IBP and Reduction to Master Integrals
The next method in our list is based on integration by parts (IBP) [15] within dimensional regularization, i.e. property (2.41). The idea is to write down various equations (2.41) for integrals of derivatives with respect to loop momenta and use this set
- PDF / 417,712 Bytes
- 29 Pages / 439.37 x 666.142 pts Page_size
- 100 Downloads / 257 Views
IBP and Reduction to Master Integrals
The next method1 in our list is based on integration by parts2 (IBP) [18] within dimensional regularization, i.e. property (2.41). The idea is to write down various equations (2.41) for integrals of derivatives with respect to loop momenta and use this set of relations between Feynman integrals in order to solve the reduction problem, i.e. to find out how a general Feynman integral of the given class can be expressed linearly in terms of some basic (master) integrals. In contrast to the evaluation of the master integrals, which is performed, at a sufficiently high level of complexity, in a Laurent expansion in ε, the reduction problem is usually solved at general d, and the expansion in ε does not provide simplifications here. The reduction to master integrals can be performed in the two different ways: one can stop the reduction when one arrives at integrals which can be expressed in terms of gamma functions at general d or to try to reduce any given integral to true master integrals. The latter variant is the reduction problem in the ultimate mathematical sense, i.e. the reduction to irreducible integrals which cannot be reduced further. For many years IBP relations were solved by hand. There is a lot of example of such solutions in the literature. To illustrate this procedure we consider in Sect. 6.1 various simple examples. There are several public codes for certain classes of Feynman integrals, with the IBP reduction done by hand. Typically, this reduction is in the first way, i.e. to Feynman integrals expressed in terms of gamma functions at general d. In the rest of this chapter, we will turn to algorithmic ways to solve IBP relations, in particular to the well-known Laporta’s algorithm [43, 44]. For algorithmic IBP reductions, the second way is typical, i.e. a reduction of any given integral to true master integrals.
1
A recent alternative review on the method of IBP can be found in [33]. For one loop, IBP was used in [36]. The crucial step—an appropriate modification of the integrand before differentiation, with an application at the two-loop level (to massless propagator diagrams)— was taken in [18] and, in a coordinate-space approach, in [71]. The case of three-loop massless propagators was treated in [18].
2
V. A. Smirnov, Analytic Tools for Feynman Integrals, Springer Tracts in Modern Physics 250, DOI: 10.1007/978-3-642-34886-0_6, © Springer-Verlag Berlin Heidelberg 2012
127
128
6 IBP and Reduction to Master Integrals
6.1 Solving IBP Relations by Hand The first example is very simple: Example 6.1 One-loop vacuum massive Feynman integrals F(a) =
(k 2
dd k . − m 2 )a
(6.1)
In this chapter, we are concentrating on the dependence of Feynman integrals on the powers of the propagators so that we will usually omit dependence on dimension, masses and external momenta. Let us forget that we know the explicit result (10.1) and try to exploit information following from IBP. Let us use the IBP identity dd k
1 ∂ ·k = 0, ∂k (k 2 − m 2 )a
(6.2)
with (∂/(∂k))
Data Loading...
