A solvable model for small- physics
[1ex] in dimensions
Dipartimento di Fisica, Università di Firenze, 50019 Sesto Fiorentino (FI), Italy;
INFN Sezione di Firenze, 50019 Sesto Fiorentino (FI), Italy.
I present a simplified model for the gluon Green’s function governing high-energy QCD dynamics, in arbitrary space-time dimensions. The BFKL integral equation (either with or without running coupling) reduces to a second order differential equation that can be solved in terms of Bessel and hypergeometric functions. Explicit expressions for the gluon density and its anomalous dimension are derived in and factorization schemes. This analysis illustrates the qualitative features of the QCD gluon density in both factorization schemes. In addition, it clarifies the mathematical properties and validates the results of the “-representation” method  proposed by M.Ciafaloni and myself for extracting resummed next-to-leading- anomalous dimensions of phenomenological relevance in the two schemes.
Small- resummations in QCD have been extensively investigated in the past years in order to improve the fixed order perturbative description of high-energy hard processes in the small- regime, where higher order perturbative corrections grow rapidly due to logarithmically enhanced contributions . Knowledge of the precise relationship between the fixed order approach — based on the collinear factorization formula and the DGLAP equation — and the small- resummed ones — based on the high-energy factorization formula  and the BFKL equation — is of course needed for a unified picture of small- physics, e.g., to provide quantitatively accurate predictions in the small- region, which will be explored by next-generation colliders.
A major aspect of this relationship is the issue of the factorization scheme employed to define parton densities and coefficient functions. Fixed order perturbative calculations mostly use the (modified) minimal subtraction () scheme in the context of dimensional regularization. On the other hand, small- resummed approaches — being based on -factorization  which involves off-shell intermediate particles with non-vanishing transverse momentum — are naturally defined in the so-called -scheme , where infra-red (IR) singularities are regularized by an off-shell probe whose non-vanishing virtuality plays the role of an IR cutoff.
The basic relations for the scheme change of anomalous dimensions and coefficient functions were obtained some time ago [6, 7] at relative leading- (LL) order, then improved to include next-to-leading- (NL) running coupling corrections  and recently extended by M.C. and myself at full NL level . The main tool of our analysis  was the generalization to dimensions of the -representation of the gluon density — a Mellin representation of the BFKL solution in which is conjugate to . While for the running-coupling BFKL equation is a differential equation in , for it becomes a finite-difference equation, whose solution, however, is not unambiguously determined and has been computed by using sometimes rather formal manipulations. Despite the sensible physical meaning of the procedure and of its results, from a mathematical point of view some steps of our method are not fully proven. It is therefore desirable to have at least an explicit example that could confirm our method of solution of the finite-difference equation, especially in view of its application to compute the anomalous dimensions at full NL accuracy.
The purpose of the present work is to devise a non-trivial, physically motivated and solvable model which: 1) by providing explicit solutions, illustrates the main qualitative features of the real QCD case; 2) can clarify the less understood aspects of the procedure developed in  and verify the correctness of its results. The model I am going to present is a generalization to arbitrary space-time dimensions of the collinear model  used in the past to study the interplay between perturbative and non-perturbative QCD dynamics at high energies. Starting from the formulation of the LL BFKL equation in dimensions, only the collinearly enhanced ( and ) contributions of the integral kernel are kept. Despite its poor phenomenological accuracy, this model contains most of the qualitative features of the real theory: it is symmetric in the gluon exchange , it generates collinear singularities in the limit, it correctly describes the leading-twist LL behaviour of the gluon density, it includes the running of the coupling. Most importantly, in contrast to the BFKL equation, the collinear model can be solved, as a 1-dimensional Schrödinger-like problem.
Sec. 2 is devoted to the definition of the model in generic number of dimensions. A preliminary study on the qualitative features of the solution of the master integral equation is presented. To this purpose, I briefly review the two types of running-coupling behaviour that are present in dimensions.
The resolution of the model in the fixed-coupling case is presented in sec. 3. The ensuing integral equation is then recast into a second order differential equation of Bessel type, whose solution provides the unintegrated gluon density. The unintegrated gluon density is first compared with the known perturbative solution  and then used to compute the integrated gluon density and anomalous dimension in both the -scheme and -scheme. The last part of this section concerns the analysis of the Mellin representation of the gluon density and its comparison with the corresponding series and integral representations derived in ref. .
Sec. 4 includes the one-loop running coupling. In this case the differential equation is solved in terms of hypergeometric functions. The analyticity properties of the solution will reveal essential in extending the unintegrated gluon density from the IR-free regime — where the coupling is bounded — to the ultra-violet (UV)-free regime — where the Landau pole renders the integral equation meaningless. The explicit results of the -dependent resummed and anomalous dimensions — which are shown to agree with the known lowest order running coupling corrections — provide a strong check for the connection between -dependence of the kernel and -dependence of the anomalous dimension argued in ref. .
A final discussion is reported in sec. 5.
I distinguish two symbols of asymptotic behaviour:
means for some finite
and non-zero , while refers to the special case .
The hypergeometric function is denoted by .
A citation like (2.3) means eq. (2.3) of ref. .
There are some change of notations between ref.  (left side) and this paper (right side):
2 Formulation of the collinear model
In this section, I define a simplified model for the gluon density in high-energy QCD with both running and frozen coupling constant. After recalling the features of the two running coupling regimes, I briefly discuss the expected qualitative behaviour of the solutions of the model.
2.1 Motivation of the model
In high-energy QCD, parton densities and anomalous dimensions are often computed in two different factorization schemes, which differ essentially by the regularization of the infra-red (IR) singularities.
In the so-called scheme , the IR regularization occurs by considering off-shell initial partons with non-vanishing virtuality , which plays the role of a momentum cut-off;
The minimal subtraction () scheme instead, is based on dimensional regularization with on-shell initial partons living in space-time dimensions, where IR singularities shows up as poles and are subtracted from the physical quantities according to the prescription.
The relation between these two schemes can be investigated by including in the defining equations for partons both off-shell initial conditions and arbitrary space-time dimensions.
As for the physical case of 4 space-time dimensions, also in generic dimensions the high energy (i.e., small-) behaviour of cross sections in QCD is governed by the gluon Green’s function (GGF) . Here is the Mellin variable conjugated to , while and are the transverse momenta of the (reggeized) gluons emerging from the impact-factors of the external particles . The GGF obeys the integral equation (in the following the dependence on the variable will always be understood)
where the kernel has been determined exactly in the leading- (LL) approximation  and can be conveniently improved to include subleading corrections (in particular the running of the coupling). Detailed studies of the ensuing solutions and physical consequences has been presented in refs.  in the LL approximation, and in refs. [1, 10] at subleading level.
It should be noted that the NL approximation limits not only the knowledge of the kernel , but also the method of solution of eq. (1). However, it would be desirable to have an exact solution of eq. (1), even with an approximate kernel, in order to understand the overall “non-perturbative” feature of the QCD gluon Green’s function. To this purpose, I consider a simplified model for whose main virtue is to provide a GGF which can be expressed in terms of known analytic functions. This toy-kernel resembles the field-theoretical one in the collinear regions and , and has already been considered in the past  in order to study the structure of high-energy QCD dynamics in 4 dimensions. In the following I generalize the collinear model to the dimensional regularized theory, with running coupling as well as with fixed coupling constant.
2.2 Definition of the model
The collinear model is defined by the collinear limit of the LL BFKL high-energy evolution kernel
() being the smallest (biggest) transverse momentum, and the dimensionful gauge coupling. By introducing the dimensionless coupling constant , the small- expansion parameter and the logarithmic variable
we can express both GGF and kernel in terms of the dimensionless quantities (the unintegrated gluon density) and defined by
where in the second line I have substituted the expression of the collinear kernel
A second way to relate this model to QCD is to compare the eigenvalue function
with the BFKL one , as in fig. 1. Clearly the two eigenvalue functions display the same qualitative behaviour in the region around and between the leading-twist poles at .
The collinear model can be easily generalized to include the running of the coupling. The small- parameter acquires a -dependence according to the evolution equation
where is the one-loop beta function coefficient ( in QCD). The solution of eq. (10) is given by
Note that in dimensional regularization () the coupling has a non-trivial -dependence also in the so-called frozen coupling case corresponding to . Substituting in place of in eq. (6), we obtain, after rearranging some terms, the generalization of the collinear model with running coupling:
2.3 Running coupling regimes
It is important at this point to realize that the running coupling behaves in two qualitatively different ways, according to whether the parameter is greater or less than 1.
When , i.e., , the running coupling is bounded, positive and increases monotonically from the IR-stable fixed point to the UV-stable fixed point , as shown in fig. 2.
When , i.e., , the running coupling starts from the positive UV-stable fixed point , then increases and diverges at the Landau point
becomes negative for and finally vanishes at . This is the situation realizing the physical limit at fixed .
In the former case, the extra-dimension parameter not only regularizes the IR singularities, but avoids also the occurrence of the Landau pole, thus allowing a formulation of the integral equation free of singularities. In practice, the strategy of dimensional regularization consists in computing the physical quantities in the “regular” regime ; the universal -singular factors are then removed into non-perturbative quantities, and finally by analytic continuation the physical case at is recovered.
2.4 Qualitative behaviour of the solutions
Before embarking upon the resolution of the collinear model equations (7,12), it is instructive to estimate the qualitative behaviour of the solutions by using well-known methods  in the context of high-energy QCD. Particularly important is the factorization property which allows one to split the unintegrated gluon density into a perturbative and a non-perturbative part, provided the “hard scale” is sufficiently large:
up to terms exponentially suppressed in (higher-twists). In turn, the perturbative factor
is given in terms of the gluon anomalous dimension determined by the small- equation
where is the eigenvalue function of the integral kernel in eq. (1).
the perturbative branch being the one with minus sign: . At large , the running coupling saturates at the UV fixed point , so that the large- behaviour of the unintegrated gluon density is given by
According to the value of we expect two kinds of asymptotic behaviour:
For the square root is real and positive, the UV regular solution corresponds to the perturbative branch of the anomalous dimension and we must reject the (UV irregular) solution which diverges more rapidly: .
For the two exponents are complex conjugate, and the gluon density becomes oscillatory at large . It is not possible to distinguish an UV regular solution, and one has to determine the coefficients by analytic continuation in from . The fixed coupling () solution belongs to this class.
The above results will be also obtained in a more rigorous way in sec. 4.1, when treating the running-coupling equation.
3 Collinear model with frozen coupling ()
Since the properties of the solution of the collinear model and its connection with the solution method of ref.  are more easily illustrated in the fixed coupling case, I start considering the integral equation (7) with .
3.1 Solution in momentum space
The presence of the exponential factor in front of the r.h.s. of eqs. (6,7) spoils scale invariance, therefore the determination of both eigenfunctions and eigenvalues of the integral operator by means of standard techniques is not possible. It turns out, however, that one can exactly solve eq. (7). In fact, by differentiating it twice with respect to , we obtain the second order differential equation (the -dependence of is understood in this section)
which can be recast in a more familiar form if we introduce the variables
In the l.h.s. of eq. (22) one recognizes the differential operator defining the Bessel functions and as solutions of the corresponding homogeneous equation.
The general solution of eq. (22) has the form
where and denote respectively the IR-regular and the UV-regular solutions of the homogeneous equation, while and are -dependent coefficients to be determined by the two conditions of continuity of and discontinuity of at :
By solving the above linear system one obtains
where is the Wronskian of the two solutions of the homogeneous equation.
It remains to determine and , each being a linear combinations of, say, and :
(the absolute normalization is irrelevant). From the asymptotic relations
it is clear that the IR-regular solution is , since it vanishes more rapidly than any linear combination containing when with . On the other hand, the UV-regular solution cannot be determined in this case of , because of the identical asymptotic behaviour (up to normalization and phase) for of all solutions in eq. (26). However, the UV-regular solution can be unambiguously determined in the formulation with running coupling (cf. sec. 4.3), and in the limit it reduces to . In conclusion
It is possible to show that in the previous equation obeys also the integral equation (7).
3.2 On-shell limit and perturbative expansion
It is important at this point to check the explicit solution in eq. (31) with known results of the literature. The perturbative expression for the GGF in dimensional regularization was given in (3.3) for an on-shell () initial gluon. In terms of the dimensionless density their result reads
for a generic integral kernel with eigenvalue function .
In words, the on-shell unintegrated gluon density is equal to the UV regular solution of the homogeneous differential equation with a proper normalization.
In order to compare the solution (33) with the perturbative expression (32), one has to expand the r.h.s. of eq. (33) in series of . By rewriting as a combination of according to eq. (29), and then using the ascending series (9.1.10)
The second term of eq. (35) provides contributions of order , each being outside the domain of the kernel and therefore out of the reach of the iterative procedure. Furthermore, this term is strongly suppressed when with respect to the perturbative one. Therefore, it is possible to correctly compute the perturbative coefficients to any order provided is sufficiently small (). In the limit the perturbative solution agrees with the exact one to all orders.222These conclusions are valid in the off-shell case () too, but for sake of simplicity they have been presented only in the on-shell case.
3.3 Integrated gluon densities
The major issue this paper is devoted to, concerns the scheme-change, namely the relation between gluon densities and anomalous dimensions in the two factorization schemes. In the collinear model, the off-shell integrated gluon density defined by
can be computed in closed form (app. A.1), and for reads
Note the remarkable fact that , like , is factorized in the - and -dependence.
The -scheme gluon is given by the limit of the above expression, yielding (app. A.2)
whence one immediately derives the -scheme anomalous dimension (a dot means -derivative)
It is interesting to note that the on-shell limit of the integrated gluon density, i.e., the gluon density in dimensional regularization
provides the same effective anomalous dimension (app. A.2)
which means that the two limiting operations and commute. Actually, in this model this is a trivial consequence of the factorized structure of the gluon density (38) in its and dependence, which causes the ratios in eqs. (40) and (42) to be -independent.
The relation with the MS-scheme anomalous dimension is obtained as follows. From the asymptotic behaviour of the on-shell gluon density (app. A.2)
one identifies the exponential in eq. (43) as the gluon density ,333Due to the particular definition of in eq. (3) which includes -dependent factors, eq. (43) defines a “modified” minimal subtraction scheme, related to the customary MS and schemes by a finite scheme change. These details are unimportant for the purpose of this paper. since it sums all and only -singular terms up to the scale . The anomalous dimension is then computed from the logarithmic derivative
The coefficient function , on the other hand, is finite in the limit, and is given by the product  , where
originates from the -dependence of the eigenvalue function. Since in this model is independent of , , and therefore , in agreement with eq. (43).
3.4 Solution in space
Having the solution of the integral equation at our disposal, we are ready to check the validity of the procedure suggested in ref. , at least in this simplified model. I start reviewing the main steps of that procedure.
1) We introduce for the unintegrated gluon density an integral representation of Mellin-type:
2) In -space, the integral equation (7) is thus recast into the finite difference equation
3) The finite difference equation (48) is solved in terms of a Laurent series in , so as to provide the following expression (cf. , sec. 2 and eqs. (C.1,C.2)) for the on-shell unintegrated gluon density:
where is a normalization factor, , , and the coefficients denote Bernoulli numbers.
4) The solution is determined by assuming the existence of a saddle point
on the real axis, whose steepest descent direction lies on the real axis.
Let us now analyze each point in turn, in the context of the collinear model.
1) Concerning the existence of a Mellin representation for the solution of the integral equation (7), the asymptotics in eq. (27) guarantee that the Mellin transform is defined in the strip for all . Explicitly, is given in terms of sums, as follows:
I will show now that, with a proper choice of the contour , only the first term of in eq. (50c) contributes to the inverse Mellin transform (47) for — the relevant region for the on-shell limit. Notice that the analytic continuation of defines a meromorphic function whose singularities are just the simple poles of at , as shown in fig. 3. Actually, is holomorphic in the whole plane , since the poles at stemming from the ratio in the first line of eq. (50c) are exactly canceled by those in the sum on the second line; furthermore, the poles at stemming from in the first line are also canceled by those in the sum on the second line.
It is convenient to compute the inverse Mellin transform separately for the and pieces. In the integral of one can close the contour path to the left (), without crossing any singularity, thus obtaining a vanishing contribution, as expected. Considering now the integral of , one is not allowed to close the contour either to the left or to the right, because the factor in front of the sum grows for , while the ratio of gamma-functions in the first term grows with for . However, by folding the contour so as to let it cross the real axis at some value (remember that has no singularity), and then computing the two contributions of eq. (50c) separately, one obtains a vanishing integral from the second line, because the contour can be closed to the left without crossing any singularities.
By performing the on-shell limit I end up with
2) It is straightforward to check that the on-shell Mellin transform in eq. (51) obeys the homogeneous difference equation
3) The third issue concerns the validity of eq. (49). By explicitly computing the integral and the derivatives of in the collinear model
the exponent within curly brackets in eq. (49) becomes ()
Apart from the irrelevant normalization factor , eq. (56) agrees with the integrand in eq. (51), when one takes into account that the asymptotic expansion in powers of of is a numeric constant ( according to the sign of ).
4) The last step is to evaluate the integral in eq. (51) in the large- limit. It turns out that, for small values of and values of , the fastest convergence contour path surrounds the interval (cf. fig. 4) at a distance decreasing with . The main contribution to the integral comes just from this region (parts B and D in fig. 4).
In the limit of vanishing , the string of poles at accumulates into a branch-cut at . In fact, while the ratio of gamma-functions is regular at also in the limit, the cotangent becomes discontinuous across the real axis with a jump equal to .
Therefore, neglecting the contributions to the integral in eq. (51) from the parts A, C and E of the contour path, the contributions of B and D amount to the integral in of the discontinuity of the integrand, which can be easily obtained by replacing with <