Variable Flavor Number Scheme for Final State Jets in DIS

TL;DR

This work develops a variable flavor number scheme (VFNS) for the endpoint region of inclusive DIS, where final states form a tightly collimated jet as $x \to 1$. By embedding the treatment of massive quarks within SCET, it achieves a continuous description across mass hierarchies, with mass effects entering exclusively through secondary radiation and via universal threshold corrections that couple the hard, jet, and PDF sectors. The authors derive explicit threshold corrections for the hard function, jet function, and PDF, including the handling of rapidity logarithms that exponentiate, and they provide all ingredients needed for N$^3$LL resummation in this regime. The framework is shown to be RG-consistent and universally applicable to other processes featuring PDFs, jets, and hard coefficients, highlighting the universality of the mass- and rapidity-logarithm structures in high-energy QCD factorization.

Abstract

We discuss massive quark effects in the endpoint region $x \to 1$ of inclusive deep inelastic scattering, where the hadronic final state is collimated and thus represents a jet. In this regime heavy quark pairs are generated via secondary radiation, i.e. due to a gluon splitting in light quark initiated contributions starting at $\mathcal{O}(α_s^2)$ in the fixed-order expansion. Based on the factorization framework for massless quarks in Soft Collinear Effective Theory (SCET), we construct a variable flavor number scheme that deals with arbitrary hierarchies between the mass scale and the kinematic scales exhibiting a continuous behavior between the massless limit for very light quarks and the decoupling limit for very heavy quarks. We show that the threshold matching corrections for all gauge invariant components at the mass scale are related to each other via consistency conditions. This is explicitly demonstrated by recalculating the known threshold correction for the parton distribution function at $\mathcal{O}(α_s^2 C_F T_F)$ within SCET. The latter contains large rapidity logarithms $\sim \ln(1-x)$ that can be summed by exponentiation. Their coefficients are universal which can be used to obtain potentially relevant higher order results for generic threshold corrections at colliders from computations in deep inelastic scattering. In particular, we extract the $\mathcal{O}(α_s^3)$ threshold correction multiplied by a single rapidity logarithm from results obtained earlier.

Figures (7)

• Figure 1: Relevant momentum modes for inclusive DIS in the endpoint region $x \rightarrow 1$ with $1-x \gg \Lambda_{\rm QCD}/Q$.
• Figure 2: Schematic picture of the multistage matching procedure for $1-x \gg \Lambda_{\rm QCD}/Q$ described in the text.
• Figure 3: Exemplary diagrams for secondary massive quark production in DIS at $\mathcal{O}(\alpha_s^2 C_F T_F)$.
• Figure 4: Illustration of the different RG setups for the hierarchy $\mu_J>\mu_m>\mu_\phi$ leading to the consistency relations mentioned in the text. We display the cases where the common renormalization scale $\mu$ satisfies (a) $\mu_m>\mu>\mu_\phi$ and (b) $\mu_J>\mu>\mu_m$.
• Figure 5: Feynman diagram for the collinear PDF function with a massless quark field in the initial state and a massive gluon at $\mathcal{O}(\alpha_s)$. The symmetric diagram and wave function corrections have to be added.