All master integrals for three-jet production at NNLO

We evaluate analytically all previously unknown nonplanar master integrals for massless five-particle scattering at two loops, using the differential equations method. A canonical form of the differential equations is obtained by identifying integrals with constant leading singularities, in $D$ space-time dimensions. These integrals evaluate to $\mathbb{Q}$-linear combinations of multiple polylogarithms of uniform weight at each order in the expansion in the dimensional regularization parameter, and are in agreement with previous conjectures for nonplanar pentagon functions. Our results provide the complete set of two-loop Feynman integrals for any massless $2\to 3$ scattering process, thereby opening up a new level of precision collider phenomenology.


INTRODUCTION
The ever improving experimental precision at the LHC challenges theoretical physicists to keep up with the accuracy of the corresponding theoretical predictions. In order for this to be possible, analytic expressions for higherloop amplitudes play a crucial role. Among the processes that are investigated at hadron colliders, jet production observables offer unique opportunities for precision measurements. In particular, the ratio of three-and two-jet cross sections gives a measure of the strong coupling constant α S (Q 2 ) at high energy scales Q 2 [1][2][3][4][5][6].
The situation was somewhat similar about fifteen years ago at NLO, when novel theoretical ideas led to what is now called the "NLO revolution" [16]. Thanks to recent progress in quantum field theory methods, we are today at the brink of an NNLO revolution.
The new ideas include cutting-edge integral reduction techniques based on finite fields and algebraic geometry [17][18][19], a systematic mathematical understanding of special functions appearing in Feynman integrals [20,21], and their computation via differential equations [22] in the canonical form [23]. The latter in fact lead to simple iterated integral solutions that have uniform transcendental weight (UT), also called pure functions.
It is particularly interesting that many properties of the integrated functions can be anticipated from properties of the simpler Feynman loop integrands through the study of the so-called leading singularities [24]. A useful conjecture [23,24] allows one to predict which Feynman integrals satisfy the canonical differential equation by analyzing their four-dimensional leading singularities. This can be done algorithmically [25].
It turns out that in complicated cases, especially when many scales are involved, the difference between treating the integrand as four-or D-dimensional can become relevant. In particular, integrands whose numerators contain Gram determinants that vanish in four dimensions may spoil the UT property. In this Letter we propose a new, refined criterion for finding the canonical form of the differential equations, and hence UT integrals. The method involves computing leading singularities in Baikov representation [26].
We apply our novel technique to the most complicated nonplanar massless five-particle integrals at NNLO. We explain how the UT basis is obtained, and derive the canonical differential equation. We determine analytically the boundary values by requiring physical consistency. The solutions are found to be in agreement with a previous conjecture for nonplanar pentagon functions, and also with a previously conjectured second entry condition [27].
This result completes the analytic calculation of all master integrals required for three-jet production at hadron colliders to NNLO in QCD. We expect that our method will have many applications for multi-jet calculations in the near future. Figure 1 shows the integral topologies needed for studying the scattering of five massless particles at two loops. The master integrals of the planar topology shown in Fig. 1a were computed in Ref. [9,28,29]. The nonplanar integral family depicted in Fig. 1b was computed in [30]. (See also [27,[31][32][33]). In this Letter, we compute the previously unknown master integrals of the doublepentagon family shown in Fig. 1c.

INTEGRAL FAMILIES
Genuine five-point functions depend on five independent Mandelstam invariants, X = {s 12 , s 23 , s 34 , s 45 , s 15 }, where s ij = 2p i · p j , and p i are massless external momenta. We also introduce the parity-odd invariant ǫ 5 as We denote the loop momenta for the double-pentagon family by k 1 and k 2 , defined as shown in Fig. 1c. The inverse propagators are where D 9 , D 10 and D 11 are irreducible scalar products (ISPs).

LEADING SINGULARITIES AND UNIFORM TRANSCENDENTAL WEIGHT INTEGRALS
The integrals of the double-pentagon family, shown in Fig. 1c, can be related through integration-by-parts relations [34][35][36] to a basis of 108 master integrals. Out of these, 9 are in the so-called top sector, namely they have all 8 possible propagators. Our goal is therefore to find 108 linearly independent UT integrals.
The integrals of the sub-topologies are already known, because they are either sub-topologies of the pentabox [9,29] and of the hexa-box [30] families, or they correspond to sectors with less than five external momenta [37,38]. In order to complete the UT basis, we begin by searching for four-dimensional d log integrals, which are closely related to UT integrals [24].
An ℓ-loop four-dimensional d log integral is an integral whose four-dimensional integrand Ω can be cast in the form where the Q-valued constants c I are the leading singularities of Ω.
In order to perform the loop integration in D = 4 − 2ǫ dimensions, where ǫ is the dimensional regulator, it is necessary to clarify how the integrand is to be defined away from four dimensions. For example, one may simply "upgrade" the loop momenta from 4-dimensional to D-dimensional (abbreviated as 4d and Dd) ones. We call this the "naïve upgrade" of a 4d integrand. While this method is quite powerful in finding a UT basis, and indeed it has already found many successful applications [23,39], the freedom involved in the upgrade can become important, especially for integrals with many kinematic scales. We first review the four-dimensional analysis, and then provide a method of fixing the freedom, while maintaining the advantages of the canonical differential equations method.
In this Letter, we use two techniques to find 4d d log integrals.
(1) The algorithm [25], which can decide if a given rational integrand can be cast in d log form (3). Starting from a generic ansatz for the numerator, this algorithm can classify all possible 4d d log integrals in a given family.
(2) Using computational algebraic geometry, we consider a generic ansatz for the numerator N even = α c α m α of the parity-even, or N odd = α c α m α /ǫ 5 of the parity-odd d log integrals. Each c α is a polynomial in s ij , and m α is a monomial in the scalar products. By requiring the 4d leading singularities of the ansatz to match a given list of rational numbers, we can use the module lift techniques [40] in computational algebraic geometry to calculate all c α and to obtain a 4d d log basis. This method usually needs only a very simple ansatz, and the module lift can then be performed through the computer algebra system Singular [41].
One interesting phenomenon is that, for the doublepentagon family, the naïve upgrade of a 4d d log integral is in general not UT. Let us take the 4d d log integrals presented in Ref. [42] as examples. The sum of the first and the fifth d log integral numerators for the doublepentagon diagram in Ref. [42], which we denote by B 1 + B 5 , does not yield a UT integral after the naïve upgrade. This can be assessed from the explicit computation of the differential equation.
The obstruction of the naïve upgrade implies that, in order to obtain UT integrals, we have to consider terms in the integrands which vanish as D = 4. These terms can be conveniently constructed from Gram determinants involving the loop momenta k 1 and k 2 , An integrand whose numerator is proportional to a combination of the different G ij explicitly vanishes in the D → 4 limit. UT integral criteria based on 4d cuts or 4d d log constructions can not detect these Gram determinants, and may yield inaccurate answers on whether an integral is UT in D dimensions or not. Instead, we develop a new D-dimensional criterion for UT integrals, based on the study of the cuts in Baikov representation. Our method analyzes the Dd leading singularities, and for a given 4d d log integral with 4d integrand N/(D 1 . . . D k ), our criterion generates a Dd integrand of the formÑ which is a UT integral candidate. Here the tilde sign denotes the naïve upgrade, andS is proportional to Gram determinants. We name Eq. (1) For any 4d double-pentagon d log in Ref. [42] we can find its refined upgrade from our Dd UT criterion. We verified that such refined upgrades are indeed UT integrals by computing the differential equation. For example, the refined upgrade of (B 1 + B 5 ) is (2) Some integrals with purely Gram determinant numerators satisfy our Dd UT criterion: Once again we verified that these integrals are indeed UT by examining the differential equation.

CRITERION FOR PURE INTEGRALS FROM D-DIMENSIONAL CUTS
In this section we present our new criterion for UT integrals based on Dd cuts in the Baikov representation [26]. As we have already seen, this new criterion is sharper than the original 4d one, as it can also detect Gram determinants to which the latter is blind.
Let us recall that in the Baikov representation [26] the propagators of a Dd Feynman integrand are taken to be integration variables (Baikov variables). The Dd leading singularities can thus be calculated easily by taking iterative residues. Then, our Dd criterion for a UT integral is to require all the residues of its Baikov representation to be rational numbers.
For the double-pentagon integral family, the standard Baikov cut analysis [43,44], based on the two-loop Baikov representation, eventually leads to complicated threefold integrals. To avoid this computational difficulty, we adopt the loop-by-loop Baikov cut analysis [45].
For a double-pentagon integral with some numerator N , for instance, the integration can be separated loopby-loop as The two-loop integral can thus be decomposed into a pentagon integral with loop momentum k 1 and external legs p 1 , p 2 , p 3 and −k 2 , and a triangle integral with loop momentum k 2 . Note that, if necessary, we might need to carry out a one-loop integrand reduction for the numerator N first, in order to make sure that the integrand contains no cross terms such as k 1 · p 4 or k 1 · p 5 . As a consequence, D 9 drops out from the integrand. We then apply the Baikov representation loop-by-loop, i.e. we change integration variables from the components of the loop momenta to 10 Baikov variables Once this is done, we can explore the Dd residues.
For instance, consider the double-pentagon integral I dp [G 12 ]. Its 4d leading singularities are all vanishing, and can therefore not determine whether I dp [G 12 ] is UT or not. Conversely, by using our Baikov cut method, having integrated out the term k 1 · p 4 , we get a Baikov integration with 10 variables. Taking the residues in z i = 0, ∀i ∈ C, where C ⊆ {1, . . . , 8}, yields integrands which do not vanish in the D → 4 limit. Using the algorithm [25], we systematically compute all possible residues of these integrands in the remaining variables, and make sure that there are no double poles. In this way we compute the leading singularities on different cuts, and find that they all evaluate to ±ǫ 5 /(s 12 − s 45 ) or zero. As a result, we see that the integral satisfies our Dd criterion. We confirmed that (9) is indeed a UT integral by explicitly computing it from differential equations. Similarly, we can use this loop-by-loop Baikov cut method to find the UT integral candidates listed in Eqs. (6) and (7), for which the 4d leading singularity calculation cannot give a definitive answer. All these candidates are subsequently proven to be UT by the differential equations.
It is worth noting that this Dd Baikov cut analysis only involves basic integrand reduction and residue computations. We expect that this method, combined with the d log construction algorithm described in [25], will prove to be a highly efficient way of determining UT integral candidates for even more complicated diagrams in the future.

MASTER INTEGRALS AND CANONICAL DIFFERENTIAL EQUATIONS
With the study of 4d d log integrals, and the novel Dd Baikov cut analysis, we constructed a candidate UT integral basis for the double-pentagon family.
Through IBPs, we find that the eight 4d d logs in Ref. [42], after our refined upgrade, together with the three Gram-determinant integrals given in Eq. (7), span a 8-dimensional linear space. By combining the algorithm described in [25] and the computational algebraic geometry method, we easily find another linearly independent integral satisfying our Dd UT criterion. This completes the basis for the double-pentagon on the top sector. Sub-sector UT integrals are either found via [25], or taken from the literature [9,29,30].
By differentiating our candidate UT basis for the double-pentagon family, we see that the differential equations are immediately in the canonical form [23] d I(s ij ; ǫ) = ǫ dÃ(s ij ) I(s ij ; ǫ) , without the need for any further basis change. This is the ultimate proof that our basis integrals are indeed UT. We wish to emphasize here that the construction of the UT basis is done at the integrand level via Baikov cut analysis, and as such does not require the a priori knowledge of the differential equations.
It is also worth mentioning that the analytic inverse of the transformation matrix between our UT basis and the "traditional" basis from Laporta algorithm was efficiently computed by means of the sparse linear algebra techniques described in [46]. Equation (10) can be further structured to the form where W k are letters of the pentagon symbol alphabet conjectured in [27], and each a k is a 108 × 108 rational number matrix. We consider the integrals in the s 12 scattering region. The latter is defined by positive s-channel energies, {s 12 , s 34 , s 45 , s 35 } ≥ 0, and negative t-channel energies, {s 23 , s 24 , s 25 , s 13 , s 14 , s 15 } ≤ 0, as well as reality of particle momenta, which translates to ∆ ≤ 0.
We choose a boundary point inside this region. We determine the boundary values of the integrals by requiring physical consistency, as described in [30]. This yields a system of equations for the boundary constants at X 0 , whose coefficients are Goncharov polylogarithms. We evaluate the latter to high precision using GiNaC [47]. The values at X 0 were validated successfully with the help of SecDec [48].
The full result for the integrals is again written in terms of Goncharov polylogarithms. For reference, we provide numerical values for all integrals at the symmetric point X 0 , as well as for an asymetric point The values, given in ancillary files, have at least 50 digit precision. Here we display the results for integral I 107 , I 107 (X 1 , ǫ) = 6.9362922441923047974i + O(ǫ) . (15) From their leading order term in ǫ of the boundary values, one can immediately write down the symbol of the integrals. This has also been computed independently in [49], and has already been employed in the computation of two-loop five-point amplitudes in N = 4 super-Yang-Mills theory [49,50] and N = 8 supergravity [51,52] at symbol level. We observe that the second entry condition conjectured in [27] is indeed satisfied. We provide the UT basis for the double-pentagon family, the A matrix of the canonical differential equation (10), and the boundary values at X 0 and X 1 in ancillary files.

DISCUSSION AND OUTLOOK
In this Letter, we computed analytically the master integrals of the last missing integral family needed for massless five-particle scattering amplitudes at two loops. We applied the canonical differential equation method [23], supplemented with a novel strategy for finding integrals evaluating to pure functions based on the analysis of Dd leading singularities in Baikov representation.
Our calculation confirms the previously conjectured pentagon functions alphabet and second entry condition [27]. Our result implies the latter is a property of individual Feynman integrals, not only of full amplitudes. It will be interesting to find a field theory explanation of this condition, perhaps along the lines of the Steinmann relations.
With our result, all master integrals relevant for threejet production at NNLO are now known analytically. Moreover, they are ready for numerical evaluation in physical scattering regions. This opens the door to computing full 2 → 3 scattering amplitudes at two loops.
We expect that our Dd Baikov cut analysis will prove to be a powerful method to find Feynman integrals evaluating to pure functions, in particular for integral families involving many scales. We expect it will have many further applications for multi-particle amplitudes, e.g. for H + 2 j and V + 2 j productions, and other multi-scale processes relevant for collider physics.

ACKNOWLEDGMENTS
We are indebted to Gudrun Heinrich and Stephan Jahn for providing numerical checks and for help with using SecDec. Y.Z. thanks Alessandro Georgoudis and Yingxuan Xu for enlightening discussions. This research received funding from the Swiss National Science Foundation (Ambizione Grant No. PZ00P2 161341), the European Research Council (ERC) under the European Union's Horizon 2020 Research, and Innovation Programme (Grant Agreement No. 725110), "Novel Structures in Scattering Amplitudes." J. H., Y. Z., and S. Z. also wish to thank the Galileo Galilei Institute for hospitality during the workshop "Amplitudes in the LHC Era."