Effects of quark anomalous magnetic moment on the thermodynamical properties and mesonic excitations of magnetized hot and dense matter in PNJL model

Abstract

Various thermodynamic quantities and the phase diagram of strongly interacting hot and dense magnetized quark matter are obtained with the 2-flavour Nambu–Jona-Lasinio model with Polyakov loop considering finite values of the anomalous magnetic moment (AMM) of the quarks. Susceptibilities associated with constituent quark mass and traced Polyakov loop are used to evaluate chiral and deconfinement transition temperatures. It is found that, inclusion of the AMM of the quarks in presence of the background magnetic field results in a substantial decrease in the chiral as well as deconfinement transition temperatures in contrast to an enhancement in the chiral transition temperature in its absence. Using standard techniques of finite temperature field theory, the two point thermo-magnetic mesonic correlation functions in the scalar ($\sigma$) and neutral pseudoscalar ($\pi ^0$) channels are evaluated to calculate the masses of $\sigma$ and $\pi ^0$ considering the AMM of the quarks.

Appendices

Appendix A: double derivatives of $\Omega$ with respect to $M, \Phi$ and ${\overline{\Phi }}$

From Eq. (10) we get

$$\begin{aligned} \frac{\partial {\Omega }}{\partial {M}}= & {} \frac{M- m}{G} - 3 \sum _{n,f,s} \frac{\left| q_f B\right| }{2\pi ^2 } \int _{0}^{\infty }dp_z \frac{M}{\omega _{nfs} }\nonumber \\&\quad \left( 1 - \frac{ s\kappa _f q_f B}{M_{nfs}} \right) \Big [1- f^+ \left( \Phi , {\bar{\Phi }}, T \right) \nonumber \\&\quad - f^- \left( {\Phi , {\bar{\Phi }}, T }\right) \Big ]. \end{aligned}$$

(A1)

Following relations can be used to arrive at the above result

$$\begin{aligned}&\frac{\partial {\omega _{nfs}}}{\partial {M}} = \frac{M}{\omega _{nfs} } \left( 1 - \frac{s\kappa _f q_f B}{M_{nfs}} \right) , \end{aligned}$$

(A2)

$$\begin{aligned}&\frac{\partial {e^{-{n}\beta \left( \omega _{nfs} {\mp }\mu _q \right) } }}{\partial {M}} = -\frac{n\beta M}{\omega _{nfs}} \left( 1 - \frac{s\kappa _f q_f B}{M_{nfs}} \right) \nonumber \\&\quad \times e^{-{n}\beta \left( \omega _{nfs} {\mp }\mu _q \right) } , \end{aligned}$$

(A3)

$$\begin{aligned}&\frac{\partial {\ln g^{(+)}}}{\partial {M}} = -\frac{3 \beta M}{\omega _{nfs}} \left( 1 - \frac{s\kappa _f q_f B}{M_{nfs}} \right) f^+ \left( \Phi , {\bar{\Phi }}, T \right) ,\end{aligned}$$

(A4)

$$\begin{aligned}&\frac{\partial {\ln g^{(-)}}}{\partial {M}} = -\frac{3 \beta M}{\omega _{nfs}} \left( 1 - \frac{s\kappa _f q_f B}{M_{nfs}} \right) f^- \left( \Phi , {\bar{\Phi }}, T \right) . \end{aligned}$$

(A5)

Note that in Eq. (A1) the medium independent term has to be regularized by introducing a field dependent cutoff (see [83] for details):

$$\begin{aligned} \Lambda _z = \sqrt{\Lambda ^2 - (2n + 1-s ) \left| q_f B\right| + 2M_{nfs} s\kappa _f q_f B - (\kappa _f q_f B)^2 }. \end{aligned}$$

(A6)

So the regularized version of Eq. (A1) is

$$\begin{aligned} \frac{\partial {\Omega }}{\partial {M}}= & {} \frac{M- m}{G}- 3 \sum _{n,f,s} \frac{\left| q_f B\right| }{2\pi ^2 } \int _0^{\Lambda _z} dp_z \frac{M}{\omega _{nfs} }\nonumber \\&\times \left( 1 - \frac{s\kappa _f q_f B}{M_{nfs}} \right) + 3 \sum _{n,f,s} \frac{\left| q_f B\right| }{2\pi ^2 } \int _{0}^{\infty }dp_z \nonumber \\&\times \frac{M}{\omega _{nfs} } \left( 1 - \frac{ s\kappa _f q_f B}{M_{nfs}} \right) \nonumber \\&\times \left[ f^+ \left( \Phi , {\bar{\Phi }}, T \right) + f^- \left( \Phi , {\bar{\Phi }}, T \right) \right] . \end{aligned}$$

(A7)

Now to evaluate the second derivative with respect to M, the following relations will be useful:

$$\begin{aligned}&\frac{M}{\omega _{nfs} }\left( 1 - \frac{s\kappa _f q_f B}{M_{nfs}} \right) \bigg |_{p_z = \Lambda _z} = \frac{M^ 2}{\Lambda _z\sqrt{\Lambda ^2 + M^2 }} \frac{s\kappa _f q_f B}{M_{nfs}}\nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \times \left( 1 - \frac{s\kappa _f q_f B}{M_{nfs}} \right) \end{aligned}$$

(A8)

$$\begin{aligned}&\frac{\partial {\Lambda _z }}{\partial {M}} = \frac{s\kappa _f q_f B}{\Lambda _z} \frac{M}{M_{nfs}} , \end{aligned}$$

(A9)

$$\begin{aligned}&\frac{\partial }{\partial M} {\left( 1 - \frac{s\kappa _f q_f B}{M_{nfs}} \right) } = \frac{1}{\omega _{nfs}}\left( 1 - \frac{s\kappa _f q_f B}{M_{nfs}} \right) \nonumber \\&\quad - \frac{M^2}{\omega _{nfs}^3} \left( 1 - \frac{s\kappa _f q_f B}{M_{nfs}} \right) ^2 + \frac{M^2 s\kappa _f q_f B}{\omega _{nfs} M_{nfs}^3}. \end{aligned}$$

(A10)

Thus we can finally write

$$\begin{aligned} \frac{\partial ^2 {\Omega }}{\partial {M }^2}= & {} \frac{1}{G} - 3 \sum _{n,f,s} \frac{\left| q_f B\right| }{2\pi ^2 } \int _{0}^{\Lambda _z } dp_z \nonumber \\&\times \left[ \frac{1}{\omega _{nfs}}\left( 1 - \frac{s\kappa _f q_f B}{M_{nfs}} \right) - \frac{M^2}{\omega _{nfs}^3} \left( 1 - \frac{s\kappa _f q_f B}{M_{nfs}} \right) ^2 \right. \nonumber \\&\left. + \frac{M^2 s\kappa _f q_f B}{\omega _{nfs} M_{nfs}^3}\right] \left[ 1 - f^+ \left( \Phi , {\bar{\Phi }}, T \right) - f^- \left( \Phi , {\bar{\Phi }}, T \right) \right] \nonumber \\&- 3 \sum _{n,f,s} \frac{\left| q_f B\right| }{2\pi ^2 } \frac{M^ 2}{\Lambda _z\sqrt{\Lambda ^2 + M^2 }} \frac{s\kappa _f q_f B}{M_{nfs}} \left( 1 - \frac{s\kappa _f q_f B}{M_{nfs}} \right) \nonumber \\&- \frac{3}{T}\sum _{n,f,s} \frac{\left| q_f B\right| }{2\pi ^2} \int _{0}^{\infty }dp_z \frac{M^2}{\omega _{nfs}^2}\left( 1 - \frac{s\kappa _f q_f B}{M_{nfs}} \right) ^2 \nonumber \\&\times \left[ \frac{e^{-{ }\beta \left( \omega _{nfs} {- }\mu _q \right) } }{g^{(+)}}\left\{ \left( \Phi \right. \right. {+} 4 {\bar{\Phi }}e^{-{ }\beta \left( \omega _{nfs} {- }\mu _q \right) } {+} 3 e^{-{2}\beta \left( \omega _{nfs} {-}\mu _q \right) } \right) \nonumber \\&-3f^+\left( \Phi , {\bar{\Phi }}, T \right) \left( \Phi + 2{\bar{\Phi }}e^{-{ }\beta \left( \omega _{nfs} {-}\mu _q \right) } \right. \nonumber \\&\left. \left. + e^{-{ 2}\beta \left( \omega _{nfs} {-}\mu _q \right) } \right) \right\} \nonumber \\&\left. + \left\{ \Phi \leftrightarrow {\bar{\Phi }}; \mu _q \rightarrow -\mu _q \right\} \right] ,\nonumber \\ \end{aligned}$$

(A11)

$$\begin{aligned} \frac{\partial ^2 {\Omega }}{\partial {\Phi }\partial {M}}= & {} 3 \sum _{n,f,s} \frac{\left| q_f B\right| }{2\pi ^2} \int _{0}^{\infty }dp_z \frac{M}{\omega _{nfs} } \left( 1 - \frac{s\kappa _f q_f B}{M_{nfs}} \right) \nonumber \\&\left[ \frac{e^{-{ }\beta \left( \omega _{nfs} {-}\mu _q \right) } }{g^{(+)}} -\frac{3f^+\left( \Phi , {\bar{\Phi }}, T \right) }{g^{(+)}} e^{-{ }\beta \left( \omega _{nfs} {-}\mu _q \right) } \right. \nonumber \\&\left. + \frac{2e^{-{ 2}\beta \left( \omega _{nfs} {+}\mu _q \right) } }{g^{(-)}} -\frac{3f^-\left( \Phi , {\bar{\Phi }}, T \right) }{g^{(-)}} e^{-{2 }\beta \left( \omega _{nfs} {+}\mu _q \right) } \right] , \nonumber \\\end{aligned}$$

(A12)

$$\begin{aligned} \frac{\partial ^2 {\Omega }}{\partial {{\bar{\Phi }}}\partial {M}}= & {} \frac{\partial ^2 {\Omega }}{\partial {\Phi }\partial {M}} \left\{ \Phi \leftrightarrow {\bar{\Phi }}; \mu _q \rightarrow -\mu _q \right\} , \end{aligned}$$

(A13)

$$\begin{aligned} \frac{\partial ^2 {\Omega }}{\partial {\Phi }^2}= & {} \left( -b_3 \Phi + \frac{b_4}{2} {\bar{\Phi }}^2\right) T^4 + 9 T \sum _{n,f,s} \frac{\left| q_f B\right| }{2\pi ^2} \int _{0}^{\infty }dp_z \nonumber \\&\times \left[ \frac{e^{-{2}\beta \left( \omega _{nfs} {-}\mu _q \right) } }{{g^{(+)}}^2 } + \frac{e^{-{4}\beta \left( \omega _{nfs} {+}\mu _q \right) } }{{g^{(-)}}^2 } \right] ,\end{aligned}$$

(A14)

$$\begin{aligned} \frac{\partial ^2 {\Omega }}{\partial {{\bar{\Phi }}}^2}= & {} \frac{\partial ^2 {\Omega }}{\partial {\Phi }^2} \left\{ \Phi \leftrightarrow {\bar{\Phi }}; \mu _q \rightarrow -\mu _q \right\} , \end{aligned}$$

(A15)

$$\begin{aligned} \frac{\partial ^2 {\Omega }}{\partial {\Phi }\partial {{\bar{\Phi }}}}= & {} \left( \frac{- b_2(T)}{2} + b_4 {\bar{\Phi }}\Phi \right) T^4 \nonumber \\&+9 T \sum _{n,f,s} \frac{\left| q_f B\right| }{2\pi ^2} \int _{0}^{\infty }dp_z \nonumber \\&\times \left[ \frac{e^{-{3}\beta \left( \omega _{nfs} {-}\mu _q \right) } }{{g^{(+)}}^2 } + \frac{e^{-{3}\beta \left( \omega _{nfs} {+}\mu _q \right) } }{{g^{(-)}}^2 } \right] . \end{aligned}$$

(A16)

Appendix B: T-derivatives of $M, \Phi , {\overline{\Phi }}$

We have

$$\begin{aligned}&\dfrac{\partial }{\partial T} \left[ e^{-{n}\beta \left( \omega _{nfs} {\mp }\mu _q \right) } \right] = n e^{-{n}\beta \left( \omega _{nfs} {\mp }\mu _q \right) } \nonumber \\&\quad \times \left[ \frac{\omega _{nfs} \mp \mu _q }{T^2} - \frac{M}{\omega _{nfs} } \left( 1 - \frac{s\kappa _f q_f B}{M_{nfs}} \right) \left( \frac{\partial {M}}{\partial {T}} \right) \right] , \end{aligned}$$

(B1)

$$\begin{aligned}&\frac{\partial {f^+\left( \Phi , {\bar{\Phi }}, T \right) }}{\partial {T}} =\left( \frac{\partial {f^+}}{\partial {M}} \right) \left( \frac{\partial {M}}{\partial {T}} \right) \nonumber \\&\quad + \left( \frac{\partial {f^+}}{\partial {\Phi }} \right) \left( \frac{\partial {\Phi }}{\partial {T}} \right) +\left( \frac{\partial {f^+}}{\partial {{\bar{\Phi }}}} \right) \left( \frac{\partial {{\bar{\Phi }}}}{\partial {T}} \right) + A^+_{M,T} \end{aligned}$$

(B2)

where

$$\begin{aligned}&\frac{\partial {f^+}}{\partial {M }} = - \frac{M}{\omega _{nfs}} \left( 1 - \frac{s\kappa _f q_f B}{M_{nfs}} \right) \frac{e^{-{ }\beta \left( \omega _{nfs} {-}\mu _q \right) } }{g^{(+)}}\nonumber \\&\quad \qquad \times \left[ \left( \Phi + 4 {\bar{\Phi }}e^{-{ }\beta \left( \omega _{nfs} {- }\mu _q \right) } + 3 e^{-{2}\beta \left( \omega _{nfs} {-}\mu _q \right) } \right) \nonumber \right. \\&\quad \qquad -3f^+\left( \Phi , {\bar{\Phi }}, T \right) \left( \Phi + 2{\bar{\Phi }}e^{-{ }\beta \left( \omega _{nfs} {-}\mu _q \right) } \right. \nonumber \\&\left. \left. \qquad \,\,\, + e^{-{ 2}\beta \left( \omega _{nfs} {-}\mu _q \right) } \right) \right] , \end{aligned}$$

(B3)

$$\begin{aligned}&\frac{\partial {f^+}}{\partial {\Phi }} = \frac{e^{-{ }\beta \left( \omega _{nfs} {-}\mu _q \right) } }{g^{(+)}} -\frac{3f^+\left( \Phi , {\bar{\Phi }}, T \right) }{g^{(+)}} e^{-{ }\beta \left( \omega _{nfs} {-}\mu _q \right) } , \nonumber \\ \end{aligned}$$

(B4)

$$\begin{aligned}&\frac{\partial {f^+}}{\partial {{\bar{\Phi }}}} = \frac{2e^{-{ 2}\beta \left( \omega _{nfs} {-}\mu _q \right) } }{g^{(+)}} -\frac{3f^+\left( \Phi , {\bar{\Phi }}, T \right) }{g^{(+)}} e^{-{2 }\beta \left( \omega _{nfs} {-}\mu _q \right) } ,\nonumber \\ \end{aligned}$$

(B5)

$$\begin{aligned}&A^+_{M,T} = \frac{e^{-{}\beta \left( \omega _{nfs} {-}\mu _q \right) } }{{g^{(+)}}^2} \frac{\left( \omega _{nfs} -\mu _q \right) }{T^2}\nonumber \\&\quad \qquad \times \left\{ \Phi + 4{\bar{\Phi }}e^{-{ }\beta \left( \omega _{nfs} {-}\mu _q \right) } + 3\left( 1 + {\bar{\Phi }}\Phi \right) e^{-{2}\beta \left( \omega _{nfs} {-}\mu _q \right) } \nonumber \right. \\&\quad \qquad \left. +4\Phi e^{-{3}\beta \left( \omega _{nfs} {-}\mu _q \right) } + {\bar{\Phi }}e^{-{4}\beta \left( \omega _{nfs} {-}\mu _q \right) } \right\} , \end{aligned}$$

(B6)

$$\begin{aligned}&\frac{\partial {f^-\left( \Phi , {\bar{\Phi }}, T \right) }}{\partial {T}} \nonumber \\&\qquad \qquad =\frac{\partial {f^+\left( \Phi , {\bar{\Phi }}, T \right) }}{\partial {T}} \left\{ \Phi \leftrightarrow {\bar{\Phi }}; \mu _q \rightarrow -\mu _q \right\} . \end{aligned}$$

(B7)

Now using the above relations and results given in Appendix 1, T-derivatives of the gap equations of $M, \Phi$ and ${\bar{\Phi }}$ can be calculated starting from Eqs. (15), (16) and (17). The expression can be written in a matrix form in the following way:

$$\begin{aligned} \begin{bmatrix} C_{MM} &{} C_{M\Phi } &{} C_{M{\bar{\Phi }}} \\ C_{\Phi M} &{} C_{\Phi \Phi } &{} C_{\Phi {\bar{\Phi }}} \\ C_{{\bar{\Phi }}M} &{} C_{{\bar{\Phi }}\Phi } &{} C_{{\bar{\Phi }}{\bar{\Phi }}} \end{bmatrix} \begin{bmatrix} \frac{1}{\Lambda } \frac{\partial {M}}{\partial {T}} \\ \frac{\partial {\Phi }}{\partial {T}} \\ \frac{\partial {{\bar{\Phi }}}}{\partial {T}} \end{bmatrix} = \begin{bmatrix} \frac{T}{\Lambda ^2} A_{M,T} \\ \frac{T^2}{\Lambda ^3} A_{\Phi ,T} \\ \frac{T^2}{\Lambda ^3} A_{{\bar{\Phi }},T} \end{bmatrix} \end{aligned}$$

(B8)

where

$$\begin{aligned} A_{M,T}= & {} -\frac{3}{T^4} \sum _{n,f,s} \frac{\left| q_f B\right| }{2\pi ^2} \int _{0}^{\infty }dp_z \frac{M}{\omega _{nfs} }\nonumber \\&\times \left( 1 - \frac{s\kappa _f q_f B}{M_{nfs}} \right) \left[ \frac{e^{-{}\beta \left( \omega _{nfs} {-}\mu _q \right) } }{{g^{(+)}}^2} \left( \omega _{nfs} -\mu _q \right) \right. \nonumber \\&\times \left\{ \Phi + 4{\bar{\Phi }}e^{-{ }\beta \left( \omega _{nfs} {-}\mu _q \right) } + 3\left( 1 + {\bar{\Phi }}\Phi \right) e^{-{2}\beta \left( \omega _{nfs} {-}\mu _q \right) } \nonumber \right. \\&\left. +4\Phi e^{-{3}\beta \left( \omega _{nfs} {-}\mu _q \right) } + {\bar{\Phi }}e^{-{4}\beta \left( \omega _{nfs} {-}\mu _q \right) } \right\} \nonumber \\&\left. + \left\{ \Phi \leftrightarrow {\bar{\Phi }}; \mu _q \rightarrow -\mu _q \right\} \right] , \end{aligned}$$

(B9)

$$\begin{aligned} A_{\Phi ,T}= & {} - \frac{T}{2}\frac{\partial {b_2(T)}}{\partial {T}} {\bar{\Phi }}- \frac{9}{T^3}\sum _{n,f,s} \frac{\left| q_f B\right| }{2\pi ^2} \int _{0}^{\infty }dp_z \nonumber \\&\times \left( \frac{e^{-{}\beta \left( \omega _{nfs} {-}\mu _q \right) } }{g^{(+)}} + \frac{e^{-{2}\beta \left( \omega _{nfs} {+}\mu _q \right) } }{g^{(-)}} \right) \nonumber \\&+ \frac{3}{T^4} \sum _{n,f,s} \frac{\left| q_f B\right| }{2\pi ^2} \int _{0}^{\infty }dp_z \left[ \frac{e^{-{}\beta \left( \omega _{nfs} {-}\mu _q \right) } }{{g^{(+)}}^2} \left( \omega _{nfs} -\mu _q \right) \right. \nonumber \\&\times \left\{ 1 - 3 {\bar{\Phi }}e^{-{2}\beta \left( \omega _{nfs} {-}\mu _q \right) } - 2 e^{-{3}\beta \left( \omega _{nfs} {-}\mu _q \right) } \right\} \nonumber \\&+ \frac{e^{-{2}\beta \left( \omega _{nfs} {+}\mu _q \right) } }{{g^{(-)}}^2} \left( \omega _{nfs} +\mu _q \right) \nonumber \\&\left. \times \left\{ 2+ 3 {\bar{\Phi }}e^{-{}\beta \left( \omega _{nfs} {+}\mu _q \right) } - e^{-{3}\beta \left( \omega _{nfs} {+}\mu _q \right) } \right\} \right] \end{aligned}$$

(B10)

$$\begin{aligned}&\mathrm{and }\nonumber \\ A_{{\bar{\Phi }},T}= & {} A_{\Phi ,T} \left\{ \Phi \leftrightarrow {\bar{\Phi }}; \mu _q \rightarrow -\mu _q \right\} .\end{aligned}$$

(B11)

During this calculation we have put a combination of T and $\Lambda$ with several quantities to make sure we get matrix with dimensionless co-efficients as introduced in Sect. 2.2.

Appendix C: $\mu _q$-derivatives of $M,\Phi ,{\overline{\Phi }}$

Similar matrix form can also be written for $\mu _q$-derivatives of the gap equations as shown below

$$\begin{aligned} \begin{bmatrix} C_{MM} &{} C_{M\Phi } &{} C_{M{\bar{\Phi }}} \\ C_{\Phi M} &{} C_{\Phi \Phi } &{} C_{\Phi {\bar{\Phi }}} \\ C_{{\bar{\Phi }}M} &{} C_{{\bar{\Phi }}\Phi } &{} C_{{\bar{\Phi }}{\bar{\Phi }}} \end{bmatrix} \begin{bmatrix} \frac{1}{\Lambda } \frac{\partial {M}}{\partial {\mu _q}} \\ \frac{\partial {\Phi }}{\partial {\mu _q}} \\ \frac{\partial {{\bar{\Phi }}}}{\partial {\mu _q}} \end{bmatrix} = \begin{bmatrix} \frac{T}{\Lambda ^2} A_{M,\mu _q} \\ \frac{T^2}{\Lambda ^3} A_{\Phi ,\mu _q} \\ \frac{T^2}{\Lambda ^3} A_{{\bar{\Phi }},\mu _q} \end{bmatrix} \end{aligned}$$

(C1)

where

$$\begin{aligned} A_{M,\mu _q}= & {} -\frac{3}{T^3} \sum _{n,f,s} \frac{\left| q_f B\right| }{2\pi ^2} \int _{0}^{\infty }dp_z \frac{M}{\omega _{nfs} }\nonumber \\&\times \left( 1 - \frac{s\kappa _f q_f B}{M_{nfs}} \right) \left[ \frac{e^{-{}\beta \left( \omega _{nfs} {-}\mu _q \right) } }{{g^{(+)}}^2} \left\{ \Phi + 4{\bar{\Phi }}e^{-{ }\beta \left( \omega _{nfs} {-}\mu _q \right) } \right. \right. \nonumber \\&+ 3\left( 1 + {\bar{\Phi }}\Phi \right) e^{-{2}\beta \left( \omega _{nfs} {-}\mu _q \right) } \nonumber \\&\left. +4\Phi e^{-{3}\beta \left( \omega _{nfs} {-}\mu _q \right) } + {\bar{\Phi }}e^{-{4}\beta \left( \omega _{nfs} {-}\mu _q \right) } \right\} \nonumber \\&\left. - \left\{ \Phi \leftrightarrow {\bar{\Phi }}; \mu _q \rightarrow -\mu _q \right\} \right] , \end{aligned}$$

(C2)

$$\begin{aligned} A_{\Phi ,\mu _q}= & {} \frac{3}{T^3} \sum _{n,f,s} \frac{\left| q_f B\right| }{2\pi ^2} \int _{0}^{\infty }dp_z \left[ \frac{e^{-{}\beta \left( \omega _{nfs} {-}\mu _q \right) } }{{g^{(+)}}^2}\right. \nonumber \\&\times \left\{ 1 - 3 {\bar{\Phi }}e^{-{2}\beta \left( \omega _{nfs} {-}\mu _q \right) } - 2 e^{-{3}\beta \left( \omega _{nfs} {-}\mu _q \right) } \right\} \nonumber \\&+ \frac{e^{-{2}\beta \left( \omega _{nfs} {+}\mu _q \right) } }{{g^{(-)}}^2}\left\{ 2+ 3 {\bar{\Phi }}e^{-{}\beta \left( \omega _{nfs} {+}\mu _q \right) } \right. \nonumber \\&\left. \left. - e^{-{3}\beta \left( \omega _{nfs} {+}\mu _q \right) } \right\} \right] , \end{aligned}$$

(C3)

$$\begin{aligned}&A_{{\bar{\Phi }},\mu _q} A_{\Phi ,\mu _q } \left\{ \Phi \leftrightarrow {\bar{\Phi }}; \mu _q \rightarrow -\mu _q \right\} . \end{aligned}$$

(C4)