

 14 Mar 1995

Perturbative study of the electroweak phase transition

Zolt'an Fodor

\Lambda

Deutsches Elektronen-Synchrotron, DESY, 22603 Hamburg, Germany

Abstract The electroweak phase transition is studied at finite temperature. The effective action is given to higher orders, including wave function correction factors and the full g4; *2 effective potential. An upper bound for the Higgs mass mH ss 70 GeV is concluded for the reliability of the perturbative approach. A gauge invariant treatment of the phase transition is presented.

1. Introduction At high temperatures (T ) the spontaneously broken electroweak symmetry is restored. Since the baryonnumber violating processes are unsuppressed at high T , there is a possibility to understand the observed baryon asymmetry within the standard model [1]. However, a departure from thermal equilibrium, a sufficiently strong first order phase transition via bubble nucleation is needed.

In Sect. 2 the finite T wave function corrections of the SU(2)-Higgs model to one-loop order [2] and the effective potential to order g4; *2 will be studied [3]. This gives a range of Higgs boson masses (mH ) for which the derivative expansion of the effective action is reliable. Sect. 3 contains the gauge-invariant treatment of the finite T electroweak effective potential [4]. This is of particular importance for comparison with lattice simulations [5], where the expectation value of \Phi \Phi y is well suited to characterize the broken phase, and the corresponding effective potential has been evaluated [6].

2. The effective action at finite temperature 2.1. The wave function correction term Consider the SU(2)-Higgs model at finite T , described by the lagrangian L = W a_*W a_*=4 + (D_\Phi )yD_\Phi +

\Lambda On leave from Institute for Theoretical Physics, E"otv"os University, Budapest, Hungary

V0('2), where V0('2) = m2'2=2 + _*'4=4, '2 = 2\Phi y\Phi . D_ and W a_* are the covariant derivative and the YangMills field strength, respectively. In this section Landau gauge is used and the effects of the three generations of fermions (mt = ftv=p2) have been included.

To get the effective action \Gamma fi[\Phi ] at finite temperature a systematic expansion is needed where in all propagators the tree-level masses are replaced by one-loop plasma masses to order g2 and *.

At one-loop order this improved perturbation theory yields the effective potential to order g3; *3=2,

Veff ('2; T ) = 12 ` 3g

2

16 +

*

2 +

1 4 f

2t ' (T 2 \Gamma T 2b )'2

+ *4 '4 \Gamma (3m3L + 6m3T + m3' + 3m3O/) T12ss ; (1) which is equivalent to the result of the ring summation [7]. Here m2L = 11g2T 2=6 + g2'2=4, m2T = g2'2=4, m2' = (3g2=16 + *=2 + f2t =4)(T 2 \Gamma T 2b ) + 3*'2, m2O/ = (3g2=16 + *=2 + f2t =4)(T 2 \Gamma T 2b ) + *'2 and T 2b = (16*v2)=(3g2 + 8* + 4f2t ).

The strength of the electroweak phase transition is rather sensitive to the nonperturbative magnetic mass of the gauge bosons. In Landau gauge the one-loop gap equations yield mT = g2T =(3ss) at ' = 0. In order to estimate its effect we will replace [8] the previous definition of mT by m2T = fl2g4T 2=(9ss2) + g2'2=4 and compute sensitive quantities for different values of fl.

Veff of (1) has degenerate local minima at ' = 0 and ' = 'c ? 0 at a critical temperature

2 Figure 1. The one-loop wave function correction ffiZ as a function of mH for different values of fl.

Tc. The evaluation of the transition rate requires knowledge of a stationary point of the free energy which interpolates between the two local minima. The effective action can be expanded in powers of derivatives, and for time-independent fields one has T \Delta \Gamma fi[\Phi ] =R

d3x[Veff ('2; T ) + (ffiIJ + ZIJ (\Phi ; T ))~r'I ~r'J =2 + : : :]. Using the inverse scalar propagator in the homogeneous scalar background field \Phi one obtains on the oneloop level ZIJ (\Phi ; T ) = Z'('2; T )P 'IJ + ZO/('2; T )P O/IJ, where Z' = T [* _m2(3=m3' + 1=m3O/)=4 \Gamma 2g2=(mO/ + mT ) + g2m2(1=m3L + 10=m3T )=16]=(4ss) and ZO/ = T [2* _m2=(m' + mO/)3 \Gamma 2g2=(mO/ + mT ) \Gamma g2=(m' + mT )]=(6ss) with P 'IJ = 'I'J ='2, P O/IJ = ffiIJ \Gamma 'I 'J ='2,

'2 = P

4

I=1 'I 'I, _m

2 = *'2 and m2 = g2'2=4.Error: /rangecheck in --repeat--
Operand stack:
-1 --nostringval--
Execution stack:
%interp_exit .runexec2 --nostringval-- --nostringval-- --nostringval-- 2 %stopped_push --nostringval-- --nostringval-- --nostringval-- false 1 %stopped_push 2 3 %oparray_pop 2 3 %oparray_pop 2 3 %oparray_pop 2 3 %oparray_pop .runexec2 --nostringval-- --nostringval-- --nostringval-- 2 %stopped_push --nostringval-- --nostringval-- --nostringval-- --nostringval--
Dictionary stack:
--dict:1100/1123(ro)(G)-- --dict:0/20(G)-- --dict:74/200(L)-- --dict:120/250(L)-- --dict:37/200(L)--
Current allocation mode is local

