A New Algorithm For The Generation Of UnitarityCompatible Integration By Parts Relations
Abstract:
Many multiloop calculations make use of integration by parts relations to reduce the large number of complicated Feynman integrals that arise in such calculations to a simpler basis of master integrals. Recently, Gluza, Kajda, and Kosower argued that the reduction to master integrals is complicated by the presence of integrals with doubled propagator denominators in the integration by parts relations and they introduced a novel reduction procedure which eliminates all such integrals from the start. Their approach has the advantage that it automatically produces integral bases which mesh well with generalized unitarity. The heart of their procedure is an algorithm which utilizes the weighty machinery of computational commutative algebra to produce complete sets of unitaritycompatible integration by parts relations. In this paper, we propose a conceptually simpler algorithm for the generation of complete sets of unitaritycompatible integration by parts relations based on recent results in the mathematical literature. A striking feature of our algorithm is that it can be described entirely in terms of straightforward linear algebra.
1 Introduction
When the technique of integration by parts in dimensions was first proposed by Tkachov and Chetyrkin [1], it represented a major breakthrough in the study of perturbative gauge theories at the multiloop level.^{1}^{1}1For the reader less familiar with integration by parts, we strongly recommend Smirnov’s excellent book on Feynman integral calculus [2]. He explains the technique and works through a number of simple (and nonsimple) examples. Their discovery was of fundamental practical importance, as it allowed researchers to perform many multiloop calculations that were previously thought to be intractable. Furthermore, the idea of the technique is simple to describe. By taking integrals of derivatives in dimensions, one generates a tower of equations for the Feynman integrals belonging to a particular topology. Then one tries to solve these equations, either by inspection or via some systematic procedure, for an independent basis of master integrals.
Unfortunately, the total number of equations generated in this way grows rapidly with the number of loops and external states in the integral topology under consideration. As a consequence, the solution of the socalled integration by parts relations is complicated for all but the simplest examples. For many years after the technique was introduced, no systematic procedure for the solution of integration by parts relations was known and it was therefore only possible to apply the method to simple integral topologies.^{2}^{2}2Of course, simple is a relative term. For a rather impressive early application of the method at the threeloop level see the long writeup for MINCER [3]. The situation changed just over a decade ago with the introduction of a Gaussian eliminationlike solution algorithm. This algorithm, due to Laporta [4], was a crucial step forward because it allowed researchers to apply the integration by parts technique to highly nontrivial problems for which an ad hoc approach would be impractical if not impossible. Although Laporta’s algorithm has been tremendously successful, it has long been known that it may lead to master integrals with doubled propagator denominators.
To be more precise, for an loop topology, let be the set of loop momenta together with a set of independent external momenta and let be a generic element of this set. Normally, one generates the set of integration by parts relations in an obvious way, considering each in turn and writing all possible equations of the form
(1) 
Crucially, the indices and satisfy certain boundary conditions; typically the irreducible numerators (here we assume that each has mass dimension two) for a given topology enter raised to, at most, some relatively small positive integer and, in some cases, certain propagator denominators are constrained to have nonnegative indices since otherwise the resulting integrals vanish in dimensional regularization. This set of equations together with the appropriate boundary conditions can then be fed into Laporta’s algorithm. The price one pays for being able to trivially generate the set of integration by parts relations in this fashion is twofold. Not only will the algorithm (Laporta or typical variation thereof [5, 6, 7]) used to solve the resulting system of equations typically have to eliminate an enormous number of spurious integrals as it attempts to solve the system, it is not straightforward to ensure that each integral in the basis ultimately returned by the algorithm has the property that .
Although there is nothing wrong with integrals that have some , they may be inconvenient for particular multiloop applications. For example, in computational approaches based on generalized unitarity, one would like to have a basis with welldefined unitarity cuts in all channels (see e.g. [8] and [9] for recent work in this direction at two loops). It is unclear how one would make sense of integrals with doubled propagator denominators in such a framework. Gluza, Kajda, and Kosower (hereafter referred to as GKK) also argued in [10] that master integrals with doubled propagator denominators can be significantly harder to expand in than those without.
The idea of the GKK procedure is relatively easy to state now that the stage is set. With the above motivation, GKK found that they could completely avoid the introduction of doubled propagators by imposing from the beginning. They observed that, generically, one expects doubled propagators for the simple reason that the derivatives in eqs. (1) act on the propagator denominators, . They also recognized that there is no good reason why one ought to consider the elements of one at a time; one can generalize eqs. (1) by replacing with a general linear combination of the elements of :
(2) 
It is also convenient at this point to combine together some of the equations by summing over :

(3)
In a nutshell, the GKK strategy is to start with all the equal to one
(4) 
and then choose the coefficients in such a way that the numerator exactly cancels all unwanted, derivativegenerated powers of the propagator denominators. In other words, for each , we demand that
(5) 
where is some polynomial built out of the irreducible numerators and Lorentz invariant combinations of the vectors in .^{3}^{3}3We write instead of so that we will ultimately arrive at the same form that GKK did in their eqs. (4.1). Each independent set of coefficients satisfying eqs. (5), upon substitution into eqs. (4), yields a unitaritycompatible integration by parts relation, by construction free of doubled propagators.
The downside of this novel approach is that now one needs to find a way to generate complete sets of coefficients and it turns out that this is not a straightforward task. Although GKK did propose a solution^{4}^{4}4Actually, GKK presented two distinct algorithms for the generation of complete sets of unitaritycompatible integration by parts relations. In what follows, when we refer to GKK’s “solution” or “solution algorithm” with no additional qualifier, we are referring to their best solution (what they call Algorithm III). to this problem in [10], they admitted that their solution was somewhat provisional and that there is likely room for improvement. The GKK solution relies heavily on the use of Gröbner bases, important constructs in computational commutative algebra which are, however, notoriously difficult to compute in practice [11].^{5}^{5}5For the reader less wellversed in computational commutative algebra, we can strongly recommend the very wellwritten and concise survey by Adams and Loustaunau [12]. Most of the relevant mathematical concepts are also defined and briefly explained in the GKK paper [10]. In this paper we propose a completely different solution to the problem of generating complete sets of unitaritycompatible integration by parts relations. As we shall see, our Algorithm 1 is based entirely on simple linear algebra and, in particular, completely avoids the use of Gröbner bases.
This article is organized as follows. In section 2 we describe more precisely the problem we wish to solve and introduce some useful notation. In section 3 we present the main result of this paper, Algorithm 1, and talk the reader through it. In Section 4, we give a detailed example of how our algorithm works in practice. In section 5 we present our conclusions and outline our plan for future research.
2 Preliminaries
In this section we take a closer look at the reduction procedure introduced by GKK and discuss its strengths and weaknesses. Our initial goal will be to precisely set up the mathematical problem that lies at the heart of the GKK procedure and discuss why (in the opinion of both GKK and the present author) the solution presented in [10] is not likely to be the best one possible. We will then explain our approach to the problem and illustrate with a very simple example what precisely Algorithm 1 is designed to do.
As we left them, eqs. (5) look rather cumbersome. We can clean them up by absorbing the numerator polynomial into the
(6) 
with the understanding that now the are dimensionful. We will have to take this into account in our search for independent solutions. To make further progress, GKK expressed (6) as a matrix equation:
(7) 
where
(8) 
Given an explicit expression for and an ordering on the set of propagator denominators, the entries of can be straightforwardly read off from (6). For an explicit example, we refer the reader to eq. (5.3) of [10]. In eq. (5.3), GKK wrote out explicitly for the planar massless double box, choosing for the ordering on the set of propagator denominators (this ordering fixes the sequence of the columns of ).^{6}^{6}6We should also point out that the precise definitions GKK made for the (what they call ) and (what they call ) do not seem to exactly reproduce eq. (5.3). However, the differences can be taken into account by appropriately rescaling the unknowns and are therefore unimportant. The point of making this rearrangement is that now the problem of determining all independent sets of coefficients looks like a wellknown, wellstudied problem from computational commutative algebra: the computation of the syzygy module^{7}^{7}7Given a set of generators for a module , , the syzygy module of is simply the set of all such that . In this paper, we will deal only with submodules of and for a field, . of a module for which one has an explicit set of generators.
Actually, as explained by GKK, it suffices to solve this problem for ideals, since given a set of generators for a module , one can easily define a set of generators for an ideal with exactly the same syzygies. Suppose the set generates . By simply taking the dot product of each with a tuple of dummy variables satisfying the relations , , we can convert our generating set for into a generating set for an ideal canonically associated to . The class of ideals canonically associated to the modules described by eqs. (6) can be taken to be homogeneous of uniform degree two. We can see this as follows. By definition, each parameter entering into the matrix of eq. (7) is of the form or and is going to have mass dimension two. Therefore, it makes sense to take every independent Lorentz product that can arise in to be an independent variable and the class of modules described by eqs. (6) to be homogeneous of uniform degree one. It then follows, after applying the canonical map described above, that the ideals of interest are homogeneous of uniform degree two. In the end, we find that each generator of has terms of the schematic form , for some field .^{8}^{8}8In this paper, is given by the field of rational functions of the auxiliary parameters in the matrix (e.g. in the case of the planar massless double box treated in detail by GKK) of eq. (7). However, for practical purposes, it is more convenient to simply assign prime numbers to the dimensionless auxiliary parameters and work over .
Before continuing, we need to introduce a little more notation and point out an important fact about the syzygies of generating sets for homogeneous ideals of uniform degree. Given an ideal generated by , we define to be the set of all syzygies of such that implies that each is a polynomial of uniform degree . We will typically refer to as the set of all degree syzygies of . It turns out that, for the case of homogeneous ideals of uniform degree, the syzygy module, , is a graded module [13]:
(9) 
This means that, for homogeneous ideals of uniform degree, it suffices to search for degree syzygies.
Determining a basis for the syzygy module of a generic ideal is known to be a very difficult problem [11] (and is very much an active area of mathematical research). Therefore, one needs a dedicated solution algorithm, tailormade for the class of ideals described above. As mentioned in the introduction, the solution algorithm employed by GKK relies heavily on the use of Gröbner bases. GKK chose the wellknown Buchberger algorithm [14] to compute their Gröbner bases. In their paper, GKK pointed out that there have been a number of recent attempts to improve on Buchberger’s algorithm (see e.g. [15] for a description of one of the most promising of these recent attempts, based on Faugère’s algorithm [16]) and concluded that their Buchbergerbased approach was not likely to be optimal. In this paper we rethink their approach at a more fundamental level.
Certainly, one could attempt to compute complete sets of syzygies using an approach based on Faugère’s algorithm [17] or some other improved algorithm for the computation of Gröbner bases. However, it is actually no longer clear that one should use Gröbner bases at all. Quite recently (after the appearance of [10]) it was shown in [13] that, remarkably, bases for the modules of syzygies of special classes of ideals can be computed using simple linear algebra. Actually, at this stage, the reader may be wondering what stopped GKK from using a linear algebrabased approach in the first place. Naïvely, linear algebra seems to offer a very easy way to compute syzygies.
To illustrate why the ideas of reference [13] are nontrivial, we consider the ideal generated by and attempt to compute its syzygies using linear algebra.^{9}^{9}9For the sake of definiteness, suppose that we are working with polynomials in . By inspection, we see that has no degree zero syzygies. By definition, a degree one syzygy of must have the form for some elements of . Starting with this ansatz, we can take the dot product with
(10) 
set the coefficient of each monomial in the sum to zero, and solve the resulting system of equations. It turns out that there is a oneparameter family of solutions which we parametrize by :
(11) 
We can arbitrarily set to obtain a basis for , . So far so good. By definition, a degree two syzygy of must have the form . Going through the same procedure, we arrive at a twoparameter family of solutions which we parametrize by and :
(12) 
By replacing with each of the standard basis vectors for in turn, we find that and form a basis for the set of solutions. In order to map these solutions back to degree two syzygies of , we partition each vector into two nonoverlapping subsets of length three without changing the overall ordering of the entries
(13) 
and then dot each resulting 3tuple into :
(14) 
This overly formal description of the mapping back to syzygies of is overkill for the example at hand but will be useful later on.
We might be tempted to conclude that we have found two new, linearly independent, degree two syzygies of . This, however, is not true. It turns out that the syzygies of eq. (14) can be expressed as multiples of . Explicitly, we have
(15) 
and
(16) 
Besides highlighting the profound difference between linear independence in a vector space and linear independence in a module, this example shows what goes wrong if one tries to compute the syzygies of homogeneous ideals of uniform degree using naïve linear algebra. One is able to compute syzygies in a straightforward manner but not a basis of linearly independent syzygies. It is therefore remarkable that, under appropriate assumptions, this obstacle is actually surmountable. In the next section, drawing upon some of the ideas introduced in [13], we present a simpler linear algebrabased alternative to the solution adopted by GKK.
3 The Algorithm
The purpose of this section is to give an explicit pseudocode detailing our solution to the problem defined in Section 2 and to thoroughly comment it. The pseudocode we present (Algorithm 1 and associated subroutines) is quite explicit and should allow the reader to fashion a crude implementation of our algorithm in Maple or Mathematica with very little effort (beyond that required to understand the algorithm in the first place). We should emphasize that we do not in any way claim that an implementation based on our pseudocode is an effective implementation. The pseudocode given below is intended to be maximally clear as opposed to maximally efficient.^{10}^{10}10For example, the first statement in the upper branch of the If statement in Algorithm 1 is completely superfluous and was included only because, in our opinion, it makes the functionality of Subroutine 1 significantly easier to understand. Before discussing the nontrivial features of Algorithm 1, we need to make a few more definitions.
First, let be the set of all monomials of degree built out of the variables .^{11}^{11}11Note that there are precisely monomials of degree built out of variables. This result follows immediately once one recognizes that the monomial counting problem is isomorphic to one of the usual “ballbox” counting problems of enumerative combinatorics. We refer the unfamiliar reader to Section 1.4 of Stanley’s textbook on the subject [18]. For example, . For definiteness, we will always order sets of monomials lexicographically. This choice of ordering is just a choice and has no deeper significance. Now, if we have in hand a sequence of homogeneous polynomials of uniform degree two, , then is defined to be the outer product of and :
(17) 
Clearly, is a set of homogeneous polynomials of uniform degree . For example, if we take it is easy to see that . The construction of can be thought of as an intermediate step towards the extraction of the degree two syzygies of . Instead of solving the system , we can construct and then solve the system . So, instead of trying to work with directly, we can work with the vector space . This natural correspondence between basis vectors of and degree syzygies of is an essential part of Algorithm 1, which is why we took the time to carefully describe it while working through the illustrative example at the end of Section 2. However, it is important to remember that this map does not necessarily yield independent elements of ; to work as advertised, our solution algorithm must be able to determine, using only linear algebra, what basis vectors of correspond to new syzygies of , linearly independent of those already determined.
We now describe how Algorithm 1 works in some detail. By assumption, the propagator denominators of the integral topology under consideration form a linearly independent set.^{12}^{12}12If this is not the case, one should first reduce to simpler integral topologies satisfying this condition before attempting to apply Algorithm 1. GKK explain this procedure in some detail in [10]. This implies that is a minimal generating set for the ideal under consideration and there are no nontrivial syzygies of degree zero. Since Algorithm 1 proceeds incrementally in the degree of the syzygies, it is convenient to define two bookkeeping lists, and , indexed by . is a basis for the vector space and is a set of linearly independent elements of which are, for , also linearly independent of all syzygies of degree in the set (we will call the elements of new degree syzygies of ). Due to the fact that has no degree zero syzygies, both lists are initialized to and is initialized to one. Since , Algorithm 1 always starts with a pass through the lower branch of the If statement. Using Subroutine 2, Algorithm 1 determines a basis for along the lines described in the example near the end of Section 2. For the sake of discussion, we assume that this basis is nontrivial. This basis, , is then used to determine both and, via Subroutine 3, . In this section we will not step into Subroutines 2 or 3. There is nothing nontrivial about them and we will in any case go through the subroutines once explicitly in Section 4.
Increment to two. Subroutine 1 lifts the syzygies in to syzygies in in every independent way possible. This is accomplished by simply multiplying each of the elements of (written out explicitly in terms of their coordinates, the elements of ) by each one of the variables in turn. The results can then be interpreted as syzygies in (written out explicitly in terms of their coordinates, the elements of ). Clearly, if is a scalar syzygy of , for each by linearity. This implies that can be interpreted as a scalar syzygy of in independent ways. Given syzygies in , Subroutine 1 produces syzygies in (some of which may be linearly dependent). As we shall see, Subroutine 1 is a crucial first step towards determining which syzygies in correspond to new degree two syzygies of and which do not.^{13}^{13}13The reader who has read reference [13] might be concerned that we have not yet spoken about the principal syzygies of our ideals. Actually, the fact that the are simply coordinate variables with no independent existence of their own (they satisfy for all and ) implies that the ideals of interest to us have no principal syzygies at all.
Since the syzygies in produced by Subroutine 1 are not guaranteed to be linearly independent, the next step is to put the output of Subroutine 1 into row echelon form, discard all rows without nonzero entries, and call the result . Since, by construction, each of the rows of is a scalar syzygy of , we could in principle rewrite of the polynomials in as linear combinations of the other . Algorithm 1 takes this fact into account by replacing the entries of which correspond to the pivot columns of with zero and calling the result . In constructing , it has isolated the subspace of which is in correspondence with the new degree two syzygies of (if any new degree two syzygies exist). Next, Algorithm 1 uses Subroutine 2 to determine a basis, , for the subspace of under consideration. However, in order to search for degree three syzygies in an analogous fashion, we need a basis for in its entirety. Therefore, Algorithm 1 sets (for the sake of discussion assuming that Subroutine 2 found new degree two syzygies). Finally, Algorithm 1 solves for the new degree two syzygies themselves by applying Subroutine 3 to . After incrementing to three, Algorithm 1 would pass through the upper branch of the If statement again in an attempt to find new degree three syzygies of .
Of course, we have been assuming throughout our discussion that . It is worth emphasizing that, the termination condition we are using for Algorithm 1 comes entirely from the physics. The fact that irreducible numerators typically have, at worst, a relatively small mass dimension is the only reason that we were able to fruitfully apply the ideas of reference [13] and construct Algorithm 1. Although we believe that the treatment given here is appropriate, some readers may prefer a more formal one. If that is the case, then we recommend reading [13]. Most of the nontrivial aspects of our pseudocode are treated there as well in the style preferred by mathematicians.
4 An Explicit Example
In this section, we show how Algorithm 1 works in practice by going through it for a particular example. It was challenging to find a physically motivated example compact enough to present in detail and, at the same time, rich enough to give the reader a good feeling for how the algorithm functions. In the end, we found that the module given by the irreducible part of the module associated (associated in the sense of the construction reviewed in Section 2) to the planar massless double box works very well. In the solution algorithm of GKK, the study of this module (hereafter referred to as ) is the first step towards the determination of the complete set^{14}^{14}14Here it is perhaps worth pointing out that, in fact, the application of our algorithm to the module associated to the planar massless double box yields more independent solutions than GKK found (working modulo reducibility as they do). We conjecture that, perhaps, GKK did not really seek the complete set of linearly independent syzygies modulo reducibility but were instead content to determine a subset sufficient for the elimination of as many irreducible numerators as possible. If our reading of GKK is correct, then the obvious question is whether discarding potentially useful linear relations between Feynman integrals of a given topology is prudent. We suspect that this is not the best strategy because, at least for the planar double box, the elimination of irreducible numerators in this manner seems to lead to a large number of irreducible integrals of simpler topology. Unfortunately, further exploration of this interesting question is beyond the scope of the present paper. of linearly independent syzygies of the module associated to the planar massless double box. In their paper, GKK assert that has just three linearly independent syzygies: one of degree one and two of degree two. This a priori knowledge of the syzygy module of will allow us to stop the example when it ceases to be interesting. Otherwise we would have to make several more trips through the While loop (as explained in [10], for the planar massless double box), each time discovering no new syzygies. For the sake of clarity, our exposition will mirror the pseudocode of Section 3 quite closely.
As a preliminary step we must derive the generators of . In Section 2, we pointed out that the ordering adopted by GKK for the propagator denominators of the planar massless double box is given by . We actually prefer the ordering and this is what we will use. Note, however, that we do adopt their ordering for the generators themselves. By definition, the generators of are the generators of the module associated to the planar massless double box by eqs. (6) (the rows of the matrix in eq. (5.3) of [10]) reduced over the propagator denominators of the massless double box. This reduction is effected by making the substitutions . We find that is generated by
(18) 
If we let , , , and take the dot product of each generator in (18) with , we arrive at the generating set for the ideal (hereafter referred to as ) canonically associated to , :
(19) 
Only the first ten rows of remain nonzero after the reduction is carried out. Provided that all propagator denominators are independent of one another (which is certainly true in our case), the we arrive at in this fashion will always be a minimal generating set for which implies that, as assumed in Algorithm 1, . Actually, for most of steps of the algorithm the explicit form of is unimportant and we suppress it, writing instead .
We initialize to one and enter the While loop. Since , the algorithm directs us to the lower branch of the If statement. , , and