Kinematical Issues in the GPD Formulation of DVCS

The kinematics used in computing deeply virtual Compton scattering makes a significant difference in terms of the widely used reduced operators that define generalized parton distributions. We analyze this difference at tree-level.

This kinematics leads to the relation t = Δ 2 = (ζ P) 2 = ζ 2 M 2 > 0, while DVCS requires t ≤ 0. Thus, this kinematics becomes consistent with DVCS if one sets the nucleon mass M to zero, and thus t → 0. In this limit Ji's ξ and Radyushkin's ζ are related by ζ = 2ξ 2+ξ , ξ = 2ζ 2−ζ . If the mass of the target nucleon may be neglected in the kinematics, P and q can also be considered Sudakov vectors, i.e., P μ /P + = n μ (+) and q μ /q − = P + q μ /P.q = n μ (−). The hadronic tensor is written by Radyushkin as follows (our notation) Despite the fact that the hadronic tensor is written in terms of physical momenta, this kinematics is accurate in leading twist only. In a frame where all physical momenta are aligned, i.e., the extreme collinear kinematics, the two formulations are equivalent. Here we are interested in the differences between the collinear kinematics and a realistic one, e.g., in JLab.

Tree-Level Calculation
We performed a tree-level calculation of DVCS, which is relevant for the hard-scattering part of the physical DVCS amplitude. It avoids computational complexity and issues of model building. Our results [3] are obtained for structureless, massless spin-1/2 hadrons and massless electrons. We expect that the differences found here will have consequences for realistic physical situations. We use the overall plus-momentum scale P + . The hadron momenta are now called k and k , which corresponds to the tree-level amplitude to be understood as the hard-scattering part of the DVCS amplitude. The hadron mass is set to 0. The following momenta are kept fixed Three types of kinematics are used (K 1 , used earlier [12], is not discussed here). K 2 Kinematics, completely collinear K 3 Non-vanishing q + and q + kinematics The Mandelstam variables in these two kinematics are the same, in particular t = 0.
To see the effect of taking t < 0 we mimic the kinematics at JLab in kinematics K 4 , where the final hadron and final photon move off the z-axis while the initial hadron and virtual photon move along the z-axis, i.e., k μ and q μ are given in Eqs. 6 and 7. The quantity ζ eff , determined by four-momentum conservation, is given by The complete amplitude is the amplitude for the physical process which splits into a leptonic and a hadronic part The operators occurring are, for massless hadrons The polarization vectors are defined in Ref. [13] and the spinors are those given in Ref. [14] 4

Results in Three Kinematics
The leptonic amplitudes are displayed in Table 1 and the hadronic parts in Table 2.
The difference between K 2 and K 3 shows the difference of the contributions from the longitudinal virtual photon between the two kinematics. The difference in the hadronic amplitudes between K 2 and K 3 is also related to the definition of the helicity of the final-state photon, and the difference with K 4 is due to the difference in t, being less than 0 in that kinematics. Now we compare the results obtained from different formulations, namely Refs. [1], [2], with our solvablemodel results. We shall show in our calculations that the terms missing in the other two are of order Δ 2 ⊥ /Q 2 . This confirms that the use of Eqs. 4 and 5 are limited to the lowest twist.
Rewriting the s-and u-channel hadronic amplitudes as we find respectively. Using the identity one finds for T s in leading order of Q A similar expression is found for T u . This expression is equivalent to those given in Refs. [1], [2]. One fully recovers in this approximation the expressions given in Refs. [1,2] for the hadronic tensors after adding the s-and u-channel parts. Upon contracting them with the polarization vectors, we obtain full agreement in the hadronic amplitudes between K 2 and K 4 in the DVCS limit except for the swap between the amplitudes with h = +1 and −1. This swap will be explained below in our complete calculation. In kinematics K 3 and K 4 there is, however, a discrepancy. To understand this we expand the momenta in Sudakov vectors, but retain the remainder, denoted as perpendicular parts. In our kinematics where q and k are aligned with the z-axis we have (k − = 0 in the massless case) In our more general kinematics K 4 we thus find for T s (see also Vanderhaegen et al. [6][7][8][9][10][11] where transversemomentum components are included) By retaining only the terms proportional to the highest power in Q, namely those proportional to q − , we obtain the approximate tensor shown in Eq. 17. However, the amplitudes being obtained by contraction with the polarization vectors are sensitive to the neglected parts as we show in Table 3, where we give the results for the complete amplitudes in kinematics K 4 using the formalisms of Ref. [1] (XJ), Ref. [2] (AVR), and our exact calculation up to the order of Δ 2 ⊥ /Q 2 . The differences are summarized in the last Section. Owing to the definition of the LF helicity, it is opposite to the IF helicity if the momentum of the photon in the final state is pointing strictly in the −z-direction. This swap explains the differences among the results in kinematics K 2 , K 3 , and K 4 at t = 0 displayed in Table 4, where we summarize our results for the complete amplitudes. The sign ± in this Table denotes the helicity of the emitted photon. Table 4 Comparison of the three approaches in three kinematics

Summary and Conclusions
By applying the GPD formulation to an exactly solvable model, namely DVCS at tree level, we have been able to pinpoint the differences between the lowest-twist formulations of Refs. [1], [2] and the exact results.
1. In a kinematics where the hadrons and the photons are aligned (fully collinear kinematics K 2 ), the three approaches give identical results in the DVCS limit Q → ∞ except for the swap between the amplitudes with h = +1 and −1. In kinematics K 4 we get agreement if we take there the limit t → 0. 2. The effect of Δ ⊥ is power suppressed as Δ 2 ⊥ /Q 2 . The amplitude with λ = 1/2, h = −1, and s = 1/2 does not vanish to this order of Δ 2 ⊥ /Q 2 , while the original leading-twist formulas cannot generate this non-zero amplitude. 3. The formulation of Ref. [1] can be used if the difference between light-front and instant-form helicities are taken into account. 4. Since both lowest-twist approximations Refs. [1], [2] are accurate to zeroth order in Δ 2 ⊥ /Q 2 , the lowest-twist formulae can be used to test the sum rules that relate the form factors to the GPDs only at t = 0.
Having said this, we note that including the transverse momentum component in the kinematic tensor part of T μν does not completely cover the higher twist effects. The inclusion of higher twist effects [6][7][8][9][10][11]15] is at the current forefront of research and it may well go beyond the handbag approximation.
We caution against using the lowest-twist formulas in the analysis of experimental data in situations where the net transverse-momentum transfer to the target is not small compared to Q. In this respect, it may be useful to note that e.g. in JLab the values of Δ 2 ⊥ /Q 2 are larger than 5-10%.
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