Condensation of interacting scalar bosons at finite temperatures

TL;DR

This paper investigates the thermodynamics of interacting scalar bosons at finite temperatures, revealing a first-order phase transition and scalar condensate formation within a specific temperature range using a mean-field approach.

Contribution

It introduces a field-theoretical model with both attractive and repulsive interactions, demonstrating the conditions for phase transition and scalar condensate formation.

Findings

First-order phase transition occurs with strong attractive interaction.

Scalar condensate appears within a finite temperature interval.

Condensed phase has a constant scalar density.

Abstract

Thermodynamical properties of an interacting system of scalar bosons at finite temperatures are studied within the framework of a field-theoretical model containing the attractive and repulsive self-interaction terms. Self-consistency relations between the effective mass and thermodynamic functions are derived in the mean-field approximation. We show that for a sufficiently strong attractive interaction a first-order phase transition develops in the system via the formation of a scalar condensate. An interesting prediction of this model is that the condensed phase appears within a finite temperature interval and is characterized by a constant scalar density of Bose particles.

Equations

$$\mathcal{L}=\frac{1}{2}\left[{\partial }_{\mu }\widehat{\text{ϕ}}(x){\partial }^{\mu }\widehat{\text{ϕ}}(x)-m^2\widehat{\text{σ}}(x)\right]+{\mathcal{L}}_{\mathrm{int}}[\widehat{\text{σ}}(x)].$$

$$⟨\hat{\text{A}}⟩=\frac{1}{Z}\mathrm{Tr}\left[e^{-\beta \left(\hat{\text{H}}-\mu \hat{\text{N}}\right)}\hat{\text{A}}\right],Z=\mathrm{Tr}\left[e^{-\beta \left(\hat{\text{H}}-\mu \hat{\text{N}}\right)}\right],$$

$${\mathcal{L}}_{\mathrm{int}}(\widehat{\text{σ}})\simeq {\mathcal{L}}_{\mathrm{int}}(\sigma )+\delta \widehat{\text{σ}}{\mathcal{L}}_{\mathrm{int}}^{\prime }(\sigma )(\text{MFA}),$$

$$\mathcal{L}\simeq \frac{1}{2}\left[{\partial }_{\mu }\widehat{\text{ϕ}}(x){\partial }^{\mu }\widehat{\text{ϕ}}(x)-M^2(\sigma ){\widehat{\text{ϕ}}}^2(x)\right]+P_{\mathrm{ex}}(\sigma ),$$

$$M^2(\sigma )=m^2+\Pi (\sigma ),$$

$$P_{\mathrm{ex}}(\sigma )={\mathcal{L}}_{\mathrm{int}}(\sigma )-\sigma {\mathcal{L}}_{\mathrm{int}}^{\prime }(\sigma )$$

$${\partial }^{\mu }{\partial }_{\mu }\widehat{\text{ϕ}}+M^2(\sigma )\widehat{\text{ϕ}}=0.$$

$$\frac{d{P}_{\mathrm{ex}}}{d\sigma }=\frac{\sigma }{2}\frac{d\Pi }{d\sigma },$$

$$\left[\widehat{\text{ϕ}}(t,\mathbf{r}),\widehat{\text{π}}(t,{\mathbf{r}}^{\prime })\right]=i{\delta }^3(\mathbf{r}-{\mathbf{r}}^{\prime }).$$

$$\hat{\text{H}}\simeq \frac{1}{2}\left[{\widehat{\text{π}}}^2(x)+\nabla \widehat{\text{ϕ}}(x)\cdot \nabla \widehat{\text{ϕ}}(x)+M^2(\sigma ){\widehat{\text{ϕ}}}^2(x)\right]-P_{\mathrm{ex}}(\sigma ).$$

$$\widehat{\text{ϕ}}(x)=g\int \frac{d^{3}k}{(2\pi {)}^3\sqrt{2{\omega }_{\mathbf{k}}}}\left[a_{\mathbf{k}}{e}^{-ik\cdot x}+a_{\mathbf{k}}^{+}{e}^{ik\cdot x}\right].$$

$$k^0={\omega }_{\mathbf{k}}=\sqrt{{\mathbf{k}}^2+M^2(\sigma )},$$

$$\left[a_{\mathbf{k}},a_{{\mathbf{k}}^{\prime }}^{+}\right]=(2\pi {)}^3\delta (\mathbf{k}-{\mathbf{k}}^{\prime }),\left[a_{\mathbf{k}},a_{{\mathbf{k}}^{\prime }}\right]=\left[a_{\mathbf{k}}^{+},a_{{\mathbf{k}}^{\prime }}^{+}\right]=0.$$

$$\mathbf{\hat{\text{$H$}}}=\mbox{$\displaystyle\int d^{\,3}x~{}\mathbf{\hat{\text{$\mathcal{H}$}}}$}=V\left[\mbox{${g}\displaystyle\int\frac{\displaystyle d^{\,3}k}{\displaystyle(2\pi)^{3}}\,\normalsize\omega_{\bm{k}}\hskip 1.0pta^{+}_{\bm{k}}\hskip 1.0pta_{\bm{k}}-P_{\rm ex}(\sigma)$}\right],$$

$$n_{\mathbf{k}}(\sigma )\equiv ⟨a_{\mathbf{k}}^{+}{a}_{\mathbf{k}}⟩=(e^{\beta {\omega }_{\mathbf{k}}}-1{)}^{-1},$$

$$\sigma ={\sigma }_{\mathrm{th}}(M,T)\text{where}{\sigma }_{\mathrm{th}}(M,T)=g\int \frac{d^{3}k}{(2\pi {)}^3}\frac{n_{\mathbf{k}}(\sigma )}{{\omega }_{\mathbf{k}}}.$$

$$P=P_{\mathrm{kin}}(M,T)+P_{\mathrm{ex}}(\sigma ).$$

$$P_{\mathrm{kin}}(M,T)=\frac{g}{3}\int \frac{d^{3}k}{(2\pi {)}^3}\frac{k^2}{{\omega }_{\mathbf{k}}}{n}_{\mathbf{k}}(\sigma ),$$

$$\epsilon =g\int \frac{d^{3}k}{(2\pi {)}^3}{\omega }_{\mathbf{k}}{n}_{\mathbf{k}}(\sigma )-P_{\mathrm{ex}}(\sigma ).$$

$$s=\frac{g}{T}\int \frac{d^{3}k}{(2\pi {)}^3}\left({\omega }_{\mathbf{k}}+\frac{k^2}{3{\omega }_{\mathbf{k}}}\right)n_{\mathbf{k}}(\sigma ).$$

$$\frac{dP}{dT}={\left(\frac{\partial P_{\mathrm{kin}}}{\partial T}\right)}_M+\left[{\left(\frac{\partial P_{\mathrm{kin}}}{\partial M}\right)}_T+\frac{d{P}_{\mathrm{ex}}}{dM}\right]\frac{dM}{dT}=s.$$

$${\left(\frac{\partial P_{\mathrm{kin}}}{\partial T}\right)}_M=s,{\left(\frac{\partial P_{\mathrm{kin}}}{\partial M}\right)}_T=-M{\sigma }_{\mathrm{th}},\frac{d{P}_{\mathrm{ex}}}{dM}=M\sigma .$$

$$\sigma \equiv ⟨{\widehat{\text{ϕ}}}^2⟩={\sigma }_{\mathrm{cond}}+{\sigma }_{\mathrm{th}}.$$

$$M^2({\sigma }_0)=m^2+\Pi ({\sigma }_0)=0.$$

$${\sigma }_{\lim }(T)\equiv {\sigma }_{\mathrm{th}}(M=0,T)=\frac{g{T}^2}{12}.$$

$${\sigma }_{\mathrm{cond}}(T)={\sigma }_0-\frac{g{T}^2}{12}>0.$$

$$T<T_{\max }=\sqrt{\frac{12{\sigma }_0}{g}},$$

$$P=\frac{g{\pi }^2}{90}{T}^4+P_{\mathrm{ex}}({\sigma }_0),\epsilon =\frac{g{\pi }^2}{30}{T}^4-P_{\mathrm{ex}}({\sigma }_0),$$

$$s=\frac{2g{\pi }^2}{45}{T}^3=\frac{2{\pi }^4}{45\zeta (3)}n\simeq 3.60n,$$

$${\mathcal{L}}_{\mathrm{int}}(\sigma )=\frac{a}{4}{\sigma }^2-\frac{b}{6}{\sigma }^3,$$

$$\Pi (\sigma )=-2{\mathcal{L}}_{\mathrm{int}}^{\prime }=-a\sigma +b{\sigma }^2,$$

Full text

I Introduction

In recent years properties of hot and dense hadronic matter have attracted considerable interest. Such matter can be produced in relativistic nucleus-nucleus collisions which are experimentally studied in many laboratories. Both, QCD-motivated effective models and lattice simulations indicate that the chiral symmetry restoration and the deconfinement phase transition (PT) should occur at high temperatures and particle densities. The properties of hadrons will be strongly modified under such conditions.

The hadron resonance gas model with excluded volume corrections is widely used in the literature (see e.g. andronic-2006 ; satarov-2009 ; vovchenko-anch-2015 ; vovchenko-2017 ; vovchenko-2017a ; anch-vovchenko-2015 ) to fit the lattice results and the experimental data from relativistic heavy-ion collisions. These calculations show that the pion densities may reach the values ${n_{\pi}\sim}(0.1-0.2)~{}\textrm{fm}^{-3}$ at temperatures ${T\sim}(140-160)~{}\textrm{MeV}$. At such high densities the interaction effects became important. Dense hadronic systems have been investigated recently within the mean-field approach, using the van der Waals vovchenko-2017a ; anch-vovchenko-2015 ; anch-2016 and Skyrme-like anch-2019 models. The pion gas with repulsive $\phi^{4}$ interaction has been considered earlier in Ref. vos-2018 .

In the present paper we study properties of interacting bosonic systems. This problem has been studied previously, starting from the pioneer works of Migdal and coworkers migdal-1972 ; migdal-1974 ; migdal-1978 ; saperstein-1990 and later by many authors using different models and methods. The formation of classical pion fields in relativistic nucleus-nucleus collisions was discussed in Refs. anselm-1991 ; blaizot-1992 ; bjorken-1992 . Pionic systems with a finite isospin chemical potential were considered within effective chiral models in Refs. mishustin-greiner-1993 ; son-2001 ; kogut-2001 ; toublan-2001 ; mammarella-2015 ; carignano-2017 . Such systems have been investigated also by lattice QCD simulations brandt-2016 ; brandt-2017 .

Below we consider a general system of interacting bosons associated with a scalar field $\phi$. Following Ref. anch-nazarenko-2006 we introduce an effective Lagrangian which contains the attractive ($\phi^{4}$) and the repulsive ($\phi^{6}$) self-interaction terms. The calculations are carried out within the mean-field approach, taking into account the thermodynamic consistency conditions. It is assumed that the system has no conserved charge and, therefore, it is characterized by a vanishing chemical potential. It will be shown that at strong enough attractive interactions, the quasiparticle’s effective mass vanishes and the system undergoes a first-order PT with formation of a scalar Bose-Einstein condensate. This phase exists within a finite temperature interval, determined by the parameters of the interaction.

II Formulation of the model

Let us consider a real (pseudo-)scalar field  \mathbf{\hat{\text{\phi}}}\hskip 0.5pt(x), which has no conserved charge. The corresponding Lagrangian can be written as ($\hbar=c=1$)

[TABLE]

Here \mathbf{\hat{\text{\sigma}}}\hskip 0.5pt(x)=\mathbf{\hat{\text{\phi}}}^{\hskip 1.0pt2}(x) is the ’scalar density’ operator, $m$ is the boson mass in the vacuum, and $\mathcal{L}_{\rm int}$ is the interaction Lagrangian. One can decompose $\mathcal{L}_{\rm int}$ in powers of \delta\hskip 1.0pt\mathbf{\hat{\text{\sigma}}}=\mathbf{\hat{\text{\sigma}}}-\sigma where \sigma=\langle\mathbf{\hat{\text{\sigma}}}\rangle is the mean value of the scalar density. Here and below, angular brackets denote the statistical averaging in the grand-canonical ensemble:

[TABLE]

where $\beta=1/T$ is the inverse temperature, $\mu$ is the chemical potential, \mathbf{\hat{\text{H}}} and \mathbf{\hat{\text{N}}} are the Hamiltonian and the particle number operators, respectively. Below we consider only the case $\mu=0$ which corresponds to a system, where the total particle number is not conserved, but is determined by the temperature (like photons or $\pi$-mesons). However, this model can be easily extended to nonzero $\mu$ .

In the following we apply the mean-field approximation (MFA), where fluctuations of the scalar field up to the first order in \delta\mathbf{\hat{\text{\sigma}}} (i.e., up to the second order in \mathbf{\hat{\text{\varphi}}}) are taken into account. Then one can write

[TABLE]

where the prime denotes the differentiation with respect to $\sigma$ . The effective Lagrangian of field excitations can be represented as

[TABLE]

where Eqs. (1) and (3) have been used. The quantity

[TABLE]

is the effective mass of bosonic quasiparticles and $\Pi(\sigma)=-2\hskip 0.5pt\mathcal{L}_{\rm int}^{\prime}\hskip 0.5pt(\sigma)$ is the ’polarization operator’. The last term in Eq. (4),

[TABLE]

is the so-called ’excess’ pressure. It will be shown that this term gives rise to the pressure shift due to interactions. Note that the argument $\sigma$ in the functions $M,\Pi,P_{\rm ex}$ is a $c$–number. In fact, Eq. (5) can be regarded as the ’gap’ equation for $M=M\hskip 0.5pt(\sigma)$, as $\sigma$ itself is a function of $M$ (see below).

It follows from Eq. (4) that the equation of motion for the operator  \mathbf{\hat{\text{\phi}}} can be written in Klein-Gordon form with the effective mass $M\hskip 0.5pt(\sigma)$

[TABLE]

Equations (5) and (6) lead to the following self-consistency relation

[TABLE]

which should hold for any $\mathcal{L}_{\rm int}$. This is another form of the corresponding relation derived in Refs. anch-1992 ; anchsu-1995 . The shift of a single-particle energy (commonly called as the effective potential) in nonrelativistic limit is equal to $U\simeq\Pi/(2m)$ and $\sigma\simeq n/m$, where $n$ is the particle number density.

III Derivation of the thermodynamic functions

Let us now introduce the momentum operator \mathbf{\hat{\text{\pi}}}(x)=\partial_{t}\,\,\mathbf{\hat{\text{\phi}}}(x) which satisfies the equal-time commutation relation

[TABLE]

In the MFA, the Hamiltonian density operator \mathbf{\hat{\text{\mathcal{H}}}}=\mathbf{\hat{\text{\pi}}}\hskip 1.0pt\partial_{t}\,\,\mathbf{\hat{\text{\phi}}}-\mathcal{L} takes the form

[TABLE]

Using solutions of the Klein-Gordon equation (7) one can represent the scalar field  \mathbf{\hat{\text{\phi}}}\hskip 1.0pt(x) as

[TABLE]

Here $g$ is the spin-isospin statistical weight,

[TABLE]

and $a_{\bm{k}},a^{+}_{\bm{k}}$ are the annihilation and creation operators, respectively. They obey the standard commutation relations:

[TABLE]

Substituting (11) into (10) one gets the Hamiltonian operator in the MFA

[TABLE]

where $V$ is the total system’s volume. Using Eqs. (2) and (14) one can calculate all thermodynamic functions of the considered system. In the MFA the equilibrium momentum distribution coincides with that of an ideal gas of bosons with the effective mass $M(\sigma)$,

[TABLE]

where $\omega_{\bm{k}}$ is given by Eq. (12).

The equation for the scalar density \sigma=\langle~{}\mathbf{\hat{\text{\phi}}}^{\,2}\rangle is obtained by direct calculation using Eqs. (11), (15). This leads to the following gap equation

[TABLE]

Note that the number density of quasiparticles, $n=g\int\frac{d^{\,3}k}{(2\pi)^{3}}\,n_{\bm{k}}(\sigma)$, does not contain $\omega_{\bm{k}}$ in the denominator. The pressure $P=\frac{T}{V}\ln{Z}$ is calculated as

[TABLE]

Here the first term is the pressure of an ideal gas of bosonic quasiparticles with mass $M$,

[TABLE]

where $n_{\bm{k}}(\sigma)$ is given in Eq. (15). To obtain $P=P\hskip 0.5pt(T)$ one should calculate $\sigma$ and $M$ as functions of $T$ by simultaneously solving the system of equations (5) and (16).

Using Eqs. (14) and (15) one can calculate the energy density \varepsilon=\langle\,\mathbf{\hat{\text{H}}}\rangle/V as

[TABLE]

Within the MFA, the entropy density $s=(\varepsilon+P)/T$ formally coincides with that for the ideal gas of quasiparticles:

[TABLE]

Here the interaction effects enter via the effective mass $M\hskip 0.5pt(\sigma)$ .

Equations (17), (20) satisfy the condition of thermodynamic consistency, $s=dP/dT$ gorenstein-1995 . Indeed, in accordance with Eq. (17), one has

[TABLE]

Here it is taken into account that 111 The second equality in Eq. (22) is obtained by direct differentiation of (18), while the last relation is obtained from (8).

[TABLE]

The terms in square brackets of Eq. (21) add to $M\hskip 0.5pt(\sigma-\sigma_{\rm th})$ which is zero both in the normal phase $(\sigma=\sigma_{\rm th})$ as well as in the phase with a condensate, where $M=0$ (see below).

IV Formation of scalar Bose condensate

When the lowest energy level with $|\bm{k}|=0$ is ’macroscopically’ occupied, one should treat this level separately, i.e. write instead of (11) the equation \mathbf{\hat{\text{\phi}}}\hskip 0.5pt(x)=\varphi+\mathbf{\hat{\text{\chi}}}(x). Here $\varphi$ is the classical part of the field operator  \mathbf{\hat{\text{\phi}}}, which is analogous to the Bose-Einstein condensate of massive particles. Below we denote such a classical field as Scalar Bose Condensate (SBC). The second term \mathbf{\hat{\text{\chi}}} is the fluctuating (quantum) part of  \mathbf{\hat{\text{\phi}}}, which satisfies the relation \langle\mathbf{\hat{\text{\chi}}}\rangle=0. In the domain with SBC, $\varphi\neq 0$ and instead of Eq. (16), we now write

[TABLE]

Here both, $\sigma_{\rm cond}=\varphi^{2}$ and \sigma_{\rm th}=\langle{\mathbf{\hat{\text{\chi}}}^{2}}\rangle, are nonzero, positive quantities.

By analogy to the case of a conserved charge ($\mu\neq 0$) we assume that the SBC occurs for states where the occupation number $n_{\bm{k}}(\sigma)$ diverges at $k\to 0$. As follows from Eq. (15), the divergence takes place at $M\hskip 0.5pt(\sigma)\to 0$ 222 In Refs. migdal-1972 ; migdal-1974 ; migdal-1978 ; saperstein-1990 the onset of pion condensation was determined by the occurrence of solutions with $\omega^{2}_{\bm{k}}\leqslant 0$. . This condition is satisfied at $\sigma=\sigma_{0}$ where $\sigma_{0}$ is the root of the equation:

[TABLE]

As we will see below, Eq. (24) gives a necessary, but in general, not sufficient condition for the SBC formation. According to Eq. (24), this condensation requires a negative $\Pi\hskip 0.5pt(\sigma)$ which implies a sufficiently strong attractive interaction of particles.

For massless quasiparticles ($\omega_{\bm{k}}=k$) Eq. (16) yields

[TABLE]

Substituting (25) into (23) gives

[TABLE]

Hence, the SBC may appear only at temperatures

[TABLE]

where $\sigma_{0}$ is found by solving Eq. (24).

In the phase with SBC, the thermodynamical quantities $P\hskip 0.5pt(T),\varepsilon\hskip 0.5pt(T),s\hskip 0.5pt(T)$ and $n\hskip 0.5pt(T)$ are given by the corresponding formulae in the preceding section with $M=0$, $\sigma=\sigma_{0}$, namely

[TABLE]

where $\zeta(x)=\sum_{k=1}^{\infty}k^{-x}$ is the Riemann function. Note that $-P_{\rm ex}(\sigma_{0})$ plays the role of an effective ’bag constant’ for this phase.

V Bosonic system with Skyrme-like interaction

The above results are valid in the MFA and they do not depend on a specific form of the interaction. For illustration, below we consider the Skyrme-like ($\phi^{4}-\phi^{6}$) interaction Lagrangian

[TABLE]

where $a,b$ are positive constants. The first and the second terms in (29) describe, respectively, the attractive and the repulsive interactions between the scalar bosons. Note that the bosonic system becomes unstable in the limit $b\to 0$ (see below). Using formulas of preceding section one has

[TABLE]

The scalar density $\sigma$ of stable matter should satisfy the condition

[TABLE]

As discussed above, the onset of condensation corresponds to the equality sign in this expression.

It is convenient to introduce parameter $\kappa=a/(2m\sqrt{b})$ to consider differen possibilities predicted by the model. They are presented in Fig. 1. In the case of ’weak attraction’, $\kappa<1$, the right hand side of Eq. (32) is positive at any $\sigma$, i.e. no SBC can be formed. In this case all thermodynamic quantities change smoothly with temperature and no PT is expected. Nevertheless, the equation of state differs significantly from the ideal gas with the vacuum masses of bosons. In the region $\kappa<1$, the equilibrium values of $\sigma$ and $M$ as functions of $T$ are found by simultaneously solving the equations

[TABLE]

where $\sigma_{\rm th}(M,T)$ is defined in Eq. (16). Then, the pressure $P(T)$ can be calculated by using Eqs. (17), (18) and (31). It is an increasing function of $T$, such that the relation $s=dP/dT>0$ is satisfied . At high temperatures $\sigma$ and $P$ increase with $T$ less rapidly than for the case of ideal gas of massless bosons 333 Note, that our model does not take into account possible excitations of bosonic resonances and neglects the deconfinement effects which become more and more important with increasing temperature. .

In the case of strong attraction, $\kappa>1$, the condition $M^{2}(\sigma)<0$ holds in the interval $\sigma_{1}<\sigma<\sigma_{2}$, where $\sigma_{1,2}$ are the two roots of the equation $M(\sigma)=0$:

[TABLE]

The real solutions of Eq. (32) exist only outside of this interval. One can easily see that the true equilibrium state is $\sigma=\sigma_{2}$ , and the root $\sigma=\sigma_{1}$ corresponds to a maximum of the thermodynamic potential $\Omega=-PV$. It is interesting to note that the lower branch of $\sigma(T)$ bends before reaching the line $\sigma_{\rm lim}(T)$ (see the curves for $\kappa=1.1$ in Fig. 1). After the system reaches the value $\sigma=\sigma_{1}$ at temperature $T=T_{1}$, it will ”roll down” into the state with a larger pressure at $\sigma=\sigma_{2}$. At fixed temperature this is only possible by creating a condensate of bosons with zero momentum. In principle, the system may reach metastable states marked in Fig. 1 by the dashed line up to the cross, but at higher temperatures the allowed states lie on the line $\sigma=\sigma_{2}$.

The thermodynamic characteristics of the ’mixed’ states, where the condensate coexists with the normal phase, can be found from Eqs. (26), (28), after substituting $\sigma_{0}=\sigma_{2}$ . In particular, the pressure and the energy density in this phase are given by

[TABLE]

As can be shown by using Eqs. (31), (35), at $T\to 0$ the pressure $P_{\rm mix}^{(2)}\simeq P_{\rm ex}(\sigma_{2})$ becomes negative at $\kappa<2/\sqrt{3}\simeq 1.155$ 444 At $\kappa>2/\sqrt{3}$, even the bosonic vacuum at $T=0$ becomes unstable with respect to formation of a classical boson field. Indeed, at such $\kappa$ the energy density of the condensate, $\varepsilon_{\rm{SBC}}(T=0)=-P_{\rm ex}(\sigma_{2})$, becomes negative. This possibility is analogous to a spontaneous creation of the vacuum condensate in the linear sigma model. We do not consider such a possibility in the present paper. . On the other hand, as discussed above, the normal states without condensate have a positive pressure at all $T$. Hence, the mixed phase becomes unfavorable at low temperatures.

The true transition point between the normal (’liquid-gas’) and the mixed phases occurs at critical temperature $T_{c}$, which is found from the Gibbs condition $P_{\rm\hskip 1.0ptlg}\hskip 0.5pt(T_{c})=P_{\rm mix}^{(2)}(T_{c})$. At $T=T_{c}$, the scalar density $\sigma$ jumps from some thermally-generated value $\sigma_{c}$ to a larger value $\sigma_{2}$ which contains the condensate $\sigma_{\rm cond}=\sigma_{2}-\sigma_{c}$. At the same time, at $T=T_{c}$ the boson effective mass drops from $M=M\hskip 0.5pt(\sigma_{c})$ to zero, see Fig. 2. As temperature grows above $T_{c}$, the condensate gets smaller and finally vanishes at $T=T_{2}$. We call this unusual behaviour as ”triangular phase diagram”.

VI Numerical results

The general formalism presented above can be used for any (pseudo)scalar particles with strong self-interaction. As an illustrative example, we consider pion-like particles with $m=m_{\pi}=140~{}\textrm{MeV}$, $g=3$, $b=25\hskip 1.0ptm^{-2}$ anch-2019 . Figures 1–3 show our numerical results for several $\kappa$ values, namely, $\kappa=0,0.7,1.0$ and $1.1$. In the latter case, $\sigma_{2}\simeq 0.312\,m^{2}$ and the SBC phase appears in the temperature interval $T_{c}<T<T_{2}$, where $T_{2}=\sqrt{12\hskip 0.5pt\sigma_{2}/g}\simeq 156~{}\textrm{MeV}$ and $T_{c}\simeq 80~{}\textrm{MeV}$ is the critical temperature of the first-order PT. At this temperature, the scalar density jumps from $\sigma_{c}\simeq 0.022\,m^{2}$ to $\sigma_{2}$ and then remains constant until $T=T_{2}$. At higher temperatures the condensate disappears, and the scalar density $\sigma$ is generated entirely by thermal excitations (see the branch starting from $(T_{2},\sigma_{2})$ in Fig. 1).

As demonstrated above, the phase transition predicted by the model (for $\kappa>1$) is rather unusual. With increasing temperature it starts at some temperature $T=T_{c}$ as the first-order transition with a jump of scalar density from $\sigma_{c}$ to $\sigma_{2}$. With further increase of temperature the scalar density remains constant, $\sigma=\sigma_{2}$, but the condensate density $\sigma_{\rm cond}=\sigma_{2}-\sigma_{\rm th}$ decreases and finally vanishes at $T=T_{2}$ where $\sigma_{\rm th}=\sigma_{2}$ . Therefore, we do not expect that higher-order fluctuations will destroy or significantly modify the behaviour of the scalar density at $T\simeq T_{c}$ . On the other hand, at $T=T_{2}$ the condensate vanishes smoothly and only thermal fluctuations remain at $T>T_{2}$ .

Finally we would like to point out that pions have a special nature as Goldstone bosons of spontaneously broken chiral symmetry, and by this reason require a special treatment, like that in the linear sigma-model Len00 . In this case the scalar condensate exists already in the vacuum at T=0 but melts at higher temperatures. In the future we are planning to study the mesonic sector of this model in more details.

VII Concluding remarks

In this paper we have presented a thermodynamically consistent model to describe dense bosonic systems at high temperatures and zero chemical potential. A central step of our approach is to solve Eq. (16) for the boson scalar density $\sigma$ as a function of temperature. We show that if the attractive mean field is so strong that the stability condition $M^{2}(\sigma)>0$ is violated, the classical scalar field (condensate) forms in the multi-boson system. Our analysis leads to the conclusion that in the presence of a condensate, the allowed states of the system satisfy the condition $M\hskip 0.5pt(\sigma)=0$, i.e., the bosonic quasiparticles are massless.

It is known that loop corrections and higher-order fluctuations may change the second-order phase transition to first order or crossover, see e.g. Kap06 ; Kad09 . As far as we know, there are no dedicated studies for the $\phi^{4}-\phi^{6}$ model considered here. However, we do not think that the above-mentioned corrections can qualitatively change the character of a strong first-order phase transition predicted by our calculations.

Of course, at high temperatures considered in this paper other hadronic degrees of freedom will be present in the system. They may add additional terms to the effective potential which are proportional to the density of these species Shu91 . These terms will reduce the meson mass $m$ and thus, the threshold for Bose condensation (see Eq. (5)).

Acknowledgements

The authors thank M. I. Gorenstein for useful discussions. The work of D. V. A. is supported by the Programs ”The structure and dynamics of statistical and quantum-field systems” and ”The dynamics of formation of spatially-heterogeneous structures in many-particle systems” of the Department of Physics and Astronomy of NAS of Ukraine. I. N. M. acknowledges the financial support from the Helmholtz International Center for FAIR, Germany. L. M. S. appreciates the support from the Frankfurt Institute for Advanced Studies. H. St. thanks for support from the J. M. Eisenberg Professor Laureatus of the Fachbereich Physik.