Spin effects in deeply virtual Compton scattering
Spin effects in deeply virtual Compton scattering
A.V. Belitsky, A. Kirchner, D. Müller
C.N. Yang Institute for Theoretical Physics
State University of New York at Stony Brook
NY 117943840, Stony Brook, USA
Institut für Theoretische Physik, Universität Regensburg
D93040 Regensburg, Germany
Fachbereich Physik, Universität Wuppertal,
D42097 Wuppertal, Germany
Abstract
We consider the azimuthal angle dependence in the cross section of the hard leptoproduction of a photon on a nucleon target. We show that this dependence allows to define observables that isolate the twisttwo and twistthree sectors in the deeply virtual Compton scattering amplitude. All twisttwo and twistthree Compton form factors can be extracted from measurements of the charge odd part of the polarized cross section and give access to all generalized parton distributions.
Talk given at the
IX International Workshop on Deep Inelastic Scattering
Bologna, 27 April  1 May 2001
We consider the azimuthal angle dependence in the cross section of the hard leptoproduction of a photon on a nucleon target. We show that this dependence allows to define observables that isolate the twisttwo and twistthree sectors in the deeply virtual Compton scattering amplitude. All twisttwo and twistthree Compton form factors can be extracted from measurements of the charge odd part of the polarized cross section and give access to all generalized parton distributions.
1 Introduction
The hard leptoproduction of a photon is a promising process to access a new type of nonperturbative distribution functions, the socalled generalized parton distributions (GPDs) [1, 2, 3]. In this process the partonic content of the hadron is studied by an electromagnetic probe and is free from the fragmentation in the final state present e.g. in the hard leptoproduction of a meson. Although the leptoproduction of a photon contains the contaminating BetheHeitler (BH) process in addition to the deeply virtual Compton scattering (DVCS), their interference gives a particular term since it provides a large number of observables that linearly depend on GPDs [2, 4, 5].
The GPDs are defined as nondiagonal matrix elements of lightray operators sandwiched between initial and final hadronic states with different momenta and possibly different quantum numbers:
(1) 
Here is a lightlike vector and denotes the Dirac and flavour structure. They depend on the (partonic) momentum fraction in the schannel, the longitudinal momentum fraction in the channel, called skewedness parameter, the momentum transfer , the resolution scale , and the helicities of the hadrons. Their evolution arises from the renormalization group equations of the lightray operators and is known to nexttoleading order (NLO) of perturbation theory.
As we see the GPDs have a complex structure. On the other side their definition implies also that they encode new information about the nonperturbative QCD sector. This information can not be obtained from other measurable nonperturbative quantities, i.e. parton densities or hadronic distribution amplitudes. The second moment of certain GPDs is directly related to gravitational form factors and could give some insight in the spin structure of the nucleon[6]. It is also very interesting, that their first moments give us partonic form factors. Moreover, the GPDs provide us with a link between different exclusive and inclusive quantities. In the forward case the nonhelicity flip GPDs reduce to the parton densities. In the ‘exclusive’ region of the phase space, i.e. , they are related to distribution amplitudes of mesonic like states, while in the ‘inclusive’ region they are analogous to parton densities. They are also related to hadronic wave functions by a DrellYanWest overlap type representation[7, 8].
2 The general azimuthal angular dependence
As mentioned before the hard leptoproduction of a photon contains two processes, the BetheHeitler and the DVCS (see Fig. 1).
The amplitude of the former one is parametrized for a nucleon target by the Pauli and Diracform factors and , which appear in the hadronic current
(2) 
where . The amplitude of the DVCS process is given in terms of the hadronic tensor that reads at twisttwo level [5]
(3) 
where ensures gauge invariance, and Analogous to Eq. (2), we decompose below the vector and axialvector amplitudes into form factors, the socalled Compton form factors (CFFs).
Let us start with the azimuthal angle dependence of the interference term for an unpolarized lepton scattering on a polarized nucleon in the target rest frame where . The interference term can be written as a contraction with a leptonic tensor :
(4) 
Here is built from the leptonic fourvectors and . The longitudinal polarization vector , having only a component, can be constructed from and . In this case the azimuthal angular dependence , where () is the azimuthal angle of the outgoing nucleon (lepton), can only arise from a contraction of with (or ) and from . Therefore, the highest moment in the Fourier series is :
(5) 
The additional dependence in the prefactor is induced by the (scaled) lepton propagators and of the BH process, where . The dimensionless Fourier coefficients depend on the beam and target polarization and , respectively. A more detailed analysis shows that () and () arise from the leading twisttwo sector, while and () appear at the twistthree level and are additionally suppressed by . The coefficient () is caused by the spinflip of the photon by two units and is, thus, related to the tensor gluon operators. Such a contribution is perturbatively suppressed by the strong coupling . In the case of a transversely polarized target, , the coefficients can be additionally decomposed with respect to their dependence:
(6) 
All the coefficients () appear in doublespin (singlespin) flip experiments and are given by the real (imaginary) part of CFFs. To extract them from the charge odd part of the cross section one has to form appropriate moments with the overall weight .
The charge even sector is given by the sum of the squared DVCS and BH term. In certain kinematical regions it is possible to access the former by subtracting the latter. The definition of the squared DVCS term
(7) 
allows an analysis analogous to the one given above:
(8) 
Here the coefficients , , and ( and ) appear at twisttwo (twistthree) level. However, and arise from the interference of the tree level contribution with the gluonic transversity and, thus, they are perturbatively suppressed by .
3 Predictions for twisttwo and three observables
We sketch now the calculation of the Fourier coefficients. First we parametrize the vector and axialvector contributions in four CCFs :
(9) 
where are vector, tensor, axialvector, and pseudoscalar Dirac bilinears and the ellipsis stand for higher twist contributions. The CCFs are given as convolutions of a hardscattering amplitude, calculable in perturbation theory, with the GPDs and read in leading order (LO)
(10) 
where the scaling variable is related to the Bjorken variable and are the charge fractions of quarks participating in the scattering. The complete NLO corrections to the twisttwo sector are worked out [9], while the twistthree sector is completely known to LO accuracy [10, 11].
The Fourier coefficients for unpolarized (unp), longitudinally (LP), and transversely polarized (TP) target are expressed in terms of combinations
(11)  
Note that these relations can be inverted to determine the set from the set . For instance, in the case of an unpolarized target the (twisttwo) and (twistthree) coefficients read
(12)  
(13) 
where with being the minimal value of . In the twistthree sector a new set of four twistthree CCFs arise. They are partly given by the twisttwo GPDs through the WandzuraWilczek relation but are also sensitive to new dynamical effects arising from antiquarkgluonquark correlations. The four twistthree coefficients are of purely kinematical origin and are given in terms of twisttwo CFFs. Up to different kinematical prefactors, relations (12,13) hold true for polarized targets too.
The squared DVCS term can be calculated in the same manner. The functional dependence at twisttwo and three levels should be universal, too.
4 Discussions and conclusions
We defined appropriate observables in the hard leptoproduction of a photon that separate the twisttwo and three sectors. We expect that higher twist contributions will only affect them by suppressed corrections. Facilities that have lepton beams of both charges can access the interference term. In combination with single () and double spin flip experiments () and by forming appropriate moments with respect to azimuthal angles one can extract all possible twisttwo and three CFFs:

unpolarized beam and unpolarized target

double spin flip with longitudinal polarized target

double spin flip with transverse polarized target

polarized beam and unpolarized target

unpolarized beam and longitudinal polarized target

unpolarized beam and transverse polarized target
Then the coefficients and of the interference and the squared DVCS term can serve for an experimental consistency check at twisttwo level while and allow the same at twistthree level.
The second step would be to extract information about the GPDs from such measurements by means of Eq. (10). In LO approximation one can directly extract the shape of the GPDs at the diagonal from the imaginary part of the CCFs. However, beyond the tree level this feature is lost. Indeed, the (magnitude of the) theoretical predictions is sensitive to special features, but, in general not to the shape of GPDs, so that experimental measurements can give constraints for models of GPDs. A crucial issue for this is the contamination of Eq. (10) by perturbative and power suppressed contributions. The former ones have been evaluated at NLO in the twisttwo sector and model dependent numerical estimates show that they can be large. The study of the latter has been just started with a consideration of the target mass corrections.
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