Thermal phase transition with full 2-loop effective potential

TL;DR

This paper develops a resummed 2-loop effective potential calculation for extended Higgs models, avoiding high-temperature expansion, and assesses its impact on phase transition strength, relevant for cosmological phenomena.

Contribution

It introduces a method to compute the 2-loop effective potential without high-T expansion and evaluates its effects on phase transition properties in extended Higgs models.

Findings

2-loop corrections weaken the phase transition but remain moderate

High-T expansion performs well except for critical temperature Tc

Strong phase transitions are still possible with large couplings

Abstract

Theories with extended Higgs sectors constructed in view of cosmological ramifications (gravitational wave signal, baryogenesis, dark matter) are often faced with conflicting requirements for their couplings; in particular those influencing the strength of a phase transition may be large. Large couplings compromise perturbative studies, as well as the high-temperature expansion that is invoked in dimensionally reduced lattice investigations. With the example of the inert doublet extension of the Standard Model (IDM), we show how a resummed 2-loop effective potential can be computed without a high-T expansion, and use the result to scrutinize its accuracy. With the exception of Tc, which is sensitive to contributions from heavy modes, the high-T expansion is found to perform well. 2-loop corrections weaken the transition in IDM, but they are moderate, whereby a strong transition remains an option.

Tables

Table 1. Table 1: Tree-level masses squared in the Feynman R ξ subscript 𝑅 𝜉 R_{\xi} gauge (cf. sec.  4 ). The thermal mass corrections δ ​ m ϕ ​ T 2 𝛿 superscript subscript 𝑚 italic-ϕ 𝑇 2 \delta m_{\mbox{\tiny\rm{$\phi T$}}}^{2} , δ ​ m χ ​ T 2 𝛿 superscript subscript 𝑚 𝜒 𝑇 2 \delta m_{\mbox{\tiny\rm{$\chi T$}}}^{2} , m E2 2 superscript subscript 𝑚 E2 2 m_{\mbox{\tiny\rm{E2}}}^{2} and m E3 2 superscript subscript 𝑚 E3 2 m_{\mbox{\tiny\rm{E3}}}^{2} are given in eqs. ( 4.3 )–( 4.3 ). By D = 4 − 2 ​ ϵ 𝐷 4 2 italic-ϵ D=4-2\epsilon we denote the dimensionality of spacetime. The fields A μ subscript 𝐴 𝜇 A_{\mu} and C μ subscript 𝐶 𝜇 C_{\mu} correspond to the gauge groups SU L L {}_{\mbox{\tiny\rm{L}}} (2) and SU(3), respectively, with c A , c ¯ A subscript 𝑐 𝐴 subscript ¯ 𝑐 𝐴 c_{\mbox{\tiny\rm{$A$}}},\bar{c}_{\mbox{\tiny\rm{$A$}}} , c C , c ¯ C subscript 𝑐 𝐶 subscript ¯ 𝑐 𝐶 c_{\mbox{\tiny\rm{$C$}}},\bar{c}_{\mbox{\tiny\rm{$C$}}} being the Faddeev-Popov ghosts.

field	vacuum mass	thermal mass	degeneracy
$h$	$m_h^2={\mu }_1^2+3{\lambda }_1{v}^2$	${\tilde{m}}_h^2=m_h^2+\delta m_{\varphi T}^2$	1
$G$	$m_G^2={\mu }_1^2+{\lambda }_1{v}^2+m_W^2$	${\tilde{m}}_G^2=m_G^2+\delta m_{\varphi T}^2$	3
$H$	$m_H^2={\mu }_2^2+\frac{1}{2}\left({\lambda }_3+{\lambda }_4+{\lambda }_5\right)v^2$	${\tilde{m}}_H^2=m_H^2+\delta m_{\chi T}^2$	1
$A$	$m_A^2={\mu }_2^2+\frac{1}{2}\left({\lambda }_3+{\lambda }_4-{\lambda }_5\right)v^2$	${\tilde{m}}_A^2=m_A^2+\delta m_{\chi T}^2$	1
$H_{\pm }$	$m_{H_{\pm }}^2={\mu }_2^2+\frac{1}{2}{\lambda }_3{v}^2$	${\tilde{m}}_{H_{\pm }}^2=m_{H_{\pm }}^2+\delta m_{\chi T}^2$	2
$A_i$	$m_W^2=\frac{1}{4}{g}^2{v}^2$	$m_W^2$	$3\left(D-1\right)$
$A_0$	$m_W^2$	${\tilde{m}}_W^2=m_W^2+m_{\text{E2}}^2$	3
$c_A,{\overline{c}}_A$	$m_W^2$	$m_W^2$	$-\mathrm{\hspace{0.17em}6}$
$C_i$	$0$	$0$	$8\left(D-1\right)$
$C_0$	$0$	$m_{\text{E3}}^2$	8
$c_C,{\overline{c}}_C$	$0$	$0$	$-\mathrm{\hspace{0.17em}16}$
$t$	$m_t^2=\frac{1}{2}{h}_t^2{v}^2$	$m_t^2$	12

Table 2. Table 2: The benchmark scenarios from ref.  [ 50 ] . The values of ( λ 3 + λ 4 + λ 5 ) / 2 subscript 𝜆 3 subscript 𝜆 4 subscript 𝜆 5 2 (\lambda_{3}+\lambda_{4}+\lambda_{5})/2 and λ 2 subscript 𝜆 2 \lambda_{2} refer to the renormalization scale μ ¯ = m Z ¯ 𝜇 subscript 𝑚 𝑍 \bar{\mu}=m_{\mbox{\tiny\rm{$Z$}}} . The smallness of ( λ 3 + λ 4 + λ 5 ) / 2 subscript 𝜆 3 subscript 𝜆 4 subscript 𝜆 5 2 (\lambda_{3}+\lambda_{4}+\lambda_{5})/2 was justified with dark matter relic density considerations, and that of λ 2 subscript 𝜆 2 \lambda_{2} with constraints from dark matter self-interactions.

scenario	$m_H/$GeV	$m_A$/GeV	$m_{H_{\pm }}$/GeV	$\left({\lambda }_3+{\lambda }_4+{\lambda }_5\right)\left(m_Z\right)/2$	${\lambda }_2\left(m_Z\right)$
BM1	66	300	300	1.07$\times {10}^{-2}$	0.01
BM2	200	400	400	1.00$\times {10}^{-2}$	0.01
BM3	5	265	265	$-$0.60$\times {10}^{-2}$	0.01

Table 3. Table 3: Values of MS ¯ ¯ MS {\overline{\mbox{\rm MS}}} couplings at the scale μ ¯ = m Z ¯ 𝜇 subscript 𝑚 𝑍 \bar{\mu}=m_{\mbox{\tiny\rm{$Z$}}} , obtained as explained in appendix  C.4 . Here “2-loop” signals that a subset of 2-loop corrections was included in μ 1 2 superscript subscript 𝜇 1 2 \mu_{1}^{2} and λ 1 subscript 𝜆 1 \lambda_{1} . For the thermal analysis the values are treated as fixed input, so more digits have been given than are physically accurate.

	BM1		BM2		BM3
	1-loop	“2-loop”	1-loop	“2-loop”	1-loop	“2-loop”
${\mu }_1^2\left(m_Z\right)/$GeV2	-6669	-6568	-8463	-8127	-7392	-7251
${\mu }_2^2\left(m_Z\right)/$GeV2	842	842	36620	36620	-1243	-1243
${\lambda }_1\left(m_Z\right)$	0.0670	0.0634	0.0671	0.0579	0.1021	0.0979
${\lambda }_2\left(m_Z\right)$	0.010	0.010	0.010	0.010	0.010	0.010
${\lambda }_3\left(m_Z\right)$	2.757	2.757	2.618	2.618	2.243	2.243
${\lambda }_4\left(m_Z\right)$	-1.368	-1.368	-1.299	-1.299	-1.127	-1.127
${\lambda }_5\left(m_Z\right)$	-1.368	-1.368	-1.299	-1.299	-1.127	-1.127
$g_2^2\left(m_Z\right)$	0.425	0.425	0.425	0.425	0.425	0.425
$g_3^2\left(m_Z\right)$	1.489	1.489	1.489	1.489	1.489	1.489
$h_t^2\left(m_Z\right)$	0.971	0.971	0.973	0.973	0.969	0.969

Table 4. Table 4: Results for the physical quantities defined in sec.  2 , as well as for the gauge dependent v min subscript 𝑣 min v_{\mbox{\scriptsize min}} evaluated in Feynman R ξ subscript 𝑅 𝜉 R_{\xi} gauge, for the benchmark scenarios listed in table  2 . The uncertainties were obtained through the scale variation in eq. ( 5.5 ). We note that the scale uncertainties, which reflect the size of higher-order corrections from large scalar couplings, completely dominate over ambiguities related to gauge dependence, whose size is indicated by the difference between v phys subscript 𝑣 phys v_{\mbox{\scriptsize phys}} and v min subscript 𝑣 min v_{\mbox{\scriptsize min}} .

			full effective potential
	$T_c/\text{GeV}$		$L/T_c^4$		$v_{\text{phys}}/T_c$		$v_{\text{min}}/T_c$
	1-loop	2-loop	1-loop	2-loop	1-loop	2-loop	1-loop	2-loop
BM1	139(14)	155(21)	0.44(1)	0.34(1)	1.14(12)	0.98(4)	1.15(12)	0.98(3)
BM2	159(13)	181(22)	0.07(7)	0.03(3)	0.39(28)	0.16(16)	0.39(28)	0.17(17)
BM3	138(8)	167(19)	0.35(3)	0.20(1)	0.96(10)	0.84(6)	0.98(10)	0.81(2)

Equations

$$V_0^{\text{mbox}}$$

$$\frac{v_{\text{mbox}phys}^2}{2}\equiv Z_{{\mu }_1^2}\Delta ⟨{\varphi }^{†}\varphi ⟩,$$

$$\phi_{\mbox{\tiny\rm{R}}}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}G_{2}+iG_{1}\\ v+h-iG_{3}\end{array}\right)\;,\quad\chi_{\mbox{\tiny\rm{R}}}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}H_{2}+iH_{1}\\ H_{0}-iH_{3}\end{array}\right)\;.$$

$$L_{\text{mbox}\mathrm{B}}({\varphi }_{\text{mbox}\mathrm{B}})$$

$$L_{\mbox{\scriptsize gauge fixing}}=\frac{1}{2\xi}\sum_{a=1}^{3}\Bigl{(}\partial_{\mu}A^{a}_{\mu}-\frac{\xi gv}{2}\,G_{a}\Bigr{)}^{2}+\frac{\xi g^{2}v}{4}\Bigl{(}\bar{c}^{a}_{\mbox{\tiny\rm{$A$}}}\,h\,c^{a}_{\mbox{\tiny\rm{$A$}}}+\epsilon^{abc}\bar{c}^{a}_{\mbox{\tiny\rm{$A$}}}\,G_{b}\,c^{c}_{\mbox{\tiny\rm{$A$}}}\Bigr{)}+\ldots\;,$$

$$\displaystyle\phi^{\dagger}_{\mbox{\tiny\rm{B}}}\phi_{\mbox{\tiny\rm{B}}}=\phi^{\dagger}\phi\,\bigl{(}1+\delta Z_{\phi}\bigr{)}\;,\quad A^{a}_{\mu{\mbox{\tiny\rm{B}}}}A^{a}_{\nu{\mbox{\tiny\rm{B}}}}=A^{a}_{\mu}A^{a}_{\nu}\,\bigl{(}1+\delta Z_{A}\bigr{)}\;,\quad\xi_{\mbox{\tiny\rm{B}}}=\xi\,\bigl{(}1+\delta Z_{\xi}\bigr{)}\;.$$

$$\delta L(\varphi )$$

$$\displaystyle\delta m_{\mbox{\scriptsize$h$}}^{2}$$

$$\displaystyle\delta m_{\mbox{\tiny\rm{$G$}}}^{2}$$

$$\displaystyle\delta m_{\mbox{\tiny\rm{$\phi T$}}}^{2}$$

$$\displaystyle\delta m_{\mbox{\tiny\rm{$\chi T$}}}^{2}$$

$$m_{\text{mbox}E2}^2$$

$$m_{\text{mbox}E3}^2$$

$$V_{0}=\frac{\mu_{1}^{2}\,h^{2}}{2}\,\Bigl{(}1+\delta Z_{\phi}+\delta Z_{\mu_{1}^{2}}\Bigr{)}+\frac{\lambda_{1}h^{4}}{4}\,\Bigl{(}1+2\delta Z_{\phi}+\delta Z_{\lambda_{1}}\Bigr{)}\;.$$

$$\frac{V_1^{(-1)}}{ϵ}$$

$$V_2$$

$$=$$

$$\frac{V_{2}^{(-2)}}{\epsilon^{2}}=\frac{3\lambda_{1}[\delta m_{\mbox{\tiny\rm{$\phi T$}}}^{2}+\mu_{1}^{2}+3\lambda{}_{1}v^{2}][\delta m_{\mbox{\tiny\rm{$\phi T$}}}^{2}-\mu_{1}^{2}-9\lambda{}_{1}v^{2}]}{4(4\pi)^{4}\epsilon^{2}}\;.$$

$$\left.\frac{V_{2}^{(-1)}}{\epsilon}\right|_{\mbox{\scriptsize non-analytic}}\;=\;-\frac{3\lambda_{1}\delta m_{\mbox{\tiny\rm{$\phi T$}}}^{2}}{2(4\pi)^{2}\epsilon}\,\biggl{[}-\frac{\widetilde{m}_{\mbox{\scriptsize$h$}}^{2}}{(4\pi)^{2}}\ln\biggl{(}\frac{\bar{\mu}^{2}}{\widetilde{m}_{\mbox{\scriptsize$h$}}^{2}}\biggr{)}+I^{(0)}_{\mbox{\tiny\rm{$T$}}}(\widetilde{m}_{\mbox{\scriptsize$h$}})\biggr{]}\;.$$

$$V_0+V_1$$

$$\frac{V_{1,{\mbox{\tiny\rm{$T$}}}}^{(-1)}}{\epsilon}=-\frac{\delta m_{\mbox{\tiny\rm{$\phi T$}}}^{2}\,(m_{\mbox{\scriptsize$h$}}^{2}+3m_{\mbox{\tiny\rm{$G$}}}^{2})+\delta m_{\mbox{\tiny\rm{$\chi T$}}}^{2}\,(m_{\mbox{\tiny\rm{$H$}}}^{2}+m_{\mbox{\tiny\rm{$A$}}}^{2}+2m_{\mbox{\tiny\rm{$H_{\pm}$}}}^{2})+3m_{\mbox{\tiny\rm{E2}}}^{2}m_{\mbox{\tiny\rm{$W$}}}^{2}}{2(4\pi)^{2}\epsilon}+(\mbox{$v$-independent})\;.$$

$$(\text{mbox}s)+(\text{mbox}v)$$

$$V_1^{(0)}$$

$$\bar{\mu }=\alpha {\bar{\mu }}_T,{\bar{\mu }}_T\equiv \pi T,\alpha \in (0.5,2.0).$$

$$J(m)\;\equiv\;{\frac{1}{2}}{\hbox{$\sum$}\!\!\!\!\!\!\!\int\,}_{\!\!\!\!\raise-2.71246pt\hbox{$\scriptstyle{P}$}}\ln\bigl{(}{P^{2}+m^{2}}\bigr{)}\;=\;\frac{m^{2}A(m)}{D}-\frac{1}{D-1}\int_{{\bf p}}\frac{p^{2}n(\omega)}{\omega}\;,$$

$$J(m)=\frac{1}{ϵ}{J}^{(-1)}(m)+J^{(0)}(m)+\mathcal{O}(ϵ).$$

$$\ln {\bar{\mu }}^2\equiv \ln {\mu }^2+\ln (4\pi )-{\gamma }_{\text{mbox}\mathrm{E}},$$

$$J^{(-1)}(m)$$

$$J^{(0)}(m)$$

$$J^{(0)}(m_b)$$

$$J^{(0)}(m_{\text{mbox}\text{sl}f})$$

$$I(m)\equiv {\sum \int }_{\text{raise}-2.71246ptP}\frac{1}{P^2+m^2}=A(m)+{\int }_{\mathbf{p}}\frac{n(\omega )}{\omega },$$

$$I(m)=\frac{1}{ϵ}{I}^{(-1)}(m)+I^{(0)}(m)+ϵI^{(1)}(m)+\mathcal{O}({ϵ}^2),$$

Full text

April 2017

{centering}

**Thermal phase transition with full 2-loop effective potential **

M. Laine, M. Meyer, G. Nardini

*AEC, Institute for Theoretical Physics, University of Bern,

Sidlerstrasse 5, CH-3012 Bern, Switzerland

Abstract

Theories with extended Higgs sectors constructed in view of cosmological ramifications (gravitational wave signal, baryogenesis, dark matter) are often faced with conflicting requirements for their couplings; in particular those influencing the strength of a phase transition may be large. Large couplings compromise perturbative studies, as well as the high-temperature expansion that is invoked in dimensionally reduced lattice investigations. With the example of the inert doublet extension of the Standard Model (IDM), we show how a resummed 2-loop effective potential can be computed without a high-$T$ expansion, and use the result to scrutinize its accuracy. With the exception of $T_{\rm c}$, which is sensitive to contributions from heavy modes, the high-$T$ expansion is found to perform well. 2-loop corrections weaken the transition in IDM, but they are moderate, whereby a strong transition remains an option.

1 Introduction

With the upcoming years of the LHC probing the Higgs mechanism, and the continued direct, indirect and collider searches for dark matter, together with the prospect of LISA probing gravitational wave backgrounds related to particle physics, it has become popular to search for a framework which may play a role in all contexts. Surprisingly, the Standard Model supplemented by an additional scalar field, for instance in the singlet, doublet, triplet, or higher representation, cannot easily be excluded from these considerations. We focus here on the doublet case, simplified further by an additional Z(2) symmetry, a framework that is generally referred to as the Inert Doublet Model (IDM) [1, 2, 3].

The original interest in the IDM came largely from the dark matter context [4, 5, 6, 7, 8, 9, 10, 11, 12, 13], which remains a viable option today (cf. e.g. refs. [14, 15, 16, 17, 18, 19, 20, 21] and references therein). Many theoretical (cf. e.g. refs. [22, 23, 24, 25, 26]) and collider (cf. e.g. refs. [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43]) constraints on the model have been considered. Furthermore, following early suggestions [44, 45, 46, 47, 48], a strong phase transition appears possible [49, 50, 51]. However the issue of large couplings emerges, for instance in some of the benchmarks of ref. [50] certain scalar couplings attain the magnitude $\lambda_{3}\simeq 3$ in a normalization in which the Standard Model Higgs self-coupling is $\lambda_{1}\simeq 0.15$.

There is a clear reason for the need for large couplings if a strong phase transition is to be present. Without any additional particles, the theory has no thermal phase transition at all (for a review, see ref. [52]). If degrees of freedom are added which are weakly coupled and massive, they can be integrated out, resulting in the same “dimensionally reduced” effective theory [53, 54] as for the Standard Model [55], and thereby with the same conclusion concerning the phase transition. To change the conclusion, we either need to add new degrees of freedom which are light around the transition point, or which come with large couplings, so that the effective couplings of the low-energy theory change by a significant amount. Light degrees of freedom could experience a transition of their own and thereby indeed influence the dynamics substantially [56, 57]; this is an interesting option but will not be considered here, given that it requires a degree of fine tuning. Thereby we are left with large couplings as the remaining avenue. It is difficult to exclude the existence of such couplings phenomenologically, given that Higgs physics does not easily avail itself to precision inspection and that constraints from fermionic processes are largely missing for the inert doublet. Large couplings do imply the presence of a nearby Landau pole and, conversely, could originate as a low-energy description of some sort of composite dynamics.

In the context of electroweak baryogenesis, a strong phase transition refers to a discontinuity $\Delta v\sim T$, where $v$ is a gauge-fixed Higgs expectation value ($v\simeq 246$ GeV at $T=0$), and $T$ is the temperature [58]. In the Higgs phase, gauge boson masses are then of order m_{\mbox{\tiny\rm{W}}}\sim gv/2\ll\pi T, where $g\sim 2/3$ is the SU${}_{\mbox{\tiny\rm{L}}}$(2) gauge coupling. In this situation a high-$T$ expansion in m_{\mbox{\tiny\rm{W}}}^{2}/(\pi T)^{2} works well.111For bosonic degrees of freedom the high-$T$ expansion also includes non-analytic terms, such as (m_{\mbox{\tiny\rm{W}}}^{2})^{3/2}/(\pi T)^{3}; however any sum-integral only generates a finite number of such terms, associated with Matsubara zero-mode contributions, so that they do not affect the convergence of the infinite series.

The high-$T$ expansion is an ingredient for instance in non-perturbative studies based on dimensional reduction (cf. e.g. refs. [59, 60, 61, 62, 63] and references therein). However, new degrees of freedom which get a mass through a large coupling $\lambda_{3}^{1/2}\sim 2$ may become heavy in the broken phase, $\lambda_{3}^{1/2}v\sim\pi T$. Given that the high-$T$ expansion is an asymptotic series, it is not clear whether it is numerically accurate in such a situation.

In order to test the convergence of the high-$T$ expansion, and of the perturbative treatment in general, a sufficient loop order is needed. Here we go to 2-loop level for the effective potential. Earlier results probing the validity of the high-$T$ expansion at 2-loop level, associated however with large vacuum masses rather than with large couplings, can be found in ref. [64]. Another related investigation, albeit restricted to an Abelian theory and without a detailed exposition of the “master” sum-integrals that appear, was presented in ref. [65].

The outline of this paper is the following. After defining the IDM and the basic observables of our interest (sec. 2), we adopt a simple procedure for implementing the thermal resummations that are necessary for a consistent computation at finite temperature (sec. 3). Intricacies related to renormalization of the effective potential in the $R_{\xi}$ gauge and in the presence of resummation are briefly reiterated (sec. 4). After illustrating our results numerically (sec. 5), we collect together our conclusions (sec. 6). In a number of appendices, the thermal 2-loop “master” sum-integrals used for representing the effective potential are computed, both without and with a high-$T$ expansion (appendix A); the 2-loop Feynman diagrams are listed in terms of these “masters” (appendix B); and 1-loop formulae for vacuum renormalization, both as concerns the initial values of renormalization group evolution and the renormalization group evolution itself, are specified (appendix C).

2 Model and observables

In the IDM [1, 2, 3], the Standard Model Higgs doublet, $\phi$, is supplemented by an additional doublet, $\chi$, which has the same gauge charges as $\phi$ but in addition displays an unbroken global Z(2) symmetry, which forbids Yukawa couplings to Standard Model fermions. The scalar potential has the form

[TABLE]

A global phase rotation permits for us to choose $\lambda_{5}$ purely real and negative. Several extensions of the IDM have also been proposed, with additional scalars and additional gauge symmetries; many lead to fascinating phenomenology but for brevity we restrict ourselves to the simplest case here, since this is sufficient for our methodological considerations.

We are interested in the behaviour of the model at finite temperature. Simple thermodynamic characteristics of a phase transition are its critical temperature ($T_{\rm c}$) and latent heat ($L$). The discontinuity of the Higgs condensate (cf. e.g. ref. [66]),

[TABLE]

where ${Z}_{\mu_{1}^{2}}$ is the renormalization factor related to the bare mass parameter $\mu_{1}^{2}$, is a gauge-independent but scale-dependent characteristic of the transition. We choose the fixed ${\overline{\mbox{\rm MS}}}$ renormalization scale \bar{\mu}=m_{\mbox{\tiny\rm{Z}}} for its definition. Denoting by $f\equiv F/V$ the free energy density and by $\Delta f$ its discontinuity across the transition (with $\Delta f=0$ precisely at $T_{\rm c}$), we can equivalently write v_{\mbox{\scriptsize phys}}^{2}/2=\partial\Delta f/\partial\mu_{1}^{2}(m_{\mbox{\tiny\rm{Z}}}). The importance of $v_{\mbox{\scriptsize phys}}$ stems from the fact that it is strongly correlated with the rate of anomalous baryon number violation [67].

Other important characteristics of the transition are its surface tension at $T_{\rm c}$, and the bubble nucleation rate in the whole metastability range. Determining these necessitates, however, the study of inhomogeneous configurations, which is a notoriously hard problem (cf. e.g. ref. [68]) and not addressed here. However our conclusions do support a low-energy effective theory approach, which can subsequently also be applied to this problem [69].

As a tool for computing $T_{\rm c}$, $L$ and $v_{\mbox{\scriptsize phys}}$ we employ the effective potential. At its minima, the effective potential equals the free energy density, $f=V(v_{\mbox{\scriptsize min}})$, up to an overall constant which drops out in $\Delta f\equiv V(v_{\mbox{\scriptsize min}})-V(0)$. The effective potential is defined through a shift of the neutral Higgs component by a constant, $v$, so that the Higgs doublets can be written as

[TABLE]

Here $h$ represents the physical Higgs boson, and $H\equiv H_{0},A\equiv H_{3}$ as well as $H_{\pm}=(H_{1}\pm iH_{2})/\sqrt{2}$ are the new scalar degrees of freedom. The Z(2) symmetry associated with $\chi$ is assumed to be unbroken, and we check this assumption a posteriori (see below). The meaning of $v$ depends on the gauge choice and also on the renormalization factors ${Z}_{\phi}$, ${Z}_{v}$ (see below). Nevertheless, as has been demonstrated within the high-$T$ expansion both for covariant [70] and $R_{\xi}$ [71] gauges, gauge independent observables can be obtained from $V(v)$, in particular $\Delta f=V(v_{\mbox{\scriptsize min}})-V(0)$, which in turn fixes $T_{\rm c}$, $L$, and $v_{\mbox{\scriptsize phys}}$ as discussed above.

3 Resummation

When we are addressing the regime v\mathop{\raise 1.29167pt\hbox{<\kern-7.5pt\raise-4.73611pt\hbox{\sim}}}T, then the masses generated by the Higgs mechanism, m_{\mbox{\tiny\rm{W}}}\sim gv/2, are of a similar magnitude or smaller than thermal “Debye masses”, $m_{\mbox{\tiny\rm{E2}}}\sim gT$. Therefore thermal masses play an important role. For the Standard Model, all thermal masses were determined in ref. [72], and a way to incorporate them at 2-loop level was worked out in ref. [73]. However, even though theoretically consistent, the procedure of ref. [73] is simple only in a setting in which a high-$T$ expansion is valid: technically it amounts to carrying out a resummation only for Matsubara zero modes, for which it is strictly necessary. In our more general setting, in which some degrees of freedom may become heavy in the broken phase, a split-up into zero and non-zero Matsubara modes is cumbersome. Therefore, we propose to implement a resummation for all modes.222Analogous procedures have been pursued in other contexts, cf. e.g. refs. [74, 75, 76, 77, 78] and references therein.

Of course, there is a price to pay for this “simplification”, discussed at the end of this section and in sec. 4.4.

The set of fields for which a resummation is needed comprises the scalar fields ($\phi,\chi$) as well as, in covariant and $R_{\xi}$ gauges, the temporal components of the gauge fields. Consider $\phi$ as an example. The original (imaginary-time) Lagrangian can be written as

[TABLE]

Here $\phi_{\mbox{\tiny\rm{B}}}$ denotes a bare and $\phi\equiv\phi_{\mbox{\tiny\rm{R}}}$ a renormalized field, and $\delta L$ contains the vacuum counterterms. Resummation can now be implemented by incorporating +\,\delta m_{\mbox{\tiny\rm{\phi T}}}^{2}\,\phi^{\dagger}\phi on par with vacuum masses in the propagators, whereas the part \delta L(\phi)-\delta m_{\mbox{\tiny\rm{\phi T}}}^{2}\,\phi^{\dagger}\phi is treated as a “counterterm”. If we choose \delta m_{\mbox{\tiny\rm{\phi T}}}^{2} properly, i.e. as the thermal mass generated for the Matsubara zero modes, then this procedure is equivalent to the approach of ref. [73], up to corrections that are of higher order in couplings than the computation at hand.

Even though the idea just introduced is simple, the devil lies in the details, particularly in the precise choice of \delta m_{\mbox{\tiny\rm{\phi T}}}^{2}. At high temperatures, the parametric form is \delta m_{\mbox{\tiny\rm{\phi T}}}^{2}\sim g^{2}T^{2}+{\mathcal{O}}(g^{4}). The factor $T^{2}$ originates from integrating out the non-zero Matsubara modes, so that \delta m_{\mbox{\tiny\rm{\phi T}}}^{2}\sim g^{2}I_{n\neq 0}(m)+{\mathcal{O}}(g^{4}), where I_{n\neq 0}(m)\equiv{{\Sigma}\!\!\!\!\raise 1.29167pt\hbox{\int}_{\mbox{\tiny\rm{{P}}}}}\!\!^{\prime}\frac{1}{P^{2}+m^{2}} is a tadpole integral omitting a Matsubara zero mode. Following a frequent convention, we make a choice in the following that \delta m_{\mbox{\tiny\rm{\phi T}}}^{2}\sim g^{2}I(0)\equiv g^{2}{{\Sigma}\!\!\!\!\raise 1.29167pt\hbox{\int}_{\mbox{\tiny\rm{{P}}}}}\!\frac{1}{P^{2}}, thereby omitting corrections of ${\mathcal{O}}(g^{4}T^{2})$ and ${\mathcal{O}}(g^{2}m^{2})$. Both approximations can be systematically lifted within the framework of dimensionally reduced theories, whereas within the approach of eq. (3.1) there is no unambiguous way to do this. We choose the simple procedure because it is sufficient for addressing the main goals of our study, namely the convergence of the high-$T$ expansion and the magnitude of 2-loop corrections. However, it should be acknowledged that this approximation is numerically questionable for the IDM: large scalar couplings imply that corrections of ${\mathcal{O}}(\lambda_{3}^{2}T^{2})$ can be significant, and a large mass parameter $\mu_{2}^{2}$ implies that mass-dependent corrections of ${\mathcal{O}}(\lambda_{3}\mu_{2}^{2})$ should be included. The omission of these corrections leads to specific problems, discussed below.

In dimensional regularization, where the space-time dimension is $D=4-2\epsilon$, the sum-integral $I(0)$ contains terms of ${\mathcal{O}}(\epsilon)$, cf. eq. (A.17). In loop diagrams $I(0)$ can be multiplied by $1/\epsilon$, and therefore ${\mathcal{O}}(\epsilon)$ contributions can give finite results. The thermal masses including these pieces are listed in eqs. (4.3)–(4.3) below.

A formal crosscheck on the consistency of the resummation carried out is that the so-called “linear terms” cancel in the 2-loop result within the high-$T$ expansion [79]. Such terms have the form $\sim g^{2}I(0)I_{n=0}(m)$, where the Matsubara zero-mode part evaluates to $I_{n=0}(m)\equiv T\int_{\bf p}\frac{1}{p^{2}+m^{2}}=-\frac{mT}{4\pi}[1+{\mathcal{O}}(\epsilon)]$. We have analytically verified the cancellation of linear terms to all orders in $\epsilon$ within our resummation. However, as alluded to above, our simple resummation does not properly capture the infrared structure of the 2-loop potential in the IDM, in which substantial corrections of ${\mathcal{O}}(\lambda_{3}\mu_{2}^{2})$ to the effective Higgs mass parameter can appear. This implies that the cancellation of linear terms is incomplete beyond the formal high-$T$ limit: a remainder $\sim\lambda_{3}[I(\mu_{2})-I(0)]I_{n=0}(m)$ is left over. In cases with \mu_{2}\mathop{\raise 1.29167pt\hbox{>\kern-7.5pt\raise-4.73611pt\hbox{\sim}}}T such terms become visible at small $v/T$ (cf. sec. 5.4).

4 Gauge fixing and renormalization

4.1 Gauge fixing

Perturbative computations in gauge theories require gauge fixing, even though physical observables are independent of it. For simplicity we employ the Feynman $R_{\xi}$ gauge in our analysis. We also omit the hypercharge U${}_{\mbox{\tiny\rm{Y}}}(1)$ coupling $g_{1}$, whose ${\mathcal{O}}(1\%)$ influence is an order of magnitude smaller than our uncertainties, and denote the weak SU${}_{\mbox{\tiny\rm{L}}}$(2) coupling by $g\equiv g_{2}$. Then the gauge fixing and Faddeev-Popov terms read

[TABLE]

where only terms coupling to scalar degrees of freedom have been shown; $G_{a}$ are Goldstone modes from eq. (2.3); and c^{a}_{\mbox{\tiny\rm{A}}},\bar{c}^{a}_{\mbox{\tiny\rm{A}}} are SU${}_{\mbox{\tiny\rm{L}}}$(2) ghost fields.

In $R_{\xi}$ gauges the parameter $v$ has two different origins: it appears as a “background field” in the gauge fixing term in eq. (4.1), and it originates from a shift of the Higgs field according to eq. (2.3). For a proper renormalization of gauge-dependent quantities, these two fields need to be kept track of and renormalized separately (cf. refs. [80, 81] and references therein). The renormalization factor related to the background field is denoted by $Z_{v}$: $v^{2}_{\mbox{\tiny\rm{B}}}=v^{2}(1+\delta Z_{v})$.

4.2 Vacuum counterterms

The bare couplings are expressed as \mu_{i{\mbox{\tiny\rm{B}}}}^{2}=\mu_{i}^{2}\,\bigl{(}1+\delta Z_{\mu^{2}_{i}}\bigr{)}, \lambda_{i{\mbox{\tiny\rm{B}}}}=\lambda_{i}\,\bigl{(}1+\delta Z_{\lambda_{i}}\bigr{)}, g^{2}_{\mbox{\tiny\rm{B}}}=g^{2}\,\bigl{(}1+\delta Z_{g^{2}}\bigr{)}, h^{2}_{t{\mbox{\tiny\rm{B}}}}=h_{t}^{2}\,\bigl{(}1+\delta Z_{h^{2}_{t}}\bigr{)}, where $h_{t}$ is the top Yukawa coupling. Because we compute the effective potential as a function of $v$ and because resummation treats $A^{a}_{0}$ and $A^{a}_{i}$ separately, we also need to renormalize certain unphysical objects, namely wave functions and the gauge fixing parameter:

[TABLE]

The renormalized gauge parameter is set to $\xi=1$. After the shift of the Higgs vacuum expectation value according to eq. (2.3), various counterterms are generated. For instance $\delta L$ from eq. (3.1) becomes

[TABLE]

Physical on-shell Green’s functions, such as those listed in appendix C.3 for purposes of vacuum renormalization, are not affected by the field renormalization constants [82].

4.3 Thermal masses

With the setup introduced, the thermal mass corrections for eq. (3.1) and table 1 read

[TABLE]

where the function $I$ is defined in eq. (A.8). We have adopted a notation in which $0_{b}$ and $0_{\mbox{\scriptsize\sl f}}$ denote vanishing masses carried by bosons and fermions, respectively; and n_{\mbox{\tiny\rm{G}}}\equiv 3 denotes the number of generations. The resulting tree-level mass spectrum is listed in table 1.

As alluded to in the paragraphs below eq. (3.1), the leading-order “massless” resummation of eqs. (4.3)–(4.3) is not sufficient for a precise determination of $T_{\rm c}$: the effective Higgs mass parameter gets large corrections of ${\mathcal{O}}(\lambda_{3}\mu_{2}^{2},\lambda_{3}^{2}T^{2})$ which are not properly accounted for. These corrections could be systematically included in a dimensionally reduced investigation [55, 64, 63], whose principal accuracy our study aims to justify.

4.4 Illustration of cancellation of divergences

The resummation introduced in eq. (3.1) modifies the divergence structure of the theory at any given loop order. Even though the changes are of higher order than the computation carried out, this leads to divergences which look worrisome at first sight. We illustrate this with the help of a single-component scalar theory,

[TABLE]

At 1-loop level the counterterms read $\delta Z_{\phi}=0$, $\delta Z_{\mu_{1}^{2}}=3\lambda_{1}/(16\pi^{2}\epsilon)$ and $\delta Z_{\lambda_{1}}=9\lambda_{1}/(16\pi^{2}\epsilon)$. The thermal mass correction is \delta m_{\mbox{\tiny\rm{\phi T}}}^{2}=3\lambda_{1}I(0), and we denote \widetilde{m}_{\mbox{\scriptsizeh}}^{2}=\mu_{1}^{2}+3\lambda_{1}v^{2}+\delta m_{\mbox{\tiny\rm{\phi T}}}^{2}.

The 1-loop effective potential is given by the function $J$ defined in eq. (A.1): V_{1}=J(\widetilde{m}_{\mbox{\scriptsizeh}}). Writing $V_{1}=\sum_{n=-1}^{\infty}V_{1}^{(n)}\epsilon^{n}$, let us consider the divergent part, given by eq. (A.4), viz.

[TABLE]

The $T$-independent divergences $\propto\mu_{1}^{2}v^{2},v^{4}$ are cancelled by the counterterms in eq. (4.10). In contrast, the $T$-dependent divergence \propto\delta m_{\mbox{\tiny\rm{\phi T}}}^{2} is only cancelled by a part of $V_{2}$, as we show below. This is the peculiarity related to thermal divergences within the resummation we have adopted: whereas vacuum divergences are cancelled by counterterms originating from lower-order diagrams ($V_{0}$), thermal divergences are cancelled by the appearance of \delta m_{\mbox{\tiny\rm{\phi T}}}^{2} within higher-order contributions ($V_{2}$).

There are three diagrams contributing to $V_{2}$ (cf. fig. 1): a “sunset” diagram containing three propagators, denoted by (sss); a “figure-8” diagram containing two propagators, denoted by (ss); and a counterterm diagram containing one propagator, denoted by (s). Making use of the notation of appendix A, the expressions read

[TABLE]

The sum of eqs. (4.13)–(4.15) contains divergences of orders $1/\epsilon^{2}$ and $1/\epsilon$, cf. eqs. (A.11), (A.33) and (A.34). Writing $V_{2}=\sum_{n=-2}^{\infty}V^{(n)}_{2}\epsilon^{n}$, the $1/\epsilon^{2}$ divergences sum up to

[TABLE]

The $T$-independent parts $\propto\mu_{1}^{2}v^{2},v^{4}$ can be taken care of by 2-loop contributions to $\delta Z_{\phi}$, $\delta Z_{\mu_{1}^{2}}$ and $\delta Z_{\lambda_{1}}$ in eq. (4.10). However, a $T$ and $v$-dependent part \sim\lambda_{1}^{2}v^{2}\delta m_{\mbox{\tiny\rm{\phi T}}}^{2}/\epsilon^{2} remains. This is only cancelled by diagrams in which the insertion \delta m_{\mbox{\tiny\rm{\phi T}}}^{2} appears inside 2-loop topologies, which are counted on par with diagrams of 3-loop order (diagrams (a) and (b) in fig. 1). There is also a $T$-dependent but $v$-independent term \sim\lambda_{1}\delta m_{\mbox{\tiny\rm{\phi T}}}^{4}/\epsilon^{2} which is cancelled by 2-loop topologies containing two insertions of \delta m_{\mbox{\tiny\rm{\phi T}}}^{2}, a contribution which is counted on par with diagrams of 4-loop order (diagram (d) in fig. 1).

As far as the divergences of order $1/\epsilon$ go, the term proportional to \delta m_{\mbox{\tiny\rm{\phi T}}}^{2} in eq. (4.15) exactly cancels against the 1-loop contribution in eq. (4.11), apart from a $v$-independent divergence \sim\delta m_{\mbox{\tiny\rm{\phi T}}}^{4}/\epsilon. This gets cancelled by a 1-loop topology dressed by two appearances of \delta m_{\mbox{\tiny\rm{\phi T}}}^{2}, which is counted on par with diagrams of 3-loop order (diagram (c) in fig. 1).

A non-trivial cancellation is observed by considering $1/\epsilon$-divergences proportional to the non-analytic structure -{\widetilde{m}_{\mbox{\scriptsizeh}}^{2}}/{(4\pi)^{2}}\ln\bigl{(}{\bar{\mu}^{2}}/{\widetilde{m}_{\mbox{\scriptsizeh}}^{2}}\bigr{)}+I^{(0)}_{\mbox{\tiny\rm{T}}}(\widetilde{m}_{\mbox{\scriptsizeh}}), originating from I^{2}(\widetilde{m}_{\mbox{\scriptsizeh}}) and H(\widetilde{m}_{\mbox{\scriptsizeh}},\widetilde{m}_{\mbox{\scriptsizeh}},\widetilde{m}_{\mbox{\scriptsizeh}}) as shown by eqs. (A.12) and (A.34), respectively. We find that vacuum parts $\propto\lambda_{1}\mu_{1}^{2},\lambda_{1}^{2}v^{2}$ cancel from the coefficient of this divergence, as is required by renormalizability. The remainder reads

[TABLE]

Again this is only cancelled by diagrams in which the insertion \delta m_{\mbox{\tiny\rm{\phi T}}}^{2} appears inside 2-loop topologies, which are counted on par with 3-loop graphs (diagrams (a) and (b) in fig. 1).

To summarize, one price we pay for the resummation introduced in sec. 3 is that ultraviolet divergences do not cancel order by order in our power counting, in which the last term of eq. (3.1) is treated as an insertion. Instead thermal divergences cancel once all insertions contributing to a given order in couplings have been accounted for. This “drawback” is compensated for by the fact that infrared sensitive terms get resummed to all orders.

5 Results

5.1 Diagrams

At tree and 1-loop levels the effective potential follows from the Lagrangian in eq. (2.1) and from 1-loop contributions in terms of the sum-integral $J$ defined in appendix A.1. With masses and degeneracies as listed in table 1, we get

[TABLE]

where the counterterms are from eq. (4.4). The 2-loop diagrams are given in appendix B.

5.2 Cancellation of divergences

The cancellation of divergences through vacuum counterterms offers for a useful crosscheck of the computation. As illustrated in sec. 4.4, the cancellation is non-trivial and incomplete in the presence of the thermal resummation introduced in eq. (3.1). We briefly summarize here the cancellations that can be observed.

First of all, vacuum (i.e. temperature independent) divergences are cancelled by counterterms of a lower loop order. Parametrically, the tree-level potential $V_{0}$ is of order $m^{4}/g^{2}$. The 1-loop contribution $V_{1}$ is of order $m^{4}$ and contains divergences. These are cancelled by tree-level counterterms $\delta Z\sim g^{2}$, which modify $V_{0}$ by effects of $\sim V_{0}\,\delta Z\sim m^{4}$. Similarly, $V_{2}$ is of order $g^{2}m^{4}$ and contains divergences. These are cancelled by contributions of $\sim g^{4}$ to $\delta Z$ appearing in $V_{0}\sim m^{4}/g^{2}$, and by 1-loop effects containing the counterterms, likewise of order $V_{1}\delta Z\sim g^{2}m^{4}$.

In contrast, thermal divergences are cancelled by higher-order effects. When the thermal masses of table 1 are used within the 1-loop expression, cf. eq. (5.1), then the divergent part of the function $J$, cf. eq. (A.4), leads to temperature-dependent divergences. Writing $V_{1}=V^{(-1)}_{1}/\epsilon+V^{(0)}_{1}+\ldots$ and denoting the thermal part by V^{(-1)}_{1,{\mbox{\tiny\rm{T}}}}, we get

[TABLE]

Recalling the divergent part of the function $I$ as given in eq. (A.11), eq. (5.2) is cancelled by the parts of $V_{2}$ given in eqs. (B.3) and (B.11) that contain the thermal counterterms:

[TABLE]

There is a $v$-independent remainder \propto\delta m_{\mbox{\tiny\rm{\phi T}}}^{4}/\epsilon left over which is fully cancelled only once the “3-loop” diagram (c) in fig. 1 is included, as discussed in sec. 4.4.

A stringent test is given by the cancellation of non-analytic divergences, of the type in eq. (4.17). We find that divergences proportional to the functions I^{(0)}_{\mbox{\tiny\rm{T}}}(\underline{m_{\mbox{\tiny\rm{W}}}}), I^{(0)}_{\mbox{\tiny\rm{T}}}(\underline{\widetilde{m}_{\mbox{\tiny\rm{W}}}}), I^{(0)}_{\mbox{\tiny\rm{T}}}(\widetilde{m}_{\mbox{\scriptsizeh}}), I^{(0)}_{\mbox{\tiny\rm{T}}}(\widetilde{m}_{\mbox{\tiny\rm{G}}}), I^{(0)}_{\mbox{\tiny\rm{T}}}(\widetilde{m}_{\mbox{\tiny\rm{H}}}), I^{(0)}_{\mbox{\tiny\rm{T}}}(\widetilde{m}_{\mbox{\tiny\rm{A}}}), I^{(0)}_{\mbox{\tiny\rm{T}}}(\widetilde{m}_{\mbox{\tiny\rm{H_{\pm}}}}), I^{(0)}_{\mbox{\tiny\rm{T}}}(m_{\mbox{\tiny\rm{W}}}) and I^{(0)}_{\mbox{\tiny\rm{T}}}(\widetilde{m}_{\mbox{\tiny\rm{W}}}) do cancel, apart from terms proportional to thermal masses, which are cancelled by higher-order contributions as discussed in sec. 4.4.

In our practical procedure, we let the divergences be cancelled by the vacuum counterterms to the extent that this happens. The remaining divergences, which are proportional to thermal masses and thereby formally of higher order, are removed by hand. Furthermore, because divergences proportional to thermal masses do cancel at higher order, we do not expand the thermal masses in $\epsilon$, but remove these divergences as a whole. This implies, for instance, that the finite part of the 1-loop contribution in eq. (5.1) becomes

[TABLE]

where the functions $J^{(-1)}$ and $J^{(0)}$ are from eqs. (A.4) and (A.5), respectively. Similarly, the 2-loop potential $V^{(2)}$ contains contributions of the types $I^{(0)}I^{(0)}$, $I^{(-1)}I^{(1)}$, $H^{(0)}$ and, from coefficients containing $D=4-2\epsilon$, $I^{(-1)}I^{(0)}$ and $H^{(-1)}$.

5.3 Fixing the couplings

For numerical evaluations we focus on three benchmark points, introduced in ref. [50]. As it turns out, this is sufficient for addressing generic issues concerning the high-$T$ expansion and the convergence of the perturbative expansion. The physical parameters associated with these benchmarks are listed in table 2.

A common feature of all the benchmark points is that the mass splittings in the inert sector are larger than would “naturally” be expected from electroweak symmetry breaking, specifically m_{\mbox{\tiny\rm{A}}}-m_{\mbox{\tiny\rm{H}}}\gg m_{\mbox{\tiny\rm{Z}}} and m_{\mbox{\tiny\rm{H_{\pm}}}}-m_{\mbox{\tiny\rm{H}}}\gg m_{\mbox{\tiny\rm{Z}}}. This assumption necessitates some of the inert scalar self-couplings to be large. In fact, the couplings are so large that 1-loop corrections to physical parameters, such as pole masses, are of order unity. The ingredient from vacuum renormalization that is relevant for our study is the determination of the values of all ${\overline{\mbox{\rm MS}}}$ parameters at some reference scale, chosen as \bar{\mu}=m_{\mbox{\tiny\rm{Z}}} in accordance with ref. [50]. The procedure that we have adopted for estimating these values is to employ a “self-consistent” prescription in order to resum a subset of higher-order corrections and thereby to delimit the magnitude of loop effects; details are deferred to appendix C.4.333To summarize briefly, the Higgs sector parameters $\mu_{1}^{2},\lambda_{1}$ are evaluated à la Coleman-Weinberg; the other couplings are evaluated iteratively such that the same values appear on both sides of the equation.

The resulting couplings are listed in table 3. For the remainder of this study, we can forget about vacuum renormalization and simply use the values in table 3 as input. We have checked that variations of the renormalization prescription, which lead to ${\mathcal{O}}(20\%)$ variations of $\mu_{1}^{2}$ and $\lambda_{1}$ for BM2, nevertheless leave our physics conclusions concerning thermal effects qualitatively intact.

An important first observation from table 3 is that the Higgs self-coupling $\lambda_{1}$ can be smaller than in the Standard Model, \lambda_{1}(m_{\mbox{\tiny\rm{Z}}})\simeq 0.07\ll 0.15. A small quartic coupling favours a strong phase transition. However, for thermal considerations, the renormalization scale permitting to avoid large logarithms differs from that used for vacuum renormalization. Specifically, thermal fluctuations introduce logarithms of the type $\ln(\bar{\mu}/(\pi T))$ [55]. Therefore at finite temperature we use

[TABLE]

The couplings are run between \bar{\mu}=m_{\mbox{\tiny\rm{Z}}} and $\bar{\mu}=\alpha\bar{\mu}_{T}$ according to 1-loop renormalization group equations as specified in appendix C.5. We believe that uncertainties from higher-order corrections to the running are of secondary importance for the qualitative issues that we are addressing, particularly the convergence of the high-$T$ expansion.

5.4 Numerical evaluation

We have evaluated the 1-loop and 2-loop effective potentials both in a closed form utilizing the high-$T$ expansion, and numerically without resorting to it.

As discussed above, the computation is theoretically consistent only in stable phases: the “symmetric phase” at high temperatures and the “Higgs phase” at low temperatures. Outside of these phases the results are gauge dependent. In addition the results contain uncancelled ultraviolet divergences at any finite loop order as discussed in secs. 4.4 and 5.2. Once the ultraviolet divergences are removed by hand, an uncancelled $\bar{\mu}$-dependence is left over. Furthermore, some masses may become tachyonic; we replace the masses squared by their absolute values in such cases. However, these ambiguities are numerically benign in comparison with the “physical” uncertainty associated with scale dependence, which is formally of higher order but in practice substantial, given the large values of some of the mass parameters and couplings. This uncertainty is estimated through the scale variation in eq. (5.5).

We assume in this paper that the Z(2) symmetry related to the Inert Doublet $\chi$ is unbroken. In order to verify the validity of this assumption, the effective mass squared of the $\chi$ field, \mu_{2}^{2}+\delta m_{\mbox{\tiny\rm{\chi T}}}^{2} in the notation of table 1, is evaluated at the critical temperature. We find \mu_{2}^{2}+\delta m_{\mbox{\tiny\rm{\chi T}}}^{2}\,\mathop{\raise 1.29167pt\hbox{>\kern-7.5pt\raise-4.73611pt\hbox{\sim}}}\,(0.6T)^{2} in all cases, justifying the assumption a posteriori, given that $0.6T\sim gT$ is parametrically a “heavy” scale, similar to $m_{\mbox{\tiny\rm{E2}}}$.

In fig. 2, the 1-loop and 2-loop effective potentials are plotted at the corresponding critical temperatures, both with and without a high-$T$ expansion. We note that the 2-loop corrections are substantial but in general they do not modify the 1-loop predictions qualitatively. For BM1 and BM3 they weaken the transition moderately, and for BM2 they remove the transition altogether for \bar{\mu}\mathop{\raise 1.29167pt\hbox{>\kern-7.5pt\raise-4.73611pt\hbox{\sim}}}\bar{\mu}_{T} (in contrast they appear to strengthen the transition for BM2 within the high-$T$ expansion, however those results are unreliable because of the large value of $\mu_{2}^{2}$). With the exception of BM2, the high-$T$ expansion is seen to work very well. However, the value of the critical temperature does change substantially through the high-$T$ expansion; this is easily understood and is elaborated upon in sec. 6.

For BM2, in which case we find a very weak transition, the problems mentioned at the end of sec. 3, associated on one hand with the breakdown of the high-$T$ expansion for heavy inert modes (cf. table 3), and on the other with infrared sensitive terms not captured by our simple thermal resummation, become visible as “linear terms” at small $v/T$. We do not elaborate on these any further, given that the transition is too weak to be of physical interest.

Physical (gauge independent) results for the quantities $T_{\rm c},L$ and $v_{\mbox{\scriptsize phys}}$, defined in sec. 2, are collected in table 4. The errors originate from a variation of $\bar{\mu}$ in the range $(0.5...2.0)\,\pi T$, cf. eq. (5.5). These results confirm the heuristic impressions visible in fig. 2. At least for BM1, the transition could be marginally strong enough to support electroweak baryogenesis.

6 Conclusions

In this paper we have developed the general technology for evaluating the full 2-loop thermal effective potential for the Higgs field, without resorting to a high-temperature expansion. The technology has been applied to the Inert Doublet Model (IDM), incorporating thermal resummation in a particularly simple way.

Even though we did not dwell on this in the text, the largest technical effort of our work went into the derivation of the formulae given in appendices A and B, and into their numerical evaluation, which poses a challenge of its own (some remarks can be found in a paragraph below eq. (A.45)). The results of appendix A are model-independent, and can be applied to any extension of the Standard Model.

A word of caution is in order on the simple resummation that we adopted (sec. 3). As explained at the end of sec. 3 and in sec. 5.4, the resummation is not suitable for a precise treatment of infrared effects; at the same time, as explained in secs. 4.4 and 5.2, it is also problematic in the ultraviolet, introducing spurious divergences which are only cancelled by higher-order diagrams. We therefore do not endorse its use for practical computations aiming at physical precision; for us it was just a simple tool permitting to compare two different computations (a full 2-loop analysis and its high-$T$ expansion). As explained below, our final conclusion concerning the high-$T$ expansion suggests the availability of other tools for addressing physical observables with good precision.

Applying the formalism to the IDM as an example, our main finding is that the high-$T$ expansion works well for describing the strength of the phase transition, despite the fact that some degrees of freedom become heavy in the Higgs phase (cf. fig. 2 and table 4). This is a welcome observation, given that it opens the avenue for dimensionally reduced lattice investigations, necessary for cases in which a good precision is needed and/or properties associated with inhomogeneous configurations are of interest. This concerns for instance the surface tension [59, 60], the bubble nucleation rate [69], and the sphaleron rate [67].

Based on the benchmark points considered, as well as on a parameter scan at 1-loop level, we find that in general the IDM transition is at most moderately strong, as long as the $\chi$-field does not become so light that it would experience a transition of its own. In other models, possessing a stronger transition, the high-$T$ expansion could fail. At the same time, we would expect smaller 2-loop corrections in those cases, given that the infrared sensitive expansion parameter is \sim g^{2}T/(\pi m_{\mbox{\tiny\rm{W}}})\sim 2gT/(\pi v).

There is one observable for which the high-$T$ expansion does not work well: the critical temperature $T_{\rm c}$ (cf. the 2-loop results in table 4). This should not come as a surprise: some of the inert scalars are heavy and/or strongly coupled, and should not be treated with the high-$T$ expansion nor with the leading-order resummation of sec. 3. Even though they have little effect on the phase transition, they “renormalize” the effective Higgs mass parameter by a large amount. Within a dimensionally reduced investigation [53, 54], these effects can be incorporated without a high-$T$ expansion [64] and including higher orders in large couplings [55], so a good precision can be expected also for $T_{\rm c}$. We believe that a detour through the dimensionally reduced description, with effects of ${\mathcal{O}}(\lambda_{3}\mu_{2}^{2})$ and ${\mathcal{O}}(\lambda_{3}^{2}T^{2})$ included in thermal masses, should be chosen even in purely perturbative studies, if numerical precision at or below the 10% level is needed.

Turning finally to cosmology, our study supports previous suggestions according to which the IDM can incorporate a phase transition marginally strong enough for baryogenesis, at least for the benchmark point BM1 (cf. table 4). The 2-loop corrections weaken the transition somewhat, but in many cases they also reduce the scale uncertainties of $L,v_{\mbox{\scriptsize phys}}$ and $v_{\mbox{\scriptsize min}}$ (cf. the panel “full effective potential” in table 4). However the problem of obtaining sufficient CP violation is not alleviated by the IDM Higgs sector which contains no new physical phases. As an example of a work-around, it has been suggested that CP violation in the interactions responsible for neutrino masses could play a role for baryogenesis (cf. e.g. ref. [83]).

Acknowledgements

This work was supported by the Swiss National Science Foundation (SNF) under grant 200020-168988, and by the Munich Institute for Astro- and Particle Physics (MIAPP) of the DFG cluster of excellence “Origin and Structure of the Universe”.

Appendix A Thermal master sum-integrals

We list here the expressions for the sum-integrals that are needed for evaluating the 2-loop effective potential. The results are divergent for $\epsilon\to 0$. For the 1-loop structures the divergences are proportional to $1/\epsilon$; the 2-loop result contains squares of 1-loop structures, so that the 1-loop structures need to be evaluated up to terms of ${\mathcal{O}}(\epsilon)$. For genuine 2-loop structures, the terms that need to be tracked are of ${\mathcal{O}}(1/\epsilon^{2})$, ${\mathcal{O}}(1/\epsilon)$, and ${\mathcal{O}}(1)$.

The master sum-integrals contain both vacuum (i.e. temperature independent) and thermal parts. In the following, the thermal corrections are given both in an exact form suitable for numerical evaluation, and analytically in a high-$T$ expansion. In the latter case, the leading contribution is of ${\mathcal{O}}(T^{4})$, and we list terms up to ${\mathcal{O}}(m^{4},g^{2}m^{2}T^{2})$, where $m^{2}$ denotes a generic mass squared and $g^{2}$ a generic coupling constant. This is consistent with a power counting m^{2}\mathop{\raise 1.29167pt\hbox{<\kern-7.5pt\raise-4.73611pt\hbox{\sim}}}g^{2}T^{2} which can be used for justifying the high-$T$ expansion. Some of the master structures are always multiplied by $\sim g^{2}m^{2}$, in which case high-$T$ expansions are given up to ${\mathcal{O}}(T^{2})$.

Thermal corrections depend on whether the particle in question is a boson or a fermion. In order to compactify the expressions, we employ an implicit notation in which the statistics is identified through the mass carried by the particle. The distribution function is denoted generically by $n$, and if the argument is “bosonic”, it is to be interpreted as the Bose distribution, $n(\omega)\to n_{\mbox{\tiny\rm{B{}}}}(\omega)\equiv 1/(e^{\omega/T}-1)$. In contrast, with a “fermionic” argument, minus the Fermi distribution is to be understood, $n(\omega)\to-n_{\mbox{\tiny\rm{F{}}}}(\omega)\equiv-1/(e^{\omega/T}+1)$. A vanishing bosonic mass is denoted by $0_{b}$ and a fermionic one by $0_{\mbox{\scriptsize\sl f}}$.

A.1 Function $J(m)$

The master sum-integral appearing in the 1-loop result is denoted by

[TABLE]

where $P=(\omega_{n},{\bf p})$; $\omega_{n}$ denote Matsubara frequencies; $p\equiv|{\bf p}|$; the vacuum function $A$ is given in eq. (C.1); $D=4-2\epsilon$; $\omega_{i}\equiv\sqrt{p^{2}+m_{i}^{2}}$; and we made use of partial integrations and properties of dimensional regularization. We write

[TABLE]

Suppressing an overall $\mu^{-2\epsilon}$, where $\mu$ is a scale parameter related to dimensional regularization, and denoting

[TABLE]

the expressions for the functions in eq. (A.2) read

[TABLE]

where I^{(0)}_{\mbox{\tiny\rm{T}}}(\underline{m}) is given in eq. (A.24). In the high-$T$ limit, the expressions depend on whether bosonic or fermionic particles are considered. Expanding up to ${\mathcal{O}}(m^{4})$, the bosonic contributions read

[TABLE]

whereas the fermionic expression is

[TABLE]

A.2 Function $I(m)$

The basic 1-loop structure appearing within the 2-loop result is denoted by

[TABLE]

where the vacuum part $A$ is from eq. (C.1). We write

[TABLE]

and subsequently separate each contribution into a vacuum and thermal part,

[TABLE]

Suppressing an overall $\mu^{-2\epsilon}$, the expressions for the functions in eq. (A.9) read

[TABLE]

These functions are related through $I^{(n)}(m)=\bar{\mu}^{2}\partial I^{(n+1)}(m)/\partial\bar{\mu}^{2}$. In the high-$T$ limit, the bosonic contributions read

[TABLE]

where the Stieltjes constant $\gamma_{1}$ is defined through $\zeta(s)=1/(s-1)+\sum_{n=0}^{\infty}\gamma_{n}(-1)^{n}(s-1)^{n}/n!$. The corresponding fermionic expressions are

[TABLE]

A.3 Function $I(\underline{m})$

The thermal 2-loop effective potential contains appearances of a “Lorentz-violating” 1-loop structure denoted by

[TABLE]

where the vacuum part $A$ is from eq. (C.1). We write

[TABLE]

and I^{(n)}(\underline{m})=I^{(n)}_{0}(\underline{m})+I^{(n)}_{\mbox{\tiny\rm{T}}}(\underline{m}). Suppressing an overall $\mu^{-2\epsilon}$, the expressions for the structures in eq. (A.21) read

[TABLE]

These functions are related through $I^{(n)}(\underline{m})=\bar{\mu}^{2}\partial I^{(n+1)}(\underline{m})/\partial\bar{\mu}^{2}$. In the high-$T$ limit, the bosonic contributions read

[TABLE]

whereas the corresponding fermionic expressions are

[TABLE]

A.4 Function $H(m_{1},m_{2},m_{3})$

Next we consider the 2-loop “sunset” sum-integral,

[TABLE]

The Matsubara sums can be carried out explicitly, and for the vacuum part the spatial integrations are also doable [84, 85, 86]. Writing the result as

[TABLE]

and omitting an overall $\mu^{-4\epsilon}$, the results read

[TABLE]

where $\omega^{p}_{i}\equiv\sqrt{p^{2}+m_{i}^{2}}$ and $B^{(0)}$ is from eq. (C.1). For finite masses a representation of the undefined functions reads [86]444The function $L$ is singular in certain limits, for instance $L(m^{2},\epsilon^{2},\epsilon^{2})=-\pi^{2}/6-2\ln^{2}(m/\epsilon)$ for $\epsilon\to 0^{+}$.

[TABLE]

The functions $H^{(n)}$ are related through $H^{(n)}(\{m_{i}\})={\frac{1}{2}}\bar{\mu}^{2}\,\partial H^{(n+1)}(\{m_{i}\})/\partial\bar{\mu}^{2}$.

Given that the function $H$ is always multiplied by $\sim g^{2}m^{2}$ in the effective potential, the order $\sim T^{2}$ in the high-$T$ is sufficient for reaching the order $\sim g^{2}m^{2}T^{2}$ for $V_{2}$. To this accuracy (cf. e.g. refs. [87, 88]),

[TABLE]

A.5 Function $H(\underline{m_{1}},m_{2},m_{3})$

The final ingredient needed is a variant of the “sunset” sum-integral in eq. (A.31), with one line weighted by an additional spatial momentum:

[TABLE]

The Matsubara sums can be carried out as before, and for the vacuum parts Lorentz symmetry allows furthermore to write

[TABLE]

Expressing the result like in eq. (A.32) and omitting an overall $\mu^{-4\epsilon}$, we get

[TABLE]

where $\omega^{p}_{i}\equiv\sqrt{p^{2}+m_{i}^{2}}$, $B^{(0)}$ is from eq. (C.1), and $R$ and $L$ are from eqs. (A.36) and (A.37), respectively. The functions are related through $H^{(n)}={\frac{1}{2}}\bar{\mu}^{2}\,\partial H^{(n+1)}/\partial\bar{\mu}^{2}$.

The numerical evaluation of eq. (A.45) is straightforward if all masses are of similar orders of magnitude. In contrast, if there is a hierarchy between the masses (cf. e.g. eq. (B.14)), care must be taken in order to avoid significance loss in the numerics. For instance, the coefficient multiplying I^{(0)}_{\mbox{\tiny\rm{T}}}(\underline{m_{2}}) has a finite limit for $m_{2}\to 0$, but many individual terms within the curly brackets diverge as $\sim 1/m_{2}^{4}$. A similar problem appears in the coefficient multiplying I^{(0)}_{\mbox{\tiny\rm{T}}}(m_{2}), even though divergences are only $\sim 1/m_{2}^{2}$ in this case. It may also be noted that the coefficients of I^{(0)}_{\mbox{\tiny\rm{T}}}(\underline{m_{2}}) and I^{(0)}_{\mbox{\tiny\rm{T}}}(m_{2}) have cusps at $m_{2}=m_{1}+m_{3}$, originating from the function $B^{(0)}$, which cancel against corresponding cusps originating from the last three rows of eq. (A.45). For a proper cancellation of the cusps, all terms involved need to be evaluated with good precision. A powerful crosscheck on the numerics is provided by the high-$T$ limit, which can be given in analytic form (cf. eqs. (A.5)–(A.5)).

In the high-$T$ limit, making use of relations determined in ref. [89],555We thank Y. Schröder for locating the necessary relations.

we get

[TABLE]

Here the so-called linear terms, of the type ${\mathcal{O}}(mT^{3})$, have been written in a $D$-dimensional form, permitting for a crosscheck of their $D$-dimensional cancellation in the full result (cf. the discussion around the end of sec. 3).

The function $H(\underline{m_{1}},m_{2},m_{3})$ always appears in a difference containing various masses, so that the leading term $\propto T^{4}$ of the high-$T$ expansion drops out from the effective potential. Moreover, at ${\mathcal{O}}(m^{2})$, only the non-analytic terms originating from Matsubara zero modes have been kept in the expressions above, given that analytic terms lead to $v$-independent structures of the type \sim g^{2}(\widetilde{m}_{\mbox{\tiny\rm{W}}}^{2}-m_{\mbox{\tiny\rm{W}}}^{2})T^{2}\sim g^{4}T^{4}.

Appendix B 2-loop diagrams

In order to list all contributions to $V_{2}$, we make use of the master sum-integrals defined in appendices A.2–A.5. The diagrams are of three types, illustrated in fig. 1. We denote the particles circling in the loops by scalar (s), vector (v), ghost (g), or fermion (f). Then the various contributions read (some $v$-independent terms have been dropped for simplicity)

[TABLE]

Appendix C Vacuum renormalization

For the thermal computations, we need to know the running couplings as functions of the ${\overline{\mbox{\rm MS}}}$ scale $\bar{\mu}$ up to a scale $\bar{\mu}\sim\pi T$, cf. eq. (5.5). These can be obtained from renormalization group equations, provided that the initial values are known at some scale \bar{\mu}\sim m_{\mbox{\tiny\rm{Z}}}. In order to obtain the latter, we compute physical pole masses and the Fermi constant in terms of the ${\overline{\mbox{\rm MS}}}$ parameters, and then invert the relations in order to express the ${\overline{\mbox{\rm MS}}}$ couplings at \bar{\mu}=m_{\mbox{\tiny\rm{Z}}} in terms of the physical ones. For the Standard Model, these relations were determined up to 1-loop level in ref. [55],666In eq. (193) of ref. [55], there is a term $-\frac{8}{3}t^{2}\ln h$ missing from within the square brackets.

and here we extend the relations to the IDM. Closely related expressions for the IDM can be found in ref. [14].

C.1 Basis functions

In order to display the results for physical quantities, we make use of standard Passarino-Veltman type functions, which we have defined in Euclidean spacetime:

[TABLE]

For $D=4-2\epsilon$ and writing $A=\sum_{n=-1}^{\infty}A^{(n)}\epsilon^{n}$ etc, the divergent parts of these functions can be expressed as

[TABLE]

The finite parts of $A$ and $B$ read

[TABLE]

Given that $B^{(0)}$ is a function of $K^{2}$ only, we use an implicit notation in which $K$ may denote either a vector or its modulus. The corresponding expressions after going to Minkowskian signature, i.e. $K\to-i\mathcal{K}$, are conventionally expressed in terms of a function $F$ defined by

[TABLE]

The (real part of) $F$ is given in eq. (C.16) below.

C.2 Gauge coupling renormalization

The first quantity needed is the initial value of the SU${}_{\mbox{\tiny\rm{L}}}$(2) gauge coupling $g_{2}^{2}$ at the scale \bar{\mu}\sim m_{\mbox{\tiny\rm{Z}}}. It can be expressed in terms of the Fermi constant. Including the contribution of the new scalar degrees of freedom through a function $\Delta$ (cf. eq. (C.15)) we get

[TABLE]

where g_{0}^{2}\equiv 4\sqrt{2}G_{\mbox{\tiny\rm{F}}}m_{\mbox{\tiny\rm{W}}}^{2}, $G_{\mbox{\tiny\rm{F}}}=1.166379\times 10^{-5}$ GeV*-2* is the Fermi constant, h\equiv m_{\mbox{\scriptsizeh}}/m_{\mbox{\tiny\rm{W}}}, t\equiv m_{\mbox{\scriptsizet}}/m_{\mbox{\tiny\rm{W}}}, the masses m_{\mbox{\tiny\rm{W}}},m_{\mbox{\scriptsizet}},m_{\mbox{\scriptsizeh}},m_{\mbox{\tiny\rm{H}}},m_{\mbox{\tiny\rm{A}}} and m_{\mbox{\tiny\rm{H_{\pm}}}} are the physical (vacuum) masses, and we have defined

[TABLE]

Here the function $F$, defined in eq. (C.13), has the real part

[TABLE]

with the special limits

[TABLE]

C.3 Pole masses and scalar coupling renormalizations

The other couplings can be expressed in terms of pole masses. For this purpose we compute the full renormalized on-shell self-energies $\Pi(K;\bar{\mu})$ of the neutral Higgs fields $h$; of the $W$ boson; of the top quark; and of the new scalars $H,A$ and $H^{\pm}$. For Standard Model particles the expressions read (here $v_{0}$ is the tree-level vacuum expectation value which can within the 1-loop expressions be approximated as v_{0}^{2}\equiv\mu_{1}^{2}(\bar{\mu})/\lambda_{1}(\bar{\mu})\approx 4m_{\mbox{\tiny\rm{W}}}^{2}/g_{0}^{2}):

[TABLE]

For $W$ only the transverse parts play a role, and $D^{(T)}$ is defined by $D_{\mu\nu}\equiv D^{(T)}\delta_{\mu\nu}+{\mathcal{O}}(K_{\mu}K_{\nu})$. In the case of the top quark the self-energy was expressed as \Pi_{t}=i\,/\!\!\!\!{K}\,\Sigma_{\mbox{\tiny\rm{V}}}+i\,/\!\!\!\!{K}\,\!\gamma_{5}\Sigma_{\mbox{\tiny\rm{A}}}+m_{\mbox{\scriptsizet}}\,\Sigma_{\mbox{\tiny\rm{S}}}; bracketing this with on-shell spinors eliminates the function $\Sigma_{\mbox{\tiny\rm{A}}}(K;\bar{\mu})$.

For the on-shell self-energies of the new scalars we obtain (denoting $n_{3}=n_{4}=-n_{5}\equiv 1$)

[TABLE]

The on-shell self-energies correct tree-level masses, which can be expressed in terms of ${\overline{\mbox{\rm MS}}}$ parameters. For instance, the physical Higgs mass squared has the form m_{\mbox{\scriptsizeh}}^{2}=-2\mu_{1}^{2}(\bar{\mu})+\mathop{\mbox{Re}}\Pi_{h}(-im_{\mbox{\scriptsizeh}};\bar{\mu}). Here we defined the pole mass m_{\mbox{\scriptsizeh}} through the projection of the complex pole to the real axis. The pole mass equation can be inverted to give

[TABLE]

Similarly, the other parameters read (always implicitly assuming \bar{\mu}\approx m_{\mbox{\tiny\rm{Z}}})

[TABLE]

where $\delta g_{2}^{2}(\bar{\mu})\equiv g_{2}^{2}(\bar{\mu})-g_{0}^{2}$ is from eq. (C.14). Following conventions in the literature, \lambda_{2}(m_{\mbox{\tiny\rm{Z}}}) and \lambda_{\mbox{\tiny\rm{H}}}(m_{\mbox{\tiny\rm{Z}}})\equiv\lambda_{3}(m_{\mbox{\tiny\rm{Z}}})+\lambda_{4}(m_{\mbox{\tiny\rm{Z}}})+\lambda_{5}(m_{\mbox{\tiny\rm{Z}}}) are used directly as input parameters.

For approximate estimates, including only the large effects from $\lambda^{2}_{3}$, $(\lambda_{3}+\lambda_{4}\pm\lambda_{5})^{2}$ and $h_{t}^{4}$, eqs. (C.25) and (C.26) can be simplified into

[TABLE]

Apart from directly approximating eqs. (C.25) and (C.26), these expressions can also be derived from the “naive” procedure of minimizing the effective potential $V_{0}+V_{1}$, and tuning \mu_{1}^{2}(m_{\mbox{\tiny\rm{Z}}}) and \lambda_{1}(m_{\mbox{\tiny\rm{Z}}}) so that the location of the minimum is at v_{\mbox{\scriptsize min}}^{2}\simeq 4m_{\mbox{\tiny\rm{W}}}^{2}/g_{2}^{2} and the second derivative at the minimum is (V_{0}+V_{1})^{\prime\prime}(v_{\mbox{\scriptsize min}})\simeq m_{\mbox{\scriptsizeh}}^{2}. This naive procedure can easily be implemented numerically and then also applied to the 2-loop potential at zero temperature.

C.4 Practical procedure

A problem with the 1-loop expressions listed in appendix C.3 is that if the couplings $\lambda_{3},\lambda_{4}$ and $\lambda_{5}$ are first determined at tree level, and these values are subsequently inserted into the 1-loop corrections, as given in eqs. (C.25)–(C.31), then the corrections are in many cases of order 100%; for instance, \lambda_{1}(m_{\mbox{\tiny\rm{Z}}}) can be driven to a negative value. If corrections are of order 100%, there is no reason to trust the results. The problem can be somewhat “regulated” by solving eqs. (C.25)–(C.31) “self-consistently”, i.e. by requiring that the couplings have the same values on both sides of the equations. In general, this reduces the magnitude of the largest coupling $\lambda_{3}$, whereby the corrections remain below 100%. A further “resummation” can be implemented by determining \mu_{1}^{2}(m_{\mbox{\tiny\rm{Z}}}) and \lambda_{1}(m_{\mbox{\tiny\rm{Z}}}) à la Coleman-Weinberg, as outlined below eq. (C.3). An advantage of this procedure is that 2-loop corrections can be partially included into \mu_{1}^{2}(m_{\mbox{\tiny\rm{Z}}}) and \lambda_{1}(m_{\mbox{\tiny\rm{Z}}}). The values listed in table 3 have been obtained by determining \mu_{1}^{2}(m_{\mbox{\tiny\rm{Z}}}) and \lambda_{1}(m_{\mbox{\tiny\rm{Z}}}) from the effective potential and the other parameters from the 1-loop pole mass relations in sec. C.3. However, we have also tested the procedure where 1-loop pole mass relations are used for all the couplings. This changes the values of \mu_{1}^{2}(m_{\mbox{\tiny\rm{Z}}}) and \lambda_{1}(m_{\mbox{\tiny\rm{Z}}}) given in table 3 by up to $\sim 20$% for BM2, and only a few % for BM1 and BM3, however our conclusions concerning thermal effects remain unchanged in all cases.

C.5 Counterterms and renormalization group equations

Finally, let us list the counterterms needed in our analysis. The notation for them was defined in sec. 4.2. We stress that the same counterterms appear both in the vacuum renormalization computations of the current section and in the thermal 2-loop effective potential given in appendix B. The results agree with ref. [14] for the counterterms that can be found there. The complete list reads

[TABLE]

The counterterm $\delta Z_{\chi}$, appearing in eq. (B.3), only contributes to thermal effects which are formally of higher order than the accuracy of the computation.

As usual, the counterterms fix the renormalization group equations as

[TABLE]

and similarly for the other couplings.