Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals

Integration-by-parts identities between loop integrals arise from the vanishing integration of total derivatives in dimensional regularization. Generic choices of total derivatives in the Baikov or parametric representations lead to identities which involve dimension shifts. These dimension shifts can be avoided by imposing a certain constraint on the total derivatives. The solutions of this constraint turn out to be a speciﬁc type of syzygies which correspond to logarithmic vector ﬁelds along the Gram determinant formed of the independent external and loop momenta. We present an explicit generating set of solutions in Baikov representation, valid for any number of loops and external momenta, obtained from the Laplace expansion of the Gram determinant. We provide a rigorous mathematical proof that this set of solutions is complete. This proof relates the logarithmic vector ﬁelds in question to ideals of submaximal minors of the Gram matrix and makes use of classical resolutions of such ideals.

Integration-by-parts identities between loop integrals arise from the vanishing integration of total derivatives in dimensional regularization. Generic choices of total derivatives in the Baikov or parametric representations lead to identities which involve dimension shifts. These dimension shifts can be avoided by imposing a certain constraint on the total derivatives. The solutions of this constraint turn out to be a specific type of syzygies which correspond to logarithmic vector fields along the Gram determinant formed of the independent external and loop momenta. We present an explicit generating set of solutions in Baikov representation, valid for any number of loops and external momenta, obtained from the Laplace expansion of the Gram determinant. We provide a rigorous mathematical proof that this set of solutions is complete. This proof relates the logarithmic vector fields in question to ideals of submaximal minors of the Gram matrix and makes use of classical resolutions of such ideals.

I. INTRODUCTION
The increasing precision of the experimental measurements of scattering processes at the Large Hadron Collider (LHC) is calling for increased precision in the theoretical prediction of cross sections. The computations of the leading-order (LO) and next-to-leading-order (NLO) contributions to the cross sections are by now automated, but for many processes the next-to-next-to-leading-order (NNLO) contribution is needed to reach the required precision.
The NNLO cross section has double-real, real-virtual and double-virtual contributions. The aim of this paper is to provide new tools for computing the latter contributions, i.e., the two-loop scattering amplitudes. Results for the latter may in turn motivate progress on the combination of all virtual and real contributions to the NNLO cross section.
A key tool in the calculation of multi-loop amplitudes are integration-by-parts (IBP) reductions. The latter arise from the vanishing integration of total derivatives in dimensional regularization, where P and the vectors v µ j are polynomial in the loop momenta i and external momenta, the D k denote inverse propagators, and the ν i are integers. The IBP identities turn out to generate a large set of linear relations between loop integrals. This then allows for most integrals to be reexpressed as a linear combination of basis integrals. In practice, the basis contains much fewer integrals than the number of integrals produced by the Feynman rules for a multi-loop amplitude.
The step of performing Gaussian elimination on the linear systems that arise from eq. (1.1) may be carried out with the Laporta algorithm [1,2], which leads in general to relations that involve integrals with squared propagators. There are various publicly available implementations of automated IBP reduction: AIR [3], FIRE [4,5], Reduze [6,7], LiteRed [8], Kira [9], as well as private implementations. A formalism for obtaining IBP reductions without squared propagators was developed in refs. [10,11]. A systematic method of deriving IBP reductions on generalized-unitarity cuts was given in ref. [12]. A recent approach [13] uses sampling over finite fields to construct the reduction coefficients. Other recent developments include software for determining a basis of integrals [14] and a D-module theory based approach for computing the number of basis integrals [15].
In this paper we study integration-by-parts identities (1.1). Generic choices of total derivatives in the Baikov or parametric representations lead to identities which involve undesirable dimension shifts. These dimension shifts can be avoided by imposing a certain constraint on the total derivatives. The solutions of this constraint turn out to be a specific type of syzygies which correspond to logarithmic vector fields along the Gram determinant formed of the independent external and loop momenta. We will present an explicit generating set of solutions in Baikov representation, valid for any number of loops and external momenta, obtained from the Laplace expansion of the Gram determinant. We will then present a rigorous mathematical proof that this set of solutions is complete. This proof relates the logarithmic vector fields in question to ideals of submaximal minors of the Gram matrix and makes use of classical resolutions of such ideals.
An important feature of the obtained generating set of syzygies is that they are guaranteed to have degree at most one. In contrast, a generating set of syzygies obtained from an S-polynomial computation would in general have higher degrees. The fact that the syzygies obtained here are of degree at most one is useful because it dramatically simplifies the computation of solutions which satisfy further constraints. For example, one may be interested in imposing the further constraint on the total derivatives that no integrals with squared propagators are encountered in the integration-by-parts identities.
This paper is organized as follows. In Sec. II we set up notation and give the Baikov representation of a generic Feynman loop integral. In Sec. III we study integrationby-parts relations on unitarity cuts and derive the syzygy equation of interest. In Sec. IV we obtain a closed-form generating set of solutions to the syzygy equation. In Sec. V we present a proof that the set of syzygies is complete. In Sec. VI we provide an example of the formalism. In Sec. VII we give our conclusions.

II. BAIKOV REPRESENTATION OF LOOP INTEGRALS
In this paper we will make use of the Baikov representation of Feynman loop integrals. As will be explained later, this parametrization is useful for our purpose of studying integration-by-parts identities (1.1) on so-called cuts where some number of propagators are put on shell, i.e. after evaluating the residue at D α = 0. Since the Baikov representation uses the inverse propagators D α as variables, it greatly facilitates the application of cuts.
In this section we fix our notations and review the Baikov representation of a general Feynman loop integral. We consider an integral with L loops and k propagators. We denote the loop momenta as 1 , . . . , L and the external momenta as p 1 , . . . , p E , p E+1 , where E thus denotes the number of linearly independent external mo-menta. Furthermore, the integrand may involve m − k irreducible scalar products-that is, polynomials in the loop momenta and external momenta which cannot be expressed as a linear combination of the inverse propagators. As will be shown below, m is a function of L and E. We apply dimensional regularization to regulate infrared and ultraviolet divergences and normalize the integral as follows, (2.1) The inverse propagators D j are of the form P 2 where P is an integer-coefficient linear combination of vectors taken from the ordered set of all independent external and loop momenta, Furthermore, the quantity N k,m in eq. (2.1) is defined as N k,m ≡ D ν k+1 k+1 · · · D νm m . We now proceed to present the Baikov representation [29] of the integral (2.1). To this end, we start by writing down the Gram matrix S of the independent external and loop momenta, 3) where the entries are given by, where v i and v j are entries of V in eq. (2.2). In addition, we let F denote the determinant of S, The entries of the upper-left E × E block of S are constructed out of the external momenta only, and it will be convenient for the following to emphasize this by relabeling these entries, Furthermore we define G as the Gram matrix of the independent external momenta, and let U denote its determinant, We remark that U is equal to the square of the volume of the parallelotope formed by the independent external momenta {p 1 , . . . , p E }. Thus, U is non-vanishing provided that p 1 , . . . , p E are not linearly dependent. The entries of the remaining blocks of S depend on the loop momenta. As S is a symmetric matrix, not all entries are independent. We can choose as a set of independent entries for example the entries of the upperright E × L block along with the upper-triangular entries of the lower-right L × L block, (2.9) Hence we find that S contains LE + L(L+1) 2 independent entries which depend on the loop momenta. From the fact that each inverse propagator D α is the square of a linear combination of the elements of V in eq. (2.2) and the fact that the elements of V are linearly independent, it follows that D α can be written as a unique linear combination of the x i,j in eq. (2.9). We therefore conclude that the combined number of propagators and irreducible scalar products in eq. (2.1) is given by the expression, Keeping the relabeling in eq. (2.6) in mind, we can write any inverse propagator D α (with α = 1, . . . , m) as an explicit linear combination of the x i,j in eq. (2.9) as follows, where A α,β and the entries of B α are integers. In writing this expression we introduced a lexicographic order on the set of elements (i, j) in eq. (2.9) and let β = 1, . . . , m denote the element label in the ordered set. The variables of the Baikov representation [29] are chosen as the inverse propagators and the irreducible scalar products, We can now present the Baikov representation of the integral in eq. (2.1). It takes the following form 1 , (2.13) 1 We remark that the Baikov representation in eq. (2.13) is consistent with that used in ref. [12]. This is a consequence of the identity det i,j=1,...,L µ i,j = F U , which in turn follows from the Schur complement theorem in linear algebra. Moreover, ref. [12] makes use of the four-dimensional helicity scheme. It is there-where the first prefactor is given by the expression, where A is the matrix defined in eq. (2.11).

III. INTEGRATION-BY-PARTS IDENTITIES ON UNITARITY CUTS
In this section we consider integration-by-parts identities (1.1) on cuts where some number of propagators are put on shell, i.e. roughly speaking 1 Di → δ(D i ). This has the advantage of reducing the linear systems to which Gauss-Jordan elimination is to be applied. As explained in ref. [12], it is possible to determine complete integration-by-parts reductions by performing the reductions on a suitably chosen spanning set of cuts and merge the information found on each cut.
The virtue of the Baikov representation (2.13) is that it makes manifest the effect of cutting propagators. Cf. refs. [11,12], we consider applying a c-fold cut (where 0 ≤ c ≤ k) to eq. (2.13). We let S cut , S uncut and S ISP denote the sets of indices labeling cut propagators, uncut propagators and irreducible scalar products respectively, and set, We will restrict the analysis to the case where the propagator powers in eq. (2.1) are equal to one, ν i = 1.
The result of applying the cut, where Γ (0) denotes a circle centered at 0 of radius > 0 to eq. (2.13) is obtained by evaluating the residue at We now turn to integration-by-parts identities evaluated on the c-fold cut S cut . Such identities correspond fore imposed as a constraint on the external momenta that they span a vector space of dimension at most four. In order words, one must have dim span{p 1 , . . . , p E } ≤ 4. Accordingly, the exponent of the Baikov polynomial to exact differential forms of degree m − c. The most general exact differential form which is of the form of the integrand of eq. (3.3) is, where the a i are polynomials in {z r1 , . . . , z rm−c }. Expanding eq. (3.4), we get an integration-by-parts identity, We observe that, for an arbitrary choice of polynomials a i (z), the two terms in the parenthesis (· · · ) in eq. (3.5) correspond to integrals in D and D − 2 dimensions, respectively. This is because the 1 F (z) factor in the second term has the effect of modifying the integration measure, thereby shifting the space-time dimension from D to D − 2.
To get the exact form in eq. (3.4) to correspond to an integration-by-parts relation in D dimensions, we require the a i (z) to be chosen such that, where b denotes a polynomial, since then the 1 F factor in eq. (3.5) cancels out, and no dimension shift occurs. Equations of the type (3.6) are known in algebraic geometry as syzygy equations (describing in our setting the polynomial relations-that is, syzygies, between F, ∂F ∂zr 1 , . . . , ∂F ∂zr m−c ). They have also been considered in the context of integration-by-parts relations in refs. [10][11][12][30][31][32]. We remark that it follows from Schreyer's theorem that a generating set of solutions of eq. (3.6) can be found algebraically by determining a Gröbner basis of the ideal generated by the above polynomials, considering the S-polynomials involved in the Buchberger test, and expressing the corresponding relations in terms of the original generators [33]. We refer to refs. [11,14] for a geometric interpretation of eq. (3.6).

IV. SYZYGY GENERATORS FROM LAPLACE EXPANSION
In this section we turn to obtaining a generating set T = g 1 , . . . , g d of syzygies g i = (a r1 , . . . , a rm−c , b) of eq. (3.6). By this we mean that T is such that any solution of eq. (3.6) can be written in the form g i p where g i ∈ T and p denotes a polynomial.
For a general polynomial F , determining a generating set of syzygies would require an S-polynomial computation. However, as we will shortly see, in the case where F is the determinant of a matrix, a generating set of syzygies can be obtained from the Laplace expansion of the determinant of F . We remark that related work has appeared in ref. [32].

A. Off-shell case
For simplicity we start with the case where no cuts are applied, c = 0. Let M = (m i,j ) i,j=1,...,n be a generic matrix, i.e. such that all entries are independent. We consider the determinant of M and perform Laplace expansion of the determinant along the ith row, (4.1) The identities with i = j follow by replacing the ith row of M by the jth row, m i,k → m j,k , as the resulting matrix clearly has a vanishing determinant.
For a symmetric matrix S = (s i,j ) i,j=1,...,n , the entries satisfy s i,j = s j,i and are thus not all independent. For this case, one obtains from the Laplace expansion the following identities, (4.2) In taking the derivatives one must take into account that the entries are not independent. To do so, we replace s j,i → s i,j with i ≤ j in S before taking derivatives and furthermore replace ∂(det S) ∂s i,k with i > k by ∂(det S) ∂s k,i . We will now apply the identity (4.2) to the Gram matrix S in eq. (2.3). However, before doing so, we note that the first E rows only contain external invariants λ i,j and entries which also appear in the last L rows by symmetry of S. Derivatives with respect to the λ i,j are not of interest in the problem at hand, since for integrationby-parts identities (1.1), only derivatives with respect to the loop momenta play a role. We therefore apply the identity (4.2) only to the last L rows of S, from which we find, where E + 1 ≤ i ≤ E + L and 1 ≤ j ≤ E + L. We can express the derivatives with respect to x i,k in terms of derivatives with respect to z α by making use of the chain rule, By splitting the sum in eq. (4.3) into sums over the first E, subsequent i − 1 − E and E + L − i + 1 terms and using that x i,k = x k,i for the former two, application of the chain rule (4.4) yields, where a i,j and b are given by the following expressions, We conclude that with a i,j and b i,j given in eq. (4.6) are solutions of eq. (3.6) in the case c = 0. We note that it follows from the relations in eqs. (2.11)-(2.12) that the derivatives ∂zα ∂x i,k are integers. Furthermore, we may use the relations to express the xvariables as a linear combination of the z-variables. This shows that the syzygies t i,j in eq. (4.7) are at most linear polynomials in the Baikov variables z α .
We emphasize that the closed-form expressions in eqs. (4.6)-(4.7) are valid for any number of loops and external legs. The only quantities that depend on the graph in question are the relations of the z-variables to the x-variables in eqs. (2.11)-(2.12). We note that the approach of using Laplacian expansion to obtain syzygies works equally well in cases where the propagators are massive, since the variables x i,j in eq. (2.4) will be independent of the internal mass parameters. These mass parameters will appear explicitly after the linear transformation from the x i,j variables to the Baikov variables z i . For an explicit example we refer to Sec. VI B.
We emphasize that the closed-form expressions allow the construction of purely D-dimensional integration-byparts identities in cases where S-polynomial based computations of syzygies are not feasible. Another important aspect of the syzygies in eqs. (4.6)-(4.7) is that they are of degree one. This would not be guaranteed for the output of an S-polynomial-based computation of the syzygy generators which in relevant examples (see below) turn out to have higher degrees. Low-degree syzygies are particularly advantageous if we are interested in imposing additional constraints on the Ansatz for the exact form in eq. (3.5). For example, we may demand that no integrals with squared propagators are encountered in the IBP identities, a i + b i z i = 0 where i = 1, . . . , k . That is, the generators of T ∩ L form a generating set of solutions of eqs. (3.6) and (4.8) [34]. The fact that the syzygies in eqs. (4.6)-(4.7) are of degree one dramatically simplifies the computation of the module intersection T ∩ L. We remark that efficient methods for computing module intersections are presented in ref. [35], and that in this reference non-trivial computations are carried out using these methods for non-planar multi-scale diagrams.
In Sec. V we give a proof that the L(L + E) syzygies in eq. (4.7) form a generating set.

B. On-shell case
We now turn to obtaining a generating set of syzygies of eq. (3.6) for a generic cut S cut = {ζ 1 , . . . , ζ c }. We start by taking the module of the syzygies in eq. (4.9) and evaluating this on the cut S cut , T = T zq=0, q∈Scut . (4.11) Now, the generators t i,j of T will not in general be solutions of eq. (3.6) because the ζ n -entries of t i,j may be nonzero on the cut. This leads us to consider the module, Z = e r1 , . . . , e rm−c , (4.12) where e ri is an (m + 1)-dimensional unit vector with 1 in the r i entry and 0 elsewhere. Namely, the generators of the intersection T ∩ Z are solutions of eq. (3.6). The module intersection can be found with Singular and in practice takes less than a second to compute.

V. PROOF OF COMPLETENESS OF SYZYGIES
In this section we show that the L(L + E) syzygies in eqs. (4.6)-(4.7) form a generating set of syzygies of eq. (3.6). In order to give a formal proof of this fact we adopt a more general setup considering polynomial logarithmic vector fields along determinants of generic (symmetric) square matrices. We reduce the problem to known resolutions of ideals of submaximal minors of such matrices.
Fix a field K. For 0 = m ∈ N denote by Y = K m affine m-space. The coordinate ring of Y is a polynomial ring A polynomial function We denote the ideal of partial derivatives of f by Then Der(− log(f )) identifies with the projection to the first m components of the syzygy module (cf. eq. (3.6)) We call χ ∈ Der(− log(f )) an Euler vector field for f if If f admits an Euler vector field, then where is the annihilator of f in Θ and can be identified with the syzygy module of J f . In the remainder of the section we specialize to the case where f is the determinant of a (symmetric) n×n matrix where 0 = n ∈ N. We write Mat n (O) for the O-module of n × n matrices with entries O, Sym n (O) (and Skw n (O)) for its submodule of (skew-)symmetric matrices and The preceding discussion applies to both these cases. The coordinate rings of X and X are polynomial rings The modules of polynomial vector fields on X and X respectively are The following result provides generators of the modules of logarithmic vector fields along f = det and f = det (cf. eq. (5.2)). Denote by M = (x i,j ) ∈ Mat n (O X ) the generic n × n matrix and by S = (x i,j ) ∈ Sym n (O X ) its symmetric counterpart. Note that det = det M , det = det S .
Assume from now on that K has characteristic different from 2 (which will be the case in our applications). Goryunov and Mond made the following observation (cf. Secs. 3.1-3.2 of ref. [39]).
where, using eq. (5.5), In particular, if det M admits an Euler vector field χ ∈ Θ, then Der(− log(det M )) is generated by χ and the image of π.
(b) If I n−1 (S) has (the maximal) codimension 3, then there is a surjective map In particular, if det S admits an Euler vector field χ ∈ Θ, then Der(− log(det S)) is generated by χ and the image of π .
Finally we specialize to the case of interest in our context.
Corollary 3. Assume that S ∈ Sym n (O) has a block form where S 1,1 is constant invertible and s i,j = x i,j = y σi,j for i ≤ j with (i, j) in block column 2. Then Der(− log(det S)) is generated by all

(5.11)
Proof. By Micali-Villamayor (see Lemma (1.1) of ref. [41]), there is an invertible matrix C such that The matrix S 0 is still constant invertible and S ≡ S 2,2 modulo the variables x i,j with (i, j) in block (1,2). The entries of S ∈ Sym n (O) are thus algebraically independent over the polynomial ring over K in these variables. By Józefiak (Thm. (2.3) of ref. [41]) it follows that I n−1 (S) = I n −1 (S ) has codimension 3. For well-definedness of π , it suffices to verify that π (A) ∈ Der(− log(det S)) if A = (δ i,k δ j, ). In this case and hence, using eq. (5.5) and Laplace expansion, Note that det S admits the Euler vector field (1 + δ i,n )y σi,n ∂ ∂y σi,n .
Returning to the setup of Sec. II, consider the matrix S in eq. (2.3) with the given block form. Its submatrix S 1,1 is the Gram matrix G in eq. (2.7) whose entries are the Mandelstam variables λ i,j in eq. (2.6) which are treated as constants in the integration and IBP reduction. As noted below eq. (2.8), U = det G is nonvanishing provided that p 1 , . . . , p E are linearly independent. Let K = Q(λ i,j ) be the field of rational functions in the Mandelstam variables over the rational numbers. Note that the characteristic of K is 0, so that the above assumption on the characteristic is satisfied. 2 Then S 1,1 is constant invertible and Corollary 3 applies. As a result the L(L + E) syzygies in eqs. (4.6)-(4.7) generate all syzygies in eq. (3.6). 2 Note that while in the actual integration the variables corre-

VI. EXAMPLES
In this section we work out explicit expressions for the syzygy generators presented in Sec. IV for three diagrams.
A. Fully massless planar double box As a simple example we consider the fully massless planar double-box diagram shown in fig. 1. In this case, the combined number of propagators and irreducible scalar products (2.10) is m = 9. In agreement with eq. (2.12), we define the z-variables as follows, setting P 1,2 ≡ p 1 + p 2 , (6.1) We choose as the set (2.2) of all independent external and loop momenta V = (p 1 , p 2 , p 3 , 1 , 2 ). The lexicographically-ordered set of elements (i, j) in eq. (2.9) then becomes, and it immediately follows that the matrices in eq. (2.11) sponding to the external momenta may take real values and the remaining ones may take complex values, the IBP relations have a generating system defined over the rationals. Both the Laplace expansion and a syzygy module computation via Gröbner basis methods lead to such a generating system. This again simplifies the computation of solutions satisfying further constraints.
are given by, and, for α = 1, . . . , 9, 4) and m α = 0. We can now use eqs. (2.11)-(2.12) to express the syzygy generators in eqs. (4.6)-(4.7) in terms of the z α , yielding, Syzygies obtained from S-polynomial-based computations are not guaranteed to be of degree one. For example, from the Singular command syz one can obtain a representation with 13 generators of up to cubic degree. More specifically, syz produces 10 generators of degree one, two generators of degree two, and one generator of degree three.
Expressions for on-shell syzygies are too lengthy to record here, but we give a few examples: on the cut S cut = {1, 4, 7} one can find a representation of T ∩ Z with 18 generators of up to cubic degree, and on the cut S cut = {2, 5, 7} a representation of T ∩ Z with 20 generators of up to cubic degree.

B. Planar double box with internal mass
As a more non-trivial example we consider a planar double-box diagram with propagators of equal mass as shown in fig. 2. As a yet more non-trivial example we consider the fully massless non-planar double-pentagon diagram shown in fig. 3.

VII. CONCLUSIONS
Integration-by-parts (IBP) identities between loop integrals arise from the vanishing integration of total derivatives in dimensional regularization. The condition that a total derivative leads to an IBP identity which does not involve dimension shifts can be stated as the syzygy equation (3.6). We presented in eqs. (4.6)-(4.7) an explicit generating set of solutions of the syzygy equation, valid for any number of loops and external momenta. In general, S-polynomial computations would be required in order to obtain the syzygies. However, as the Baikov polynomial (2.5) is the determinant of a matrix, a generating set of syzygies can be obtained from the Laplace expansion of the determinant. Moreover, we showed that the syzygies needed for IBP identities evaluated on a generalized-unitarity cut can be obtained immediately from eqs. (4.6)-(4.7) by a straightforward module intersection computation.
We emphasize that the closed-form expressions in eqs. (4.6)-(4.7) are valid for any number of loops and external legs. The only quantities that depend on the graph in question are the relations of the z-variables to the xvariables in eqs. (2.11)-(2.12). In particular, the closedform expressions allow the construction of purely Ddimensional IBP identities in cases where S-polynomial based computations of syzygies are not feasible. An example of the latter is the non-planar double-pentagon diagram considered in Sec. VI C.
Moreover, an important feature of the syzygies eqs. (4.6)-(4.7) is that they are guaranteed to have degree at most one. In contrast, a generating set of syzygies obtained from an S-polynomial computation would in general have higher degrees. The fact that the syzygies obtained here are of degree at most one is useful because as a result the computation of solutions which satisfy further constraints is dramatically simplified. For example, one may be interested in imposing the further constraint on the total derivatives that no integrals with squared propagators are encountered in the IBP identities.
It is worth pointing out that ref. [42] makes use of syzygies to construct IBP identities which involve arbitrary numerator powers. These are then solved as difference equations to obtain the IBP reductions. Based on preliminary tests of several two-loop examples, the syzygies from Laplacian expansion considered here can also produce recursive relations similar to the relations in ref. [42]. This direction merits further investigation.