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 μ. 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.

Detecting and mapping the induced seismicity of a planned EGS in Helsinki/Espoo, Finland

“…The resulting integrals can be performed numerically at ease, or even analytically for m = 0 (see appendix D of [71] for details), to reach the final…”

“…After the above process, we are left with a relatively high number of independent master integrals (of order 100) to evaluate, which one may try to reduce using various novel methods, including but not limited to fourdimensional IBP identities recently generalized to the T = 0, µ ̸ = realm in [71]. Finally, once the seemingly irreducible masters have been identified, the remaining task is to evaluate them starting from the ones that factorize into lower-loop-order entities.…”

“…2 above. We furthermore assume that no fermionic propagator is raised to a power higher than unity, although this restriction can be relaxed if proper care is taken in the calculation [70,71].…”