Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes
Abstract:
We study the factorization of soft and collinear singularities in dimensionally regularized fixedangle scattering amplitudes in massless gauge theories. Our factorization is based on replacing the hard massless partons by lightlike Wilson lines, and defining gaugeinvariant jet and soft functions in dimensional regularization. In this scheme the factorized amplitude admits a powerful symmetry: it is invariant under rescaling of individual Wilsonline velocities. This symmetry is broken by cusp singularities in both the soft and the eikonal jet functions. We show that the cancellation of these cusp anomalies in any multileg amplitude imposes allorder constraints on the kinematic dependence of the corresponding soft anomalous dimension, relating it to the cusp anomalous dimension. For amplitudes with two or three hard partons the solution is unique: the constraints fully determine the kinematic dependence of the soft function. For amplitudes with four or more hard partons we present a minimal solution where the soft anomalous dimension is a sum over colour dipoles, multiplied by the cusp anomalous dimension. In this case additional contributions to the soft anomalous dimension at three loops or beyond are not excluded, but they are constrained to be functions of conformal cross ratios of kinematic variables.
DFTT–29/2008
1 Introduction
Studies of infrared and collinear singularities of fixedangle scattering amplitudes in massless gauge theories have a long history (for early results, see for example [1] and [2]), and they have led to remarkable insights into the allorder structure of the perturbative expansion.
These studies are not motivated by a purely theoretical interest: in fact, a detailed understanding of the longdistance singularity structure of QCD amplitudes is a crucial element in predicting highenergy collider cross sections. Indeed, the calculation of observable cross sections involves intricate cancellations of soft and collinear singularities between real and virtual corrections (see for example [3]). Furthermore, with a precise knowledge of singularities one can predict dominant higher–order corrections, and in many occasions resum certain classes of logarithmically enhanced contributions to all orders [4, 5].
Our understanding of long–distance singularities is based on the ideas of factorization and universality [6]. Fixedangle scattering amplitudes are functions of Lorentzinvariant combinations of external momenta, which are assumed to be uniformly much larger than the relevant infrared cutoff, typically given by the scale of confinement; one expects then that exchanges of virtual particles with vanishingly small energies, or with vanishing transverse momenta with respect to given external legs, should decouple from hard exchanges, which happen at much shorter distances. Such a decoupling is far from apparent in Feynman diagram calculations, but it can indeed be proven to all orders in perturbation theory, once gauge invariance is enforced by means of appropriate Ward identities [4].
The result has a simple structure, explained in detail in Section 2. Briefly, multileg amplitudes can be organized as vectors, in a vector space spanned by the irreducible representations of the gauge group that can be constructed with the given external particles; such a vector can be shown to have a factorized structure: each external leg is dressed by virtual collinear emissions, building up a coloursinglet ‘jet’ function; soft gluons exchanged at wide angles are assigned to a separate factor, which is a matrix, mixing the available colour representations; this matrix of ‘soft’ functions is then contracted with a vector of hard scattering coefficients, which contain no infrared or collinear singularities.
Given this factorized structure, one may immediately deduce that the various factors in the amplitude obey simple evolution equations, embodying the consequences of renormalization group as well as gauge invariance [7] (for reviews of this viewpoint, see [8, 9]). Evolution equations of this type were derived for the first time for the form factors of elementary fields, with a variety of methods [10, 11, 12], and were later extended to Wilson loops [13, 14, 15, 16, 17, 18, 19, 20] and to cross sections and amplitudes of phenomenological interest [21, 22, 23, 24]. Solving these equations leads to the exponentiation of all infrared and collinear singularities. The singularities in the exponent are generated by integrals over the scale of the running coupling of specific anomalous dimensions, which can be computed orderbyorder in perturbation theory. A significant step was taken in [25], where the evolution equation for the Sudakov form factor was solved in dimensional regularization. Within this framework, infrared and collinear poles are generated by integration over the scale of the dimensional version of the running coupling; the results of exponentiation can then be directly compared with finiteorder Feynman diagram calculations; the Landau pole is also regulated by dimensional continuation, so that resummed amplitudes can be computed as analytic functions of the coupling at a fixed scale and of the dimension of spacetime [26]. This approach was extended to multileg amplitudes in [27], confirming earlier predictions [28] on the structure of singularities at NNLO.
In recent years, the development of novel and advanced techniques for highorder calculations in QCD and in general gauge theories has stimulated further investigation of the exponentiation of infrared and collinear singularities. In particular, great theoretical effort has been made to further our understanding of amplitudes in supersymmetric gauge theories, and most notably in the maximally supersymmetric YangMills theory. This theory is of special interest for several reasons: it is quantum conformal invariant; in the planar limit, it is expected to have a simple, theoretically accessible strongcoupling limit, because of its connection with string theory through the AdSCFT correspondence [29]; finally, its amplitudes and anomalous dimensions are of practical relevance, since they have nontrivial relations with the corresponding quantities in QCD (see for example Refs. [30, 31], and recent studies of the Regge limit [32, 33]). Explicit calculations for the fourpoint function in superYangMills (SYM) theory have led to an allorder conjecture [34], suggesting that nonsingular terms exponentiate together with infrared and collinear poles, at least for the class of maximally helicity violating (MHV) amplitudes. While this conjecture has now been shown to fail starting with the twoloop sixpoint function [35], it is clear that SYM perturbative amplitudes must have a remarkably simple allorder structure, which may well be brought under full theoretical control in the near future. A step in this direction was taken with the discovery of a surprising duality between scattering amplitudes in momentum space and expectation values of Wilson loops taken in an auxiliary coordinate space [36]. Further, remarkable progress was made at strong coupling in Ref. [37], where a calculation of the fourpoint amplitude was performed, by adapting string techniques to dimensional regularization. This allowed a direct comparison with resummed perturbative calculations, finding an exact matching in the structure of longdistance singularities. Recent results in this fastdeveloping field are reviewed in Ref. [38].
Most of the calculations just described have been carried out in the planar limit^{1}^{1}1An exception is Ref. [55], which studies the leading infrared singularities in subleading colour components of the SYM gluongluon scattering amplitude to threeloop order., which has special simplifying properties. In this limit, soft contributions to multileg amplitudes can be further factorized into a product of ‘wedges’, each one proportional to a form factor, since soft exchanges can only take place between adjacent external legs. In essence, in the planar limit amplitudes can have only a single colour structure, so that the soft anomalous dimension matrix must be proportional to the unit matrix. All soft and collinear singularities are then determined by just two colourdiagonal functions: the cusp anomalous dimension [16, 17, 18, 19, 20], and a subleading function [40], responsible for single soft or collinear poles.
It is of great interest to push our understanding of infrared singularities in terms of a limited set of anomalous dimensions beyond the planar limit. Indeed, from a theoretical point of view, only at nonplanar level one begins to see the intricate pattern of colour correlations that are characteristic of nonabelian gauge theories: only at this level spacetime and colour degrees of freedom become explicitly correlated. Furthermore, coloursubleading contributions in QCD have important phenomenological effects on resummed hadronic cross sections, beginning at the nexttoleading logarithmic order, and the understanding of subleading poles would also play an important role in the development of infrared and collinear subtraction schemes at higher orders in perturbation theory. Finally, recent work [40, 43, 44, 45, 46, 47, 48] has highlighted new properties of the functions that generate infrared and collinear enhancements in gauge theory amplitudes and cross sections in the case of two hard partons, leading to a better understanding of the process–dependence of soft radiation, to the discovery of allorder connections between different physical processes, and to the possibility of performing internal resummations of running–coupling corrections within the Sudakov exponent. It would be very interesting to extend these studies to general colour configurations.
Soft anomalous dimension matrices for multiparticle scattering have also been intensively studied in recent years. A complete oneloop calculation for the simplest nontrivial case of scattering was carried out originally in [49]. More recently, the calculation was reproduced in a physically motivated, dipolebased formalism in [50]: an interesting observation there was that the anomalous dimension matrix for gluongluon scattering displays an unexplained symmetry relating kinematic invariants with the number of colours . A different symmetry property was observed by [51], where it was noted that all oneloop anomalous dimension matrices are complex symmetric matrices in a suitably chosen orthonormal basis. This property was later explicitly verified with the calculation of the matrices for all processes at one loop [52, 53], and very recently proven [54].
Finally, a remarkable result was derived in [56], where it was shown that soft anomalous dimension matrices at two loops, with any number of external legs, are proportional to their oneloop value, with the proportionality constant given by the twoloop coefficient of the cusp anomalous dimension. This is of course a great reduction in the number of possible degrees of freedom, since a priori each matrix element could have acquired an independent twoloop correction. The fact that the correlation between colour and kinematic dependence in the soft function does not get more complex at two loops as compared to one loop, calls for a deeper explanation. At present it is not known whether this remarkable property remains valid at higher orders.
In this paper, we begin to tackle this question. In Section 2 we develop in detail the factorization of soft and collinear singularities for fixed–angle scattering amplitudes, following the approach of Ref. [40]. There are two main differences between our factorization and earlier calculations of soft matrices. First, we employ dimensional regularization as the unique infrared and collinear regulator: thus, for example, in contrast with Ref. [56] we do not tilt the Wilson lines off the light cone to regulate collinear poles. While this approach makes explicit loop calculations slightly more delicate, it has the advantage that Wilson line correlators are given by pure counterterms to all orders in perturbation theory, and they do not depend on any mass scales. Second, instead of using the jet definition as the squareroot of the Sudakov form factor as in Refs. [27, 56], we define each jet by introducing a separate auxiliary vector , as suggested in early work on Sudakov factorization [57]. This will allow us to conveniently trace the effect of rescaling of the Wilson–line velocities.
Section 3 studies the kinematic dependence of the eikonal functions that enter the factorization of multiparton amplitudes. First, in Section 3.1, we consider eikonal jets, and we determine their kinematic dependence to all orders in perturbation theory in terms of the cusp anomalous dimension. This simple result follows from the fact that the eikonal jet is defined as a correlator of semiinfinite Wilson lines (see (5) below) one of which goes along the lightlike direction defined by the momentum of an external hard parton. Any such correlator of semiinfinite Wilson lines is classically invariant under rescaling of any of the corresponding velocity vectors (independently of whether they are lightlike or not): this invariance is a property of the eikonal Feynman rules. In the presence of cusps with lightlike rays, however, the renormalization procedure breaks this invariance: the counter terms include double poles, corresponding to overlapping ultraviolet and collinear singularities, along with single poles that carry explicit dependence on the normalization of the lightlike velocity vectors. Thus, the renormalized correlators, which do retain their invariance under rescaling of any nonlightlike Wilsonline velocity vector, acquire a dependence on the normalization of the lightlike ones. The origin of this anomaly is well understood [13, 14, 15, 16]: the violation of classical rescaling invariance is governed by the cusp anomalous dimension, and we will refer to it as the ‘cusp anomaly’.
In Section 3.2 we extend the analysis to soft gluon functions. To deal with the general multileg case, we examine combinations of soft and jet correlators where the cusp anomaly cancels, so that rescaling invariance must be recovered. We find that this strongly constrains the dependence of the soft anomalous dimension on the kinematics of the scattering process, and eventually also on the colour degrees of freedom.
Section 4 deals with the simple case of amplitudes with only two hard coloured partons. In Section 4.1 we develop the consequences of the new constraints for the case of the Sudakov form factor, and show that the complete dependence of the corresponding eikonal function on the kinematics is indeed governed by the cusp anomalous dimension. In Section 4.2 we analyse the kinematic dependence of the partonic jet function and contrast it with the eikonal case.
In Section 5 we return to the case of generic multiparton fixedangle scattering amplitudes and study the impact of the new constraints on the soft function. We show that while these constraints are insufficient to fully determine the functional dependence on the kinematic variables, they admit a remarkably simple solution, where the soft anomalous dimension matrix at any order in perturbation theory is proportional to the oneloop result. The solution corresponds to a sum over all colour dipoles, which correlate the kinematic dependence to the colour degrees of freedom, multiplied by the cusp anomalous dimension. This formula is consistent with the result of Ref. [56] at two loops and generalizes it to all orders. We also discuss possible sources of further corrections. We conclude in Section 6 by summarizing our results, while two appendices discuss concrete examples. In Appendix A we describe the special case of amplitudes with three hard partons, where the sumoverdipoles formula is the unique solution to the new constraints. Appendix B studies scattering of quarks at one loop, describing the way in which conformal cross ratios are formed through a sum over diagrams.
2 Factorization of fixed–angle scattering amplitudes
We begin by describing the factorization of a general fixed–angle massless gauge theory amplitude into soft, hard and jet functions. We follow the notations of Ref. [40] and generalize the definition of the soft function given there to the case of multileg amplitudes. The amplitude describes the scattering of hard massless gauge particles (plus any number of colour–singlet particles) so it is characterized by colour indices , , belonging to arbitrary representations of the gauge group. The representation content of the amplitude is collectively denoted by . Such a coloured object can be decomposed into components by picking a basis of independent colour tensors with the same index structure. We denote these tensors by , where and is the number of irreducible representations of the gauge group that can be constructed with the given particles. We write then
(1) 
with being the renormalization scale and , where is the dimension of spacetime. General factorization arguments guarantee that the colour components of the amplitude may be written in a factorized form. Following [2, 49, 27, 56, 40], we write
(2)  
where the hard function , like the amplitude itself, is a vector in the colour space described above; the soft function is a matrix in this space, while the jet functions and do not carry any colour index. A sum over is assumed on the r.h.s. The soft matrix and the jet functions and contain all infrared and collinear singularities of the amplitude, while the hard functions are independent of . Each of the functions appearing in Eq. (2) is separately gauge invariant and admits an operator definition given below. These definitions also clarify the choice of the arguments of each function. In particular, we have exhibited here the fact that the eikonal functions and depend on the dimensionless fourvelocities associated with external particles, rather than the particle momenta . The velocities are defined by scaling the momenta according to , where the magnitude of is unimportant so long as this substitution^{2}^{2}2As soon as the variables are used in the hard functions or in the partonic jet functions , which do depend on physical scales, needs to be specified. We shall avoid that, except in Section 4.2 where we relate the normalization of the partonic jet to the Sudakov form factor. is restricted to the eikonal functions. The fixedangle assumption implies that all scalar products () are of order .
The definitions of the soft and jet functions all involve Wilson lines, which we write as
(3) 
In terms of these Wilson–line operators, one may then define the ‘partonic jet’ functions (for, say, an outgoing quark with momentum ) as
(4) 
The function represents a transition amplitude connecting the vacuum and a oneparticle state. The eikonal line simulates interactions with fast partons moving in different directions: the direction is arbitrary, but off the lightcone (in order to avoid spurious collinear singularities). Since eikonal Feynman rules are invariant under rescalings of the eikonal vector , and this invariance is not broken by the cusp anomaly for , can depend on the vectors and only through the argument given in Eq. (4)^{3}^{3}3For later convenience factors of have been introduced into the arguments of the jet functions.. To avoid any ambiguity with respect to unitarity phases associated with the first argument of we shall choose such that . Note that this can be done so long as one retains the vectors corresponding to different partons independent of each other.
The factorization formula (2) also requires to introduction of the eikonal approximation to the partonic jet , which we call the ‘eikonal jet’. It is defined by
(5) 
Both the partonic jet (4) and the eikonal jet (5) have infrared divergences, as well as collinear divergences associated to their lightlike leg; thus, they display double poles orderbyorder in perturbation theory. The doublepole singularities are however the same, since in the infrared region correctly approximates : singular contributions to the two functions differ only by hard collinear radiation.
It is important to note that the eikonal jet depends on the renormalization scale only through the coupling: indeed, diagram by diagram is given by integrals with no dimensionful parameter. Such integrals vanish identically in dimensional regularization, but since this trivial result involves cancellations between ultraviolet and infrared singularities, upon renormalization becomes nontrivial: the contribution of each graph equals minus the corresponding ultraviolet counterterm. As a consequence, using a minimal subtraction scheme, the result for at each order in is a sum of poles in , without any nonnegative powers. These properties are not special to the jet function, but apply to any eikonal function not involving dimensionful parameters, provided it is defined in dimensional regularization and in a minimal subtraction scheme.
The final ingredient in Eq. (2) is the soft matrix. It is constructed by taking the eikonal approximation for all soft exchanges. Since soft gluons are insensitive to the structure of hard collinear emissions, they couple effectively to Wilson lines in the colour representation of the corresponding hard external parton. Such exchanges mix the colour components of the amplitude, so one is led to define
(6) 
where for simplicity of notation we have defined all eikonal lines as outgoing. Note that in our definition we keep all Wilson lines on the lightcone. As a consequence, also the soft matrix is a pure counterterm in dimensional regularization and it depends on the renormalization scale only through the coupling; furthermore, the homogeneity of the eikonal Feynman rules with respect to rescalings of the eikonal vectors would suggest that can depend on only through homogeneous ratios invariant under such rescalings. As described in [40], this is not true: indeed, rescaling invariance is broken by the cusp anomaly, so that the soft matrix acquires nontrivial dependence on the scalar products . This observation will be central to our arguments in the rest of the paper.
The soft matrix, Eq. (6), displays both infrared and collinear poles. One must then correct the factorization formula in order to avoid double counting of the infraredcollinear region for each external leg. This is achieved in Eq. (2) by dividing by an eikonal jet for each external leg, thus removing from its eikonal part, which is already accounted for in . One may then observe that the ratios are free of infrared poles, and thus contain only single collinear poles at each order in perturbation theory. Similarly, the ‘reduced’ soft matrix
(7) 
is free of collinear poles, and contains only infrared singularities originating from soft gluon radiation at large angles with respect to all external legs. This means that the effects of the cusp anomaly, which is the source of double infraredcollinear poles, must cancel in . More generally, invariance under rescaling of each individual lightlike eikonal velocity,
(8) 
which is broken separately in and in , must be recovered in their ratio, Eq. (7). Indeed, in the factorized amplitude (2) the dependence on the normalizations of the vectors appears only though the eikonal functions contained in Eq. (7), so the invariance of the amplitude as a whole with respect to such rescaling amounts to invariance of . The immediate consequence is that can only depend on arguments that are simultaneously homogeneus in and in . Given the different functional dependencies of and , this can be achieved only if depends on kinematics only through the variables^{4}^{4}4Following [28] we keep track of the unitarity phase by writing where if and are both initialstate partons or are both final–state partons, and otherwise.
(9) 
In Section 3.2 we will explore further consequences of this constraint on the functional dependence of the reduced soft matrix.
Finally, it is important to control the ultraviolet behavior of the jet and soft functions thus introduced. All these functions are multiplicatively renormalizable [14, 15]; there is however an important difference between eikonal operators involving lightlike Wilson lines and partonic amplitudes. First, as already mentioned, the ultraviolet divergence of eikonal operators is directly related to their infrared singularities. Moreover, anomalous dimensions of operators involving cusped Wilson lines with lightlike segments are themselves divergent, due to the overlapping of collinear and ultraviolet poles. These divergences are controlled by the the cusp anomalous dimension [16, 17, 18, 19, 20]. Let us then write down renormalization group equations for the various functions defined above.
The partonic jet does not involve any lightlike Wilson line, and therefore does not have a cusp anomaly. Its anomalous dimension is finite, and one may write
(10) 
In contrast, for the eikonal jet we write
(11) 
In both cases the index is kept as a reminder that the jet function for a given parton carries information not only on the kinematics, but also on the parton spin, flavor and colour, while the eikonal jet depends on the colour representation only.
For the soft matrix multiplicative renormalizability must be understood in the sense of matrix multiplication [15], and one writes
(12) 
where will be referred to as the ‘soft anomalous dimension’; it is similar to the ‘cross anomalous dimension’ of Ref. [15, 22], taken in the limit where all the lines are lightlike. In the following sections we shall examine the dependence of the anomalous dimensions defined above on the kinematic variables as well as on the colour degrees of freedom.
3 On the kinematic dependence of eikonal functions
We now discuss the properties of the eikonal functions and taking into account their gauge invariance, their renormalization group evolution and their independence of any dimensionful kinematic scale. By considering the effect of velocity rescaling we deduce that the kinematic dependence of these functions is tightly connected with cusp singularities. We first illustrate this for the eikonal jet function , and then we move on to consider the central object of our work, the soft anomalous dimension matrix introduced in Eq. (6).
3.1 Explicit solution for the eikonal jet
Let us consider first the anomalous dimension of the eikonal jet in Eq. (11). One observes that the homogeneity of eikonal Feynman rules under the rescaling in Eq. (8) would forbid any dependence on , were it not for the cusp singularity. One expects then that the full dependence of should be proportional to the cusp anomalous dimension, and this is indeed the case as we now explicitly show.
Our starting point is Eq. (11); in dimensional regularization, the statement that is a pure counterterm implies that it can depend on only through the running coupling; one may then solve Eq. (11) as
(13) 
Next, we observe that the eikonal jet must obey an evolution equation of the same form as the Sudakov form factor itself, a socalled ‘K+G’ equation; a similar observation was made in Ref. [40] concerning the eikonal approximation to the form factor. Following the standard reasoning, one rewrites the anomalous dimension as a sum of a singular term, generated by the cusp singularity, and a residual finite function that contains the kinematic dependence. We write then
(14) 
Here we have introduced the cusp anomalous dimension , for an eikonal line in the representation of parton , and a remainder function . The normalization of the singular term on the r.h.s. of Eq. (14) is one half of the corresponding term in the Sudakov form factor, since the form factor is comprised of two jets^{5}^{5}5Indeed an alternative definition of the partonic jet function, which was used for example in Refs. [27, 56], is based on taking the square root of the Sudakov form factor..
Upon inserting Eq. (14) into Eq. (13), and changing the order of integration, one readily arrives at
(15) 
which is analogous to the expression for the Sudakov form factor, as given in Eq. (57) below (or in Eq. (2.11) in Ref. [40]), with the physical scale of the form factor, , replaced here by the renormalization point . The finite function has no explicit dependence in a minimal subtraction scheme, since is a pure counterterm.
We are now going to show that Eq. (15) can be further simplified, since the dependence on the kinematic variable in the function can be completely solved for. In order to do that, we use the results of Ref. [40] for the dependence of the eikonal jet, which is given by
(16) 
Clearly, Eq. (15) and Eq. (16) are compatible only if is a linear function of . Indeed, by taking the derivative of Eq. (15) with respect to , and using Eq. (16), one gets
(17) 
for any . Therefore
(18) 
which we integrate to get
(19) 
where is a constant of integration, free of any kinematic dependence. Using Eq. (19) in Eq. (14) we finally get
(20)  
We can now write down our final expression for the eikonal jet, using Eq. (15). We obtain
(21) 
where, as anticipated, the kinematic dependence of the eikonal jet is explicitly written, to all orders in perturbation theory, in terms of the cusp anomalous dimension. We observe that the cusp anomalous dimension simultaneously controls the double poles and the kinematic dependence of the single poles. In the following sections we will see that this property holds also in the more complex soft functions.
Returning to the comparison with the Sudakov form factor case (see Eq. (57)), we now see that the physical scale is replaced here by . It is important to note that Eq. (21) can also be expressed as
(22) 
exhibiting the fact that dependence appears only through the dimensional running coupling.
Finally we emphasize that the above result for the eikonal jet holds for quarks as well as for gluons. In fact, the dependence of Eq. (21) on the colour representation of the parton appear only though the two functions and . Moreover, the nonAbelian exponentiation theorem [41] implies that the colour structure of these functions is ‘maximally nonAbelian’. Up to three loops, this implies, in particular (see e.g. Refs. [18, 20]) that the cusp anomalous dimension depends on the representation only through an overall multiplicative factor, the total colour charge, given by the quadratic Casimir in the representation of parton ,
(23) 
where is the unit matrix and is a generator in the corresponding representation^{6}^{6}6 should be interpreted as follows: for a final–state quark or an initial–state antiquark: ; for a final–state antiquark or an initial–state quark: ; for a gluon: . For the index runs from 1 to , and specifically for quarks and for gluons. In our normalization .. Casimir scaling, namely the universality of between quarks and gluons, has been explicitly verified in recent years by threeloop calculation of the QCD splitting functions in Ref. [58]. Starting at four loops, however, the colour structure in the exponent may not be expressible in terms of quadratic Casimirs. In general, higher Casimir contributions do appear in QCD calculations at this order, for example in the QCD beta function [59, 60], where one finds coloursinglet contributions constructed of traces of products of four generators. The potential appearance of such higher Casimir terms in the exponent, despite the nonAbelian exponentiation theorem [41], was first observed in Ref. [42], where it was proposed to describe the colour structure of the exponent by ‘colour connected webs’, giving a more precise meaning to the notion ‘maximally nonAbelian’. Recently it has been argued [39], based on different theoretical considerations^{7}^{7}7We thank Juan Maldacena for pointing this out to us., that such terms may indeed appear in the cusp anomalous dimension starting at four loops.
Let us therefore write in full generality,
(24) 
where is given by (23), , and . Note that is a universal function of the coupling, strictly independent of the representation of the parton . This function is known [58] up to three loops in QCD. In contrast, the residual term represents (yet unknown) potential contributions which violate Casimir scaling; it depends on the representation in a more complicated way, for example through terms that involve irreducible combinations of four colour generators. The particular way in which the cusp anomalous dimension depends on the representation will not be important for most of what follows, but it will be used in Section 5 for constructing an explicit expression for the soft anomalous dimension.
3.2 Factorization constraints for soft anomalous dimension matrices
Having established the simple result in Eq. (21) for the eikonal jet, where the kinematic dependence is determined by the cusp anomalous dimension, one may wonder if the same is true for the soft function. In other words, one may ask whether the full dependence of in Eq. (12) on is associated with the cusps.
One may try to apply the rescaling argument, arguing that if not for the cusp singularities should have been invariant with respect to independent rescalings of each . One realises however that if the number of hard external lines is , it is possible to construct homogeneous conformal cross–ratios such as
(26) 
which are inherently invariant with respect to rescalings of each individual velocity, thus evading this argument. Kinematic dependence does not necessarily lead to violation of the rescaling–invariance property, and therefore might not be associated with the cusp singularities. It is important to note, though, that for there are no such ratios and the argument does hold. One should expect, therefore, that at least for the full kinematic dependence should be controlled by to all orders. We shall see below that this is indeed the case.
The observation that allows us to make a step forward is that the soft function, for any number of legs, can be indirectly constrained by considering the kinematic dependence of the reduced soft function, defined in Eq. (7). Here the cusp singularity itself cancels out, and yet, as we will see, it leaves its trace through the dependence on the kinematics. To proceed, consider the renormalization group equation for the reduced soft matrix , which reads
(27) 
where , in contrast to and , is free of singularities. Its invariance with respect to scaling of each individual velocity is manifest in its functional dependence on the velocities only through the ratios , defined in Eq. (9).
Given the definition of the reduced soft matrix in Eq. (7), one easily sees that the various eikonal anomalous dimensions are related by
(28) 
where, as above, . In words, pole terms must cancel on the righthand side of Eq. (28), and the functional dependence on eikonal vectors must be arranged so as to reconstruct functions of in order to be consistent with the lefthand side. Substituting Eq. (20) into Eq. (28) we get an explicit expression for . In terms of the dimensional running coupling, we can write
(29) 
This immediately implies that

singular terms in must be confined to diagonal matrix elements, and must be determined by the cusp anomalous dimension according to
(31)
To illustrate how these features arise, in Appendix B we perform an explicit oneloop calculation of a quarkantiquark scattering amplitude. Note in particular that a given diagram violates rescaling invariance also in offdiagonal terms, but this violation is eliminated upon taking the sum of all diagrams, which is where conformal cross ratios like are formed. This is a consequence of gauge invariance.
Returning to the general case, in Section 5 we will give an explicit formula that satisfies the requirements outlined above. Our goal here is to first formulate the requirements in a compact and general way. To this end, let us consider the derivative of Eq. (28) (or Eq. (29)) with respect to . Noting that the dependence appears only through the eikonal jet functions, and using the results of Section 3.1, which imply
(32) 
we obtain a simple result for the dependence of ,
(33) 
This result can be turned into an equation for the dependence of the anomalous dimension matrix on its proper arguments, , just using the chain rule. Indeed, for any function depending on only through , one finds
(34) 
We conclude that
(35) 
As expected, the first equation in (35) states that offdiagonal matrix elements of the soft anomalous dimension matrix should be logarithmic functions of homogeneous conformal cross–ratios of ’s, such as . For diagonal terms, the second equation in (35) states that inhomogeneous terms are allowed, but they must be proportional to the cusp anomalous dimension in the colour representation of parton . We will explore the consequences of these constraints in the following sections, beginning with the case of twoparton amplitudes.
4 Twoparton amplitudes
In this section we consider in some detail the consequences of the new constraints in the simplest case of amplitudes with two hard coloured partons. We choose to analyse in Section 4.1 the special case of the spacelike Sudakov form factor of a quark, but the results apply, with minor modifications, to any amplitude with two hard partons. In Section 4.2 we consider the partonic jet function, which is an important building block in the factorization formula, Eq. (2), for any amplitude. We use there the results of Section 4.1 to constrain the kinematic dependence of the partonic jet, which is significantly more involved than that of the eikonal jet considered above.
4.1 The case of the Sudakov form factor
Let us consider the implications of the factorization constraints derived above on the simplest fixedangle scattering amplitude, the Sudakov form factor. We will see that the constraints of Eq. (35) lead to a refinement of the results of Ref. [40], since the kinematic dependence of the Sudakov soft function can be explicitly determined in terms of the cusp anomalous dimension.
As for any amplitude, our starting point is the factorization formula of Eq. (2), which here takes the form
(36)  
where . For definiteness, we will consider the spacelike form factor, .
In the case of the form factor, the soft function is simply the eikonal correlator defined by two Wilson lines running along the classical lightlike parton trajectories, with velocities given by and . Thus
(37) 
To determine the kinematic dependence of , we consider the reduced soft function , which is given by
(38) 
where, as before, and ; the latter is specific to the spacelike momentum configuration where so the phase in Eq. (9) is absent.
The reduced soft function obeys the renormalization group equation
(39) 
which leads to exponentiation. Since is a pure counterterm, one simply gets
(40) 
in analogy with Eq. (13) for the eikonal jet.
Factorization now requires the anomalous dimension to be a linear function of . Indeed, Eq. (35) in this case reads
(41) 
which integrates to
(42) 
where is introduced as a constant of integration, and does not depend on . As expected, the dependence of on the kinematic variable is very simple, and is fully determined by the cusp anomalous dimension.
Note that, similarly to what was done for the jet function in Eq. (25), we may extract from the anomalous dimension a factor of the Casimir operator of the relevant representation, defining
(43) 
where as usual , , and is a universal function of the coupling, independent of the colour representation. The oneloop result quoted here will be determined in Eq. (56) below.
Using Eq. (42) and Eq. (40), we can now write down an explicit expression for , where the kinematic dependence is completely solved. We find
(44) 
Finally, using the definition of in Eq. (38) we obtain an explicit result for the original soft function ,
(45) 
where we defined
(46) 
combining the two constants of integration introduced in Eqs. (19) and (42). Eq. (45) is intuitively appealing: the spacelike or timelike nature of the eikonal form factor is associated with the explicit logarithm multiplying the cusp anomalous dimension, just as is the case for the full form factor.
We conclude this section by briefly comparing our results with those reported in Ref. [40]. In order to do so, consider the anomalous dimension of the soft function , defined by
(47) 
Using Eq. (45), we obtain an explicit expression for . At the scale , it reads
(48)  
which is analogous to the result for the anomalous dimension of the eikonal jet in Eq. (20). One may of course verify the consistency of the various renormalization group equations corresponding to Eq. (38), observing that
(49) 
where logarithms of different arguments on the righthand side nicely combine to form a logarithm of the scale–invariant ratio , as expected.