Hard Exclusive Reactions and the Structure of Hadrons
Abstract
We outline in detail the properties of generalized parton distributions (GPDs), which contain new information on the structure of hadrons and which enter the description of hard exclusive reactions. We highlight the physics content of the GPDs and discuss the quark GPDs in the large limit and within the context of the chiral quarksoliton model. Guided by this physics, we then present a general parametrization for these GPDs. Subsequently we discuss how these GPDs enter in a wide variety of hard electroproduction processes and how they can be accessed from them. We consider in detail deeply virtual Compton scattering and the hard electroproduction of mesons. We identify a list of key observables which are sensitive to the various hadron structure aspects contained in the GPDs and which can be addressed by present and future experiments.
Institut für Theoretische Physik II, RuhrUniversität Bochum, D44780 Bochum, Germany
Petersburg Nuclear Physics Institute, 188350, Gatchina, Russia
Institut für Kernphysik, Johannes GutenbergUniversität, D55099 Mainz, Germany
Prog. Part. Nucl. Phys. 47 (2001) 401  515
Contents
 1 INTRODUCTION

2 GENERALIZED PARTON DISTRIBUTIONS (GPDs)
 2.1 Definitions, link with ordinary parton distributions and nucleon form factors
 2.2 Polynomial conditions, Dterm
 2.3 Angular momentum sum rule
 2.4 Chiral properties of GPDs
 2.5 Twist3 GPDs
 2.6 GPDs for transitions : SU(3) relations and beyond
 2.7 GPDs for transition
 2.8 Further generalizations: GPDs for transitions
 3 GPDs IN THE LARGE LIMIT
 4 GPDs FROM HARD ELECTROPRODUCTION REACTIONS
 5 CONCLUSIONS, PERSPECTIVES AND KEY EXPERIMENTS
1 Introduction
The famous Lagrangian of Quantum Chromodynamics (QCD):
(1) 
contains, in principle, all phenomena of hadronic and nuclear physics ranging from the physics of pions to the properties of heavy nuclei. The main difficulty and challenge in the derivation of strong interaction phenomena from the QCD Lagrangian (1) is that the theory is formulated in terms of quark and gluon degrees of freedom whereas phenomenologically one deals with hadronic degrees of freedom. Understanding of how colorless hadrons are built out of colored degrees of freedom (quarks and gluons) would allow us to make predictions for strong interaction phenomena directly from the Lagrangian (1). The physics of hadronization of quarks and gluons is governed by such phenomena as confinement and spontaneous chiral symmetry breaking which are in turn related to the nontrivial structure of the QCD vacuum. It implies that the studies of hadronization processes provide us with valuable information on the fundamental questions of the vacuum structure of nonabelian gauge theories.
The quark and gluon structure of hadrons can be best revealed with the help of weakly interacting probes, such as (provided by Nature) photons and , bosons. One needs probes which are weakly coupled to quarks in order to “select” a well defined QCD operator expressed in terms of quark and gluon degrees of freedom of the Lagrangian (1). By measuring the reaction of a hadron to such a probe, one measures the matrix element of the welldefined quarkgluon operator over the hadron state revealing the quarkgluon structure of the hadron.
Historically the famous experiment of Otto Stern et al. [Fri33] measuring the anomalous magnetic moment of the proton revealed for the first time the nontrivial (now we say quarkgluon) structure of the proton. In modern language we can say that in this experiment with the help of the low energy photon probe (weakly interacting to quarks) one selects the QCD operator , then one couples it to the proton, , and extracts the structural information encoded in the values of hadron form factors. The experimental observation of a large Pauli form factor ^{1}^{1}1 The irony is that this form factor carries the name of W. Pauli who tried to talk out O. Stern of his experiment, because for all theoreticians at that time there were no doubts that the proton is a point like particle. See the nice historical overview in Ref. [Dre00]. of the proton [Fri33] has created the physics of hadrons as strongly interacting many body systems.
Are we limited to explore the structure of hadrons by QCD operators created by photons and bosons? Can we find other weak coupling mechanism to select more sophisticated QCD operators to explore the structure of hadrons? Actually weak interactions as such are an inherent property of QCD due to the phenomenon of asymptotic freedom meaning that at short distances the interactions between quarks and gluons become weak. This implies that if one manages to create a small size configuration of quarks and gluons it can be used as a new probe of hadronic structure. The possibility to create small size configurations of quarks and gluons is provided by hard reactions, such as deep inelastic scattering (DIS), semiinclusive DIS, DrellYan processes, hard exclusive reactions, etc.
The common important feature of hard reactions is the possibility to separate clearly the perturbative and nonperturbative stages of the interactions, this is the socalled factorization property. Qualitatively speaking, the presence of a hard probe allows us to create small size quark, antiquark and gluon configurations whose interactions are described by means of perturbation theory due to the asymptotic freedom of QCD. The nonperturbative stage of such a reaction describes how a given hadron reacts to this configuration, or how this probe is transformed into hadrons.
In the present work we deal with hard exclusive reactions of the type:
(2) 
in which a photon with high energy and large virtuality scatters off the hadronic target and produces a meson (or a low mass mesonic cluster) or a real photon , and a lowmass hadronic state (where the mass is small compared to ). The all order factorization theorems for such kind of reactions have been proven in Ref. [Col97] (meson production) Refs. [Ji98a, Col99, Rad98] (real photon production) . These theorems assert that the amplitude of the reactions (2) can be written in the form (we show this for the case of hard meson production [Col97]):
(3) 
where is a generalized parton distribution (GPD), [Bar82, Dit88, Mul94, Ji97b, Rad97, Col97] (see details in Sec. 2), is the fraction of the target momentum carried by the interacting parton, is the distribution amplitude (DA) of the meson, and is a hardscattering coefficient, computable as a power series in the strong coupling constant . The amplitude depends also on the Bjorken variable , and on the momentum transfer squared which is assumed to be much smaller than the hard scale . In Eq. (3) the GPDs and the meson DAs contain the nonperturbative physics and the perturbative one. The proof of cancellation of the soft gluon interactions in the processes (2) is intimately related to the fact that the final meson arises from a quarkantiquark (gluon) pair generated by the hard interaction. Thus the pair starts as a small size configuration and only substantially later grows to a normal hadronic size, i.e. into a meson.
Qualitatively one can say that the reactions (2) allow one to perform a “microsurgery” of a nucleon by removing in a controlled way a quark of one flavor and spin and implanting instead another quark (in general with a different flavor and spin). It is illustrated in Fig. 1 for the case of the deeply virtual Compton scattering (DVCS) and in Fig. 2
for hard exclusive meson production (HMP). The lower blobs in both figures correspond to GPDs. It is important that according to the QCD factorization theorem in both types of processes the same universal distribution functions enter. This allows us to relate various hard processes to each other. Additionally the detected final state can be used as a filter for the spin, flavor, Cparity, etc. of the removed and implanted quarks.
In the processes as in Eq. (2) the short distance stage of the reaction described by in Eq. (3) corresponds to the interaction of a parton with a highly virtual photon. This stage is described by perturbative QCD. The corresponding hard scattering coefficients have been computed to the NLO order in Refs. [Ji98a, Bel98b, Man98b] for DVCS and recently in Ref. [Bel01c] for the hard pion production. Also the perturbative evolution of generalized parton distributions is elaborated to the NLO order [Bel99, Bel00d, Bel00e]. This shows that the perturbative QCD calculations of the hard scattering coefficients and evolution kernels are under theoretical control. In this work we shall not review the corresponding perturbative calculations, for a review see Refs. [Ji98b, Rad01b], and for our analysis we will always stick to the LO expressions.
We can say that the hard scattering part (“handbag part” of Fig. 1 and the part denoted as in Fig. 2) of the process “creates” a welldefined QCD operator which is “placed” into the target nucleon and the outgoing meson. We concentrate in present work mostly on studies of such nonperturbative objects entering the factorization theorem (3). They can be generically described by the following matrix elements:
(4) 
where the operators are on the lightcone, i.e. . Depending on the initial and the final states and , the following nomenclature of the nonperturbative matrix elements exists:

one particle state – parton distributions in the hadron . These objects enter the description of hard inclusive and semiinclusive reactions.

vacuum, one particle state – distribution amplitudes or lightcone wave functions of a hadron . They enter the description of, say, hadronic form factors at large momentum transfer as well as hard exclusive production of mesons.

Both and are one particle states with different momenta, i.e. – generalized parton distributions. They enter the description of hard exclusive production of mesons and deeply virtual Compton scattering.

vacuum, = many particle state – generalized distribution amplitudes. They enter the description of hard exclusive multimeson production or transition form factors between multimeson states at large momentum transfer.

one particle state, many particle state, e.g. , etc. – can be called many body generalized parton distributions. They enter the description of semiexclusive production of mesons and semiexclusive DVCS.

many particle state – can be called interference parton distributions. An interesting object which has not yet been considered in the literature.
Although all these objects have different physical meaning, their unifying feature is that they describe the response of hadronic states to quarkgluon configurations with small transverse size, which have an extension along the lightcone direction. This can also be rephrased by saying that one probes the transition from one hadronic state to another by lowenergy extended objects–“QCD string operators” (4). From such a point of view, the studies of hard exclusive reactions allow us to extend considerably the number of lowenergy fundamental probes that can be used to study hadrons. An advantage of hard exclusive reactions is that in this case the number of probes with different quantum numbers is very large and also that these reactions probe nondiagonal transitions in flavor and spin spaces. To great extent, the properties of these new probes have not been studied, especially such aspects as chiral perturbation theory for these probes^{2}^{2}2See however works in this direction [Pol99a, Leh01, Tho00, Pob01, Che01, Arn01] which we do not review here., the large limit, the influence of spontaneous breaking of the chiral symmetry, etc. In the present work we attempt to address these questions.
As an illustration of the wider scope of hard exclusive reactions for studies of hadronic structure we give here a qualitative discussion to what type of usual parton distributions these reactions are sensitive. We shall see that the hard exclusive reactions allow us (leaving aside truly nonforward effects) to probe flavor and Cparity combinations of usual parton distributions which are not accessible in inclusive measurements. The amplitude of a hard exclusive meson production on the nucleon depends on the nucleon generalized parton distributions. In the limiting case of zero “skewedness” (i.e. when the momentum transfer in Figs. 1,2) the latter are reduced to usual parton distributions in the nucleon. In this sense it holds that the hard exclusive meson production is, among other things, sensitive to the usual parton distributions in the nucleon. In Tables 13 [Fra01] we give examples of hard exclusive production processes of mesons with various quantum numbers on a proton target and list the corresponding parton distributions to which the amplitudes are sensitive. In these tables the non empty entries mean that the given combination of parton distributions can be probed in the production of a mesonic system with the given quantum numbers.
The physics content of the GPDs is of course not reduced to the forward quark distributions. In fact, the “truly nonforward” effects play a crucial role in the physics of hard exclusive processes (2), i.e. HMP as well as DVCS. These effects also contain new information about the structure of mesons and baryons which is not accessible by standard hard inclusive reactions.
The paper is organized as follows. In section 2 we review the formalism of generalized parton distributions. We concentrate on such properties of quark GPDs which are related to the structure of the nucleon leaving aside a discussion of the perturbative evolution which has been extensively reviewed in Refs. [Ji98b, Rad01b]. Section 3 is devoted to studies of quark GPDs in the limit of a large number of colors . In this chapter we also review and present new calculations of GPDs in the chiral quarksoliton model. The basic structures of GPDs discussed in the sections 2 and 3 are then used in chapter 4 in order to construct a phenomenological parametrization of the GPDs. These parametrizations are then used to make predictions for a wide variety of hard electroproduction processes. Since our main objective is to investigate how to reveal the basic features of the nucleon structure in these reactions we restrict ourselves in the calculation of the hard interaction kernels to the leading perturbative order for observables. Studies of the NLO effects on observables can be found in Refs. [Bel98a, Bel00a, Bel01c] ^{3}^{3}3In these papers it was found that for some of observables the NLO corrections can be rather large. This implies that more detail studies of the NLO effects are needed.. In particular, we perform detailed studies for DVCS observables in Sec. 4.3 at twist2 and twist3 level. We demonstrate how various DVCS observables are sensitive to parameters of GPDs, e.g. to the quark total angular momentum contribution to the nucleon spin. In Sec. 4.5 we study hard meson production discussing in detail for each specific process the information about nucleon structure one accesses. In all calculations we restrict ourselves to the region of not too small (valence region), where one is mainly sensitive to the quark GPDs. Discussions of various aspects of hard exclusive meson production at small can be found e.g. in Refs. [Bro94, Fra96, Abr97, Mar98b, Iva98, Iva99, Shu99, Cle00] and of DVCS at small in Ref. [Fra98b, Fra99a]. We note that in the hard exclusive processes at small the GPDs can be related to the usual parton distributions in a model independent way.
The GPDs have also been discussed within the context of wide angle Compton scattering [Rad98, Die99] where the GPDs are related to soft overlap contributions to elastic and generalized form factors (for a review of such processes see Ref. [Rad01b]). Additionally the GPDs enter the description of the diffractive photoproduction of heavy quarkonia, see e.g. Refs. [Rys97, Fra98c, Fra99c, Mar99], as well as in diffractive jets [Gol98] and in jets processes [Fra93, Fra00b, Bra01, Cher01]. We do not review these applications in the present work.
Finally, we summarize our main findings in section 5 and list a number of key experiments to access the GPDs and to extract their physics content.
2 GENERALIZED PARTON DISTRIBUTIONS (GPDs)
In this section we consider basic properties of the generalized parton distributions (GPDs). We shall pay special attention to such properties of GPDs which are related to the nonperturbative structure of hadrons and the QCD vacuum. We shall not discuss the perturbative evolution of the GPDs, as this issue is excellently reviewed in [Ji98b, Rad01b]. Our aim will be to cover topics which are complementary to the ones discussed in those reviews [Ji98b, Rad01b].
2.1 Definitions, link with ordinary parton distributions and nucleon form factors
The factorization theorem for hard exclusive reactions allows us to describe the wide class of such reactions in terms of universal generalized parton distributions. This nucleon structure information can be parametrized, at leading twist–2 level, in terms of four (quark chirality conserving) generalized structure functions. These functions are the GPDs denoted by (we do not consider chirally odd GPDs, which are discussed in Refs. [Hoo98, Die01b]) which depend upon three variables : , and . The lightcone momentum ^{4}^{4}4using the definition for the lightcone components fraction is defined (see Figs. 1 and 2) by , where is the quark loop momentum and is the average nucleon momentum (, where are the initial (final) nucleon fourmomenta respectively). The skewedness variable is defined by , where is the overall momentum transfer in the process, and where in the Bjorken limit. Furthermore, the third variable entering the GPDs is given by the Mandelstam invariant , being the total squared momentum transfer to the nucleon. ^{5}^{5}5In what follows we shall omit the dependence of GPDs if the corresponding quantity is assumed to be defined at . In a frame where the virtual photon momentum and the average nucleon momentum are collinear along the axis and in opposite direction, one can parametrize the nonperturbative object in the lower blobs of Figs. 1 and 2 as (we follow Ref. [Ji97b]) ^{6}^{6}6In all nonlocal expressions we always assume the gauge link ensuring the color gauge invariance of expressions.:
(5)  
where is the quark field
of flavor , the nucleon spinor
and the nucleon mass.
The lhs of Eq. (5) can be interpreted as a Fourier
integral along the lightcone distance of a quarkquark
correlation function, representing the process where
a quark is taken out of the
initial nucleon (having momentum ) at the spacetime point , and
is put back in the final nucleon (having momentum ) at the spacetime
point . This process takes place at equal lightcone time () and at zero transverse separation () between
the quarks. The resulting onedimensional Fourier integral along the
lightcone distance is with respect to the quark lightcone
momentum .
The rhs of Eq. (5) parametrizes this
nonperturbative object in terms of four GPDs, according to whether
they correspond to a vector operator or
an axialvector operator at the
quark level. The vector operator corresponds at the nucleon side
to a vector transition (parametrized by the function , for a quark
of flavor ) and
a tensor transition (parametrized by the function ).
The axialvector operator corresponds at the nucleon side
to an axialvector transition (function )
and a pseudoscalar transition (function ).
In Fig. 1, the variable runs from 1 to 1.
Therefore, the momentum fractions ( or ) of the
active quarks can either be positive or negative. Since positive
(negative) momentum fractions correspond to quarks (antiquarks), it
has been noted in [Rad96a] that in this way, one can
identify two regions for the GPDs :
when both partons represent quarks, whereas for
both partons represent antiquarks. In these regions,
the GPDs are the generalizations of the usual parton distributions from
DIS. Actually, in the forward direction, the GPDs and
reduce to the quark density distribution and
quark helicity distribution respectively, obtained from DIS :
(6)  
(7) 
The functions and are not measurable
through DIS because the associated tensors
in Eq. (5) vanish in the forward limit ().
Therefore, and are new leading twist functions, which
are accessible through the
hard exclusive electroproduction reactions, discussed in the following.
In the region , one parton connected to the lower
blob in Fig. 1 represents a
quark and the other one an antiquark. In this region, the GPDs
behave like a meson distribution amplitude and contain completely new
information about nucleon structure, because the region
is absent in DIS, which corresponds to the limit
.
Besides coinciding with the quark distributions at vanishing momentum
transfer, the generalized parton distributions have interesting links with other
nucleon structure quantities. The first moments of the GPDs are related to
the elastic form factors
of the nucleon through model independent sum rules.
By integrating Eq. (5) over , one
obtains for any
the following relations for a particular quark flavor [Ji97b] :
(8)  
(9)  
(10)  
(11) 
where represents the elastic Dirac form factor for the quark flavor in the nucleon. When referring to the quark form factors in the following, we understand them in our notation to be for the proton, e.g. . In this notation, the quark form factor is normalized at as so as to yield the normalization of 2 for the quark distribution in the proton, whereas the quark form factor is normalized at as so as to yield the normalization of 1 for the quark distribution in the proton. These elastic form factors for one quark flavor in the proton, are then related to the physical nucleon form factors (restricting oneself to the and quark flavors), using isospin symmetry, as :
(12) 
where and are the proton and neutron electromagnetic form factors respectively, with and . In Eq. (12) is the strangeness form factor of the nucleon. Relations similar to Eq. (12) hold for the Pauli form factors . For the axial vector form factors one uses the isospin decomposition :
(13) 
where are the isovector (isoscalar) axial form factors of the nucleon respectively. Similar relations exist for . The isovector axial form factor is known from experiment, with . The induced pseudoscalar form factor contains an important pion pole contribution, through the partial conservation of the axial current (PCAC), as will be discussed in more details below.
An interesting connection of GPDs to lightcone wave functions of the nucleon was considered in Refs. [Die01a, Bro01]. In these papers an exact representation of generalized parton distributions for unpolarised and polarized quarks and gluons as an overlap of the nucleon lightcone wave functions has been constructed. Using such an overlap representation of the GPDs model calculations have been presented in Ref. [Cho01, Tib01, Burk01].
2.2 Polynomial conditions, Dterm
One of the nontrivial properties of the generalized parton distributions is the polynomiality of their Mellin moments which follows from the Lorentz invariance of nucleon matrix elements [Ji97b]. Indeed the th Mellin moment of GPDs corresponds to the nucleon matrix element of the twist2, spin local operator. Lorentz invariance then dictates that the Mellin moments of GPDs should be polynomials maximally of the order , i.e. the polynomiality property implies that [Ji97b]^{7}^{7}7The general method for counting of generalized form factors of twist2 operators can be found in Ref. [Ji01]:
(14)  
Note that the corresponding polynomials contain only even powers of the skewedness parameter . This follows from the time reversal invariance, see Ref. [Man98a, Ji98b]. This fact implies that the highest power of is for odd (singlet GPDs ) and for even (nonsinglet GPDs). Furthermore due to the fact that the nucleon has spin , the coefficients in front of the highest power of for the singlet functions and are related to each other [Ji97b, Ji98b]:
(15) 
The polynomiality conditions (14) strongly restrict the class of functions of two variables and . For example the conditions (14) imply that GPDs should satisfy the following integral constrains:
(16)  
Note that the skewedness parameter enters the lhs of this equation, whereas the rhs of the equation is independent. Therefore this independence of the above integrals is a criterion of whether the functions , satisfy the polynomiality conditions (14). Simultaneously these integrals are generating functions for the highest coefficients . In addition, the condition (16) shows that there are nontrivial functional relations between the functions and .
An elegant possibility to implement the polynomiality conditions (14) for the GPDs is to use the double distributions [Mul94, Rad96a, Rad97]. A detailed discussion of the double distributions has been given in the review of Ref. [Rad01b]. In this case the generalized distributions are obtained as a one–dimensional section of the two–variable double distributions (Recently in Refs. [Bel00c, Ter01] the inversion formula has been discussed):
(17) 
and an analogous formula for the GPD :
(18) 
Obviously, the double distribution function should satisfy the condition:
(19) 
in order to reproduce the forward limit (6) for the GPD . It is easy to check that the GPDs obtained by reduction from the double distributions satisfy the polynomiality conditions (14) but always lead to , i.e. the highest power of for the singlet GPDs is absent. In other words the parametrization of the singlet GPDs in terms of double distributions is not complete. It can be completed by adding the socalled Dterm to Eq. (17) [Pol99b]:
(20) 
Here is an odd function (as it contributes only to the singlet GPDs) having a support . In the Mellin moments, the Dterm generates the highest power of :
(21) 
Note that for both GPDs and the absolute value of the Dterm is the same, it contributes to both functions with opposite sign. The latter feature follows from the relation (15). We shall see in the section 3.2 that estimates of the Dterm in the chiral quark–soliton model gives . Therefore in the calculations below we shall assume that:
(22) 
where is the number of active flavors and is the flavor singlet Dterm.
The Dterm evolves with the change of the renormalization scale according to the ERBL evolution equation [Efr80, Lep79]. Hence it is useful to decompose the Dterm in a Gegenbauer series (eigenfunctions of the LO ERBL evolution equation):
(23)  
(24) 
where represents the gluon Dterm. Because the Dterm is a singlet quantity therefore the quark Dterm is mixed under evolution with the gluon Dterm . Asymptotically (for a renormalization scale ) we obtain:
(25)  
(26) 
where . Furthermore, the scale independent constant is a nonperturbative parameter characterizing the Dterm at a low normalization point, with from Eq. (23) and being the first Gegenbauer coefficient of the gluon Dterm (24). We see that the Dterm survives in the limit and therefore the complete form^{8}^{8}8In the literature, to the best of our knowledge, only an incomplete form of the asymptotic singlet GPDs was presented. of the singlet quark GPDs at an asymptotically large scale is the following:
(27) 
The asymptotic form of the corresponding gluon GPD is the following:
(28) 
Analogously, one can obtain the asymptotic form of the singlet GPD :
(29) 
The asymptotic form of the corresponding gluon GPD is the following:
(30) 
Furthermore, note that asymptotically the GPD is completely determined by the Dterm. Note that all expressions for the asymptotic singlet GPDs depend on the scale independent (conserved) constant . From this point of view this constant is as fundamental as other conserved characteristics of the nucleon, such as the total momentum or total angular momentum.
Up to now we considered the polynomiality properties of the GPDs and . For the quark helicity dependent GPDs and the polynomiality conditions are very similar to those for and , see Eq. (14). The only difference is that in the case of the quark helicity dependent GPDs the highest powers of the polynomial in is for the singlet case and for the nonsinglet one. This implies that the Dterm is absent for the GPDs and .
2.3 Angular momentum sum rule
The second Mellin moments of the quark helicity independent GPDs are given by the nucleon form factors of the symmetric energy momentum tensor. The quark part of the symmetric energy momentum tensor is related to the quark angular momentum operator by:
(31) 
This relation implies that the forward nucleon matrix element of the angular momentum operator can be related to the form factor of the symmetric energy momentum tensor. This results in the sum rule relating the second Mellin moment of the GPDs to the angular momentum carried by the quarks in the nucleon [Ji97b]:
(32) 
In Eq. (32) is the fraction of the nucleon angular momentum carried by a quark of the flavor (i.e. the sum of spin and orbital angular momentum). The closest analogy for this sum rule is the relation between the magnetic moment of the nucleon and the eletromagnetic (e.m.) form factors:
(33) 
In this case the magnetic moment is given by the “forward” form factor plus the “nonforward” form factor . In analogy to Eq. (31) the relation between the magnetic moment operator and the e.m. current: contains explicitly the coordinate operator . A detailed derivation and discussion of the angular momentum sum rule (32) can be found in Refs. [Ji97b, Ji98b].
Let us discuss here the role played by the Dterm in the angular momentum sum rule. Unfortunately it is very hard to find an observable in which the GPDs and enter as a sum. Also we should keep in mind that in a hard exclusive process, we have kinematically and that the GPDs , enter observables with different kinematical factors. Therefore, we shall discuss the angular momentum sum rule (32) separately for the functions and . We rewrite Ji’s sum rule (32) in the following equivalent way [Ji97a]
(34) 
Here is the first Gegenbauer coefficient in the expansion of the Dterm (23), and is the momentum fraction carried by the quarks and antiquarks in the nucleon:
(35) 
We see that the sum rule (34) is sensitive only to the combination . What can we say about this combination? Asymptotically one has that [Ji96]. At some finite scale, the value of which one would like to extract from the data should be compared with the contribution of the Dterm on the rhs of the sum rule (34). We shall see below that estimates in the chiral quarksoliton model for the first Gegenbauer coefficient of the Dterm give the value , which leads at to a value of for the second term in Eq. (34). Therefore at accessible values of the extraction of the angular momentum carried by the quarks from the observables requires (among other things) an accurate knowledge of the Dterm. The size of the Dterm can be determined from the observables which are sensitive to the real part of the amplitude of the corresponding process. An example of such an observable is the DVCS charge asymmetry (accessed by reversing the charge of the lepton beam), see a detailed discussion in Sec. 4.3.4.
2.4 Chiral properties of GPDs
Here we shall argue that the spontaneously broken chiral symmetry of QCD plays an important role in determining the properties of the generalized parton distributions. We shall illustrate this on examples of the Dterm and the GPD .
2.4.1 Spontaneously broken chiral symmetry and the Dterm
First we will discuss how the physics of spontaneously broken chiral symmetry plays an important role in determining the size and the sign of the Dterm. In particular, this role is seen clearly in the case of the GPDs in the pion. In this case the value of the coefficient in the parametrization of the pion Dterm (23) can be computed in a model independent way and it is strictly nonzero. To compute the pion Dterm we use the softpion theorem for the singlet GPD in the pion derived in [Pol99a]. This softpion theorem has been derived using the fact that the pion is a (pseudo)Goldstone boson of the spontaneously broken chiral symmetry. The theorem states that the singlet GPD in the pion vanishes for and momentum transfer squared (corresponding to to a pion with vanishing four momentum):
(36) 
Hence (see also [Pol99b]),
(37) 
Evaluating Eq. (37) at , determines :
(38) 
being the fraction of the momentum carried by the quarks and antiquarks in the pion. As the highest power in in Eq. (37) (i.e. the term in ) originates solely from the Dterm, one easily obtains the following expression for pion Dterm :
(39) 
We see therefore that the first Gegenbauer coefficient of the pion Dterm is negative and strictly nonzero. Its value is fixed by the chiral relations in terms of the momentum fraction carried by the partons in the pion.
The nucleon Dterm is not fixed by general principles. However one may expect that the contribution of the pion cloud of the nucleon can be significant. Indeed in the chiral quarksoliton model, which emphasizes the role of broken chiral symmetry in the nucleon structure, the Dterm is large and has the sign of the pion Dterm. Quantitatively for the coefficients (see Eq. 23), the estimate which is based on the calculation of GPDs in the chiral quark soliton model [Pet98] at a low normalization point GeV, gives [Kiv01b] (see also Sec. 3.2.1) :
(40) 
and higher moments (denoted by the ellipses in Eq. (23)) are small. Notice the negative sign of the Gegenbauer coefficients for the Dterm in Eq. (40), as obtained in the chiral quarksoliton model. The coincidence of the signs in the nucleon and pion Dterms hints that the Dterm in the nucleon is intimately related to the spontaneous breaking of the chiral symmetry. The chiral contributions of the long range twopion exchange shown in Fig. 3 to the GPDs of the nucleon are sizeable giving large and negative contribution to the leading Gegenbauer coefficient of the nucleon Dterm . The sign is related to the negative sign of the pion Dterm which is fixed by the soft pion theorem (39).
2.4.2 Pion pole contribution to the GPD
The role of the spontaneously broken chiral symmetry in the structure of GPDs is seen particularly clearly in the case of the quark helicity dependent GPD . We remind that this GPD satisfies the sum rule of Eq. (11), in terms of the pseudoscalar nucleon nucleon form factor . It is well known (see e.g. Ref. [Adl68]) that due to the spontaneously broken chiral symmetry this form factor at small is dominated by the contribution of the pion pole of the form:
(41) 
where is the nucleon isovector axial charge and is the Pauli matrix in the flavor space. The presence of the chiral singularity on the rhs of the sum rule (11) implies that one should also expect the presence of the chiral singularity in the GPD at small momentum transfer squared [Fra98a]. The form of this singularity has been specified in Refs. [Man99a, Fra99a, Pen00a] to be:
(42) 
where is the pion distribution amplitude entering e.g. description of the pion e.m. form factor at large momentum transfer and the hard reaction . In Ref. [Pen00a] in the framework of the chiral quarksoliton model Eq. (42) and the deviations from it have been computed. The corresponding results are discussed in Sec. 3.2.
The presence of the pion pole singularity in the GPD leads, in particular, to a strong dependence of the differential cross section of, for example, hard production on the polarization of the target [Fra99a]. The dependence of the exclusive cross section on the transverse polarization of the proton target has the following dominant structure ^{9}^{9}9Detailed expressions and quantitative estimates are shown in Sec. 4.5.2:
(43)  
where is the azimuthal angle between lepton plane and the plane spanned by 3vectors of the virtual photon and produced meson. The specific dependence of the cross section is due to the contribution of the chiral singularity (pion pole) to the “truly nonforward” GPD whose presence is dictated by the chiral dynamics of QCD.
2.5 Twist3 GPDs
Recently the DVCS amplitude has been computed in Refs. [Ani00, Pen00b, Bel00b, Rad00, Rad01b] including the terms of the order . The inclusion of such terms is mandatory to ensure the electromagnetic gauge invariance of the DVCS amplitude to the order . At the order the DVCS amplitude depends on a set of new generalized parton distributions. Let us introduce generic generalized “vector” and “axial” distributions^{10}^{10}10In this section we do not specify the flavor index as all formulae here are valid for each flavor separately.:
(44) 
If in the above equations the index is projected onto the “plus” lightcone direction (i.e. ) we reproduce the twist2 GPDs, see definition (5). The case of , according to the general twist counting rules of Ref. [Jaf91] corresponds to the twist4 GPDs which contribute to the DVCS amplitude at the order of . The twist3 GPDs correspond to . For a detailed discussion of the DVCS observables to the twist3 accuracy see Sec. 4.3.3. Here we first discuss the general properties of the twist3 GPDs and then will show that they can be related to the twist2 GPDs in the socalled WandzuraWilczek approximation.
2.5.1 General properties of the twist3 GPDs
Several properties of the twist3 GPDs can be derived without invoking any approximation.
Forward limit: In the forward limit the “vector” GPD can be parametrized as follows:
(45) 
where is the unpolarized forward quark (antiquark for ) distribution of twist2, is the twist4 quark distribution, see [Jaf91]. Also we introduced the lightcone vectors and , the former has only “plus” nonzero component and the latter only “minus” component . As it could be expected, the twist3 part of disappears in the forward limit, because unpolarized parton densities of twist3 are absent.
The forward limit for the twist3 part of the “axial” GPD is nontrivial: