# Psi-Pr-96-19 A simple method for multi-leg loop calculations

###### Abstract

In this paper, I present a technique to simplify the tensorial reduction of one-loop integrals with arbitrary internal masses, but at least two massless external legs. By applying the method to rank tensor integrals, one ends up with at most rank 1 tensor functions with the initial number of denominators, plus tensor integrals with less denominators and rank . To illustrate the algorithm, I explicitly compute diagrams contributing to processes of physical interest and show how the usual numerical instabilities due to the appearance of Gram determinants can be controlled.

## 1 Introduction

Present and future experiments in high energy physics require an increasing understanding of multi-particle final states. Idealized calculations, in which heavy particles are treated as being on-shell, are often not accurate enough to match the experimental needs. In fact, the produced heavy particles always decay, giving a multi-body final state as an observable signal.

At first, investigations of multi-particle processes can be
performed at the tree level, but, in a second stage, also radiative
corrections have to be taken into account. At LEP2, for instance, a complete
electroweak one-loop calculation to off-shell four-fermion production
beyond the leading log approximation is still missing, and this reflects on
the accuracy in the mass measurement. Besides, next-to-leading
order QCD calculations of multi-jet final states are becoming unavoidable
to control the background in the search of new signals.
Therefore, it is clear that multi-leg loop calculations
are playing a role of increasing importance and that any progress
achieved in this field is welcome.

One of the major problems in loop calculations is the reduction (in dimensions) of rank tensor -points integrals as follows

(1) |

The standard reduction method [1] consists in decomposing the tensorial structure in terms of momenta and metric tensor. For example, a rank two 5-point one-loop integral can be expressed in terms of linearly independent scalar form factors

(2) | |||||

Analogously, the rank three 5-point function gives terms.

The freedom in choosing the basis for the tensorial decomposition can be used to define suitable combinations of such that [2]. In any case, in order to express the coefficients of the decomposition in terms of scalar loop functions, one has to solve algebraical systems obtained by multiplying both sides of eq. (1) by (or linear combinations of them) and the metric tensor. This causes the appearance of Gram determinants

into the denominator, that can give rise to numerical instabilities. In fact, they vanish in collinear regions of the phase space where the cross section is well defined, so that large numerical cancellations are expected in the numerator in order to keep everything finite.

Different approaches also exist in which the form factors are identified with integrals in a different number of dimensions [3] or directly formulated in the Feynman parameter space [4]. In both cases, Gram determinants still appear at some stage in the reduction.

Recently, a new method has been proposed by Campbell, Glover
and Miller [5], in which the coefficients of the reduction are
built up by combining the scalar integrals in
groupings that are well behaved in the limit . This
somehow solves the problem of the Gram determinants, but to actually
construct the well behaved groupings can be cumbersome in general
- non QCD like - cases, especially when the rank of the appearing tensor
integrals is large. Furthermore, the number of the basic functions
increases, and additional scalar integrals must be evaluated
in or higher dimensions.

In this paper, I propose a method to simplify Feynman diagrams,
in which no evaluation of new functions is required, apart from the
usual scalar integrals (new functions may appear when ,
but their computation is trivial). Although, in general, the appearance
in the denominator of quantities that can vanish in some corner of the
phase space cannot be completely avoided, a better control
of those singularities is possible with respect to the traditional
methods of refs. [1, 2].
The procedure is somehow complementary to the technique of
Campbell, Glover and Miller, and can be used to simplify the problem
before applying their method.

A trivial example serves to illustrate the basic idea. Consider the tensorial reduction of the following quantity

(3) |

One can certainly use eq. (2), but the equation

immediately gives the desired answer in terms of just two scalar functions

(4) |

Therefore, by using in a diagram the algebra of the matrices to reconstruct denominators, one gets a simpler tensorial structure.

This technique of reconstructing denominators has been already
used in the literature for the evaluation of specific integrals
[6, 7].
In this paper, I show that a reduction based on this procedure can
always be worked out for
generic one-loop integrals with arbitrary internal masses, but at least
two massless external legs.

In the next section, I present the method, with the help two 4-dimensional examples. In section 3 and appendix A, I describe the extension to dimensions and give explicit results for cases of physical interest.

## 2 The Method

After the decay of the intermediate heavy particles, a generic -point loop diagram can be written as follows (see fig. 1-a)

(5) |

where are the external momenta of the diagram, the momenta in the loop functions - as defined in eq. (1) - and traces over matrices, which contain ’s. In most practical cases, the external momenta are massless, in fact they can only be photons, gluons or light fermions. The appearance of traces in eq. (5) is a general feature. In fact, if the external particles of the diagram are fermions, traces naturally appear, while the polarization vectors of photons and gluons can always be written in terms of spinor strings using the following representation [8]

(6) |

In standard calculations, each term in eq. (5) is rewritten as follows

(7) |

and the tensors

(8) |

decomposed using a basis of independent four-vectors, as described in the introduction. Instead, I propose to use -algebra and spinor manipulations to move all ’s close to each other, and subsequently apply in order to rewrite each trace as

(9) |

where contain at most one , do not contain and the coefficients and are functions of the scalar products between and the external momenta of the diagram . Since a single appears in the right side of eq. (9), just a rank 1 tensor decomposition is formally left. Anyway, there is no gain in applying the described procedure if the coefficients and are generic. In fact, if scalar products of the kind appear and is not equal to one of the momenta in the loop functions, the same kind of tensorial decomposition necessary for computing eq. (8) is hidden in eq. (9). On the other hand, if only powers of and appear, it is always possible to express at least one of them in terms of denominators. In this case, starting from -point rank tensor integrals, the algorithm gives at most rank 1 -point functions, plus -point rank tensor integrals with and .

Naturally the question arises whether
the coefficients and can
be made to have such a nice feature. To answer that question, while keeping
everything transparent, I will assume, for the rest of this section,
four-dimensional space time, namely ultraviolet finite integrals,
with four-dimensional regularization of infrared and collinear
divergences. The extension to dimensions will
be considered in the next section. In the following,
I introduce all notations and necessary formulae.

Basic objects are strings of matrices between massless spinors. I will use the notation

(10) |

The Weyl spinors are defined as

(11) |

where denotes the spinor momentum and are chirality projectors obeying

(12) |

Note that, using , one can always rewrite any spinorial string with spinors on the left and spinors on the right, so that the notation in eq. (10) is completely general.

The completeness relations for massless spinors read

(13) |

from which one can easily get similar relations for generic momenta and [9]

(14) |

where is an arbitrary four-vector and

(15) |

The following identities hold for spinor strings [11]

(16) |

being the string obtained from by reversing the order of the matrices.

The following Kahane-Chisholm identities [12] are also very useful

(17) |

where and stand for strings with an odd and even number of matrices, respectively. Finally, one can prove [11]

(18) |

Armed with the previous formulae, I will show that, in four dimensions, the coefficients and in eq. (9) can be made to depend only on powers of and on the condition that at least two momenta in the set (say and ) are massless, namely coincide with some of the external massless momenta of the diagram, as in fig. 1-b and 1-c.

In fact, the following identity

(19) |

Figure 1: Generic one-loop multi-leg process (a). The blob stands for the sum of all possible one-loop diagrams. Two classes of diagrams are also shown, for which the reduction method is applicable (b and c).

allows to “extract” from the traces

(20) | |||||

where represent generic strings of matrices. In the first two terms, the dependence is already extracted out. The other two traces can be broken with the help of eq. (13)

(21) | |||||

By iteratively applying the above procedure together with the identities (derived again by means of eq. (13) and eq. (14))

(22) | |||||

it is easy to see that the resulting
coefficients and only depend on
powers of , , and
.
The vector is arbitrary (and not necessarily massless), but
different from and . By identifying it with a third
momentum in the set
^{1}^{1}1This identification is possible only for -point loop
functions with . However, in the next section, I will show that
the method still works when .,
and have the suitable structure to get
the desired simplifications in the tensorial decomposition.

In the computation of physical processes, the one-loop diagrams with highest always lie in the class of the corrections connecting two external legs (see again fig. 1-b and 1-c). That guarantees at least two massless momenta in the loop functions and, therefore, that the described reduction method is applicable at least for the most complicated diagrams appearing in the calculation.

Finally, one should notice that structures that can vanish appear in the denominator of eqs. (20) and (22). However, as we will see, often in practical calculations there is no need to extract the dependence as described, since the reconstruction of denominators naturally takes place without introducing too many factors in the denominators.

In the following two examples, I show the method at work in four dimensions.

Example 1

I shall compute, in the renormalizable gauge, the 6-point diagram of fig. 2, relevant for studying electroweak corrections at LEP2.

The spinorial structure in the numerator of the diagram reads

(23) | |||||

from which the scattering amplitude can be obtained after division by the denominators and integration over