Momentum expansions in finite-density perturbative calculations

Abstract

Complex-valued Feynman integrals in the imaginary time formalism and zero-temperature limit suffer from particular types of infrared divergences that can not be regulated by integration dimension alone. Related problems leading to integration order dependent results are even further pronounced in the presence of additional scales such as external momenta. This plays a noticeable role in systems featuring fermionic degrees of freedom such as cold Quantum Chromodynamics, where loop integrals are complexified by chemical potential(s). Working in the limit of vanishing temperature, we utilize novel complex-valued extensions to bubble Feynman integrals and study momentum expansions of fermionic loop integrals. The expansions are then used to illustrate the mechanisms of manifested discrepancies between orders of integration, associated with the residue theorem. Finally, we address the issues by introducing a representation avoiding the observed ambiguity and briefly overview classes of integrals insensitive to problems from external momenta.

Figures (2)

• Figure 1: Left: The line integral representation of the contour $\gamma$ is described with blue solid lines. It resembles two solid lines parallel to the real axis apart from the two removed points signified by the red dots along the imaginary axis. The thinly dotted black line indicates the positions of the poles arising from the Fermi-Dirac distributions function. Note in particular that $p_0=i\mu$ is not included in these poles. Right: The addition of the two pairs of dashed line integrals does not change the result of the integration. The dashed blue lines are placed at infinitely far along the real axis where the integrand is assumed to systematically vanish. The dashed purple lines are running in opposite directions along the imaginary axis, and therefore canceling out, provided that the integrated function is holomorphic in the integration contours. This facilitates the contour to be used for summation of the residues corresponding to the Fermi-Dirac distribution function. The image is adapted from Osterman:2023tntOstermanthesis.
• Figure 2: In the strict residue approach at $T=0$ the thermally suppressed part of image \ref{['fig:contouroverlap']} is removed, and thus only the upper line integral -- above the black dotted poles of the Fermi-Dirac distribution function -- remains, indicated by the blue solid line above. We emphasize that the blue solid line includes every point along its length, in particular the previously excluded point $\text{Re}(p_0)=0$. The line integral is studied in conjunction with an added semicircular arc -- the red line in image above -- covering the upper half of complex plane, specifically above the line parallel to real axis $\text{Im}(p_0) = i\mu + i\eta$ and therefore avoiding the black dotted line. Here $\eta > T$ is an arbitrarily small regulator needed to treat the low-temperature limit properly, and can be removed completely after the vanishing-temperature limit has been taken. The residue theorem is then studied within the corresponding closed contour.