Vector boson pair production at one loop: analytic results for the process $q \bar{q} \ell \bar\ell \ell^\prime \bar{\ell}^\prime g$

We present compact analytic results for the one-loop amplitude for the process $0 \rightarrow q \bar{q} \ell \bar\ell \ell^\prime \bar{\ell}^\prime g$, relevant for both the production of a pair of $Z$ and $W$-bosons in association with a jet. We focus on the gauge-invariant contribution mediated by a loop of quarks. We explicitly include all effects of the loop-quark mass $m$, appropriate for the production of a pair of $Z$-bosons. In the limit $m \to 0$, our results are also applicable to the production of $W$-boson pairs, mediated by a loop of massless quarks. Implemented in a numerical code, the results are fast. The calculation uses novel advancements in spinor-helicity simplification techniques, for the first time applied beyond five-point massless kinematics. We make use of primary decompositions from algebraic-geometry, which now involve non-radical ideals, and $p$-adic numbers from number theory. We show how to infer whether numerator polynomials belong to symbolic powers of non-radical ideals through numerical evaluations.


Introduction
In many respects the numerical calculation of one-loop amplitudes, both in the standard model and in proposed models beyond the standard model, is a solved problem. Following Passarino and Veltman [1] the problem is separated into the calculation of scalar one-loop integrals and the calculation of the coefficients with which these integrals appear in the particular amplitude at hand. The needed finite scalar integrals are provided in ref. [2,3], whereas the needed singular integrals are provided in ref. [4]. Techniques based on numerical unitarity [5,6] as well as methods based on iterative calculation of Feynman diagrams [7][8][9] have been automated into sophisticated tools, which give reliable numerical results for the coefficients of the scalar integrals.
On the other hand, analytic unitarity techniques have also matured so they can give analytic results following an automatic recipe [6,[10][11][12][13]. However, the resultant analytical expressions are often complicated so that the singularity structure of the amplitude is hard to divine and numerical evaluations are suboptimal. In this paper we push the analytic techniques a step further by obtaining simpler analytic expressions where the form of the answer, especially with regard to the singularity structure, is manifest. The possible benefits of utilizing the amplitude in such a form are: • the simpler form may lead to faster numerical evaluation; • the singularity structure is manifest, with physical poles of as low degree as possible; • the consequent analytic form leads to improved stablility of numerical evaluation; • the numerical behaviour around singular points can be improved by analytic expansions, if necessary.
The issue of stable evaluation of the amplitude is particular pressing for the case of vector boson pair production. Cuts on the transverse momenta of the decay products of the vector bosons do not exclude the region where the vector sum of the transverse momenta of all the uncoloured particles in the final state is equal to zero. In this kinematic region the amplitudes contain soft and collinear singularities. This issue is especially important in the context of next-to-next-to-leading order calculations since cancellations between real and virtual diagrams occur in the region of zero transverse momentum.
A compact representation of scattering amplitudes is provided in principle by the spinor-helicity formalism [14][15][16]. These are especially convenient for the case at hand because the factors associated with vector boson decays are simple and the whole family of diboson processes may be described by appropriate dressings of a core set of amplitudes [17]. However, since the spinor products are not all independent, spinor product expressions are not straightforward to simplify. In particular, the application of momentum conservation and Schouten identities leads to many equivalent representations of the same amplitude. The application of systematic reduction techniques (such as momentum twistors or Gröbner basis reduction) does not necessarily result in simpler expressions.
A number of strategies are available to facilitate simplification. The method of momentum twistors [18,19] allows one to write spinor expressions contributing to an n-point amplitude in a unique form in terms of 3n − 10 independent variables. However it is cumbersome to revert to simple spinor expressions as the number of external legs grows. Alternatively, simpler expressions may be obtained by reconstructing multivariate polynomials and rational functions from their evaluation over finite fields [20][21][22][23][24].
Our analytic results will be simplified using large-precision floating point arithmetic and fitting in singular limits [25], as well as using p-adic numbers and technology from algebraic-geometry [26]. The expressions we provide are explicitly rational, i.e. no square roots are present, and contain poles of the lowest degree possible.

Motivation
Our motivation for this paper is twofold. First, we want to investigate and extend the practical limits on the numerical simplification techniques, alluded to above and to be explained in more detail in Section 3. In particular, it is interesting to investigate their feasibility in high multiplicity settings, where new poles appear in the master-integral coefficients. For this purpose, we choose a subset of the diagrams contributing to the one-loop 7-point process qq ¯ ¯ g, namely the diagrams including a quark loop. Second, vector boson pair production is an important process, to which these amplitudes contribute. For example, the ZZ final state is one of the decay channels of the Higgs boson, and onshell and offshell calculations are of great interest. The amplitudes we consider contribute to, and to the related processes obtained by crossing the coloured partons. This amplitude receives contributions both at tree level and at one loop. In this paper we report on the one-loop amplitude mediated by quarks with a common mass, m. In the limit m → 0 our results can also be used for the massless quark-loop contributions to the process, We note that results for this process, in the limit of massless quarks circulating in the loop, have been calculated in analytic form previously [27], but the resulting amplitudes were only distributed in the form of computer code since they were not sufficiently compact to present otherwise.
In this paper we will present simpler formulae for all the ingredients necessary to assemble the quark-loop amplitude for the process in Eq. (1.1). In addition, we have added the simplified results to the code MCFM [28][29][30], which we will demonstrate leads to improvements in speed.

Plan of this paper
In Section 2 we review the spinor notation and use it to present results for the lowest order amplitude. In Section 3 we review and build upon the algebro-geometric methods of ref. [26], with additional details worked out in appendix C. Section 4 (Appendix A) presents the analytic results for the coefficients of box, triangle and bubble scalar integrals coming from quark-loop box (triangle diagrams) contributing to the process in Eq. (1.1). Timing results from the numerical implementation of our calculation are described in Section 5 and our conclusions are given in Section 6.

Spinor notation
We begin this section by introducing the notation which we shall use to present the treegraph results for the Z-pair production amplitude, as well as for the one-loop results to be presented in Section 4. All results are presented using the standard notation for the kinematic invariants of the process, and the Gram determinant, The Weyl spinor λ A is a two dimensional complex vector and its complex conjugate is denoted asλȦ. The spinorial inner product between to Weyl spinors λ 1 and λ 2 is written as, where ε is the totally antisymmetric tensor in two dimensions, For a light-like momentum p, (i.e.
For the conjugate spinor we have that, The corresponding results for the spinor products are We also defineΓ (note the tilde) to imply anti-symmetrization, Lastly, we point out that Eq. (2.12) and Eq. (2.13) imply definitions for non-fully contracted spinor strings. These un-contracted spinor strings are useful to express generators with open indices for some ideals in Section 3.3.
The basic amplitude which we calculate is the one for the process involving four leptons, two quarks and one gluon, with all particles outgoing; the superscript denotes the helicity and the subscript shows the type of particle. The couplings required to reconstruct the physical amplitude will be given later.
The colour matrix t B is normalized such that tr(t A t B ) = δ AB . The indices i 1 andī 2 denote the colours of the quark and anti-quark line, i 1 ,ī 2 = {1, 2, 3}. The strong and electromagnetic couplings are denoted by g s and e. Written in this form A tree,B 7 is exactly the amplitude where the production of both pairs of leptons (off a unit electric charge quark line) is mediated by virtual photons. The appropriate coupling factors and propagators for Z boson production will be added below. The result for the reduced tree amplitude is, This amplitude was first presented in ref. [17] although our labelling of the lepton momenta, see Eq. (2.14), differs from the notation in that paper. The remaining tree amplitudes are obtained by symmetry operations. These correspond to flipping the helicities of the leptons, e.g.
flipping the helicities of the quarks, e.g., and reversing the helicity of the gluon, e.g.,

Restoring the couplings
We define the left-and right-handed couplings of a Z boson to quarks and leptons by, where Q q is the charge of the quark (in units of the positron charge) and τ q = +1 for up-type quarks and τ q = −1 for down-type quarks. The full tree amplitude for the ZZ case is then, where M is the (complex) mass of the vector boson. Dressed this way, these amplitudes account for the effect of virtual photons as well as Z bosons.

Geometric Reconstruction
Any one-loop amplitude can be written as a sum of scalar master integrals with definite coefficients and rational terms [1]. Therefore, we write, The exact definitions of the scalar integrals D 0 , C 0 , and B 0 are given in Appendix B. For the case at hand, the rational term is fully determined by the mass dependence of the coefficients [12]. In this section, we deal with the problem of simplifying the coefficients of the master integrals: d {i×j×k} , c {i×j} and b {i} . Even though the coefficients are obtained through analytic unitarity methods, we phrase the simplification problem as a reconstruction problem from numerical samples. In this context, we treat all three types of coefficients on the same footing, thus we give them a generic name C i . These coefficients C i are rational functions of the external kinematics. We write where (λ,λ) denote the set of right-and left-handed Weyl spinors, which we treat as independent. That is, we consider the C i in the analytical continuation to complex momenta. The exponents q ij are integers and they are allowed to take negative values, thus denoting common factors in the numerator. From a geometric perspective, since these rational coefficients are functions of many complex variables, they are evaluated in a multi-dimensional space analogous to the complex plane. Their poles and zeros will be surfaces (a.k.a. varieties) of one less dimension than the full space. Leveraging the geometric picture, in this section we describe the approach employed to simplify the amplitudes presented in this paper.
In our theoretical approach we make use of elements of both the algebro-geometric reconstruction procedure of ref. [26] and of the iterated, in-limit reconstruction strategy of ref. [25]. On a more practical level, we rely on the computer algebra system Singular [31], through the Python interface syngular [32], on the multi-precision floating-point arithmetic package mpmath [33] and on an in-house Python implementation of p-adic numbers (Q p ) with variable size mantissa and explicit precision tracking.
We begin in Section 3.1 with a brief review of the most important algebro-geometric concepts, namely ideals and varieties in spinor space. The aim here is mainly to set up notation and recall concepts; for a more comprehensive discussion we refer the reader to ref. [26], and references therein. In Section 3.2, we present the poles and zeros of the coefficients as varieties in spinor space, with an associated degree of divergence or vanishing. Then, in Section 3.3 we consider intersections of these varieties, and their decompositions into irreducible components. Numerical evaluations close to these new, lower-dimensional varieties are used to obtain constraints on the numerator structure and, in particular, on the possible partial fraction decompositions. Finally, we sample the coefficients near singular varieties to rationally reconstruct the remaining free parameters of the ansatz.

Ideals and varieties in spinor space
For the purposes of algebro-geometric computations, let us begin by considering polynomials in the components of the Weyl spinors of the massless external legs. Mathematically, we say that these polynomials belong to the polynomial ring defined as, where the spinors are understood to be taken component-wise and we employ standard spinor-helicity notation. We stress that, in the analytic continuation to complex momenta, the spinors are independent. The field F is taken to be either the complex numbers (C) or the p-adic numbers (Q p ). In practice, we usually work over C and all geometric considerations are always understood to be over the complex numbers, as it is often required for the field to be algebraically closed. However, in some circumstances evaluations over Q p can be very useful since the scale hierarchy is easier to control. In order to discuss the geometric properties of the amplitude coefficients, we introduce the algebraic concept of an ideal, which is denoted by a pair of angle brackets 1 . An ideal is defined to be the set of all polynomial linear combinations of an initial set of polynomials which are called generators. As algebraic objects, ideals have well defined algebraic operations, addition, multiplication etc. In particular, a physically important ideal of S n is that of momentum conservation, For the spinors to describe a physically meaningful phase space, they must statisfy momentum conservation. That is, any polynomial belonging to the momentum conservation ideal J Λn has to be considered a rewriting of zero. For this reason, it is convenient to introduce the quotient ring, This is the set of all polynomials in spinor components where any pair of polynomials differing by a member of J Λn is considered to be equivalent. That is, the elements of R n are not polynomials, but equivalence classes of polynomials. For example, the following are six different ways to write the same element of R 7 , Just like we can define ideals of S n , we can also define ideals of R n . In fact, there exists a one-to-one map between ideals of R n and ideals of S n that contain J Λn , given by, for an arbitrary set of generators p 1 , . . . , p k . The numerators N i and the denominators j D q ij j of the coefficients C i belong to R n . Therefore, the coefficients C i belong to the field of fractions of R n , denoted as F F (R n ) . However, note that it is not entirely trivial to define a field of fractions over a quotient ring (see ref. [34,Chapter 5]). In particular, R n needs to be a so-called integral domain, and R 3 is not 2 . For n ≥ 4, R n is an integral domain and F F (R n ) is well defined. We will work in R 7 .
The key geometric concept that we consider is that of a variety. We denote a variety U associated to an ideal J as U = V (J). For an ideal J of S n , V (J) is defined as the set of points (λ,λ) ∈ F 4n such that the generators of J evaluate to zero. Through the correspondence of Eq. (3.7), the same definition applies to varieties associated to ideals of R n . Note that in the latter case, all varieties will be sub-varieties of V (J Λn ). Similarly, to every variety U we can associate an ideal J, denoted as J = I(U ), and defined as the set of polynomials which vanish on U . Varieties and ideals have well defined dimensions. For example, dim(V (J Λn )) = (4n − 4), as 4 constraints are imposed in F 4n . We also define codimension as the complement of dimension w.r.t. the dimension of the full space, i.e. codim(V (J Λn )) = 4. Note that the codimension need not always match the number of generators. If it is possible to find a set of generators with as few elements as the codimension, then we describe the ideal as of maximal codimension. As the quotient of a polynomial ring by a maximal codimension ideal, R n is a Cohen-Macaulay ring [26]. This property of R n has a number of useful implications which we will review shortly. In the context of R n , as . The poles of the rational coefficients are codimension one varieties.
It turns out that the order of a pole is not necessarily well defined on all varieties. If a variety is comprised of multiple irreducible components, which we call branches, then the degree of divergence need not be the same for all of them. Therefore, it is important to be able to identify the branches of reducible varieties. Any variety admits a unique minimal decomposition, where n B (U ) denotes the number of branches U k . If U itself is irreducible then we simply have n B (U ) = 1. To compute a minimal decompositions of a variety, one can rely on the corresponding concept for ideals, that is on a so-called minimal primary decomposition. Given any ideal J, in analogy to Eq. (3.8), we can write, where n Q (J) denotes the number of primary ideals Q l in the primary decomposition. Each primary ideal Q l has an associated prime ideal P l = √ Q l , where the root denotes the ideal radical. One says that Q l is P l -primary. Now let U = V (J). The subset of prime ideals P l such that P l = V (U k ) for some k is called the set of minimal associated primes. It has n B (U ) elements and we denote it as minAssoc(J). The complement of this subset is the set of so-called embedded components. An embedded prime P l is such that V (P l ) is redundant (and thus absent) in Eq. (3.8), but such that Q l is not redundant in Eq. (3.9). The number of embedded components is n Q (J)−n B (U ). A first useful consequence of R n being a Cohen-Macaulay ring is that maximal codimension ideals in R n are equi-dimensional, i.e. they are free of embedded components [26], (3.10) It is also useful to note that S n , J n and R n are symmetric under permutations of the external legs and a swap of left-and right-handed spinors. We can make use of these properties to aid the computation of primary decompositions.
Since the degree of divergence of a rational fraction of R n when considered simultaneously near a pair of poles is often less than the sum of the divergences near each pole separately, it is useful to consider elements of R n vanishing to a given order on a given variety. Given an irreducible variety V (Q l ) and a numerator N i which vanishes to order κ on V (Q l ), the Zariski-Nagata theorem [35][36][37] tells us that N i has to belong to the κ th symbolic power of the associated prime ideal P l , denoted as P κ l . This power κ can reliably be identified numerically [26]. To see why a refined notion of power is needed, let us consider the standard ideal power P κ l of a prime ideal P l . It is defined through repeated ideal multiplication and it may not be a primary ideal, i.e. P κ l may involve embedded components. These embedded components imply additional non-trivial vanishing properties on sub-varieties of V (Q l ). In contrast, the symbolic power P κ l is defined as the P l -primary component of P κ l . A second useful consequence of R n being a Cohen-Macaulay ring is that symbolic powers of maximal codimension ideals in R n coincide with the standard powers, (3.11)

Poles and zeros as codimension-one varieties
We begin by determining the denominators of the coefficients C i , which belong to F F (R 7 ). Poles and zeros of these rational functions are irreducible varieties of codimension one, with an associated degree of vanishing or divergence. we observe that the poles needed for the one-loop amplitudes under our consideration in this paper are of the form, where the indices a, b, c, d, e are assumed to be distinct and in the set {1, . . . , 7}. All associated codimension-one ideals, that is ideals of the form D j R 7 , are prime, meaning that the corresponding varieties V ( D j R 7 ) are irreducible. Then, it follows that the order of the poles and zeros can be determined by either a pair of high-precision floating point evaluations in the limit approaching the variety, as by ref. [25], or by a single p-adic evaluation at a point close to the variety, as by ref. [26]. With this procedure we obtain the least common denominators (LCD), We stress that since the pole orders are determined numerically, it follows that all spurious poles are automatically removed and the physical ones are of as low degree as posisble.
Common factors in the numerator (q ij < 0) are also obtained in this way.

Partial fractions and numerators from codimension-two varieties
To proceed in the simplification we aim to constrain the numerators N i . For this purpose we study of their behaviour on codimension-two varieties. In particular, it is convenient to consider the behaviour of the numerators on those codimension-two varieties that originate from the intersection of varieties of codimension one corresponding to poles of the coefficients. We can now assume we have access to numerical evaluations of the numerators N i , as the denominators have been determined in Section 3.2 from the study of varieties of codimension one. The following discussion is applicable to any quotient ring R n with a multiplicity n bigger than four 3 , thus we will omit the ideal subscripts. Given a pair of distinct poles (D α , D β ), we consider the intersection of the associated varieties, which is equivalent to the variety associated to the sum of the ideals, (3.14) Contrary to the ideals of codimension one introduced in the previous section, these ideals D α , D β of codimension two do not always correspond to irreducible varieties. As the numerator may vanish to different orders on the different branches of V ( D α , D β ), we compute the primary decompositions, as by Eq. (3.9). They read, where, by Eq. (3.10), we have That is, we are in a special situation where the minimal decomposition of the associated variety reads For each irreducible variety V (Q l ) we then generate either a single or a pair of nearby phase-space points, depending on whether the field F is taken to be Q p or C respectively, and thus obtain the degree of vanishing of the numerators N i . Given a prime ideal P l = p 1 , . . . , p r , q 1 , . . . , q 4 , where J Λ 7 = q 1 , . . . , q 4 , the phase-space point we require is a set (η ( ) ,η ( ) ) ∈ F 28 such that, where κ q is the working precision (κ q κ i ), and κ i is the largest integer such that p i ∈ P κ i l .
Usually, but not always, one has κ i = 1. Such a phase-space point can be built with the method described in ref. [26,Section 3], which is also easily adapted to F = C. The parameter denotes a small quantity with respect to the absolute value of the chosen field. For each variety V (Q l ) this results in a constraint on the numerator of the form To achieve compact representations of the coefficients, as well as to be able to reconstruct one pole residue at a time, we wish to interpret as many as possible of these constraints on the numerators in terms of a partial fraction decomposition of C i . To illustrate this let us now restrict our discussion to those codimension-two ideals D α , D β which are radical, that is to those ideals such that Q l = P l for all l. The extension of the following reasoning to the generic case is addressed in appendix C. Let κ be the largest power such that N i vanishes to order κ on all branches By Eq. (3.11) we can then explicitly expand the symbolic power as, This can be interpreted in terms of a partial-fraction decomposition of the rational function, which is manifestly free from spurious poles. This decomposition is maximal in the sense that it is not possible to replace κ with κ + 1. However, it may still be possible to further refine the decomposition as some N ik may be proportional to D α or D β . How to obtain and interpret these additional constraints is addressed in appendix C. We perform this analysis for the vast majority of pairs of poles, (D α ,D β ). The required primary decompositions are presented in Section 3.3.1. In some rare cases, the partial fraction decomposition is expected from the structure of unitarity cuts, and this analysis is thus not needed. For example, one of the two bubble coefficients in Section 4.3 involves two three-mass Gram-determinant poles, but these are clearly associated to different triple cuts and thus separable.
Since the degree of vanishing of N i need not be uniform on all branches of the ideal, it follows that not all constraints can be interpreted this way. The extra degree of vanishing of the numerator beyond κ on a particular branch is then purely a statement about the structure of the numerators N ik . For example, we observe that the integral coefficients presented in Section 4 often diverge less strongly on one of the two branches of , which are given in Eq. (3.30). Therefore, besides a partialfraction decomposition, we obtain further information on the numerators. For instance, if the degree of divergence is lower on the second branch than on the first one, the numerators corresponding to the leading poles of 7|Γ 34|56 |7 and [7|Γ 34|56 |7] may contain contractions of the third generatorΓ 12|34|56 given in the second primary ideal of Eq. (3.30). Thus, the primary decompositions allow one to uncover numerator structures with specific vanishing properties in certain regions of phase space.
So far we have achieved a systematic partial fraction decomposition between pairs of poles, and a way to identify new spinor structures for the numerators. The next step is to combine all the decompositions and numerator constraints together, so that the resulting expression is simple. However, note that in general not all ideal-membership constraints to which the common numerator N i is subject carry over to the individual N ik . For example, if N i ∈ J 1 ∩ J 2 , with J 1 = D α , D β κ 1 and J 2 = D γ , D δ κ 2 , then we could attempt a partial fraction decomposition of the form However, this may or may not be a valid decomposition of the coefficient C i . To see this, let us consider the simplified case κ 1 = κ 2 = 1. We can refer back to Eq. (3.22) and use the first J 1 constraint, that is, we can write To then achieve the decomposition of Eq. (3.22) we need N i0 ∈ J 2 and N i1 ∈ J 2 . Yet, we are guaranteed that this is true only if J 1 ∩ J 2 = J 1 · J 2 , which is not the case in general. On the other hand, failure of the intersection to equal the product does not automatically imply that an ansatz of the form of Eq. (3.22) has to fail. This makes it highly non-trivial to make use of multiple constraints while performing a partial-fraction decomposition. In fact, it is not hard to find cases where, given a certain partial-fraction decomposition, it becomes impossible to make all other numerator constraints manifest. This can lead to spurious singular behaviour even in the absence of spurious poles. That is, spurious divergences on codimension-two varieties are possible even in the absence of spurious divergences on codimension-one varieties. A well known example of this type of spurious singularities is the appearance of s ij poles in lieu of ij and/or [ij] poles in gauge-theory amplitudes. An example of a partial fraction decomposition which makes it impossible to manifest all numerator constraints is given in Section 3.3.2. Because of this subtlety, which warrants further investigation in the future, for the time being we take an heuristic guess-and-check approach when combining multiple constraints. That is, we assume that multiple constraints can be naively combined, for example as in Eq. (3.23), and attempt to fit the free coefficients in the ansatz for the numerators with the in-limit reconstruction strategy of ref. [25]. If this fails, we relax one or more constraints until the reconstruction succeeds.

Codimension-two primary decompositions
In this sub-section we present several primary decompositions which were used in the present computation. Unless otherwise stated, all ideals are understood to be taken in the quotient ring R 7 . The following is not meant to be a complete list of primary decompositions for codimension-two ideals at seven point. In fact, including spinor strings of the form a|Γ bc |d] and three-particle Mandelstam invariants s abc one obtains hundreds of distinct codimensiontwo varieties, which is beyond the current scope. The notation employed is as in Eq. (3.15), that is, the left-hand side consists of a reducible ideal and the right-hand side gives its decomposition as an intersection of primary ideals. Whenever a primary ideal Q l does not correspond to its associated prime P l , i.e. if Q l = √ Q l , then we also write an equation of the form √ Q l = P l , with P l explicitly given by a set of generators. There are five independent ideals generated by pairs of two-particle invariants. Only one is not prime, and it splits into a "collinear" branch and a "soft" branch, We note that compared to the codimension-two five-point primary decompositions presented in ref. [26], the ideal generated by all angle brackets is absent. This is explained by the following observation, which can be easily checked up to very high multiplicity. There are also five independent ideals generated by a two-particle invariant and a parity-invariant spinor string,  In the above 17 , |7 7| is clearly not radical, as it contains the outer product of |7 with itself, i.e. it is primary but not prime. The associated prime is simply the soft ideal, We also make extensive use of two primary decompositions of codimension-two ideals generated by pairs of spinor chains, In particular, the latter provides significant constraints, both in terms of partial fractions and in terms of numerator spinor structures, since the involved polynomials are of high degree. We recall that the tilde denotes anti-symmetrization, as defined in Eq. (2.13). One of the primaries is again not radical, and it has the same associated prime as Eq. (3.28), Lastly, we consider a few codimension two ideals involving three-mass Gram determinants. These are often not radical, but generally they are primary, 12 , ∆ 3 (1, 2, 3, 4) = 12 , (s 567 − s 34 ) 2 , where we have made explicit the existence of a perfect-square polynomial in the ideals. Then, the radicals can be shown to be,  The attentive reader may recognize that 1|Γ 34 |2] is not one of the poles listed in Eq. (3.12), however it is often a zero of the residue of the three-mass Gram pole, hence we choose to include it here. Finally, there is also strong evidence for the following primary decomposition The primality of all prime ideals can be proven via the test presented in Appendix B.3 of ref. [26], except the very last one of Eq. (3.34), which remains to be proven. It is also possible to explicitly check that the intersection of the primaries equals the reducible ideal, and whenever a primary is not radical one can check that √ Q l = P l by verifying dim(Q l ) = dim(P l ) and that Q l /P ∞ l = 1 , where the latter operation denotes ideal saturation. In the ancillary files we provide a Python script that performs these checks.

Example of simultaneous constraints and spurious singularities
Let us now consider an example to illustrate the subtetly with combining multiple numerator constraints when interpreting some in terms of partial-fraction decompositions. We take a part of the integral coefficientc (2) 12×56 from Eq. (4.28), namely, The numerator belongs to the following two ideals: and J 2 = 12 , [12] . That is, the numerator N belongs to J 1 ∩ J 2 . One can recognize J 1 as the second primary ideal in the primary decomposition of Eq. (3.30). At the same time, N does not belong to the first primary ideal in that same decomposition. Therefore, the two poles 7|Γ 34|56 |7 and [7|Γ 34|56 |7] cannot be separated without introducing spurious poles 4 . However, the numerator does belong to the ideal 12 , [12] , even if this is perhaps not entirely manifest. It becomes apparent by expanding the above as, which can then be written as a partial fraction decomposition of the form, . (3.37) Note that now it has become impossible to have aΓ 12|34|56 factor in either numerator, as their mass dimensions are not sufficient. We can see this might happen as product and intersection are not equal for the ideals J 1 and J 2 . These kinds of consideration have implications on the stability of the expressions. For instance, the latter partial-fraction form of Eq. (3.37) is potentially unstable near the variety associated to the second primary ideal in the primary decomposition of Eq. (3.30) (double-pole cancelling to give a simple pole in the sum of the fractions).

Integral Coefficients
In this section we present results for the one-loop master-integral coefficients for the process under consideration, starting for the six one-loop diagrams shown in Fig. 1 which contribute to doubly-resonant Z-boson pair production in association with a jet. These results have been calculated using standard analytic techniques [6,[10][11][12][13] and subsequently simplified using the methods of Section 3. The one-loop colour amplitudes have the decomposition, where the SU (3) colour matrix in the fundamental representation, t B , is normalized such that tr t A t B = δ AB . The quark and antiquark colour indices are i 1 andī 2 . The colour stripped amplitude can be expressed in terms of scalar integrals and rational terms r.
The definitions of the scalar integrals are given in Appendix B. As in Eq. (2.14) we present results for one specific helicity choice, Other helicities are obtained by permutation of the arguments. The tree amplitude that we presented in Eqs. (2.15,2.16) was exactly the amplitude for production of a gluon and two lepton pairs off a massless, unit-charge, quark line. The production of the two lepton pairs was mediated by a virtual photon, and the propagators and couplings appropriate for Z-boson exchange were added subsequently as detailed in Eq. (2.23). For the quark loop amplitudes, because of the mass running in the quark loop, we need to maintain helicity information. Therefore the amplitudes that we present below generalize the coupling of the "photon" to the massive quark line as follows, The amplitude for true photon exchange is recovered by setting v L = v R = 1. Given this helicity information it is straightforward to add the couplings and propagators appropriate for ZZ production.
In several cases we observe that, while overall a given coefficient C may be of mixed symmetry under a certain swap operation S, some of its poles are actually either fully symmetric or anti-symmetric. In these cases, we find it convenient to split the coefficients in their symmetric and anti-symmetric parts with respect to S as,

Results for box coefficients
We begin with a decomposition of the box coefficients in terms of the mass m and the left-handed and right-handed couplings of the vector bosons to the quark loop i, j and k thus represent the outgoing momenta at three of the four corners of the box. Analogous expansions will follow for triangle and bubble coefficients.

Results for {12 × 34 × 56} box
The result for the quark mass-independent piece of this box is, . (4.14) All the coefficients for this box are fully symmetric under the exchange 3 ↔ 5, 4 ↔ 6, which is a symmetry of the relevant diagrams, Figs. 1(a) and 1(b). The mass-independent piece is determined by Eq. (4.7), The coefficient proportional to m 2 has certain elements in common with the suitably permuted d  Because a rank two box integral is cut constructible, the vanishing of the rational piece requires the following relationship, The c coefficients in this Eq. (4.20) are presented in subsection 4.2.

Results for triangle coefficients
The triangle coefficients have no terms which are quartic in the mass of the quark,

Results for {12 × 56} triangle
The mass-independent term is obtained by exchange, The m 2 piece contains both symmetric and anti-symmetric parts,

Results for {12 × 34} triangle
The results for this triangle are obtained by exchange, . (4.30)

Results for {7 × 12} triangle
The mass-independent part of the coefficient is determined by infrared relations that ensure the cancellation of 1/ poles in the massless case. In terms of the box integral coefficients defined above we have, The results for this triangle are obtained by exchange, (4.39)

Results for bubble coefficients
The bubble coefficients are all independent of the mass. Consequently the bubble integrals do not contibute to the LR structure. In addition since the full result for the amplitude is UV finite we have that

Results for remaining bubble coefficients
The remaining bubble coefficients are all obtained by exchange, . (4.44)

Results for rational term
The rational piece is determined by the triangle coefficients proportional to m 2 and the unique box coefficient proportional to m 4 , (see Eqs. (4.11,4.19,4.14))

Numerical implementation
We can directly compare our analytic results against those obtained using Recola2 [9], with the model file 'SM_FERM' that only computes the effects of fermion loops. In order to perform a direct comparison we must also account for additional contributions to the amplitude from triangle diagrams in which the Z bosons are not both attached to the quark loop. Representative diagrams for these three processes are shown in Fig. 2 and explicit results for the contributions are given in Appendix A.
We have performed a comparison for the ZZ+jet process, obtaining perfect agreement. For the purposes of this comparison we have used a physical value of the top-quark mass and set the bottom quark mass to zero. We observe that the analytic amplitudes presented here are more than an order of magnitude faster than their Recola2 counterparts.

Conclusions
In this paper, we have presented compact analytical expressions for the rational coefficients of the master integrals of the one-loop QCD helicity amplitudes for the production of a pair of vector bosons in association with a jet. We have focused on the contribution mediated by a closed quark loop, and retained full dependence on the quark mass. The results are expressed in spinor-helicity variables by factoring out propagators involving masses and by expressing the results in terms of the massless decay products of the vector bosons, which are therefore considered fully off mass shell.
Due to the large number of scales involved in 7-point phase space and the fact that the rational coefficients are ratios of polynomials subject to constraints, namely momentumconservation and Schouten identities, simplifying the analytical expressions is a complex task. To tackle it, we rely and expand upon recent advances in spinor-helicity simplification techniques based on algebraic geometry and numerical sampling in singular limits. In particular, for the first time beyond five-point amplitudes, we systematically identify irreducible varieties in spinor space, thus quantitatively identifying the pole structure of the amplitude in the analytical continuation to complex momenta. We also observe that some of the involved ideals are not radical, meaning special care is required in making the connection between numerical evaluations and membership to symbolic powers. We employ floating-point and p-adic evaluations close to these irreducible varieties to infer membership of numerator polynomials to symbolic powers of prime ideals. Subsequently, this allows to identify possible partial-fraction decompositions as well as new numerator structures. We fit the numerators by sampling near singular varieties.
The usefulness of compact analytical expressions, and related simplification techniques, goes beyond that of mere theoretical understanding. As the computational load on the Worldwide LHC Computing Grid is predicted to fall short of the demands in the near future [38], speeding up matrix-element providers by using simplified analytic expressions [39] would improve the performance of event generators and aid LHC data analysis. Further-more, in light of phenomenological applications, and in particular for the numerical stability in singular regions, it would be interesting to understand the interplay between the primary decompositions in complexified momentum space and the real subset of the latter needed for physical kinematics.

A.2 Higgs contribution
We now address another component of the vector boson pair production amplitude, which is the piece containing an intermediate Higgs boson as shown in Fig. 2(b). This result has been known for almost 35 years [42,43]. The relevant amplitude is, This process is of interest since it is one of the simplest processes to illustrate the fundamental role of the Higgs boson in cancelling bad high energy behaviour [44]. The amplitude for the production of a Higgs boson by two offshell gluons has been given for example in refs. [45,46]. Using that result and attaching the decays of the Z bosons we obtain,

A.3 Single resonant axial anomaly
This contribution corresponds to diagrams such as the one in Fig. 2(c). As in the doubleresonant case, the result for this contribution can be obtained by contracting the appropriate currents with the known result from refs. [40,41]. In this case there are two essential amplitudes, We note that, even accounting for coupling changes, the two are not related by a 5 ↔ 6 interchange because one of the terms flips sign. These amplitudes depend explicitly on the charges of the decay products (q 34 , q 56 ) and their left-(l 34 , l 56 ) and right-handed (r 34 , r 56 ) couplings to Z bosons. The combination of couplings and propagator factors is, The loop factor is F = F 1 (p 12 , p 3456 , m t ) − F 1 (p 12 , p 3456 , m b ) where F 1 has already been specified in Eq. (A.6). Remaining amplitudes are obtained by symmetry operations. These correspond to flipping the helicities of the quarks, e.g., flipping the helicities of the leptons, e.g.

B Integrals
This section gives the precise definition of the scalar integrals. We define the denominators of the integrals as follows, Either the denominators all have a common non-zero mass, for the case of ZZ production, or the mass can be taken to be zero, m = 0 for the case of W W production. In the latter case we ignore the contributions of top and bottom loops. The momenta running through the propagators are, The p i are the external momenta, whereas the q i are the off-set momenta in the propagators.
In terms of these denominators the integrals are, , andμ is an arbitrary mass scale.

C Asymmetric approaches and ring extensions
In Section 3.3.1 we presented several primary decompositions, some of which involve ideals which are not radical, while Eq. (3.20), which is the basis for the partial-fraction decomposition of Eq. (3.22), requires the involved ideal D α , D β to be radical. A natural question which arises is then how to generalize this to non-radical ideals. Furthermore, even when the condition on Eq. (3.20) is satisfied, i.e. when the ideal is radical, the phase-space point chosen in Eq. (3.18) is a very specific one, which satisfies D α ∼ D β ∼ . In this case D α , D β vanish symmetrically. It was shown in ref. [25] that by allowing unequal (i.e. asymmetric) degrees of vanishing it is possible glean more information on the numerators. In this appendix, we address both points in unified way with algebraic geometry. In particular, we are going to reduce the asymmetric case to the symmetric one and, in doing so, provide an algebraic interpretation to evaluations in asymmetric approaches. First of all, let us generalize Eq. (3.20) to a special class of non-radical ideals D α , D β , specifically to those whose primary components are (symbolic) powers of their associated primes 5 . To achieve this, let us recall that the κ th symbolic power Q κ of any P -primary ideal Q can be defined as the P -primary component of Q κ , that is, where Q κ is P -primary and the Q em. 1≤i≤m are embedded. See, for instance, ref. [47,Lemma 1.18] for why this is a valid definition. It follows that, We denote the exponent as s because it is the saturation index of P in Q. Now, let the primary decomposition of D α , D β read, and define κ such that, κ = min κ l : N i vanishes to order s l · κ l on V (Q l ) .

(C.4)
We have then achieved a generalization of Eq. (3.20), it now reads, This generalization is, however, insufficient: not a single non-radical primary ideal obtained in Section 3.3.1 is a power of the associated prime. Since the prime ideal P l = Q l is unique, if no positive integer s l exists such that Q l = P s l l , then it is not possible to find the desired prime ideal P l in R n such that the condition on Eq. (C.5) is satisfied. Nevertheless, even if such a P l does not exist in R n , we can achieve this by extending the ring in which the ideal is defined by allowing it to include roots of polynomials.
For our purposes, it suffices to consider a ring extension involving a single s th -root. This can be achieved by extending the quotient ring R n of Eq. (3.5) by a single variable x, and by taking the quotient with respect to an ideal defining x s as a member q of R n . That is, we define the extended ring R n as, where R n [x] denotes the ring of polynomials in x with coefficients in R n . Given this definition, R n is a quotient ring of (an extension of) a quotient ring. Nevertheless, we can also regard R n as a simple quotient ring, just like R n . In fact, by the Third Isomorphism Theorem [34], we have, . (C.7) That is, R n is isomorphic to the quotient of the (extended) polynomial ring S n [x], by an ideal whose generators define both momentum conservation and x s as a polynomial in S n . Since this ideal is of maximal codimension, by the same reasoning which makes R n a Cohen-Macaulay ring [26], R n is also Cohen-Macaulay. Effectively, we have extended R n by s √ q. Therefore, with a slight abuse of notation, let us simply denote x as s √ q. In the following, R n will denote different extensions of R n , but it will always be clear which one is being considered at any one time depending on the polynomial appearing under the root.
We are now in a position to build an ideal P of R n such that, (P ) s ∩ R n = Q , (C.8) even if no prime ideal P of R n exists such that P s = Q. Note that using the symbolic power instead of the standard power in Eq. (C.8) would again be redundant: either the symbolic power coincides with the standard power, or the embedded components in the primary decomposition of the standard power must become redundant in the intersection with R n 6 . By the same reasoning, given Eq. (C.8), it can be shown that, As the numerators N i belong to the quotient ring R n , we are always free to add the intersection with R n to a membership statement of the form N i ∈ P sκ . Thus, assuming we can find an appropriate P and up to a suitable re-scaling of the symbolic power, we are now in a position to numerically obtain information about membership to Q κ , independently of whether Q is the power of a prime ideal in R n . That is, we have completely generalized Eq. (C.5). Furthermore, by extending the reasoning to R n , we have provided an interpretation to the κ th symbolic power of a non-radical ideal I = l Q l free of embedded components as the set of polynomials vanishing to degree s l κ on the varieties V (P l ) in R n , where the relation between Q l and P l is given by Eq. (C.8).
Let us now consider applications to the problem at hand. Starting from the first nonradical ideal that we encountered in Eqs. (3.27) and (3.28), for the "sharp" prime ideal in the ring extension we can write, 17 , |7 R 7 , (C. 10) such that taking the second power we obtain, Then, intersecting with R n , we have, that is, we have effectively removed the generator involving the radical. This is an explicit example of the form of Eq. (C.8). From this construction we observe that the phase-space 6 To show this, one has to remember that Q is a primary ideal, and that inclusion is preserverd in the intersection with a subring, i.e. (P ) s ⊂ Q ,em.
i ∩ Rn. point required to infer membership to symbolic powers of the ideal of Eq. (C.10) is such that, 17 ∼ , |7 ∼ ⇒ 17 ∼ 2 , |7 ∼ , (C.13) while a standard "symmetric" approach to V |7 reads, 17 ∼ , |7 ∼ . (C.14) An analogous construction can be followed for the ideals in Eqs. (3.32) and (3.33), for example we can write, 12 , (s 567 − s 34 ) R 7 , (C. 15) where we stress that the different meaning of R 7 between Eq. (C.15) and Eq. (C.10). In this case, we also have that the ideal is of maximal codimension, thus its symbolic powers must coincide with normal powers. Once again, we have constructed the desired ideal, Finally, as promised, the same approach can also be employed independently of whether an ideal is radical or not, in order to obtain futher data regarding the pole structure of the integral coefficients. For instance, let us reconsider the primary decomposition of Eq. (3.30 where we verified again in R 7 the equality as well as the primality of the ideals in the RHS. We can than combine constraints from the simultaneous membership to a symbolic power of 7|Γ 34|56 |7 , [7|Γ 34|56 |7] R 7 and 7|Γ 34|56 |7 , [7|Γ 34|56 |7] R 7 to obtain refined partial fraction decompositions. To see this in practice, let us refer back to section 4. It can be seen that a number of coefficients, such as d Their numerator in least common denominator form belongs to 7|Γ 34|56 |7 , [7|Γ 34|56 |7] R 7 , therefore we can write them as, . (C. 19) However, by probing them in the asymmetric approach we also obtain the constraint, As this is a maximal codimension ideal the symbolic power coincides with the normal power. Thus, we conclude that a more accurate representation is, . (C.21)