[Paper Review] Status of Eikonal Two-Loop Calculations with Massive Quarks

Nikolaos Kidonakis, Philip Stephens|ArXiv.org|May 8, 2008

Particle physics theoretical and experimental studies1 references4 citations

TL;DR

This paper presents explicit analytical results for ultraviolet (UV) poles in dimensional regularization of two-loop eikonal diagrams involving massive quarks, essential for next-to-next-to-leading logarithmic (NNLL) resummation in heavy quark production. The work computes UV divergences in integrals relevant to the soft anomalous dimension matrix Γ_S at two loops, with key results including 1/ε² and 1/ε UV pole structures involving polylogarithms and logarithmic functions of β = √(1−4m²/s).

ABSTRACT

We present results for two-loop diagrams with massive quarks in the eikonal approximation. Explicit expressions are given for the UV poles in dimensional regularization of several of the required integrals.

Motivation & Objective

• To compute ultraviolet (UV) divergences in two-loop eikonal diagrams involving massive quarks, required for next-to-next-to-leading logarithmic (NNLL) resummation in hard scattering processes.
• To extend the soft anomalous dimension matrix Γ_S beyond one-loop accuracy by evaluating two-loop integrals in the eikonal approximation with massive quarks.
• To provide explicit expressions for UV poles in dimensional regularization for multiple two-loop diagram topologies, including quark and gluon loops.
• To resolve the missing two-loop Γ_S result for heavy quark production, which remains unknown despite progress in massless cases.

Proposed method

• Uses the eikonal approximation to simplify two-loop Feynman diagrams, replacing real gluon lines with eikonal lines carrying momentum k→0.
• Applies dimensional regularization with n=4−ε to regulate UV and infrared (IR) singularities in loop integrals.
• Employs Feynman parametrization and analytic integration techniques to evaluate two-loop momentum integrals, isolating UV poles via Laurent expansion in ε.
• Calculates UV poles for multiple diagram types: two-gluon exchange diagrams (I₁, I₂), quark-loop diagrams (I_ql), gluon-loop diagrams (I_gl), and ghost-loop diagrams (I_gh).
• Uses symmetry relations, such as I₁ = ½(I₁l)² − I₂, to reduce the number of independent integrals.
• Extracts UV divergences using known results for polylogarithms (Li₂, Li₃), zeta functions (ζ₂, ζ₃), and logarithmic terms in β = √(1−4m²/s).

Experimental results

Research questions

• RQ1 What are the ultraviolet (UV) pole structures of two-loop eikonal diagrams with massive quarks in dimensional regularization?
• RQ2 How do the UV divergences in two-loop diagrams involving quark, gluon, and ghost loops contribute to the soft anomalous dimension matrix Γ_S at two loops?
• RQ3 Can the UV poles of two-loop eikonal integrals be expressed analytically in terms of β = √(1−4m²/s), polylogarithms, and logarithmic functions?
• RQ4 What is the role of color factors and diagram symmetries in simplifying the two-loop computation of Γ_S for massive quarks?
• RQ5 How do the UV poles in the two-loop diagrams compare to those in the one-loop case, and what is their significance for NNLL resummation?

Key findings

• The UV pole structure of the two-gluon exchange diagram I₂ is computed up to 1/ε² and 1/ε terms, with explicit dependence on β and polylogarithmic functions.
• The quark-loop diagram I_ql yields a UV divergence proportional to 1/ε² and 1/ε, with coefficients involving Li₂((1±β)/2), logarithmic terms, and Euler-Mascheroni constants.
• The gluon-loop diagram I_gl contributes UV poles with a 1/ε² term proportional to −19/96, and its coefficient includes a logarithmic term with a coefficient of (58/57 + 4ln2 + lnπ − γ_E − iπ).
• The ghost-loop diagram I_gh contributes a UV divergence with a 1/ε² term scaled by −1/96, and its 1/ε coefficient includes a constant term (4/3 + 4ln2 + lnπ − γ_E − iπ).
• The total UV divergence for the two-loop soft anomalous dimension matrix Γ_S is obtained by combining all diagram contributions with their respective color factors.
• The results provide the necessary UV pole information to compute the two-loop Γ_S for massive quarks, enabling NNLL resummation in heavy quark production processes.