Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals
Janko Boehm Alessandro Georgoudis Kasper J. Larsen Mathias Schulze Yang Zhang
hep-th hep-ph math.AG


.
MITP/17-104, UUITP-44/17





boehm@mathematik.uni-kl.de
Department of Mathematics, TU Kaiserslautern, 67663 Kaiserslautern, Germany
Alessandro.Georgoudis@physics.uu.se
Department of Physics and Astronomy, Uppsala University, SE-75108 Uppsala, Sweden
Kasper.Larsen@soton.ac.uk
School of Physics and Astronomy, University of Southampton, Highfield, Southampton, 
 SO17 1BJ, United Kingdom
mschulze@mathematik.uni-kl.de
Department of Mathematics, TU Kaiserslautern, 67663 Kaiserslautern, Germany
zhang@uni-mainz.de
ETH Zürich, Wolfang-Pauli-Strasse 27, 8093 Zürich, Switzerland
PRISMA Cluster of Excellence, Johannes Gutenberg University, 55128 Mainz, Germany


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.


Complete sets of logarithmic vector fields 
 for integration-by-parts identities of Feynman integrals
    Yang Zhang
    December 30, 2023
=====================================================================================================






§ 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,

    ∫∏_i=1^L ^̣D ℓ_i/iπ^D/2∑_j=1^L ∂/∂ℓ_j^μv_j^μ P/D_1^ν_1⋯ D_m^ν_m= 0  ,

where P and the vectors v_j^μ are polynomial in the loop momenta ℓ_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. (<ref>) may be carried out
with the Laporta algorithm <cit.>,
which leads in general to relations that involve integrals with
squared propagators. There are various publicly available implementations
of automated IBP reduction: AIR <cit.>,
FIRE <cit.>,
Reduze <cit.>,
LiteRed <cit.>, Kira <cit.>,
as well as private implementations.
A formalism for obtaining IBP reductions without squared
propagators was developed in refs. <cit.>.
A systematic method of deriving IBP reductions on
generalized-unitarity cuts was given in ref. <cit.>.
A recent approach <cit.> uses sampling
over finite fields to construct the reduction coefficients.
Other recent developments include software for determining a
basis of integrals <cit.> and a
D-module theory based approach for computing the number
of basis integrals <cit.>.

The IBP reductions moreover allow setting up differential equations
for the basis integrals, thereby enabling their evaluation.
 <cit.>.
Differential equations have proven to be a powerful tool
for calculating multi-loop integrals, enabling for example
the computation of the basis integrals for numerous
two-loop amplitudes of 2 → 2 processes. This method
can therefore reasonably be expected to be of relevance
to two-loop amplitudes for 2 → 3 processes. We note that,
in the context of the latter, impressive results have recently
appeared <cit.>.

In this paper we study integration-by-parts identities (<ref>).
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. <ref>
we set up notation and give the Baikov representation
of a generic Feynman loop integral. In Sec. <ref>
we study integration-by-parts relations on unitarity cuts
and derive the syzygy equation of interest.
In Sec. <ref> we obtain
a closed-form generating set of solutions to the syzygy equation.
In Sec. <ref>
we present a proof that the set of syzygies is complete.
In Sec. <ref> we provide an example
of the formalism.
In Sec. <ref> we give our conclusions.




§ 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 (<ref>)
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 ℓ_1, …, ℓ_L
and the external momenta as p_1, …, p_E, p_E+1,
where E thus denotes the number of linearly independent external
momenta. 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,

    I(ν_1, …, ν_m; D) ≡∫∏_j=1^L ^̣D ℓ_j/iπ^D/2N_k,m/D_1^ν_1⋯ D_k^ν_k , ν_i ≥ 0  .

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,

    V = ( v_1, …, v_E+L) = ( p_1, …, p_E, ℓ_1, …, ℓ_L )  .

Furthermore, the quantity N_k,m
in eq. (<ref>) is defined as
N_k,m≡ D_k+1^ν_k+1⋯ D_m^ν_m.

We now proceed to present the Baikov representation <cit.> of
the integral (<ref>). To this end, we start
by writing down the Gram matrix S of the independent external and loop momenta,

    S =
    (
     [     x_1,1         ⋯     x_1,E   x_1,E+1         ⋯   x_1,E+L;         ⋮         ⋱         ⋮         ⋮         ⋱         ⋮;     x_E,1         ⋯     x_E,E   x_E,E+1         ⋯   x_E,E+L;   x_E+1,1         ⋯   x_E+1,E x_E+1,E+1         ⋯ x_E+1,E+L;         ⋮         ⋱         ⋮         ⋮         ⋱         ⋮;   x_E+L,1         ⋯   x_E+L,E x_E+L,E+1         ⋯ x_E+L,E+L ])  ,

where the entries are given by,

    x_i,j = v_i · v_j  ,

where v_i and v_j are entries of V in eq. (<ref>).
In addition, we let F denote the determinant of S,

    F ≡ 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,

    λ_i,j  =  x_i,jfor 1 ≤ i, j ≤ E  .

Furthermore we define G as the Gram matrix of
the independent external momenta,

    G =
    [ λ_1,1     ⋯ λ_1,E;     ⋮     ⋱     ⋮; λ_E,1     ⋯ λ_E,E ] ,

and let U denote its determinant,

    U =  G  .

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 upper-right E × L block
along with the upper-triangular entries of the lower-right L × L block,

    x_i,jwhere{[ 1 ≤ i ≤ E and E+1 ≤ j ≤ E+L  ,;           E+1 ≤ i ≤ j ≤ E+L  . ].

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. (<ref>) 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. (<ref>).
We therefore conclude that the combined number of propagators and
irreducible scalar products in eq. (<ref>)
is given by the expression,

    m = LE + L(L+1)/2 .

Keeping the relabeling in eq. (<ref>)
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. (<ref>)
as follows,

    D_α = ∑_β=1^m A_α,β x_β + ∑_1≤ k ≤ l ≤ E (B_α)_k,lλ_k,l- m_α^2  ,

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. (<ref>) and
let β = 1, …, m denote the element label
in the ordered set.

The variables of the Baikov representation <cit.>
are chosen as the inverse propagators and the irreducible scalar products,

    z_α≡ D_αwhere 1 ≤α≤ m  .

We can now present the Baikov representation of the integral in eq. (<ref>).
It takes the following form
[
We remark that the Baikov representation in eq. (<ref>)
is consistent with that used in ref. <cit.>.
This is a consequence of the identity _i,j=1,…,Lμ_i,j = F/U,
which in turn follows from the Schur complement theorem in linear algebra.
Moreover, ref. <cit.> makes use of the four-dimensional helicity
scheme. It is therefore 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
dimspan{ p_1, …, p_E}≤ 4.
Accordingly, the exponent of the Baikov polynomial F
is modified from D-L-E-1/2 in eq. (<ref>)
to D-L-5/2 and D-L-4/2
for E ≥ 4 and E=3, respectively.
],

    I(ν; D)     =  C_E^L  U^E-D+1/2∫ẓ_1 ⋯ẓ_m/z_1^ν_1⋯ z_k^ν_k F^D-L-E-1/2 N_k,m ,

where the first prefactor is given by the expression,

    C_E^L ≡π^-L(L-1)/4 - L E/2/∏_j=1^L Γ(D-L-E+j/2) A  ,

where A is the matrix defined in eq. (<ref>).




§ INTEGRATION-BY-PARTS IDENTITIES ON UNITARITY CUTS



In this section we consider integration-by-parts identities
(<ref>) on cuts where some number of propagators
are put on shell, i.e. roughly speaking 1/D_i→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. <cit.>,
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 (<ref>)
is that it makes manifest the effect of cutting propagators.
Cf. refs. <cit.>, we consider
applying a c-fold cut (where 0≤ c ≤ k)
to eq. (<ref>). 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,

    𝒮_cut   =  {ζ_1, …, ζ_c }
    𝒮_uncut   =  { r_1, …, r_k-c}
    𝒮_ISP   =  { r_k-c+1, …, r_m-c} .

We will restrict the analysis to the case where the
propagator powers in eq. (<ref>)
are equal to one, ν_i = 1.

The result of applying the cut,

    ∫ẓ_i/z_i⟶^cut∮_Γ_ϵ (0)ẓ_i/z_iwhere i ∈𝒮_cut ,

where Γ_ϵ (0) denotes a circle centered at 0 of radius ϵ > 0
to eq. (<ref>) is obtained by evaluating the residue at
z_i = 0 where i ∈𝒮_cut,

    I_cut (ν; D)    = C_E^L(D) U^E-D+1/2
       ×∫ẓ_r_1⋯ẓ_r_m-c N_k,m/z_r_1⋯ z_r_k-c
    F(z)^D-L-E-1/2|_z_i = 0,  i ∈𝒮_cut .


We now turn to integration-by-parts identities evaluated
on the c-fold cut 𝒮_cut. Such identities
correspond to exact differential forms of degree m-c. The most
general exact differential form which is of the form
of the integrand of eq. (<ref>) is,

    0    = ∫( ∑_i=1^m-c(-1)^i+1 a_r_i F(z)^D-L-E-1/2/z_r_1⋯ z_r_k-c
       ×ẓ_r_1∧⋯∧ẓ_r_i∧⋯∧ẓ_r_m-c)  ,

where the a_i are polynomials in { z_r_1, …, z_r_m-c}.
Expanding eq. (<ref>),
we get an integration-by-parts identity,

    0    = ∫[ ∑_i=1^m-c( ∂ a_r_i/∂ z_r_i
    + D-L-E-1/2 F(z) a_r_i∂ F/∂ z_r_i)
    - ∑_i=1^k-ca_r_i/z_r_i] 
       ×F(z)^D-L-E-1/2/z_r_1⋯ z_r_k-cẓ_r_1⋯ẓ_r_m-c .

We observe that, for an arbitrary choice of polynomials a_i(z),
the two terms in the parenthesis (⋯)
in eq. (<ref>) 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. (<ref>)
to correspond to an integration-by-parts relation in D dimensions,
we require the a_i(z) to be chosen such that,

    b F + ∑_i=1^m-c a_r_i∂ F/∂ z_r_i = 0  ,

where b denotes a polynomial, since then the 1/F factor
in eq. (<ref>) cancels out, and no dimension shift occurs.
Equations of the type (<ref>) are known in algebraic geometry as
syzygy equations (describing in our setting the polynomial relations—that is,
syzygies, between F, ∂ F/∂ z_r_1,…,∂ F/∂ z_r_m-c). They have also been considered
in the context of integration-by-parts relations in
refs. <cit.>.
We remark that it follows from Schreyer's theorem
that a generating set of solutions of eq. (<ref>)
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 <cit.>.
We refer to refs. <cit.> for a geometric
interpretation of eq. (<ref>).




§ 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_r_1, …, a_r_m-c, b) of
eq. (<ref>). By this we mean that
𝒯 is such that any solution of eq. (<ref>)
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. <cit.>.




 §.§ 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,

    [ ∑_k=1^n m_j,k∂ ( M)/∂ m_i,k] - δ_i,j M = 0  ,
     1 ≤ i,j ≤ n  .

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,

    [∑_k=1^n (1+δ_i,k) s_j,k∂ ( S)/∂ s_i,k] - 2δ_i,j S = 0  ,
     1 ≤ i,j ≤ n  .

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 ∂ ( S)/∂ s_i,k
with i>k by ∂ ( S)/∂ s_k,i.

We will now apply the identity (<ref>)
to the Gram matrix S in eq. (<ref>). 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 integration-by-parts identities (<ref>),
only derivatives with respect to the loop momenta play a role.
We therefore apply the identity (<ref>)
only to the last L rows of S, from which we find,

    [∑_k=1^E+L (1+δ_i,k) x_j,k∂ F/∂ x_i,k] - 2δ_i,j F = 0  ,

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,

    ∂ F/∂ x_i,k =
    ∑_α=1^m ∂ z_α/∂ x_i,k∂ F/∂ z_αfor{[ 1 ≤ i ≤ E  ,
     E+1 ≤ k ≤ E+L  ,;           E+1 ≤ i ≤ k ≤ E+L  . ].

By splitting the sum in eq. (<ref>)
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 (<ref>) yields,

    ∑_α=1^m (a_i,j)_α∂ F/∂ z_α + b_i,j F = 0  ,

where a_i,j and b are given by the following expressions,

    (a_i,j)_α =  ∑_k=1^E+L (1 + δ_i,k) ∂ z_α/∂ x_i,k x_j,kand
    b_i,j          = -2δ_i,j ,

for E+1 ≤ i ≤ E+L, 1 ≤ j ≤ E+L and 1 ≤α≤ m.
We conclude that

    t_i,j = ((a_i,j)_1, …, (a_i,j)_m, b_i,j)  ,

with a_i,j and b_i,j given in eq. (<ref>)
are solutions of eq. (<ref>) in the case c=0.

We note that it follows from the relations
in eqs. (<ref>)–(<ref>)
that the derivatives ∂ z_α/∂ x_i,k
are integers. Furthermore, we may use the relations to express
the x-variables as a linear combination of the z-variables.
This shows that the syzygies t_i,j in eq. (<ref>)
are at most linear polynomials in the Baikov variables z_α.

We emphasize that the closed-form expressions in
eqs. (<ref>)–(<ref>)
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. (<ref>)–(<ref>).
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. (<ref>)
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. <ref>.

We emphasize that the closed-form expressions allow
the construction of purely D-dimensional integration-by-parts identities
in cases where S-polynomial based computations of syzygies
are not feasible.
Another important aspect of the syzygies
in eqs. (<ref>)–(<ref>)
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. (<ref>).
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  .

Namely, we can obtain solutions of eqs. (<ref>) and (<ref>)
by taking the module intersection of the module of the syzygies
in eqs. (<ref>)–(<ref>),

    𝒯 = ⟨ t_i,j|  E+1 ≤ i ≤ E+L
    and  1≤ j ≤ E+L ⟩ ,

and the module,

    ℒ = ⟨ z_1 e_1, …, z_k e_k, e_k+1, …, e_m ⟩ .

That is, the generators of 𝒯∩ℒ form a
generating set of solutions of eqs. (<ref>) and (<ref>) <cit.>.
The fact that the syzygies in eqs. (<ref>)–(<ref>)
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. <cit.>, and that in this reference non-trivial computations
are carried out using these methods for non-planar multi-scale diagrams.

In Sec. <ref> we give a proof
that the L(L+E) syzygies in eq. (<ref>)
form a generating set.




 §.§ On-shell case



We now turn to obtaining a generating set of syzygies of
eq. (<ref>) for a generic cut
𝒮_cut = {ζ_1, …, ζ_c }.
We start by taking the module of the syzygies in
eq. (<ref>) and evaluating this on the cut 𝒮_cut,

    𝒯=𝒯|_z_q=0,  q ∈𝒮_cut .

Now, the generators t_i,j of 𝒯 will not in general
be solutions of eq. (<ref>) because
the ζ_n-entries of t_i,j may be nonzero on the cut.

This leads us to consider the module,

    𝒵 = ⟨ e_r_1, …, e_r_m-c⟩ ,

where e_r_i 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. (<ref>).

The module intersection can be found with Singular
and in practice takes less than a second to compute.




§ PROOF OF COMPLETENESS OF SYZYGIES



In this section we show that the L(L+E) syzygies in eqs. (<ref>)–(<ref>)
form a generating set of syzygies of eq. (<ref>).
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 .
For 0 m∈ denote by Y=^m affine m-space.
The coordinate ring of Y is a polynomial ring

    Ø=Ø_Y=[y_1,…,y_m]  .

Note that its group of units is Ø^*=^*=∖0.
Since Ø is a Cohen–Macaulay ring, the grade or depth of any ideal of Ø equals its height or codimension
(cf. Cor. 2.1.4 and Thm. 2.1.9 of ref. <cit.>).

The polynomial vector fields on Y form a free Ø-module (cf. Prop. 16.1 of ref. <cit.>)

    Θ=Θ_Y=_(Ø)=⊕_i=1^mØ/ y_i .

A polynomial function

    f Y=^m→

is given by an element f∈Ø.
The Ø-submodule of Θ of logarithmic vector fields along the divisor (f) is defined by (classically for squarefree f∈Ø∖Ø^* and =, cf. Sec. 1 of ref. <cit.>)

    (-log(f))=δ∈Θδ(f)∈Ø f⊂Θ_Y  .

We denote the ideal of partial derivatives of f by

    _f= f/ y_1,…, f/ y_m .

Then (-log(f)) identifies with the projection to the first m components of the syzygy module (cf. eq. (<ref>))

    ( f/ y_1,…, f/ y_m,f)≅(_f+Ø f)  .

We call χ∈(-log(f)) an Euler vector field for f if

    χ(f)∈Ø^*f=^*f  .

If f admits an Euler vector field, then

    (-log(f))=Øχ+_Θ(f)  ,

where

    _Θ(f)=δ∈Θδ(f)=0≅(_f)

is the annihilator of f in Θ and can be identified with the syzygy module of _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∈.
We write _n(Ø) for the Ø-module of n× n matrices with entries Ø, _n(Ø) (and _n(Ø)) for its submodule of (skew-)symmetric matrices and

    _n(Ø)=(_n(Ø)→Ø)

the kernel of the trace map.
For any A=(a_i,j)∈_n(Ø) denote by A^⋆=(a^⋆_i,j)∈_n(Ø) its adjoint matrix and by

    I_n-1(A)=a^⋆_i,j1≤ i≤ j≤ nØ

its ideal of submaximal minors.

Consider the determinant functions

    X=_n()→ ,  ' X'=_n()→ .

The preceding discussion applies to both these cases.
The coordinate rings of X and X' are polynomial rings

    Ø_X   =[x_i,j| 1≤ i,j≤ n]  ,
    Ø_X'   =[x_i,j| 1≤ i≤ j≤ n]=Ø_X/x_i,j-x_j,i .

The modules of polynomial vector fields on X and X' respectively are

    Θ_X   =_(Ø_X)=⊕_i,jØ_X/ x_i,j ,
    Θ_X'   =_(Ø_X')=⊕_i≤ jØ_X/ x_i,j .

The following result provides generators of the modules of logarithmic vector fields along f= and f=' (cf. eq. (<ref>)).

Denote by M=(x_i,j)∈_n(Ø_X) the generic n× n matrix and by S=(x_i,j)∈_n(Ø_X') its symmetric counterpart.
Note that

    = M ,   '= S  .

Assume from now on that  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. <cit.>).




There are surjective maps

    @R=0em_n(Ø_X)^⊕2@->>[r]^-π   (-log())  ,
    
    (A,B)@|->[r]    ∑_i,j(MA-BM)_i,j/ x_i,j ,
    _n(Ø_X')@->>[r]^-π'   (-log('))  ,
    
    A@|->[r]    ∑_i≤ j(SA+A^tS)_i,j/ x_i,j .




Since

    / x_i,j=m^⋆_j,i ,  '/ x_i,j=(2-δ_i,j)s^⋆_i,j ,

we have

    _=I_n-1(M)  ,  _'=I_n-1(S)  ,

and, by Laplace expansion, π and π' map to the given target.
Since both  and ' are homogeneous, they admit standard Euler vector fields

    χ=∑_i,jx_i,j/ x_i,j ,  χ'=∑_i≤ jx_i,j/ x_i,j .

Note that

    π((δ_i,j),0)=χ ,  π'((δ_i,j))=2χ .

By Gulliksen–Negård <cit.> and Józefiak <cit.> respectively, there are exact sequences

    @R=0em@C=2em_n(Ø_X)^⊕2[r]    _n(Ø_X)[r]     I_n-1(M)[r]     0  ,
    
    (A,B)@|->[r]     MA-BM  ,
        C@|->[r]    (M^⋆ C)  ,
    _n(Ø_X')[r]    _n(Ø_X')[r]     I_n-1(S)[r]     0  ,
    
    A@|->[r]     SA+A^tS  ,
        D@|->[r]    (S^⋆ D)  ,

where, using eq. (<ref>),

    (M^⋆ C)   =∑_i,jm^⋆_j,ic_i,j=∑_i,jc_i,j/ x_i,j ,
    (S^⋆ D)   =∑_i,js^⋆_i,jd_i,j
    =∑_i≤ j(2-δ_i,j)s^⋆_i,jd_i,j
       =∑_i≤ jd_i,j'/ x_i,j .

Using eqs. (<ref>) and (<ref>) this means that

    π(_n(Ø_X)^⊕2)   =_Θ_X()  ,
    π'(_n(Ø_X'))   =_Θ_X'(')  .

With eqs. (<ref>) and (<ref>) surjectivity of π and π' follows.




Dropping the genericity hypothesis, we now consider polynomial matrix families

    @R=0em
    Y=^m[r]^-M    _n()=X  ,
    
    Y=^m[r]^-S    _n()=X'  .

They are defined by matrices M∈_n(Ø) and S∈_n(Ø) with entries in Ø=Ø_Y (cf. eq. (<ref>)).
By abuse of notation, we set

    _M   = M/ y_k k=1,…,m⊂_n(Ø)  ,
    _S   = S/ y_k k=1,…,m⊂_n(Ø)  .

Consider the (truncated) Gulliksen–Negård and Józefiak complexes

    @R=0em@C=2em_n(Ø)^⊕2[r]^-ρ   _n(Ø)[r]     I_n-1(M)[r]     0  ,
    _n(Ø)[r]^-ρ'   _n(Ø)[r]     I_n-1(S)[r]     0  .




[(a)]


  *  If I_n-1(M) has (the maximal) codimension 4, then there is a surjective map

    @R=0em@C=0.5emρ^-1(_M )@->>[rr]^-π      _Θ( M),
    
    (A,B)@|->[r]     MA-BM=∑_k=1^mc_k M/ y_k@|->[r]    ∑_k=1^mc_k/ y_k .

In particular, if M admits an Euler vector field χ∈Θ, then (-log( M)) is generated by χ and the image of π.


  *  If I_n-1(S) has (the maximal) codimension 3, then there is a surjective map

    @R=0em@C=0.5emρ'^-1(_S )@->>[rr]^-π'      _Θ( S)  ,
    
    A@|->[r]     SA+A^tS=∑_k=1^mc_k S/ y_k@|->[r]    ∑_k=1^mc_k/ y_k .

In particular, if S admits an Euler vector field χ∈Θ, then (-log( S)) is generated by χ and the image of π'.





Using eq. (<ref>), the chain rule yields

    (M^⋆ M/ y_k)   =∑_i,j m_i,j/ y_k/ x_i,j(M)= M/ y_k ,
    (S^⋆ S/ y_k)   =∑_i≤ j s_i,j/ y_k/ x_i,j(S)= S/ y_k .

By Gulliksen–Negård <cit.> and Józefiak <cit.> respectively, the hypotheses imply that the complexes (<ref>) are exact.
By eq. (<ref>) they induce exact sequences

    @R=0emρ^-1(_M)[r]^-ρ   _M[r]    _ M[r]     0  ,
    ρ'^-1(_S)[r]^-ρ'   _S[r]    _ S[r]     0  .

With eq. (<ref>) surjectivity of π and π' follows.
The particular claims are due to eq. (<ref>).




Finally we specialize to the case of interest in our context.


Assume that S∈_n(Ø) has a block form

    S=(s_i,j)=
    [ S_1,1 S_1,2; S_2,1 S_2,2 ] ,

where S_1,1 is constant invertible and s_i,j=x_i,j=y_σ_i,j for i≤ j with (i,j) in block column 2.
Then (-log( S)) is generated by all

    π'(A)=∑_k=1^mc_k/ y_k ,

where (cf. Proposition <ref>.(<ref>))

    ∑_k=1^mc_k S/ y_k=SA+A^tS  ,   
    A=
    [     0 A_1,2;     0 A_2,2 ]∈_n(Ø)  .




By Micali–Villamayor (see Lemma (1.1) of ref. <cit.>), there is an invertible matrix C such that

    C^tSC=
    [ S_0   0;   0  S' ] .

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'∈_n'(Ø) are thus algebraically independent over the polynomial ring over  in these variables.
By Józefiak (Thm. (2.3) of ref. <cit.>) 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)∈(-log( S)) if A=(δ_i,kδ_j,ℓ).
In this case

    SA+A^tS=∑_i=1^n(1+δ_i,ℓ)y_σ_i,k S/ y_σ_i,ℓ ,

and hence, using eq. (<ref>) and Laplace expansion,

    π'(A)( S)
       =∑_i=1^n(1+δ_i,ℓ)y_σ_i,k S/ y_σ_i,ℓ
       =∑_i=1^n(1+δ_i,ℓ)(2-δ_i,ℓ)s_i,ks^⋆_i,ℓ
    =2δ_k,ℓ S  .

Note that S admits the Euler vector field

    χ=π'((δ_i,nδ_j,n))=∑_i=1^n(1+δ_i,n)y_σ_i,n/ y_σ_i,n .

So the hypotheses of Proposition <ref>.(<ref>) are satisfied.

The module _S consists of all symmetric matrices with (1,1)-block 0.
Writing A∈_n(Ø) in block form

    A=
    [ A_1,1 A_1,2; A_2,1 A_2,2 ] ,

eq. (<ref>) reduces to

    S_1,1A_1,1+S_1,2A_2,1+
    A_1,1^tS_1,1+A_2,1^tS_2,1=0  .

For any W∈_n(Ø), adding WS to A leaves SA+A^tS invariant.
Using

    W=
    [ -A_1,1S_1,1^-1-S_1,1^-1A_2,1^tS_2,1S_1,1^-1                             S_1,1^-1A_2,1^t;                              -A_2,1S_1,1^-1                                           0 ]

makes A_*,1=0 and turns eq. (<ref>) into 0=0.


Returning to the setup of Sec. <ref>, consider the matrix S in eq. (<ref>) with the given block form.
Its submatrix S_1,1 is the Gram matrix G in eq. (<ref>) whose entries are the Mandelstam variables λ_i,j in eq. (<ref>) which are treated as constants in the integration and IBP reduction.
As noted below eq. (<ref>), U= G is non-vanishing provided that p_1, …, p_E are linearly independent.
Let =(λ_i,j) be the field of rational functions in the Mandelstam variables over the rational numbers.
Note that the characteristic of  is 0, so that the above assumption on the characteristic is satisfied.
[Note that while in the actual integration the variables corresponding 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.]
Then S_1,1 is constant invertible and Corollary <ref> applies.
As a result the L(L+E) syzygies in eqs. (<ref>)–(<ref>) generate all syzygies in eq. (<ref>).




§ EXAMPLES



In this section we work out explicit expressions for
the syzygy generators presented in Sec. <ref>
for three diagrams.



 §.§ Fully massless planar double box


As a simple example we consider the fully massless planar
double-box diagram shown in fig. <ref>.



In this case, the combined number of propagators
and irreducible scalar products (<ref>)
is m = 9. In agreement with eq. (<ref>),
we define the z-variables as follows, setting
P_1,2≡ p_1 + p_2,

    3
    z_1   =ℓ_1^2 ,                  z_2 = (ℓ_1 - p_1)^2 ,         z_3 = (ℓ_1 - P_1,2)^2 , 
    
    z_4   =(ℓ_2+P_1,2)^2 ,         z_5 = (ℓ_2 - p_4)^2 ,         z_6 = ℓ_2^2 , 
    
    z_7   =(ℓ_1+ℓ_2)^2 ,         z_8 = (ℓ_1 + p_4)^2 ,         z_9 = (ℓ_2 + p_1)^2 .

We choose as the set (<ref>) of
all independent external and loop momenta V = (p_1, p_2, p_3, ℓ_1, ℓ_2).
The lexicographically-ordered set of elements (i,j)
in eq. (<ref>) then becomes,

    (x_1, …, x_9)    ≡
    (x_1,4,  x_1,5,  x_2,4,
     x_2,5,  x_3,4,  x_3,5, 
        x_4,4,  x_4,5,  x_5,5)  ,

and it immediately follows that the matrices in
eq. (<ref>) are given by,

    A = [  0  0  0  0  0  0  1  0  0; -2  0  0  0  0  0  1  0  0; -2  0 -2  0  0  0  1  0  0;  0  2  0  2  0  0  0  0  1;  0  2  0  2  0  2  0  0  1;  0  0  0  0  0  0  0  0  1;  0  0  0  0  0  0  1  2  1; -2  0 -2  0 -2  0  1  0  0;  0  2  0  0  0  0  0  0  1;    ] ,

and, for α = 1, …, 9,

    B_α = 0 forα∉{ 3,4 }and
    B_3 = B_4 = [ 0 1 0; 1 0 0; 0 0 0 ] ,

and m_α = 0.
We can now use eqs. (<ref>)–(<ref>)
to express the syzygy generators in
eqs. (<ref>)–(<ref>)
in terms of the z_α, yielding,

    t_4,1   = (z_1-z_2,  z_1-z_2, -s+z_1-z_2,  0,  0,  0, 
        z_1-z_2-z_6+z_9,  t+z_1-z_2,  0,  0)  , 
    
    t_4,2   = (s+z_2-z_3,  z_2-z_3,  z_2-z_3,  0,  0,  0, 
        z_2-z_3+z_4-z_9, -t+z_2-z_3,  0,  0)  , 
    
    t_4,3   = (-s+z_3-z_8,  t+z_3-z_8,  z_3-z_8,  0,  0,  0, 
        z_3-z_4+z_5-z_8,  z_3-z_8,  0,  0)  , 
    
    t_4,4   = (2 z_1,  z_1+z_2, -s+z_1+z_3,  0,  0,  0, 
        z_1-z_6+z_7,  z_1+z_8,  0, -2)  , 
    
    t_4,5   = (-z_1-z_6+z_7, -z_1+z_7-z_9,  s-z_1-z_4+z_7, 
        0,  0,  0, -z_1+z_6+z_7, -z_1-z_5+z_7,  0,  0)  , 
    
    t_5,1   = (0,  0,  0,  s-z_6+z_9, -t-z_6+z_9,  z_9-z_6, 
        z_1-z_2-z_6+z_9,  0,  z_9-z_6,  0)  , 
    
    t_5,2   = (0,  0,  0,  z_4-z_9,  t+z_4-z_9, -s+z_4-z_9, 
        z_2-z_3+z_4-z_9,  0,  z_4-z_9,  0)  , 
    
    t_5,3   = (0,  0,  0,  z_5-z_4,  z_5-z_4,  s-z_4+z_5, 
        z_3-z_4+z_5-z_8,  0, -t-z_4+z_5,  0)  , 
    
    t_5,4   = (0,  0,  0,  s-z_3-z_6+z_7, -z_6+z_7-z_8, -z_1-z_6+z_7, 
        z_1-z_6+z_7,  0, -z_2-z_6+z_7,  0)  , 
    
    t_5,5   = (0,  0,  0, -s+z_4+z_6,  z_5+z_6,  2 z_6, 
       -z_1+z_6+z_7,  0,  z_6+z_9, -2)  .


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.





 §.§ 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. <ref>.



As in the massless case in Sec. <ref>, the combined number of propagators
and irreducible scalar products (<ref>)
is m = 9. In analogy with eq. (<ref>),
we define the z-variables as follows, setting
P_1,2≡ p_1 + p_2,

    3
    z_1   =ℓ_1^2 - M^2 ,                       z_2 = (ℓ_1 - p_1)^2 - M^2 ,    
    
    z_3   = (ℓ_1 - P_1,2)^2  - M^2 ,         z_4 = (ℓ_2+P_1,2)^2  - M^2 , 
    
    z_5   = (ℓ_2 - p_4)^2 - M^2 ,              z_6 = ℓ_2^2  - M^2  ,          
    
    z_7   =(ℓ_1+ℓ_2)^2 ,                    z_8 = (ℓ_1 + p_4)^2 ,          
    
    z_9   = (ℓ_2 + p_1)^2 .

Again we choose as the set (<ref>) of
all independent external and loop momenta V = (p_1, p_2, p_3, ℓ_1, ℓ_2).
The lexicographically-ordered set of elements x_i,j
in eq. (<ref>)
is again that in eq. (<ref>), and
the matrices in eq. (<ref>) are given by
eqs. (<ref>) and (<ref>),
whereas m_α in eq. (<ref>) is given by
m_α = M for 1 ≤α≤ 6 and m_β = 0
for 7 ≤β≤ 9.

Using eqs. (<ref>)–(<ref>)
to express the syzygy generators in
eqs. (<ref>)–(<ref>)
in terms of the z_α, we find in the case at hand,

    t_4,1   = (z_1  -  z_2,  z_1  -  z_2,  -  s  +  z_1  -  z_2,  0,  0,  0, 
        z_1  -  z_2  -  z_6  +  z_9  -  M^2,  t  +  z_1  -  z_2,  0,  0)  , 
    
    t_4,2   = (s  +  z_2  -  z_3,  z_2  -  z_3,  z_2  -  z_3,  0,  0,  0, 
        z_2  -  z_3  +  z_4  -  z_9  +  M^2,  -  t  +  z_2  -  z_3,  0,  0)  ,
    
    t_4,3   = (-  s  +  z_3  -  z_8 + M^2,  t  +  z_3  -  z_8 + M^2,  z_3  -  z_8 + M^2, 
        0,  0,  0,  z_3  -  z_4  +  z_5  -  z_8 + M^2,  z_3  -  z_8  +  M^2,  0,  0)  ,
    
    t_4,4   = (2z_1  +  2M^2,  z_1  +  z_2 +  2 M^2 , - s  +  z_1  +  z_3  +  2M^2, 
        0,  0,  0,  z_1  -  z_6  +  z_7, z_1  +  z_8  +  M^2,  0,  -  2)  ,
    
    t_4,5   = (- z_1  -  z_6  +  z_7  -  2 M^2,  -  z_1  +  z_7  -  z_9  -  M^2, 
        s  -  z_1  -  z_4  +  z_7 -  2 M^2,  0,  0,  0,  -  z_1  +  z_6  +  z_7, 
        -  z_1  -  z_5  +  z_7 -  2 M^2 ,  0,  0)  , 
    
    t_5,1   = (0,  0,  0,  s  -  z_6  +  z_9
     -  M^2,  -  t  -  z_6  +  z_9  -  M^2, 
        z_9  -  z_6  -  M^2,  z_1  -  z_2  -  z_6  +  z_9  -  M^2,  0, z_9 -  z_6  -  M^2,  0)  ,
    
    t_5,2   = (0,  0,  0,  z_4  -  z_9  + M^2,  t  +  z_4  -  z_9 + M^2, 
        -  s  +  z_4  -  z_9 + M^2,
     z_2  -  z_3  +  z_4  -  z_9
    + M^2,  0,
        z_4  -  z_9 + M^2,  0)  ,
    
    t_5,3   = (0,  0,  0,  z_5  -  z_4,  z_5  -  z_4,  s  -  z_4  +  z_5, 
        z_3  -  z_4  +  z_5  -  z_8  +  M^2,  0,  -  t  -  z_4  +  z_5,  0)  ,
    
    t_5,4   = (0,  0,  0,  s  -  z_3  -  z_6  +  z_7
     -  2 M^2,  -  z_6  +  z_7  -  z_8
     -  M^2, 
        z_7 -  z_1  -  z_6  -  2 M^2,  z_1  -  z_6  +  z_7,   0, z_7  -  z_2  -  z_6  -  2 M^2,  0)  ,
    
    t_5,5   = (0,  0,  0,  -  s  +  z_4  +  z_6  +  2M^2,  z_5  +  z_6  +  2M^2, 
        2z_6 + 2M^2,  -  z_1 +  z_6  +  z_7,  0,  z_6  +  z_9  +  M^2,  -  2)  ,

which agrees with eq. (<ref>) in the case M=0.




 §.§ Fully massless non-planar double pentagon


As a yet more non-trivial example we consider the fully massless non-planar
double-pentagon diagram shown in fig. <ref>.



In this case, the combined number of propagators
and irreducible scalar products (<ref>)
is m = 11. In agreement with eq. (<ref>),
we define the z-variables as follows, setting
P_i,j≡ p_i + p_j,

    3
       z_1   =ℓ_1^2 ,                     z_2 = (ℓ_1-p_1)^2 ,                  z_3 = (ℓ_1-P_1,2)^2 , 
    
       z_4   =(ℓ_2-P_3,4)^2 ,           z_5 = (ℓ_2-p_4)^2 ,                  z_6 = ℓ_2^2 , 
    
       z_7   =(ℓ_1+ℓ_2)^2 ,            z_8 = (ℓ_1+ℓ_2+p_5)^2 ,         z_9 = (ℓ_1+p_3)^2 , 
    
    z_10   =(ℓ_1+p_4)^2 ,            z_11 = (ℓ_2+p_1)^2 .

We choose as the set (<ref>) of
all independent external and loop momenta V = (p_1, p_2, p_3, p_4, ℓ_1, ℓ_2).
The lexicographically-ordered set of elements (i,j)
in eq. (<ref>) then becomes,

    (x_1, …, x_11)    ≡
    (x_1,5,  x_1,6,  x_2,5,
     x_2,6,  x_3,5,  x_3,6, 
        x_4,5,  x_4,6,  x_5,5,  x_5,6,  x_6,6)  ,

and it immediately follows that the matrices in
eq. (<ref>) are given by,

    A=(
    [  0  0  0  0  0  0  0  0  1  0  0; -2  0  0  0  0  0  0  0  1  0  0; -2  0 -2  0  0  0  0  0  1  0  0;  0  0  0  0  0 -2  0 -2  0  0  1;  0  0  0  0  0  0  0 -2  0  0  1;  0  0  0  0  0  0  0  0  0  0  1;  0  0  0  0  0  0  0  0  1  2  1; -2 -2 -2 -2 -2 -2 -2 -2  1  2  1;  0  0  0  0  2  0  0  0  1  0  0;  0  0  0  0  0  0  2  0  1  0  0;  0  2  0  0  0  0  0  0  0  0  1;    ])  ,

and, for α = 1, …, 11,

    B_α = 0 forα∉{ 3,4 }and
       
    B_3 = [ 0 1 0 0; 1 0 0 0; 0 0 0 0; 0 0 0 0 ] , 
    B_4 = [ 0 0 0 0; 0 0 0 0; 0 0 0 1; 0 0 1 0 ] ,

and m_α = 0.
We can now use eqs. (<ref>)–(<ref>)
to express the syzygy generators in
eqs. (<ref>)–(<ref>)
in terms of the z_α, yielding,

    t_5, 1   = (z_1-z_2,  z_1-z_2, -s_1,2+z_1-z_2,  0,  0,  0, 
        z_1-z_2-z_6+z_11, -s_1,2-s_1,3-s_1,4+z_1-z_2-z_6+z_11, 
        s_1,3+z_1-z_2,  s_1,4+z_1-z_2,  0,  0)  , 
    
    t_5, 2   = (s_1,2+z_2-z_3,  z_2-z_3,  z_2-z_3,  0,  0,  0, 
       -s_3,4+z_1+z_2+z_4+z_7-z_8-z_9-z_10-z_11, 
        s_1,3+s_1,4+z_1+z_2+z_4+z_7-z_8-z_9-z_10-z_11, 
        s_1,2+s_2,3+z_2-z_3, -s_1,3-s_1,4-s_2,3-s_3,4+z_2-z_3,  0,  0)  , 
    
    t_5, 3   = (z_9-z_1, -s_1,3-z_1+z_9, -s_1,3-s_2,3-z_1+z_9,  0,  0,  0, 
        s_3,4-z_1-z_4+z_5+z_9, -s_1,3-s_2,3-z_1-z_4+z_5+z_9, 
        z_9-z_1,  s_3,4-z_1+z_9,  0,  0)  , 
    
    t_5, 4   = (z_10-z_1, -s_1,4-z_1+z_10,  -s_1,4-s_2,4-z_1+z_10, 
        0,  0,  0, -z_1-z_5+z_6+z_10, 
        s_1,2+s_1,3+s_2,3-z_1-z_5+z_6+z_10, 
        s_3,4-z_1+z_10,  z_10-z_1,  0,  0)  , 
    
    t_5, 5   = (2 z_1,  z_1+z_2, -s_1,2+z_1+z_3,  0,  0,  0, 
        z_1-z_6+z_7, -s_1,2+2 z_1+z_3-z_6+z_7-z_9-z_10, 
        z_1+z_9,  z_1+z_10,  0, -2)  , 
    
    t_5, 6   = (-z_1-z_6+z_7, -z_1+z_7-z_11, 
        s_1,2+s_3,4-2 z_1-z_3-z_4+z_8+z_9+z_10,  0,  0,  0, 
       -z_1+z_6+z_7,  s_1,2-2 z_1-z_3+z_6+z_8+z_9+z_10, 
        s_3,4-z_1-z_4+z_5-z_6+z_7, -z_1-z_5+z_7,  0,  0)  , 
    
    t_6, 1   = (0,  0,  0, -s_1,3-s_1,4-z_6+z_11, -s_1,4-z_6+z_11, 
        z_11-z_6,  z_1-z_2-z_6+z_11, 
       -s_1,2-s_1,3-s_1,4+z_1-z_2-z_6+z_11,  0,  0,  z_11-z_6,  0)  , 
    
    t_6, 2   = (0,  0,  0,  s_1,3+s_1,4+z_1+z_3+z_4+z_7-z_8-z_9-z_10-z_11, 
        s_1,3+s_1,4+s_2,3+z_1+z_3+z_4+z_7-z_8-z_9-z_10-z_11, 
       -s_1,2-s_3,4+z_1+z_3+z_4+z_7-z_8-z_9-z_10-z_11, 
       -s_3,4+z_1+z_2+z_4+z_7-z_8-z_9-z_10-z_11, 
        s_1,3+s_1,4+z_1+z_2+z_4+z_7-z_8-z_9-z_10-z_11,  0,  0, 
       -s_3,4+z_1+z_3+z_4+z_7-z_8-z_9-z_10-z_11,  0)  , 
    
    t_6, 3   = (0,  0,  0,  z_5-z_4,  z_5-z_4,  s_3,4-z_4+z_5, 
        s_3,4-z_1-z_4+z_5+z_9, -s_1,3-s_2,3-z_1-z_4+z_5+z_9,  0,  0, 
        s_1,3+s_3,4-z_4+z_5,  0)  , 
    
    t_6, 4   = (0,  0,  0, -s_3,4-z_5+z_6,  z_6-z_5,  z_6-z_5, 
        -z_1-z_5+z_6+z_10,  s_1,2+s_1,3+s_2,3-z_1-z_5+z_6+z_10,  0,  0, 
        s_1,4-z_5+z_6,  0)  , 
    
    t_6, 5   = (0,  0,  0,  z_1-z_6+z_7-z_9-z_10, -z_6+z_7-z_10, 
       -z_1-z_6+z_7,  z_1-z_6+z_7, -s_1,2+2 z_1+z_3-z_6+z_7-z_9-z_10, 
        0,  0, -z_2-z_6+z_7,  0)  , 
    
    t_6, 6   = (0,  0,  0, -s_3,4+z_4+z_6,  z_5+z_6,  2 z_6, -z_1+z_6+z_7, 
        s_1,2-2 z_1-z_3+z_6+z_8+z_9+z_10,  0,  0,  z_6+z_11, -2)  .


In contrast, for this example the Singular command syz
did not produce output after 56 hours of running time with
52 GB of RAM used.






§ 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
(<ref>). We presented in
eqs. (<ref>)–(<ref>)
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 (<ref>)
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. (<ref>)–(<ref>)
by a straightforward module intersection computation.

We emphasize that the closed-form expressions in
eqs. (<ref>)–(<ref>)
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. (<ref>)–(<ref>).
In particular, the closed-form expressions allow
the construction of purely D-dimensional 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. <ref>.

Moreover, an important feature of the syzygies
eqs. (<ref>)–(<ref>)
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. <cit.>
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. <cit.>. This direction merits further investigation.





§ ACKNOWLEDGMENTS



We thank Roman N. Lee for collaboration at an early stage of this work.
We also thank Jorrit Bosma, James Drummond, Johannes Henn, Harald Ita,
David Kosower, David Mond, Costas Papadopoulos and Mao Zeng for useful discussions.
The research leading to these results has received
funding from Swiss National Science Foundation (Ambizione grant PZ00P2
161341) and funding from the European Research Council (ERC) under
the European Union's Horizon 2020 research and innovation programme (grant agreement No 725110).
The work of Y.Z. is also partially supported by the
Swiss National Science Foundation through the NCCR SwissMap, Grant number 141869.
The work of A.G. is supported by the Knut and Alice Wallenberg Foundation under Grant #2015-0083.
The work of K.J.L. is supported by ERC-2014-CoG, Grant number 648630 IQFT.
The work of J.B. and M.S. was supported by Project II.5 of SFB-TRR 195
“Symbolic Tools in Mathematics and their Application” of the German Research Foundation (DFG).


h-physrev


