Charge and isospin fluctuations in a non-ideal pion gas with dynamically fixed particle number

Abstract

We study the behavior of the non-ideal pion gas with the dynamically fixed number of particles, formed on an intermediate stage in ultra-relativistic heavy-ion collisions. The pion spectrum is calculated within the self-consistent Hartree approximation. General expressions are derived for cross-covariances of the number of various particle species in the pion gas of an arbitrary isospin composition. The behavior of the cross-variances is analyzed for the temperature approaching from above the maximal critical temperature of the Bose–Einstein condensation for the pion species $a=\pm ,0$, i.e. for $T>\max T_\mathrm{cr}^a$. It is shown that in the case of the system with equal averaged numbers of isospin species, the variance of the charge, $Q=N_{+}-N_{-}$, diverges at $T\rightarrow T_\mathrm{cr}=T_\mathrm{cr}^a$, whereas variances of the total particle number, $N=N_{+} + N_{-} + N_{0}$, and of a relative abundance of charged and neutral pions, $G=(N_{+}+N_{-})/2 - N_{0}$, remain finite in the critical point. Then, fluctuations are studied in the pion gas with small isospin imbalance $0<|G|\ll N$ and $0<|Q|\ll N$  and shifts of the effective masses, chemical potentials, and values of critical temperatures are calculated for various pion species, and the highest critical temperature, $\text{ max }T_\mathrm{cr}^{{a}}$ is found, above which the pion system exists in the non-condensed phase. Various pion cross variances are calculated for $T>\text{ max }T_\mathrm{cr}^{{a}}$, which prove to be strongly dependent on the isospin composition of the system, whereas the variances of N and G are found to be independent on the isospin imbalance up to the term linear in G/N and Q/N.

Appendices

Appendix A: First and second T-derivatives of $m^*-\mu$ in isospin-symmetrical system

The dependence of $m^*$ and $\mu$ on the temperature and the density in the isospin-symmetrical gas is determined by set of equations (31). Differentiating them with respect to T at fixed n we obtain the set of equations

$$\begin{aligned} \Big (2m^*-\frac{\partial \Pi }{\partial m^*}\Big )\frac{\partial m^*}{\partial T}\Big |_n- \frac{\partial \Pi }{\partial \mu }\frac{\partial \mu }{\partial T}\Big |_n&=\frac{\partial \Pi }{\partial T} \,, \nonumber \\ \frac{\partial h}{\partial m^*}\frac{\partial m^*}{\partial T}\Big |_n + \frac{\partial h}{\partial \mu }\frac{\partial \mu }{\partial T}\Big |_n&=-\frac{\partial h}{\partial T} \,, \end{aligned}$$

(A1)

where quantities h and

$$\begin{aligned} \Pi =10\lambda m^{*2} d(m^{*2},\mu ,T) \end{aligned}$$

are treated as functions of variables $m^{*2}$, $\mu$ and T and, the partial derivatives with respect to one of these variables are taken at the fixed other two. The solution of Eqs. (A1) is as follows:

$$\begin{aligned} \frac{\partial \mu }{\partial T}\Big |_n&= \frac{ \Big (\frac{\partial \Pi }{\partial m^{*2}} -1 \Big )\frac{\partial h}{\partial T} - \frac{\partial h}{\partial m^{*2}} \frac{\partial \Pi }{\partial T} }{\Big (1-\frac{\partial \Pi }{\partial m^{*2}}\Big )\frac{\partial h}{\partial \mu } +\frac{\partial \Pi }{\partial \mu }\frac{\partial h}{\partial m^{*2}}} \,, \end{aligned}$$

(A2)

$$\begin{aligned} \frac{\partial m^{*}}{\partial T}\Big |_n&= \frac{ \frac{1}{2m^*}\frac{\partial h}{\partial \mu } \frac{\partial \Pi }{\partial T} -\frac{1}{2m^*}\frac{\partial \Pi }{\partial \mu }\frac{\partial h}{\partial T} }{\Big (1-\frac{\partial \Pi }{\partial m^{*2}}\Big )\frac{\partial h}{\partial \mu } +\frac{\partial \Pi }{\partial \mu }\frac{\partial h}{\partial m^{*2}}} \,. \end{aligned}$$

(A3)

Derivatives of h and $\Pi$ with respect to T can be expressed through the quantities $I_n$, or through $I_3$, d and ${\tilde{d}}$, with the help of Eq. (23) as

$$\begin{aligned} \frac{\partial h}{\partial T}&= \frac{n - \mu m^{*2} I_1 +m^{*3} I_3}{T}\nonumber \\&=\frac{m^*-\mu }{T} m^{*2} I_3 +\frac{n-2\mu m^{*2}({\tilde{d}}+d)}{T} \,, \nonumber \\ \frac{\partial \Pi }{\partial T}&= 5\lambda \frac{m^{*2} I_1-\mu m^{*}I_3}{T} \nonumber \\&=5\lambda m^{*2}\frac{m^*-\mu }{T} I_3+\frac{10\lambda m^{*2}}{T}({\tilde{d}}+d) \, , \end{aligned}$$

(A4)

and for the derivatives with respect to $m^{*2}$ and $\mu$ we have

$$\begin{aligned} \frac{\partial \Pi }{\partial m^{*2}}&= -\frac{5}{2}\lambda I_2=-\frac{5}{2} \lambda \big ( {I_3} +2\,({\tilde{d}} - d)\big ) \,,\,\,\nonumber \\ \frac{\partial \Pi }{\partial \mu }&= 5\lambda m^* I_3 \,, \quad \frac{\partial h}{\partial m^{*2}} = -\frac{m^*}{2} I_3 \,,\,\,\nonumber \\ \frac{\partial h}{\partial \mu }&= m^{*2}I_1=m^{*2}\big ( I_3 + 2\,({\tilde{d}} + d)\big )\,. \end{aligned}$$

(A5)

Now, using these expressions we can simplify the denominators in Eqs. (A2) and (A3) as

$$\begin{aligned}&\Big (1-\frac{\partial \Pi }{\partial m^{*2}}\Big )\frac{\partial h}{\partial \mu } +\frac{\partial \Pi }{\partial \mu }\frac{\partial h}{\partial m^{*2}} = \frac{\partial h}{\partial \mu }+5C m^{*2} I_1 \nonumber \\&\quad \quad =m^{*2} \big ( I_3 + 2\,({\tilde{d}} + d)\big )(1+5C)\,, \end{aligned}$$

(A6)

where the quantity

$$\begin{aligned} C=\frac{\lambda }{2}\left( I_2-\frac{I_3^2}{I_1}\right) =2\lambda \frac{{\tilde{d}}}{d } \frac{d\,\,I_3 +({\tilde{d}}^2-d^2)}{I_3 +2({\tilde{d}}+d)}\,, \end{aligned}$$

(A7)

wherefrom for $T\rightarrow T_\mathrm{cr}$ using (25) we get

$$\begin{aligned} C(T_\mathrm{cr})=2\lambda {{\tilde{d}}_\mathrm{cr}} \,. \end{aligned}$$

(A8)

From Eq. (A5) we find another useful relations for the derivatives of $\Pi$ and h:

$$\begin{aligned}&\frac{\partial \Pi }{\partial m^*}+\frac{\partial \Pi }{\partial \mu } = -10\lambda \,{m^{*}}{({\tilde{d}} - d)}\,, \nonumber \\&\frac{\partial h}{\partial m^*}+\frac{\partial h}{\partial \mu } =2{m^{*2}}({\tilde{d}}+d)\,. \end{aligned}$$

(A9)

These expressions are finite at $T=T_\mathrm{cr}$. Oppositely, differences of these derivatives are divergent at $T\rightarrow T_\mathrm{cr}$ and the leading terms are

$$\begin{aligned}&\Big (\frac{\partial \Pi }{\partial \mu } - \frac{\partial \Pi }{\partial m^*}\Big )\Big |_{T\rightarrow T_\mathrm{cr}} \rightarrow 10\lambda \mu _\mathrm{cr} I_3(T\rightarrow T_\mathrm{cr})\,,\ \nonumber \\&\Big (\frac{\partial h}{\partial \mu } - \frac{\partial h}{\partial m^*}\Big )\Big |_{T\rightarrow T_\mathrm{cr}} \rightarrow 2 \mu _\mathrm{cr}^2 I_3 (T\rightarrow T_\mathrm{cr})\,. \end{aligned}$$

(A10)

Now we can express temperature derivatives of $m^*$ and $\mu$ as

$$\begin{aligned} \frac{\partial \mu }{\partial T}\Big |_n&= \frac{-n\chi _\mu (T)}{TI_1(1+5C)} \,,\,\, \frac{\partial m^{*}}{\partial T}\Big |_n = \frac{-n\chi _m(T)}{T m^{*2}I_1(1+5C)} \,, \end{aligned}$$

(A11)

where

$$\begin{aligned} \chi _\mu (T)&= \frac{5}{2}\lambda {I_3} \Bigg (1-2(\mu +m^*)\frac{m^{*2}({\tilde{d}}+d)}{n}\Bigg ) \end{aligned}$$

(A12)

$$\begin{aligned}&\quad \quad \quad +\Bigg (1 +5\,\lambda {({\tilde{d}} - d)}\Bigg ) \nonumber \\&\quad \quad \quad \times \Bigg ( \frac{m^{*2}(m^*-\mu ) I_3}{n} + 1 - \frac{2\mu m^{*2}}{n}({\tilde{d}}+d)\Bigg )\,, \nonumber \\ \chi _m(T)&= \frac{5}{2}\lambda {I_3}\Bigg (1-\frac{4m^{*3}}{n}{({\tilde{d}}+d})\Bigg )\,. \end{aligned}$$

(A13)

At $T\rightarrow T_\mathrm{cr}$ we have $\chi _m=\chi _\mu \propto I_3$ since $(m-\mu )I_3\rightarrow 0$ in view of (19). Therefore both derivatives (A11) equal to each other and are finite,

$$\begin{aligned} \beta&=\frac{\partial \mu }{\partial T}\Big |_{n, T_\mathrm{cr}} = \frac{\partial m^{*}}{\partial T}\Big |_{n, T_\mathrm{cr}}\nonumber \\&= -\frac{5\lambda n}{2T_\mathrm{cr} \mu _\mathrm{cr}^2} \frac{1-\frac{4\mu _\mathrm{cr}^3}{n}({\tilde{d}}_\mathrm{cr} + d_\mathrm{cr})}{1+10\lambda {\tilde{d}}_\mathrm{cr}}\,. \end{aligned}$$

(A14)

We see that at $T\rightarrow T_\mathrm{cr}$ the divergency in denominator ($\propto I_1$) is canceled by the divergency in numerator ($\propto I_3$).

Note that for the ideal gas, when $\lambda =0$, both derivatives (A14) vanish. Thus, the finiteness of the derivatives (A11) is another manifestation of the effect of the self-consistent account of the interaction. Using Eq. (27) we can write the expansion of the coefficient $\beta$ for $\mu _\mathrm{cr}\ll T_\mathrm{cr}$  as

$$\begin{aligned} \beta&= -\frac{5\lambda n}{2T_\mathrm{cr}\mu _\mathrm{cr}^2}\Bigg \{ 1-\frac{\zeta (\frac{3}{2})t_\mathrm{cr}^{3/2}}{2(2\pi )^{3/2}}\frac{m^{*3}}{n} \Bigg (6 +\frac{15}{4}\frac{\zeta (\frac{5}{2})t_\mathrm{cr}}{\zeta (\frac{3}{2})} \nonumber \\&+5\lambda \frac{n}{\mu _\mathrm{cr}^3}\Bigg [1+\frac{9}{8} \frac{\zeta (\frac{5}{2})t_\mathrm{cr}}{\zeta (\frac{3}{2})} -\frac{3\zeta ({\textstyle \frac{3}{2}})t_\mathrm{cr}^{3/2}}{(2\pi )^{3/2}}\Bigg (\frac{\mu _\mathrm{cr}^3}{n} + \frac{5}{6}\lambda \Bigg ) \Bigg ]\Bigg )\Bigg \} \nonumber \\&+O(t_c^{7/2})\,. \end{aligned}$$

(A15)

We turn now to the combinations of partial derivatives appearing in Eq. (56). From Eq. (A11) we can construct

$$\begin{aligned}&\frac{\partial (\mu -m)}{\partial T}\Big |_n =-\frac{n\chi (T)}{T m^{*2}I_1(1+5C)}\,, \end{aligned}$$

(A16)

where the function $\chi (T)$ is

$$\begin{aligned} \chi (T)&= \chi _\mu (T) - \chi _m(T) \nonumber \\&\quad = \Big (1+5\lambda ({\tilde{d}}-d)\Big ) \frac{m^{*2}(m^*-\mu ) I_3}{n} \nonumber \\&\quad +\Big (1+10\lambda {\tilde{d}}\Big ) \Big (1-\frac{2\mu m^{*2}}{n}({\tilde{d}}+d)\Big )\nonumber \\&\quad +10\lambda \frac{m^{*2}({\tilde{d}}+d)^2}{n} \,. \end{aligned}$$

(A17)

We observe that the terms in $\chi _\mu$ and $\chi _m$ divergent at $T\rightarrow T_\mathrm{cr}$ cancel each other exactly and, therefore, in the limit $T\rightarrow T_\mathrm{cr}$ the function $\chi$ reduces to the finite quantity

$$\begin{aligned} \chi _\mathrm{cr} =1+{5\lambda }({\tilde{d}}_\mathrm{cr}-d_\mathrm{cr}) -\frac{2\mu _\mathrm{cr}^3}{n}({\tilde{d}}_\mathrm{cr}+d_\mathrm{cr})(1- 10\lambda d_\mathrm{cr})\,. \end{aligned}$$

(A18)

The lower limit of $\chi _\mathrm{cr}$ is realized for $n\rightarrow 0$ when $d_\mathrm{cr}\rightarrow 0$ and ${\tilde{d}}_\mathrm{cr}/d_\mathrm{cr}\rightarrow \frac{1}{2}$ and using Eq. (62) we have $\chi _\mathrm{cr}(n\rightarrow 0) \rightarrow \frac{1}{2}$ . The quantity $\chi _\mathrm{cr}$ is illustrated in Fig. 10. As we see, it exhibits a rather weak dependence on the coupling constant $\lambda$ and on the pion density n.

Fig. 10

Quantities $\chi _\mathrm{cr}$ and $\frac{T_\mathrm{cr}\alpha }{\mu _\mathrm{cr}}$ given in Eqs. (A18) and (A28), respectively, as functions of a density n for three values of the coupling parameter $\lambda$

Since quantity C remains finite at $T=T_\mathrm{cr}$ and $I_1$ diverges at $T\rightarrow T_\mathrm{cr}$ the derivative ${\partial (\mu -m^*)}/{\partial T}$ at $T=T_\mathrm{cr}$ vanishes. The same can be seen also directly from (A14). Thus, if we want to estimate the dependence of $\mu -m^*$ on the temperature for T close to $T_\mathrm{cr}$ we have to calculate the second derivatives.

After differentiating Eq. (A1) second time with respect to T, we obtain

$$\begin{aligned} \Big (2m^*-\frac{\partial \Pi }{\partial m^*}\Big )\frac{\partial ^2 m^*}{\partial T^2}\Big |_n- \frac{\partial \Pi }{\partial \mu }\frac{\partial ^2 \mu }{\partial T^2}\Big |_n&=\frac{\partial ^2\Pi }{\partial T^2} + A\,, \nonumber \\ \frac{\partial h}{\partial m^*}\frac{\partial ^2 m^*}{\partial T^2}\Big |_n + \frac{\partial h}{\partial \mu }\frac{\partial ^2 \mu }{\partial T^2}\Big |_n&=-\frac{\partial ^2 h}{\partial T^2}-m^{*}B\,, \end{aligned}$$

(A19)

where

$$\begin{aligned}&A = {\hat{D}}_T\frac{\partial \Pi }{\partial T}+{\hat{D}}_T^2(\Pi -m^{*2})\,, \nonumber \\&m^{*} B = {\hat{D}}_T\frac{\partial h}{\partial T} + {\hat{D}}_T^2 h \end{aligned}$$

(A20)

with ${\hat{D}}_T$ standing for the differential operator

$$\begin{aligned}&{\hat{D}}_T = \frac{\partial (\mu -m^*)}{\partial T}{\hat{D}}_- +\frac{\partial (\mu +m^*)}{\partial T}{\hat{D}}_+\,, \nonumber \\&\quad {\hat{D}}_{\pm } = \frac{1}{2}\Big (\frac{\partial }{\partial \mu }\pm \frac{\partial }{\partial m^*}\Big ) \,. \end{aligned}$$

(A21)

From Eq. (A4) second derivatives with respect to T can be written as

$$\begin{aligned} \frac{\partial ^2\Pi }{\partial T^2}&= -\frac{1}{T}\frac{\partial \Pi }{\partial T} +\frac{5\lambda }{T}\Big (m^{*2}\frac{\partial I_1}{\partial T} - \mu m^{*}\frac{\partial I_3}{\partial T}\Big ) \,, \nonumber \\ \frac{\partial ^2h}{\partial T^2}&= -\frac{1}{T}\frac{\partial h}{\partial T} +\frac{m^{*2}}{T}\Big (m^{*}\frac{\partial I_3}{\partial T} - \mu \frac{\partial I_1}{\partial T}\Big ) \,. \end{aligned}$$

(A22)

The solution of the system (A19) for the second derivatives, $\partial ^2 m^*/\partial T^2$ and $\partial ^2\mu /\partial T^2$, can be cast in the same form as for the first derivatives, Eqs. (A2) and (A3) with the replacement of $\partial h/\partial T$ and $\partial \Pi /\partial T$ through $\frac{\partial ^2 h}{\partial T^2} + m^{*}B$ and $\frac{\partial ^2\Pi }{\partial T^2} + A$, repsectively. Now using (A6), (A9), and (A10) we obtain

$$\begin{aligned} \frac{\partial ^2(m^*-\mu )}{\partial T^2}\Big |_n&= \frac{1+5\lambda ({\tilde{d}}-d)}{m^* I_1(1+5C)} \Big (\frac{1}{m^{*}}\frac{\partial ^2 h}{\partial T^2}+B\Big ) \nonumber \\&+ \frac{({\tilde{d}}+d)}{ m^* I_1(1+5C)} \Big (\frac{\partial ^2\Pi }{\partial T^2}+A\Big )\,. \end{aligned}$$

(A23)

The full evaluation of this expression is very cumbersome because of the necessity to calculate additional derivatives in Eqs. (A20) and (A22). However, the task is simplified, if we are interested in the value of this quantity at $T\rightarrow T_\mathrm{cr}$. In this case the quantity (A23) does not tend to zero only, if the divergency of $I_1(T\rightarrow T_\mathrm{cr})$ in the denominator is compensated by another divergent term. Since quantities d and ${\tilde{d}}$ are finite at $T_\mathrm{cr}$ the divergence has to be in the second derivatives with respect to the temperature and/or in the quantities A and B. To isolate these terms we can use the following considerations. First, we note that the quantities $\Pi$, $m^*$ and h, which derivatives enter in Eqs. (A20) and (A22) are finite at $T_\mathrm{cr}$. The partial derivative with respect to the temperature cannot increase the degree of the divergence of the integrals, since they lead to the multiplication of an integrand by the quantity $(\omega -\mu )\, f(\omega -\mu )$, where f is the Bose–Einstein distribution, being regular in the limit $k\rightarrow 0$ and $\mu \rightarrow m^*$. Similarly, the differential operator ${\hat{D}}_+$ does not lead to an enhancement of the divergence power of integrals, e.g., see Eq (A9), since the differentiation of the distribution function f is accompanied by the factor $\frac{1}{2}(m^*-\omega )/\omega$ killing additional divergencies at $k\rightarrow 0$. Thus, only terms with the ${\hat{D}}_-$ operator can be potentially divergent and, therefore, survive in (A23) at $T_\mathrm{cr}$. An additional analysis shows that the terms linear in ${\hat{D}}_-$ produce at the end terms proportional to $\frac{I_3}{I_1}\frac{\partial (\mu -m^*)}{\partial T}\propto I_3/I_1^2$, which vanish at $T\rightarrow T_\mathrm{cr}$. Concluding, we see that in the limit $T\rightarrow T_\mathrm{cr}$ we may keep in Eq. (A23) only the terms quadratic in ${\hat{D}}_-$. As the result we find

$$\begin{aligned}&\frac{\partial ^2(m^*-\mu )}{\partial T^2}\Big |_{n, T\rightarrow T_\mathrm{cr}} \rightarrow \frac{1}{m^*} \Big [\frac{\partial (\mu -m^*)}{\partial T}\Big ]^2 \nonumber \\&\quad \times \left( \frac{ 1+5\lambda ({\tilde{d}}-d) }{m^* I_1(1+5C)}{\hat{D}}_-^2 h + \frac{({\tilde{d}}+d)}{ I_1(1+5C)} {\hat{D}}_-^2\Pi \right) _{T_\mathrm{cr}} \,, \end{aligned}$$

(A24)

or after account for Eq. (A10) we have

$$\begin{aligned} \frac{\partial ^2(m^*-\mu )}{\partial T^2}\Big |_{n,T_\mathrm{cr}}&= \frac{n^2\chi _\mathrm{cr}^2\,}{T_\mathrm{cr}^2 \mu _\mathrm{cr}^4 \big (1+10\lambda {\tilde{d}}\big )^2} \Big [\frac{{\hat{D}}_-I_3}{I_1^3}\Big ]\Big |_{T\rightarrow T_\mathrm{cr}} \,, \end{aligned}$$

(A25)

where

$$\begin{aligned} \Big [\frac{{\hat{D}}_-I_3}{I_1^3}\Big ]\Big |_{T\rightarrow T_\mathrm{cr}}=\frac{4\pi \mu _\mathrm{cr}}{\, T_\mathrm{cr}^2}\,. \end{aligned}$$

(A26)

Combining all terms we write the expansion of $m^*-\mu$ for T close to $T_\mathrm{cr}$ as

$$\begin{aligned} m^*-\mu \approx \frac{\alpha }{2\mu _\mathrm{cr}} (T-T_\mathrm{cr})^2 \,, \end{aligned}$$

(A27)

with

$$\begin{aligned} \alpha = \frac{4\pi n^2\chi _\mathrm{cr}^2\,}{\mu _\mathrm{cr}^{2} T_\mathrm{cr}^4 \big (1+10\lambda {\tilde{d}}\big )^2} \,. \end{aligned}$$

(A28)

The dependence of the quantity $T_\mathrm{cr}\,\alpha /\mu _\mathrm{cr}$ on a density is shown in Fig. 10. We see that this product is finite for $n\rightarrow 0$.

Appendix B: $\varpi _Q$ in isospin-symmetrical system

In Sect. 4 we have obtained that the variance of the total charge, $\varpi _Q$, diverges at $T\rightarrow T_\mathrm{cr}$ in the isospin-symmetrical system even with taking into account of a strong pion-pion interaction within $\lambda \phi ^4$ model. In this Appendix we study, under which conditions this divergency can be eliminated. For this we rewrite the variances derived in Sect. 3, as the derivatives of the densities with respect to chemical potentials through the derivatives of the chemical potentials with respect to densities. Below we do not indicate explicitly the dependence on the number of $\pi ^0$ mesons, assuming that it is fixed.

The chemical potentials $\mu _a$ and the particle densities are connected by Eq. (4). We consider this relation as an equation for $n_a(\mu _+,\mu _-)$. Then we can calculate derivatives in (15). On the other hand, the system of equations (4) implicitly defines functions $\mu _a=\mu _a(n_+,n_-)$. The relation among the partial derivatives of direct and inverse function implies in this case

$$\begin{aligned} \delta _{ab}=\sum _{c=\pm }\frac{\partial n_a}{\partial \mu _c}\frac{\partial \mu _c}{\partial n_b}\,,\quad a,b=\pm \,. \end{aligned}$$

(B1)

Terms with $\pi ^0$ vanish here, as we assume that $n_{\pi ^0}$ is fixed. Then the letter relation can be written as inversion of a $2\times 2$ matrix,

$$\begin{aligned} \frac{\partial n_a}{\partial \mu _b}=[M^{-1}]_{ab},\quad M=\left[ \begin{array}{c@{\quad }c} \frac{\partial \mu _+}{\partial n_+} &{} \frac{\partial \mu _+}{\partial n_-}\\ \frac{\partial \mu _-}{\partial n_+} &{} \frac{\partial \mu _-}{\partial n_-} \end{array}\right] \,, \end{aligned}$$

(B2)

or explicitly as

$$\begin{aligned} \frac{\partial n_\pm }{\partial \mu _\pm }&=\frac{1}{E}\frac{\partial \mu _\mp }{\partial n_\mp } \,,\quad \frac{\partial n_\pm }{\partial \mu _\mp }=-\frac{1}{E}\frac{\partial \mu _\pm }{\partial n_\mp }\,, \nonumber \\ E&=\frac{\partial \mu _+}{\partial n_+}\frac{\partial \mu _-}{\partial n_-} -\frac{\partial \mu _+}{\partial n_-}\frac{\partial \mu _-}{\partial n_+}\,. \end{aligned}$$

(B3)

Using these relations the charge variance (45) can be written as

$$\begin{aligned} \varpi _Q =&3\frac{T}{n}\frac{1}{E}\Big (\frac{\partial \mu _-}{\partial n_-} +\frac{\partial \mu _+}{\partial n_+} +\frac{\partial \mu _+}{\partial n_-} +\frac{\partial \mu _-}{\partial n_+} \Big ) \nonumber \\ =&3\frac{T}{n}\frac{1}{E}\Big (\frac{\partial }{\partial n_+} +\frac{\partial }{\partial n_-}\Big )( \mu _+ + \mu _- )\,. \end{aligned}$$

(B4)

Now we can introduce densities $n_\mathrm{ch}=n_++n_-$ and $n_Q=n_+-n_-$ and the corresponding derivatives

$$\begin{aligned} \frac{\partial }{\partial n_\pm }=\frac{\partial }{\partial n_\mathrm{ch}} \pm \frac{\partial }{\partial n_Q}\,. \end{aligned}$$

(B5)

So, the quantity E in (B3) can be rewritten as

$$\begin{aligned} E&= \frac{\partial (\mu _+ - \mu _-)}{\partial n_Q} \frac{\partial (\mu _+ + \mu _-)}{\partial n_\mathrm{ch}} -\frac{\partial (\mu _+ - \mu _-)}{\partial n_\mathrm{ch}}\frac{\partial (\mu _+ + \mu _-)}{\partial n_Q} \,. \end{aligned}$$

(B6)

Since we consider fluctuations in the isospin-symmetrical system we need these derivatives for $n_Q=0$ and $n_\mathrm{ch}=\frac{2}{3} n$. Since $\mu _+ - \mu _-$ and $\mu _+ + \mu _-$ are odd and even functions of $n_Q$, respectively, the last term in (B6) vanishes.¹ Then from Eqs. (B4) and (B6) we find a simple relation

$$\begin{aligned} \varpi _Q&=6\frac{T}{n}\Big (\frac{\partial (\mu _+ - \mu _-)}{\partial n_Q} \Big |_{n_Q=0}\Big )^{-1}\,. \end{aligned}$$

(B7)

In order to proceed further we calculate the partial derivatives in Eq. (B7). We start with the equation relating the charge density and the chemical potentials,

$$\begin{aligned} n_\mathrm{Q} =\big [ f_+ \big ]_p - \big [ f_- \big ]_p \,. \end{aligned}$$

(B8)

We introduce here a short notation for the momentum integration $\big [\dots \big ]_p =\intop \frac{\mathrm{d}^3 p}{(2\pi )^3}(\dots )$, and the Bose–Einstein occupation factor

$$\begin{aligned} f_\pm =\big (e^{(\omega _\pm (p,n,n_\mathrm{ch},n_Q) - \mu _\pm (n,n_\mathrm{ch},n_Q))/T}-1\big )^{-1}\,, \end{aligned}$$

(B9)

where $\omega _\pm (p,n,n_\mathrm{ch},n_Q)$ is the spectrum of a $\pi ^\pm$ meson, which in general case can depend on the total density of pions, n, the density of charged pions, $n_\mathrm{ch}$, and the charge density $n_Q$. Differentiating Eq. (B8) with respect to $n_Q$ keeping other densities fixed we get

$$\begin{aligned} 1&= \Big [ \frac{f_-(1+f_-)}{T} \frac{\partial (\omega _- - \mu _-)}{\partial n_{Q}} \Big ]_p \nonumber \\&\quad \quad - \Big [ \frac{f_+(1+f_+)}{T} \frac{\partial (\omega _+ - \mu _+)}{\partial n_{Q}} \Big ]_p\,. \end{aligned}$$

(B10)

Hence for $n_Q=0$ we obtain

$$\begin{aligned} \frac{\partial ( \mu _+ - \mu _-)}{\partial n_{Q}}\Big |_{n_Q=0}&=\frac{T}{\big [ f(1+f) \big ]_p} \nonumber \\&+\frac{ \big [ f(1+f) \frac{\partial (\omega _+ -\omega _- )}{\partial n_{Q}}\Big |_{n_Q=0} \big ]_p}{{\big [ f(1+f) \big ]_p}} \,. \end{aligned}$$

(B11)

Substituting this relation in Eq. (B7) and taking into account that $\big [ f(1+f)\big ]_p =I_1 T$, we obtain the relation

$$\begin{aligned} \varpi _Q=6\frac{T}{n}\frac{I_1}{1 +\frac{1}{T}\Big [ f(1+f)\frac{\partial (\omega _+ -\omega _- )}{\partial n_{Q}}\Big |_{n_Q=0} \Big ]_p } \,, \end{aligned}$$

(B12)

which can be rewritten as Eq. (48).

Alternatively to Eq. (48), which explicitly depends on the model for particle spectra, we can rewrite $\varpi _Q$ through the free-energy density $F(n_+,n_-,T)$. The chemical potentials as functions of densities can be obtained as partial derivatives of the free energy, $\mu _\pm =\frac{\partial F}{\partial n_\pm }$, at fixed T and V. Then we have $\mu _+ - \mu _-=2\frac{\partial F}{\partial n_\mathrm{Q}}$ and therefore Eq. (B7) takes the form

$$\begin{aligned} \varpi _Q&=3\frac{T}{n}\Big (\frac{\partial ^2 F}{\partial n_{Q}^2}\Big |_{n_Q=0} \Big )^{-1}\,. \end{aligned}$$

(B13)

Appendix C: $I_n (T\rightarrow T_\mathrm{cr}^{{\tilde{a}}})$ at non-zero isospin imbalance

In this Appendix we consider pion gas with a small isospin imbalance, $\delta n_Q\ne 0$ or $\delta n_G\ne 0$, so the critical temperature of the BEC is largest for the species ${\tilde{a}}$, i.e., $T_\mathrm{cr}^{{\tilde{a}}}=\max _a\{ T_\mathrm{cr}^{a}\}$. We are interested in quantities $I_n^b(T\rightarrow T_\mathrm{cr}^{{\tilde{a}}})$ for $b\ne {\tilde{a}}$ in the case of very small imbalance $|\delta n_{Q,G}|\ll n$, i.e. when $|T_\mathrm{cr}^{b,{\tilde{a}}}-T_\mathrm{cr}|\ll T_\mathrm{cr}$, where $T_\mathrm{cr}$ is the critical temperature for the isospin-symmetrical system with the density n. The derived results are needed for expansion of Eqs. (85), (92), and (98). To find leading and next-to-leading terms in the $\delta n_{Q,G}$ expansion of $I_n^{b}(T_\mathrm{cr}^{{\tilde{a}}})$ for small $\delta n_{Q,G}$, we first separate the divergent part (19), determined by the small momenta in the integrals (18). Then we have

$$\begin{aligned} I_n^{b}(T_\mathrm{cr}^{{\tilde{a}}}) \approx \frac{T_\mathrm{cr}^{{\tilde{a}}}}{2^{3/2}\pi \sqrt{m_b^*(T_\mathrm{cr}^{{\tilde{a}}})[m_b^*(T_\mathrm{cr}^{{\tilde{a}}})-\mu _b(T_\mathrm{cr}^{{\tilde{a}}})}]} + \delta I_{\mathrm{cr}, n}\,, \end{aligned}$$

(C1)

where in the regular term $\delta I_{\mathrm{cr}, n}$ we can take all quantities in the isospin symmetrical limit.

1.1 1. Variation $\delta n_Q$ at $\delta n_G=0$

Here we consider variations $\delta n_Q\ne 0$ at $\delta n_G=0$, studied in Sect. 5.2, point (i). The condition $\delta n_G=0$ implies $\delta n_+ = -\delta n_-$ and along this line, as we see in Fig. 8 (b,c), the maximal critical temperature is realized for the most abundant charged species, i.e. $T_\mathrm{cr}^{+} \lessgtr T_\mathrm{cr}^{-}$ provided $\delta n_+ \lessgtr \delta n_-$ , respectively. To be specific consider $\delta n_Q>0$, then the critical temperature of positively charged pions, $T_\mathrm{cr}^{+}$, is the highest one.

Our aim here is to find quantities $I_n^{0,-}( T_\mathrm{cr}^{+})$ for $\delta n_Q\ll n$ and, therefore, for $\delta T_\mathrm{cr}^{+}=T_\mathrm{cr}^{+} - T_\mathrm{cr}\ll T_\mathrm{cr}$, where $T_\mathrm{cr}$ is the critical temperature of the isospin-symmetrical pion gas with density n. To use Eq. (C1) we have to evaluate the differences $m_a^*(T_\mathrm{cr}^{+}) - \mu _a(T_\mathrm{cr}^{+})$ for $a=0,-$. Taking into account (60), we find

$$\begin{aligned}&m_{-}^*(T_\mathrm{cr}^{+}) - \mu _{-}(T_\mathrm{cr}^{+})=m_{+}^*(T_\mathrm{cr}^{+})-\mu _{+}(T_\mathrm{cr}^{+}) \nonumber \\&\quad +\mu _{+}(T_\mathrm{cr}^{+}) - \mu _{-}(T_\mathrm{cr}^{+}) \approx 2\delta \mu _{+}(T_\mathrm{cr}^{+}) \approx \frac{\delta n_Q}{\mu _\mathrm{cr}^2 I_1(T_\mathrm{cr}^{+})}, \nonumber \\&m_{0}^*(T_\mathrm{cr}^{+}) - \mu _{0}(T_\mathrm{cr}^{+}) = m_{+}^*(T_\mathrm{cr}^{+})-\mu _{+}(T_\mathrm{cr}^{+}) \nonumber \\&\quad +\mu _{+}(T_\mathrm{cr}^{+}) - \mu _{0}(T_\mathrm{cr}^{+}) \approx \delta \mu _{+}(T_\mathrm{cr}^{+}) \approx \frac{\delta n_Q}{2\mu _\mathrm{cr}^2 I_1(T_\mathrm{cr}^{+})}, \end{aligned}$$

(C2)

where we used that the effective masses do not depend on $\delta n_Q$, i.e. $m^*_{-,0}(T_\mathrm{cr}^{+})=m^*_{+}(T_\mathrm{cr}^{+})$ and corrections to the chemical potentials are given by Eq. (60). Also we can write the expansion for the effective mass and the chemical potential

$$\begin{aligned} m^*_{-}(T_\mathrm{cr}^{+})&\approx m^*_{0}(T_\mathrm{cr}^{+}) \approx m^*_{+}(T_\mathrm{cr}^{+}) = \mu _+(T_\mathrm{cr}^{+})\,, \end{aligned}$$

(C3)

$$\begin{aligned} \mu _+(T_\mathrm{cr}^{+})&\approx \mu (T_\mathrm{cr}^{+}) + \frac{\delta n_Q}{2\mu _\mathrm{cr}^2I_1(T_\mathrm{cr}^{+})} \nonumber \\&\approx \mu _\mathrm{cr}+\beta \, T_\mathrm{cr}\frac{\delta n_Q}{2n}\frac{\eta _{T_\mathrm{cr}}^{(Q)}}{\eta _n} + \frac{\delta n_Q}{2\mu _\mathrm{cr}^2I_1(T_\mathrm{cr}^{+})} \,, \end{aligned}$$

(C4)

where we used Eqs. (A14) and that the critical temperature $T_\mathrm{cr}^+$ is shifted with respect to $T_\mathrm{cr}$ according to Eq. (61),

$$\begin{aligned} T_\mathrm{cr}^+\approx T_\mathrm{cr}\Big (1+\frac{\delta n_Q}{2n}\frac{\eta _{T_\mathrm{cr}}^{(Q)}}{\eta _n}\Big ) \,. \end{aligned}$$

(C5)

In Eqs. (C2) and (C4) the quantity $I_1(T_\mathrm{cr}^{+})$ is calculated with effective masses and chemical potentials computed for the isospin-symmetrical pion gas but at the temperature $T_\mathrm{cr}^+$, i.e. $m^*(T_\mathrm{cr}^{+})$ and $\mu (T_\mathrm{cr}^{+})$, respectively.

To evaluate $I_1(T_\mathrm{cr}^{+})$ we use (C1), where we replace $m_a^*$ and $\mu _a$ to $m^*$ and $\mu$, respectively and take into account that

$$\begin{aligned} m^*(T_\mathrm{cr}^+)\approx \mu _\mathrm{cr} +\beta \, T_\mathrm{cr} \frac{\delta n_Q}{2n}\frac{\eta _{T_\mathrm{cr}}^{(Q)}}{\eta _n} \end{aligned}$$

(C6)

according to Eqs. (A14) and (61). The mass difference in the numerator of the singular term in $I_1(T_\mathrm{cr}^{+})$ can be rewritten as

$$\begin{aligned}&m^*(T_\mathrm{cr}^{+})-\mu (T_\mathrm{cr}^{+})=m^*(T_\mathrm{cr}+\delta T_\mathrm{cr}^{+})-\mu (T_\mathrm{cr}+\delta T_\mathrm{cr}^{+}) \nonumber \\&\quad \approx \frac{1}{2}\frac{\alpha }{\mu _\mathrm{cr}}\,[\delta T_\mathrm{cr}^{+}]^2 \approx \frac{\alpha }{2\mu _\mathrm{cr}}\, \frac{(\delta n_Q)^2}{4n^2}\frac{\eta _{T_\mathrm{cr}}^{(Q)2}}{\eta _n^2} T_\mathrm{cr}^2\,, \end{aligned}$$

(C7)

where we used Eq. (A27) with the coefficient $\alpha$ given in Eq. (A28). Thus, using Eqs. (C5), (C6), and (C7) we obtain

$$\begin{aligned} I_1(T_\mathrm{cr}^{+})&\approx \frac{1}{\pi \alpha ^{1/2}_\mathrm{cr}} \Big (\frac{n}{\delta n_Q}\frac{\eta _n}{\eta _{T_c}^{(Q)}} +\frac{1}{2}-\frac{\beta }{4}\frac{T_\mathrm{cr}}{\mu _\mathrm{cr}}\Big ) + \delta I_{\mathrm{cr},1}\,. \end{aligned}$$

(C8)

Now taking $I_n^{a}(T_\mathrm{cr}^{+})$ from (C1) we can write for $a=``-''$:

$$\begin{aligned} I_n^{-}(T_c^{+})&\approx \beta ^{(Q)} \frac{n}{\delta n_Q} \Big ( 1 +\frac{\delta n_Q}{2n}\frac{\eta _{T_c}^{(Q)}}{\eta _n} \nonumber \\&\times \Big (\frac{3}{2}-\frac{3}{4}\beta \frac{T_\mathrm{cr}}{\mu _\mathrm{cr}} + \pi \alpha ^{1/2}_\mathrm{cr}\delta I_{\mathrm{cr},1}\Big ) \Big ) + \delta I_{\mathrm{cr},n}\,, \end{aligned}$$

(C9)

where

$$\begin{aligned} \beta ^{(Q)} = \sqrt{\frac{\mu _\mathrm{cr}T_\mathrm{cr}^2}{ (2\pi )^{3}\alpha ^{1/2}_\mathrm{cr} n} \frac{\eta _n}{\eta _{T_c}^{(Q)}} }\,. \end{aligned}$$

Analogously, for neutral pions we get

$$\begin{aligned} I_n^{(0)}(T_c^{(+)})\approx \sqrt{2} [I_n^{(-)}(T_c^{(+)}) -\delta I_{\mathrm{cr},n}] +\delta I_{\mathrm{cr},n}\,. \end{aligned}$$

(C10)

Some comments about parameters of our expansions are in order. Expressions for shifts of pion effective masses and chemical potentials obtained in Sect. 5.1 are derived as expansions in $\delta n_a$ up to linear terms $O(\delta n_a)$, see Eqs. (53) and (55). The results are valid for any temperature $T\ge \max _a \{T_\mathrm{cr}^a\}$. The difference between the effective mass and the chemical potential, e.g. in Eq. (C2), is $\propto \delta n_Q/I_1(T)$. Formally for arbitrary temperatures $T>T_\mathrm{cr}^+$ this result is of the order $O(\delta n_Q)$. However, for $T\rightarrow T_\mathrm{cr}^+=T_\mathrm{cr}+O(\delta n_Q)$ the quantity $I_1$ behaves at the leading order like $I(T_\mathrm{cr}^+)\propto 1/\delta n_Q$, see Eq. (C8). Therefore, the expansion (C2) is effectively of the order $O(\delta n_Q^2)$. The same expansion order is explicitly seen in the difference $m^*-\mu$ at $T_\mathrm{cr}^+$ for the isospin symmetric medium, Eq. (C7), which is based on the expansion (A27) independently on the $\delta n_Q$ and that $\delta T_\mathrm{cr}^a\propto \delta n_Q$. The final expressions of this section (C8), (C9), and (C10) hold up to terms linear in $\delta n_Q$.

1.2 2. Variation $\delta n_G$ at $\delta n_Q =0$

(a) Let now $\delta n_Q=0$ and $\delta n_G<0$. In this case, the maximal is the critical temperature of the BEC for neutral pions, $T_\mathrm{cr}^{0}$. In order to expand relations (92) in small quantity $-\delta n_G\ll n$ we need the corresponding expansions of $I_n^{+}(T_\mathrm{cr}^0)$. We can use a relation analogous to (C1) and expand the mass-chemical potential difference in the denominator with the help of Eq. (63) as follows

$$\begin{aligned}&m_{+}^*(T_\mathrm{cr}^{0}) - \mu _{+}(T_\mathrm{cr}^{0}) \approx m_0^*(T_\mathrm{cr}^{0})+3\delta m^*_+(T_\mathrm{cr}^{0}) \nonumber \\&\quad \quad - \mu _0(T_\mathrm{cr}^{0}) - 3\delta \mu _+(T_\mathrm{cr}^{0})= 3\big (\delta m^*_+(T_\mathrm{cr}^{0}) -\delta \mu _+(T_\mathrm{cr}^{0})\big ) \nonumber \\&\quad \quad = m^{*}(T_\mathrm{cr}^{0})\frac{\delta n_G}{ n} \big (\frac{1}{2}\eta _m^{(G)}(T_\mathrm{cr}^{0}) -\eta _\mu ^{(G)}(T_\mathrm{cr}^{0})\big )\,. \end{aligned}$$

(C11)

We observe that although each of the quantities $\eta _m^{(G)}(T_\mathrm{cr}^{0})$ and $\eta _\mu ^{(G)}(T_\mathrm{cr}^{0})$ remains finite, if $T_\mathrm{cr}^0\rightarrow T_\mathrm{cr}$, see Eq. (63), their difference in (C11) vanishes and hence integrals $I_n^+(T_\mathrm{cr}^0)$ are enhanced, when $\delta n_G\rightarrow 0$ and $T_\mathrm{cr}^0\rightarrow T_\mathrm{cr}$. Being interested only in terms $\sim O(\delta n_G)$ in the small-$\delta n_G$ expansion, we can write

$$\begin{aligned} m_{+}^*(T_\mathrm{cr}^{0}) - \mu _{+}(T_\mathrm{cr}^{0})&\approx -\frac{\delta n_G}{m^{*2} I_1(T_\mathrm{cr}^{0}) } \frac{1 +\lambda (I_2 -I_3)}{1+2C}\Big |_{T_\mathrm{cr}} \nonumber \\&= -\frac{\delta n_G}{\mu _\mathrm{cr}^2 I_1(T_\mathrm{cr}^{0})}\frac{\eta _{T_\mathrm{cr}}^{(G)}}{\eta _{T_\mathrm{cr}}^{(Q)}}\,, \end{aligned}$$

(C12)

where in the last equation we used Eqs. (23), (64), (65), and (A8). Next-to-leading terms in these expansions are of the order $O([\delta n_G/ I_1(T_\mathrm{cr}^{0})]^2)$. For the integral $I_1(T_\mathrm{cr}^{0})$ we can use the expansion (C1), where we replace $m_a^*$ and $\mu _a$ by $m^*$ and $\mu$, respectively. According to Eq. (64) we have

$$\begin{aligned} T_\mathrm{cr}^0=T_\mathrm{cr}\Big (1-\frac{2}{3}\frac{\delta n_G}{n} \frac{\eta _{T_\mathrm{cr}}^{(G)}}{\eta _n}\Big ) + O\big ((\delta n_G)^2\big ) \,. \end{aligned}$$

(C13)

The expression for the $m-\mu$ difference can be written in analogy to Eq. (C7) using Eqs. (A27) and (C13):

$$\begin{aligned}&m^*(T_\mathrm{cr}^{0})-\mu (T_\mathrm{cr}^{0}) \approx \frac{\alpha _\mathrm{cr}}{2\mu _\mathrm{cr}}\,\frac{4}{9} \frac{(\delta n_G)^2}{n^2}\frac{\eta _{T_\mathrm{cr}}^{(G)2}}{\eta _n^2} T_\mathrm{cr}^2 \,. \end{aligned}$$

(C14)

Since, as we show below, $I_1(T_\mathrm{cr}^0)\propto 1/\delta n_G$, the expansion (C12) is of the same quadratic order in $\delta n_G$ as expansion (C14). Additionally, to get the expansion $I_1(T_\mathrm{cr}^{0})$ we need the expansion for the effective mass,

$$\begin{aligned} m^*(T_\mathrm{cr}^0)&= m^*(T_\mathrm{cr})+\beta \delta T_\mathrm{cr}^0 + O\big ((\delta T_\mathrm{cr}^0)^2\big ) \nonumber \\&= \mu _\mathrm{cr} -\beta \, T_\mathrm{cr} \frac{2}{3}\frac{\delta n_G}{n}\frac{\eta _{T_\mathrm{cr}}^{(Q)}}{\eta _n} +O\big ((\delta n_G)^2\big ) \end{aligned}$$

(C15)

where we used (A14) and (C13).

Thus, we obtain

$$\begin{aligned} I_1(T_\mathrm{cr}^{0}) =&-\frac{1}{\pi \alpha ^{1/2}_\mathrm{cr}} \Big (\frac{3n}{4\delta n_G}\frac{\eta _n}{\eta _{T_\mathrm{cr}}^{(G)}} -\frac{1}{2} +\frac{\beta }{4} \frac{T_\mathrm{cr}}{\mu _\mathrm{cr}} \Big ) \nonumber \\&\quad \quad + \delta I_{\mathrm{cr},1} + O(\delta n_G)\,. \end{aligned}$$

(C16)

To derive the expansion of $I_n^{+}(T_\mathrm{cr}^{0})$ we also need the expansion for the effective mass $m_+^*(T_\mathrm{cr}^0)$, which we obtain using relations (63),

$$\begin{aligned} m_{+}^*(T_\mathrm{cr}^{0})&\approx m_0^*(T_\mathrm{cr}^{0})+3\delta m^*_+(T_\mathrm{cr}^{0}) \nonumber \\&=\mu (T_\mathrm{cr}^{0})-2\delta \mu _+(T_\mathrm{cr}^{0}) + 3\delta m^*_+(T_\mathrm{cr}^{0}) \nonumber \\&\approx \mu _\mathrm{cr}+\beta \delta T_\mathrm{cr}^0-2\delta \mu _+(T_\mathrm{cr}^{0}) + 3\delta m^*_+(T_\mathrm{cr}^{0}) \nonumber \\&\approx \mu _\mathrm{cr}\Bigg (1-\beta \frac{2}{3}\frac{\delta n_G}{n}\frac{\eta _{T_\mathrm{cr}}^{(G)}}{\eta _n}\frac{T_\mathrm{cr}}{\mu _\mathrm{cr}} \nonumber \\&+ \frac{\delta n_G}{3n}\Bigg (\frac{3}{2}\eta _m^{(G)}(T_\mathrm{cr})-2\eta _\mu ^{(G)}(T_\mathrm{cr})\Bigg ) \Bigg ) \nonumber \\&\approx \mu _\mathrm{cr}\Bigg (1-\beta \frac{2}{3}\frac{\delta n_G}{n}\frac{\eta _{T_\mathrm{cr}}^{(G)}}{\eta _n}\frac{T_\mathrm{cr}}{\mu _\mathrm{cr}} + \frac{\lambda \delta n_G}{3\mu _\mathrm{cr}^{3}(1+2C)} \Big )\nonumber \\&\quad + O\Bigg ((\delta n_G)^2\Bigg ). \end{aligned}$$

(C17)

The expansion for the chemical potential $\mu _\mathrm{cr,0}$ is obtained using Eqs. (63) and (C13),

$$\begin{aligned} \mu _\mathrm{cr,0}&= \mu _0(T_\mathrm{cr}^0) = \mu (T_\mathrm{cr}^0)+\delta \mu _0(T_\mathrm{cr}^0) \nonumber \\&\simeq \mu _\mathrm{cr}+\beta \delta T_\mathrm{cr}^0 - \mu _\mathrm{cr}\frac{2}{3} \frac{\delta n_G}{n}\eta _\mu ^{(G)}(T_\mathrm{cr}) +O\big ((\delta T_\mathrm{cr}^0)^2,(\delta n_G)^2\big ) \nonumber \\&\approx \mu _\mathrm{cr}\Big (1 -\beta \frac{2}{3}\frac{\delta n_G}{n}\frac{\eta _{T_\mathrm{cr}}^{(G)}}{\eta _n}\frac{T_\mathrm{cr}}{\mu _\mathrm{cr}} -\frac{2\lambda \delta n_G}{3\mu _\mathrm{cr}^{3}(1+4\lambda {\tilde{d}}_\mathrm{cr})} \Big )+ O\big ((\delta n_G)^2\big )\,. \end{aligned}$$

(C18)

Now substituting Eqs. (C12), (C13), and (C17) in Eq. (C1) we can expand

$$\begin{aligned}&I_n^{+}(T_\mathrm{cr}^0) \approx -\beta ^{(G)}\frac{n}{\delta n_G} \Big [1 -\frac{\lambda \delta n_G}{6\mu _\mathrm{cr}^3(1+4\lambda {\tilde{d}}_\mathrm{cr})} \nonumber \\&- \frac{2\delta n_G}{3n}\frac{\eta _{T_\mathrm{cr}}^{(G)}}{\eta _n} \Big (\frac{3}{2} -\frac{3}{4}\beta \frac{T_\mathrm{cr}}{\mu _\mathrm{cr}} + \pi \alpha ^{1/2}_\mathrm{cr}\delta I_\mathrm{cr 1}(T_\mathrm{cr}) \Big ) \Big ] \nonumber \\&+ \delta I_{\mathrm{cr},n} + O(\delta n_G) \,, \end{aligned}$$

(C19)

where

$$\begin{aligned} \beta ^{(G)} = \beta ^{(Q)} \frac{\sqrt{3}}{2}\frac{\eta ^{(Q)}_{T_\mathrm{cr}}}{\eta ^{(G)}_{T_\mathrm{cr}}}\,. \end{aligned}$$

(C20)

(b) Now we consider the case $\delta n_G>0$. The maximal critical temperature is now $T_\mathrm{cr}^{+}$. To expand relations (98) we have to find expansions of $I_n^{0}(T_\mathrm{cr}^+)$ for $\delta n_G\ll n$. For this we need the expansion for the critical temperature,

$$\begin{aligned} T_\mathrm{cr}^+=T_\mathrm{cr}\Big (1+ \frac{1}{3}\frac{\delta n_G}{n} \frac{\eta _{T_\mathrm{cr}}^{(G)}}{\eta _n} \Big ) +O\big ((\delta n_G)^2\big )\,, \end{aligned}$$

(C21)

and for the mass difference

$$\begin{aligned} m_{0}^*(T_\mathrm{cr}^{+}) - \mu _{0}(T_\mathrm{cr}^{+})&\approx 3\big (\delta \mu _+(T_\mathrm{cr}^+)- \delta m_+^* (T_\mathrm{cr}^+)\big ) \nonumber \\&= \frac{\delta n_G}{\mu _\mathrm{cr}^2 I_1(T_\mathrm{cr}^{+})}\frac{\eta _{T_\mathrm{cr}}^{(G)}}{\eta _{T_\mathrm{cr}}^{(Q)}}\,, \end{aligned}$$

(C22)

which we derived in a similar way as in Eqs. (C11) and (C12). To expand $I_1(T_\mathrm{cr}^{+})$ we need the difference

$$\begin{aligned}&m^*(T_\mathrm{cr}^{+})-\mu (T_\mathrm{cr}^{+})\approx \frac{\alpha }{2\mu _\mathrm{cr}}\, \frac{(\delta n_G)^2}{9n^2}\frac{\eta _{T_\mathrm{cr}}^{(G)2}}{\eta _n^2} T_\mathrm{cr}^2\,, \end{aligned}$$

(C23)

obtained using Eqs. (A27) and (C21), and the relation for the effective mass,

$$\begin{aligned} m^*(T_\mathrm{cr}^+)\approx \mu _\mathrm{cr} +\beta \, T_\mathrm{cr} \frac{1}{3}\frac{\delta n_G}{n}\frac{\eta _{T_\mathrm{cr}}^{(Q)}}{\eta _n} +O\big ((\delta n_G)^2\big )\,, \end{aligned}$$

(C24)

where we used Eq. (A14) and (C21). As we have argued above for the cases described by Eqs. (C2) and (C7) and Eqs. (C12) and (C14), the differences between an effective mass and chemical potentials in Eqs. (C22) and (C23) prove to be of the order $(\delta n_G)^2$, that is seen after taking into account that $I_1(T_\mathrm{cr}^{+})\propto 1/\delta n_G$.

Now substituting Eqs. (C23) and (C21) in Eq. (C1) with the pion mass and the chemical potential taken as in the isospin symmetrical matter, we obtain

$$\begin{aligned} I_1(T_\mathrm{cr}^{+}) \approx&\frac{1}{\pi \alpha ^{1/2}_\mathrm{cr}} \Big (\frac{3n}{2\delta n_G}\frac{\eta _n}{\eta _{T_\mathrm{cr}}^{(G)}} +\frac{1}{2} -\frac{\beta }{4}\frac{T_\mathrm{cr}}{\mu _\mathrm{cr}}\Big ) \nonumber \\&+ \delta I_\mathrm{cr,1} +O(\delta n_G) \,. \end{aligned}$$

(C25)

Now using this result we can evaluate Eq. (C22) and substitute it in Eq. (C1) together with the effective mass

$$\begin{aligned} m_{0}^*(T_\mathrm{cr}^{+}) =&\mu _\mathrm{cr}\Big (1 +\beta \frac{1}{3}\frac{\delta n_G}{n}\frac{\eta _{T_\mathrm{cr}}^{(G)}}{\eta _n}\frac{T_\mathrm{cr}}{\mu _\mathrm{cr}} \nonumber \\&\quad - \frac{\delta n_G}{3n}\Big (\frac{3}{2}\eta _m^{(G)}(T_\mathrm{cr})-\eta _\mu ^{(G)}(T_\mathrm{cr})\Big ) \Big ) +O\big ((\delta n_G)^2\big ) \nonumber \\ =&\mu _\mathrm{cr}\Big (1+\beta \frac{1}{3}\frac{\delta n_G}{n}\frac{\eta _{T_\mathrm{cr}}^{(G)}}{\eta _n}\frac{T_\mathrm{cr}}{\mu _\mathrm{cr}} - \frac{\lambda \delta n_G}{3\mu _\mathrm{cr}^{3}(1+2C)} \Big ) \nonumber \\&\quad +O\big ((\delta n_G)^2\big )\,, \end{aligned}$$

(C26)

and the chemical potential

$$\begin{aligned} \mu _\mathrm{cr,+} =&\mu _\mathrm{cr}\Big (1+\beta \frac{1}{3}\frac{\delta n_G}{n}\frac{\eta _{T_\mathrm{cr}}^{(G)}}{\eta _n}\frac{T_\mathrm{cr}}{\mu _\mathrm{cr}} + \frac{\lambda \delta n_G }{3\mu _\mathrm{cr}^{3}(1+2C)} \Big ) \nonumber \\&\quad +O\big ((\delta n_G)^2\big )\,, \end{aligned}$$

(C27)

obtained in the same way as Eqs. (C18) and (C18), and the critical temperature (C21). Finally we obtain

$$\begin{aligned} I_n^{0}(T_\mathrm{cr}^+)&\approx \sqrt{2} \beta ^{(G)}\frac{n}{\delta n_G} \Big [1 + \frac{\lambda \delta n_G}{6\mu _\mathrm{cr}^{3}(1+2C)} \nonumber \\&+ \frac{\delta n_G}{3n}\frac{\eta _{T_\mathrm{cr}}^{(G)}}{\eta _n} \Big (\frac{3}{2} -\frac{3}{4}\beta \frac{T_\mathrm{cr}}{\mu _{cr}} + \pi \alpha ^{1/2}_\mathrm{cr} \delta I_\mathrm{cr,1} \Big ) \Big ] \nonumber \\&+ \delta I_{\mathrm{cr},n} +O\big (\delta n_G\big ) \,. \end{aligned}$$

(C28)