Fractional anomalous determinants and the chiral phase transition
Abstract
At high temperature instantons form a dilute gas, so in QCD-like theories the breaking of the anomalous U(1)A symmetry is given by integral powers of the anomalous determinant, ∼ (det Φ)Q, where Φ∼ q̅L qR is bilinear in the quark fields, and with untwisted boundary conditions, the topological charge, Q, is an integer. A syncretic model is constructed, which is manifestly "beyond Landau". In the chiral limit, at temperatures above the chiral phase transition, T > Tχ, only integral powers of the anomalous determinant appear. Below Tχ, following 't Hooft et al. I assume that the topological charge Q is fractional, as an integer times 1/Nc, where Nc is the number of colors. I suggest that consequently, fractional powers of the anomalous determinant appear in the chiral effective Lagrangian. For Nf degenerate flavors, this generalizes the Witten-Veneziano term, valid for small Nf/Nc, to arbitrary Nf/Nc. In this model the chiral phase transition is generically of second order. The two exceptions are for one flavor, where it is probably crossover, and three flavors, where it could well be weakly first order. This can be tested in lattice QCD with 2+1 flavors by comparing the (known) temperature dependence of the difference of the πa and a0a propagators, to the chiral condensate of the strange quark, between Tχ and ∼ 2 Tχ. Analogous measurements are possible for one to four degenerate flavors about Tχ. Lastly, I propose an operator for baryon number in the symmetric phase.
Related
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BibTeX
@misc{pisarski2026fractional,
title = {{Fractional anomalous determinants and the chiral phase transition}},
author = {Robert D. Pisarski},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00472},
eprint = {2610.00472},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00472},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fractional anomalous determinants and the chiral phase transition
AU - Robert D. Pisarski
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00472
UR - https://arxiv.org/abs/2610.00472
AB - At high temperature instantons form a dilute gas, so in QCD-like theories the breaking of the anomalous U(1)_A symmetry is given by integral powers of the anomalous determinant, ∼ (det Φ)^Q, where Φ∼ q̅_L q_R is bilinear in the quark fields, and with untwisted boundary conditions, the topological charge, Q, is an integer. A syncretic model is constructed, which is manifestly "beyond Landau". In the chiral limit, at temperatures above the chiral phase transition, T > T_χ, only integral powers of the anomalous determinant appear. Below T_χ, following 't Hooft et al. I assume that the topological charge Q is fractional, as an integer times 1/N_c, where N_c is the number of colors. I suggest that consequently, fractional powers of the anomalous determinant appear in the chiral effective Lagrangian. For N_f degenerate flavors, this generalizes the Witten-Veneziano term, valid for small N_f/N_c, to arbitrary N_f/N_c. In this model the chiral phase transition is generically of second order. The two exceptions are for one flavor, where it is probably crossover, and three flavors, where it could well be weakly first order. This can be tested in lattice QCD with 2+1 flavors by comparing the (known) temperature dependence of the difference of the π^a and a_0^a propagators, to the chiral condensate of the strange quark, between T_χ and ∼ 2 T_χ. Analogous measurements are possible for one to four degenerate flavors about T_χ. Lastly, I propose an operator for baryon number in the symmetric phase.
ER -