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

II. Interacting charged scalars under rigid rotation

A. The free propagator

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

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

with the metric

$\begin{array}{c}{g}_{\mu \nu }=\left(\begin{array}{cccc}1-r^2{\text{Ω}}^2& y\text{Ω}& -x\text{Ω}& 0\\ y\text{Ω}& -1& 0& 0\\ -x\text{Ω}& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right)(II.2)\end{array}$

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 x^µ = (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

$\phi (\text{x})\to {\text{e}}^{-\text{i}\alpha }\phi (\text{x}),{\phi }^{\star }(\text{x})\to {\text{e}}^{+\text{i}\alpha }{\phi }^{\star }(\text{x})( III.3)$

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

$\begin{array}{c}L={\left|\left({\partial }_0-i\mu -i\text{Ω}{L}_z\right)\phi \right|}^2-|\nabla \phi {|}^2-m^2|\phi {|}^2-\lambda \mid \phi (II.4)\end{array}$

L = |(∂₀ − iμ − iΩL_z)φ|² − |∇φ|² − m²|φ|² − λ|φ|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 real fields 1 and 2 appearing in  = 1/(1+i2) and perform the shift φi → Φi + φi with $\text{Φ}=\left(\begin{array}{c}v\\ 0\end{array}\right)and$ v = const. We arrive at

$\begin{array}{l}ℒ={\displaystyle \sum }_{i=0}^4{ℒ}_i,\text{ (III}\text{.5)}\\ \text{ with}\\ {ℒ}_0 =\frac{1}{2}\left({\mu }^2-m^2\right)v^2-\frac{\lambda }{4}{v}^4,\\ {\text{L}}_1 ={\left({\mu }^2-{\text{m}}^2\right)}_{\text{v}\phi 1}-\mu \text{v}{\partial }_0{\phi }_2-\lambda {\text{v}}^3{\phi }_1+\text{i}\mu \text{ΩvLZ}\phi 2,\\ {ℒ}_2 =\frac{1}{2}\{{\left({\partial }_0{\phi }_1\right)}^2+{\left({\partial }_0{\phi }_2\right)}^2-{\left({\nabla }_{{\phi }_1}\right)}^2-{\left({\nabla }_{{\phi }_2}\right)}^2+\left({\mu }^2-m^2\right)\left({\phi }_1^2+{\phi }_2^2\right)+2\mu \left({\phi }_2{\partial }_0{\phi }_1-{\phi }_1{\partial }_0{\phi }_2\right)-\lambda \left(3v^2{\phi }_1^2+v^2{\phi }_2^2\right)-\text{Ω}2[( {\text{L}}_{\text{Z}}\\ {{\phi }_1)}^2+{\left({\text{L}}_{\text{Z}}{\phi }_2\right)}^2]-2\text{iΩ}\left[{\left({\partial }_0{\phi }_1+\mu {\phi }_2\right)}^{{\text{L}}_{\text{Z}}{\phi }_1}\right]-2\text{iΩ}{\left[{\left({\partial }_0{\phi }_2-\mu {\phi }_1\right)}_{{\text{L}}_{\text{Z}}{\phi }_2}\right]}^{\}},\\ {\text{L}}_3 =-\lambda \text{v}{\phi }_1\left({\phi }_1^2+{\phi }_2^2\right),\\ {ℒ}_4 =-\frac{\lambda }{4}{\left({\phi }_1^2+{\phi }_2^2\right)}^2\text{ (III}\text{.6)}\end{array}$

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

$\begin{array}{c}{\mathcal{V}}_{\text{cl}}(v)\equiv -{ℒ}_0=\frac{1}{2}\left(m^2-{\mu }^2\right)v^2+\frac{\lambda }{4}{v}^4(II.7)\end{array}$

The free propagator arises from the quadratic term L2 in the fluctuating fields φ₁ and φ₂. To derive the free propagator in the momentum space, we use the Fourier- Bessel transformation

$\begin{array}{c}{\phi }_i(x)=\sqrt{\frac{\beta }{V}}{\displaystyle \sum }_{n,\ell ,k}{e}^{i\left({\omega }_n\tau +\ell \varphi +k_{z}z\right)}{J}_{\ell }\left(k_{\perp }r\right){\tilde{\phi }}_i(k)(II.8)\end{array}$

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 L_z, continuous momentum k_z, and ${\text{k}}_{\perp }\equiv \left|k_{\perp }\right|\equiv {\left({\text{k}}_{\text{x}}^2+{\text{k}}_{\text{y}}^2\right)}^{1/2}$ 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

$\begin{array}{c}\underset{X}{{\displaystyle \int }}\equiv \underset{0}{\overset{\beta }{{\displaystyle \int }}}d\tau \underset{0}{\overset{\infty }{{\displaystyle \int }}}rdr\underset{0}{\overset{2\pi }{{\displaystyle \int }}}d\varphi \underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}dz(II.9)\end{array}$

we arrive after some manipulations at

$\begin{array}{c}\underset{X}{{\displaystyle \int }}{ℒ}_2=-\frac{V}{2}{\displaystyle \sum }_{n,\ell ,k}\left({\tilde{\phi }}_1(-k){\tilde{\phi }}_2(-k)\right)\left({\beta }^2{\mathcal{D}}_{\ell }^{-1}(k)\right)\left(\begin{array}{c}{\tilde{\phi }}_1(k)\\ {\tilde{\phi }}_2(k)\end{array}\right),(II.10)\end{array}$

with the free propagator

$\begin{array}{c}{\beta }^2{\mathcal{D}}_{\ell }^{-1}(k)=\left(\begin{array}{cc}{\left({\omega }_n+i\ell \text{Ω}\right)}^2+{\omega }_1^2-{\mu }^2& -2\mu \left({\omega }_n+i\ell \text{Ω}\right)\\ 2\mu \left({\omega }_n+i\ell \text{Ω}\right)& {\left({\omega }_n+i\ell \text{Ω}\right)}^2+{\omega }_2^2-{\mu }^2\end{array}\right)(II.11)\end{array}$

Here, ${\omega }_i^2\equiv k^2+{\text{m}}_i^2,\text{i}=1,2$ with ${\text{m}}_1^2(\text{v})\equiv 3\lambda {\text{v}}^2+{\text{m}}^2$ and ${\text{m}}_2^2(\text{v})\equiv \lambda {\text{v}}^2+{\text{m}}^2$ the corresponding masses to two fields φ 1 and φ2. In cylinder coordinate system, we have $k^2\equiv k_{\perp }^2+{\text{k}}_{\text{Z}. }^2$ . 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

$\mathcal{V}=-\frac{T}{V}\text{ln}\mathcal{Z}, (II.12)$

with

$\text{ln}\mathcal{Z}=-\frac{1}{2}\text{lndet}\left({\beta }^2{\mathcal{D}}_{\ell }^{-1}(k)\right)$ (II.13) Let us first focus on ln Z with Z the partition function of this model. Plugging Dℓ-1 from (II.11) into (II.13), we arrive first at

$\begin{array}{c}lnZ=-\frac{1}{2}{\displaystyle \sum }_{e=\pm }{\displaystyle \sum }_{n,\ell ,k}ln\left|{\beta }^2\left[{\left({ϵ}_k^e\right)}^2+{\left({\omega }_n+i\ell \text{Ω}\right)}^2\right]\right|(II.14)\end{array}$

with ${ϵ}_k^{\pm }$ given by

$\begin{array}{c}{ϵ}_k^{\pm }\equiv {\left(E_k^2+{\mu }^2\mp \sqrt{4{\mu }^2{E}_k^2+\delta M^4}\right)}^{1/2}(II.15)\\ {\text{E}}_k^2=k^2+{\text{M}}^2, and \end{array}$

$M^2\equiv \frac{1}{2}\left(m_1^2+m_2^2\right),\delta M^2\equiv \frac{1}{2}\left(m_1^2-m_2^2\right)\text{ (II}\text{.16)}$

Following standard steps, it is possible to show that

$\begin{array}{l}\text{ln}\mathcal{Z}=-\frac{1}{4}{\displaystyle \sum }_{e=\pm }{\displaystyle \sum }_{n,\ell ,k}\{\text{ln}\left({\beta }^2\left[{\omega }_n^2+{\left({ϵ}_k^e+\ell \text{Ω}\right)}^2\right]\right)\\ +\text{ln}\left({\beta }^2\left[{\omega }_n^2+{\left({ϵ}_k^e-\ell \text{Ω}\right)}^2\right]\right)\}(\text{II}.17)\end{array}$

Performing the Matsubara sum with

$\begin{array}{c} {\displaystyle \sum }_{n=-\infty }^{+\infty }ln\left({(2\pi n)}^2+{\eta }^2\right)=\eta +2ln\left(1-e^{-\eta }\right)(II.18)\end{array}$

n= -∞

we arrive at

$\begin{array}{l}\text{ln}\mathcal{Z}=-\frac{V}{2}{\displaystyle \sum }_{e=\pm }{\displaystyle \sum }_{\ell }{{\displaystyle \int }}^{\text{}} d\tilde{k}\{\beta {ϵ}_k^e\\ +\text{ln}\left(1-e^{-\beta \left({ϵ}_k^e+\ell \text{Ω}\right)}\right)+\text{ln}\left(1-e^{-\beta \left({ϵ}_k^e-\ell \text{Ω}\right)}\right)\}(II.19)\end{array}$

where the summation over k is replaced with the inte- gration over k in the cylinder coordinate system, ${\displaystyle \sum }_k\to V{\displaystyle \sum }_{n,\ell }{{\displaystyle \int }}^{\text{}} d\tilde{k}, with {{\displaystyle \int }}^{\text{}} d\tilde{k}\equiv {{\displaystyle \int }}^{\text{}} \frac{k_{\perp }d{k}_{\perp }d{k}_z}{{(2\pi )}^3}$ (II.20)

Here, k⊥ ≡ |k⊥ |. Using (II.12), the thermodynamic po- tential V is given by

V = Vvac + VT, (II.21)

with the vacuum part

${\text{V}}_{\text{vac}}\equiv \frac{1}{2}{{{\displaystyle \sum }}^{\text{}}}_{\ell }{{\displaystyle \int }}^{\text{}}d\tilde{k}\left({ϵ}_k^{+}+{ϵ}_k^{-}\right)\text{ (II}\text{.22)}$

and the matter (thermal) part

$\begin{array}{l}{\mathcal{V}}_T=\frac{T}{2}{{{\displaystyle \sum }}^{\text{}}}_{e=\pm }{{{\displaystyle \sum }}^{\text{}}}_{\ell }{{\displaystyle \int }}^{\text{}}d\tilde{k}\{\text{ln}\left(1-e^{-\beta \left({ϵ}_k^e+\ell \text{Ω}\right)}\right)\text{ (II}\text{.23)}\\ +\text{ln}\left(1-e^{-\beta \left({ϵ}_k^e-\ell \text{Ω}\right)}\right)\}\end{array}$

Adding V with V_cl (v) from (II.7), to include the zero mode contribution, we obtain the full thermodynamic potential Vtot,

$\begin{array}{l}{\mathcal{V}}_{tot }=\frac{1}{2}\left(m^2-{\mu }^2\right)v^2+\frac{\lambda }{4}{v}^4+\frac{1}{2}{\displaystyle \sum }_{\ell }{{\displaystyle \int }}^{\text{}} d\tilde{k}\left({ϵ}_k^{+}+{ϵ}_k^{-}\right)\text{ (II}\text{.24)}\\ +\frac{T}{2}{\displaystyle \sum }_{e=\pm }{\displaystyle \sum }_{\ell \ne 0}{{\displaystyle \int }}^{\text{}} d\tilde{k}\{\text{ln}\left(1-e^{-\beta \left({ϵ}_k^e+\ell \text{Ω}\right)}\right)\\ +\text{ln}\left(1-e^{-\beta \left({ϵ}_k^e-\ell \text{Ω}\right)}\right)\}\end{array}$

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, V_cl possesses one single min- imum at ${\overline{\upsilon }}_0=0$ and the system is in its symmetric phase.

In this case, ${\text{m}}_1^2(\text{v}-\text{o})={\text{m}}_2^2(\text{v}-\text{o})={\text{m}}^2,\delta {\text{M}}^2=\text{o}$ and ${ϵ}_{\text{k}}^{\pm }$ is given by

${ϵ}_k^{\pm }=\sqrt{k^2+m^2}\mp \mu \text{ (II}\text{.25)}$

Here, m is a mass gap and $\Delta ϵ\text{k}\equiv ϵ{\text{k}}^{-}-ϵ\stackrel{+}{k}=2\mu \frac{1}{2}$ . In Figur $\mu =0.6\text{MeV}(\mu <\text{m})$ e is plotted for generic mass m=1MeV and chemical potential ${\text{m}}_2^2(\text{v}-\text{b})={\mu }^2$ .

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

${\overline{\nu }}_b=\pm \sqrt{\frac{{\mu }^2-m^2}{\lambda }}$

The masses and ${\text{m}}_2^2(\text{v}-\text{b})={\mu }^2$ . We thus have ${\text{M}}^2=2{\mu }^2-{\text{m}}^2$ and $\delta {\text{M}}^2={\mu }^2-{\text{m}}^2$ leading to

$\begin{array}{c}{ϵ}_k^{\pm }=\sqrt{k^2+\left(3{\mu }^2-m^2\right)\mp \sqrt{4{\mu }^2{k}^2+\left(3{\mu }^2-m^2\right)}}(II.26)\end{array}$

In Figure $1( \text{b}),\in \underset{k}{\pm }$ is plotted for generic $\mu =1.1\text{MeV}$ and $\text{m}=1\text{MeV}(\mu >\text{m})$ . As it is shown, whereas $ϵ{\text{k}}^{-}$ is quadratic in $\text{k}\equiv \left|k\right|,ϵ\stackrel{+}{k}\sim \text{o}$ for k ~ 0. This behavior indicates the presence of a massless Goldstone mode. By expanding ${ϵ}_k^{\pm }$ in the orders of k ~ 0, we obtain

$\begin{array}{rr}\hfill {ϵ}_k^{-}& \hfill \simeq \sqrt{2\left(3{\mu }^2-m^2\right)}+\frac{5{\mu }^2-m^2}{2\sqrt{2{\left(3{\mu }^2-m^2\right)}^3}}{k}^2\\ \hfill {ϵ}_k^{+}& \hfill \simeq \sqrt{\frac{{\mu }^2-m^2}{3{\mu }^2-m^2}}\left|k\right|.\end{array} (II.27)$

Figure 1: (color online). The k dependence of the energy dispersion ϵk±k from (II.25) and (II.26) in the U(1) symmetric phase (panel a) and the symmetry-broken phase (panel b), characterized by µ < m and µ > m, respectively. As demonstrated, in the symmetry-broken phase, there is a massless Goldstone mode. These findings remain unchanged regardless of any rotation.

According to these results, ${ϵ}_k^{+}$ and $\ell \text{Ω}$ correspond to phonon and roton modes in the symmetry-broken phase $m<\mu$ , respectively.

As it is shown in this section, ${{ϵ}_k^{\pm }|}_{\lambda =0,\mu \ne 0}=\sqrt{k^2+m^2}\mp \mu$ appears in the ther- mal part of the effective potential VT from (II.23) and does not modify neither ${\text{m}}_i^2(\text{v})$ nor the energy dispersion $\text{Tc}={\left(\frac{a^2{\pi }^2\Omega }{2\lambda \zeta (3)}\right)}^{1/3}$ . Hence, a comparison with analogous results for nonrotating bosons [42] shows that rigid rotation has no effect on the behavior of ${ϵ}_k^{\pm }$ at $\text{k}\sim 0$ .

D. Two special cases

In what follows, we consider two special cases $\lambda =0,\mu \ne 0$ and $\ne 0,\mu =0$ :

Case 1: For the special case of noninteracting rotating Bose gas with $\lambda =0$ and $\mu \ne 0$ , we have ${\text{m}}_1={\text{m}}_2=\text{m}$ ,

${\text{E}}_{\text{k}}^2=k^2+{\text{m}}^2$ , and $\delta \text{M}=\text{o}$ . We thus have

${\text{m}}_2^2(\text{v})=\lambda {\text{v}}^2-{\text{c}}^2\text{ (II}\text{.28)}$

and therefore

${{\mathcal{V}}_{tot }|}_{\lambda =0,\mu \ne 0}=\frac{1}{2}\left(m^2-{\mu }^2\right)v^2+{{{\displaystyle \sum }}^{\text{}}}_{\ell }{{\displaystyle \int }}^{\text{}}d\tilde{k}\{E_k$

$+T^{[}\text{ln}(1-e^{-\beta (Ek-\mu eff))}+\text{ln}{\left(1-e^{-\beta (Ek+\mu eff))}\right)}^{]\}}\text{ (II}\text{.29)}$

with μ eff ≡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

λ≠0 and μ=0. In this case, $\in \stackrel{\pm }{k}$ are given by

${ϵ}_k^{+}=\sqrt{k^2+m_2^2}={\omega }_2$

${ϵ}_k^{-}=\sqrt{k^2+m_1^2}={\omega }_1\text{ (II}\text{.30)}$

Plugging (II.30) into (II.24) and choosing μ=0 and ${\text{m}}^2=-{\text{c}}^2$ with ${\text{c}}^2>0$ , 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

in ${\text{V}}_T^{\pm }$ from(35). Having in mind that the arguments of ln(1 — e^{-β(ωi ℓΩ)}) are to be positive, the summation over ℓ in ln(1 — e^{-β(ωi -ℓΩ)}) is over ℓ ∈ (—∞, —1] and in ln(1 — e^{-β(ωi +ℓΩ)}) is over ℓ ∈ [1, ∞) [30]. Performing a change ℓ → —ℓ, we thus have

${\mathcal{V}}_i^{+}=T{{{\displaystyle \sum }}^{\text{}}}_{\ell =1}^{\infty }{{\displaystyle \int }}^{\text{}}d\tilde{k}\text{ln}\left(1-e^{-\beta \left({\omega }_i+\ell \text{Ω}\right)}\right)={\mathcal{V}}_i^{-}\text{ (II}\text{.36)}$

Hence, the final form of VT from (II.34) reads

$\begin{array}{c}{\mathcal{V}}_T=T{\displaystyle \sum }_{i=1,2}{\displaystyle \sum }_{\ell =1}^{\infty }{{\displaystyle \int }}^{\text{}} d\tilde{k}ln\left(1-e^{-\beta \left({\omega }_i+\ell \text{Ω}\right)}\right)(II.37)\end{array}$

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

$\widetilde{{ℒ}_0}=\frac{1}{2}{m}_0^2\left({\phi }_1+v\right)v,( III. 1)$

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

$v_0^2\equiv \frac{a^2}{\lambda }. (III.2)$

At this minimum, the masses of ${\text{m}}_1^2(\text{v})=3\lambda {\text{v}}^2-{\text{c}}^2$ and ${\text{m}}_2^2(\text{v})$ are given by

${\text{m}}_1^2\left({\text{v}}_0\right)=3{\text{a}}^2-{\text{c}}^2, {\text{m}}_2^2\left({\text{v}}_0\right)={\text{a}}^2-{\text{c}}^2\text{ (III}\text{.3)}$

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 $\begin{array}{l}{\text{v}}_{\text{min}}^2( \text{T},\Omega )=\{\frac{(a^2}{\lambda }{(1-\text{t}3)}^{\text{'}}\text{t}<1,\\ (0,\text{t}\ge 1\end{array}$ and ${\text{m}}_2^2(\text{v})$ into it, the high-temperature expansion of Va reads

$\begin{array}{l}{\mathcal{V}}_a(v,T,\Omega )=-\frac{a^2{v}^2}{2}\left(1-\frac{2\lambda T^3\zeta (3)}{a^2{\pi }^2\Omega }\right)+\frac{\lambda v^4}{4}\text{ (III}\text{.4)}\\ -\frac{2T^5\zeta (5)}{{\pi }^2\Omega }-\frac{c^2{T}^3\zeta (3)}{2{\pi }^2\Omega }+\cdots \end{array}$

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

$\text{Tc}={\left(\frac{a^2{\pi }^2\text{Ω}}{2\lambda \zeta (3)}\right)}^{1/3} (III.5)$

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

$\begin{array}{l}{\text{v}}_{\text{min}}^2( \text{T},\text{Ω})=\{\frac{(a^2}{\lambda }{(1-\text{t}3)}^{\text{'}}\text{t}<1,\text{ (III}\text{.6)}\\ (0,\text{t}\ge 1\end{array}$

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 $m_1^2\left(v_0\right)=3\lambda v_0^2-c^2=$ = 3λv0(2)0 — c2 =

${\text{m}}_{\sigma }^2$ and ${\text{m}}_2^2\left({\text{v}}_0\right)=\lambda {\text{v}}_O^2-{\text{c}}^2={\text{m}}_{\pi }^2$ with v₀ the classical minimum from (III.2). Moreover, we choose m0 in (III.1) equal to m_π. For m_σ = 400 MeV, and m_π = 140 MeV, we obtain

$\text{c}={\left(\frac{m_{\sigma }^2-3m_{\pi }^2}{2}\right)}^{1/2}\simeq 225\text{MeV}.(\text{III}.7)$

Moreover, $\text{a}={\left({\text{c}}_2+{\text{m}}_{\pi }^2\right)}^{1/2}\simeq 265\text{MeV}$ . We also choose λ = 0.5. Using these quantities the function

$=-\frac{a^2{v}^2}{2}(1-\text{t}3)+\frac{\lambda v^4}{4},(III.8)$

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) (Figure 7).

Figure 2: (color online). The v dependence of ∆Va from (III.8) is plotted at t = 0.6, 0.8, 1, 1.2. At t < 1 the global U(1) symmetry is broken and ∆Va possesses nontrivial minima at v2min = a2 (1 - t3)/λ. At t = 1 the symmetry is restored and at t ≥ 1 a single minimum at vmin = 0 appears (see (III.6)).

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 P> (v, T,Ω), we have

$\begin{array}{l}{\text{P}}_{<}(\text{vmin},\text{T},\text{Ω})=-\frac{a^4}{2\lambda }{\text{t}}^3+\frac{c^2{T}^3\zeta (3)}{2{\pi }^2\text{Ω}}+\frac{2T^5\zeta (5)}{{\pi }^2\text{Ω}}+\frac{a^4}{4\lambda }{\text{t}}^6,\text{P}>(0, \text{T},\text{Ω})\\ =\frac{c^2{T}^3\zeta (3)}{2{\pi }^2\text{Ω}}+\frac{2T^5\zeta (5)}{{\pi }^2\text{Ω}}-\frac{a^4}{4\lambda }\text{ (III}\text{.9)}\end{array}$

$\begin{array}{l}{\text{P}}_{<}(\text{vmin},\text{T},\text{Ω})=-\frac{a^4}{2\lambda }{\text{t}}^3+\frac{c^2{T}^3\zeta (3)}{2{\pi }^2\text{Ω}}+\frac{2T^5\zeta (5)}{{\pi }^2\text{Ω}}+\frac{a^4}{4\lambda }{\text{t}}^6,\text{P}>(0, \text{T},\text{Ω})\\ =\frac{c^2{T}^3\zeta (3)}{2{\pi }^2\text{Ω}}+\frac{2T^5\zeta (5)}{{\pi }^2\text{Ω}}-\frac{a^4}{4\lambda }\text{ (III}\text{.9)}\end{array}$

Here, we have added a term -a^4/4λ to P_in order to guarantee ${\text{P}}_{<}\left({\stackrel{2}{v}}_{min }^2,0,\text{Ω}\right)=0$ and P_<=P> at the the transition temperature T_c. At T=T_c, the pressure is given by

$\begin{array}{l}{P}_c\left(V_{\text{min}}^2,T_c,{\text{Ω}}^{\text{'}}\right)=P_{>}\left(0,T_c,\text{Ω}\right)\\ =-\frac{a^4}{4\lambda }+\frac{a^2{c}^2}{\underset{\_}{4\lambda }}+\frac{a^{10/3}{\text{Π}}^{4/3}{\text{Ω}}^{2/3}\zeta (5)}{2^{2/3}{\text{Λ}}^{5/3}{[\zeta (3)]}^{5/3}}( III.10 )\end{array}$

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 T = Tc, Tc = Tc, $\text{T}=\text{Tc},{\frac{d{P}_{<}}{dT}|}_{\text{Tc}}={\frac{d{P}_{>}}{dT}|}_{\text{Tc}}$ (III.11)

the heat capacity (d² P/dT²) is discontinuous

$\begin{array}{c}\frac{d^2{P}_{<}}{d{T}^2}{I}_{T_c}-\frac{d^2{P}_{>}}{d{T}^2}{I}_{T_c}=\frac{9c^{8/3}{[\zeta (3)]}^{2/3}}{2^{1/3}{\pi }^{4/3}{\lambda }^{1/3}{\text{Ω}}^{2/3}}(III.12)\end{array}$

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, ${\text{a}}^2={\text{c}}^2+{\text{m}}_{\text{O}}^2$ min from (III.6) into ${\text{m}}_1^2(\text{v})=3\lambda {\text{v}}^2-{\text{c}}^2$ and ${\text{m}}_2^2(\text{v})=\lambda {\text{v}}^2-{\text{c}}^{2,}$ we arrive at

$\begin{array}{c}\begin{array}{rrr}\hfill m_1(v\text{min})=& \hfill 3-a^2\left(1-t^3\right)-c^2,& \hfill t<1\\ \hfill & \hfill c^2,& \hfill t\ge 1\\ \hfill m_2(v\text{min})=& \hfill & \hfill a^2\left(1-t^3\right)-c^2,\\ \hfill -c^2,& \hfill & \hfill t<1\end{array}\end{array}\text{ (III}\text{.13)}$

Hence, as it turns out, at $\text{t}\ge 1$ , after the symmetry is restored, ${\text{m}}_1^2$ and ${\text{m}}_2^2$ become negative. Contrary to our expectation, for a = c, i.e., in the chiral limit mo =o, the Goldstone boson ' 2 acquires a negative mass $-c^2{t}^3$ in the symmetry-broken phase at $t<1$ . In what follows, we compute the one-loop tadpole diagram contributions to masses m_1 and m_2. 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 mo =0.

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

To calculate the one-loop corrections to m_{_1} and m₂, let us consider L₄ from (II.4). Three vertices, corre-sponding to three terms in ${ℒ}_4=-\frac{\lambda }{4}{\left({\phi }_1^2+{\phi }_2^2\right)}^2$ , are to be considered in this computation (Figure 3),

Figure 3: Three vertices arising from L4 from (II.4). Dashed and solid lines correspond to φ1 and φ2 fields, respectively.

$\begin{array}{rrrrr}\hfill -{-}_4^{\text{'}4}1& \hfill \to & \hfill & \hfill -& \hfill \\ \hfill -4_2^{\text{'}4}& \hfill \to & \hfill & \hfill -4& \hfill \\ \hfill -\frac{1}{2}{\phi }_1^2{\phi }_2^2& \hfill & \hfill \to & \hfill -\frac{1}{2}& \hfill \end{array}\text{ (III}\text{.14)}$

They lead to two different tadpole contributions to $\langle \Omega \left|\text{T}\left({ }^{\text{'}}1(\text{x}){ }^{\text{'}}1(\text{y})\right)\right|\Omega \rangle$ and $D_{\ell }({\omega }_n,{\omega }_i)\equiv \frac{1}{{({\omega }_n-i\ell \Omega )}^2+{\omega }_i^2}$ that correct m_{_1} and m_{_2} 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 to whether the internal loop is built from ' 1 or ' 2 (Figure 4 , where $\Pi \text{ij}$ are plotted). Hence, according to this notation, the one-loop perturbative corrections to ${\text{m}}_1^2$ and ${\text{m}}_2^2$ arise from

Figure 4: The tadpole diagrams contributing to the one-loop corrections of m12 and m22. Dashed and solid lines correspond to φ1 and φ2 fields, respectively.

${\text{m}}_1^2(\text{v})\to {\text{m}}_1^2(\text{v})+\Pi 11+\Pi 12;$

${\text{m}}_2^2(\text{v})\to {\text{m}}_2^2(\text{v})+\Pi 21+\Pi 22\text{ (III}\text{.15)}$

At this stage, we introduce

${\text{Π}}_i\left(T,\text{Ω},m_i\right)\equiv T{\displaystyle \sum }_{n=-\infty }^{\infty }{\displaystyle \sum }_{\ell =-\infty }^{\infty }{{\displaystyle \int }}^{\text{}} d\tilde{k}{D}_{\ell }\left({\omega }_n,{\omega }_i\right)\text{ (III}\text{.16)}$

with free boson propagator

$D_{\ell }\left({\omega }_n,{\omega }_i\right)\equiv \frac{1}{{\left({\omega }_n-i\ell \text{Ω}\right)}^2+{\omega }_i^2}\text{ (III}\text{.17)}$

arising from (II.11) with =0. Here, ${!}_i^2=k_{\perp }^2+{\text{k}}_Z^2+{\text{m}}_i^2$ and i = 1; 2 . Using this notation, it turns out that

П 11 = 3 П 1 ; П 11 = 12

П 22 = 3 П 1 ; П 21 = П 1. (III.18)

Hence, the perturbative corrections of masses are given by

${\text{m}}_1^2(\text{v})\to {\text{m}}_1^2(\text{v})+3{\text{Π}}_1+{\text{Π}}_2;$

${\text{m}}_2^2(\text{v})\to {\text{m}}_2^2(\text{v})+3{\Pi }_2+{\Pi }_1\text{ (III}\text{.19)}$

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

${{{\displaystyle \sum }}^{\text{}}}_n{D}_{\ell }\left({\omega }_n,{\omega }_i\right)=\frac{1}{2T{\omega }_i}\left[n_b\left({\omega }_i+\ell \Omega \right)+n_b\left({\omega }_i-\ell \Omega \right)+1\right]\text{ (III}\text{.20)}$

where ${\text{n}}_{\text{b}}(!)\equiv 1=(\text{e}\beta \omega -1)$ 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

$\Pi \text{i},{\Pi }_i^{mat }=\frac{1}{2}{{{\displaystyle \sum }}^{\text{}}}_{e=\pm }{{{\displaystyle \sum }}^{\text{}}}_{\ell \ne 0}{{\displaystyle \int }}^{\text{}}d\tilde{k}\frac{n_b\left({\omega }_i+e\ell \Omega \right)}{{\omega }_i}\text{ (III}\text{.21)}$

Having in mind that in $\text{nb}(!\text{i}\pm \text{‘}\Omega )$ , we must have $\text{e}\beta (\omega \text{i}\pm \ell \Omega )-1>0$ , it is possible to limit the summation over. We thus obtain $\Pi_{i}^{\mathrm{mat}}=\sum_{\ell=1}^{\infty} \int d \tilde{k} \frac{n_{b}\left(\omega_{i}+\ell \Omega\right)}{\omega_{i}}$. (III.22)

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 ${\text{k}}_{\perp }$ and ${\text{k}}_{\text{z}}$ , we use

${\Pi }_i^{\text{mat}}=\frac{\lambda T^3\zeta (3)}{2{\pi }^2\Omega }+\cdots \text{ (III}\text{.23)}$

and arrive first at

${\Pi }_i^{\text{mat}}=T{\displaystyle \sum }_{\ell =1}^{\infty }{{\displaystyle \int }}^{\text{}} d\tilde{k}\frac{1}{{\omega }_i}\frac{d}{d{\omega }_i}\text{ln}\left(1-e^{-\beta \left({\omega }_i+\ell \Omega \right)}\right)\text{ (III}\text{.24)}$

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

${\Pi }_i^{\text{mat}}=-T{\displaystyle \sum }_{\ell =1}^{\infty }{\displaystyle \sum }_{j=1}^{\infty }{{\displaystyle \int }}^{\text{}} d\tilde{k}\frac{1}{{\omega }_i}\frac{d}{d{\omega }_i}\frac{e^{-\beta {\omega }_{i}j}{e}^{-\beta \ell \Omega j}}{j}$

$={\displaystyle \sum _{\ell =1}^{\infty }{\displaystyle \sum _{j=1}^{\infty }{\displaystyle \int d}}}\overrightarrow{k}\frac{e^{-\beta {\omega }_j}{e}^{-\beta {\Omega }_{\ell j}}}{{\omega }_{\ell }}\text{ (III}\text{.25)}$

The summation over ` can be performed by making use of (A.4). Assuming Ω<1 and using (A.5), ${\text{Π}}_i^{mat }$ reads

$\begin{array}{c}{\text{Π}}_i^{\text{mat}}=\frac{ }{\text{Ω}}{\displaystyle \sum }_{j=1}^{\infty }\frac{1}{j}{{\displaystyle \int }}^{\text{}} d\tilde{k}\frac{e^{-\beta {\omega }_{i}j}}{{\omega }_i}\end{array}\text{ (III}\text{.26)}$

Following the method presented in Appendix B, we fi- nally arrive at ${\text{Π}}_i^{mat }=\frac{\lambda T^3\zeta (3)}{2{\pi }^2\text{Ω}}+\cdots$ . (III.27)

The first term in (III.27) is analogous to the thermal mass $\lambda {\text{T}}^2$ /3 in a nonrotating interacting Bose gas [40] and the ellipsis includes higher order corrections of ${\text{Π}}_i^{mat }$ in $\beta \text{mi}$ .

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

$\begin{array}{rr}\hfill & \hfill {\text{Π}}_1^{\text{mat}}={\text{Π}}_2^{\text{mat}}=\frac{\lambda T^3\zeta (3)}{2{\pi }^2\text{Ω}},( III. 28) and therefore \\ \hfill & \hfill {\text{m}}_1^2(\text{v})\to {\text{m}}_1^2(\text{v})+4{\text{Π}}_1^{{\text{m}}^{\text{a}}\text{at}}={\text{m}}_1^2(\text{v})+{\text{a}}^2\text{t}3\\ \hfill & \hfill {\text{m}}_2^2(\text{v})\to {\text{m}}_2^2(\text{v})+4{\text{Π}}_2^{{\text{m}}^{\text{m}}\text{at}}={\text{m}}_2^2(\text{v})+{\text{a}}^2{\text{t}}^3\end{array}\text{ (III}\text{.28)}$

$\begin{array}{l}{\text{m}}_1^2(\text{v})\to {\text{m}}_1^2(\text{v})+4{\text{Π}}_1^{{\text{m}}^{\text{a}}\text{at}}={\text{m}}_1^2(\text{v})+{\text{a}}^2\text{t}3\\ {\text{m}}_2^2(\text{v})\to {\text{m}}_2^2(\text{v})+4{\text{Π}}_2^{{\text{m}}^{\text{m}}\text{at}}={\text{m}}_2^2(\text{v})+{\text{a}}^2{\text{t}}^3\text{ (III}\text{.29)}\end{array}$

with $\text{t}=\text{T}/\text{Tc}$ 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 ${\text{m}}_1^2\left({\text{v}}_{min }^2\right)$ and ${\text{m}}_2^2\left({\text{v}}_{min }^2\right)$ , according to (III.29), we obtain

$\begin{array}{l}{m}_1^2\left(v_{min }\right) =\{\begin{array}{ll}2c^2\left(1-t^3\right)+3m_0^2\left(1-\frac{2t^3}{3}\right),\hfill & t<1,\hfill \\ c^2\left(t^3-1\right)+m_0^2{t}^3,\hfill & t\ge 1,\hfill \end{array}\text{ (III}\text{.30)}\\ m_2^2\left(v_{min }\right)=\{\begin{array}{ll}{m}_0^2,\hfill & \text{t}<1,\hfill \\ c^2\left(t^3-1\right)+m_0^2{t}^3,\hfill & \text{t}\ge 1,\hfill \end{array}\end{array}$

where $a^2=c^2+m_0^2$ is used. Assuming $m_0=0,m_2$ van- ishes at $t<1$ . This indicates that the Goldstone theorem is valid when the thermal mass corrections to $m_1^2$ and $m_2^2$ are taken into account. Moreover, we observe that ${\text{m}}_1^2(\text{vmin})={\text{m}}_2^2(\text{vmin})$ in the symmetry-restored phase at $\text{t}\ge 1$ . In Figure 5, the t dependence of m_1^2 (vmin) and m_2^2 (vmin) from (III.30) is plotted. These masses are identified with ${\text{m}}_{\sigma }^2$ and ${\text{m}}_{\pi }^2$ , respectively. We use $\text{c}\simeq 0.225\text{GeV}$ from (III.7) and mo $=0.140\text{GeV}$ , as de- scribed in Sec. III B and observe that in the symmetry- broken phase, at $\text{t}<1, {\text{m}}_{\sigma }$ decreases with increasing temperature, while $\text{m}\pi$ remains constant. As expected, at symmetry-restored phase at $\text{t}\ge 1, {\text{m}}_{\sigma }$ 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 ${\text{t}}^2$ to ${\text{t}}^3$ (see [40]).

Figure 5: (color online). The t dependence of m12 and m222 from (III.30) at vm(2)min from (III.6) is plotted. These masses are iden- tified with σ and π meson masses. In the symmetry-broken phase, at t < 1, mσ decreases with increasing temperature, while mπ remains constant. At symmetry-restored phase at t ≥ 1, mσ and mπ are equal and increase with increasing t.

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 $\ell \in (-\infty ,+\infty )$ in this expression. This sum is divergent and need an appropriate regularization. To perform the summation over $\ell$ , we use

${\displaystyle \sum }_{\ell =-\infty }^{\infty }1=\underset{x\to 0}{\text{lim}}{\displaystyle \sum }_{\ell =-\infty }^{\infty }{e}^{-{\ell }^{2}x}$

$={\text{lim}}_{x\to 0}\left(1+2{{{\displaystyle \sum }}^{\text{}}}_{\ell =1}^{\infty }{e}^{-{\ell }^{2}x}\right)=1+{\text{lim}}_{x\to 0}\frac{1}{1-e^{-x}}$

= 1 + divergent term. (III.31)

Neglecting the divergent term, we obtain

$\begin{array}{c}{\mathcal{V}}_{\text{vac}}=\frac{1}{2}{{\displaystyle \int }}^{\text{}} d\tilde{k}\left({\omega }_1+{\omega }_2\right)\end{array}\text{ (III}\text{.32)}$

The above regularization guarantees that rotation does not alter Vvac. To perform the integration over ${\text{k}}_{\perp }$ and ${\text{k}}_{\text{z}}$ , let us consider the integral

$I(m)\equiv \frac{{\overline{\mu }}^{\epsilon }}{2}{{\displaystyle \int }}^{\text{}} d\tilde{k}{\left(k^2+m^2\right)}^{1/2}\text{ (III}\text{.33)}$

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

$\begin{array}{l}\Phi (m,d,n)={{\displaystyle \int }}^{\text{}} \frac{d^{d}k}{{(2\pi )}^d}\frac{1}{{\left(k^2+m^2\right)}^n}\\ =\frac{1}{{(4\pi )}^{d/2}}\frac{\Gamma (n-d/2)}{\Gamma (n)}\frac{1}{{\left(m^2\right)}^{n-d/2}}\text{ (III}\text{.34)}\end{array}$

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

$\begin{array}{c}I(m)=-\frac{m^4}{64{\pi }^2}\left(\frac{2}{ϵ}+\frac{3}{2}-{\gamma }_E-\text{ln}\frac{m^2}{4\pi {\overline{\mu }}^2}\right)\end{array}\text{ (III}\text{.35)}$

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

$\begin{array}{l}Vvac =\text{I}\left( {\text{m}}_1\right)+\text{I}\left( {\text{m}}_2\right)\\ =-\frac{\left(m_1^4+m_2^4\right)}{64{\pi }^2}\left(\frac{2}{ϵ}+\frac{3}{2}-{\gamma }_E\right)+\frac{m_1^4}{64{\pi }^2}\text{ln}\frac{m_1^2}{4\pi {\overline{\mu }}^2}\text{ (III}\text{.36)}\\ +\frac{m_2^4}{64{\pi }^2}\text{ln}\frac{m_2^2}{4\pi {\overline{\mu }}^2}\end{array}$

In what follows, we regularize this potential by following the method presented in [43]. To do this, we first define

$\text{Vb}\equiv {\text{V}}_{\text{cl}}+{\text{V}}_{\text{vac}}+\text{VCT (III}\text{.37)}$

with V_cl from (II.32) with c^2 replaced with ${\text{a}}^2={\text{c}}^2+{\text{m}}_{\text{O}}^2$ and V vac from (III.36). The counterterm potential is given by

${\mathcal{V}}_{\text{CT}}=\frac{A{v}^2}{2}+\frac{B{v}^4}{4}+C\text{ (III}\text{.38)}$

The coefficients A and B are determined by utilizing two prescriptions

$\begin{array}{c}{\frac{\partial {\mathcal{V}}_b}{\partial v}|}_{v_0^2}=0,{\frac{{\partial }^2{\mathcal{V}}_b}{\partial v^2}|}_{v_0^2}=m_1^2\left(v_0\right)\end{array}\text{ (III}\text{.39)}$

Here, ${\text{v}}_{\text{O}}^2$ from (III.2) is the classical minimum and ${\text{m}}_1^2(\text{vo})$ 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

$\begin{array}{l}A=-\frac{m_0^2}{2}+\frac{3c^2\lambda }{8{\pi }^2}+\frac{c^2\lambda {\gamma }_E}{4{\pi }^2}+\frac{5m_0^2\lambda }{8{\pi }^2}-\frac{c^2\lambda }{2{\pi }^2ϵ}\\ +\frac{c^2\lambda }{16{\pi }^2}\text{ln}\left(\frac{m_0^2}{4\pi {\overline{\mu }}^2}\right)+\frac{3c^2\lambda }{16{\pi }^2}\text{ln}\left(\frac{2c^2+3m_0^2}{4\pi {\overline{\mu }}^2}\right)\\ B=\frac{m_0^2\lambda }{2a^2}-\frac{5{\lambda }^2{\gamma }_E}{8{\pi }^2}+\frac{5{\lambda }^2}{4{\pi }^2ϵ}-\frac{{\lambda }^2}{16{\pi }^2}\text{ln}\left(\frac{m_0^2}{4\pi {\overline{\mu }}^2}\right)\text{ (III}\text{.40)}\\ -\frac{9{\lambda }^2}{16{\pi }^2}\text{ln}\left(\frac{2c^2+3m_0^2}{4\pi {\overline{\mu }}^2}\right)\end{array}$

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

$\begin{array}{l}C=\frac{c^4}{16{\pi }^2ϵ}-\frac{c^4}{64{\pi }^2}\text{ln}\left(\frac{m_0^2}{4\pi {\overline{\mu }}^2}\right)-\frac{c^4}{64{\pi }^2}\text{ln}\left(\frac{2c^2+3m_0^2}{4\pi {\overline{\mu }}^2}\right)\text{ (III}\text{.41)}\\ +\frac{3c^4}{64{\pi }^2}-\frac{c^4{\gamma }_E}{32{\pi }^2}\end{array}$

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

${\mathcal{V}}_b=-\frac{a^2{v}^2}{2}+\frac{\lambda v^4}{4}-\frac{m_0^2{v}^2}{4}+\frac{3c^2\lambda v^2}{8{\pi }^2}+\frac{5m_0^2\lambda v^2}{16{\pi }^2}$

$\begin{array}{l}-\frac{15{\lambda }^2{v}^4}{64{\pi }^2}+\frac{m_0^2\lambda v^4}{8a^2}\\ +\frac{m_1^4}{64{\pi }^2}\text{ln}\left(\frac{m_1^2}{2c^2+3m_0^2}\right)+\frac{m_2^4}{64{\pi }^2}\text{ln}\left(\frac{m_2^2}{m_0^2}\right)\text{ (III}\text{.32)}\end{array}$

As mentioned earlier, the energy scale $\overline{\mu }$ does not appear in the final expression of Vb . Additionally, a nonzero mo 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 φ1 and φ2, three type of vertices appear in the $\lambda$ ( ${\phi }^{\star }\phi$ ) model (Figure 3 ). We thus have four different types of ring diagrams:

Type A: A ring with N insertions of Π2 and N propagators $D\ell (\omega n,\omega 1)$ propagators, $V_r^A$ ing,

Type B: A ring with N insertions of $\text{Π}1$ and N propagators $D\ell (\omega \text{n}\omega 2)$ propagators, ${\text{V}}_{ring }^{\text{B}}$ ,

Type C: A ring with r insertions of $\text{Π}2$ and s insertions of $\text{Π}1$ with N propagators $\text{D}\ell (\omega \text{n}\omega 2),\stackrel{\text{C}}{{\text{V}}_{ring }}$ .

Here, $\text{r}\ge 1$ and $\text{r}+\text{s}=\text{N}$ .

Type D: A ring with r insertions of $\text{Π}1$ and s insertions of $\text{Π}2$ with N propagators De $(\omega \text{n}\omega 1)$ , ${\text{V}}_{ring }^{\text{D}}$ .

Similar to the previous case, $\text{r}\ge 1$ and $\text{r}+\text{s}=\text{N}$ .