Spontaneous U(1) Symmetry Breaking and Phase Transitions in Rotating Interacting Bose Gases

I. Introduction

II. Interacting Charged Scalars Under Rigid Rotation

A. The Free Propagator

We start with the Lagrangian density of a charged scalar field φ

L = gµν ∂µ φ⋆∂ν φ − m² φ⋆ φ − λ(φ⋆ φ)², (II.1)

with the metric

$g_{\mu \nu }=\left(\begin{array}{cccc}1-r^2{\Omega }^2& y\Omega & -x\Omega & 0\\ y\Omega & -1& 0& 0\\ -x\Omega & 0& -1& 0\\ 0& 0& 0& -1\end{array}\right),$ (II.2)

describing a rigid rotation. Here, m is the rest mass and 0 < λ < 1 is the coupling constant, describing the strength of the interaction. The spacetime coordinate is described by xµ = (t, x, y, z) and r² ≡ x² + y². More- over, Ω is the constant angular velocity of a rigid rotation around the z-axis. The above Lagrangian is invariant un- der global U(1) transformation

φ(x) → e-iαφ(x), φ⋆ (x) → e+iαφ⋆ (x), (II.3)

with α a real constant phase. Plugging the metric into (II.1), we obtain

L = |(∂0 − iµ − iΩLz)φ|² − |▽φ|² − m² |φ|² − λ|φ(|4II,.4)

where the chemical potential µ corresponding to the global U(1) symmetry (II.3) is introduced. The z- component of the angular momentum, Lz, is defined by Lz = i(y∂x − x∂y). To investigate the spontaneous breaking of U(1) symmetry, we rewrite L in terms of and perform the shift φi → Φi + φi with

The classical part of the Lagrangian, L0, defines the clas- sical (zero mode) potential

The free propagator arises from the quadratic term L2 in the fluctuating fields φ 1 and φ2. To derive the free propagator in the momentum space, we use the Fourier- Bessel transformation

with i = 1, 2. The cylindrical symmetry is implemented by introducing the cylinder coordinate system described by x^µ = (t, x, y, z) = (t, r cosϕ, r sinϕ, z), with r the ra- dial coordinate, ϕ the azimuthal angle, and z the height of the cylinder. The conjugate momenta, corresponding to these coordinates at finite temperature T, are given by the bosonic Matsubara frequency ωn = 2πnT, dis- crete quantum number ℓ, which is the eigenvalue of Lz, continuous momentum kz, and k⊥ ≡ |k⊥ | ≡ (k_x² + k_y²)^1/2 in cylindrical coordinates. The Bessel function Jℓ (k⊥ r) captures the radial dependence in this transformation and τ ≡ it. Plugging (II.8) into L2 and performing an integration over cylindrical coordinates, according to

we arrive after some manipulations at

with the free propagator

Here, ωi² = k² + mi², i = 1, 2, with m1²(v) ≡ 3λv² + m² and m2²(v) ≡ λv² + m², the corresponding masses to two fields φ 1 and φ2. In cylinder coordinate system, we have k² ≡ k1² + kz². In Sec. III, we break the global U(1) symmetry by choosing m² = −c² with c² > 0 and show that after considering the quantum corrections, φ2 become a massless Goldstone mode.

A comparison with similar results for a nonrotating charged Bose gas at T and µ shows that while ℓΩ is said to play a role analogous to that of the chemical potential µ [23], the manner in which it is incorporated into the free propagator and the thermodynamic potential differs significantly (as discussed below).

B. The Thermodynamic Potential

To derive the thermodynamic potential V, corre- sponding to this model, we follow the standard procedure and define this potential by

C. Spontaneous Breaking of Global U(1) Symmetry

Let us consider the classical potential (II.7). Assum- ing m² > µ², the coefficient of v² in this expression is positive and, as it turns out, Vcl possesses one single min- imum at 0 = 0 and the system is in its symmetric phase.

Here, m is a mass gap and ∆ϵk ≡ ϵk- ⁻ ϵk⁺ = 2µ. In Figure 1(a), ${\mathrm{ϵ}}_{\mathrm{k}}^{\mathrm{\pm }}$ is plotted for generic mass m = 1 MeV and chemical potential µ = 0.6 MeV (µ < m).

In the symmetry-broken phase characterized by m² < µ², however, extremizing Vcl yields a maximum at va = 0 and two minima at

The masses m1²(v-b) = 3µ² − 2m² and m2²(v-b) = µ². We thus have M² = 2µ² − m² and δM² = µ² − m² leading to

According to these results, ϵk⁺ and ϵk^- correspond to phonon and roton modes in the symmetry-broken phase m < µ, respectively.

As it is shown in this section, ℓΩ appears in the ther- mal part of the effective potential VT from (II.23) and does not modify neither m1²(v) nor the energy dispersion ϵk^±. Hence, a comparison with analogous results for nonrotating bosons [42] shows that rigid rotation has no

effect on the behavior of ϵk^± at k ~ 0.

D. Two Special Cases

In what follows, we consider two special cases λ = 0, µ ≠ 0 and λ ≠ 0,µ = 0:

Case 1: For the special case of noninteracting rotating Bose gas with λ = 0 and µ ≠ 0, we have m1 = m2 = m,

with µeff ≡ µ + ℓΩ. This potential is exactly the same potential arising in [30]. Using this potential, the effect of rotation on the BE condensation of a relativistic free Bose gas is studied.

Case 2:  Another important case is characterized by

Plugging (II.30) into (II.24) and choosing µ = 0 and m² = —c² with c² > 0, the total thermodynamic po- tential is given by

Here, ωi, i = 1, 2 are given in (II.30). Let us notice that in (II.35), the ℓ = 0 contribution is excluded, because the zero mode contribution is already captured by Vcl from (II.32). It is possible to limit the integration over ℓ

III. Spontaneous Breaking of Global U(1) Symmetry in a Rigidly Rotating Bose Gas

A. The Critical Temperature of U(1) Phase Transition; Analytical Result

In this section, we study the effect of rigid rotation on the spontaneous breaking of global U(1) symmetry in an interacting charged Bose gas. Before starting, we add a new term

to L from (II.5). This leads to an additional mass term in the classical potential Vcl. We define a new mass a² ≡ c² + m0², which replaces c² in (II.32). Minimizing the resulting expression, the (classical) minimum of Vcl is thus given by

At this minimum, the masses of m1²(v) = 3λv² — c² and m2²(v) = λv² — c² are given by

For m0 = 0, we have m2 = 0 and φ2 becomes a massless Goldstone mode. The position of this (classical) mini- mum changes, once the contribution of the thermal part of the thermodynamic potential, VT, is considered. To show this, we first define Va ≡ Vcl + VT and use the high-temperature expansion of VT by making use of the results presented in Appendix A. Considering only the first two terms of (A.13) and plugging the definitions of m1²(v) and m2²(v) into it, the high-temperature expansion of Va reads

Setting the coefficient of v² equal to zero, the critical tem- perature of global U(1) phase transition is determined,

. (III.5) In [30], the BE transition in a noninteracting Bose gas under rigid rotation is studied. It is shown that in nonrel- ativistic regime Tc ∝ Ω^2/5 and in ultrarelativistic regime Tc ∝ Ω ^1/4. In the present case of interacting Bose gas, similar to that noninteracting cases, the critical temper- ature increases with increasing Ω.

Introducing the reduced temperature t = T/Tc, with Tc = Tc (Ω) from (III.5), and minimizing Va from (III.4) with respect to v, the new nontrivial minimum is given by

When comparing with a similar result for a nonrotating charged Bose gas [40], it turns out that the power of t in (III.6) changes once the gas is subjected to small rotation. In Sec. IV, we numerically study the effect of rotation on the spontaneous breaking of global U(1) symmetry. For this purpose, we employ a phenomenological model that includes σ and π mesons, replacing φ 1 and φ2 fields in the above computation. We set m1²(v0) = 3λv0² — c2 =

with v0 the classical minimum from (III.2). Moreover, we choose m0 in (III.1) equal to mπ. For mσ = 400 MeV, and mπ = 140 MeV, we obtain

is plotted in Figure 2 at t = 0.6, 0.8 in the symmetry-broken phase and t = 1.2 in the symmetry-restored phase. At t = 1 a phase transition from the symmetry-broken phase to a symmetry-restored phase occurs. Let us notice, that the effect of rotation consists of changing the power of t in (III.6) and (III.8) from t² to t³. This is apart from the Ω dependence of the critical temperature Tc from (III.5) (see Figure 7).

The result indicates a continuous phase transition from a symmetry-broken phase at t < 1 to a symmetry- restored phase at t ≥ 1. To scrutinize this conclusion, let us consider the pressure P arising from Va from (III.4).

It is given by P = —Va. Denoting the pressures below and above Tc with P< (v, T,Ω) and

Here, we have added a term —a⁴ /4λ to P< and P> in order to guarantee P< (vmin², 0, Ω) = 0 and P< = P> at the the transition temperature Tc. At T = Tc, the pressure is given by

For m0 = 0 (or a = c), the first two terms cancel, resulting in an increase in pressure as Ω increases. Moreover, whereas the entropy (dP/dT) is continuous at (III.11)

the heat capacity (d² P/dT²) is discontinuous

Hence, according to Ehrenfest classification, this is a sec- ond order phase transition. In comparison to the non- rotating case [40], although rotation alters the critical temperature, the order of the phase transition remains unchanged. It is noteworthy that the discontinuity in the heat capacity decreases with increasing Ω.

Plugging at this stage, vmin² from (III.6) into , we arrive at

Hence, as it turns out, at t ≥ 1, after the symmetry is restored, m1² and m2² become negative. Contrary to our expectation, for a = c, i.e., in the chiral limit m0 = 0, the Goldstone boson ‘2 acquires a negative mass -c²t² in the symmetry-broken phase at t < 1. In what follows, we compute the one-loop tadpole diagram contributions to masses m1 and m2. We show, in particular, that by con- sidering the thermal mass, the one-loop corrected mass of the Goldstone mode ‘2 vanishes in chiral limit m0 = 0.

B. One-Loop Corrections to m1 (v) and m2 (v)

To calculate the one-loop corrections to m1 and m2, let us consider L4 from (II.4). Three vertices, corre- be considered in this computation (see Figure 3),

They lead to two different tadpole contributions to ⟨Ω|T (‘1 (x)’1 (y))|Ω⟩ and ⟨Ω|T (‘2 (x)’2 (y))|Ω⟩ that correct m1 and m2 perturbatively. They are denoted by Πij with the first index, i = 1; 2, corresponds to whether ‘1 or ‘2 are in the external legs, and the second index j = 1; 2 to whether the internal loop is built from ‘1 or ‘2 (see Figure 4, where Πij are plotted). Hence, according to this notation, the one-loop perturbative corrections to m1² and m2² arise from

with free boson propagator

arising from (II.11) with = 0. Here, and i = 1; 2. Using this notation, it turns out that

Π 11 = 3Π 1 ; Π 12 = Π2 ;

Π22 = 3Π2 ; Π21 = Π 1. (III.18)

Hence, the perturbative corrections of masses are given by

To evaluate Πi from (III.16), we follow the same steps as presented in [30]. The Matsubara summation is eval- uated with

where nb (!) ≡ 1= (eβω - 1) is the BE distribution func- tion. In what follows, we insert (III.20) into (III.16) and focus only on the matter (T and Ω dependent) part of Πi,

(III.21) Having in mind that in nb (!i ± `Ω), we must have eβ(ωi ±ℓΩ) - 1 > 0, it is possible to limit the summation over `. We thus obtain

Let us notice that in the term including nb (!i - `Ω) an additional shift ` → -` is performed. To carry out the summation over ` and eventually the integration over k⊥ and kz, we use

and arrive first at

(III.24)

Using, at this stage, (A.2), we then obtain

(III.25) The summation over ` can be performed by making use of (A.4). Assuming Ω < 1 and using (A.5), Π1^mat reads

Following the method presented in Appendix B, we fi- nally arrive at

(III.27) The first term in (III.27) is analogous to the thermal mass λT² /3 in a nonrotating interacting Bose gas [40] and the ellipsis includes higher order corrections of Π1^mat in βmi.

At high temperature, it is enough to consider only the first term in (III.27), which is independent of mi. We thus have

(III.29)

with t = T/Tc and Tc from (III.5).

C. Goldstone Theorem

Let us consider again the result presented in (III.13). Adding the contribution of thermal mass (III.28) to

where a² = c² + m0² is used. Assuming m0 = 0, m2 van- ishes at t < 1. This indicates that the Goldstone theorem is valid when the thermal mass corrections to m1² and m2² are taken into account. Moreover, we observe that m1²(vmin) = m2²(vmin) in the symmetry-restored phase at t ≥ 1. In Figure 5, the t dependence of m1²(vmin) and m2²(vmin) from (III.30) is plotted. These masses are identified with mσ² and mπ², respectively. We use c ≃ 0.225 GeV from (III.7) and m0 = 0.140 GeV, as de- scribed in Sec. III B and observe that in the symmetry- broken phase, at t < 1, mσ decreases with increasing temperature, while mπ remains constant. As expected, at symmetry-restored phase at t ≥ 1, mσ and mπ are equal and increase with increasing temperature. It is noteworthy that the effect of rotation, apart from affect- ing the value of the critical temperature Tc from (III.5), consists of changing the power of t in (III.30) from t² to t³ (see [40]).

D. Vacuum Potential

In what follows, we compute the contribution of the vacuum part of the thermodynamic potential, Vvac from (II.33) to Vtot. Let us first consider the summation over ℓ ∈ (-∞, +∞) in this expression. This sum is divergent and need an appropriate regularization. To perform the summation over ℓ, we use

= 1 + divergent term. (III.31)

Neglecting the divergent term, we obtain

(III.32)

The above regularization guarantees that rotation does not alter Vvac. To perform the integration over k⊥ and kz, let us consider the integral

(III.33) with ϵ = 3-d. Here, d is the dimension of spacetime and $\overline{\mu }$ denotes an appropriate energy scale. Later, we show that can be eliminated from the computation. Utilizing

to perform a d dimensional regularization, we obtain for Φ(m,3 - ϵ,-1/2), (In Appendix C, we derive (III.24) in cylinder coordinate system).

The vacuum part of the thermodynamic potential (III.32) is thus given by

Vvac = I(m1) + I(m2)

In what follows, we regularize this potential by following

the method presented in [43]. To do this, we first define Vb ≡ Vcl + Vvac + VCT, (III.37)

with Vcl from (II.32) with c² replaced with a² = c² + m0² and Vvac from (III.36). The counterterm potential is given by

(III.38) The coefficients A and B are determined by utilizing two prescriptions

Here, v0² from (III.2) is the classical minimum and m1²(v0) from (III.3). Let us note that the first prescription guar- antees that the position of the classical minimum does not change by considering the vacuum part of the poten- tial. The term C in (III.38) includes all terms which are independent of v. Using (III.39), we arrive at

Plugging A and B from (III.40) into VCT from (III.38) and choosing

the counterterm potential from (III.38) is determined. These counterterms eliminate the divergent terms in the vacuum potential, as expected. The total potential Vb from (III.37) is thus given by

As mentioned earlier, the energy scale $\overline{\mu }$ does not appear in the final expression of Vb. Additionally, a nonzero m0 is necessary to specifically regularize the last term in Vb from (III.42).

E. Ring Potential

We finally consider the nonperturbative ring poten- tial Vring. As mentioned in the previous paragraphs, the Lagrangian is written in terms of φ 1 and φ2, three type of vertices appear in the λ(φ⋆ φ) model (see Figure 3). We thus have four different types of ring diagrams:

- Type A:  A ring with N insertions of Π2 and N propagators Dℓ (ωn,ω 1) propagators, $V_{ring}^A$,

- Type B:  A ring with N insertions of Π 1 and N propagators Dℓ (ωn,ω2) propagators, $V_{ring}^B$,

- Type C: A ring with r insertions of Π2 and s insertions of Π 1 with N propagators Dℓ (ωn,ω2), $V_{ring}^C$.

Here, r ≥ 1 and r + s = N.

- Type D: A ring with r insertions of Π 1 and s insertions of Π2 with N propagators Dℓ (ωn,ω 1), $V_{ring}^D$.

Similar to the previous case, r ≥ 1 and r + s = N.

Here, Πi (T,Ω, mi) and Dℓ (ωn,ωi), i = 1, 2 are defined in (III.16) and (III.17), respectively. In Figure 6, these different types of ring potentials are demonstrated. The full contribution of the ring potential is given by

(III.43)

Following standard field theoretical method, it is possi- ble to determine the combinatorial factors leading to the standard form of the ring potential [40]. In Appendix D, we outline the derivation of $V_{ring}^I$, I = A, ···, D. They

are given by

Here, (i, j) = (2, 1) and (i, j) = (1, 2) correspond to $V_{ring}^A$ and $V_{ring}^B$, respectively. Plugging Dj from (III.17) into (III.45) and focusing on n = 0 as well as ℓ ≠ 0 contri- butions in the summation over n and ℓ, we arrive first at

into (III.46), the integration over k⊥ and kz can be car- ried out by making used of (A.9). To limit the summation over ℓ from below, we use the fact that the summand is even in ℓ. To perform the integration over k⊥ and kz, we use the Mellin transformation of ($u_j^2$) ^—N,

where

$I(\Omega )\equiv {\displaystyle \sum _{\mathcal{l}=1}^{\infty }}{e}^{{\mathcal{l}}^2{\Omega }^{2}t}.$ (III.50) To evaluate the summation over ℓ, we expand $e^{{\mathcal{l}}^2{\Omega }^{2}t}$ in a Taylor expansion and obtain

$I(\Omega )={\displaystyle \sum _{r=0}^{\infty }}{\displaystyle \frac{({\Omega }^{2}t{)}^r}{r!}}\zeta (-2r),$ (III.51) with ${\sum }_{\mathcal{l}=1}^{\infty }{\mathcal{l}}^{2r}=\zeta (-2r) \mathrm{a}\mathrm{n}\mathrm{d} \zeta (z) \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{R}\mathrm{i}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{n} \zeta -$ function. Since for r ∈ N, we have ζ(—2r) = 0, the only nonvanishing contribution to the summation over r $\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{e}\mathrm{s} \mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m} r=0.\mathrm{W}\mathrm{e}\mathrm{t}\mathrm{h}\mathrm{u}\mathrm{s}\mathrm{u}\mathrm{s}\mathrm{e}\zeta (0)=-{\displaystyle \frac{1}{2}}\mathrm{t}\mathrm{o}\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{a}\mathrm{t}$

$I(\Omega )=-{\displaystyle \frac{1}{2}}.$ (III.52)

Plugging this result into (III.49), using

${\int }_0^{\infty }dt{t}^{N-5/2}{e}^{-m_j^{2}t}={(m_j^2)}^{-j+3/2}\Gamma (j-3/2),$(III.53)

and performing the summation over N, we arrive at ${\mathcal{V}}_{\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}^{(i,j)}={\displaystyle \frac{T}{24\pi }}(2{(m_j^2+{\Pi }_i)}^{3/2}-2m_j^3-3m_j{\Pi }_i).$(III.54)

We arrive eventually at

Here, (i, j) = (2, 1) corresponds to $V_{ring}^C$ and (i, j) = (1, 2) to $V_{ring}^D$. Plugging Di from (III.17) into (III.56) and focusing on n = 0 and ℓ $\ne$ 0 contributions in the summa- tion over n and ℓ, we obtain

$V_{\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}^{(i,j)}=T{\displaystyle \sum _{\mathcal{l}=1}^{\infty }}\int d\tilde{k}{\displaystyle \sum _{N=2}^{\infty }}{\displaystyle \sum _{r=1}^N}{\displaystyle \frac{(-1{)}^N(N-r)!(r-1)!}{N!}}$

$\times {\Pi }_i^r{\Pi }_j^{N-r}(u_i^2{)}^{-N},(\mathrm{I}\mathrm{I}\mathrm{I}.57)$

where $u_j^2$ is defined below (III.46). Following, at this stage, the same steps as described in previous paragraph, we arrive first at

$V_{\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}^{(i,j)}={\displaystyle \frac{m_i^{3}T}{16{\pi }^{3/2}}}{\displaystyle \sum _{N=2}^{\infty }}{\displaystyle \sum _{r=1}^N}{\displaystyle \frac{(-1{)}^N}{N!}}{\displaystyle \frac{(N-r)!(r-1)!}{\Gamma (N)}}$

To perform the summation over N and r, we use the relation

${\displaystyle \sum _{N=2}^{\infty }}{\displaystyle \sum _{r=1}^N}f(N,r)={\displaystyle \sum _{N=2}^{\infty }}f(N,N)+{\displaystyle \sum _{r=1}^{\infty }}{\displaystyle \sum _{N=r+1}^{\infty }}f(N,r).$(III.59)

We thus obtain

$V_{\mathrm{ring}}^{(i,j)}$, = V (i) + V (i,j), (III.60)

with

$V^{(i)}\equiv {\displaystyle \frac{m_i^{3}T}{16{\pi }^{3/2}}}{\displaystyle \sum _{N=2}^{\infty }}{\displaystyle \frac{(-1{)}^N}{N}}{\displaystyle \frac{\Gamma (N-3/2){\Pi }_i^N(m_i^2{)}^{-N}}{\Gamma (N)}}$

$V^{(i,j)}\equiv {\displaystyle \frac{m_i^{3}T}{16{\pi }^{3/2}}}{\displaystyle \sum _{r=1}^{\infty }}{\displaystyle \sum _{N=r+1}^{\infty }}{\displaystyle \frac{(-1{)}^N}{N!}}{\displaystyle \frac{(N-r)!(r-1)!}{\Gamma (N)}}$

For V (i), the summation over N can be carried out and yields

$V^{(i)}={\displaystyle \frac{T}{24}}(2{(m_i^2+{\Pi }_i)}^{3/2}-2m_i^3-3m_i{\Pi }_i).$(III.62)

As concerns V (i,j), we perform the summation over N and arrive at

$V^{(i,j)}={\displaystyle \sum _{r=1}^{\infty }}{\displaystyle \frac{(-1{)}^{r+1}}{r}}{\displaystyle \frac{\Gamma (r-1/2)}{\Gamma (r+2)}}{\Pi }_i^r{\Pi }_j{(m_i^2)}^{-r-1}$

${\times }_3{F}_2((\mathrm{1,2},r-1/2);(r+1,r+2);-{\displaystyle \frac{{\Pi }_j}{m_i^2}}),$(III.63)

where pFq (a;b;z) is the generalized hypergeometric function having the following series expansion

${}_q{}^{}F_p^{}(\mathit{a};\mathit{b};z)={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{(a_1{)}_k\dots (a_p{)}_k}{(b_1{)}_k\dots (b_q{)}_k}}{\displaystyle \frac{z^k}{k!}}.$ (III.64) Here, a = (a1, ···, ap), b = (b1, ···, bq) are vectors with p and q components. Moreover, (ai)k ≡ Γ(ai + k)/Γ(ai) is the Pochhammer symbol. For our purposes, it is suffi- cient to focus on the contribution at r = 1 in (III.63).

$V^{(i,j)}{|}_{r=1}={\displaystyle \frac{T}{24\pi }}(2{(m_i^2+{\Pi }_j)}^{3/2}-2m_i^3-3m_i{\Pi }_j){\displaystyle \frac{{\Pi }_i}{{\Pi }_j}}.$(III.65)

Having in mind that the one-loop contribution to the self- energy Πi, which is determined in Sec. III B is of order O(λ), the contributions corresponding to r ≥ 2 are of order O(λ²) and can be neglected at this stage. We thus have

The final result for Vring is given by plugging $V_{ring}^I$, I = A, ···, D from (III.55) and (III.65) into (III.43),

Focusing only on the first perturbative correction to Πi and using ${\Pi }_i^{mat}$, i = 1, 2 from (III.28), the above results is simplified as

${\mathcal{V}}_{\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}}\approx {\displaystyle \frac{T}{8\pi }}{\displaystyle \sum _{i=1}^2}(2{(m_i^2+{\Pi }^{\mathrm{m}\mathrm{a}\mathrm{t}})}^{3/2}-2m_i^3-3m_i{\Pi }^{\mathrm{m}\mathrm{a}\mathrm{t}}),$(III.68)

$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}{\mathsf{\Pi }}^{\mathrm{m}\mathrm{a}\mathrm{t}}\equiv {\mathsf{\Pi }}_1^{\mathrm{m}\mathrm{a}\mathrm{t}}={\mathsf{\Pi }}_2^{\mathrm{m}}\mathrm{a}\mathrm{t}={\displaystyle \frac{\lambda T^3\zeta (3)}{2{\pi }^2\Omega }}.$

F. Summary of Analytical Results in Sec. III

In this section, we summarize the main findings. According to these results, the total thermodynamic potential of a rigidly rotating Bose gas, Vtot, including the classical potential Vcl from (II.32) with c² replaced with a², the vacuum potential (II.33), the thermal part (II.34), and the ring potential (III.43) is given by

Vtot = Vcl + Vvac + VT + Vring, (III.69) with

Here, a² = c² + m0², m1²(v) = 3λv² − c² and m2²(v) = λv² − c², and Π^mat = λT³ ζ(3)/2π² Ω. We notice that the logarithmic terms appearing in Vvac from (III.42) are skipped in (III.70).

In the next section, we study the effect of rotation on the formation of condensate and the critical temperature of the global U(1) phase transition. To this purpose, we compare our results with the results arising from the full thermodynamic potential of a nonrotating Bose gas. According to [40], it is given by (Subscripts (0) correspond to Ω = 0.)

$V_{tot}^{\left(0\right)}$ = Vcl + Vvac + $V_T^{\left(0\right)}$ + $V_{\mathrm{ring}}^{\left(0\right)}$, (III.71) where Vcl and Vvac are given in (III.70), while $V_T^{\left(0\right)}$ and $V_{\mathrm{ring}}^{\left(0\right)}$ read

with the one-loop self-energy correction Π0^mat = λT² /3 [40] and mi², i = 1, 2 given as above.

IV. Numerical Results

In this section, we explore the effect of rotation on different quantities related to the spontaneous breaking

of global U(1) symmetry. To this purpose, we consider different parts of Vtot from (III.69).

In Sec. III A, we derived the minimum of the potential

Va including Vcl and VT. We arrived at vmin² (T,Ω) from

(III.6). Replacing VT with $V_T^{\left(0\right)}$ from (III.72) for a nonrotating Bose gas and following the same steps leading from (III.4) to (III.6), we arrive at the critical temperature

${{\mathsf{T}}_{\mathsf{c}}}^{(0)}={({\displaystyle \frac{3a^2}{\lambda }})}^{1/2},$(IV.1)

and the T dependent minima

with the reduced temperature t0 = T/Tc⁽⁰⁾ and Tc⁽⁰⁾ from (IV.1). In Figure 7, vmin² is plotted for Ω = 0 [see (IV.2)] and Ω $\ne$ 0 [see (III.6)] as function of the corresponding reduced temperature t0 and t. The difference between these two plots arises mainly from different exponents of the corresponding reduced temperatures t0 and t in (IV.2) and (III.6). The reason of this difference lies in dif- ferent results for the high-temperature expansion of $V_T^{\left(0\right)}$ for Ω = 0 [see (III.72)] and VT for Ω $\ne$ 0 [see (III.70)].

Let us consider Vtot − Vring = Vcl + Vvac + VT from (III.69). By minimizing this potential with respect to v, and solving the resulting gap equation,

${\displaystyle \frac{d}{dv}}{({\mathrm{V}}_{\mathrm{t}\mathrm{o}\mathrm{t}}-{\mathrm{V}}_{\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{g}})}_{{\mid }_{{\overline{v}}_{\mathrm{m}\mathrm{i}\mathrm{n}}}}=0,(\mathrm{I}\mathrm{V}.3)$

it is possible to determine numerically the T dependence the minima, denoted by v-min (T,Ω), for fixed Ω. To this purpose, we use the quantities a ≃ 0.265 GeV, c ≃ 0.225 GeV, and λ = 0.5 given in (III.7). In Figure 8, the T/Tc⁽⁰⁾ dependence of $\overline{v}$min is demonstrated for βΩ = 0.1, 0.2, 0.3

(dashed, dotted, and dotted-dashed curves). The results are then compared with the corresponding minima for a nonrotating Bose gas (red solid curve). The latter is determined by minimizing the combination $V_{\mathrm{tot}}^{\left(0\right)}$ — $V_{\mathrm{ring}}^{\left(0\right)}$, according to

${\displaystyle \frac{d}{dv}}({\mathrm{V}}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{(0)}{=\mathrm{V}}_{\mathrm{r}\mathrm{i}\mathrm{m}.\mathrm{a}}^{(0)}{)}_{{\displaystyle \genfrac{}{}{0pt}{}{\mathrm{t}}{,{\overline{v}}_{\mathrm{m}\mathrm{i}\mathrm{n}}}}}^{\mathrm{t}}=0,(\mathrm{I}\mathrm{V}.4)$

with $V_{\mathrm{tot}}^{\left(0\right)}$ from (III.71). In both cases, Tc⁽⁰⁾ ≃ 0.681 GeVis the critical temperature of the spontaneous U(1) sym- metry breaking in a nonrotating Bose gas. (The critical temperature is the temperature at which the condensate $\overline{v}$min vanishes.)

These results indicate that rotation lowers the critical temperature of the phase transition. However, as shown in Figure 8, Tc increases with increasing Ω. It is also im- portant to note that this same trend is observed in a noninteracting Bose gas under rigid rotation [30].

To answer the question whether the transition is con- tinuous or discontinuous, we have to explore the shape of the potential, the value of its first and second order derivatives at temperatures below and above the critical temperature, Tc. Using the numerical values for the set of free parameters a, c, and λ as mentioned above, the transitions turns out to be continuous not only for Ω = 0 but also for Ω $\ne$ 0.

To explore the effect of the ring potential on the tem- perature dependence of the condensate v-min, we solved numerically the gap equation

for a rotating and a nonrotating Bose gas, respectively. The corresponding results are demonstrated in Figure 9. Because of the specific form of the ring potentials Vring and $V_{\mathrm{ring}}^{\left(0\right)}$ from (III.70) and (III.73), including in particular (mi² + Π^mat)3/2, there is a certain value of v below which the potential is undefined (imaginary). Let us denote this value by $\overline{v}$⋆. In both rotating and non- rotating cases v⋆ ≃ 0.319 GeV. As it is shown in Figure 9, the minima decrease with increasing temperature and converge towards $\overline{v}$⋆. Let us denote the temperature at which $\overline{v}$min = v⋆ with T⋆ for Ω $\ne$ 0 and T⋆⁽⁰⁾ for Ω = 0. For Ω = 0, T⋆⁽⁰⁾ ≃ 0.300 GeV, and as it is shown in Figure 9, the transition to v⋆ is discontinuous (red circles). For Ω $\ne$ 0, however, T⋆ < T⋆⁽⁰⁾ and increases with increasing βΩ, similar to the results presented in Figure 8. Moreover, in contrast to the case of Ω = 0, the transition to $\overline{v}$⋆ for all values of βΩ $\ne$ 0 is continuous.

In Figure 10, the phase diagram Tc-Ω is plotted for two cases: The blue solid curve demonstrates Tc from (III.5) arising from Vcl + VT. Red dots denote the Ω depen- dence of Tc arising from the potential Vtot − Vring. A comparison between these data reveals the effect of Vvac in increasing Tc. Apart from the Ω dependence of Tc, the Ω dependence of T⋆ is demonstrated in Figure 10. It arises by adding the ring contribution to Vcl + VT + Vvac, as described above. According to the results demonstrated in Figure 10, considering Vring decreases Tc. But, simi- lar to Tc, T⋆ also increases with increasing Ω. It should be emphasized that the transition shown in Figure 8 is a crossover, since $\overline{v}$⋆ $\ne$ 0.

In Sec. III B, the masses m1², i = 1, 2 including the one-loop correction are determined [see (III.29)]. Identi- fying m1² with mσ² and m2² with mπ², we arrive at

$\begin{array}{cc}{\mathsf{m}}_{\overline{n}}^2(\mathsf{v})=3\lambda {\mathsf{v}}^2-c^2+{\mathsf{a}}^2{\mathsf{t}}^3,& \\ {\mathsf{m}}_{\overline{n}}^2(\mathsf{v})=\lambda {\mathsf{v}}^2-c^2+{\mathsf{a}}^2{\mathsf{t}}^3.& (10.7)\end{array}$

Using the data for $\overline{v}$min² arising from the solution of the gap equation (IV.3) and (IV.5), and evaluating mσ²(v²) and mπ² (v²) from (IV.7) at m2 in for a fixed βΩ, the t = T/Tc dependence of mσ² and mπ² is determined. In Figure 11(a), the dependence of mσ²($\overline{v}$²min) and mπ² ($\overline{v}$²min) with vmin arising from (IV.3) on the reduced tempera- ture t = T/Tc is plotted for fixed βΩ = 0.1. Here, the contribution of the ring potential is not taken into ac- count. Hence, a continuous phase transition occurs with the critical temperature Tc ∼ 0.399 GeV for Ω = 0.1 GeV. In contrast, in Figure 11(b), mσ² and mπ² are determined by plugging the data of $\overline{v}$min arising from (IV.5), with Vtot including the ring potential. Hence, the difference between the plots demonstrated in Figs. 11(a) and 11(b) arises from the contribution of the nonperturbative ring potential. As we have mentioned above, when the ring potential is taken into account, the data demonstrated in Figure 9 do not describe a true transition, since $\overline{v}$⋆ is not zero. The reduced temperature in Figure 11(b) is thus defined by t⋆ ≡ T/T⋆, where, according to the data pre- sented in Figure 10 T⋆ ∼ 0.278 GeV for Ω = 0.1.

Let us compare the results demonstrated in Figure 11(a) with that in Figure 5. In both cases, before the phase transition at t < 1, $m_{\sigma }^2$ decreases with increasing t. Moreover, whereas in Figure 5, $m_{\pi }^2$ remains constant, it slightly decreases once the Vvac contribution is taken into account.

After the transition, at t ≥ 1, $m_{\sigma }^2$ becomes equal to $m_{\pi }^2$ and they both increase with increasing t. It is straight- forward to verify this statement using equation (IV.7). Given that, in this case, the minima of the potential at t ≥ 1 are zero, it follows that both masses are equal, specifically $m_{\sigma }^2$(0) = $m_{\pi }^2$(0), once we substitute $\overline{v}$min = 0 into (IV.7).

This behavior is expected from the case of Ω = 0 and in the framework of fermionic Nambu-Jona–Lasinio (NJL) model: As noted in [45], in the symmetry-broken phase, $m_{\sigma }^2$ > $m_{\pi }^2$. As the transition temperature is approached, $m_{\sigma }^2$ decreases, and at a certain dissociation temperature Tdiss, the masses mσ and mπ become de-generate. This temperature is characterized by mσ (Tdiss) = 2mπ (Tdiss). (IV.8)

As it is described in [45], σ mesons dissociates into two pions because of the appearance of an s-channel pole in the scattering amplitude π + π → π + π. In this process a σ meson is coupled to two pions via a quark triangle. In the symmetry-restored phase, at t ≥ 1, mσ becomes equal to mπ. They both increase with increasing T [45,46].

In Table 1, the σ dissociation temperatures are listed for Ω = 0, 0.1, 0.2, 0.3 GeV. The data in the second (third) column correspond to Tdiss ($T_{\mathrm{diss}}^{\star }$) for the case when $\overline{v}$min is the solution of (IV.3) [(IV.5)] for Ω $\ne$ 0 and (IV.4) [(IV.6)] for Ω = 0. Comparing Tdiss and $T_{\mathrm{diss}}^{\star }$ with Tc and T⋆ shows that Tdiss < Tc and similarly $T_{\mathrm{diss}}^{\star }$ < T⋆. The property Tdiss $\ne$ Tc is because we are working with mπ $\ne$ 0. Let us notice that, as aforementioned, the σ dissociation temperature is originally introduced in a fermionic NJL model [45]. In this model, nonvanishing mπ $\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{a}\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}\widetilde{{\mathrm{m}}_0},\mathrm{a}\mathrm{n}\mathrm{d}$ $\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\tilde{m}$0 $\ne$ 0 implies a crossover transition charac- terized by Tdiss $\ne$ Tc. It seems that in the bosonic model studied in the present work, a nonvanishing pion mass leads similarly to Tdiss $\ne$ Tc.

The behavior demonstrated in Figure 11(a) changes once the contribution of the ring potential is taken into account. As it is shown in Figure 11(b), in the symmetry- broken phase at t⋆ < 1, mσ decreases slightly with T, while mπ increases with T. Moreover, in contrast to the case in which Vring is not taken into account, mσ and mπ are not equal at t ≥ 1. This observation highlights the ef- fect of nonperturbative ring contributions on the relation between mσ and mπ, mainly in the symmetry-restored phase. This behavior is directly related to the fact that the effect illustrated in Figure 9 is a crossover once the ring contribution is considered: Plugging u-⋆ into (IV.7), the masses of σ and π mesons are given by

Their difference is thus given by and remains constant in t. This fact can be observed in Figure 11(b) at t⋆ ≥ 1.

V. Summary and Conclusions

In this paper, we extended the study of the effects of rigid rotation on BE condensation of a free Bose gas in [30], to a self-interacting charged Bose gas under rigid rotation. In the first part, we considered the Lagrangian density of a complex scalar field 夕 with mass m, in the presence of chemical potential μ and angular velocity Ω. The interaction was introduced through a λ(夕⋆ 夕) term. This Lagrangian is invariant under global U(1) transfor- mation. To investigate the spontaneous breaking of this symmetry, we chose a fixed minimum with a real com- ponent u, and evaluated the original Lagrangian around this minimum to derive a classical potential. Then, we applied an appropriate Bessel-Fourier transformation to determine the free propagator of this model, expressed in terms of two masses m1 and m2, corresponding to the two components of the complex field 夕. These masses depend explicitly on u,λ, and m, and played a crucial role when the spontaneous breaking of U(1) symmetry was consid- ered in a realistic model that includes σ and π mesons. Using the free boson propagator of this model, we derived the thermodynamic potential of self-interacting Bose gas at finite temperature T. This potential consists of a vac- uum and a thermal part. Along with the classical poten- tial, this forms the total thermodynamic potential of this model V_tot from (II.24). This potential is expressed in terms of the energy dispersion relation ${\in }_k^{\pm }$ from (II.15), and explicitly depends on lΩ. A novel result presented here is that, although lΩ appears to resemble a chemical potential in combination with ${\in }_k^{\pm }$ in Vtot, the chemical potential μ affects ${\in }_k^{\pm }$ in a nontrivial manner. The effec-tive chemical potential μeff = μ + lΩ appears solely in a noninteracting Bose gas under rotation (see the special case 1 in Sec. II D and compare the thermodynamic po- tential with that appearing in [30]).

For λ,μ $\ne$ 0, we explored two cases μ > m and μ < m. The former corresponds to the phase where U(1) symme- try is broken, while the latter describes the symmetry- restored phase. By expanding the two branches of the energy dispersion relation around k ~ 0 in the symmetry- broken phase, we identified ∈k⁺ and ∈k- as phonon and roton, with the latter representing a massless Goldstone mode. Upon comparison with analogous results for a nonrotating and self-interacting Bose gas, we found that rigid rotation does not alter the behavior of ${\in }_k^{\pm }$ at k ~ 0. This is mainly because rotation appears in terms of lΩ within Vtot, rather than directly affecting ${\in }_k^{\pm }$.

In the second part of this paper, we examined the effect of rigid rotation on the spontaneous breaking of U(1) symmetry in an interacting Bose gas at μ = 0 (see Sec. III). In this case, where m² < 0, we replaced m² with −c², where c² > 0. By introducing an additional term to the original Lagrangian, we defined a new mass, a²= c² + m0². We demonstrated that the minimum of the classical potential is nonzero, indicating a sponta- neous breaking of U(1) symmetry. We then addressed the question about the position of this minimum, specif- ically its dependence on T and Ω, after accounting for the thermal part of the effective potential combined with the classical potential. To investigate this, we performed a high-temperature expansion of the thermal part of the potential, utilizing a method originally introduced in [30]. This approach enabled us to sum over the angular mo- mentum quantum numbers ℓ for small values of βΩ, al- lowing us to derive both the critical temperature of the phase transition Tc and the dependencies of the mini- mum of the potential on T and Ω. At this stage, we have Tc ∝ λ ^-1/3, which is in contrast to the Tc⁽⁰⁾ ∝ λ ^-1/2 for a nonrotating Bose gas. In addition, Tc ∝ Ω ^1/3. Let us remind that the critical temperature of a BEC transi- tion for a noninteracting Bose gas in nonrelativistic and ultrarelativistic limits are Tc ∝ Ω^2/5 and Tc ∝ Ω ^1/4, re- spectively [30]. This demonstrates the effect of rotation in changing the critical exponents of different quantities in the symmetry-broken phase.

We defined a reduced temperature t = T/Tc, and showed that in the symmetry-broken phase, the minimum men- tioned above depends on (1-t³), while for a nonrotating Bose gas this dependence is (1 - ${t_0}^2$), where t0 = T/Tc⁽⁰⁾. In the symmetry-restored phase, this minimum vanishes. This indicates a continuous phase transition in both non- rotating and rotating Bose gases. Plugging these minima into ${m_1}^2$(v) and ${m_2}^2$(v), it turned out that at t ≥ 1, i.e., in the symmetry-restored phase m1 and m2 are imaginary. Since, according to our arguments in Sec. III, m2 is the mass of a Goldstone mode, we expect that in the chiral limit, i.e., when m0 = 0, it vanishes in the symmetry- broken phase at t < 1. However, as it is shown in (III.13), ${m_2}^2$ < 0 in this phase.

To resolve this issue, we followed the method used in [40] and added the thermal part of one-loop self-energy diagram to the above results. In contrast to the case of nonrotating bosons, where the thermal mass square is proportional to λT², for rotating bosons it is proportional to λT³ /Ω. To arrive at this result, a summation over ℓ was necessary. This was performed by utilizing a method originally introduced in [30]. Adding this perturbative contribution to ${m_i}^2$, i = 1, 2 at t < 1 and t ≥ 1, we showed that the Goldstone theorem is satisfied in the chiral limit [see Sec. III C].

In Secs. III D and III E, we added the vacuum and nonperturbative ring potentials to the classical and ther- mal potentials. The main novelty of these sections lies in the final results for these two parts of the total potential, specifically the method we employed to sum over ℓ. Ac- cording to this method the vacuum part of the potential for a rigidly rotating Bose gas is the same as that for a nonrotating gas. We followed the method described in [43] to dimensionally regularize the vacuum potential. As concerns the ring potential, we present a novel method to compute this nonperturbative contribution to the ther- modynamic potential. In particular, we summed over ℓ by performing a ζ-function regularization. In Sec. III F, we presented a summary of these results.

In Sec. IV, we used the total thermodynamic poten- tial presented in Sec. III to study the effect of rotation on the spontaneous U(1) symmetry breaking of a realistic model including σ and π mesons. Fixing free parameters mσ, mπ, and λ, and identifying m1 and m2 with the me- son masses mσ and mπ, we obtained numerical values for c and a (see Sec. III A). First, we determined the T de- pendence of the minima of the total thermodynamic po- tential, excluding the ring contribution. According to the results presented in Figure 8, rotation decreases the critical temperature of the U(1) phase transition. Additionally, it is shown that Tc increases with increasing Ω. In [30], it is shown that the critical temperature of the BEC in a noninteracting Bose gas under rotation behaves in the same manner. This phenomenon indicates that rotation enhances the condensation. Recently, a similar result was observed in [47], where it is demonstrated that the inter- play between rotation and magnetic fields significantly increases the critical temperature of the superconducting phase transition.

To explore the effect of nonperturbative ring poten- tial, we numerically solved the gap equation correspond- ing to the total thermodynamic potential and determined its minima $\overline{v}$min. Because of the specific form of the ring potential, there was a certain $\overline{v}$⋆ through which all the curves v-min (T,Ωf), independent of the chosen Ωf, con- verge (see Figure 9). Moreover, the transition for Ω = 0 turned out to be discontinuous, while it is continuous for all Ω $\ne$ 0. As it is demonstrated in Figure (10), T⋆ increases with increasing Ω.

Finally, we determined the T dependence of the masses mσ and mπ mesons for a fixed value of Ω. To achieve this, we utilized (IV.7) along with $\overline{v}$min, which is derived from Figs. 8and 9. The plot shown in Figure 11(a), based on the total potential excluding the ring contribu- tion, is representative of the T dependence of mσ and mπ (see e.g., [46]). However, when we include the ring contribution, the shape of the plots changes, especially at T > T⋆. The reason is that considering the ring potential changes the order of the phase transition from a second order transition to continuous (for Ω $\ne$ 0) or discontinu- ous (for Ω = 0) a crossover. In this context, we numer- ically determined the σ dissociation temperature Tdiss, which may serve as an indicator for type of the transi- tion into the symmetry-restored phase. We showed that Tdiss < Tc and $T_{\mathrm{diss}}^{\star }$ < T⋆, as expected from a crossover transition [46].

It would be intriguing to extend the above findings, in particular those from Sec. III, to the case of nonvanish- ing chemical potential. In [48], the kaon condensation in a certain color-flavor locked phase (CFL) of quark mat- ter is studied at nonzero temperature. This is a state of matter which is believed to exist in quark matter at large densities and low temperatures. Large densities at which the color superconducting CFL phase is built are expected to exist in the interior of neutron stars. One of the main characteristic of these compact stars, apart from densities, is their large angular velocities. It is not clear how a rigid rotation, like that used in the present paper, may affect the formation of pseudo-Goldstone bosons and the critical temperature of the BE condensation in this nontrivial environment. We postpone the study of this problem to our future publication.

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