The diamond rule for multi-loop Feynman diagrams

An important aspect of improving perturbative predictions in high energy physics is efficiently reducing dimensionally regularised Feynman integrals through integration by parts (IBP) relations. The well-known triangle rule has been used to achieve simple reduction schemes. In this work we introduce an extensible, multi-loop version of the triangle rule, which we refer to as the diamond rule. Such a structure appears frequently in higher-loop calculations. We derive an explicit solution for the recursion, which prevents spurious poles in intermediate steps of the computations. Applications for massless propagator type diagrams at three, four, and five loops are discussed.


Introduction
Reducing complexities of Feynman integrals through integration by parts (IBP) relations [1,2] is an important component of modern multi-loop calculations.Finding more efficient reduction methods allows the computation of higher order terms in perturbative expansions which in turn aids in providing a better quantitative understanding of ongoing experiments.Since the 1980s, the so-called triangle rule [1,2] has been used for removing a propagator line from diagrams corresponding to a certain class of integrals.Any topology that has the following substructure can be simplified using the triangle rule: where D is the dimension which is set to 4 − 2 [3,4], and b, c 1 , c 2 are positive integers.The diagram corresponding to this integral is shown in Fig. 1.We write out the IBP relation ∂ ∂kµ k µ F = 0 (where the derivative must be performed before the integration) to obtain where A + i , B − , and C − i are operators acting on an integral that increase the power a i by one, decrease the power b by one, and decrease the power c i by one, respectively.Numerators that are expressed in dot products of k and an external line, contribute as a constant N to the rule.The rule of the triangle can be recursively applied to remove one of the propagators associated with k, p 1 , or p 2 from the system.
The recursion in the triangle rule can be explicitly 'solved' [5], such that the solution is expressed as a linear combination of integrals for which either b, c 1 , or c 2 is 0. The advantage of the summed system over the recursion is that it generates fewer intermediate terms and it cannot have spurious poles: terms in which the factor D + N − a 1 − a 2 − 2b becomes proportional to more than once during the full recursion.
In this work we introduce a more general class of diagrams that can be reduced using an extension of the triangle rule.We call this rule the diamond rule.We will show that the diamond rule 1) can be extended to any number of loops, 2) allows for a complete set of irreducible numerators that only contributes as a constant and 3) can be explicitly 'solved' to prevent spurious poles.
It should be noted that throughout this paper we mean by reduction the removal of a single line.This does not necessarily mean that the remaining diagram(s) will be trivial.An example of this is the four-loop ladder diagram, which can be reduced twice with the rule of the triangle, after which one of the remaining diagrams involves a master integral and needs a complete reduction scheme of 14 steps in which all numerators are removed and the power of all denominators is lowered to one successively.In Section 2 the diamond rule is derived.Section 3 shows the explicit summation formula and Section 4 shows examples.Finally, Section 5 gives a conclusion and discussion.

Diamond rule
Consider the following family of Feynman integrals in D-dimensions arising from the (L + S)-loop diagram in Fig. 2: The diagram consists of (L + 1) paths from the top vertex T to the bottom vertex B with an external connection in between, and S lines without external connections.The upper, lower, and external lines of the diamond are represented by red with dashed lines, green with double lines, and blue with thick lines, respectively.The lines without external connections, we call spectator lines.In principle any pair of spectator lines can be seen as a two point function which can be reduced to a single line by integration.This line would then have a power that is not an integer.Depending on the complete framework of the reductions this may or may not be desirable.Hence we leave the number of spectators arbitrary.In any case, the contribution of the spectators is a constant (see below), which allows us to characterise integrals in the family only by 2(L + 1) indices a i and b i , and not by s i .Without loss of generality, we assign loop momenta k i to the lower lines of the diamond as well as l i to the spectator lines, except the last diamond line which is fixed by momentum conservation: In contrast, we do not require any constraints on the momentum conservation at the top vertex in the arguments below, hence any number of external lines can be attached to this point.In the middle of the diamond, external lines with momentum p i are attached by three-point vertices.The upper lines in the diamond may have masses m i , whereas the lower lines in the diamond and the spectator lines have to be massless.In addition, we allow arbitrary tensor structures of k i and l i with homogeneous degrees N i and R i , respectively, in the numerator.
Constructing the IBP identity corresponding to the operator straightforwardly gives the following operator identity: Here A + i and B − i are understood as operators increasing a i and decreasing b i by one, respectively, when acting on F ({a i }, {b i }).Note that operators changing the spectator indices s i are absent in the identity.
For a typical usage of Eq. ( 6), one may identify a diamond structure as a subgraph in a larger graph.If the line with the momentum p i has the same mass m i as the corresponding upper line, the term (p 2 i + m 2 i ) in the identity reads as an operator C − i decreasing the corresponding index c i of the power of the propagator (p 2 i + m 2 i ) −ci in the larger graph by one.Applying the rule where The above diamond rule contains the conventional triangle rule as a special case.For the one-loop case L = 1 and S = 0, the two lower lines may be identified as a single line and the triangle integral in Eq. ( 1) can be reproduced.Correspondingly, the IBP identity (7) becomes Eq.(2).

Summation rule
We now derive an explicit summation formula for the recursion in the diamond rule.First, we consider the possible connectivities.If we allow for some external lines to be directly connected to each other, we get at least one triangle that can be used for the triangle rule: suppose the external momenta of k i and of k j are connected and identified with p ij , then this triangle is k i , k j , p ij .In this case, the triangle rule generates fewer terms and is preferred to the diamond rule.Thus, we only consider the case where the diamond does not have direct connections of external lines.
We follow the same procedure as outlined in [5].First, we rewrite Eq. ( 7) as: where E is the operator (L + S)D We identify the first class with the label +, since it increases E by 1, and the latter with the label −, since it decreases E by 1.The remaining part of the derivation is analogous to the one in [5].
Finally, we obtain the explicit summation formula: where Because the Pochhammer symbol that depends on E only appears once in each term, powers of 1/ 2 or higher cannot occur.Thus, the explicit summation formula for the diamond rule does not have spurious poles.

Examples
Several examples of diamond structures are displayed in Fig. 3.The role of each line in the diamond rule is highlighted by different colors and shapes.Red dashed lines, green double lines, and blue thick lines represent upper, lower, and external lines of the diamond, respectively.Label T represents the top vertex, and B the bottom vertex.In Fig. 3a a four-loop diagram is displayed.For this diagram, the line of either p 5 , p 6 , p 7 , p 8 , p 9 , or p 10 can be removed by recursive use of the diamond rule or by the explicit formula given in the previous section.The irreducible numerators of this diagram are selected as Q•p 8 , Q•p 10 , p 5 •p 10 , and p 5 •p 7 , such that they adhere to the tensorial structure in the diamond rule.The last numerator, p 5 •p 7 , lies outside of the diamond and does not interfere with the rule.
If, in this figure, we draw an additional line from the top (T) to the bottom (B) vertex, we obtain the simplest nontrivial propagator topology with a spectator line.As a five-loop diagram it is unique.
In Fig. 3b the three-loop master topology NO is displayed.Q•p 5 is chosen as irreducible numerator.One of the lines attached to the diamond is actually an off-shell external line.In general, if the line with momentum p L+1 is one of the external momenta of the larger graph, the factor (p   constant with respect to the loop integration and has no role for reducing the complexity of the integral.As a result, the rule ( 7) is not applicable to remove one of the internal lines.Even for such cases, one can still find a useful rule by shifting a L+1 → a L+1 − 1: which decreases at least a L+1 or b L+1 by one.Repeated use of this rule from positive integer a L+1 and b L+1 reduces a L+1 or b L+1 to 1.For the NO topology, this variant yields the rule to reduce the line p 1 to 1 in Mincer [6,7]. 2n Fig. 3c we show two five-loop topologies for which the diamond rule can eliminate one line.The first diagram is unique in the sense that it is the simplest diagram for which L = 3, S = 0.The second diagram is a typical representative of the 29 five-loop topologies with L = 2, S = 0 and all three p-momenta of the diamond internal.

Conclusion and discussion
We have indicated an extensible, multi-loop topology substructure that can be reduced efficiently.We call the corresponding reduction formula the diamond rule.Additionally, we have derived an explicit summation formula for the recursion in the diamond rule, which avoids spurious poles.
For parametric reduction applications such as the Mincer program, an implementation of the diamond rule would be faster than automatically generated reduction rules.This is already the case with the summed triangle as it is used in the Mincer program.It allows the program to avoid spurious poles altogether and hence it can run at a fixed precision in powers of .It is currently not clear whether Laporta approaches [8] as used in systems such as AIR [9], Reduze [10,11], and FIRE [12,13,14] benefit from applications of the diamond rule.
It is important to note that the tensorial structure for the irreducible numerators should be adhered to.If one chooses numerators of the form (p i − p j ) 2 instead of dot products p i •p j , extra terms are introduced to the diamond rule.It is unclear to us whether the performance gain of using the rule is greater than the cost of rewriting the numerators to dot products in software such as LiteRed [15,16].In the Mincer approach with its dot products in the numerators, the diamond rule fits in perfectly.
Neither at the four-loop level, nor at the five-loop level we have found structures that can be reduced by a single IBP identity, apart from the diamond rule.

Figure 1 :
Figure 1: A triangle subtopology where the loop momentum k is assigned to the central line.p 1 and p 2 are external momenta.a 1 , a 2 , b, c 1 , and c 2 represent the powers of their associated propagators.

Figure 2 :
Figure 2: (L + S)-loop diamond-shaped diagram.(L + 1)-lines have external connections and S-lines do not.Red with dashed lines, green with double lines, and blue with thick lines represent upper, lower, and external lines of the diamond, respectively.Label T represents the top vertex, and B the bottom vertex.k i , p i , and l i are momenta, and a i , b i , and s i are the powers of their associated propagators.

L+1i=1
(b i +c i ) of integrals appearing in the right-hand side, at the cost of increasing L+1 i=1 a i .Starting from positive integer indices b i and c i , one can repeatedly use the rule until one of either b i or c i is reduced to zero. 1 and (a) b is the rising Pochhammer symbol Γ(a + b)/Γ(a).The first term decreases the power b r to 0, and the second term decreases c r to 0. The only significant difference between the two terms is the +1 in the Pochhammer symbol.

Figure 3 :
Figure3: Two topologies with highlighted diamond structures.Red with dashed lines, green with double lines, and blue with thick lines represent upper, lower, external lines of the diamond, respectively.Label T represents the top vertex, and B the bottom vertex.(a) shows a four-loop topology which can be completely reduced.(b) shows the three-loop NO master topology, for which a modified form of the diamond rule can be applied to lower the power of line p 1 to 1. (c) shows five-loop topologies, which the diamond rule can be applied to.