---
abstract: 'Direct evaluation of the 1-loop fluctuation determinant of non-static degrees of freedom in a complete static background is advocated to be more efficient for the determination of the effective three-dimensional model of the electroweak phase transition than the one-by-one evaluation of Feynman diagrams. The relation of the couplings and fields of the effective model to those of the four-dimensional finite temperature system is determined in the general ’t Hooft gauge with full implementation of renormalisation effects. Only field renormalisation constants display dependence on the gauge fixing parameter. Characteristics of the electroweak transition are computed from the effective theory in Lorentz-gauge. The dependence of various physical observables on the three-dimensional gauge fixing parameter is investigated.'
author:
- |
    [A. Jakovác$^{1}$ and A. Patkós$^{2}$]{}\
    [Department of Atomic Physics]{}\
    [Eötvös University, Budapest, Hungary]{}\
title: |
    Finite Temperature Reduction\
    of the SU(2) Higgs-Model\
    with Complete Static Background\
---

A new wave of investigations of finite temperature gauge theories is driven by the challenge of the matter-antimatter asymmetry of the Universe. Anomalous baryon number violating processes thermally excited near the electroweak phase transition certainly have had impact on any [*a priori*]{} asymmetry. Additional non-equilibrium and CP-violating effects, occuring during the transition, might have contributed to the generation of the present day value of the symmetry.

Temperature introduces a natural mass-scale into the relevant field theory. It builds up a hierarchy among the fluctuations, which should be exploited in the evaluation of the partition function. Heavy modes with non-zero Matsubara index are important for the accurate determination of the couplings between the (almost) T-scale independent static modes, which drive the phase transition. This physical picture is the content of the dimensional reduction [@pisarski; @landsman] of finite temperature field theories. The validity of the assumed mass-hierarchy should be checked carefully after each reduction step.

A correctly reduced 3-d effective model offers important advantages from the point of view of the application of standard methods of statistical physics to the electroweak phase transition [@arnold1; @march; @gleiser]. Also lattice simulations are greatly facilitated if the full 4-d system is replaced by the coresponding 3-d effective model [@kajantie; @farakos1; @karsch], since the extreme weak coupling situation makes the simulation of the 4-d system a particularly involved task [@bunk; @montvay].

We emphasize, that for the success of the above strategies the most faithful possible mapping of the 4-d couplings on the temperature dependent 3-d ones is essential. For instance, in the renormalisation group flow of the\
3-d model [*dim 6*]{} operators might play important role. The determination of their weights in the Lagrangian of the effective model with help of the usual Feynman diagram technique requires calculations of increased complexity.

The first complete determination of the reduced model up to [*dim 4*]{} operators in the 1-loop approximation, including field renormalisation effects has been published very recently [@farakos2]. The authors evaluate all relevant Feynman diagrams with two and four, zero-momentum external field insertions. The computation has been performed in the Landau gauge using dimensional regularisation, followed by the application of the $\overline {{\rm MS}}$ renormalisation scheme.

In this note we present evidence that the evaluation of the functional fluctuation determinant in a complete static background ($A_i^a({\bf
x}), A_{4}^{a}({\bf x}), \Phi ({\bf x})$) offers a simpler and more compact calculational scheme. It allows the unified determination of all renormalisation constants of the 4-d theory, and in principle it is easily extendable also to the computation of the higher dimensional operators. (After the completion of our investigation we received a paper by Chapman [@chapman], where an analogous calculation has been performed for SU(N) pure gauge theories up to [*dim 6*]{} operators. The calculational technique, however, was fully different.)

Since the method of symbolic evaluation of the functional determinant with constant complete background is of equal difficulty for any member of a certain gauge class, without any extra complication one is able to study the dependence of the action of the effective theory on the gauge fixing parameter. Specifically, we shall perform the reduction with general ’t Hooft gauge fixing, applying 3-d momentum cut-off regularisation. The normalisation of the scalar potential piece of the effective action will be fixed by imposing Linde’s conditions [@linde].

We shall show, that the effective theory and the expressions of the 3-d couplings do not depend on the gauge fixing parameter. The 1-loop effective potential of the 3-d theory will be determined next in the general (three-dimensional) Lorentz gauge, and the dependence of the critical data ($T_c$, order parameter discontinuity, etc...) on the parameter of the 3-d gauge fixing be discussed. This point essentially follows [@arnold2], going beyond it in the implementation of the detailed relation between the couplings of the 3-d theory to the 4-d ones, and the numerical evaluation of the physical characteristics of the transition, not restricting the discussion to analytic perturbative considerations. 1truecm 1. The model under consideration is the SU(2) gauge+scalar theory at finite temperature $$S=\int_{0}^{\beta}d\tau \int d^{3}x\bigl [{1\over 4}F_{mn}^{a}F_{mn}^{a}+
{1\over 2}(D_{m}\Phi)^{\dagger}(D_{m}\Phi)+V(\Phi )\bigr ],$$ $$V(\Phi )= {1\over 2}m^{2}\Phi^{\dagger}\Phi\
+{\lambda\over 24}(\Phi^{\dagger}\Phi)^{2},$$ $$D_{m}\Phi =(\partial_{m}+igA_{m}^{a}\tau^{a}/2)\Phi,$$ m=1,...,4; a=1,2,3. (In eqs. (1-3) the renormalised parameters appear, the counterterms are not displayed explicitly, also Euclidean metrics is understood). The 1-loop integration over non-static modes will be peformed with full background, that is all fields are split into a non-zero static and a non-static part: $$\begin{aligned}
&
A_m=A_m({\bf x})+a_m({\bf x},\tau),\nonumber\\
&
\Phi=\left(\matrix{ 0 \cr \Phi_{0}({\bf x})\cr}\right)
          +\left(\matrix{\xi_{1}({\bf x},\tau )+i\xi_{2}({\bf x},\tau )\cr
           \xi_{3}({\bf x},\tau )+i\xi_{4}({\bf x},\tau )\cr}
\right).
\label{eq4}\end{aligned}$$

We shall demonstrate that the full renormalised reduced action can be recovered by choosing the static background [*constant*]{} (with the most general orientation in the isospace). Upon substituting the decomposition (4) into (1) one separates terms containing the non-static fields up to second power, for the 1-loop integration. The piece depending only on the constant background takes the form: $$U^{(0)}=\beta V\bigl [{1\over 4}g^2(A_i\times A_j)^2+{1\over 2}
g^2(A_i\times A_4)^2+{1\over 8}g^2(A_i^2+A_4^2)\Phi_0^{\dagger}
\Phi_0+V(\Phi_0)\bigr ]$$ (i=1,2,3). The part quadratic in the non-static components will not be displayed explicitly, since its expression is lengthy and not enlightening. The only important point for us is, that the fluctuations are characterised by a $16\times 16$ matrix, because the 12 gauge field components and 4 real Higgs scalar components are fully coupled in the most general constant background.

The gauge fixing conditions imposed on the fluctuations $a_{m},
\xi_{\alpha} $ are $$\begin{aligned}
&
F^1 =(D_{\mu}(A)a_{\mu})^{1}-\alpha{g\Phi_0\over 2}\xi_2,\nonumber \\ &
F^2 =(D_{\mu}(A)a_{\mu})^2-\alpha{g\Phi_0\over 2}\xi_1,\nonumber \\ &
F^3 =(D_{\mu}(A)a_{\mu})^3+\alpha{g\Phi_0\over 2}\xi_4,\end{aligned}$$ ($D_{\mu}(A)$ is the covariant derivative in the background field $A$, $\alpha$ is the gauge fixing parameter). The corresponding Faddeev-Popov determinant is $$\det \{ [K^2 +g^2(A_i^2+A_4^2)+{\alpha \over 4}g^2\Phi_0^2 ]\delta^{a,b}-
2ig\epsilon^{abc}K_mA_m^c-g^2A_m^aA_m^b)\},$$ where $K^2 ={\bf k}^2+\omega_n^2$.

Since the distinguishing difference of the proposed method relative to the conventional Feynman diagrams consists of the explicit evaluation of the fluctuation determinant in constant background, we are going to elaborate on certain technical details of this computation.

The general structure of the 1-loop non-static contribution to the reduced action looks very simple $$\begin{aligned}
&
U^{(1)}[A_i ,A_4 ,\Phi_0 ]=
T\sum_{n\neq 0}\int{d^3k\over (2\pi)^3}\bigl\{{1\over 2}\ln [K^{32}+
\alpha^2 a_2 K^{30}+\alpha^3a_4K^{28}+...]\nonumber \\ &
{}~~~~~~~~~~~~~~~~~~~~~-\ln [K^6+...]\bigr\}\end{aligned}$$ (The second logarithm stands for the contribution of the Faddeev-Popov determinant). The coefficients $a_2 ,a_4$ depend on the background fields qudratically ($\Phi_0^2, (\hat K_m A_m)^2, A_m^2)$ and quartically $(\Phi_0^4 , (\hat K_m A_m)^2\Phi_0^2,
(\hat K_m A_m)^4,...)$, respectively. Here $\hat K_m$ is the Euclidean unit vector pointing in the direction of $K_m$. If one restricts the calculation to finding the coefficients in the effective action of all operators up to [*dim 4*]{}, one expands the first logarithm in (8) up to terms $a_2^2, a_4$, and similar expansion is applied to the contribution of the Faddeev-Popov determinant. After throwing away (divergent) field independent terms, the coefficients we are interested in, turn out to be proportional to various infrared safe sum-integrals of the type $$T\sum_{n\neq 0}\int{d^3k\over (2\pi)^3}\bigl( {1\over K^2}, {{\bf
k}^2\over K^4},{\omega_n^2\over K^4}, {1\over K^4},...\bigr ).$$ In these integrals, where it is necessary, a three-dimensional ultraviolet cut-off has been introduced.

The complete evaluation of the fluctuation determinant with help of symbolic programming met considerable computer memory problems. Fortunately, in the basis, where the gauge field components are explicitly separated into longitudinal and transversal ones, the diagonal elements are ${\cal O}(K^2)$, while the off-diagonals are at most ${\cal O}(K)$. Therefore, for the first few leading K-powers in the argument of the first logarithm of (8) it is convenient to use a decompositon of the determinant of a certain $n\times n$ matrix $M$ into a sum of subsequent contributions containig products of decreasing number of diagonal elements:

$$\det M=\prod\limits_{i=1}^n M_{ii}\,\,\, + \sum\limits_{k_1>k_2}^n\!
{\rm det}_2 N(k)\!\!\!\prod\limits_{i\neq k_1,k_2}^n\!\! M_{ii} \,\,\,
+\! \sum\limits_{k_1>k_2>k_3}^n\!\!\!\!
{\rm det}_3 N(k)\!\!\!\prod\limits_{i\neq k_1\dots k_3}^n\!\!\!\! M_{ii}
\,\,\,+\dots,$$ where $$N(k)_{ij}=M_{k_i k_j}(1-\delta_{k_i k_j})$$ Therefore, for $a_2$ and $a_4$ it is sufficient to consider the set of $2\times 2$ up to $4 \times 4$ minors of the full $16\times 16$ matrix, corresponding to the first four terms of the above decomposition.

The resulting cut-off regularised expression is the central computational result of the present note: $$\begin{aligned}
&
U^{(1)}[A_i, A_4, \Phi_0]=\beta V\bigl\{{1\over 2}\Phi_0^2\bigl [
({3g^2\over 16}
+{\lambda\over 12})T^2-({9g^2\over 4}+\lambda)
{\Lambda T\over 2\pi^2}\nonumber \\ &
+{m^2\over 8\pi^2}({3\alpha\over 4}g^2+\lambda)(1-D_0)\bigr ]
+{1\over 96\pi^2}\Phi_0^4(1-D_0)({27g^4\over 16}+{3\alpha g^2\lambda\over 8}
+\lambda^2)+{17g^4\over 192\pi^2}A_4^4
\nonumber \\ &
+{1\over 2}A_4^2({5g^2\over 6}T^2+{m^2g^2\over 8\pi^2}-
{5g^2\Lambda T\over 2\pi^2})
+{1\over 64\pi^2}g^2A_4^2\Phi_0^2((9-
{5\alpha \over 4})g^2+\lambda-{9+3\alpha\over 4}g^2D_0)\nonumber \\ &
{1\over 64\pi^2}g^2A_i^2\Phi_0^2({17\alpha g^2\over 12}-
{\lambda\over 3}-{9+3\alpha\over 4}g^2D_0)+{1\over 12\pi^2}g^4
(A_i\times A_4)^2({163\over 24}-\alpha-{43\over 8}D_0)\nonumber \\ &
+{1\over 24\pi^2}g^4(A_i\times A_j)^2({509\over 120}+\alpha-{43\over 8}
D_0)\bigr\}
\label{eq9}\end{aligned}$$ with the logarithmically divergent quantity: $D_0=\ln{\Lambda\over T}-\ln 2\pi +\gamma_{Euler}.$

It is important to call the attention of the reader to the inconvenient fact that blindly following the calculational scheme outlined above, two additional terms would appear in (10): $$U^{(1)}_{fake}=-\beta V({17g^4\over 960\pi^2}(A_i^2)^2+
{m^2g^2\over 48\pi^2}A_i^2).$$ These terms clearly violate the invariance of the final reduced theory under spatial gauge transformations. In small periodic volumes this is the actual physical situation, since then $A_i$ is a physical degree of freedom on the same footing as $A_4$ (spacelike Polyakov loops are also observables). Then (11) should be added to (10) for fields fulfilling the restriction $A_i<< 2\pi /L$ (L is the linear spatial dimension of the system). These terms would be in complete formal analogy with the terms representing $A_4$ in (10).

However, the expansion of the logarithms and the limit $V\rightarrow \infty$ are not interchangeable: $$\lim_{V\rightarrow\infty}\sum_{n\neq 0}\int_k\ln (1+f(A_i))\neq
\sum_{m=1}^{\infty}\lim_{V\rightarrow\infty}\sum_{n\neq 0}\int_k {(-1)^m\over
m}f(A_i)^m.$$ This can be shown the cleanest way for “quasi-abelian” configurations $(A_i^a=A_i\delta_{a3}, A_4=\Phi =0)$, when the left hand side of (12) is a well-known periodic function of $A_i$ with period $2\pi/L$ [@weiss; @gross]. The difference between the two sides of (12) is exactly given by (11), therefore we are led to the prescription to subtract it from the complete result of the calculation. Only by following this careful consideration one arrives at (10). This expression is the starting point for the discussion of the renormalisation of the effective action.

The key observation is the renormalisation invariance of $gA_i$ in background gauges [@abbott]. Exploiting this circumstance one can find all field renormalisations from appropriately selected terms of the sum $U^{(0)}+U^{(1)}$. The coefficient of $g^2A_i^2\Phi_0^2/8$ determines $Z_{\Phi}$ , that of $g^2(A_i\times A_0)^2/2$ leads to $Z_{A_0}$, and finally $g^2(A_i\times A_j)^2/2$ to $Z_{A_i}=Z_{g}^{-1}$: $$\begin{aligned}
&
Z_{\Phi} = 1+{17\alpha \over 192\pi^2}g^2-{1\over 48\pi^2}\lambda
-{9+3\alpha\over 64\pi^2}g^2D_0,\nonumber \\ &
Z_{A_0}=1-{163g^2\over 288\pi^2}+{\alpha g^2\over 12\pi^2}+
{43g^2\over 96\pi^2}D_0,\nonumber \\ &
Z_{A_i}=1-{509g^2\over 1440\pi^2}-{\alpha g^2\over 12\pi^2}+{43g^2\over
96\pi^2}D_0.\end{aligned}$$ After renormalisation, these three terms are completed in view of the minimal coupling principle into the full kinetic terms of the corresponding fields, varying in space $$\begin{aligned}
&
L_{3-d}^{(kin)}={1\over 4}F_{ij}^{a}F_{ij}^a+{1\over 2}(D_i(A)A_4)^2+
{1\over 2}(D_i(A)\Phi)^{\dagger}(D_i(A)\Phi),\nonumber \\ &
D_i(A){ A_4}=\partial_i A_0+g(A_i\times A_4).\end{aligned}$$ The second step is the renormalisation of the Higgs potential. Taking into account the effect of $Z_{\Phi}$ the regularised expression goes over into $$\begin{aligned}
&
U_{Higgs}^{(reg)}={1\over 2}\Phi^{\dagger} \Phi\bigl\{({3g^2\over 16}+
{\lambda\over 12})T^2-{1\over 8\pi^2}({3g^2\over 4}+\lambda)
(\Lambda^2-4\Lambda T)\nonumber \\ &
{}~~~~~~~~~~+m^2\bigl [1+{\lambda\over 6\pi^2}-{\alpha g^2\over 12\pi^2}+
{D_0\over 8\pi^2}({9g^2\over 4}-\lambda )\bigr ]\bigr\}\nonumber \\ &
{}~~~~+{1\over 24}(\Phi^{\dagger}\Phi)^2\bigl [\lambda+({27\over 8}g^4+
{\lambda^2\over 4}-{4\alpha g^2\lambda\over 3}){1\over 8\pi^2}\nonumber \\ &
{}~~~~~~~~+{D_0\over 4\pi^2}({9g^2\lambda\over 4}-{27g^4\over 16}-
\lambda^2)\bigr ].\end{aligned}$$ It is reassuring, that the cut-off dependences of $m^2(\Lambda )$ and $\lambda (\Lambda )$ correctly reproduce the 1-loop $\beta$-functions of the SU(2) gauge+scalar theory (this is also true for $g^2(\Lambda )$ as can be seen from (13)) [@arnold1]. For this result it is essential to employ in the Higgs potential the correct renormalisation of the $\Phi$-field.

The renormalisation conditions we have applied to the temperature-independent part of the potential, were the Linde-type conditions, used also in our previous publication [@jako]: $${dU_{Higgs}(T-indep.)\over d\Phi_0}=0,~~~~~{d^2U_{Higgs}(T-indep.)
\over d\Phi_0^2}=m_H^2(T=0),
{}~~~~\Phi_0=v_0$$ ($v_0$ is the $T=0$ expectation value of the Higgs field). The details of the corresponding subtraction procedure were discussed in [@jako] for the thermal static gauge. Here we give the final result from the analysis done along the same lines, just for the ’t Hooft gauge: $$\begin{aligned}
&
U_{Higgs}^{(1-loop)}={1\over 2}\Phi^{\dagger}\Phi\bigl [m^2+({3g^2\over 16}
+{\lambda\over 12})T^2-{\Lambda T\over 2\pi^2}({9g^2\over 4}+\lambda )\bigr ]
+{\lambda\over 24}(\Phi^{\dagger}\Phi)^2\nonumber \\ &
{}~~~~~~~~-{1\over 64\pi^2}\sum_jn_j[m_j^4(\Phi )(\ln{m_j^2(v_0)\over T^2}+
{3\over 2})-2m_j^2(v_0)m_j^2(\Phi )],\end{aligned}$$ where the index j runs over all formal degrees of freedom: j=4-d transversal (T), Higgs (H), 4-d longitudinal (L), pseudo-Goldstone (G). The corresponding quantities appearing in (17) are: $$\begin{aligned}
&
n_T=9,~~~~~m_T^2={g^2\over 4}\Phi^{\dagger} \Phi,\nonumber \\ &
n_L=-3,~~~~~m_L^2={3g^2\over 4}\Phi^{\dagger}\Phi,\nonumber \\ &
n_H=1,~~~~~m_H^2=m^2+{\lambda\over 2}\Phi^{\dagger}\Phi,\nonumber \\ &
n_G=3,~~~~~m_G^2=-m^2+({3g^2\over 4}-{\lambda\over 6})\Phi^{\dagger}\Phi.\end{aligned}$$ One has to emphasize that these formal degrees of freedom are not the diagonal modes of the coupled fluctuation matrix, therefore the masses do not correspond to any actual thermal mass. Especially, $m_G^2>0$ for the range of the $\Phi$ values between 0 and $v_0$.

The renormalisation leads also to finite rescalings of the $A_4-\Phi$ interaction and of the $A_4$-potential, due to wave function renormalisations: $$\begin{aligned}
&
U^{(1-loop)}[A_4,\Phi]={17g^4\over 192\pi^2}(A_4^2)^2(1-{153g^2\over
180\pi^2}+{2\alpha g^2\over 3\pi^2})+{1\over 8}g^2A_4^2\Phi^{\dagger}\Phi
(1+{7g^2\over 10\pi^2}+{\lambda\over 6\pi^2})\nonumber \\ &
{}~~~~~~~+{1\over 2}A_4^2({5g^2\over 6}T^2+{m^2g^2\over 8\pi^2}-
{5g^2\over 2\pi^2}\Lambda T)(1-{153g^2\over 360\pi^2}+
{\alpha g^2\over 3\pi^2}).\end{aligned}$$ The renormalised Euclidean Lagrangian density is the sum of (14),(17) and (19). The linear divergences induced for the mass terms of $A_4$ and $\Phi$ are necessary for the mass renormalisations of the 3-d theory at 1-loop.

It is important to note, that the effective theory shows dependence on the gauge fixing parameter only in (19). Clearly, a 2-loop computation of the reduced potential will also give ${\cal O}(g^4)$ contributions, therefore the present expressions of the corrections in (19) cannot be considered final. Omitting these incomplete corrections, we summarize the effective theory and the relations of the 3-d and 4-d couplings in the formulae below. It is obvious that these relations do not involve the gauge fixing parameter. $$\begin{aligned}
&
L_{3-d}={1\over 4}F_{ij}^aF_{ij}^a+{1\over 2}(D_i(A)A_0)^2+{1\over 2}(D_i(A)
\Phi)^{\dagger}(D_i(A)\Phi)\nonumber \\ &
+{1\over 2}m^2(T)\Phi^{\dagger}\Phi+{\hat\lambda\over 24}(\Phi^{\dagger}\Phi)^2
+{1\over 2}m_D^2A_4^2+{17g^4\over 192\pi^2}(A_4^2)^2+{\rm 3-d~ct.-terms}\end{aligned}$$ where $$\begin{aligned}
&
m^2(T)=\hat m^2(T)+({3g^2\over 16}+{\lambda\over 12})T^2,\nonumber \\ &
\hat m^2(T)=m^2(1-{1\over 32\pi^2}[\lambda\ln{\lambda v_0^2\over 3T^2}+
(\lambda -{9g^2\over 2})\ln{3g^2v_0^2\over 4T^2}+
{27g^4\over 4\lambda}+5\lambda-{45g^2\over 4}]),\nonumber \\ &
\hat\lambda =\lambda -{3\over 8\pi^2}[{9g^4\over 16}\ln{g^2v_0^2\over 4T^2}
+{\lambda^2\over 4}\ln{\lambda v_0^2\over 3T^2}+({\lambda^2\over 12}-
{3g^2\lambda\over 4})\ln{3g^2v_0^2\over 4T^2}
+{3\over 2} ({9g^4\over 16}+{\lambda^2\over 3}-
{3g^2\lambda\over 4})],\nonumber \\ &
m_D^2={5\over 6}g^2T^2.\end{aligned}$$ 1truecm

2\. The most adequate physical test for the effective theory seems to be the analysis of the electroweak phase transition with help of the effective Higgs potential calculated from the 3-d theory in the general 3-d Lorentz-gauge. This has been discussed already by Arnold [@arnold2], therefore we can start from his formula for the effective potential adapted to the SU(2) case: $$\begin{aligned}
&
U_{eff}^{(1-loop)}(\Phi_0)={1\over 2}m^2(T)\Phi_0^2+{\hat\lambda\over 24}
\Phi_0^4-{T\over 12\pi}\bigl\{6({g^2\over 4}\Phi_0^2)^{3/2}+
3({5g^2\over 6}T^2+{1\over 4}g^2\Phi_0^2)^{3/2}\nonumber \\ &
+(m^2(T)+{\hat\lambda\over 2}
\Phi_0^2)^{3/2}
+3[{1\over 2}(m_{\chi}+|m_{\chi}|
(m_{\chi}^2-\alpha g^2\Phi_0^2)^{1/2})]^{3/2}\nonumber \\ &
+3[{1\over 2}(m_{\chi}^2-|m_{\chi}|
(m_{\chi}^2-\alpha g^2\Phi_0^2)^{1/2})]^{3/2}\bigr \}.\end{aligned}$$ In (22) the abbreviation $$m_{\chi}^{2}=m(T)^2+{\hat\lambda\over 6}\Phi_0^2$$ is introduced and $\alpha$ is now the gauge fixing parameter of the 3-d Lorentz gauge class.

It has been argued in [@arnold2] that perturbative expansion of (22) leads to a unique barrier temperature, independent of $\alpha$. We concentrate here on $T_c$ (the transition temperature) and some further physical characteristics of the transition, which will be determined numerically. Also we use the detailed relationship between 3-d and 4-d couplings, which were not taken into account in previous analyses.

$T_c$ has been determined for two characteristic values of the Higgs mass: 60 and 80 GeV. Also the order parameter discontinuity and the surface tension between coexisting phases have been evaluated (the latter in the thin wall approximation). The gauge dependence of these quantities has been tested by varying $\alpha$ in the interval (0,1). We have restricted this interval further by requiring $U_{eff}$ to be real in the interval of more direct physical interest, $\Phi\in(0,\Phi_{min}(T_c)).$ Clearly, for large enough $\alpha$ the last two terms of (22) become complex at fixed $\Phi_0$. It is also obvious that the limiting value of $\alpha$ found in this way will depend sensitively on the Higgs-mass $(\lambda)$ (c.f. (23)). For $m_H(T=0)=60$ GeV $\alpha \leq 0.3$, for $m_H(T=0)=80$ GeV $\alpha\leq 0.7$ was found to be the “upper bound” of its allowed range of variation.

In the Table we display the relevant physical quantities, which show remarkable stability, but definitely depend on $\alpha$. It is interesting to note that in the same quantities calculated from an effective potential determined in the 3-d analogue of ’t Hooft’s gauge, more important variation can observed, namely of the same order of magnitude as the difference found between 1-loop and 2-loop calculations performed in the 4-d, finite T theory [@fodor; @arnold3]. (For a criticism concerning the physical interpretation of the effective potential determined in the ’t Hooft gauge, see [@arnold2].) 1truecm 3. In conclusion, we have determined in a cut-off regularized calculation the relationship of the effective 3-d theory to those of the original 4-d system in the general ’t Hooft gauge, subject to the renormalisation conditions (16). It has been demonstrated that the evaluation of the determinant of non-static fluctuations in a complete constant background is sufficient for the full specification of the effective theory at 1-loop. The effective action proved to be independent of the gauge fixing parameter and invariant under spatial gauge transformations. The 1-loop analysis of the electroweak phase transition in the effective model has been shown to be rather insensitive to the actual choice of the gauge fixing parameter in a general Lorentz gauge.

The method presented here is of considerable calculational advantage over the direct enumeration and evaluation of Feynman diagrams at 1-loop level. Prospective further advantages will be explored in connection of 2-loop reduction of the standard model at high T, in the near future. 1truecm [**Acknowledgements**]{}

The authors are grateful to Z. Fodor, K. Kajantie, M. Laine and M. Shaposhnikov for important suggestions during the progress of this calculation. 1truecm [**Table Caption**]{}

Dependence of some physical observables of the electroweak phase transition on the parameter of the 3-d Lorentz gauge fixing $\alpha$ for two typical values of the Higgs mass ($m_H$). In subsequent columns the transition temperature ($T_c$), the order parameter discontinuity ($\Phi_c$), the mass of the magnetic gauge quanta $(m_W)$ at $T_c$ and the surface tension ($\sigma$) are shown. $v_0$ is the vacuum expectation value of the Higgs-field. 1truecm

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