MPP200361
hepph/0308246
Separation of soft and collinear singularities
[.5em] from oneloop point integrals
Stefan Dittmaier
[.5cm] MaxPlanckInstitut für Physik (WernerHeisenbergInstitut)
Föhringer Ring 6, D80805 München, Germany
Abstract:
The soft and collinear singularities of general scalar and tensor oneloop point integrals are worked out explicitly. As a result a simple explicit formula is given that expresses the singular part in terms of 3point integrals. Apart from predicting the singularities, this result can be used to transfer singular oneloop integrals from one regularization scheme to another or to subtract soft and collinear singularities from oneloop Feynman diagrams directly in momentum space.
August 2003
1 Introduction
Many interesting highenergy processes at future colliders, such as the LHC and an linear collider, lead to final states with more than two particles, rendering precise predictions much more complicated than for particle reactions. The necessary calculation of radiative corrections bears additional complications and requires a further development of calculational techniques, as recently reviewed in Ref. [1].
Full oneloop calculations for processes with more than two finalstate particles require, for instance, the evaluation of scalar and tensor point integrals. For several approaches have been proposed in the literature [2, 3, 4, 5]. In this context the two main complications concern a numerically stable evaluation of tensor integrals on the one hand and a proper separation of infrared (soft and collinear) singularities on the other. In Ref. [5] the direct reduction of scalar 5point to 4point integrals, as proposed in Ref. [2], has been extended to tensor integrals. The nice feature in this approach is the avoidance of leading inverse Gram determinants, which necessarily appear in the wellknown Passarino–Veltman reduction [6] and lead to numerical instabilities at the phasespace boundary.
In this paper we focus on the treatment of infrared (IR) or socalled “mass singularities” at one loop. According to Kinoshita [7], such mass singularities arise from two configurations that both lead to logarithmic singularities. Collinear singularities appear if a massless external particle splits into two massless internal particles of a loop diagram, and soft singularities arise if two external (onshell) particles exchange a massless particle. If the involved particles are not precisely massless, the corresponding singularities show up as large logarithms involving the small masses, like where is the electron mass and a large scale. If the masses involved in the singular configurations are exactly zero, the singularities appear as regularized divergences, such as poles where in dimensional regularization. In both cases an analytical control over such terms is highly desirable, either in order to perform cancellations at the analytical level or to carry out resummations. In the following we show how to extract mass singularities from general tensor point integrals and finally give an explicit result for the singular parts in terms of related 3point integrals. Moreover, we describe several ways how this result can be exploited and give some examples illustrating the easy use of our final result. The method to derive the general result of this paper was already applied in Ref. [8] to specific 5point integrals, which appear in the nexttoleading order QCD corrections to the processes .
Mass singularities of the virtual and real radiative corrections are intrinsically connected in field theory. In fact, the famous Kinoshita–Lee–Nauenberg (KLN) theorem [7, 9] states that the singularities completely cancel in sufficiently inclusive quantities. The result of this paper allows for a simple analytical handling of the mass singularities of the virtual oneloop corrections. The singular structure of the real corrections induced by oneparton emission (which corresponds to the oneloop level) can be easily read from socalled subtraction formalisms [10] which are designed for separating these singularities. Using these results and the KLN theorem, the oneloop singular structure of complete QCD and SUSYQCD amplitudes has been derived in a closed form in Ref. [11]. Note that the attitude of this paper is rather different, since the oneloop integrals are inspected themselves, eventually leading to a prescription for extracting the singularities diagram by diagram.
The paper is organized as follows: In Section 2 we set our conventions and describe the situations in which mass singularities appear at one loop. Section 3 contains the actual separation of the mass singularities and our final result at the end of the section. In Section 4 we describe various ways to make use of the result and present some explicit applications. Section 5 contains a short summary. In the appendices we present a proof of an auxiliary identity used in Section 3 and provide a list of masssingular scalar 3point integrals that frequently appear in applications.
2 Oneloop point integrals and mass singularities
We consider the general oneloop point integrals
(2.1) 
with the denominator factors
(2.2) 
A diagrammatic illustration is shown in Figure 1.
Note that we do not set the momentum to zero, as it is often done by convention, but keep this variable in order to facilitate a generic treatment of related integrals. In particular, with this convention all are invariant under exchange of any two propagator denominators , or equivalently of two pairs of . We follow the usual convention to denote point integrals with as
(2.3) 
Whenever the index on momenta or masses exceeds the range , it is understood as modulo , i.e. and are equivalent, etc. For later use, we introduce the variables
(2.4) 
We are interested in “mass” singularities that appear if combinations of external squared momenta, , and internal masses become small, but do not consider singular configurations that are related to specific or isolated points in phase space, such as thresholds or forward scattering. We can, thus, distinguish two sets of parameters: one set that comprises all quantities , with fixed nonzero values, and another set of those quantities that are considered to be small, i.e. which formally tend to zero. In order to simplify the notation, we define an operation, indicated by a caret over a quantity , which implies that all small quantities are set to zero in .
As shown by Kinoshita [7], “mass” (or “IR”) singularities can appear in oneloop diagrams in the following two situations:

An external line with a lightlike momentum (e.g. a massless external onshell particle) is attached to two massless propagators, i.e. there is an with
(2.5) The singularity is logarithmic and originates from integration momenta with
(2.6) where is an arbitrary real variable. Since the momentum on line is then collinear to the external momentum , such singularities are called collinear singularities.

A massless particle is exchanged between two onshell particles, i.e. there is an with
(2.7) The singularity is also logarithmic and originates from integration momenta with
(2.8) i.e. the momentum transfer of the th propagator tends to zero. Therefore, these singularities are called soft singularities.
In the following we focus on integrals with and express the singular structure of oneloop point integrals in terms of 3point integrals which are easily calculated with standard techniques, as for instance described in Refs. [4, 6, 12]. Of course, the same is true for the cases , i.e. for tadpole and selfenergy integrals, which are even simpler.
3 Separation of mass singularities
In this section, we first consider the asymptotic behaviour of the denominator of the integrand in Eq. (2.1) in the individual collinear and soft limits. Based on these partial results we derive a simple expression that resembles the whole integrand in all singular regions. Applying the loop integration to this expression directly leads to our main result which expresses the singular structure of an arbitrary point integral (2.1) with in terms of 3point integrals.
3.1 Asymptotic behaviour in collinear regions
For an integration momentum in the collinear domain, as specified in Eq. (2.6), the two propagator denominators and tend to zero (), and the behave as
(3.1) 
The collinear limit is mass singular if the external momentum squared and the two masses , are small. In this limit the two propagator denominators , tend to zero, but the others remain finite (for ):
(3.2) 
Note that the variable is the only integration variable that is not fixed by the collinear limit. The product of all regular propagators can be decomposed into a sum over these propagators via taking the partial fraction,
(3.3) 
The coefficients are functions of the variables and alone and thus fixed by the external kinematics. The explicit result for reads
(3.4) 
as proven in App. A. The collinear singularity arises from the region where the propagator denominators and both become small with no preference to any of the two. In order to reveal this equivalence, we rewrite using
(3.5) 
and
(3.6) 
Inserting these relations, we get
(3.7) 
in which the equivalence of the th and th propagators is evident.
Multiplying Eq. (3.3) with yields a relation, which is valid in the collinear limit, between the product of all propagators and a linear combination of products involving only three propagators,
(3.8) 
Thus, the collinear singularity associated with the propagators , in an point integral is expressed in terms of a sum of 3point integrals involving the th, the th, and any other line of the diagram.
3.2 Asymptotic behaviour in soft regions
The soft singularity connected with a massless propagator arises from momenta , where . The other denominators tend to a regular limit in this case,
(3.9) 
and the product of all propagators behaves like
(3.10) 
We still have to consider the possibility that one or both ends of the soft line is part of a collinear configuration treated above. If this is the case, the soft limit can be reached as limiting case of a collinear limit. Assuming again as the soft line, the two “degenerate” collinear limits are and . Both lead to , but in the former case lines and correspond to a collinear configuration, in the latter lines and . It is quite easy to see that the soft asymptotic behaviour (3.10) is already correctly included in the collinear behaviour (3.8) in either case, because
(3.11) 
3.3 Final result
From the above considerations it is clear that we obtain an expression for the asymptotic behaviour of the product of all propagator denominators in all collinear and soft regions upon adding the asymptotic expressions of all collinear and soft regions, which can be read from Eqs. (3.8) and (3.10), and carefully avoiding doublecounting of soft asymptotic terms. To this end, we define
(3.12) 
With this definition we can write down the asymptotic behaviour valid for all collinear and soft regions as
(3.13) 
Obviously each soft part is included by the terms exactly once, and the collinear contributions from and are omitted if they are already covered by the soft terms . Integrating Eq. (3.13) over on the l.h.s. yields the scalar integral and on the r.h.s. a linear combination of scalar 3point integrals , which has exactly the same structure of collinear and soft singularities as . An analogous relation is obtained for tensor integrals if the additional factor is included in the integration, since this factor does not lead to additional singularities. Note that the asymptotic relation (3.13), which describes the leading behaviour, is in fact sufficient to extract all mass singularities from the oneloop integral, since the degree of the singularities is logarithmic. In summary the complete masssingular part of a general oneloop tensor point function reads
(3.14)  
The sum over and runs over all subdiagrams whose scalar integral develops a collinear or soft singularity. A tensor integral is, however, not necessarily mass singular if the related scalar integral develops such a singularity. For such tensor integrals the regular 3point integrals on the r.h.s. of Eq. (3.14) could be dropped. We note that for , artificial ultraviolet singularities appear in the tensor 3point integrals on the r.h.s. of Eq. (3.14). These can be regularized in dimensional regularization and easily separated from the mass singularities (see, e.g., the appendix of Ref. [5]).
In order to render the above result more useful, we present a list of masssingular functions in App. B. This paper, thus, contains the needed ingredients to predict the mass singularities of most scalar point functions occurring in practice. To obtain the singularities of tensor integrals, only the 3point tensor integrals have to be derived, which can be easily inferred with the wellknown Passarino–Veltman algorithm [6] (see also Refs. [4, 5]).
4 Discussion and applications
4.1 Possible applications of the final result
The relation (3.14) can be exploited in various directions:

As pointed out in the previous section, the mass singularities of arbitrary point integrals can be easily derived from 3point functions. This statement is true in any regularization scheme, i.e. for any point integral all smallmass logarithms and/or poles in in dimensional regularization can be easily inferred.

The singular integral can be used to translate any IRdivergent point integral from one regularization scheme to another. To this end, the regularizationschemeindependent difference
(4.1) is considered. For the translation from one scheme to the other only the singular part , and thus the relevant 3point integrals, have to be known in the two regularization schemes.

The trick described in the last item has been used in Ref. [8] to translate dimensional 5point integrals into a mass regularization with , in order to make use of the direct reduction [2, 5] of 5point to 4point integrals, which works in four spacetime dimensions. In this context it was observed that the formal relation between 5point and 4point integrals, which was derived in four dimensions, is also valid in dimensions up to terms, since the extraction of the singularities works in any regularization scheme with the same linear combination of 3point integrals. From the results of this paper we conclude that this statement generalizes to arbitrary point integrals, i.e. the reduction of an point integral to 4point integrals works in dimensions in precisely the same way as in four dimensions [up to terms of ], without the appearance of extra terms.

Since Eq. (3.14) has been derived in momentum space, it could also be used as local counterterm in the momentumspace integral, i.e. taking the difference in Eq. (4.1) before the integration over the loop momentum , the integral becomes IR (soft and collinear) finite and can be evaluated without IR regulator. The loop integration of the subtracted part is extremely simple, because it involves only 3point functions, and can be added again after the integration of the difference. This procedure could be very useful in purely numerical approaches to loop integrals, as e.g. described in Ref. [13].
4.2 Sudakov limit of the oneloop box integral
As a simple application, we consider the box integral in the socalled Sudakov limit, where all external squared momenta and internal masses are considered to be much smaller than the two Mandelstam variables
(4.2) 
In this limit, there are four regions for soft singularities, and Eq. (3.14) yields
(4.3)  
For the scalar integral this is in agreement with Eq. (57) of Ref. [14], where the remaining finite contribution was derived as well. In Ref. [14] also tensor integrals up to rank 4 have been considered; the singularities predicted by Eq. (4.3) have been checked against these results. The soft singularities in the functions on the r.h.s. of Eq. (4.3) arise from integration momenta , , , , respectively. The singular terms in the tensor coefficients of the functions can be related to the respective scalar functions rather easily. For instance, shifting the integration momentum in the first function on the r.h.s. of Eq. (4.3), powercounting in the shifted momentum shows that terms with in the numerator are not mass singular. Thus, in the first tensor function only covariants built of the momentum alone receive singular coefficients that are all proportional to the respective function. The same reasoning applies to the other three functions. In summary, the masssingular terms of the tensor 4point functions in the Sudakov limit are given by
where the sign indicates that regular terms have been dropped, i.e. Eq. (3.14) is not applied literally.
4.3 Singular structure of some 5point integrals
(i) A 5point integral for the process
In Ref. [8] the three different types of IRsingular 5point integrals that appear in the nexttoleading order (NLO) QCD correction to have been calculated in dimensional regularization. One of the corresponding pentagon diagrams is shown on the l.h.s. of Figure 2.