Key publications

We complete the analytic calculation of the full set of two-loop Feynman integrals required for computation of massless five-particle scattering amplitudes. We employ the method of canonical differential equations to construct a minimal basis set of transcendental functions, pentagon functions, which is sufficient to express all planar and nonplanar massless five-point two-loop Feynman integrals in the whole physical phase space. We find analytic expressions for pentagon functions which are manifestly free of unphysical branch cuts. We present a public library for numerical evaluation of pentagon functions suitable for immediate phenomenological applications.

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 QCD corrections to processes such as the production of a <math display="inline"><mrow><mi>H</mi><mo>/</mo><mi>Z</mi><mo>/</mo><mi>W</mi></mrow></math> 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.

In the well-studied genus zero case, bases of d log integrands with integer leading singularities define Feynman integrals that automatically satisfy differential equations in canonical form. Such integrand bases can be constructed without input from the differential equations and without explicit involvement of dimensional regularization parameter <math display="inline"><mi>ε</mi></math>. We propose a generalization of this construction to genus one geometry arising from the appearance of elliptic curves. We argue that a particular choice of algebraic 1-forms of the second kind that avoids derivatives is crucial. We observe that the corresponding Feynman integrals satisfy a special form of differential equations that has not been previously reported, and that their solutions order by order in <math display="inline"><mi>ε</mi></math> yield pure functions. We conjecture that our integrand-level construction universally leads to such differential equations.

We present the analytic form of all leading-color two-loop five-parton helicity amplitudes in QCD. The results are analytically reconstructed from exact numerical evaluations over finite fields. Combining a judicious choice of variables with a new approach to the treatment of particle states in D dimensions for the numerical evaluation of amplitudes, we obtain the analytic expressions with a modest computational effort. Their systematic simplification using multivariate partial-fraction decomposition leads to a particularly compact form. Our results provide all two-loop amplitudes required for the calculation of next-to-next-to-leading order QCD corrections to the production of three jets at hadron colliders in the leading-color approximation.

We compute the two-loop helicity amplitudes for the production of three photons at hadron colliders in QCD at leading-color. Using the two-loop numerical unitarity method coupled with analytic reconstruction techniques, we obtain the decomposition of the two-loop amplitudes in terms of master integrals in analytic form. These expressions are valid to all orders in the dimensional regulator. We use them to compute the two-loop finite remainders, which are given in a form that can be efficiently evaluated across the whole physical phase space. We further package these results in a public code which assembles the helicity-summed squared two-loop remainders, whose numerical stability across phase-space is demonstrated. This is the first time that a five-point two-loop process is publicly available for immediate phenomenological applications.

We compute the two-loop helicity amplitudes for the scattering of five gluons, including all contributions beyond the leading-color approximation. The analytic expressions are represented as linear combinations of transcendental functions with rational coefficients, which we reconstruct from finite-field samples obtained with the numerical unitarity method. Guided by the requirement of removing unphysical singularities, we find a remarkably compact generating set of rational coefficients, which we display entirely in the manuscript. We implement our results in a public code, which provides efficient and reliable numerical evaluations for phenomenological applications.

We present next-to-next-to-leading-order (NNLO) QCD corrections to the production of three isolated photons in hadronic collisions at the fully differential level. We employ qT subtraction within Matrix and an efficient implementation of analytic two-loop amplitudes in the leading-colour approximation to achieve the first on-the-fly calculation for this process at NNLO accuracy. Numerical results are presented for proton–proton collisions at energies ranging from 7 TeV to 100 TeV. We find full agreement with the 8 TeV results of Ref. [1] and confirm that NNLO corrections are indispensable to describe ATLAS 8 TeV data. In addition, we demonstrate the significance of NNLO corrections for future precision studies of triphoton production at higher collision energies.

We consider the production of a <math display="inline"><mi>Z</mi></math> boson decaying to leptons in association with a bottom-quark pair in hadronic collisions. For the first time, we compute predictions at next-to-next-to-leading order (NNLO) in QCD, and we combine them with the all-order radiative corrections from a parton-shower simulation (<math display="inline"><mrow><mi>NNLO</mi><mo>+</mo><mi>PS</mi></mrow></math>). Our method represents the first approach to <math display="inline"><mrow><mi>NNLO</mi><mo>+</mo><mi>PS</mi></mrow></math> event generation applicable to processes featuring a color singlet and a heavy-quark pair in the final state. The novel two-loop corrections are computed for massless bottom quarks, and the leading mass corrections are restored through a small-mass expansion. The calculation is carried out in the four-flavor scheme, and we find that the sizable NNLO QCD corrections lift the long-standing tension between lower-order predictions in four- and five-flavor schemes. Our predictions are compared to a CMS measurement for <math display="inline"><mi>Z</mi></math> boson plus <math display="inline"><mi>b</mi></math>-jet production, achieving an excellent description of the data.

We present the analytic form of the two-loop four-graviton scattering amplitudes in Einstein gravity. To remove ultraviolet divergences we include counterterms quadratic and cubic in the Riemann curvature tensor. The two-loop numerical unitarity approach is used to deal with the challenging momentum dependence of the interactions. We exploit the algebraic properties of the integrand of the amplitude in order to reduce it to a minimal basis of Feynman integrals. Analytic expressions are obtained from numerical evaluations of the amplitude. Finally, we show that four-graviton scattering observables depend on fewer couplings than naïvely expected.