Integrating by parts at finite density

Abstract

Both nonzero temperature and chemical potentials break the Lorentz symmetry present in vacuum quantum field theory by singling out the rest frame of the heat bath. This leads to complications in the application of thermal perturbation theory, including the appearance of novel infrared divergences in loop integrals and an apparent absence of four-dimensional integration-by-parts (IBP) identities, vital for high-order computations. Here, we propose a new strategy that enables the use of IBP techniques in the evaluation of Feynman integrals, in particular vacuum or bubble diagrams, in the limit of vanishing temperature $T$ but nonzero chemical potentials $\mu$. The central elements of the new setup include a contour representation for the temporal momentum integral, the use of a small but nonzero $T$ as an IR regulator, and the systematic application of both temporal and spatial differential operators in the generation of linear relations among the loop integrals of interest. The relations we derive contain novel inhomogeneous terms featuring differentiated Fermi-Dirac distribution functions, which severely complicate calculations at nonzero temperature, but are shown to reduce to solvable lower-dimensional objects as $T$ tends to zero. Pedagogical example computations are kept at the one- and two-loop levels, but the application of the new method to higher-order calculations is discussed in some detail.

48 pages, 1 figure, structural revisions and clarifications added, matches published version (v3)

References in corpus (9)

• Cold Quark Matter
• The five-loop beta function of Yang-Mills theory with fermions
• Ab-initio QCD calculations impact the inference of the neutron-star-matter equation of state
• Bayesian uncertainty quantification of perturbative QCD input to the neutron-star equation of state
• On high-order perturbative calculations at finite density
• A QCD Debye mass in a broad temperature range
• Augmenting the residue theorem with boundary terms in finite-density calculations
• Degenerate fermionic matter at N${}^3$LO: Quantum Electrodynamics
• Recursion-free solution for two-loop vacuum integrals with "collinear" masses

Cited by in corpus (4)

• Equation of State of Cold Quark Matter to $O({\alpha }_s^3\ln {\alpha }_s)$
• Factorizing two-loop vacuum sum-integrals
• Thermodynamics of strongly magnetized dense quark matter from hard dense loop perturbation theory
• Momentum expansions in finite-density perturbative calculations