Generalized parton distributions and Deeply Virtual Compton Scattering in Color Glass Condensate model
aa r X i v : . [ h e p - ph ] J un Generalized parton distributions and Deeply Virtual Compton Scattering in Color GlassCondensate model
K. Goeke, ∗ V. Guzey, † and M. Siddikov ‡ Institut für Theoretische Physik II, Ruhr-Universität-Bochum, D-44780 Bochum, Germany Theory Center, Jefferson Lab, Newport News, VA 23606, USA Theoretical Physics Department, Uzbekistan National University, Tashkent 700174, Uzbekistan (Dated: November 2, 2018)Within the framework of the Color Glass Condensate model, we evaluate quark and gluon Generalized PartonDistributions (GPDs) and the cross section of Deeply Virtual Compton Scattering (DVCS) in the small- x B region. We demonstrate that the DVCS cross section becomes independent of energy in the limit of very small x B , which clearly indicates saturation of the DVCS cross section. Our predictions for the GPDs and the DVCScross section at high-energies can be tested at the future Electron-Ion Collider and in ultra-peripheral nucleus-nucleus collisions at the LHC. PACS numbers: 12.38.Mh,13.60.Fz,13.85.Fb,24.85.+p,25.20.Dc
I. INTRODUCTION
During the last decade hard exclusive reactions, such as Deeply Virtual Compton Scattering (DVCS), γ ∗ ( q ) + p → γ ( q ′ ) + p ′ ,have been a subject of intensive theoretical and experimental studies [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. A particularinterest has been attached to the generalized Bjorken kinematics, − q = Q large ,W = ( P + q ) large ,x B = Q P · q = const ,t = ∆ = ( P ′ − P ) ≪ Q , (1)where q is the momentum of the virtual photon; P is the initial momentum of the target hadron; P ′ is the final momentum of thetarget, and t is the momentum transfer.In this kinematics the DVCS amplitude is factorized [7, 8] into the convolution of the perturbative coefficient function withnonperturbative Generalized Parton Distributions (GPDs) of the target. Recently, the leading-twist dominance (validity of thecollinear QCD factorization) in DVCS on the proton target was demonstrated by the Hall A collaboration at Jefferson Labora-tory [16], already at rather low values of Q , . ≤ Q ≤ . GeV .However it turns out that in experiments with nuclei the virtuality Q is not always very large, and one cannot say how accuratethe predictions based on factorization are, or, in other words, how large the higher-twist corrections are. One of the examples,where this approach cannot be applied, is DVCS on the nuclei measured by HERMES collaboration in DESY [17]. Due tosmall- Q ∼ − one has to use other effective models, e.g. Generalized Vector Meson Dominance model (GVMD) [18].At very large W (very small values of x B ), the perturbative collinear factorization is expected to break down due to highdensities of the partons [19]. Even for relatively large values of Q when the running coupling constant of the strong interactions α s ( Q ) is small, the effective expansion parameter α s ( Q ) g ( x, Q ) , where g ( x, Q ) is the gluon density in the target, becomeslarge. This invalidates the perturbative expansion leading to the collinear factorization. Since in heavy nuclei the parton densitiesare enhanced by the atomic number A compared to those in the nucleon, the onset of the effects associated with high partondensities may take place at the values of x B , which will be already achieved at the future Electron-Ion Collider (EIC).In this paper we use the framework of the Color Glass Condensate (CGC) model offered in [20, 21] (see also recent re-views [22, 23, 24, 25]). We generalize the formalism of the CGC model to exclusive reactions and evaluate Generalized PartonDistributions (GPDs) and the DVCS amplitude at small- x B . We find that for DVCS off heavy nuclei, the DVCS cross section is ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] virtually x B -independent, i.e. the DVCS cross section saturates in the small- x B limit. The general saturation property built-ininto this model is an essentially nonperturbative effect, which complies with the general Froissart (unitarity) bound [26, 27, 28] F ( x, Q ) ≤ ln n ( s ) ∼ ln n (cid:18) x (cid:19) , (2)where n = 3 for DIS on nucleons; n = 1 for DIS on heavy nuclear targets. This should be compared to the Froissart bound forthe case of hadron-hadron scattering, σ ≤ ln s .For comparison, estimates of the x -dependence based on the perturbative evolution equations do not possess saturation: TheDGLAP predicts a fast growing x -dependence [29, 30] x g ( x, Q ) ∼ exp s π α s ( Q ) ln Q Q ln 1 x ! , (3)and the BFKL framework [22, 31, 32, 33, 34] predicts the power-growing x -dependence x g ( x, Q ) ∼ (cid:18) x (cid:19) α s ( Q ) ln 2 . (4)The crucial parameter of the CGC model is the saturation scale Q s ( x, A ) , which gives the threshold for transition to saturationregime. The saturation scale Q s ( x, A ) comes into play as a universal parameter in many tasks. For example, in CGC explanationof the geometric scaling [35] in DIS data from HERA, the structure function F ( x, Q ) is represented as a function of only onevariable, i.e. F ( x, Q ) = f (cid:16) Q Q s ( x ) (cid:17) .The paper is organized as follows. In Section II A we give a brief overview of the model used for evaluations. In particular,we generalize the original framework of [20, 21] to the finite nucleus case in order to consider the off-forward matrix elements.In Section III we evaluate the quark GPDs, and in Section IV we evaluate the DVCS amplitude. In Section V we present ourresults and draw conclusions. II. GENERALIZED PARTON DISTRIBUTIONS IN THE COLOR GLASS CONDENSATE MODELA. Overview of the Color Glass Condensate model
The basic assumption of CGC is that one can separate the partons into fast ( x B ∼ ) and slow ( x B ≪ ) ones, accordingto their light-cone fraction p + . The former are considered as classical “sources”, and the latter are the dynamical degrees offreedom in the model. In the leading order over α s ( Q ) one has just ordinary Yang-Mills equations for the gluon fields, in NLOone has a standard loop expansion. It is assumed that dynamics of the “fast” partons does not depend on the “slow” partons;thus the configurations of the fast partons are random and one must average over all possible configurations of these “sources” J aµ ( x ) = δ µ + ρ a ( x ) , where a is a color index, and x is a coordinate. The weight functional W [ ρ ] encodes the dynamics of the“fast” subsystem and comes as an external parameter in the model. There are no restrictions on this functional except for theobvious gauge and Lorentz invariance. An additional requirement of color neutrality, Z d x D ρ a ( ~x ) ρ b ( ~ E = 0 , (5)was introduced in [36]. It reflects the fact that the physical states are colorless.If we define x as a scale which separates “fast” and “slow” partons, then the dependence of the functional W [ ρ ] on the scale x will be described by a kind of “renormgroup equation” ∂W [ ρ ; τ ] ∂τ = 12 Z d~xd~y δδρ a ( ~x ) χ ( ~x, ~y ) δδρ b ( ~y ) W [ ρ ; τ ] , (6)where τ = ln (cid:16) x (cid:17) and χ ( ~x, ~y ) is a complicated functional of the field ρ .While in the general case this equation has not been solved so far, there are known solutions for some special (asymptotic)cases. Conventionally W [ ρ ] is chosen in a Gaussian form [20, 21, 23] W [ ρ ] = N exp (cid:18) − Z d~xd~y ρ a ( ~x ) ρ a ( ~y ) λ ( ~x, ~y ) (cid:19) , (7)where N is the normalization factor fixed from the condition R D ρW [ ρ ] = 1 and the function λ ( ~x, ~y ) is either a constant ora function fixed with some additional assumptions. Physically the function λ ( ~x, ~y ) describes correlation of partons inside thetarget. It is obvious that in the infinite nuclear matter it may depend only on the relative distance, i.e. λ ( ~x, ~y ) = µ A ( ~x − ~y ) . (8)In the general case, the shape of the function µ A ( ~r ) is unknown. However, the color neutrality condition (5) and the re-quirement that in the low parton density limit the model should reproduce BFKL predictions (4), fix the short-distance andlarge-distance behaviour. It was proposed in [23] that one can use the interpolationParameterization I: µ A ( ~r ) = Z d k (2 π ) µ ( k ) e − i~k~r = Z d k (2 π ) e − i~k~r k ⊥ π (cid:16) Q s ( x ) k ⊥ (cid:17) γ (cid:16) Q s ( x ) k ⊥ (cid:17) γ , (9)where γ = q ζ (3) ≈ . is a numerical coefficient.There are also simpler versions of the model [20, 21], which neglect correlation of the partons, i.e.Parameterization II: λ ( ~x, ~y ) = δ ( ~x − ~y ) λ A ( x − ) , (10)where λ A ( x − ) is some function [47]. In subsequent sections we will consider first evaluation with a simple parameteriza-tion (10), and after that discuss, how the results change for the parameterization (9).The choice of the Gaussian parameterization (7) enables us to evaluate all the results analytically. Notice however, that (7) isexplicitly C -even, i.e. the number of quarks is equal to the number of antiquarks inside any target in this model. This agreeswith experimental fact that the quark and anti-quark parton densities are approximately equal at small x B . On the other hand, C -parity of (7) implies that the model does not distinguish matter and antimatter and is not applicable to evaluation of somequantities. For example, the baryon number and electric charge of the target are exactly zero, since they are due to the valencequarks.An interesting generalization of the Gaussian parameterization (7) was discussed in [38]. In particular, it was found thatfor the model of k ≫ independent noninteracting quarks the distribution is indeed Gaussian, and the first correction isproportional to ∼ d abc R d x ρ a ( ~x ) ρ b ( ~x ) ρ c ( ~x ) , where d abc is defined from the anticommutator of the generators T a of thegroup, { T a , T b } = 2 d abc T c . However, for the DVCS and singlet GPDs discussed in this paper the C -odd correction does notcontribute.It is well-known that at high-energies, the real part of scattering amplitudes is suppressed by the slow energy dependenceof the amplitude compared to the imaginary part [39, 40, 41]. Therefore, it is sufficient to consider only the imaginary part.Actually, as we shall show in Sect. IV, the real part of the DVCS amplitude in the CGC model is exactly zero.The generating functional of the model has a form [48] Z [ j ] = Z Dρ W [ ρ ] R DAδ ( A + ) e iS [ A,ρ ] − R dxj · A R DAδ ( A + ) e iS [ A,ρ ] , (11)where S [ A, ρ ] = S [ A ] + R d~xρ a ( ~x ) A a − ( ~x ) and we used light-cone gauge n · A = 0 , n = 0 . In order to restore the explicitgauge invariance of the action S [ A, ρ ] , the interaction term R d~xρ a ( ~x ) A − ( ~x ) is sometimes replaced with T r R d ~xρ ( ~x ) W [ A, ~x ] ,where W [ A, ~x ] = P exp ig Z x + −∞ dζA + ( ζ ) ! (12)is the Wilson link. B. Finite nucleus
Since in this paper we are interested in DVCS–off-forward reaction, we can no longer use the infinite nuclear matter approx-imation. Indeed, the DVCS cross section off a nuclear target rapidly decreases as one increases the momentum transfer t . As aresult, the sizable cross-sections exist only for | t | ∼ /R A , where R A is the nuclear radius. In the infinite nuclear matter, all theoff-forward cross-sections vanish [49]. This means that we have to take into account the off-forward kinematics from the verybeginning. If the coordinate of the nucleus center of mass is ~X , then the weight functional W [ ρ ] may be chosen as W ρ [ ρ, X ] = exp ( − Z d xθ ( | ~x ⊥ − ~X ⊥ | < R A ) ρ a ( ~x − ~X ) ρ a ( ~x − ~X ) λ A ( x − − X − ) ) , (13)where we extracted the “zero mode” (integration over the nucleus center of mass) explicitly according to standard technique [42]and introduced an explicit cutoff factor θ ( | ~x ⊥ − ~X ⊥ | < R A ) which forbids the color condensate ρ a ( ~x ) from outside of thenucleus. The cutoff in x − is provided by the factor λ A ( x − − X − ) . The interaction of gluons with the condensate is alsomodified by this cutoff factor: S [ A, ρ ] = S [ A ] + T r Z d xθ ( | ~x ⊥ − ~X ⊥ | < R A ) ρ ( ~x ) A − ( ~x ) (14)for the linear interaction, or S [ A, ρ ] = S [ A ] + T r Z d xθ ( | ~x ⊥ | < R A ) ρ ( ~x ) W [ A ]( ~x ) (15)for the interaction via Wilson link (12). The generating functional (11) takes the form Z [ j ] = Z DρdX e i~ ∆ ~X W [ ρ, X ] R DAδ ( A + ) e iS [ A,ρ ] − R dxj · A R DAδ ( A + ) e iS [ A,ρ ] . (16)Notice that the formal introduction of the θ -functions is equivalent to the redefinition of the functional integral: Z Dρ := Y x dρ ( x − , | ~x ⊥ − ~X ⊥ | < R A ) d X . (17)Indeed, configurations with ρ ( | ~x ⊥ | > R A ) = 0 do not interact with anything and thus contribute only to the normalizationconstant.Since the coupling constant α s is small, we can take the integral over the gluon field A µ in (11) in the saddle-point approxi-mation. In the leading order, the gluon field A µ is just the solution of the equation of motion D µ F νµa ( x ) = δ ν, + δ ( x − ) ρ a ( x ⊥ ) , (18)where ρ a ( x ⊥ ) is the arbitrary external field, and an additional gauge constraint A + = 0 is implied. Notice that we do notimpose any conditions onto the gluonic fields A aµ at the large distance | ~x | > R A . The solution of the equation (18) is [23] A µ = U (cid:18) ˜ A µ + ig ∂ µ (cid:19) U † , (19)where [50] ˜ A µ = δ µ + α ( x − , x ⊥ ) , (20) U = P exp ( ig Z x − −∞ dz − α a ( z − , x ⊥ ) T a ) , (21) α ( x − , x ⊥ ) = 1 − ∂ ⊥ ˜ ρ = Z d y ⊥ π ln 1( x ⊥ − y ⊥ ) Λ QCD ˜ ρ ( x − , y ⊥ ) , (22) ˜ ρ ( x − , x ⊥ ) = U † ( x − , x ⊥ ) ρ ( x − , x ⊥ ) U ( x − , x ⊥ ) , (23)and T a are the generators of the color group.The analytical solution (19) enables us to evaluate different correlators. A straightforward evaluation of the h ρρ i -correlatorwith the weight function (13) yields h P ′ | ρ ( ~x ) ρ ( ~y ) | P i = ¯ P + Z d X e i~ ∆ ~X θ ( | ~x ⊥ − ~X ⊥ | < R A ) λ A ( x − − X − ) δ ( ~x − ~y ) = f (∆) ¯ P + e − i~ ∆ ~x δ ( ~x − ~y ) , (24)where f (∆) = (cid:18) ˜ λ (∆ + ) ≡ Z dx − λ ( x − ) e − ix − ∆ + (cid:19) f ⊥ (∆ ⊥ ) πR A , (25)and f ⊥ (∆ ⊥ ) = 1 πR A Z d x ⊥ θ ( | ~x ⊥ | < R A ) e i~ ∆ ⊥ ~x ⊥ = J (∆ ⊥ R A )∆ ⊥ R A . (26)We can see that for any fixed nonzero ∆ ⊥ = 0 the result vanishes in the R A → ∞ limit in agreement with discussion at thebeginning of this section. The evaluation of the gluonic GPDs defined as [11] x H g ( x, ξ, t ) = 1¯ P + Z dz − e ix ¯ P + (cid:28) P ′ (cid:12)(cid:12)(cid:12)(cid:12) F a + k (cid:18) − z − (cid:19) F k,a + (cid:18) − z − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) P (cid:29) (27)is done in quasiclassical approximation, x H g ( x, ξ, t ) ≈ Z d X e i~ ∆ ~X Z dz − e ix ¯ P + z − F a + k (cid:18) − z − − ~X (cid:19) F k,a + (cid:18) z − − ~X (cid:19) , (28)where F µν in the rhs of (28) corresponds to the classical solution found in previous subsection. Evaluation of (28) [23] gives [51] x H g ( x, ξ, t ) = ( N c − P + Z d X e i~ ∆ ~X (cid:18)Z d ˜∆ e − i ˜∆ ~X (cid:0) − ∂ r ⊥ (cid:1) ˜ γ A ( x − , ~r ⊥ ; ˜∆) (cid:19) × exp − g N c ˜ f (cid:16) ~ , ~r − ~X (cid:17) + ˜ f (cid:16) ~ , − ~r − ~X (cid:17) − ˜ f (cid:16) ~r, − ~X (cid:17) r ⊥ ≈ /Q , (29)where ˜ f ( ~r , ~r ) = Z d ˜∆(2 π ) e − i ˜∆ ~r Z + ∞−∞ dz − ˜ γ A ( z − , ~r ; ˜∆) , (30)and ˜ γ A ( x − , ~r ⊥ ) is defined as f (∆) ¯ P + Z d k (2 π ) e − ix ( k +∆ / e iy ( k − ∆ / (cid:0) k ⊥ − ∆ ⊥ (cid:1) (cid:0) k ⊥ + ∆ ⊥ (cid:1) = δ ( x − − y − )˜ γ A ( x − , ~x ⊥ − ~y ⊥ ) e i~ ∆( ~x + ~y ) / . (31)As one can see from (29), the gluon GPD H g ( x, ξ, t ) has a trivial x -dependence /x for all ( ξ, t ) , since x does not enter theright-hand side of Eq. (29). Physically, the exponent in (29) takes into account nonlinear in α s effects in the model. C. Alternative kernel
In this section we discuss how all the previous formulae change with an alternative weight function (9). The weight functionalin this case should be written as W ρ [ ρ, X ] = N exp ( − Z d x θ (cid:16)(cid:12)(cid:12)(cid:12) ~x ⊥ − ~X ⊥ (cid:12)(cid:12)(cid:12) < R A (cid:17) ρ a ( ~x − ~X ) ρ a ( ~y − ~X ) µ A ( ~x − ~y ) ) , (32)where the function µ A ( ~z ) describes correlation of hadrons inside the nuclei and was defined in (9). Performing the evaluation aswas discussed in Sect. II B, we obtain h P ′ | ρ ( ~x ) ρ ( ~y ) | P i = Z dXe i~ ∆ ~X θ (cid:16)(cid:12)(cid:12)(cid:12) ~x ⊥ − ~X ⊥ (cid:12)(cid:12)(cid:12) ≤ R A (cid:17) θ (cid:16)(cid:12)(cid:12)(cid:12) ~y ⊥ − ~X ⊥ (cid:12)(cid:12)(cid:12) ≤ R A (cid:17) µ A ( ~x ⊥ − ~y ⊥ ) == f ( ~x − ~y, ∆) e i~ ∆ ~x + ~y µ A ( ~x − ~y ) = Z d k (2 π ) ˜ µ A ( ~k ) e − ix ( k +∆ / e iy ( k − ∆ / , (33)where ˜ µ A ( ~k ) = Z d ρe − i~k~ρ f ( ~ρ, ∆) µ A ( ~ρ ) , (34) f ( ~ρ, ∆) = Z d Xe i~ ∆ ~X θ (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) ~ρ − ~X ⊥ (cid:12)(cid:12)(cid:12)(cid:12) ≤ R A (cid:19) θ (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) ~ρ ~X ⊥ (cid:12)(cid:12)(cid:12)(cid:12) ≤ R A (cid:19) = (35) = Z d k (2 π ) e − i~k~ρ φ ~k + ~ ∆2 ! φ ~k − ~ ∆2 ! ,φ ( ~k ) = πR A J ( kR A ) kR A . (36)From (33) we can see that in the finite nuclei the color neutrality condition (5) implies that we have to identify µ A from (9)with ˜ µ A ( r ) . For ˜ γ A ( x − , ~r ⊥ ) we can immediately obtain h P ′ | α ( ~x ) α ( ~y ) | P i = ... = Z d k (2 π ) ˜ µ A ( k ) e − ix ( k +∆ / e iy ( k − ∆ / (cid:0) k ⊥ − ∆ ⊥ (cid:1) (cid:0) k ⊥ + ∆ ⊥ (cid:1) = δ ( x − − y − )˜ γ A ( x − , ~x ⊥ − ~y ⊥ ) e i~ ∆( ~x + ~y ) / . (37)Thus we can see that this kernel differs from the previous one only by an additional factor ˜ µ ( ~k ) in the integrand. D. Quark propagator in CGC field
Although for the evaluation of the DVCS amplitude one may use the color dipole approximation, in this paper we evaluate theGPDs and Compton amplitudes directly. In the diagrammatic language this corresponds to summation of all [52] the multigluondiagrams, whereas the color dipole approach assumes either only Born term contribution or Eikonal approximation, as is shownon the Fig. 1. γ∗ γ∗ γ∗ γ∗
Figure 1: Diagrams contributing to DVCS in color dipole approximation. See [43] for an example of DIS evaluation in this approach.
For the evaluation of the quark GPDs in the leading order over α s ( Q ) , we need to evaluate the quark propagator in theclassical gluonic field found in the previous section. To this end, we consider only the zero width limit, ρ = δ ( x − ) ρ ( ~x ⊥ ) . (38)Beyond this limit, equations with explicit x − -dependence become much more complicated. Physically, the use of (38) in theoff-forward kinematics is justified, since the light-cone fractions of the partons are small, i.e. x, ξ ≪ .The basic idea is that for x − = 0 the field ρ ( ~x ) = 0 and we have just vacuum equations, gluon field A µ reduces to a puregauge. It is possible to choose the gauge in such a way that for x − < the field disappears, A µ = 0 , and for x − > it is a puregauge, A µ = ig U ∂ µ U † and thus the wave function of the quark has a form ψ ps ( x ) = u s ( p ) e − ipx , x − < R d p ′ δ ( p − p ′ ) P s ′ C ss ′ ( p, p ′ ) u s ′ ( p ′ ) e − ipx , x − > , (39)where u s ( p ) is a free Dirac spinor, and the matrix C ss ′ ( p, p ′ ) is found from the continuity at the point x − = 0 . One subtle pointis that the Dirac operator has the form i ˆ D = i∂ − γ − + ... , (40)and matrix γ − is singular, because it is proportional to the light-cone projector Λ ( − ) . This implies that the continuity conditionmust be imposed not on the function ψ ps ( x ) as a whole, as in [20, 21, 23], but rather only on the component [53] ψ ( − ) ps ( x ) =Λ ( − ) ψ ps ( x ) . The final result for the wave function is ψ ps ( x ) = θ ( − x − ) u s ( p ) e − ip · x + θ ( x − ) U ( x ⊥ ) Z d k (2 π ) δ ( k − − p − ) δ (cid:18) k + − k ⊥ + p p − (cid:19) × (cid:18)Z d ze i ( p ⊥ − k ⊥ ) · z U † ( z ) (cid:19) e − ik · x (cid:18) γ k − √ k ⊥ + M ) (cid:19) Λ ( − ) u s ( p ) , (41)where M is the mass of the quark. The evaluation of the quark propagator according to S ( x, y ) = Z d p (2 π ) P s ψ ps ( x ) ¯ ψ ps ( y ) p − M + i , (42)yields S ( x, y ) − S ( x − y ) == , x − < , y − < (cid:0) U ( x ⊥ ) U † ( y ⊥ ) − (cid:1) S ( x − y ) , x − > , y − > R d p (2 π ) p − M + i R d k (2 π ) exp n i (cid:16) k ⊥ + M p − y − + p − y + − k ⊥ y ⊥ − p · x (cid:17)o × R d ze − i ( p ⊥ − k ⊥ ) z (ˆ p + M )Λ (+) (cid:16) γ p − √ ( M − ˆ k ⊥ ) (cid:17) (cid:0) U ( z ) U † ( y ⊥ ) − (cid:1) , x − < , y − > R d p (2 π ) p − M + i R d k (2 π ) exp n − i (cid:16) k ⊥ + M p − x − + p − x + − k ⊥ x ⊥ − p · y (cid:17)o × R d ze i ( p ⊥ − k ⊥ ) z (cid:16) γ p − √ (ˆ k ⊥ + M ) (cid:17) Λ ( − ) (ˆ p + M ) (cid:0) U ( x ⊥ ) U † ( z ) − (cid:1) , x − > , y − < (43)where S ( x − y ) is the free propagator S ( x − y ) = Z d p (2 π ) e − ip ( x − y ) ˆ p − M + i . (44)It might be checked that the propagator S ( x, y ) satisfies the equation ( i ˆ D − M ) S ( x, y ) = δ ( x − y ) as well as reduces to S ( x − y ) in the U → limit. III. UNINTEGRATED QUARK GPDS
In this section we evaluate unintegrated quark GPDs defined via the following matrix element (we assume that the target hasspin ) H ( x, ξ, ~ ∆ ⊥ , ~k ⊥ ) = (45) = Z dz − π Z d r ⊥ e − i~k ⊥ ~r ⊥ e ix ¯ P + z − (cid:28) P ′ (cid:12)(cid:12)(cid:12)(cid:12) ¯ ψ (cid:18) − z − − ~r ⊥ (cid:19) γ + ψ (cid:18) z − ~r ⊥ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) P (cid:29) . In the forward limit ( ~ ∆ ⊥ → , ξ → ) the function H (cid:16) x, ξ, ~ ∆ ⊥ , ~k ⊥ (cid:17) reduces to unintegrated parton distribution q (cid:16) x, ~k ⊥ (cid:17) ,and when integrated over ~k ⊥ , it gives ordinary GPDs. In the quasiclassical approximation, (45) reduces to H ( x, ξ, ~ ∆ ⊥ , ~k ⊥ ) = (46) = Z dz − π e ix ¯ P + z − Z d r ⊥ e − i~k ⊥ ~r ⊥ i ¯ P + Z d Xe − i~ ∆ ~X (cid:28) T r (cid:20) γ + S (cid:18) − z − − ~r ⊥ − ~X, z − ~r ⊥ − ~X (cid:19)(cid:21)(cid:29) , where here and below angular brackets without explicit initial and final states h ... i are the short-hand notation for averaging(integration) over all possible configurations ρ ( x ) , i.e. D ˆ O E := R D ρ W [ ρ ] O ( ρ ) . Substituting the propagator (43) and takingthe integral over each domain, one obtains the final result H (cid:16) x, ξ, ~ ∆ ⊥ , ~k ⊥ (cid:17) = H (+ − ) + H ( − +) (47)where H + − = 2 N c Z d κ ⊥ (2 π ) ˜ γ (cid:18) κ ⊥ + ∆ ⊥ , κ ⊥ − ∆ ⊥ (cid:19) × M − (cid:16) ~k + ~ ∆ ⊥ (cid:17) · (cid:16) ~k − ~κ ⊥ (cid:17) ( x − ξ ) (cid:18)(cid:16) ~k − ~κ ⊥ (cid:17) + M (cid:19) − ( x + ξ ) (cid:18)(cid:16) ~k + ~ ∆ ⊥ (cid:17) + M (cid:19) ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x − ξx + ξ (cid:16) ~k − ~κ ⊥ (cid:17) + M (cid:16) ~k + ~ ∆ ⊥ (cid:17) + M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (48) H − + = 2 N c Z d κ ⊥ (2 π ) ˜ γ (cid:18) κ ⊥ + ∆ ⊥ , κ ⊥ − ∆ ⊥ (cid:19) × M − (cid:16) ~k − ~ ∆2 (cid:17) · ( ~k ⊥ − ~κ ⊥ )( x + ξ ) (cid:18)(cid:16) ~k ⊥ − ~κ ⊥ (cid:17) + M (cid:19) − ( x − ξ ) (cid:18)(cid:16) ~k − ~ ∆2 (cid:17) + M (cid:19) ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x + ξx − ξ (cid:16) ~k ⊥ − ~κ ⊥ (cid:17) + M (cid:16) ~k − ~ ∆2 (cid:17) + M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (49)the superscript signs ( ± , ± ) refer to different integration domains over ( dX − , dz − ) in (46), and function ˜ γ (cid:16) ~κ − ~ ∆ ⊥ , ~κ + ~ ∆ ⊥ (cid:17) is defined as ˜ γ (cid:18) κ ⊥ + ∆ ⊥ , κ ⊥ − ∆ ⊥ (cid:19) := Z d ρd X (2 π ) e i ∆ ⊥ X ⊥ + iκ ⊥ ρ D U † (cid:16) X + ρ (cid:17) U (cid:16) X − ρ (cid:17)E (50)Evaluation of this quantity (see Sect.B for details) yields ˜ γ ~κ − ~ ∆ ⊥ , ~κ + ~ ∆ ⊥ ! = (51) Z d r e i~κ~r Z d X ⊥ e i~ ∆ ⊥ ~X ⊥ exp − g N c ˜ f (cid:16) ~ , ~r − ~X (cid:17) + ˜ f (cid:16) ~ , − ~r − ~X (cid:17) − ˜ f (cid:16) ~r, − ~X (cid:17) . Notice that GPD (47) is antisymmetric, i.e. H ( − x, ξ ) = − H ( − x, ξ ) . Also, (47) is not required to satisfy polynomiality sincethe original model is valid only for x ≪ .One of the subtle points of the result (47) is the logarithmic behaviour ∼ ln | x ± ξ | in the vicinity of the points x ∼ ± ξ .Physically, in these points one of the quarks has a zero light-cone fraction and becomes especially sensitive to the details of themodel. However, since we are in a saturation regime, factorization formula does not work and we expect that such behaviourshould not cause any physical problems. In Appendix A we give details of evaluation of (47), and in particular discuss thelogarithmic singularities. IV. DVCS AMPLITUDE
In this section we evaluate the DVCS amplitude directly (not using factorization). The first reason for this is that the GPDsevaluated in the previous section are valid only for the small x ≪ whereas the convolution formula which follows fromfactorization implies integration over the light-cone fraction over the region − < x < . The second reason is that, as wediscussed in Sect. (I), in the saturation (high-density) regime the convolution formula becomes invalid.The starting point of our derivation is the definition of the DVCS amplitude A µν = − i Z d z h P ′ | J ν (0) J µ ( z ) | P i A e − iq · z . (52)In the quasiclassical approximation the matrix element h P ′ | J ν (0) J µ ( z ) | P i A is reduced to h P ′ | J ν (0) J µ ( z ) | P i A = Z d Xe i~ ∆ ~X D P ′ (cid:12)(cid:12)(cid:12) J ν ( − ~X ) J µ ( z − ~X ) (cid:12)(cid:12)(cid:12) P E = − Z d Xe i~ ∆ ~X h T r [ γ µ S ( z − X, − X ) γ ν S ( − X, z − X )] i (53)where S ( x, y ) is the propagator (43). Substituting (43) into (53) and taking the integrals, we may reduce the DVCS amplitudeto the form A µν = i M A π Z d p (2 π ) d k (2 π ) Θ (cid:16) − q − ≤ p − ≤ q − (cid:17) q + (( p − ) − ( q − ) /
4) + k ⊥ + M q − − i γ ~k − ~ ∆ ⊥ , ~k + ~ ∆ ⊥ ! × q + − ∆ + ) (( p − ) − ( q − ) /
4) + p − ~p ⊥ ~ ∆ ⊥ + q − (cid:16) ~p ⊥ + M + ~ ∆ ⊥ (cid:17) × N c δ µ + δ ν + (cid:16) M − ~k ⊥ − ~p ⊥ (cid:17) M − ~p ⊥ + ~ ∆ ⊥ ! +8 N c δ µ ⊥ δ ν ⊥ h p µ ⊥ p ν ⊥ (cid:0) ( q − ) − p − ) (cid:1) + p − g µν (cid:16) p − ( M + p ⊥ ) − q − ~p ⊥ · ~ ∆ ⊥ (cid:17)i + 32 N c δ µ − δ ν − (cid:18) ( p − ) − ( q − ) (cid:19) + 8 N c ( δ µ + δ ν − + δ ν + δ µ − ) (cid:18) ( p − ) − ( q − ) (cid:19) M − ~p ⊥ − ~k ⊥ + ~ ∆ ⊥ !! . (54)One interesting point is that the real part of (54) is exactly zero. Indeed, taking the imaginary part of the first ratio containing − i and using x − i P (cid:18) x (cid:19) + iπδ ( x ) , (55)we can immediately find that the argument of δ -function is zero only for | p − | = q − s k ⊥ + M ) Q ≥ q − , (56)i.e. outside the integration domain. For comparison, from phenomenology it is known that the high-energy amplitude getsdominant contribution from the imaginary part. V. RESULTS FOR GPDS AND DVCS CROSS-SECTIONS
In this section we present results of the numerical evaluation of the GPDs and DVCS cross-sections. In subsection (V A) weconsider first the results with a simpler parameterization (10), and after that in subsection (V B) with a more realistic parameter-ization (9).
A. Results with parameterization II
As one can see from (29), for both parameterizations I and II the x -dependence of the gluon GPD H gA ( x, ξ, t ) is trivial–just /x for all ( ξ, t ) . For quark GPDs H gA ( x, ξ, t ) the x -dependence is more complicated, however in the forward case the partondistribution q A ( x ) has also a simple /x -dependence. For better understanding, we prefer to discuss out results for the gluonsin terms of the ratio H gA ( x, ξ, t ) /g A ( x ) , which measures the off-forward effects, and g A ( x ) is the forward gluon PDF evaluatedin the same model.In Figure 2 we plot the ξ and t -dependence of the ratio H gA ( x, ξ, t ) /g A ( x ) for different nuclei. In the left panel of Figure 2we plot the t -dependence of the ratio H gA ( x, ξ, t ) /g A ( x ) in nuclei for ξ = 0 . We can see that H g ( x, ξ, t ) is decreasing as afunction of t . For the sake of comparison, on the same plot we also plotted in grey lines the nuclear form factors in conventionalexponential parameterization, F A ( t ) = exp (cid:16) R A t (cid:17) , and for radius R A we used R A = 1 . f m × A / . We can see that to agood extent the t -dependence of the GPDs is similar to that of the form factors.In the right panel of Figure 2 we plot the ξ -dependence of the gluon GPDs in nuclei. We can see that in the small- ξ region H g ( x, ξ, t ) is independent of the skewedness ξ . This results is quite easy to understand: in the ultrarelativistic limit the nucleusin laboratory frame is squeezed to an infinitely thin “pancake", so the condensate distribution along the x − -axis is stronglypeaked around x − ≈ , λ A ( x − ) ∼ δ ( x − ) . As a consequence, the gluon GPD which is proportional to the Fourier of λ A ( x − ) ,0almost does not depend on ∆ + ∼ ξ . The only exception is the region of sufficiently large ξ ∼ . , where the ξ -dependence ismainly a kinematical effect–the increase of H g ( x, ξ, t ) is due to decreasing ∆ ⊥ at fixed t . However, these values of ξ ∼ . aretoo large, and our extrapolation of the model becomes unreliable. Ca Zr Pb g A (x, ξ ,t)/g A (x), Q =1 GeV , ξ =0 Ca Zr Pb -1 -2 -3 -4 ξ g A (x, ξ ,t)/g A (x), Q =1 GeV , t=-0.01 GeV Figure 2: Left plot: t -dependence of the gluon distribution for different nuclei. ξ = 0 , Q = 1 GeV . For comparison, we also plotted in greylines the nuclear formfactor in the simplest exponential parameterization F A ( t ) = exp “ R A t ” . Right plot: ξ -dependence for the same nucleifor fixed t = − .
01 GeV , Q = 1 GeV . We do not plot the x -dependence of the gluon GPD H g , which is according to (29) just a trivial /x for all ( ξ, t ) . In Figure 3 we plot the x -, ξ - and t -dependence of the quark GPD H A ( x, ξ, t ) in nuclei. As one can see from (47), in theforward limit the quark distributions have a very simple x -dependence, H A ( x, , ≡ q A ( x ) ∼ /x. For better legibility, weprefer to discuss out results for the quarks in terms of the ratio H A ( x, ξ, t ) /q A ( x ) , which measures off-forward effects.From the left panel in Figure 3 we can see that for x ≪ ξ the GPD H A ( x, ξ, t ) is decreasing approximately as H A ( x, ξ, t ) ∼ x and as a result the ratio H A ( x, ξ, t ) /q A ( x ) behaves approximately as H A ( x, ξ, t ) /q A ( x ) ∼ x . For x ≫ ξ, H A ( x, ξ, t ) ≈ q A ( x ) F A ( t ) , and the ratio is a constant. In the point x = ξ we have a singularity ∼ ln | x − ξ | , which was mentioned at the endof Section III and discussed in details in Appendix A.From the middle panel in Figure 3 we can see that as a function of ξ the generalized quark distribution is a constant for ξ ≪ x ,but is a decreasing function for ξ ≫ x .From the right panel in Figure 3 we can see the t -dependence of the GPD H A ( x, ξ, t ) . For the sake of comparison, on thesame plot we also plotted in grey lines the nuclear form factors in the frequently used exponential parameterization, F A ( t ) =exp (cid:16) R A t (cid:17) . We can see that H A ( x, ξ, t ) is decreasing a bit faster than F A ( t ) . x = ξ Ca Zr Pb
101 10 -1 -2 -3 -4 x10 -1 -2 -3 -4 -5 -6 H qA (x, ξ =0.001, t=-0.01 GeV )/q A (x) x = ξ Ca Zr Pb ξ max (t)10 -2 -3 -4 ξ -1 -2 -3 -4 -5 H qA (x ,ξ , t=-0.01 GeV )/q A (x), x=10 -3 Ca Zr Pb q A (x, ξ ,t)/q A (x), Q =1 GeV , ξ =0 Figure 3: Left plot: x -dependence of the quark GPD H A ( x, ξ, t ) . ξ = 10 − , Q = 1 GeV . Middle plot: ξ -dependence of the same GPD H A ( x, ξ, t ) for fixed x = 10 − , Q = 1 GeV , ξ max = p − t/ (4 M − t ) . Right plot: t -dependence of the same GPD H A ( x, ξ, t ) . Forcomparison, we also plotted in grey lines the nuclear formfactor in the the simplest exponential parameterization F A ( t ) = exp “ R A t ” . In Figure 4 we plot the ξ - and t -dependence of the differential DVCS cross-section dσ/dt for fixed Q and different nuclei.1From the left part of Figure 4 we can see that the cross-section is growing when ξ is decreasing, but at some ξ , which we call ξ sat ( Q , A ) , we have a qualitative transition to the saturation. The value ξ sat ( Q , A ) depends on the external kinematics. Therelatively large value ξ sat ∼ . is due to the small value of Q = 1 GeV , ξ sat is decreasing when Q increases.From the right plot on Figure 4 we can see the t -dependence of the differential cross-section dσ/dt . For the sake of com-parison, on the same plot we also plotted in grey lines the nuclear form factors in conventional exponential parameterization, F A ( t ) = exp (cid:16) R A t (cid:17) . We can see that dσ/dt is decreasing a bit faster than F A ( t ) . A=40A=90A=208 ξ max (t)10 -2 -3 -4 ξ d σ /dt , nb/GeV Ca Zr Pb -t, GeV σ dt ( d σ dt ) t=0 , Q =1 GeV , ξ =10 -4 Figure 4: Left plot: ξ -dependence of the differential DVCS cross-section in the CGC model for different nuclei. Kinematic is chosen as Q = 1 GeV , t = − .
01 GeV . Right plot: t -dependence of the DVCS cross-section at fixed ξ = 10 − . On the right plot, we also plottedin grey lines what one would have with the simplest factorized t -dependence of the DVCS amplitude and exponential parameterization for theformfactor: dσdt ∼ F A ( t ) ∼ exp “ R A t ” B. Results with parameterization I
In this section we discuss the results of Color Glass Condensate model in parameterization (9). The crucial point is that thismodel explicitly contains the saturation scale Q s and as a consequence we can apply it only to the kinematics where saturationis present. In our evaluations we used for Q s the parameterization from [24], where Q s ( A ) is found as a solution of the equation Q s ( A ) = α s ( Q ) N c µ A ln Q s ( A )Λ QCD ! . (57)This equation has real solutions only for A & A min ( Q ) ∼ for Q ∼ , and A min ( Q ) is a growing function of Q . In Figures 5 and 6 we plot the ξ and t -dependence of the gluon and quark distributions H A ( x, ξ, t ) /q A ( x ) , and in Figure 7 weplot the ξ - and t -dependence of the differential DVCS cross-section dσ/dt . We can see that qualitatively the behaviour is thesame as in the previous section, although absolute values differ. C. Comparison to DVCS cross section in GVMD model
In the Figure 8 we compare predictions for the DVCS cross-section with our earlier result [18] obtained in GeneralizedVector Dominance Model (GVMD). We can see the difference in predictions of GVMD and CGC models: In contrast to thesaturation behavior in CGC, the GVMD cross-section is slowly growing as ξ − α when ξ is decreasing. Nevertheless in the region − ≤ ξ ≤ − predictions of both models have comparable values. VI. CONCLUSION
In this paper we considered Generalized Parton Distributions (GPDs) and Deeply Virtual Compton Scattering (DVCS) ampli-tudes in the Color Glass Condensate model. We modified the original formulation of [20, 21] to off-forward kinematics of hardexclusive reactions, which provided the necessary framework for the calculation of GPDs and the DVCS amplitude.2 Ta Pb g A (x, ξ ,t)/g A (x), Q =1 GeV , ξ =0 Ta Pb -1 -2 -3 -4 ξ g A (x, ξ ,t)/g A (x), Q =1 GeV , t=-0.01 GeV Figure 5: Left plot: t -dependence of the gluon distribution for different nuclei. ξ = 0 , Q = 1 GeV . Right plot: ξ -dependence for the samenuclei for fixed t = − .
01 GeV , Q = 1 GeV . We do not plot the x -dependence of the gluon GPD H g , which is according to (29) just atrivial /x for all ( ξ, t ) . On the left plot, we also plotted in grey lines the nuclear formfactor in the the simplest exponential parameterization F A ( t ) = exp “ R A t ” . x = ξ Ta Pb
101 10 -1 -2 -3 -4 x10 -1 -2 -3 -4 -5 -6 H qA (x, ξ =0.001, t=-0.01 GeV )/q A (x) x = ξ Ta Pb ξ max (t)10 -2 -3 -4 ξ -1 -2 -3 -4 -5 H qA (x ,ξ , t=-0.01 GeV )/q A (x), x=10 -3 Ta Pb q A (x, ξ ,t)/q A (x), Q =1 GeV , ξ =0 Figure 6: Left plot: x -dependence of the quark GPD H A ( x, ξ, t ) . ξ = 10 − , Q = 1 GeV . Middle plot: ξ -dependence of the same GPD H A ( x, ξ, t ) for fixed x = 10 − , Q = 1 GeV . Right plot: t -dependence of the same GPD H A ( x, ξ, t ) . On the left plot, we also plotted ingrey lines the nuclear formfactor in the the simplest exponential parameterization F A ( t ) = exp “ R A t ” . We evaluated the quark and gluon GPDs in this model and studied their dependence on variables x, ξ, t . We found that thegluon GPD H g in this model has a simple x -dependence H g ( x ) ∼ /x for all ( ξ, t ) . Similar /x -behaviour was observed forthe quark GPDs H q in the x ≫ ξ region and in the forward limit ( t = 0 ). Both the quark and gluon GPDs are decreasing as afunction of momentum transfer t , and the quark GPD is decreasing a bit faster than gluon GPD.Without assuming the validity of the collinear factorization, we evaluated the DVCS cross-sections in the small- ξ region onthe large nuclei. We found that in this region the DVCS cross-sections are almost independent of ξ . This is a manifestation of thegeneral saturation property inherent to the CGC model. As far as absolute values are concerned, we found that the predictions ofCGC in the relevant range of ξ are comparable with predictions of other models, e.g. GVMD. Currently there is no experimentaldata available for DVCS cross-section in this kinematics.The present calculation should be important for a wide range of the future experiments. For example, gluon GPDs in the small- x region may be used for evaluation of the heavy vector meson production in ultraperipheral collisions at the LHC [45, 46]. Acknowledgments
We would like to thank P. Pobylitsa and M. Strikman for useful discussions. The work has been partially supported bythe Collaborative Research Center Bonn-Bochum-Giessen of the DFG, by the I3HP European Project (6-th Framework), by3 Ta Pb ξ max (t)10 -2 -3 -4 ξ d σ /dt , nb/GeV Ta Pb σ dt ( d σ dt ) t=0 , Q =1 GeV , ξ =10 -4 Figure 7: Left plot: ξ -dependence of the differential DVCS cross-section in the CGC model for different nuclei. Kinematic is chosen as Q = 1 GeV , t = − .
01 GeV . Right plot: t -dependence of the DVCS cross-section at fixed ξ = 10 − . On the right plot, we also plottedin grey lines what one would have with the simplest factorized t -dependence of the DVCS amplitude and exponential parameterization for theformfactor: dσdt ∼ F A ( t ) ∼ exp “ R A t ” Parameterization IParameterization IIGVMD ξ max (t)10 -2 -3 -4 ξ d σ /dt , nb/GeV , t=-0.01 GeV , Q =1 GeV Figure 8: Comparison of ξ -dependence of the DVCS cross-section in different models. Solid curve corresponds to parameterization I fromEq. (9), dashed curve corresponds to parameterization II from Eq. (10), dot-dashed corresponds to Generalized Vector Meson Dominance(GVMD) from [18]. Kinematic is chosen as Q = 1 GeV , t = − .
01 GeV , nucleus A = 208 (lead). the Verbundforschung “Hadrons and Nuclei” of the BMBF, by the Graduate College Dortmund-Bochum of the DFG, by theCOSY-Project Juelich, and by the AvH-Kovalevskaja Funds (M.Polyakov). Notice:
Authored by Jefferson Science Associates, LLC under U.S. DOE Contract No. DE-AC05-06OR23177. The U.S.Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce this manuscript for U.S.Government purposes.
Appendix A: DETAILS OF EVALUATION OF (46)
In this section we evaluate the unintegrated GPD (45), which in quasiclassical approximation was reduced to H ( x, ξ, ~ ∆ ⊥ , ~k ⊥ ) == i ¯ P + Z dz − π Z d r ⊥ e − i~k ⊥ ~r ⊥ Z d Xe − i~ ∆ ~X (cid:28) T r (cid:20) γ + S (cid:18) − z − − ~r ⊥ − ~X, z − ~r ⊥ − ~X (cid:19)(cid:21)(cid:29) = i ¯ P + Z d ξ (2 π ) d ξ (2 π ) e ip · ξ − ip · ξ (cid:10) T r (cid:2) γ + S ( ξ ; ξ ) (cid:3)(cid:11) , (A1)4where we changed the integration variables according to ~ξ = − z − − ~r ⊥ − ~X , (A2) ~ξ = z − ~r ⊥ − ~X , (A3)and introduced shorthand notations ~p = x ¯ P + + ~k ⊥ − ~ ∆2 , (A4) ~p = x ¯ P + + ~k ⊥ + ~ ∆2 . (A5)Now we have to consider separately the first case ξ − > , ξ − < and the second case ξ − < , ξ − > . All the other regionsare just the vacuum contributions ∼ δ (∆ ⊥ ) and must be omitted. For the sake of brevity we will refer to the contribution ofthe first region as H + − , and to the second one as H − + .For θ -functions of arguments ± ξ , we will use an integral representation θ ( ± ξ ) = 12 πi Z ∞−∞ dα e ± iαξ α − i − πi Z ∞−∞ dα e ∓ iαξ α + i . (A6)
1. Evaluation of H + − In the first case we have explicitly H + − = − i ¯ P + Z d ξ (2 π ) d ξ (2 π ) e ip · ξ − ip · ξ Z dα dα (2 π ) e i ( α ξ − − α ξ − ) ( α − i α − i × Z d p (2 π ) p − M + i Z d q (2 π ) exp (cid:26) − i (cid:18) q ⊥ + M p − ξ − − q ⊥ · ξ ⊥ − p + ξ − + ~p ⊥ ξ ⊥ (cid:19)(cid:27) × Z d z e i ( p ⊥ − q ⊥ ) z T r (cid:20) γ + (cid:18) γ p − √ q ⊥ + M ) (cid:19) Λ ( − ) (ˆ p + M ) (cid:21) (cid:10) U † ( z ) U ( ξ ⊥ ) (cid:11) . (A7)Now evaluate each of the integrals: Z dξ − dξ − (2 π ) e ip +1 ξ − − ip +2 ξ − e i ( α ξ − − α ξ − ) exp (cid:18) − i q ⊥ + M p − ξ − + ip + ξ − (cid:19) = δ (cid:18) α + p +1 − q ⊥ + M p − (cid:19) δ (cid:0) α + p +2 − p + (cid:1) , (A8) Z d p ⊥ (2 π ) Z d z e i ( p ⊥ − q ⊥ ) z Z d ξ ⊥ d ξ ⊥ (2 π ) e − ip ⊥ ξ ⊥ + ip ⊥ ξ ⊥ e iq ⊥ ξ ⊥ − ip ⊥ ξ ⊥ (cid:10) U ( ξ ⊥ ) U † ( z ) (cid:11) = Z d z e i ( p ⊥ − q ⊥ ) z Z d ξ ⊥ (2 π ) e − i ( p ⊥ − q ⊥ ) ξ ⊥ (cid:10) U † ( z ) U ( ξ ⊥ ) (cid:11) | p ⊥ = p ⊥ . (A9)Now change the dummy integration variables ~ξ ⊥ and ~z to ~X ⊥ and ~ρ ⊥ : ξ ⊥ := X ⊥ − ρ ⊥ , (A10) z := X ⊥ + ρ ⊥ , ⇒ Z d z e i ( p ⊥ − q ⊥ ) z Z d ξ ⊥ d ξ ⊥ (2 π ) e − ip ⊥ ξ ⊥ + ip ⊥ ξ ⊥ e iq ⊥ ξ ⊥ − ip ⊥ ξ ⊥ (cid:10) U ( ξ ⊥ ) U † ( z ) (cid:11) = ˜ γ (cid:18) k ⊥ − q ⊥ + ∆ ⊥ , k ⊥ − q ⊥ − ∆ ⊥ (cid:19) , (A11)5where ˜ γ was defined in (50). It is convenient to make a shift of the dummy integration variable according to Z d q ⊥ (2 π ) → Z d κ ⊥ (2 π ) where ~κ ⊥ = ~k ⊥ − ~q ⊥ . (A12) ⇒ H + − ( x, ξ, t, k ⊥ ) = − iN c Z d κ ⊥ (2 π ) ˜ γ (cid:18) κ ⊥ + ∆ ⊥ , κ ⊥ − ∆ ⊥ (cid:19) Z dp + dp − (2 π ) p + p − − p ⊥ − M + i × p + − p +2 − i ( k ⊥ − κ ⊥ ) + M p − − p +1 − i M − ~p ⊥ ( ~k ⊥ − ~κ ⊥ ) p − (A13)Now we take the integrals over p + , p − in (A13). The first integral is taken over p + , the result is Z dp + (2 π ) 12 p + p − − p ⊥ − M + i p + − p +2 − i iθ ( p − )2 p +2 p − − ( p ⊥ ) − M + i . (A14)Integration over p − yields H + − ( x, ξ, t, k ⊥ ) = 2 N c Z d κ ⊥ (2 π ) ˜ γ (cid:18) κ ⊥ + ∆ ⊥ , κ ⊥ − ∆ ⊥ (cid:19) × M − ~p ⊥ · ( ~k ⊥ − ~κ ⊥ )( x − ξ )(( ~k ⊥ − ~κ ⊥ ) + M ) − ( x + ξ )( p ⊥ + M ) ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x − ξx + ξ (( ~k ⊥ − ~κ ⊥ ) + M )( p ⊥ + M ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 2 N c Z d κ ⊥ (2 π ) ˜ γ (cid:18) κ ⊥ + ∆ ⊥ , κ ⊥ − ∆ ⊥ (cid:19) × M − (cid:16) ~k + ~ ∆ ⊥ (cid:17) · (cid:16) ~k − ~κ ⊥ (cid:17) ( x − ξ ) (cid:18)(cid:16) ~k − ~κ ⊥ (cid:17) + M (cid:19) − ( x + ξ ) (cid:18)(cid:16) ~k + ~ ∆ ⊥ (cid:17) + M (cid:19) ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x − ξx + ξ (cid:16) ~k − ~κ ⊥ (cid:17) + M (cid:16) ~k + ~ ∆ ⊥ (cid:17) + M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (A15)
2. Evaluation of H − + In complete analogy we evaluate the term H − + : H − + = − i Z d ξ (2 π ) d ξ (2 π ) e ip · ξ − ip · ξ Z dα dα (2 π ) e i ( α ξ − − α ξ − ) ( α + i α + i × Z d p (2 π ) p − M + i Z d q (2 π ) exp (cid:26) i (cid:18) q ⊥ + M p − ξ − − q ⊥ · ξ ⊥ − p + ξ − + ~p ⊥ ξ ⊥ (cid:19)(cid:27) × Z d z e − i ( p ⊥ − q ⊥ ) z T r (cid:20) γ + (ˆ p + M )Λ (+) (cid:18) γ p − √ M − ˆ k ⊥ ) (cid:19)(cid:21) (cid:10) U † ( ξ ⊥ ) U ( z ) (cid:11) (A16)Now take the integrals term-by-term in complete analogy with the previous case Z dξ − dξ − (2 π ) e ip +1 ξ − − ip +2 ξ − e i ( α ξ − − α ξ − ) exp (cid:18) i q ⊥ + M p − ξ − − ip + ξ − (cid:19) = δ (cid:0) α + p +1 − p + (cid:1) δ (cid:18) α + p +2 − q ⊥ + M p − (cid:19) , (A17) Z d p ⊥ (2 π ) Z d z e − i ( p ⊥ − q ⊥ ) z Z d ξ ⊥ d ξ ⊥ (2 π ) e − ip ⊥ ξ ⊥ + ip ⊥ ξ ⊥ e − iq ⊥ ξ ⊥ + ip ⊥ ξ ⊥ (cid:10) U † ( ξ ⊥ ) U ( z ) (cid:11) = Z d z e − i ( p ⊥ − q ⊥ ) z Z d ξ ⊥ (2 π ) e i ( p ⊥ − q ⊥ ) ξ ⊥ (cid:10) U † ( ξ ⊥ ) U ( z ) (cid:11) | p ⊥ = p ⊥ , (A18)6Now change the dummy integration variables ~ξ ⊥ , ~z to ~X, ~ρ ⊥ according to ξ ⊥ := X ⊥ + ρ ⊥ , (A19) z := X ⊥ − ρ ⊥ . (A20) ⇒ Z d p ⊥ (2 π ) Z d z e − i ( p ⊥ − q ⊥ ) z Z d ξ ⊥ d ξ ⊥ (2 π ) e − ip ⊥ ξ ⊥ + ip ⊥ ξ ⊥ e − iq ⊥ ξ ⊥ + ip ⊥ ξ ⊥ (cid:10) U † ( ξ ⊥ ) U ( z ) (cid:11) = ˜ γ (cid:18) k ⊥ − q ⊥ + ∆ ⊥ , k ⊥ − q ⊥ − ∆ ⊥ (cid:19) (A21)where function ˜ γ was defined in (50). Now shift the dummy integration variable according to Z d q ⊥ (2 π ) → Z d κ ⊥ (2 π ) where ~κ ⊥ = ~k ⊥ − ~q ⊥ ⇒ H − + ( x, ξ, t, k ⊥ ) = + iN c Z d κ ⊥ (2 π ) ˜ γ (cid:18) κ ⊥ + ∆ ⊥ , κ ⊥ − ∆ ⊥ (cid:19) Z dp + dp − (2 π ) p + p − − p ⊥ − M + i × p + − p +1 + i ( k ⊥ − κ ⊥ ) + M p − − p +2 + i M − ~p ⊥ ( ~k ⊥ − ~κ ⊥ ) p − . (A22)First take the integral over the p + : Z dp + (2 π ) 12 p + p − − p ⊥ − M + i p + − p +1 + i − iθ ( − p − )2 p +1 p − − ( p ⊥ ) − M + i , (A23)next take the integral over p − : ⇒ H − + = 2 N c Z d κ ⊥ (2 π ) ˜ γ (cid:18) κ ⊥ + ∆ ⊥ , κ ⊥ − ∆ ⊥ (cid:19) × M − ~p ⊥ · ( ~k ⊥ − ~κ ⊥ )( x + ξ )(( ~k ⊥ − ~κ ⊥ ) + M ) − ( x − ξ )(( ~p ⊥ ) + M ) ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x + ξx − ξ ( ~k ⊥ − ~κ ⊥ ) + M ( ~p ⊥ ) + M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 2 N c Z d κ ⊥ (2 π ) ˜ γ (cid:18) κ ⊥ + ∆ ⊥ , κ ⊥ − ∆ ⊥ (cid:19) × M − (cid:16) ~k − ~ ∆2 (cid:17) · ( ~k ⊥ − ~κ ⊥ )( x + ξ ) (cid:18)(cid:16) ~k ⊥ − ~κ ⊥ (cid:17) + M (cid:19) − ( x − ξ ) (cid:18)(cid:16) ~k − ~ ∆2 (cid:17) + M (cid:19) ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x + ξx − ξ ( ~k ⊥ − ~κ ⊥ ) + M (cid:16) ~k − ~ ∆2 (cid:17) + M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (A24)In summary, we have H + − = 2 N c Z d κ ⊥ (2 π ) ˜ γ (cid:18) κ ⊥ + ∆ ⊥ , κ ⊥ − ∆ ⊥ (cid:19) × M − (cid:16) ~k + ~ ∆ ⊥ (cid:17) · (cid:16) ~k − ~κ ⊥ (cid:17) ( x − ξ ) (cid:18)(cid:16) ~k − ~κ ⊥ (cid:17) + M (cid:19) − ( x + ξ ) (cid:18)(cid:16) ~k + ~ ∆ ⊥ (cid:17) + M (cid:19) ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x − ξx + ξ (cid:16) ~k − ~κ ⊥ (cid:17) + M (cid:16) ~k + ~ ∆ ⊥ (cid:17) + M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (A25) H − + = 2 N c Z d κ ⊥ (2 π ) ˜ γ (cid:18) κ ⊥ + ∆ ⊥ , κ ⊥ − ∆ ⊥ (cid:19) × M − (cid:16) ~k − ~ ∆2 (cid:17) · ( ~k ⊥ − ~κ ⊥ )( x + ξ ) (cid:18)(cid:16) ~k ⊥ − ~κ ⊥ (cid:17) + M (cid:19) − ( x − ξ ) (cid:18)(cid:16) ~k − ~ ∆2 (cid:17) + M (cid:19) ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x + ξx − ξ (cid:16) ~k ⊥ − ~κ ⊥ (cid:17) + M (cid:16) ~k − ~ ∆2 (cid:17) + M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (A26)7Notice that the sum H (cid:16) x, ξ, t, ~k ⊥ (cid:17) = H + − (cid:16) x, ξ, t, ~k ⊥ (cid:17) + H − + (cid:16) x, ξ, t, ~k ⊥ (cid:17) (A27)is antisymmetric w.r.t. the inversion of the light-cone fraction x → − x, i.e. H (cid:16) − x, ξ, t, ~k ⊥ (cid:17) = − H (cid:16) x, ξ, t, ~k ⊥ (cid:17) . We can see that in the points x = ± ξ the result (A27) has logarithmic divergences ∼ ln | x ∓ ξ | . Physically, in this points oneof the quarks has a zero light-cone fraction, and as a consequence (A27) becomes very sensitive to the details of short-distancestructure of the model. When we evaluated (A27), we integrated over p ± up to infinity. Rigorously speaking, this contradicts thebasic assumptions of the model, in particular, (38), which is valid only when the moments of the active partons are much smallerthan the moment of the whole nucleus. However, since for p +1 , = 0 the integrals were convergent (the dominant contributioncomes from the region where the model is valid), we could ignore such an explicit cutoffs. Notice that in evaluation of thephysical DVCS amplitude (54) the cutoffs | p − | ≤ q − / were provided by the external kinematics. Generalization of (38) to themore realistic color source is a much more complicated task. Appendix B: ˙ U † U ¸ CORRELATOR IN FINITE NUCLEI.
As we have seen in the previous section, as well as we will see in the next section, physical observables depend on thecorrelator (cid:10) P ′ (cid:12)(cid:12) U † ( x ) U ( y ) (cid:12)(cid:12) P (cid:11) ≈ ¯ P + Z d X e i~ ∆ ~X T r (cid:0) U † ( x − X ) U ( y − X ) (cid:1) . (B1)Notice that the weight functional W is expressed in terms of the field ρ, i.e. the correlator is essentially nonlinear object.In the finite nucleus evaluation of this object slightly differs from the original derivation given in [20, 21]. However, since theweight functional W [ ρ ] is Gaussian, the total result can be expressed in terms of the elementary correlator[54] h P ′ | ρρ | P i . The final result of our evaluation is S ( x, y ) = (cid:10) P ′ (cid:12)(cid:12) U † ( x ⊥ ) U ( y ⊥ ) (cid:12)(cid:12) P (cid:11) = e i~ ∆ ~x ⊥ + ~y ⊥ Z d X e i~ ∆ ⊥ ~X × (B2) × exp − g N c ˜ f (cid:16) ~ , ~x ⊥ − ~y ⊥ − ~X (cid:17) + ˜ f (cid:16) ~ , − ~x ⊥ − ~y ⊥ − ~X (cid:17) − ˜ f (cid:16) ~x ⊥ − ~y ⊥ , − ~X (cid:17) , where ˜ f ( ~r , ~r ) = R d ˜∆(2 π ) e − i ˜∆ ~r R + ∞−∞ dz − ˜ γ A ( z − , ~r ; ˜∆) . Indeed, using definition U ( x ) = P exp (cid:18) ig Z + ∞−∞ dz − α a ( z − , ~x ⊥ ) T a (cid:19) , (B3)we may notice that• Only the even powers of α give nonzero contribution to (B2)• The first term (zero order in α ) is proportional to δ (∆) and vanishes in the off-forward limit.Contribution of the second-order term gives − g N c e i~ ∆ ⊥ ~x ⊥ + ~y ⊥ (cid:18) cos (cid:18) ~ ∆ ⊥ ~x ⊥ − ~y ⊥ (cid:19) Z dz − ˜ γ A ( z − ,~ ⊥ ) − Z dz − ˜ γ A ( z − , ~x ⊥ − ~y ⊥ ; ∆) (cid:19) . (B4)It is very convenient to introduce temporary notation R dz − ˜ γ A ( z − , ~r ⊥ ; ∆) = f ( ~r ⊥ ; ∆) . In this notation (B4) reduces to − g N c e i~ ∆ ⊥ ~x ⊥ + ~y ⊥ cos ~ ∆ ⊥ ~r ⊥ ! f ( ~ ⊥ ; ∆) − f ( ~r ⊥ ; ∆) ! , (B5)8where we used notation ~r = ~x ⊥ − ~y ⊥ . Evaluation of the higher-order contributions is a bit more tricky. First we have to notice that the Gaussian form of W [ ρ ] enables us to introduce a sort of Wick theorem for evaluation of the multileg correlators. After that, we have to make Fouriertransformation of each correlator, take the integral over d X ⊥ and make the Fourier back to coordinate space. Performing suchprocedure step-by-step, contribution of the n -th order term after some manipulations may be reduced to n X m =0 min ( m, n − m ) X k =0 ( − n − m g n N nc k !( m − k )!(2 n − m − k )! Z d ∆ ⊥ (2 π ) ... Z d ∆ ⊥ n (2 π ) δ ~ ∆ ⊥ − n X i =0 ~ ∆ ⊥ i ! × [ m − k ] Y i =1 f ( ~ , ~ ∆ ⊥ i ) e i~x~ ∆ ⊥ i n Y i =[ m + k ]+1 f ( ~ , ~ ∆ ⊥ i ) e i~y ~ ∆ ⊥ i [ m + k ] Y i =[ m − k ]+1 f ( r ⊥ , ~ ∆ ⊥ i ) = e i~ ∆ ~x ⊥ + ~y ⊥ n X m =0 min ( m, n − m ) X k =0 ( − n − m g n N nc k !( m − k )!(2 n − m − k )! × Z d ∆ ⊥ (2 π ) ... Z d ∆ ⊥ n (2 π ) δ ( ~ ∆ ⊥ − n X i =0 ~ ∆ ⊥ i ) × [ m − k ] Y i =1 f ( ~ , ~ ∆ ⊥ i ) e i~r~ ∆ ⊥ i / n Y i =[ m + k ]+1 f ( ~ , ~ ∆ ⊥ i ) e − i~r~ ∆ ⊥ i / [ m + k ] Y i =[ m − k ]+1 f ( r ⊥ , ~ ∆ ⊥ i ) . (B6)Now we replace back δ ( ~ ∆ ⊥ − P ni =0 ~ ∆ ⊥ i ) = R d X e i~ ∆ ⊥ ~X ⊥ Q ni =1 e − i~ ∆ ⊥ i ~X ⊥ and reduce (B6) to e i~ ∆ ~x ⊥ + ~y ⊥ Z d X e i~ ∆ ⊥ ~X ⊥ n X m =0 min ( m, n − m ) X k =0 ( − n − m g n N nc k !( m − k )!(2 n − m − k )! × [ m − k ] Y i =1 Z d ∆ ⊥ i (2 π ) f ( ~ , ~ ∆ ⊥ i ) e i~r~ ∆ ⊥ i / n Y i =[ m + k ]+1 Z d ∆ ⊥ i (2 π ) f ( ~ , ~ ∆ ⊥ i ) e − i~r~ ∆ ⊥ i / × [ m + k ] Y i =[ m − k ]+1 Z d ∆ ⊥ i (2 π ) f ( r ⊥ , ~ ∆ ⊥ i ) = e i~ ∆ ~x ⊥ + ~y ⊥ Z d X e i~ ∆ ⊥ ~X n X m =0 min ( m, n − m ) X k =0 ( − n − m g n N nc k !( m − k )!(2 n − m − k )! × ˜ f [ m − k ] (cid:18) ~ , ~r − ~X (cid:19) ˜ f [ m + k ] (cid:18) ~ , − ~r − ~X (cid:19) ˜ f k (cid:16) ~ , − ~X (cid:17) = e i~ ∆ ~x ⊥ + ~y ⊥ Z d X e i~ ∆ ⊥ ~X exp − g N c ˜ f (cid:16) ~ , ~r − ~X (cid:17) + ˜ f (cid:16) ~ , − ~r − ~X (cid:17) − ˜ f (cid:16) ~r, − ~X (cid:17) , (B7)in agreement with (B2).For evaluation of the complicated objects like (cid:10) P ′ (cid:12)(cid:12) Φ[ ρ ] U † ( x ⊥ ) U ( y ⊥ ) (cid:12)(cid:12) P (cid:11) (see e.g. Gluon distributions) we can use aquasiclassical formula9 D P ′ (cid:12)(cid:12)(cid:12) ˆ A ( x, y ) ˆ B ( x, y ) (cid:12)(cid:12)(cid:12) P E = e i~ ∆ ~x + ~y ¯ P + Z d X e i~ ∆ ~X × (cid:18)Z d ∆ (2 π ) e − i~ ∆ ~X (cid:28) P + ∆ (cid:12)(cid:12)(cid:12)(cid:12) ˆ A (cid:18) ~r , − ~r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) P (cid:29)(cid:19) (cid:18)Z d ∆ (2 π ) e − i~ ∆ ~X (cid:28) P + ∆ (cid:12)(cid:12)(cid:12)(cid:12) ˆ B (cid:18) ~r , − ~r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) P (cid:29)(cid:19) = e i~ ∆ ~x + ~y ¯ P + Z d ∆ (2 π ) d ∆ (2 π ) (2 π ) δ (∆ − ∆ − ∆ ) (cid:28) P + ∆ (cid:12)(cid:12)(cid:12)(cid:12) ˆ A (cid:18) ~r , − ~r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) P (cid:29) (cid:28) P + ∆ (cid:12)(cid:12)(cid:12)(cid:12) ˆ B (cid:18) ~r , − ~r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) P (cid:29) = e i~ ∆ ~x + ~y Z d X e i~ ∆ ~X A cl (cid:18) ~r − ~X, − ~r − ~X (cid:19) B cl (cid:18) ~r − ~X, − ~r − ~X (cid:19) , (B8)where ~r = ~x − ~y. 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