How can two-loop Feynman integrals for vector boson fusion be evaluated?

What the VBF calculation must deliver

Vector boson fusion is the second most dominant Higgs-boson production mode at the Large Hadron Collider, and its signature of two high-energy forward jets makes it useful for precision Standard Model studies and for searches for anomalous Higgs couplings to electroweak vector bosons. With conventional VBF cuts the fiducial cross section is dominated by factorisable contributions with no colour exchange between the two incoming partons. The paper states that this is no longer clear once the cuts are relaxed, which is what motivates the non-factorisable QCD corrections that first appear at next-to-next-to-leading order (NNLO).

Those corrections are harder than the factorisable ones because they involve the full five-point kinematics of the VBF process. The paper notes that no exact NNLO prediction is yet available, and that existing NNLO QCD predictions for the non-factorisable part use the eikonal approximation, which the authors describe as sufficient for conventional VBF cuts but not necessarily for more general phase-space regions. The obstacle is the complicated five-point two-loop Feynman integrals in the non-factorisable topologies.

Two topologies and 177 master integrals

The paper treats two five-point two-loop families: a planar penta-box (PB) and a non-planar hexa-box (HB). Each has five external legs, one of them massive, and eight propagators, two of which are massive. These are two of the four integral families arising in the computation; the two double-pentagon families shown in the second row of the paper's Figure 1 are left for future work.

Using integration-by-parts reduction with Laporta's algorithm, implemented in Reduze 2 and in a private finite-field code called Finred, the authors reduce the PB family to 111 master integrals and the HB family to 121, of which 66 are not shared with the penta-box. The total number of different master integrals is therefore 177. The paper also includes a non-IBP relation, derived by examining Symanzik polynomials, that relates a two-point to a one-point subtopology. Because several subtopologies of the two families are related by permutations of external massless momenta, the number of integrals posing genuine challenges is significantly smaller than 177.

Why seven variables make this hard

Scattering kinematics are described by seven independent Lorentz invariants: the internal mass squared, s1, s12, s23, s34, s45 and s15. The presence of both an internal and an external mass scale is what makes this a seven-variable problem, comparable to several recent two-loop computations cited by the authors. The integrals are computed in all 24 permutations of the massless legs, which lets the authors eliminate redundancies between permutations in their special-function representation; for the VBF application it is then sufficient to evaluate them in one specific 2-to-3 scattering channel, the s45 channel with particles 4 and 5 incoming.

The target form is an epsilon-factorised differential equation, in which the derivative of the master-integral vector equals epsilon times a sum of one-forms, called letters, each multiplied by a rational matrix. Once such a system and boundary values are known, it can be solved order by order in epsilon in terms of iterated integrals. Finding a basis that produces this form is the central technical task, and the paper works bottom-up in the number of propagators, starting each sector on the maximal cut where all propagators are put on-shell.

Nested square roots still admit d log forms

The basis change to the canonical basis is algebraic for both topologies, but one subsector of the penta-box requires nested square roots. The authors note that nested square roots appeared in a similar topology for ttH production, where not all differential forms could be expressed as d log forms, and that whether this is possible is not straightforward to see because nested square roots complicate the algebra and the derivation of the equations. For the topology considered here, they find that the differential equations can be expressed entirely in d log form, and they describe in detail how the corresponding letters are determined.

The paper's strategies combine several ingredients. Integrals with constant leading singularities at epsilon equal zero are often good canonical candidates on the maximal cut, analysed with the DlogBasis package in loop-by-loop Baikov representations. Several five-point masters are instead constructed using Gram-determinant numerators, which vanish at epsilon equal zero and can often be related to a dimension shift of sub-loops. For sectors with massless box subloops the authors use a building-block method, normalising the leading singularities of the one-loop box subdiagram to a constant. For letters with nested square roots they generalise a factorisation-based algorithm and also adapt letters from earlier work.

One function basis for all permutations

After deriving the differential equations, the authors express the epsilon-expanded integrals for all permutations of the massless legs in terms of a common basis of algebraically independent functions. They point out that the same transcendental function basis may also be used to compute two-loop QCD corrections to the Higgs-Strahlung process, and they expect the results to allow compact amplitude representations and analytical cancellations of ultraviolet and infrared poles in finite remainders.

The evaluation itself is numerical: the authors solve the differential equations numerically, which benefits from the simpler form produced by the analytical preparation. The paper reports values for the almost-canonical top-sector master integrals at one phase-space point, with the epsilon^3 and epsilon^4 coefficients given to high precision; several entries are zero through epsilon^3, and two hexa-box entries are small negative numbers at epsilon^4. These are values at a single chosen point, not a scan over phase space.

What remains before an exact NNLO result

The paper is explicit that it calculates only two of the two-loop families required for VBF at NNLO. The complete two-loop amplitudes involve two additional double-pentagon families, which the authors expect to be more difficult, in particular encompassing further nested square roots and elliptic structures. The exact NNLO prediction is therefore not delivered here; what is delivered is the integral technology for two of the four families, together with the strategies for handling nested square roots.

The authors also state that this is the first time a canonical basis has been constructed for non-planar Feynman integrals depending on seven scales. That claim is the paper's own framing of its novelty, and it is scoped to the non-planar seven-scale case rather than to canonical bases in general. The integral basis results are provided in computer-readable form in the ancillary files accompanying the article.

Where the nested-root strategy matters most

The most transferable part of the paper is not the specific penta-box or hexa-box result but the machinery for nested square roots: how to find canonical master integrals, how to derive d log letters, and how to perform analytic continuation when the argument of a square root itself contains a square root. For analytic continuation, the authors use the minimal polynomial of the function and its discriminant to build a polynomial whose zeros must include any branch-cut crossing, then test those points numerically and multiply by appropriate signs.

That machinery is what makes the numerical integration of the differential equations feasible here, and the authors expect it to be more generally applicable to similar problems, which they expect to become increasingly common for integrals with massive propagators. The practical priority for a reader is therefore the split between what is finished and what is not: two of four families are reduced to canonical form and evaluated numerically, the nested-root case is handled in d log form for this topology, and the remaining double-pentagon families, with their expected elliptic structures, are the next obstacle.

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