[Paper Review] Soft-virtual correction and threshold resummation for $n$-colorless particles to fourth order in QCD: Part I

Taushif Ahmed, A. H. Ajjath+4 more|arXiv (Cornell University)|Oct 6, 2020

Particle physics theoretical and experimental studies4 citations

TL;DR

This paper presents a universal soft-collinear distribution operator to compute next-to-next-to-next-to-next-to-leading order (N⁴LO) soft-virtual corrections in QCD for $2\to n$ processes with colorless final states at hadron colliders. By leveraging factorization, renormalization group evolution, and universal soft gluon contributions, the method enables automated computation of N⁴LO soft-virtual cross-sections once virtual amplitudes are known, with explicit results provided for Higgs and Drell-Yan production up to N³LL accuracy in threshold resummation.

ABSTRACT

We present the general form of a universal soft-collinear distribution operator to compute the soft-virtual cross-section to next-to-next-to-next-to-next-to-leading order (N$^4$LO) in QCD for a process with any number of final state colorless particles in hadron colliders. By acting this universal operator on the pure virtual corrections, which need to be computed explicitly for a process, the soft-virtual cross-section can be obtained. The operator is constructed by exploiting the factorization and renormalization group evolution of amplitudes in QCD, and the universality of soft gluon contributions. We also provide the hard coefficient to perform the threshold resummation to next-to-next-to-next-to-leading logarithmic (N$^3$LL) accuracy. Furthermore, we present the approximate analytical results of the soft-virtual cross-sections at N$^4$LO and N$^3$LL for the Higgs boson production through gluon fusion and bottom quark annihilation, and also for the Drell-Yan production at the hadronic collider.

Motivation & Objective

• To develop a universal formalism for computing soft-virtual corrections to $2\to n$ QCD processes at N⁴LO for colorless final states in hadron colliders.
• To provide a systematic method that combines factorization, renormalization group evolution, and universal soft gluon contributions to automate N⁴LO cross-section computation.
• To derive the hard coefficient for threshold resummation up to next-to-next-to-next-to-leading logarithmic (N³LL) accuracy.
• To present approximate analytical results for soft-virtual cross-sections at N⁴LO and N³LL for Higgs boson production via gluon fusion and bottom quark annihilation, and Drell-Yan production.
• To validate the universality of soft-collinear distributions via Casimir scaling and extend its applicability to four-loop order.

Proposed method

• The authors construct a universal soft-collinear distribution operator based on the factorization of QCD amplitudes and their renormalization group evolution.
• The operator acts on pure virtual corrections to generate the full soft-virtual cross-section at N⁴LO, enabling automated computation for any $2\to n$ process with colorless final states.
• The method relies on the universality of soft gluon contributions and the structure of infrared singularities in QCD amplitudes.
• The framework incorporates the complete four-loop cusp anomalous dimension and conjectured relations between virtual, soft, and collinear anomalous dimensions.
• The threshold resummation is performed using a hard coefficient derived from the universal structure, with logarithmic terms organized to N³LL accuracy.
• The results are expressed in terms of kinematic variables $\omega = 1 - \hat{t}/\hat{s}$, logarithms $L_\omega = \ln(1-\omega)$, and zeta values $\zeta_n$.

Experimental results

Research questions

• RQ1 Can a universal soft-collinear distribution operator be constructed to compute N⁴LO soft-virtual corrections for $2\to n$ QCD processes with colorless final states?
• RQ2 To what extent can the soft-virtual cross-section be computed automatically once the virtual amplitudes are known?
• RQ3 What is the structure of the hard coefficient required for threshold resummation to N³LL accuracy in such processes?
• RQ4 How do the soft-virtual corrections scale with color Casimirs $C_F$, $C_A$, and $n_f$ at four loops?
• RQ5 Does Casimir scaling hold for the soft-collinear distribution beyond three loops, and how is it reflected in the analytic structure of the cross-section?

Key findings

• The paper derives a universal soft-collinear distribution operator that enables automated N⁴LO soft-virtual cross-section computation for $2\to n$ QCD processes with colorless final states.
• The soft-virtual cross-section for Higgs boson production via gluon fusion and bottom quark annihilation is computed analytically at N⁴LO, with explicit dependence on $\omega$, $\zeta_2$, $\zeta_3$, $\zeta_5$, and logarithmic terms.
• For Drell-Yan production, the N⁴LO soft-virtual cross-section is derived in terms of $\omega$, $L_\omega$, and multiple zeta values, including $\zeta_2^3$, $\zeta_3^2$, and $\zeta_5$.
• The hard coefficient for threshold resummation is determined to N³LL accuracy, with explicit analytic expressions in terms of $\beta_0$, $\beta_1$, $C_F$, $C_A$, $n_f$, and $\zeta$-values.
• The results confirm Casimir scaling for the soft-collinear distribution at four loops, with the $C_F/C_A$ ratio governing the relation between Drell-Yan and Higgs production cross-sections.
• The framework provides a systematic and universal method to compute dominant higher-order corrections in high-precision QCD calculations, particularly for processes with multi-leg final states.