Generalized Parton Distributions and deep virtual Compton scattering

Progress in Particle and Nuclear Physics 61 (2008) 89–105 www.elsevier.com/locate/ppnp

Review

Generalized Parton Distributions and deep virtual Compton scattering M. Guidal ∗ Institut de Physique Nucl´eaire d’Orsay, 91405 Orsay, France

Abstract We give a pedagogical introduction to the field of Generalized Parton Distributions and review shortly the experimental situation and perspective for Deep Virtual Compton Scattering. c 2007 Elsevier B.V. All rights reserved.

Keywords: Nucleon structure; Parton Distributions; Compton scattering

1. Introduction Forty years after the discovery of quarks in the nucleon, the precise way they compose the nucleon remains a largely unveiled mystery. Why do they remain confined within a 1 fermi cube or so volume? How are they (spatially, momentum, . . . ) distributed in this volume? How do they contribute to the global properties (charge, magnetism, spin, . . . ) of the nucleon? The purpose of this short write-up is to review some aspects of the physics of the Generalized Parton Distributions (GPDs) which, to this day, might actually provide the most complete information accessible on the structure of the nucleon. As a few examples, the GPDs describe the (correlated) spatial and momentum distributions of the quarks in the nucleon (including the polarization aspects), its quark-antiquark content, a way to access the orbital momentum of the quarks, etc. . . . In the next section, the general formalism of the GPDs will be introduced. It will be done so in a relatively intuitive and pedagogical way, in the spirit of a school, in order to enlight and stress out the logic and the main reasoning that lie behind the formalism which can be rather heavy at some instances. Excellent, quasi-exhaustive (and lengthy) reviews already exist on the GPD formalism (see Refs. [1–3], for example) and we refer the reader to them for more details. In the subsequent section, we will review the experimental situation concerning the DVCS (“Deep Virtual Compton Scattering”) process which is the “golden” channel to access GPDs and we will outline the rich perspectives which lie ahead of us. ∗ Tel.: +33 01 69 15 73 21; fax: +33 01 69 15 64 70.

E-mail address: [email protected]. c 2007 Elsevier B.V. All rights reserved. 0146-6410/$ - see front matter doi:10.1016/j.ppnp.2007.12.022

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Fig. 1. Deep Inelastic Scattering: at high virtuality of the photon exchanged between the electron and the nucleon, the photon interacts with a single quark of the nucleon. The hit quark hadronizes and escapes the nucleon leaving the final hadronic state undetermined. Only the scattered electron is detected in a DIS experiment.

2. Generalized Parton Distributions Most of what we know on the structure of the nucleon has come from the scattering of high energy leptons. This is due to their structureless nature, their well-known and quantified electromagnetic interaction with matter and their immunity to the strong interaction. This scattering can be inclusive, i.e. only the scattered lepton is detected and the target nucleon (or what remains of it) is unobserved (l N → l X in a concise notation), or exclusive in which case the final hadronic state of the reaction is fully determined (the elastic process l N → l N is the simplest one). In the following, we will, in relatively simple terms, show how these experiments can be interpreted theoretically in terms of the relativistic quantum field theory of QCD (“Quantum Chromo-Dynamics”) and how the recent concept of the Generalized Parton Distributions emerged. 2.1. Inclusive scattering, Parton Distributions and non-local forward matrix elements The first electron inclusive scattering experiments on the nucleon, eN → eX , generically called “Deep Inelastic Scattering” (DIS), have started in the late 60s at the Stanford Linear Accelerator Center (SLAC) laboratory. Let us define the Lorentz invariant variable Q 2 as the squared four-momentum transferred to the nucleon by the electron beam (Q 2 = −(e − e0 )2 ), where e and e0 are, respectively, the incident and scattered electron 4-momenta. Then, at high Q 2 , the process can be depicted by Fig. 1, where, the electron is seen to interact with a single quark of the nucleon via the exchange of a virtual photon. Behind this picture is a key concept of QCD: the notion of “factorization” where one can separate a simple “hard”, pointlike, perturbatively calculable subprocess from the “soft” full non-perturbative complexity of QCD which is then absorbed or parametrized in terms of generic structure functions. In DIS, Lorentz and gauge invariance arguments lead to four structure functions f 1,2 and g1,2 . At leading twist,1 there are only two independent functions, f 1 and g1 which, since one has an elastic scattering between two pointlike objects (the electron and the quark), depend, at leading order, on only one variable x, the fraction of the nucleon’s momentum carried by the struck quark. f 1 (x) and g1 (x) are related, respectively, to the unpolarized and polarized x-momentum distributions of the partons q(x) and ∆q(x) (called, respectively, the unpolarized and polarized Parton Distributions Functions – PDFs –): 1 The formal definition of twist is dimension−spin. For instance, a ψγ ¯ µ ψ operator, that we will often meet in the following, where ψ is a spinor field and γ the standard Dirac matrix, has twist-2: since a spinor has dimension 32 , the operator has dimension 3 and, due to the γ µ matrix which 1 expansion. carries one Lorentz index µ, it is a spin 1 operator (i.e. vector). In a general fashion, twist allows to classify the order of terms in a Q

1. In other words, higher twists with respect to the leading twists are suppressed by Q

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Fig. 2. The optical theorem: the cross-section of the DIS process is equal to the imaginary part of the forward amplitude of the (doubly virtual) Compton process on a quark.

Fig. 3. Illustration of the non-local forward matrix element h p|ψ¯ q (0)Oψq (y)| pi| y + =Ey =0 . ⊥

f 1 (x) =

1X 2 e q(x) 2 q q

g1 (x) =

1X 2 e ∆q(x), 2 q q

(1)

where eq2 is the electrical charge of the quark of flavour q (q = u, d and s). These structure functions correspond to matrix elements of QCD. To see this precisely, let us use the optical theorem which identifies the DIS cross-section to the imaginary part of the forward amplitude of the doubly virtual Compton process, as illustrated on Fig. 2. Now, the QCD matrix element corresponding to this forward Compton process is shown on Fig. 3 and, formally, in quantum field theory terms, it can be written as h p|ψ¯ q (0)Oψq (y)| pi y + =Ey =0 , (2) ⊥

where ψq is the quark field of flavor q and p represents the initial (and final, since it is the same) nucleon momentum. We use a frame where the initial and √final nucleons are collinear along the z-axis and we define the light-cone components a ± by a ± ≡ (a 0 ± a 3 )/ 2. The motivation for adopting this system of coordinates is that, among other aspects, particles going at the speed of light (such as quarks and real photons that we will often meet in the following) along the z-axis have only one 4-vector component a + or a − according to their direction along the z-axis. One has thus to treat, in general, operators and 4-vectors in only one dimension. For instance, the matrix element of Eq. (2) is evaluated only along the light-cone segment of length y − , as the (y + = yE⊥ = 0) subscript indicates. We will illustrate this concept of the light-cone frame in Section 2.3 with a simple picture for the case of the DVCS process. In Eq. (2), the ψ¯ q and ψq spinors being, respectively, 1 × 4 and 4 × 1 (“number of lines” × “number of columns”) quantities, the operator O is a 4 × 4 matrix and can be decomposed on the basis of the 16 independent matrices of Dirac’s theory: 1, γ 5 , γ µ , γ µ γ 5 and σ µν = 2i [γ µ , γ ν ] (respectively, the scalar, pseudoscalar, vector, axial and tensor components),2 the µ and ν indices ranging from 0 to 3. We speak about a “non-local” “forward” matrix element. The term nonlocal (more precisely here, bilocal) refers to the fact that two space–time points (0 and y) are involved and forward means that the nucleon has not changed 2 For a spinorial field ψ, the 16 Dirac bilinear covariants are ψψ, ¯ ¯ µ ψ, ψσ ¯ µν ψ, ψγ ¯ 5 ψ and ψγ ¯ 5 γ µ ψ which Lorentz transform as, ψγ respectively, scalars, vectors, tensors, pseudo-scalars and pseudo-vectors (axial).

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Fig. 4. Elastic scattering: at high virtuality of the photon exchanged between the electron and the nucleon, the photon interacts with a single quark which remains in the nucleon. The nucleon has changed its momentum in the process but it remains a nucleon, unlike the DIS process where it has been “smashed into pieces”.

momentum. The matrix element simply expresses the probability amplitude to create/find a quark at a certain space–time point 0 in a nucleon with momentum p and to create/find the same one (since the process is symmetrical) at another space–time point y in the nucleon, this latter having kept the same momentum p. Due to the symmetry of the process, the Fourier transform of this matrix element is also simply interpretable as the probability (i.e. the square of the amplitude probability) to find a quark with the momentum fraction x in a nucleon, irrespective of its position. Precisely, the two parton distribution functions q(x) and ∆q(x), which are defined in a momentum space, correspond respectively to the Fourier transform of the vector and axial matrix elements: Z p+ + − q(x) = dy − eix p y h p|ψ¯ q (0)γ + ψq (y)| pi , 4π y + =Ey⊥ =0 Z p+ + − ∆q(x) = dy − eix p y h pSk |ψ¯ q (0)γ + γ5 ψq (y)| pSk i , (3) 4π y + =Ey⊥ =0 where p represents the nucleon momentum and Sk is the longitudinal nucleon spin projection. Among the 16 Dirac bilinear covariants, only the vector (γ µ ) and axial (γ µ γ5 ) structures are helicity conserving and leading twist contributions. One should keep in mind however that the tensor (σ µν ) structure, although helicity nonconserving, contributes also to the leading twist and gives rise to so-called transversity distributions. We cannot be exhaustive in this document and, in the following, we will limit our discussions to the helicity conserving quantities, which are the most simply accessible and, as a corollary, the most studied experimentally. 2.2. Elastic scattering, form factors and local nonforward matrix elements The first elastic electron–nucleon scattering experiments were carried out at Stanford in the 50s by Hofstadter and collaborators, using the electron accelerators which were the predecessors of the SLAC machine, at Stanford. At high Q 2 , the eN → eN process can be illustrated by Fig. 4. Again, similarly to the previous section, one can identify a QCD matrix element with this process. It is represented on Fig. 5 and it can be written as h p 0 |ψ¯ q (0)Oψq (0)| pi.

(4)

One speaks here about a “local” “nonforward” matrix element. The term local refers to the fact that only 1 space–time point (0) is involved, and nonforward means that the nucleon has changed its momentum. In simple terms again, this matrix element expresses the probability amplitude to create/find a quark at a certain space–time point 0 in a nucleon with momentum p and to create another one at the same space–time point 0 in the nucleon, this latter having however changed its momentum to p 0 . In an appropriate reference frame, the so-called

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Fig. 5. Illustration of the local nonforward matrix element h p 0 |ψ¯ q (0)Oψq (0)| pi.

Fig. 6. Illustration of the nonlocal nonforward matrix element h p 0 |ψ¯ q (0)Oψq (y)| pi.

Breit frame, where the energy transfer by the virtual photon is zero, this is also simply interpretable as the probability (i.e. the square of the probability amplitude) to find a quark at a certain space point in the nucleon, irrespective of its momentum. In momentum space, the vector and axial matrix elements give then rise to the so-called “form factors” – FFs – (no Fourier transform is required since the operator is local): ∆ν q q h p 0 |ψ¯ q (0)γ + ψq (0)| pi = F1 (t) N¯ ( p 0 )γ + N ( p) + F2 (t) N¯ ( p 0 )iσ +ν N ( p) 2m N ∆+ q q N ( p). h p 0 |ψ¯ q (0)γ + γ5 ψq (0)| pi = G A (t) N¯ ( p 0 )γ + γ5 N ( p) + G P (t) N¯ ( p 0 )γ5 2m N

(5)

This is the most general Lorentz invariant decomposition in momentum space of this matrix element that one can write, using the available quantities: the γ µ and σ µν matrices for the vector current 3 (and the γ 5 and γ µ γ 5 matrices for the axial current) and the only independent 4-vectors p µ and ∆µ . q q q q The complex structure of the nucleon is absorbed in F1 (t), F2 (t), G A (t) and G P (t), respectively the Dirac, Pauli, axial and induced pseudo-scalar form factors, which are here defined for each quark flavour (q = u, d and s). They depend on the only independent variable available in this elastic process: t = ∆2 (where ∆ = p 0 − p, i.e. the transfer q q 4-momentum between the final and initial nucleon) and which is, in this case, equal to Q 2 . F1 and F2 can simply be q q rewritten as a linear combination of G E and G M , which are the electric and magnetic form factors. These latter are then simply interpreted, via a Fourier transform as the spatial density of electric and magnetic charges in the nucleon. 2.3. Exclusive scattering, generalized Parton Distributions and nonlocal nonforward matrix elements Then, only in the last decade, came, with Mueller et al. [4], Ji [5] and Radyushkin [6], the idea to generalize these operators and to introduce the notion of “nonlocal” “nonforward” operators. These can be illustrated by Fig. 6 and can be written as h p 0 |ψ¯ q (0)Oψq (y)| pi,

(6)

where two space points are involved and the momenta of the initial and final nucleons are different. 3 There is in principle, for the vector matrix element, another structure available proportional to the identity matrix, as seen in the previous footnote. However, this latter structure, because of current conservation and the Gordon identity, can be absorbed into the other two (vector and tensor) structures. As a reminder, the Gordon identity is: N ( p 0 )σ µν ( p 0 − p)ν N ( p) = N ( p 0 )[2Mγ µ − ( p 0 + p)µ ]N ( p), which, basically, allows to write the tensor structure in terms of the scalar and vector structures and vice-versa.

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Fig. 7. The “handbag” diagrams for DVCS (left) and DVMP (right). The factorization theorems state that these are the dominant processes at sufficiently high virtuality of the initial virtual photon: it is the same quark that has been hit by the virtual photon that radiates the final photon (DVCS) or ends up in the final meson (DVMP).

Taking the Fourier transform of the vector and axial matrix elements yields then Z P+ + − dy − eix P y h p 0 |ψ¯ q (0)γ + ψq (y)| pi 2π y + =Ey⊥ =0 ∆ν = H q (x, ξ, t) N¯ ( p 0 )γ + N ( p) + E q (x, ξ, t) N¯ ( p 0 )iσ +ν N ( p) 2m N Z P+ + − dy − eix P y h p 0 |ψ¯ q (0)γ + γ 5 ψq (y)| pi 2π y + =Ey⊥ =0 ∆+ = H˜ q (x, ξ, t) N¯ ( p 0 )γ + γ5 N ( p) + E˜ q (x, ξ, t) N¯ ( p 0 )γ5 N ( p). 2m N 0

(7)

P is the average nucleon 4-momentum: P = p+2 p and ∆ = p 0 − p, the transfer 4-momentum between the final and initial nucleon. As mentioned earlier, we use the light-cone frame, i.e. the (initial and final) nucleons go, at the speed of light, along the positive z-axis. A few paragraphs below, we are going to illustrate this frame and the associated variables for a particular physical process (ep → epγ ) but let us for the moment define “formally” the variables. x +ξ represent the + momentum fraction of the incident quark k and −2ξ the + momentum fraction of the transfer ∆. These momentum fractions could be defined relative to the initial nucleon momentum p such that k + = x p + and ∆+ = −2ξ p + but, in order to keep a symmetrical notation with respect to x and follow Ji’s notation [5], they are here defined relative to P, the average nucleon momentum: k + = x P + and ∆+ = −2ξ P + . In this frame, x + ξ is then the + momentum fraction (of the average nucleon momentum) carried by the initial quark and x −ξ is the + momentum fraction (of the average nucleon momentum) carried by the final quark going back in the nucleon. t, the squared 4-momentum transfer between the final nucleon and the initial one, is defined as ∆2 . In comparison to −2ξ which refers to purely longitudinal momentum transfer, t, the squared 4-momentum transfer between the final nucleon and the initial one, contains also a transverse momentum transfer component (called ∆⊥ ). A precise definition of these variables and 4-vectors can be found, for instance, in Eqs. (5)–(11) of Ref. [7]. H q (x, ξ, t), E q (x, ξ, t), H˜ q (x, ξ, t) and E˜ q (x, ξ, t) are the so-called Generalized Parton Distributions and parametrize, as we shall see, in a more complete way than the PDFs accessed in DIS and the FFs accessed in elastic scattering, the complex unknown structure of the nucleon. The GPDs are typically accessed in exclusive photon or meson electroproduction on the nucleon. Based on the notion of QCD factorization that was introduced in Section 2.1, the idea is to “plug” a “hard”, short distance, perturbatively calculable process on top of the “blob” of Fig. 6. Fig. 7 illustrates the leading order exclusive photon (DVCS) and meson (DVMP for “Deep Virtual Meson Production”) electroproduction on the nucleon accessing the GPDs. The factorization proof for DVCS was shown by Ji and Radyushkin [5,6] and for DVMP by Collins et al. [8]. As mentioned earlier, let us now illustrate the light-cone frame and the relevant variables for the DVCS process. Fig. 8 depicts the DVCS process in the light-cone frame. The (initial and final) nucleons go, at the speed of light, along the positive z-axis and the (initial and final) photons go along the negative z-axis. This means that P, p and p 0 have only a + component and the final real photon only a − component. The initial virtual photon is a bit different. It clearly has a − component since it goes along the negative z direction ; however, it also has a + component because,

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Fig. 8. The lightcone frame: the proton(s) go(es) along the positive z direction at the speed of light (i.e. along the + axis) while the final real photon goes along the negative z direction at the speed of light (i.e. along the − axis) and the initial virtual photon goes along the negative z direction faster than light (i.e. it has both + and − components). Table 1 The three families of operators that we have discussed with O = γ + or γ + γ 5 Operator in space coordinates

Nature of the matrix element

Associated structure functions in momentum coordinates

¯ h p|ψ(0)Oψ(y)| pi ¯ h p 0 |ψ(0)Oψ(0)| pi ¯ h p 0 |ψ(0)Oψ(y)| pi

Nonlocal Forward Local Nonforward Nonlocal Nonforward

f 1 (x), g1 (x) F1 (t), F2 (t), G A (t), G P (t) ˜ H (x, ξ, t), E(x, ξ, t), H˜ (x, ξ, t), E(x, ξ, t)

due to its virtuality Q 2 (which can be interpreted as its negative mass squared), it can go faster than light and therefore be out of the light-cone ! In this frame, the difference of + momentum fraction between the initial and final quarks, which is −2ξ is therefore brought by the + component of the virtual photon. Coming back to GPDs, intuitively, they represent the probability amplitude of finding a quark in the nucleon with a + momentum fraction x + ξ and of putting it back into the nucleon with a + momentum fraction x − ξ plus some transverse momentum “kick”, which is represented by t (or ∆2⊥ ). As can be seen from Eq. (7), H and E are independent of the quark helicity and are therefore called unpolarized GPDs, whereas H˜ and E˜ are helicity dependent and are called polarized GPDs. Also, note that the GPDs H q , E q , H˜ q , E˜ q are defined for a single quark flavour (q = u, d and s). In this short write-up, we only discuss “quark” GPD. However, one can also define, in a very similar fashion, gluonic GPDs corresponding to the operators h p 0 |G +µ (0)G + µ (y)| pi h p 0 |G +µ (0)G˜ + µ (y)| pi,

(8)

where G µν is the gluon field tensor and G˜ µν = 12  µναβ G αβ its dual. Essentially for reasons of space constraints, we do not discuss them in detail here. To finish this section, we summarize in Table 1, the line of the reasoning and the progression that we have followed since the beginning of this section. 2.4. Properties and physics content of the GPDs 2.4.1. Link with PDFs and FFs Since the matrix element of Eq. (6) (Fig. 6) is a generalization of the matrix elements of Eqs. (2) (Fig. 3) and (4) (Fig. 5), the GPDs are not completely unknown and have links with the PDFs and the FFs. From the optical theorem (Fig. 2), due to the symmetry of the forward Compton process where there is no longitudinal momentum transfer at the quark level (ξ = 0) and no momentum transfer at the nucleon level either (t = 0), one immediately deduces that H q (x, 0, 0) = q(x),

H˜ q (x, 0, 0) = ∆q(x).

(9)

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Fig. 9. Illustration of the sum rule of Eq. (11) linking the first x moment of the GPDs to FFs. The integration on the x momentum leads to the “merging” (via the δ function) of the 0 and y space points.

Similarly: H g (x, 0, 0) = xg(x),

H˜ g (x, 0, 0) = x∆g(x),

(10)

where g(x) and ∆g(x) are the forward unpolarized and polarized, respectively, gluon densities. At finite momentum transfer, there are model independent sum rules which relate the first moments of the GPDs to the elastic form factors of Eq. (5): Z +1 Z +1 q q q dx H (x, ξ, t) = F1 (t), dx E q (x, ξ, t) = F2 (t), −1

Z

−1

+1 −1

q

dx H˜ q (x, ξ, t) = G A (t),

Z

+1

−1

q

dx E˜ q (x, ξ, t) = G P (t).

(11)

with F1 , F2 , G A and G P defined as in Eq. (5). The idea behind these sum rules is simply that Z Z dy − ix y − 0 ¯ e h p |ψq (0)Oψq (y)| pi = h p 0 |ψ¯ q (0)Oψq (0)| pi y + =Ey =0 , (12) dx ⊥ 2π y + =Ey⊥ =0 R − where the trivial Fourier Transform dxeix y = 2π δ(y − ) is used. Fig. 9 illustrates this sum rule. These relations between GPDs and FFs have given rise to the idea [9–11] that the t dependence of the GPDs could be related, via a Fourier transform, to the transverse spatial distribution of the partons in the nucleon, in a similar way as the FFs can be related to spatial densities. For ξ = 0 (where t = −∆2⊥ ), one can thus get an impact parameter version of GPDs through a Fourier integral in transverse momentum ∆⊥ : Z 2 d ∆⊥ ib⊥ ∆⊥ q q H (x, b⊥ ) = e H (x, −∆2⊥ ). (13) (2π )2 One sees then how the information contained in a traditional PDF, such as the ones measured in DIS, and the information contained in a FF, as measured in elastic lepton-nucleon scattering, are now combined and correlated in the GPD description [12]. Thus, at ξ = 0, the GPD(x, 0, t) can be interpreted as the probability amplitude of finding in a nucleon a parton with longitudinal momentum fraction x at a given transverse impact parameter b⊥ (which is the conjugate variable of ∆⊥ ). Fig. 10 illustrates how this “3-dimensional” image of the nucleon could look like (according to the particular “VGG” GPD model [7,13]). 2.4.2. Polynomiality There is actually a general rule on all the x moments of GPDs because of Lorentz invariance. The so-called polynomiality rule states that their nth x moment must be a polynomial in ξ of order n (for n even) or n + 1 (for n odd); thus, for the H GPD: Z 1 if n even: x n H (x, ξ, t)dx = a0 + a2 ξ 2 + a4 ξ 4 + · · · + an ξ n −1

Z

1

if n odd:

x n H (x, ξ, t)dx = a0 + a2 ξ 2 + a4 ξ 4 + · · · + an+1 ξ n+1

−1

and similarly for E.

(14)

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Fig. 10. “Tri-dimensional” view of the nucleon: the GPD H u (x, ξ, t) as a function of the longitudinal momentum fraction x and the transverse impact parameter b⊥ (the conjugate variable of t) at ξ = 0 according to the VGG model [7,13].

Let us first note that in this equation only even power of ξ appear. This a consequence of the time reversal invariance 0 + + which states that: H (x, −ξ, t) = H (x, ξ, t) (the usual, simple, argument for this is that ξ ∝ ∆ = (( pp+−pp)0 )+ reverses P+ sign when exchanging p with p 0 , which is, in effect, the operation of time reversal). This is, in particular, one of the reasons why in Eq. (11), there is no dependence on ξ . There are similar rules for the E, H˜ and E˜ GPDs with a few nuances: for the GPD E, the an+1 coefficient is the ˜ the maximum ξ power in Eq. (14) is n same as for H except that it has the opposite sign. And for the H˜ and E, (instead of n + 1). We will give the reasons for this in the following. Let us first concentrate on H and show what is the origin of this polynomiality rule. The idea behind the sum rule of Eq. (14) comes from some general properties of Fourier transforms: multiplying the Fourier transform of a function by x n means to take its nth derivative (F T [ f (n) (x)] = y n F T [ f (x)](y), where f (n) denotes the nth derivative).4 And, more precisely, the nth moment of a function, i.e. Rintegrating over x in addition to multiplying by x n , is proportional to the nth derivative of its Fourier transform at 0: x n f (x)dx = F T (n) (0). To come back to GPDs, this all means that the nth x moments of the GPDs (bilocal operators) are related to the nth derivatives of local operators: Z + n+1 Z n (P ) − ix P + y − 0 ¯ + dx x dy e h p |ψq (0)γ ψq (y)| pi 2π y + =Ey⊥ =0 ↔+ = h p 0 |ψ¯ q (0)γ + (i D )n ψq (0)| pi , (15) y + =Ey⊥ =0

↔ ← − − → where D = 21 [ D − D ] is the covariant derivative. Eq. (12) is just the particular case of Eq. (15) for n = 0. Using Lorentz symmetry, it remains to write, in momentum space, the most general form of the operator ↔+ 0 h p |ψ¯ q (0)γ + (i D )n ψq (0)| pi| y + =Ey⊥ =0 using, like in Eq. (5), the available quantities: the γ µ , σ µν and identity matrices and the only independent 4-vectors p µ and ∆µ , keeping in mind that n + 1 Lorentz indices must now ↔+

be present (1 for the γ matrix and n for (i D )n ). To keep the general “intuitive” spirit of this text, we don’t want to write here this general formula, which can be found in Refs. [2] – formula (35) – or [14] – formula (4) –, suffice it to say that, in addition to the vector ( N¯ ( p 0 )γ µ N ( p)) and tensor ( N¯ ( p 0 )iσ µν ∆ν N ( p)) parts that can already be found in Eq. (5), there is now also a scalar part which is proportional to N¯ ( p 0 )N ( p) (proportional to the identity matrix). To 4 For sake of simplicity, we omit here and in the following formula, normalization factors of the type (2iπ ).

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obtain the n + 1 Lorentz indices with this latter structure, it has to contain, among other terms and without any reason to exclude this one in particular, a term of the form ∆n+1 , or equivalently, when projecting on the + components, (2ξ )n+1 , in accordance with Eq. (14). ˜ These GPDs are respectively associated with the pseudoscalar N¯ ( p 0 )γ 5 N ( p) There is no such term for H˜ and E. 0 µ 5 ¯ and axial N ( p )γ γ N ( p) matrix elements which are the only two available Dirac bilinear covariant associated with a γ 5 matrix. There is no other independent “γ 5 ” term, like the scalar term could be independent of the vector and tensor terms for the H and E GPDs. Therefore, the maximum ξ power in Eq. (14) for H˜ and E˜ is n. 2.4.3. Link with orbital momentum Let us look at the second x moment of GPDs. This will serve to illustrate some of the points made in the previous section and, in particular, to derive a relation between GPDs and the orbital momentum contribution of the quarks to the spin of the nucleon. According to Eq. (15), one has (taking n = 1): Z Z ↔+ (P + )2 + − dx x dy − eix P y h p 0 |ψ¯ q (0)γ + ψq (y)| pi = h p 0 |ψ¯ q (0)γ + (i D )ψq (0)| pi . 2π y + =Ey⊥ =0 y + =Ey⊥ =0 (16) ↔+

Let us write the h p 0 |ψ¯ q (0)γ + (i D )ψq (0)| pi| y + =Ey⊥ =0 operator in terms of GPDs. We simply have to take the derivative of Eq. (7) with respect to y − : Z ↔+ P+ − ix P + y − 0 ¯ + dy e h p |ψq (0)γ (i D )ψq (y)| pi 2π y + =Ey⊥ =0 Z + 2 (P ) − ix P + y − 0 ¯ + dy e h p |ψq (0)γ ψq (y)| pi = x 2π y + =Ey⊥ =0   ∆ν = x H q (x, ξ, t) N¯ ( p 0 )γ + N ( p) + x E q (x, ξ, t) N¯ ( p 0 )iσ +ν N ( p) P + 2m N   +  q  P ¯ 0 q q 0 +ν ∆ν ¯ = x H (x, ξ, t) N ( p )N ( p) + x H (x, ξ, t) + E (x, ξ, t) N ( p )iσ N ( p) P + , (17) mN 2m N where for the last line, we used the Gordon identity, that we have already met in Section 2.2, and which allows us to write the vector structure in terms of a scalar and a tensor one. Now, let us write, in momentum space, independently of any GPD definition, the most general form of the local ↔+ operator h p 0 |ψ¯ q (0)γ + (i D )n ψq (0)| pi| y + =Ey⊥ =0 using the available quantities: the γ µ , σ µν and identity matrices and the only independent 4-vectors p µ and ∆µ . One way to do this is ↔ν 0 µ ¯ h p = p + ∆|ψq (0)γ (i D )ψq (0)| pi y + =Ey⊥ =0

 1 ∆α N ( p) + C(t)(∆µ ∆ν − ∆2 g µν ) = N¯ ( p 0 ) A(t)γ µ P ν + B(t)P µ iσ να 2m N mN   0 µ ν µ να ∆α µ ν 2 µν 1 ¯ = N ( p ) A(t)P P + (A(t) + B(t))P iσ + C(t)(∆ ∆ − ∆ g ) N ( p), 2m N mN 

(18)

where for the last line, we have used once again the Gordon identity. For sake of clarity, we don’t write this explicitly, but this formula must be symmetrized with respect to µ and ν. Now, taking the + components yields: Z Z (P + )2 − ix P + y − 0 ¯ + dy e h p |ψq (0)γ ψq (y)| pi dx x 2π y + =Ey⊥ =0  +h  i P + ¯ 0 2 +α ∆α A(t) + 4ξ C(t) + [A(t) + B(t)] iσ N ( p). (19) = P N(p ) mN 2m N

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One can then readily identify the scalar and tensor structures of Eqs. (19) and (18), from which we derive: Z x H (x, ξ, t)dx = A(t) + 4ξ 2 C(t) Z x E(x, ξ, t)dx = B(t) − 4ξ 2 C(t) Z ⇒ x [H (x, ξ, t) + E(x, ξ, t)] dx = A(t) + B(t).

(20)

This illustrates simply, for the n = 1 case, the polynomiality rule that we have discussed in Section 2.4.2. In particular, we recognize the maximal ξ power of 2 which is “allowed” for the moment of H and E and whose origin is the scalar term of Eq. (18). Let us also note that this ξ 2 term comes with a different sign for H and for E, as was mentioned in the previous section. This arises from the simple fact that, when one uses the Gordon identity to write the scalar part (associated to the maximal ξ power) in terms of a vector and a tensor structure, these tho latter structures appear with an opposite sign to each other (N ( p 0 )σ µν ( p 0 − p)ν N ( p) = N ( p 0 )[2Mγ µ − ( p 0 + p)µ ]N ( p)). ↔ν A key thing now is to understand that the operator ψ¯ q (0)γ µ (i D )ψq (y) that we have just studied is actually nothing less than the energy momentum tensor T µν . Indeed, the operator product expansion (OPE) allows to develop, for small lightcone distances y, a bilocal operator in a series of local operators, such as (for, say, a bilocal vector operator) ↔+

ψ¯ q (0)γ µ ψq (y) = ψ¯ q (0)γ µ ψq (0) + y − ψ¯ q (0)γ µ i D ψq (0) + · · · .

(21)

The vector structure couples to spin-1 probes (γ , W , Z ) to which are associated the electromagnetic form factors (see Eq. (5)) while the tensor structure couples to spin-2 probes (graviton. . . or 2-photons. . . like DVCS !) to which are associated energy–momentum tensor form factors.5 The A(t), B(t) and C(t) factors are therefore the form factors of the energy momentum tensor. The link between these form factors (and therefore the GPDs) is now clear given that it is well known that the operator for angular momentum is directly related to the energy momentum tensor Z h p| JE| pi = h p| d3 yE yE × pE| pi, (22) where pi = T 0i . We have already seen (and used) the basic property of Fourier transforms that multiplying a function by a variable comes to taking the derivative of its Fourier transform with respect to the conjugate variable. Therefore, to connect Eq. (18), which gives the energy momentum tensor, with Eq. (22), one has to multiply6 Eq. (18) by y, i.e. take the derivative with respect to ∆ = p 0 − p, and take the forward limit (∆ → 0). Only the A and B factors from Eq. (18) remain then and one has straightforwardly 1 [A(0) + B(0)]. (23) 2 With Eq. (22), we therefore have the sum rule, first derived by Ji [5], linking the second moment of the GPDs to the orbital angular momentum carried by the quarks: Z   1 +1 dx x H q (x, ξ, t = 0) + E q (x, ξ, t = 0) . (24) Jq = 2 −1 J=

More generally, in Ref. [5], it was shown that there exists a (colour) gauge-invariant decomposition of the nucleon spin 1 = Jq + Jg , 2

(25) ν

5 As a general remark, the ψ¯ (0)(0)γ µ (i↔ D )ψq (y) operator is of twist 2 according to the definition of twist that we have given in Section 2.1. q ↔ν

↔ν

Indeed, on the one hand, the dimension of the ψ¯ q (0)(0)γ µ (i D )ψq (y) operator is 4 since the 2 spinors have dimension 2 × 32 and the i D operator ↔ν

has dimension 1 and, on the other hand, the structure γ µ (i D ) has spin 2 since it has two Lorentz indices µ and ν. 4 − 2 = 2 ! The energy momentum tensor is, by definition, the spin-2 twist-2 operator. 6 More precisely, take the cross product.

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Fig. 11. The cross-section of the ep → epγ reaction is proportional to the squared amplitude: |M DV C S + M B H |2 .

where Jq and Jg are respectively the total quark and gluon spin contributions to the nucleon total angular momentum, with the total quark spin contribution Jq which decomposes as Jq =

∆Σ + Lq , 2

(26)

where ∆Σ 2 and L q are respectively the quark spin and quark orbital contributions to the nucleon spin. Since ∆Σ has been measured through polarized DIS experiments (it is 25% ± 10%, for a scale of Q 2 = 2 GeV2 ) and Jg is currently being measured at COMPASS and RHIC, a measurement of the sum rule of Eq. (24) in terms of the GPD’s provides a model independent way of determining the quark orbital contribution to the nucleon spin, and therefore complete the “spin-puzzle”. 3. Experimental situation and perspectives on DVCS In this section, we shortly review the experimental situation related to the DVCS process. Due to space constraints, we cannot cover the experimental situation for DVMP. In order to fully describe the 3-body final state reaction ep → epγ , there are four independent variables.7 They are usually chosen as Q 2 , x B , −t and Φ which are, respectively, the standard squared electron four-momentum transfer, Q2 0 the Bjorken variable defined as 2m p (E−E 0 ) (where E is the beam energy and E is the scattered electron energy), the squared four-momentum transfer between the incoming virtual photon and the outgoing real photon and Φ the azimuthal angle between the electron scattering plane and the hadronic production plane. At some points, the centre 2 B of mass energy of the (γ ∗ p) system W which is equal to Q 2 ( 1−x x B ) + m p is also used. Only these past five years, experimental data which can lend themselves to GPDs interpretation and of sufficient precision have been obtained. The first of these observables is the Beam Spin Asymmetry (BSA) of the ep → epγ process, which is the difference of the (beam) polarized cross-sections divided by their sum. It is an observable which is relatively straightforward to extract experimentally since, in a first order approximation, normalization factors such as the efficiency/acceptance of the detector and, more generally, many sources of systematic errors cancel in the ratio. This asymmetry arises from the interference of the “pure” DVCS process (when the outgoing photon is emitted by the nucleon) and the Bethe-Heitler – BH – process (when the outgoing photon is radiated by the incoming or scattered lepton), see Fig. 11. Polarization effects are, in a general fashion, sensitive to the imaginary part of amplitudes. The BH being purely real, the imaginary part of the ep → epγ process arises solely from the DVCS contribution. This kind of observable has therefore a strong sensitivity to DVCS, even if the strength of this latter process is much smaller than the BH one, which is generally the case in the kinematics presently explored experimentally. For the BSA, a shape close to a sin Φ reminiscent of the standard 5th response function of 2-body exclusive reactions is expected. Fig. 12 shows the first measurement ever of the Beam Spin Asymmetry for DVCS on the proton carried out by the HERMES collaboration, a few years ago, with the 27 GeV positron beam of the DESY 7 There are actually five variables to fully describe the kinematics of the reaction but there is a trivial symmetry, in the case of unpolarized nucleons, with respect to the azimuthal angle of the scattered electron

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Fig. 12. The proton DVCS beam asymmetry as a function of the azimuthal angle Φ as measured by HERMES [15]. Average kinematics are: hx B i = 0.11, hQ 2 i = 2.6 GeV2 and h−ti = 0.27 GeV2 . The dashed curve is a sin Φ fit whereas the solid curve is the theoretical GPD calculation of the VGG model, including twist-3 effects (see Ref. [16]).

laboratory [15]. At HERMES, the average kinematics is hx B i = 0.11, hQ 2 i = 2.6 GeV2 and h−ti = 0.27 GeV2 for which an amplitude of 0.23 for the sin Φ moment is extracted from the fit. The figure shows also the comparison to the theoretical prediction of the VGG model [7,13], already mentioned in the previous section. There is a very correct agreement; the discrepancies between the theoretical curve and the data can be attributed, in part, to the large kinematic range over which the experimental data have been integrated, and where the calculation can vary significantly, but also to higher twist corrections not taken into account (so far, only twist-3 corrections are relatively under theoretical control for DVCS, keeping in mind that the leading twist is twist-2 as we saw in Section 2; one can refer for instance to Refs. [16–18] for discussions on twist-3 accuracy). Let us also mention that the DVCS reaction at HERMES is identified by detecting the scattered lepton (positron) and the outgoing photon from which the missing mass of the nondetected proton is calculated. Due to the limited resolution of the HERMES detector, the selected peak around the proton mass is −1.5 < M X < 1.7 GeV, which means that contributions to this asymmetry from nucleon resonant states as well, cannot be excluded. This same observable, i.e. the DVCS BSA on the proton, has been measured at JLab [19], quasi-simultaneously, with a 4.2 GeV electron beam and the 4π CLAS detector of the Hall B of JLab. Due to the lower beam energy compared to HERMES, the kinematic range accessed at JLab is different: hx B i = 0.19, hQ 2 i = 1.25 GeV2 and h−ti = 0.19 GeV2 . In this case, the DVCS reaction was identified by detecting the scattered lepton and the recoil proton. The missing mass of the photon was then calculated. The contamination by ep → epπ 0 events could be estimated to some extent and subtracted bin per bin. Fig. 13 shows the CLAS measured asymmetry along with the theoretical calculations (predictions) of the VGG model. The different sign of the CLAS BSA relative to HERMES is due to the use of electron beams in the former case compared to positron beams in the latter. Again, the discrepancies between the experimental data and the theoretical calculations can be assigned to the fact that the theory is calculated at a single, well-defined, kinematic point whereas data have been integrated over several variables and wide ranges. Furthermore, Next to Leading Order as well as higher twists corrections which may be important at these rather low Q 2 values, still need to be quantified. It should be noted at this stage that it is not trivial to produce asymmetries of the order of 20%–30% and the fact that some GPD models could predict such amplitudes was taken at that time as an extremely encouraging sign that one was indeed “seeing” the handbag process and, therefore, GPDs. Let us also mention that this DVCS BSA on the proton has also been measured at higher energies (although not published yet), using beams of 4.8 GeV and 5.75

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Fig. 13. The proton DVCS beam asymmetry as a function of the azimuthal angle Φ as measured by CLAS [19]. Average kinematics are: hx B i = 0.19, hQ 2 i = 1.25 GeV2 and h−ti = 0.19 GeV2 . The shaded regions are error ranges for sin Φ and sin 2Φ fits. Calculations are: leading twist without ξ dependence [7] (dashed curve), leading twist with ξ dependence [7] (dotted curve) and including twist-3 [16] (solid curve).

Fig. 14. The figure on the top shows the difference of (beam) polarized cross-sections for DVCS on the proton, as a function of the Φ angle, measured by the JLab Hall A collaboration [24]. The average kinematics is hx B i = 0.36, hQ 2 i = 2.3 GeV2 and h−ti = 0.28 GeV2 . The figure on the bottom shows the total (i.e. unpolarized) cross-section as a function of Φ. The red curves show a fit to the data. The BH contribution is represented by the green curve. The difference between the data and the BH is attributed to the DVCS whose twist-3 contribution is estimated by the dot-dashed curve (i.e. it is very small).

GeV [20], by the CLAS collaboration and that a first exploratory measurement of the DVCS BSA on the neutron has been carried out by the JLab Hall A collaboration [21]. The first DVCS cross-sections on the proton have been measured in two very different kinematical regimes: at high energy (30 < W < 140 GeV, 2 < Q 2 < 100 GeV2 ), by the H1 and ZEUS collaborations [22,23] and at lower energies (W ≈ 2 GeV) by the Hall A collaboration of JLab [24]. At large W (i.e. low x B ), the DVCS process is sensitive mostly to “gluon” GPDs which we cannot cover in this short article, as already mentioned. Let us concentrate on the lower energies, in the valence region, where the first ever 4-fold dσ (polarized and unpolarized) differential cross-sections dx dQ 2 dtdΦ (i.e. without any integration over an independent B

variable) was released by the JLab Hall A collaboration [24]. Fig. 14 shows these data for hQ 2 i = 2.3 GeV2 . The particular shape in Φ of the unpolarized cross-section is typical of the BH process. In this case, the final state photon is radiated by the beam or the scattered electron and, more precisely, in a very peaked cone along their

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Fig. 15. The figure shows the Q 2 dependence of the imaginary part of the proton DVCS amplitude extracted from the polarized cross-section shown in Fig. 14 which was measured at three different Q 2 values.

Fig. 16. The proton DVCS beam charge asymmetry as a function of Φ, measured by the HERMES collaboration. The solid curve shows a fit to the data: A + Bcos Φ + Ccos 2Φ + Dcos 3Φ (the constant term and the cos 3Φ term are compatible with zero). The “pure” cos Φ component is shown by the dashed curve.

direction. This means that this outgoing photon can barely be out of the electron-proton plane. In other words, the BH process is strongly peaked around Φ = 0◦ where, a priori, it completely dominates the cross-section. The difference of polarized cross-sections, like, in general, most of polarization observables, allows us to extract the imaginary part of the DVCS amplitude. This observable has been measured for two other Q 2 values and thus has permitted to extract the Q 2 dependence of this imaginary part of the amplitude, predicted to be Q 2 independent at leading order. This is presented on Fig. 15, where it can be seen that it seems to follow this “scaling” law and, therefore, to confirm that one indeed accesses the leading twist handbag process at the JLab kinematics. Of course, the Q 2 domain covered by the Hall A experiment to study this Q 2 dependence is quite limited and this conclusion, although clearly very encouraging, should be taken with caution. Besides the BSA and cross sections just mentioned, two other observables relative to DVCS and GPDs have been measured: the DVCS beam charge asymmetry (see Fig. 16), which is sensitive to the real part of the amplitude and which has been measured on the proton by the HERMES collaboration [25] and the proton target asymmetry (see Fig. 17) by the CLAS collaboration [26].

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Fig. 17. The proton DVCS target spin asymmetry as a function of Φ, measured by the CLAS collaboration. The solid curve shows a fit to the data: Asin Φ + Bsin 2Φ. The sin 2Φ term is compatible with zero. The dashed curve is the VGG model prediction, based on the contributions of H and H˜ , with twist-3 contribution included. The dotted curve shows the asymmetry when H˜ = 0 and therefore shows the sensitivity of this observable to H˜ .

All these experimental results are very encouraging in the sense that the observed signals are, for the most part, in relatively good agreement (in amplitude and in shape) with the theoretical calculations. Let us stress that most of the calculations shown were predictions and were published before the experimental results. However, it should be clear that a detailed mapping and extraction of the GPDs require many more data, in quantity and in quality, to really constrain the models. Experimentally, at short term, numerous quality data are expected from JLab, with the 6 GeV beam for many channels and observables: for DVCS in particular, several experiments are planned for the next few years in Hall A [27] and B [28,29]. They aim at increasing (at least doubling) the statistics (and therefore, in particular, refining the binning), explore new kinematic domains for the (polarized and unpolarized) cross-sections (and beam spin asymmetries) and also measure new observables such as the (longitudinal) target spin cross-sections and asymmetries and the double target-beam spin asymmetries, which provide a particular sensitivity to the H˜ GPD. HERMES, with a 27 GeV beam, is currently taking data with a newly installed recoil detector which will ensure the exclusivity of the reaction. New beam charge asymmetries and beam spin asymmetries should be available in the next couple of years in a lower x B domain than at JLab [30]. After 2010, the COMPASS experiment at CERN also intends to study GPDs with a 200 GeV muon beam. Similarly to HERMES, a dedicated recoil detector will have to be installed in order to detect all the particles of the final state DVCS or DVMP reaction and ensure the exclusivity of the process. It will have the unique feature of accessing very small x B at sufficiently large Q 2 . In the longer term (>2013), the upgraded JLab [31], with a 12 GeV beam promises to yield a wealth of new experimental data, allowing us to reach new kinematic domains, in particular, the higher Q 2 regime, which is crucial for the understanding of preasymptotic effects and higher twists. Two experiments for DVCS [32,33] and one for (pseudoscalar) mesons [34], with the 12 GeV beam, have already been approved by the JLab PAC. 4. Conclusions In this short write-up, we have introduced in the first section, in a hopefully pedagogical way, the formalism of the Generalized Parton Distributions and attempted, given that it is difficult to be original on the subject, to give an intuitive “flavour” for the derivation of their main model-independent properties. We then reviewed the experimental situation on DVCS and the associated perspectives. Theoretically, the challenges are numerous, especially now that the data start to massively come out : refining, completing, revisiting the parametrizations of GPDs, calculating, understanding the higher twists effects, developing the calculations of (moments of) GPDs on the lattice, controlling the chiral extrapolation towards the physical pion mass, etc. In conclusion, in establishing this 3-dimensional momentum–space correlation of the partons in the nucleon, a tremendous amount of work is lying in front of us. . . but, probably, also a tremendous reward in our understanding of QCD.

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Acknowledgements The author thanks B. Pire, M. Vanderhaeghen, V. Burkert and M. Garc¸on for insightful comments. 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]

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