Comparison of 4d and 3d Lattice Results for the Electroweak Phase Transition

TL;DR

This paper compares 4d and 3d lattice results for the electroweak phase transition, providing a non-perturbative check of dimensional reduction despite large uncertainties.

Contribution

It offers the first direct comparison between 4d and 3d lattice results for the electroweak phase transition, testing the validity of dimensional reduction.

Findings

Results are compatible within error margins.

Provides evidence supporting dimensional reduction.

Highlights limitations due to large error bars.

Abstract

We compare 4d lattice results for the finite temperature phase transition in the SU(2)+Higgs model with 3d lattice results for the phase transition in the corresponding dimensionally reduced effective theory. While the large errorbars and the lack of a relation of the 4d lattice gauge coupling to continuum physics prevent rigorous conclusions, the results are nevertheless compatible. This provides a direct non-perturbative check of dimensional reduction in the present context.

Tables

$g_R^2\left(m_W^{-1}\right)$	$m_H/$GeV	$T_c$/GeV	$L/T_c^4$	$\sigma /T_c^3$	$\tilde{v}/T_c$
0.585(10)	34(1)	73.0(14)	0.28(12)	0.053(5)	1.65(1)
		($L_t=\infty$)	($L_t=4$)	($L_t=2$)	($L_t=4$)

$g^2\left(m_W\right)$	$m_H/$GeV	$T_c$/GeV	$L/T_c^4$	$\sigma /T_c^3$	$v/T_c$
0.420	29.1(5)	76.8(5)	0.200(7)	[0.071(3)]	1.74(3)
0.420	54.4(5)	132.6(5)	0.0294(7)	0.0017(4)	0.626(8)
0.420	64.3(5)	151.2(5)	0.0194(12)	?	0.529(20)

$g^2\left(m_W\right)$	$m_H/$GeV	$T_c$/GeV	$L/T_c^4$	$\sigma /T_c^3$	$v/T_c$
0.570	34.0(5)	76.7(5)	0.19(1)	0.048(12)	1.43(4)
0.585	34.0(5)	75.7(5)	0.19(1)	0.050(12)	1.44(4)
0.600	34.0(5)	74.9(5)	0.20(1)	0.052(13)	1.45(4)

Equations

$$O_6=c\frac{{\varphi }^6}{T^2}.$$

$$m_3^2\sim -\frac{{\bar{m}}_H^2}{2}+c{\hat{g}}_3^2{T}_c^2=0,$$

$$\frac{dy}{dT}\sim\frac{d}{dT}\biggl{(}\frac{-\bar{m}_{H}^{2}/2+c\hat{g}_{3}^{2}T^{2}}{\hat{g}_{3}^{4}T^{2}}\biggr{)}\approx\frac{\bar{m}_{H}^{2}}{g_{3}^{4}T_{c}}\approx\frac{2c}{g_{3}^{2}},$$

Full text

HD-THEP-96-08

                                                                                                                hep-lat/9604011

                                                                                                                April 12, 1996

1. Due to its effect on the baryon number of the Universe, the electroweak phase transition in the Standard Model and beyond is of considerable interest (for a review, see [1]). The first question to be asked is whether the phase transition is strongly enough of first order for cosmological consequences. It turns out that even this simple question is difficult to answer reliably, due to the infrared (IR) problem in finite temperature field theory. This has led to several lattice investigations of the problem [2–12].

There are two different lattice approaches available. First, there are the finite temperature 4d simulations of the bosonic SU(2)+Higgs theory [2–6]. Second, there are 3d simulations [7–12] of an effective theory obtained with dimensional reduction [13, 14] from the original theory. Recently, the extrapolation to the continuum limit has been investigated both in 4d [6] and in 3d [8]. Hence, a comparison of the results should become possible, allowing to check non-perturbatively the validity of the perturbative dimensional reduction used in the derivation of the effective 3d theory. We will make the comparison in a part of the parameter space where corrections to the 3d theory are largest and where the transition is strong enough to be accessible also to the 4d simulations. However, the corrections are still very small and hence the 4d and 3d results turn out to agree within errorbars.

2. Let us start by briefly reviewing the main features of the two approaches. In principle, the more straightforward way of solving the problem is the 4d approach, in which the original theory is put as such on the lattice. The parameters of the lattice action are fixed from masses and from a suitably defined coupling constant measured at zero temperature. Then the lattice shape is changed so as to go finite temperature.

Unfortunately, there are a few problems with the 4d approach. First, one cannot put the chiral fermions appearing e.g. in the Standard Model on lattice, and hence one can only study the bosonic sector of the theory. This is unacceptable, since especially the heavy top quark is numerically important for the electroweak phase transition. Second, it is not known how the renormalized lattice gauge coupling $g_{R}^{2}$ of the 4d simulations, defined in terms of a static potential measured by Wilson loops, is related to continuum physics. Hence one does not know e.g. the $\overline{\mbox{\rm MS}}$ gauge coupling $g^{2}(\mu)$ at $\mu=m_{W}$ to which some given 4d simulation corresponds. Third, there are many different length scales in the problem — $(2\pi T)^{-1},m_{W}^{-1}(T)$ and $m_{H}^{-1}(T)$ — making the simulations technically demanding. In particular, for a large pole Higgs mass $m_{H}$ the inverse correlation length $m_{H}^{-1}(T)$ at the phase transition point becomes much larger than $(2\pi T)^{-1}$, necessitating very large lattices.

The 3d approach overcomes most of the direct problems of the 4d approach. Chiral fermions appear only in the perturbative dimensional reduction step and hence the final effective theory is purely bosonic. The 3d gauge coupling of the effective theory is in direct perturbative relation to the original zero temperature 4d gauge coupling in $\overline{\mbox{\rm MS}}$ or any other desired scheme (in the Standard Model, it is related directly to the muon lifetime). Finally, dimensional reduction removes the smallest length scale $(2\pi T)^{-1}$ from the problem, making the simulations technically less demanding. Hence one can easily go to larger Higgs masses. In practice, 4d simulations have concentrated on $m_{H}=34$ GeV, whereas in 3d the continuum limit has been taken up to $m_{H}\approx 70$ GeV.

On the other hand, the 3d approach of course relies on perturbation theory in the derivation of the effective theory. While the perturbation theory used is free of IR-problems and is hence expected to be as accurate as perturbation theory at zero temperature, it might nevertheless be useful to check its accuracy non-perturbatively. In particular, to obtain the simplest possible effective theory, one neglects higher-dimensional radiatively generated operators with small coefficients, for instance of the type

[TABLE]

In principle, there might also be non-local operators generated at high loop orders. The perturbative estimates made in [14] indicate that the effects of higher-order operators should be on the 1% level (for other estimates, see [13]).

The problems and benefits of the two approaches suggest that one should take a bosonic theory resembling the true original theory, and study the accuracy of dimensional reduction at a Higgs mass accessible to both ways of making simulations. Then one can use the 3d theory to include fermions and to go to higher Higgs masses.

To investigate the accuracy of dimensional reduction, we will in this paper compare the results of 4d and 3d simulations for the bosonic SU(2)+Higgs model at the zero temperature pole Higgs mass of about 34 GeV. Hence the U(1) subgroup and the fermions of the Standard Model are neglected. Since the Higgs mass is small, the transition is strong and the vacuum expectation value $v/T_{c}=\langle\phi\rangle/T_{c}$ of the Higgs field is large in the broken phase. Consequently, the comparison should be sensitive to higher dimensional operators of the type in eq. (1).

3. The 4d results relevant for the comparison are taken from [5, 6]. For the zero temperature parameters, an extrapolation to the continuum limit has there been taken. The relation of the Higgs mass to the W mass is $m_{H}/m_{W}=0.422(11)$, which for $m_{W}=80.22$ GeV gives $m_{H}=33.9(9)$ GeV (the number in parentheses is the error of the last shown digit). We shall assume $m_{H}=34(1)$ GeV. The renormalized gauge coupling defined in [4] has the value $g_{R}^{2}(m_{W}^{-1})=0.585(10)$. In principle, to get the corresponding continuum coupling one should calculate the static potential from which $g_{R}^{2}$ is extracted also in the $\overline{\mbox{\rm MS}}$ scheme. This kind of a calculation is not available; nevertheless, one might expect that $g_{R}^{2}(m_{W}^{-1})=0.585(10)$ roughly corresponds to the $\overline{\mbox{\rm MS}}$ running coupling $g^{2}(m_{W})\sim 0.585$, since loop corrections should be of the relative magnitude $g^{2}/(16\pi^{2})$. We shall in the following consider the possibilities that $g^{2}(m_{W})=0.570,0.585,0.600$.

For the properties of the phase transition, the extrapolation to the continuum limit is not quite perfect yet. The critical temperature has been extrapolated to be $T_{c}/m_{H}=2.147(40)$, but the fit is much affected by the point where the temporal extension of the lattice is $L_{t}=2$, relatively far from the continuum limit $L_{t}=\infty$. The latent heat has been measured to be $L/T_{c}^{4}=0.240(34)$ at $L_{t}=2$ and $L/T_{c}^{4}=0.28(12)$ at $L_{t}=4$. For the surface tension, values are given only for $L_{t}=2$; there two different methods give $\sigma/T_{c}^{3}=0.053(5)$ and $\sigma/T_{c}^{3}=0.065(10)$ [5]. For the order parameter $v/T_{c}$ the number 1.65(1) can be read from fig. 9 of [6], but there are large systematic errors since the relation of the lattice observable to continuum values is unclear (it should be noted that $v$ is defined as $v^{2}\propto\langle\phi^{\dagger}\phi\rangle$, leading to a gauge independent but scheme dependent quantity). The 4d results used for the comparison are summarized in Table 1.

4. The 3d results relevant for the comparison can be read from Table 10 in [8]. In particular, the second block gives the lattice values for the relevant dimensionless 3d observables, and the third block gives the corresponding 4d physical quantities. The results of the third block are reproduced in Table 2.

Two things should be noted from Table 2, relevant for the comparison with 4d lattice results. First, the pole Higgs masses studied do not include 34 GeV. For the sake of the comparison, one should hence make an interpolation to this Higgs mass. Second, the results shown correspond to the $\overline{\mbox{\rm MS}}$ gauge coupling $g^{2}(m_{W})=0.420$ (this value is close to that in the Standard Model). Since 4d simulations were made with a different gauge coupling, one also has to study how the results depend on $g^{2}$. Fortunately, the scaling with $g^{2}$ can easily be studied since the same 3d results describe simultaneously a large class of different 4d theories.

We shall first explain the precise way of doing the interpolation of $m_{H}$ and the scaling of $g^{2}$, which was used for deriving the numerical values below. Then we also explain an approximate procedure for getting a rough estimate of the numbers without having to resort to the precise form of the analytical formulae [14] of dimensional reduction.

5. The precise way of doing the interpolation of $m_{H}$ and change in $g^{2}$ is based on the dimensionless quantities $x,y_{c},\Delta\ell_{3},\ell_{3}^{b},\sigma_{3}$ of the 3d theory and on their relation to 4d physics as explained in Sec. 11 of [8]. It should be noted that the values measured for these quantities in 3d are independent of the value of $g^{2}$ and of the formulae for dimensional reduction.

First, we need an interpolation of $y_{c},\Delta\ell_{3},\ell_{3}^{b},\sigma_{3}$ to all $x$. Since there are only three lattice points and since they are rather far from each other, we use perturbation theory to get a better fit. For $y_{c}$ which is close to zero at the critical point, we subtract from the lattice value the 2-loop perturbative value and then fit a parabola to the difference. For the other quantities, we calculate the ratio of the lattice and perturbative values and fit a parabola to this. With this procedure, one should get a reasonable interpolation for the critical curve $y=y_{c}(x)$ and for the values of $\Delta\ell_{3},\ell_{3}^{b}$ along it.

For the surface tension, there is only data point ($m_{H}=54.4$ GeV; the tree-level Higgs mass is then $m_{H}^{*}=60$ GeV) in the continuum limit in Table 10 of [8], and this is pretty far from the point $m_{H}=34$ GeV. Hence, to get some kind of a comparison also for the surface tension, we include the value of $\sigma_{3}$ at $m_{H}=29.1$ GeV and an estimate of the surface tension at $m_{H}^{*}=70$ GeV from [10] into the fit, although these values are not extrapolations to the continuum limit. The point $m_{H}^{*}=70$ GeV does not affect the result practically at all.

Next, one needs the relations to 4d. We fix $m_{H}=34$ GeV and $g^{2}(m_{W})=0.570$, 0.585, 0.600. For each $g^{2}$, we vary $T$ and draw the corresponding curve $x(m_{H},T),y(m_{H},T)$ in the $(x,y)$-plane. The value of $T$ for which this curve crosses the critical curve $y=y_{c}(x)$ gives the critical temperature. Using the 3d gauge coupling $g_{3}^{2}(m_{H},T_{c})$, the derivatives $dx/dT,dy/dT$ at $T_{c}$, the slope of the critical curve $dy_{c}/dx$, and the values of $\Delta\ell_{3},\ell_{3}^{b}$ and $\sigma_{3}$ at $x_{c}$, one can then calculate $v/T_{c},\sigma/T_{c}^{3}$ and $L/T_{c}^{4}$ from eqs. (11.5), (11.6) and (11.7) of [8].

With the method explained, we get the results in Table 3. We have not added any errors from the interpolation to the values of $T_{c},L/T_{c}^{4}$ or $v/T_{c}$. The error of $T_{c}$ is mainly determined by the uncertainty in the Higgs mass, and might be slightly underestimated. For $\sigma/T_{c}^{3}$ we give just the percentual error of the $m_{H}=54.4$ GeV continuum limit; if there were simulations at the pole mass $m_{H}=34$ GeV (in which case the tree-level mass parameter is $m_{H}^{*}\approx 41$ GeV) the errors would be considerably smaller.

6. For a rough estimate of the numbers of Table 3, one can use a simplified procedure. First, just interpolate the values in Table 2 to $m_{H}=34$ GeV with the method explained above, giving $T_{c}=88.4(5)$ GeV, $L/T_{c}^{4}=0.118(5),\sigma/T_{c}^{3}=0.031(8),v/T_{c}=1.30(3)$. Then look from Sec. 11 of [8] how the values scale with $\hat{g}_{3}^{2}\equiv g_{3}^{2}/T_{c}\approx g^{2}(7T_{c})$. The critical temperature is determined approximately from

[TABLE]

where $m_{3}^{2}$ is the scalar mass parameter in the dimensionally reduced 3d theory, $c$ is a constant, and $\bar{m}_{H}^{2}$ is the running $\overline{\mbox{\rm MS}}$ Higgs mass at a scale of order $7T_{c}$. It follows from eq. (2) that $T_{c}\propto\hat{g}_{3}^{-1}$. From eqs. (11.5) and (11.6) one can see that $v/T_{c}$ roughly scales as $\hat{g}_{3}$ and $\sigma/T_{c}^{3}$ scales as $\hat{g}_{3}^{4}$. To get the scaling of $L/T_{c}^{4}$, one can neglect the latter term in eq. (11.7) since $dx/dT$ is non-zero only through logarithmic one-loop corrections to the dimensional reduction of the coupling constants. Using $y=m_{3}^{2}/g_{3}^{4}$ one then gets

[TABLE]

where the logarithmic running of $\hat{g}_{3}^{2}$ and $\bar{m}_{H}^{2}$ with temperature was neglected and eq. (2) was used. Using eq. (11.7), one finally gets $L/T_{c}^{4}\propto\hat{g}_{3}^{4}$.

The values of $\hat{g}_{3}^{2}$ are $\hat{g}_{3}^{2}=0.384,0.521$ for $g^{2}(m_{W})=0.420,0.585$, respectively. This gives $T_{c}=75.9(5)$ GeV, $L/T_{c}^{4}=0.22(1),\sigma/T_{c}^{3}=0.057(15),v/T_{c}=1.51(4)$ for $g^{2}(m_{W})=0.585$. These values are in rough agreement with the second row of Table 3.

7. Let us finally compare the 4d results in Table 1 and the 3d results in Table 3. For the critical temperature, the 3d values are a bit higher, but compatible within errorbars for $g^{2}(m_{W})=0.600$. This might be an indication of the correct value of $g^{2}(m_{W})$. A higher value of $g^{2}$ also brings the other variables closer to each other. One should also remember that there may be small errors e.g. in the interpolation. In addition, it should be reiterated that the extrapolation of the 4d critical temperature to the continuum limit might allow for improvement.

The values of latent heat are clearly compatible, but the errorbars in 4d are rather large. The values of surface tension agree, as well, but the errorbars are again large, and neither the 4d nor the 3d values represent the continuum limit at this $m_{H}$. In $v/T_{c}$ there is some discrepancy which is not surprising since the definitions of $v/T$ are different in 4d and 3d.

We conclude that at the present level of accuracy, the results agree. This gives non-perturbative evidence for the applicability of dimensional reduction in the present context.

Clearly, it would be most desirable to improve the quality of the comparison. Especially the extrapolation to the continuum limit should be made more precise. In the 4d case $T_{c}$ relies perhaps too much on $L_{t}=2$, and for $L/T_{c}^{4}$ and $\sigma/T_{c}^{3}$ a fit to $L_{t}=\infty$ is lacking altogether. In the 3d case the fit to the continuum limit for $\sigma/T_{c}^{3}$ at $m_{H}=54.4$ GeV ($m_{H}^{*}=60$ GeV) contains large errors since the transition is already rather weak there; for $m_{H}=29.1$ GeV ($m_{H}^{*}=35$ GeV) the continuum limit is much easier but was not studied in [8]. Of course, it would also be most important to get some direct evidence on the value of $g^{2}(m_{W})$ to which the 4d simulations correspond. A comparison of finite temperature correlation lengths, which seems not to be available at the moment, would be welcome. Finally, all the comparisons would be more straightforward and the errors related to interpolation removed if lattice simulations were made with the same pole Higgs masses. Once the errors have been reduced, the effects of higher-order operators can be reliably estimated.

Acknowledgements. This investigation was prompted and improved by discussions with K.Kajantie, A.Laser, M.G.Schmidt and M.Shaposhnikov. I also thank Z.Fodor and K.Rummukainen for useful discussions. The investigation was partially supported by the University of Helsinki.