Abstract The Local Unitarity (LU) formalism provides a constructive, integrand-level realisation of the Kinoshita-Lee-Nauenberg (KLN) theorem, by combining loop and phase-space integrals appearing in scattering cross-sections in such a way that their final-state infrared singularities cancel before integration. Supplemented with localised ultraviolet renormalisation, it enables the direct Monte Carlo integration of cross sections at arbitrary perturbative order in four-dimensional spacetime. In this paper, we present its application to the next-to-next-to-leading order (NNLO) QCD total cross sections for heavy-quark pair production in direct photon fusion, involving the contribution from 138 distinct forward-scattering diagrams where external photons couple only to heavy quarks. By combining NNLO QCD with next-to-leading order (NLO) electroweak (EW) corrections and next-to-leading power (NLP) Coulomb resummation, we obtain state-of-the-art predictions for top-, bottom-, and charm-quark production in ultraperipheral hadron collisions and at e + e − colliders.

We have extended our previous computations of the even-N moments of the flavour-singlet four-loop splitting functions to N = 12 for the pure-singlet quark case and N = 10 for all other cases. These results, obtained using physical quantities in inclusive deep-inelastic scattering, have been and will be used to validate conceptionally much more challenging determinations of these splitting functions from off-shell operator matrix elements (OMEs). For the quark-gluon and gluon-gluon splitting functions, which have yet to be computed to higher N using OMEs, we construct approximations based on our moments and endpoint constraints, where we present new large-x results for the gluon-gluon case. These approximations facilitate an approximate N^3LO evolution of parton distributions which are sufficiently accurate outside the region of small momentum fractions x.

We report results for the lowest even-N moments of the flavor-nonsinglet structure functions F_2 and F_L in QCD at the fourth order in the perturbative expansion in the strong coupling constant α_s. Our results are presented in numerical form and we compare them with the leading and subleading terms of the threshold expansion for large values of N, which corresponds to the limit x → 1.

A bstract The Local Unitarity (LU) representation of differential cross-sections locally realises the cancellations of infrared singularities predicted by the Kinoshita-Lee-Nauenberg theorem. In this work we solve the two remaining challenges to enable practical higher-loop computations within the LU formalism. The first concerns the generalisation of the LU representation to graphs with raised propagators. The solution to this problem results in a generalisation of distributional Cutkosky rules. The second concerns the regularisation of ultraviolet and spurious soft singularities, solved using a fully automated and local renormalisation procedure based on Bogoliubov’s R -operation. We detail an all-order construction for the hybrid $$ \overline{\textrm{MS}} $$ MS ¯ and On-Shell scheme whose only analytic input is single-scale vacuum diagrams. We validate this novel technology by providing (semi-)inclusive results for two multi-leg processes at NLO, study limits of individual supergraphs up to N3LO and present the first physical NNLO cross-sections computed fully numerically in momentum-space, namely for the processes γ ∗ → jj and $$ {\gamma}^{\ast}\to t\overline{t} $$ γ ∗ → t t ¯ .

We have computed the four lowest even-N moments of all four splitting functions for the evolution of flavour-singlet parton densities of hadrons at the fourth order in the strong coupling constant alpha_s. The perturbative expansion of these moments, and hence of the splitting functions for momentum fractions x >~ 0.1, is found to be well behaved with relative alpha_s-coefficients of order one and sub-percent effects on the scale derivatives of the quark and gluon distributions at alpha_s ~< 0.2. More intricate computations, including other approaches such as the operator-product expansion, are required to cover the full x-range relevant to LHC analyses. Our results are presented analytically for a general gauge group for detailed checks and validations of such future calculations.

We propose a novel representation of differential scattering cross-sections that locally realises the direct cancellation of infrared singularities exhibited by its so-called real-emission and virtual degrees of freedom. We take advantage of the Loop-Tree Duality representation of each individual forward-scattering diagram and we prove that the ensuing expression is locally free of infrared divergences, applies at any perturbative order and for any process without initial-state collinear singularities. Divergences for loop momenta with large magnitudes are regulated using local ultraviolet counterterms that reproduce the usual Lagrangian renormalisation procedure of quantum field theories. Our representation is especially suited for a numerical implementation and we demonstrate its practical potential by computing fully numerically and without any IR counterterm the next-to-leading order accurate differential cross-section for the process e+e− → $$ d\overline{d} $$ . We also show first results beyond next-to-leading order by computing interference terms part of the N4LO-accurate inclusive cross-section of a 1 → 2 + X scalar scattering process.

FORM is a symbolic manipulation system, which is especially advantageous for handling gigantic expressions with many small terms, as often occurs in real problems in perturbative quantum field theory. In this work we describe some main features of FORM, such as the preprocessor and $-variables with emphasizing on benefit of metaprogramming, and introduce a new feature: a topology generator.

Since modern-day multi-loop Feynman diagram computations often require manipulating billions of terms, taking up terabytes of memory, a powerful symbolic manipulation toolkit (SMT) is essential. The de facto solution is Form, but it has several shortcomings. In this work we present reForm, a new SMT in early development, that will handle the same workload as Form but does not have its shortcomings. We showcase some features of reForm, including Python and C APIs. Finally, we provide benchmarks for polynomial GCD computations, which show that reForm often outperforms its competitors. A link to the source code of the technical preview version is provided.

We introduce a novel construction of a contour deformation within the framework of Loop-Tree Duality for the numerical computation of loop integrals featuring threshold singularities in momentum space. The functional form of our contour deformation automatically satisfies all constraints without the need for fine-tuning. We demonstrate that our construction is systematic and efficient by applying it to more than 100 examples of finite scalar integrals featuring up to six loops. We also showcase a first step towards handling non-integrable singularities by applying our work to one-loop infrared divergent scalar integrals and to the one-loop amplitude for the ordered production of two and three photons. This requires the combination of our contour deformation with local counterterms that regulate soft, collinear and ultraviolet divergences. This work is an important step towards computing higher-order corrections to relevant scattering cross-sections in a fully numerical fashion.

We apply a fully automated extension of the R* operation capable of calculating higher-loop anomalous dimensions of n-point Green's functions of arbitrary, possibly nonrenormalizable, local quantum field theories. We focus on the case of the CP-violating Weinberg operator of the Standard Model effective field theory whose anomalous dimension is so far known only at one loop. We calculate the two-loop anomalous dimension in full QCD and the three-loop anomalous dimensions in the limit of pure Yang-Mills theory. We find sizable two-loop and large three-loop corrections, due to the appearance of a new quartic group invariant. We discuss phenomenological implications for electric dipole moments and future applications of the method.

Loop-Tree Duality (LTD) is a framework in which the energy components of all loop momenta of a Feynman integral are integrated out using residue theorem, resulting in a sum over tree-like structures. Originally, the LTD expression exhibits cancellations of non-causal thresholds between summands, also known as dual cancellations. As a result, the expression exhibits numerical instabilities in the vicinity of non-causal thresholds and for large loop momenta. In this work we derive a novel, generically applicable, Manifestly Causal LTD (cLTD) representation whose only thresholds are causal thresholds, i.e. it manifestly realizes dual cancellations. Consequently, this result also serves as a general proof for dual cancellations. We show that LTD, cLTD, and the expression stemming from Time Ordered Perturbation Theory (TOPT) are locally equivalent. TOPT and cLTD both feature only causal threshold singularities, however LTD features better scaling with the number of propagators. On top of the new theoretical perspectives offered by our representation, it has the useful property that the ultraviolet (UV) behaviour of the original 4D integrand is maintained for every summand. We show that the resulting LTD integrand expression is completely stable in the UV region which is key for practical applications of LTD to the computation of amplitudes and cross sections. We present explicit examples of the LTD expression for a variety of up to four-loop integrals and show that its increased computational complexity can be efficiently mitigated by optimising its numerical implementation. Finally, we provide computer code that automatically generates the LTD expression for an arbitrary topology.

Loop-tree duality (LTD) offers a promising avenue to numerically integrate multiloop integrals directly in momentum space. It is well established at one loop, but there have been only sparse numerical results at two loops. We provide a formal derivation for a novel multiloop LTD expression and study its threshold singularity structure. We apply our findings numerically to a diverse set of up to four-loop finite topologies with kinematics for which no contour deformation is needed. We also lay down the ground work for constructing such a deformation. Our results serve as an important stepping stone towards a generalized and efficient numerical implementation of LTD, which is applicable to the computation of virtual corrections.

We present the first calculations of next-to-next-to-next-to-next-to-leading order (N4LO) contributions to anomalous dimensions of spin-N twist-2 operators in perturbative QCD. Specifically, we have obtained the respective non-singlet quark–quark anomalous dimensions at N=2 and N=3 to the fifth order in the strong coupling αs. These results set the scale for the N4LO contributions to the evolution of the non-singlet quark distributions of hadrons outside the small-x region, and facilitate a first approximate determination of the five-loop cusp anomalous dimension. While the N4LO coefficients are larger than expected from the lower-order results, their inclusion stabilizes the perturbative expansions for three or more light flavours at a sub-percent accuracy for αs<0.3.

Some of the difficulties faced when calculating multi-loop amplitudes with several mass scales are reviewed. We then focus on one particular difficulty, the evaluation of the Feynman integrals, and introduce the program pySecDec which can be used to numerically compute such integrals. Some of the new features and in particular the sector symmetry finder, which can help to reduce the number of sectors to be numerically integrated after sector decomposition, are described.

We report on recent progress on the flavour non-singlet splitting functions in perturbative QCD. The~exact four-loop (N^3LO) contribution to these functions has been obtained in the planar limit of a large number of colours. Phenomenologically sufficient approximate expressions have been obtained for the parts not exactly known so far. Both cases include results for the four-loop cusp and virtual anomalous dimensions which are relevant well beyond the evolution of non-singlet quark distributions, for which an accuracy of (well) below 1% has now been been reached.

We report on recent progress on the splitting functions for the evolution of parton distributions and related quantities, the (lightlike) cusp anomalous dimensions, in perturbative QCD. New results are presented for the four-loop (next-to-next-to-next-to-leading order, N$^3$LO) contributions to the flavour-singlet splitting functions and the gluon cusp anomalous dimension. We present first results, the moments $N=2$ and $N=3$, for the five-loop (N$^4$LO) non-singlet splitting functions.

We sketch how the R*-operation can be used to compute the poles of Feynman diagrams. Next, we identify computational difficulties when performing five-loop calculations. We provide four solutions that drastically reduce the number of generated terms.

We sketch how the R*-operation can be used to compute the pole terms of Feynman diagrams. We identify computational difficulties when performing five-loop calculations, and provide four solutions that drastically reduce the number of terms that are generated. Using these methods, we have computed the beta function for Yang-Mills theory with fermions, the R-ratio in electron-positron annihilation, and Higgs decays to quarks and gluons at five-loop accuracy. The results for the beta function and Higgs decay width to gluons in the heavy-top limit are briefly discussed. There is no need for six-loop extensions of these calculations in the near future.

We have computed the contributions of the quartic Casimir invariants to the four-loop anomalous dimensions of twist-2 spin-N operators at N≤16. The results provide new information on the structure of the next-to-next-to-next-to-leading order (N3LO) splitting functions Pik(3)(x) for the evolution of parton distributions, and facilitate approximate expressions which include the quartic-Casimir contributions to the (light-like) gluon cusp anomalous dimension. These quantities turn out to be closely related, by a generalization of the lower-order 'Casimir scaling', to the corresponding quark results. Using these findings, we present an approximate result for the four-loop gluon cusp anomalous dimension in QCD which is sufficient for phenomenological applications.

We sketch how the $R^*$-operation can be used to compute the pole terms of Feynman diagrams. As an example, we briefly discuss the computatation of the Higgs decay to gluons. Next, we illustrate how \MSbar{} counterterms can be obtained with the $R^*$-operation. Finally, we describe how the UV counterterm of any $L$-loop Feynman diagram with arbitrary local operators can be computed from $L-1$-loop massless propagator diagrams.