Supported in part by the National Natural Science Foundation of China (12235016, 12221005, 11725523, 11735007), the Strategic Priority Research Program of Chinese Academy of Sciences (XDB34030000, XDPB15), start-up funding from the University of Chinese Academy of Sciences (UCAS), and the Fundamental Research Funds for the Central Universities

^{3}and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd

The quark anomalous magnetic moment (AMM) is dynamically generated through spontaneous chiral symmetry breaking. It has been revealed that, even though its exact form is still unknown, the quark AMM is essential to exploring quark matter properties and QCD phase structure under external magnetic fields. In this study, we take three different forms of the quark AMM and investigate its influence on the chiral phase transition under a magnetic field. In general, a negative (positive) quark AMM acts as a magnetic-catalyzer (magnetic-inhibitor) for chiral symmetry breaking. It is found that a constant quark AMM drives an unexpected 1st order chiral phase transition, a quark AMM proportional to the chiral condensate flips the sign on the chiral condensate, and a quark AMM proportional to the square of the chiral condensate suppresses the magnetic enhancement in the chiral condensate at finite temperatures while retaining the chiral crossover phase transition. We also evaluate the intrinsic temperature dependence of the effective AMM form by fitting the effective model result of the chiral condensate to lattice QCD data, which may have a nontrivial correlation with the chiral phase transition.

Article funded by SCOAP^{3}

Exploring the properties of magnetized hot/dense quark matter is a subject of great interest in high energy nuclear physics relevant to neutron stars and quark-gluon plasma created in non-central relativistic heavy ion collisions. In such a thermomagnetic system, striking phenomena emerge, which are expected to introduce a new aspect of the nonperturbative feature of QCD. In particular, the influence of an external magnetic field on the QCD phase transition is an important phenomenon in understanding quark matter under extreme conditions.

The nonperturbative phenomenon of dynamical chiral symmetry breaking is crucially affected by external magnetic fields in hot/dense QCD matter. At low temperatures, when chiral symmetry is broken, the magnetic field enhances the chiral condensate, which acts as a catalyzer for spontaneous chiral symmetry breaking. This magnetic catalysis (MC) behavior has been observed in the Nambu-Jona-Lasinio (NJL) model [

Effective model analyses provide a clear interpretation of MC: magnetic dimensional reduction induces the enhancement of chiral symmetry breaking, and then MC is realized in the vacuum. In contrast to the case at low temperatures, inverse MC (IMC) arises at approximately the chiral phase transition temperature, which has been observed by lattice QCD simulations [

Recently, the anomalous magnetic moment (AMM) of quarks has attracted considerable interest for offering new insight into QCD matter under a magnetic field. In the perturbative framework of massless QCD, the AMM contribution for quarks is prohibited owing to the presence of chiral symmetry. However, when chiral symmetry is dynamically broken in the low energy regime of QCD, the quarks possess AMM terms. It has been shown in [

Although the AMM of quarks is expected to be a key factor in investigating thermomagnetic QCD matter, its expression remains obscure in the low energy regime of QCD. In vacuum, the AMM of quarks is conventionally treated as a constant value, and the influence of the constant AMM on the chiral phase transition has been investigated [

In this study, we explore the effective form of the quark AMM, which can properly describe thermomagnetic QCD matter. Considering the uncertainty of the explicit expression of the quark AMM, we make assumptions about its effective form. First, we regard the AMM as a constant value for the sake of simplicity. Next, we suppose that the AMM depends on the scalar meson field serving as the chiral order parameter. We deduce the effective form of the AMM from the findings in [

This paper is organized as follows. In Sec. II, we introduce the NJL model, considering the effective interaction of the quark AMM, and assume several forms of the quark AMM. Then, we evaluate the influence of the quark AMM on the chiral phase transition under a constant magnetic field and compare the NJL results with the lattice QCD observations in Sec. III. Finally, a summary and discussion are given in Sec. IV.

The regular NJL model consists of four-quark point interactions, which can describe chiral symmetry breaking in vacuum and symmetry restoration at finite-temperatures. As a minimal extension, we work on the NJL model involving the AMM term to explore the effective form of the quark AMM and its influence on the chiral phase transition. We suppose that the chiral condensate and quark AMM take the isospin symmetric form to simply consider the correlation between spontaneous chiral symmetry breaking and the AMM contribution.

In this section, we first briefly introduce the NJL framework. Then, we show several forms of the quark AMM in the NJL-model description.

The two-flavor NJL model involving the AMM term is written as

where

Introducing the auxiliary scalar, pseodoscalar fields

where

where

Here, ^{
①
}:

where Λ is the ultraviolet momentum cutoff.

The expectation value of the auxiliary scalar field

Indeed,

As shown in the NJL Lagrangian in Eq. (1), the AMM interaction term for quarks can be expressed as

In the Bethe-Salpeter approach for the quark-photon vertex without external magnetic field correction, it has been argued in [

Though the explicit form of

Furthermore, when we consider quark matter under substantial magnetic fields, the dynamical generation of the quark AMM becomes more intricate owing to the magnetic field correlation to chiral symmetry breaking. Indeed, the external magnetic field induces the dynamical generation of the AMM, which can be called the magnetic-dependent AMM. In the NJL model with tensor interactions [

In addition, we also raise an alternative possibility for the expression of the magnetic-dependent AMM. In the QED framework, it was found in [

Given the above facts, we consider the following three cases separately to find the effective form of the quark AMM in the low energy regime of QCD^{
②
}:

(a)

(b)

(c)

where

In the case of (a), we assume that the AMM

Note that the NJL model including the magnetic-dependent AMM has previously been discussed to study the AMM contribution in meson masses and magnetic susceptibility [

In this study, we investigate the influence of the quark AMM on the chiral phase transition to find the effective form of

In this section, we numerically evaluate the AMM dependence on the chiral condensate (quark condensate) based on the NJL model. Comparing the (subtracted) quark condensate with recent lattice QCD data [^{
③
}, we restrict the effective form of the quark AMM in the low energy dynamics of QCD under the magnetic field and discuss the nontrivial correlation between the quark AMM interaction and chiral phase transition.

The model parameters are fixed as

This NJL model experiences a chiral crossover at

Below, we investigate the influence of the AMM on the chiral condensate by taking the following three forms of the quark AMM separately: (a)

In ^{
④
}.

(color online) Constant quark AMM effect on the chiral condensate at

In the absence of the AMM, the chiral condensate

The constant AMM unexpectedly triggers the first order phase transition even in the zero temperature system (

Panel (a) of

Subsequently, we consider the case of a positive constant quark AMM,

Incidentally, the behavior of the phase transition can be directly observed from the effective potential in Eq. (3). Panel (b) of ^{
⑤
}

Next, we move onto the thermal system.

(color online) Chiral phase transition under external magnetic fields with a constant quark AMM in the case of

As mentioned above, in the case of zero-temperature, the small-constant AMM corresponding to

(color online) Similar to

Note that the chiral phase transitions including the constant AMM effect at finite temperature have already been discussed in the NJL [

Owing to the presence of the constant AMM, the phase structure becomes rich. However, the result from lattice QCD simulations in a thermomagnetic system yields chiral crossover [

In this subsection, we consider the magnetic-dependent quark AMM proportional to the chiral condensate:

(color online) Similar to

For

Note that the magnetic-dependent AMM for

We also evaluate the thermal effect on the chiral condensate including the magnetic-dependent AMM effect. As shown in panel (a) of

(color online) (a) Thermal effect on the chiral condensate as a function of the magnetic field for

For

(color online) (a) Thermal effect on the chiral condensate as a function of the magnetic field for

Although the flip in the sign of the chiral condensate may possibly induce intriguing phenomena in meson dynamics, the flipped-chiral condensate (quark condensate) has not been observed in lattice QCD simulations [

Now, we take the magnetic-dependent AMM for ^{
⑥
}.

(color online) Similar to

However, when the AMM parameter ^{
⑦
}. Consequently, in panel (a) of Fig. (7), the chiral condensate in the case of

We next discuss the vacuum stability at finite temperatures, as well as the chiral condensate. Indeed, for

(color online) (a) Thermal effect on the chiral condensate for

For

(color online) Thermal effect on vacuum instability for

In the case of the magnetic-dependent AMM proportional (^{
⑧
}, the effective potential maintains the stable structure while the unexpected behavior does not emerge in the chiral condensate, such as the induced-first order phase transition and the flip in the sign of

Although the proposed extra mechanisms, such as neutral pion fluctuation [

The quark condensate (chiral condensate) involves ultraviolet divergence at the zero-temperature part and should be renormalized as a finite quantity. To eliminate the divergence arising in the quark condensate under the magnetic field, we use the following dimensionless quantity of the subtracted quark condensate:

(color online) Comparison of the subtracted quark condensate between the NJL result and lattice QCD data at

(color online) Comparison of the subtracted quark condensate of the NJL result and lattice QCD data [

In the previous subsections, we attempt to reveal the effective form of the quark AMM through a comparison of the order of the chiral phase transition observed in the lattice QCD simulation. We find that the quark AMM depending on the square of the chiral condensate is an applicable effective form to describe the phase transition order of thermomagnetic QCD. Although its form potentially suppresses the magnetic enhancement in the chiral condensate as shown in

(color online) Temperature dependence of the AMM parameter

Using the temperature-dependent AMM parameter ^{
⑨
}. This figure shows that the pseudocritical temperature decreases with the development of the magnetic field, and IMC is surely provided by the NJL analysis with the parameter ^{
⑩
} This indicates that the magnetic-dependent AMM with the intrinsic temperature dependence (

(color online) Chiral phase diagram under a constant magnetic field based on the NJL model with the fitting-parameter

In this study, we explore the effective form of the AMM in thermomagnetic QCD vacuum. Employing the NJL model with the effective interaction of the quark AMM, we discuss the influence of the AMM on the chiral condensate in the following three forms: (a)

Our findings in the three forms can be summarized as follows:

(a) For

(b) For

(c) For

Our findings indicate that the magnetic-dependent AMM form

As mentioned above, the subtracted quark condensate involving the AMM contribution can qualitatively fit the lattice observation but does not perfectly agree with the lattice data, especially at high temperatures. The IMC behavior observed in the lattice QCD simulations cannot be completely explained by only the effective quark AMM (

Before concluding this study, we comment on the implications of the magnetic-dependent AMM. A recent lattice QCD simulation also exhibited the magnetic effect on the meson masses at zero-temperature [

Furthermore, magnetic susceptibility in the thermomagnetic QCD vacuum has recently been revealed via lattice QCD simulation [

Moreover, magnetic susceptibility is closely linked with the spin polarization condensate [

Though an extra mechanism, such as the use of a coupling

The predicted magnetic-dependent AMM interaction could be applied to high-dense matter physics with magnetic fields. In particular, a strong magnetic field is generated in neutron stars or magnetars. The mechanism of the the generation of a strong magnetic field has not yet been clarified. In Refs. [

In this study, we suppose that the chiral condensate and AMM take the isospin symmetric form:

Here, we discuss the regularization scheme dependence of the first order phase transition induced by a constant AMM at zero temperature. As stated in Sec. III.A, the induced-first order phase transition is actually observed in the NJL model based on both the Lorentzian and Woods-Saxon type form factors [

The Pauli-Villars regularization scheme has previously been applied to avoid the unphysical oscillation of the chiral condensate in two-flavor NJL analysis including a constant AMM [

In

(color online) Similar to

The magnetic dependence of observables within the NJL model significantly depends on an ultraviolet-regularization scheme. By using a magnetic field dependent regulator, oscillations occur in the chiral condensate as a function of the magnetic field, which contradict with the lattice observation. To avoid the nonphysical oscillation, the Magnetic Field Independent Regularization (MFIR) has been developed at zero temperature and zero zero baryonic density in [

One may think that more higher order terms of

In [

The regularization scheme dependence has also been addressed in the two-flavor gauged NJL model including the one-loop contribution for the photon-quark-antiquark vertex function, where the quark AMM is dynamically generated through the coupled gap equations among

One would think that the choice of the regularization scheme may also be relevant to the induced-first order phase transition. To address the regularization scheme dependence on the phase transition, we also take the Pauli-Villars regularization scheme and investigate the constant AMM effect on the phase transition in Appendix A. We find that the chiral condensate certainly undergoes the jump even in the case of the Pauli-Villars regularization scheme. This implies that the choice of the regularization scheme would be irrelevant to the induced-first order phase transition.

For

For

For the constrained parameter space

To evaluate the psuedocrtical temperature from the inflection point of the quark condensate with respect to temperature, we have used the alternative expression of the subtracted quark condensate,

The tendency of the inverse magnetic catalysis is more prominent in the other lattice observation [