Using Dimensional Reduction for Hadronic Collisions

We discuss how to apply regularization by dimensional reduction for computing hadronic cross sections at next-to-leading order. We analyze the infrared singularity structure, demonstrate that there are no problems with factorization, and show how to use dimensional reduction in conjunction with standard parton distribution functions. We clarify that different versions of dimensional reduction with different infrared and factorization behaviour have been used in the literature. Finally, we give transition rules for translating the various parts of next-to-leading order cross sections from dimensional reduction to other regularization schemes.


Introduction
Recently progress on the understanding of regularization by dimensional reduction has been achieved in three directions. First, a mathematically consistent definition avoiding the problem found in Refs. [1,2] was formulated, and a succinct method to check the symmetry properties of dimensional reduction was developed [3], leading to the verification of supersymmetry in important cases at the two-loop level [4]. Second, explicit calculations demonstrated how dimensional reduction can be applied to multiloop calculations and how renormalization has to be carried out in a non-supersymmetric context [5]. This provides the basis of transition rules between various definitions of parameters such as α s or m b and is useful to derive the GUT-scale values of these parameters from the experimental values [6]. Third, an obstacle in the application of dimensional reduction to hadronic processes was removed [7] by the resolution of the factorization problem of dimensional reduction found in Refs. [8][9][10].
The purpose of the present article is to further elaborate on the application of dimensional reduction to hadronic processes. In Ref. [7] we restricted ourselves to the case considered in Ref. [8], the real corrections to the process gg → tt, and showed that, despite first appearances, in the collinear limit these real corrections factorize into products of splitting functions and leading-order cross sections.
Here we will consider real and virtual NLO QCD corrections to arbitrary hadronic 2 → (n − 2) processes with massless or massive partons. We will discuss the infrared singularity structure and the associated regularization-scheme dependence of all these corrections, provide transition rules between the schemes and show that all singularities factorize. In this way we show that the framework of dimensional reduction is completely consistent with factorization, and we show how this scheme can be used to compute hadronic processes in practice.
One of the main points of this article is the distinction of two different versions of dimensional reduction that have been used in the literature. One of the reasons why the factorization problem of Refs. [8][9][10] has remained unsolved for so long is that these two versions have mainly been applied by two different communities. The version used in Refs. [8][9][10] is the same as the one defined in Refs. [3,11,12] and is the one mainly used in the context of supersymmetry. The version used in Refs. [13][14][15], which was denoted by dr and is actually equivalent to the four-dimensional helicity (fdh) scheme [16] at one-loop, is mainly used in the context of QCD. For the latter version, the infrared singularity structure and transition rules have already been derived [13][14][15]. We denote these two versions by dred and fdh. They differ in their treatment of external particles, in a way analogous to the difference of the "conventional" and " 't Hooft Veltman" versions of dimensional regularization, cdr and hv.
In the main part of the present article we will provide results and transition rules for all these four regularization schemes, keeping in mind that the results for the fdh, cdr and hv schemes can already be found in Refs. [13][14][15], while the results for dred are new. The results for the infrared singularity structure in cdr, hv, fdh and in dred are developed in Section 2 and 3, respectively. The practical application of dred and transition rules are discussed in Section 4. Appendix A provides explicit results for all relevant splitting functions, and in Appendix B we provide three explicit examples of NLO computations in dred.

Elements and scheme dependences of hadronic cross sections
We consider n-parton processes with up to two hadrons in the initial state at nextto-leading (NLO) order in QCD. The partons can be either massless quarks q or gluons g or massive partons such as heavy quarks Q, gluinos, or squarks. In our equations we will restrict ourselves to the most interesting case of two initial-state partons as the simpler cases can be obtained by straightforward modifications. The cross sections of such processes can be written as dσ H 1 (K 1 )H 2 (K 2 ) → a 3 . . . a n = a 1 ,a 2 1 0 × dσ a 1 (x 1 K 1 )a 2 (x 2 K 2 ); a 3 . . . a n , where H 1,2 are the initial-state hadrons, K 1,2 their momenta and a i (i = 3 . . . n) the final-state partons. The sums run over all possible flavours of the initial-state partons a 1,2 of the hard partonic cross section dσ, and f a i /H i (x) denote the appropriate parton distribution functions (PDF). In the computation of hadronic cross sections, three scheme choices have to be made: the choice of the renormalization scheme, the factorization scheme, and the regularization scheme. The functional dependence of the hard cross section dσ on input parameters like α s , particle masses, etc, depends on the renormalization and the factorization schemes.
The choice of the renormalization scheme is equivalent to a precise definition of the input parameters entering the computation, in particular of α s , particle masses and other coupling constants. Once it has been fixed and renormalization has been carried out accordingly, all off-shell Green functions are finite and unambiguously defined. Changing the renormalization scheme changes both the functional form of dσ and the numerical values of the input parameters, such that dσ is renormalization-scheme independent up to terms which are formally of higher order than NLO. Common renormalization schemes are the MS-scheme for α s , the DR-scheme for supersymmetric parameters, or the on-shell scheme for masses. 1 In the following we are not concerned with the renormalization-scheme dependence and assume that some renormalization scheme has been fixed.
The choice of the factorization scheme is equivalent to a precise definition of the parton distribution functions. Both the functional form of the hard cross section and the numerical values of the parton distribution functions depend on this choice, but this dependence cancels in the full hadronic cross section up to terms which are formally of higher order than NLO. The major part of the following considerations is independent of the factorization scheme, and we simply assume that some factorization scheme has been fixed. At the end we will specialize to the important case of the MS-factorization scheme. In that case, the parton distribution functions f a i /H i can be taken for instance from the well-known MRST or CTEQ sets [17,18].
Our main focus is the influence of the regularization scheme (rs). After removing the regularization, all quantities appearing in Eq. (1) are rs independent, but the hard partonic cross section is a sum of rs dependent parts. It is commonly written as where the rs dependence is explicitly indicated. The lowest-order, or Born cross section dσ Born is finite, and in the limit where the regularization is removed its rs dependence vanishes. The three NLO contributions are the real and virtual corrections dσ real , dσ virt and the collinear counterterm dσ coll , which subtracts initial-state collinear singularities. All NLO contributions involve collinear and/or soft singularities and depend on the rs in their finite and their divergent parts. There are no ultraviolet singularities and associated rs dependences in dσ virt , because these are eliminated by renormalization and by fixing the renormalization scheme.

Variants of dimensional regularization and dimensional reduction
In all dimensional schemes space-time is continued from 4 to D dimensions, where D = 4 − 2ǫ is an arbitrary complex number. In this way momentum integrals become well-defined and ultraviolet and infrared singularities appear as 1/ǫ kpoles as ǫ → 0. Gluon fields are treated differently in dimensional regularization and dimensional reduction. In the former, gluons are treated as D-dimensional as well; in the latter, gluons are treated as 4-dimensional. Both choices have certain advantages. The purely D-dimensional treatment of all objects leads to simpler expressions, but it breaks supersymmetry owing to the different number of degrees of freedom of the gluon and the gluino. The 4dimensional treatment of the gluon is better compatible with supersymmetry and it is more amenable to helicity methods, which are commonly used to simplify QCD higher-order computations.
In order to formulate the two schemes one needs to distinguish three spaces: • the original 4-dimensional space (4S).
• the formally D-dimensional space for momenta and momentum integrals. This space is actually an infinite-dimensional vector space with certain D-dimensional properties [19,20], and is sometimes called "quasi-Ddimensional space" (QDS). The space 4S is therefore a subspace of QDS.
• the formally 4-dimensional space for e.g. gluons in dimensional reduction. This space has to be a superspace of QDS in order for the dimensionally reduced theory to be gauge invariant. Hence it cannot be identified with the original 4S -it can only be constructed as a "quasi-4-dimensional space" (Q4S) [3,21] with certain 4-dimensional properties. In practice the distinction between Q4S and 4S often does not matter, but it is important in the definition of the different versions of dimensional reduction, see below, and to avoid the inconsistency uncovered in Ref. [2].
These three spaces are characterized by their metric tensors, which we denote by g µν (for Q4S),ĝ µν (for QDS), andḡ µν (for 4S). The dimensionalities of the spaces are expressed by the following equations: The following projection relations express that 4S is a subspace of QDS and QDS is a subspace of Q4S: It is useful to introduce the orthogonal complement to QDS. This is a 4−D = 2ǫdimensional space with metric tensorg µν , which satisfies g µν =ĝ µν +g µν , Within this framework it is now possible to precisely state the calculational rules of dimensional regularization and dimensional reduction. Since momenta are always treated in D dimensions, it only needs to be specified how gluons (or other vector fields) are treated. More precisely, it needs to be specified which metric tensors are used in gluon propagator numerators and in gluon polarization sums. Particularly important for the understanding of factorization is the treatment of gluon polarization sums in squared matrix elements. Without regularization, such polarization sums can be written as cdr hv fdh dred internal gluonĝ µνĝµν g µν g µν external gluonĝ µνḡµνḡµν g µν Table 1: Treatment of internal and external gluons in the four different rs, i.e.
prescription for which metric tensor is to be used in propagator numerators and polarization sums. For the definition of "internal" and "external" see text.
where k is the gluon momentum and n is a gauge vector such that nk = 0. With regularization, the metric tensor in this polarization is replaced by either g µν , g µν ,g µν , orḡ µν . It is not strictly necessary to regularize all gluons. Only gluons that appear inside a divergent loop or phase space integral ("internal") need to be regularized; for all other gluons ("external") regularization is optional. The precise definitions of "internal/external" in this context are as follows: "Internal gluons" are defined as either virtual gluons that are part of a one-particle irreducible loop diagram or, for real correction diagrams, gluons in the initial or final state that are collinear or soft. "External gluons" are defined as all other gluons. Now, since external gluons do not have to be treated in the same way as internal ones, it is in fact possible to distinguish two variants of each regularization. The two variants of dimensional regularization are: • cdr ("conventional dimensional regularization"): Here internal and external gluons (and other vector fields) are all treated as D-dimensional.
Note that the above definition of internal gluons in phase space integrals is necessary for unitarity but leads to complications in the treatment of phase space integrals in schemes where internal and external gluons are treated differently. The two analogous variants of dimensional reduction are: • dred ("original/old dimensional reduction"): Internal and external gluons are all treated as quasi-4-dimensional.
Note that the version of dimensional reduction denoted by dr e.g. in Refs. [13,15] is equivalent to fdh at the one-loop level (see e.g. Refs. [13,22]). 2 The infrared properties of the three schemes cdr, hv, dr (or equivalently fdh) have been studied and compared in Ref. [15] and found to be consistent with factorization.
An apparent inconsistency between dimensional reduction and factorization has been identified in Refs. [8,10], but in these references the version dred has been used. In Ref. [7] it was found that factorization holds as expected in dred if external quasi-4-dimensional gluons are decomposed into D dimensional gauge fields and (4 − D) dimensional "ǫ-scalars", which are treated as separate partons. Technically, this decomposition amounts to replacing for squared matrix elements, where the different gluon types g,ĝ,g are denoted by the same symbols as the associated metric tensors. The algebraic expressions for the partonic processes involving g,ĝ, org are defined by the values of the corresponding gluon polarization sums. These read g : g : g : Eq. (9) follows trivially from Eq. (5). The decomposition of gluons into their D-dimensional and ǫ-scalar part in dred is also relevant for the renormalization of UV divergences. In order to make Green functions with external ǫ-scalars finite, the renormalization constants for ǫ-scalar couplings in general have to be different from the corresponding gluon couplings. For example, in pure QCD, the couplings α s and α e for the quarkantiquark-gluon and the quark-antiquark-ǫ-scalar vertices receive different counterterms δα s = δα e even if α s = α e at tree level [5,24].

Splittings in the four schemes
An essential part of the rs dependence of NLO contributions is related to the rs dependence of the splittings i → jk of one parton i into two collinear partons j, k. The rs dependence of real corrections is related to the splitting functions P RS i→jk ; the rs dependence of virtual corrections is related to constants γ RS (i) [13], which in turn can be derived from the P RS i→jk via unitarity [15]. In this section we explain the rs dependence of the splitting functions and correspondingly of the γ RS (i). The full results can be found in Appendix A. Figure 1 shows the most interesting case of a gluon splitting into two collinear gluons. According to the definition given above, the two collinear gluons j and k are treated as "internal", and the virtual gluon i as "external". The appropriate treatment of the gluons in the four rs can be read off from Table 1 and is displayed in the figure. Two simple observations allow an easy comparison of the four cases. First, the projection of a D-dimensional onto a strictly 4-dimensional parent gluon does not change the structure of the result of the splitting functions. And second, the result in dred should be decomposed according to Eq. (9) into four splittingsĝ →ĝĝ,ĝ →gg,g →ĝg,g →gĝ. 3 Then the result in cdr is identical to the dred result forĝ →ĝĝ, and all scheme differences can be explained in the following way: • The splitting g → gg is identical in the cdr and hv schemes. Formally, this is expressed in the equality for the splitting functions defined for z < 1.
• In the fdh scheme the outgoing gluons are treated as quasi-4-dimensional. The resulting additional term can be interpreted as being due to the splittingĝ →gg as already discussed in Ref. [15]. Hence, • In the dred scheme the parent gluon is also treated as quasi-4-dimensional, and therefore the two additional splittingsg →ĝg andg →gĝ are possible.
In the spirit of our discussion around Eq. (9) we do not combine the dred splitting functions into a single one.
The splitting functions involving quarks are related in a similar way. Via unitarity, the rs dependence of the constants γ(i) follows from the splitting functions [15] and can thus be explained in an analogous way: • The γ(i) in cdr and hv are the same, • The additional terms in the fdh scheme are due to the splittingsĝ →gg and q → qg: • In dred one has to distinguish γ constants forĝ,g, and q. The ones corresponding toĝ and q are the same as the ones in the fdh scheme; the one forg is related to the additional splittingsg →ĝg andg →gĝ: These relations form the basis for understanding the rs dependence of NLO contributions and in particular the difference between dred and the other schemes.
In the subsequent sections we will see that additional rs dependences arise from the crossing of the splitting functions to initial-state parton splitting and from the rs dependence of the LO matrix element.

CDR, HV, FDH
Our starting point is the decomposition, Eq. (2), of the hard partonic cross section, and we are mainly interested in the rs dependence of the separate terms contributing to dσ. In this section we will restrict ourselves to the well-known cases of cdr, hv and fdh. The rs dependence of quantities will be indicated by a subscript RS * , the star reminding us that we consider cdr, hv and fdh, but not (yet) dred.

Born term
We consider the partonic process where a i and p i denote the flavour and the momentum of parton i respectively. The l-loop correction to the rs dependent squared matrix element for the process given in Eq.
. . a n (p n )) or by M (l) RS * (a 1 . . . a n ) for short. For the cross section we need the averaged squared matrix elements where ω RS * (a i ) denotes the rs dependent number of degrees of freedom of a parton with flavour a i and s 12 ≡ 2 (p 1 · p 2 ) in the case of massless incoming partons. The Born cross section is obtained by integrating the squared and averaged tree-level matrix element over the (n − 2) parton phase space dΦ n−2 (p 1 , p 2 ; p 3 . . . p n ) multiplied by a measurement function for an infrared-safe quantity and a symmetry factor. The latter two are always implicitly understood in our notation and we simply write The rs dependence in Eq. (21) is due to O(ǫ) terms in M (0) RS * . Since we consider an infrared-finite quantity, the phase-space integration does not introduce any poles. Therefore, we can take the limit ǫ → 0 and dσ Born (a 1 . . . a n ) ≡ dσ is rs independent, as indicated by the absence of the subscript RS * .

Virtual corrections
For the virtual corrections we need M (1) RS * , the interference terms of the one-loop amplitude and the tree-level amplitude. The structure of the singular terms of M (1) RS * is well known [14,25]. For the fully renormalized matrix element, it is given by where the sums i and j are over all initial or final state partons and we introduced The soft and collinear poles are contained in the terms proportional to M (0) RS * and M ij RS * . The latter are the colour-linked Born squared matrix elements introduced in Ref. [26] and correspond to the square of the colour-correlated tree amplitudes with a T i · T j insertion, used in Ref. [27]. If particles i and j are massless, we have and Eq. (23) reduces to the well-known expression for the singularities of oneloop QCD amplitudes [25]. If one or both of the particles i, j are massive, these expressions have to be generalized [14], but the structure of the singularities remains as in Eq. (23). The rs dependence of Eq. (23) is contained in the constants γ RS * (a i )/ǫ, as well as in M NS (a 1 . . . a n ), is in general very complicated, but is finite and, after taking the limit D → 4, rs independent.
The essential, non-trivial part of the rs dependence is due to the γ RS * (a i )/ǫ terms. These are closely related to collinear singularities due to self-energy insertions on external legs and depend on the flavour a i of leg i. The rs dependence of γ RS * has been given in Ref. [13]. Via unitarity it is related to parton splittings a i →anything [15], as summarized in Section 1.3.
In the present paper we determine γ RS * by insisting that the sum rules hold in all rs to all orders in ǫ. This is a slightly different approach compared to Ref. [13] and simply amounts to a shift of finite terms between M (1) NS and the γ RS * terms in Eq. (23). Neglecting O(ǫ 2 ) terms, we find with β 0 = (11N c − 4 T F N F )/3 and T F = 1/2. For heavy quarks the result is rs independent [14] and we have γ(Q) = C F . The final virtual corrections dσ virt RS * are obtained as RS * (a 1 . . . a n ) .
Since this phase space integration does not give rise to any pole in ǫ, taking the limit D → 4 in M NS is justified.

Real corrections
For the real corrections dσ real RS * to the partonic process a 1 a 2 → a 3 . . . a n we have to consider contributions from all 2 → (n − 1) processes that are obtained by a split of any of the outgoing partons. We have to evaluate the corresponding squared matrix elements M (0) RS (a 1 , a 2 ;ā 3 . . .ā n+1 ) and integrate them over the (n− 1) parton phase space dΦ n−1 (p 1 , In Eq. (28) we denote byā i , i ∈ {3 . . . n + 1} the flavour of the outgoing partons, and as indicated by ā i , we have to sum over all relevant processes.
As is well known, the matrix elements can develop singularities in regions of the phase space where a parton becomes soft or two partons become collinear. The integration over dΦ n−1 (p 1 . . . p n+1 ) in this region then results in 1/ǫ 2 and 1/ǫ poles. Thus the rs dependence of the matrix elements which manifests itself in the O(ǫ) terms of M (0) RS results in differences in the O(1/ǫ) and in the finite terms of the real corrections. 4 In order to deal with the phase-space integration at NLO one often uses either phase-space slicing [28] or subtraction [26,29], and several general procedures have been developed [27,30,31]. For our purposes it is sufficient to know that they all rely on the same main points. Using the results given below it will be obvious how any of these procedures can be applied in the context of dred.
The first point is that in any of the singular regions the matrix elements take a simple form and can be written as a factor containing the kinematic singularity times a reduced (colour-linked) tree-level matrix element, associated with a 2 → (n − 2) process. Secondly, the phase space is factorized according to The factor with the kinematic singularity is integrated analytically over dΦ rad , producing the poles in analytic form. These poles will be multiplied by the reduced (colour-linked) matrix element and are to be integrated over a slightly modified (n − 2) parton phase space It is therefore not surprising that the real corrections have a similar structure as the virtual corrections, Eqs. (23) and (27). We will now look at all three potentially singular regions in turn.
Soft Region: In the limit where gluon g k (or another massless gauge boson) becomes soft we have where we introduced g 2 s = 4π α s . It is understood that the set of partons {a 3 . . . a n } is equal to the set {ā 3 . . . g k . . .ā n+1 } with g k removed. Similar comments apply to the analogous equations below. The phase space integration of Eq. (29) leads to dΦ rad s ij /(s ik s jk ) and results in poles that cancel the corresponding poles in V(i, j). Due to the measurement function implicitly included in dΦ n−2 , the remaining integration does not result in any singularities. The scheme dependence enters only through M ij RS * and after summation over all real processes trivially cancels between the real and virtual corrections.
Final-State Collinear Region: In the limit where two outgoing partonsā k and a l become collinear we have . . a n ).
As detailed in Appendix A, P < RS * (kl) * →kl is the rs dependent splitting function defined for z < 1 with p k → z(p k + p l ) and p l → (1 − z)(p k + p l ). In Eq. (30) the flavours a k andā l are fixed. This uniquely determines the flavour of the parent parton a (kl) . It is understood that if the split is flavour forbidden we set P < RS * (kl) * →kl = 0. To avoid a proliferation of subscripts, we denote the flavour of the partons in the splitting functions simply by (kl) etc. rather than a (kl) . The parent parton is slightly off shell as indicated by the notation (kl) * .
Contrary to the soft limit, in the collinear limit there are two sources of rs dependence. Apart from the trivial dependence through M (0) RS * (a 1 . . . a n ), the prefactor P < RS * (kl) * →kl is also rs dependent. Its rs dependence can be found in Section 1.3 and in Appendix A. Since the z dependence in Eq. (30) is entirely in the prefactor, the integration dΦ rad P < RS * (kl) * →kl /s kl can be performed separately. The terms related to the collinear singularities due to the splitting of parton a i = a (kl) schematically can be written as where the sum over all possible splittings, a k , is due to the sum over the relevant real processes, Eq. (28). Note that in the sum a k ∈ {g, q,q} a sum over the N F massless quark flavours is implicitly understood. After this sum the integrand is symmetric with respect to z ↔ 1 − z. Hence the integration can be restricted to the region z > 1/2 and the potential singularity at z = 1 is regularized with the usual +prescription. The z integration in Eq. (31) results in a rs dependence of the singular prefactor that multiplies the Born term. In fact, the factor in the second line of Eq. (31) is equal to (−γ RS * (a i )). Therefore, after summing up the contributions of all final state partons, i ∈ {3 . . . n} these terms precisely cancel the singularity and the rs dependence of those virtual terms displayed in the first line of Eq. (23) that are associated with outgoing partons, i ≥ 3 [15].

Initial-State Collinear Region:
Finally we turn to the case of an outgoing partonā k becoming collinear to the incoming parton a 1 (or a 2 ). There are some important differences with respect to Eq. (30). To start with, the collinear limit has to be written with the spin/colour summed/averaged matrix elements M . . . a n ) .
Contrary to Eq. (30), the z dependence in Eq. (32) is not restricted to the prefactor and, therefore, the z-integration results in a more complicated structure. In Eq. (32) the flavours a 1 andā k are fixed and uniquely determine the flavour of parton a (1k) , which is slightly off shell. The splitting functions of Eqs. (30) and (32), with initial/final-state off-shell parton, are related by a crossing relation where #f denotes the number of crossed fermions. We remark that the wellknown crossing symmetry P < CDR l→(lk) * k = P < CDR l * →(lk)k does not hold in all rs, and in general we have with ∆ FDH = 0 and ∆ HV = 0. The explicit form of ∆ FDH and ∆ HV can easily be found using Eq. (33) and the results in Appendix A. However, as will be discussed in Section 2.4, the distinction made in Eq. (34) is ultimately not required. Using Eqs. (34) and (86)  we can write the initial state collinear term for parton 1 schematically as . . . a n ) .
The first term on the r.h.s. of Eq. (35) is due to the δ(1 − z) term present in Eq. (86). Together with the corresponding term for the second incoming parton, dσ real,2 RS * , this results in a term that precisely cancels the singularity and the rs dependence of those virtual terms displayed in the first line of Eq. (23) that are associated with incoming partons i ≤ 2. The remaining terms given in the second line on the r.h.s. of Eq. (35) are associated with collinear counterterms.

Collinear counterterm
In the sum of the virtual and real corrections, dσ virt RS * + dσ real RS * , all singularities and rs dependences cancel, apart from the terms given in the last line of Eq. (35). These are cancelled by the collinear counterterm dσ coll RS * . While the divergent parts of the collinear counterterms are completely determined, there is some freedom in how to specify the finite parts of dσ coll RS * . Any specific choice of the finite parts of dσ coll RS * is equivalent to the definition of a particular factorization scheme. Leaving the factorization scheme open, we can write dσ coll RS * ,FS (a 1 , a 2 ; . . . a n ) = α s 2π RS * (a i (z p 1 ), a 2 (p 2 ); . . . a n ) RS * (a 1 (p 1 ), a i (z p 2 ); . . . a n ) , where the sum is over all possible splittings of the incoming partons, and the index i is defined in analogy to Eqs. (30) and (31). The X FS l→ik are the finite (i.e. ǫ-independent), rs independent terms which define the factorization scheme. The formulas in Sections 2.2 and 2.3 show that in this way the hard (subtracted) partonic cross section dσ FS (a 1 . . . a n ) = dσ Born RS * (a 1 . . . a n ) + dσ virt RS * (a 1 . . . a n ) + dσ real RS * (a 1 . . . a n ) + dσ coll RS * ,FS (a 1 . . . a n )

D→4
is finite and rs independent as indicated by the absence of the subscript RS. In this approach the most natural factorization scheme would correspond to setting all X FS = 0. In principle, this particular factorization scheme is as good as any other, and it could be used in practice in Eq. (1) in conjunction with parton distribution functions f a i /H determined in the same scheme. In practice, however, parton distribution functions such as the standard MRST or CTEQ sets [17,18] are mainly available in the MS factorization scheme, which is different. The MS factorization scheme is defined by using rs=cdr and replacing the square bracket in Eq. (36) by The minimal subtraction procedure corresponds to setting ǫ → 0 in the splitting functions P CDR 1→ik . Thus, even if we use cdr (and even though ∆ CDR 1→ik = 0) the MS scheme does not correspond to X FS = 0 but to where P MS ǫ 1→ik (z) ≡ P CDR 1→ik (z) −[P CDR 1→ik (z)] D→4 denote the O(ǫ) terms of the splitting functions in cdr.
In the evaluation of the real corrections, Eq. (35), as well as in the collinear counterterm, Eq. (36), we made the distinction between P < RS * l * →ik and P < RS * l→i * k , the splitting functions appropriate for an outgoing and incoming split, respectively. In a general rs, these two splitting functions differ as indicated in Eq. (34), hence the presence of the ∆ terms in Eqs. (35) and (36). While the expressions given in Eqs. (35) and (36) are those that naturally arise in the calculation, we note that the ∆ terms cancel in the sum of dσ real RS * + dσ coll RS * ,FS . Thus we can drop ∆ RS * 1→(1k)k in Eq. (35) (and the corresponding term in dσ real,2 RS * ) if we also drop ∆ RS * 1→ik (and ∆ RS * 2→ik ) in Eq. (36). Ultimately, the distinction between P < RS * l * →ik and P < RS * l→i * k is not needed.

Dimensional Reduction
In this section we show how the structure described in Section 2 can be generalized to include dred. As mentioned in the introduction, the key point is to split the gluon into a D-dimensional gluonĝ and a (4−D)-dimensional ǫ-scalarg by setting g =ĝ +g and using Eq. (9). Often it is sufficient to perform this split for only one gluon as in Eq. (9); in general, if a process contains #g gluons, g i 1 . . . g i #g , we can decompose the matrix element in dred into 2 #g terms according to We consider the two partonsĝ andg to be two different partons and consequently regard the r.h.s of Eq. (40) as a sum over the squared matrix elements of 2 #g different processes. To bring our notation in line with the previous section, we will write Eq. (9) and Eq. (40) as respectively, where it is understood that if a i = g we sum over the two terms a i ∈ {ĝ,g} whereas if a i = q there is only one term in the sumȃ i ∈ {q}.
For the spin summed/averaged matrix elements the relation equivalent to Eq. (42) reads M DRED (a 1 , a 2 ; . . . a n ) = M DRED (ȃ 1 ,ȃ 2 ; . . .ȃ n ) . (43) The explicit expressions for ω RS (a i ) are given in the Appendix in Eq. (85). We stress that while the split g =ĝ +g is conceptually simple, it seems to complicate practical computations. As we will see in the later sections, however, in an explicit computation of a physical process in dred it is only required at a very limited number of steps. In particular, it will turn out that no PDF for the unphysical ǫ-scalarg will be required.
In the present section we will use the split to understand the infrared structure of matrix elements in dred. In fact, it is straightforward to see that Eqs. (23), (30), and (32) for the collinear singularities of virtual and real corrections hold in the same form in dred for the individual processes with split partons, i.e. if we replace RS * → DRED and a i →ȃ i in these equations. However, our main interest are the infrared properties and rs dependences of matrix elements for full gluons; therefore we will carry out the sums over {ȃ i } wherever possible.

Born term
The full tree-level matrix element in dred is equal to the one in fdh and hv and can be obtained from the cdr result simply by setting D → 4 M (0) DRED (a 1 . . . a n ) = M

Virtual corrections
The structure of the virtual corrections in dred is analogous to Eq. (23) for each individual dred process with split partonsȃ i . Hence, by summing over all processes as in Eq. (42) we obtain The non-trivial structure of Eq. (46) is most essential for the γ terms, since γ DRED (ĝ) = γ DRED (g). In fact, as discussed in Section 1.3 and Appendix A, the γ forĝ and q match the ones in the fdh scheme, while the one forg is new and different: On the other hand, V(i, j) is not affected if aĝ is replaced by ag, and the Born terms in the last line on the r.h.s. of Eq. (46) can be combined in a straightforward way. Hence one can immediately obtain the result for either the fully or the partially combined process, M where M ij DRED (a 1 . . . a n ) = M ij FDH (a 1 . . . a n ) and M NS (a 1 . . . a n ) is the rs independent term appearing also in Eq. (23).
We recall that M DRED denotes the fully renormalized one-loop matrix element and Eq. (46) does not contain any ultraviolet singularities. In dred this implies that off-shell Green functions are finite also if externalg are present. This requires that couplings involvingĝ and couplings involvingg in general renormalize differently [24]. As a result, the renormalization procedure in non-supersymmetric theories can be slightly more involved in dred. We refer to Appendix B for examples.

Real corrections
The calculation of the real corrections in dred follows the same pattern as in the other schemes discussed in Section 2.3. However, there are some important differences which we will consider for the three singular regions in turn.
Soft Region: In close analogy to Eq. (29), we have to consider the limit of the real matrix element when a gluon becomes soft. In dred this soft gluon can be either aĝ or ag. The soft limit is governed by eikonal factors of the form p µ i p ν j /(s ik s jk ), contracted with the corresponding polarization sum of the soft gluong k . The polarization sums in Eq. (10) then show that a softĝ k leads to the same limit as a soft full gluon g k , while a soft ǫ-scalarg k leads to zero. Hence, In analogy to the behaviour of V(i, j) in the virtual corrections, the soft limit Eq. (49) does not require the split g k =ĝ k +g k . Thus, with respect to the soft limit, dred is equivalent to hv and fdh.

Final-State Collinear Region:
Here the split g =ĝ +g is essential. According to the main result of Ref. [7], the key equation for the collinear limit, Eq. (30), has to be modified in dred if the parent parton a (kl) is a gluon. In this case the  a 2 ; . . .ȃ (kl) (p k + p l ) . . . a n ).
We remark that Eq. (51) leads us to consider splitting functions withȃ (kl) =ĝ andȃ (kl) =g in dred. They are given in Appendix A. Eq. (51) remains true if all partons a j are replaced byȃ j , however, only forȃ i =ȃ (kl) the split is strictly required. Thus, Eq. (31) is modified to This is illustrated in Figure 4, where we consider the particularly interesting case of an outgoing splitting gluon. For each choiceȃ i ∈ {ĝ,g} we have to sum over all splittings a k ∈ {g, q,q}. Thus, the two terms of Figure 2 become the four terms of Figure 4.

Initial-State Collinear Region:
As for the final-state collinear region, the key point is the necessary split g =ĝ +g in the factorization of the collinear limit. In dred, Eq. (32) has to be generalized to a 1 (p 1 ), a 2 ; . . .ā k (p k ) . . .ā n+1 ) . . . a n ) . + + + Figure 5: Illustration of Eq. (54) for an incoming splitting gluon with partons represented as in Figure 4. The sum over all relevant real processes, a k , together with the sum due to the splitȃ (1k) ∈ {ĝ,g} gives rise to four terms.
Again, as far as the collinear limit is concerned, not all gluons in Eq. (53) have to be split. Only for the virtual partonȃ (1k) the split is essential. Note that in dred the crossed splitting functions satisfy the crossing relation Eq. (34) without ∆ terms. Hence, the initial state collinear term for parton 1 can be written as . . . a n ) .
As illustrated in Figure 5, in the sum overȃ (1k) in Eq. (54) it is essential that we treatĝ andg as separate partons, whereas a k ∈ {g, q,q}. For an incoming gluon, the three terms of Figure 3 are generalized in dred to the four terms of Figure 5.

D→4
is equal to the one in the other schemes given in Eq. (37). This shows in particular that it is possible to realize the MS factorization scheme in dred in the same way as in cdr, hv, or the fdh scheme. In order to make this result explicit we close the section by providing the full form of the appropriate collinear counterterm, valid in all rs, where the sum a k runs over a k ∈ {g, q,q} in all rs, whereas the sum ȃ i runs over inȃ i ∈ {g, q,q} in cdr,hv, fdh, and overȃ i ∈ {ĝ,g, q,q} in dred. Also, in dred we define 5 in analogy to Eq. (56).

Applying DRED
In the previous two sections we discussed how the singularities and rs dependence between the various parts of Eq. (2) cancel, and we found that the subtracted partonic cross sections given in Eqs. (37) and (57) are finite and rs independent. This is precisely what we wanted to achieve. However, there is still one conceptual issue to be addressed. The question is whether in the convolution of subtracted partonic cross sections with PDF we need to distinguish betweenĝ andg in DRED. We will show that this is not the case. This will also entail that no PDF for finding an unphysicalg in a hadron will be required.
Once this issue is clarified, we will summarize our results and give transition rules between the various rs separately for all parts of the subtracted finite partonic cross sections.

Parton distribution functions in DRED
The results Eqs. (37) and (57) have been given without taking into account that the subtracted partonic cross sections have to be multiplied by PDF. In dred it might seem natural to distinguish partonic cross sections withĝ org in the initial state and convolute them with different PDF. We will show that this is not required and that we can use Eq. (57) for initial state full gluons g, convoluted with just one PDF even in DRED. In particular, there is no need to introduce unphysical PDF for finding ag in a hadron.
In the strict spirit of dred it is correct to consider independent PDF forĝ andg and write a hadronic cross section as a sum of the form (60) The partonic cross sections dσ(g 1 ) can be constructed in the same way as Eq. (57). They are individually finite and satisfy dσ FS (ȃ 1 ,ȃ 2 ; . . . a n ) = dσ FS (a 1 , a 2 ; . . . a n ).
All the PDF in Eq. (60) would be obtained by performing a fit at one particular factorization scale µ 0 and then using Altarelli-Parisi equations to evolve them to any other scale µ.
The central point is that the unphysical PDF fg /H is of the order ǫ, and hence its contributions to both the hadronic cross section, Eq. (60), and to the evolution of the other PDF are of the order ǫ and thus negligible.
In order to prove this we start by noting that since in other regularization schemes one gluon PDF f g/H is sufficient it is possible to arrange the fit in dred such that fg /H (µ 0 ) = 0 at the starting scale µ 0 . The evolution is given by the Altarelli-Parisi equations, generalized to includeg: where we have suppressed the µ dependence of α s and f a i /H and have used the short-hand notation P i→j ≡ P DRED i→j(ij) (z/ξ). The evolution of fg /H (z) gets contributions from P q→g × f q/H , Pĝ →g × fĝ /H and Pg →g × fg /H . They are O(ǫ) × 1, O(ǫ) × 1 and 1 × O(ǫ), respectively, confirming that fg /H (µ) = O(ǫ) for all scales µ. This in turn implies that the contribution to the evolution of f q/H and fĝ /H due to fg /H is also O(ǫ). The situation is in fact very similar to the contributions due to quarks if there were N F = ǫ flavours. Finally, dσ FS (ȃ 1 ,ȃ 2 ; . . . a n ) as appearing in Eqs. (60) and (61) is finite for all initial statesȃ 1 ,ȃ 2 separately. Hence the contribution of the unphysical PDF fg /H in Eq. (60) is O(ǫ).
This confirms that dred can be used throughout all parts of calculations for hadronic cross sections at one loop, without the need for unphysical PDF. It is correct to consider only one gluon PDF f g/H and the combined cross sections dσ (a 1 , a 2 ; . . . a n ) from Eq. (57) also in dred. In other words, Eq. (1) is correct in all schemes, including dred, if the sums over all parton types include only the full gluon g and quarks q,q (and possibly further, massive partons).

Summary of practical computations in DRED
Let us finally summarize how to do a next-to-leading order calculation in practice in dred or any other scheme. The main point is that only for the ultraviolet renormalization and the collinear counterterm a split of g intoĝ +g is required.
Virtual Corrections: To obtain the virtual corrections we start by computing M (1) RS (a 1 . . . a n ) with a i ∈ {g, q,q}. For the actual calculation of the one-loop diagrams we do not need to split the process into many different parts as in Eq. (46), but can compute directly with g. The structure of the ultraviolet counterterms depends on the rs and the symmetries of the underlying theory. If dred is used in a supersymmetric context, even the counterterms can typically be computed without the split, and usual multiplicative renormalization is sufficient to generate the counterterms. Using hv or cdr in supersymmetric theories leads to the complication of non-multiplicative, supersymmetry-restoring counterterms. In non-supersymmetric theories, determining counterterms in dred requires the split g =ĝ +g.
Once we have the renormalized one-loop matrix element, M RS (a 1 . . . a n ) we obtain dσ virt RS by integration over the phase space. Again, a split of dσ virt RS as in Eq. (61) is not required. The split g =ĝ +g for external gluons becomes useful if we want to express the singularity structure or rs dependence in a simple way as done in the next subsection.
Real Corrections: The real corrections in dred can be obtained in a straightforward way by directly integrating the 4-dimensional tree-level matrix elements containing only 4-dimensional gluons g and (anti)quarks. Likewise, in cdr we have to integrate the D-dimensional tree-level matrix elements. Regarding the real corrections in hv and fdh we remind the reader of a subtlety related to unitarity (see also Ref. [15]). At first sight it might appear that there is no difference in dσ real RS for schemes where the tree-level matrix elements are evaluated in four dimensions, e.g. between dσ real HV and dσ real FDH . However, this is not correct. In order to maintain unitarity, in the singular regions initial and final state partons have to be treated in the same way as partons in a closed loop, i.e. as "internal". Thus in hv and fdh it is not correct to simply integrate the corresponding four-dimensional real tree-level matrix elements over the phase space. In particular, Eqs. (30) and (32) contain O(ǫ) terms in fdh and hv and result in finite differences between dσ real HV and dσ real FDH , even though the tree-level matrix elements agree.
In principle this procedure leads to the ∆ terms in Eq. (35) because in the hv and fdh schemes incoming and outgoing splittings differ. As discussed in Section 2.4 it is possible to redefine the results by ignoring the ∆ terms if the same is done in the collinear counterterms.
Collinear Counterterm: The collinear counterterm is given by Eq. (58). It realizes the MS factorization scheme independent of the rs used for the computation. For dred, we stress that for the term given explicitly on the r.h.s. of Eq. (58) the partons a 1 . . . a n as well as a k are neverĝ org separately, but can always be combined to g. For the virtual partonȃ i , however, it is important to treatĝ andg separately [7]. This requires the use of the splitting functions given in Eqs. (98) and (99). For the hv and fdh schemes, the ∆ terms can be ignored in accordance with the computation of the real corrections.

Translation rules between different schemes
In the following we describe how the results in the various rs are related, making use of the split g =ĝ +g as appropriate. We will focus on the virtual corrections; similar results for the real corrections and the collinear counterterms can be trivially obtained from Eqns.  The only difference between the hv and fdh scheme on the other hand does come from the differences of the γ RS terms, which have been explained in Section 1.3 and quantitatively given in Eq. (26). For a process with #g gluons, #q massless (anti)quarks and #Q massive (anti)quarks the difference is In the second line the influence of the different γ RS for all parton types is made explicit, in the third line the result in brought into a compact form. The difference between fdh and dred is obtained by taking the difference between Eqs. (48) and (23). We can bring it into a simple form by using that we can write the tree-level quantities in fdh in a dred-like form, The difference is then governed by the factors γ DRED (ĝ) − γ FDH (g) and γ DRED (g) − γ FDH (g). As explained in Section 1.3 the first of these vanishes. The second is non-zero, and is present for everyg in the initial or final state, see Eq. (48). Exploiting also that at leading order α s = α e , we obtain Again, in the second line the influence of the different γ RS relevant forg is made explicit. In the third line we used that for processes with at least oneg, i.e. with #g({ȃ}) ≥ 1 we have M  (66) ignore O(ǫ) terms and are given for pure QCD processes. However, they can easily be generalized to other processes, involving e.g. photons or massive partons, simply by using the corresponding explicit expressions for γ RS .

Conclusions
The main result presented in this paper is that dred can be used for the calculation of cross sections at NLO, even for processes with hadrons in the initial state. Problems related to factorization, as reported in the literature [8][9][10], can be avoided by taking into account the appropriate, generalized factorization in dred [7]. We have shown explicitly how to use dred together with an arbitrary factorization scheme. In particular, the conventional PDF [17,18] in the MSfactorization scheme can be used. Also we have given explicit rules on how to transform separately the various parts of the hard partonic cross section, Eq. (2), from dred to other rs. This completes the previously known set of transition rules between cdr, hv and fdh [13][14][15]. It is thus possible to use different rs for different parts of the calculation which might help simplifying the explicit computations. In this context we also reiterate the distinction between fdh and dred. According to the definitions of the rs given in Section 1.2, at one loop fdh is equivalent to the scheme dr used e.g. in Refs. [13][14][15] but differs from dred used e.g. in Refs. [3,11,12].
The salient feature of a consistent use of dred is the split g =ĝ +g. In practice, this split does not significantly complicate calculations. It is needed mainly for the correct treatment of the collinear limit of squared matrix elements. Thus it affects the collinear counterterm and the phase-space integration over the singular, collinear region. The modifications regarding the former are shown in Eqs. (35) and (54). For the phase-space integration, the usual procedures have to be slightly modified. For the method presented in Ref. [30] for example, the collinear singularities in the real corrections are singled out using distributions, enforcing the collinear limit of the real matrix element squared. If this method is to be used together with dred this simply means that the proper collinear limit, Eqs. (51) and (53), has to be taken. For the dipole subtraction method [27] additional dipoles withg are required. These can be obtained making minor modifications of existing dipoles, similar to the corresponding adaptation to fdh [32].
In some cases, the split g =ĝ +g is also required for the ultraviolet counterterms, since e.g. the couplingsĝqq andgqq renormalize differently. This seems to be a disadvantage of dred. On the other hand, one of the advantages of dred is that in supersymmetric theories no supersymmetry-restoring counterterms are required (in many practical cases; for a recent discussion see Ref. [3]). In this case, also couplings withĝ andg renormalize identically and renormalization is actually simpler in dred than e.g. in cdr. This facilitates the use of the DR-scheme for supersymmetric parameters which is used in a wide variety of calculations [33].
Thus, dred is a rs which is well compatible with supersymmetry and which can be realized with minimal modifications compared to cdr and used for an arbitrary cross section at NLO. In the past, following the examples of e.g. Refs. [9,34], many predictions for supersymmetric processes at hadron colliders were calculated using cdr in spite of the required supersymmetry-restoring counterterms. In the future, similar calculations can alternatively be carried out using dred, which can lead to simplifications with the present, better understanding of dred.
the spirit of dred, we can further disentangle the splitting operator and write The splitting functions P < RS g * →gg can be read off of Eqs. (70) -(74) simply by dropping the polarization sum. Performing similar calculations for all other possible splits we find the following results: The results for hv are always identical to the ones for cdr, The results particular to dred are given by The remaining splitting functions can be obtained by P < RS i * →kl (z) = P < RS i * →lk (1 − z). The splitting functions appropriate for the split of an incoming parton P < RS i→k * l (z) can be obtained through the crossing relation Eq. (33) with We note that in cdr and in dred there is no difference between the splitting functions for incoming and outgoing partons, i.e. Eq. (34) holds with all ∆ terms equal to zero. Thus, Eqs. (81) and (82) and Eqs. (83) and (84) are not independent. For the hv and fdh scheme the ∆ terms do not vanish. In the fdh scheme, for example, the different coefficient of the three terms on the r.h.s. of Eq. (76) lead to ∆ FDH g→gg = 0. In the hv scheme, the different ω HV (g) = ω CDR (g), together with Eq. (80), are the origin of the non-vanishing ∆ HV terms.
The splitting functions P < RS i * →kl are defined only for z < 1. We define the full splitting functions through the relation where we made use of the standard +prescription. The factors γ RS (a i ) and thus P RS i→kl are determined by requiring that the momentum sum rules are satisfied in all the schemes cdr, hv, fdh, i.e. also taking into account terms of higher-order in ǫ if appropriate. Eq. (87) determines γ RS * (g) and Eq. (88) determines γ RS * (q). For dred, the sum rules given in Eqs. (87) and (88) have to be generalized in an obvious way, since we also have to take into accountg: As before, Eqs. (89) and (90) determine γ DRED (ĝ) and γ DRED (q) respectively, while Eq. (91) determines γ DRED (g). The results are and γ HV (a i ) = γ CDR (a i ). Expanding these results and taking into account all terms to O(ǫ) we obtain the results given in Eq. (26). We note that these results are also consistent with the quark-number conservation sum rule as well as with Eqs. (31) and (52). In fdh and dred they also satisfy the supersymmetric relation if we set N c = C F = 2 T F N F . Finally we mention that the dred splitting function used in Eqs. (54) and (55) are defined as Using the explicit results above we find for the splitting functions used in Eq. (53).

B Examples
The process gg → qq with massless quarks has been computed long ago at one loop [35] in cdr and was one of the processes used to determine the relations between the hv and fdh scheme [13]. The one-loop matrix elements M (1) RS * (g, g; q,q) were found to be related as given in Eq. (23). This and the related process with massive quarks was also at the centre of claims regarding problems with factorization in dred [8][9][10]. The factorization issue related to the real corrections for these processes has been solved in Ref. [7]. Here we focus on some aspects related to issues with dred, starting with the virtual corrections.
The calculation of the one-loop diagrams is straightforward and we stress once more that there is no need to disentangle g intoĝ +g in the explicit calculation this allows us to perform the calculation without specifying a renormalization scheme for the unphysical gluons. The counterterm M (1,ct) DRED (ĝ,ĝ; q,q) is simply given by 2δZ DRED g M (0) DRED (ĝ,ĝ; q,q). As illustrated in Figure 6, the counterterm M in dred is related to the other schemes as given in Eq. (66). The calculation of the real matrix elements is trivial. They are simply the four-dimensional results, Eq. (44), and the corresponding real cross section can be obtained in dred by integrating these matrix elements over the phase space. The only remaining and main issue is the factorization of the initial state collinear singularities. According to our discussion, Eq. (58), it is clear that we will have to add dσ coll DRED,MS (g, g; q,q) = (109) α s 2π c Γ ǫ dz P DRED g→ĝg (z) − P MS ǫ g→gg (z) dσ (0) DRED (ĝ(z p 1 ), g(p 2 ); q,q) DRED (q(z p 1 ), g(p 2 ); g, q) + P DRED g→qq (z) − P MS ǫ g→qq (z) dσ (0) DRED (q(z p 1 ), g(p 2 );q, g) + P DRED g→gg (z) dσ (0) DRED (g(z p 1 ), g(p 2 ); q,q) + {1 ↔ 2} .
The conversion to the MS scheme requires the terms P MS ǫ g→gg ≡ P CDR g→gg − P CDR g→gg D→4 = ǫ It is the last term on the r.h.s. of Eq. (109) that is non-standard and deserves special mention since it resolves the issue regarding the seemingly non-factorizing corrections in dred.
Let us add a comment on why the factorization problem of Refs. [8][9][10] was found in the context of the process with massive quarks, discussed below, rather than the one with massless quarks. The reason is the fact that in the present, massless case, the dred cross sections for theg andĝ initial states happen to be equal, dσ DRED (ĝ(p 1 ), g(p 2 ); q,q) = dσ (0) DRED (g(p 1 ), g(p 2 ); q,q).
Hence, in Eq. (109) one can combine the terms in the first and the last line to P DRED g→gg (z) dσ (0) DRED (g(zp 1 ), g(p 2 ); q,q), and the process is seen to factorize even without distinguishing betweenĝ andg.

B.2 gg → QQ
Problems to reconcile factorization with dred were first mentioned in the context of this process with massive final state quarks [8]. As explained in Ref. [7] and the present paper, the factorization problem disappears ifĝ andg are treated as separate partons in formulas such as Eqs. (53) and (54) or in the last line of Eq. (109). The reason why the apparent problem has been found only in the massive process gg → QQ is not related to quark masses but to Eq. (112), which happens to hold in the massless case.
With this in mind, the massive process can be treated in the same way as the massless one. The only additional complication in the case of massive quarks is to consider the rs dependence of the mass renormalization Z RS m and external wave-function renormalization Z RS Q for massive quark lines. The rs dependence of these renormalization factors has been considered before (see e.g. Ref. [36]) and, using for example the pole scheme to define the mass of the heavy quark m, can be summarized as follows: and Z RS Q = Z RS m . As mentioned in the main text, γ(Q) = C F is rs independent [14]. We have verified by explicit calculation that using the rs dependent coupling renormalization and Eqs. (114) and (115) for the mass counterterms and wave-function renormalization, the rs dependence of the virtual corrections take the form as given in Eq. (46). To use dred throughout in the calculation of this process we simply have to use the correct rs dependent collinear counterterm as given in Eq. (109) and fold the hard partonic cross sections with the standard PDF in the MS factorization scheme.

B.3 gg → h
In this example we consider the production of a Higgs h through gluon fusion in a hadronic collision. While this process is relatively simple at one-loop it is complicated enough to illustrate all main points discussed in the main text. The interaction of the Higgs with gluons is given by the Lagrangian where the coupling g h has mass dimension −1. In dred we have to distinguish between the coupling forĝĝh, denoted by g h and the coupling forggh, denoted byg h . At tree level the two couplings are the same, but they differ at higher orders.
The only process that contributes at tree level is g(p 1 ) g(p 2 ) → h. The matrix elements are given by M (0) RS (g,g; h) = g 2 h ω RS (g) DRED (g,ĝ; h) = 0. For the calculation of the O(α s ) corrections to g g → h the distinction between g h andg h at tree level will be relevant for the renormalization. Importantly, however, for the non-trivial part of the explicit calculation of the virtual and real corrections we can set g h =g h and we do not have to distinguish betweenĝ and g in loop diagrams.
First we discuss the virtual corrections and how their rs dependences arise. The explicit calculation of the two non-vanishing one-loop diagrams in Feynman gauge results in the following unrenormalized one-loop matrix elements: with ∆ virt CDR = ∆ virt HV = 0, ∆ virt FDH = N c /3 and ∆ virt DRED = 2N F T F /3. Thus, after adding the counterterms, these expressions are in agreement with the general formula Eq. (46), with the finite rs independent part given by M To summarize the calculation in dred: Both the one-loop diagrams and the real corrections and phase space integrals can be computed in a straightforward way, using only full gluons g. The split g =ĝ +g has to be used in the evaluation of the collinear counterterms dσ coll and in the computation of the UV counterterms. In the end, the cross sectionsσ(gg),σ(gq),σ(qq), as well asσ(qg),σ(gq),σ(qg),σ(qq) are rs independent. The hadronic cross section is obtained by convoluting them with the standard parton distribution functions obtained in the MS factorization scheme.