Instantons in the quark plasma at finite temperature and quark chemical potential beyond the one-loop dilute instanton gas

A. A. Abrikosov Jr. (Nucl. Phys. B 218 (1983) 459) computed one-loop instanton effects in a dense, heated quark plasma: constructing the fermion propagator and zero-mode projector in a self-dual gauge field at nonzero quark chemical potential $\mu$ and temperature $T$, he found that the medium generates an additional one-loop term in the instanton action that suppresses large pseudoparticles so the gas stays dilute, and evaluated the instanton contribution to the thermodynamic functions — small through the quark-mass product but enhanced by the vacuum quark condensate. The explicit medium suppression factor $\exp(-N_f\mu^2\rho^2)$ (equivalently an action correction $\Delta S_{\mathrm{inst}}=N_f\mu^2\rho^2$, with $N_f$ the number of light quark flavors and $\rho$ the instanton size) is reproduced in Schäfer (1998, Eq. (8)) and Schäfer (2002, Eq. (7)) and rests on the finite-density zero-mode results of Abrikosov (1983) (Nucl. Phys. B 218, 459; companion paper Yad. Fiz. 37, 772 (1983) [Sov. J. Nucl. Phys. 37, 459]); the coefficient of the small-$\rho T$ temperature term in the 1983 action has not been independently confirmed. Beyond this one-loop dilute-gas treatment lie open aspects documented in the later literature (Schäfer 1998; Schäfer 2002) — the breakdown of the treatment when the instanton volume fraction approaches saturation, the crossover regimes $T\gtrsim\mu$ and $\rho T\gtrsim1$, the true chemical-potential dependence of the condensate, many-instanton interactions, and renormalization-scale and truncation uncertainties — which define an open problem resolved so far only at its two controlled ends: high temperature (Debye-screening suppression of instantons, confirmed by lattice QCD as dilute-instanton-gas behavior above roughly $2T_c$) and asymptotically large density (power suppression in $\Lambda_{\mathrm{QCD}}/\mu$, with $\Lambda_{\mathrm{QCD}}$ the intrinsic QCD scale, and the 1983 finite-density zero mode as a standard technical input). The surviving open core is the moderate-density regime $\mu\sim\Lambda_{\mathrm{QCD}}$, where large instantons dominate, finite-size effects are marginal, the chiral condensate melts, and no controlled or first-principles determination exists — chiefly because finite-baryon-density lattice QCD is blocked by the sign problem.

Background

An instanton is a self-dual solution of the Euclidean Yang–Mills equations (the BPST solution, found by Belavin, Polyakov, Schwartz and Tyupkin) carrying topological charge $Q=1$; in QCD it describes tunneling between topologically distinct vacua. 't Hooft showed that the massless Dirac operator has exact chiral zero modes in an instanton background, so summing over a dilute instanton ensemble produces a multi-fermion interaction (the 't Hooft vertex) proportional to the product of the quark masses $\prod_f m_f$; this vertex breaks the axial $U(1)_A$ symmetry and accounts for much of the anomalous chiral physics of light hadrons. In the semiclassical, dilute instanton gas approximation the instanton density per unit volume and size $\rho$ is $d(\rho)\sim\prod_f(m_f\rho)\big(8\pi^2/g^2(\rho)\big)^{2N_c}\exp[-8\pi^2/g^2(\rho)]$, with $g^2(\rho)$ the running coupling at the instanton size and $N_c$ the number of colors; the gas is dilute when the four-volume fraction $x$ occupied by pseudoparticles (instantons and anti-instantons) is small, while at larger $x$ one speaks of an interacting instanton liquid.

At nonzero temperature $T$ Euclidean time is compactified with period $\beta=1/T$, and the corresponding periodic instanton is called a caloron. Pisarski and Yaffe (1980), and the review by Gross, Pisarski and Yaffe (1981), evaluated the one-loop instanton density at finite temperature and found that the quark-gluon plasma Debye-screens the gluoelectric (color-electric) fields: the density acquires a factor $\exp[-\frac{2N_c+N_f}{3}(\pi\rho T)^2-B(\pi\rho T)]$, with $B$ a slowly varying function, so large instantons ($\rho\gtrsim1/T$) are exponentially suppressed and instanton effects die off at high $T$. Lattice simulations confirm that the topological susceptibility (the fluctuation measure of the topological charge) drops above the chiral transition temperature $T_c$ and approaches the dilute-instanton-gas prediction for $T\gtrsim2T_c$.

At nonzero quark chemical potential $\mu$ — the variable conjugate to quark number, appropriate for cold dense quark matter such as neutron-star interiors — the Euclidean fermion determinant acquires a complex weight (the origin of the finite-density sign problem that still blocks direct lattice simulation at physical baryon density), so the instanton calculation must be reorganized. A. A. Abrikosov Jr. (1983) constructed the fermion propagator and the fermion zero-mode projector in a single self-dual field at $\mu\neq0$ and $T\neq0$, keeping the non-diagonal elements of the quark density matrix, and computed a medium-induced one-loop correction to the single-instanton action — the explicit suppression factor $\exp(-N_f\mu^2\rho^2)$, equivalently $\Delta S_{\mathrm{inst}}=N_f\mu^2\rho^2$, as reproduced in Schäfer (1998, Eq. (8)) and Schäfer (2002, Eq. (7)) — a result obtained independently by Shuryak (1982) in his study of instantons in the quark-gluon plasma. Large pseudoparticles are thereby suppressed and the gas remains dilute; the instanton contribution to the thermodynamic potential is proportional to the occupied four-volume fraction $x$ and to $\prod_f m_f$, but is enhanced when the current quark masses are replaced by effective masses involving the chiral condensate $\langle\bar q q\rangle$ (the quark–antiquark expectation value that signals spontaneous chiral symmetry breaking). The treatment was explicitly one-loop and dilute; outside the large-$\mu$ regime the instanton rate depends on the true in-medium dependence of $\langle\bar q q\rangle$ on the chemical potential, which remained undetermined (Schäfer 1998).

The question of what instantons do at high baryon density re-emerged in the late 1990s in the context of color superconductivity: in dense quark matter, quarks near the Fermi surface form Cooper pairs, and for three flavors the expected ground state is the color-flavor locked (CFL) phase, in which color and flavor symmetry are locked and chiral symmetry is broken by diquark rather than quark–antiquark condensates. In this framework the instanton-induced interaction between quarks near the Fermi surface is a leading driver of the pairing gap, while at asymptotically large $\mu$ the instanton density is suppressed by a power of $\Lambda_{\mathrm{QCD}}/\mu$ (with $\Lambda_{\mathrm{QCD}}$ the intrinsic QCD scale) because perturbative screening cuts off instanton sizes $\rho\gtrsim1/\mu$. Between the two controlled limits — high $T$ and asymptotically large $\mu$ — lies the regime the 1983 computation could not reach, and it is there that the question posed by its limitations still stands.

Problem Statement

Determine, beyond the fermion one-loop dilute-instanton-gas approximation, the instanton ensemble of quark matter at nonzero quark chemical potential $\mu$ and temperature $T$, and its consequences for the thermodynamic functions and for chiral order. This single objective, posed by the limitations of the 1983 one-loop dilute-gas computation as documented in the later literature (Schäfer 1998; Schäfer 2002), decomposes into scientifically linked components: (a) the non-dilute regime, including many-instanton interactions — what happens to the semiclassical ensemble when the instanton four-volume fraction $x$ is not small or the medium-induced action correction $\Delta S_{\mathrm{inst}}$ is not small, so that the classical self-dual field may be strongly distorted and the dilute-gas weights fail, and what the resulting thermodynamic consequences are; (b) the crossover regimes $T\gtrsim\mu$ and $\rho T\gtrsim1$ ($\rho$ the instanton size; equivalently, Fermi momentum times instanton size of order one), where neither the density-dominated nor the temperature-dominated expansions apply; (c) the true dependence of the quark condensate $\langle\bar q q\rangle$ on baryon asymmetry — as opposed to its dependence on a vacuum renormalization scale — and its feedback on the instanton weights; and (d) the quantification of truncation and renormalization-scale uncertainties and of the neglected beyond-one-loop corrections (the one-loop coupling $g^2(\rho)$, the running coupling evaluated at the instanton size, and the saddle-point, i.e. steepest-descent, evaluation of the instanton-size integrals). The limiting regimes in which the dilute gas and weak coupling are controlled are already settled — high-temperature suppression (Pisarski–Yaffe/Gross–Pisarski–Yaffe; lattice dilute-instanton-gas behavior above roughly $2T_c$, with $T_c$ the QCD chiral transition temperature) and the asymptotically large-density power suppression in $\Lambda_{\mathrm{QCD}}/\mu$ ($\Lambda_{\mathrm{QCD}}$ the intrinsic QCD scale) — so the open target is the intermediate, moderate-density region $\mu\sim\Lambda_{\mathrm{QCD}}$ (the neutron-star-density regime $\mu\simeq0.5$ GeV, at roughly $(5\!-\!10)$ times nuclear matter density, analyzed by Schäfer (2002)) and the $T$–$\mu$ crossover, where large instantons dominate, finite-size effects are marginal, and no controlled or first-principles determination exists. The open target is: a controlled determination of the instanton ensemble in this intermediate, moderate-density window ($\mu\sim\Lambda_{\mathrm{QCD}}$, the neutron-star-density anchor $\mu\simeq0.5$ GeV) and the $T$–$\mu$ crossover, and of its consequences for the thermodynamic functions (the equation of state) and for chiral order (the in-medium condensate $\langle\bar q q\rangle$), with the systematic uncertainties of the result quantified and shown to be under control.

References

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