All Two-Loop Feynman Integrals for Five-Point One-Mass Scattering

We compute the complete set of two-loop master integrals for the scattering of four massless particles and a massive one. Our results are ready for phenomenological applications, removing a major obstacle to the computation of complete next-to-next-to-leading order (NNLO) QCD corrections to processes such as the production of a $H/Z/W$ boson in association with two jets at the LHC. Furthermore, they open the door to new investigations into the structure of quantum-field theories and provide precious analytic data for studying the mathematical properties of Feynman integrals.

Feynman integrals play a central role in obtaining precise predictions in quantum-field theory (QFT).They are also of great mathematical interest, giving rise to noteworthy classes of special functions.Advances in the calculation of Feynman integrals have led to new insights into the mathematical properties of these functions, as well as to new results in formal studies of QFTs and (beyond) Standard-Model phenomenology.While the calculation of two-loop five-point Feynman integrals is an active area of research [1][2][3][4][5][6][7][8][9][10][11][12], a complete set of integrals is only available for massless particles [1][2][3][4][5][6][7].In this letter, we advance the state of the art by computing all Feynman integrals necessary to describe the scattering of four massless particles and a massive one at two loops.
We obtain a representation for the Feynman integrals in dimensional regularization that exhibits their analytic structure and allows for a stable and efficient numerical evaluation.This is achieved by finding bases of pure integrals [13] that satisfy differential equations (DEs) [14][15][16][17][18] in canonical form [19], explicitly displaying all singularities of the integrals.Despite recent progress [5,[20][21][22][23][24][25], finding a pure basis is still a major bottleneck.We followed the approach of refs.[3,6], building on modular arithmetic to bypass intermediate computational bottlenecks [26,27].Using Chen iterated integrals [28], we solve the DEs at each order in the dimensional regulator in terms of a minimal set of functions, called (onemass) pentagon functions.We demonstrate that this solution is efficient and numerically stable over phase space and therefore ready for phenomenological application.A C++ library [29] for the evaluation of the pentagon functions makes our results accessible to the whole community.Previous approaches for constructing such solutions [7,[30][31][32] relied on the possibility of representing them through multiple polylogarithms (MPLs) [33], but it is generally unclear if such a representation exists [34].We show that this is not required, and only one evaluation of the integrals at a precision comparable to that at which we evaluate the pentagon functions is sufficient.Non-planar integrals introduce added complexity: some exhibit non-analytic behavior or a logarithmic singularity within the physical scattering region.We isolate this behavior in the pentagon functions, and extend the numerical methods of refs.[7,32] to deal with it.
Our results open the door to new explorations in many different directions.On the analytic side, this is the first complete set of two-loop integrals allowing us to explore the (extended) Steinmann relations [35][36][37][38][39][40][41].Moreover, these integrals make it possible to study how unexplained observations of analytic cancellations in gauge-theory amplitudes [31,[42][43][44] extend beyond the leading-color approximation.In formal studies of QFTs, they have already been central in bootstrapping results in N = 4 super Yang-Mills theory (sYM), leading to new conjectures [45,46].Finally, all two-loop Feynman integrals for the production of a massive boson in association with two jets at hadron colliders are now readily available, removing a main bottleneck for these important processes in LHC physics for which more precise theoretical predictions are urgently needed.Indeed, it is now possible to go beyond the leading-color approximation for W + 2-jet processes [31,43,[47][48][49], and obtain two-loop results for crucial processes such as H + 2 jets which are intrinsically non-planar.

I Notation and Conventions
We define the momenta of external particles as p i , i = 1, . . ., 5, where p 2 1 ̸ = 0 and p 2 i = 0 for i = 2, . . ., 5. For a fixed ordering of the massless legs, there are three planar penta-box (PB) families, three non-planar hexa-box (HB) families and two non-planar double-pentagon (DP) families that we depict in fig. 1, as well as a factorizable planar topology.The factorizable, PB and HB families have already been studied in the literature [2,6,[8][9][10].
Here we define the DP families.Integrals in these families can generically be written as where we set D = 4−2ϵ, and ⃗ ν is a vector of integers with the restriction that ν 9 , ν 10 , ν 11 ≤ 0. Explicit expressions for the ρ i are given in ancillary files [50].
There are six independent variables s ij = (p i + p j ) 2 , which we choose to be Together with the parity-odd object they fully specify a point in the five-particle phase space.Singularities of Feynman integrals are located at zeroes of certain determinants, see e.g.refs.[51][52][53][54][55]. Three cases play a special role here: the three and five-point Gram determinants where G(q 1 , . . ., q n ) = 2{q i • q j } i,j∈{1,...,n} , and the polynomial [9] Σ While ∆ 5 = tr 2 5 , relating tr 5 to √ ∆ 5 precisely is a subtle issue.We adopt the convention of ref. [9] to only use √ ∆ 5 in the pure integrals' definitions.Fig. 1 shows a fixed ordering of the massless legs, but we consider the set of integrals closed under all permutations of these legs.While ∆ 5 is invariant under these permutations, there are three different permutations of ∆ 3 , denoted ∆ and ∆ 5 are given in ancillary files [50].
Let ⃗ g τ denote a vector whose entries form a pure [13] basis of master integrals for a family of integrals τ .It satisfies a DE in canonical form [19] d⃗ where the W i are the letters of the (symbol) alphabet [33] associated with ⃗ g τ .While the W i are algebraic functions of ⃗ s, the matrices M i are matrices of rational numbers.Finding a pure basis is still the most challenging aspect in obtaining DEs in canonical form.We construct educated guesses for pure bases following the ideas of refs.[4][5][6]9], and test candidate bases by evaluating their derivatives at numerical points and verifying the factorization of ϵ.
Once a pure basis is found, we proceed as in section 4 of ref. [6] to determine that the alphabet for the DPmz and DPzz families is contained within the one obtained in ref. [9].DPmz and DPzz have 62 and 74 letters respectively.As in ref. [6], we fit the matrices M i from numerical evaluations on a finite field.Our results for the pure bases, the alphabet (closed under all permutations of the massless legs), and the analytic DEs can be found in ancillary files [50].Some pure integrals were simplified with ideas from ref. [64].

III Construction of One-Mass Pentagon Functions
The (one-mass) pentagon functions are a basis of special functions to express all one-and two-loop five-point integrals with an external massive leg, up to the order in ϵ required to compute two-loop corrections.The oneloop and planar two-loop integrals have been previously considered in ref. [32].Here we discuss a novel approach, suitable for the non-planar families, that overcomes bottlenecks of previous strategies.
We start by considering the ϵ expansion of the master integrals for each family τ in fig. 1.To cover all integrals relevant for an amplitude, we consider all 4! permutations σ of the massless momenta, denoting the corresponding master integrals by ⃗ g τ,σ .We normalize the ⃗ g τ,σ so that they have the expansion τ,σ (⃗ s) .
We obtain the DEs for all ⃗ g τ,σ by permuting those in the standard ordering, and use them to write the ⃗ g τ,σ (⃗ s) as Q−linear combinations of Chen iterated integrals [28], and boundary values at a point ⃗ s 0 , ⃗ g τ,σ (⃗ s 0 ).The path γ connects ⃗ s 0 and ⃗ s, and the iteration starts with [] ⃗ s0 = 1.The number of integrations w is called the transcendental weight.Up to two loops, it suffices to restrict our focus to w ≤ 4. We choose [32] which is in the physical s 45 channel: particles 4 and 5 are in the initial state, and the remaining particles are in the final state.The ⃗ g τ,σ (⃗ s 0 ) are rational numbers and are determined by the first-entry condition [65] up to an overall normalization.For w ≥ 1, we compute the ⃗ g τ,σ (⃗ s 0 ) with 60-digit precision using AMFlow [66], interfaced to FiniteFlow [59] and LiteRed [60,61].
The structure of the iterated integrals in eq. ( 8) is very well understood [28]: integrals involving different letters are linearly independent, and products of integrals are controlled by a shuffle algebra.This enables an algorithmic construction of a minimal set of functions in which to express the ⃗ g (w) τ,σ (⃗ s): the one-mass pentagon functions.To construct this minimal set, we follow the procedure in refs [7,32].We begin by considering the ⃗ g (w) τ,σ (⃗ s) at symbol level [33] select a minimal set of the ⃗ g τ,σ (⃗ s) from the symbol-level solutions, starting at weight 1 and proceeding iteratively up to weight 4. To this end, we consider the set of coefficients ⃗ g (w) τ,σ (⃗ s) for all τ and σ, as well as all weight-w products of lower-weight functions, and select a subset of linearly independent elements, preferring products of lower-weight pentagon functions.This minimizes the number of irreducible functions: functions which cannot be expressed in terms of products of lowerweight functions.In this way, at each weight we construct a set of algebraically independent ⃗ g (w) τ,σ (⃗ s), which we call the pentagon functions and denote by {f (w) i }.We find 11 irreducible functions of weight 1, 35 of weight 2, 217 of weight 3 and 1028 of weight 4. The total number is substantially lower than 2304, the number of independent master integrals.
We now turn to expressing the ⃗ g (w) τ,σ (⃗ s) in terms of pentagon functions.The approach that we use bypasses the determination of the relations between the boundary values ⃗ g (w) τ,σ (⃗ s 0 ), a bottleneck of previous approaches.To this end, we make the following ansatz: we assume that the ⃗ g (w) τ,σ are graded polynomials in the pentagon functions and two transcendental constants, ζ 2 and ζ 3 , over the field of rational numbers.Let us consider a weight-2 coefficient g τ,σ as an example.Our ansatz is g (2)  τ,σ = i a i τ,σ f i f We determine the rational numbers a i τ,σ and a i,j τ,σ from symbol-level manipulations, and the ãτ,σ by numerically evaluating the coefficients on both sides of the equation at ⃗ s 0 .In this way, we explicitly write all ⃗ g (w) τ,σ (⃗ s) in terms of ζ values and a minimal set of pentagon functions, for which we have an iterated-integral representation and a boundary condition valid to 60 digits.
We emphasize that the ansatz of eq. ( 10) implies nontrivial polynomial relations among the ⃗ g τ,σ (⃗ s 0 ).Previous approaches [7,31,32] to determine these relations required PSLQ [67] studies of high-precision numerical evaluations (O(3000) digits) that were obtained from MPL expressions [8].Bypassing the need for highprecision evaluations, which are not available for nonplanar topologies, is therefore a substantial improvement.
To validate our results, we checked that they agree with those of refs.[6,32] for the factorizable and PB families.For HB families, we compared our results with evaluations from DiffExp [68] starting at the boundary values of ref. [9].We find numerical agreement with the results of ref. [10] for the HBmzz family, but were unable to evaluate their results for HBzzz and HBzmz.For the two DP families, we started from an evaluation of the pentagon functions, and used DiffExp to verify that the integrals are regular at all Euclidean spurious singularities.We further checked selected permutations of the DP integrals against AMFlow at random phase-space points.

IV Numerical Evaluation and Analytic Structure
The algorithm given above leaves freedom in the definition of pentagon functions.We leverage it to build a basis of functions that can be evaluated in an efficient and stable way, and is informed by the singularities that are expected in physical quantities.Non-planar families bring considerable new challenges, highlighted below.
The general approach to the evaluation of pentagon functions follows refs.[7,30,32,69].We focus on the phase-space region corresponding to the s 45 channel defined below eq.( 9), which is sufficient for hadron-collider processes (other 2 → 3 channels are obtained through appropriate permutations [32]).We construct the path γ in eq. ( 8) so that it never leaves the s 45 channel, following the algorithm of ref. [32].Up to weight 2, we obtain TABLE I: Non-differentiable integral functions due to divergent integrands in eq. ( 11) at singularity of ∂ t log(W [t]), which cause discontinuities (right column).expressions in terms of logarithms and dilogarithms [70] with no logarithmic branch points within the s 45 channel.At weights 3 and 4 we construct one-fold integral representations [69] and perform the integration numerically.We refer to ref. [32] for a thorough discussion, and highlight here novel cases where the one-fold integral representation has a singularity at some point on γ.
We exemplify our approach with weight-three pentagon functions, but generalization to weight four is straightforward.The one-fold integral representations of the pentagon functions are combinations of terms of the form For simplicity, we parametrize γ in terms of t so that ∂ t log(W [t]) diverges at t = 0 (if it does diverge on γ).
To construct a numerically stable algorithm for evaluating the pentagon functions, we consider the analytic structure of eq. ( 11) in detail.In many instances the singularity at t = 0 cancels in the sum of the contributions of the form of eq. ( 11).We arrange these cancellations analytically as in refs.[7,32], and such pentagon functions are infinitely differentiable in the physical region.The novel behavior in this work is a feature of five-point onemass non-planar pentagon functions, related to Σ (i) 5 = 0 surfaces.All cases are summarized in table I, and can be organized in terms of the local behavior of h and W .In all cases the integrands diverge.For cases a and b, the singularity is integrable.In case a, W = Σ (i̸ =3) 5 ; in case b, d log W is odd with respect to Σ (3) 5 and, as h 0 is known, we handle the integrable singularity with an analytic subtraction.In case c, W = Σ (3) 5 and the integral has a logarithmic singularity.We introduce a subtraction term and analytically integrate the 1/t singularity, resulting in a numerical integration over the remaining integrable singularity.We discuss the analytic continuation across Σ Integrable singularities in eq. ( 11) complicate the application of the numerical algorithms of refs.[7,32].Since all problematic cases have W = Σ  of pentagon functions with this behavior as F Σ5 .Integrating over the integrable singularities in cases a and c demands higher intermediate precision, at the cost of performance.This motivates the construction of the pentagon functions so that the set F Σ5 is as small as possible, which also isolates the logarithmic singularity at Σ 5 = 0 (case c) in as few functions as possible.More generally, we can organize the construction of the pentagon functions to make analytic properties manifest.For instance, a conjecture about the absence of the letter ∆ 5 in the properly defined finite remainders of scattering amplitudes has been put forward for the massless case (see [71][72][73] for possible explanations), and we thus isolate the ∆ 5 dependence in as few functions as possible.
The numerical evaluation of the one-mass pentagon functions is available through the C++ library PentagonFunctions++ [29].To assess its numerical performance, we evaluate all functions on a sample of 105 points as in ref. [32].We present the distributions of correct digits with standard intermediate precision for the functions in F Σ5 (blue line in fig.2a) and the ones that do not involve the letters Σ (i) erated by phase-space points that use integration paths which intersect the surfaces Σ (i) 5 = 0.As expected, it is absent for the orange line as well as in fig.2b, where higher intermediate precision is employed to rescue the cases contributing to the blue peak mentioned above.Given the evaluation times of a few seconds per point and the overall good numerical stability, our results are suitable for immediate phenomenological applications.

V Discussion and Outlook
We complete the calculation of all two-loop five-point integrals with massless propagators and a single massive external leg.Our results allow us to study their analytic structure, and to efficiently evaluate them numerically through weight 4, as needed for current phenomenological applications.The numerical evaluation is readily available as a C++ library [29].
The new algorithm we present for constructing pentagon functions provides a substantial improvement over previous ones, only relying on the knowledge of pure bases and the evaluation of the functions at a single point to moderate precision [12,66].This robust algorithm will certainly find applications in other classes of integrals.
Our results open door to further explorations of the analytic structure of this class of integrals, for instance along the direction of refs.[53][54][55].An undoubtedly important question to explore in more detail is the presence of logarithmic singularities within the physical region associated with the letter Σ (i) 5 for some master integrals.Furthermore, our results can be used in the very active area of bootstrapping in N = 4 sYM (see e.g.refs.[45,46,[74][75][76]). Finally, they will be central to the calculation of the NNLO corrections to processes such as the production of a massive boson in association with two jets at hadron colliders, as well as the ongoing N 3 LO calculations [77,78] of processes involving a massive external particle and three massless ones.

FIG. 1 :
FIG.1: Two-loop five-point one-mass families.The thick external line denotes the massive external leg.
appendix A. The distinguished role played by Σ (3) 5 is a consequence of working in the s 45 channel.
Functions F Σ 5 in blue and FΣ 5 in orange.Dashed line represents the latter's cumulative distribution.