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Chiral anomaly: from vacuum to Columbia plot

Francesco Giacosa, Győző Kovács, Péter Kovács, Robert D. Pisarski, Fabian Rennecke

TL;DR

The paper investigates how the $U(1)_A$ axial anomaly influences the finite-temperature chiral transition in QCD-like theories. It uses the extended linear sigma model ($\mathrm{eLSM}$) with determinant-based anomaly terms, including the square of the $tHooft$ determinant and the polydeterminant extension, to analyze vacuum phenomenology and the Columbia plot. A key finding is that the $Q=2$ square-determinant term can enlarge the crossover region and shrink the first-order domain, with an effective coupling $\xi_{\rm eff}$ capturing the combined impact of multiple anomaly operators. The polydeterminant extension opens new interaction channels among distinct meson multiplets and glueball-like states, providing a broader framework for exploring chiral dynamics at finite temperature.

Abstract

We use a low-energy effective approach, the extended linear sigma model, to study realizations of the $U(1)_A$ anomaly with different operators, linear and quadratic in the 't Hooft determinant. After discussing the parameterization in agreement with vacuum's phenomenology, we investigate the influence of these different anomaly terms on the Columbia plot: the square of the 't Hooft determinant favors a cross-over for small quark masses. Finally, we also discuss the extension of the 't Hooft determinant to cases in which different mesonic multiplets interact with each other. Novel chiral anomalous interaction terms involving excited (pseudo)scalar states, pseudovector, and pseudotensor mesons are expressed via a mathematical extension of the determinant, denoted as a polydeterminant.

Chiral anomaly: from vacuum to Columbia plot

TL;DR

The paper investigates how the axial anomaly influences the finite-temperature chiral transition in QCD-like theories. It uses the extended linear sigma model () with determinant-based anomaly terms, including the square of the determinant and the polydeterminant extension, to analyze vacuum phenomenology and the Columbia plot. A key finding is that the square-determinant term can enlarge the crossover region and shrink the first-order domain, with an effective coupling capturing the combined impact of multiple anomaly operators. The polydeterminant extension opens new interaction channels among distinct meson multiplets and glueball-like states, providing a broader framework for exploring chiral dynamics at finite temperature.

Abstract

We use a low-energy effective approach, the extended linear sigma model, to study realizations of the anomaly with different operators, linear and quadratic in the 't Hooft determinant. After discussing the parameterization in agreement with vacuum's phenomenology, we investigate the influence of these different anomaly terms on the Columbia plot: the square of the 't Hooft determinant favors a cross-over for small quark masses. Finally, we also discuss the extension of the 't Hooft determinant to cases in which different mesonic multiplets interact with each other. Novel chiral anomalous interaction terms involving excited (pseudo)scalar states, pseudovector, and pseudotensor mesons are expressed via a mathematical extension of the determinant, denoted as a polydeterminant.
Paper Structure (7 sections, 7 equations, 2 figures)

This paper contains 7 sections, 7 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Anomalous interactions and the parametrisation
  3. 't Hooft anomaly terms within the eLSM
  4. Parametrization of the eLSM
  5. Extending the anomaly terms within the eLSM: the polydeterminant
  6. The Columbia plot
  7. Conclusions

Figures (2)

  • Figure 1: The different parameter sets in the parameters subspace spanned by the $\xi_1$, $\xi_1^1$, and $\xi_2^\mathrm{gen}=\xi_2,\xi_2^\pm$ are aligned on the $F(\phi_N,\phi_S)$ surface $\xi_1 + \alpha\,\xi_1^1+\beta\,\xi_2^\mathrm{gen}=\bar{\xi}_\mathrm{eff}$ determined by the constraint in Eq. \ref{['Eq:chieff']}. The numerical values are $\alpha=(0.13^2 + 0.14^2)/2$ and $\beta=0.13^2\cdot0.14/\sqrt2$, while $\bar{\xi}_\mathrm{eff}=1.4$ is used, but it is not fixed during the parametrization procedure.
  • Figure 2: The Columbia plot in the plane of the normalized explicit symmetry breaking parameters with parameters including $\xi_1$ (left), $\xi_2$ (right), and both (center).