It is by now well known that, at subleading power in scale ratios, factorization theorems for high-energy cross sections and decay amplitudes contain endpoint-divergent convolution integrals. The presence of endpoint divergences hints at a violation of simple scale separation. At the technical level, they indicate an unexpected failure of dimensional regularization and the MS subtraction scheme. In this paper we start a detailed discussion of factorization at subleading power within the framework of soft-collinear effective theory. As a concrete example, we factorize the decay amplitude for the radiative Higgs-boson decay h → γγ mediated by a b-quark loop, for which endpoint-divergent convolution integrals require both dimensional and rapidity regulators. We derive a factorization theorem for the decay amplitude in terms of bare Wilson coefficients and operator matrix elements. We show that endpoint divergences caused by rapidity divergences cancel to all orders of perturbation theory, while endpoint divergences that are regularized dimensionally can be removed by rearranging the terms in the factorization theorem. We use our result to resum the leading double-logarithmic corrections of order α n s ln 2n+2 (−M 2 h /m 2 b ) to the decay amplitude to all orders of perturbation theory.

Building on the recent derivation of a bare factorization theorem for the b-quark induced contribution to the h → γγ decay amplitude based on soft-collinear effective theory, we derive the first renormalized factorization theorem for a process described at subleading power in scale ratios, where λ = mb/Mh « 1 in our case. We prove two refactorization conditions for a matching coefficient and an operator matrix element in the endpoint region, where they exhibit singularities giving rise to divergent convolution integrals. The refactorization conditions ensure that the dependence of the decay amplitude on the rapidity regulator, which regularizes the endpoint singularities, cancels out to all orders of perturbation theory. We establish the renormalized form of the factorization formula, proving that extra contributions arising from the fact that “endpoint regularization” does not commute with renormalization can be absorbed, to all orders, by a redefinition of one of the matching coefficients. We derive the renormalization-group evolution equation satisfied by all quantities in the factorization formula and use them to predict the large logarithms of order $$ {\alpha \alpha}_s^2{L}^k $$ αα s 2 L k in the three-loop decay amplitude, where $$ L=\ln \left(-{M}_h^2/{m}_b^2\right) $$ L = ln − M h 2 / m b 2 and k = 6, 5, 4, 3. We find perfect agreement with existing numerical results for the amplitude and analytical results for the three-loop contributions involving a massless quark loop. On the other hand, we disagree with the results of previous attempts to predict the series of subleading logarithms $$ \sim {\alpha \alpha}_s^n{L}^{2n+1} $$ ∼ αα s n L 2 n + 1 .

Soft functions defined in terms of matrix elements of soft fields dressed by Wilson lines are central components of factorization theorems for cross sections and decay rates in collider and heavy-quark physics. While in many cases the relevant soft functions are defined in terms of gluon operators, at subleading order in power counting soft functions containing quark fields appear. We present a detailed discussion of the properties of the soft-quark soft function consisting of a quark propagator dressed by two finite-length Wilson lines connecting at one point. This function enters in the factorization theorem for the Higgs-boson decay amplitude of the h → γγ process mediated by light-quark loops. We perform the renormalization of this soft function at one-loop order, present a conjecture for its two-loop anomalous dimension and discuss solutions to its renormalization-group evolution equation in momentum space, in Laplace space and in the "diagonal space", where the evolution is strictly local in the momentum variable.

We present a fully differential next-to-next-to-leading order calculation of charm-quark production in charged-current deep-inelastic scattering, with full charm-quark mass dependence. The next-to-next-toleading order corrections in perturbative quantum chromodynamics are found to be comparable in size to the next-to-leading order corrections in certain kinematic regions. We compare our predictions with data on dimuon production in (anti)neutrino scattering from a heavy nucleus. Our results can be used to improve the extraction of the parton distribution function of a strange quark in the nucleon. DOI: 10.1103/PhysRevLett.116.212002 Introduction.-Charm-quark (c) production in deepinelastic scattering (DIS) of a neutrino from a heavy nucleus provides direct access to the strange-quark (s) content of the nucleon. At lowest order, the relevant partonic process is neutrino interaction with a strange quark, νs → cl, mediated by weak vector boson W exchange. Another source of constraints is charm-quark production in association with a W boson at hadron colliders, gs → cW. The DIS data determine parton distribution functions (PDFs) in the nucleon whose detailed understanding is vital for precise predictions at the Large Hadron Collider (LHC). The strangequark PDF can play an important role in LHC phenomenology, contributing, for example, to the total PDF uncertainty in W or Z boson production [1,2], and to systematic uncertainties in precise measurements of the W boson mass and weak-mixing angle [3][4][5]. It is estimated that the PDF uncertainty of the strange quark alone could lead to an uncertainty of about 10 MeV on the W boson mass measurement at the LHC [6]. From the theoretical point of view, it is important to test whether the strange PDFs are suppressed compared to those of other light sea quarks, related to the larger mass of the strange quark, as suggested by various models [7][8][9], and to establish whether there is a difference between the strange-and antistrange-quark PDFs.In this Letter, we report on a complete calculation at next-to-next-to-leading order (NNLO) in pertubative quantum chromodynamics (QCD) of charm-quark production in DIS of a neutrino from a nucleon. Our calculation is based on a phase-space slicing method and uses a fully differential Monte Carlo integration. It maintains the exact mass dependence and all kinematic information at the parton level. The NNLO corrections can change the cross sections by up to 10%, depending on the kinematic region considered. Our results show that the next-to-leading-order (NLO) predictions underestimate the perturbative uncertainties