Abstract
We consider elastic quarkquark scattering at high energy with fixed momentum transfer and perform factorization of softgluon exchanges into a vacuum expectation value of Wilson lines. Taking into account nonperturbative corrections whose structure is predicted from infrared renormalon analysis, we represent the scattering amplitude as an asymptotic series. In the region of small momentum transfer, where the nonperturbative corrections are dominant, the scattering amplitude is Gaussian distribution in with a slope depending on a nonperturbative scale. A nonperturbative origin of the soft pomeron is thus identified.
ITP–SB–95–19
May, 1995
The soft pomeron and nonperturbative corrections
[5mm] in quarkquark scattering.
Irina A. Korchemskaya ^{*}^{*}*I; on leave from the Moscow Energy Institute, Moscow, Russia
Institute for Theoretical Physics,
State University of New York at Stony Brook,
Stony Brook, New York 11794 – 3840, U.S.A.
and
Universitá di Parma and INFN,
Gruppo Collegato di Parma, I–43100 Parma, Italy
1. Introduction
Regge theory [1] explains a large class of experimental results for hadronic scattering amplitudes at high energy and fixed transferred momentum . After almost 35 years, however, it remains a challenge to understand Regge theory within the framework of fundamental quantum field theory. One such attempt led to the development of the BFKL pomeron [2], which describes the compound state of two reggeized gluons with vacuum quantum numbers. Found in the leading logarithmic approximation, the BKFL pomeron, sometimes called the hard pomeron, leads to scattering amplitudes which violate the Froissart bound. Recently [4], [5], [6] it was shown that in generalized leading logarithmic approximation QCD is described by an effective 2dimensional field theory, which is equivalent to an XXX Heisenberg magnet of spin . This theory takes into account the propagation of infinite numbers of interacting Reggeons in the channel and restores the unitary of the Smatrix.
This paper is devoted to the “soft” pomeron [7] , [8]. While this idea is phenomenologically very successful [9], and we know from experiment the pomeron trajectory, , a deep understanding how the pomeron appears in QCD is still lacking. It is widely believed that the soft pomeron has a nonperturbative origin. It was proposed in [10] to include nonperturbative effects through a modification of the gluon propagator. Another ideas, in particular the application of the method of the stochastic vacuum models were discussed in [11]. In [12] the high energy interactions were described in terms of a twodimensional sigmamodel action. The approach developed here follows from a different perspective. We shall exploit the viewpoint that perturbation theory itself can predict the form of nonperturbative corrections, and show that ambiguities of the perturbative series caused by infrared renormalons allow one to identify the structure of nonperturbative corrections. In case of annihilation, which admits the operator product expansion, the nonperturbative corrections can be parameterized by local vacuum condensates. Further analysis [14] revealed that the perturbative series is not Borel summable. The singularities of the Borel transform, which are called IR renormalons [13] imply that the physical quantity will be well defined if the ambiguity caused by IR renormalons is compensated by an ambiguity in the definition of local vacuum condensates. In [15] the idea of IR renormalons was generalized to hadronic processes (jet cross sections, inclusive DrellYan leptonpair production) to which the operator product expansion is not applicable. It was found that nonperturbative corrections are parameterized by new parameters, associated with vacuum expectation values of nonlocal operators involving Wilson lines and the gluon field strength. The strength of the nonperturbative corrections is determined by the position of the leading renormalon. The relation between infrared renormalons and power corrections was also discussed in [16], [17], [18], [19], [20].
Analyzing highenergy hadronic scattering, we consider hadrons as consisting of partons. The soft pomeron is thought to couple to the valence quarks only. This additive quark rule is supported by much experimental data. Quarkquark scattering naturally involves into consideration the Wilson lines. In nonperturbative QCD the Wilson line appears, for example, in the representation of the quark propagator as a sum over random paths between points x and y [22],[23] and it takes into account the interactions of gauge field with the color current created by the quark moving along the path. The remarkable fact, however, is that in high energy scattering, , the quarks move along the straight lines. In perturbative QCD the Wilson lines describes the infrared asymptotics of the quark propagator [21]. Moreover one can expand the Wilson line in powers of gauge fields and reproduce the eikonal approximation for the interaction vertices of quark with soft gluons.
Our strategy is the following. We start with factorization of the soft gluon exchanges into a vacuum expectation value of Wilson lines and represent the quarkquark scattering amplitude as an expectation value of a Fourier transformed Wilson line, evaluated along an integration path which consists of two semiclassical quark trajectories, separated by an impact parameter in the transverse direction [24]. For the case in which the quarks are described by lightlike Wilson lines the similar formula was proposed by Nachtmann [25]. We shall go on and find the expression for the quarkquark elastic scattering amplitude, which resembles an eikonal formula of Cheng and Wu [3]. We shall derive it, however, from the renormalization properties of the cross singularities of Wilson loops [24].
Calculated to the lowest order of perturbation theory the quarkquark scattering amplitude gets large perturbative corrections such as . Resummation of these Sudakov corrections can be performed using an “evolution equation” technique [26], [27]. In the present paper, we generalize the method [24] to perform the resummation of both kind of corrections: perturbative and nonperturbative in the impact parameter space. The perturbative Sudakov corrections come from very soft virtual gluons with transverse momenta much smaller than the hadronic scale. In this region the nonperturbative corrections are very large. As we will show below, they have the form of power corrections , where is a parameter characterizing the nonperturbative interactions, while the perturbative corrections behave as . The dependence of the scattering amplitude on comes from the dependence of the Wilson lines on the impact parameter . This means that we expect a more or less realistic prediction for the dependence of the scattering amplitude, and our soft pomeron predicts a linear Regge trajectory. It is problematic to find the intercept of the soft pomeron from our model and we can only say that it is close to unity.
The paper is organized as follows. In Section 2 we review the application of the Wilson loop formalism to quarkquark scattering [24],[28]. In Section 3 we show that the Wilson loop in perturbation theory contains an ambiguous contribution to power corrections caused by infrared renormalons. In Section 4 using the prediction for the asymptotics of the cross anomalous dimension [28] in all orders of perturbation theory we include the nonperturbative corrections in the expression for the scattering amplitude, and represent the scattering amplitude as a Mellin integral. In Section 5 we calculate the scattering amplitude for the case of a frozen coupling constant. In Section 6 we consider the case of a running coupling constant and represent the scattering amplitude as an asymptotic series. Here we discuss the origin of the asymptotic expansion as a consequence of the presence of a singularity in the running coupling constant. We also derive the main results of this analysis: a Gaussian distribution over transferred momenta in the scattering amplitude, shrinkage of the distribution with increasing energy, and a crossover region in the differential crosssection. Section 7 contains concluding remarks.
2. Scattering amplitude in perturbative QCD
We consider near forward elastic quarkquark scattering at high energy and fixed transferred momentum in the following kinematics:
Here is the invariant energy of quarks with mass , is the transferred momentum and is an IR cutoff. Let us explain the origin of this kinematics. The quarkquark scattering should be thought of as embedded in a physical process involving hadronhadron elastic or inelastic scattering. Considered in isolation, the qq scattering amplitude has IR divergences and an IR cutoff is necessary. In exclusive and inclusive physical processes involving the qq scattering, the IR divergences are canceled and the IR cutoff is replaced by a dynamically generated transverse momentum scale of the hadronic state. Therefore for a purely perturbative calculation we should choose . To include nonperturbative effects, however, we find it useful to relax this condition such as . As a particular example of hadronic state one can consider the perturbative onium state [29] built from heavy quarks with mass . In this case the mass has a meaning of the transverse size of the hadron.
In the center of mass frame of the incoming quarks, quark momenta have the following lightcone components: and . In the limit the angle between quark velocities become large and both quark move close to the “+” and “” lightcone directions. The components of the total momentum transfer are and . Thus, in the limit we can neglect the longitudinal components of transferred momentum and put . Since the transferred momentum is much smaller than the energies of incoming quarks, the quarks interact each other by exchanging soft gluons in the channel with total momentum . Interacting with each of soft gluons quark does not alter its velocity in the limit and thus the only effect of its interaction is the appearance of an additional phase in the quark wave function. This phase, the socalled eikonal phase, is equal to a Wilson line evaluated along the classical trajectory of quark in the direction of the quark velocity. We combine the eikonal phases of both quarks and obtain the representation for the scattering amplitude as [24]:
(2.1) 
where the line function is given by
(2.2) 
Here the line function contains color indices of both incoming and outgoing quarks. The two Wilson lines are defined in the fundamental representation of the SU(N) gauge group and evaluated along infinite paths in the direction of the quark velocities and . The integration paths are separated by impact vector in the transverse direction, .
The scattering amplitude (2.1) depends on the quark velocities and , the transferred momentum and the IR cutoff . These variables give rise to only two scalar dimensionless invariants: and , as explicitly indicated in (2.1). The dependence of the amplitude comes from the dependence on the angle between quark fourvelocities and , defined in Minkowski spacetime as while its dependence is related to the dependence of the line function on the impact vector . In the limit of highenergy quarkquark scattering we have
(2.3) 
The line function is divergent for . This divergence, the so called cross divergence, has an ultraviolet origin, because for the integration paths of Wilson lines (See Fig. 1a) cross each other. According to the general analysis in [30], the Wilson line of Fig. 1a is mixed under renormalization with the Wilson line of Fig. 1b. As a consequence, the renormalized line functions and satisfy the following renormalization group equation:
(2.4) 
where and is the renormalization scale. Here, is the cross anomalous dimension, which is a gaugeinvariant matrix, depending only on the coupling constant and the angle between the lines at the cross point. Solving the RG equation for with the boundary conditions and and identifying the UV cutoff with we find the following expression for the scattering amplitude [24], [28]:
(2.5) 
where and are elements of the matrix
(2.6) 
We conclude that the asymptotic behavior of the scattering amplitude is governed by the matrix of the cross anomalous dimension . The expression (2.6) takes into account not only all and corrections but nonperturbative corrections as well. As follows from (2.6), the dependence originates from the evolution of the coupling constant.
The oneloop expression for the matrix cross anomalous dimension is
(2.7) 
In the large limit we have the following expressions for the eigenvalues of the matrix :
(2.8) 
(2.9) 
The scattering amplitude can be decomposed into singlet and octet invariant amplitudes corresponding to exchanges in the channel with quantum numbers of the vacuum and the gluon, respectively:
(2.10) 
Using the oneloop expression (2.7) for the matrix we find the following expressions for the invariant amplitudes:
(2.11) 
(2.12) 
where
(2.13) 
In the leading and approximation the result for the invariant scattering amplitude has the standard reggeized form [31]:
(2.14) 
where corrections drastically change the behavior of the scattering amplitude. Indeed, the functions (2.13) have a Reggelike behavior, and for we have . Thus, the highenergy behavior of the invariant scattering amplitudes (2.11), (2.12) is dominated by the contribution of , and as a consequence the amplitude of the octet exchange is suppressed by the factor compared to the amplitude of the singlet exchange. Therefore we will start from the main result of perturbative QCD: . The nonleading
(2.15) 
(2.16) 
3. IR renormalons in Wilson loops
Expression (2.13) for the scattering amplitude was found by summing all large Sudakov logarithms which were artificially extracted from the uniquely defined scattering amplitude (2.1). As a result the perturbative expansion for the scattering amplitude, as we will show in this section, is not well defined, and has ambiguities associated with IR renormalons. To restore the uniqueness of the physical quantity, the perturbative expression should be supplemented by nonperturbative corrections. However, we have to pay for this by introducing a new scale, which characterizes the size of nonperturbative effects. We do not know how to evaluate the nonperturbative corrections but we may predict in general their structure and dependence on impact parameter by exploiting the idea of IR renormalons.
The occurrence of infrared renormalons in the amplitude may be seen explicitly in an “improved” calculation of the Wilson line (WL) expectation value by replacing by the running coupling constant [15]. As an example, we consider the oneloop calculation of of Fig. 2 in Feynman gauge, and find the structure of the nonperturbative correction for this particular diagram.
(3.1) 
where , . After integration over and in (3.1) we get the vacuum average of the WL as an integral over the gluon transverse momenta:
(3.2) 
Notice that after integration over , even for fixed coupling constant this expression contains infrared poles in . To regularize IR divergences we have introduced dimensional regularization, with scale . The IR cutoff (see Sect. 2) is neglected for this argument, and we may assume for a moment that . The cross divergence does not appear for nonzero impact parameter , because regularizes the gluon propagator at short distances. Let us now review how IR renormalons appear [15].
After substitution of the relation into (3.2), and after integration over transverse momenta, we get
(3.3) 
We notice that with . We thus identify the righthand side of (3.3) as the Borel representation
(3.4) 
of , with
(3.5) 
The limit produces a singularity in due to IR divergences in (3.1), because we have neglected the IR cutoff . Away from we put and find that the function has singularities generated by function at . These are the infrared renormalons. The result of integration in (3.4) depends on the regularization prescription. That is, the factor induces an ambiguity in the scattering amplitude at the level of the power corrections . This fact, together with the observation that the infrared renormalons come from region of small gluon momenta where the coupling constant becomes large and where we expect the appearance of nonperturbative corrections, implies that the nonperturbative corrections are power corrections. The first renormalon gives a contribution at the level . Thus, for the WL to be well defined, nonperturbative effects should contribute at the same level. Hence the nonperturbative correction has the following form:
(3.6) 
where is some parameter characterizing their size.
Let us consider the color matrix structure of nonperturbative corrections. The direct product of the gauge group generators can be decomposed into the sum of invariant tensors
Therefore the structure , which corresponds to the particular configuration of Fig. 2, reproduces the first two elements of the matrix cross anomalous dimensions (2.7). Notice that the structure containing the dependence on is factorized after integrating over and Wilson line parameters and over . Evidently the angle and group structure of the oneloop diagrams of reproduce the other two elements of the matrix cross anomalous dimension (2.7). Therefore we conclude that the factorized matrix structure of nonperturbative corrections is exactly .
In summary, we may represent as a sum of perturbative and nonperturbative terms as
(3.7) 
where and is IR cutoff. We must emphasize that we choose to be able to consider the influence of nonperturbative effects. In general the nonperturbative parameter depends on the IR cutoff . We recall that in this section and afterward is fundamental QCD scale.
4. Full scattering amplitude
It is now natural to generalize the exponentiated expressions (2.13), (2.15), (2.16) to include the nonperturbative corrections. The Wilson lines (2.2) are defined beyond perturbation theory. In perturbation theory, however, we know the renormalization group equation (2.4) for the Wilson line and its solution. That is why it is natural to consider the exponentiation of nonperturbative corrections as well as perturbative. In fact at oneloop level, the matrix structures of perturbative and nonperturbative corrections are the same. The origin of this property is the following.
As was shown in the previous section, in Feynman diagrams contributing to the Wilson loop nonperturbative corrections come from the integration over small transverse momenta. At the same time, the  dependence appears when one integrates over small angles between gluon momenta and quark momenta . Since integrations over small angles are completely independent of the integration over small transverse momenta, the  dependence of both perturbative and nonperturbative results coincides. In the paper [28] it was shown that higher order corrections preserve the asymptotic behavior of the eigenvalues of the matrix cross anomalous dimension. Together, these arguments imply that nonperturbative corrections have and asymptotics. Therefore the generalization can be performed in the same manner as the oneloop example before (3.7)
(4.1) 
For small values of the scattering amplitude gets its main contribution from the perturbative region. At large the nonperturbative corrections become important.
For invariant amplitudes we find expressions like (2.17) and (2.18):
(4.2) 
(4.3) 
We expect that the condition is conserved. Moreover, this condition is provided explicitly by the structure in the argument of the exponent, because in high energy limit .
Now we come to the main problem of our paper : how to evaluate the integral over impact parameter in the expression for (4.1). We will use the method of representation of the Fourier transformation via a Mellin transformation, which was so elegantly applied to the DrellYan process [32]. We should mention also the papers [33], which contain some interesting calculations based on Mellin transformation technique. Let us briefly formulate this method.
5. Scattering amplitude for a frozen coupling constant
First we consider the situation when the coupling constant does not run. For this case, one will be able to get an exact answer for the scattering amplitude. Let us rewrite expression (4.6) for taking into account that . We get
(5.1) 
It is convenient to denote
(5.2) 
From the beginning let us integrate in space, and after that concentrate our efforts on the integration over . Evidently, is proportional the Gammafunction. Substituting the expression obtained for in the expression (4.5) for the scattering amplitude, in which it is useful to shift the integration variable , one finds
(5.3) 
Note, if we deal with only nonperturbative corrections () the expression (5.3) is simplified. Imagining the contour to envelope the right half plane and summing over all residues of we find
(5.4) 
This answer coincides, of course, with the result of direct integration over impact parameter in the expression (4.1) for , and sheds some light on the occurance of the Gaussian distribution on transferred momentum.
Let us continue and recall the definition of the confluent hypergeometrical function (CHF) by means of Mellin integral representation (see Appendix). Taking into account the remarkable property of the CHF (A.2), and identifying , , , one obtains
(5.5) 
A more detailed form of this expression, using definition (5.2), is
(5.6) 
This is the exact expression for the scattering amplitude in the case when the coupling constant does not run. Let us find the asymptotics of (5.5) using the properties of the CHF (A.3), (A.4). There are two limiting cases to consider: and , and we certainly have a critical value of transferred momentum such that
(5.7) 
which corresponds to . At this value there is a crossover between rapid Gaussian decrease over in the scattering amplitude and slower Regge dependence in .

1. , which corresponds to .
Substituting (A.3) in (5.5) we get
(5.8) 
where . The result is a Gaussian distribution over the transferred momentum.

2. , which corresponds to .
Substituting (A.4) in (5.5) we get
(5.9) 
The result has the standard reggeized form with Regge trajectory
6. The scattering amplitude as an asymptotic series: Running coupling constant.
Let us substitute the expression for the running coupling constant, in (4.6) and integrate over . Then
(6.1) 
To evaluate a double integral like (4.5), Collins and Soper proposed in [32] a saddle point approximation. For the DrellYan process this method is applicable because in the high energy limit the nonperturbative corrections play no role, and the saddle point lies in the perturbative region. In quarkquark scattering, however, the application of this method is doubtful from the beginning because the factor , which appears in the argument of the exponent in (6.1), approaches zero in the high energy limit. Moreover, the position of the saddle point is very sensitive to the nonperturbative parameter . The saddle point may lie in perturbative region or out of it. That is why we suggest a method of evaluating the integral based on the sum of the contributions from all poles in .
It is useful now to introduce new variables,
(6.2) 
Using an representation for
(6.3) 
The integral is not well defined at because the expression
In the expression (4.5) for we shift variable to and substitute the result (6.3) for . We get,
(6.4) 
We notice that this expression is similar to (5.3) except for the integration over . Using the definition of the CHF by means of Mellin integral representation (A.1), taking into account the property (A.2) of the CHF and identifying , , , one obtains
(6.5) 
As in Section 5 we concentrate our attention on the two limiting cases. The critical value of the transferred momentum is determined as before by (5.7).

1. , which corresponds to .
Substituting (A.3) in (6.5) we get
(6.6) 
where
(6.7) 
(6.8) 
One can easily determine corrections by keeping more terms in the expansion of the CHF. Consider the integral (6.7) and represent it as an asymptotic series. In this case, one can turn the contour from to or change variable to . After expanding the Gamma function as a series in powers of and integrating over we get
(6.9) 
We now have the expression (6.9) for in terms of an asymptotic series. The first terms of this series yield accurate value for when
(6.10) 
Note that in high energy limit and therefore we didn’t include in the condition (6.10). differs from only by an additional power of in the integral. It means that the contributions from are suppressed compared to those from . Evidently the same argument can be applied to ,… Therefore we conclude that the scattering amplitude can be approximated under the condition (6.10) by the first terms of the asymptotic series:
(6.11) 
A more detailed form, using definition (6.2) is
(6.12) 
Therefore we have the Gaussian distribution over transferred momentum. Let us investigate in more detail the first exponent in (6.12). Using the expression (2.9) for the eigenvalue one can rewrite it as
(6.13) 
This means that by increasing energy the bulk of the diffraction peak, which is concentrated for
becomes narrower. This phenomena is called shrinkage, which we have thus derived from IR renormalon analysis. The slope of the soft gluon trajectory is
(6.14) 
and we can estimate the value of nonperturbative parameter using the experimental result for the slope: [35]. It turn out that .
Let us now substitute (6.12) into expression (4.2) for invariant vacuum amplitude and evaluate the differential cross section as . Taking into account that at high energy and factor , we find that the differential cross section is given by:
(6.15) 
Notice that the exponential factor in the expression (6.15) has a very interesting feature : the nonperturbative parameter has penetrated into the perturbative expansion. This factor should determine the intercept of the soft pomeron. However, based on our analysis we can only estimate it qualitatively. All that we can say is that the intercept depends on the nonperturbative parameter , depends slightly on the energy through , and at high energy is close to 1. Moreover, the cross section (6.15) cannot grow faster than at the limit . We conclude thus that our soft pomeron satisfies the Froissart bound.