Beyond-one-loop interaction of instantons with the finite-temperature quark plasma in QCD

The published abstract of Abrikosov's 1981 letter on the instanton in the heated quark plasma establishes that, "independently of the temperature, in leading order of perturbation theory the instantons do not interact with the dense plasma of light quarks"; beyond leading order, whether higher-loop finite-temperature corrections produce such an interaction, suppress large instantons, or generate Debye-type screening is left open by the published result. Subsequent literature has settled the one-loop story: the Gross-Pisarski-Yaffe thermal determinant does suppress instantons at finite $T$ with a Debye-screening interpretation, and at quark chemical potential $\mu$ a one-loop Pauli-blocking factor $e^{-N_f\mu^2\rho^2}$ arises from the fermion zero modes (zero-energy solutions of the Dirac equation in the instanton background), as established by Abrikosov's own 1983 paper and adopted in the modern dense-QCD literature. What has never been computed in four-dimensional QCD is the two-loop finite-temperature correction to the caloron effective action and instanton size distribution - the leading beyond-one-loop instanton-medium interaction, with higher orders entering the required answer only where they are parametrically enhanced relative to it - even though modern work repeatedly identifies it as needed: lattice QCD finds the high-temperature instanton density roughly ten times larger than the one-loop dilute-instanton-gas prediction, and axion-cosmology applications depend on precisely this regime. The open problem is to determine that correction.

Background

An instanton is a self-dual solution of the classical Euclidean Yang-Mills equations (the Belavin-Polyakov-Schwartz-Tyupkin solution) carrying topological charge $Q=1$; it mediates quantum tunneling between gauge vacua of different winding number, and the fermion zero modes (zero-energy solutions of the Dirac equation) localized in its core generate the anomalous multi-fermion interaction known as the 't Hooft vertex, the microscopic origin of the $U(1)_A$ anomaly and of anomalous chiral processes in electroweak and QCD plasmas. At finite temperature $T$, Euclidean time is compactified to a circle of circumference $\beta=1/T$ with periodic boundary conditions for the gauge fields and antiperiodic boundary conditions for the quarks, so that fields are expanded in discrete thermal frequencies (Matsubara frequencies), $\omega_n=2\pi nT$ for bosons and $(2n+1)\pi T$ for fermions; the instanton generalizes to the Harrington-Shepard caloron, a periodic self-dual solution. In the dilute instanton gas approximation (DIGA), instantons of size $\rho$ are treated as a non-interacting gas with a size distribution $n(\rho,T)\propto d(\rho)\exp[-S_{\mathrm{eff}}(\rho,T)]$, where $S_{\mathrm{eff}}=8\pi^2/g^2(\rho)+\Delta S(\rho,T)$ is the classical action (with $g$ the QCD gauge coupling, running with the scale $\rho$) plus the medium correction $\Delta S$ from quantum fluctuations. Integrating $n(\rho,T)$ over $\rho$ gives the high-temperature topological susceptibility $\chi_{\mathrm{top}}(T)$, the variance of the total topological charge, which is the key input to the cosmological history of the QCD axion (a hypothetical light pseudoscalar particle).

The one-loop medium correction at finite temperature was computed by Pisarski and Yaffe and summarized by Gross, Pisarski and Yaffe:

$$S(\lambda)=\lambda^2(2N_c+N_f)/3+12A(\lambda)(1+(N_c-N_f)/6),\quad \lambda=\pi\rho T,$$

where $N_c$ is the number of colors, $N_f$ the number of quark flavors, and $A(\lambda)$ a known function with $A(\lambda)\approx-\lambda^2/36$ at small $\lambda$. This factor exponentially suppresses large instantons at high $T$. Because instantons carry long-range color-electric fields, the dominant $\rho^2m_D^2$-type piece is interpretable as Debye screening - the screening of static electric color fields in the plasma, with electric screening mass $m_D^2=(N_c/3+N_f/6)g^2T^2$ at one loop. Korthals Altes and Sastre later rewrote this one-loop thermal determinant in compact form and clarified its infrared structure: the gluonic (periodic) determinant is logarithmically infrared divergent at large $\rho T$, while the fermionic (antiperiodic) determinant is infrared finite.

Abrikosov's 1981 letter approached the same medium from a different decomposition of the thermal fermion propagator: in a thermal ensemble at temperature $T$ and quark chemical potential $\mu$, the free propagator splits into a vacuum part and an on-shell medium part whose phase-space weights are Fermi-Dirac occupation factors. Computing this medium part in the caloron background at one loop in SU(2) QCD with one massless quark flavor (plus the first nontrivial small-quark-mass term), he found that the quark number density equals that of the plasma without the caloron and concluded that instantons do not interact with the on-shell quark plasma at one loop, independently of $T$. The analysis is restricted to one loop and to the dilute-gas setting; since the published result holds only "in leading order of perturbation theory", whether two-loop and higher finite-temperature corrections generate an instanton-plasma interaction remains open.

The subsequent arc frames the open question. Abrikosov's own two-loop computation in a cold dense plasma at $T=0$ found logarithmically enhanced interaction terms that decrease the effective action as the instanton size grows. His 1983 long paper established that at $\mu\neq 0$ the fermion zero modes produce a one-loop Pauli-blocking suppression (the filled Fermi sea blocks occupation of the zero-mode fermion states) $\Delta S=N_f\mu^2\rho^2$, with total small-$\rho T$ action $8\pi^2/g^2(\rho)+N_f\mu^2\rho^2+(\pi^2/18)\rho^2T^2(11N_c+7N_f-6)$, the last term being the small-$\lambda$ limit of the Gross-Pisarski-Yaffe factor. Morris, Ross and Sachrajda computed the two-loop instanton density at $T=0$ and found it renormalization-scheme dependent. Modern work - Boccaletti and Nogradi's precision re-evaluation of the one-loop finite-$T$ distribution, Pisarski and Rennecke's observation that the lattice instanton density exceeds the one-loop dilute-gas result by a prefactor of order ten that a complete two-loop thermal computation might ameliorate, and Högl and Moore's one-loop caloron-gas treatment of the finite-$T$ topological susceptibility with heavy quarks - all operates at one loop. Whether the instanton-quark-plasma interaction survives beyond one loop at finite temperature, and with what sign, remains unknown.

Problem Statement

Abrikosov (1981) concluded, per the published abstract, that "independently of the temperature, in leading order of perturbation theory the instantons do not interact with the dense plasma of light quarks"; the full text derives this at one loop in SU(2) QCD with one massless quark flavor, from the on-shell medium part of the thermal fermion propagator in a caloron background leaving the quark number density unchanged. Beyond leading order, whether two-loop and higher finite-temperature corrections generate a net interaction between instantons and the heated plasma is left open. The one-loop part of the question has since been settled - the Gross-Pisarski-Yaffe determinant suppresses instantons at finite $T$ (with a Debye-screening interpretation of the dominant $\rho^2m_D^2$-type term, where $\rho$ is the instanton size and $m_D$ the electric color-screening mass), and a finite quark chemical potential $\mu$ adds the Pauli-blocking factor $e^{-N_f\mu^2\rho^2}$ ($N_f$ being the number of quark flavors) - but the beyond-one-loop part has never been computed in four-dimensional QCD.

Determine the two-loop finite-temperature correction to the caloron effective action and to the instanton size distribution $n(\rho,T)$ of a thermal quark-gluon medium; higher-order terms enter the required answer only where they are parametrically enhanced relative to the two-loop result. This determination includes the interaction between the caloron and the thermal quark medium, understood as the $(\rho,T)$-dependent part of the two-loop correction to the medium term $\Delta S(\rho,T)$ of the effective action, and must establish whether that interaction survives beyond one loop and whether it suppresses large instantons, acting as Debye-type screening beyond the one-loop factor, or instead decreases the effective action with growing instanton size $\rho$, as Abrikosov's two-loop analysis of the cold dense plasma suggests.

The answer should be stated in a definite renormalization scheme together with its scheme dependence (the $T=0$ two-loop density of Morris-Ross-Sachrajda is already scheme dependent), should specify its domain of validity in $\rho T$ (the semiclassical dilute-gas regime versus the infrared-sensitive large-instanton regime), and should be consistent with the known one-loop infrared structure (logarithmic divergence of the periodic gluon determinant, finiteness of the antiperiodic fermion determinant). The two-loop correction to $\Delta S(\rho,T)$ and to $n(\rho,T)$, together with the sign of the interaction, is subject to these scheme, domain-of-validity, and infrared-consistency conditions.

References

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