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The Off-forward Quark-Quark Correlation Function

The properties of the non-forward quark-quark correlation function are examined. We derive constraints on the correlation function from the transformation properties of the fundamental fields of QCD occurring in its definition. We further develop a method to construct an ansatz for this correlator. We present the complete leading order set of generalized parton distributions in terms of the amplitudes of the ansatz. Finally we conclude that the number of independent generalized parton helicity changing distributions is four.


I. INTRODUCTION
In quantum field theory the non-perturbative nature of composite particles is described by matrix elements of operators of the fundamental fields of the theory evaluated between the initial and the final state of the particle under consideration. Accordingly, in QCD, the information on the internal partonic structure of hadrons is entirely given by matrix elements of all possible quark and gluon operators evaluated between hadronic states, forward and non-forward.
Forward matrix elements of quark and gluon operators, i.e. operators between hadronic states of equal momenta, are investigated in hard inclusive processes, while non-forward matrix elements, i.e. operators between hadronic states of different momenta, have been investigated in recent years in the context of non-forward high-energy exclusive processes such as Compton scattering in the deeply virtual kinematical limit ( DVCS ) and hard diffractive vector-meson production. Experimental results [1,2,3,4] indicate the feasibility of measuring these processes much more precisely with dedicated experiments in the future.
In this article we will focus on hadronic matrix elements of bilinear quark operators, since these are the ones involved in the description of dominant contributions to non-forward hard processes. We consider their Fourier transforms, i.e. the quark-quark correlation functions, which contain all information concerning the non-perturbative nature of hadrons. From non-forward quark-quark correlation functions, for convenience, the so called generalized parton distributions (GPDs) are defined. [5,6] Following and generalizing the method developed for the ordinary forward parton distributions [7,8], we will formulate the most general ansatz for the quark-quark correlation functions starting from general principles, and we will then analyze leading order GPDs by projecting the ansatz with different Dirac matrices. In this way we will be able to express the GPDs in terms of the amplitudes entering the ansatz and establish a formal method for the determination and classification of the independent quark GPDs.
The number of independent GPDs occurring in a given Dirac projection is not selfevident. For instance it has been debated in the literature [9,10] whether there are two or four independent quark helicity changing GPDs corresponding to the forward transversity distribution. With the method developed in this article we will show unambiguously that there are indeed four independent helicity changing GPDs at leading twist.
The outline of the work is as follows: in Section II we define the non-forward quarkquark correlation function and we introduce a new method to build an ansatz for it. We also derive constraints on the non-forward quark-quark correlator imposed by the hermiticity properties of the quark fields and their well-known behavior under parity and time reversal operations. In Section III we relate the non-forward correlation function to leading twist quark GPDs by tracing the ansatz with various Dirac matrices and by integrating over quark momentum components. Different Dirac structures probe different spin properties of the hadrons. Finally, in Section IV we draw conclusions and discuss the outlook for this subject.

II. THE NON-FORWARD QUARK-QUARK CORRELATION FUNCTION
We define the non-forward quark-quark correlation function Φ iΛ ′ , jΛ (k, k ′ , P, P ′ ), depending on the hadron and quark momenta (see Fig. 1 for notation), by Fourier transforming the hadronic matrix elements of quark-quark operators We assume the hadron is a spin 1 2 particle, say a nucleonic target, which is in an eigenstate of light-cone helicity characterized by the initial and final light-cone helicity, Λ and Λ ′ respectively, which are defined from the spin vectors S µ and S ′ µ where m is the hadron mass, and v ′ µ is the null vector on the light-cone v ′ µ = [0, 1, 0 T ].
Throughout the paper we use the component notation u = [u + , u − , u T ] with u ± = (u 0 ± u 3 )/ √ 2 and the transverse part u T = (u 1 , u 2 ). Explicit representations of spinors for lightcone helicity eigenstates are given for instance by Kogut and Soper [13] or Brodsky and Lepage [14]. The non-forward correlation function in Eq. (1) represents a transition matrix element of quark-quark operators and not an expectation value like the forward quark-quark correlation function [15]. Therefore it provides more general information about hadrons compared to the ordinary quark-quark correlation function.
The known properties of the quark-quark correlation function determine the structure of an ansatz, which will be the starting point for our analysis. The quark-quark correlation Diagrammatic representation of the correlation function Φ iΛ ′ , jΛ (k, k ′ , P, P ′ ). From an initial hadron with momentum P a quark with momentum k is taken out, and reinserted with momentum k ′ = k + ∆ to form the final state hadron with changed momentum P ′ = P + ∆.
function in the helicity basis, defined in Eq. (1), is a 4 ×4 matrix in the partonic Dirac space, labelled by i and j, and a 2 × 2 matrix in the hadronic helicity space, labelled by Λ and Λ ′ .
Therefore we formulate an ansatz given by the product of a partonic and a hadronic sector separately. The partonic sector is spanned by the setΓ ij of the 16 independent 4×4 partonic Dirac matrices, and the hadronic sector is represented by all possible independent spinorial products,ū k (P ′ , Λ ′ ) Γ kl u l (P, Λ), evaluated between final and initial light-cone helicity states.
Since the correlation function in Eq. (1) is a scalar in Lorentz space, we saturate the open indices of the tensorial product of hadronic and partonic sectors with all possible independent tensors, t µ 1 ···µpν 1 ···ν h , constructed from the kinematical variablesk = (k + k ′ )/2,P = (P + P ′ )/2, and ∆ = P ′ − P , the metric tensor g αβ and one antisymmetric tensor ǫ αβρσ , leading to the ansatz of the form In the hadronic sector the spinorial products (spinor indices suppressed) occur. From the Gordon identities we can deduce that only the three spinor products in (4) are independent. These spinorial products are evaluated in a specific frame in Appendix A.
The correlation function (1) fulfills the following constraints derived from properties of the Dirac quark fields and of the hadronic states under hermitian conjugation, parity and time reversal transformations (cf. [8,16]) where the notationũ = (u 0 , − u) for momenta and spin vectors is used. These constraints can be implemented in building the ansatz for the correlation function. As far as the time reversal constraint is concerned, Collins has shown that the time reversal constraint is not applicable if a Wilson line is inserted in the quark correlation function. Therefore imposing the time reversal constraint corresponds to defining the correlator without a Wilson line. [12] Taking into account all the above information leads to the most general ansatz for Φ.
The lengthy expressions are an intermediate result of our investigation and are explicitly given in Appendix B.

III. GENERALIZED PARTON DISTRIBUTIONS
The generalized parton distributions Φ iΛ ′ ,jΛ (x, ξ; t) are obtained as traces of the quarkquark correlation function with the Dirac matrices Γ, integrated over the transverse and minus components of the quark momentum, k T andk, respectively, where x and ξ are the light-cone momentum fractions x ≡k + /P + and ξ ≡ −∆ + /(2P + ) and the Mandelstam variable t = ∆ 2 denotes the momentum transfer squared. The projections Λ ′ ,Λ carry only hadron helicity indices since the parton Dirac indices have been saturated in taking the trace. Following [7,8] we rewrite the integral in Eq. (11) with respect to the covariant integration variables, σ = 2P ·k and τ =k 2 , and the azimuthal angle φ The projections of the quark-quark correlation function Φ with the various Dirac matrices Γ in Eq. (12) correspond to the different generalized distribution functions and determine also which distribution functions occur in a non-forward hard process at different orders in 1/P + , P + scaling with the hard scale in the process. In particular the leading order ( twist 2 ) distribution functions [7,8,17] are obtained by projecting out the ansatz for the correlation function with the Dirac matrices γ + , γ + γ 5 and i σ +i γ 5 .

A. Unpolarized parton distribution
Let us consider the unpolarized generalized distribution functions of the proton. Thus we project the non-forward quark-quark operator with the matrix γ + . Following [19] using the notation of [18] the GPDs H(x, ξ; t) and E(x, ξ; t) are defined by Evaluating the quantity Φ with t 0 defined as and a phase η given by Substituting the expression of the ansatz (B7) and (B8) for the helicity non-flip and helicity flip correlation functions, Φ ++ or Φ −− , and Φ −+ or Φ +− , respectively, in (12) and tracing with γ + we have a set of two equations for the two unknown distribution functions H and E, which is solvable and gives the two unpolarized generalized parton distribution functions where we have introduced the function A (1) (x, ξ; t) and A (3) (x, ξ; t) defined through the coefficients d 10 ] (18) and The expressions for the generalized distribution functions H and E in Eq. (17) do not contain any term in the amplitudes d Helicity conservation in Eq. (20) then shows a link between the quark and nucleon helicity degrees of freedom. Through the projection of the quark-quark correlation function with the matrix γ + the nucleon helicity does not flip and the only possible hadron helicity combinations in the forward limit are + + and − −.

B. Polarized parton distributions
The polarized quark distributions, H(x, ξ; t) and E(x, ξ; t), are defined by the Fourier transforms of the axial vector matrix element For the different proton helicity combinations we now find and Substituting the ansätze (B7) and (B8) for the correlation functions we obtain a set of two equations in the two unknown functionsH andẼ where we have introduced the function B (1) (x, ξ; t) and B (3) (x, ξ; t) defined as and respectively.

C. Parton helicity changing distributions
There are also twist 2 generalized distributions that change the helicity of the active parton. The corresponding quark distributions are constructed from the operatorψ q i σ +i γ 5 ψ q .
By counting the helicity amplitudes Hoodbhoy and Ji [9] introduce two independent quark helicity changing generalized distributions corresponding to hadron helicity flip and non-flip terms. However, Diehl [10] claims that there are four independent parton helicity changing generalized distributions defined by In Eq. (27) the Lorentz index i takes the values 1,2). In the following we show that the same conclusion concerning the number of independent helicity changing GPDs is obtained by tracing the ansatz for the quark-quark correlation given in Eq. (B7) with the matrix ψ q i σ +i γ 5 . Following Diehl [10] we define By comparing the definitions for G i Λ ′ Λ in Eq. (28) and the projections of the non-forward correlation function Φ with matrix i σ i+ γ 5 , the GPDs, H T ,H T , E T , andẼ T , arise. In fact the trace of the ansatz for the non-forward quark-quark correlation function with the matrix i σ i+ γ 5 gives a system of four linear independent equations where C (1) i , expressed in terms of the coefficients of the ansatz, reads and D (1) i reads wherek and 46 + x d 49 ) Inserting in Eq. (27) the results for the spinorial products, given in the Appendix A, we obtain the four possible helicity combinations Since we are only interested in the number of independent GPDs we refrain from isolating Note that in Eq. (11) we integrate over d 2k T dk − and thus we consider only distribution functions which do not depend on the transverse momentum of the quarks k T . We remark that, in principle by having an ansatz for the off-forward quark-quark correlator, one could extract generalized profile functions which depend additionally on the transverse momentum k T of quarks. For instance the investigation ofk T -depending ordinary parton distributions has been extensively carried out by many groups theoretically and experimental investiga-tions are currently under way. For non-forward processes no formalism for the systematic study ofk T -dependence has yet been attempted and an experimental program onk T effects seems far beyond present abilities. In the foreseeable future there are good prospects to acquire some knowledge on GPDs depending on (x, ξ, t; Q 2 ), as well as possibly additional k T -dependence.
We have worked out a powerful method of analysis which in the present paper was applied completely to the leading twist level. The same method can be implemented to investigate twist 3 and twist 4 generalized distribution functions. For instance one could expect that useful relations between leading and next to leading order generalized distributions could emerge as suggested by similar experience in the forward case. In this sense the present work represents a valuable starting point for further investigations.

APPENDIX A: SPINORIAL PRODUCTS
The independent spinorial products of ansatz (3) listed in Eq. (4) are most easily evaluated by using explicit expressions for light-cone helicity spinors [14] u LC (P, +) = 1 With this parametrization one obtains for the spinorial products in the helicity non-flip case, i.e. Λ ′ = Λ = ±1ū The spinorial products if the helicity is flipped and Λ ′ = −Λ = 1 arē where the phase factor is given as and which contains the implicit definition of the quantity t 0 . Studying the form factor decomposition of the tensor current of the proton [10], we need additionally the following Dirac bilinears. For the helicity non-flip case, i. e., Λ ′ = Λ = ±1, we havē The results for the helicity flip case with Λ ′ = −Λ = +1 arē and for Λ ′ = −Λ = −1 where the phase is again given as in Eq. (A7). Note that latin indices i, j = 1, 2, while greek indices run from 0 to 3 and ǫ +1−2 = 1, since we have adopted the convention ǫ 0123 = 1.

RELATION FUNCTION
For the sake of convenience, in view of the numerous coefficients, we now switch over from explicitly displaying the hadron helicity indices to a more dense notation using the index (κ) defined as (1) = + +, (2) = + −, (3) = − +, and (4) = − − and suppressing spinor indices ij. Moreover contractions of momenta with Levi-Civita tensors will be written in the compact notation ǫ αβγδ a α b β c γ d δ ≡ ǫ abcd .
In order to write the ansatz for the quark-quark correlation function as indicated in Eq. (3) we now multiply the independent spinorial products, given in Eqs. (A4), (A5) and (A6), with the 16 independent partonic Dirac matrices and saturate the free indices with the following tensors ( indices α and β are intended saturated with spinorial products, while indices µ, ν with indices from partonic Dirac matrices ): The method introduced in Eq. m depending implicitly on all scalar products which can be formed from the momentum vectorsk,P and ∆ For the helicity non-flipped case with κ = 1, 4 the ansatz reads  κ = 1, 4 ), the ansatz for the quark-quark correlation function can be defined as real by multiplying the pure imaginary amplitudes a κ m by the imaginary unity i If the hadron helicity is flipped from Eqs. (A5) and (A6) one deduces that the ansatz has a factor −η (η * ), given in Eq. (16), which reflects the difference in phase of the initial and final hadronic spin. From the method introduced in Eq. (3) the ansatz for the off-diagonal components of the quark-quark correlation function ( κ = 2, 3 ) reads Because of the relations in Eq. (B9), (B10) and (B11) some of the amplitudes are zero. We refrain from rewriting the lengthy expression of the off-diagonal ansatz.