Augmenting the residue theorem with boundary terms in finite-density calculations

Abstract

At zero temperature and finite chemical potential, $d$-dimensional loop integrals with complex-valued integrands in the imaginary-time formalism yield results dependent on the integration order. We observe this even with the simplest one-loop dimensionally regularized integrals. Computing such integrals by evaluating the spatial ${\mathrm{d}}^{d}p$ integral before the temporal $\mathrm{d}{p}_0$ integral yields results consistent with those obtained at small but nonvanishing temperatures. Computing the temporal integral first by applying the residue theorem to the integrand yields a different answer. The same holds for general complexified propagators. In this work we aim to understand the theoretical background behind this difference, in order to fully enable the powerful techniques of residue calculus in applications. We cast the difference into the form of a derivative term related to Dirac deltas, and further demonstrate how the difference originates from the zero-temperature limit of the Fermi-Dirac occupation functions treated as complex-valued functions. We also discuss a generalization to propagators raised to non-integer powers.

arXiv comment13 pages, 2 figures. Multiple minor changes to phrases and conventions following reviewer comments

References in corpus (12)

• Direct Proof Of Tree-Level Recursion Relation In Yang-Mills Theory
• Higgs boson production at hadron colliders in NNLO QCD
• High-precision QCD at hadron colliders: electroweak gauge boson rapidity distributions at NNLO
• The Four-Loop QCD Beta-Function and Anomalous Dimensions
• Cold Quark Matter
• QCD Decoupling at Four Loops
• Four-Loop Decoupling Relations for the Strong Coupling
• The pressure of QCD at finite temperatures and chemical potentials
• Soft interactions in cold quark matter
• Cold quark matter at N3LO: soft contributions
• On high-order perturbative calculations at finite density
• IBP methods at finite temperature

Cited by in corpus (6)

• Equation of State of Cold Quark Matter to $O({\alpha }_s^3\ln {\alpha }_s)$
• Degenerate fermionic matter at N${}^3$LO: Quantum Electrodynamics
• Integrating by parts at finite density
• Thermodynamics of strongly magnetized dense quark matter from hard dense loop perturbation theory
• Momentum expansions in finite-density perturbative calculations
• Subtleties in the calculation of correlation functions for hot and dense systems