The QCD Axion in Hot and Dense Matter ^†

Abstract

Within the framework of the two-flavor Nambu–Jona-Lasinio model, we study the quantum chromodynamics axion properties at finite temperature and chemical potential. Our analysis shows that the axion properties are strongly influenced by the critical behavior of the chiral phase transition. In particular, the axion mass follows the response of the chiral condensate to temperature and chemical potential, decreasing monotonically as either parameter increases. Moreover, we observe that at relatively low temperatures, the axion self-coupling constant exhibits a sharp peak near the critical point when the chemical potential increases, reaching more than twice its vacuum value. Beyond the critical point, the self-coupling rapidly decreases to values much smaller than in vacuum, eventually approaching zero at high chemical potential. These results suggest that the significant enhancement of the axion self-coupling in dense matter near the chiral phase transition may favor the formation or amplification of an axion Bose–Einstein condensate in compact astrophysical objects.

1. Introduction

The quantum chromodynamics (QCD) axion arises as the Nambu–Goldstone boson [1,2] associated with the spontaneous breaking of the Peccei-Quinn (PQ) symmetry [3,4], originally proposed as a natural solution to the strong CP problem. However, due to the non-perturbative effects of QCD, the axion acquires a mass [5,6]. Beyond its role in particle physics, the axion has profound implications in astrophysics, where it may influence supernova explosions and protoneutron star evolution [7,8,9,10,11,12]. They can be produced via the Primakoff process, where high-energy photons scatter off atomic nuclei to generate pseudoscalar particles. In addition, axion emission offers an efficient cooling mechanism for compact stars, acting as a complement to the conventional neutrino- and photon-driven cooling channels [13,14,15,16,17]. In this work, we aim to investigate how the properties of the QCD axion evolve with variations in quark chemical potential and temperature, which are of particular relevance to astrophysical environments characterized by finite baryon density. At finite chemical potential, however, lattice QCD simulations with three colors encounter severe difficulties due to the notorious sign problem [18], which poses a major obstacle to first-principle numerical studies of baryonic matter. Furthermore, because the QCD coupling constant increases with energy and becomes large in the low-energy regime, the artificial omission of higher-order terms in conventional perturbative approaches leads to thermodynamic inconsistencies, which are particularly problematic when describing the equation of state of quark matter at low densities [19]. This limitation highlights the need to employ non-perturbative approaches, such as effective QCD models, to investigate the properties of the QCD axion in hot and dense strongly interacting matter.

Among such approaches, the Nambu–Jona-Lasinio (NJL) model has proven to be a versatile framework for describing the QCD phase structure and the properties of strongly interacting quark matter [20,21,22]. It has successfully reproduced both zero-temperature [23] and finite-temperature [24] results for isospin-imbalanced matter, showing consistency with lattice QCD and chiral perturbation theory (CHPT). Notably, the NJL model captures the peak structure of the energy density normalized to its Stefan–Boltzmann limit [23], in agreement with CHPT predictions [25] and lattice results [26], and reproduces the critical point of the normal to pion superfluid phase transition at ${\mu }_I=m_{\pi }$ [27,28,29,30]. Building on these successes, the axion field has been incorporated into the NJL Lagrangian through the $U{\left(1\right)}_A$ anomaly term, enabling direct investigation of axion properties in hot and/or dense media [31,32,33,34,35,36,37,38,39,40,41,42]. Within this framework, one can calculate topological observables such as the susceptibility, as well as axion properties including its mass and self-coupling, across the chiral phase transition. This provides a valuable nonperturbative tool for exploring axion physics in regimes where lattice methods fail. We emphasize that recent NJL-based studies have primarily focused on finite temperature at vanishing baryon chemical potential [31] or in the presence of external magnetic fields [42]. In contrast, the present work goes a step further by incorporating the quark chemical potential into the NJL Lagrangian. This allows us to examine the impact of finite baryon density on the properties of QCD axions. Our results show that the properties of QCD axions are strongly modified in hot and dense matter, with particularly pronounced effects near the chiral phase transition.

2. NJL Model

In this work, we employ the two-flavor NJL model, taking into account the effects of both temperature and chemical potential. The NJL model Lagrangian density, incorporating the axion field, can be expressed as

$$\begin{array}{c}\hfill \mathcal{L}=\overline{q}\left(i{\gamma }^{\mu }{\partial }_{\mu }+\mu {\gamma }_0-m_0\right)q+{\mathcal{L}}_{\mathrm{int}},\end{array}$$

(1)

where q represents the matrix of light quark fields, $\mu$ and $m_0$ denote the quark chemical potential and the degenerate current quark mass, respectively. Furthermore, the interaction term ${\mathcal{L}}_{\mathrm{int}}$ is given by [31]

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{L}}_{\mathrm{int}}=& G_1\left[\left(\overline{q}{\tau }_{a}q\right)\left(\overline{q}{\tau }_{a}q\right)+\left(\overline{q}i{\tau }_a{\gamma }_{5}q\right)\left(\overline{q}i{\tau }_a{\gamma }_{5}q\right)\right]\hfill \\ & +8G_2\left[e^{i\theta }det\left({\overline{q}}_R{q}_L\right)+e^{-i\theta }det\left({\overline{q}}_L{q}_R\right)\right],\hfill \end{array}\end{array}$$

(2)

which can be derived through a chiral rotation of the quark fields in the path integral [20,43,44]. In this expression, ${\tau }_a$ denote matrices in flavor space with $a=0,1,2,3$; specifically, ${\tau }_0$ is the identity matrix, while ${\tau }_{1,2,3}$ correspond to the Pauli matrices. The coupling constant $G_1$ governs the $U{\left(1\right)}_A$-invariant interaction, whereas $G_2$ controls the strength of the $U{\left(1\right)}_A$-breaking term.

After using the following mean-field approximation,

$$\begin{array}{ccc}\hfill {\left(\overline{q}q\right)}^2& \approx & 2\left(\overline{q}q\right)\langle \overline{q}q\rangle -{\langle \overline{q}q\rangle }^2,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill {\left(\overline{q}i{\tau }_a{\gamma }_{5}q\right)}^2& \approx & 2\left(\overline{q}i{\tau }_a{\gamma }_{5}q\right)\langle \overline{q}i{\tau }_a{\gamma }_{5}q\rangle -{\langle \overline{q}i{\tau }_a{\gamma }_{5}q\rangle }^2,\hfill \end{array}$$

(4)

one can obtain the one-loop level expression for the thermodynamic potential as [45,46]

$$\begin{array}{c}\hfill \mathsf{\Omega }={\mathsf{\Omega }}_{\mathrm{mf}}+{\mathsf{\Omega }}_q,\end{array}$$

(5)

in which the mean-field contribution, ${\mathsf{\Omega }}_{\mathrm{mf}}$, is explicitly given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathsf{\Omega }}_{\mathrm{mf}}=& -G_2\left({\eta }^2-{\sigma }^2\right)cos\theta +G_1\left({\eta }^2+{\sigma }^2\right)-2G_2\sigma \eta sin\theta .\hfill \end{array}\end{array}$$

(6)

In the above, the condensates are defined as $\sigma =\langle \overline{q}q\rangle$ and $\eta =〈\overline{q}i{\gamma }_{5}q〉$. Moreover, the quark loop contribution ${\mathsf{\Omega }}_q$ takes the form

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathsf{\Omega }}_q=& -2N_{c}T\sum _{f=u,d}\int {\displaystyle \frac{d^{3}p}{{\left(2\pi \right)}^3}}\{{\displaystyle \frac{E_p}{T}}\hfill \\ & +ln\left[1+e^{-\left(E_p-{\mu }_f\right)/T}\right]+ln\left[1+e^{-\left(E_p+{\mu }_f\right)/T}\right]\},\hfill \end{array}\end{array}$$

(7)

where the dispersion relations of quarks are given by

$$\begin{array}{c}\hfill E_p=\sqrt{p^2+{\Delta }^2},{\Delta }^2={\left(m_0+{\alpha }_0\right)}^2+{\beta }_0^2,\end{array}$$

(8)

with

$$\begin{array}{c}\hfill \begin{array}{cc}& {\alpha }_0=-2\left(G_1+G_{2}cos\theta \right)\sigma +2G_2\eta sin\theta ,\hfill \\ & {\beta }_0=-2\left(G_1-G_{2}cos\theta \right)\eta +2G_2\sigma sin\theta .\hfill \end{array}\end{array}$$

(9)

To determine the ground state of the system at finite temperature and chemical potential, we minimize the thermodynamic potential, as expressed in Equation (5), with respect to the condensates $\sigma$ and $\eta$. This procedure yields the following gap equations

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \mathsf{\Omega }}{\partial \sigma }}{|}_{\sigma =\overline{\sigma }}=0,{\displaystyle \frac{\partial \mathsf{\Omega }}{\partial \eta }}{|}_{\eta =\overline{\eta }}=0,\end{array}$$

(10)

where the values $\sigma =\overline{\sigma }$ and $\eta =\overline{\eta }$ represent the condensates corresponding to the global minimum of the thermodynamic potential.

3. Results and Discussions

The first term on the right-hand side of Equation (7) contains an ultraviolet divergence, which is regulated by a momentum cutoff $p=\mathsf{\Lambda }$. Consequently, the two-flavor NJL model contains three parameters: the current quark mass m, the coupling constant G, and the cutoff $\mathsf{\Lambda }$. These are fixed by reproducing key low-energy observables, namely the pion mass $m_{\pi }=140.2$ MeV, the pion decay constant $f_{\pi }=92.6$ MeV, and the quark condensate in the vacuum ${\sigma }_0=2{(-241.5\mathrm{MeV})}^3$. The resulting parameter set is $\mathsf{\Lambda }=590\mathrm{MeV}$, $G_0=2.435/{\mathsf{\Lambda }}^2$, $G_1=(1-c)G_0$, $G_2=c{G}_0$, $c=0.2$, $m_0=6\mathrm{MeV}$ [31]. The thermodynamic properties of the system can therefore be determined by numerically solving Equation (10) for given values of the chemical potential and temperature.

3.1. Chiral Condensate

The chiral condensate is commonly used as an order parameter to characterize the chiral phase transition, and it can be obtained numerically via the gap equations. In Figure 1, we plot the chiral condensate, normalized to its vacuum value, as a function of the temperature for $\mu =0,100,200,300$ MeV (left panel) and of the chemical potential for $T=0,30,60,150$ MeV (right panel). At vanishing chemical potential, the condensate decreases smoothly with temperature, indicating a chiral crossover. As the chemical potential increases, the drop becomes sharper and shifts to lower temperatures, reflecting the interplay between thermal and density effects in driving chiral symmetry restoration. Conversely, at zero temperature the condensate remains nearly constant at small chemical potentials and then exhibits a sudden drop near the critical region, characteristic of a first-order chiral phase transition. With increasing temperature, this discontinuity gradually weakens and eventually turns into a smooth crossover, consistent with the expected phase structure of QCD matter.

3.2. Axion Mass

In dense media, the effective potential and mass of the QCD axion depend on both temperature and chemical potential. In particular, the axion mass squared can be expressed as [46]

$$\begin{array}{c}\hfill m_a^2={\left.{\displaystyle \frac{{\mathrm{d}}^2\mathcal{V}(a,T,\mu )}{\mathrm{d}{a}^2}}\right|}_{a=0},\end{array}$$

(11)

where $\mathcal{V}(a,T,\mu )=\mathsf{\Omega }(T,\mu ,\sigma =\overline{\sigma },\eta =\overline{\eta })$ represents the effective potential for the QCD axion at finite temperature and chemical potential, and $f_a$ is the axion decay constant.

As shown in Figure 2, we plot the axion mass, normalized to its vacuum value, as a function of temperature (left panel) and chemical potential (right panel), for the same parameter sets in Figure 1. Both panels in Figure 2 show that the axion mass decreases with increasing temperature and/or chemical potential, indicating that thermal and density effects tend to suppress the axion mass. A comparison with Figure 1 shows that the evolution of the axion mass with respect to both temperature (left panel) and chemical potential (right panel) closely follows the behavior of the chiral condensate. This is consistent with the fact that the axion mass is determined by Equation (11). Furthermore, the dependence of the axion mass on chemical potential agrees with the results of CHPT calculations at vanishing temperature [47]. However, as pointed out in Ref. [45], CHPT results at finite chemical potential and high temperature cannot be considered reliable, since they do not incorporate the effects of QCD phase transitions in this regime.

3.3. Axion Self-Coupling Constant

The axion self-coupling plays a significant role in the formation of both Bose-Einstein condensates [48] and axion stars [49], and can be obtained by

$$\begin{array}{c}\hfill \lambda ={\left.{\displaystyle \frac{{\mathrm{d}}^4\mathcal{V}(a,T,\mu )}{\mathrm{d}{a}^4}}\right|}_{a=0}.\end{array}$$

(12)

In Figure 3, we plot the normalized axion self-coupling constant $\lambda /{\lambda }_0$ versus temperature (left panel) and chemical potential (right panel), for the same parameter sets in Figure 1. At vanishing chemical potential, the self-coupling remains close to its vacuum value at low temperatures and then gradually decreases as the system approaches the chiral crossover. With increasing chemical potential, a pronounced peak develops near the critical region, where the self-coupling can exceed twice its vacuum value before rapidly dropping to much smaller values at higher densities. A similar behavior is observed in the right panel: at zero temperature, the self-coupling exhibits a sharp enhancement around the first-order chiral transition, while at higher temperatures this peak becomes smoother and eventually turns into a continuous suppression. These results indicate that the axion self-coupling is strongly affected by the chiral phase structure, being significantly enhanced near the critical region but suppressed in the high-temperature or high-density limits.

4. Conclusions

In this work, we have investigated the effects of temperature and chemical potential on axion properties within the framework of the two-flavor NJL model, where the axion field is incorporated through the PQ mechanism associated with the $U{\left(1\right)}_A$ symmetry breaking term. Our results show that the chiral phase transition is of first order at low temperature and high chemical potential, while at higher temperatures and lower chemical potentials it becomes a smooth crossover, as indicated by the behavior of the chiral condensate. Correspondingly, the axion mass decreases monotonically with increasing temperature and chemical potential, closely following the evolution of the condensate. In contrast, the axion self-coupling constant exhibits a pronounced enhancement near the critical region at low temperatures and high densities, reaching values significantly above its vacuum counterpart before dropping sharply at larger chemical potentials. These findings highlight the profound influence of the QCD phase structure on axion physics, showing that dense matter near the chiral phase transition can strongly affect axion properties in hot and dense environments. Further quantitative studies that include the effects of isospin density and magnetic fields are needed to refine and extend these results.