We study the thermodynamics of the symmetry-improved Cornwall-Jackiw-Tomboulis formalism and apply it to a low-energy effective theory of QCD. In the symmetry-improved formulation, Ward-Takahashi identities are restored by auxiliary sources whose values are fixed self-consistently by the equilibrium state. While this construction improves the symmetry properties of the loopwise truncated two-particle-irreducible (2PI) theory, it also makes the thermodynamic interpretation of the pressure nontrivial. We formulate several pressure prescriptions, including the conventional vacuum-subtracted pressure, a source-matched subtraction, and a pulled-back pressure in which the explicit source-induced energy shift is removed. Using the three-flavor linear sigma model with quarks, we analyze the equation of state, isentropic trajectories, adiabatic sound velocity, and trace anomaly across the chiral transition. We find that the global thermodynamic structure is stable under the different prescriptions, while quantitative differences are concentrated near the crossover and first-order transition region. These results establish a practical framework for constructing thermodynamically consistent observables in symmetry-improved 2PI approaches.

China

Yuepeng Guan

Mamiya Kawaguchi

Shinya Matsuzaki

Japan

Akio Tomiya

Thermodynamics in symmetry-improved Cornwall-Jackiw-Tomboulis formalism: Application to the low-energy effective theory of QCD

We propose a new scenario to realize the Coleman-Weinberg (CW) type chiral phase transition in the QCD thermal history. This scenario predicts a heavy axionlike particle (ALP) with mass ~ 5 MeV, consistently with the current experimental and cosmological bounds. The chiral phase transition is evaluated by monitoring ordinary QCD setup in a view of a two-flavor Nambu-Jona-Lasinio model including a simplified meson fluctuation contribution. The present work thus can open a new window to search for the ALP associated with the QCD phase transition epoch of the thermal history. The new QCD cosmological scenario potentially predicts rich epochs around the QCD scale: a mini-inflation; a nonperturbative preheating and/or reheating, which can provide characteristic gravitational wave and primordial black hole productions. This proposal is based on a generic classification of the order of the chiral phase transition at the level of the mean field approximation in view of the scale violation classes: the soft-scale breaking term and the CW-type scale anomaly term, in or off the medium with or without chemical potentials. On this theoretical ground, we also revisit existing scenarios which undergo the supercooling chiral phase transition, such as nearly scale-invariant QCD and QCD with a large baryon chemical potential.

Zheng-liang Jiang

He-Xu Zhang

Axionlike particle-assisted supercooling chiral phase transition in QCD: identifying Coleman-Weinberg type-chiral phase transition in QCD-like scenarios

We present charmonium spectral functions extracted from Euclidean-time correlation functions using sparse modeling (SpM). SpM solves inverse problems by considering only the sparsity of the target solution. To assess the applicability of the method, we first test it with mock data designed to mimic charmonium correlation functions. We demonstrate that while resonance peaks in the spectral functions can be reconstructed using this method, transport peaks are difficult to resolve without introducing further assumptions beyond sparsity. We then apply the method to charmonium correlation functions obtained from lattice quantum chromodynamics (QCD) at temperatures below and above the critical temperature. The results are found to be qualitatively consistent with those obtained using the maximum entropy method, although the transport peak is not clearly resolved. This indicates that, even when relying solely on the assumption of sparsity, the method can capture some relevant features of the underlying physics.

Junichi Takahashi

Hiroshi Ohno

Sparse modeling study of extracting charmonium spectral functions from lattice QCD at finite temperature

We study the Columbia plot for the chiral phase transition in the framework of a three-flavor linear sigma model based on the Cornwall-Jackiw-Tomboulis (CJT) formalism. The conventional CJT approach with the Hartree truncation suffers from artificial chiral breaking, leading to the violation of the Nambu-Goldstone theorem and the (anomalous) chiral Ward-Takahashi identities. We apply the symmetry-improved CJT formalism to resolve this issue. We observe a first-order phase transition and a tricritical point in the light-quark mass regime, which is relatively independent of the size of the sigma meson, in contrast to the conventional CJT approach. The tricritical point, found on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>s</mi></msub></math> axis, is at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>m</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>tri</mi></mrow></msubsup><mo>/</mo><msubsup><mrow><mi>m</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>phys</mi></mrow></msubsup><mo>=</mo><mn>0.175</mn></mrow></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>m</mi><mi>s</mi><mrow><mi>phys</mi></mrow></msubsup></math> being the physical strange quark mass in real-life QCD. The critical pion mass in the three-flavor symmetric limit, on the second-order boundary, is measured at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>π</mi></mrow></msub><mo>∼</mo><mn>52.4</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>, with the critical temperature <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>∼</mo><mn>51.7</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>.

Columbia plot based on the symmetry-improved Cornwall-Jackiw-Tomboulis formalism in the linear sigma model

The decrease of the chiral pseudocritical temperature <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mrow><mi>p</mi><mi>c</mi></mrow></msub></math> with an applied strong magnetic field has been extensively investigated by various QCD low-energy effective models and lattice QCD at the physical point. We find that this decreasing feature may not hold in the case with a weak magnetic field and still depends on quark masses: when the quark masses get smaller, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mrow><mi>p</mi><mi>c</mi></mrow></msub></math> turns to increase with the weak magnetic field. This happens due to the significant electromagnetic scale anomaly contribution in the thermomagnetic medium. We demonstrate this salient feature by employing the Nambu-Jona-Lasinio model with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> quark flavors including the electromagnetic scale anomaly contribution. We observe that at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">(</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>0</mn><mi>c</mi></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>s</mi><mi>c</mi></mrow></msub><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>20</mn><mo stretchy="false">)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math> for the isospin symmetric mass for up and down quarks, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mn>0</mn></msub></math>, and the strange quark mass, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>s</mi></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mrow><mi>p</mi><mi>c</mi></mrow></msub></math> decreases with the magnetic field if the quark masses exceed the critical values, and increases as the quark masses become smaller. Related cosmological implications, arising when the supercooled electroweak phase transition or dark QCD cosmological phase transition is considered along with a primordial magnetic field, are also briefly addressed.

Yuanyuan Wang

Implications of the electromagnetic scale anomaly to the QCD chiral phase transition in the smaller quark mass regime: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mrow><mi>p</mi><mi>c</mi></mrow></msub></math> does not drop with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>e</mi><mi>B</mi></math>

We discuss the thermal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> phase transition in QCD at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>θ</mi><mo>=</mo><mi>π</mi></math> under a weak magnetic field background, where the electromagnetic scale anomaly gets significant. To be explicit, we work on a two-flavor Nambu–Jona-Lasinio model at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>θ</mi><mo>=</mo><mi>π</mi></math> in the mean field approximation, including the electromagnetic-scale anomaly term. We find that the thermal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> phase transition becomes first order and the strength of the first order gets more prominent as the magnetic field increases. The associated potential barrier is thermally created by the electromagnetic scale anomaly and gives rise to criticality due to the induced potential of a nonperturbative form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>∼</mo><mfrac><mrow><mo stretchy="false">|</mo><mi>e</mi><mi>B</mi><msup><mo stretchy="false">|</mo><mn>3</mn></msup></mrow><msub><mi>f</mi><mi>π</mi></msub></mfrac><mfrac><mrow><mo stretchy="false">|</mo><mi>P</mi><mo stretchy="false">|</mo></mrow><mrow><msup><mi>P</mi><mn>2</mn></msup><mo stretchy="false">+</mo><msubsup><mi>m</mi><mn>0</mn><mn>2</mn></msubsup></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>e</mi><mi>B</mi></math> denotes the magnetic field strength; <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>P</mi></math> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math>-order parameter, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mn>0</mn></msub></math> is the isospin-symmetric current-quark mass.

First-order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> phase transition in two-flavor QCD at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>θ</mi><mo>=</mo><mi>π</mi></math> under electromagnetic scale anomaly via a Nambu–Jona-Lasinio description

We investigate the stability of topological charge under gradient flow taking the admissibility condition into account. For the SU(2) Wilson gauge theory with $\textit{β}$ = 2$\textit{.}$45 and $\textit{L}$ = 12 , we numerically show that the gradient flows with the Iwasaki and DBW2 gauge actions stabilize the topological sectors significantly, and they have qualitatively different behaviors compared with the Wilson and tree-level Symanzik flows. By considering the classical continuum limit of the flow actions, we discuss that the coefficient of dimension-6 operators has to be positive for stabilizing the one-instanton configuration, and the Iwasaki and DBW2 actions satisfy this criterion while the Wilson and Symanzik actions do not. Moreover, we observe that the DBW2 flow stabilizes the topological sectors at the very early stage of the flow ( $$ \hat{t} $$ ≈ 0$\textit{.}$5–1), suggesting that a further systematic investigation of the DBW2 flow is warranted to confirm its computational efficiency in determining the gauge topology.

Yuya Tanizaki

Hiromasa Watanabe

Lattice gradient flows (de-)stabilizing topological sectors

We perform a functional renormalization group analysis of a four-fermion model with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>P</mi></math> violation to explore the nonperturbative infrared dynamics of quantum chromodynamics (QCD) within the Wilsonian renormalization group framework, particularly in the context of spontaneous <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math>-violation models. Our analysis of the fixed-point structure reveals that, in the large-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>c</mi></msub></math> limit, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math>-violating <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>θ</mi><mo stretchy="false">¯</mo></mover></math> parameter is dynamically induced and approaches <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>π</mi><mo>·</mo><mo stretchy="false">(</mo><msub><mi>N</mi><mi>f</mi></msub><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo></math> (where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub></math> is the number of flavors) as the system enters the chirally broken phase. This behavior arises due to criticality and the large anomalous dimensions of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo><mi>A</mi></msub></math>-violating four-fermion couplings. Furthermore, this trend appears to persist beyond the leading large-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>c</mi></msub></math> approximation, provided that the infrared dynamics of QCD remains dominated by the scalar condensate of the quark bilinear, as expected. Notably, our findings highlight that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math>-violating four-fermion interactions, which are perturbatively irrelevant, can become relevant in the chirally broken phase through nonperturbative effects, with potential implications for spontaneous <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math>-violation scenarios.

Linlin Huang

Yadikaer Maitiniyazi

Masatoshi Yamada

Functional renormalization group study of a four-fermion model with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> violation

We propose gauge covariant neural networks along with a specialized training algorithm for lattice QCD, designed to handle realistic quarks and gluons in four-dimensional space-time. We show that the smearing procedure can be interpreted as an extended version of residual neural networks with fixed parameters. To demonstrate the applicability of our neural networks, we develop a self-learning hybrid Monte Carlo algorithm in the context of two-color QCD, yielding outcomes consistent with those from the conventional hybrid Monte Carlo approach.

Yuki Nagai

Gauge covariant neural network for quarks and gluons

We argue that the axionic domain-wall with a QCD bias may be incompatible with the NANOGrav 15-year data on a stochastic gravitational wave (GW) background, when the domain wall network collapses in the hot-quantum chromodynamics (QCD) induced local <math><mi>C</mi><mi>P</mi></math>-odd domain. This is due to the drastic suppression of the QCD bias set by the QCD topological susceptibility in the presence of the <math><mi>C</mi><mi>P</mi></math>-odd domain with nonzero <math><mi>θ</mi></math> parameter of order one which the QCD sphaleron could generate. We quantify the effect on the GW signals by working on a low-energy effective model of Nambu-Jona-Lasinio type in the mean field approximation. We find that only at <math><mi>θ</mi><mo>=</mo><mi>π</mi></math>, the QCD bias tends to get significantly large enough due to the criticality of the thermal <math><mi>C</mi><mi>P</mi></math> restoration, which would, however, give too big signal strengths to be consistent with the NANOGrav 15-year data and would also be subject to the strength of the phase transition at the criticality.