Nuclear chromodynamics: Novel nuclear phenomena predicted by QCD

Progress in Particle and Nuclear Physics 74 (2014) 1–34

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Review

Nuclear chromodynamics: Novel nuclear phenomena predicted by QCD Bernard L.G. Bakker a,∗ , Chueng-Ryong Ji b a

Department of Physics and Astrophysics, Vrije Universiteit, De Boelelaan 1081, NL-1081 HV Amsterdam, The Netherlands

b

Department of Physics, North Carolina State University, Raleigh, NC 27695-8202, USA

article

info

Keywords: Nuclear physics Quantum chromodynamics Quark models

abstract With the acceptance of QCD as the fundamental theory of strong interactions, one of the basic problems in the analysis of nuclear phenomena became how to consistently account for the effects of the underlying quark/gluon structure of nucleons and nuclei. Besides providing more detailed understanding of conventional nuclear physics, QCD may also point to novel phenomena accessible by new or upgraded nuclear experimental facilities. We review several interesting applications of QCD to nuclear physics. © 2013 Elsevier B.V. All rights reserved.

Contents 1. 2.

3.

4.

5.

∗

Introduction............................................................................................................................................................................................. Reduced form factors: link between nucleons and quarks .................................................................................................................. 2.1. Factorization of coherent pion photoproduction on the deuteron.......................................................................................... 2.2. Kinematics and factorization ..................................................................................................................................................... 2.2.1. π 0 photoproduction kinematics ................................................................................................................................. 2.2.2. Improved RNA factorization ....................................................................................................................................... 2.3. Comparison with experiment .................................................................................................................................................... 2.4. Discussion.................................................................................................................................................................................... Nucleon–nucleon interactions ............................................................................................................................................................... 3.1. Multi-quark states ...................................................................................................................................................................... 3.2. Quark interchange between MIT bags....................................................................................................................................... 3.3. Multi-quark bags ........................................................................................................................................................................ 3.3.1. P-matrix analysis ......................................................................................................................................................... 3.4. Quark-compound bag model ..................................................................................................................................................... 3.5. Microscopic cluster models........................................................................................................................................................ Multi-quark QCD evolutions .................................................................................................................................................................. 4.1. Color, Isospin, and spin states .................................................................................................................................................... 4.2. Orbital states ............................................................................................................................................................................... 4.3. Antisymmetrization.................................................................................................................................................................... 4.4. Multi-quark application ............................................................................................................................................................. 4.5. Six-quark evolution .................................................................................................................................................................... 4.6. Nine-quark color singlets ........................................................................................................................................................... Model-independent predictions from angular conditions ................................................................................................................... 5.1. Light-front treatment ................................................................................................................................................................. 5.2. Frame relations and general angular condition ........................................................................................................................ 5.2.1. Relations among helicity amplitudes .........................................................................................................................

Corresponding author. E-mail address: [email protected] (B.L.G. Bakker).

0146-6410/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ppnp.2013.10.001

2 3 3 7 7 8 9 9 10 11 11 11 12 12 15 16 16 17 18 19 20 21 22 22 23 23

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5.2.2. Deriving the light-front to Breit relation.................................................................................................................... Light-front discrete symmetry................................................................................................................................................... 5.3.1. Light-front parity ......................................................................................................................................................... 5.3.2. Light-front time reversal ............................................................................................................................................. 5.3.3. The x-Breit frame ......................................................................................................................................................... 5.4. Consequences.............................................................................................................................................................................. 5.4.1. Light-front parity and the angular condition............................................................................................................. 5.4.2. The angular condition for deuterons .......................................................................................................................... 5.4.3. A consequence of the angular condition for deuterons ............................................................................................ 5.4.4. The angular condition for N − ∆ transitions............................................................................................................. 5.4.5. Equivalence of leading powers in Breit and Light-front frames ............................................................................... 5.5. Summary ..................................................................................................................................................................................... Conclusion ............................................................................................................................................................................................... Acknowledgments .................................................................................................................................................................................. References................................................................................................................................................................................................ 5.3.

6.

25 26 26 27 27 28 28 29 29 30 30 32 32 33 33

1. Introduction This review is concerned with the novel QCD phenomena that can be observed in nuclear experiments. The 12 GeV upgrade of the continuous electron beam accelerator facility (CEBAF) in Jefferson Laboratory (JLab) provides an opportunity to investigate novel nuclear phenomena predicted by QCD. In particular, a JLab collaboration proposed to measure the deuteron tensor structure function which may provide a probe of exotic QCD effects due to hidden color in a six-quark configuration [1]. It is also interesting to note the recent measurements of electron scattering from high-momentum nucleons in nuclei performed at Hall C(E02-019) in JLab [2], which may allow for an improved determination of the strength of two- and three-nucleon short-range correlations for several nuclei. This result may indicate the existence of hidden color in multi-quark configurations. These new developments in experimental facilities motivate us to review a number of applications of QCD to nuclear physics which is the main concern of nuclear chromodynamics (NCD) [3]. Its goal is to give a fundamental description of nuclear dynamics and nuclear properties in terms of quark and gluon fields at short distance, and to obtain a synthesis at long distances with the normal nucleon, isobar, and meson degrees of freedom. NCD provides an important testing ground for coherent effects in QCD and nuclear effects at the interface between perturbative and nonperturbative dynamics. Among the areas of interest are: 1. The representation of the nuclear force in terms of quark and gluon subprocesses [3]. The nuclear force between nucleons can in principle be represented at a fundamental level in QCD in terms of quark interchange (equivalent at large distances to pion and other meson exchange) and multiple-gluon exchange. Although calculations from first principles are still too complicated, many of the basic features of the nuclear force can be understood from the underlying QCD substructure. 2. The composition of the nucleon and nuclear states in terms of quark and gluon quanta. The light-front quantization formalism provides a consistent relativistic Fock-state momentum-space representation of multi-quark and gluon colorsinglet bound states. 3. Color-coherence effects in high-momentum exclusive reactions in nuclei and new color-singlet multi-quark states. 4. The use of reduced nuclear amplitudes in order to obtain a consistent and covariant identification of the effects of nucleon compositeness in nuclear reactions. 5. Model-independent predictions of the kinematic domain where the scaling laws obtained from perturbative QCD (pQCD) can be applied to nuclear amplitudes. NCD may imply in some cases a breakdown of traditional nuclear-physics concepts. For example, one can identify where off-shell effects modify traditional nuclear physics-formulas, such as the impulse approximation for elastic nuclear form factors. At high momentum transfer, nuclear amplitudes are predicted to have a power-law fall off in QCD in contrast to the Gaussian or exponential fall off usually assumed in nuclear physics. In QCD, the fundamental degrees of freedom of nuclei as well as hadrons are postulated to be the spin-l/2 quark and spin-1 gluon quanta. Nuclear systems are identified as color-singlet composites of quark and gluon fields, beginning with the six-quark Fock component of the deuteron. An immediate consequence is that nuclear states are a mixture of several color representations which cannot be described solely in terms of the conventional nucleon, meson, and isobar degrees of freedom: there must also exist hidden-color multi-quark wave-function components, i.e., nuclear states which are not separable at large distances into the usual color-singlet nucleon clusters. One of the basic problems in the analysis of nuclear scattering amplitudes is how to consistently account for the effects of the underlying quark/gluon component structure of nucleons. In the next section, Section 2, we discuss the idea of the reduced nuclear amplitudes to the coherent pion photoproduction on the deuteron. In Section 3, the NCD is applied to the nucleon–nucleon interactions. An attempt was made to reconcile the quark picture of hadrons with phenomenological theories of the strong interaction. In Section 4, we discuss the QCD evolution of multi-quark systems. In particular, we pay attention to the six- and nine-quark hidden-color configurations. In Section 5, we discuss the consequences of angular momentum conservation, the so-called angular conditions, for the deuteron electromagnetic form factors and the nucleon–delta transition form factors. Conclusions follow in Section 6.

B.L.G. Bakker, C.-R. Ji / Progress in Particle and Nuclear Physics 74 (2014) 1–34

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2. Reduced form factors: link between nucleons and quarks One of the basic problems in the analysis of nuclear scattering amplitudes is how to consistently account for the effects of the underlying quark/gluon component structure of nucleons. Traditional methods based on the use of an effective nucleon/meson local Lagrangian field theory are not really applicable, giving the wrong dynamical dependence in virtually every kinematic variable for composite hadrons. The inclusion of ad hoc vertex form factors is unsatisfactory since one must understand the off-shell dependence in each leg while retaining gauge invariance; such methods have little predictive power. On the other hand, the explicit evaluation of the multi-quark hard-scattering amplitudes needed to predict the normalization and angular dependence for a nuclear process, even at the leading order of the QCD coupling constant αs requires the consideration of millions of Feynman diagrams. Beyond leading order one must include contributions of non-valence Fock states wave functions, and a rapidly expanding number of radiative corrections and loop diagrams. The reduced amplitude method [4], although not an exact replacement for a full QCD calculation, provides a simple method for identifying the dynamical effects of nuclear substructure, consistent with covariance, QCD scaling laws and gauge invariance. The basic idea has already been introduced for the reduced deuteron form factor. More generally, if the nuclear binding is neglected, then the light-cone nuclear wave-function can be written as a cluster decomposition of collinear nucleons: ψq/A = ψN /A ΠN Ψq/N where each nucleon has 1/A of the nuclear momentum. A large momentum transfer nucleon amplitude then contains as a factor the probability amplitude for each nucleon to remain intact after absorbing l/A of the respective nuclear momentum transfer. Each probability amplitude can be identified with the respective nucleon form factor F (tˆi = A12 tA ), where tA is the square of the transferred momentum to the nucleus with mass number A. Thus for any exclusive nuclear scattering process, the reduced nuclear amplitude can be defined as m=

M

ΠiA=1 FN

(tˆi )

.

(1)

The QCD scaling law for the reduced nuclear amplitude m is then identical to that of nuclei with point-like nuclear components: e.g., the reduced nuclear form factors obey fA (Q 2 ) =

FA (Q 2 )

 ∼

[FN (Q 2 /A2 )]A

1

A−1

Q2

.

(2)

Comparisons with experiment and predictions for leading logarithmic corrections to this result are given in Ref. [4]. In the case of photo- (or electro-) disintegration of the deuteron, one has mγ d→np =

Mγ d→np Fn (tn )Fp (tp )

∼

1 pT

f (θcm ),

(3)

i.e., the same elementary scaling behavior as for Mγ M →qq¯ . Comparison with experiment is encouraging, showing that the pQCD scaling regime begins at Q 2 ≥ 1 GeV2 as was in the case of Q 2 fd (Q 2 ). Detailed comparisons and a model for the angular dependence and the virtual photon-mass dependence of deuteron electro-disintegration are discussed in Ref. [4]. Other potentially useful checks of QCD scaling of reduced amplitudes are 2 mpp→dπ + ∼ p− T f (t /s),

(4)

mpd→3 H π + ∼ pT f (t /s), −4

4 mπ d→π d ∼ p− T f (t /s).

It is also possible to use these QCD scaling laws for the reduced amplitude as a parametrization for the background for detecting possible new dibaryon resonance states. 2.1. Factorization of coherent pion photoproduction on the deuteron The predictions of pQCD for pion photoproduction on the deuteron γ D → π 0 D have been given [5] at large momentum transfer using the reduced amplitude formalism. The cluster decomposition of the deuteron wave function at small binding only allows the nuclear coherent process to proceed if each nucleon absorbs an equal fraction of the overall momentum transfer. Furthermore, each nucleon must scatter while remaining close to its mass shell. Thus, the nuclear photoproduction amplitude Mγ D→π 0 D (u, t ) factorizes as a product of three factors: (1) the nucleon photoproduction amplitude Mγ N1 →π 0 N1 (u/4, t /4) at half of the overall momentum transfer, (2) a nucleon form factor FN2 (t /4) at half the overall momentum transfer, and (3) the reduced deuteron form factor fd (t ), which according to pQCD, has the same monopole fall-off as a meson form factor. A comparison with the recent JLAB data for γ D → π 0 D of Meekins et al. [6] and the available γ p → π 0 p data [7–11] shows good agreement between the pQCD prediction and experiment over a large range of momentum transfers and center-of-mass angles. The reduced amplitude prediction is consistent with the constituent counting rule p11 T Mγ D→π 0 D → F (θcm ) at large momentum transfer. This is found to be consistent with measurements for photon lab energies Eγ > 3 GeV at θcm = 90° and 136°. The predictions of QCD for nuclear reactions are most easily described in terms of light-front (LF) wave functions defined at equal LF time τ = t + z /c [12]. The deuteron eigenstate can be projected on the complete set of baryon number B = 2,

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isospin I = 0, spin J = 1, Jz = 0, ±1 color-singlet eigenstates of the free QCD Hamiltonian, beginning with the six+ quark Fock states. Each Fock state is weighted by an amplitude which depends on the LF momentum fractions xi = k+ i /p and on the relative transverse momenta ⃗ k⊥i . There are five different linear combinations of six color-triplet quarks which make an overall color-singlet, only one of which corresponds to the conventional proton and neutron three-quark clusters. Thus, the QCD decomposition includes four six-quark unconventional states with ‘‘hidden color’’ [13]. The spacelike form factors Fλλ′ (Q 2 ) measured in elastic lepton–deuteron scattering for various initial and final deuteron helicities have exact representations as overlap integrals of the LF wave functions constructed in the Drell–Yan–West frame [14,15], where q+ = 0 and Q 2 = −q2 = q2⊥ . At large momentum transfer, the leading-twist elastic deuteron form factors can be written in a factorized form Fλλ′ (Q 2 ) =

1

 0

5  i =1

1

 dxi 0

5 

dyj φλ′ (xi , Q )THλλ (xi , yi , Q )φλ (yj , Q ), ′

(5)

j =1

where the notation dxi indicates that the integral is done subject to the condition i xi = 1, the φλ (xi , Q ) are the deuteron distribution amplitudes, defined as the integral of the six-quark LF wave functions integrated in transverse momentum up to ′ the factorization scale Q , and THλλ is the hard scattering amplitude for scattering six collinear quarks from the initial to final deuteron directions. A sum over the contributing color-singlet states is assumed. Because the photon and exchanged-gluon couplings conserve the quark chiralities and the distribution amplitudes project out Lz = 0 components of initial and final wave functions, the dominant form factors at large momentum transfer are the hadron-helicity conserving amplitudes. The QCD evolution for the distribution amplitudes has been analyzed [13,16].



5

The hard-scattering amplitude scales as αs /Q 2 at leading order corresponding to five gluons exchanged between the six propagating valence quarks. Higher order diagrams involving additional gluon exchanges and loops give NLO corrections of higher order in αs . Thus, the nominal behavior of the helicity conserving deuteron form factors is 1/Q 10 , modulo the logarithmic corrections from the running of the QCD coupling and the anomalous dimensions from the evolution of distribution amplitudes. In fact, the measurement [17] of the high Q 2 ≥ 5 GeV2 helicity-conserving deuteron form factor  A(Q 2 ) appears consistent with the Q 10 A(Q 2 ) scaling predicted by perturbative QCD and constituent counting rules [18]. The analogous factorization formulas for deuteron photodisintegration and pion photoproduction predicts the nominal scaling law s11 ddtσ (γ D → np) ∼ const and s13 ddtσ (γ D → π 0 D) ∼ const at high energies and fixed θcm . Comparison with the data shows this prediction is only successful at the largest momentum transfers [19–21]. This is not unexpected, since the presence of the large nuclear mass can be expected to delay the onset of leading-twist scaling. The above discussion does not take into account a simplifying feature of nuclear dynamics—the very weak binding of the deuteron state. The cluster decomposition theorem [22] states that in the zero binding limit (B.E . → 0), the LF wave function of the deuteron must reduce to a convolution of on-shell color-singlet nucleon wave functions:



D lim ψuududd (xi , ⃗k⊥i , λi ) =

B.E .→0

1



 dz

p

d2 ℓ⊥ ψ d (z , ℓ⊥ )ψuud (xi /z , ⃗ k⊥i + (xi /z )ℓ⊥ , λi )

0 n × ψudd (xi /(1 − z ), ⃗k⊥i − [xi /(1 − z )]ℓ⊥ , λi ),

(6)

where ψ (z , ℓ⊥ ) is the reduced ‘‘body’’ LF wave function of the deuteron in terms of its nucleon components. Applying this cluster decomposition to an exclusive process involving the deuteron, one can derive a corresponding reduced nuclear amplitude (RNA) [4,16,23]. Moreover, at zero binding, one may take ψ d (z , ℓ⊥ ) → δ(z − mp /(mp + mn )) × δ 2 (ℓ⊥ ). In effect, each nucleon carries half of the deuteron’s four-momentum. Thus, in the weak nuclear binding limit, the deuteron form factor reduces to the overlap of nucleon wave functions at half d

2

of the momentum transfer, and FD (Q 2 ) → fd (Q 2 )FN2 ( Q4 ) where the reduced form factor fd (Q 2 ) is computed from the overlap of the reduced deuteron wave functions [23]. The reduced deuteron form factor resembles that of a spin-one meson form factor since its nucleonic substructure has been factored out. The pQCD predicts the nominal scaling Q 2 fd (Q 2 ) ∼ const [16]. The measurements of the deuteron form factor show that this scaling is in fact well satisfied at spacelike Q 2 ≥ 1 GeV2 [17]. The reduced amplitude factorization is evident in the representative QCD diagram of Fig. 1. Half of the incident photon’s momentum is carried over to the spectator nucleon by the exchanged gluon. The struck quark propagator is off shell with high virtuality [x1 (pd + q) + q/2]2 ∼ (1 + 2x1 )q2 /4 ∼ q2 /3 (using x1 ∼ 1/6) which provides the hard scale for the reduced form factor fd (Q 2 ). Fig. 2 shows a similar diagram with quark interchange, which is consistent with the color-singlet clustered structure of the weak-binding amplitude. Now, consider a similar analysis of pion photoproduction on the deuteron γ D → π 0 D at weak binding. The cluster decomposition of the deuteron wave function at small binding only allows this process to proceed if each nucleon absorbs an equal fraction of the overall momentum transfer. Furthermore, each nucleon must scatter while remaining close to its mass shell. Thus, we expect the photoproduction amplitude to factor as

Mγ D→π 0 D (u, t ) = C (u, t )Mγ N1 →π 0 N1 (u/4, t /4)FN2 (t /4).

(7)

Note that the on-shell condition requires the center-of-mass angle of pion photoproduction on the nucleon N1 to be commensurate with the center-of-mass angle of pion photoproduction on the deuteron.

B.L.G. Bakker, C.-R. Ji / Progress in Particle and Nuclear Physics 74 (2014) 1–34

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Fig. 1. Illustration of the basic QCD mechanism in which the nuclear amplitude for elastic electron deuteron scattering ℓD → ℓD factorizes as a product of two on-shell nucleon amplitudes. The propagator of the hard quark line labeled Pq is incorporated into the reduced form factor fd .

Fig. 2. Illustration of the basic QCD mechanism in which the nuclear amplitude for elastic electron deuteron scattering ℓD → ℓD factorizes as a product of two on-shell nucleon amplitudes. The quark interchange allows the amplitude to proceed when the deuteron wave function contains only color-singlet clusters.

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B.L.G. Bakker, C.-R. Ji / Progress in Particle and Nuclear Physics 74 (2014) 1–34

Fig. 3. Illustration of the basic QCD mechanism in which the nuclear amplitude for γ D → π 0 D factorizes as a product of two on-shell nucleon amplitudes. The propagator of the hard quark line labeled pq is incorporated into the reduced form factor fd .

A representative QCD diagram illustrating the essential features of pion photoproduction on a deuteron is shown in Fig. 3. The exchanged gluon carries half of the momentum transfer to the spectator nucleon. Thus as in the case of the deuteron form factor the nuclear amplitude contains an extra quark propagator at an approximate virtuality t /3 in addition to the on-shell nucleon amplitudes. Thus taking this graph as representative, we can identify C (u, t ) = C ′ fd (t ), where the constant C ′ is expected to be close to unity. This correspondence is also shown in Fig. 4 which includes a quark interchange to account for the color-singlet cluster structure. This structure predicts the reduced amplitude scaling

Mγ D→π 0 D (u, t ) = C ′ fd (t )Mγ N1 →π 0 N1 (u/4, t /4)FN2 (t /4).

(8)

A comparison with elastic electron scattering then yields the following proportionality of amplitude ratios:

Mγ D→π 0 D MeD→eD

= C′

Mγ p→π 0 p Mep→ep

.

(9)

More details of the derivation of Eq. (8) will be presented in the following subsection. The normalization is fixed by the requirement that this factorization yields the same result as the full counting rules for M in the asymptotic limit. Fixing the normalization at a non-asymptotic energy can be a poor approximation, as can be seen in a previous analysis [6]. The new factored form, Eq. (8), differs significantly from the older reduced nuclear amplitude factorization [4], for which

Mγolder (u, t ) ≃ mγ d→π 0 d (u, t )FN2 (t /4). D→π 0 D

(10)

Here, mγ d→π 0 d is the reduced amplitude; it scales the same as mγ ρ→π 0 ρ at fixed angles since the nucleons of the reduced deuteron d are effectively point-like. The advantages of this reduction are that some nonperturbative physics is included via the nucleon form factors and that systematic extension to many nuclear processes is possible [4]. The new factorization given by Eq. (8) is an improvement because it includes nonperturbative effects in the pion production process itself. JLAB experimental data [6] on π 0 photoproduction from a deuteron target, up to a photon lab energy Elab = 4 GeV, were presented as an example inconsistent with both constituent-counting rules (CCR) [18] and RNA [4] predictions. While the data at θcm = 136° are consistent with the CCR, predicted as s−13 scaling for the differential cross section dσ /dt, the data at θcm = 90° exhibit a large disagreement with this prediction. Also, the data at both angles were interpreted [6] as being inconsistent with the RNA approach. This is in sharp contrast to other measurements of the deuteron electric form factor A(Q 2 ), that are consistent with both the CCR and RNA predictions in a similar four-momentum transfer range 2 GeV2 ≤ Q 2 ≤ 6 GeV2 [17].

B.L.G. Bakker, C.-R. Ji / Progress in Particle and Nuclear Physics 74 (2014) 1–34

7

Fig. 4. Illustration of the basic QCD mechanism in which the nuclear amplitude for γ D → π 0 D factorizes as a product of two on-shell nucleon amplitudes. The quark interchange allows the amplitude to proceed when the deuteron wave function contains only color-singlet clusters.

One potential explanation for this disagreement is odderon exchange [24]. Because the odderon has zero isospin and is odd under charge conjugation, such an exchange is allowed in the t channel of π 0 photoproduction. However, it is shown that the improved factorization given by Eqs. (8) and (9) is in good agreement with the JLAB data [6] for γ D → π 0 D and the available γ p → π 0 p data [7–11] as well as the existing eD → eD and ep → ep data. There is thus no need to invoke any additional anomalous contribution to understand the data [6]. The results for γ D → π 0 D were given in Ref. [5]. The π 0 transverse momentum PT dependence of the amplitude Mγ D→π 0 D in the c.m. frame was analyzed and the scaling of the predicted Mγ D→π 0 D was found to be consistent with the CCR prediction of PT−11 when the photon lab energy is only a few GeV for both values of θcm , 90° and 136°. 2.2. Kinematics and factorization 2.2.1. π 0 photoproduction kinematics The Mandelstam [25] variables of the γ D → π 0 D process are given by s = (qγ + pD )2 ,

t = (qγ − qπ )2 ,

u = (pD − qπ )2 ,

(11)

where pD = a=1 pa is the momentum of the target deuteron and qγ , qπ and pa are the momenta of the photon, pion and a’th quark of the deuteron. In the γ − D center-of-momentum frame (CMF), where experimental results are reported, these variables are related to the photon energy and pion momentum by

6

s = (Eγ CM +



m2D + Eγ2 CM )2 ,



t = m2π − 2Eγ CM ( m2π + q2CM − |qCM | cos θCM ),



u = m2D + m2π − 2( m2D + q2CM



(12)

m2π + q2CM + Eγ CM |qCM | cos θCM ),

with mD the deuteron mass and θCM the angle between the photon and the π 0 in the  CMF. Here, the photon energy and the √ magnitude of the π 0 momentum are given by Eγ CM = (s − m2D )/2 s and |qCM | = (s + m2π − m2D )2 /4s − m2π , respectively. The transverse momentum of the π 0 is then given by PT = |qCM π | sin θCM and, if all the masses are neglected, P⊥ ≈

√

tu/s. The

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B.L.G. Bakker, C.-R. Ji / Progress in Particle and Nuclear Physics 74 (2014) 1–34

Mandelstam variables sN , tN and uN of the process γ N → π 0 N can also be defined, with the deuteron momentum in Eq. (11) 3 0 replaced by the nucleon momentum pN = a=1 pa . In the γ − N CMF of the γ N → π N process, the photon energy and the

√

magnitude of the π 0 momentum are given by (Eγ CM )N = (sN − m2N )/2 sN and (qCM )N =

 (sN + m2π − m2N )2 /4sN − m2π ,

respectively, with the nucleon mass being mN . One can find the magnitude of the invariant amplitude Mγ D→π 0 D (u, t ) directly from the experimental differential cross section data by using

 |Mγ D→π 0 D (u, t )| = 4(s − m2D ) π

dσ dt

(γ D → π 0 D).

(13)

Similarly, we obtain the invariant amplitude |Mγ N →π 0 N (uN , tN )| from the available data for γ p → π 0 p [7–11]. The proton data and the factorization formula (8) can then be used to predict |Mγ D→π 0 D (u, t )|. We take the generic nucleon form

 −2

factor FN (t ) to be 1 − t /(0.71 GeV2 ) and the reduced deuteron form factor [26] fd (t ) ≈ 2.14/ 1 − t /(0.28 GeV2 ) , as determined by the analyses of the elastic deuteron form factors [27]. As one can see in the previous analysis [16], the experimental data for |t | ≤ 2 GeV2 are better described without the logarithmic corrections. The normalization constant C ′ in Eq. (8) is then fixed by the largest Elab data point of γ D → π 0 D amplitude [6] at θcm = 90° and obtained as C ′ ≈ 0.8.







2.2.2. Improved RNA factorization The factorization given by Eq. (8) can be derived in analogy with earlier work on the deuteron form factor [23]. The first step is to replace at the quark level the electromagnetic vertex γ q → q with a photoproduction amplitude γ q → π 0 q. Let Mγ q→π 0 q (qγ , qπ , pa ) be the amplitude for this subprocess, with qγ , qπ , and pa the momenta of the photon, pion, and a’th quark, respectively. The full amplitude for the hadronic process γ D → π 0 D can be transcribed from Eq. (2.10) of [23], with insertion of Mγ q→π 0 q , as

Mγ D→π 0 D (u, t ) =

6  

[dx]i [d2 k⊥ ]i ΨD∗ (xi , ⃗k⊥i + (δia − xi )⃗q⊥ )

a =1

× Mγ q→π 0 q (qγ , qπ , pa )ΨD (xi , ⃗k⊥i ), where ΨD is the valence wave function, q ≡ qγ − qπ is the momentum transfer, and     6 6 6 6     d 2 k ⊥i dxi 2 3 2 ⃗ , [d k⊥ ]i = 16π δ k⊥i . xi [dx]i = δ 1 − xi 16π 3 i =1 i=1 i =1 i=1

(14)

(15)

The reference frame has been chosen such that q+ ≡ q0 + q3 = 0. The deuteron wave function factorizes in the manner described by Eq. (2.23) of Ref. [23], which reads 6 

 3  6 

ΨD (xi , ⃗k⊥i + (δia − xi )⃗q⊥ ) = ∗

a=1

a=1 b=4

×

1 q2⊥

+

 6  3  a =4 b =1

xa 1 − xa

V (xi , (δia − xi )⃗ q⊥ ; xj , [yδja + (1 − y)δjb − xj ]⃗ q⊥ )

× ψN (zi , ⃗k⊥′ i + (δia − zi )yq⃗⊥ )ψN (zj , ⃗k⊥′ j + (δjb − zj )(1 − y)⃗q⊥ )ψ d (0),

(16)

where ψ (0) is the body wave function of the deuteron at the origin and d

y=

3 

xi ,

⃗l⊥ =

i=1

zj =

xj 1−y

3 

⃗k⊥i ,

zi =

i =1

,

xi y

,

⃗k⊥′ i = ⃗k⊥i − zi⃗l⊥ , (17)

⃗k⊥′ j = ⃗k⊥j + zj⃗l⊥ .

In the weak binding limit, the value of y is approximately 1/2, ⃗l⊥ is approximately zero, and the kernel V contributes only a constant. The deuteron amplitude (14) reduces to the analog of Eq. (2.24) in Ref. [23]

Mγ D→π 0 D (u, t ) =

C q2⊥

 |ψ (0)| d

2

3  

  ⃗⊥ q [dz ]i [d2 k′⊥ ]i ψN∗ zi , ⃗k⊥′ i + (δia − zi ) 2

a =1

× Mγ q→π 0 q (qγ , qπ , pa )ψN (zi , ⃗k⊥′ i )

6   b=4

 × ψN (zj , ⃗k⊥′ j ) + (a ↔ b) .

[dz ]j [d k⊥ ]j ψN 2 ′

∗



⃗⊥ q zj , ⃗ k⊥′ j + (δjb − zj )



2

(18)

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9

This result does not quite factorize because the quark amplitude Mγ q→π 0 q depends on the full qγ and qπ , whereas the individual nucleons experience momentum transfers of (qγ − qπ )/2. To relate this quark amplitude to the one for a subprocess involving only a nucleon, one may use the spin-averaged form of the amplitude obtained by Carlson and Wakely [28]

|Mγ q→π 0 q (ˆu, tˆ)|2 ∼ tˆ

sˆ2 + uˆ 2 sˆ2 uˆ 2

,

(19)

where sˆ = (qγ + pa )2 , tˆ = (qγ − qπ )2 = t, and uˆ = (pa − qπ )2 . For photoproduction from a single nucleon, embedded in the deuteron, we have instead the quark-level invariants sˆN = (qγ /2 + pa )2 ,

tˆN = (qγ /2 − qπ /2)2 = tˆ/4,

uˆ N = (pa − qπ /2)2 .

(20)

The quark momentum is the same in both cases simply because it is the same quark. In the zero-mass limit, we have sˆN = sˆ/2 and uˆ N = uˆ /2. This (with Eq. (19)) leaves |Mγ q→π 0 q (ˆu, tˆ)|2 ≃ |Mγ q→π 0 q (ˆuN , tˆN )|2 . Furthermore, the above values

of sˆN , tˆN , uˆ N correspond to using qγ /2 and qπ /2 in evaluating the proton photoproduction amplitude. With these values the factorization can now be completed to obtain Eq. (8). A similar derivation can be constructed for a relation between the amplitudes of eD → eD and eN → eN processes, from which one can prove Eq. (9). 2.3. Comparison with experiment

From the recent JLAB γ D → π 0 D data [6], we computed the corresponding invariant amplitudes using Eq. (13) both for θcm = 90° and 136°. We then used the factorization formula given by Eq. (8) to predict |Mγ D→π 0 D | with input from the

available γ p → π 0 p data [7–11]. The results are presented in Figs. 5 and 6. In Fig. 5, the normalization of our prediction is fixed (at C ′ = 0.8) by the overlapping data point at Elab = 4 GeV, which is the highest photon lab energy used in the JLAB γ D → π 0 D experiment [6]. It is interesting to find that the general trend of our prediction (the open circles) is very similar to that of the direct result from the JLAB data [6], shown as filled circles. The prediction is remarkably consistent with the CCR prediction. In addition, our ‘‘prediction’’ in the Elab overlap region, denoted by crosses, mimics the shape of the direct result. The crosses are systematically above all the filled circles by 50% or more (on a linear scale). This difference could be absorbed into the determination of the normalization; however, the factorization is expected to be less accurate at these lower energies, and normalization is best done at the highest available energy. Also, one should note that there is a resonance contribution in the γ p → π 0 p data [7], in the region of 700 MeV ≤ Elab ≤ 800 MeV, which could bias a normalization done at lower energies. In Fig. 6, we use input from the proton data [7–11] in the vicinity of θcm = 136°. Using the same procedure for fixing the normalization, we find that our prediction is nicely connected to the direct calculation from the JLAB data [6], again shown as filled circles. The prefactor PT11 is computed at θcm = 136° for all data points. Due to the variation of θcm values in the available data, the consistency with the CCR is not as good as in the previous Fig. 5. Nevertheless, we can still see in the behavior of our prediction that scaling of PT−11 begins already at a few GeV. 2.4. Discussion We discussed the predictions of pQCD for coherent photoproduction on the deuteron γ D → π 0 D at large momentum transfer using a new form of reduced amplitude factorization displayed in Eq. (14). The underlying principle of the analysis is the cluster decomposition theorem for the deuteron wave function at small binding: the nuclear coherent process can proceed only if each nucleon absorbs an equal fraction of the overall momentum transfer. Furthermore, each nucleon must scatter while remaining close to its mass shell. Thus the nuclear photoproduction amplitude Mγ D→π 0 D (u, t ) factorizes as a product of three factors: (1) the nucleon photoproduction amplitude Mγ N1 →π 0 N1 (u/4, t /4) at half of the overall momentum transfer and at the same overall center-of-mass angle, (2) a nucleon form factor FN2 (t /4) at half the overall momentum transfer, and (3) the reduced deuteron form factor fd (t ), which according to the pQCD, has the same monopole fall-off as a meson form factor. The on-shell condition requires the center of mass angle of pion photoproduction on the nucleon N1 to be commensurate with the center-of-mass angle of pion photoproduction on the deuteron. The reduced amplitude prediction is consistent with the constituent counting rule p11 T Mγ D→π 0 D → F (θcm ) at large momentum transfer.

A comparison with the JLAB data for γ D → π 0 D of Meekins et al. [6] and the available γ p → π 0 p data [7–11] shows good agreement between the pQCD prediction and experiment over a large range of momentum transfers and center-of-mass angles. The scaling given by

Mγ D→π 0 D MeD→eD

= C′

Mγ p→π 0 p Mep→ep

(21)

was also consistent with experiment. The constant C ′ was found to be close to 1, suggesting similar underlying hardscattering contributions. No anomalous contributions such as might derive from odderon exchange were required.

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Fig. 5. PT11 |Mγ D→π 0 D | versus photon lab energy Elab at θcm = 90°. The filled circles are obtained directly from the JLAB γ D → π 0 D data [6], and the crosses

and open circles are our predictions from the γ p → π 0 p data presented in Ref. [7] and Refs. [8,9]. Note that an open circle is overlaid on top of a filled circle for the overlapping data point at Elab = 4 GeV.

Fig. 6. PT11 (136)|Mγ D→π 0 D | versus photon lab energy Elab with PT11 (136) defined by the PT value computed at θcm = 136°. This definition is due to the fact

that the θcm values of the γ p → π 0 p data are near 136° but not exactly equal. The filled circles are obtained directly from the JLAB γ D → π 0 D data [6], and the other symbols are our predictions based on γ p → π 0 p data at angles near θcm = 136°, presented in Refs. [7–9].

3. Nucleon–nucleon interactions Perhaps the most important gross feature of the nuclear force is that it produces saturation: already for rather small values of the nucleon number A, the volume of a nucleus is proportional to A, which means that the density is independent of the number of nucleons and its radius proportional to the cubic root of A. Very soon after it was discovered, this property was thought to be caused by repulsion between nucleons at small distances, roughly of the order of the nuclear radius divided by A1/3 [29]. In models of the nucleon–nucleon interaction, repulsion was implemented first by a hard core in the potentials, but later replaced in the sophisticated one-boson exchange (OBE) models [30] by vector-meson exchanges. Such OBE potentials could only be made consistent with the data on nucleon–nucleon scattering, if cut offs in momentum space were introduced. The latter could be taken as a signal that nucleons were not to be understood as point-like sources and sinks of mesons, but rather as extended objects. This idea fits very well with the (constituent) quark model and equally well with the more refined framework of NCD. In particular, there is a stream of works [31–33] regarding on the quark picture for two nucleon systems and the roles of quark symmetry to the nucleon interaction. Although these works were based on the nonrelativistic Hamiltonian model, they provide solutions of 6-quark systems dynamically and show the origin of the short-range repulsive force from the quark symmetry viewpoint. Recent lattice QCD calculations [34] have revealed such symmetry is indeed realized in QCD. In this section, two attempts to try and find ways to reconcile the quark picture of hadrons with phenomenological theories of the strong interaction are discussed. The first one [35] is about an improvement of the OBE models by

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11

(i) explaining meson exchanges as quark interchange and (ii) by calculating the cut-off parameters from the quark wave functions of the nucleons. The second one [36,37] is related to the idea that at short distances, where the nucleons strongly overlap, a description of two-nucleon correlations in terms of six-quark clusters should be the most natural way to describe the dynamics. Before discussing those two efforts of detailed model building, we review some work devoted to multi-quark systems per sé, that has relevance for the nuclear interaction. 3.1. Multi-quark states A particular interesting suggestion was made by Harvey [38,39]. Considering a system of six quarks of two flavors u and d, and three colors, one could construct a tower of states that are fully anti-symmetric and colorless, i.e., belong to the [222] color symmetry. Such states could be expanded in terms of di-baryon states, i.e., states that can be described as consisting of colorless three-quark clusters, say, NN , ∆∆, and the combination of CC , where a state C is a three-quark color-octet state. Harvey’s analysis showed that the CC would be dominant at small intercluster distances in for instance the lowest isospin-0, spin-1 state, relevant for the deuteron and the dominant low-energy nucleon–nucleon scattering channels 1 S0 and 3 S1 −3 D1 . In Ref. [40], a one-gluon exchange potential combined with a linear or harmonic confining potential was used to fit the nucleon and ∆ baryon-number-1 states. Using the potential with the parameters calibrated this way, he determined an effective nucleon–nucleon interaction. It was found to be attractive, thus being in contradiction with the data on nucleon–nucleon scattering. Nevertheless, the idea that at short distances a description of the hadronic interactions in terms of colorless degrees of freedom would be unreasonable, turned out to be fruitful. 3.2. Quark interchange between MIT bags A model developed by Weber et al. for the nucleon–nucleon interaction was based on the idea that the interaction would be initiated by the production of a quark-anti-quark pair which at larger distances would hadronize as a meson. In this way, the popular OBE models could perhaps be understood to emerge from quark- and gluon-interchange between nucleons. The nucleons themselves were supposed to be modeled by bags which provided confinement. The connection between quark interchange and meson exchange was made using a Fierz transformation to expand a quark-exchange matrix element into a set of quark-anti-quark matrix elements of various spin and isospin quantum number. The latter were to be linked to mesons with the same quantum numbers. A distinctive feature of this approach is the fact that it describes the meson–nucleon–nucleon vertex functions from a unified point of view: These functions are defined as matrix elements of the Dirac invariants I , γ µ , iγ5 , and γ5 γ µ , combined with the isospin matrices for iso-vector exchanges. Consequently, they can be expressed in terms of integrals of the bagmodel quark wave functions R0 (r ) and R1 (r ), which occur in the MIT-bag spinor

ψs (r ) =

R0 ( r ) −iR1 (r )σ · r





χs ,

(22)

χs being the Pauli spinor with magnetic quantum number s = ±1/2. The bag wave functions were taken from Ref. [41]. The vertex form factors obtained this way turned out to be soft, i.e., if they were fitted to either a monopole or a dipole form, Λ2 /(q2 + Λ2 ) or [Λ2 /(q2 + Λ2 )]2 , the regulators Λ found were considerably smaller, e.g. in the range 0.5 − 0.65 GeV/c for the monopole compared to the range 1.2–2.5 GeV/c for monopole regulators in phenomenological OBE potentials [30]. Owing to its unified point of view, this model has essentially two parameters besides the physical masses of the exchanged meson, namely the effective quark-mass mQ = 0.108 GeV/c 2 on which the bag spectroscopy depends, and the effective quark–gluon coupling. This can be compared with the purely phenomenological OBE potentials, where the regulators in all the vertex functions are free parameters, to be fitted to the data. The phenomenological success of this approach turned out to be somewhat limited. On the one hand, it is remarkable that the bag model plus quark exchange mechanism could at all produce a OBE-potential that gives values for the phaseshifts in nucleon–nucleon scattering below 350 MeV laboratory energy that are in qualitative agreement with the data, at least for the lower partial waves, the 3 P0 and 3 P1 waves being the exception. (See Fig. 7 for the S-waves.) On the other hand, a good description of the strong short-range repulsion seemed to be missing, which can be seen in the S-wave phase shifts, which change sign at too low energy (1 S0 ) or at too high energy (3 S1 ). This feature could be a consequence of the soft form factors, but might also be due to the fact that in this approach the two-baryon systems at small distances are described by two bags with the quantum numbers of the nucleon, exchanging quarks, neglecting multi-quark configurations. 3.3. Multi-quark bags The fact that colored objects are confined in nature leads in a natural way to the conjecture that measurable quantities of for instance nuclei must necessarily be amenable to a description in terms of colorless degrees of freedom only. In the case of individual nucleons or mesons, this idea appeared to be rather barren, as it requires all properties of these hadrons to be taken as primitive features, not open to further analysis. The history of deep-inelastic lepton scattering off nucleons shows

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Fig. 7.

1

S0 and 3 S1 phase shifts in Weber’s model.

that the application of quark (parton) models, and later the formalism of pQCD, appeared to be very fruitful. Nevertheless, one might wonder if systems consisting of several, or even many nucleons would yield to a description in terms of subnucleon degrees of freedom. At the same time that Harvey published his papers on multi-quark states, Jaffe and Low [36] made the observation that eigenstates of a multi-quark bag that can decay into colorless hadrons should appear as poles in the so-called P-matrix, rather than poles in the S-matrix. (We shall discuss the P-matrix shortly.) This idea was elaborated by Simonov [37] in a simple and explicit model, dubbed the quark-compound bag (QCB) model. A certain degree of agnosticism concerning the bag-model states turned out to be allowed: Only the radius b of the compound bag and the positions and residues of the P-matrix poles are needed in this model. This economy of the QCB model could be compared with the strong model-dependence of quark-potential models for mesons and baryons, which lead to a proliferation even in the three-quark and quark-anti-quark sectors. Another attractive feature of the QCB model is that it explains in a very economical way how the repulsive core in the nucleon–nucleon interaction may arise. Let us recall that in OBE models the repulsive core is caused by heavy-meson exchange, mainly ρ and ω mesons. To describe the data, this picture needs to be extended to inter-nucleon distances of the order of 0.3–0.5 fm, i.e., far into the region where nucleons with a radius of approximately 0.8 fm strongly overlap. Now, in such circumstances the probability that the colliding nucleons would retain their identities as colorless three-quark clusters should be low. This expectation was of course strengthened by Harvey’s analysis of six-quark states. Thus, there were several reasons to try and formulate the nucleon–nucleon scattering in terms of the QCB model. Before describing the model, let us recall some features of the P-matrix analysis. 3.3.1. P-matrix analysis In Ref. [42], a review of P-matrix methods in hadronic scattering is given. The P-matrix is given in the simple case of S-wave scattering of two spinless particles with one open channel by Pb (k) =

  = k cot[kb + δ(k)], u(b) dr r =b 1

du 

(23)

if beyond the radius b there is no interaction. The quantity δ(k) is the S-wave phase shift at a CMF momentum k. Clearly, Pb (k) will have poles at values of k where the cotangent diverges: kn b + δ(kn ) = nπ ,

n = 1, 2, . . . .

(24)

According to Jaffe and Low [36], if an appropriate choice is made for b, these poles, denoted as primitives, are the energies of the multi-quark confined states and their residues are proportional to the projections of the primitive wave functions on the open (color singlet) channels. The assumption that outside b there is no interaction cannot be maintained in realistic applications. In practice, see Ref. [42] and references therein, one chooses the radius b such that one may believe that asymptotic freedom is valid within b, and uses a realistic hadronic interaction in the peripheral domain r > b. Thus, the P-matrix method does not provide a very strong connection between the properties of multi-quark confined states and the nuclear phase shifts, but its underlying philosophy connects them in such a way that an attempt to build a model based on it could succeed. 3.4. Quark-compound bag model Simonov’s ideas [37] were elaborated by his collaborators [43]. Here, we briefly mention the derivation of the QCB interaction within the framework of the cluster approach [44] given in Ref. [45]. Following Ref. [46], the total wave function

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13

is split into two parts:

|ψ⟩ = |ψh ⟩ + |ψq ⟩,

(25)

where |ψh ⟩, the hadronic or conventional part, contains the properly anti-symmetrized clusters that are also present at asymptotic distances and |ψq ⟩ is a linear combination of six-quark confined states. The latter are eigenstates of the full Hamiltonian, subject to a confining boundary condition at r = b:

|ψq ⟩ =

 ν

(H − Eν )|ψν ⟩ = 0.

aν |ψν ⟩,

(26)

Projecting the Schrödinger equation on the hadronic and QCB parts, an effective energy-dependent interaction in the hadronic channel is obtained by elimination of the QCB-part à la Feshbach [47] VNQN (r ′ , r ; E ) =

 ν

VNν (r ′ ; E )

1 E − Eν

Vν N (r ; E ).

(27)

The form factors Vν N (r ; E ) can be written in terms of the wave functions of the bag states ψν and their nucleon–nucleon components. If a hard wall is assumed at r = b, one finds [45] Vν N (r ; E ) =

 1  ℓ cν δ(r − b) + (Eν − E )xℓν ηνℓ (r ) Yjls (ˆr ). r ℓ

(28)

√

The coefficients cνℓ fix the residues of the P-matrix poles, the factors xℓν are equal to 10 times the fractional-parentage coefficients gνNN ℓ , and the functions ηνℓ (r ) are the bag-model wave functions. The quantum numbers j, ℓ, s refer to the NNstate. In Ref. [45], it is shown that for a successful application to NN scattering and the description of the deuteron it suffices to retain just the lowest bag state explicitly, while the effect of the higher ones can be parametrized as follows: ′ ′ ∞  Vℓ Vℓ

Nν

ν=2

νN

E − Eν

ζℓ ζℓ ′

→−

MN

δ(r ′ − b)δ(r − b).

(29)

The occurrence of the delta functions causes a discontinuity in the derivatives of the radial wave functions at r = b. In the peripheral region, r > b, there exists an interaction that acts on the colorless degrees of freedom only. In Ref. [45], the Paris potential was chosen. The connection between the inner part of ψh and its peripheral part is obtained by matching at r = b, taking into account the discontinuity just mentioned. Fitting the phase shifts to the partial-wave analysis [48] the parameters are determined. This model was denoted as QCB86. The most important ones, namely the primitive energy and the bag radius were found to be for the 1 S0 channel E1 = 712(±12) MeV,

b = 1.00 fm,

√

s1 = 2.4885 GeV.

(30)

The bag wave function is the spherical Bessel function that has its first node at r = b. The fractional-parentage coefficient can be found in the literature. This model produces the low-energy observables very well, in particular the scattering length and the effective range, see Fig. 8. In the 3 S1 −3 D1 channel one finds E1 = 629(±32) MeV,

b = 1.00 fm,

√

s1 = 2.4269 GeV.

(31)

In this channel the binding energy of the deuteron is also included in the fit. For the phase-shift results, see Fig. 9. The predictions of this model are the wave functions and consequently also the form factors of the deuteron. Starting with the wave functions, an important quantity is the value of the admixture of the compound bag in the deuteron: PQCB =

 aν (E )2 ν

∥ψ∥

.

(32)

For the model described above, a value PQCB = 1.8% is found. This small amount is not surprising, because the deuteron is a loosely bound nucleus, in which a considerable part of the spatial probability distribution is outside the range of the nuclear force. Recall that the charge radius as derived from the form factor data is rch = 1.97 fm. The deuteron wave functions exhibits nodes that are characteristic for the solutions of a Schrödinger equation with an eliminated channel, in our case the CC channel, see Fig. 10.

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Fig. 8.

Fig. 9.

3

1

S0 Phase Shift in the QCB86 model.

S1 and 3 D1 Phase Shifts in the QCB86 model.

Interestingly, the deuteron form factors calculated in this model are very close to the data, even reproducing the node in the form factor B(q2 ) in the correct place. If instead of the QCB model, the Paris potential would have been used also in the inner region, the fit to the form factors would have been much worse at q2 > 1 GeV2 , see Fig. 11.

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15

Fig. 10. r-space S and D wave functions of the deuteron in the QCB86 model.

Fig. 11. Form factors of the deuteron in the QCB86 model.

3.5. Microscopic cluster models Given the success of the phenomenological QCB model, one may wonder if an approach that treats the multi-quark states at a microscopic level would not be more informative than the more or less ‘‘agnostic’’ QCB. Already at the time of development of the QCB model, some microscopic calculations of the low-lying six-quark states were performed. Mulders et al. [49] found the values M (S − 0, T + 1) = 2.243 GeV/c 2 and M (S = 1, T = 0) = 2.164 GeV/c 2 . These values are low compared with the values found in Ref. [45], which are closer to the values found in Ref. [50]. A modern treatment of the multi-quark spectroscopy is reviewed in Ref. [51]. The author concludes that ‘‘constituent quark models remain in the front of investigation’’. He quotes work by the Bonn group [52] on mesons and baryons, to continue with ‘‘The case of multi-quarks is of course much more difficult, with the mixing of confined channels and

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hadron–hadron components in the wave function’’. This comment hits the nail on the head, and in the QCB model an attempt is made to describe this mixing. Still, there exists many references to derivations of the interaction between baryons from quark models. In particular, work based on the resonating-group method was reviewed in Ref. [53]. 4. Multi-quark QCD evolutions The short-distance behavior of a hadron wave function can be computed systematically in pQCD [54]. The leading behavior of the hadron wave function at large momentum transfer or short distances is controlled through an evolution equation with an irreducible hard-scattering kernel which, in lowest   order, is identical to the gluon-exchange potential. Since the running coupling constant αs (Q 2 ) = 4π / β ln(Q 2 /Λ2 ) (β = 11 − 23 nf , where nf is the number of flavors) is small for large momentum transfer Q , a perturbative calculation of the short-distance part of the wave function can be justified. The anomalous dimensions of the multi-quark amplitude can also be predicted by the operator-product expansion and the renormalization group [55]. A particularly convenient and physical formalism for studying processes with large momentum transfer is the LF quantization, discussed in Ref. [54]. A systematic analysis of exclusive processes and hadron distribution amplitudes has been given [56], including a complete classification of the proton, neutron, delta states, etc. Here, we present a general method for solving the QCD evolution equations which govern relativistic multi-quark wave functions. A LF basis of completely antisymmetric wave functions is used in multi-quark systems. In the case of threequark systems, it provides a general covariant classification of baryonic states. The spin–orbit mixing generated by the QCD evolution kernel can be computed in the basis of completely antisymmetric representations. Using this method, one can obtain a distinctive classification of nucleon and ∆ wave functions [56]. The corresponding Q 2 dependence of the baryon distribution amplitudes distinguishes the nucleon and ∆ form factors. A systematic expansion to the multiquark antisymmetric representations such as six- and nine-quark systems can be explored. In the six-quark systems, the eigensolutions in the physical two-cluster basis of SU (3)C dibaryon (NN , N ∆, ∆∆) and hidden-color (CC ) components can be found. The CC components are expected to dominate the short-distance behavior of the six-quark systems and provide a constraint on the short-range correlations between two clusters. We also present the number of hidden-color components in the nine-quark systems and discuss its implication in the experimental measurements of electron scattering from highmomentum nucleons in nuclei. A fermionic system in QCD is classified by the assignment of four quantum numbers: color (C ), isospin (T ), spin (S), and orbital (O). Each quantum sector of the wave function can be classified using irreducible representations of permutation symmetry denoted by Young diagrams. The explicit construction of totally antisymmetric representations in terms of an orbital index-power basis is described below. 4.1. Color, Isospin, and spin states We can classify the states with quantum numbers C , T , and S by the group G = SU (3)C × SU (2)T × SU (2)S without loss of generality. Each quantum state with given C , T , and S is a basis vector of an irreducible representation of G, and each irreducible representation is denoted by the corresponding Young diagram. Once a Young diagram is given, the explicit representation can be constructed from its permutation symmetry and Schmidt orthogonalization. All physical baryons are color-singlet states. The corresponding Young diagram is given by r y b

(33)

in SU(3)C . Thus, the explicit color representation of the baryon is fixed: r 1 y = √ (ryb + ybr + bry − byr − rby − yrb) 6 b 1

≡ √ ϵijk , 6

(34)

where the completely antisymmetric Cartesian tensor ϵijk (i, j, and k correspond to one of r , y, and b) defines the color-singlet representation. The quantum state (color in this case) of the first, second, and third quark is represented by the first, second, and third location of every term in Eq. (34). Hereafter, we will use this convention for each quantum number unless we specifically denote the particle number. The classification of the baryon into N and ∆ is given by the isospin label, namely, T = 12 or 32 , and the corresponding Young diagrams are

and

(35)

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17

for N and ∆, respectively. The mixed symmetry

(36) has two orthogonal permutation symmetries, represented by two different Yamanouchi labels 1 2 3

1 3 2 .

and

(37)

As an example, we present the explicit representation of Tz =

u u d

=

 1 3 =     2     1 2 = 3

( T , Tz )

 =

1 1

,



2 2

√1 2

√1 6

1 2

for T =

1 2

or

3 : 2

(duu − udu), (duu + udu − 2uud),

,

(38)

1 u u d = √ (uud + udu + duu), 3

( T , Tz )

 =

3 1

,



2 2

,

(39)

where (T , Tz ) = ( 21 , 21 ) and ( 32 , 21 ) correspond to p and ∆+ , respectively.

The spin states of the three-quark system are classified by the Young diagrams for S = 12 and 32 . The explicit representations are obtained from the isospin representations with the replacement of u and d by ↑ and ↓. 4.2. Orbital states The orbital states are normally defined by the quantum numbers of angular momentum L and Lz . In the LF formalism, the quark distribution amplitude φ(xi , Q ) is defined by



φ(xi , Q ) = ln

Q2

−3CF /2β 

Q

  2  d2 ⃗ k⊥i 16π 3 i =1

Λ2

 16π δ

3 2

3 

 ⃗k⊥i ψ (Q ) (xi , ⃗k⊥i ),

(40)

i=1

, ψ (Q ) (xi , ⃗k⊥i ) is the wave function of three quarks which have longitudinal-momentum fractions xi = ki /( i=1 ki ) and transverse momenta ⃗ k⊥i . In this definition, the Lz = 0 projection defines the amplitude for finding the constituents collinear up to the scale Q . We use as a basis for the orbital dependence of φ(xi , Q ) the index-power space n n n representations x11 x22 x33 with n = n1 +n2 +n3 . The total power is analogous, as far as permutation symmetry is concerned, to

where CF = +

3

4 3

+

the angular momentum L for the nonrelativistic system. In the QCD evolution equation, the minimal anomalous dimensions

γn which determine hadronic amplitudes at very short distances are associated with small values of n; only the smallest powers of xi are important for probing the short-distance behavior of φ(xi , Q ). In this power space, the orbital states are determined by filling up the possible Young diagrams with the powers of xi . For example, if we consider n = n1 + n2 + n3 = 0 case, then the only possible Young diagram

gives the representation

0 0 0 = 1.

(41)

For the n = 1 case, the possible diagrams and representations are 1 0 0 1 = √ (x1 + x2 + x3 ), 3

 1 3 =     2

0 0 =  1 

  1 2 = 3

√1 2

(42)

(x1 − x2 ), (43)

√1 6

(x1 + x2 − 2x3 ).

However, the representations given by Eqs. (42) and (43) are not independent of the representation given by Eq. (41) because 3 of the conservation of momentum i=1 xi = 1. Generally, the orbital representations can overlap each other between the

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same diagrams. Thus, we use the Gram–Schmidt orthogonalization procedure and normalize the states by the following rule between the orbital representations φn (xi , Q0 ) and φm (xi , Q0 ) with the same Young diagram:

⟨φm (xi , Q0 )|φn (xi , Q0 )⟩ =



[dx]ω(xi )φm∗ (xi , Q0 )φn (xi , Q0 )

= δmn ,

(44)

where

 [dx] = dx1 dx2 dx3 δ 1 −

3 

 xi

,

ω(xi ) = x1 x2 x3 .

(45)

i =1

After orthonormalization, we obtain the basis set of orbital states. The explicit representations and Young diagrams up to n = 2 were presented in Ref. [56]. We note that the orbital representations in power space are independent of any dynamics, and any model-dependent representation can be projected onto this representation. A state which has arbitrary angular momentum L can be projected on the corresponding index-power space. 4.3. Antisymmetrization In the previous two subsections, we showed that the quantum states for each C , T , S, and O quantum number are explicitly represented by the permutation symmetry given by the Young diagrams. The completely antisymmetric representation of a fermionic system is obtained by the inner product of C , T , S, and O quantum states represented by the corresponding Young diagrams. As an example, let us construct the antisymmetric representation of the excited state of the baryon with (S , Sz ) = ( 32 , 12 ). For this state, the C and T representations are given by Eqs. (34) and (38), respectively, and the S representation is given by Eq. (39) with the replacement of u and d by ↑ and ↓. To construct the completely antisymmetric representations, we combine the possible orbital symmetries as given by the Clebsch–Gordan series of the permutation group S3 . In this case, the only possible orbital Young diagram is

.

(46)

The lowest state is 0 1(n = 1) and the representation is given in Table 1 of Ref. [56]. Considering the Clebsch–Gordan coefficients of the permutation group given by 2

1

1 3 × 1 3 + √1 2 2 2

= √

2

1 2 × 1 2, 3 3

(47)

we obtain the completely antisymmetric representation given by r

× u u × ↑ ↑ ↓S× 0 0

= y b

CTSO

d

1

T

C

(48) O

√ N

=

18

ϵijk (↑↑↓ + ↑↓↑ + ↓↑↑)[duu(2x1 − x2 − x3 ) + udu(−x1 + 2x2 − x3 ) + uud(−x1 − x2 + 2x3 )],

where N = 21 × 5! and ϵijk is defined by Eq. (34). In a similar way, we can classify all possible three-quark states and obtain the explicit antisymmetric representations. The classification and the representations of the baryon system up to the power n = 2 were presented in Table 1 of Ref. [56]. Using the antisymmetric representations of the three-quark system, we now discuss the QCD evolution of the baryonic wave function at short distances. The QCD evolution equation for the three-quark distribution amplitude φ(x, Q ) with Lz = 0 is given by

 x1 x2 x3

∂ 3CF + ∂ξ 2β



˜ x, Q ) = φ(

CB

β

1



˜ y, Q ), [dy]V (x, y)φ(

(49)

0

˜ x, Q ) and the variable ξ are defined by where the reduced amplitude φ( ˜ x, Q ), φ(x, Q ) = x1 x2 x3 φ(

(50)

and

β ξ (Q ) = 4π 2



Q2 Q02

dk2 k2

αs (k ) ∼ 2



ln(Q 2 /Λ2 ) ln(Q02 /Λ2 )



.

(51)

B.L.G. Bakker, C.-R. Ji / Progress in Particle and Nuclear Physics 74 (2014) 1–34

19

The color factor CB = (nc + 1)/2nc = 23 is fixed. The evolution kernel V (x, y) is the sum over interactions between quark pairs i, j due to exchange of a single gluon: V (x, y) = 2x1 x2 x3



θ (yi − xi )δ(xk − yk )

i̸=j

yj xj

 δ hi h¯ j

∆ + xi + xj yi − xi



= V (y, x),

(52)

where δhi h¯ j = 1(0) when the helicities of the quark pairs i, j are antiparallel (parallel). The infrared singularity at xi = yi is

˜ y, Q ) = φ( ˜ y, Q ) − φ( ˜ x, Q ) reflecting the fact that the baryon is a color singlet. canceled by 1φ( The evolution equation (49) has a general solution of the form φ(x, Q ) = x1 x2 x3

∞  n =0



An φ˜ n (x) ln

Q2

Λ2

−γn

,

(53)

where γn and φ˜ n satisfy

 x1 x2 x3

  1 CB − γn φ˜ n (x) = [dy]V (x, y)φ˜ n (y). 2β β 0

3CF

(54)

The γn are the anomalous dimensions corresponding to the three-quark eigensolutions φ˜ n . The result of γn and φ˜ n with a definite classification of baryon systems has been presented in Ref. [56]. This method provides a general technique which predicts the short-distance behavior and classifies the spectrum of relativistic many-fermion systems. It may be regarded as a fundamental method for studying short-distance dynamics in the multi-quark systems of nuclear physics as discussed in the following subsections. 4.4. Multi-quark application The short-distance behavior of multi-quark wave functions can be systematically computed in pQCD. A simple illustration may be found in Ref. [57], where the wave function of a four-quark color-singlet bound state in SU (2)C has been analyzed as an analog to the six-quark problem in QCD. The QCD evolution equation was solved for the multi-quark distribution amplitude at short distances in the basis of completely anti-symmetrized quark representations. The eigensolutions of the evolution kernel correspond to a spectrum of candidate states of the relativistic multi-quark system. The four-quark antisymmetric representations are then connected to the eigensolutions to the physical two-cluster basis of SU (2)C dibaryon (NN , N ∆, ∆∆) and hidden-color (CC ) components. It provides constraints on the effective nuclear potential between two clusters. Anomalous states are also found in the spectrum which cannot exist without substantial hidden-color degrees of freedom. As technical details are presented in Ref. [57], a given four-quark antisymmetric representation (A) can be decomposed onto two clusters (A1 ⊗ A2 ) using the following steps: 1. Represent the four-quark antisymmetric representation as an inner product form A = C × T × S × O. 2. Decompose each four-quark representation C , T , S, and O as an outer product of 2 two-quark representations using fractional parentage coefficients, e.g., C = C1 ⊗ C2 . 3. Recombine the representations as an inner product: A = (C1 ⊗ C2 ) × (T1 ⊗ T2 ) × (S1 ⊗ S2 ) × (O1 ⊗ O2 ). 4. Commute the order of inner product and outer product, gathering together representations of the same cluster: A = (C1 × T1 × S1 × O1 ) ⊗ (C2 × T2 × S2 × O2 ) ≡ A1 ⊗ A2 . 5. It is sufficient to consider only the coefficient of the symmetric orbitals O1 and O2 to classify the clusters such as NN , N ∆ and ∆∆. With this method, the four-quark eigensolutions can be expanded on the physical basis of effective clusters which are the analogs of the NN , ∆∆, N ∆, and CC states in QCD. By analyzing the behavior of φ(xi , Q ) at large Q , one can predict the effective potential between two clusters. For example, it was found that one of the hidden-color states has a large projection on the eigensolution with leading anomalous dimension (dominant at short distances), whereas the states analogous to NN and ∆∆ in QCD have an almost negligible leading component. This implies that the effective potential tends to be repulsive between color-singlet clusters and attractive between colored clusters at short distance. Two other types of four-quark states were also found in SU (2)C , which cannot be identified with dibaryon degrees of freedom. One of these states has equal NN , ∆∆ and CC components. The other state is an anomalous hidden-color two-cluster system orthogonal to the usual hidden-color state which has the unusual feature that it has very small projection on the eigensolutions which dominate at short distance, i.e., the effective potential between the colorful clusters of the anomalous hidden-color state tends to be repulsive. One may speculate that the analogous anomalous states in QCD could be quasistable non-nucleonic nuclear systems, possibly related to the anomalous phenomena apparently observed in nuclear collisions [58,59]. These results also give some support to the conjecture that multi-quark hidden-color components exist in ordinary nuclei [59].

20

B.L.G. Bakker, C.-R. Ji / Progress in Particle and Nuclear Physics 74 (2014) 1–34

The results in Ref. [57] represented a first attempt to extract exact results for the composition and interactions of multiquark nuclear systems at short distances. Although just the four-quark bound state in SU (2)C was analyzed in Ref. [57] for simplicity, many of the derived properties were indeed extended to six-quark states in QCD as discussed in Ref. [13]. In particular, since the leading eigensolution at high-momentum transfer has 80% hidden-color probability, we expect a transition of the ordinary nuclear state to non-nucleonic degrees of freedom as one evolves from long to short distances. The set of eigensolutions of the evolution equation represent all the possible degrees of freedom of the multi-quark boundstate system since its kernel has the same invariances and symmetries of the full QCD Hamiltonian. We thus expect that the eigensolutions of the evolution kernel which are dominantly hidden-color to correspond to actual states and excitations of ordinary nuclei. A careful experimental search for these exotic resonances should be made. Possible channels where signals for such states may be observed include Compton scattering and pion photo-production on a deuteron target at large angles as well as the electron scattering from high-momentum nucleons in nuclei. 4.5. Six-quark evolution Six-quark states can be classified by their symmetries under SU (3)C (color), SU (2)T (isospin), SU (2)S (spin), and spatial symmetry. Since the physical states are color singlets, the Young symmetry of the color-singlet states of the six-quark system is [222] or

. Since three colors are shared by six quarks, there are five independent color-singlet states corresponding

to five different Yamanouchi labels of [222] symmetry. The explicit representations of the five independent color-singlet states and their correspondence to Yamanouchi labels are given in the Appendix of Ref. [13]. Harvey [38,40,39] has classified the color-singlet six-quark states in terms of a physical cluster decomposition. In this classification, the physical deuteron state (i.e., a bound state of two color-singlet clusters) is represented as a linear combination of several different kinds of totally antisymmetric color-singlet six-quark states. For example, the two wellseparated nucleons |NN ⟩ are given by

|NN ⟩ = 1/3|[6]{33}⟩ + 2/3|[42]{33}⟩ − 2/3|[42]{51}⟩,

(55)

where [] and {} represent the orbital and spin–isospin symmetry and color symmetry [222] is abbreviated. However, this classification itself does not include the dynamics of strong interactions between the constituents. In other words, the dynamics between the quarks inside the deuteron is not included. Thus, the dynamical evolution equation of six-quark systems was formulated [13] and solved to give the general form of the quark distribution amplitude φd (xi , Q ):

φd (xi , Q ) = (CTS )φ(xi , Q ),

(56)

where (CTS ) is a tensor representation obtained from the Young symmetry of SU (3)C , SU (2)T , and SU (2)S , and the orbital distribution amplitude is given by

φ(xi , Q ) = x1 x2 x3 x4 x5 x6

∞ 

 

an φ˜ n (xi ) ln

n =0

Q2

−γn

Λ2

.

(57)

One may then project Eq. (55) to momentum space:

φNN (xi , Q ) =

1 3

2

2

3

3

φ[6]{33} (xi , Q ) + φ[42]{33} (xi , Q ) − φ[42]{51} (xi , Q ).

(58)

In the limit Q → ∞, the dependence on Q is determined by the leading anomalous dimension; all other terms which have nonleading anomalous dimensions are suppressed by logarithmic damping factors. The orbital symmetry of the eigensolution which has the leading anomalous dimension cannot be [42] but is [6]. This means only the first term of Eq. (58) survives at the large-Q limit. The NN amplitude itself is not sufficient. One can show that an 80% hidden-color state is necessary to saturate the normalization of the six-quark amplitude when six quarks approach the same position in impact space. One may call this new degree of freedom an anomalous state since it does not correspond to the usual nucleonic degrees of freedom of the nucleus. The physical implication of the anomalous state was discussed in the previous toy model analysis [57]. The QCD predictions for high-Q behavior of the deuteron form factor and the form of the deuteron distribution amplitude at short distances were presented in Ref. [16]. The fact that the six-quark state is 80% hidden color at small transverse separation implies that the deuteron form factors cannot be described at large Q by meson–nucleon degrees of freedom alone, and that the nucleon–nucleon potential is repulsive at short distances. Since the basic scale of QCD, ΛQCD , is phenomenologically of the order of a few hundred MeV or less, QCD predicts a transition from the traditional meson and nucleon degrees of freedom of nuclear physics to quark and gluon degrees of freedom at inter-nucleon separations of a fm or less. In this respect, it may be important to realize the 12 GeV upgrade of JLab as a viable opportunity to investigate novel nuclear phenomena predicted by QCD. As an example, a JLab collaboration proposed to measure the deuteron tensor structure function b1 [1]. This leading twist tensor structure function of spin-1 hadrons provides a tool to study partonic effects, while also being sensitive to coherent nuclear properties in the simplest nuclear system. Although shadowing

B.L.G. Bakker, C.-R. Ji / Progress in Particle and Nuclear Physics 74 (2014) 1–34

21

effects are expected to dominate this structure function at low values of the Bjorken scaling variable x, it may provide a probe of exotic QCD effects due to hidden color in 6-quark configuration at larger values of x. Since the deuteron wave function is relatively well known, any novel effects are expected to be readily observable. All available models predict a small or vanishing value of b1 at moderate x. However, the first measurement of b1 at HERMES revealed a crossover to an anomalously large negative value in the region 0.2 < x < 0.5, albeit with relatively large experimental uncertainty. The proposal [1] describes an inclusive measurement of the deuteron tensor asymmetry in the region 0.15 < x < 0.45, for 0.8 < Q 2 < 5.0 GeV2 . It might be possible to determine b1 with sufficient precision to discriminate between conventional nuclear models, and the more exotic behavior which is hinted at by the HERMES data. This measurement will provide access to the tensor quark polarization, and allow a test of the Close–Kumano sum rule [60], which vanishes in the absence of tensor polarization of the quark sea. 4.6. Nine-quark color singlets In this subsection, we provide the counting of the number of hidden-color states in nine-quark systems. It is well known that the baryon multiplets can be constructed from the three triplets (quarks) in SU (3): i.e.,

⊗

⊗

=

⊕

⊕

⊕

(59)

or

{3} ⊗ {3} ⊗ {3} = {10} ⊕ {8}S ⊕ {8}A ⊕ {1}.

(60)

Applying it in the color degrees of freedom, i.e. SU (3)C , one can see that only one color singlet appears in the three-quark system, or baryon. However, more color singlets independent of each other can be formed as the number of quarks gets increased. In the six-quark system such as the deuteron, five color singlets are formed owing to the fact that the three color degrees of freedom are shared by the six quarks in the system. In other words, not only the singlet and singlet but also the octet and octet can form a six-quark color-singlet bound-states. Similar to Eq. (59), one may explicitly get

⊗

⊗

⊕9

⊗

⊗

⊗

⊕ 15

=

⊕5

⊕ 16

⊕5

(61)

or

{3} ⊗ {3} ⊗ {3} ⊗ {3} ⊗ {3} ⊗ {3} = 729 = {28} ⊕ 5{35} ⊕ 9{27} ⊕ 15{10} ⊕ 16{8} ⊕ 5{1}.

(62)

Among the five color-singlet states in Eq. (62), just one of them stems from the two color-singlet clusters of three-quark system such as N and ∆ while the remaining four states correspond to the two color-octet clusters denoted as CC or hiddencolor states. Now, let us count here how many hidden-color states are available in the nine-quark system such as the 3 He nucleus. Extending the same color algebra, we get

⊗

⊗

⊗

⊗

⊕8

⊕ 162

⊕ 163

⊗

⊕ 43

⊕ 33

⊗

⊗

⊗

=

⊕ 32

⊕ 74

⊕ 27 ⊕ 120

⊕ 135

⊕ 37

(63)

or

{3} ⊗ {3} ⊗ {3} ⊗ {3} ⊗ {3} ⊗ {3} ⊗ {3} ⊗ {3} ⊗ {3} = 19683 = {55} ⊕ 27{81} ⊕ 8{80} ⊕ 43{64} ⊕ 32{35} ⊕ 120{35} ⊕ 162{27} ⊕ 33{28} ⊕ 74{10} ⊕ 135{10} ⊕ 163{8} ⊕ 37{1}.

(64)

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B.L.G. Bakker, C.-R. Ji / Progress in Particle and Nuclear Physics 74 (2014) 1–34

This shows a remarkable proliferation of the hidden color degrees of freedom in the nine-quark system compared to the six-quark system, i.e. 37 − 1 = 36 = 9(5 − 1). Here, we find thirty six hidden-color states in the nine-quark system since only one of the thirty seven color-singlet states stem from the three color-singlet clusters of three-quark systems. Compared to just four hidden-color states in the six-quark system, we now find nine times more hidden-color states in the nine-quark system. It amounts to almost an order of magnitude increase in the hidden-color sectors. It is interesting to note the recent measurements of electron scattering from high-momentum nucleons in nuclei performed at JLab Hall C(E02-019) [2]. These data allowed for an improved determination of the strength of two- and threenucleon correlations for several nuclei, including light nuclei where clustering effects can be examined. At x > 2, where three-nucleon short-range correlations (3N-SRCs) are expected to dominate, their A/3 He ratios were significantly higher than the previous CLAS data and suggested that contributions from 3N-SRCs in heavy nuclei are larger than previously believed. Interestingly, there were large differences between the 4 He/3 He ratios from E02-019 (Q 2 ≈ 2.9 GeV2 ) and those from CLAS (⟨Q 2 ⟩ ≈ 1.6 GeV2 ). If these results indeed indicate a large dependence on Q 2 for the x > 2 plateau region, then it might be associated with the dramatic increase of the hidden color degrees of freedom that we found in the three-nucleon systems. It will be interesting to see if this ratio difference between Hall C and CLAS is truly due to the Q 2 difference between these two experiments. This may be tested in the upcoming experiments to be performed after the 12 GeV upgrade in JLab. 5. Model-independent predictions from angular conditions A relativistic treatment is one of the essential ingredients that should be incorporated in describing hadronic systems. The hadrons have an intrinsically relativistic nature since QCD, governing the quarks and gluons inside the hadrons, has a priori a strong interaction coupling and the characteristic momenta of quarks and gluons are of the same order, or even very much larger, than the masses of the particles involved. It has also been realized that a parametrization of nuclear reactions in terms of non-relativistic wave functions must fail. In principle, a manifestly covariant framework such as the Bethe–Salpeter approach and its covariant equivalents can be taken for the description of hadrons. However, in practice, such tools are difficult to use, because of the relative-time dependence and the difficulty of systematically including higherorder kernels. A different and more intuitive framework is the relativistic Hamiltonian approach. With the advances in the Hamiltonian renormalization program, a promising technique to implement the relativistic treatment of hadrons appears to be light-front dynamics (LFD), in which a Fock-space expansion of bound states is made at equal LF time τ = t + z /c. The reasons that make LFD so attractive to solve bound-state problems in field theory make it also useful for a relativistic description of nuclear systems. In this section, we review the work in collaboration with Carl Carlson [61] for the application of LFD to analyze the current matrix elements in the general collinear (Breit) frames and find the relation between the ordinary (or canonical) helicity amplitudes and the LF helicity amplitudes. Using the conservation of angular momentum, we derive a general angular condition which should be satisfied by the LF helicity amplitudes for any spin system. In addition, we obtain the LF parity and time-reversal relations for the LF helicity amplitudes. Applying these relations to the spin-1 form-factor analysis, we note that the general angular condition relating the five helicity amplitudes is reduced to the usual angular condition relating the four helicity amplitudes owing to the LF time-reversal condition. We make some comments on the consequences of the angular condition for the analysis of the high-Q 2 deuteron electromagnetic form factors, and we further apply the general angular condition to the electromagnetic transition between spin-1/2 and spin-3/2 systems and find a relation useful for the analysis of the N − ∆ transition form factors. We also discuss the scaling law and the subleading power corrections in the Breit and LF frames. 5.1. Light-front treatment The LF quantization [62,63] was already applied successfully in the context of current algebra [64] and the parton model [65] in the past. For the analysis of exclusive processes involving hadrons, the framework of LF quantization [12] is also one of the most popular formulations. In particular, the LF or Drell–Yan–West (q+ = q0 + q3 = 0) frame has been extensively used in the calculation of various electroweak form factors and decay processes [66–68]. In this frame [69], one can derive a first-principles formulation for the exclusive amplitudes by judiciously choosing the component of the LF current. As an example, only the parton-number-conserving (valence) Fock-state contribution is needed in the q+ = 0 frame when a ‘‘good’’ component of the current, J + or J⊥ = (Jx , Jy ), is used for the spacelike electromagnetic form-factor calculation of pseudoscalar mesons. One does not need to suffer from complicated vacuum fluctuations in the equal-τ = t + z /c formulation owing to the rational dispersion relation in LFD. The zero-mode contribution may also be avoided in the Drell–Yan–West (DYW) frame by using the plus component of the current [70]. The pQCD factorization theorem for the exclusive amplitudes at asymptotically large momentum transfer can also be proved in LFD formulated in the DYW frame. However, caution is needed in applying the established Drell–Yan–West formalism to other frames, because the current components mix under transformations of the reference frame [71]. Especially, for spin systems, the LF helicity states are in general different from the ordinary (or canonical) helicity states which may be more appropriate degrees of freedom to discuss angular-momentum conservation. As the spin of the system becomes larger, the number of current matrix elements gets larger than the number of physical form factors and the conditions that the current matrix elements must satisfy are essential to test the underlying theoretical

B.L.G. Bakker, C.-R. Ji / Progress in Particle and Nuclear Physics 74 (2014) 1–34

23

model for the hadrons. Thus, it is crucial to find the relations between the ordinary helicity amplitudes and the LF helicity amplitudes in the frame in which they are computed. Here, we use the general collinear frames which cover both Breit and target-rest frames to find the relations between the canonical helicity amplitudes and the LF helicity amplitudes. Using the conservation of angular momentum, we derive a general angular condition which can be applied to any spin system. The relations among the LF helicity amplitudes are further constrained by LF parity and time-reversal considerations. For example, the spin-1 form-factor analysis involves nine LF helicity amplitudes although there are only three physical form factors. Thus, there must be six conditions for the helicity amplitudes. Using the LF parity relation, one can reduce the number of helicity amplitudes down to five. The general angular condition gives one relation among the five LF helicity amplitudes, leaving four of them independent. One more relation comes by applying the LF time-reversal relation, also having the effect that the general angular condition can be reduced to the usual angular condition relating only four helicity amplitudes. Consequently, only three helicity amplitudes are independent of each other, as it should be, because there are only three physical form factors in spin-1 systems. We also apply the general angular condition to the electromagnetic transition between spin-1/2 and spin-3/2 systems and find the relation among the helicity amplitudes that can be used in the analysis of the N − ∆ transition. In particular, the angular condition provides a strong constraint to the N − ∆ transition indicating that the suppression of the helicity-flip amplitude with respect to the helicity-non-flip amplitude for the momentum transfer Q , in pQCD is of order m/Q or M /Q rather than ΛQCD /Q , where both the nucleon mass m and the ∆ mass M are much larger than the QCD scale ΛQCD . Thus, one may expect that the applicability of leading pQCD could be postponed to a larger Q 2 region than one might naively. The same consideration may apply for the deuteron form-factor analysis using the spin-1 angular condition. Finally, we present further discussions on the scaling law and the subleading power corrections in the Breit and LF frames. 5.2. Frame relations and general angular condition Our subject is relations among matrix elements or helicity amplitudes for the process γ ∗ (q) + h(p) → h′ (p′ ), where γ ∗ is an off-shell photon of momentum q, and h and h′ are hadrons with momenta p and p′ , respectively. (Results will be easily extendable to the case of other incoming vector bosons.) Calculations may be done in the LF frame, which is characterized by having q+ ≡ q0 + q3 = 0, or may be done in the Breit frame, which is characterized by having the photon and hadron 3-momenta along a single line. Each frame has its advantages. In the light front frame with q+ = 0, and for matrix elements of the current component J + , the photon only couples to forward moving constituents (quarks) of the hadrons and never produces a quark–antiquark pair. Thus one only needs wave functions for hadrons turning into constituents going forward in (LF) time, and can develop a simple parton picture of the interaction. On the other hand, the Breit frame, being a collinear frame, makes it easy to add up the helicities of the incoming and outgoing particles and to count the number of independent non-zero amplitudes. By transforming efficiently back and forth one can realize the advantages of both frames. Hence our first goal in this Section will be to find the relations between the LF and Breit frame helicity amplitudes, and then to use those relations to derive in a transparent way the general relation among the LF amplitudes that is usually referred to as the ‘‘angular condition’’. 5.2.1. Relations among helicity amplitudes Connecting LF and Breit helicity amplitudes is facilitated by finding frames that are both simultaneously realized. One excellent and easy example is the particular LF frame where the target is at rest. This is also a Breit frame, since with the target 3-momentum zero, the remaining momenta must lie along the same line. We are perhaps extending the idea of a Breit frame, but are doing so in a way that leaves invariant the Breit frame helicity amplitude. That is, one normally thinks of a Breit frame as one where the incoming and outgoing hadron have oppositely directed momenta, along the same line. Sometimes one specifies that this line is the z-axis. However, since helicities are unaffected by rotations [72], one can choose any line at will. Further, helicities are unaffected by collinear boosts [72] that do not change the particle’s momentum direction. One can also boost along the direction of motion until one of the hadrons is at rest, provided one defines the positive helicity direction for the particle at rest to be parallel to the momentum the particle would have in a conventional Breit frame. With this natural helicity-direction choice, the Breit frame helicity amplitude in a target-rest frame is (if we use relativistic normalization conventions, as we shall always do) precisely the same as the Breit-frame amplitude in a conventional Breit frame with the same helicity labels. Another useful example is a Breit frame with the incoming and outgoing hadrons moving in the negative and positive x-directions, adjusted to have equal incoming and outgoing energies. In this case, q0 and q3 are individually zero, so that q+ = 0 and we also have a LF frame. Even in a frame that is simultaneously LF and Breit, the connection between the two types of helicity amplitudes can be a bit involved. This is so, because the definitions of the LF and canonical helicity states are not the same and the general conversion between them for a moving state involves a rotation by an angle that is not trivial to determine. We use the rest of this subsection to define our notation, state the main result for the LF to Breit and vice-versa helicity amplitude conversion formulas, and show how one obtains the general angular condition from this result. Then in the next subsection, we will give the details of the derivation.

24

B.L.G. Bakker, C.-R. Ji / Progress in Particle and Nuclear Physics 74 (2014) 1–34

For LF amplitudes one uses LF helicity states, which for a momentum p are defined by taking a state at rest with the spin projection along the z-direction equal to the desired helicity, then boosting in the z-direction to get the desired p+ , and then doing a LF transverse boost to get the desired transverse momentum p⊥ . We call this state

|p, λ⟩L ,

(65)

and give its definition by a formula in the next subsection. The spin of the particle, j, is understood but not usually written, λ is the LF helicity of the particle, and the normalization is L

+ + 2 ⟨p2 , λ2 |p1 , λ1 ⟩L = (2π )3 2p+ 1 δ(p1 − p2 )δ (p2⊥ − p1⊥ )δλ1 λ2 .

(66)

The LF helicity amplitude GL is a matrix element of the electromagnetic current J ν given by GνLλ′ λ = L ⟨p′ , λ′ |J ν |p, λ⟩L .

(67)

In the Breit frame, we use ordinary helicity states, which are defined by starting with a state at rest having a spin projection along the z-direction equal to the desired helicity, then boosting in the z-direction to get the desired |p|, and then rotating to get the momentum and spin projection in the desired direction. (We shall generally keep our momenta in the x − z plane, so we do not need to worry about the distinction between, for example, the Jacob–Wick [72] helicity states and the Wick states [73] defined somewhat later.) The state will now be denoted,

|p, µ⟩B ,

(68)

where µ is the helicity or spin projection in the direction of motion, and the subscript ‘‘B’’ reminds us which frame we use these states in. Except for momenta directly along the z-axis, the LF helicity and regular helicity states are not the same, but if the 4-momenta are the same they must be related by a rotation. The Breit frame helicity amplitude GB is, GνBµ′ µ = B ⟨p′ , µ′ |J ν |p, µ⟩B ,

(69)

ν

where J is the same electromagnetic current. Our main result is the relation between GL and GB , which is j′

GνBµ′ µ = dλ′ µ′ (−θ ′ ) GνLλ′ λ dµλ (θ ). j

(70)

A sum on repeated helicity indices is implied. The d-functions are the usual representations of rotations about the y-axis for particles whose spins are given by the superscript. The angles are given by tan

θ 2

=

Q+ Q− − Q 2 − M 2 + m2

(71)

2mQ

and tan

θ′ 2

=−

Q+ Q− − Q 2 + M 2 − m2 2MQ

,

(72)

where m is the mass of the incoming hadron, M is the mass of the outgoing hadron, Q = Q 2 = −q2 = −(p − p′ )2 ,

Q± = Q 2 + (M ± m)2



1/2



Q 2 , and

,

(73)

and we assume that q is spacelike (negative, in our metric). For the elastic case, M = m, the angles θ and −θ are the same. (It may seem peculiar to have a minus sign inserted twice, as it appears in Eqs. (70) and (72), but it will seem more sensible when one sees in the next subsection how these angles arise.) In the Breit frame, since it is collinear, the sum of spin projections along the direction of motion must be conserved, so that if λγ is the helicity of the photon, ′

2

λγ = µ + µ′ .

(74)

Even if the photon is off-shell, it cannot have more than one unit of helicity in magnitude. Hence there is a constraint on the Breit frame helicity amplitude, GνBµ′ µ = 0

if |µ′ + µ| ≥ 2.

(75)

This induces a constraint on the light front amplitudes, and this constraint is what is called the angular condition, given generally by [74] dλ′ µ′ (−θ ′ ) GνLλ′ λ dµλ (θ ) = 0 j′

j

if |µ′ + µ| ≥ 2,

(76)

and most often applied when the Lorentz index ν is +. One sees that the result follows from angular momentum conservation and the limited helicity of the photon. We will check in Section 5.4 that upon expressing the d-functions in terms of Q 2 and mass, one obtains the known angular condition for electron-deuteron elastic scattering, and we shall also obtain the angular condition for the N − ∆(1232) electromagnetic transition in terms of Q 2 and the masses.

B.L.G. Bakker, C.-R. Ji / Progress in Particle and Nuclear Physics 74 (2014) 1–34

25

Fig. 12. Photon with q+ = 0 absorbed by a particle at rest. Two choices for the target spin axis are indicated. In the Breit frame (B), the helicity state positive direction is opposite to the direction of the incoming photon. The LF state (L), in this case, is identical to the rest state quantized along the positive z-axis. The angle θ between the photon 3-momentum direction and the (negative) z-axis is also the angle between the two choices of spin quantization axis.

5.2.2. Deriving the light-front to Breit relation We gave two examples of frames that were simultaneously Breit and LF frames. It turns out that half the work we need to do is very easy in one of these frames, and that visualizing one ensuing equality is quite easy in the other. Clearly, one can write GνBµ′ µ = B ⟨p′ , µ′ |p′ , λ′ ⟩L GνLλ′ λ L ⟨p, λ|p, µ⟩B ,

(77)

so that the problem reduces to finding the overlaps of the LF helicity and ordinary helicity amplitudes. We shall start using the LF frame with the target at rest. For the initial state, being at rest, the LF helicity state is identical to the state with spin quantized in the positive z-direction,

|p, λ⟩L = |rest, λ⟩z .

(78)

The helicity state, however, should be quantized along a direction antiparallel to the momentum of the entering photon; see Fig. 12. The photon four-momentum is given by q = (q+ , q− , q⊥ ) =



0,

Q 2 + M 2 − m2 m

 ,Q,0 ,

(79)

and it makes an angle θ with the z-axis, where tan θ =

2mQ Q2

+ M 2 − m2

,

(80)

equivalent to the half-angle version given earlier, Eq. (71). With θ taken positive, it is also the rotation angle from the Breit frame helicity state to the LF state,

|p, µ⟩L = Ry (θ )|p, µ⟩B ,

(81)

which leads to L

⟨p, λ|p, µ⟩B = djλµ (−θ ) = djµλ (θ ).

(82)

For the outgoing hadron, the helicity state is (see Fig. 12 to get the angle),

|p′ , µ′ ⟩B = Ry (π − θ )e−iK3 ξ |rest, µ′ ⟩z ,

(83)

where K3 is the boost operator for the z-direction and ξ is a rapidity given in terms of the energy E of the outgoing hadron, ′

ξ = arccosh

E′ M

= arccosh

Q 2 + M 2 + m2 2mM

.

(84)

The LF state is given by first boosting to the correct (p′ )+ = m for the outgoing hadron (a boost in the negative z-direction, if Q ̸= 0), followed by a boost to get the correct transverse momentum, which is the same as for the photon. One has

|p′ , λ′ ⟩L = e−iQE1 /m e−iK3 ξ |rest, λ′ ⟩z , ′

(85)

where

ξ ′ = −arccosh

M 2 + m2

2mM and E1 is the LF transverse boost E 1 = K1 + J 2 .

(86)

(87)

26

B.L.G. Bakker, C.-R. Ji / Progress in Particle and Nuclear Physics 74 (2014) 1–34

Thus the overlap is

⟨p′ , µ′ |p′ , λ′ ⟩L = z ⟨rest, µ′ |eiK3 ξ eiJ2 (π−θ) e−iQE1 /m e−iK3 ξ |rest, λ′ ⟩z ′

B

′

≡ z ⟨rest, µ′ |Ry (−θ ′ )|rest, λ′ ⟩z = djµ′ λ′ (−θ ′ ),

(88)

where we know the product of the four operators can only be a rotation because the rest four momentum is undisturbed. Consistent with our previous choice, we define θ ′ as the angle rotating from the rest state connected to the Breit frame helicity state to the corresponding state connected to the LF state. Our method for finding θ ′ is to choose a representation for the operators, namely J2 =

1 2

σ2 ,

K3 =

i 2

σ3 ,

and E1 =

1 2

(iσ1 + σ2 ),

(89)

where the σi are the usual 2 × 2 Pauli matrices, and then to multiply the operators out explicitly. The result is tan θ ′ = −

2MQ Q 2 − M 2 + m2

,

(90)

equivalent to the useful half-angle version given earlier, Eq. (72). Putting the pieces together gives the LF to Breit frame helicity amplitude conversion formula, quoted in Eq. (70). The inverse of this relation follows using j

j

dµλ1 (θ )dµλ (θ ) = δλ1 λ ,

(91)

and is j′

GνLλ′ λ = dλ′ µ′ (−θ ′ ) GνBµ′ µ dµλ (θ ). j

(92)

5.3. Light-front discrete symmetry The discrete symmetries of parity inversion and time reversal are not compatible with the LF requirement that q+ = 0. However, putting all momenta in the x − z plane, we can compound the usual parity and time reversal operators with 180° rotations about the y-axis to produce useful and applicable LF parity and time reversal operators [75]. 5.3.1. Light-front parity Let P be the ordinary unitary parity operator that takes ⃗ x → −⃗ x and t → t. Define the LF parity operator by [75,72] YP = Ry (π )P.

(93)

Since YP commutes with operators E1 and K3 , one has that YP acting on a LF state gives + + YP |p, λ⟩L = YP e−iE1 p⊥ /p e−iK3 ξ |rest, λ⟩z = e−iE1 p⊥ /p e−iK3 ξ YP |rest, λ⟩z .

(94)

Further, j

YP |rest, λ⟩z = ηP Ry (π )|rest, λ⟩z = ηP |rest, λ′ ⟩z dλ′ λ (π ),

(95)

j

where ηP is the intrinsic parity of the state. Then using dλ′ λ (π ) = (−1)j+λ δλ′ ,−λ , one gets for the states YP |p, λ⟩L = ηP (−1)j+λ |p, −λ⟩L .

(96)

+

For the current component J , since YP is unitary, one finds L

⟨p′ , λ′ |J + |p, λ⟩L = L ⟨YP (p′ , λ′ )|YP J + YPĎ YP |p, λ⟩L ′

′

= ηP′ ηP (−1)j −j+λ −λ L ⟨p′ , −λ′ |J + |p, −λ⟩L .

(97)

Hence, the parity relation for LF helicity amplitudes is ′ ′ G+ = ηP′ ηP (−1)j −j+λ −λ G+ . L,−λ′ ,−λ Lλ′ λ

(98)

The parity relation for the usual (Breit frame) helicity amplitudes is known [72], and is usually given in terms of amplitudes with definite photon helicity, which we define in Section 5.4.5. We shall only note that we can derive the relation from the LF result just above, and quote for completeness, ′ λγ −λγ GB,−µ′ ,−µ = ηP′ ηP (−1)j +j GBµ′ µ .

(99)

B.L.G. Bakker, C.-R. Ji / Progress in Particle and Nuclear Physics 74 (2014) 1–34

27

5.3.2. Light-front time reversal Let T be the ordinary time reversal operator which takes t → −t and ⃗ x → ⃗ x and which is antiunitary. By known arguments, time reversal acting on a state at rest reverses the spin projection, and one has

T|rest, λ⟩z = (−1)j−λ |rest, −λ⟩z .

(100)

Recall that this can be proven by starting with T|rest, λ⟩z = ηT (λ)|rest, −λ⟩z , and using that the states with different λ are related by the angular momentum raising and lowering operators J± . One then shows that ηT (λ) changes sign as the spin projection changes by one unit by considering how T commutes with the raising and lowering operators. That only leaves ηT (j) to be fixed. Since T is antiunitary, one can change ηT (j) by changing the phase of the state, and one chooses the phase of the state so that ηT (j) is one. Define a LF time reversal operator by YT = Ry (π )T,

(101)

YT |rest, λ⟩z = |rest, λ⟩z .

(102)

giving

This also works for moving LF states. Since YT is antiunitary, YT iK3 YT−1 = iK3

and YT iE1 YT−1 = iE1 ,

(103)

from which we see, + YT |p, λ⟩L = YT e−iE1 p⊥ /p e−iK3 ξ |rest, λ⟩z = |p, λ⟩L .

(104)

We use time reversal first to show that the LF amplitudes are real for the current component J + , still remembering that YT is antiunitary, L

⟨p′ , λ′ |J + |p, λ⟩L = L ⟨YT (p′ , λ′ )|YT J + YT−1 YT |p, λ⟩∗L = L ⟨p′ , λ′ |J + |p, λ⟩∗L ,

(105)

or, G+ = (G+ )∗ Lλ′ λ Lλ′ λ

(106)

for momenta in the x − z plane. In general, it is not useful to reverse the initial and final states because the particles are different. But for the elastic case we can further use time reversal to relate amplitudes with interchanged helicity. First note that for the LF frame with the target at rest, the initial and final particles have the same p+ , so to get a state with the final momentum requires just the transverse boost,

|p′ , λ⟩L = e−iQE1 /p |p, λ⟩L . +

(107)

Beginning by applying the previous time reversal result to the elastic case, and recalling that E1 commutes with ‘‘+’’ components of four-vectors, leads to G+ = L ⟨p′ , λ′ |J + |p, λ⟩L = L ⟨p, λ|J + |p′ , λ′ ⟩L Lλ′ λ

= L ⟨p, λ|J + e−iQE1 /p |p, λ′ ⟩L = L ⟨p, λ|e−iQE1 /p J + |p, λ′ ⟩L +

+

= L ⟨p, λ|e−iJ3 π e+iQE1 /p eiJ3 π J + |p, λ′ ⟩L +

= (−1)λ −λ L ⟨p′ , λ|J + |p, λ′ ⟩L . ′

(108)

Thus when the incoming and outgoing particles have the same identity, time reversal gives G+ = (−1)λ −λ G+ . Lλ′ λ Lλλ′ ′

(109)

Similarly to the close of the previous subsection, we record the time reversal result for the helicity amplitudes in the Breit frame, for identical incoming and outgoing particles, λγ

λγ

GBµ′ µ = (−1)µ −µ GBµµ′ . ′

(110)

5.3.3. The x-Breit frame Note that θ = −θ ′ for the equal mass case. While some of the transformations are easy in the target rest frame, where we did our calculations, visualizing this result is not. For this purpose, the x-Breit frame, where the incoming and outgoing particles are both along the x-direction, works well.

28

B.L.G. Bakker, C.-R. Ji / Progress in Particle and Nuclear Physics 74 (2014) 1–34

Fig. 13. Momenta and spin directions for LF helicity states in the x-Breit frame. The momenta are in the ±x-direction and the spin directions for the LF states are indicated by the doubled lines.

The momenta are, in (p0 , p1 , p2 , p3 ) notation,



p = ( m2 + Q 2 /4, −Q /2, 0, 0),



p′ = ( m2 + Q 2 /4, Q /2, 0, 0), q = (0, Q , 0, 0),

(111)

and the incoming states are defined by,

|p, λ⟩L = eiE1 Q /2p e−iK3 ξ1 |rest, λ⟩z , +

|p, λ⟩B = Ry (−π/2)e−iK3 ξ |rest, λ⟩z .

(112)

The outgoing states have the same longitudinal boosts, but have opposite transformations for getting the transverse momentum. The boost parameters are not the same. The transformation with ξ gives a momentum along the z-direction with the final energy; the transformation with ξ1 gives a momentum along the z-direction with the final p+ , but with energy m2 +Q 2 /8 and pz different from the final ones. From the kinematics given by Eq. (111), we find ξ1 = arccosh √ m

√ ξ = arccosh

m2 +Q 2 /4

and

m2 +Q 2 /4 . m

Formally, one defines the angle θ from L

⟨p, λ|p, µ⟩B = z ⟨rest, λ|e−iJ2 θ |rest, µ⟩z ,

(113)

with a corresponding equation involving the final states and angle θ ′ . Using the representation given earlier in Eq. (89), we find + e−iσ2 θ/2 = e−σ3 ξ1 /2 e(σ1 −iσ2 )Q /4p eiσ2 π/4 eσ3 ξ /2 .

(114)

Conjugating the above equation with σ3 (i.e., taking σ3 . . . σ3 ) gives + ′ e+iσ2 θ/2 = e−σ3 ξ1 /2 e−(σ1 −iσ2 )Q /4p e−iσ2 π/4 eσ3 ξ /2 = e−iσ2 θ /2 ,

(115)

and θ = −θ ′ . Pictorially, we draw the momenta in Fig. 13, and for the helicity states the particle spins point along the direction of the momenta. The incoming and outgoing light front states both start with a boost in the z-direction, and then receive symmetrically opposite transverse boosts which rotate the spin vectors in opposite directions by the same amount. The angles θ and θ ′ are indicated in the figure. One can see both that the size of the angles should be the same and that the senses should be opposite. 5.4. Consequences 5.4.1. Light-front parity and the angular condition The general angular condition for current component J + reads j′

j

dλ′ µ′ (−θ ′ ) G+ d (θ ) = 0 Lλ′ λ µλ

for |µ′ + µ| ≥ 2.

(116)

Say that µ + µ ≥ 2. By changing the sign of both µ and µ it looks like we could get another angular condition, ′

j′

′

j

dλ′ ,−µ′ (−θ ′ ) G+ d (θ ) = 0. Lλ′ λ −µ,λ However, using the first of the identities j

′ j

(117)

j

′ j

dm′ m (θ ) = (−1)m−m d−m′ ,−m (θ ) = d−m,−m′ (θ ) = (−1)m−m dmm′ (θ ) (118) ′ and the LF parity relation, Eq. (98), one can show by a series of reversible steps that each angular condition with µ+µ ≤ −2 is equivalent to one with µ + µ′ ≥ 2. Hence, we only need to consider cases where µ + µ′ ≥ 2.

B.L.G. Bakker, C.-R. Ji / Progress in Particle and Nuclear Physics 74 (2014) 1–34

29

5.4.2. The angular condition for deuterons We shall implement the general angular condition in a couple of special cases, rewriting the angular dependence in terms of Q 2 and masses. For the deuteron, the angular condition comes only from µ = µ′ = 1 and we have d1 (θ ) = 0. d1λ′ 1 (−θ ′ )G+ Lλ′ λ 1λ

(119)

For the equal mass case, the arguments of the d-functions are the same, 2md

tan θ = − tan θ ′ =

,

Q (md is the deuteron mass). Using LF parity, Eq. (98), and the d-function identities, Eq. (118), one gets G+ L++



d111

2

 2   +  1  1   1 2 + + 1 1 1 + d11,−1 − GL+ − G+ = 0. L+0 d10 d11 − d1,−1 + GL+− 2d11 d1,−1 − GL d10

(120)

(121)

+ Substituting for the d1 s and tan θ , and using the LF time reversal result G+ L+0 = −GL0+ , leads to the angular condition in its known form [76,77],

(2η + 1)G+ L++ +

+ + 8ηG+ L+ + GL+− − GL = 0,



(122)

For the record, we have removed an overall factor, 1/2(1 + η). where η = Q / We [78] obtained two constraints on the deuteron helicity amplitudes by noting that there were five amplitudes, and that all five could be derived from three independent form factors. Both constraints we called angular conditions. They appeared differently in different frames; our Drell–Yan–West frame results can be most directly compared to the present results. The + constraint we call ‘‘AC1’’ is, for momenta in the xz- plane, just G+ L0+ + GL+0 = 0. In the present review this follows from LF time reversal invariance. Our constraint ‘‘AC2’’ is then precisely the same as the angular condition here. 2

4m2d .

5.4.3. A consequence of the angular condition for deuterons The pQCD predicts, as we shall review below, that the hadron helicity conserving amplitude G+ L00 is the leading amplitude at high Q and that G+ L +0 =

aΛQCD + GL00 , Q

G+ L+− =



bΛQCD Q

2

G+ L00

(123)

to leading order in 1/Q . No statement is initially made about the size of a and b. One may go further, following Chung et al. [79] or Brodsky and Hiller [80] (who interestingly mention the work of Carlson and Gross [81] in this regard), to argue that the scale of QCD is given by ΛQCD and that we can implement this in the LF frame by saying that a, b = O (1).

(124)

A consequence of this, written in terms of the deuteron charge, magnetic and quadrupole form factors [82], is that to good approximation one gets the ‘‘universal ratios’’ [80],

 GC : GQ : GM =

2 3

 η − 1 : 1 : −2.

(125)

This agrees with the leading power of Q 2 result [81] that GC = (2/3)ηGQ , but goes beyond it and also gives a prediction for GM . We have so far in this subsection used only three LF helicity amplitudes. There are more that are not zero, and we find a difficulty when we discuss a fourth. Amplitude G+ L++ is related to the others by the angular condition quoted above. Also, the 2 perturbative QCD arguments that give the scaling behavior of the other helicity amplitudes give for G+ L++ at very high Q , G+ L++ =



c ΛQCD

2

Q

G+ L00 .

(126)

(Helicity is conserved, but other spin dependent rules [81] dictate an asymptotic suppression of G+ L++ by two powers of Q . This is also consistent with a naturalness condition discussed in Ref. [78].) The angular condition to leading order now reads, 1+

√ aΛQCD 2

md

−

1 2



c ΛQCD md

2

= 0.

(127)

The hypothesis that ΛQCD sets the scale of the subleading amplitudes would suggest that c as well as a is of O (1). Given the angular condition result just above, this cannot be right; at least one of a and c must be O(md /ΛQCD ) ≈ 20. Hence the hypothesis is not generally workable, and one needs to consider thinking the same about the next-to-leading corrections in the ‘‘universal ratios’’ expression, Eq. (125). This indicates that the leading pQCD prediction of the deuteron exclusive amplitudes may be postponed to a larger Q 2 region and the 12 GeV upgrade in JLab is anticipated to shed some light on the predictability of pQCD.

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B.L.G. Bakker, C.-R. Ji / Progress in Particle and Nuclear Physics 74 (2014) 1–34

5.4.4. The angular condition for N − ∆ transitions The γ ∗ N → ∆(1232) transition is an important reaction that involves final and initial states with different spins and masses. This makes working out the angular condition more involved technically, but not unduly so, as we shall demonstrate. There is one angular condition, 1/2

3/2

d 0 = dλ′ ,3/2 (−θ ′ ) G+ (θ ) Lλ′ λ 1/2,λ

 3/2 1/2 3/2 1/2 = G+ L,−3/2,1/2 −d3/2,3/2 d1/2,−1/2 + d−3/2,3/2 d1/2,1/2   3/2 1/2 3/2 1/2 + G+ d d + d d L,−1/2,1/2 1/2,3/2 1/2,−1/2 −1/2,3/2 1/2,1/2   3/2 1/2 3/2 1/2 + + GL,1/2,1/2 −d−1/2,3/2 d1/2,−1/2 + d1/2,3/2 d1/2,1/2   3/2 1/2 3/2 1/2 + G+ d d + d d L,3/2,1/2 −3/2,3/2 1/2,−1/2 3/2,3/2 1/2,1/2 , 

(128)

using the LF parity. Explicit substitution for the d-functions yields 2

0 = − cos

θ′ 2

θ′

sin

2

√ −

3 G+ L,−1/2,1/2

− G+ L,3/2,1/2



cos

θ

 G+ L,−3/2,1/2

2

 tan

θ 2

+ tan

θ

 − tan

2

cot

√

′

+

2

θ

θ′ 2

3G+ L,1/2,1/2

+ tan

2

θ′



2

 1 − tan

θ 2

tan

θ′



2

 ′ θ′ θ 2 θ cot + tan tan . 2

2

(129)

2

Finally, removing the overall factors and substituting for the trigonometric functions gives the angular condition for the N − ∆ transition, 0 = (M − m)(M 2 − m2 ) + mQ 2 G+ L,−3/2,1/2 +





√

+

√

3MQ (M − m)G+ L,−1/2,1/2

+ 2 3MQ 2 G+ L,1/2,1/2 + Q Q − m(M − m) GL,3/2,1/2 ,





(130)

where m is the nucleon mass and M is the ∆ mass. For the record, we have removed another overall factor, Q+ (Q+ − Q− )/(2mM 2 Q 2 ). + 4 The asymptotic scaling rules, cited in the next subsection, say that G+ L,1/2,1/2 goes like 1/Q at high Q , that GL,3/2,1/2 and + 5 6 G+ L,−1/2,1/2 go like 1/Q , and that GL,−3/2,1/2 goes like 1/Q . If we write

G+ L,3/2,1/2 =

bΛQCD + GL,1/2,1/2 Q

(131)

modulo logarithms at high Q , then the leading Q part of the angular condition says

√ 3+

bΛQCD M

= 0.

(132)

√ Since the value of b must be O( 3M /ΛQCD ), where M is the ∆ mass, the fate of pQCD predictions in the N − ∆ transition may be similar to the case of deuteron discussed previously. Again, the 12 GeV upgrade in JLab is anticipated to reveal the predictability of pQCD. 5.4.5. Equivalence of leading powers in Breit and Light-front frames The idea of ‘‘good currents’’ and ‘‘bad currents’’ is native to the LF frame. In analyzing the power-law scaling behavior at high Q 2 , for a given helicity amplitude, it is often thought to be safest to stay in the LF frame and use only ‘‘good currents’’. We shall here derive the Breit frame helicity amplitude scaling behaviors from their LF counterparts. Note here that q+ = 0 both in the LF frame and the Breit frame that we discuss in this subsection. All the q+ = 0 frames are related to each other only by the kinematical operators that keep the LF time τ intact. We will find, nicely enough, that the scaling behaviors are the same as one would have found using the Breit frame only. That is, one can get the correct leading power scaling behavior from a Breit frame analysis alone. For a LF helicity amplitude, the scaling behavior at high Q is [83] G+ ∝ Lλ′ λ

 2(n−1)+|λ′ −λmin |+|λ−λmin | m Q

,

(133)

B.L.G. Bakker, C.-R. Ji / Progress in Particle and Nuclear Physics 74 (2014) 1–34

31

where n is the number of quarks in the state, m is a mass scale, and λmin is the minimum helicity of the incoming or outgoing state (i.e., 0 or 1/2 for bosons or fermions, respectively). Regarding the Breit frame, we have thus far given its helicity amplitudes in terms of GνB where ν is a Lorentz index. It is usual to substitute a photon helicity index for the Lorentz index, using (for incoming photons) λγ

GBµ′ µ = ϵν (q, λγ ) GνBµ′ µ ,

(134)

with polarizations (in (t , x, y, z )-type notation)

√ ϵ± = ϵ(q, λγ = ±) = (0, ± cos θ , −i, ± sin θ )/ 2 ϵ0 = ϵ(q, λγ = 0) = (csc θ , cos θ , 0, − cos2 θ csc θ ),

(135)

where θ is the angle between q and the negative z-direction, as shown in Fig. 12. One can work out that

  1 η ≡ (1, 0, 0, −1) = sin θ ϵ0 − √ (ϵ+ − ϵ− ) ,

(136)

2

so that Gν=+ B..

= sin θ



G0B..



 1  − − √ G+ , B.. − GB.. 2

(137)

where the superscripts on GB will in the rest of this subsection refer to photon helicity unless explicitly stated otherwise. With the general relation, Eq. (70), this gives directly the expression we use to obtain the scaling behavior of the Breit amplitudes,

 ′ 1  + j G0Bµ′ µ − √ GBµ′ µ − G− = csc θ djλ′ µ′ (−θ ′ )G+ d (θ ). ′ Bµ µ Lλ′ λ µλ 2

(138)

We can select terms on the left-hand-side by choice of µ′ and µ, since Breit amplitudes are non-zero only for λγ = µ′ + µ. The d-functions can be written in terms of sines and cosines of half angles, so we record that at high Q , sin

sin

θ

=

2

θ′ 2

 3

m

+O

Q

=−

m

M Q

and

Q

cos

θ 2

 3 m

+O

and

Q

cos

 2 =1+O θ′

m Q

 2 =1+O

2

m Q

,

(139)

where m inside the O symbol is a generic mass scale. Eq. (139) can be derived from Eqs. (71) and (72), or equivalently from Eqs. (80) and (90). The d-functions can be expanded as [84] j

dµλ (θ ) = a1

 cos

θ

2j−|µ−λ|  sin

2

θ

|µ−λ|

2

    θ 2j−|µ−λ|−2 θ |µ−λ|+2 + a2 cos sin + ···, 2

2

(140)

where a1 , a2 , . . . are numerical coefficients with a1 ̸= 0. Thus for large Q , j

dµλ (θ ) ∝

 |µ−λ| m

 |µ−λ|+2 m

+O

Q

Q

.

(141)

On the right-hand-side of Eq. (138), there is no term that falls slower than the term that has λ′ = λ = λmin , and 2(n−1) G+ . Thus, the Breit amplitude’s leading fall off at high Q is L,λmin ,λmin ∝ (m/Q ) λγ GBµ′ µ

 −1+2(n−1)+|µ′ −λmin |+|µ−λmin | ∝

m Q

.

(142)

These are the same results one can get by directly analyzing amplitudes in the Breit frame for various photon helicities [85]. By way of example, we will give the Breit and LF frame helicity amplitudes for elastic electron–nucleon scattering. In terms of the standard Dirac, Pauli, and Sachs form factors, one may work out G+ L++ G+ L−+

 ≡L  ≡L





1  1 p ,  J + p, 2 2 ′







= 2p+ F1 (q2 ), L

1  1 p , −  J + p, 2 2 ′



= 2p+ L

Q 2m

F2 (q2 ),

(143)

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for the LF states and

   √  1   = 2QGM (q2 ), ϵ · J p , B +   2 2 B      1 1 ′   ≡ B p , −  ϵ0 · J p, = 2mGE (q2 ),

G+ B++ ≡ G0B−+



p′ ,

1

2

2

(144)

B

for the Breit frame states. The scaling rules predict that GM , GE , and F1 scale as 1/Q 4 and that F2 scales as 1/Q 6 , consistent with the relations GM = F1 + F2 ,

GE = F1 +

q2 4m2

F2 .

(145)

(There are data [86] and commentary [87] on the helicity flip scaling results.) 5.5. Summary The purpose of the present Section has been largely kinematical. We have examined the relationship between the helicity amplitudes in the Breit and LF frames. One particular result has been a clear view of where the angular condition comes from. The angular condition is a constraint on LF helicity amplitudes. It follows from applying angular momentum conservation in the Breit frame, where the application of angular momentum conservation to the helicity amplitudes is elementary. One consequence is that an amplitude must be zero if it requires the photon to have more than one unit magnitude of helicity, and this statement cast in terms of LF amplitudes is the angular condition [74,76]. Another set of constraints follows from parity and time-reversal invariance. Neither of these symmetries can be used directly on the LF because the LF has a preferred spatial direction. However, each of them can be modified to give a valid symmetry operation (at least for strong and electromagnetic interactions) for the LF [75]. The general angular condition appears compactly in terms of d-functions, the representations of the rotation operators. It can be rewritten in terms of masses and momentum transfer and the result is model-independent. We discussed its consequence in two exclusive processes, ed → ed and eN → e∆. The power-law scaling of the helicity amplitudes can be analyzed, with a definite power of 1/Q given in terms of the number of constituents in the wave function and in terms of the helicities of the incoming and outgoing states. In both cases, the angular condition reveals that the mass scale associated with the asymptotic power-law fall off of non-leading amplitudes must generally be of the order of the hadron masses such as the deuteron mass and the ∆ mass which are typically much larger than the QCD scale parameter ΛQCD . It had been hoped that the LF was a favored frame where the non-leading amplitudes would have small numerical coefficients: asymptotically of order ΛQCD /Q , to an appropriate power, times the leading amplitude, consistent with the leading pQCD prediction. As the angular condition contradicts this, it appears that the leading pQCD prediction on both processes may be postponed to a larger Q 2 region than one might have hoped. In this respect, the 12 GeV upgrade in JLab is a timely progress to examine stringently the predictability of pQCD. 6. Conclusion In this review, we have indicated several phenomena that go beyond what can be described in terms of the baryon–meson picture of nuclear physics. We have stressed that if one accounts for the substructure of hadrons in terms of quarks and gluons, new phenomena must occur. An elaboration of this idea was given in Section 2, where the link between the traditional nuclear physics and the quark–gluon picture was determined and denoted by the name of ‘‘reduced nuclear amplitudes’’. In Section 3, we discussed some ideas concerning a modification of the nucleon–meson picture of the nuclear forces. The occurrence of colored multi-quark substructures in nuclei, was found to have consequences for the understanding of the nucleon–nucleon potential at short distances, which means distances of the same order as the size of the nucleons themselves. Two main points were found, (i) the composite nature of the nucleons, which naturally leads to the idea of strong form factors for the meson–nucleon vertices, can be used to estimate these form factors in the quark–gluon picture, and (ii) the occurrence of ‘‘exotic’’ channels, say the colorless combination of two colored clusters, CC , leads in a natural way to an energy-dependent effective, ‘‘optical’’, potential which incorporates the short-range effective repulsion that is responsible for the saturation of the nuclear force as an effect of the leaking of the nucleon–nucleon channel into the CC channel. Such multi-quark states must have their own evolution, which was discussed in Section 4. Combined with the idea of reduced nuclear amplitudes, this work leads to predictions of the asymptotic behavior of nuclear observables like electromagnetic form factors, which can be and have been tested in experiments. The deuteron tensor structure function b1 could be sensitive to hidden color degrees of freedom at large x. The order of magnitude increase in the nine-quark hidden color degrees of freedom may be behind the significant enhancement of the A/3 He ratio as Q 2 gets larger. The idea that scattering amplitudes for particles with spin can depend on a fixed number of independent scalar amplitudes gives rise to the concept of angular conditions (ACs), the subject of Section 5. The number of angular conditions

B.L.G. Bakker, C.-R. Ji / Progress in Particle and Nuclear Physics 74 (2014) 1–34

33

follows from the number of independent scalars and the number of spin- or helicity-amplitudes. The ACs are of course framedependent, because the amplitudes are. This means that the information that can be extracted from these conditions does depend on the frame in which they are evaluated. The coefficients involved in the ACs depend on the kinematics and thus their dependence on a hard scale, say Q , may be studied. This leads to model-independent estimates of the Q -dependence of the scalar functions, which also can be tested in experiments. Acknowledgments We are indebted to a number of our collaborators in completing this review. In particular, we thank Stan Brodsky, John Hiller, Jerry Miller, Hans Weber and Carl Carlson for their collaborations on the works presented in this review. This work was supported partially through GAUSTEQ (Germany and US Nuclear Theory Exchange Program for QCD Studies of Hadrons and Nuclei) under contract number DE-SC0006758 and the U.S. Department of energy (DE-FG02-03ER41260). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]

[36]

[37]

[38] [39] [40] [41] [42]

K. Allada, et al., A Proposal to Jefferson Lab PAC-40. Update to PR12-11-110. N. Fomin, et al., Phys. Rev. Lett. 108 (2012) 092502. S.J. Brodsky, C.-R. Ji, Lecture Notes in Phys. 248 (1986) 153; Comments Nucl. Part. Phys. 13 (1984) 213. S.J. Brodsky, J.R. Hiller, Phys. Rev. C 28 (1983) 475. 30 (1984) 412E. S.J. Brodsky, J.R. Hiller, C.-R. Ji, G.A. Miller, Phys. Rev. C 64 (2001) 055204. hep-ph/0105259. D.G. Meekins, et al., Phys. Rev. C 60 (1999) 052201. A. Imanishi, et al., Phys. Rev. Lett. 54 (1985) 2497. M.A. Shupe, et al., Phys. Rev. Lett. 40 (1978) 271. G.C. Bolon, et al., Phys. Rev. Lett. 18 (1967) 926. R.L. Anderson, et al., Phys. Rev. Lett. 30 (1973) 627. R.L. Anderson, et al., Phys. Rev. Lett. 26 (1971) 30. S.J. Brodsky, H.C. Pauli, S.S. Pinsky, Phys. Rep. 301 (1998) 299. C.-R. Ji, S.J. Brodsky, Phys. Rev. D 34 (1986) 1460. S.D. Drell, T.M. Yan, Phys. Rev. Lett. 24 (1970) 181. G. West, Phys. Rev. Lett. 24 (1970) 855. S.J. Brodsky, C.-R. Ji, G.P. Lepage, Phys. Rev. Lett. 51 (1983) 83. L.C. Alexa, et al., Phys. Rev. Lett. 82 (1999) 1374. S.J. Brodsky, G.R. Farrar, Phys. Rev. Lett. 31 (1973) 1153; Phys. Rev. D 11 (1975) 1309; V.A. Matveev, R.M. Muradyan, A.V. Tavkhelidze, Lett. Nuovo Cimento 7 (1973) 719. J. Napolitano, et al., Phys. Rev. Lett. 61 (1988) 2530; S.J. Freedman, et al., Phys. Rev. C 48 (1993) 1864. J.E. Belz, et al., Phys. Rev. Lett. 74 (1995) 646. C.W. Bochna, et al., Phys. Rev. Lett. 81 (1998) 4576. J.M. Namyslowski, IFT/80/3-WARSAW Invited Contribution to 9th Int. Conf. on Few Body Problems, Eugene, Oregon, August 17–23, 1980. S.J. Brodsky, C.-R. Ji, Phys. Rev. D 33 (1986) 2653. I.F. Ginzburg, D.Yu. Ivanov, V.G. Serbo, Phys. Atomic Nuclei 56 (1993) 1474; Yad. Fiz. 56 (1993) 45; E.R. Berger, O. Nachtmann, Nucl. Phys. Proc. Suppl. 79 (1999) 352. S. Mandelstam, Phys. Rev. 112 (1958) 1344. S.J. Brodsky, B.T. Chertok, Phys. Rev. Lett. 37 (1976) 269; Phys. Rev. D 14 (1976) 3003. R.G. Arnold, et al., Phys. Rev. Lett. 35 (1975) 776. C.E. Carlson, A.B. Wakely, Phys. Rev. D 48 (1993) 2000. S.A. Moszkowski, in: S. Flügge (Ed.), Enc. of Physics, Vol. XXXIX, Springer-Verlag, Berlin, 1975, p. 411. K. Erkelenz, Phys. Rep. 13 (1974) 191; K. Holinde, Phys. Rep. 68 (1981) 121. J.E.T. Ribeiro, Z. Phys. C 5 (1980) 27. M. Oka, K. Yazaki, Phys. Lett. B 90 (1980) 41. A. Faessler, F. Fernandez, G. Lubeck, K. Shimizu, Phys. Lett. B 112 (1982) 201. N. Ishii, S. Aoki, T. Hatsuda, Phys. Rev. Lett. 99 (2007) 022001. H.J. Weber, Z. Phys. A 297 (1980) 261; H.J. Weber, J.N. Maslow, Z. Phys. A 297 (1980) 271; H.J. Weber, Z. Phys. A 301 (1981) 261; B.L.G. Bakker, J.N. Maslow, H.J. Weber, Phys. Lett. B 104 (1981) 349; B.L.G. Bakker, M. Bozoian, J.N. Maslow, H.J. Weber, Phys. Rev. C 25 (1982) 1134. R.L. Jaffe, F.E. Low, Phys. Rev. D 19 (1979) 2105; R.L. Jaffe, in: W. Pfeil, Proceedings of the 1981 International Symposium on Lepton and Photon Interactions at High Energies, eConf C810824(1981)395; F.E. Low, in: A. Zichichi (Ed.), Pointlike Structure inside and Outside Hadrons, Proceedings of the 17th International School of Subnuclear Physics, Erice, 1979, Plenum, New york, 1979, p. 155; R.L. Jaffe, in: A.H. Guth, K. Huang, R.L. Jaffe (Eds.), Asymptotic Realms in Physics: A Festschrift for Francis Low, MIT Press, Cambridge, 1983. Yu.A. Simonov, Phys. Lett. B 107 (1981) 1; Yu.A. Simonov, Sov. J. Nucl. Phys. 7 (1981) 722; Yu.S. Kalashnikova, et al., Phys. Lett. B 155 (1985) 217. M. Harvey, Nuclear Phys. A 352 (1981) 301; M. Harvey, Nuclear Phys. A 481 (1988) 834. Erratum. M. Harvey, Nuclear Phys. A 424 (1984) 428. M. Harvey, Nuclear Phys. A 352 (1981) 326. T. deGrand, R. Jaffe, K. Johnson, J. Kiskis, Phys. Rev. D 12 (1975) 2060. B.L.G. Bakker, P.J. Mulders, Adv. Nucl. Phys. 17 (1986) 1.

34

B.L.G. Bakker, C.-R. Ji / Progress in Particle and Nuclear Physics 74 (2014) 1–34

[43] I.L. Grach, et al., Z. Phys. C 21 (1984) 229; I.M. Narodetskii, in: L.D. Faddev, T.I. Kopaleishvili (Eds.), Few-Body Problems in Physics, World Scientific, Singapore, 1985, p. 55; I.M. Narodetskii, in: Gy. Bencze, P. Doleschall, and J. Revai (Eds.), Proc. Xth Eur. Symp. on the Dynamics of Few-Body Systems (Balatonfüred, 1985), p. 23; B.L.G. Bakker, in: Gy. Bencze, P. Doleschall, and J. Revai (Eds.), Proc. Xth Eur. Symp. on the Dynamics of Few-Body Systems (Balatonfüred, 1985), p. 37; Yu.S. Kalashnikova, et al., Z. Phys. A 323 (1986) 205; A.I. Veselov, et al., Sov. J. Nucl. Phys. 44 (1985) 14; B.L.G. Bakker, in: C. Ciofi degli Atti, O. Benhar, G. Salmè (Eds.), Proc. of the European Workshop on Few-Body Physics, Springer-Verlag, Wien, 1986; I.L. Grach, et al., Z. Phys. C 38 (1988) 427. [44] K. Wildermuth, Y.C. Tang, A Unified Theory of the Nucleus, Vieweg, Braunschweig, 1977. [45] H. Dijk, B.L.G. Bakker, Nuclear Phys. A 494 (1989) 438. [46] Yu.A. Simononv, Sov. J. Nucl. Phys. 36 (1982) 422. [47] H. Feshbach, Ann. Phys., NY 5 (1958) 357. [48] R.A. Arndt, et al., Phys. Rev. D 35 (1987) 128. [49] P.J. Mulders, A.T. Aerts, J.J. de Swart, Phys. Rev. D 21 (1980) 2653. [50] Yu.S. Kalashnikova, I.M. Narodetskii, Yu.A. Simonov, Sov. J. Nucl. Phys. 46 (1987) 68; C.W. Wong, Prog. Part. Nucl. Phys. 8 (1982) 223; M. Oka, K. Yazaki, in: W. Weise (Ed.), Quarks in Nuclei, World Scientific, Singapore, 1984, p. 489. [51] J.-M. Richard, EPJ Web Conf. 3 (2010) 01015. [52] B. Metsch, Eur. Phys. J. A 35 (2008) 275. [53] Y. Fujiwara, Y. Suzuki, C. Nakamoto, Prog. Part. Nucl. Phys. 58 (2007) 439. [54] G.P. Lepage, S.J. Brodsky, Phys. Rev. D 22 (1980) 2157. [55] M. Peskin, Phys. Lett. B 88 (1979) 128; S.J. Brodsky, Y. Frishman, G.P. Lepage, C. Sachrajda, Phys. Lett. B 91 (1980) 239; A.V. Efremov, A.V. Radyushkin, Phys. Lett. B 94 (1980) 245. [56] S.J. Brodsky, C.-R. Ji, Phys. Rev. D 33 (1986) 1951. [57] S.J. Brodsky, C.-R. Ji, Phys. Rev. D 33 (1986) 1406. [58] B. Judek, Can. J. Phys. 46 (1968) 343. 50 (1972) 2082; E.M. Friedlander, et al., Phys. Rev. Lett. 45 (1980) 1084. [59] R. Jaffe, Nature 296 (1982) 305; E. Gabathuler, Prog. Part. Nucl. Phys. 13 (1985) 329. [60] F.E. Close, S. Kumano, Phys. Rev. D 42 (1990) 2377. [61] C.E. Carlson, C.-R. Ji, Phys. Rev. D 67 (2003) 116002. [62] P.A.M. Dirac, Rev. Modern Phys. 21 (1949) 392. [63] P.J. Steinhardt, Ann. Phys. 128 (1980) 425. [64] S. Fubini, G. Furlan, Physics 1 (1965) 229; S. Weinberg, Phys. Rev. 150 (1966) 1313; J. Jersak, J. Stern, Nuclear Phys. B 7 (1968) 413; H. Leutwyler, in: G. Höhler (Ed.), Springer Tracts in Modern Physics, Vol. 50, Berlin, 1969. [65] J.D. Bjorken, Phys. Rev. 179 (1969) 1547; S.D. Drell, D. Levy, T.M. Yan, Phys. Rev. 187 (1969) 2159; S.D. Drell, D. Levy, T.M. Yan, Phys. Rev. D 1 (1970) 1035. [66] W. Jaus, Phys. Rev. D 44 (1991) 2851. [67] H.-M. Choi, C.-R. Ji, Phys. Rev. D 59 (1999) 074015; Phys. Rev. D 56 (1997) 6010. [68] H.-M. Choi, C.-R. Ji, Phys. Lett. B 460 (1999) 461; Phys. Rev. D 59 (1999) 034001. [69] S.D. Drell, T.M. Yan, Phys. Rev. Lett. 24 (1970) 181; G. West, Phys. Rev. Lett. 24 (1970). [70] H.-M. Choi, C.-R. Ji, Phys. Rev. D 58 (1998) 071901. [71] C.-R. Ji, C. Mitchell, Phys. Rev. D 62 (2000) 085020. [72] M. Jacob, G.C. Wick, Ann. Physics 7 (1959) 404; Ann. Physics 281 (2000) 774. [73] G.C. Wick, Ann. Physics 18 (1962) 65. [74] S. Capstick, B.D. Keister, Phys. Rev. D 51 (1995) 3598. arXiv:nucl-th/9411016. [75] D.E. Soper, Phys. Rev. D 5 (1972) 1956. [76] H. Osborn, Nuclear Phys. B 38 (1972) 429; H. Leutwyler, J. Stern, Ann. Physics 112 (1978) 94; I.L. Grach, L.A. Kondratyuk, Sov. J. Nucl. Phys. 39 (1984) 198; Yad. Fiz. 39 (1984) 316; B.D. Keister, Phys. Rev. D 49 (1994) 1500. arXiv:hep-ph/9303264. [77] C.E. Carlson, J.R. Hiller, R.J. Holt, Annu. Rev. Nucl. Part. Sci. 47 (1997) 395. [78] B.L.G. Bakker, C.R. Ji, Phys. Rev. D 65 (2002) 073002. arXiv:hep-ph/0109005. [79] P.L. Chung, W.N. Polyzou, F. Coester, B.D. Keister, Phys. Rev. C 37 (1988) 2000. [80] S.J. Brodsky, J.R. Hiller, Phys. Rev. D 46 (1992) 2141. [81] C.E. Carlson, F. Gross, Phys. Rev. Lett. 53 (1984) 127. [82] See for example R.G. Arnold, C.E. Carlson, F. Gross, Phys. Rev. C 21 (1980) 1426. [83] J.F. Gunion, P. Nason, R. Blankenbecler, Phys. Rev. D 29 (1984) 2491; C.E. Carlson, Int. J. Mod. Phys. E 1 (1992) 525. [84] A.R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1957. [85] The Breit frame result can be gotten using techniques shown in [81]. [86] O. Gayou, et al., (Jefferson Lab Hall A Collaboration), Phys. Rev. Lett. 88 (2002) 092301. arXiv:nucl-ex/0111010; M.K. Jones, et al., (Jefferson Lab Hall A Collaboration), Phys. Rev. Lett. 84 (2000) 1398. arXiv:nucl-ex/9910005. [87] J.P. Ralston, P. Jain, arXiv:hep-ph/0207129; J.P. Ralston, P. Jain, R.V. Buniy, AIP Conf. Proc. 549 (2000) 302. arXiv:hep-ph/0206074; G.A. Miller, M.R. Frank, Phys. Rev. C 65 (2002) 065205. arXiv:nucl-th/0201021; S.J. Brodsky, Talk at Workshop on Exclusive Processes at High Momentum Transfer, Newport News, Virginia, 15–18 May 2002. arXiv:hep-ph/0208158; A.V. Belitsky, X. Ji, F. Yuan, Phys. Rev. Lett. 91 (2003) 092003; X. Ji, J.-P. Ma, F. Yuan, Phys. Rev. Lett. 90 (2003) 241601.