The chiral anomaly arises from the Adler-Bell-Jackiw (ABJ) equation when applied to QCD. The ABJ equation was originally written as a description of anomalies in electromagnetism, involving the electromagnetic tensor $F_{\mu \nu }$. The chiral anomaly is invoked to explain the non-conservation of the axial vector current [31] . In what follows, ${\alpha }_s$ is the strong force coupling. The non-vanishing contravariant derivative of the current in Equation (2) is given by

${\partial }^{\mu }{J}_{\mu }^5=2i{m}_Q\bar{Q}{\gamma }_{5}Q+\frac{{\alpha }_s}{4\pi }{T}_{R}G\tilde{G}$ (5)

where $Q$ represents the quark, $T_R=\frac{1}{2}$, and $G$ is the gluon field-strength

tensor. There are two contributions to Equation (2.1): the first term is due to the non-zero quark mass, which arises from the spontaneously broken symmetry; the second term is the ABJ equation for QCD [31] . We may assume the form for $J_{\mu }^5$ in Equation (2). The derivation of the form factor in the following subsection, in the model of axions from quark matter that we present in this paper, arises from taking the matrix elements. In the context of QCD, as we discuss here, ABJ is a QCD perturbation in powers of ${\alpha }_s$. The chiral anomaly of QCD and the spontaneously broken symmetry are invoked, by the ABJ equation, to explain the non-conservation of the axion current.

In order to obtain the theoretical emissivities, luminosities, and energy fluxes for axions from neutron stars, we need to obtain the form factor, thereby computing the axion-quark coupling. In our calculations, we may consider the customary momentum transfer

$q=p_1-p_2,$ (6)

and the customary center-of-mass (CM) energy

$s={\left(p_1+p_2\right)}^2$ (7)

The vertex function, ${\text{Λ}}_{Q,\mu }$ may be expressed in terms of the form factors [32] , which are functions of the momentum transfer $q^2$, as shown below

${\text{Λ}}_{Q,\mu }={\gamma }_{\mu }{\gamma }_5{F}_1^5\left(q^2\right)+\frac{1}{2m_Q}{p}_{\mu }{\gamma }_5{F}_3^5\left(q^2\right).$ (8)

The form factor $F_3^5$ can be shown to be complex in general. We may consider the case $q^2\gg 4m^2$. In Ref. [32] , the renormalization has been taken into account in the derivation of the form factors. The real part of the form factor $F_3^5$ [32] is

$\begin{array}{c}\mathrm{Re}{F}_3^5\left(q^2\right)=\frac{{\alpha }_s^2}{32{\pi }^2}\left(\frac{2}{q^4}\right)\left[{\log }^2\left(q^2\right)-6\log \left(q^2\right)-2\zeta \left(2\right)\right)\\ +\frac{2}{{\left(q^2\right)}^2}\left[-32\log \left(q^2\right)+8\zeta \left(2\right)+12{\log }^2\left(q^2\right)\right].\end{array}$ (9)

The imaginary part of the form factor may be written as [32] :

$\mathrm{Im}{F}_3^5\left(q^2\right)=\frac{4/\pi }{q^4}\left[3-\log \left(q^2\right)\right]+\frac{1/\pi }{q^4}\left[32-24\log \left(q^2\right)\right]$ (10)

We may in general consider the amplitude $\left|F_3^5\left(q^2\right)\right|$. The axion-quark coupling can be written, in a form which uses the form factors, from invoking the Goldberger-Treiman relation [31] :

$g_{aQ\bar{Q}}\left(q^2\right)=\frac{1}{2f_a}{q}^2\left|F_3^5\left(q^2\right)\right|.$ (11)

The derivation is completely analogous to the pion-nucleon coupling via a triangle diagram.

The tree-level s-channel (via gluon exchange) coupling can be safely neglected in considering the coupling of quarks to axions. The form factor for the tree-level coupling can be written as

$\left|F_3^5\right|=\frac{1}{q^4}{f}_a^2{\alpha }_s^4$ (12)

We take $q=100$ MeV, $f_a\simeq 100$ GeV, ${\alpha }_s\simeq {10}^{-2}$, $F_3^5\simeq {10}^{-18}$ for tree-level

coupling. In this case, ${F_3^5|}_{\text{tree}}\ll {F_3^5|}_{\text{two-loop}}$. We do not consider the one-loop

coupling, because it concerns the axial vector current. The tree-level coupling is proportional to ${\alpha }_s^2$, whereas the two-loop coupling is proportional ${\alpha }_s^4$. The form-factor for the two-loop coupling may outweign the dependence on ${\alpha }_s^4$, such that the ${\alpha }_s^2$ term may be safely neglected. Note that $\alpha <1$.

The emissivity is a useful way to quantify the energy emitted by axions from neutron stars, as well as an intermediate calculation to calculate the flux, gamma-ray SED, and gamma-ray energy flux. We can derive the axion emissivity proceeding first from the axion-quark coupling, which can be written by using the form factors and by invoking the Goldberger-Treiman relation [31] :

$g_{aQ\bar{Q}}\left(q^2\right)=\frac{1}{2f_a}{q}^2\left|F_3^5\left(q^2\right)\right|$ (13)

Expressed in terms of the coupling to $g_{aq\bar{q}}$, we may write the volume emissivity ${ϵ}_a$ of the quark matter to axions as a phase-space integral over the quark momenta:

${ϵ}_a=\frac{1}{n}{\displaystyle {\int }_0^{p_F}\text{d}{\Omega }_1\text{d}{p}_1{p}_1^2}{\displaystyle {\int }_0^{p_F}\text{d}{\Omega }_2\text{d}{p}_2{p}_2^2}$ (14)

$\times {\displaystyle \int \text{d}\omega }{\omega }^2{g}_{aQ\bar{Q}}^2{\left(q^2\right)}^2{f}_1\left(\omega ;\mu ,T\right)f_2\left(\omega ;\mu ,T\right)\delta \left(\omega -\sqrt{p_1^2+m_Q^2}-\sqrt{p_2^2+m_Q^2}\right).$ (15)

where $\omega$ is the quark energy, $p_{F,Q}$ is the quark Fermi momentum, and $f_i\left(E;\mu ,T\right)$ is the Fermi-Dirac distribution for the quark chemical potential $\mu$ at neutron star temperature $T$.

We may use the energy-time uncertainty relation to derive a timescale for the process of axion emission:

$\text{Δ}t\simeq \frac{1}{\text{Δ}E}.$ (16)

We may invoke this uncertainty relation, as the neutron star of quark matter is a Fermi-Dirac system of a quantum phase. Further, the timescale for axion emission of 30 ms is of order of the timescale of the binary inspiral, and is thus relevant for the current study, whereas the gravitational wave signal lasted approximately 100 s. Let us consider the defined quarks in the neutron star core. If we may naïvely take the position uncertainty $\text{Δ}x=0.1$ fm, then we have $\text{Δ}p\simeq {10}^{-13}$ MeV. Assuming energy-time uncertainty relation, one may obtain a time uncertainty of 0.030 s for the processes in the medium. Although axions would be produced quite quickly, the detection is not instantaneous. The NS has to be close enough, and the detection efficiency is not high enough.

$f_i\left(E;\mu ,T\right)=\frac{1}{1+\text{exp}\left(\frac{E-\mu }{T}\right)}$ (17)

We evaluate the width for the $q\bar{q}$ state, to a first approximation, as [31] :

$\text{Γ}\left(q\bar{q}\to a\right)=\frac{16{\pi }^2{\alpha }_s^2}{m_a^2}{\left|\psi \left(0\right)\right|}^2$ (18)

where $\psi$ represents the axion wavefunction ( Figure 2 ).

The axion-quark-antiquark coupling can be written in terms of the Goldberger-Treiman relation:

$g_{aq\bar{q}}=\frac{q^2}{2f_a}\left|F_3^5\left(q^2\right)\right|$ (19)

The cross-section amplitude may be written in the form:

$\sigma \left(q\bar{q}\to a\right)=\frac{12{\pi }^2}{M}\text{Γ}\left(a\to q\bar{q}\right)\delta \left(M^2-s\right).$ (20)

In the model that we consider, the neutron star is may be considered to be in a QCD phase of quark matter, or a mixed phase of hadronic and quark matter [33] . This phase of quark matter is considered to be color superconducting. In addition, it is possible to consider the “CFL” phase where strange quarks, in addition to up and down quarks, participate in Cooper pairing mechanism [34] . This phase is superfluid and exhibits broken chiral symmetry. The neutral quark (NQ), the gapless CFL, and the gapless 2SC phases are candidates for stellar matter [1] .

We may assume a number density of $n=0.15{\text{fm}}^{-3}$ within the neutron star/ quark star and a quark chemical potential of $\mu =400$ MeV in Ref. [27] . We also cite the more recent reference of 2014 [2] , which yields a similar density of $\rho \simeq 0.15{\text{fm}}^{-3}$. We assume a quark matter or a mixed quark-hadron phase for the neutron star. We assume a Fermi momentum $p_F$ which is given by:

$p_F={\left(3{\pi }^{2}n\right)}^{1/3}$ (21)

which is calculated to be 323.81 MeV from the density that we cite.

The temperature of the QCD phase of quark matter depends on the exact description of the neutron star in the QCD phase diagram. In deriving the prediction for gamma-ray signal below, we assume $T=90$ MeV.

The photon flux can be derived in a stepwise procedure in a Monte Carlo simulation, largely relying on the results of Section 2.3. First, we integrate over momentum space.

$\mathcal{D}v={\text{d}}^3{p}_1{\text{d}}^3{p}_2\frac{{\omega }^2}{{\left[1+{\text{e}}^{\left(E-\mu \right)/T}\right]}^2}{g}_{aqq}^2$ (22)

Next, we define the neutron star volume, assuming a radius of 10 km.

$V_{NS}=\frac{4\pi }{3}{\left({10}^6\right)}^3{\text{cm}}^3$ (23)

We provide a delta function, with respect to the kinetic energy $\omega$, invoking the $u$ and $d$ momenta and masses, $m_u$ and $m_d$.

$\delta \left(\omega -\sqrt{p_1^2+m_u^2}-\sqrt{p_2^2+m_d^2}\right)$ (24)

The axion-quark-quark coupling is derived in terms of the real and imaginary components of the form factor $F_3^5$,

$g_{aqq}=\frac{1}{2}\sqrt{\mathrm{Re}{\left(F_3^5\right)}^2+\mathrm{Im}{\left(F_3^5\right)}^2}\frac{q^2}{f_a}$ (25)

We write the axion wavefunction as follows.

${\psi }_0={10}^{-10}{\text{cm}}^{-3/2}$ (26)

We derive the following variable $\lambda$ as such:

$\lambda =\frac{16\pi }{3}{\alpha }_s^2{\left({\psi }_0^2\right)}^{-1}$ (27)

The cross section is written as shown below.

$\sigma =12{\pi }^2\frac{\lambda }{\sqrt{s}}$ (28)

The derived derivative is defined in terms of $\text{d}v$ and the number density of quarks, $\rho$.

$\mathcal{D}ϵ=\sigma \frac{\mathcal{D}v}{{\rho }^2}$ (29)

Finally, we obtain the differential photon flux:

$E\frac{\text{d}\Phi }{\text{d}E}=2\frac{m_a^5}{4\pi d^2}{\displaystyle {\int }_0^{p_F}\mathcal{D}ϵV_{NS}\lambda \text{Δ}t\delta \left(E-\omega /2\right)}$ (30)