###### Abstract

I give an efficient algorithm for the reduction of multi-leg one-loop integrals of rank one. The method combines the basic ideas of the spinor algebra approach with the dual vector approach and is applicable to box integrals or higher point integrals, if at least one external leg is massless. This method does not introduce Gram determinants in the denominator. It completes an algorithm recently given by R. Pittau.

NIKHEF-98-033

Reduction of multi-leg loop integrals

[.3cm]

[1cm]

NIKHEF, P.O. Box 41882, NL - 1009 DB Amsterdam, The Netherlands

PACS : 12.38.Bx, 11.15.Bt

Keywords : Perturbative calculations, one-loop integrals

email address :

## 1 Introduction

Next-to-leading order calculations have become nowadays the standard
for perturbative calculations in high energy physics.
One of the major obstacles is the tedious reduction of tensor
loop integrals (e.g. integrals where the loop momentum appears in the
numerator) to standard scalar integrals. General algorithm have been
known for a long time (like the Passarino-Veltman algorithm [1], [2] or
the Feynman parameter space technique [3]). Although they solve the
problem in principle, the resulting expressions are usually quite long
and involve the appearence of unphysical singularities in the form of
Gram determinants in the denominator. These singularities usually drop
out only after combination of many different structured terms.

A variety of alternative approaches exists:
J.M. Campbell, E.W.N. Glover and D.J. Miller [8] introduced
scalar integral functions in higher dimensions and absorbed the Gram
determinants in a linear combination of these.
D.B. Melrose [4] and independently W.L. van Neerven and J.A.M. Vermaseren [5]
have used the fact, that in four dimensions there are maximally four
linearly independent vectors, to derive a relation between a scalar
pentagon integral and scalar box integrals.
The derivation of van Neerven and Vermaseren was based
on dual vectors. The dual vector approach was further used by
G.J. van Oldenborgh and J.A.M. Vermaseren [6] and A. Signer [9].
Recently R. Pittau [7] has given
an algorithm, which makes use of the spinor algebra, to reduce tensor
integrals to scalar and rank one integrals.
In this paper I give a formula which allows the reduction
of the remaining rank one integrals to scalar integrals. It combines the
basic ideas
of the spinor
algebra approach and the dual vector approach. This formula completes
the algorithm given by R. Pittau.

This paper is organized as follows: The following section introduces the
necesarry notation. In section 3 I derive the basic
formula. The complete algorithm is given in section 4. An example is worked
out in section 5. Section 6 contains the conclusions.

## 2 Notation

For an introduction to spinor products I refer to [11]. A typical one-loop integral has the form

(1) |

where the and the are external massless momenta and the denote the internal propagators. The loop momentum is allowed to appear more than once in each sandwich . This form is general: After the decay of (possible) intermediate heavy particles all external particles can be taken to be gluons, photons or light fermions. The polarization vectors of gluons and photons are expressed in the above form using the spinor helicity method [10]:

(2) |

where is an arbitrary null reference momentum. It will be usefull to introduce a slight generalization of spinor products: Consider first elements of the form . They form an algebra, which I will denote by . It is convenient to think of the space of ket-spinors generated by the spinors obtained from all external momenta as an left -module . The dual space of bra-spinors is denoted by and is a right -module. The spaces and decompose into a direct sum of subspaces of positive and negative helicity:

(3) |

Non-vanishing spinor products are only obtained between elements of and or between elements of and . They are denoted by

(4) |

where and . Multiplication of a spinor with an element of changes the helicity, e.g. if and then . It is convenient to define a “degree” of these objects. Assume that the spinor corresponds to the external momentum and that

(5) |

Then we define and

(6) |

A similar definition applies to bra-spinors. Changing the helicity or flipping a spinor from bra to ket (or vice versa) does not change the degree. It follows that for arbitrary spinors , , and

(7) |

For these formulae coincide with the usual ones for spinor products.

I take the sign of the antisymmetric tensor as
.
It is useful to introduce the following short-hand notation:

## 3 Derivation of the basic formula

Combining the basic ideas of reduction methods based on dual vectors and on the spinor algebra, I derive
a formula, which allows an efficient reduction of higher point integrals.

The kinematics are shown in fig.1.

The are supposed to be external momenta whereas is taken as a loop momentum. I assume that the are linearly independent. This means that the method applies to pentagon or higher point integrals. A similar formula for box integrals is given later on. Let us introduce the notation

(8) |

It is convenient to set and , so that one basically always deals with five external momenta , even for hexagon or higher point integrals. We have

(9) |

It is convenient to work in a scheme where the Dirac matrices are in four dimensions (like the four-dimensional helicity scheme [12]). If the integral is finite we can take all vectors in four dimensions. The case of a divergent integral needs some special care: I follow the approach by G. Mahlon [13] and split the loop momentum into a four-dimensional part and a -dimensional part

(10) |

It is further assumed that a four-dimensional vector is orthogonal to a -dimensional one. I start from the dual vectors written as

(11) |

and the Schouten identity

(12) |

This equation is related to four dimensions, therefore appears on the left hand side. Since and are four-dimensional objects, the scalar products of with or vanish and I dropped therefore the superscripts. Now

(13) | |||||

where denotes a trace of Dirac matrices with the insertion of a projection operator . Since

(14) |

and each terms has one scalar product symmetric in two indices, these terms drop out when contracted with the antisymmetric tensor. Working out the r.h.s of eq. (12) yields

(15) | |||||

Writing , one sees that the first two terms cancel propagators, whereas the last term yields a contribution of

(16) | |||||

The r.h.s. of eq. (16) has been arranged such that no constant term survives. One finally obtains

(17) | |||||

This equation holds if the Dirac matrices and the external momenta are in four dimensions. The loop momentum may be in dimensions, allowing the application also to divergent integrals. A careful inspection of the basic equation (12) shows that the term on the left-hand side has to be in four dimensions, whereas the manipulations on the right-hand side involved only scalarproducts which may be continued into dimensions as long as the external momenta stay in four dimensions. The last term of eq. (17) is a correction term, which takes care of divergent integrals. If the integral is finite, may be taken to be four dimensional and the last term vanishes. It will turn out that the correction term is irrelevant in the application of eq. (17) to pentagon or higher point integrals. The equation above may be rewritten as :

with , where the internal propagators and masses are denoted by and , respectively. depends only on the pinched integral under consideration and is given by

(19) |

Starting from the pentagon integral the internal propagator is first pinched. This gives a box integral and the value of is calculated from that box integral according to eq. (19) and according to the kinematics shown in fig. 2.