Hard loops, soft loops, and high density effective field theory

Abstract

We study several issues related to the use of effective field theories in QCD at large baryon density. We show that the power counting is complicated by the appearance of two scales inside loop integrals. Hard dense loops involve the large scale µ 2 and lead to phenomena such as screening and damping at the scale gµ. Soft loops only involve small scales and lead to superfluidity and non-Fermi liquid behavior at exponentially small scales. Four-fermion operators in the effective theory are suppressed by powers of 1/µ, but they get enhanced by hard loops. As a consequence their contribution to the pairing gap is only suppressed by powers of the coupling constant, and not powers of 1/µ. We determine the coefficients of four-fermion operators in the effective theory by matching quark-quark scattering amplitudes. Finally, we introduce a perturbative scheme for computing corrections to the gap parameter in the superfluid phase.

General effective action for high-density quark matter

“…We also have to clarify how the patches covering the Fermi surfaces introduced in Refs. [14,15,16,17,18,19] arise within our effective theory. It is obvious that the radial dimension Λ of a patch is related to the quark cut-off Λ q .…”

“…In the following, we shall show that the action of the high-density effective theory as discussed in Refs. [14,15,16,17,18,19] is contained in our effective action (53). To this end, however, we shall employ the choice (25) and (42) for the projectors for quark and gluon modes, and not Eq.…”

“…As in Refs. [14,15,16,17,18,19], the quark mass will be set to zero, m = 0. We also have to clarify how the patches covering the Fermi surfaces introduced in Refs.…”

“…This suggests that the most efficient way to compute properties such as the color-superconducting gap parameter is via an effective theory for quark modes near the Fermi surface. Such an effective theory has been originally proposed by Hong [14,15] and was subsequently refined by others [16,17,18,19]. At this point it is worth reviewing the standard approach to derive an effective theory [20,21].…”

Effective Theories of Dense and Very Dense Matter

“…The logarithmic dependence on the cutoff is absorbed by the BCS four-fermion operator. A simple matching calculation gives 68]. The solution to the gap equation was found by Son [70] ∆…”