Instantons in cold dense quark matter: self-consistent medium back-reaction, the interacting many-instanton ensemble, and quantitative control beyond leading-logarithmic accuracy

A 1981 two-loop calculation by A. A. Abrikosov (Nucl. Phys. B 182, 441) evaluated, using exact quark Green functions in the instanton background, the correction to the BPST instanton action from a cold ($T\to 0$) degenerate plasma of light quarks. Its logarithmically enhanced medium terms make the effective action decrease with the instanton radius $\rho$, so the plasma does not suppress instantons at two-loop order; large instantons are strongly distorted by the medium polarization, none larger than a Debye-type radius survive as classical configurations, and the $\rho$-integral is dominated by large, strongly interacting pseudoparticles, invalidating the dilute instanton gas approximation. The paper left open the self-consistent gauge field including the medium back-reaction, the many-instanton phase, control beyond leading-logarithmic accuracy, and the collective consequences of the predicted instanton-induced attraction between real quarks. The subsequent literature reversed the central two-loop conclusion: the author's own 1983 sequel found that medium fermion zero modes suppress large pseudoparticles and restore a dilute gas, while modern treatments establish a Gaussian high-density suppression of the instanton size distribution and identify the instanton-induced attraction as a pairing mechanism of color superconductivity. What survives unsolved is the quantitative instanton content of dense quark matter: the self-consistent back-reacted configuration, the interacting instanton ensemble, and reliable size distributions and thermodynamic predictions, at moderate densities where large instantons dominate and current semiclassical treatments disagree at the factor-of-few level.

Background

An instanton is a topologically nontrivial classical solution of the Euclidean Yang-Mills equations: the simplest one, the BPST instanton (Belavin, Polyakov, Schwartz and Tyupkin, 1975), is (anti-)self-dual (its field strength satisfies $F=\pm\,{}^{*}\!F$), carries topological charge $\pm 1$, and is parametrized by a size $\rho$ and a position. In quantum chromodynamics (QCD) instantons mediate tunnelling between topologically distinct vacua and induce, once one integrates over a dilute gas of them, the celebrated 't Hooft determinant vertex ('t Hooft, 1976) — an effective multi-fermion interaction that breaks the anomalous $U(1)_A$ symmetry and shapes chiral physics. The dilute instanton gas approximation (the assumption that pseudoparticles are well separated relative to their sizes, so their mutual interactions can be neglected) is the workhorse of this semiclassical picture. A basic obstruction is the infrared problem of the size integral: the one-loop instanton weight contains the running coupling $8\pi^2/g^2(\rho)\simeq-(11N_{\mathrm{c}}-2N_{\mathrm{f}})\ln(\rho\Lambda)/3$ (for $N_{\mathrm{c}}$ colors, $N_{\mathrm{f}}$ flavors, QCD scale $\Lambda$), which decreases with $\rho$, so the integral over instanton sizes diverges at large $\rho$ — large instantons are formally favored and the dilute gas assumption breaks down.

Whether a dense fermionic medium changes this was a live question around 1980. The thermal case was settled by Gross, Pisarski and Yaffe (Rev. Mod. Phys. 53 (1981) 43): a sufficiently hot plasma Debye-screens color-electric fields (static screening with length set by the Debye mass, of order $gT$), which suppresses large-scale instantons; the full temperature dependence of the instanton density was computed and instanton effects became reliably calculable at high temperature. The cold, dense analog — a degenerate plasma of light quarks at $T\to 0$ and quark chemical potential $\mu$ (the Fermi momentum, i.e. the baryon-number fugacity setting the density of occupied quark states) — was addressed by A. A. Abrikosov (Jr.), the particle physicist son of the condensed-matter Nobel laureate of the same name, not the laureate himself. After a 1980 one-loop study, the 1981 two-loop paper (the source of this problem) computed the instanton-action correction due to real, on-shell medium quarks using the exact quark Green function in the instanton background. The one-loop medium shift vanishes; at two loops two logarithmically enhanced medium terms enter — a plasmon (ring-diagram) term and a zero-momentum gluon-exchange term between the color currents the instanton induces in the populated quark states — both negative, contributing to the effective instanton action $S_{\mathrm{inst}}(\mu,\rho)$ powers of $\mu\rho$ times logarithms. The published conclusions are: the action falls abruptly as the instanton radius increases; effects of instanton suppression by the plasma are absent; large instantons are strongly distorted by the plasma (the medium polarization energy exceeds the classical field energy beyond an outer, Debye-like scale of order $1/(g\mu)$ and inside an inner scale of the same parametric size), so there are no instantons greater than the Debye radius; and because the $\rho$-integral is then dominated by large, strongly interacting pseudoparticles, the dilute gas approximation fails. The same analysis found the instanton-mediated interaction between occupied quark states to be attractive, strongest for oppositely directed momenta, hinting at collective phenomena in dense quark matter.

The paper acknowledged what its framework could not deliver: the self-consistent classical gauge field including the back-reaction of the polarized medium (the BPST ansatz ceases to satisfy the classical equations once the medium polarization is strong), the many-pseudoparticle phase under medium effects, quantitative control beyond leading-logarithmic two-loop accuracy, and extensions to finite temperature and dynamical (time-dependent) media. Within two years the same author revised the central physics: constructing the fermion propagator in self-dual fields at nonzero density and temperature and keeping the fermion zero modes (normalizable solutions of the Dirac equation in the instanton background that exist for topological reasons) in the medium produced an additional one-loop term in the instanton action that suppresses large pseudoparticles and makes the gas dilute — reversing the two-loop no-suppression conclusion (announced in JETP Lett. 35 (1982) 156; full sequel in Nucl. Phys. B 218 (1983) 459, doi:10.1016/0550-3213(83)90375-9). The later literature (interacting-instanton phenomenology, instanton-induced pairing and color superconductivity, and controlled high-density computations) descends from these questions.

Problem Statement

The problem is the quantitative instanton content of a cold, dense (degenerate) quark plasma — QCD at $T\to 0$ with quark chemical potential $\mu$ — comprising three open directions left unresolved by the 1981 two-loop analysis. They share a common objective, a first-principles account of instanton effects in dense quark matter: the single-pseudoparticle building block, its assembly into an ensemble, and the accuracy that makes the resulting predictions reliable.

1. Self-consistent configuration with medium back-reaction. Determine the classical gauge-field configuration that solves the Yang-Mills equations including the polarization of the degenerate quark medium (the induced color current computed from the exact quark propagator in that same background), or establish that no such configuration with unit topological charge and finite action exists. The 1981 two-loop analysis found strong medium distortion of large instantons at and beyond Debye-type scales of order $1/(g\mu)$; the author's 1982/1983 zero-mode treatment reversed the central no-suppression conclusion, and the status of the distortion claims remains unsettled in the accessible record, while the later consensus treats BPST-type configurations of size $\rho\lesssim 1/\mu$ with a Gaussian-suppressed measure as the semiclassical building blocks.

2. The interacting many-instanton phase at finite density. Characterize the ensemble of pseudoparticles once medium-induced modifications and instanton interactions are included: the phase that replaces (or validates) the dilute gas, and the domain of validity of the dilute instanton gas approximation. Exact multi-instanton solutions in vacuum rely on (anti-)self-duality, which the medium breaks, so this component is not reducible to known vacuum multi-instanton constructions.

3. Quantitative control beyond leading-logarithmic accuracy. Compute the instanton size distribution $n(\rho,\mu)$ — the number of instantons per unit four-volume per unit interval of the size $\rho$ at quark chemical potential $\mu$ — with accuracy beyond the leading-log two-loop treatment — at least through the next-to-leading (nonlogarithmic) order omitted there — and its consequences for thermodynamic and anomalous observables of dense quark matter (the instanton contribution to the free energy and equation of state, $U(1)_A$-breaking quantities, and pairing-related observables). At moderate density — where large instantons dominate the size integral and current semiclassical treatments disagree at the factor-of-few level (Schäfer, Phys. Rev. D 65 (2002) 094033; Alford, Schmitt, Rajagopal and Schäfer, Rev. Mod. Phys. 80 (2008) 1455) — no reliable determination of $n(\rho,\mu)$ and its thermodynamic consequences exists, and the recognized remedies (two-color lattice proxies, phenomenological screening constraints) are uncontrolled extrapolations to $N_{\mathrm{c}}=3$ QCD.

The three components are separate open directions; none of them follows from the others. The source paper's further outlook — finite temperature, dynamical (time-dependent) media, and collective phenomena seeded by the instanton-induced attraction of real quarks — is treated as context rather than an open deliverable here, because the finite-density-and-temperature propagator was constructed in the author's 1983 sequel, the thermal instanton problem was developed into a controlled separate field, and the instanton-attraction question grew into the color-superconductivity program.

References

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