On high-order perturbative calculations at finite density

Abstract

We discuss the prospects of performing high-order perturbative calculations in systems characterized by a vanishing temperature but finite density. In particular, we show that the determination of generic Feynman integrals containing fermionic chemical potentials can be reduced to the evaluation of three-dimensional phase space integrals over vacuum on-shell amplitudes - a result reminiscent of a previously proposed "naive real-time formalism" for vacuum diagrams. Applications of these rules are discussed in the context of the thermodynamics of cold and dense QCD, where it is argued that they facilitate an extension of the Equation of State of cold quark matter to higher perturbative orders.Comment: 22 pages, 3 figures; v2: discussion extended and references adde

Quark Matter Equation of State from Perturbative QCD

“…A crucial aid in this process turns out to be an analytic tool dubbed 'cutting rules'. These rules were first discussed in [15] but fully proven only recently [38], and provide a systematic and easily automatizable way of performing all q 0 integrations in the individual graphs (see also a related result presented in [39]). This procedure, which is summarized in fig.…”

“…Of the two terms to be computed, it suffices to consider the resummed version, as its naive counterpart can be obtained from this expression in a straightforward manner. As we shall show in detail in a forthcoming publication [37], the most nontrivial part of the calculation boils down to determining the two-loop HTL gluon polarization tensor and the Next-to-Leading Order asymptotic mass parameter of the gluon field. This is, however, a substantial challenge that in particular requires carrying out a separate computation in resummed perturbation theory.…”

“…This is, however, a substantial challenge that in particular requires carrying out a separate computation in resummed perturbation theory. The details of this exercise are left to be thoroughly explained in our later publication [37].…”

“…This procedure, which is summarized in fig. 5 and thoroughly explained in [38], reduces the evaluation of a scalarized Euclidean Feynman diagram at finite chemical potential into a number of three-dimensional phase space integrals over on-shell amplitudes, where the entire µ-dependence resides in the 3d integration measure.…”

“…An illustration of the cutting procedure of [38] being applied to a two-loop vacuum graph. Here, the solid line denotes a scalarized fermion propagator containing a chemical potential, the dotted line a boson propagator, and the dashed line a fermion propagator evaluated at µ = 0.…”