Deeply Virtual Pseudoscalar Meson Production at Jeﬀerson Lab and Transversity GPDs

The cross section of the exclusive π 0 and η electroproduction reaction ep → e (cid:1) p (cid:1) π 0 /η was measured at Jeﬀerson Lab with a 5.75-GeV electron beam and the CLAS detector. Diﬀerential cross sections d 4 σ/dtdQ 2 dx B dφ π and structure functions σ T + (cid:5)σ L ,σ TT and σ LT as functions of t were obtained over a wide range of Q 2 and x B . The data are compared with the GPD based theoretical models. Analyses ﬁnd that a large dominance of transverse processes is necessary to explain the experimental results. Generalized form factors of the transversity GPDs (cid:3) H T (cid:4) π,η and (cid:3) ¯ E T (cid:4) π,η were directly extracted from the experimental observables for the ﬁrst time. It was found that GPD ¯ E T dominates in pseudoscalar meson production. The combined π 0 and η data opens the way for the ﬂavor decomposition of the transversity GPDs. The ﬁrst ever evaluation of this decomposition was demonstrated.


Introduction
Understanding nucleon structure in terms of the fundamental degrees of freedom of Quantum Chromodynamics (QCD) is one of the main goals in the theory of strong interactions. In recent years it became clear that exclusive reactions may provide information about hadron structure encoded in so-called Generalized Parton Distributions 1, 2 (GPDs). For each quark flavor q there are eight GPDs. Four correspond to parton helicity-conserving (chiral-even) processes, denoted by H q ,H q , E q and E q , and four correspond to parton helicity-flip (chiral-odd) processes, 3 The GPDs depend on three kinematic variables: x, ξ and t. In a symmetric frame, x is the average longitudinal momentum fraction of the struck parton before and after the hard interaction and ξ (skewness) is half of the longitudinal momentum fraction transferred to the struck parton. The skewness can be expressed in terms of the Bjorken variable x B as ξ x B /(2 − x B ). Here x B = Q 2 /(2p · q) and t = (p − p ) 2 , where p and p are the initial and final four-momenta of the nucleon.
When the theoretical calculations for longitudinal virtual photons were compared with the JLab π 0 data 5, 6 they were found to underestimate the measured cross sections by more than an order of magnitude in their accessible kinematic regions. The failure to describe the experimental results with quark helicity-conserving operators stimulated a consideration of the role of the chiral-odd quark helicity-flip processes. Pseudoscalar meson electroproduction, and in particular π 0 production in the reaction ep → e p π 0 , was identified 7-9 as especially sensitive to the quark helicityflip subprocesses. During the past few years, two parallel theoretical approaches (GL) 7, 10 and (GK) 8,9 have been developed utilizing the chiral-odd GPDs in the calculation of pseudoscalar meson electroproduction. The GL and GK approaches, though employing different models of GPDs, lead to transverse photon amplitudes that are much larger than the longitudinal amplitudes.

Definition of Structure Functions
The unpolarized reduced meson cross section is described by 4 structure functions References 9, 10 obtain the following relations for unpolarized structure functions: Here m is the mass of the proton, t = t−t min , where |t min | is the minimum value of |t| corresponding to θ π = 0, k (Q 2 , x B ) is a phase space factor andĒ T = 2 H T + E T . The brackets H T and Ē T denote the convolution of the elementary process γ * q → qπ 0 with the GPDs H T andĒ T . We call them generalized form factors.

Experimental Data
Cross section of the reaction ep → epπ 0 measured by the CLAS collaboration at Jlab at 1800 kinematic points in bins of Q 2 , x B , t and φ were published in Refs. 5 and 6. Structure functions σ U = σ T + σ L , σ LT and σ T T have been obtained. These functions were compared with the predictions of the GPD models. 9, 10 CLAS confirmed that the measured unseparated cross sections are much larger than expected from leading-twist handbag calculations which are dominated by longitudinal photons. The same conclusion can be made in an almost model-independent way by noting that the structure functions σ U and |σ T T | are comparable to each other while |σ LT | is quite small (see Fig.1). Cross section and structure functions for ep → eηp were also obtained in parallel. The comparison of the π 0 and preliminary η structure functions is shown in Fig. 1. σ U drops by a factor of 2.5 for η in comparison with π 0 and σ T T drops by a factor of 10. The GK GPD model 9 (curves) follows the experimental data. The inclusion of η data into consideration strengthens the statement about the transversity GPD dominance in the pseudoscalar electroproduction process.

Generalized Form Factors
The squared magnitudes of the generalized form factors | H T | 2 and Ē T 2 may be directly extracted from the experimental data (see Eqs. 2 and 4 ) in the framework of GPD models.

Flavor Decomposition
In electroproduction the GPDs F i appears in the following combinations The q andq GPDs contribute in the quark combinations F q i − Fq i . Hence there is no contribution from the strange quarks if we assume that F s i Fs i . For flavor decomposition we have to take into account the decay constants f π and f η , the chiral condensate constants µ π 0 =2.57 GeV, µ 1 =0.958 GeV and µ 8 =2.32 GeV, and the contribution from singlet and octet η states. 9 where the mixing angles are: θ 8  and e d = − 1 3 we will end up with equations Experimentally we have access only to the | F π i | 2 and | F η i | 2 (see Eq. 5). The final and simular equations for Ē T . The solution of these equations will lead to the flavor decomposition of the transversity GPDs H T u and H T d as well as Ē T u and Ē T d . However the convolution integrals have real and imaginary parts. So it is impossible to solve these equations unambiguously with only two equations in hands. As a guidance we can estimate the form factors if we suppose that the relative phase ∆φ between H T u and H T d equals 0 or 180 degrees. Ignoring an overall phase, the form factors are then real and we arbitrarily choose the solution with H T u and Ē T u positive. Fig. 3

Conclusion
Differential cross sections of exclusive π 0 and η electroproduction have been obtained in the few-GeV region at more than 1800 kinematic points in bins of Q 2 , x B , t and φ π . Virtual photon structure functions σ U , σ T T and dσ LT have been obtained. It is found that σ U and σ T T are comparable in magnitude with each other, while σ LT is very much smaller than either. Generalized form factors of the transversity GPDs H T π,η and Ē T π,η were directly extracted from the experimental observables for the first time. It was found that the GPDĒ T dominates in pseudoscalar meson production. The combined π 0 and η data opens the way for the flavor decomposition of the transversity GPDs. Within some simplifying assumptions, the decomposition has been demonstrated for the first time.