Feynman integral reduction using Gröbner bases

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SI-HEP-2022-30

P3H-22-101

FEYNMAN INTEGRAL REDUCTION USING GRÖBNER BASES

MOHAMED BARAKAT, ROBIN BRÜSER, CLAUS FIEKER, TOBIAS HUBER, AND JAN PICLUM

ABSTRACT. We investigate the reduction of Feynman integrals to master integrals using Gröb-

ner bases in a rational double-shift algebra Yin which the integration-by-parts (IBP) relations

form a left ideal. The problem of reducing a given family of integrals to master integrals can

then be solved once and for all by computing the Gröbner basis of the left ideal formed by

the IBP relations. We demonstrate this explicitly for several examples. We introduce so-called

ﬁrst-order normal-form IBP relations which we obtain by reducing the shift operators in Y

modulo the Gröbner basis of the left ideal of IBP relations. For more complicated cases, where

the Gröbner basis is computationally expensive, we develop an ansatz based on linear algebra

over a function ﬁeld to obtain the normal-form IBP relations.

CONTENTS

1. Introduction 1

2. The left ideal of IBP relations in the rational double-shift algebra 3

3. Gröbner bases in the noncommutative double-shift algebras 7

4. Sectors and symmetries of loop integrals 9

5. Examples 11

6. The special IBP relations 19

7. The Linear Algebra Ansatz 20

8. Conclusion 22

Acknowledgments 23

Appendix A. Mathematical details 23

References 24

1. INTRODUCTION

The LHC has been running for more than a decade now and has produced numerous in-

teresting results, among them the discovery of the Higgs boson, precision measurements of

Standard Model parameters like the top-quark mass, and searches for physics beyond the Stan-

dard Model. On the theoretical side, all of these studies require precise evaluations of signal

and background processes (see for example [1]). In perturbative quantum ﬁeld theory this

entails the calculation of Feynman integrals with many loops and often with many kinematic

invariants like masses and scalar products of external momenta.

A Feynman diagram corresponds in a well-deﬁned way to a loop integral of the form

I(z1, . . . , zn) = Zddℓ1· · · ddℓL

Pz1

1· · · Pzn

,(1.1)

where dis the space-time dimension in dimensional regularization and ℓiwith i= 1, . . . , L

are the loop momenta. The propagators (more precisely, propagator denominators) Piwith

i= 1, . . . , n are usually of the form m2

i−p2

i, where miis a particle mass and piis a linear

combination of the Lloop momenta and Eexternal momenta k1, . . . , kE. They are deduced

from the process under consideration and, without loss of generality, can be assumed to be

linearly independent. We call the set {I(z1, . . . , zn)|zi∈Z}with given propagators Pia

2010 Mathematics Subject Classiﬁcation. 13P10, 16D25, 16Z05, 81Q30, 81T18.

Key words and phrases. Loop integrals, rational double shift algebra, integration-by-parts reduction, commu-

tative and noncommutative Gröbner bases, computer algebra.

arXiv:2210.05347v2 [hep-ph] 31 May 2023

2 M. BARAKAT, R. BRÜSER, C. FIEKER, T. HUBER, AND J. PICLUM

family of integrals. The integers ziare called the indices of the integral. A given physical pro-

cess is usually expressed in terms of integrals of several different families. Note that we have

suppressed the integral’s dependence on the kinematic invariants, since we are here mostly

concerned with the dependence on the indices. For a modern account on the calculation of

loop integrals see for example [2].

In a typical calculation, one has to evaluate thousands of loop integrals belonging to several

integral families. An indispensable tool in the calculation of multiloop integrals is therefore the

integration-by-parts (IBP) method [3,4], which provides relations between loop integrals with

different propagator and numerator powers. These recurrence relations are called IBP relations

and can be used to express all integrals of a given family in terms of a small number of so-called

master integrals. Nowadays this is usually applied in the form of Laporta’s algorithm [5],

which solves the system of linear equations generated by plugging in numerical values for the

indices z1, . . . , zn.

The performance of Laporta’s algorithm can be greatly improved by using modular arith-

metic [6,7]. This avoids huge intermediate expressions during the calculation and allows for

more efﬁcient parallelization. There are many public and private codes to perform the integra-

tion by parts reduction, for instance AIR [8], FIRE [9–11], Reduze [12,13], Kira [14,15], and

FINITEFLOW [16].

In order to generate a linear system of equations for Laporta’s algorithm one has to specialize

a set of IBP relations to a range of indices z1, . . . , zn. Typically this linear system contains a

large number of integrals (as unknowns) that are oftentimes not directly needed but must be

included to ensure the full reduction of the desired integrals. This problem can be (partially)

avoided by starting with a set of so-called unitarity-compatible IBP relations based on syzygies

[17–22] (over a polynomial ring). We refer to these IBP relations in Section 6as special IBP

relations. They reduce the size of the linear system and improve the performance. Additional

new ideas towards a more direct reduction procedure have been developed: They rely on

algebraic geometry [23–25] and intersection theory [26–34].

One limitation of Laporta’s algorithm is that the reduction is only found for a given list of

integrals. Thus, when additional integrals are needed at a later point, the program has to be

run again. This can be overcome by deriving a full solution to the system of IBP recurrence

relations. Since a parametric solution by hand is clearly not feasible for multiscale problems,

it would be desirable to have an algorithmic way of solving the IBP relations once and for all.

LiteRed [35] is a publicly available program that performs this task using tailored heuristics

that is able to reduce several complicated integral families. In our work, we investigate the

application of Gröbner bases to the solution of IBP recurrence relations.

Previous application of Gröbner bases in this context can be found in [36–41]. In [36], the

IBP relations are ﬁrst transformed into a system of partial differential equations for which the

Gröbner basis is then computed. This method requires that all propagators have different, non-

zero masses and that no external momentum squared is equal to one of the masses squared.

Thus, it is not always possible to apply the result to cases with zero or equal masses or on-

shell momenta, since such limits can be singular. Reference [38] uses a modiﬁed version

of Buchberger’s algorithm to obtain so-called sector bases [39]. Finally we would like to

emphasize that – in contrast to previous noncommutative Gröbner basis approaches – we work

in the rational double-shift algebra deﬁned in Section 2.

This paper is organized as follows. In Section 2we deﬁne the (rational) double-shift algebra

in which the IBP relations form a left ideal. In Section 3we introduce the ﬁrst-order normal-

form IBP relations and highlight in which sense they differ from other well-known sets of

IBP relations. Furthermore we introduce the notion of a ﬁrst-order family, i.e., an integral

family for which the left ideal of IBP relations is generated by the ﬁrst-order normal-form

IBP relations. In Section 4we deﬁne the notion of formally scaleless monomials and relate

FEYNMAN INTEGRAL REDUCTION USING GRÖBNER BASES 3

them to scaleless sectors. In Section 5we demonstrate on selected examples of ﬁrst-order

families the computation of Gröbner bases, normal-form IBP relations, and the detection of

scaleless sectors using Gröbner basis reductions. Section 6recalls the construction of special

IBP relations using syzygies in a polynomial ring. The special IBP relations turn out to provide

a more efﬁcient set of generators as a starting point for the Linear Algebra Ansatz in Section 7

to compute the normal-form IBP relations without precomputing a Gröbner basis. Finally we

conclude in Section 8.

2. THE LEFT IDEAL OF IBP RELATIONS IN THE RATIONAL DOUBLE-SHIFT ALGEBRA

2.1. Notation. Denote by pi(for a symbol pand a natural number i) the column vector of d′

indeterminates

pi=





pd′−1







(2.1)

and deﬁne the Lorentz invariant quadratic expression

pi·pj=p0

ip0

j−

d′−1

µ=1

pµ

ipµ

j.(2.2)

For p=ℓthe vectors ℓ1, . . . , ℓLwill refer to the Lloop momenta. For p=kthe vectors

k1, . . . , kEwill refer to the Eexternal momenta.

Consider the polynomial algebra Q[d, m2

i]with coefﬁcients in the ﬁeld Qof rational num-

bers. Its elements are polynomial expressions in the dimension symbol dand the symbols of

squared masses m2

i. Deﬁne the ﬁeld of rational functions

F:=Q(d, m2

i):=Z

NZ, N ∈Q[d, m2

i], N ̸= 0

(2.3)

where the numerators Zand nonzero denominators Nare polynomials in Q[d, m2

i].

Consider the Lorentz invariant expressions that are polynomial expressions in the scalar

products of the L+Emomenta ℓ1, . . . , ℓL, k1, . . . , kEwith coefﬁcients in F. Each such expres-

sion can be written as a polynomial in the n=L(L+1)

2+LE propagators P1, . . . , Pnand so-

called extra Lorentz invariants S1, . . . , Sqwith coefﬁcients in F. The extra Lorentz invariants

are constructed from the external momenta in such a way that the set {P1, . . . , Pn, S1, . . . , Sq}

is algebraically independent over F. This means that P1, . . . , Pn, S1, . . . , Sqgenerate a poly-

nomial algebra over F, which we denote by

T=F[S1, . . . , Sq][P1, . . . , Pn].(2.4)

Since from some point on we do not need the special form of the Pi’s and Sj’s we replace

them by symbols D1, . . . , Dnand s1, . . . , sq, respectively. Likewise we replace the polynomial

algebra Tby the isomorphic polynomial algebra

R=F[s1, . . . , sq][D1, . . . , Dn].(2.5)

For more mathematical details on the construction of the polynomial algebras Tand Rsee

Appendix A.

The IBP relations are obtained from the fact that the operator ∂

∂ℓµ

ivµ

iturns the loop inte-

grand into a divergence, i.e., annihilates the loop integral in dimensional regularization. More

4 M. BARAKAT, R. BRÜSER, C. FIEKER, T. HUBER, AND J. PICLUM

precisely:

0 = Zddℓ1· · · ddℓL

∂

∂ℓµ

ivµ

Pz1

1· · · Pzn

n

(2.6)

=Zddℓ1· · · ddℓL∂vµ

∂ℓµ

i1

Pz1

1· · · Pzn

+(2.7)

Zddℓ1· · · ddℓLvµ

∂

∂ℓµ

i1

Pz1

1· · · Pzn

,(2.8)

where

vi=Cj

iBj

(2.9)

for Bj∈ {ℓ1, . . . , ℓL, k1, . . . , kE}with coefﬁcients (column) vector

C= (Cj

i)i=1,...,L,j=1,...,L+E∈TL(L+E)×1.(2.10)

The standard IBP relations are obtained by Crunning through the standard basis of TL(L+E)×1

(see (2.26) below).

In the following we will rewrite the expression ∂vµ

∂ℓµ

iin (2.7) and the differential operator

vµ

i∂

∂ℓµ

iin (2.8) in terms of the ring R. To this end we deﬁne the IBP-generating matrix1as

the product matrix

E=Ei

j,c

(2.11)

:=J·IL⊗ℓ1· · · ℓLk1· · · kE

| {z }

∈TLd′×L(L+E)

=∂Pc

∂ℓµ

Bµ

j∈Tn×L(L+E)⊂e

Tn×L(L+E),

where J:=∂Pc

∂ℓµ

i∈e

Tn×Ld′is the Jacobian matrix of the propagators, and where e

Tis deﬁned

in Appendix A. Like the propagators, and unlike the Jacobian matrix, the entries of the IBP-

generating matrix belong to the subring Tand can therefore be effectively rewritten as matrices

over R∼

=Tusing the subalgebra membership algorithm. The latter can be replaced by simple

linear algebra due to the afﬁne nature of Pias expressions in pi·pj. The dimensions of Eare

already independent of d′. However, its entries as expressions in the generators of the subring

Tformally still depend on d′. But once Eis rewritten as a matrix over R, the initial dependency

of E ∈ Rn×L(L+E)on the dimension d′disappears2.

For the coefﬁcients vector C∈RL(L+E)×1consider the Jacobian

JC:= ∂Cj

∂Dc!∈RL(L+E)×n

(2.12)

and the square matrix

EC:=EJC∈Rn×n.(2.13)

The divergence summand in (2.7) becomes

∂vµ

∂ℓµ

=d·Ci

i+ tr EC:=d·

i=1

i+ tr EC∈R.(2.7’)

1The name is motivated by equation (2.24).

2Physically, d′should be thought of as the symbolic regularizing dimension drather than an integer.

FEYNMAN INTEGRAL REDUCTION USING GRÖBNER BASES 5

Furthermore, the second summand (2.8) becomes

vµ

∂

∂ℓµ

=Cj

iBµ

∂Db

∂ℓµ

i∂

∂Db

=Ei

j,bCj

∂

∂Db

b=1 L+E

j=1

i=1

j,bCj

i!∂

∂Db

.(2.8’)

To determine the action of the differential operation vµ

i∂

∂ℓµ

ion D−zc

cwe use ∂

∂DbD−zc

−zcδb

cD−(zc+1)

c= (−zcD−1

c)δb

cD−zc

cresulting in

vµ

∂

∂ℓµ

iD−zc

c=−zcD−1

cEi

j,cCj

iD−zc

c:=−zcD−1

c L+E

j=1

i=1

j,cCj

i!D−zc

c.(2.14)

The next section introduces the shift algebra which contains the IBP relations as shift operators.

2.2. The (rational) double-shift algebra. The IBP relations can be understood as shift oper-

ators acting on the polynomial algebra

A:=F[s1, . . . , sq][a1, . . . , an](2.15)

by shifts, and therefore as elements of the double-shift algebra

Ypol :=A⟨D1, D−

1, . . . , Dn, D−

n⟩(2.16)

with the relations (no summation over repeated indices)

(2.17) [ai, Dj] = δijDi,[ai, D−

j] = −δijD−

i, DiD−

i= 1,

[ai, aj]=[Di, Dj]=[D−

i, D−

j]=[Di, D−

j] = 0,

and partial right action

I(...,zi, . . .)•Di=I(...,zi−1, . . .),(2.18)

I(...,zi, . . .)

| {z }

not scaleless

•D−

i=I(...,zi+ 1, . . .),(2.19)

I(...,zi, . . .)•ai=ziI(...,zi, . . .).(2.20)

The preﬁx “double” refers to the simultaneous occurrence of both the lowering operators Di

and the raising operators D−

The action is partial since D−

icannot be applied to a scaleless integral.3Our choice of the

right action will be justiﬁed in Remark 2.3 and the deﬁnition of scaleless integrals is deferred

to Section 4.

One can extend the action to the rational function ﬁeld

K:= Frac A=F(s1, . . . , sq)(a1, . . . , an):=Z

NZ, N ∈A, N ̸= 0,(2.21)

yielding the rational double-shift algebra

Y:=K⟨D1, D−

1, . . . , Dn, D−

n⟩.(2.22)

The partial right action is extended via

I(z1, . . . , zn)•Z(a1, . . . , an)

N(a1, . . . , an)=Z(z1, . . . , zn)

N(z1, . . . , zn)I(z1, . . . , zn),(2.23)

where Z(a1, . . . , an)and N(a1, . . . , an)̸= 0 are polynomial expressions in the ai’s with coefﬁ-

cients in rational expressions of the kinematic invariants F(s1, . . . , sq)whenever N(z1, . . . , zn)

is nonzero.

3Alternatively, one could rephrase such partial actions of algebras as actions of associated algebroids.

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SI-HEP-2022-30P3H-22-101FEYNMANINTEGRALREDUCTIONUSINGGRÖBNERBASESMOHAMEDBARAKAT,ROBINBRÜSER,CLAUSFIEKER,TOBIASHUBER,ANDJANPICLUMABSTRACT.WeinvestigatethereductionofFeynmanintegralstomasterintegralsusingGröb-nerbasesinarationaldouble-shiftalgebraYinwhichtheintegration-by-parts(IBP)relationsformalefti...

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