Correlation $ \chi^{BQ}_{11} $ and quadratic fluctuations $ \chi^B_2,\ \chi^Q_2,\ \chi^T_2 $ of baryon number $\textit{B}$, electric charge $\textit{Q,}$ and temperature $\textit{T}$ are investigated in a two-flavor Polyakov loop extended Nambu-Jona-Lasinio (PNJL) model at finite temperature and magnetic field. The inverse magnetic catalysis (IMC) effect is introduced through magnetic-field-dependent parameters $ G(eB) $ and $ T_0(eB) $ , and we compare the results in scenarios with and without the IMC effect. Under a nonvanishing magnetic field, correlation $ \chi^{BQ}_{11} $ and fluctuations $ \chi^B_2,\ \chi^Q_2,\ \chi^T_2 $ increase with temperature and then exhibit a peak around the pseudocritical temperatures of chiral restoration and deconfinement phase transitions in the cases with and without the IMC effect. The correlation and fluctuations along the phase transition line under an external magnetic field are characterized by scaled correlation ${\hat {\chi}}_{11}^{BQ}={\chi_{11}^{BQ}(eB,T_{pc}^c(eB))}/{\chi_{11}^{BQ}(eB=0,T_{pc}^c(eB=0))}$ and scaled fluctuations ${\hat {\chi}}_2^{B(Q,T)}={\chi_2^{B(Q,T)}(eB,T_{pc}^c(eB))}/{\chi_2^{B(Q,T)}(eB=0,T_{pc}^c(eB=0))}$ at pseudocritical temperature $ T_{pc}^c $ of chiral restoration phase transition. $ {\hat {\chi}}_{11}^{BQ},\ {\hat {\chi}}_2^{B} $ , and $ {\hat {\chi}}_2^{Q} $ increase with the magnetic field, and the inclusion of the IMC effect enhances their values somewhat. However, $ {\hat {\chi}}_2^{T} $ is altered by the IMC effect. Without the IMC effect, $ {\hat \chi}^T_2 $ increases slightly and then decreases with the magnetic field. Considering the IMC effect using $ G(eB) $ , $ {\hat \chi}^T_2 $ monotonically increases with the magnetic field, and that using $ T_0(eB) $ is a nonmonotonic function of the magnetic field.

China

Shijun Mao

Correlations and fluctuations in a magnetized PNJL model with and without the inverse magnetic catalysis effect * Supported by the National Natural Science Foundation of China (12275204)

We investigate the chiral restoration and deconfinement phase transitions under an external magnetic field in the frame of a Pauli-Villars regularized Polyakov loop extended Nambu-Jona-Lasinio model. A running Polyakov loop scale parameter <math><msub><mi>T</mi><mn>0</mn></msub><mo>(</mo><mi>e</mi><mi>B</mi><mo>)</mo></math> is introduced to mimic the reaction of the gluon sector to the presence of magnetic fields. It is found that a decreasing <math><msub><mi>T</mi><mn>0</mn></msub><mo>(</mo><mi>e</mi><mi>B</mi><mo>)</mo></math> with magnetic fields can realize the inverse magnetic catalysis phenomena of chiral condensates of <math><mi>u</mi></math> and <math><mi>d</mi></math> quarks, increase of Polyakov loop, and the reduction of pseudocritical temperatures of chiral restoration and deconfinement phase transitions.

Reduction of pseudocritical temperatures of chiral restoration and deconfinement phase transitions in a magnetized PNJL model

In the framework of the Nambu–Jona-Lasinio model beyond mean field approximation, the effects of pion fluctuations on (inverse) magnetic catalysis and magnetic susceptibility are studied. The negative magnetic susceptibility at low temperature is observed when contributions from both neutral and charged pions are taken into account. In the weak field approximation, it is observed that at finite temperature, the magnetic inhibition effect in the chiral limit, resulting from the difference between the transverse and longitudinal velocities of neutral pions, converts to weak magnetic catalysis when considering a nonzero current quark mass. Moreover, the magnetic catalysis is amplified by the charged pions. Therefore, no inverse magnetic catalysis is observed when considering pion fluctuations.

Jie Mei

Rui Wen

Mei Huang

Kun Xu

Magnetic catalysis and diamagnetism from pion fluctuations

Mass spectra and Mott transitions of pions <math><mo>(</mo><msup><mi>π</mi><mn>0</mn></msup><mo>,</mo><msup><mi>π</mi><mo>±</mo></msup><mo>)</mo></math> at finite temperature and magnetic field are investigated in a two-flavor Nambu–Jona-Lasinio model, and we focus on the inverse magnetic catalysis (IMC) effect and current quark mass (CQM) effect. Due to the dimension reduction of the constituent quarks, the pion masses jump at their Mott transitions, which is independent of the IMC effect and CQM effect. We consider the IMC effect by using a magnetic-field-dependent coupling constant, which is a monotonic decreasing function of magnetic field. With the IMC effect, the Mott transition temperature of <math><msup><mi>π</mi><mn>0</mn></msup></math> mesons, <math><msubsup><mi>T</mi><mi>m</mi><mn>0</mn></msubsup></math>, is a monotonic decreasing function of magnetic field. For charged pions, <math><msup><mi>π</mi><mo>±</mo></msup></math>, the Mott transition temperature <math><msubsup><mi>T</mi><mi>m</mi><mo>+</mo></msubsup></math> increases quickly in the weak magnetic field region and then decreases with increasing magnetic field, which is accompanied with some oscillations. Comparing with the case without IMC, <math><msubsup><mi>T</mi><mi>m</mi><mn>0</mn></msubsup></math> and <math><msubsup><mi>T</mi><mi>m</mi><mo>+</mo></msubsup></math> are lower when including the IMC effect. The CQM effect is considered by varying parameter <math><msub><mi>m</mi><mn>0</mn></msub></math> in the nonchiral limit. For <math><msup><mi>π</mi><mn>0</mn></msup></math> mesons, <math><msubsup><mi>T</mi><mi>m</mi><mn>0</mn></msubsup></math> is not a monotonic function of magnetic field with low <math><msub><mi>m</mi><mn>0</mn></msub></math>, but it is a monotonic decreasing function with larger <math><msub><mi>m</mi><mn>0</mn></msub></math>. In the weak magnetic field region, <math><msubsup><mi>T</mi><mi>m</mi><mn>0</mn></msubsup></math> is higher for larger <math><msub><mi>m</mi><mn>0</mn></msub></math>, but in the strong magnetic field region, it is lower for larger <math><msub><mi>m</mi><mn>0</mn></msub></math>. For <math><msup><mi>π</mi><mo>+</mo></msup></math> mesons, <math><msubsup><mi>T</mi><mi>m</mi><mo>+</mo></msubsup></math> is only quantitatively modified by the current quark mass effect, and it becomes higher with larger <math><msub><mi>m</mi><mn>0</mn></msub></math>.

Luyang Li

Inverse magnetic catalysis effect and current quark mass effect on mass spectra and Mott transitions of pions under external magnetic field

The mass spectra of neutral mesons <math><msub><mi>K</mi><mn>0</mn></msub><mo>,</mo><msub><mi>π</mi><mn>0</mn></msub><mo>,</mo><mi>η</mi><mo>,</mo><msup><mi>η</mi><mo>′</mo></msup></math> on the temperature-quark chemical potential (<math><mrow><mi>T</mi><mtext>−</mtext><mi>μ</mi></mrow></math>) plane under a constant magnetic field is investigated in the <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>3</mn><mo>)</mo></math> Nambu-Jona-Lasinio model. As a Goldstone boson of chiral symmetry breaking, the mass of <math><msub><mi>K</mi><mn>0</mn></msub></math> meson generally increases with temperature and/or quark chemical potential, and we observe two kinds of mass jumps of the <math><msub><mi>K</mi><mn>0</mn></msub></math> meson in medium, one of which is induced by the mass jump of constituent quarks and the other by the magnetic field. Due to the breaking of isospin symmetry between <math><mi>u</mi></math> and <math><mi>d</mi></math> quarks in the presence of magnetic fields, a flavorless neutral meson mixing involving the <math><msub><mi>π</mi><mn>0</mn></msub><mo>−</mo><mi>η</mi><mo>−</mo><msup><mi>η</mi><mo>′</mo></msup></math> mesons occurs (in the isospin symmetric case the mixing involves only <math><mi>η</mi><mo>−</mo><msup><mi>η</mi><mo>′</mo></msup></math>) and a rich structure is found in the mass spectra of these mesons. For instance, the <math><msub><mi>π</mi><mn>0</mn></msub></math> mass is influenced by the strange quark. A change in the slope of the <math><mi>μ</mi></math> dependency of the <math><msub><mi>π</mi><mn>0</mn></msub></math> mass at high <math><mi>μ</mi></math> and vanishing T can be associated with the crossing of the threshold of two times of strange quark mass. In the finite T case, this behavior is changed into the appearance of a critical chemical potential at which a jump over the threshold of <math><mn>2</mn><msub><mi>m</mi><mi>s</mi></msub></math> occurs. The mass ordering of <math><msub><mi>π</mi><mn>0</mn></msub><mo>,</mo><mi>η</mi><mo>,</mo><msup><mi>η</mi><mo>′</mo></msup></math> mesons varies in medium, due to their mass jumps, which are induced by the mass jump of constituent quarks or the magnetic field.

Tao Xia

Mass spectra of neutral mesons <math><msub><mi>K</mi><mn>0</mn></msub><mo>,</mo><msub><mi>π</mi><mn>0</mn></msub><mo>,</mo><mi>η</mi><mo>,</mo><msup><mi>η</mi><mo>′</mo></msup></math> at finite magnetic field, temperature and quark chemical potential

The effect of quark anomalous magnetic moment (AMM) to chiral restoration and deconfinement phase transitions under magnetic fields is investigated in a Pauli-Villars regularized Polyakov-extended Nambu–Jona-Lasinio model. A linear-in-<math><mi>B</mi></math> term for quark anomalous magnetic moment is introduced to the Lagrangian density of our model, and it plays the role of inverse catalysis to the phase transitions. With fixed magnetic field, the critical temperature decreases with quark AMM. When fixing quark AMM, the critical temperature increases with magnetic field for a small quark AMM, but decreases with magnetic field for a large quark AMM. The critical temperature of chiral restoration and deconfinement phase transitions is determined by the two competing factors, the catalysis effect of magnetic field and inverse catalysis of quark anomalous magnetic moment.

Inverse catalysis effect of the quark anomalous magnetic moment to chiral restoration and deconfinement phase transitions

We study the quark-antiquark scattering phase shift and meson spectral function in the pion superfluid described by the Nambu-Jona-Lasinio model. Meson mixing in the pion superfluid dramatically changes the full scattering phase shift and significantly broadens the spectral function of some collective modes.

Jin Hu

Quark-antiquark scattering phase shift and meson spectral function in pion superfluid Supported by the NSFC (11775165) and Fundamental Research Funds for the Central Universities

We study how the mass and magnetic moment of the quarks are dynamically generated in nonequilibrium quark matter. We derive the equal-time transport and constraint equations for the quark Wigner function in a magnetized quark model and solve them in the semi-classical expansion. The quark mass and magnetic moment are self-consistently coupled to the Wigner function and controlled by the kinetic equations. While the quark mass is dynamically generated at the classical level, the quark magnetic moment is a pure quantum effect, induced by the quark spin interaction with the external magnetic field.

Germany

Dirk H. Rischke

Dynamically generated magnetic moment in the Wigner-function formalism

We study inverse magnetic catalysis in the Nambu–Jona-Lasinio model beyond mean field approximation. The feed-down from mesons to quarks is embedded in an effective coupling constant at finite temperature and magnetic field. While the magnetic catalysis is still the dominant effect at low temperature, the meson dressed quark mass drops down with increasing magnetic field at high temperature due to the dimension reduction of the Goldstone mode in the Pauli–Villars regularization scheme.