q qc ¯ c¯ of the pion and/or the ρ wave function [21]. 6. Nuclear effects The LCWF formalism can be extended beyond the single hadron into nuclear systems, in particular few nucleon systems and scattering processes. The nucleon–nucleon interaction and the scattering process are governed by the QCD counting rules [22] which predict that the cross section at large momentum transfer scales as f ( st ) dσ (AB → C D) = n−2 (8) dt s where n = n A + n B + n C + n D is the total number of interacting elementary fields. For N N scattering this dependence is s −10 which was verified by experiments. The cross section for the d(γ , p)n photodisintegration is expected to scale as s −11 . This was recently observed for proton transverse momenta pT > 1.1 GeV/c [23]. Analysis of old data indicates an s −22 dependence for d(d, n) 3 He and d(d, p) 3H measured at c.m.s. scattering angle θcm = 60◦ in the interval of the deuteron beam energy 0.5−1.2 GeV and ∼s −16 for d p → d p scattering at θcm = 125◦–135◦ at beam energies 0.5–5 GeV [24]. The anomalous spin asymmetry in pp scattering [25] has been interpreted [26] as the contribution from a higher Fock state: |uuduudcc ¯ that couples to the J = S = 1 initial state thus affecting the spin asymmetry. The electromagnetic moments of the deuteron have been calculated in the LCWF framework [27]. It was shown that a part of the quadrupole moment may not be due to the D state component of the nuclear wave function but to relativistic recoil corrections. This may solve a discrepancy between the measured quadrupole moment and that calculated with conventional nuclear theory. The deuteron form factor at large momentum transfer is predicted to become symmetric among the possible color-singlet components (orthogonal to N N and ∆∆). This six-quark component of the deuteron wave function is referred to as the “hidden color” component [28]. It is expected to manifest in high momentum transfer photodisintegration of the deuteron and was searched for in studies of pion absorption in 3 He [29]. 1.3. Factorization and hard processes Factorization theorems allow the separation of soft non-perturbative dynamics from that of the hard perturbatively calculable process. Not every process is factorizable; for example, it
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fails for hard diffractive hadron–hadron scattering. Factorization theorems have been proved rigorously for processes such as diffractive hard Deep Inelastic Scattering (DIS) [30], exclusive DIS processes, and meson production [31]. For processes where factorization is applicable there is a scale Λ such that Fock state components with mass Mn2 < Λ2 are considered soft and have to be evaluated by non-perturbative methods, while those with Mn2 > Λ2 contribute to large momentum transfer hard processes which can then be evaluated perturbatively. As an example, in Eq. (7) factorization is applied: φπ (u, Q) contains the non-perturbative distribution amplitude for the quarks and gluons in the pion up to a scale Q 2 . All momentum transfers higher than Q 2 are included in the scattering amplitude TH which can be computed perturbatively. The commonly used value of Λ ∼ 1 GeV is somewhat arbitrary and may need experimental input for better determination. The non-perturbative distribution amplitudes must be derived with some assumptions and rely on models. At very low Q 2 QCD sum-rules were used to develop forms of the distribution amplitude for mesons and hadrons [32], Section 3. These forms have been extended into the perturbative domain by adding perturbative calculations [33]. For high Q 2 the known asymptotic forms of the distribution amplitudes [6], Section 3 is reproduced. Evolution equations of the distribution amplitude with Q 2 allow prediction of the distribution amplitude at some scale Q 2 if it is known at some other value Q 20 [6]. For the inclusive parton distributions the well known DGLAP [34] and BFKL [35] evolution equations allow prediction of the parton distribution at some scale Q 2 if it is known at some other value Q 20 . 1.4. Diffraction The diffractive process is the coherent scattering from an object with a size comparable to the wavelength of the scattered projectile. In these processes the target and/or the projectile usually remain intact. It is observed in the scattering of light, of electrons and nuclear projectiles off nuclei and at high energy scattering of hadrons and photons. The process is mediated by boson exchange, such as virtual photons, pions or Pomerons. The angular distribution of the scattered projectile is characterized by diffraction minima. Bessel functions describe the angular distribution for scattering of light and approximately for scattering of electrons and α particles off nuclei and nucleons. Diffractive elastic and inelastic electron scattering has been used extensively to measure the charge and magnetic form factors of nucleons and nuclei. The positions of the minima are inversely proportional to the scatterer size. The angular distributions are usually replaced through Fourier transform by distribution of the four-momentum transfer t. As the cross section for the process may drop many orders of magnitude before reaching the first minimum, frequently only the region of low t-distribution is observed [36]. For small t, a series expansion 2 of the Fourier transform leads to the approximated form: e−bt where b = R3 and R is the scatterer radius. It can also be related to the combined size of the scatterer and projectile, see Section 2.2. As the diffractive interaction is confined to small values of t, it is characterized by small transfer of energy between the interacting particles. For high energy hadronic diffractive interactions the exchanged boson is the Pomeron (P), described as a two-gluon system with vacuum quantum numbers [37]. Depending on the momentum transfer, there are “soft” and “hard” Pomerons. In addition to the usual DIS variables Q 2 , W 2 , x B J and y, the variables used to describe the diffractive process in ep → ep X interaction are: xP =
M 2 + Q2 q · ( p − p ) X2 , q·p W + Q2
(9)
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β=
xBJ Q2 Q2 = . 2q · ( p − p ) xP Q 2 + M X2
(10)
Here q, p, p are the momenta of the virtual photon and of the proton before and after scattering, respectively. x P represents the fraction of the proton momentum carried by the Pomeron. β is 2 analogous to x B J = 2Qp·q and represents the fraction of the Pomeron momentum carried out by the quarks. M X is the invariant mass of the hadronic final state. The approximations are valid for small values of t and large W . Diffractive processes are identified experimentally by observing the intact scattered proton or by the presence of large rapidity gaps [38]. A major project is measurements of the diffractive structure functions F2D . They are obtained from measurements of diffractive cross sections in a way similar to derivation of the total structure functions F2 from the total DIS cross section. F2D are related to the diffractive parton distribution through generalized DGLAP evolution equations. They are also used to measure the Pomeron trajectory αP used for parametrization of the diffractive cross sections. More exclusive processes being studied are diffractive jet production, vector meson production, charm production and DVCS. Some of these processes will be addressed in more detail in the present work. 2. Color transparency The phenomenon of color transparency is intimately related to the description of hadrons in terms of LCWFs. The lowest (valence) Fock component of the LCWF is made of the minimal partonic configuration consistent with the hadron quantum numbers: a q q¯ for mesons and qqq for baryons. The distances between these partons must be small, an interpretation being that otherwise gluons may fill the space. Due to the small size, compared with the regular hadronic size, these states are referred to as “Point Like Configurations” (PLC). At short ranges the relative momentum is large and hence the mass of this component is large, far off-shell. A hadron moving with momentum P continuously fluctuates between the various Fock states with fluctuation times of the order of: τ ∼ 2P/(M 2 − m 2 )
(11)
2 +m 2 k⊥i i
where M 2 = u(1−u) is the invariant mass of the Fock state, m i the parton mass and m the regular mass of the hadron. During this time all the phases between the states remain approximately constant; this condition is referred to as the “Frozen Approximation”. When an interaction involving a large momentum transfer Q occurs, the related short wavelength naturally selects the Fock component with small separation, of the order of 1/Q, and large internal momentum, the PLC. This also follows from the dimensional counting rules [22, 39] as the cross section for the more complex Fock states will be suppressed by 1/Q 2 for each additional parton. It is therefore expected that such interactions will be useful for studying the properties of the PLC. Color transparency (CT) is the name given to the prediction that the color fields of QCD cancel for physically small color-singlet systems of quarks and gluons [40]. This color neutrality (or color screening) should lead to the suppression of initial and final state interactions of smallsized systems formed in hard processes [41]. The cross section for these interactions is expected to have the dependence: σ (b2 ) ∝ b2
(12)
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where b is the transverse size of the PLC. For a recent review of the subject see [42]. Measurements of color transparency are important for clarifying the dynamics of bound states in QCD. The most straightforward approach is to study interactions between high energy hadrons and nuclei and to demonstrate that the interactions of a PLC with the nucleons in a nucleus are suppressed compared to those of ordinary hadrons. This requires identifying observables which depend explicitly on the cross-sections of the PLC. There are three necessary ingredients in such an experiment: 1. An interaction that selects the PLC. 2. Kinematic conditions such that the PLC fulfils the frozen approximation. 3. An observable sensitive to the PLC cross section. It should be stressed that determination of whether each of these conditions is fulfilled is subject to significant uncertainties. As noted above, high momentum transfer reactions are expected to select the PLC, and it remains to be determined how high the momentum transfer should be. We discuss below in detail high momentum transfer (e, e p) and ( p, 2 p) reactions, formation of vector mesons and pion dissociation to dijets. A crucial condition is that the PLC will not expand during the interaction with the nucleus. This expansion can be caused by fluctuation to other Fock states or, as the PLC is not an eigenstate of the Hamiltonian it can be regarded as a wave packet that undergoes time evolution. This is controlled by the PLC lifetime (Eq. (11)) which can be translated to the distance the PLC will travel during this time and is called the coherence length, c : c ∼ 2P/(M 2 − m 2 ).
(13)
Here the relevant value of M and the expansion time and process are model dependent. For deep inelastic scattering experiments this will be given by: ∼ DIS c
2ν (Q 2
+ Mq2q¯ )
(14)
where ν is the energy of the virtual photon. The mass of the q q¯ system, Mq q¯ , selected by DIS with momentum transfer Q is of the order of Q, thus c ∼ 1/2M p x B J . This relation is referred to as the Ioffe time [43]. The requirement is that c R A , the nuclear radius. The typical observables used as signature for existence of the PLC are comparisons between the cross section for interaction of the PLC with the nucleus and that expected from a normalsize hadron. The applicable calculation of both of these quantities has some uncertainties. These observables are looked for when conditions (1) and (2) are expected to be fulfilled or as function of a parameter which controls these conditions so as to observe the onset of color transparency. Observation of this onset is particularly important in view of the uncertainties involved in determining the exact conditions and signature for the color transparency phenomenon. 2.1. Quasi-elastic scattering As was pointed out in [41], for a high momentum transfer p– p scattering to proceed the momentum should be transferred to all the quarks involved, hence the internal distances should be of the order of the momentum transfer. This is also reflected in the dimensional counting rules [22], where the cross section for such a process should scale as s −10 . One then expects that in ( p, 2 p) reactions on nuclei the initial and final state interactions of the incoming and outgoing protons will be strongly reduced as Q 2 increases. Experimentally, this was studied [44]
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Fig. 1. Schematic drawing of E850 EVA solenoidal detector, [45].
by measuring the cross section on several nuclear targets and different beam momenta. The authors determine the transparency T defined as: T =
(dσ/dt)( p − p elastic in nucleus) . (dσ/dt)( p − p elastic in hydrogen)
(15)
The detector was made of a scintillation counter backed with a two-layer 1.5 radiation-length lead–scintillator sandwich. This counter, spanning about 2/3 of the total solid angle identified charged tracks produced out of the two-body scattering plane. The momentum of only one of the outgoing protons was measured. The data was analyzed by reconstructing the momentum of the struck proton and using the free p– p cross section to calculate T . After corrections for background and experimental acceptance, values of T were extracted with normalization to the number of “nuclear protons” (5.1 for aluminum). The results show an increase of T with incident momentum up to about 10 GeV/c but it drops at higher momenta. This result as well as the anomalous spin asymmetry observed in pp scattering at similar center of mass energies [25] have been interpreted [26] as the contribution from a resonance produced by a higher Fock state. The authors predict that CT will reappear at higher energies. At the moment, the experimental results as far as color transparency is concerned remain inconclusive. The experiment was later repeated with an improved apparatus, the EVA detector having good tracking and momentum analysis capabilities for both protons. The EVA (BNL-E850) experiment embedded carbon (C), CH2 , and CD2 targets inside a 2 m diameter, 3 m long superconducting solenoid with a 0.8 T field as shown in Fig. 1. The annular space between the pole piece and the solenoid allowed particles scattered near 90◦c.m. to reach the detectors outside the solenoid. For tracking purposes the targets were surrounded by four concentric cylinders each fabricated from four layers of thin wall straw tubes filled with a 50–50 mixture of argon–ethane. The momentum resolution was σ ( p)/ p 7%. The tracks scattered near 90◦ c.m. passed through the annulus between the solenoid and the steel pole piece until they reached the two fan-shaped scintillation counter arrays, H1 and H2, and the two larger straw tube cylinders, C3 and C4. The detector was configured to have acceptance for c.m. scattering angles in the range ∼86◦ < θc.m. ≤ 90◦.
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Fig. 2. Nuclear transparencies measured by BNL experiments 834 [44] and 850 [45].
From the momentum measurement of the two scattered protons the authors deduce the momentum of the struck proton and define an effective beam momentum Peff in its rest frame. The results were recently summarized [45] and compared with the transparency T = 0.2 expected from the Glauber model. The results are found to be consistent with the earlier ones, reproducing the rise and fall of the transparency for incident momentum above 10 GeV/c, translated to Q 2 ∼ 8 GeV/c, see Fig. 2. Studies of color transparency using the A( p, 2 p) reaction are particularly difficult. It is necessary that the conditions for color transparency be fulfilled for both incoming and the two outgoing protons. The incoming proton should maintain its PLC configuration as it penetrates the nucleus to the point where it strikes a nuclear proton which is, too, in a PLC configuration so that the high momentum transfer scattering can occur. Then the two outgoing protons should stay in that state as they move through the nuclear medium. The probability for the scattering is already small as determined by the s −10 dependence, but this does not include the probability for all associated coherence lengths to be sufficiently long. Better conditions can be expected in studies of the A(e, e p) reactions. The interaction is by absorption of a high Q 2 virtual photon which fluctuates to a q q¯ PLC system with coherence length defined by Eq. (14). If this is sufficiently long it will be necessary that the fluctuation time of the outgoing (PLC) proton for growing into a normal size be sufficiently long too. This fluctuation time is referred to as the “formation length” f and it is governed by the same considerations as for the coherence length, Eq. (13). An experimental study of the A(e, e p) reaction was performed by the NE-18 collaboration at SLAC [46]. The cross sections were measured on several nuclear targets with transfer momenta Q 2 = 1, 3, 5, 6.8 (GeV/c)2 . The outgoing electron and proton were detected with magnetic spectrometers with 0.15% momentum resolution. This allowed the reconstruction of the struck proton momentum and the missing energy which was required to be well below the pion production threshold. The transparency was determined by taking the ratio of the experimental results to PWIA calculations. The expected color transparency signature was that
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Fig. 3. Nuclear transparencies measured by the A(e, e p) experiments at SLAC [46], open symbols and JLAB [47] closed symbols. The transparency at lowest Q 2 is from a Bates experiment [48]. The solid lines are Glauber model calculations [49].
the transparency will grow with Q 2 , as was observed by the BNL experiments in this Q 2 range. Furthermore, when the A-dependence is studied in fixed Q 2 values and parametrized as: 2 T ∝ Aα(Q ) the (usually negative) value of α is expected to increase with Q 2 approaching zero (T = 1) for full transparency. The experimental results showed none of these signatures: the Q 2 dependence was consistent with being flat and the value of α remained constant α ∼ −0.2 as expected from simple Glauber model calculations. A similar experiment was carried out at the Jefferson Laboratory [47] in which the momentum transfer range was extended to 8.1 (GeV/c)2 and with improved statistical and systematic precision. The results were consistent with those of the SLAC experiment and showed no Q 2 dependence of the transparency or of the A-dependence parameter α. The results of both experiments are summarized in Fig. 3. The results of the ( p, 2 p) and (e, e p) experiments were very puzzling and resulted in many theoretical papers with a variety of interpretations. We may attempt to determine what can be expected from these experiments based on the concept of coherence length, Eq. (13). For Q 2 = 8 (GeV/c)2 , the maximum value for the (e, e p) experiments and where the maximal transparency is seen in the ( p, 2 p) experiments, the energy of the scattered proton ∼Q 2 /2m p and its momentum is ∼5 GeV/c. To use Eq. (13) we must use a mass M for the excited, smallsize proton and this is subject to a large uncertainty. The lowest one and the one most frequently used is that of the Roper resonance, [42]. The resulting coherence length is c ∼ 2 fm which is smaller even than the radius of 12 C, the lightest nucleus in these experiments (except for the deuteron, for which these concepts are not applicable). Thus, the conditions for CT are not fulfilled and what we can expect to see is the onset of CT which may be even more interesting than having full CT. The two reactions suffer from a kinematic constraint that relates Q 2 and the energy of the scattered proton. The cross section for the ( p, 2 p) reaction has a s −10 dependence. The reaction would occur mainly on protons with longitudinal Fermi momentum in the beam direction that would reduce
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291
Fig. 4. Nuclear transparency for 12 C(e, e p) quasielastic scattering: SLAC [46] (open symbols) and JLAB [47] (closed symbols). The transparency at lowest Q 2 is from a Bates experiment [48]. The thin solid curve is from Glauber model calculations [49]. The thick solid curve is a Glauber calculation of Ref. [52]. The dot–dashed, dotted, and dashed curves are color transparency predictions from Refs. [52–54], and [55], respectively.
s. It will map high nuclear momentum components with protons far off shell. For the (e, e p) reaction the cross section has a s −6 dependence hence it may be mapping lower momentum components and protons more on shell. This may affect the probability for observation of CT in these two reactions. If the probability for the outgoing proton to avoid final state interaction is p then for the (e, e p) experiment it will be p and for the ( p, 2 p) experiment p2 if we assume that the higher momentum incoming proton in the ( p, 2 p) reaction does not interact in the nuclear medium at all. If it does, the probability can be closer to p3 . In any event, we can expect a steeper increase of transparency with Q 2 for the ( p, 2 p) experiment than for the (e, e p) experiment. It now becomes a quantitative issue of whether the slope observed in the ( p, 2 p) experiment can be consistent with the results of the (e, e p) experiment. There are large theoretical uncertainties related to the formation and expansion of the PLC; the coherence length being just a simplified parameter. There are smaller but accumulating uncertainties related to how to handle the internal momentum in the nucleus, Glauber and PWIA calculations and the nucleon–nucleon cross section in the nuclear medium compared with free nucleons. For example, the oscillations in the free nucleon–nucleon cross section are expected to be filtered out by the nuclear medium [50]. For reviews of these works see [50,51]. The early experimental results had significant uncertainties such that a variety of models could be accommodated and it was possible to reach a conclusion that there is no clear inconsistency between the (e, e p) and ( p, 2 p) experiments [50,51]. The more recent and precise results make this conclusion less clear as demonstrated in Fig. 4 taken from [47]. The authors show that calculations that included CT effects consistent with the ( p, 2 p) results (dashed and dotted lines) were not consistent with the (e, e p) results. In fact some calculations claim that substantial CT effects will be seen only for Q 2 ∼ 20 (GeV/c)2 [56]. In conclusion it cannot be confirmed that color transparency has been observed in these experiments. 2.2. Vector meson production Studies of vector meson (VM) electroproduction in DIS are related to color transparency through two aspects. One is the slope of the diffractive t-distribution (t: the momentum transfer
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to the target) which provides a measure of the size of the q q¯ system at large Q 2 . This is measured on a nucleon target. The second is measurements of the transparency: ratios of the cross sections on nuclei to the cross section on the nucleon, similar to the approach in quasielastic scattering discussed in Section 2.1. The cross section for leptoproduction of VM was calculated in [16]. The process is described as three sequential steps: 1. The virtual photon breaks up into a q q¯ pair with a lifetime τi given by Eq. (14) and Mq2q¯ =
2 + m2 k⊥ . u(1 − u)
(16)
Here m is the current quark mass and k⊥ the quark transverse momentum. This estimate is valid for the production of a longitudinally polarized vector meson which is expected to dominate at large Q 2 . 2. The q q¯ pair scatters off the target proton. 3. The q q¯ pair then lives a time τ f (and travels distance f ) determined by τf ∼ f ∼
2ν Mq2q¯
− m 2V
before the final state vector meson with mass m V is formed. The amplitude for this process M is written as a product of three factors: (i) the photon wave γ function ψλ1 λ2 (k⊥ , u) giving the amplitude for the virtual photon to break into a quark–antiquark pair; (ii) the hard scattering amplitude T H of the quark–antiquark pair on the target; and (iii) the
, u ) giving the amplitude for the scattered quark–antiquark pair of VM wave function ψλV1∗λ2 (k⊥ flavor f to become a vector meson: Mf =
1 1 d2 k ⊥ d2 k
⊥ Nc du du ψλV1∗λ2 (k⊥ , u )TλH1 λ2 (k⊥ , u ; k⊥ , u) (16π 3 )2 0 λ ,λ 1
2
γ × ψλ1 λ2 (k⊥ , u)
0
(17)
where λ1 and λ2 are the helicities of the quark–antiquark pair which are conserved during the scattering off the target and Nc is the number of colors necessary for normalization. For the VM the authors use a distribution amplitude similar to that of the pion (Section 3). The LCWF of the photon is discussed in Section 4. The authors then deduce a cross section for t = 0 with values in a range of about a factor 3 depending on which LCWF is used for the VM. This sensitivity is interesting and may be used to actually measure the vector meson LCWF. The numerical agreement between the theoretical estimates and the measurements is reasonable for some parameters but if one wants to use this comparison, variations of the cross section (e.g. as function of Q 2 ) rather the absolute values should be used. The t-dependence of the cross section for diffractive VM production is expected to have the form: dσ/dt ∝ ebt where the slope b is related to the size of the system. This relation was further examined in Ref. [57] with the result that it should have the form: b=
1 1 2 R + r 2 3 8
(18)
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293
Fig. 5. Slope parameter of the t-distribution for ρ, φ and J /ψ vector mesons as a function of Q 2 + m 2V , [60].
where r∼
6
(19)
Q 2 + m 2V
is the (excited) ρ 0 size and R is the proton radius [57]. This result implies that the diffraction cone, as described by the parameter b should shrink as Q 2 is increasing. It is also telling us that the dependence will be on Q 2 + m 2V which can be verified by production measurements of various vector mesons. An additional contribution to b may come from the form factor of the exchanged boson, Section 4.2.1. There have been several experimental studies of the t-dependence of VM production on the proton: ρ production [58], ρ and φ production [59] and ρ, φ and J/Ψ production [60]. The results from the three experiments are consistent with each other. Recent results are summarized in Fig. 5 [60] where the parameter b is plotted as a function of Q 2 + m 2V . It is seen that on this scale the slope is approximately universal for all the VM. At large Q 2 + m 2V the value of r (Eq. (19)) is very small, and b ∼ 4.5 (GeV/c)−2 is dominated by R and consistent with Eq. (18). The size of the q q¯ system is then small compared to this value. Experiments of VM production in nuclei [61–63] were discussed and interpreted without relating to the VM size effects discussed above. This is unfortunate as it is likely that with so many uncertainties and model dependence, combining the experimental information on VM production on the proton and on nuclei can improve our understanding of the process. Two different VM production mechanisms in nuclei are considered. One is the incoherent process where VM are produced on individual nucleons and as result the nucleus breaks apart. This is similar to quasi-elastic scattering where the produced VM must go through the nuclear medium. In the coherent process the VM are produced on the nucleus as a whole and it remains intact. Both of these processes were studied experimentally and theoretically.
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0 Fig. 6. Nuclear transparencies Trinc A measured for incoherent ρ production by Fermilab experiment E665 [61]. The calculations are from [64]; solid line with full calculations, dashed lines with frozen approximation.
An important difference between VM production and quasi-elastic scattering in nuclei is that the kinematic constraints discussed for quasi-elastic scattering do not exist, or are at least much weaker for VM production. One then can vary almost independently the coherence and formation lengths. Both coherent and incoherent processes were discussed in great detail in [64–66] where the authors pay particular attention to the identification of coherence length effects and CT effects. In [64] the authors calculate the VM production cross section on the nucleon using a similar approach to that of [16]. They use a photon LCWF in coordinate space modified to avoid end-point singularities. For the VM they use a boosted Gaussian wave function and for the interaction a phenomenological dipole cross section σqq ¯ . The resulting amplitude for the process is: 1 du d2r ΨV∗ (r , u) σqq r , s) Ψγ ∗ (r , u, Q 2 ). (20) Mγ ∗ N→V N (s, Q 2 ) = ¯ ( 0
They deduce the t-dependence by fitting data with an expression similar to Eq. (18) and with this fit obtain good agreement with measured ρ 0 and φ production on the proton. For VM incoherent production in nuclei the authors [64] use Eq. (20) modified by a Green function to describe propagation of the q q¯ in the nuclear medium attenuated by a simplified dipole cross section. The results then depend on the size of the coherence length. For large coherence lengths (c , f R A ) the frozen approximation is valid, and the transparency is calculated with only Q 2 dependence showing clean CT effect. The calculations are compared with the experimental results of the E665 collaboration [61] that measured ρ 0 production in DIS of 470 GeV/c muons on hydrogen, deuterium, carbon, calcium and lead. At these high energies, with ν > 100 GeV, the coherence lengths are large: c , f R A . The results are shown in Fig. 6. The increasing transparency with Q 2 shows that the size of the produced q q¯ system that eventually couples into the ρ 0 is smaller for larger Q 2 and interacts weakly in the nucleus. At very small x B j this effect may be partially offset by nuclear shadowing [69]. The deviation from the frozen approximation for Pb at large Q 2 comes from the reduction of c which is no longer large compared with the Pb radius. In this experiment there are no complications due to mixing of coherence length and CT effects.
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295
Fig. 7. t-dependence of the ρ 0 production on H and 14 N [62] used for separation of coherent and incoherent cross sections.
Fig. 8. Nuclear transparency as a function of Q 2 in specific coherence length bins for incoherent ρ 0 production on nitrogen. The straight line is the result of the common fit of the Q 2 -dependence [62].
The situation is very different in the lower energy measurements of the HERMES collaboration [62,63] where 9 < ν < 20 GeV. The ρ 0 production was studied on H and 14 N. To separate the coherent and incoherent cross sections the authors used the t-distributions. The excess yield at small t for 14 N, showing the larger nuclear radius, is taken as the coherent production, Fig. 7. At these energies both c and f are short and changes of transparency with Q 2 reflect combined changes in c , f and size of the q q¯ system. In order to be more selective the authors plot the transparency as a function of Q 2 in bins of 0.1 fm in c = 2ν/(Q 2 + Mρ2 ), Fig. 8. The results show an increase of the transparency with Q 2 even with fixed c and this is taken as a signal of CT. It should be noted that for fixed c the value of ν increases with Q 2 , and
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f = 2ν/(Mρ2 − Mρ2 ) ∼ 2ν/1.51 GeV−1 (as used by the authors of [62]) increases accordingly. Thus we may be observing a combined effect of shrinking q q¯ and increasing f with Q 2 . The authors of the theoretical calculations [64] reproduce the experimental results. Their calculations show that for fixed Q 2 the transparency decreases with c . The given interpretation is that increasing c results in a longer path of the q q¯ in the nucleus i.e. suppression of transparency. This effect is claimed to be stronger than that of the increasing f . The experimental results were not presented in this way. From Fig. 8 one can see that for Q 2 = 2 the transparency is constant at T = 0.5 ± 0.1 over the range 1.35 fm < c < 2.35 fm. Coherent VM production on nuclei can provide additional information and, under some circumstances allow disentangling the CT and coherence-length effects. In the absence of any CT effect the coherent production cross section is expected to fall with Q 2 because of the coherent nuclear form factor. To estimate the various CT and coherence-length effects, we first consider the extreme case of high energies such that c , f > R A . The VM is then produced coherently over the whole nuclear volume and the nuclear production amplitude is predicted to be A times the nucleon production amplitude: MA = AMN , σ A ∝ A2 . If Q 2 is large enough such that the interaction of the q q¯ in the nuclear medium vanishes, the cross section will be given by: dσ N 2 2 dσ A ∝ A FA (t) dt dt
(21)
where FA2 (t) is the nuclear form factor squared (neglecting r , Eqs. (18) and (19)): FA (t)2 ∼ ebt ,
b=
R 2 , 3
When integrating over t: 3 A2 A2 = 2 (dσ/dt)dt ∝ b R0 A2/3 σ A ∝ A4/3,
T ∝ A1/3.
R = R0 A1/3 .
(22)
(23)
(24)
If Q 2 is not large enough but the energy is high the onset of CT can be observed as Q 2 increases. This is seen in Fig. 9 where the calculations of [64] with the frozen approximation are in reasonable agreement with the data. Calculation of the transparency for fixed Q 2 as a function of ν [64,65] show increased transparency (as coherence lengths increase) and saturation where 1 the coherence lengths exceed the nuclear size. The saturated transparencies scale as A 3 . This is indeed seen in the E665 data of Fig. 9 and the calculations of [64], 10. For lower energies the situation is more complex. The HERMES collaboration shows transparency increasing with Q 2 for fixed c [63]. This may be due to CT or to the increase in ν and coherence lengths with increasing Q 2 or, most probably, combination of the effects. In summary, studies of coherent and incoherent VM production on nuclei provide us with interesting information about CT and coherence lengths. Through a combination of measurements the effects may be studied separately. At high energies, where coherence length effects are minimal the CT effect can be observed through the increase of transparency with Q 2 in both coherent and incoherent production. At lower energies one must rely on calculations that incorporate all the effects and can consistently reproduce the experimental results. We finally mention an experiment in which the A-dependence of J/ψ photoproduction was measured [67]. Fermilab experiment E691 measured the relative cross section for J/ψ
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297
Fig. 9. Nuclear transparencies for coherent ρ 0 production measured by Fermilab experiment E665 [61]. The calculations are from [64]; solid line with full calculations, dashed lines with frozen approximation.
Fig. 10. Nuclear transparencies for coherent ρ 0 production as function of energy and Q 2 [64]; solid line with full calculations, dashed lines without gluon shadowing.
production by real photons with energies of 90–190 GeV incident on H, Be, Fe and Pb targets. They separated the contributions of coherent and incoherent yield using the same method as done in the E665 and HERMES experiments (see Fig. 7). The measured cross sections were parametrized: σ = σ0 Aα with the results: αcoh = 1.40 ± 0.06 ± 0.04, αincoh = 0.94 ± 0.02 ± 0.02. The result for coherent production is consistent with the expectations for fully coherent and transparent production as in Eq. (24). We note that at these high energies the coherence lengths are long. From Fig. 5 we can estimate that the J/ψ is photoproduced in a compact (b ∼ 4 GeV−2 ) cc¯ configuration, similar to high Q 2 ρ 0 production. 2.3. Pion diffractive dissociation to dijets The idea to use diffractive dissociation of pions to dijets in order to observe CT effects and study the pion wave function was proposed by Bertsch et al. [68]. The authors point out that when a high energy pion hits a nuclear target, all the large-size Fock components of the pion wave function will be absorbed in the nucleus. The small-size components will interact very weakly thus the nucleus will be transparent to these components. As the pion hitting the target fluctuates
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Fig. 11. Pion dissociation to q q¯ via two-gluon exchange.
between the various Fock components, only the small-size valence q q¯ state will emerge. As this state is no longer an eigenstate of the pion Hamiltonian it will dissociate and the quark and antiquark will each hadronize into a jet. In this process the nucleus acts as a filter absorbing all Fock components except for the valence q q¯ component. These ideas were further developed by Frankfurt et al. [51,69,70] who calculated the cross section for dijets with large transverse momentum kt . The pion fluctuates into a large kt q q¯ pair long before it hits the target nucleus and interacts via two-gluon exchange, Fig. 11. The mass of the system is given by Eq. (16) and for a high incident momentum (500 GeV/c) the coherence length (Eq. (13)) is long. This high kt system is a PLC and interacts weakly in the nuclear medium. The authors calculate the cross section for this process in leading twist. For the dijet production amplitude on a nucleon they take: 2 f (b ) ei κt ·b (25) M(N) = d2 bψπ (u, b) 2 here b is the transverse size of the q q¯ system, f (b2) is its forward scattering amplitude normalized according to the optical theorem Im f (b2 ) = s σ (b2 ), and t = 0. The factor e−i κt ·b accounts for the final hadronic wave function of two jets with a high relative momentum. is the q q¯ LCWF in coordinate space, b being the transverse separation of the quarks. ψπ (u, b) The assumed b2 dependence of the interaction σ (b2 ) allows an evaluation of Eq. (25) in terms of the momentum-space Fourier transformed ψ˜ π (u, κt ) by expressing b2 as b2 = −∇κ2t . Then M(N) = i
1 s σ (−∇κ2t )ψ˜ π (u, κt ). 2
(26)
The authors calculate Eq. (26) using two possible pion LCWFs (Section 3) and a phenomenological cross section σ (b2 ) and obtain: 8 d3 σ N 1 m N |M N |2 −6 GeV
= ≈ 2.6 GeV φ 2 (u) (27) κt (2π)4 16 m 2 + P 2 s 2 dudM J2 d2 PNt N N φ(u) is the pion distribution amplitude (Section 3) and there is a factor 2 different normalization depending on which one is used. For the nuclear cross section the authors use the frozen approximation and nuclear multiple scattering. The results M(A)/AM(N) are shown in Fig. 12.
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299
Fig. 12. Ratios of nuclear to nucleon amplitudes [69]. Solid line: exact evaluation, dashed line: higher twist expansion.
As can be seen the ratio increases with kt , reaches unity for kt ∼ 1.5 GeV/c and overshoots this value. This overshoot is attributed to higher twist effects which are studied by the authors [69, 70]. For M(A)/AM(N) ∼ 1 d4 σ A dudM J2 d2 qt
=
d4 σ N dudM J2 d2 qt
FA2 (t)A2
(28)
in striking contrast to Glauber calculations. When integrating over t the results of Eq. (24) hold. The experimental study of this process was performed by the Fermilab E791 collaboration [71]. A-dependence of the diffractive dissociation of 500 GeV/c pions into dijets, scattering coherently from carbon and platinum targets was measured. 2 × 1010 π − nucleus interactions were recorded using an open geometry spectrometer [72], see Fig. 13. The segmented target consisted of one platinum foil and four diamond foils, each foil approximately 0.4% of an interaction length thick (0.5 mm for platinum and 1.6 mm for carbon). Six planes of silicon microstrip detectors (SMD) and eight proportional wire chambers (PWC) were used to track the beam particles. The downstream detector consisted of 17 planes of SMDs for vertex detection, 35 drift chamber planes, two PWCs, two magnets for momentum analysis, ˇ two multi-cell threshold Cerenkov counters, electromagnetic and hadronic calorimeters for
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Fig. 13. The spectrometer used by the E791 collaboration [72].
electron identification and for online triggering, and two planes of shielded scintillators for muon detection. The trigger required a beam particle, an interaction in the target and a very loose transverse energy trigger, based on the energy deposited in the calorimeters. The data analysis required that at least 90% of the beam momentum is carried by charged particles. This reduced the effects of unobserved neutral particles. The JADE jet-finding algorithm [73] was used to identify two-jet events made of charged particles. The algorithm’s cutoff parameter (m cut ) was optimized using Monte Carlo simulations. For each two-jet event the transverse momentum of each jet with respect to the beam axis (kt ), the transverse momentum of the dijet system with respect to the beam axis (qt ), and the dijet invariant mass, M J were calculated. A minimum kt of 1.2 GeV/c was required. Selection of diffractive dijets was based on the qt2 distributions of the selected dijet events shown in Fig. 14. The diffractive peaks at low qt2 were simulated by dN/dqt2 ∝ exp(−bqt2) with b inversely proportional to the nucleus’s radius (2.44 fm for carbon and 5.27 fm for platinum). Because theory predicts that the A-dependence varies with kt [69], the analysis is carried out in three kt regions: 1.25 GeV/c ≤ kt ≤ 1.5 GeV/c, 1.5 GeV/c < kt ≤ 2.0 GeV/c, and 2.0 GeV/c < kt ≤ 2.5 GeV/c. The distributions were fit by Monte Carlo simulations of sums of qt2 distributions of dijet events produced coherently and incoherently from nuclear targets and of background. The generated events were passed through detector simulation, reconstructed and analyzed using the same programs used for the data. The A-dependence of the diffractive cross sections was derived from the fits in each kt range. The exponents of σ = σ0 Aα are listed in Table 1 and are close to the expectations, Eq. (24). The conditions of this experiment are such that the onset of CT was expected to be seen as an increase of α with increasing kt . In fact the values of α do not show this but the values are in good agreement with what is expected from CT. So apparently the onset is at lower values of kt . For the measured kt range the corresponding transverse size is ∼0.1 fm. We should note that assuming M 2 ∼ 4kt2 we get in the low kt region M 2 < 9 GeV2 resulting in c ∼ 22 fm, R A . At the high kt range we have M 2 > 16 GeV2 and c ∼ 12 fm, ∼2R A . It may be that in the high kt range some CT effects are lost due to reduced c , compensating the expected rise in α. We
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301
Table 1 The exponent in σ ∝ Aα , experimental results for coherent dissociation and the color-transparency (CT) predictions [69] kt bin (GeV/c)
α
αstat
αsys
α
α(CT)
1.25–1.5 1.5–2.0 2.0–2.5
1.64 1.52 1.55
±0.05 ±0.09 ±0.11
+0.04–0.11 ±0.08 ±0.12
+0.06–0.12 ±0.12 ±0.16
1.25 1.45 1.60
Fig. 14. qt2 distributions of dijets with 1.5 ≤ kt ≤ 2.0 GeV/c for the platinum and carbon targets. The lines are fits of the MC simulations to the data: coherent nuclear dissociation (dotted line), coherent nucleon/incoherent nuclear dissociation (dashed line), background (dashed–dotted line) and total fit (solid line).
note also that in their more recent work [70] the authors carried out more detailed calculations and predicted a value α = 1.54. This process was calculated also by Nikolaev et al. [74] who include higher twist corrections. They calculate the α dependence and their results are very similar to those shown in Table 1 as derived from [69]. In summary of this section we may conclude that color transparency was well demonstrated in vector meson electroproduction and in diffractive dissociation of the pion to dijets. It was not unambiguously verified for the proton. It is important to understand the experimental results for the proton: why (e, e p) experiments show no sign of CT and why ( p, 2 p) experiments show a rise and fall of transparency, strongly deviating from Glauber calculations and at the same time not reproducing the expected CT signature. It can be expected that if the effect exists in the q q¯ system it should also exist for the qqq system. One could argue that the probability to find a q q¯ at short distances is higher than that to find a qqq in short distances. If we interpret these systems as the valence components of their respective LCWFs, this may indicate that the contribution of the valence component to the total LCWF may be different for mesons and baryons. The difficulties encountered in understanding the anomalous spin effects in pp scattering [25,26] leave this as an open question. For observation of CT with protons there might also be the problem of choosing the sensitive process: reaction, momentum transfer etc. that would select a proton in a PLC state and the observable that would identify it as such. It may be that diffractive dissociation of protons or perhaps baryon photoproduction would show this effect. Following the example
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of the pion one could consider proton diffractive dissociation to three jets in proton–nucleus interactions. This may be achieved at RHIC and in fixed target experiments at the Bevatron. High momentum transfer electroproduction of mesons off nuclei: (e, e π), (e, e k) should provide additional confirmation of the observations of CT for mesons, and perhaps some understanding of why the (e, e p) experiments did not show CT effects. 3. The pion light-cone wave function 3.1. Theoretical predictions The quantity usually used for description of the pion internal structure is the distribution amplitude, the probability amplitude to find the q q¯ system in the pion with longitudinal momentum fractions u and 1 − u. It is given by an integral of the pion LCWF over the transverse momentum (see Eq. (4)) up to a scale Q 2 : Q2 2 2 φq q/π (u, Q ) ∼ ψq q/π (29) ¯ ¯ (u, k˜t )d k˜t . 0
Because of the relation between the distribution amplitude and the LCWF it is common to find in the literature the distribution amplitude being referred to as a wave function. The distribution amplitude φπ (u, Q 2 ) was shown to be only weakly dependent on Q 2 and an evolution equation was derived. The most general solution of the evolution equation yields the distribution amplitude expansion [6]: −γn Q2 3/2 2 an Cn (2u − 1) ln 2 (30) φπ (u, Q ) = u(1 − u) Λ n≥0 where Cn are the Gegenbauer polynomials and γn the anomalous dimensions: n+1 2δh h¯ CF 1 γn = − 1+4 . β k (n + 1)(n + 2) 2
(31)
C F = (n 2c − 1)/2n c = 43 for n c = 3 colors and β = 11 − 23 n f = 29 3 for n f = 2 flavors. δh h¯ = 1 (0) when the constituent’s helicities are antiparallel (parallel). If it is known for a given Q 20 , φπ (u, Q 20 ) can be used to calculate the coefficients an and obtain φπ (u, Q 2 ) for any Q 2 . φπ (u, Q 20 ) can be either measured or calculated at some Q 20 where the least modeldependence is expected. For Q 2 → ∞ perturbative QCD (pQCD) is expected to hold and only the n = 0, a0 term survives (γ0 < γn for all n > 0). For pions δh h¯ = 1 hence γ0 = 0 (Eq. (31)) and the high Q 2 asymptotic distribution amplitude is [6,68,75]: π φAsy (u) = a0 u(1 − u).
(32)
We note here that for a vector meson with parallel helicities, δh h¯ = 0 and the asymptotic distribution amplitude becomes: −C F /β Q2 VM φAsy (u, Q) = a0 u(1 − u) ln 2 . (33) Λ The normalization coefficient a0 = 6 if the integral of the distribution amplitude is unity. Alternatively, it is given by normalizing ψq q/π at the origin yielding [6]: ¯
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1
a0 = 6 0
φπ (u, Q)du =
√ 3 fπ
303
(34)
with the pion decay constant f π . At the other end of the scale, Q 2 ≤ 1 GeV2 it is much more difficult to predict φ as it is dominated by non-perturbative interactions. This was addressed by Chernyak and Zhitnitsky (CZ) who proposed [32] to use QCD sum rules for that purpose. They evaluated sum rules that allow the determination of moments of the distribution amplitude. With their longitudinal momentum fraction ζ = u 1 − u 2 = 2u − 1, they obtained for the moments: 1 n ζ = φ(ζ )ζ n dζ (35) −1
values derived from phenomenological analysis: ζ 2 µ0 = 0.46,
ζ 4 µ0 = 0.30,
µ0 = 500 MeV
(36)
at the scale of 500 MeV. At this low scale they estimate that only the pion and the A1 resonance contribute to the sum rules and only the lowest moments can be practically determined. They search for a function that will fulfil these requirements as well as normalization to unity and end-point requirement. Their model wave function is: φCZ (ζ, µ0 ) =
15 (1 − ζ 2 )ζ 2 , 4
µ0 = 500 MeV
(37)
ζ 4 = 0.24.
(38)
with moments: ζ 0 = 1.0,
ζ 2 = 0.43,
In terms of u: φCZ (u) = 30u(1 − u)(1 − 2u)2 .
(39)
This function corresponds to a2 = 2/3, a4 = 0 in Eq. (30). The distribution amplitudes (squared) are plotted in Fig. 15 and as can be seen there is big difference between the two. One is faced with the question of how high Q 2 should be in order for the distribution amplitude to resemble the asymptotic one. This may be extended to the question of where asymptotia starts and for what Q 2 values pQCD calculations apply. The statement in [32] is that due to the slow, logarithmic evolution of the distribution amplitudes, resemblance to the asymptotic one will be reached only for Q 2 ≥ 100 GeV2 . Other distribution amplitudes were proposed by several authors [76–78]. It becomes a challenge to experimentalists to measure the distribution amplitude at any given scale. 3.2. Distribution amplitudes and form factors Traditionally, studies of the pion LCWF and distribution amplitude were done by measurements of the electromagnetic form factors of the pion. For the charged pion at very low Q 2 the measurements utilize the π ± e → π ± e elastic scattering. This was done using a pion beam scattering from a heavy nuclear target. At very forward angles the scattering is electromagnetic and the cross section measures the form factor much like for electron scattering on nuclei [79]. For higher Q 2 , the process of electron scattering from a virtual pion, p(e, e π + )n (Fig. 16(a))
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Fig. 15. Two predictions of the pion distribution amplitude. Asymptotic function: dashed line, and CZ function: solid line.
Fig. 16. (a) Measurement of the charged pion form factor by electron scattering off a virtual pion. (b) Measurement of the neutral pion form factor by e+ e− annihilation.
was used. This process is more complicated and its analysis is model dependent [80]. The tchannel scattering on an almost on-shell virtual pion has to be selected. This is done by selecting a quasi-free kinematic configuration for the experiment. Selection of longitudinal virtual photons is done using the “Rosenbluth separation” method where electrons are detected at several angles while keeping Q 2 fixed. The cross section for exchange of a longitudinal virtual photon is related to the form factor: σL ∼ −
tgπ2 N N (t) 2 2 F (Q ) (t − m 2π )2 π
(40)
where t is the squared momentum transferred to the nucleon and gπ2 N N is the π N N coupling constant. The form factor, in turn, is related to the distribution amplitude through: 1 pquark du du φπ (u , µ2 ) T (u , u, Q 2 , µ2 ) φπ (u, µ2 ), u, u = . (41) Fπ (Q 2 ) = pπ 0
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Fig. 17. The space-like form factor of the π + meson. The data are from Refs. [83–85]. See the text for description of the calculated curves.
T (u , u, Q 2 , µ2 ) is the transition amplitude that has to be calculated from the diagram in Fig. 16(a) and should include treatment of the off-shell pion. This becomes increasingly uncertain as Q 2 increases. The experimental situation and comparisons with calculations are shown in Fig. 17 taken from [39]. The lines marked CZ and ASY were calculated using a simplified form of Eq. (41): 1 16πC F αs (Q 2 ) 2 φπ (u) (42) |I | with I = du Fπ (Q 2 ) = 2 Q u 0 and both disagree with the data. Various corrections were applied, mostly to account for soft parts of the integrals which are particularly significant near the end-points. The line marked LS comes from using φCZ and applying Sudakov resummation of the transverse degrees of freedom [81]. The line marked SR comes from direct calculation of the form factor from QCD sum rules without using the distribution amplitude at all [82]. While the latter results seem to be in relatively good agreement with the data, they scale as Q −4 at higher Q 2 and the agreement in the measured Q 2 region is generally taken as accidental. Calculations of asymptotic form factors are discussed in [86]. For more detailed discussion and references see [39]. Turning to the neutral pion, the π 0 -photon transition form factor, Fig. 16(b) is expected to be a better test of the pion distribution amplitude as it is, to lowest order, a pure QED process. The measurements are mostly from e+ e− colliders [87] where the measured process is: e+ e− → e+ e− γ ∗ γ ∗ ,
γ ∗γ ∗ → π 0.
(43)
In the experiment one of the scattered electrons is detected and used to measure the virtuality Q 2 of the associated photon. A soft, quasi-real photon is assumed to be emitted by the undetected electron. The π 0 is detected through its decay π 0 → γ γ . The cross section is measured as a function of Q 2 . In Fig. 18 we show the experimental results together with two calculations [88, 89]. The calculations use both the asymptotic and the Chernyak–Zhitnitsky distribution amplitudes. The calculations by Jakob et al. [88] get satisfactory agreement when using the asymptotic distribution amplitude but a large overestimate with the Chernyak–Zhitnitsky distribution amplitude. The calculations by Cao et al. [89] underestimate the data for both distribution amplitudes. Other calculations [90] found moderate agreement with the asymptotic
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Fig. 18. Experimental values of the π 0 -photon transition form factor [87]. The solid lines are calculations using the asymptotic distribution amplitude and the dashed lines are with the Chernyak–Zhitnitsky distribution amplitude. Left: Ref. [88], right: Ref. [89].
Fig. 19. Electron–positron pair annihilating to form a photon which then decays to a pion pair. The pion form factor is represented by the circle.
distribution amplitude for Q 2 > 3 GeV2 . It appears that the asymptotic distribution amplitude gives better agreement than the Chernyak–Zhitnitsky distribution amplitude. This is in contrast with the charged pion where, while inconclusive, the apparent agreement with the Chernyak–Zhitnitsky distribution amplitude seemed better [39]. The problems can be traced to the inherent difficulty to deduce a functional form from an integral, which has to be done for these measurements. The form factors are related to the integral over the distribution amplitude and the scattering matrix element (Eq. (41)) and their sensitivity to the shape of the distribution amplitude is low. In addition there are the various model-dependent ingredients with different prescriptions for their calculation. A differential measurement of the LCWF is therefore necessary. Finally, there are the time-like form factors measured at positive four-momentum transfer: q 2 > 0 as opposed to the previously discussed space-like form factors. The calculation of time-like form factors is done in the same way as for the space-like form factors including the same distribution amplitudes [91]. The authors discuss the analytical continuity of the functions, αs (Q 2 ) and the propagators and compute the ratio of time-like to space-like form factors. This ratio is asymptotically unity but in the range of 15 ≤ |Q 2 | ≤ 40 GeV2 the ratio is about 2.0 with 10% variation depending on the distribution amplitude being used. The charged pion time-like form factor has been measured through the process: e+ e− → + π π − , Fig. 19. The cross section for this process is related to the form factor by O’Connel
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Fig. 20. Left: Charged pion time-like form factors measured in electron–positron pair annihilation [93]. Right: Calculated values of the pion time-like form factor using the asymptotic [6,75], Chernyak–Zhitnitsky [32] and the FHZ [76] distribution amplitudes.
et al. [92]: α 2 π (s − 4m 2π )3/2 |Fπ (s)|2 . (44) 3 s 5/2 Where |Fπ (s)| is the time-like form factor and the expression preceding it in Eq. (44) represent the cross section for the e+ e− annihilation and direct coupling of the photon to the pions, calculated in QED. An option to extend this approach to π + π − electroproduction in ep collision is discussed in Section 4.4. Results [93] from the e+ e− measurements are shown in Fig. 20 (left). In this energy range the form factor is dominated by the ρ and ρ resonances. In Fig. 20 (right) we see the calculations of [91]. If we take the highest non-zero values in Fig. 20 (left) we get Q 2 Fπ ∼ 0.4 for Q 2 ∼ 4 which would be in agreement with the values calculated with any of the distribution amplitudes, Fig. 20 (right). The calculations in the region of 4 < Q 2 < 15 show a decrease of Q 2 Fπ when the asymptotic distribution amplitude is used but an increase when the other distribution amplitudes are used. Above this all distribution amplitudes give a flat dependence and they differ only in the magnitude. At s = 9.6 GeV2 a value of the time-like form factor Q 2 Fπ = 0.94 ± 0.0077 was deduced from the ratio (J/ψ → π + π − )/(J/ψ → e+ e− ) [94]. This value is about 50% larger than the value calculated using the Chernyak–Zhitnitsky distribution amplitude and almost a factor 5 larger than the one calculated using the asymptotic distribution amplitude. There is a measured value for the space-like form factor at this energy: Q 2 Fπ = 0.68±0.19 [85]. These two experimental results would be consistent with the predicted ratio of 2.0. However, it will take a very special mechanism to have the value of time-like Q 2 Fπ change from 0.4 to 0.94 over such a narrow Q 2 range. It is therefore of particular interest to extend measurements of the pion time-like form factor to cover the 4 < Q 2 < 15 range. σ =
3.3. Measurement of the pion LCWF by diffractive dissociation to dijets 3.3.1. Concept of the experiment The discussion in Section 2.3 and in particular Eq. (27), taken from [69], shows that under certain circumstances the cross section for diffractive dissociation of a pion to dijets is
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proportional to the square of the distribution amplitude. This would mean that the u-dependence of the cross section is the same as that of the distribution amplitude squared and thus can be a direct measurement of it. The concept of the measurement presented here is the following: a high energy pion dissociates diffractively while interacting with a heavy nuclear target. The first (valence) Fock component dominates at large Q 2 ; the other terms are suppressed by powers of 1/Q 2 for each additional parton, according to counting rules [22,39]. This is a coherent process in which the quark and antiquark break apart and hadronize into two jets. If in this fragmentation process the quark momentum is transferred to the jet, measurement of the jet momentum gives the quark (and antiquark) momentum. Thus: pjet1 . (45) u measured = pjet1 + pjet2 We consider now the circumstances under which this can be expected to be valid. We note that Eq. (26) requires double differentiation with respect to kt of the LCWF, ψ(u, kt ), not the distribution amplitude φ(u) for which φas and φCZ were proposed. In their evaluation of the cross section Frankfurt et al. make some assumptions. The pion LCWF is taken to have the form: 2 ψ˜ π(L ) (u, κt2 ) = B(L 2 )
φ(u) kt2
+ µ2
(46)
where the parameters L and µ account for non-perturbative effects. It is necessary that kt2 µ2 . They estimate µ2 ∼ 0.1 GeV2 so a requirement that kt > 1 GeV should be imposed in the experiment. The actual form of the LCWF presented in Eq. (46) may not be appropriate for low kt . Another important assumption is that quark momenta are not modified by nuclear interactions; i.e., that color transparency is satisfied. This has been verified in the experiment described in Section 2.3 which is the same as the one used to measure the LCWF by diffractive dissociation to dijets discussed below [96]. The experiment and analysis were carried out by the Fermilab E791 Collaboration. The diffractive dissociation of high momentum pions into two jets can be described by factoring out the perturbative high momentum transfer process from the soft nonperturbative part [31]. This factorization allows for definition of the virtuality of the process, Q 2 as the masssquared of the dijets. This is analogous to Q 2 being the mass of the virtual photon in DIS. From simple kinematics and assuming that the masses of the jets are small compared with the mass of the dijet, the virtuality and mass-squared of the dijets are given by: 2 = Q 2 ∼ MDJ
kt2 , u(1 − u)
(47)
where kt is the transverse momentum of each jet and reflects the intrinsic transverse momentum of the valence quark or antiquark. By studying the momentum distribution for various kt bins, one can observe changes in the apparent fractions of asymptotic and Chernyak–Zhitnitsky contributions to the pion distribution amplitude. The basic assumption is that the momentum carried by the dissociating q q¯ is transferred to the dijets. Distortions of the momentum distribution can occur in the fragmentation of the quarks to jets and then in the detection process. This was examined by Monte Carlo (MC) simulations of the q q¯ momentum distribution according to the asymptotic and the Chernyak–Zhitnitsky distribution amplitudes (squared). The MC samples were allowed to hadronize through the LUND PYTHIA–JETSET model [95] and then passed through simulation of the experimental
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Fig. 21. Monte Carlo simulations of q q¯ momentum distributions as predicted by the two distribution amplitudes squared, 2 is the asymptotic function at the quark level (left) and of the reconstructed distributions of dijets as detected (right). φas 2 is the Chernyak–Zhitnitsky function (squared). The dijet mass used in the simulation is 6 GeV/c2 and (squared) and φcz the plots are for 1.5 GeV/c ≤ kt ≤ 2.5 GeV/c.
apparatus (Fig. 13) to account for the effect of unmeasured neutrals and other experimental distortions. The results are shown in Fig. 21 where the initial distributions at the quark level are compared with the final distributions of the detected dijets, including distortions in the hadronization process and influence due to experimental acceptance. As can be seen, the qualitative features of the two distributions are retained. 3.3.2. Longitudinal momentum distribution Under the assumptions discussed in the previous section, measurement of the longitudinal momentum distribution will be a measurement of the distribution amplitude. Results of experimental studies of the pion distribution amplitude were recently published by the Fermilab E791 Collaboration [96]. More details on the experimental system are given in Section 2.3. In the experiment, diffractive dissociation of 500 GeV/c negative pions interacting with carbon and platinum targets was measured. Diffractive dijets were required to carry the full beam momentum.
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Fig. 22. The u distribution of diffractive dijets from the platinum target for 1.25 ≤ kt ≤ 1.5 GeV/c (left) and for 1.5 ≤ kt ≤ 2.5 GeV/c (right). The solid line is a fit to a combination of the asymptotic and CZ distribution amplitudes. The dashed line shows the contribution from the asymptotic function and the dotted line that of the CZ function.
They were identified through the e−bqt dependence of their yield (qt2 is the square of the trans2
verse momentum transferred to the nucleus and b = R3 where R is the nuclear radius). For measurement of the wave function the most forward events (qt2 < 0.015 GeV/c2 ) from the platinum target were used, see Fig. 14. For these events, the value of u was computed from the measured longitudinal momenta of the jets. The analysis was carried out in two windows of transverse momentum kt : 1.25 GeV/c ≤ kt ≤ 1.5 GeV/c and 1.5 GeV/c ≤ kt ≤ 2.5 GeV/c. The resulting u distributions are shown in Fig. 22. In order to get a measure of the correspondence between the experimental results and the calculated distribution amplitudes, the results were fit with a linear combination of squares of the two distribution amplitudes after smearing, as shown on the right side of Fig. 21. This assumes an incoherent combination of the two distribution amplitudes and that the evolution of the Chernyak–Zhitnitsky function is slow (as stated in [32]). The results for the higher kt window show that the asymptotic distribution amplitude describes the data very well. Hence, for kt > 1.5 GeV/c, which translates to Q 2 ∼ 10 (GeV/c)2 , the pQCD approach that led to construction of the asymptotic distribution amplitude is reasonable. The distribution in the lower window is consistent with a significant contribution from the Chernyak–Zhitnitsky distribution amplitude or may indicate contributions due to other nonperturbative effects. The quantity measured in this experiment, the distribution of longitudinal momentum within a kt window, is not exactly the distribution amplitude. The latter is an integral over kt with a lower limit of zero, covering the low Q 2 non-perturbative region (Eq. (4)). The results can be regarded instead as representing the square of the light-cone wave function averaged over kt in the window: ψq2q¯ (u, kt ). With the measured kt -dependence described in Section 3.3.4 the average values are kt = 1.34 GeV/c and 1.75 GeV/c for the low and high kt windows, respectively: ψq2q¯ (u, 1.34) and ψq2q¯ (u, 1.75) were measured. Alternatively, the results for each window can be related to the difference of distribution amplitudes: 2 k 2 2 2 ψ(u, k )d k (48) t t = |φ(u, k 2 ) − φ(u, k 1 )| . 2
k1
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3.3.3. Coefficients of the Gegenbauer polynomials The results published by Fermilab E791 collaboration [96] were fitted to a linear combination 2 and φ 2 , Fig. 22. For φ of φAsy CZ the authors used the function of Eq. (37) [32] that was proposed CZ for a scale of 500 MeV. One could evolve the function to the higher Q 2 values of the measurement and attempts to do so were done [102,103,78]. However, the evolved function has to be subjected to the distortions that the E791 authors calculated in their Monte Carlo simulations and this was not done in [102,103,78]. Furthermore, φCZ is a model distribution amplitude and other distribution amplitudes were proposed. Indeed, the conclusion that was derived from the E791 results is that in the higher kt range the asymptotic distribution describes the data well while in the lower kt range there are significant non-perturbative contributions, not necessarily in the form of φCZ . We make here an attempt at deriving the actual structure of φ 2 from the data. If Eq. (27) holds and the cross section is proportional to the square of the distribution amplitude, φπ2 (u), the experimental results can be used to determine the coefficients of the Gegenbauer polynomials in the expansion of φπ (u, Q 2 ), Eq. (30). One cannot fit directly the results shown in Fig. 22 as these results are distorted by the hadronization process and experimental acceptance. It is also not practical to carry out simulations of the distortions for a large variety of distribution amplitudes. We adopt here a simpler approach, albeit somewhat less precise than that used by E791. In order to carry out such a fit we use results of the Monte Carlo simulations shown in Fig. 21. We define the Acceptance as the ratio of the reconstructed event distribution (right side of Fig. 21) and the generated distribution (left side of Fig. 21): Acceptance = reconstructed/generated. To first order the Acceptance should not depend on the distribution. For the Chernyak–Zhitnitsky distribution there is a singularity in the Acceptance near u = 0.5. This can be avoided by using only fully charged generated events, the distribution of which is finite near u = 0.5. It turns out that the Acceptance calculated for such events is nearly the same for the asymptotic and the Chernyak–Zhitnitsky distributions. We then use the Acceptance derived from the asymptotic distribution. For the high kt range it is nearly flat in the central u region and drops near the end points. For the low kt range it drops gradually from u = 0.5 towards the end points. We will here, as did the E791 authors, avoid the regions of u < 0.1, u > 0.9. In these regions jet momenta can be too low for reliable detection. By dividing the experimental results of Fig. 22 by the Acceptance calculated for the two kt regions the Acceptance-corrected distributions, which should be proportional to the cross sections, are obtained. The results are fitted to the expression: 2 dσ 3/2 3/2 ∝ φπ2 (u, Q 2 ) = N · u 2 (1 − u)2 1.0 + a2 C2 (2u − 1) + a4 C4 (2u − 1) (49) du where N is a normalization constant and Cn are the Gegenbauer polynomials. It is assumed that Cn = 0 for n > 4. In Fig. 23 we show the Acceptance-corrected distributions with the fits to Eq. (49). We note that, due to the flat Acceptance in the high kt range the Acceptancecorrected distribution is very similar to the uncorrected one. On the other hand, the drop of the Acceptance towards the end points for the low kt range makes the corrected values be higher than the uncorrected ones. The results of the fits are that for the high kt region a2 = a4 = 0 within errors. For the low kt region the coefficients are: a2 = 0.30 ± 0.05, a4 = (0.5 ± 0.1) × 10−2 . The results for the high kt region confirm the conclusion of the E791 authors, namely that for 2 describes the data well. For the results in the low k region we should note that this region φAsy t the quoted errors represent statistical errors only resulting from the fit procedure. There should be also the same systematic uncertainties discussed in [96] and perhaps some uncertainties due to the Acceptance calculations. Nevertheless, the results using some variations of the Acceptance
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Fig. 23. The Acceptance-corrected u distributions of diffractive dijets obtained by applying correction to the E791 results [96]. The distributions are for 1.25 ≤ kt ≤ 1.5 GeV/c (left) and for 1.5 ≤ kt ≤ 2.5 GeV/c (right). The solid line is a fit to a combination of Gegenbauer polynomials, Eq. (49).
were very stable. The fact that a4 = 0, which seems to be essential for a reasonable fit, indicates a distribution amplitude that is different from φCZ as defined in Eq. (37) which contains only a a2 term [32]. 3.3.4. Transverse momentum distribution As discussed in Section 2.3, derivation of the cross section for diffractive dissociation [69] is based on the double-differentiation of the LCWF with respect to kt (Eq. (26)). More specifically: 2 2 dσ 2 2 2 ∂ ∝ |α (k )x G(u, k )| ψ(u, k ) (50) s t N t , t 2 2 dkt ∂kt with x N = 2kt2 /s and G N the gluon distribution function in the nucleon. This doubledifferentiation leads to a prediction of the kt dependence of the cross section. By comparing the measured and predicted kt distributions it is possible to test to what extent the assumptions used in deriving the cross section are correct with sensitivity to both the LCWF and the interaction. When applying Eq. (26) to the pion LCWF given by Eq. (46) the differentiation with respect to kt does not modify the u-dependence if kt2 µ2 . An additional kt dependence comes from the 1
gluon distribution in the nucleon. With αs (kt2 )x N G(u, kt2 ) ∼ kt2 [97] this yields: M(N) ∝
xN GN kt4
,
dσ dkt2
∝
(x N G N )2 kt8
,
dσ ∝ kt−6 dkt
(51)
and the u-dependence is the same as for φ 2 (u), Eq. (27). The experimental results are shown in Fig. 24 where they are compared with several fits. An attempt to fit the data over the whole kt dσ range to a power-law dependence: dk ∝ ktn resulted in n = −9.2 ± 0.4(stat) ± 0.3(sys), much t larger than expected from Eq. (51). This result is dominated by the low kt high statistics region. It can be seen that for the larger kt the slope changes and when only the kt > 1.8 GeV region is fit to a power-law the result is n = −6.5 ± 2.0, consistent with the predictions, Fig. 24(a, b).
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Fig. 24. Comparison of the experimental kt distribution [96] with fits derived from: (a) Gaussian LCWF [98] for low kt dσ ∝ k n , as expected from perturbative calculations, for high k ; (b) Two-term Singletand a power law dependence: dk t t t Model wave function [99] for low kt and a power law for high kt .
This is also consistent with the conclusions of Sections 3.3.2 and 3.3.3 that the asymptotic distribution amplitude agrees with the data only for the large kt region. Both results indicate that for kt > 1.5 GeV/c, which translates to Q 2 > 10 (GeV/c)2 the asymptotic distribution amplitude and the pQCD calculations are applicable. Below this value non-perturbative effects may play a significant role. This answers the question raised at the end of Section 3.1 concerning the Q 2 value where the distribution amplitude may resemble the asymptotic one. Naturally, the transition between the two regions is not sharp. The region of 1.0 ≤ kt ≤ 1.8 GeV/c may be a transition region where we can still apply pQCD techniques and in particular Eq. (26) but must use LCWFs that better describe the non-perturbative structure of the pion. In Fig. 24(a) we show the results [96] of fitting the low kt region with the cross section derived with Eq. (26) and 2 the non-perturbative Gaussian function: ψ ∼ e−βkt [98], resulting in β = 1.78 ± 0.1. Modeldependent values for β in the range of 0.9–4.0 were predicted [98]. This fit, although resulting in the parameter β being consistent with theoretical expectations, is not very satisfactory. As seen in Fig. 24(a) the curved shape of the theoretical calculation is not observed in the data. In Fig. 24(b) we show the results of fitting the low kt region using the non-perturbative two-term Coulomb wave function [99] which describes the L z = Sz = 0 component of the u d¯ wave function: ψ(u, k⊥ ) ≡ Ψu d¯ (u, k⊥ ; ↑↓). [100,101]: ϕ(k z , k⊥ ) , ψ(u, k⊥ ) = √ u(1 − u)
ϕ( p) =
A pa2
+
2 p 2
+
B pb2
+ p 2
2
(52)
where A, B, pa , pb are parameters of fitting this function to the numerically calculated LCWF. The agreement with this function is very good. These comparisons show that the transition region may be described using non-perturbative LCWFs. Since this is a relatively large kt region for non-perturbative interactions, it is actually the high momentum tail of the functions that is being compared. The observed sensitivity shows that the kt distribution is useful in studying wave functions in this transition region.
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Fig. 25. Diagram of diffractive dissociation of a pion to two jets used for the calculations by Chernyak [103] and by Braun et al. [102,106].
3.3.5. Has E791 measured the pion distribution amplitude? Following publication of the E791 results [96] several theoretical papers were published discussing the question of whether they can indeed be taken as measurement of the pion distribution amplitude. The subject was also discussed in several conferences [104]. We bring here a brief summary of the main points that were raised and add some comments. The main questions that were discussed are: • Is the cross section for the process indeed proportional to φ(u)2 as claimed in Eq. (27) [69]? • Are the results precise enough to distinguish between φAsy (u) and other forms of φ(u)? Nikolaev et al. [74] calculate the cross section for diffractive dissociation of pions to dijets using pQCD methods. They show that the cross section is proportional to φ 2 (u) and to the unintegrated gluon structure function of the nucleon. They disagree with Frankfurt et al. [69] who used the integrated gluon structure function. They calculate higher-twist effects which contain some u-dependence but show that in nuclear medium they are suppressed. As a result, when the measurements are done in a heavy nuclear target the cross section is proportional to φ(u)2 and can be used to determine it. Hence their response to the first question is positive. Concerning the shape of φ(u) they propose a soft model distribution amplitude that has a different mathematical form than that of φAsy (u) but has a very similar u-dependence. Because of this similarity they conclude that the E791 results are consistent with their calculations as well. They are also able to reproduce the kt and A dependence observed in the experiment. V. Chernyak [103–105] calculates the process described in Fig. 25. The lower blob in the diagram represents the skewed gluon distribution of the nucleon. The upper blob represents the hard kernel of the amplitude that consists of 31 connected Born diagrams. Nuclear effects and the quark transverse momenta are ignored. Calculations of these diagrams lead to an expression for the amplitude which is not proportional to φ(u) but rather to a sum of four integrals over φ(u) multiplied by expressions that contain u-dependence. His conclusion is that the cross section depends on φ 2 (u) in a complicated way hence measurement of the cross section cannot provide a measurement of φ 2 (u). Chernyak disagrees with the authors of [74] as they ignore the contributions where the jet momenta differ significantly from the quark momenta. He agrees that making this assumption will lead to proportionality of the cross section and φ 2 (u). He also disagrees with the authors of [69] that ignored contributions from diagrams that are, according to their evaluation of the E791 conditions, suppressed by Sudakov form factors. Following these arguments Chernyak applies his calculations to φAsy (u) and to φCZ (u) which he evolves to the scale of 2 GeV. He does it by treating the pion as free q q¯ and does not use the logarithmic
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Fig. 26. Left: the Chernyak–Zhitnitsky distribution amplitude evolved to a scale of 2 GeV (Eq. (53), solid line) compared with φAsy (u) and φCZ (u), [103]. In the figure x has the same meaning as u. Right: the cross section calculated in [103] compared with experimental results. Solid line: using φCZ (u) evolved to 2 GeV, dashed line: using φAsy (u). The parameter y is the longitudinal momentum fraction of the jets which is not taken to be the same as u.
evolution. It should be noted that with this definition the four-momentum squared of the process becomes: p2 = Mq2q¯ =
kt2 u(1−u)
and not p2 = m 2π . He obtains the form:
φCZ (u, 2 GeV) = 15u(1 − u)[(2u − 1)2 + 0.2].
(53)
This function is shown in Fig. 26 (left). An attempt to fit this function to the experimental data presented in Fig. 23 failed, yielding χ 2 ∼ 5. The author then modifies the function to account for final state interaction of the outgoing jets. This disregards the color transparency conditions under which the measurement was done. The statement is: “The detailed consideration of nuclear effects is out the scope of this paper” [104]. The results of his cross section calculations are shown in Fig. 26 (right) compared with the experimental results in the high kt region. He concludes that the precision of the experimental results is insufficient for discrimination between these functions. Braun et al. [102,106] carry out calculations along the same lines as Chernyak (Fig. 25) and display all 31 Born diagrams that go in the calculations. There are some technical differences such as parametrization of the gluon distribution. They find that coherent jet production is dominated by gluon contributions and that collinear factorization is violated near the end-points and when the quark and jet momenta differ significantly. However, the amplitude near the end points resembles φAsy(u) and when the quark and jet momenta are not very different the cross section becomes proportional to φ 2 (u). The factorization is restored at energies much higher than the E791 energy. Like Chernyak, Braun et al. do the calculations for a nucleon target and not a heavy nucleus such as was used in E791. They state, however, that in a heavy nucleus color filtering will cause suppression of the nonfactorizable contributions. Bakulev et al. [78] use the CLEO data [87] to fit coefficients of Gegenbauer polynomial expansion, Eq. (30). They obtain results for a2 and a4 for several scales and conclude that the CLEO data are inconsistent with both φAsy (u) and φCZ (u). They obtain optimal coefficients opt opt a2 = 0.188, a4 = −0.13 and these values represent a distribution amplitude developed by
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them using nonlocal QCD sum rules at a scale of 1 GeV2 . They then use the method of [106] to calculate a cross section which turns out to be in good agreement with the E791 results. Comments The calculations carried out by Chernyak and by Braun et al., done for a nucleon target, indicate that the cross section for diffractive dijet production is not simply proportional to φ 2 as suggested in Eq. (27). They also reach the associated conclusion that the E791 results are not sufficiently precise for discriminating between the φAsy (u) and φCZ (u) evolved to the scale of 2 GeV. It should be noted that φCZ (u, 2 GeV) is significantly different from the φCZ (u, 0.5 GeV) (Fig. 26, left), not as would be expected from a slow, logarithmic evolution. The cross section 2 (u) (Fig. 26, right) which would show calculated using φAsy (u) is very similar in shape to φAsy that for this distribution amplitude the cross section is not far from being proportional to φ 2 (u). It appears that the conclusions of these works result from a very rapid evolution of φCZ (u) and distortions that affect φCZ (u) much more than φAsy(u) resulting in similar final results. The E791 results were obtained using a heavy nucleus and color transparency conditions were verified. Under these conditions the pion fluctuated to its PLC state before interacting with the target nucleus. According to [69,74,106] it may then be assumed that the cross section is in fact proportional to φ 2 (u). There are distortions which were taken into account by E791 using the PYTHIA simulations. Naturally, every experimental measurement has limits on its discriminating power between distribution amplitudes that are not very different from each other such as the one suggested in [74] which almost coincides with φAsy(u) or φCZ (u, 2 GeV) which is not very different from it. We should finally note that any comparison between theoretical calculations and experimental data should take into account experimental distortions. This can be done either, as done by E791, by subjecting the theoretical calculations to all the distortions that affect the experimental data or, as attempted to do in this work (Section 3.3.3), correct the experimental data for these distortions thus making the results readily comparable with theoretical calculations. 4. The photon light-cone wave function 4.1. Predictions of the photon LCWF The photon light-cone wave function can be described in a way similar to that of the pion except that it has two major components: the electromagnetic and the hadronic. Being a gauge field capable of point-like coupling it has also a point bare-photon component. Consequently, it can be expanded in terms of Fock states: (Λ) (Λ) ¯ ¯ |ψγ = |γ p + ψll/γ ¯ (u i , k ⊥i , λi )|l l + ψl lγ ¯ /γ (u i , k ⊥i , λi )|l lγ + · · · (Λ) (Λ) ¯ + ψq q¯ g/γ (u i , k⊥i , λi )|q qg ¯ + ··· + ψq q/γ ¯ (u i , k ⊥i , λi )|q q
(54)
¯ are lepton pairs with photons in where |γ p describes the point bare-photon and |l +l − , |l lγ the higher Fock states of the electromagnetic component. Similarly, |q q, ¯ |q qg ¯ are the q q¯ pairs with gluons in the higher Fock states of the hadronic component. Each of these states is a sum over the relevant helicity components. The wave function is very rich: it can be studied for real photons, for virtual photons of various virtualities, for transverse and longitudinal photons and the hadronic component may be decomposed according to the quark’s flavor. The pointbare photon does not have internal structure, it can only Compton-scatter. The LCWF for the lowest Fock states is described [16] through the interactions shown in Fig. 27. In Fig. 27(a) the
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Fig. 27. Diagrams for photon dissociation: (a) represents the diagrams leading to the photon electromagnetic dissociation, (b) is for dissociation to q q. ¯
electromagnetic component of the photon interacts via one photon exchange and dissociates to two leptons (e+ e− , µ+ µ− , τ + τ − ). In Fig. 27(b) the hadronic component interacts via two gluon exchange and dissociates into a q q¯ which then hadronize to a dijet. The form of the photon LCWF is given by Brodsky et al. [16]: ψλλ1 λ2 (kt , u) = −ee f
f λ1 (k)λ · λ f λ2 (q − k) 2 +m 2 √ k⊥ 2 u(1 − u) Q + u(1−u)
(55)
where λ is the polarization vector and f λ , m, λ1 , λ2 , ee f are the fermion distributions, masses, helicities and charges, respectively. Q 2 is the photon virtuality. Similar expressions are given in [107,108]. The predicted LCWFs for the electromagnetic component are based on quantum electrodynamics and can be considered precise. Those for the hadronic component are model dependent. The wave function for kt2 Λ2QCD is expected to be similar for the electromagnetic and hadronic components. The probability distribution amplitude Φ 2 is obtained from the trace of this function. For transversely polarized photon the result is: Φ 2f f¯/γ ∗ ∼ T
2 1 m 2 + kt2 [u 2 + (1 − u)2 ] Tr ψ 2 = 4 [kt2 + a 2 ]2 µ=1
(56)
and for longitudinally polarized photons: Φ 2f f¯/γ ∗ ∼ L
Q 2 [u 2 (1 − u)2 ] [kt2 + a 2 ]2
(57)
where f f¯ stands for l l¯ or q q¯ and a 2 = m 2 + Q 2 u(1 − u). While leptons usually satisfy m l2 kt2 this is not always so for quarks. For a real photon and light quarks (as for the electromagnetic component) the result is: Φ 2f f¯/γ ∼
1 kt2
[u 2 + (1 − u)2 ].
(58)
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Fig. 28. The photon wave function for transverse virtual photons (Q 2 = 5 (GeV/c)2 ) and massless quarks/leptons (solid line), real photons and massless quarks/leptons (dashed line), real photons and charm quarks (dotted line) and the asymptotic function (dashed–dotted line). The functions are arbitrarily normalized at u = 0.5.
For massive quarks: Φq2q/γ ∼ ¯
m 2q + kt2 [u 2 + (1 − u)2 ] [m 2q + kt2 ]2
(59)
and for virtual photons and light quarks: Φ 2f f¯/γ ∗ ∼ T
kt2 [u 2 + (1 − u)2 ] . [kt2 + Q 2 u(1 − u)]2
(60)
As in the case of the pion, the photon LCWF in the nonperturbative region is model dependent and Q 2 values for transition to the perturbative regime have to be determined. The actual lightk2
t cone wave functions of the photon are very interesting and subtle. At small invariant mass u(1−u) , the mixing of the photon with vector meson physics is inevitable. Balitsky et al. [109] predict for real photons a distribution amplitude identical to the pion asymptotic function [6]:
Φq2q/γ ∼ u 2 (1 − u)2 ¯
(61)
and presumably can be valid for kt > 1.5 GeV/c (as found for the pion). Examples of Φ 2 calculated using Eq. (56) and the asymptotic LCWF are shown in Fig. 28. Instanton models are now providing a guide to the two-quark valence wave function where the quark and antiquark helicities are parallel, a component proportional to the dynamical quark mass. The distribution amplitudes for these components, integrated over kt as in Eq. (4), were calculated by Petrov et al. [110] using the instanton model and are shown in Fig. 29.
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Fig. 29. The photon wave functions, Petrov et al. [110] Real photons (Solid line), virtual photons Q 2 = 250 MeV2 (dashed line), Q 2 = 500 MeV2 (dotted line).
4.2. Cross section of coherent dijet electroproduction As will be discussed in Sections 4.6–4.8 experiments were done in order to study the photon LCWF by measurements of diffractive dissociation of the photon to dimuons and dijets, similar to studies of the pion. The cross section for photon diffractive dissociation (DD) was calculated by several authors. In [112] this cross section is given in terms of the “dipole cross section”: 1 dσDD 2 (Q ) = du d2r |ΨγT∗,L (r , u, Q 2 )|2 σdipole(r , u) (62) dt 0 which is very similar to Eq. (20). ΨγT∗,L is the transverse/longitudinal wave function in coordinate space. Various models were proposed for the dipole cross section σdipole(r , u). A calculation specific for the photon diffractive dissociation to q q¯ followed by hadronization to dijets is given in [113]. The authors give expressions of cross sections for longitudinal and transverse photons in terms of the integrated gluon distribution of the proton for which they use some parametrizations. They give numerical predictions for the cross section dependence on various variables. In [114] the authors follow these calculations and give predictions for the azimuthal angular distributions (angle φ between lepton plane and dijet plane) of the dijets that originate from q q¯ fragmentation: kt2 − αem 1 + (1 − y)2 γ ∗ p e p γ∗p M2 dσ D,T − 2(1 − y) cos 2φ dσ D,T dσ D = 2 2 k 2 yQ π 1 − 2 Mt 2 ∗ ∗ γ p γ p + (1 − y)dσ D,L + (2 − y) (1 − y) cos φdσ D,I (63)
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Fig. 30. The φ dependence of total ep—cross section with different cuts in k 2 : 1 GeV2 < k 2 < 2 GeV2 , 2 GeV2 < k 2 , 5 GeV2 < k 2 (from top to bottom), Ref. [114]. γ∗p
γ∗p
γ∗p
where σ D,T , σ D,L , σ D,I are the transverse, longitudinal and interference cross sections. While a purely longitudinal cross section has no angular dependence, a purely transverse cross section does have such dependence resulting from interference of the two helicity amplitudes. The distribution peaks at π/2 for kt2 > 2 GeV2 , Fig. 30. This gives a unique signature which will be exploited in the experimental studies of this process (Section 4.8). The authors show that for other mechanisms such as boson–gluon fusion the azimuthal distribution has a minimum at π/2. A detailed discussion of these processes including evaluation of bounds on the cross sections is given in [115] where similar angular distributions are derived. In [116,117] the authors calculate the cross section for dissociation to q qg ¯ which is the dominant background for studies aimed at the q q¯ dissociation. These calculations help in determining the region of phase space where the signal to background ratio can be optimized. Another prediction of the cross section was recently published by Braun et al. [118] and the results are shown in Fig. 31. The authors calculate the cross sections for real photons and the other kinematic conditions as for the experiment described in Section 4.8. They show cross sections and angular distributions depending on the helicity of the Fock state. Each of the helicity states is expected to dominate in a different kt region so they can be distinguished experimentally. Their angular distribution peaks at π/2, similar to the observation in [114] for virtual photons, for both helicity states. More recently the group extended their calculation to virtual photons with kinematical conditions very similar to those used in the experiments described in Section 4.8 [119]. Their results are shown in Fig. 32 where, again, the meaning of z is the same as u. The solid and dashed lines are the results using the CTEQ6L and MRST2001LO parton distributions, respectively. The calculations are (top to bottom) for kt = 1.25, 1.5, 1.75 GeV/c. The condition β > 0.5 is similar to the one applied in the experimental analysis in order to enrich the fraction of q q¯ dijets. The effect of this condition on the u distribution is consequence of the associated reduction in phase space.
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Fig. 31. The cross section for dijet production by real photons predicted by Braun et al., Ref. [118]. The meaning of z is the same as u. The solid/dashed lines are for different cut-offs and mass parameters.
4.2.1. Cross section derived from the LCWF Here we present a calculation of the cross section for diffractive dissociation of the photon to dijets in a simplified approach that makes the role of the photon LCWF transparent. The aim is to compare the u-dependence of the cross section to that of the LCWF so that the ability to determine the LCWF from cross section measurements, as done for the pion, can be assessed. The derivations are done within DGLAP in the kinematical region kt2 m 2ρ . These calculations are equivalent to those of [113] in the double-leading-log approximation. The general relation of cross section and transition amplitude: dσ παs2 2 = |A|2 F2g 2 (16s)2 dt du dkt
(64)
where A is the amplitude of the process and is independent of t. The form factor F2g is given by: F2g = (1 − t/m 2 (2g))−2
(65)
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Fig. 32. The cross section for dijet production by virtual photons predicted by Braun et al., Ref. [119]. The meaning of z is the same as u. The solid and dashed lines are the results using CTEQ6L and MRST2001LO parton distributions, respectively. The calculations are (top to bottom) for kt = 1.25, 1.5, 1.75 GeV/c. The condition β > 0.5 is applied in (a) and not in (b).
where t = tmin − qt2 , qt is the momentum transferred to target. The two-gluon mass m(2g) was estimated to be ≈1 GeV [111]. For two-gluon exchange the amplitude is given by replacing σdipole ∝ b2 by −∇κ2t as done in Section 2.3, Eq. (26). Due to the polarization states we have here the trace of the second derivative of the LCWF. For longitudinally polarized photons: |A L |2 = (4/3)2 e2 eq2 (s · xg)2 · |Tr ∇κ2t ψ L |2
(66)
and for transversely polarized photons: |A T |2 =
1 (4/3)2 e2 eq2 (s · xg)2 · |Tr ∇κ2t ψ T |2 2
(67)
where ψ L and ψ T are the longitudinal and transverse LCWFs, respectively, following the prescription for photon LCWF as suggested by Brodsky and Lepage [6]. s is the total energy, (xg) the gluon distribution and eq the quark charge. Using Eq. (55) we can write the longitudinal LCWF: ψL =
Q · u(1 − u) a 2 + kt2
(68)
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where a 2 = Q 2 u(1 − u) + m 2q . Derivation of the transverse LCWF is more complex due to the two components. Using the relation: σi σ j = δi j + i i j k it can be written as: ψµT =
σµ m q + σz kµ (2u − 1) + i µlz ktl a 2 + kt2
.
(69)
Differentiation yields: ∇κ2t ∇κ2t
kµ 8kµ a 2 = − a 2 + kt2 (a 2 + kt2 )3 1 a2
+
kt2
=−
4(a 2 − kt2 ) (a 2 + kt2 )3
(70) .
(71)
We then get for the cross sections at t = 0: 2 2 2 2 2 dσ L παs2 2 2 2 2 u (1 − u) (a − k t ) e = Q (xg) · e q 9 dt du dkt2 (a 2 + kt2 )6
(72)
m 2q (a 2 − kt2 )2 + 2kt2 [u 2 + (1 − u)2 ] · 4(a 2)2 dσT παs2 2 2 2 e = (xg) · . e q 18 dt du dkt2 (a 2 + kt2 )6
(73)
and
For numerical calculations we use m q = 0. This is a reasonable approximation except for heavy quarks and near the end-points of the u distributions. We also take (xg) as constant to avoid using a non-transparent parametrization. The resulting u-distribution is shown in Fig. 33 for fixed values of Q 2 and M 2 and compared with Φ 2 . As can be seen the distributions are not the same, the cross section for this process is not proportional to Φ 2 . Experimental results should be compared with the predicted cross section which is sensitive to the u-dependence of the LCWF. For massive quarks it turns out that while the cross section is insensitive to the quark mass, the LCWF changes significantly (Fig. 28). The predictions shown in Figs. 31 and 33 are quite different and demonstrate the variety of expectations with which the experimental results will have to be compared. For the M 2 dependence of the cross section we use the relations: a 2 ± kt2 = u(1 − u)(Q 2 ± M 2 ) and d(kt2 ) = u(1 − u)d(M 2 ). For t = 0: 2 2 2 παs2 2 2 2 1 dσ L 2 (Q − M ) e = Q (xg) · · e q 9 dt du dM 2 (Q 2 + M 2 )6 u(1 − u) 4 2 2 2 dσT παs2 2 2 2 8Q M [u + (1 − u) ] e = (xg) · . e q dt du dM 2 18 (Q 2 + M 2 )6 u 2 (1 − u)2
(74) (75)
These results are equivalent to those of [113] in the double-leading-log approximation, the main difference being the explicit dependence on u. Note that for longitudinal photons the cross section is zero for Q 2 = M 2 . This was also found by the authors of Ref. [113] in the doubleleading-log approximation. The authors then add a non perturbative parametrization of the BFKL Pomeron which causes some shifts of the position of the minimal cross section. The ratio between longitudinal and transverse cross sections is given by: u 2 (1 − u)2 (a 2 − kt2 )2 dσ L = 2Q 2 2 . dσT 2kt [u 2 + (1 − u)2 ] · 4a 4
(76)
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Fig. 33. The cross section for dijet production by transverse photons at fixed Q 2 , M 2 (solid line) and the perturbative LCWF squared (dashed line). The two functions are normalized at u = 0.5.
For u = 0.5: 1 kt2 dσ L = dσT 8 Q2
2 Q2 0.25 2 − 1 kt
(77)
2
and is a function of Q2 . Numerically it can be seen that the longitudinal cross section is expected kt to be smaller by more than an order of magnitude than the transverse one, in agreement with the results of [113]. The correction due to m q = 0 may be non-negligible in particular near the dip region of the ratio σ L /σT . This region will be sensitive also to NLO corrections. We mention here two other parameters used in the literature for expressing these processes. If we go to the q q¯ center of mass system and neglect the quark mass, then using Eq. (2) we find the relation: 1 1 + cos θ ∗ (78) ; u(1 − u) = sin2 θ ∗ u= 2 4 where θ ∗ is the angle between the quark and the virtual photon in the center of mass system. The u-dependence is then translated to an angular distribution of the quark in the center of mass system. Another parameter is defined by: β=
Q2 Q 2 + M X2
(79)
where M X is the mass of the diffractive system. This parameter becomes very useful when trying to isolate dijets originating from q q¯ hadronization from those coming from q qg ¯ and other configurations. It is found that at large β the dominant contribution comes from the q q¯ component. This parameter is therefore useful for studying the LCWF of the q q¯ component of the photon. In this respect it is instructive to express Eqs. (74) and (75) in terms of β:
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Fig. 34. The q q¯ dissociation cross section induced by longitudinal photons divided by that for transverse photons, both calculated at u = 0.5. The minimum is not zero due to finite resolution.
παs2 2 2 dσ L eq (xg)2 · = e dt du dβ 9 dσT παs2 2 2 = e eq (xg)2 · dt du dβ 18
β2 1 (2β − 1)2 · 4 Q u(1 − u) β2 u 2 + (1 − u)2 β(1 − β) 2 . Q4 u (1 − u)2
(80) (81)
The ratio R = dσ L /dσT for u = 0.5 is shown in Fig. 34. The dip at β = 0.5 reflects the Q 2 = M 2 region seen in Eqs. (74) and (75). The minimum is not zero due to finite resolution. This region is sensitive to higher order contributions which will cause modifications in the minimum region [114]. As can be seen, in order to study the q q¯ LCWF one should be in a relatively large β to have q q¯ dominance but not too large so that the transverse photon dominate and the cos 2φ signature (Eq. (63)) can be used. 4.3. Cross section of coherent photoproduction of dimuons derived from the LCWF This is the well known Bethe–Heitler process the cross section for which is a text-book matter [120]. Nevertheless, we bring here a short derivation of this cross section in the LCWF language to emphasize the common features of this process and the one described in the previous section. We follow the same way as done for the dijet production so the common LCWF connection is evident. There are only two differences between the present derivation and that for the dijets: αs should be replaced by αEM and since the interaction is of one-photon exchange, rather than two gluon (Fig. 27), the amplitude is given by the trace of the first derivative of the LCWF, not the second. The rest follows the same order as in Section 4.2.1. For photoproduction we have only transversely polarized photons: 2 ∂ 2 T ψµ (82) |A T | ∝ Tr ∂kµ
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Fig. 35. The photon dissociation to q q¯ followed by hadronization to two pions.
where ψ T is the transverse LCWF: σµ m l + σz kµ (2u − 1) + i µlz ktl
ψµT =
a 2 + kt2
and a 2 = Q 2 u(1 − u) + m l2 . Differentiation yields: ∂ kµ 2 4kt4 2(a 2 − k 2 ) Tr = + 2 2 ∂kµ a 2 + kt (a 2 + kt )4 (a 2 + kt2 )3 ∂ 1 2 4kt2 = . Tr 2 2 2 ∂k a + k (a + k 2 )4 µ
t
(83)
(84) (85)
t
We then get for the cross sections at t = 0: 4m l2 kt2 + 2(kt4 + a 4 )[u 2 + (1 − u)2 ] dσT ∝ . dt du dkt2 (a 2 + kt2 )4
(86)
For real photons and using m l = 0 → a = 0: dσT dt du dkt2
∝
2[u 2 + (1 − u)2 ] kt4
∼
Φ2 kt2
.
(87)
Φ is the probability amplitude obtained from the trace of the LCWF for transversely polarized photons, Eq. (56) and the cross section is directly proportional to Φ 2 . 4.4. Cross section for dipion production The processes of photo- or electro-production of two pions may be considered as a special case of the photon dissociation to dijets when each jet consists of one pion (Fig. 35). This is a very exclusive process where the pion form factor and quantum numbers most probably affect the results. There may be consequences on the ratio of longitudinal/transverse cross sections and for the u-distribution. Calculation of many of these effects may be model dependent. At low π–π masses, Mππ , there will be influence of the tails of the ρ resonance and perhaps of higher resonances. Measurement of the cross section can be used to evaluate the pion time-like form
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factor, see Section 3.2. The cross section for this process is expected to be proportional to the time-like form factor: σ (γ ∗ + p → 2π + p) ∝ |Fπ |2 σ (γ ∗ + p → X + p)
(88)
where the denominator can be taken from parametrization of measurements and the results may have to be normalized to those obtained from e+ e− measurements [93]. It may be possible to use these measurements to extend measurements of the pion time-like form factor into the 4 < Q 2 < 15 region where there is great sensitivity to the pion light-cone wave function (Section 3.2). From azimuthal angular distribution as the one described in Section 4.2, Eq. (63) it may be possible to determine the contributions from longitudinal and transverse photons which may be quite different from that for dijets. With this determined measurements of the longitudinal momentum fraction u can be compared with the predicted cross sections for dijets. This comparison may lead to conclusions on the extent to which the two processes are similar. This will have implication on our understanding of the hadronization process. The t-distribution for this reaction may be considered in the same way as for vector meson production (Section 2.2). The slope of the distribution measures the combined size of the q q¯ system and the proton, Eq. (18) and it will allow measurement of the q q¯ size also in the hadronization to continuum states. The cross section for 2π production is expected to be suppressed by |Fπ |2 compared with dijet production. We can expect |Fπ |2 ∝ 1/kt4 or ∝ 1/M 4 2 dependence of the cross section is hence expected to fall rapidly or ∝ 1/(Q 2 + M 2 )2 . The Mππ with the mass and this reduces the possibility for studies of this process at large masses. Studies of this process for relatively low masses, just above the ρ, ρ , f 2 resonances will result in a relatively large value of β and thus be dominanted by q q¯ produced from longitudinal photons (Fig. 34). 4.5. The Odderon The exclusive dipion reaction is considered as one of the ways to hunt for the elusive Odderon [121,122]. The Odderon is described as a d-coupled three-gluon color singlet C = −1 object [123]. The Odderon and the Pomeron are expected to have very similar intercepts with values near 1.0. Several attempts to observe the Odderon have so far failed. These were aimed at observing processes which the Pomeron cannot mediate while the Odderon can. Since the photon has C-parity C = −1, it can couple with the C = +1 Pomeron only to C = −1 mesons while the C = +1 mesons are expected to be produced by Odderon exchange. Following specific predictions [124], the H1 collaboration studied the photoproduction of the C = +1 π 0 and f 2 mesons. They obtained only upper limits for the cross sections which were lower than the predicted ones, [125,126]. It has been suggested [127] that as the amplitude for these processes is very weak one should look at signatures that are linear with this amplitude rather than quadratic, such as inteference effects. The Odderon–Pomeron interference can present such a case. Since the charge parity of dipions is: C(π + π − ) = (−) where is their relative angular momentum, both Pomeron and Odderon can contribute to their production and the interference will show up. The signature would be the charge asymmetry: A=
u(+) − u(−) u(+) + u(−)
(89)
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Fig. 36. The ZEUS detector at HERA, DESY. For description see [128].
which would be studied as a function of Mππ . This asymmetry was calculated for dipion electroproduction [121] and photoproduction [122]. It turns out that the asymmetry is proportional to the difference of the Pomeron and Odderon intercepts. As this is expected to be rather small the calculations utilize the effects of resonances, in particular that of the f 2 . Modeldependent asymmetries of the order a of a few percent are predicted in the f 2 mass region. 4.6. Measurement of the electromagnetic component of real photon LCWF Measurements of the photon light-cone wave function are being carried out at the DESY accelerator in the collision of 27.5 GeV/c electrons (or positrons) with 920 GeV/c protons producing real or virtual photons. The measurements were done using the ZEUS detector (Fig. 36) [128]. Measurement of the electromagnetic component of the real photon LCWF was done using the exclusive ep → eµ+ µ− p photoproduction process. Preliminary results were reported recently [129]. 4.6.1. Data analysis The integrated luminosity for the results was 55.4 ±1.3 pb−1 . Events of the exclusive reaction were triggered using the muon chambers. Each event was required to have two high quality tracks fitted to the vertex and matched to energy deposits in the calorimeter. It was required that no signal from the scattered positron or from proton dissociation be recorded. The LCWF was measured as the distribution of the longitudinal light-cone momentum fraction as defined in Eq. (2) (see Fig. 37): u=
E 1 + pz1 E 1 + pz1 + E 2 + pz2
(90)
where E 1 , p1 , E 2 , p2 are the energy and momentum of each muon. The kinematic region was defined by the following selection criteria: the invariant mass of the dimuon system 4 < Mµµ < 15 GeV, the γ p center of mass energy 30 < W < 170 GeV, the square of the four momentum exchanged at the proton vertex |t| < 0.5 GeV2 (to select diffractive events), 0.1 < u < 0.9 (to avoid regions of very low momenta where the acceptance is not well defined) and k T > 1.2 GeV (to select a hard process). The acceptance corrections and resolutions were determined using the
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Fig. 37. Definition of the kinematic variables in the γ → µ+ µ− process.
dedicated Monte Carlo GRAPE generator [130] for dilepton production in ep collisions, based on exact matrix elements. The program is interfaced to PYTHIA to generate a complete final state. The simulation included both elastic and proton-dissociative events. All generated events were passed through a detector simulation based on GEANT 3.13. 4.6.2. Results and discussion The measured k T , W, t, Mµµ distributions are presented in Fig. 38. The data are compared with GRAPE Monte Carlo simulation normalized to the data. Good agreement is shown between data and simulation. The measured differential cross section dσ/du is presented in Fig. 39 and compared with the theoretical curve (BFGMS) [16]. For this purpose Eq. (56) was adapted to the muon’s analysis: 2 Φµµ/γ =
u 2 + (1 − u)2 . 2 u (1 − u) − m 2 ]2 [Mµµ µ
(91)
The LCWF was normalized to the data. The shape of the calculated electromagnetic component of the real photon light-cone wave function is in good agreement with the experimental result. The measured cross section was also compared to the GRAPE simulation normalized to the luminosity. The agreement between data and simulation is within the systematic uncertainties. This measurement serves as a “Standard Candle” and normalization for the hadronic LCWF. It also provides the first proof that diffractive dissociation of particles can be reliably used to measure their LCWF. Furthermore it gives support for the method used in previous measurements of the pion LCWF [96] and possible future applications to other hadrons [131]. 4.7. Measurement of the hadronic component of the photon LCWF by exclusive dipion electroproduction Measurement of exclusive diffractive electroproduction of π + π − pairs was carried out by the ZEUS collaboration. Preliminary results were reported recently [133]. The concept of the measurement is the same as described in Sections 3.3 and 4.6. Viewed in the proton rest frame the photon fluctuates to its Fock states long before the interaction when it dissociates diffractively. The first (valence) Fock component dominates while the other terms are suppressed
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Fig. 38. Number of events reconstructed in the γ → µ+ µ− process for the kinematic region 4 < Mµµ < 15 GeV, 30 < W < 170 GeV/c |t| < 0.5 (GeV/c)2 and k T > 1.2 GeV/c plotted against k T , W, |t| and Mµµ . The data distributions are shown as the points with statistical errors only. The solid lines show the prediction of the GRAPE generator summing elastic and proton-diffractive components.
according to counting rules [22]. In this exclusive process the quarks break apart and hadronize into two jets which may be a π + π − pair. Measurement of their momenta gives the quark momenta. 4.7.1. Data analysis The data were collected at the HERA ep collider during 1999–2000 with the ZEUS [128] detector. At that time HERA operated at proton energy of 920 GeV and at positron energy of 27.5 GeV. The integrated luminosity used was 66.3 ± 1.7 pb−1 . Each event was demanded to have two high quality tracks from the central tracking detector, fitted to the vertex and matched to energy deposits in the uranium calorimeter. Pions were identified using a neural network algorithm. The total charge of the two tracks was required to be zero. The energy E and momentum z-component pz of the scattered positron and two pions were required to satisfy the relation 40 GeV ≤ E − pz ≤ 60 GeV. To select exclusive events, the total energy deposited in the calorimeter not associated with the scattered positron or the pions was required to be less than 300 MeV. In order to limit the contribution from proton-dissociative events, ep → eπ + π − N, the energy deposited in the forward plug calorimeter, located close to the beam pipe, was required to be less than 1 GeV.
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Fig. 39. Differential cross section dσ/du for the γ → µ+ µ− process measured for 30 < W < 170 GeV, 4 < Mµµ < 15 GeV, k T > 1.2 GeV/c and −t < 0.5 (GeV/c)2 . The inner error bars show the statistical uncertainty; the outer error bars show the statistical and systematics added in quadrature. The data points are compared to the prediction of LCWF theory [16]. The theory is normalized to data.
The kinematic variables are the same as shown in Fig. 37, replacing µ by π. These are W , the γ ∗ p center-of-mass energy, Q 2 , the photon virtuality, t the square of the four momentum exchanged at the proton vertex, and u, the longitudinal momentum-fraction carried by the pion. The kinematic region for this measurement is defined as: 2 < Q 2 < 20 GeV2 , 1.2 < Mππ < 5 GeV, 40 < W < 120 GeV, |t| < 0.5 GeV2 , 0.1 < u < 0.9 (to avoid regions of low acceptance). At that stage a remaining contamination of about 8% from proton dissociative events was not subtracted. The acceptance corrections and resolution effects were determined using the dedicated Monte Carlo ZEUSVM generator [132]. 4.7.2. Results Differential cross sections as a function of Mππ , t and u are presented in Figs. 40, 41, 42 respectively. The systematic uncertainties are dominated by the uncertainties related to the trigger selection and identification of the scattered positron. As can be seen from Eqs. (74) and (75) the predicted dependence of the cross sections on 2 are strongly coupled. Consequently, there is no simple prediction for the mass Q 2 and Mππ 2 is very different from Q 2 or a well defined value of Q 2 is dependence unless either Mππ selected. In the ZEUS experiment Q 2 and the mass have similar values and it is impractical to select a narrow Q 2 range because of limited statistics. The authors therefore chose to make n with a result: n ≈ 4.5. This distribution, when a power-law fit to the mass distribution: 1/Mππ ∗ divided by the mass dependence of the γ p → q q¯ cross section is proportional to the pion time-like form factor squared. It can be used to extend the measurements of this form factor from the highest available value at 2 GeV [93] up to about 4 GeV. Such results will provide an alternative measurement of this form factor from the evaluation at the J/ψ mass deduced from the (J/ψ → π + π − )/(J/ψ → e+ e− ) [94] which is in great disagreement with theoretical predictions [91].
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Fig. 40. The measured differential cross section dσ/dMπ π with Q 2 ≈ 7 (GeV/c)2 . The solid line indicates a fit to the form 1/Mπn π and n ≈ 4.5.
Fig. 41. The measured differential cross section dσ/dt for Mπ π > 1.5 GeV, Q 2 = 8.5 (GeV/c)2 and W = 76 GeV. The solid line indicates a fit to the form ebt .
Fig. 42. The differential cross section dσ/du measured in two Q 2 intervals: 2 < Q 2 < 5 GeV2 (left) and 5 < Q 2 < 20 GeV2 (right). The inner error-bars show the statistical uncertainties; the outer error-bars show the statistical and systematic uncertainties added in quadrature. The data points are compared to the LCWF predictions, Eqs. (74) and (75).
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The measured t distribution, Fig. 41 is fitted with the standard exponential form dσ/dt ∝ ebt . −2 The resulting slope b = −4.81 ± 0.42(stat)+0.05 is in good agreement with the −0.44 (sys) GeV values observed for vector-meson production, Fig. 5. The slope of the t-distribution is considered as a measure of the combined size of the q q¯ system and the proton: b = 13 R 2 + 18 r 2 where r ∼ 6 2 is the size of the q q¯ system before it hadronizes to a vector meson or, in this case, Q 2 +m V
to two pions in the continuum and R is the proton radius [57]. The results show that the q q¯ size in hadronization to continuum states is similar to that obtained for vector mesons. The u distribution is presented in Fig. 42 for two Q 2 intervals. Its shape is compared to the LCWF predictions for transverse and longitudinal photons, Eqs. (74) and (75). Such a comparison is legitimate if the pion quantum numbers do not affect their angular (u) distributions for a given photon polarization. The normalization for the longitudinal LCWF prediction was determined from a fit to the data. For the transverse LCWF prediction it was fixed to be the same value as that for the longitudinal prediction at u = 0.5. The u distribution is similar in the two Q 2 intervals. Indeed, Eqs. (74) and (75) predict a Q 2 -independent u-dependence provided the relative contributions from longitudinal and transverse photons do not change. We note, however, that the measured distribution in the low Q 2 region (Fig. 42 left) is more irregular than the one for the higher Q 2 range. This irregularity can be traced to the fact that the low Q 2 range is close to the average value of the dipion mass squared. In fact, for this range β = 0.52 which is where σ L /σT ∼ 0 in leading order, Fig. 34. Higher order effects are expected to play a role here and the pure longitudinal fluctuations may be shadowed. By contrast, for the high Q 2 range β = 0.75 which is a “safe” region. The results are consistent with the LCWF predictions for longitudinally polarized photons. The dominance of longitudinal photons has also been observed in the diffractive electroproduction of ρ 0 mesons [134]. The agreement between the measured u distributions and the predictions lends support to the assumption that non-resonant di-pion production is sensitive to the q q¯ component of the light-cone wave function of the virtual photon. This shows that for the phase space parameters of this study the LCWF predicted by perturbative QCD, Eq. (55) is correct. This process of exclusive diffractive electroproduction of pion pairs can be correctly described as resulting from a longitudinal photon fluctuating to a q q¯ pair which in turn hadronizes to a π + π − pair, Fig. 35. The agreement of the u-distribution (or angular distribution, cf. Eq. (78)) measured with pions in the continuum and in the resonance region above the ρ with calculations made at the parton level lends support to the picture that there is parton/hadron duality, [136]. 4.8. Measurement of the hadronic component of the photon LCWF by exclusive dijet electroproduction Measurement of exclusive diffractive electroproduction of dijets is being carried out by the ZEUS collaboration following the same concepts as in Sections 3.3, 4.6 and 4.7. The data used in this analysis is similar to that described in Section 4.7. The initial selection criteria are similar in the requirements of diffractive and exclusive events. The main difference is the way the events are selected in order to identify exclusive diffractive dijets. Furthermore, it is necessary to select those dijets likely to be produced from transverse photons fluctuating to q q¯ pairs. As discussed in Section 4.2.1 these conditions can be met by selecting large β to have q q¯ dominance but not so large as to have transverse photon dominance. Under these conditions it may be possible to observe the cos 2φ dependence of the dijet angular distribution. This would serve as an experimental signature for dijets originating from a transverse photon fluctuating in a q q¯ Fock state.
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These requirements and the difficulties inherent in jet identification dictated somewhat different phase space conditions than those of the dipion study. Here both Q 2 and the dijet mass MDJ are larger and a kt cut has to be applied in order to separate the jets. The jets were identified in the γ ∗ − P center of mass system using a “kt clustering” algorithm [135]. The selection parameter ycut that optimizes dijet selection was chosen by extensive Monte Carlo simulations. This analysis is still in progress and final results are expected soon. 5. Summary and future directions Understanding the internal structure of hadrons is one of the most challenging goals of particle and nuclear physics. In this work we concentrated on two related aspects of the internal hadronic structure: studies of the phenomenon of color transparency and measuring the valence light-cone wave function. All these studies are done in large momentum transfer processes. The CT effect requires small configurations hence a large internal momentum reached through large momentum transfer. Studies of the LCWF were done through diffractive dissociation to large transverse momentum dijets. The measurement of the photon electromagnetic LCWF through dimuon photoproduction presents a test of the concept behind these measurements. The good agreement of the results with QED show that diffractive dissociation is a good tool to study LCWFs. Measurements of the pion LCWF show that for kt > 1.5 GeV/c the LCWF derived in pQCD describes well the measured longitudinal and transverse momentum distributions. This sets a scale for applicability of the asymptotic pQCD calculations. These measurements also show a transition region from perturbative to non-perturbative regimes at lower transverse momenta. Measurements of N N scattering, d(γ , p)n photodisintegration [23], and perhaps d(d, n) 3 He and d(d, p) 3 H reactions [24] are consistent with QCD counting rules [22] at transverse momenta similar to those observed in the pion LCWF measurements. Measurements of the photon hadronic LCWF are under way at HERA and results for the photon dissociation to π + π − pairs have recently become available. They show a longitudinal photon fluctuating to a q q¯ pair which in turn hadronizes to a π + π − pair. The u-distribution is described correctly by the LCWF predicted by perturbative QCD. This agreement, observed in the continuum and the resonance region above the ρ lends support to the picture that there is parton/hadron duality. More results of these studies, including charge-asymmentry measurements for observation of Pomeron–Odderon interference are expected to be available in the near future. Results for measurements of the photon dissociation to dijets are also expected to be available soon. The main achievements summarized in this work are: • Color transparency was observed in vector meson production and, more directly, through the A-dependence of the cross section for diffractive dissociation of pions to dijets. • Direct measurement of distribution amplitudes by diffractive dissociation was demonstrated. • The distribution amplitude for pions was determined in a Q 2 region that allows observation of the transition from nonperturbative to perturbative regimes. • These measurements show that for Q 2 > 10 (GeV/c)2 perturbative calculations are applicable in these systems. • The distribution amplitude for the elecromagnetic component of real photons was measured through diffractive dissociation of real photons to µ+ µ− . The results, being in agreement with QED predictions, demonstrate the applicability of the diffractive dissociation method to distribution amplitude measurements.
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Fig. 43. Schematic description of measurements of the proton and antiproton LCWF at the Bevatron collider. Roman pots are used to identify the intact proton while the antiproton dissociates to three jets and similarly detecting the intact antiproton for proton dissociation.
• The distribution amplitude for longitudinal virtual photons fluctuating to q q¯ was measured through diffractive dissociation of virtual photons to π + π − . This also supports the concept of parton/hadron duality. • The distribution amplitude for transverse virtual photons fluctuating to q q¯ was measured through diffractive dissociation of virtual photons to dijets. Measurements of LCWFs were so far done only in q q¯ systems: the pion and the photon. The phenomenon of CT was unumbigously observed only in mesons: in vector meson production and in diffractive dissociation of pions. Attempts to observe CT with protons, while showing significant deviations from expectations based on Glauber calculations, did not produce the results expected from color transparency. This, together with difficulties in the interpretation of spin effects in proton–proton scattering, make it very important to carry out studies of CT and measurements of the LCWF of baryons. Following the observation of CT with pions, large momentum transfer (e, e π) studies should show the effect too. The (e, e k) process should be investigated as well. Studies of CT for the proton via proton dissociation to three jets in interaction with heavy nuclei can be carried out at RHIC. A natural extension of the method used to measure the LCWFs of pions and photons is the study of momentum distributions in diffractive dissociation of protons into three jets. A valence proton asymptotic distribution amplitude for three quarks is given in [6]: −2/3β Q2 ; φqqq/ p (u i , Q ) = C · (u 1 · u 2 · u 3 ) ln 2 Λ 2
u i = 1.
(92)
This shows an expected distribution that peaks when the jets have similar momenta. However, this symmetric form may be more appropriate for the ∆ resonance that has a symmetric valence quark state. A detailed discussion of the nucleon and ∆ distribution amplitudes is given in [137]. Also, following measurements of the proton spin structure functions, the possibility that orbital angular momentum has to be included in the proton distribution amplitude was considered [138]. A schematic description of such an experiment at the Fermilab Tevatron collider is shown in Fig. 43. In the p p¯ collision one would look for the dissociation of the proton while the antiproton remains intact and vice versa. For this purpose the Roman pots can be used detecting one of the particles while three jets originating from dissociation of the second particle are detected in the main detector (CDF or D0). One could also envision studying the proton dissociation to three jets in HERA. Such measurements can be eventually extended to the LHC.
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Other extensions of the same principle are measurement of the kaon LCWF which can be done in the same fashion as done for the pion provided a fixed target experiment with a kaon beam can be realized at the Tevatron. For the photon, the results presented here are only a beginning of a large study that can be realized at HERA. Measurements of the photon strange LCWF can be done by tagging the dijets with strange mesons. Measurements of the photon charmed LCWF can be done by tagging the dijets with charm mesons. The very clean D ∗ signals obtained by both ZEUS and H1 collaborations can be utilized for this purpose. This will have a heavy price in statistics but if all the data collected in both HERA I and HERA II runs is combined it may be sufficient. Another study which can be done in HERA II is the spin LCWF of the photon and spin-asymmetry production of dijets and dipions by utilizing data taken with polarized electron and positron beams. Studies of photon dissociation to three jets can yield information on the second Fock state of the photon LCWF. One can look forward to a wealth of information that will enrich our understanding of hadronic structure. Acknowledgements I would like to thank Drs. S. Brodsky and L. Frankfurt for many valuable discussions and Dr. S. Brodsky for carefully reading this manuscript. Comments made by Drs. M. Diehl, B. Kopeliovich and G. Miller are greatly appreciated. I also want to acknowledge the efforts of the E791 Collaboration and particularly my graduate student Dr. Ruth Weiss-Babai for the data on diffractive dissociation of pions. I acknowledge the efforts of the ZEUS collaboration and particularly Dr. H. Abramowicz, Dr. I. Cohen and graduate students E. Nevo, J. Szuba and J. Ukleja for the data on diffractive dissociation of photons. This work was supported in part by the US–Israel Binational Science Foundation and by the Israel Science Foundation. References [1] [2] [3] [4] [5] [6]
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See e.g. V.B. Berestetskii, E.M. Lifshitz, L.P. Pitaevskii, Quantum Electrodynamics, Pergamon Press, 1982. Ph. Hagler et al., Phys. Lett. B 535 (2002) 117. I.F. Ginzburg, I.P. Ivanov, N.N. Nikolaev, Eur. Phys. J. C 32 (Direct) (2003) s23. L. Lukaszuk, B. Nicolescu, Lett. Nuovo Cimento 8 (1973) 405; D. Joyson et al., Nuovo Cimento A 30 (1975) 345. M. Rueter, H.G. Dosch, O. Nachtmann, Phys. Rev. D 59 (1999) 014018; E.R. Berger et al., Eur. Phys. J. C 9 (1999) 491; E.R. Berger et al., Eur. Phys. J. C 14 (2000) 673. T. Golling, H1 Collaboration, DIS 2001 IX International Workshop on Deep Inelastic Scattering, Bologna, April, 2001. J. Olson, H1 Collaboration, New Trends in High Energy Physics, Talta, Crimea, 2001. S.J. Brodsky, J. Rathsman, C. Merino, Phys. Lett. B 461 (1999) 114. ZEUS Collaboration, U. Holm et al., The ZEUS Detector DESY 1993 http://www-zeus.desy.de/bluebook/bluebook.html (unpublished). ZEUS Collaboration, J.M. Ukleja, in: Proceedings of the HEP2003 Europhysics Conference July 17th–23rd 2003 Aachen, Germany, European Physical Journal 33 (2004) s506; ZEUS Collaboration, J.M. Ukleja, in: D. Bruncko, J. Ferenecei, P. Strizenec (Eds.) Proceedings of the XII International Workshop on Deep Inelastic Scattering, Strbske Pleso, Slovakia, April 2004, p. 554. T. Abe et al., in: T.A. Doyle et al. (Eds.), Proc. Workshop on Monte Carlo Generators for HERA Physics, DESY, 1999. D. Ashery, Comments Modern Phys. 2A (2002) 235. K. Muchorowski, Ph.D. Thesis, University of Warsaw, 1998 unpublished. H1 Collaboration, ZEUS Collaboration, N. Coppola, Proceedings of 32nd International Conference on HighEnergy Physics, ICHEP 04, 16–22 August 2004, Beijing, China, 2004. hep-ex/0410057. ZEUS Collaboration, Eur. Phys. J. C 12 (2000) 393. S.D. Ellis, D.E. Soper, Phys. Rev. D 48 (1993) 3160. F.E. Close, N. Isgur, Phys. Lett. B 509 (2001) 81. V.M. Braun et al., Nuclear Phys. B 553 (1999) 355. X. Ji, J.-P. Ma, F. Yuan, Nuclear Phys. B 652 (2003) 383.
Review
High momentum diffractive processes and hadronic structure Daniel Ashery School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Israel Received 14 April 2005
Abstract High momentum diffractive processes can be used to study the internal structure of hadrons. This structure is described using the light-cone formalism and the concept of light-cone wave functions (LCWFs). These exclusive observables are related to the more inclusive ones, the structure functions and form factors. Experimental methods to measure the LCWFs through measurements of form factors have proven rather insensitive and alternative approaches are necessary. Measurements of LCWFs using diffractive dissociation are differential and determine the momentum distributions of the valence partons in the hadron. They provide wave function information at the amplitude level. The concept of these measurements relies on the factorization of soft and hard processes. Recent measurements of the pion and the photon LCWFs by diffractive dissociation are described. The measurement of the pion LCWF has been extensively and critically discussed by several authors. These discussions are summarized and conclusions are drawn. These experiments allow observation of the transition from perturbative to non-perturbative QCD regimes. This transition is related to the soft components of the LCWF and to chiral symmetry breaking. The phenomenon of color transparency (CT) is an important component of the experimental approach and is closely related to the LCWF description of the hadronic structure. Observation of this effect in diffractive dissociation of pions interacting with nuclear targets is described and compared with other studies of CT. We finally discuss several future directions in this field. c 2006 Published by Elsevier B.V. Keywords: Hadronic structure; Light-cone wave functions
Contents 1.
Introduction......................................................................................................................280 E-mail address: [email protected].
c 2006 Published by Elsevier B.V. 0146-6410/$ - see front matter doi:10.1016/j.ppnp.2005.08.003
280
2.
3.
4.
5.
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1.1. Hadronic structure................................................................................................... 280 1.2. Applications of the LCWF formalism ........................................................................ 282 1.3. Factorization and hard processes ............................................................................... 284 1.4. Diffraction ............................................................................................................. 285 Color transparency ............................................................................................................ 286 2.1. Quasi-elastic scattering ............................................................................................ 287 2.2. Vector meson production ......................................................................................... 291 2.3. Pion diffractive dissociation to dijets ......................................................................... 297 The pion light-cone wave function ...................................................................................... 302 3.1. Theoretical predictions ............................................................................................ 302 3.2. Distribution amplitudes and form factors ................................................................... 303 3.3. Measurement of the pion LCWF by diffractive dissociation to dijets ............................. 307 3.3.1. Concept of the experiment ........................................................................... 307 3.3.2. Longitudinal momentum distribution ............................................................ 309 3.3.3. Coefficients of the Gegenbauer polynomials .................................................. 311 3.3.4. Transverse momentum distribution ............................................................... 312 3.3.5. Has E791 measured the pion distribution amplitude? ...................................... 314 The photon light-cone wave function ................................................................................... 316 4.1. Predictions of the photon LCWF............................................................................... 316 4.2. Cross section of coherent dijet electroproduction ........................................................ 319 4.2.1. Cross section derived from the LCWF........................................................... 321 4.3. Cross section of coherent photoproduction of dimuons derived from the LCWF ............. 325 4.4. Cross section for dipion production ........................................................................... 326 4.5. The Odderon .......................................................................................................... 327 4.6. Measurement of the electromagnetic component of real photon LCWF ......................... 328 4.6.1. Data analysis.............................................................................................. 328 4.6.2. Results and discussion................................................................................. 329 4.7. Measurement of the hadronic component of the photon LCWF by exclusive dipion electroproduction .................................................................................................... 329 4.7.1. Data analysis.............................................................................................. 330 4.7.2. Results ...................................................................................................... 331 4.8. Measurement of the hadronic component of the photon LCWF by exclusive dijet electroproduction .................................................................................................... 333 Summary and future directions ........................................................................................... 334 Acknowledgements ........................................................................................................... 336 References ....................................................................................................................... 336
1. Introduction 1.1. Hadronic structure The most challenging nonperturbative problem in Quantum Chromodynamics is to determine the structure of hadrons in terms of their quark and gluon degrees of freedom. This understanding is a very important infrastructure in the interpretation of many processes. As is the case for atoms and nuclei, the structure should be described through wave functions that are, in turn, solutions of the appropriate Hamiltonian equation. The wave function should provide the probability amplitude to find the parton in a given position in coordinate space or a given momentum in momentum space. The differences between the Hamiltonian of an atomic or nuclear system and that of a hadron stem from three main reasons:
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• The interaction between quarks and gluons is not well understood. • The number of partons is, in principle, unbounded while the number of electrons or nucleons is finite. • It should be fully relativistic. Several approaches have been developed to tackle these problems, such as using equaltime quantization motivated by non-relativistic potentials or applying the Bethe–Salpeter formalism [1]. However, these calculations have proven to be extremely complex as they require understanding of the solutions over all space and time and boosting which is by itself very hard to do. Hence, their applicability is limited. A powerful method for describing the bound state structure of relativistic composite systems in quantum field theory is the light-cone quantization and Fock expansion [2]. The light-cone QCD wave functions are constructed from the QCD light-cone Hamiltonian: HLC = P + P − − P⊥2 where P ± = P 0 ± P z with P µ the momentum operators [3]. The LCWF ψh for a hadron h QCD with mass Mh satisfies the relation: HLC |ψh = Mh2 |ψh . It has been shown [2] that both the Hamiltonian and all amplitudes are boost, rotational invariant and independent of P µ . These features make these wave functions frame independent. Experimental measurements of LCWFs can therefore provide very important information on the fundamental hadronic interactions described by the Hamiltonian. The LCWFs are expanded in terms of a complete basis of Fock states having increasing complexity [3]. For example, the negative pion has the Fock expansion: |ψπ − = n | π − |n n
) = ψd(Λ ¯ u/π ¯ (u i , k ⊥i, λi )|ud (Λ) (u i , k⊥i, λi )|udg ¯ + ··· +ψ d ug/π ¯
(1)
with longitudinal light-cone momentum fractions: ui =
ki+ ki0 + kiz = , p+ p0 + p z
n
u i = 1,
(2)
i=1
relative transverse momenta k⊥i ,
n
k⊥i = 0 ⊥
(3)
i=1
and spin projections λi . The index i runs over the partons contained in the relevant Fock state. The light-cone momentum fractions of the constituents u i and the transverse momenta k⊥i appear as the momentum coordinates of the LCWF. The light-cone wave function for each quark–gluon combination n: ψn/H (u i , k⊥i , λi ) interpolates between the hadron H and its quark and gluon degrees of freedom. The first term in the expansion is referred to as the valence Fock state, as it relates to the hadronic description in the constituent quark model. This state has a minimal gluon content, must have a small size and hence a large mass. This is further discussed in Section 2. The higher terms are related to the sea components of the hadronic structure, are larger and lighter. It has been shown that the valence Fock state may account for about 25% of the total LCWF and once determined, it is possible to build the rest of the light-cone wave function [4,5].
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The essential part of the wave function is the hadronic distribution amplitude φ(u i , Q 2 ). It describes the probability amplitude to find a q q¯ or a qqq system of the respective lowest-order Fock state carrying fractional momenta u i with ni=1 u i = 1. It is given by an integral of the LCWF over the non-perturbative low-mass region up to a scale Q 2 [6]: 3 kt2 2 2 2 ψ(u i , kti )δ k ti Θ Q − (4) φ(u i , Q ) ∼ d2 k ti . u(1 − u) 0 i=1 i An evolution equation for φ(u i , Q 2 ) from one value of Q 2 to another is given in [6]. 1.2. Applications of the LCWF formalism The structure of LCWFs has been predicted under certain conditions for a variety of hadrons. Some of these predictions and their applications will be discussed in Sections 3 and 4. Here we briefly mention several applications as examples of the large variety of processes where this formalism can be applied. We will point the interested reader to relevant references. 1. DLCQ All the applications need some sort of LCWF. The most promising method for calculating the LCWF of hadrons from the LC hamiltonian is the so-called Discretized Light-Cone Quantization (DLCQ). This is a very difficult and challenging project given the unbounded number of constituents. The method was developed [7] in (1+1) dimensions but is being extended to (3+1) dimensions. Basically it involves discretizing the LC momenta u and k⊥ and applying a cut-off on the invariant mass of the partons in the Fock expansion. This leads to diagonalization of finite matrices and may be simplified by using known physical constraints. Example of applications of (3 + 1) DLCQ include LCWFs and spectrum in the Yukawa theory [8] and the electron magnetic moment [9]. 2. Nucleon structure Given model LCWF, various properties of the nucleon have been calculated. These include the magnetic moments of the proton and the neutron and the axial vector coupling constant measured in neutron β decay [2]. Calculations of the electromagnetic form factor of baryons have been performed already in one of the early works [6]. Parton distribution functions can be calculated from LCWFs. Since they represent the probability to find any parton with fractional momentum x B j they can be obtained by integrating the LCWF over all Fock states and all variables keeping a given fractional momentum u = x B j [2]: 2 (Q) δ(x B j − u i ) (5) d[µn ] ψn/ h (u i , k⊥i , λi ) G a/ h (x B j , Q) = n
where
i
du i d2 k⊥i . . . , d[µn ] . . . = λi ∈n
2
du i d k⊥i
= δ 1−
Nn j =1
uj δ
(2)
Nn j =1
with k⊥ j
du 1 . . . du Nn d2 k⊥1 . . . d2 k⊥Nn .
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The parton distribution functions are usually identified with structure functions F2 (x, Q 2 ) measured in Deep Inelastic Scattering (DIS). It has been shown, however [10] that gluon exchange between the fast, outgoing partons and target spectators causes destructive interference of diffractive channels and thus affects the leading twist structure functions. Use of Eq. (5) can be extended by demanding certain flavor or certain helicity. It is then possible to reproduce measurements of the proton spin structure with an LCWF containing qqq and qqqg Fock states. The process of Deeply Virtual Compton Scattering (DVCS) from a proton is very sensitive to the quark structure of the proton. The photon scatters off a quark thus modifying its momentum as well as the momenta of the spectators. The amplitude for the process is then calculated from the overlap of the initial and the modified LCWFs [11]. The process of high momentum transfer real photon Compton scattering on nucleons, γ N → γ N was calculated [12] using distribution amplitudes that were calculated from QCD sum rules. For some distributions the calculations are able to reproduce experimental results for unpolarized photons and make predictions for several polarized photon scatterings. 3. Mesons Many properties of the π meson were calculated in the seminal work of Brodsky and Lepage [6]. Some of these will be discussed in greater detail in Section 3. The LCWF for the lowest (valence) Fock state is normalized using the measured pion decay constant f π ≈ 93 MeV: fπ du d2k⊥ (Λ) (6) ψd u¯ (u, k⊥ ) = √ . 16π 3 2 3 It is then used to calculate the electromagnetic form factor assuming several forms for the LCWF, given by: 1 1 1 Fπ (Q 2 ) = du dy φπ∗ (y, Q) TH (u, y, Q) φπ (u, Q) 1 + O (7) Q 0 0 with TH the scattering amplitude for the form factor but with the pions replaced by collinear q q¯ pairs. The distribution amplitude [6] φπ (u, Q) is the probability amplitude for finding the q q¯ pair in the pion with u q = u and u q¯ = 1 − u (Eq. (4)). The dimensional counting rules for form factors at large momentum transfer and other hard exclusive processes and the near-conformal behavior of the LCWFs at short distances can also be derived using AdS/CFT duality [13]. Heavy meson decay is a large momentum transfer exclusive process and factorization into perturbative and nonperturbative contributions can be applied, Section 1.3. In calculations of the B → Alν decay [14] Fock state expansion of the initial and final state were used. The amplitude for this process has contributions from two sources. One is the n = 0 parton-numberconserving amplitude with quarks changing flavor. The second is a n = −2 contribution in which a quark and an antiquark from the initial hadron high Fock state annihilate to the leptonic current creating lepton pairs. Similarly, exclusive hadronic decays such as B → π + π − were calculated [15] in pQCD with assumed structure of both the B and π LCWFs. Vector meson (VM) production was treated extensively. In [16] the authors calculate the cross section for diffractive leptoproduction of VM. They make full use of the LCWFs of both the virtual photon and the VM. This is further discussed in Section 2.2. 5. Higher Fock states While the lowest Fock state of hadrons, q q¯ or qqq describes the valence configuration the higher Fock states describe the quark and gluon sea. The hadron fluctuates between the various
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states which live times of the order of
P Mh2 −M 2
where P is the momentum, M and Mh are the Fock
state mass and hadronic mass, respectively. The higher Fock states may be probed in a variety of processes. Deep Inelastic Scattering (DIS) of leptons has been a major tool for studies of nucleon structure. The measured structure functions are a reflection of the whole Fock state expansion. The observation of a contribution from strange quarks to the nucleon spin is a result of probing the higher Fock state components. Similarly, measurements of the charm structure functions [17] reveal the charm components in the higher Fock states with lifetimes of the order of ∼ m1cc¯ . These components are also referred to as the “intrinsic strange” and “intrinsic charm” sea. The experimental results were analysed along these lines and were found [18] to be consistent with about 1% probability for intrinsic charm in the proton. Experimental observations show large asymmetry in charm production between the “leading” charm mesons (that contain a quark present in the beam) and non-leading charm mesons [19]. This effect was predicted by various models including one relating it to the intrinsic charm [20]. The decay J/ψ → ρπ has been interpreted as decay of the cc¯ into a high Fock state q qc ¯ c¯ of the pion and/or the ρ wave function [21]. 6. Nuclear effects The LCWF formalism can be extended beyond the single hadron into nuclear systems, in particular few nucleon systems and scattering processes. The nucleon–nucleon interaction and the scattering process are governed by the QCD counting rules [22] which predict that the cross section at large momentum transfer scales as f ( st ) dσ (AB → C D) = n−2 (8) dt s where n = n A + n B + n C + n D is the total number of interacting elementary fields. For N N scattering this dependence is s −10 which was verified by experiments. The cross section for the d(γ , p)n photodisintegration is expected to scale as s −11 . This was recently observed for proton transverse momenta pT > 1.1 GeV/c [23]. Analysis of old data indicates an s −22 dependence for d(d, n) 3 He and d(d, p) 3H measured at c.m.s. scattering angle θcm = 60◦ in the interval of the deuteron beam energy 0.5−1.2 GeV and ∼s −16 for d p → d p scattering at θcm = 125◦–135◦ at beam energies 0.5–5 GeV [24]. The anomalous spin asymmetry in pp scattering [25] has been interpreted [26] as the contribution from a higher Fock state: |uuduudcc ¯ that couples to the J = S = 1 initial state thus affecting the spin asymmetry. The electromagnetic moments of the deuteron have been calculated in the LCWF framework [27]. It was shown that a part of the quadrupole moment may not be due to the D state component of the nuclear wave function but to relativistic recoil corrections. This may solve a discrepancy between the measured quadrupole moment and that calculated with conventional nuclear theory. The deuteron form factor at large momentum transfer is predicted to become symmetric among the possible color-singlet components (orthogonal to N N and ∆∆). This six-quark component of the deuteron wave function is referred to as the “hidden color” component [28]. It is expected to manifest in high momentum transfer photodisintegration of the deuteron and was searched for in studies of pion absorption in 3 He [29]. 1.3. Factorization and hard processes Factorization theorems allow the separation of soft non-perturbative dynamics from that of the hard perturbatively calculable process. Not every process is factorizable; for example, it
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fails for hard diffractive hadron–hadron scattering. Factorization theorems have been proved rigorously for processes such as diffractive hard Deep Inelastic Scattering (DIS) [30], exclusive DIS processes, and meson production [31]. For processes where factorization is applicable there is a scale Λ such that Fock state components with mass Mn2 < Λ2 are considered soft and have to be evaluated by non-perturbative methods, while those with Mn2 > Λ2 contribute to large momentum transfer hard processes which can then be evaluated perturbatively. As an example, in Eq. (7) factorization is applied: φπ (u, Q) contains the non-perturbative distribution amplitude for the quarks and gluons in the pion up to a scale Q 2 . All momentum transfers higher than Q 2 are included in the scattering amplitude TH which can be computed perturbatively. The commonly used value of Λ ∼ 1 GeV is somewhat arbitrary and may need experimental input for better determination. The non-perturbative distribution amplitudes must be derived with some assumptions and rely on models. At very low Q 2 QCD sum-rules were used to develop forms of the distribution amplitude for mesons and hadrons [32], Section 3. These forms have been extended into the perturbative domain by adding perturbative calculations [33]. For high Q 2 the known asymptotic forms of the distribution amplitudes [6], Section 3 is reproduced. Evolution equations of the distribution amplitude with Q 2 allow prediction of the distribution amplitude at some scale Q 2 if it is known at some other value Q 20 [6]. For the inclusive parton distributions the well known DGLAP [34] and BFKL [35] evolution equations allow prediction of the parton distribution at some scale Q 2 if it is known at some other value Q 20 . 1.4. Diffraction The diffractive process is the coherent scattering from an object with a size comparable to the wavelength of the scattered projectile. In these processes the target and/or the projectile usually remain intact. It is observed in the scattering of light, of electrons and nuclear projectiles off nuclei and at high energy scattering of hadrons and photons. The process is mediated by boson exchange, such as virtual photons, pions or Pomerons. The angular distribution of the scattered projectile is characterized by diffraction minima. Bessel functions describe the angular distribution for scattering of light and approximately for scattering of electrons and α particles off nuclei and nucleons. Diffractive elastic and inelastic electron scattering has been used extensively to measure the charge and magnetic form factors of nucleons and nuclei. The positions of the minima are inversely proportional to the scatterer size. The angular distributions are usually replaced through Fourier transform by distribution of the four-momentum transfer t. As the cross section for the process may drop many orders of magnitude before reaching the first minimum, frequently only the region of low t-distribution is observed [36]. For small t, a series expansion 2 of the Fourier transform leads to the approximated form: e−bt where b = R3 and R is the scatterer radius. It can also be related to the combined size of the scatterer and projectile, see Section 2.2. As the diffractive interaction is confined to small values of t, it is characterized by small transfer of energy between the interacting particles. For high energy hadronic diffractive interactions the exchanged boson is the Pomeron (P), described as a two-gluon system with vacuum quantum numbers [37]. Depending on the momentum transfer, there are “soft” and “hard” Pomerons. In addition to the usual DIS variables Q 2 , W 2 , x B J and y, the variables used to describe the diffractive process in ep → ep X interaction are: xP =
M 2 + Q2 q · ( p − p ) X2 , q·p W + Q2
(9)
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β=
xBJ Q2 Q2 = . 2q · ( p − p ) xP Q 2 + M X2
(10)
Here q, p, p are the momenta of the virtual photon and of the proton before and after scattering, respectively. x P represents the fraction of the proton momentum carried by the Pomeron. β is 2 analogous to x B J = 2Qp·q and represents the fraction of the Pomeron momentum carried out by the quarks. M X is the invariant mass of the hadronic final state. The approximations are valid for small values of t and large W . Diffractive processes are identified experimentally by observing the intact scattered proton or by the presence of large rapidity gaps [38]. A major project is measurements of the diffractive structure functions F2D . They are obtained from measurements of diffractive cross sections in a way similar to derivation of the total structure functions F2 from the total DIS cross section. F2D are related to the diffractive parton distribution through generalized DGLAP evolution equations. They are also used to measure the Pomeron trajectory αP used for parametrization of the diffractive cross sections. More exclusive processes being studied are diffractive jet production, vector meson production, charm production and DVCS. Some of these processes will be addressed in more detail in the present work. 2. Color transparency The phenomenon of color transparency is intimately related to the description of hadrons in terms of LCWFs. The lowest (valence) Fock component of the LCWF is made of the minimal partonic configuration consistent with the hadron quantum numbers: a q q¯ for mesons and qqq for baryons. The distances between these partons must be small, an interpretation being that otherwise gluons may fill the space. Due to the small size, compared with the regular hadronic size, these states are referred to as “Point Like Configurations” (PLC). At short ranges the relative momentum is large and hence the mass of this component is large, far off-shell. A hadron moving with momentum P continuously fluctuates between the various Fock states with fluctuation times of the order of: τ ∼ 2P/(M 2 − m 2 )
(11)
2 +m 2 k⊥i i
where M 2 = u(1−u) is the invariant mass of the Fock state, m i the parton mass and m the regular mass of the hadron. During this time all the phases between the states remain approximately constant; this condition is referred to as the “Frozen Approximation”. When an interaction involving a large momentum transfer Q occurs, the related short wavelength naturally selects the Fock component with small separation, of the order of 1/Q, and large internal momentum, the PLC. This also follows from the dimensional counting rules [22, 39] as the cross section for the more complex Fock states will be suppressed by 1/Q 2 for each additional parton. It is therefore expected that such interactions will be useful for studying the properties of the PLC. Color transparency (CT) is the name given to the prediction that the color fields of QCD cancel for physically small color-singlet systems of quarks and gluons [40]. This color neutrality (or color screening) should lead to the suppression of initial and final state interactions of smallsized systems formed in hard processes [41]. The cross section for these interactions is expected to have the dependence: σ (b2 ) ∝ b2
(12)
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where b is the transverse size of the PLC. For a recent review of the subject see [42]. Measurements of color transparency are important for clarifying the dynamics of bound states in QCD. The most straightforward approach is to study interactions between high energy hadrons and nuclei and to demonstrate that the interactions of a PLC with the nucleons in a nucleus are suppressed compared to those of ordinary hadrons. This requires identifying observables which depend explicitly on the cross-sections of the PLC. There are three necessary ingredients in such an experiment: 1. An interaction that selects the PLC. 2. Kinematic conditions such that the PLC fulfils the frozen approximation. 3. An observable sensitive to the PLC cross section. It should be stressed that determination of whether each of these conditions is fulfilled is subject to significant uncertainties. As noted above, high momentum transfer reactions are expected to select the PLC, and it remains to be determined how high the momentum transfer should be. We discuss below in detail high momentum transfer (e, e p) and ( p, 2 p) reactions, formation of vector mesons and pion dissociation to dijets. A crucial condition is that the PLC will not expand during the interaction with the nucleus. This expansion can be caused by fluctuation to other Fock states or, as the PLC is not an eigenstate of the Hamiltonian it can be regarded as a wave packet that undergoes time evolution. This is controlled by the PLC lifetime (Eq. (11)) which can be translated to the distance the PLC will travel during this time and is called the coherence length, c : c ∼ 2P/(M 2 − m 2 ).
(13)
Here the relevant value of M and the expansion time and process are model dependent. For deep inelastic scattering experiments this will be given by: ∼ DIS c
2ν (Q 2
+ Mq2q¯ )
(14)
where ν is the energy of the virtual photon. The mass of the q q¯ system, Mq q¯ , selected by DIS with momentum transfer Q is of the order of Q, thus c ∼ 1/2M p x B J . This relation is referred to as the Ioffe time [43]. The requirement is that c R A , the nuclear radius. The typical observables used as signature for existence of the PLC are comparisons between the cross section for interaction of the PLC with the nucleus and that expected from a normalsize hadron. The applicable calculation of both of these quantities has some uncertainties. These observables are looked for when conditions (1) and (2) are expected to be fulfilled or as function of a parameter which controls these conditions so as to observe the onset of color transparency. Observation of this onset is particularly important in view of the uncertainties involved in determining the exact conditions and signature for the color transparency phenomenon. 2.1. Quasi-elastic scattering As was pointed out in [41], for a high momentum transfer p– p scattering to proceed the momentum should be transferred to all the quarks involved, hence the internal distances should be of the order of the momentum transfer. This is also reflected in the dimensional counting rules [22], where the cross section for such a process should scale as s −10 . One then expects that in ( p, 2 p) reactions on nuclei the initial and final state interactions of the incoming and outgoing protons will be strongly reduced as Q 2 increases. Experimentally, this was studied [44]
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Fig. 1. Schematic drawing of E850 EVA solenoidal detector, [45].
by measuring the cross section on several nuclear targets and different beam momenta. The authors determine the transparency T defined as: T =
(dσ/dt)( p − p elastic in nucleus) . (dσ/dt)( p − p elastic in hydrogen)
(15)
The detector was made of a scintillation counter backed with a two-layer 1.5 radiation-length lead–scintillator sandwich. This counter, spanning about 2/3 of the total solid angle identified charged tracks produced out of the two-body scattering plane. The momentum of only one of the outgoing protons was measured. The data was analyzed by reconstructing the momentum of the struck proton and using the free p– p cross section to calculate T . After corrections for background and experimental acceptance, values of T were extracted with normalization to the number of “nuclear protons” (5.1 for aluminum). The results show an increase of T with incident momentum up to about 10 GeV/c but it drops at higher momenta. This result as well as the anomalous spin asymmetry observed in pp scattering at similar center of mass energies [25] have been interpreted [26] as the contribution from a resonance produced by a higher Fock state. The authors predict that CT will reappear at higher energies. At the moment, the experimental results as far as color transparency is concerned remain inconclusive. The experiment was later repeated with an improved apparatus, the EVA detector having good tracking and momentum analysis capabilities for both protons. The EVA (BNL-E850) experiment embedded carbon (C), CH2 , and CD2 targets inside a 2 m diameter, 3 m long superconducting solenoid with a 0.8 T field as shown in Fig. 1. The annular space between the pole piece and the solenoid allowed particles scattered near 90◦c.m. to reach the detectors outside the solenoid. For tracking purposes the targets were surrounded by four concentric cylinders each fabricated from four layers of thin wall straw tubes filled with a 50–50 mixture of argon–ethane. The momentum resolution was σ ( p)/ p 7%. The tracks scattered near 90◦ c.m. passed through the annulus between the solenoid and the steel pole piece until they reached the two fan-shaped scintillation counter arrays, H1 and H2, and the two larger straw tube cylinders, C3 and C4. The detector was configured to have acceptance for c.m. scattering angles in the range ∼86◦ < θc.m. ≤ 90◦.
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Fig. 2. Nuclear transparencies measured by BNL experiments 834 [44] and 850 [45].
From the momentum measurement of the two scattered protons the authors deduce the momentum of the struck proton and define an effective beam momentum Peff in its rest frame. The results were recently summarized [45] and compared with the transparency T = 0.2 expected from the Glauber model. The results are found to be consistent with the earlier ones, reproducing the rise and fall of the transparency for incident momentum above 10 GeV/c, translated to Q 2 ∼ 8 GeV/c, see Fig. 2. Studies of color transparency using the A( p, 2 p) reaction are particularly difficult. It is necessary that the conditions for color transparency be fulfilled for both incoming and the two outgoing protons. The incoming proton should maintain its PLC configuration as it penetrates the nucleus to the point where it strikes a nuclear proton which is, too, in a PLC configuration so that the high momentum transfer scattering can occur. Then the two outgoing protons should stay in that state as they move through the nuclear medium. The probability for the scattering is already small as determined by the s −10 dependence, but this does not include the probability for all associated coherence lengths to be sufficiently long. Better conditions can be expected in studies of the A(e, e p) reactions. The interaction is by absorption of a high Q 2 virtual photon which fluctuates to a q q¯ PLC system with coherence length defined by Eq. (14). If this is sufficiently long it will be necessary that the fluctuation time of the outgoing (PLC) proton for growing into a normal size be sufficiently long too. This fluctuation time is referred to as the “formation length” f and it is governed by the same considerations as for the coherence length, Eq. (13). An experimental study of the A(e, e p) reaction was performed by the NE-18 collaboration at SLAC [46]. The cross sections were measured on several nuclear targets with transfer momenta Q 2 = 1, 3, 5, 6.8 (GeV/c)2 . The outgoing electron and proton were detected with magnetic spectrometers with 0.15% momentum resolution. This allowed the reconstruction of the struck proton momentum and the missing energy which was required to be well below the pion production threshold. The transparency was determined by taking the ratio of the experimental results to PWIA calculations. The expected color transparency signature was that
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Fig. 3. Nuclear transparencies measured by the A(e, e p) experiments at SLAC [46], open symbols and JLAB [47] closed symbols. The transparency at lowest Q 2 is from a Bates experiment [48]. The solid lines are Glauber model calculations [49].
the transparency will grow with Q 2 , as was observed by the BNL experiments in this Q 2 range. Furthermore, when the A-dependence is studied in fixed Q 2 values and parametrized as: 2 T ∝ Aα(Q ) the (usually negative) value of α is expected to increase with Q 2 approaching zero (T = 1) for full transparency. The experimental results showed none of these signatures: the Q 2 dependence was consistent with being flat and the value of α remained constant α ∼ −0.2 as expected from simple Glauber model calculations. A similar experiment was carried out at the Jefferson Laboratory [47] in which the momentum transfer range was extended to 8.1 (GeV/c)2 and with improved statistical and systematic precision. The results were consistent with those of the SLAC experiment and showed no Q 2 dependence of the transparency or of the A-dependence parameter α. The results of both experiments are summarized in Fig. 3. The results of the ( p, 2 p) and (e, e p) experiments were very puzzling and resulted in many theoretical papers with a variety of interpretations. We may attempt to determine what can be expected from these experiments based on the concept of coherence length, Eq. (13). For Q 2 = 8 (GeV/c)2 , the maximum value for the (e, e p) experiments and where the maximal transparency is seen in the ( p, 2 p) experiments, the energy of the scattered proton ∼Q 2 /2m p and its momentum is ∼5 GeV/c. To use Eq. (13) we must use a mass M for the excited, smallsize proton and this is subject to a large uncertainty. The lowest one and the one most frequently used is that of the Roper resonance, [42]. The resulting coherence length is c ∼ 2 fm which is smaller even than the radius of 12 C, the lightest nucleus in these experiments (except for the deuteron, for which these concepts are not applicable). Thus, the conditions for CT are not fulfilled and what we can expect to see is the onset of CT which may be even more interesting than having full CT. The two reactions suffer from a kinematic constraint that relates Q 2 and the energy of the scattered proton. The cross section for the ( p, 2 p) reaction has a s −10 dependence. The reaction would occur mainly on protons with longitudinal Fermi momentum in the beam direction that would reduce
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291
Fig. 4. Nuclear transparency for 12 C(e, e p) quasielastic scattering: SLAC [46] (open symbols) and JLAB [47] (closed symbols). The transparency at lowest Q 2 is from a Bates experiment [48]. The thin solid curve is from Glauber model calculations [49]. The thick solid curve is a Glauber calculation of Ref. [52]. The dot–dashed, dotted, and dashed curves are color transparency predictions from Refs. [52–54], and [55], respectively.
s. It will map high nuclear momentum components with protons far off shell. For the (e, e p) reaction the cross section has a s −6 dependence hence it may be mapping lower momentum components and protons more on shell. This may affect the probability for observation of CT in these two reactions. If the probability for the outgoing proton to avoid final state interaction is p then for the (e, e p) experiment it will be p and for the ( p, 2 p) experiment p2 if we assume that the higher momentum incoming proton in the ( p, 2 p) reaction does not interact in the nuclear medium at all. If it does, the probability can be closer to p3 . In any event, we can expect a steeper increase of transparency with Q 2 for the ( p, 2 p) experiment than for the (e, e p) experiment. It now becomes a quantitative issue of whether the slope observed in the ( p, 2 p) experiment can be consistent with the results of the (e, e p) experiment. There are large theoretical uncertainties related to the formation and expansion of the PLC; the coherence length being just a simplified parameter. There are smaller but accumulating uncertainties related to how to handle the internal momentum in the nucleus, Glauber and PWIA calculations and the nucleon–nucleon cross section in the nuclear medium compared with free nucleons. For example, the oscillations in the free nucleon–nucleon cross section are expected to be filtered out by the nuclear medium [50]. For reviews of these works see [50,51]. The early experimental results had significant uncertainties such that a variety of models could be accommodated and it was possible to reach a conclusion that there is no clear inconsistency between the (e, e p) and ( p, 2 p) experiments [50,51]. The more recent and precise results make this conclusion less clear as demonstrated in Fig. 4 taken from [47]. The authors show that calculations that included CT effects consistent with the ( p, 2 p) results (dashed and dotted lines) were not consistent with the (e, e p) results. In fact some calculations claim that substantial CT effects will be seen only for Q 2 ∼ 20 (GeV/c)2 [56]. In conclusion it cannot be confirmed that color transparency has been observed in these experiments. 2.2. Vector meson production Studies of vector meson (VM) electroproduction in DIS are related to color transparency through two aspects. One is the slope of the diffractive t-distribution (t: the momentum transfer
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to the target) which provides a measure of the size of the q q¯ system at large Q 2 . This is measured on a nucleon target. The second is measurements of the transparency: ratios of the cross sections on nuclei to the cross section on the nucleon, similar to the approach in quasielastic scattering discussed in Section 2.1. The cross section for leptoproduction of VM was calculated in [16]. The process is described as three sequential steps: 1. The virtual photon breaks up into a q q¯ pair with a lifetime τi given by Eq. (14) and Mq2q¯ =
2 + m2 k⊥ . u(1 − u)
(16)
Here m is the current quark mass and k⊥ the quark transverse momentum. This estimate is valid for the production of a longitudinally polarized vector meson which is expected to dominate at large Q 2 . 2. The q q¯ pair scatters off the target proton. 3. The q q¯ pair then lives a time τ f (and travels distance f ) determined by τf ∼ f ∼
2ν Mq2q¯
− m 2V
before the final state vector meson with mass m V is formed. The amplitude for this process M is written as a product of three factors: (i) the photon wave γ function ψλ1 λ2 (k⊥ , u) giving the amplitude for the virtual photon to break into a quark–antiquark pair; (ii) the hard scattering amplitude T H of the quark–antiquark pair on the target; and (iii) the
, u ) giving the amplitude for the scattered quark–antiquark pair of VM wave function ψλV1∗λ2 (k⊥ flavor f to become a vector meson: Mf =
1 1 d2 k ⊥ d2 k
⊥ Nc du du ψλV1∗λ2 (k⊥ , u )TλH1 λ2 (k⊥ , u ; k⊥ , u) (16π 3 )2 0 λ ,λ 1
2
γ × ψλ1 λ2 (k⊥ , u)
0
(17)
where λ1 and λ2 are the helicities of the quark–antiquark pair which are conserved during the scattering off the target and Nc is the number of colors necessary for normalization. For the VM the authors use a distribution amplitude similar to that of the pion (Section 3). The LCWF of the photon is discussed in Section 4. The authors then deduce a cross section for t = 0 with values in a range of about a factor 3 depending on which LCWF is used for the VM. This sensitivity is interesting and may be used to actually measure the vector meson LCWF. The numerical agreement between the theoretical estimates and the measurements is reasonable for some parameters but if one wants to use this comparison, variations of the cross section (e.g. as function of Q 2 ) rather the absolute values should be used. The t-dependence of the cross section for diffractive VM production is expected to have the form: dσ/dt ∝ ebt where the slope b is related to the size of the system. This relation was further examined in Ref. [57] with the result that it should have the form: b=
1 1 2 R + r 2 3 8
(18)
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293
Fig. 5. Slope parameter of the t-distribution for ρ, φ and J /ψ vector mesons as a function of Q 2 + m 2V , [60].
where r∼
6
(19)
Q 2 + m 2V
is the (excited) ρ 0 size and R is the proton radius [57]. This result implies that the diffraction cone, as described by the parameter b should shrink as Q 2 is increasing. It is also telling us that the dependence will be on Q 2 + m 2V which can be verified by production measurements of various vector mesons. An additional contribution to b may come from the form factor of the exchanged boson, Section 4.2.1. There have been several experimental studies of the t-dependence of VM production on the proton: ρ production [58], ρ and φ production [59] and ρ, φ and J/Ψ production [60]. The results from the three experiments are consistent with each other. Recent results are summarized in Fig. 5 [60] where the parameter b is plotted as a function of Q 2 + m 2V . It is seen that on this scale the slope is approximately universal for all the VM. At large Q 2 + m 2V the value of r (Eq. (19)) is very small, and b ∼ 4.5 (GeV/c)−2 is dominated by R and consistent with Eq. (18). The size of the q q¯ system is then small compared to this value. Experiments of VM production in nuclei [61–63] were discussed and interpreted without relating to the VM size effects discussed above. This is unfortunate as it is likely that with so many uncertainties and model dependence, combining the experimental information on VM production on the proton and on nuclei can improve our understanding of the process. Two different VM production mechanisms in nuclei are considered. One is the incoherent process where VM are produced on individual nucleons and as result the nucleus breaks apart. This is similar to quasi-elastic scattering where the produced VM must go through the nuclear medium. In the coherent process the VM are produced on the nucleus as a whole and it remains intact. Both of these processes were studied experimentally and theoretically.
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0 Fig. 6. Nuclear transparencies Trinc A measured for incoherent ρ production by Fermilab experiment E665 [61]. The calculations are from [64]; solid line with full calculations, dashed lines with frozen approximation.
An important difference between VM production and quasi-elastic scattering in nuclei is that the kinematic constraints discussed for quasi-elastic scattering do not exist, or are at least much weaker for VM production. One then can vary almost independently the coherence and formation lengths. Both coherent and incoherent processes were discussed in great detail in [64–66] where the authors pay particular attention to the identification of coherence length effects and CT effects. In [64] the authors calculate the VM production cross section on the nucleon using a similar approach to that of [16]. They use a photon LCWF in coordinate space modified to avoid end-point singularities. For the VM they use a boosted Gaussian wave function and for the interaction a phenomenological dipole cross section σqq ¯ . The resulting amplitude for the process is: 1 du d2r ΨV∗ (r , u) σqq r , s) Ψγ ∗ (r , u, Q 2 ). (20) Mγ ∗ N→V N (s, Q 2 ) = ¯ ( 0
They deduce the t-dependence by fitting data with an expression similar to Eq. (18) and with this fit obtain good agreement with measured ρ 0 and φ production on the proton. For VM incoherent production in nuclei the authors [64] use Eq. (20) modified by a Green function to describe propagation of the q q¯ in the nuclear medium attenuated by a simplified dipole cross section. The results then depend on the size of the coherence length. For large coherence lengths (c , f R A ) the frozen approximation is valid, and the transparency is calculated with only Q 2 dependence showing clean CT effect. The calculations are compared with the experimental results of the E665 collaboration [61] that measured ρ 0 production in DIS of 470 GeV/c muons on hydrogen, deuterium, carbon, calcium and lead. At these high energies, with ν > 100 GeV, the coherence lengths are large: c , f R A . The results are shown in Fig. 6. The increasing transparency with Q 2 shows that the size of the produced q q¯ system that eventually couples into the ρ 0 is smaller for larger Q 2 and interacts weakly in the nucleus. At very small x B j this effect may be partially offset by nuclear shadowing [69]. The deviation from the frozen approximation for Pb at large Q 2 comes from the reduction of c which is no longer large compared with the Pb radius. In this experiment there are no complications due to mixing of coherence length and CT effects.
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295
Fig. 7. t-dependence of the ρ 0 production on H and 14 N [62] used for separation of coherent and incoherent cross sections.
Fig. 8. Nuclear transparency as a function of Q 2 in specific coherence length bins for incoherent ρ 0 production on nitrogen. The straight line is the result of the common fit of the Q 2 -dependence [62].
The situation is very different in the lower energy measurements of the HERMES collaboration [62,63] where 9 < ν < 20 GeV. The ρ 0 production was studied on H and 14 N. To separate the coherent and incoherent cross sections the authors used the t-distributions. The excess yield at small t for 14 N, showing the larger nuclear radius, is taken as the coherent production, Fig. 7. At these energies both c and f are short and changes of transparency with Q 2 reflect combined changes in c , f and size of the q q¯ system. In order to be more selective the authors plot the transparency as a function of Q 2 in bins of 0.1 fm in c = 2ν/(Q 2 + Mρ2 ), Fig. 8. The results show an increase of the transparency with Q 2 even with fixed c and this is taken as a signal of CT. It should be noted that for fixed c the value of ν increases with Q 2 , and
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f = 2ν/(Mρ2 − Mρ2 ) ∼ 2ν/1.51 GeV−1 (as used by the authors of [62]) increases accordingly. Thus we may be observing a combined effect of shrinking q q¯ and increasing f with Q 2 . The authors of the theoretical calculations [64] reproduce the experimental results. Their calculations show that for fixed Q 2 the transparency decreases with c . The given interpretation is that increasing c results in a longer path of the q q¯ in the nucleus i.e. suppression of transparency. This effect is claimed to be stronger than that of the increasing f . The experimental results were not presented in this way. From Fig. 8 one can see that for Q 2 = 2 the transparency is constant at T = 0.5 ± 0.1 over the range 1.35 fm < c < 2.35 fm. Coherent VM production on nuclei can provide additional information and, under some circumstances allow disentangling the CT and coherence-length effects. In the absence of any CT effect the coherent production cross section is expected to fall with Q 2 because of the coherent nuclear form factor. To estimate the various CT and coherence-length effects, we first consider the extreme case of high energies such that c , f > R A . The VM is then produced coherently over the whole nuclear volume and the nuclear production amplitude is predicted to be A times the nucleon production amplitude: MA = AMN , σ A ∝ A2 . If Q 2 is large enough such that the interaction of the q q¯ in the nuclear medium vanishes, the cross section will be given by: dσ N 2 2 dσ A ∝ A FA (t) dt dt
(21)
where FA2 (t) is the nuclear form factor squared (neglecting r , Eqs. (18) and (19)): FA (t)2 ∼ ebt ,
b=
R 2 , 3
When integrating over t: 3 A2 A2 = 2 (dσ/dt)dt ∝ b R0 A2/3 σ A ∝ A4/3,
T ∝ A1/3.
R = R0 A1/3 .
(22)
(23)
(24)
If Q 2 is not large enough but the energy is high the onset of CT can be observed as Q 2 increases. This is seen in Fig. 9 where the calculations of [64] with the frozen approximation are in reasonable agreement with the data. Calculation of the transparency for fixed Q 2 as a function of ν [64,65] show increased transparency (as coherence lengths increase) and saturation where 1 the coherence lengths exceed the nuclear size. The saturated transparencies scale as A 3 . This is indeed seen in the E665 data of Fig. 9 and the calculations of [64], 10. For lower energies the situation is more complex. The HERMES collaboration shows transparency increasing with Q 2 for fixed c [63]. This may be due to CT or to the increase in ν and coherence lengths with increasing Q 2 or, most probably, combination of the effects. In summary, studies of coherent and incoherent VM production on nuclei provide us with interesting information about CT and coherence lengths. Through a combination of measurements the effects may be studied separately. At high energies, where coherence length effects are minimal the CT effect can be observed through the increase of transparency with Q 2 in both coherent and incoherent production. At lower energies one must rely on calculations that incorporate all the effects and can consistently reproduce the experimental results. We finally mention an experiment in which the A-dependence of J/ψ photoproduction was measured [67]. Fermilab experiment E691 measured the relative cross section for J/ψ
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297
Fig. 9. Nuclear transparencies for coherent ρ 0 production measured by Fermilab experiment E665 [61]. The calculations are from [64]; solid line with full calculations, dashed lines with frozen approximation.
Fig. 10. Nuclear transparencies for coherent ρ 0 production as function of energy and Q 2 [64]; solid line with full calculations, dashed lines without gluon shadowing.
production by real photons with energies of 90–190 GeV incident on H, Be, Fe and Pb targets. They separated the contributions of coherent and incoherent yield using the same method as done in the E665 and HERMES experiments (see Fig. 7). The measured cross sections were parametrized: σ = σ0 Aα with the results: αcoh = 1.40 ± 0.06 ± 0.04, αincoh = 0.94 ± 0.02 ± 0.02. The result for coherent production is consistent with the expectations for fully coherent and transparent production as in Eq. (24). We note that at these high energies the coherence lengths are long. From Fig. 5 we can estimate that the J/ψ is photoproduced in a compact (b ∼ 4 GeV−2 ) cc¯ configuration, similar to high Q 2 ρ 0 production. 2.3. Pion diffractive dissociation to dijets The idea to use diffractive dissociation of pions to dijets in order to observe CT effects and study the pion wave function was proposed by Bertsch et al. [68]. The authors point out that when a high energy pion hits a nuclear target, all the large-size Fock components of the pion wave function will be absorbed in the nucleus. The small-size components will interact very weakly thus the nucleus will be transparent to these components. As the pion hitting the target fluctuates
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Fig. 11. Pion dissociation to q q¯ via two-gluon exchange.
between the various Fock components, only the small-size valence q q¯ state will emerge. As this state is no longer an eigenstate of the pion Hamiltonian it will dissociate and the quark and antiquark will each hadronize into a jet. In this process the nucleus acts as a filter absorbing all Fock components except for the valence q q¯ component. These ideas were further developed by Frankfurt et al. [51,69,70] who calculated the cross section for dijets with large transverse momentum kt . The pion fluctuates into a large kt q q¯ pair long before it hits the target nucleus and interacts via two-gluon exchange, Fig. 11. The mass of the system is given by Eq. (16) and for a high incident momentum (500 GeV/c) the coherence length (Eq. (13)) is long. This high kt system is a PLC and interacts weakly in the nuclear medium. The authors calculate the cross section for this process in leading twist. For the dijet production amplitude on a nucleon they take: 2 f (b ) ei κt ·b (25) M(N) = d2 bψπ (u, b) 2 here b is the transverse size of the q q¯ system, f (b2) is its forward scattering amplitude normalized according to the optical theorem Im f (b2 ) = s σ (b2 ), and t = 0. The factor e−i κt ·b accounts for the final hadronic wave function of two jets with a high relative momentum. is the q q¯ LCWF in coordinate space, b being the transverse separation of the quarks. ψπ (u, b) The assumed b2 dependence of the interaction σ (b2 ) allows an evaluation of Eq. (25) in terms of the momentum-space Fourier transformed ψ˜ π (u, κt ) by expressing b2 as b2 = −∇κ2t . Then M(N) = i
1 s σ (−∇κ2t )ψ˜ π (u, κt ). 2
(26)
The authors calculate Eq. (26) using two possible pion LCWFs (Section 3) and a phenomenological cross section σ (b2 ) and obtain: 8 d3 σ N 1 m N |M N |2 −6 GeV
= ≈ 2.6 GeV φ 2 (u) (27) κt (2π)4 16 m 2 + P 2 s 2 dudM J2 d2 PNt N N φ(u) is the pion distribution amplitude (Section 3) and there is a factor 2 different normalization depending on which one is used. For the nuclear cross section the authors use the frozen approximation and nuclear multiple scattering. The results M(A)/AM(N) are shown in Fig. 12.
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299
Fig. 12. Ratios of nuclear to nucleon amplitudes [69]. Solid line: exact evaluation, dashed line: higher twist expansion.
As can be seen the ratio increases with kt , reaches unity for kt ∼ 1.5 GeV/c and overshoots this value. This overshoot is attributed to higher twist effects which are studied by the authors [69, 70]. For M(A)/AM(N) ∼ 1 d4 σ A dudM J2 d2 qt
=
d4 σ N dudM J2 d2 qt
FA2 (t)A2
(28)
in striking contrast to Glauber calculations. When integrating over t the results of Eq. (24) hold. The experimental study of this process was performed by the Fermilab E791 collaboration [71]. A-dependence of the diffractive dissociation of 500 GeV/c pions into dijets, scattering coherently from carbon and platinum targets was measured. 2 × 1010 π − nucleus interactions were recorded using an open geometry spectrometer [72], see Fig. 13. The segmented target consisted of one platinum foil and four diamond foils, each foil approximately 0.4% of an interaction length thick (0.5 mm for platinum and 1.6 mm for carbon). Six planes of silicon microstrip detectors (SMD) and eight proportional wire chambers (PWC) were used to track the beam particles. The downstream detector consisted of 17 planes of SMDs for vertex detection, 35 drift chamber planes, two PWCs, two magnets for momentum analysis, ˇ two multi-cell threshold Cerenkov counters, electromagnetic and hadronic calorimeters for
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Fig. 13. The spectrometer used by the E791 collaboration [72].
electron identification and for online triggering, and two planes of shielded scintillators for muon detection. The trigger required a beam particle, an interaction in the target and a very loose transverse energy trigger, based on the energy deposited in the calorimeters. The data analysis required that at least 90% of the beam momentum is carried by charged particles. This reduced the effects of unobserved neutral particles. The JADE jet-finding algorithm [73] was used to identify two-jet events made of charged particles. The algorithm’s cutoff parameter (m cut ) was optimized using Monte Carlo simulations. For each two-jet event the transverse momentum of each jet with respect to the beam axis (kt ), the transverse momentum of the dijet system with respect to the beam axis (qt ), and the dijet invariant mass, M J were calculated. A minimum kt of 1.2 GeV/c was required. Selection of diffractive dijets was based on the qt2 distributions of the selected dijet events shown in Fig. 14. The diffractive peaks at low qt2 were simulated by dN/dqt2 ∝ exp(−bqt2) with b inversely proportional to the nucleus’s radius (2.44 fm for carbon and 5.27 fm for platinum). Because theory predicts that the A-dependence varies with kt [69], the analysis is carried out in three kt regions: 1.25 GeV/c ≤ kt ≤ 1.5 GeV/c, 1.5 GeV/c < kt ≤ 2.0 GeV/c, and 2.0 GeV/c < kt ≤ 2.5 GeV/c. The distributions were fit by Monte Carlo simulations of sums of qt2 distributions of dijet events produced coherently and incoherently from nuclear targets and of background. The generated events were passed through detector simulation, reconstructed and analyzed using the same programs used for the data. The A-dependence of the diffractive cross sections was derived from the fits in each kt range. The exponents of σ = σ0 Aα are listed in Table 1 and are close to the expectations, Eq. (24). The conditions of this experiment are such that the onset of CT was expected to be seen as an increase of α with increasing kt . In fact the values of α do not show this but the values are in good agreement with what is expected from CT. So apparently the onset is at lower values of kt . For the measured kt range the corresponding transverse size is ∼0.1 fm. We should note that assuming M 2 ∼ 4kt2 we get in the low kt region M 2 < 9 GeV2 resulting in c ∼ 22 fm, R A . At the high kt range we have M 2 > 16 GeV2 and c ∼ 12 fm, ∼2R A . It may be that in the high kt range some CT effects are lost due to reduced c , compensating the expected rise in α. We
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301
Table 1 The exponent in σ ∝ Aα , experimental results for coherent dissociation and the color-transparency (CT) predictions [69] kt bin (GeV/c)
α
αstat
αsys
α
α(CT)
1.25–1.5 1.5–2.0 2.0–2.5
1.64 1.52 1.55
±0.05 ±0.09 ±0.11
+0.04–0.11 ±0.08 ±0.12
+0.06–0.12 ±0.12 ±0.16
1.25 1.45 1.60
Fig. 14. qt2 distributions of dijets with 1.5 ≤ kt ≤ 2.0 GeV/c for the platinum and carbon targets. The lines are fits of the MC simulations to the data: coherent nuclear dissociation (dotted line), coherent nucleon/incoherent nuclear dissociation (dashed line), background (dashed–dotted line) and total fit (solid line).
note also that in their more recent work [70] the authors carried out more detailed calculations and predicted a value α = 1.54. This process was calculated also by Nikolaev et al. [74] who include higher twist corrections. They calculate the α dependence and their results are very similar to those shown in Table 1 as derived from [69]. In summary of this section we may conclude that color transparency was well demonstrated in vector meson electroproduction and in diffractive dissociation of the pion to dijets. It was not unambiguously verified for the proton. It is important to understand the experimental results for the proton: why (e, e p) experiments show no sign of CT and why ( p, 2 p) experiments show a rise and fall of transparency, strongly deviating from Glauber calculations and at the same time not reproducing the expected CT signature. It can be expected that if the effect exists in the q q¯ system it should also exist for the qqq system. One could argue that the probability to find a q q¯ at short distances is higher than that to find a qqq in short distances. If we interpret these systems as the valence components of their respective LCWFs, this may indicate that the contribution of the valence component to the total LCWF may be different for mesons and baryons. The difficulties encountered in understanding the anomalous spin effects in pp scattering [25,26] leave this as an open question. For observation of CT with protons there might also be the problem of choosing the sensitive process: reaction, momentum transfer etc. that would select a proton in a PLC state and the observable that would identify it as such. It may be that diffractive dissociation of protons or perhaps baryon photoproduction would show this effect. Following the example
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of the pion one could consider proton diffractive dissociation to three jets in proton–nucleus interactions. This may be achieved at RHIC and in fixed target experiments at the Bevatron. High momentum transfer electroproduction of mesons off nuclei: (e, e π), (e, e k) should provide additional confirmation of the observations of CT for mesons, and perhaps some understanding of why the (e, e p) experiments did not show CT effects. 3. The pion light-cone wave function 3.1. Theoretical predictions The quantity usually used for description of the pion internal structure is the distribution amplitude, the probability amplitude to find the q q¯ system in the pion with longitudinal momentum fractions u and 1 − u. It is given by an integral of the pion LCWF over the transverse momentum (see Eq. (4)) up to a scale Q 2 : Q2 2 2 φq q/π (u, Q ) ∼ ψq q/π (29) ¯ ¯ (u, k˜t )d k˜t . 0
Because of the relation between the distribution amplitude and the LCWF it is common to find in the literature the distribution amplitude being referred to as a wave function. The distribution amplitude φπ (u, Q 2 ) was shown to be only weakly dependent on Q 2 and an evolution equation was derived. The most general solution of the evolution equation yields the distribution amplitude expansion [6]: −γn Q2 3/2 2 an Cn (2u − 1) ln 2 (30) φπ (u, Q ) = u(1 − u) Λ n≥0 where Cn are the Gegenbauer polynomials and γn the anomalous dimensions: n+1 2δh h¯ CF 1 γn = − 1+4 . β k (n + 1)(n + 2) 2
(31)
C F = (n 2c − 1)/2n c = 43 for n c = 3 colors and β = 11 − 23 n f = 29 3 for n f = 2 flavors. δh h¯ = 1 (0) when the constituent’s helicities are antiparallel (parallel). If it is known for a given Q 20 , φπ (u, Q 20 ) can be used to calculate the coefficients an and obtain φπ (u, Q 2 ) for any Q 2 . φπ (u, Q 20 ) can be either measured or calculated at some Q 20 where the least modeldependence is expected. For Q 2 → ∞ perturbative QCD (pQCD) is expected to hold and only the n = 0, a0 term survives (γ0 < γn for all n > 0). For pions δh h¯ = 1 hence γ0 = 0 (Eq. (31)) and the high Q 2 asymptotic distribution amplitude is [6,68,75]: π φAsy (u) = a0 u(1 − u).
(32)
We note here that for a vector meson with parallel helicities, δh h¯ = 0 and the asymptotic distribution amplitude becomes: −C F /β Q2 VM φAsy (u, Q) = a0 u(1 − u) ln 2 . (33) Λ The normalization coefficient a0 = 6 if the integral of the distribution amplitude is unity. Alternatively, it is given by normalizing ψq q/π at the origin yielding [6]: ¯
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1
a0 = 6 0
φπ (u, Q)du =
√ 3 fπ
303
(34)
with the pion decay constant f π . At the other end of the scale, Q 2 ≤ 1 GeV2 it is much more difficult to predict φ as it is dominated by non-perturbative interactions. This was addressed by Chernyak and Zhitnitsky (CZ) who proposed [32] to use QCD sum rules for that purpose. They evaluated sum rules that allow the determination of moments of the distribution amplitude. With their longitudinal momentum fraction ζ = u 1 − u 2 = 2u − 1, they obtained for the moments: 1 n ζ = φ(ζ )ζ n dζ (35) −1
values derived from phenomenological analysis: ζ 2 µ0 = 0.46,
ζ 4 µ0 = 0.30,
µ0 = 500 MeV
(36)
at the scale of 500 MeV. At this low scale they estimate that only the pion and the A1 resonance contribute to the sum rules and only the lowest moments can be practically determined. They search for a function that will fulfil these requirements as well as normalization to unity and end-point requirement. Their model wave function is: φCZ (ζ, µ0 ) =
15 (1 − ζ 2 )ζ 2 , 4
µ0 = 500 MeV
(37)
ζ 4 = 0.24.
(38)
with moments: ζ 0 = 1.0,
ζ 2 = 0.43,
In terms of u: φCZ (u) = 30u(1 − u)(1 − 2u)2 .
(39)
This function corresponds to a2 = 2/3, a4 = 0 in Eq. (30). The distribution amplitudes (squared) are plotted in Fig. 15 and as can be seen there is big difference between the two. One is faced with the question of how high Q 2 should be in order for the distribution amplitude to resemble the asymptotic one. This may be extended to the question of where asymptotia starts and for what Q 2 values pQCD calculations apply. The statement in [32] is that due to the slow, logarithmic evolution of the distribution amplitudes, resemblance to the asymptotic one will be reached only for Q 2 ≥ 100 GeV2 . Other distribution amplitudes were proposed by several authors [76–78]. It becomes a challenge to experimentalists to measure the distribution amplitude at any given scale. 3.2. Distribution amplitudes and form factors Traditionally, studies of the pion LCWF and distribution amplitude were done by measurements of the electromagnetic form factors of the pion. For the charged pion at very low Q 2 the measurements utilize the π ± e → π ± e elastic scattering. This was done using a pion beam scattering from a heavy nuclear target. At very forward angles the scattering is electromagnetic and the cross section measures the form factor much like for electron scattering on nuclei [79]. For higher Q 2 , the process of electron scattering from a virtual pion, p(e, e π + )n (Fig. 16(a))
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Fig. 15. Two predictions of the pion distribution amplitude. Asymptotic function: dashed line, and CZ function: solid line.
Fig. 16. (a) Measurement of the charged pion form factor by electron scattering off a virtual pion. (b) Measurement of the neutral pion form factor by e+ e− annihilation.
was used. This process is more complicated and its analysis is model dependent [80]. The tchannel scattering on an almost on-shell virtual pion has to be selected. This is done by selecting a quasi-free kinematic configuration for the experiment. Selection of longitudinal virtual photons is done using the “Rosenbluth separation” method where electrons are detected at several angles while keeping Q 2 fixed. The cross section for exchange of a longitudinal virtual photon is related to the form factor: σL ∼ −
tgπ2 N N (t) 2 2 F (Q ) (t − m 2π )2 π
(40)
where t is the squared momentum transferred to the nucleon and gπ2 N N is the π N N coupling constant. The form factor, in turn, is related to the distribution amplitude through: 1 pquark du du φπ (u , µ2 ) T (u , u, Q 2 , µ2 ) φπ (u, µ2 ), u, u = . (41) Fπ (Q 2 ) = pπ 0
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305
Fig. 17. The space-like form factor of the π + meson. The data are from Refs. [83–85]. See the text for description of the calculated curves.
T (u , u, Q 2 , µ2 ) is the transition amplitude that has to be calculated from the diagram in Fig. 16(a) and should include treatment of the off-shell pion. This becomes increasingly uncertain as Q 2 increases. The experimental situation and comparisons with calculations are shown in Fig. 17 taken from [39]. The lines marked CZ and ASY were calculated using a simplified form of Eq. (41): 1 16πC F αs (Q 2 ) 2 φπ (u) (42) |I | with I = du Fπ (Q 2 ) = 2 Q u 0 and both disagree with the data. Various corrections were applied, mostly to account for soft parts of the integrals which are particularly significant near the end-points. The line marked LS comes from using φCZ and applying Sudakov resummation of the transverse degrees of freedom [81]. The line marked SR comes from direct calculation of the form factor from QCD sum rules without using the distribution amplitude at all [82]. While the latter results seem to be in relatively good agreement with the data, they scale as Q −4 at higher Q 2 and the agreement in the measured Q 2 region is generally taken as accidental. Calculations of asymptotic form factors are discussed in [86]. For more detailed discussion and references see [39]. Turning to the neutral pion, the π 0 -photon transition form factor, Fig. 16(b) is expected to be a better test of the pion distribution amplitude as it is, to lowest order, a pure QED process. The measurements are mostly from e+ e− colliders [87] where the measured process is: e+ e− → e+ e− γ ∗ γ ∗ ,
γ ∗γ ∗ → π 0.
(43)
In the experiment one of the scattered electrons is detected and used to measure the virtuality Q 2 of the associated photon. A soft, quasi-real photon is assumed to be emitted by the undetected electron. The π 0 is detected through its decay π 0 → γ γ . The cross section is measured as a function of Q 2 . In Fig. 18 we show the experimental results together with two calculations [88, 89]. The calculations use both the asymptotic and the Chernyak–Zhitnitsky distribution amplitudes. The calculations by Jakob et al. [88] get satisfactory agreement when using the asymptotic distribution amplitude but a large overestimate with the Chernyak–Zhitnitsky distribution amplitude. The calculations by Cao et al. [89] underestimate the data for both distribution amplitudes. Other calculations [90] found moderate agreement with the asymptotic
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Fig. 18. Experimental values of the π 0 -photon transition form factor [87]. The solid lines are calculations using the asymptotic distribution amplitude and the dashed lines are with the Chernyak–Zhitnitsky distribution amplitude. Left: Ref. [88], right: Ref. [89].
Fig. 19. Electron–positron pair annihilating to form a photon which then decays to a pion pair. The pion form factor is represented by the circle.
distribution amplitude for Q 2 > 3 GeV2 . It appears that the asymptotic distribution amplitude gives better agreement than the Chernyak–Zhitnitsky distribution amplitude. This is in contrast with the charged pion where, while inconclusive, the apparent agreement with the Chernyak–Zhitnitsky distribution amplitude seemed better [39]. The problems can be traced to the inherent difficulty to deduce a functional form from an integral, which has to be done for these measurements. The form factors are related to the integral over the distribution amplitude and the scattering matrix element (Eq. (41)) and their sensitivity to the shape of the distribution amplitude is low. In addition there are the various model-dependent ingredients with different prescriptions for their calculation. A differential measurement of the LCWF is therefore necessary. Finally, there are the time-like form factors measured at positive four-momentum transfer: q 2 > 0 as opposed to the previously discussed space-like form factors. The calculation of time-like form factors is done in the same way as for the space-like form factors including the same distribution amplitudes [91]. The authors discuss the analytical continuity of the functions, αs (Q 2 ) and the propagators and compute the ratio of time-like to space-like form factors. This ratio is asymptotically unity but in the range of 15 ≤ |Q 2 | ≤ 40 GeV2 the ratio is about 2.0 with 10% variation depending on the distribution amplitude being used. The charged pion time-like form factor has been measured through the process: e+ e− → + π π − , Fig. 19. The cross section for this process is related to the form factor by O’Connel
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Fig. 20. Left: Charged pion time-like form factors measured in electron–positron pair annihilation [93]. Right: Calculated values of the pion time-like form factor using the asymptotic [6,75], Chernyak–Zhitnitsky [32] and the FHZ [76] distribution amplitudes.
et al. [92]: α 2 π (s − 4m 2π )3/2 |Fπ (s)|2 . (44) 3 s 5/2 Where |Fπ (s)| is the time-like form factor and the expression preceding it in Eq. (44) represent the cross section for the e+ e− annihilation and direct coupling of the photon to the pions, calculated in QED. An option to extend this approach to π + π − electroproduction in ep collision is discussed in Section 4.4. Results [93] from the e+ e− measurements are shown in Fig. 20 (left). In this energy range the form factor is dominated by the ρ and ρ resonances. In Fig. 20 (right) we see the calculations of [91]. If we take the highest non-zero values in Fig. 20 (left) we get Q 2 Fπ ∼ 0.4 for Q 2 ∼ 4 which would be in agreement with the values calculated with any of the distribution amplitudes, Fig. 20 (right). The calculations in the region of 4 < Q 2 < 15 show a decrease of Q 2 Fπ when the asymptotic distribution amplitude is used but an increase when the other distribution amplitudes are used. Above this all distribution amplitudes give a flat dependence and they differ only in the magnitude. At s = 9.6 GeV2 a value of the time-like form factor Q 2 Fπ = 0.94 ± 0.0077 was deduced from the ratio (J/ψ → π + π − )/(J/ψ → e+ e− ) [94]. This value is about 50% larger than the value calculated using the Chernyak–Zhitnitsky distribution amplitude and almost a factor 5 larger than the one calculated using the asymptotic distribution amplitude. There is a measured value for the space-like form factor at this energy: Q 2 Fπ = 0.68±0.19 [85]. These two experimental results would be consistent with the predicted ratio of 2.0. However, it will take a very special mechanism to have the value of time-like Q 2 Fπ change from 0.4 to 0.94 over such a narrow Q 2 range. It is therefore of particular interest to extend measurements of the pion time-like form factor to cover the 4 < Q 2 < 15 range. σ =
3.3. Measurement of the pion LCWF by diffractive dissociation to dijets 3.3.1. Concept of the experiment The discussion in Section 2.3 and in particular Eq. (27), taken from [69], shows that under certain circumstances the cross section for diffractive dissociation of a pion to dijets is
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proportional to the square of the distribution amplitude. This would mean that the u-dependence of the cross section is the same as that of the distribution amplitude squared and thus can be a direct measurement of it. The concept of the measurement presented here is the following: a high energy pion dissociates diffractively while interacting with a heavy nuclear target. The first (valence) Fock component dominates at large Q 2 ; the other terms are suppressed by powers of 1/Q 2 for each additional parton, according to counting rules [22,39]. This is a coherent process in which the quark and antiquark break apart and hadronize into two jets. If in this fragmentation process the quark momentum is transferred to the jet, measurement of the jet momentum gives the quark (and antiquark) momentum. Thus: pjet1 . (45) u measured = pjet1 + pjet2 We consider now the circumstances under which this can be expected to be valid. We note that Eq. (26) requires double differentiation with respect to kt of the LCWF, ψ(u, kt ), not the distribution amplitude φ(u) for which φas and φCZ were proposed. In their evaluation of the cross section Frankfurt et al. make some assumptions. The pion LCWF is taken to have the form: 2 ψ˜ π(L ) (u, κt2 ) = B(L 2 )
φ(u) kt2
+ µ2
(46)
where the parameters L and µ account for non-perturbative effects. It is necessary that kt2 µ2 . They estimate µ2 ∼ 0.1 GeV2 so a requirement that kt > 1 GeV should be imposed in the experiment. The actual form of the LCWF presented in Eq. (46) may not be appropriate for low kt . Another important assumption is that quark momenta are not modified by nuclear interactions; i.e., that color transparency is satisfied. This has been verified in the experiment described in Section 2.3 which is the same as the one used to measure the LCWF by diffractive dissociation to dijets discussed below [96]. The experiment and analysis were carried out by the Fermilab E791 Collaboration. The diffractive dissociation of high momentum pions into two jets can be described by factoring out the perturbative high momentum transfer process from the soft nonperturbative part [31]. This factorization allows for definition of the virtuality of the process, Q 2 as the masssquared of the dijets. This is analogous to Q 2 being the mass of the virtual photon in DIS. From simple kinematics and assuming that the masses of the jets are small compared with the mass of the dijet, the virtuality and mass-squared of the dijets are given by: 2 = Q 2 ∼ MDJ
kt2 , u(1 − u)
(47)
where kt is the transverse momentum of each jet and reflects the intrinsic transverse momentum of the valence quark or antiquark. By studying the momentum distribution for various kt bins, one can observe changes in the apparent fractions of asymptotic and Chernyak–Zhitnitsky contributions to the pion distribution amplitude. The basic assumption is that the momentum carried by the dissociating q q¯ is transferred to the dijets. Distortions of the momentum distribution can occur in the fragmentation of the quarks to jets and then in the detection process. This was examined by Monte Carlo (MC) simulations of the q q¯ momentum distribution according to the asymptotic and the Chernyak–Zhitnitsky distribution amplitudes (squared). The MC samples were allowed to hadronize through the LUND PYTHIA–JETSET model [95] and then passed through simulation of the experimental
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Fig. 21. Monte Carlo simulations of q q¯ momentum distributions as predicted by the two distribution amplitudes squared, 2 is the asymptotic function at the quark level (left) and of the reconstructed distributions of dijets as detected (right). φas 2 is the Chernyak–Zhitnitsky function (squared). The dijet mass used in the simulation is 6 GeV/c2 and (squared) and φcz the plots are for 1.5 GeV/c ≤ kt ≤ 2.5 GeV/c.
apparatus (Fig. 13) to account for the effect of unmeasured neutrals and other experimental distortions. The results are shown in Fig. 21 where the initial distributions at the quark level are compared with the final distributions of the detected dijets, including distortions in the hadronization process and influence due to experimental acceptance. As can be seen, the qualitative features of the two distributions are retained. 3.3.2. Longitudinal momentum distribution Under the assumptions discussed in the previous section, measurement of the longitudinal momentum distribution will be a measurement of the distribution amplitude. Results of experimental studies of the pion distribution amplitude were recently published by the Fermilab E791 Collaboration [96]. More details on the experimental system are given in Section 2.3. In the experiment, diffractive dissociation of 500 GeV/c negative pions interacting with carbon and platinum targets was measured. Diffractive dijets were required to carry the full beam momentum.
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Fig. 22. The u distribution of diffractive dijets from the platinum target for 1.25 ≤ kt ≤ 1.5 GeV/c (left) and for 1.5 ≤ kt ≤ 2.5 GeV/c (right). The solid line is a fit to a combination of the asymptotic and CZ distribution amplitudes. The dashed line shows the contribution from the asymptotic function and the dotted line that of the CZ function.
They were identified through the e−bqt dependence of their yield (qt2 is the square of the trans2
verse momentum transferred to the nucleus and b = R3 where R is the nuclear radius). For measurement of the wave function the most forward events (qt2 < 0.015 GeV/c2 ) from the platinum target were used, see Fig. 14. For these events, the value of u was computed from the measured longitudinal momenta of the jets. The analysis was carried out in two windows of transverse momentum kt : 1.25 GeV/c ≤ kt ≤ 1.5 GeV/c and 1.5 GeV/c ≤ kt ≤ 2.5 GeV/c. The resulting u distributions are shown in Fig. 22. In order to get a measure of the correspondence between the experimental results and the calculated distribution amplitudes, the results were fit with a linear combination of squares of the two distribution amplitudes after smearing, as shown on the right side of Fig. 21. This assumes an incoherent combination of the two distribution amplitudes and that the evolution of the Chernyak–Zhitnitsky function is slow (as stated in [32]). The results for the higher kt window show that the asymptotic distribution amplitude describes the data very well. Hence, for kt > 1.5 GeV/c, which translates to Q 2 ∼ 10 (GeV/c)2 , the pQCD approach that led to construction of the asymptotic distribution amplitude is reasonable. The distribution in the lower window is consistent with a significant contribution from the Chernyak–Zhitnitsky distribution amplitude or may indicate contributions due to other nonperturbative effects. The quantity measured in this experiment, the distribution of longitudinal momentum within a kt window, is not exactly the distribution amplitude. The latter is an integral over kt with a lower limit of zero, covering the low Q 2 non-perturbative region (Eq. (4)). The results can be regarded instead as representing the square of the light-cone wave function averaged over kt in the window: ψq2q¯ (u, kt ). With the measured kt -dependence described in Section 3.3.4 the average values are kt = 1.34 GeV/c and 1.75 GeV/c for the low and high kt windows, respectively: ψq2q¯ (u, 1.34) and ψq2q¯ (u, 1.75) were measured. Alternatively, the results for each window can be related to the difference of distribution amplitudes: 2 k 2 2 2 ψ(u, k )d k (48) t t = |φ(u, k 2 ) − φ(u, k 1 )| . 2
k1
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3.3.3. Coefficients of the Gegenbauer polynomials The results published by Fermilab E791 collaboration [96] were fitted to a linear combination 2 and φ 2 , Fig. 22. For φ of φAsy CZ the authors used the function of Eq. (37) [32] that was proposed CZ for a scale of 500 MeV. One could evolve the function to the higher Q 2 values of the measurement and attempts to do so were done [102,103,78]. However, the evolved function has to be subjected to the distortions that the E791 authors calculated in their Monte Carlo simulations and this was not done in [102,103,78]. Furthermore, φCZ is a model distribution amplitude and other distribution amplitudes were proposed. Indeed, the conclusion that was derived from the E791 results is that in the higher kt range the asymptotic distribution describes the data well while in the lower kt range there are significant non-perturbative contributions, not necessarily in the form of φCZ . We make here an attempt at deriving the actual structure of φ 2 from the data. If Eq. (27) holds and the cross section is proportional to the square of the distribution amplitude, φπ2 (u), the experimental results can be used to determine the coefficients of the Gegenbauer polynomials in the expansion of φπ (u, Q 2 ), Eq. (30). One cannot fit directly the results shown in Fig. 22 as these results are distorted by the hadronization process and experimental acceptance. It is also not practical to carry out simulations of the distortions for a large variety of distribution amplitudes. We adopt here a simpler approach, albeit somewhat less precise than that used by E791. In order to carry out such a fit we use results of the Monte Carlo simulations shown in Fig. 21. We define the Acceptance as the ratio of the reconstructed event distribution (right side of Fig. 21) and the generated distribution (left side of Fig. 21): Acceptance = reconstructed/generated. To first order the Acceptance should not depend on the distribution. For the Chernyak–Zhitnitsky distribution there is a singularity in the Acceptance near u = 0.5. This can be avoided by using only fully charged generated events, the distribution of which is finite near u = 0.5. It turns out that the Acceptance calculated for such events is nearly the same for the asymptotic and the Chernyak–Zhitnitsky distributions. We then use the Acceptance derived from the asymptotic distribution. For the high kt range it is nearly flat in the central u region and drops near the end points. For the low kt range it drops gradually from u = 0.5 towards the end points. We will here, as did the E791 authors, avoid the regions of u < 0.1, u > 0.9. In these regions jet momenta can be too low for reliable detection. By dividing the experimental results of Fig. 22 by the Acceptance calculated for the two kt regions the Acceptance-corrected distributions, which should be proportional to the cross sections, are obtained. The results are fitted to the expression: 2 dσ 3/2 3/2 ∝ φπ2 (u, Q 2 ) = N · u 2 (1 − u)2 1.0 + a2 C2 (2u − 1) + a4 C4 (2u − 1) (49) du where N is a normalization constant and Cn are the Gegenbauer polynomials. It is assumed that Cn = 0 for n > 4. In Fig. 23 we show the Acceptance-corrected distributions with the fits to Eq. (49). We note that, due to the flat Acceptance in the high kt range the Acceptancecorrected distribution is very similar to the uncorrected one. On the other hand, the drop of the Acceptance towards the end points for the low kt range makes the corrected values be higher than the uncorrected ones. The results of the fits are that for the high kt region a2 = a4 = 0 within errors. For the low kt region the coefficients are: a2 = 0.30 ± 0.05, a4 = (0.5 ± 0.1) × 10−2 . The results for the high kt region confirm the conclusion of the E791 authors, namely that for 2 describes the data well. For the results in the low k region we should note that this region φAsy t the quoted errors represent statistical errors only resulting from the fit procedure. There should be also the same systematic uncertainties discussed in [96] and perhaps some uncertainties due to the Acceptance calculations. Nevertheless, the results using some variations of the Acceptance
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Fig. 23. The Acceptance-corrected u distributions of diffractive dijets obtained by applying correction to the E791 results [96]. The distributions are for 1.25 ≤ kt ≤ 1.5 GeV/c (left) and for 1.5 ≤ kt ≤ 2.5 GeV/c (right). The solid line is a fit to a combination of Gegenbauer polynomials, Eq. (49).
were very stable. The fact that a4 = 0, which seems to be essential for a reasonable fit, indicates a distribution amplitude that is different from φCZ as defined in Eq. (37) which contains only a a2 term [32]. 3.3.4. Transverse momentum distribution As discussed in Section 2.3, derivation of the cross section for diffractive dissociation [69] is based on the double-differentiation of the LCWF with respect to kt (Eq. (26)). More specifically: 2 2 dσ 2 2 2 ∂ ∝ |α (k )x G(u, k )| ψ(u, k ) (50) s t N t , t 2 2 dkt ∂kt with x N = 2kt2 /s and G N the gluon distribution function in the nucleon. This doubledifferentiation leads to a prediction of the kt dependence of the cross section. By comparing the measured and predicted kt distributions it is possible to test to what extent the assumptions used in deriving the cross section are correct with sensitivity to both the LCWF and the interaction. When applying Eq. (26) to the pion LCWF given by Eq. (46) the differentiation with respect to kt does not modify the u-dependence if kt2 µ2 . An additional kt dependence comes from the 1
gluon distribution in the nucleon. With αs (kt2 )x N G(u, kt2 ) ∼ kt2 [97] this yields: M(N) ∝
xN GN kt4
,
dσ dkt2
∝
(x N G N )2 kt8
,
dσ ∝ kt−6 dkt
(51)
and the u-dependence is the same as for φ 2 (u), Eq. (27). The experimental results are shown in Fig. 24 where they are compared with several fits. An attempt to fit the data over the whole kt dσ range to a power-law dependence: dk ∝ ktn resulted in n = −9.2 ± 0.4(stat) ± 0.3(sys), much t larger than expected from Eq. (51). This result is dominated by the low kt high statistics region. It can be seen that for the larger kt the slope changes and when only the kt > 1.8 GeV region is fit to a power-law the result is n = −6.5 ± 2.0, consistent with the predictions, Fig. 24(a, b).
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Fig. 24. Comparison of the experimental kt distribution [96] with fits derived from: (a) Gaussian LCWF [98] for low kt dσ ∝ k n , as expected from perturbative calculations, for high k ; (b) Two-term Singletand a power law dependence: dk t t t Model wave function [99] for low kt and a power law for high kt .
This is also consistent with the conclusions of Sections 3.3.2 and 3.3.3 that the asymptotic distribution amplitude agrees with the data only for the large kt region. Both results indicate that for kt > 1.5 GeV/c, which translates to Q 2 > 10 (GeV/c)2 the asymptotic distribution amplitude and the pQCD calculations are applicable. Below this value non-perturbative effects may play a significant role. This answers the question raised at the end of Section 3.1 concerning the Q 2 value where the distribution amplitude may resemble the asymptotic one. Naturally, the transition between the two regions is not sharp. The region of 1.0 ≤ kt ≤ 1.8 GeV/c may be a transition region where we can still apply pQCD techniques and in particular Eq. (26) but must use LCWFs that better describe the non-perturbative structure of the pion. In Fig. 24(a) we show the results [96] of fitting the low kt region with the cross section derived with Eq. (26) and 2 the non-perturbative Gaussian function: ψ ∼ e−βkt [98], resulting in β = 1.78 ± 0.1. Modeldependent values for β in the range of 0.9–4.0 were predicted [98]. This fit, although resulting in the parameter β being consistent with theoretical expectations, is not very satisfactory. As seen in Fig. 24(a) the curved shape of the theoretical calculation is not observed in the data. In Fig. 24(b) we show the results of fitting the low kt region using the non-perturbative two-term Coulomb wave function [99] which describes the L z = Sz = 0 component of the u d¯ wave function: ψ(u, k⊥ ) ≡ Ψu d¯ (u, k⊥ ; ↑↓). [100,101]: ϕ(k z , k⊥ ) , ψ(u, k⊥ ) = √ u(1 − u)
ϕ( p) =
A pa2
+
2 p 2
+
B pb2
+ p 2
2
(52)
where A, B, pa , pb are parameters of fitting this function to the numerically calculated LCWF. The agreement with this function is very good. These comparisons show that the transition region may be described using non-perturbative LCWFs. Since this is a relatively large kt region for non-perturbative interactions, it is actually the high momentum tail of the functions that is being compared. The observed sensitivity shows that the kt distribution is useful in studying wave functions in this transition region.
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Fig. 25. Diagram of diffractive dissociation of a pion to two jets used for the calculations by Chernyak [103] and by Braun et al. [102,106].
3.3.5. Has E791 measured the pion distribution amplitude? Following publication of the E791 results [96] several theoretical papers were published discussing the question of whether they can indeed be taken as measurement of the pion distribution amplitude. The subject was also discussed in several conferences [104]. We bring here a brief summary of the main points that were raised and add some comments. The main questions that were discussed are: • Is the cross section for the process indeed proportional to φ(u)2 as claimed in Eq. (27) [69]? • Are the results precise enough to distinguish between φAsy (u) and other forms of φ(u)? Nikolaev et al. [74] calculate the cross section for diffractive dissociation of pions to dijets using pQCD methods. They show that the cross section is proportional to φ 2 (u) and to the unintegrated gluon structure function of the nucleon. They disagree with Frankfurt et al. [69] who used the integrated gluon structure function. They calculate higher-twist effects which contain some u-dependence but show that in nuclear medium they are suppressed. As a result, when the measurements are done in a heavy nuclear target the cross section is proportional to φ(u)2 and can be used to determine it. Hence their response to the first question is positive. Concerning the shape of φ(u) they propose a soft model distribution amplitude that has a different mathematical form than that of φAsy (u) but has a very similar u-dependence. Because of this similarity they conclude that the E791 results are consistent with their calculations as well. They are also able to reproduce the kt and A dependence observed in the experiment. V. Chernyak [103–105] calculates the process described in Fig. 25. The lower blob in the diagram represents the skewed gluon distribution of the nucleon. The upper blob represents the hard kernel of the amplitude that consists of 31 connected Born diagrams. Nuclear effects and the quark transverse momenta are ignored. Calculations of these diagrams lead to an expression for the amplitude which is not proportional to φ(u) but rather to a sum of four integrals over φ(u) multiplied by expressions that contain u-dependence. His conclusion is that the cross section depends on φ 2 (u) in a complicated way hence measurement of the cross section cannot provide a measurement of φ 2 (u). Chernyak disagrees with the authors of [74] as they ignore the contributions where the jet momenta differ significantly from the quark momenta. He agrees that making this assumption will lead to proportionality of the cross section and φ 2 (u). He also disagrees with the authors of [69] that ignored contributions from diagrams that are, according to their evaluation of the E791 conditions, suppressed by Sudakov form factors. Following these arguments Chernyak applies his calculations to φAsy (u) and to φCZ (u) which he evolves to the scale of 2 GeV. He does it by treating the pion as free q q¯ and does not use the logarithmic
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Fig. 26. Left: the Chernyak–Zhitnitsky distribution amplitude evolved to a scale of 2 GeV (Eq. (53), solid line) compared with φAsy (u) and φCZ (u), [103]. In the figure x has the same meaning as u. Right: the cross section calculated in [103] compared with experimental results. Solid line: using φCZ (u) evolved to 2 GeV, dashed line: using φAsy (u). The parameter y is the longitudinal momentum fraction of the jets which is not taken to be the same as u.
evolution. It should be noted that with this definition the four-momentum squared of the process becomes: p2 = Mq2q¯ =
kt2 u(1−u)
and not p2 = m 2π . He obtains the form:
φCZ (u, 2 GeV) = 15u(1 − u)[(2u − 1)2 + 0.2].
(53)
This function is shown in Fig. 26 (left). An attempt to fit this function to the experimental data presented in Fig. 23 failed, yielding χ 2 ∼ 5. The author then modifies the function to account for final state interaction of the outgoing jets. This disregards the color transparency conditions under which the measurement was done. The statement is: “The detailed consideration of nuclear effects is out the scope of this paper” [104]. The results of his cross section calculations are shown in Fig. 26 (right) compared with the experimental results in the high kt region. He concludes that the precision of the experimental results is insufficient for discrimination between these functions. Braun et al. [102,106] carry out calculations along the same lines as Chernyak (Fig. 25) and display all 31 Born diagrams that go in the calculations. There are some technical differences such as parametrization of the gluon distribution. They find that coherent jet production is dominated by gluon contributions and that collinear factorization is violated near the end-points and when the quark and jet momenta differ significantly. However, the amplitude near the end points resembles φAsy(u) and when the quark and jet momenta are not very different the cross section becomes proportional to φ 2 (u). The factorization is restored at energies much higher than the E791 energy. Like Chernyak, Braun et al. do the calculations for a nucleon target and not a heavy nucleus such as was used in E791. They state, however, that in a heavy nucleus color filtering will cause suppression of the nonfactorizable contributions. Bakulev et al. [78] use the CLEO data [87] to fit coefficients of Gegenbauer polynomial expansion, Eq. (30). They obtain results for a2 and a4 for several scales and conclude that the CLEO data are inconsistent with both φAsy (u) and φCZ (u). They obtain optimal coefficients opt opt a2 = 0.188, a4 = −0.13 and these values represent a distribution amplitude developed by
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them using nonlocal QCD sum rules at a scale of 1 GeV2 . They then use the method of [106] to calculate a cross section which turns out to be in good agreement with the E791 results. Comments The calculations carried out by Chernyak and by Braun et al., done for a nucleon target, indicate that the cross section for diffractive dijet production is not simply proportional to φ 2 as suggested in Eq. (27). They also reach the associated conclusion that the E791 results are not sufficiently precise for discriminating between the φAsy (u) and φCZ (u) evolved to the scale of 2 GeV. It should be noted that φCZ (u, 2 GeV) is significantly different from the φCZ (u, 0.5 GeV) (Fig. 26, left), not as would be expected from a slow, logarithmic evolution. The cross section 2 (u) (Fig. 26, right) which would show calculated using φAsy (u) is very similar in shape to φAsy that for this distribution amplitude the cross section is not far from being proportional to φ 2 (u). It appears that the conclusions of these works result from a very rapid evolution of φCZ (u) and distortions that affect φCZ (u) much more than φAsy(u) resulting in similar final results. The E791 results were obtained using a heavy nucleus and color transparency conditions were verified. Under these conditions the pion fluctuated to its PLC state before interacting with the target nucleus. According to [69,74,106] it may then be assumed that the cross section is in fact proportional to φ 2 (u). There are distortions which were taken into account by E791 using the PYTHIA simulations. Naturally, every experimental measurement has limits on its discriminating power between distribution amplitudes that are not very different from each other such as the one suggested in [74] which almost coincides with φAsy(u) or φCZ (u, 2 GeV) which is not very different from it. We should finally note that any comparison between theoretical calculations and experimental data should take into account experimental distortions. This can be done either, as done by E791, by subjecting the theoretical calculations to all the distortions that affect the experimental data or, as attempted to do in this work (Section 3.3.3), correct the experimental data for these distortions thus making the results readily comparable with theoretical calculations. 4. The photon light-cone wave function 4.1. Predictions of the photon LCWF The photon light-cone wave function can be described in a way similar to that of the pion except that it has two major components: the electromagnetic and the hadronic. Being a gauge field capable of point-like coupling it has also a point bare-photon component. Consequently, it can be expanded in terms of Fock states: (Λ) (Λ) ¯ ¯ |ψγ = |γ p + ψll/γ ¯ (u i , k ⊥i , λi )|l l + ψl lγ ¯ /γ (u i , k ⊥i , λi )|l lγ + · · · (Λ) (Λ) ¯ + ψq q¯ g/γ (u i , k⊥i , λi )|q qg ¯ + ··· + ψq q/γ ¯ (u i , k ⊥i , λi )|q q
(54)
¯ are lepton pairs with photons in where |γ p describes the point bare-photon and |l +l − , |l lγ the higher Fock states of the electromagnetic component. Similarly, |q q, ¯ |q qg ¯ are the q q¯ pairs with gluons in the higher Fock states of the hadronic component. Each of these states is a sum over the relevant helicity components. The wave function is very rich: it can be studied for real photons, for virtual photons of various virtualities, for transverse and longitudinal photons and the hadronic component may be decomposed according to the quark’s flavor. The pointbare photon does not have internal structure, it can only Compton-scatter. The LCWF for the lowest Fock states is described [16] through the interactions shown in Fig. 27. In Fig. 27(a) the
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Fig. 27. Diagrams for photon dissociation: (a) represents the diagrams leading to the photon electromagnetic dissociation, (b) is for dissociation to q q. ¯
electromagnetic component of the photon interacts via one photon exchange and dissociates to two leptons (e+ e− , µ+ µ− , τ + τ − ). In Fig. 27(b) the hadronic component interacts via two gluon exchange and dissociates into a q q¯ which then hadronize to a dijet. The form of the photon LCWF is given by Brodsky et al. [16]: ψλλ1 λ2 (kt , u) = −ee f
f λ1 (k)λ · λ f λ2 (q − k) 2 +m 2 √ k⊥ 2 u(1 − u) Q + u(1−u)
(55)
where λ is the polarization vector and f λ , m, λ1 , λ2 , ee f are the fermion distributions, masses, helicities and charges, respectively. Q 2 is the photon virtuality. Similar expressions are given in [107,108]. The predicted LCWFs for the electromagnetic component are based on quantum electrodynamics and can be considered precise. Those for the hadronic component are model dependent. The wave function for kt2 Λ2QCD is expected to be similar for the electromagnetic and hadronic components. The probability distribution amplitude Φ 2 is obtained from the trace of this function. For transversely polarized photon the result is: Φ 2f f¯/γ ∗ ∼ T
2 1 m 2 + kt2 [u 2 + (1 − u)2 ] Tr ψ 2 = 4 [kt2 + a 2 ]2 µ=1
(56)
and for longitudinally polarized photons: Φ 2f f¯/γ ∗ ∼ L
Q 2 [u 2 (1 − u)2 ] [kt2 + a 2 ]2
(57)
where f f¯ stands for l l¯ or q q¯ and a 2 = m 2 + Q 2 u(1 − u). While leptons usually satisfy m l2 kt2 this is not always so for quarks. For a real photon and light quarks (as for the electromagnetic component) the result is: Φ 2f f¯/γ ∼
1 kt2
[u 2 + (1 − u)2 ].
(58)
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Fig. 28. The photon wave function for transverse virtual photons (Q 2 = 5 (GeV/c)2 ) and massless quarks/leptons (solid line), real photons and massless quarks/leptons (dashed line), real photons and charm quarks (dotted line) and the asymptotic function (dashed–dotted line). The functions are arbitrarily normalized at u = 0.5.
For massive quarks: Φq2q/γ ∼ ¯
m 2q + kt2 [u 2 + (1 − u)2 ] [m 2q + kt2 ]2
(59)
and for virtual photons and light quarks: Φ 2f f¯/γ ∗ ∼ T
kt2 [u 2 + (1 − u)2 ] . [kt2 + Q 2 u(1 − u)]2
(60)
As in the case of the pion, the photon LCWF in the nonperturbative region is model dependent and Q 2 values for transition to the perturbative regime have to be determined. The actual lightk2
t cone wave functions of the photon are very interesting and subtle. At small invariant mass u(1−u) , the mixing of the photon with vector meson physics is inevitable. Balitsky et al. [109] predict for real photons a distribution amplitude identical to the pion asymptotic function [6]:
Φq2q/γ ∼ u 2 (1 − u)2 ¯
(61)
and presumably can be valid for kt > 1.5 GeV/c (as found for the pion). Examples of Φ 2 calculated using Eq. (56) and the asymptotic LCWF are shown in Fig. 28. Instanton models are now providing a guide to the two-quark valence wave function where the quark and antiquark helicities are parallel, a component proportional to the dynamical quark mass. The distribution amplitudes for these components, integrated over kt as in Eq. (4), were calculated by Petrov et al. [110] using the instanton model and are shown in Fig. 29.
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Fig. 29. The photon wave functions, Petrov et al. [110] Real photons (Solid line), virtual photons Q 2 = 250 MeV2 (dashed line), Q 2 = 500 MeV2 (dotted line).
4.2. Cross section of coherent dijet electroproduction As will be discussed in Sections 4.6–4.8 experiments were done in order to study the photon LCWF by measurements of diffractive dissociation of the photon to dimuons and dijets, similar to studies of the pion. The cross section for photon diffractive dissociation (DD) was calculated by several authors. In [112] this cross section is given in terms of the “dipole cross section”: 1 dσDD 2 (Q ) = du d2r |ΨγT∗,L (r , u, Q 2 )|2 σdipole(r , u) (62) dt 0 which is very similar to Eq. (20). ΨγT∗,L is the transverse/longitudinal wave function in coordinate space. Various models were proposed for the dipole cross section σdipole(r , u). A calculation specific for the photon diffractive dissociation to q q¯ followed by hadronization to dijets is given in [113]. The authors give expressions of cross sections for longitudinal and transverse photons in terms of the integrated gluon distribution of the proton for which they use some parametrizations. They give numerical predictions for the cross section dependence on various variables. In [114] the authors follow these calculations and give predictions for the azimuthal angular distributions (angle φ between lepton plane and dijet plane) of the dijets that originate from q q¯ fragmentation: kt2 − αem 1 + (1 − y)2 γ ∗ p e p γ∗p M2 dσ D,T − 2(1 − y) cos 2φ dσ D,T dσ D = 2 2 k 2 yQ π 1 − 2 Mt 2 ∗ ∗ γ p γ p + (1 − y)dσ D,L + (2 − y) (1 − y) cos φdσ D,I (63)
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Fig. 30. The φ dependence of total ep—cross section with different cuts in k 2 : 1 GeV2 < k 2 < 2 GeV2 , 2 GeV2 < k 2 , 5 GeV2 < k 2 (from top to bottom), Ref. [114]. γ∗p
γ∗p
γ∗p
where σ D,T , σ D,L , σ D,I are the transverse, longitudinal and interference cross sections. While a purely longitudinal cross section has no angular dependence, a purely transverse cross section does have such dependence resulting from interference of the two helicity amplitudes. The distribution peaks at π/2 for kt2 > 2 GeV2 , Fig. 30. This gives a unique signature which will be exploited in the experimental studies of this process (Section 4.8). The authors show that for other mechanisms such as boson–gluon fusion the azimuthal distribution has a minimum at π/2. A detailed discussion of these processes including evaluation of bounds on the cross sections is given in [115] where similar angular distributions are derived. In [116,117] the authors calculate the cross section for dissociation to q qg ¯ which is the dominant background for studies aimed at the q q¯ dissociation. These calculations help in determining the region of phase space where the signal to background ratio can be optimized. Another prediction of the cross section was recently published by Braun et al. [118] and the results are shown in Fig. 31. The authors calculate the cross sections for real photons and the other kinematic conditions as for the experiment described in Section 4.8. They show cross sections and angular distributions depending on the helicity of the Fock state. Each of the helicity states is expected to dominate in a different kt region so they can be distinguished experimentally. Their angular distribution peaks at π/2, similar to the observation in [114] for virtual photons, for both helicity states. More recently the group extended their calculation to virtual photons with kinematical conditions very similar to those used in the experiments described in Section 4.8 [119]. Their results are shown in Fig. 32 where, again, the meaning of z is the same as u. The solid and dashed lines are the results using the CTEQ6L and MRST2001LO parton distributions, respectively. The calculations are (top to bottom) for kt = 1.25, 1.5, 1.75 GeV/c. The condition β > 0.5 is similar to the one applied in the experimental analysis in order to enrich the fraction of q q¯ dijets. The effect of this condition on the u distribution is consequence of the associated reduction in phase space.
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Fig. 31. The cross section for dijet production by real photons predicted by Braun et al., Ref. [118]. The meaning of z is the same as u. The solid/dashed lines are for different cut-offs and mass parameters.
4.2.1. Cross section derived from the LCWF Here we present a calculation of the cross section for diffractive dissociation of the photon to dijets in a simplified approach that makes the role of the photon LCWF transparent. The aim is to compare the u-dependence of the cross section to that of the LCWF so that the ability to determine the LCWF from cross section measurements, as done for the pion, can be assessed. The derivations are done within DGLAP in the kinematical region kt2 m 2ρ . These calculations are equivalent to those of [113] in the double-leading-log approximation. The general relation of cross section and transition amplitude: dσ παs2 2 = |A|2 F2g 2 (16s)2 dt du dkt
(64)
where A is the amplitude of the process and is independent of t. The form factor F2g is given by: F2g = (1 − t/m 2 (2g))−2
(65)
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Fig. 32. The cross section for dijet production by virtual photons predicted by Braun et al., Ref. [119]. The meaning of z is the same as u. The solid and dashed lines are the results using CTEQ6L and MRST2001LO parton distributions, respectively. The calculations are (top to bottom) for kt = 1.25, 1.5, 1.75 GeV/c. The condition β > 0.5 is applied in (a) and not in (b).
where t = tmin − qt2 , qt is the momentum transferred to target. The two-gluon mass m(2g) was estimated to be ≈1 GeV [111]. For two-gluon exchange the amplitude is given by replacing σdipole ∝ b2 by −∇κ2t as done in Section 2.3, Eq. (26). Due to the polarization states we have here the trace of the second derivative of the LCWF. For longitudinally polarized photons: |A L |2 = (4/3)2 e2 eq2 (s · xg)2 · |Tr ∇κ2t ψ L |2
(66)
and for transversely polarized photons: |A T |2 =
1 (4/3)2 e2 eq2 (s · xg)2 · |Tr ∇κ2t ψ T |2 2
(67)
where ψ L and ψ T are the longitudinal and transverse LCWFs, respectively, following the prescription for photon LCWF as suggested by Brodsky and Lepage [6]. s is the total energy, (xg) the gluon distribution and eq the quark charge. Using Eq. (55) we can write the longitudinal LCWF: ψL =
Q · u(1 − u) a 2 + kt2
(68)
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where a 2 = Q 2 u(1 − u) + m 2q . Derivation of the transverse LCWF is more complex due to the two components. Using the relation: σi σ j = δi j + i i j k it can be written as: ψµT =
σµ m q + σz kµ (2u − 1) + i µlz ktl a 2 + kt2
.
(69)
Differentiation yields: ∇κ2t ∇κ2t
kµ 8kµ a 2 = − a 2 + kt2 (a 2 + kt2 )3 1 a2
+
kt2
=−
4(a 2 − kt2 ) (a 2 + kt2 )3
(70) .
(71)
We then get for the cross sections at t = 0: 2 2 2 2 2 dσ L παs2 2 2 2 2 u (1 − u) (a − k t ) e = Q (xg) · e q 9 dt du dkt2 (a 2 + kt2 )6
(72)
m 2q (a 2 − kt2 )2 + 2kt2 [u 2 + (1 − u)2 ] · 4(a 2)2 dσT παs2 2 2 2 e = (xg) · . e q 18 dt du dkt2 (a 2 + kt2 )6
(73)
and
For numerical calculations we use m q = 0. This is a reasonable approximation except for heavy quarks and near the end-points of the u distributions. We also take (xg) as constant to avoid using a non-transparent parametrization. The resulting u-distribution is shown in Fig. 33 for fixed values of Q 2 and M 2 and compared with Φ 2 . As can be seen the distributions are not the same, the cross section for this process is not proportional to Φ 2 . Experimental results should be compared with the predicted cross section which is sensitive to the u-dependence of the LCWF. For massive quarks it turns out that while the cross section is insensitive to the quark mass, the LCWF changes significantly (Fig. 28). The predictions shown in Figs. 31 and 33 are quite different and demonstrate the variety of expectations with which the experimental results will have to be compared. For the M 2 dependence of the cross section we use the relations: a 2 ± kt2 = u(1 − u)(Q 2 ± M 2 ) and d(kt2 ) = u(1 − u)d(M 2 ). For t = 0: 2 2 2 παs2 2 2 2 1 dσ L 2 (Q − M ) e = Q (xg) · · e q 9 dt du dM 2 (Q 2 + M 2 )6 u(1 − u) 4 2 2 2 dσT παs2 2 2 2 8Q M [u + (1 − u) ] e = (xg) · . e q dt du dM 2 18 (Q 2 + M 2 )6 u 2 (1 − u)2
(74) (75)
These results are equivalent to those of [113] in the double-leading-log approximation, the main difference being the explicit dependence on u. Note that for longitudinal photons the cross section is zero for Q 2 = M 2 . This was also found by the authors of Ref. [113] in the doubleleading-log approximation. The authors then add a non perturbative parametrization of the BFKL Pomeron which causes some shifts of the position of the minimal cross section. The ratio between longitudinal and transverse cross sections is given by: u 2 (1 − u)2 (a 2 − kt2 )2 dσ L = 2Q 2 2 . dσT 2kt [u 2 + (1 − u)2 ] · 4a 4
(76)
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Fig. 33. The cross section for dijet production by transverse photons at fixed Q 2 , M 2 (solid line) and the perturbative LCWF squared (dashed line). The two functions are normalized at u = 0.5.
For u = 0.5: 1 kt2 dσ L = dσT 8 Q2
2 Q2 0.25 2 − 1 kt
(77)
2
and is a function of Q2 . Numerically it can be seen that the longitudinal cross section is expected kt to be smaller by more than an order of magnitude than the transverse one, in agreement with the results of [113]. The correction due to m q = 0 may be non-negligible in particular near the dip region of the ratio σ L /σT . This region will be sensitive also to NLO corrections. We mention here two other parameters used in the literature for expressing these processes. If we go to the q q¯ center of mass system and neglect the quark mass, then using Eq. (2) we find the relation: 1 1 + cos θ ∗ (78) ; u(1 − u) = sin2 θ ∗ u= 2 4 where θ ∗ is the angle between the quark and the virtual photon in the center of mass system. The u-dependence is then translated to an angular distribution of the quark in the center of mass system. Another parameter is defined by: β=
Q2 Q 2 + M X2
(79)
where M X is the mass of the diffractive system. This parameter becomes very useful when trying to isolate dijets originating from q q¯ hadronization from those coming from q qg ¯ and other configurations. It is found that at large β the dominant contribution comes from the q q¯ component. This parameter is therefore useful for studying the LCWF of the q q¯ component of the photon. In this respect it is instructive to express Eqs. (74) and (75) in terms of β:
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Fig. 34. The q q¯ dissociation cross section induced by longitudinal photons divided by that for transverse photons, both calculated at u = 0.5. The minimum is not zero due to finite resolution.
παs2 2 2 dσ L eq (xg)2 · = e dt du dβ 9 dσT παs2 2 2 = e eq (xg)2 · dt du dβ 18
β2 1 (2β − 1)2 · 4 Q u(1 − u) β2 u 2 + (1 − u)2 β(1 − β) 2 . Q4 u (1 − u)2
(80) (81)
The ratio R = dσ L /dσT for u = 0.5 is shown in Fig. 34. The dip at β = 0.5 reflects the Q 2 = M 2 region seen in Eqs. (74) and (75). The minimum is not zero due to finite resolution. This region is sensitive to higher order contributions which will cause modifications in the minimum region [114]. As can be seen, in order to study the q q¯ LCWF one should be in a relatively large β to have q q¯ dominance but not too large so that the transverse photon dominate and the cos 2φ signature (Eq. (63)) can be used. 4.3. Cross section of coherent photoproduction of dimuons derived from the LCWF This is the well known Bethe–Heitler process the cross section for which is a text-book matter [120]. Nevertheless, we bring here a short derivation of this cross section in the LCWF language to emphasize the common features of this process and the one described in the previous section. We follow the same way as done for the dijet production so the common LCWF connection is evident. There are only two differences between the present derivation and that for the dijets: αs should be replaced by αEM and since the interaction is of one-photon exchange, rather than two gluon (Fig. 27), the amplitude is given by the trace of the first derivative of the LCWF, not the second. The rest follows the same order as in Section 4.2.1. For photoproduction we have only transversely polarized photons: 2 ∂ 2 T ψµ (82) |A T | ∝ Tr ∂kµ
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Fig. 35. The photon dissociation to q q¯ followed by hadronization to two pions.
where ψ T is the transverse LCWF: σµ m l + σz kµ (2u − 1) + i µlz ktl
ψµT =
a 2 + kt2
and a 2 = Q 2 u(1 − u) + m l2 . Differentiation yields: ∂ kµ 2 4kt4 2(a 2 − k 2 ) Tr = + 2 2 ∂kµ a 2 + kt (a 2 + kt )4 (a 2 + kt2 )3 ∂ 1 2 4kt2 = . Tr 2 2 2 ∂k a + k (a + k 2 )4 µ
t
(83)
(84) (85)
t
We then get for the cross sections at t = 0: 4m l2 kt2 + 2(kt4 + a 4 )[u 2 + (1 − u)2 ] dσT ∝ . dt du dkt2 (a 2 + kt2 )4
(86)
For real photons and using m l = 0 → a = 0: dσT dt du dkt2
∝
2[u 2 + (1 − u)2 ] kt4
∼
Φ2 kt2
.
(87)
Φ is the probability amplitude obtained from the trace of the LCWF for transversely polarized photons, Eq. (56) and the cross section is directly proportional to Φ 2 . 4.4. Cross section for dipion production The processes of photo- or electro-production of two pions may be considered as a special case of the photon dissociation to dijets when each jet consists of one pion (Fig. 35). This is a very exclusive process where the pion form factor and quantum numbers most probably affect the results. There may be consequences on the ratio of longitudinal/transverse cross sections and for the u-distribution. Calculation of many of these effects may be model dependent. At low π–π masses, Mππ , there will be influence of the tails of the ρ resonance and perhaps of higher resonances. Measurement of the cross section can be used to evaluate the pion time-like form
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factor, see Section 3.2. The cross section for this process is expected to be proportional to the time-like form factor: σ (γ ∗ + p → 2π + p) ∝ |Fπ |2 σ (γ ∗ + p → X + p)
(88)
where the denominator can be taken from parametrization of measurements and the results may have to be normalized to those obtained from e+ e− measurements [93]. It may be possible to use these measurements to extend measurements of the pion time-like form factor into the 4 < Q 2 < 15 region where there is great sensitivity to the pion light-cone wave function (Section 3.2). From azimuthal angular distribution as the one described in Section 4.2, Eq. (63) it may be possible to determine the contributions from longitudinal and transverse photons which may be quite different from that for dijets. With this determined measurements of the longitudinal momentum fraction u can be compared with the predicted cross sections for dijets. This comparison may lead to conclusions on the extent to which the two processes are similar. This will have implication on our understanding of the hadronization process. The t-distribution for this reaction may be considered in the same way as for vector meson production (Section 2.2). The slope of the distribution measures the combined size of the q q¯ system and the proton, Eq. (18) and it will allow measurement of the q q¯ size also in the hadronization to continuum states. The cross section for 2π production is expected to be suppressed by |Fπ |2 compared with dijet production. We can expect |Fπ |2 ∝ 1/kt4 or ∝ 1/M 4 2 dependence of the cross section is hence expected to fall rapidly or ∝ 1/(Q 2 + M 2 )2 . The Mππ with the mass and this reduces the possibility for studies of this process at large masses. Studies of this process for relatively low masses, just above the ρ, ρ , f 2 resonances will result in a relatively large value of β and thus be dominanted by q q¯ produced from longitudinal photons (Fig. 34). 4.5. The Odderon The exclusive dipion reaction is considered as one of the ways to hunt for the elusive Odderon [121,122]. The Odderon is described as a d-coupled three-gluon color singlet C = −1 object [123]. The Odderon and the Pomeron are expected to have very similar intercepts with values near 1.0. Several attempts to observe the Odderon have so far failed. These were aimed at observing processes which the Pomeron cannot mediate while the Odderon can. Since the photon has C-parity C = −1, it can couple with the C = +1 Pomeron only to C = −1 mesons while the C = +1 mesons are expected to be produced by Odderon exchange. Following specific predictions [124], the H1 collaboration studied the photoproduction of the C = +1 π 0 and f 2 mesons. They obtained only upper limits for the cross sections which were lower than the predicted ones, [125,126]. It has been suggested [127] that as the amplitude for these processes is very weak one should look at signatures that are linear with this amplitude rather than quadratic, such as inteference effects. The Odderon–Pomeron interference can present such a case. Since the charge parity of dipions is: C(π + π − ) = (−) where is their relative angular momentum, both Pomeron and Odderon can contribute to their production and the interference will show up. The signature would be the charge asymmetry: A=
u(+) − u(−) u(+) + u(−)
(89)
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Fig. 36. The ZEUS detector at HERA, DESY. For description see [128].
which would be studied as a function of Mππ . This asymmetry was calculated for dipion electroproduction [121] and photoproduction [122]. It turns out that the asymmetry is proportional to the difference of the Pomeron and Odderon intercepts. As this is expected to be rather small the calculations utilize the effects of resonances, in particular that of the f 2 . Modeldependent asymmetries of the order a of a few percent are predicted in the f 2 mass region. 4.6. Measurement of the electromagnetic component of real photon LCWF Measurements of the photon light-cone wave function are being carried out at the DESY accelerator in the collision of 27.5 GeV/c electrons (or positrons) with 920 GeV/c protons producing real or virtual photons. The measurements were done using the ZEUS detector (Fig. 36) [128]. Measurement of the electromagnetic component of the real photon LCWF was done using the exclusive ep → eµ+ µ− p photoproduction process. Preliminary results were reported recently [129]. 4.6.1. Data analysis The integrated luminosity for the results was 55.4 ±1.3 pb−1 . Events of the exclusive reaction were triggered using the muon chambers. Each event was required to have two high quality tracks fitted to the vertex and matched to energy deposits in the calorimeter. It was required that no signal from the scattered positron or from proton dissociation be recorded. The LCWF was measured as the distribution of the longitudinal light-cone momentum fraction as defined in Eq. (2) (see Fig. 37): u=
E 1 + pz1 E 1 + pz1 + E 2 + pz2
(90)
where E 1 , p1 , E 2 , p2 are the energy and momentum of each muon. The kinematic region was defined by the following selection criteria: the invariant mass of the dimuon system 4 < Mµµ < 15 GeV, the γ p center of mass energy 30 < W < 170 GeV, the square of the four momentum exchanged at the proton vertex |t| < 0.5 GeV2 (to select diffractive events), 0.1 < u < 0.9 (to avoid regions of very low momenta where the acceptance is not well defined) and k T > 1.2 GeV (to select a hard process). The acceptance corrections and resolutions were determined using the
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Fig. 37. Definition of the kinematic variables in the γ → µ+ µ− process.
dedicated Monte Carlo GRAPE generator [130] for dilepton production in ep collisions, based on exact matrix elements. The program is interfaced to PYTHIA to generate a complete final state. The simulation included both elastic and proton-dissociative events. All generated events were passed through a detector simulation based on GEANT 3.13. 4.6.2. Results and discussion The measured k T , W, t, Mµµ distributions are presented in Fig. 38. The data are compared with GRAPE Monte Carlo simulation normalized to the data. Good agreement is shown between data and simulation. The measured differential cross section dσ/du is presented in Fig. 39 and compared with the theoretical curve (BFGMS) [16]. For this purpose Eq. (56) was adapted to the muon’s analysis: 2 Φµµ/γ =
u 2 + (1 − u)2 . 2 u (1 − u) − m 2 ]2 [Mµµ µ
(91)
The LCWF was normalized to the data. The shape of the calculated electromagnetic component of the real photon light-cone wave function is in good agreement with the experimental result. The measured cross section was also compared to the GRAPE simulation normalized to the luminosity. The agreement between data and simulation is within the systematic uncertainties. This measurement serves as a “Standard Candle” and normalization for the hadronic LCWF. It also provides the first proof that diffractive dissociation of particles can be reliably used to measure their LCWF. Furthermore it gives support for the method used in previous measurements of the pion LCWF [96] and possible future applications to other hadrons [131]. 4.7. Measurement of the hadronic component of the photon LCWF by exclusive dipion electroproduction Measurement of exclusive diffractive electroproduction of π + π − pairs was carried out by the ZEUS collaboration. Preliminary results were reported recently [133]. The concept of the measurement is the same as described in Sections 3.3 and 4.6. Viewed in the proton rest frame the photon fluctuates to its Fock states long before the interaction when it dissociates diffractively. The first (valence) Fock component dominates while the other terms are suppressed
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Fig. 38. Number of events reconstructed in the γ → µ+ µ− process for the kinematic region 4 < Mµµ < 15 GeV, 30 < W < 170 GeV/c |t| < 0.5 (GeV/c)2 and k T > 1.2 GeV/c plotted against k T , W, |t| and Mµµ . The data distributions are shown as the points with statistical errors only. The solid lines show the prediction of the GRAPE generator summing elastic and proton-diffractive components.
according to counting rules [22]. In this exclusive process the quarks break apart and hadronize into two jets which may be a π + π − pair. Measurement of their momenta gives the quark momenta. 4.7.1. Data analysis The data were collected at the HERA ep collider during 1999–2000 with the ZEUS [128] detector. At that time HERA operated at proton energy of 920 GeV and at positron energy of 27.5 GeV. The integrated luminosity used was 66.3 ± 1.7 pb−1 . Each event was demanded to have two high quality tracks from the central tracking detector, fitted to the vertex and matched to energy deposits in the uranium calorimeter. Pions were identified using a neural network algorithm. The total charge of the two tracks was required to be zero. The energy E and momentum z-component pz of the scattered positron and two pions were required to satisfy the relation 40 GeV ≤ E − pz ≤ 60 GeV. To select exclusive events, the total energy deposited in the calorimeter not associated with the scattered positron or the pions was required to be less than 300 MeV. In order to limit the contribution from proton-dissociative events, ep → eπ + π − N, the energy deposited in the forward plug calorimeter, located close to the beam pipe, was required to be less than 1 GeV.
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Fig. 39. Differential cross section dσ/du for the γ → µ+ µ− process measured for 30 < W < 170 GeV, 4 < Mµµ < 15 GeV, k T > 1.2 GeV/c and −t < 0.5 (GeV/c)2 . The inner error bars show the statistical uncertainty; the outer error bars show the statistical and systematics added in quadrature. The data points are compared to the prediction of LCWF theory [16]. The theory is normalized to data.
The kinematic variables are the same as shown in Fig. 37, replacing µ by π. These are W , the γ ∗ p center-of-mass energy, Q 2 , the photon virtuality, t the square of the four momentum exchanged at the proton vertex, and u, the longitudinal momentum-fraction carried by the pion. The kinematic region for this measurement is defined as: 2 < Q 2 < 20 GeV2 , 1.2 < Mππ < 5 GeV, 40 < W < 120 GeV, |t| < 0.5 GeV2 , 0.1 < u < 0.9 (to avoid regions of low acceptance). At that stage a remaining contamination of about 8% from proton dissociative events was not subtracted. The acceptance corrections and resolution effects were determined using the dedicated Monte Carlo ZEUSVM generator [132]. 4.7.2. Results Differential cross sections as a function of Mππ , t and u are presented in Figs. 40, 41, 42 respectively. The systematic uncertainties are dominated by the uncertainties related to the trigger selection and identification of the scattered positron. As can be seen from Eqs. (74) and (75) the predicted dependence of the cross sections on 2 are strongly coupled. Consequently, there is no simple prediction for the mass Q 2 and Mππ 2 is very different from Q 2 or a well defined value of Q 2 is dependence unless either Mππ selected. In the ZEUS experiment Q 2 and the mass have similar values and it is impractical to select a narrow Q 2 range because of limited statistics. The authors therefore chose to make n with a result: n ≈ 4.5. This distribution, when a power-law fit to the mass distribution: 1/Mππ ∗ divided by the mass dependence of the γ p → q q¯ cross section is proportional to the pion time-like form factor squared. It can be used to extend the measurements of this form factor from the highest available value at 2 GeV [93] up to about 4 GeV. Such results will provide an alternative measurement of this form factor from the evaluation at the J/ψ mass deduced from the (J/ψ → π + π − )/(J/ψ → e+ e− ) [94] which is in great disagreement with theoretical predictions [91].
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Fig. 40. The measured differential cross section dσ/dMπ π with Q 2 ≈ 7 (GeV/c)2 . The solid line indicates a fit to the form 1/Mπn π and n ≈ 4.5.
Fig. 41. The measured differential cross section dσ/dt for Mπ π > 1.5 GeV, Q 2 = 8.5 (GeV/c)2 and W = 76 GeV. The solid line indicates a fit to the form ebt .
Fig. 42. The differential cross section dσ/du measured in two Q 2 intervals: 2 < Q 2 < 5 GeV2 (left) and 5 < Q 2 < 20 GeV2 (right). The inner error-bars show the statistical uncertainties; the outer error-bars show the statistical and systematic uncertainties added in quadrature. The data points are compared to the LCWF predictions, Eqs. (74) and (75).
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The measured t distribution, Fig. 41 is fitted with the standard exponential form dσ/dt ∝ ebt . −2 The resulting slope b = −4.81 ± 0.42(stat)+0.05 is in good agreement with the −0.44 (sys) GeV values observed for vector-meson production, Fig. 5. The slope of the t-distribution is considered as a measure of the combined size of the q q¯ system and the proton: b = 13 R 2 + 18 r 2 where r ∼ 6 2 is the size of the q q¯ system before it hadronizes to a vector meson or, in this case, Q 2 +m V
to two pions in the continuum and R is the proton radius [57]. The results show that the q q¯ size in hadronization to continuum states is similar to that obtained for vector mesons. The u distribution is presented in Fig. 42 for two Q 2 intervals. Its shape is compared to the LCWF predictions for transverse and longitudinal photons, Eqs. (74) and (75). Such a comparison is legitimate if the pion quantum numbers do not affect their angular (u) distributions for a given photon polarization. The normalization for the longitudinal LCWF prediction was determined from a fit to the data. For the transverse LCWF prediction it was fixed to be the same value as that for the longitudinal prediction at u = 0.5. The u distribution is similar in the two Q 2 intervals. Indeed, Eqs. (74) and (75) predict a Q 2 -independent u-dependence provided the relative contributions from longitudinal and transverse photons do not change. We note, however, that the measured distribution in the low Q 2 region (Fig. 42 left) is more irregular than the one for the higher Q 2 range. This irregularity can be traced to the fact that the low Q 2 range is close to the average value of the dipion mass squared. In fact, for this range β = 0.52 which is where σ L /σT ∼ 0 in leading order, Fig. 34. Higher order effects are expected to play a role here and the pure longitudinal fluctuations may be shadowed. By contrast, for the high Q 2 range β = 0.75 which is a “safe” region. The results are consistent with the LCWF predictions for longitudinally polarized photons. The dominance of longitudinal photons has also been observed in the diffractive electroproduction of ρ 0 mesons [134]. The agreement between the measured u distributions and the predictions lends support to the assumption that non-resonant di-pion production is sensitive to the q q¯ component of the light-cone wave function of the virtual photon. This shows that for the phase space parameters of this study the LCWF predicted by perturbative QCD, Eq. (55) is correct. This process of exclusive diffractive electroproduction of pion pairs can be correctly described as resulting from a longitudinal photon fluctuating to a q q¯ pair which in turn hadronizes to a π + π − pair, Fig. 35. The agreement of the u-distribution (or angular distribution, cf. Eq. (78)) measured with pions in the continuum and in the resonance region above the ρ with calculations made at the parton level lends support to the picture that there is parton/hadron duality, [136]. 4.8. Measurement of the hadronic component of the photon LCWF by exclusive dijet electroproduction Measurement of exclusive diffractive electroproduction of dijets is being carried out by the ZEUS collaboration following the same concepts as in Sections 3.3, 4.6 and 4.7. The data used in this analysis is similar to that described in Section 4.7. The initial selection criteria are similar in the requirements of diffractive and exclusive events. The main difference is the way the events are selected in order to identify exclusive diffractive dijets. Furthermore, it is necessary to select those dijets likely to be produced from transverse photons fluctuating to q q¯ pairs. As discussed in Section 4.2.1 these conditions can be met by selecting large β to have q q¯ dominance but not so large as to have transverse photon dominance. Under these conditions it may be possible to observe the cos 2φ dependence of the dijet angular distribution. This would serve as an experimental signature for dijets originating from a transverse photon fluctuating in a q q¯ Fock state.
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These requirements and the difficulties inherent in jet identification dictated somewhat different phase space conditions than those of the dipion study. Here both Q 2 and the dijet mass MDJ are larger and a kt cut has to be applied in order to separate the jets. The jets were identified in the γ ∗ − P center of mass system using a “kt clustering” algorithm [135]. The selection parameter ycut that optimizes dijet selection was chosen by extensive Monte Carlo simulations. This analysis is still in progress and final results are expected soon. 5. Summary and future directions Understanding the internal structure of hadrons is one of the most challenging goals of particle and nuclear physics. In this work we concentrated on two related aspects of the internal hadronic structure: studies of the phenomenon of color transparency and measuring the valence light-cone wave function. All these studies are done in large momentum transfer processes. The CT effect requires small configurations hence a large internal momentum reached through large momentum transfer. Studies of the LCWF were done through diffractive dissociation to large transverse momentum dijets. The measurement of the photon electromagnetic LCWF through dimuon photoproduction presents a test of the concept behind these measurements. The good agreement of the results with QED show that diffractive dissociation is a good tool to study LCWFs. Measurements of the pion LCWF show that for kt > 1.5 GeV/c the LCWF derived in pQCD describes well the measured longitudinal and transverse momentum distributions. This sets a scale for applicability of the asymptotic pQCD calculations. These measurements also show a transition region from perturbative to non-perturbative regimes at lower transverse momenta. Measurements of N N scattering, d(γ , p)n photodisintegration [23], and perhaps d(d, n) 3 He and d(d, p) 3 H reactions [24] are consistent with QCD counting rules [22] at transverse momenta similar to those observed in the pion LCWF measurements. Measurements of the photon hadronic LCWF are under way at HERA and results for the photon dissociation to π + π − pairs have recently become available. They show a longitudinal photon fluctuating to a q q¯ pair which in turn hadronizes to a π + π − pair. The u-distribution is described correctly by the LCWF predicted by perturbative QCD. This agreement, observed in the continuum and the resonance region above the ρ lends support to the picture that there is parton/hadron duality. More results of these studies, including charge-asymmentry measurements for observation of Pomeron–Odderon interference are expected to be available in the near future. Results for measurements of the photon dissociation to dijets are also expected to be available soon. The main achievements summarized in this work are: • Color transparency was observed in vector meson production and, more directly, through the A-dependence of the cross section for diffractive dissociation of pions to dijets. • Direct measurement of distribution amplitudes by diffractive dissociation was demonstrated. • The distribution amplitude for pions was determined in a Q 2 region that allows observation of the transition from nonperturbative to perturbative regimes. • These measurements show that for Q 2 > 10 (GeV/c)2 perturbative calculations are applicable in these systems. • The distribution amplitude for the elecromagnetic component of real photons was measured through diffractive dissociation of real photons to µ+ µ− . The results, being in agreement with QED predictions, demonstrate the applicability of the diffractive dissociation method to distribution amplitude measurements.
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Fig. 43. Schematic description of measurements of the proton and antiproton LCWF at the Bevatron collider. Roman pots are used to identify the intact proton while the antiproton dissociates to three jets and similarly detecting the intact antiproton for proton dissociation.
• The distribution amplitude for longitudinal virtual photons fluctuating to q q¯ was measured through diffractive dissociation of virtual photons to π + π − . This also supports the concept of parton/hadron duality. • The distribution amplitude for transverse virtual photons fluctuating to q q¯ was measured through diffractive dissociation of virtual photons to dijets. Measurements of LCWFs were so far done only in q q¯ systems: the pion and the photon. The phenomenon of CT was unumbigously observed only in mesons: in vector meson production and in diffractive dissociation of pions. Attempts to observe CT with protons, while showing significant deviations from expectations based on Glauber calculations, did not produce the results expected from color transparency. This, together with difficulties in the interpretation of spin effects in proton–proton scattering, make it very important to carry out studies of CT and measurements of the LCWF of baryons. Following the observation of CT with pions, large momentum transfer (e, e π) studies should show the effect too. The (e, e k) process should be investigated as well. Studies of CT for the proton via proton dissociation to three jets in interaction with heavy nuclei can be carried out at RHIC. A natural extension of the method used to measure the LCWFs of pions and photons is the study of momentum distributions in diffractive dissociation of protons into three jets. A valence proton asymptotic distribution amplitude for three quarks is given in [6]: −2/3β Q2 ; φqqq/ p (u i , Q ) = C · (u 1 · u 2 · u 3 ) ln 2 Λ 2
u i = 1.
(92)
This shows an expected distribution that peaks when the jets have similar momenta. However, this symmetric form may be more appropriate for the ∆ resonance that has a symmetric valence quark state. A detailed discussion of the nucleon and ∆ distribution amplitudes is given in [137]. Also, following measurements of the proton spin structure functions, the possibility that orbital angular momentum has to be included in the proton distribution amplitude was considered [138]. A schematic description of such an experiment at the Fermilab Tevatron collider is shown in Fig. 43. In the p p¯ collision one would look for the dissociation of the proton while the antiproton remains intact and vice versa. For this purpose the Roman pots can be used detecting one of the particles while three jets originating from dissociation of the second particle are detected in the main detector (CDF or D0). One could also envision studying the proton dissociation to three jets in HERA. Such measurements can be eventually extended to the LHC.
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Other extensions of the same principle are measurement of the kaon LCWF which can be done in the same fashion as done for the pion provided a fixed target experiment with a kaon beam can be realized at the Tevatron. For the photon, the results presented here are only a beginning of a large study that can be realized at HERA. Measurements of the photon strange LCWF can be done by tagging the dijets with strange mesons. Measurements of the photon charmed LCWF can be done by tagging the dijets with charm mesons. The very clean D ∗ signals obtained by both ZEUS and H1 collaborations can be utilized for this purpose. This will have a heavy price in statistics but if all the data collected in both HERA I and HERA II runs is combined it may be sufficient. Another study which can be done in HERA II is the spin LCWF of the photon and spin-asymmetry production of dijets and dipions by utilizing data taken with polarized electron and positron beams. Studies of photon dissociation to three jets can yield information on the second Fock state of the photon LCWF. One can look forward to a wealth of information that will enrich our understanding of hadronic structure. Acknowledgements I would like to thank Drs. S. Brodsky and L. Frankfurt for many valuable discussions and Dr. S. Brodsky for carefully reading this manuscript. Comments made by Drs. M. Diehl, B. Kopeliovich and G. Miller are greatly appreciated. I also want to acknowledge the efforts of the E791 Collaboration and particularly my graduate student Dr. Ruth Weiss-Babai for the data on diffractive dissociation of pions. I acknowledge the efforts of the ZEUS collaboration and particularly Dr. H. Abramowicz, Dr. I. Cohen and graduate students E. Nevo, J. Szuba and J. Ukleja for the data on diffractive dissociation of photons. This work was supported in part by the US–Israel Binational Science Foundation and by the Israel Science Foundation. References [1] [2] [3] [4] [5] [6]
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