Collinear Subtractions Research Articles

Can overline{mathrm{MS}} parton distributions be negative?

It is common lore that Parton Distribution Functions (PDFs) in the overline{mathrm{MS}} factorization scheme can become negative beyond leading order due to the collinear subtraction which is needed in order to define partonic cross sections. We show that this is in fact not the case and next-to-leading order (NLO) overline{mathrm{MS}} PDFs are actually positive in the perturbative regime. In order to prove this, we modify the subtraction prescription, and perform the collinear subtraction in such a way that partonic cross sections remain positive. This defines a factorization scheme in which PDFs are positive. We then show that positivity of the PDFs is preserved when transforming from this scheme to overline{mathrm{MS}} , provided only the strong coupling is in the perturbative regime, such that the NLO scheme change is smaller than the LO term.

Nested Soft–Collinear Subtractions in NNLO QCD Computations

Nested Soft–Collinear Subtractions in NNLO QCD Computations

Towards gauge theories in four dimensions

The abundance of infrared singularities in gauge theories due to unresolved emission of massless particles (soft and collinear) represents the main difficulty in perturbative calculations. They are typically regularized in dimensional regularization, and their subtraction is usually achieved independently for virtual and real corrections. In this paper, we introduce a new method based on the loop-tree duality (LTD) theorem to accomplish the summation over degenerate infrared states directly at the integrand level such that the cancellation of the infrared divergences is achieved simultaneously, and apply it to reference examples as a proof of concept. Ultraviolet divergences, which are the consequence of the point-like nature of the theory, are also reinterpreted physically in this framework. The proposed method opens the intriguing possibility of carrying out purely four-dimensional implementations of higher-order perturbative calculations at next-to-leading order (NLO) and beyond free of soft and final-state collinear subtractions.

Top-quark initiated processes at high-energy hadron colliders

In hadronic collisions at high energies, the top-quark may be treated as a parton inside a hadron. Top-quark initiated processes become increasingly important since the top-quark luminosity can reach a few percent of the bottom-quark luminosity. In the production of a heavy particle $H$ with mass $m_H > m_t$, treating the top-quark as a parton allows us to resum large logarithms $\log(m_{H}^{2}/m_{t}^{2}$) arising from collinear splitting in the initial state. We quantify the effect of collinear resummation at the 14-TeV LHC and a future 100-TeV hadron collider, focusing on the top-quark open-flavor process $gg\to t\bar t H$ in comparison with $t\bar t \to H$ and $tg\rightarrow tH$ at the leading order (LO) in QCD. We employ top-quark parton distribution functions with appropriate collinear subtraction and power counting. We find that (1) Collinear resummation enhances the inclusive production of a heavy particle with $m_H\approx$ 5 TeV (0.5 TeV) by more than a factor of two compared to the open-flavor process at a 100-TeV (14-TeV) collider; (2) Top-quark mass effects are important for scales $m_H$ near the top-quark threshold, where the cross section is largest. We advocate a modification of the ACOT factorization scheme, dubbed m-ACOT, to consistently treat heavy-quark masses in hadronic collisions; (3) The scale uncertainty of the total cross section in m-ACOT is of about 20 percent at the LO. While a higher-order calculation is indispensable for a precise prediction, the LO cross section is well described by the process $t\bar t\to H$ using an effective factorization scale significantly lower than $m_H$. We illustrate our results by the example of a heavy spin-0 particle. Our main results also apply to the production of particles with spin-1 and 2.

Scale dependence and collinear subtraction terms for Higgs production in gluon fusion at N3LO

The full, explicit, scale dependence of the inclusive N3LO cross section for single Higgs hadroproduction is obtained by calculating the convolutions of collinear splitting kernels with lower-order partonic cross sections. We provide results for all convolutions of splitting kernels and lower-order partonic cross sections to the order in epsilon needed for the full N3LO computation, as well as their expansions around the soft limit. We also discuss the size of the total scale uncertainty at N3LO that can be anticipated with existing information.

Collinear subtractions in hadroproduction of heavy quarks

We present a detailed discussion of the collinear subtraction terms needed to establish a massive variable-flavour-number scheme for the one-particle inclusive production of heavy quarks in hadronic collisions. The subtraction terms are computed by convoluting appropriate partonic cross sections with perturbative parton distribution and fragmentation functions relying on the method of mass factorization. We find (with one minor exception) complete agreement with the subtraction terms obtained in a previous publication by comparing the zero-mass limit of a fixed-order calculation with the genuine massles results in the MSbar scheme. This presentation will be useful for extending the massive variable-flavour-number scheme to other processes.

General subtraction method for numerical calculation of one-loop QCD matrix elements

We present a subtraction scheme for eliminating the ultraviolet, soft, and collinear divergences in the numerical calculation of an arbitrary one-loop QCD amplitude with an arbitrary number of external legs. The subtractions consist of local counter terms in the space of the four-dimensional loop momentum. The ultraviolet subtraction terms reproduce MSbar renormalization. The key point in the method for the soft and collinear subtractions is that, although the subtraction terms are defined graph-by-graph and the matrix element is also calculated graph-by-graph, the sum over graphs of the integral of each the subtraction term can be evaluated analytically and provides the well known simple pole structure that arises from subtractions from real emission graphs, but with the opposite sign.

Transversity distribution does not contribute to hard exclusive electroproduction of mesons

We show that in hard exclusive electroproduction, ep-->eVp, the leading-twist hard-scattering coefficient for the production of a transversely polarized vector meson V vanishes to all orders of perturbation theory. This implies that this process cannot be used to measure the skewed transversity distribution of quarks in a hadron. In contrast, a recent calculation obtained a non-zero value at NLO. We show that this calculation is incorrect because it failed to include the necessary collinear subtractions. Our method of proof also applies to other processes whose hard-scattering coefficients are constrained by chirality and helicity conservation, and thus validates helicity selection rules based on these symmetries.

Factorization of the Drell-Yan cross section in perturbation theory.

We develop, through detailed one- and two-loop examples, a procedure for expressing the leading-twist Drell-Yan cross section in terms of the factored form proposed by Collins, Soper, and Sterman. We then show that this factorization program can be implemented to all orders in perturbation theory. The factored cross section takes the form of an on-shell qq\ifmmode\bar\else\textasciimacron\fi{} annihilation cross section (less certain collinear subtractions) convolved with structure functions for the incoming hadrons. The structure functions contain all the collinear singularities and spectator interactions. They are shown to be simply related to those that occur in deep-inelastic lepton scattering. We also demonstrate that the qq\ifmmode\bar\else\textasciimacron\fi{} cross section minus the collinear subtractions is free of infrared singularities if one integrates over the transverse momentum of the lepton pair.