Real Compton Scattering via Color Dipoles

# Real Compton Scattering via Color Dipoles

## Abstract

We study photoabsorption reaction and real Compton scattering (RCS) within the color dipole model. We rely on a photon wave function derived in the instanton vacuum model, and on the energy dependent phenomenological elastic dipole amplitude. Data for the photoabsorption cross section at high energies agree with our parameter free calculations. We also provide predictions for the differential RCS cross section. Although no data for small angle Compton scattering are available so far, this process can be measured in ultra-peripheral hadronic and nuclear collisions at the LHC.

## I Introduction

Compton scattering, , and related photoabsorption reaction, have been a subject of intensive theoretical and experimental investigation (1); (2); (3); (4); (5); (6); (7); (8); (9); (10); (11); (12); (13); (14); (15). While in the case of deeply-virtual Compton scattering (DVCS), where the initial photon is highly-virtual, the QCD factorization is proven (5); (8); (7) and the amplitude can be expressed in terms of the generalized parton distributions (GPD) (1); (2); (3); (4); (5); (6); (7); (8); (9); (10); (11); (12); (13); (14); (15), in the case of real Compton scattering (RCS) the available theoretical tools are rather undeveloped.

On the one hand, as it has been shown in (16); (17), for large momentum transfer it is possible to factorize the RCS amplitude (18); (19) and express it in terms of the distribution amplitudes of the proton. On the other hand, it is possible to express the amplitude of the process via the minus 1st-moment of GPDs at zero skewedness (5); (20); (21).

The traditional sources of quasi-real and virtual photons, the electron beams, with very high collisions energies are expected to be available in near future. The new projects of LHeC (22); (23) and EIC (24); (25) are currently under intensive discussion. Besides the electron beams, one can also use beams of charged hadrons. Provided that the transverse overlap of the colliding hadrons is small, i.e. the transverse distance between the colliding centers is larger than the sizes of the colliding particles, , the electromagnetic interaction between colliding particles becomes the dominant mechanism. Such processes called ultra-peripheral collisions (UPC) can be studied in , and collisions. In particular, one can access RCS in the reaction

 A1+A2 →A1+γ+A2. (1)

The typical virtualities of the intermediate photon are controlled by the formfactors of the colliding particles, and are small:

 ⟨Q2γ∗⟩≲3R2A∼0.1GeV2A2/3. (2)

Thus, is of the order of the soft hadronic scale, so the intermediate photon can be treated as a free Weizsäcker-Williams one, i.e. the amplitude of the process (1) can be described in terms of RCS.

These processes at the LHC will allow to study RCS at very high energies. The possibility of observation of such processes experimentally has been demonstrated by the STAR (26); (27); (28) and PHENIX (29) experiments at RHIC. It is expected that at LHC photon-proton collisions at energies up to  GeV can be observed (30). In this paper we concentrate on RCS on a proton target. Nuclear effects will be discussed elsewhere.

In what follows we employ the color dipole approach introduced in (31); (32). The central objects of the model are the dipole scattering amplitude,  and the light-cone quark distribution functions of the photon. While pQCD predicts the dipole amplitude only for small-size dipoles, several successful phenomenological parameterizations for the large-size dipoles are known. Relying on the photon wave function evaluated in the instanton vacuum model (33), which is valid for any , one can extend the applicability of the model to the case of the processes with real photons (34). In this paper we are going to consider the real photoabsorption, , and the RCS.

## Ii Color dipole model

The color dipole model is valid only at sufficiently high energies, where the dominant contribution to the Compton amplitude comes from gluonic exchanges. Then the general expression for the Compton amplitude in the color dipole model has the form,

where is the photon polarization vector; are the light-cone fractional momenta of the quark and antiquark, are the transverse distances in the final and initial dipoles respectively; is the momentum transfer in the Compton scattering, is the scattering amplitude for the dipole state which also implicitly depends on , c.m. energy squared, and is the wavefunction of the photon in the polarization state (33).

At high energies in the small angle approximation, , the quark separation and fractional momenta are preserved, so,

Generally, the amplitude is a nonperturbative object, with asymptotic behavior for small  controlled by pQCD (31):

 fN¯qq(→r,→Δ,β)∼r2,

up to slowly varying corrections .

Calculation of the RCS differential cross section also involves the real part of scattering amplitude, whose relation to the imaginary part is quite straightforward. According to (35), if the limit exists and is finite, then the real part and imaginary parts of the forward amplitude are related as

 Ref(Δ=0)=sαtan[π2(α−1+∂∂lns)]Imf(Δ=0)sα. (5)

In the model under consideration the imaginary part of the forward dipole amplitude indeed has a power dependence on energy, , so (5) simplifies to

 ReAImA =tan(π2(α−1))≡ϵ. (6)

This fixes the phase of the forward Compton amplitude, which we retain for nonzero momentum transfers assuming for the real and imaginary parts similar dependences on. Finally we arrive at,

 Aμν≈(ϵ+i)e(i)μ(q′)e(j)ν(q)∫d2r∫dβ¯Ψ(i)γ(β,r)Ψ(j)γ(β,r)ImfN¯qq(→r,→Δ,β,s), (7)

For the cross-section of unpolarized Compton scattering, from (7) we obtain,

 dσγpeldt = 1+ϵ216π∑ij∣∣A(ij)μν∣∣2= (8) = 1+ϵ216π∑ij∣∣∣∫d2r∫dβ¯Ψ(i)γ(β,r)Ψ(j)γ(β,r)ImfN¯qq(→r,→Δ,β)∣∣∣2.

The imaginary part of the forward amplitude (7) gives the total photoabsorption cross-section,

 σγptot=116π∫dβd2r∣∣Ψγ(β,r)∣∣2ImfN¯qq(→r,→Δ,β). (9)

Formulas (8,9) are used further for numerical calculations.

## Iii Wavefunctions from the instanton vacuum

In this section we present briefly some details of the wavefunction evaluation in the instanton vacuum model (see (36); (37); (38) and references therein). The central object of the model is the partition function of the light quarks, which has the form

 Z[v]=∫dλD¯ψDψDΦeiS[λ,v,¯ψ,ψ,Φ], (10)

where the effective action is defined as (38); (41)

 S[λ,v,¯ψ,ψ,Φ]=∫d4x(NVlnλ+2Φ2(x)−¯ψ(^p+^v−m−c¯Lf⊗Φ⋅Γm⊗fL)ψ).

Here and are the fields of constituent quarks and mesons respectively, is the density of the instanton gas, is the external vector current corresponding to the photon, is the gauge factor,

 L(x,z)=Pexp(i∫xzdζμvμ(ζ)), (11)

which provides the gauge covariance of the action, and the nonlinear term in explicit form is

 −c¯ψ¯Lf⊗Φ⋅Γm⊗fLψ≡−c∫d4xd4yd4z¯ψ(x)¯L(x−z)~f(x−z)(∑iΦi(z)Γm,i)~f(z−y)L(z−y)ψ(y) (12)

where is one of the matrices, , , and is the Fourier transform of the zero-mode profile.

In the leading order in , we have the same Feynman rules as in the perturbative theory, but with momentum-dependent quark mass in the quark propagator

 S(p) = 1^p−μ(p)+i0. (13)

The mass of the constituent quark has a form

 μ(p)=m+Mf2(p),

where  MeV is the current quark mass,  MeV is the dynamical mass generated by the interaction with the instanton vacuum background. Due to the presence of instantons the coupling of a vector current to a quark is also modified,

 ^v ≡ vμγμ→^V=^v+^Vnonl, ^Vnonl ≈ −2Mf(p)df(p)dpμvμ(q)+O(q2), (14)

where is the momentum of the incoming quark, and is the momentum of the photon. Notice that for an arbitrary photon momentum the expression for depends on the choice of the path in (11) and as a result one can find in the literature different expressions used for evaluations (33); (39); (40); (41). In the limit the function falls off as so for large , where is the mean instanton size, the mass of the quark and vector current interaction vertex . However we would like to emphasize that the wavefunction gets contribution from both the soft and the hard parts, so even in the large- limit the instanton vacuum function is different from the well-known perturbative result.

We have to evaluate the wavefunctions associated with the following matrix elements:

 IΓ(β,→r) = ∫dz2πei(β+12)q⋅z⟨0∣∣∣¯ψ(−z2n−→r2)Γψ(z2n+→r2)∣∣∣γ(q)⟩, (15)

where is one of the matrices In the leading order in one can easily obtain

 IΓ=∫d4p(2π)4ei→p⊥→r⊥δ(p+−(β+12)q+)Tr(S(p)^VS(p+q)Γ). (16)

The evaluation of (16) is quite tedious but straightforward. Details of this evaluation may be found in (33).

The overlap of the initial and final photon wavefunctions in (8) was evaluated according to

 Ψ(i)∗(β,r,Q2=0)Ψ(i)(β,r,Q2)=∑ΓI∗Γ(β,r∗,0)IΓ(β,r,Q2), (17)

where summation is done over possible polarization states . In the final state we should use , which is related to the reference frame with in which the components (16) were evaluated.

## Iv Numerical results

### iv.1 Photoabsorption

The Bjorken variable used in DIS, , is not appropriate at small photon virtualities, since it does not have the meaning of a fractional quark momentum any more, and may be very small even at low energies. In particular, for RCS the Bjorken variable defined in this way would be zero. Therefore, one should rely on the phenomenological dipole cross section which depends on energy, rather than . We use the -dependent dipole cross section proposed in (42), which saturates at large separations in analogy to the -dependent one proposed in (43). Correspondingly, for the elastic dipole amplitude we employ the model developed in (44); (45); (46); (34),

 ImfN¯qq(→r,→Δ,β,s) = σ0(s)4exp[−(B(s)2+R20(s)16)→Δ2⊥]× (18) × ⎛⎝e−iβ→r⋅→Δ+ei(1−β)→r⋅→Δ−2ei(12−β)→r⋅→Δe−r2R20(s)⎞⎠,

where are the phenomenological parameters known from DIS and scattering data.

We employ the -dependent parametrization of the dipole cross section suitable for soft processes (42)

 σ¯qq(r,s) = σ0(s)(1−e−r2/R20(s)), (19) σ0(s) = σπp(s)(1+38R20(s)r2π), (20) R0(s) = 0.88fm×(s0s)0.14, (21)

where For the pion cross section we use the parametrization and fit of (47), namely its Pomeron part,

 σπp(s)=23.6(ss0)0.079mb. (22)

The parameter in Eq. (18), is related to the -slope of the differential cross section of elastic scattering (44); (46); (45),

 B(s)=Bπpel(s)−13⟨r2ch⟩π−18R20(s) (23)

Here we rely on the Regge factorization and use , with  GeV fm, GeV, and  GeV.

This parametrization may be used in Eqs. (8-9) only at very high energies where in terms of the Regge theory, the Pomeron term in the cross section dominates. So far only two data points shown in Fig. 1 are available for from the H1 and ZEUS experiments (48), and our parameter free calculation agrees well with these data.

In order to extend the model down to smaller values of , where more data are available, we added the Reggeon contribution, which was fitted to the photoabsorption data in (47)

 σ(R)γp(s)=129nb×s−0.4525. (24)

Besides, the Pomeron part of the dipole cross section parametrized as in Eqs. (19-21) exposes some problems at low energies. Indeed, as one can see from (21), the saturation radius grows and may substantially exceed the confinement radius. In order to regularize the low-energy behavior of , we modify Eq. (21) as follows,

 R0(s)⇒~R0(s)=0.88fm(s0s+s1)0.14. (25)

Fit to low-energy photoabsorption data allows to fix this parameter at  GeV  (see Fig. (1) for more details). Since further evaluations are done in the LHC energy range, the difference between (21) and (25) is negligible, as one can see from Fig. (1). Indeed, both parameterizations coincide for GeV. In the right pane of the Fig. (1) the contributions of the color dipole (9) and Reggeon (24) terms are displayed separately.

### iv.2 Compton scattering

Using parameterization (19-22), we calculate the elastic RCS differential cross-section as

 dσγpeldt = 1+ϵ216π∑ij∣∣∣∫d2r∫dβ¯Ψ(i)γ(β,r)Ψ(j)γ(β,r)ImfN¯qq(→r,→Δ,β)+16πσ(IR)γp(s)eBIR(s)t∣∣∣2. (26)

Since extraction of reggeon parameters from experimental data yields huge uncertainties (52), in this paper we rely on the -dominance of the pomeron (53) and take the reggeon slope as

 BIR(s)=B1+2α′IR(0)ln(sμ20), (27)

where GeV, GeV, and the results for the differential cross-section are presented in the Figure 2. As one can see from the left pane, for  GeV the cross-section rises with energy for small , but falls at GeV. This corresponds to the Regge predicted energy dependence . However, a word of caution is in order here, since the linear -dependence of the Pomeron trajectory may not continue at large , and indeed data indicate that levels off (49). On the right pane of the Figure 2 our predictions for the -dependence of the cross section are plotted for different energies. The cross section demonstrates a shrinkage of diffraction cone with energy in accordance with the Regge theory.

The RCS cross-section has been measured so far only at Jefferson Lab (JLAB) at energies  (50). In Figure 3 we compare predictions of the color dipole model with experimental data. Since these data also have relatively large 2 GeV (wide-angle Compton scattering), calculations in the dipole approach go beyond the kinematics of validity of the model. Indeed, Eqn. (18), lead to the RCS cross-section which decreases exponentially at fixed , while the general pQCD analysis (51) predicts behavior.

As was discussed in the introduction, there are two approaches which are used to describe the wide-angle Compton scattering. The first one is valid for large (18); (19) and expresses the amplitude via the distribution amplitude of three valence quarks in the proton. The RCS cross-section in this approach was studied in (16); (17) and it was found that evaluation with widely used distribution amplitudes also underestimates the JLAB data (50). Another description expresses the RCS amplitude via the moment of GPDs at zero skewedness,  (5); (20); (21). This approach is able to describe the existing JLAB data. However the -dependence of the cross-section in this approach depends on the model for the GPD used in the evaluation.

Since the experimental counting rate also includes the flux of quasi-real photons, we present in Fig. 4 the two-dimensional product of the flux and the differential RCS cross section,

 d2σpp→ppγdkdt=dNγdkdσγp→γpdt, (28)

where is the absolute value of the wave vector of the quasireal photon, and photon flux is given, e.g., in (30).

## V Summary

In this paper we evaluated the photoabsorption and RCS cross-sections within the color dipole model. We employed a photon wave function calculated in the instanton-vacuum model and incorporating nonperturbative effects (33). For the dipole cross section we relied on an energy dependent parametrization, because Bjorken is not a proper variable at low photon virtualities.

We found that the model describes available data for the photoabsorption cross section from the ZEUS and H1 data quite well, justifying application of the color dipole for processes with real photons. Applicability of the dipole model was extended down to smaller energies, GeV, by freezing the saturation radius Eq. (21) for the energy dependent parametrization (42), to make sure that it does not exceed the confinement radius.

We also evaluated the RCS cross-sections and made predictions for the energy range to be accessed in ultra-peripheral collisions at LHC (Figure 2).

## Acknowlegments

This work was supported in part by Fondecyt (Chile) grants 1090291, 1090073 , and by DFG (Germany) grant PI182/3-1.

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