Magnetized QCD Matter

Abstract

Over the past decade, the study of QCD matter under the influence of magnetic fields has received widespread attention, yet several unresolved puzzles remain, including: inverse magnetic catalysis (IMC), diamagnetism (negative magnetic susceptibility) at low temperature and the nonmonotonic behavior of charged pion mass under magnetic fields. We present a brief overview of these unresolved challenges and discuss potential solutions.

1. Introduction

Strong magnetic fields exist in the early universe and neutron stars and can be created in heavy ion collisions. Primordial magnetic fields with magnitude ${10}^{20–23}$ Gauss can be generated in the early universe through mechanisms driven by a chiral anomaly [1,2]. Neutron stars, among the most magnetized objects in the universe, exhibit magnetic field strengths typically ranging from ${10}^{11}$ to ${10}^{13}$ Gauss, while magnetars, a special class of neutron stars, can possess even stronger fields, reaching ${10}^{14}$ to ${10}^{15}$ Gauss. A magnetic field with a magnitude of ${10}^{18–19}$ Gauss can be generated through non-central heavy ion collisions [3,4]. As a result, the study of Quantum Chromodynamic (QCD) matter under the influence of an external magnetic field has garnered wide interest.

QCD matter under an external magnetic field exhibits a range of intriguing phenomena. These include: (1) the Chiral Magnetic Effect (CME), describing the generation of an electric current along the direction of the applied magnetic field due to the chirality imbalance [5,6,7,8]; (2) magnetic catalysis (MC), characterized by an enhancement in chiral symmetry breaking with an increasing magnetic field [9,10,11]; (3) inverse magnetic catalysis (IMC), where the critical temperature of the chiral phase transition decreases as the magnetic field increases [12,13,14]; (4) diamagnetism (negative magnetic susceptibility) at low temperature and paramagnetism (positive magnetic susceptibility) at high temperature [15]; and (5) the nonmonotonic behavior of charged pion mass under a magnetic field [16].

Some of the above properties, e.g., inverse magnetic catalysis (IMC), diamagnetism (negative magnetic susceptibility) at low temperature and the nonmonotonic behavior of charged pion mass under a magnetic field, were not anticipated within effective theories, such as the Nambu–Jona-Lasinio model. In this article, we provide a brief overview of the progress made in understanding these puzzles from the perspective of effective theories.

It is worth mentioning that there are other reviews on magnetized QCD matter such as Refs. [17,18,19,20,21] and the references therein.

2. Puzzle-1: Diamagnetism at Low Temperature

Magnetic susceptibility is a fundamental property of a material, and it measures the amount of magnetization of a material there is in response to an applied external magnetic field. Following [22,23], magnetic susceptibility $\chi$ is defined as follows:

$$\chi \left(T\right)=-{\displaystyle \frac{{\partial }^2\mathrm{\Omega }(T,eB)}{\partial {\left(eB\right)}^2}}{|}_{eB=0}\left(T\right)-\chi (T=0),$$

(1)

where $\mathrm{\Omega }$ is the grand canonical potential, and $\chi (T=0)=0$ is chosen to eliminate unobservable vacuum magnetic susceptibility.

The lattice QCD calculation in [23,24,25,26,27,28] showed that at low temperature, magnetic susceptibility is negative; thus the magnetized QCD matter exhibits diamagnetism, and at high temperature, magnetic susceptibility is positive; thus the magnetized QCD matter exhibits paramagnetism.

Paramagnetism is straightforward to understand, as quarks tend to align their spins with the magnetic field, leading to polarization. However diamagnetism at low temperature is different from the effective model predictions in the Nambu–Jona-Lasinio (NJL) model and linear-$\sigma$ model in mean-field approximation [29,30]. When meson fluctuations are considered, diamagnetism at low temperature can be obtained from pion contributions [31,32,33].

For example, in [34], considering the two-flavor NJL model beyond mean-field approximation. The Lagrangian of the SU(2) NJL model in the presence of a uniform magnetic field takes the form of

$$\begin{array}{c}\hfill \mathcal{L}=\overline{\psi }\left(i{\gamma }^{\mu }{D}_{\mu }-\widehat{m}\right)\psi +G\left[{\left(\overline{\psi }\psi \right)}^2+{\left(\overline{\psi }i{\gamma }_5\overrightarrow{\tau }\psi \right)}^2\right].\end{array}$$

(2)

Here $\psi ={(u,d)}^T$ is a two-flavor quark field, $\widehat{m}=\mathrm{diag}(m_u,m_d)$ with $m_u=m_d=m_0$ is the quark current mass matrix, and ${\tau }_i$ is the i-th component of Pauli matrices and the covariant derivative $D_{\mu }={\partial }_{\mu }-iQ{A}_{\mu }$ coupling quarks with electric charge $Q=\mathrm{diag}(2/3e,-1/3e)$ to a gauge field $\mathbf{B}=\nabla \times \mathbf{A}$ with $A_{\mu }=(0,0,B{x}_1,0)$ in the Landau gauge.

By taking into account the meson fluctuations, thermodynamic potential [35] has the form of

$$\begin{array}{c}\hfill \mathrm{\Omega }={\displaystyle \frac{{(m_q-m_0)}^2}{4G}}+{\mathrm{\Omega }}_q+\sum _M{\mathrm{\Omega }}_M,\end{array}$$

(3)

where $m_q$ is the dynamically generated constituent quark mass including the contribution from chiral condensation, and M is for neutral ${\pi }^0$ and charged pions ${\pi }^{\pm }$. $\mathrm{\Omega }$ includes the contribution from quarks

$$\begin{array}{c}\hfill {\mathrm{\Omega }}_q(T,B)={\mathrm{Tr}}_{\{c,f,s,x\}}ln\left({\displaystyle \frac{1}{T}}{S}^{-1}(x,x)\right)\end{array}$$

(4)

and mesons (mainly pions)

$$\begin{array}{c}\hfill {\mathrm{\Omega }}_{\pi }=\int {\displaystyle \frac{d^{3}k}{{\left(2\pi \right)}^3}}\left[{\displaystyle \frac{E_{\pi }}{2}}+Tln\left(1-e^{-E_{\pi }/T}\right)\right],\end{array}$$

(5)

with the energy dispersion relation of pions given by the following:

$$\begin{array}{c}\hfill E_{\pi }=\sqrt{m_{\pi ,pole}^2+{(\mathbf{v}·\mathbf{k})}^2}.\end{array}$$

(6)

Here $\mathbf{v}$ is the propagating velocity [36,37,38,39] or the ratio of pole mass over screening mass, and its i-th component has the expression of

$$\begin{array}{c}\hfill v_i={\displaystyle \frac{m_{M,pole,i}}{m_{M,scr,i}}}\end{array}$$

(7)

where the pole mass $m_{pole}$ (static solution, setting $q_1=q_2=q_3=0$) defined through $1-2G{\mathrm{\Pi }}_M(q_0^2=m_{pole}^2,0)=0$ and the screening masses $m_{scr,i}$ in $q^i$ (setting $q_0=0$, and $q_j=0$ for $j\ne i$) through $1-2G{\mathrm{\Pi }}_M(0,q_i^2=-m_{scr,i}^2)=0$, with ${\mathrm{\Pi }}_M$ being the one quark loop polarization function of pions.

The energy dispersion relation for a neutral pion is

$$\begin{array}{c}\hfill E_{{\pi }_0}=\sqrt{m_{pole,{\pi }_0}^2+v_{\perp }^2{k}_{\perp }^2+v_{\Vert }^2{k}_3^2},\end{array}$$

(8)

where $v_{\Vert },v_{\perp }$ represents the propagating velocity in the parallel and transverse directions, respectively, and the energy dispersion relation of charged pions is given by

$$\begin{array}{c}\hfill E_{{\pi }_{\pm }}=\sqrt{m_{{\pi }_{\pm }}^2+v_{\perp }^2(2n+1)\left|eB\right|+v_{\Vert }^2{k}_3^2}.\end{array}$$

(9)

It was obtained in [34] that the neutral pion mass decreases invisibly with the magnetic field, and charged pion mass increases with the magnetic field. It is easy to understand the case for neutral pions, which are neutral but composite particles made of quark–antiquarks, and under a magnetic field, the polarization of quarks will induce a tiny change in the neutral pion mass under a magnetic field. When considering the effects of both neutral and charged pion fluctuations within the framework of the NJL model in [34], it is observed in Figure 1 that neutral pion fluctuations do not contribute, while charged pion fluctuations are responsible for the negative magnetic susceptibility (diamagnetism) at low temperatures. The quark’s contribution is dominant for positive magnetic susceptibility (paramagnetism) in the high-temperature region. This result is qualitatively in agreement with the lattice result.

Therefore, puzzle-1 on diamagnetism at low temperature can be understood as follows: Below the critical temperature, pions have a dominant degree of freedom, and charged spin-0 pion fluctuations generate a magnetic moment in the opposite direction due to their orbital rotation under the influence of the Lorentz force in a magnetic field, thus exhibiting Landau diamagnetism. As temperature increases, the dominant contribution comes from the quark’s degree of freedom, leading to the prevalence of Pauli paramagnetism.

3. Puzzle-2: Inverse Magnetic Catalysis

The calculations in the effective chiral models, for example, the NJL model, showed that chiral condensation increases with the magnetic field in the vacuum, and the enhancement in chiral symmetry breaking by the magnetic field is called the MC [9,10,11], which can be explained by magnetic dimensional reduction.

It is naturally expected that the MC effect should cause an increase in the chiral critical temperature with the magnetic field. However, the lattice results in [12,13,14] showed the opposite results, i.e., the chiral critical temperature decreased with the magnetic field, which is called the IMC effect around the critical temperature.

The IMC effect has been extensively studied over the past decade [23,24,25,26,27,28,29,30,31,32,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63]. A deep understanding of an extra mechanism and the interplay between QCD and the electromagnetic field is needed to realize the IMC effect, and various efforts have been made to understand IMC, including studies of the following: (1) magnetic inhibition induced by neutral pion fluctuations [64]; (2) chirality imbalance between left-handed and right-handed quarks induced by a sphaleron transition [40] or instanton–anti-instanton molecule [41]; (3) the anomalous magnetic moment (AMM) of quarks [29,30,44]; and (4) the running coupling constant changing with the magnetic field [65,66].

3.1. Magnetic Inhibition Induced by Neutral Pion Fluctuations

It is believed in [64,67,68] that neutral pion fluctuations are responsible for the IMC effect in the chiral limit. The authors of ref. [34] carefully investigated the magnetic inhibition induced by neutral and charged pion fluctuations in the framework of the NJL model beyond mean-field approximation. The results of pion fluctuations on $T_c$ are summarized in Figure 2. It was shown that for finite current quark mass or explicit chiral symmetry breaking, in the case of only neutral pion fluctuations, magnetic inhibition is not enough to achieve the IMC effect, and the critical temperature still increases with the magnetic field, and the result is almost the same as that in mean-field approximation. When considering the contribution from charged pion fluctuations, the critical temperature increases more quickly than that with only neutral pion contribution.

3.2. IMC by Chirality Imbalance

IMC can be easily realized if there is a chirality imbalance as shown in Figure 3a between left-handed and right-handed quarks induced by a sphaleron transition [40] with an extra contribution in the Lagrangian

$${\mathcal{L}}_{\mu 5}={\mu }_5\overline{\psi }{\gamma }^0{\gamma }^5\psi$$

(10)

or by instanton–anti-instanton molecule-induced effective repulsive axial vector interaction in the Lagrangian as in [41]:

$${\mathcal{L}}_{\mathrm{A}}=G_A{\left(\overline{\psi }{\gamma }^{\mu }{\gamma }^5\psi \right)}^2,$$

(11)

which spontaneously induces an effective chiral chemical potential

$${\mu }_5=-2G_A\langle \overline{\psi }{\gamma }^0{\gamma }^5\psi \rangle$$

(12)

and modifies the energy dispersion relation

$${\omega }_{sk}=\sqrt{{\sigma }^2+{\left[|\overrightarrow{p}|+s{\mu }_5\mathrm{sgn}\left(p_z\right)\right]}^2}$$

(13)

with $\sigma$ being chiral condensation and spin factors $s=\pm 1$, ${\overrightarrow{p}}^2=p_z^2+2\left|q_{f}B\right|k$, and $k=0,1,2,\dots$ is a non-negative integer number labeling the Landau levels.

3.3. IMC by Quarks’ Anomalous Magnetic Moment

The IMC effect can also be realized by taking into account the anomalous magnetic moment of quarks [29,30,44] as shown in Figure 3b. The anomalous magnetic moments (AMMs) of quarks can be generated by spontaneous chiral symmetry breaking [69,70,71]. The effect of quarks’ AMM $\kappa$ in a magnetic field can be introduced through effective interaction

$${\mathcal{L}}_{\mathrm{AMM}}={\displaystyle \frac{1}{2}}\overline{\psi }\left({\kappa }_f{q}_f{F}_{\mu \nu }{\sigma }^{\mu \nu }\right)\psi$$

(14)

where ${\sigma }^{\mu \nu }={\displaystyle \frac{i}{2}}[{\gamma }^{\mu },{\gamma }^{\nu }]$, and the electromagnetic charge for quark flavor $q_u=+2e/3$ and $q_d=-e/3$. Correspondingly, the energy dispersion relation of up- and down-quarks in the external magnetic field with the quark AMM is modified as

$$\begin{array}{c}\hfill E_{q_f}^{(l,s)}=\sqrt{p_3^2+{\left[{\left\{|q_{f}B|(2l+1-s{\xi }_f)+M^2\right\}}^{1/2}-s{\kappa }_f{q}_{f}B\right]}^2}\end{array}$$

(15)

where l is the Landau level, and $s=\pm 1$ is the spin-up/down of quarks.

The IMC effect at finite temperature was observed in Refs. [29,30] and is shown in Figure 4a. Quantitative quarks’ AMM parameter ${\kappa }_f\sim {\sigma }_f^2$ proportional to the square of the chiral condensate was subtracted in [44] from the lattice results in [16]. It is noticed that here, the quark AMM is proportional to ${\sigma }^2$, and this naturally arises for a constituent quark in the process of spontaneous chiral symmetry breaking and dynamical mass generation.

3.4. IMC by Running Coupling Constant

IMC can also be realized when considering a running coupling constant that changes with the magnetic field [49,65,66]

$${\displaystyle \frac{G_S\left(\xi \right)}{G_S\left(0\right)}}={\displaystyle \frac{1+a{\xi }^2+b{\xi }^3}{1+c{\xi }^2+d{\xi }^4}},$$

(16)

where $G_S\left(0\right)=G_S$, $\xi ={\displaystyle \frac{eB}{{\mathrm{\Lambda }}_{QCD}^2}}$ and ${\mathrm{\Lambda }}_{QCD}=300$ MeV. The fitted parameters are chosen as $a=0.014056, b=0.00532074, c=0.0281766, d=0.00161148$ in [49] from lattice QCD calculations to produce the IMC effect, as shown in Figure 4b.

4. Puzzle-3: Nonmonotonic Charged Pion Mass

Hadron properties should also change in the presence of a magnetic field. For relativistic point-like charged particles, their energy under an external magnetic field takes the form of ${\epsilon }_{n,s_z}^2\left(p_z\right)=p_z^2+(2n-2\mathrm{sgn}\left(q\right)s_z+1)\left|qB\right|+m^2$, which results in the linear rising behavior of charged pion mass $m_{{\pi }^{\pm }}^2\left(B\right)=m_{{\pi }^{\pm }}^2(B=0)+eB$, while a charge-neutral point-particle pion would not be affected by a magnetic field, and its mass remains constant.

Both model calculations and lattice QCD simulations have demonstrated that hadron properties under a magnetic field deviate from those expected for point-like particles [37,38,50,56,72,73,74,75,76,77,78,79,80]. Notably, the lattice QCD calculations in [81] revealed intriguing properties of pion masses under a magnetic field. The mass of the neutral pion decreases with increasing magnetic field strength and eventually saturates, while the mass of the charged pion exhibits nonmonotonic dependence on the magnetic field: it increases with the magnetic field up to $eB\sim 0.6{\mathrm{GeV}}^2$, after which it begins to decrease as the magnetic field continues to strengthen.

It is well-established that hadrons are not point-like particles but are composed of quarks. As a result, quark polarization due to a magnetic field is expected to influence the hadron spectrum. Within the framework of the NJL model, incorporating magnetized quark loop effects modifies the linear rising behavior of the charged pion mass as a function of the magnetic field [37,38,50,72,73,74,75,76,77,78,79].

The nonmonotonic behavior of the charged pion mass remains one of the most challenging puzzles. To date, only two studies have successfully reported this nonmonotonic behavior: one considers quarks’ AMM in [30] in the framework of the NJL model, and the other considers the vector form factor in the framework of DSE [82].

In the NJL model, mesons are represented as a quark–antiquark bound state or resonance and can be obtained from the quark–antiquark scattering amplitude by summing up quark loops through random phase approximation (RPA). The meson polarization function has the expression of

$$\begin{array}{c}\hfill {\mathrm{\Pi }}_M(x,y)=i\int {\displaystyle \frac{d^{4}k}{{\left(2\pi \right)}^4}}{\mathrm{Tr}}_{\{c,f,s\}}\left[{\mathrm{\Gamma }}_M^{\ast }S(x,y){\mathrm{\Gamma }}_{M}S(y,x)\right]\end{array}$$

(17)

where M is the meson (here, we only focus on pion) vertex

$${\mathrm{\Gamma }}_M=\left\{\begin{array}{c}i{\gamma }_5{\tau }_{+},\hspace{1em}\hfill \\ i{\gamma }_5{\tau }_{-},\hfill \\ i{\gamma }_5{\tau }_3,\hfill \end{array}\right.\begin{array}{c}M={\pi }_{+}\hfill \\ M={\pi }_{-}\hfill \\ M={\pi }_0,\hfill \end{array},$$

(18)

where ${\tau }_{\pm }=({\tau }_1\pm i{\tau }_2)/\sqrt{2}$. Here $S_f$ is the quark propagator, and translational invariance is broken by the magnetic field; thus the 4-momentum is not representative of a good quantum number. One can introduce a new set of “good” quantum numbers, i.e., the conserved Ritus momentum $\overline{p}=(p_0,0,-s_f\sqrt{2n|q_{f}B|},p_3)$ [83], with n being the quark Landau level and $s_f=\mathrm{sign}\left(q_{f}B\right)$ the quark sign factor. In the Ritus scheme, the quark propagator with flavor f in coordinate space [84] can be expressed as

$$\begin{array}{cc}\hfill S_f(x,y)& =\sum _n\int {\displaystyle \frac{d^3\tilde{p}}{{\left(2\pi \right)}^3}}{e}^{-i\tilde{p}·(x-y)}{P}_n(x_1,p_2)D_f\left(\overline{p}\right)P_n(y_1,p_2)\hfill \\ \hfill P_n(x_1,p_2)& ={\displaystyle \frac{1}{2}}\left[g_n^{s_f}(x_1,p_2)+I_n{g}_{n-1}^{s_f}(x_1,p_2)\right]+{\displaystyle \frac{i{s}_f}{2}}\left[g_n^{s_f}(x_1,p_2)-I_n{g}_{n-1}^{s_f}(x_1,p_2)\right]{\gamma }_1{\gamma }_2\hfill \\ \hfill D_f\left(\overline{p}\right)& ={\displaystyle \frac{1}{\gamma ·\overline{p}-m_q}},\hfill \end{array}$$

(19)

where the Fourier-transformed momentum is $\tilde{p}=(p_0,0,p_2,p_3)$, and $I_n=1-{\delta }_{n0}$, and $g_n^{s_f}(x_1,p_2)={\varphi }_n(x_1-s_f{p}_2/|q_{f}B\left|\right)$ is expressed by the Hermite polynomial $H_n\left(\zeta \right)$

$$\begin{array}{c}\hfill {\varphi }_n\left(\zeta \right)=(2^{n}n!\sqrt{\pi }|q_{f}B{{|}^{-1/2})}^{-1/2}{e}^{-{\zeta }^2\left|q_{f}B\right|/2}{H}_n\left(\zeta \sqrt{|q_{f}B|}\right).\end{array}$$

(20)

By inserting the quark propagator $S_f$ into the pion polarization ${\mathrm{\Pi }}_M$, one can get the pion propagator

$$\begin{array}{c}\hfill U_M\left(q\right)={\displaystyle \frac{2G}{1-2G{\mathrm{\Pi }}_M\left(q\right)}},\end{array}$$

(21)

and the pole mass $m_M$ is determined by

$$\begin{array}{c}\hfill 1-2G{\mathrm{\Pi }}_M(m_M-i{\mathrm{\Gamma }}_M/2,\mathbf{0})=0.\end{array}$$

(22)

With the above framework, one can calculate the pion mass with quark energy dispersion from any interaction in the Lagrangian and replace $m_q$ in $D_f\left(\overline{p}\right)={\displaystyle \frac{1}{\gamma ·\overline{p}-m_q}}$.

If there is one mechanism that can simultaneously explain the three aforementioned puzzles, i.e., diamagnetism, IMC and charged pion mass, it would be highly favored. As discussed in Section 2, the diamagnetism observed at low temperatures can be attributed to pion fluctuations, which can be separated. Consequently, the proposed mechanism must account for both the IMC effect and the nonmonotonic behavior of the charged pion mass. Among the four mechanisms discussed in Section 3, the running coupling constant alone fails to reproduce the nonmonotonic behavior of the charged pion mass, and the quark’s AMM can simultaneously realize IMC and the nonmonotonic behavior of the charged pion mass [30], as shown in Figure 5a, but the value of the charged pion mass is much smaller than the lattice results.

Here we discuss whether it is possible to obtain the nonmonotonic behavior of the charged pion mass from the chirality imbalance. We introduce an effective repulsive axial vector interaction in the Lagrangian as in [41]:

$${\mathcal{L}}_{\mathrm{A}}=G_A\overline{\psi }{\gamma }^{\mu }{\gamma }^5\psi ,$$

(23)

which will spontaneously induce an effective chiral chemical potential

$${\mu }_5=-2G_A\langle \overline{\psi }{\gamma }^0{\gamma }^5\psi \rangle .$$

(24)

To obtain pion mass, one has to replace

$$D_f\left(\overline{p}\right)={\displaystyle \frac{1}{\gamma ·\overline{p}-m_q}}⟶D_f\left(\overline{p}\right)={\displaystyle \frac{1}{\gamma ·\overline{p}-m_q+{\mu }_5{\gamma }_0{\gamma }_5}}$$

(25)

Detailed calculations will be presented in our upcoming work [85]. Here we only show the effective ${\mu }_5$ extracted from the lattice data of the charged pion mass in [81] in Figure 5b.

5. Results

In this article, we present a short review on three puzzles related to QCD matter under a magnetic field, including: inverse magnetic catalysis, diamagnetism (negative magnetic susceptibility) at low temperature and the nonmonotonic behavior of charged pion mass under a magnetic field.

Diamagnetism at low temperatures can be attributed to and understood by considering pion fluctuations, which represent the dominant degrees of freedom in the chiral symmetry broken phase. It is noticed that neutral pion fluctuations do not induce diamagnetism, while charged spin-0 pion fluctuations generate a magnetic moment in the opposite direction due to their orbital rotation under the influence of the Lorentz force in a magnetic field, thus exhibiting Landau diamagnetism, which is responsible for negative magnetic susceptibility (diamagnetism) at low temperatures. At high temperature, the dominant contribution comes from the quark degree of freedom, leading to the prevalence of Pauli paramagnetism.

Then we need to find one mechanism to simultaneously explain IMC and the nonmonotonic behavior of the charged pion mass; potential candidates consider the quark’s AMM or an extra interaction in the axial vector channel, inducing a similar ${\mu }_5$ term. But the quantitative value of the charged pion mass from the quark’s AMM is much smaller than the lattice results, so a deeper understanding is needed of the physics regarding the appearance of ${\mu }_5$ at zero temperature with a finite magnetic field. It is also possible that the quark’s AMM and the interaction in the axial vector channel are related, which deserves further studies in the future.