Soft factors for double parton scattering at NNLO
We show at NNLO that the soft factors for double parton scattering (DPS) for both integrated and unintegrated kinematics, can be presented entirely in the terms of the soft factor for single Drell-Yan process, i.e. the transverse momentum dependent (TMD) soft factor. Using the linearity of the logarithm of TMD soft factor in rapidity divergences, we decompose the DPS soft factor matrices into a product of matrices with rapidity divergences in given sectors, and thus, define individual double parton distributions at NNLO. The rapidity anomalous dimension matrices for double parton distributions are presented in the terms of TMD rapidity anomalous dimension. The analysis is done using the generating function approach to web diagrams. Significant part of the result is obtained from the symmetry properties of web diagrams without referring to explicit expressions or a particular rapidity regularization scheme. Additionally, we present NNLO expression for the web diagram generating function for Wilson lines with two light-like directions.
Institut für Theoretische Physik, Universität Regensburg,
D-93040 Regensburg, Germany \emailAddaleksey.email@example.com
The effects of double parton scattering (DPS), i.e. the scattering with two partons of a hadron participating in the hard subprocess, are usually expected to be small in comparison to a single parton scattering contribution. However, at very high energies the effect of multiple parton interactions increases and presents an important part of the total cross section, see e.g. [1, 2, 3]. It is known that DPS processes can form strong background for Higgs searches [4, 5], as well as, be dominant channel for particular reactions, e.g. in the double Drell-Yan process . Therefore, the practical interest to DPS processes is constantly increasing.
From the theoretical side, the DPS processes are studied rather weakly. One of the reasons is the cumbersome kinematic structure of DPS. The double parton distributions (DPDs), the analogs of parton distribution functions for DPS, are functions of many variables: two momentum fractions and three transverse coordinates (or one transverse coordinate in the integrated case), say nothing of dependencies on two factorization scales. Additionally, DPDs have reach polarization and color structure, and even the leading order factorization formula for unpolarized and integrated double Drell-Yan involves more than dozen presumably independent DPDs [7, 8]. Nonetheless, during recent years there was significant progress in the theoretical understanding of DPS processes, due to the formulation of appropriate factorization theorems [7, 9, 8].
Apart from increased number of various functions, the DPS factorization theorems resemble the factorization formula for transverse momentum dependent (TMD) processes, see e.g. . It is not accidental since the dominant field modes are the same for TMD processes and DPS processes. This analogy grants the possibility to re-use the TMD experience during consideration of DPDs. For example, at NLO all the evolution properties for DPDs can be presented via corresponding evolution properties of TMD distributions [11, 7, 8].
In this work, we concentrate on the study of DPS soft factors, which are essential part of DPS factorization theorems. Soft factors represent the underlying interaction of soft gluons and contain the mixture of rapidity divergences related to both hadrons. This substructure should be decomposed into the parts with rapidity divergences belonging to a given hadron. Only after such decomposition a finite, i.e. meaningful, parton distributions can be defined. Naturally, the decomposition introduces the rapidity parameter. The anomalous dimension for the rapidity parameter scaling also can be deduced from the soft factor. Therefore, the study of the soft factor is an important part of the study of DPS factorization theorems.
At NLO the soft factors are nearly trivial objects. This order is given by single gluon exchange diagrams only. Therefore, DPS soft factors at NLO scatter into NLO TMD soft factors [7, 8]. At NNLO many non-trivial aspects of perturbative expansion arise. The most important one is that simultaneous interaction of several Wilson lines becomes possible, and thus, one can expect highly interesting dynamics. However, the difficulty of consideration also grows. For example, the properties of TMD soft factor, although known for a long time, have been explicitly demonstrated at NNLO only recently .
The soft factors for DPS are rather involved objects composed of four half-infinite light-like cusps of Wilson lines positioned at four different points in the transverse plane and connected in all possible ways. Consideration of such an object within a usual diagrammatic is a serious calculation problem, mostly due to confusing combinatoric of the color flow. The structure of perturbative series is exceptionally simplified within the generating function approach for web diagrams, formulated in [14, 13]. Within this approach, one should calculate the generating function, which is unique for a given geometry (in the case of TMD-like soft factors, the only important point is two light-like directions). Various matrix elements such as TMD soft factor, DPS soft factors, are obtained by a projection operation on the generating function. In this way, the usually difficult diagrammatic combinatoric is reduced to a couple of lines of simple algebraic manipulations.
One of the most attractive features of the generating function approach is an efficient organization of the expression. In particularly, it allows to avoid the calculation of whole sectors of diagrams, showing their equivalence with lower perturbative orders. As we demonstrate in this work, the consideration of the generating function at NNLO immediately shows the possibility to present any DPS soft factor in terms of TMD soft factors at this order. This fact is not trivial, since the NNLO expression contains products of TMD soft factor, but does not contain new functions. On the level of diagrams it implies that diagrams in particular combinations cancel each other, while in other combinations scatter into one-loop integrals. To find such combinations can be a tricky task, but they reveal automatically within the generating function approach.
In this paper we demonstrate that at NNLO DPS soft factors are given by a simple combination of TMD soft factors. Having at hands DPS soft factors at NNLO we study the structure of rapidity divergences and present the rapidity evolution equations at NNLO. It appears that some important results can be obtained without referring to expressions for diagrams. For example, the factorization of rapidity divergences for DPD soft factor at NNLO appears to be direct consequence of the rapidity factorization for TMD soft factor. We also perform the explicit calculation within the -regularization scheme  and confirm the results of the general analysis. The expression for the NNLO generating function for web diagrams presented here for the first time can be also used in other applications.
In the section 2 we review the derivation of factorization formula for double Drell-Yan process following articles [15, 7, 9, 16, 8]. This section is mostly needed to introduce the compact notation and necessary details about DPS soft factors. In the section 3.1 we give a short introduction to generating function approach for web diagrams. In sections 3.2 and 3.3 we discuss the details of evaluation of the generating function for DPS soft factors at NLO and NNLO, respectively. The particular form of projecting operators for DPS soft factors is given in section 3.4. In sections 3.5,3.6,3.7 we perform the projection operations and obtain the expression for DPS soft factors in terms of TMD soft factors. The origin of the simple structure of DPS soft factors is discussed in 3.6. In section 4 we discuss the influence of NNLO expression on the DPS factorization theorem. In particular, we show the factorization of rapidity divergences and define individual DPDs in sec.4.1. The rapidity evolution equations at NNLO are given in sec.4.2. The consideration is done in the most general case of unintegrated DPS. The important case of integrated DPS is obtained from these results and presented in the sec.5.
Technical details of the evaluation are collected in the set of appendices. In the appendix A we compare our notation with the notation used in  and . The explicit expressions for diagrams, as well as, their analysis are given in appendix B. The expressions for basic loop integrals that participate in the generating function are collected in appendix C.
2 Factorization of double parton scattering
In this section we review some aspects of the double parton scattering factorization. The main aim of this section is to introduce notation and make connections with previous works. The consideration presented here is very superficial, and is an extraction from [15, 7, 9, 16, 8], to which we refer for the proofs. To get access to the leading order factorized cross-section we use the SCET II technique. The consideration is in many aspects similar to the consideration of Drell-Yan process at moderate transverse momentum [8, 17, 18, 19, 10] (the so-called TMD factorization). Within this context, our main attention is devoted to the geometry of the double-parton scattering and to the color flow. Thus, we skip many important questions of DPS redirecting the reader to the literature [15, 7, 9, 16, 8].
2.1 Leading order factorization for double-Drell-Yan
where is a leptonic tensor and is the quark-to-vector boson current. Throughout the text an index enclosed in curly brackets denotes the set of indices of the same kind, e.g. here . As usual, we define two light-like vectors and along the largest components of and correspondingly, with . The vector decomposition reads , where are transverse components () and . The phase-space element denotes the complete phase-space of produced bosons, i.e.
with , (in the reference frame), is center-mass-energy and is the rapidity of produced boson. The large components of momenta are , where is a generic large scale. One of the coordinates, say , can be set to zero due to translation invariance, but we keep it explicit for homogeneity of notation and for later convenience. Also in the following we often use the shorthand notation for set of arguments as (the order indices is important). In the following, although we introduce the notation convenient for our study, we try to be close to the notation and normalizations of .
Integrated double Drell-Yan process attracts even more practical interest. The integrated cross-section has the phase-space element . It can be obtained from the unintegrated cross-section (2.1) by the integration over the transverse momenta . Consequently the expressions for the integrated anomalous dimensions, soft factors and over elements can be obtained from the unintegrated ones. In the following sections we consider only the general case of unintegrated kinematics. The expressions for the integrated case are collected in the section 5.
where are color indices, is spinor index, is a (soft) Wilson line from the point to , and field is the ”large” component of quark field along vector . The explicit definition of Wilson line is
The relation inverse to (2) is obtained by applying corresponding ”large-component” projector
and similar for anti-quark and components. Here is the projector in -direction. The Wilson line has the same formal definition as , but instead of soft gluons it consists of collinear ones.
Substituting the field decomposition (2) into the matrix element (2.1) one obtains a large set of terms. The central point of the SCET approach is that at the leading order of factorization and in the absence of Glauber interaction (which has been proved in ), the field does not interact with soft-gluons, and soft-gluon can be split up into separate matrix element. Then the cross-section is presented in the form
where we extract the Lorentz structures from currents and absorb them into the tensor . The symbol denotes a light-like cusp of half-infinite Wilson lines located at position ,
Note, that . The dots in (2.1) denote the terms suppressed by powers of [21, 22]. The hard matching coefficients of the vector currents to SCET fields are hidden inside the function . The separation of the hard part introduces the renomalization scales ’s for each hard sub-process. In the following hard renormalization scales are taken equal to for brevity.
Soft radiation does not resolve collinear scales, therefore, soft Wilson lines can be expanded at light-cone origin, , where is the transverse component of ;
The ”large” components of quark fields couple only to the hadron with corresponding momentum, i.e. couples to hadron , while couples to hadron ;
The ”large” component of the quark field does not resolve the scales in perpendicular direction, therefore, it can be expanded in that direction, i.e. .
Using these assumptions one can compute the leading contribution to double-Drell-Yan process. The factorized expression contains various terms with usual parton distributions, double-parton-distribution and combination that mix with each other within the operator-product expansion (see  for the leading order analysis). The separation of these terms from each other is an involved procedure (for theoretical development see [23, 24, 7, 16]). In this article we are interested in the study of soft factors responsible only for multi-parton scattering. Therefore, we skip the discussion on the mixture between various matrix elements and consider only the DPS contribution.
The DPS part of the cross-section corresponds to terms with simultaneous radiation of two distinct partons from the hadron. Applying leading order factorization restrictions to expression (2.1) and extracting the DPS contributions we obtain
here we have suppressed arguments of functions for brevity. The functions are soft factors and given by expressions
where and (note, that order of arguments and indices of the function is opposite to their order in the matrix element, i.e. graphical). The “arrow” notation is the visual representation of color-flow between light-cone infinities, i.e. if one writes all indices related to as down indices and all indices related as up indices, the arrows indicate the order of connection, see fig.1. The functions are double parton distributions (DPDs) and given by expressions
The similarity sign implies possible normalization factor, which depends on the spinor structure. The distributions are obtained by changing components , , and hadron states , e.g
The following steps of classification consists in the Fiertz decomposition of spinor and color structures. As a result of this procedure one gets a large set of various DPDs with different polarization properties [7, 8, 25]. However, the details of Lorentz structure are inessential for the study of soft factor, while the color structure should be considered in details.
In fact, the factorization theorems (2.1,5) are not complete, in the sense that they consist of individually singular objects (DPDs and soft factors). They suffer from rapidity divergences, and are not entirely defined. Moreover, the soft factors mix the rapidity divergences related to different sectors of integration. The standard procedure implies that a soft factor can be presented as a product of factors with rapidity divergences from different momentum sectors. Then combining these factors with appropriate parton distributions one defines an ”individual” parton distribution, which are finite and can be used in the phenomenology. Generally, it is unclear (although always implied) whenever it is possible or not to perform the rapidity-factorization procedure and define non-singular DPDs. In the section.4 we demonstrate that such procedure can be done at least at NNLO.
2.2 Color decomposition
The color structure of factorized expressions (2.1,5) is rather cumbersome. The notation introduced in (2.1-2.1) specially visualizes the color flow. The indices and denote the color adjusted to the antiquark and quark respectively. The subindex of color index designates the position of field in transverse plane (see fig.1). In this way, the gauge transformation transforms DPD such that it is left with respect to indices and right with respect to indices . For example,
where all matrices are located at light-cone infinities. Consequently, the soft factor transforms in conjugated way by eight matrices .
In a non-singular gauge the transformation at light-cone infinites can be reduced to unity. In this way, DPDs and soft factors are gauge invariant objects independently. However, there is the global rotation of quarks that still transform DPDs. Since global is a symmetry of QCD, the DPD matrix elements select only the singlet contributions. There are two singlets in that can be extracted as following
here we use the normalization for singlet parts suggested in , which is different from the normalization used in . The singlet parts of conjugated distributions are defined in similar manner. Substituting these expressions into (2.1,5) we obtain
where the DPDs and are 2-component vectors , and soft factors are -matrices, which explicit form we present later. The notation (2.2) implies the presentation of as a “column”, while as a “row”. This defines the order of matrix multiplication in the following sections.
Before we proceed to the definition of soft-factor matrices, let us discuss the symmetries of components. The obvious symmetry of a soft factor is the exchange of ’s order under the sign of T-ordering. It implies
As the consequence of the Lorentz invariance, the directions and can be exchanged within the soft factor independently of the rest expression. Therefore, a soft factor is equal the soft factor with all arrows turned upside-down,
where for . Due to these symmetries the soft factors are related to each other. There are only two independent matrices and (as discussed later these two soft factors are also related by (34)). The rest soft factors are expressed as
The soft factor matrix has four components, the Wilson lines of which are connected either by ’s, or by generators. In turn, the product of generators can be expressed as products of ’s using Fiertz identities. For practical reasons, it is convenient to consider the products of Wilson lines contracted by ’s only. There are five independent structures that can appear
A visual representation of these soft-factors is given in fig.2. The normalization is chosen such that at the leading perturbative order all soft factors are unity,
The topology of the components is different, so the components form two Wilson loops, while are single Wilson loops.
Further simplification of the structure can be made using features of the soft factor geometry special for the Drell-Yan process. The Wilson lines in the matrix element (8) are all positioned on the past light cone. Therefore, the distance between any two fields within (8) is space-like (or light-like if fields belong to the same Wilson line). It allows to rewrite the T-ordered product of Wilson lines as a usual product of Wilson lines, using the micro-causality relation. However, it is more convenient to organize Wilson lines as a single T-ordered product. We have
Such presentation is distinctive feature of Drell-Yan kinematic, and is not possible for, say, double semi-inclusive deep-inelastic scattering (SIDIS).
The representation (30) suggests higher symmetry of soft factor. Namely, the arguments can be freely exchanged preserving the topology of color-connection. Therefore, we need to conciser only soft-factors of different topology: a single Wilson-loop (we choose ), and a double Wilson-loop (we choose ). The rest are related to the chosen in the following way
We also find that the soft factor can be expressed via as
Therefore, we can consider only the case of , while the results for other channels can be obtained by permuting vectors .
Here, for compactness we use the shorthand notation for the argument . The rest of the soft factor matrices can be obtained via (20-23,34). At the leading order of perturbation theory soft factor matrices reduces to identity matrices
3 Evaluation of soft factors
3.1 Generating function for web diagrams
The straightforward evaluation of functions requires a calculation of many diagrams, most of which are equivalent under permutation of parameters and change of color factors. Such a consideration would be very inefficient and contains many potential places for a mistake. A more effective approach is to evaluate the generating function for web diagrams, which is common for all soft factors, and project out the appropriate soft factor. The theoretical description of the approach can be found in [14, 13]. In this section we describe only the basics of generating function approach needed for this particular calculation.
The generating function approach is based on the well-known fact that the perturbative series for vacuum average of some operator sources is an exponent of the connected diagrams. This property immediately leads to exponentiation theorem for Wilson lines for Abelian gauge theories . For a non-Abelian gauge theories one has an additional difficulty coming from the necessity to disentangle the color structure. The disentangling can be done in the general form . In this way, one sees that significant part of diagrams that appear in the usual perturbation expansion (as well as, in the classical Wilson loop exponentiation diagrammatic [26, 27]) are composed from the smaller-loop diagrams.
The power of the generating functions approach is that evaluated ones the generating function can be easily used to obtain the perturbative expression for any color topology. Therefore, the generating function that we present later can be used to obtain all DPS soft factors (24-28), as well as, TMD soft factor and soft factors for multi parton scattering with six, and more operators . Moreover, the approach allows one to consider the exponentiated expression directly in the matrix form.
The starting point of the construction is to carry out the color structure of the Wilson line. The effective way to do so is to introduce the ”scalar-reduction” of a Wilson line connecting points and 
where is a gauge-group generators, are c-number variables, and is functional of gauge fields. The particular form of needed for our calculation is given in (3.1), while the general form can be found in [14, 13]. In this expression the matrix structure is carried by the first exponent in the product. The first exponent does not contain any fields and thus, does not participate in the function integration. The Wilson line flowing in the opposite direction can be presented in the form
where we have used that operator is anti-hermitian .
Within the considered task, we have a simple geometry of Wilson lines. All of them are straight (along or ), and continue from to infinity (or in opposite direction). Let us enumerate these segments by number (for DPS soft factor ). The ’th Wilson line can be presented in the form
where ( or ), the variable denoted the direction if the color flow to (from) light-cone infinity. The operator , which discribes a half infinite Wilson-line is given by