Scalar one-loop integrals for QCD

Abstract: We construct a basis set of infra-red and/or collinearly divergent scalar one-loop integrals and give analytic formulas, for tadpole, bubble, triangle and box integrals, regulating the divergences (ultra-violet, infra-red or collinear) by regularization in $D=4-2\epsilon$ dimensions. For scalar triangle integrals we give results for our basis set containing 6 divergent integrals. For scalar box integrals we give results for our basis set containing 16 divergent integrals. We provide analytic results for the 5 divergent box integrals in the basis set which are missing in the literature. Building on the work of van Oldenborgh, a general, publicly available code has been constructed, which calculates both finite and divergent one-loop integrals. The code returns the coefficients of $1/\epsilon^2,1/\epsilon^1$ and $1/\epsilon^0$ as complex numbers for an arbitrary tadpole, bubble, triangle or box integral.

• The paper formulates a comprehensive basis set of divergent scalar integrals covering tadpole, bubble, triangle, and box topologies.
• It delivers analytical solutions for key integrals using dimensional regularization in D=4-2ε dimensions, critical for high-precision QCD calculations.
• The research introduces a publicly accessible numerical code that automates the computation of both finite and divergent terms in one-loop integrals.

An Expert Review of "Scalar One-Loop Integrals for QCD"

The paper "Scalar One-Loop Integrals for QCD" by R. Keith Ellis and Giulia Zanderighi addresses a fundamental aspect of perturbative quantum field theory concerning the evaluation of scalar one-loop integrals. Such evaluations are crucial for advancing the precision of theoretical predictions in Quantum Chromodynamics (QCD), particularly for processes explored at large collider facilities like the LHC, where precision measurements of hard scattering processes at next-to-leading order (NLO) are essential.

This research provides a comprehensive treatment of scalar one-loop integrals, which are instrumental in calculating one-loop amplitudes, especially in scenarios where massive or massless fermions and gluons are involved. By identifying a basis set of divergent scalar integrals, the paper lays the groundwork for practical and theoretical evaluations of loop integrals required for various types of Feynman diagrams — tadpole, bubble, triangle, and box topologies.

Key Contributions:

• Basis Set Formulation: The authors construct a basis set of infra-red (IR) and collinearly divergent integrals, addressing various scalar diagrams. For triangle and box integrals, they provide results for six and sixteen divergent integrals, respectively. This basis set is pivotal for simplifying and computing complex diagrams often encountered in higher-order QCD calculations.
• Analytical Results: The paper delivers analytical solutions for the basis set, particularly emphasizing box integrals, where five novel divergent cases are solved. The calculations employ dimensional regularization in $D=4-2\epsilon$ dimensions, a method chosen for its effectiveness in handling divergences operating within massless and high-energy limits.
• Numerical Code Development: Building upon existing literature, the authors have expanded and assembled a generalized, publicly accessible code. This code is designed to automate the calculation of one-loop integrals, outputting both finite and divergent terms as coefficients of the series expansion in $\epsilon$.

Analytical Insights and Implications:

The coherent organization of divergent integrals and their results in a dimensional regularization framework foster a more systematic understanding of IR and collinear divergences in loop-integral calculations within the scope of QCD processes. These developments indicate that one-loop scalar integrals can now be seamlessly and uniformly applied across diverse scenarios involving massive or massless particles, establishing a bridge between theoretical advancements and practical computational methods.

Future Developments:

The research, while offering comprehensive coverage of one-loop integrals, also hints at further potential explorations. The transition from one-loop to multi-loop calculations, possibly incorporating aspects of electroweak corrections or extending beyond scalar integrals to tensorial counterparts, could form the backbone of subsequent studies. Additionally, the generalization towards integrals involving complex masses could aid in exploring unstable particle amplitudes, expanding the utility and adaptability of the foundational code crafted in this study.

Conclusion:

Ellis and Zanderighi's contribution through "Scalar One-Loop Integrals for QCD" not only navigates existing gaps in the literature concerning specific divergent integrals but also proposes an invaluable computational toolset essential for high-precision QCD computations. Their analytic and numeric methodologies firmly enhance the ability of physicists to tackle complex scattering processes, marking significant progress towards more accurate and reliable predictions necessitating scalar one-loop integral evaluations at NLO and even higher orders.