After having shown that LI-ids are just a particular set of IBP-ids, a question may arise: does the IBP mechanism generate all the meaningful relations for a given topology?
At least it is not true for the discrete ones, like redefinition of loop momenta, partial frac-tioning or factorization of graphs, due to the fact that IBP-ids originates from a continuous and connected group, whilst the swapping loop momenta and the rewriting of fractions are encoded by discrete rules.
Even restricting the domain of the question to Lie groups the answer is not clear: on one hand it seems that a group of relations among FIs, called Larin identities, is not related to the IBP-ids, and on the other hand the IBP-ids are infinite in number for a given topology, so new structure may arise with higher values of the parameters.
Another important aspect of the investigated relations is that they are not all independent:
as seen with LI-ids and IBP-ids, a huge part of the relations is redundant.
In any case, despite the fact that the question on the origin of relations among FIs is very significant as a theoretical aspect, another important question is: in the case that all “continuous”
relations are generated by IBP mechanism, is this useful to compute all such relations with only this one mechanism?
At present days most of the recursive work, like the determination of the relations among FIs, is assigned to computers. If the same relations that could be obtained by immediate evaluation (like the LI-ids, related to manifest invariance of the expressions) were obtained by iterated com-putation of more complex structures (like IBP, involving differentiation and algebraic rearranging of the terms) the only result, especially for bigger topologies, will be a really huge demand of resources in terms of computational time and computer hardware.
To avoid unnecessary waste of resources, all current programs for the evaluation of relations among FIs (like Reduze [46, 47], or FIRE [51]) use different approaches to the generation of relations, like IBP, Lorentz invariance or change-of-coordinates invariance.
Feynman integrals evaluation
Two ways of evaluating Feynman integrals are presented. At first, integral evaluation via Feynman parameters [20, 44] for sunset topologies is used to find a generic ex-pression for sunset and bubble integrals [41]. The subsequent part presents a detailed discussion of the method of differential equations for Feynman integrals [4, 9, 20], as well as of the fixing of boundary conditions [4, 9–12], and of the existence and uniqueness of the solutions [4, 9, 20]. Finally, the notion of canonical system [9, 10]
is introduced starting from -factorized systems.
2.1 Master integrals and Laporta algorithm
Using the relations investigated in the previous chapter it is possible to express the FIs of a specific topology in terms of either other FIs of the same topology or of FIs belonging to the subtopology tree, or even of FIs of different topologies. Usually, the number of such fundamental FIs is greatly inferior to the total number of initial integrals, and the knowledge of this relatively restricted set of functions allows to obtain, with just algebraic operations, the values of all the others. These “fundamental” FIs are called master integrals (MI) of the system.
The amount of MIs for a given topology may vary drastically from several, to one, to none:
in this last case, the topology is said to be reducible, and can be completely expressed in terms of MIs belonging to different topologies (usually located in the subtopology tree of the orginal FI).
Unfortunately, it has not yet discovered a criterion able to determine if, starting from a FI or a set of FIs, a given set of MIs is truly a basis (i.e. is a maximal set of independent functions with respect to all the possible algebraic relations among FIs). Nevertheless, starting from even thousands of FIs, the number of MIs is usually about tens, so in any case an enormous simplification of the calculations needed has been performed. It is even possible that some MIs that appear to be independent are in fact related via identities (usually, IBP-ids) only derived at higher orders; it would not be surprising, since relations like IBP-ids are infinite in number and not fully understood.
19
Fortunately, at least it has been proven that the number of MIs for a problem starting from a Feynman diagram is always finite (see [52]).
The set of MIs is not fixed or unique, and the choice of a suitable basis is not an easy problem:
it is necessary to avoid uselessly complicated FIs, and to find expression as regular as possible for D → 4 to obtain a smart form for the FI under investigation. This is not an easy task to complete without some sort of automated classification of the “complexity” of a FI, due to the overwhelming number of function to examine. Moreover, once the basis has been chosen, the relations among FIs must be solved in order to express all the other integrals in terms of the MIs. A parallel problem is the fact that not all the relations determined before are independent, so the problem to select the easiest set of independent ones has to be considered.
The Laporta algorithm introduces a solution for the problem (for a detailed discussion, see [17]): the core of the procedure is the determination of a “weight” function for FIs, an increasing function of the exponents ni and mj, such as FIs with higher powers have higher weight; once this is done the most weighty FIs are expressed in terms of the less weighty ones, trying to minimize the total weight of the independent FIs. In this way, a set of MIs is determined.
Once a set of MIs is fixed, their evaluation can be performed in two different ways.
• Integral evaluation: for simple integrals a direct calculation can be performed, usually thanks to Feynman parametrization. Feynman parametrization is often useful also on more complex integrals to investigate some properties of the FIs without direct evaluation, like properties of their series expansion.
• Differential evaluation: solving differential relations among MIs (determined differen-tiating with respect to external parameters and simplifying the results using algebraic relations, like IBP-ids) and imposing boundary conditions a series expansion in = 4−D2 of the integrals is obtained.