Hot electroweak matter near to the endpoint of the phase transition

a r X i v :h e p -l a t /9709020v 1 9 S e p 19971HUB–EP–97/59September 9,1997Hot electroweak matter near to the endpoint of the phase transitionM.G¨u rtler a ,E.-M.Ilgenfritz b ,A.Schiller a ∗,and C.Strecha aa Institut f¨u r Theoretische Physik,Universit¨a t Leipzig,Augustusplatz 10-11,D-04109Leipzig,Germany bInstitut f¨u r Physik,Humboldt-Universit¨a t zu Berlin,Invalidenstr.110,D-10115Berlin,GermanyThe electroweak phase transition is investigated near to its endpoint in the framework of an effective three–dimensional model.We measure the very weak interface tension with the tunneling correlation length method.First results for the mass spectrum and the corresponding wave functions in the symmetric phase are presented.1.The modelThe model under consideration is the 3d SU (2)–Higgs model with the lattice action S=βGp1−12tr(Φ+x U x,µΦx +µ)+ xρ2x +βR (ρ2x −1)2,(1)ρ2x =(1/2)tr(Φ+x Φx ).Our conventions and up-date details can be found in [1].Relations be-tween the parameters of this effective model and the physical quantities Higgs mass and tempera-ture are derived in [2].2.Interface tensionThe interface tension αcharacterises the strength of a first order phase transition.We apply the tunnelling correlation method [3]to determine its value for a Higgs mass of about 70GeV by studying the inverse correlation length m lat =m gap a (given by the correlation function of the Higgs operator ρ2x along z )in an elongatedlattice L 2×L z as function of the transverse area a 2L 2.The scaling of m gap at the critical point is predicted for a φ4-model as m gap aL ∝x exp (−x )with x =αa 2L 2/T c [3]by considering quadratic fluctuations around the kink solution in a semi-classical expansion.2Figure 2.Interface tension for several Higgsmasses;for references see[5,6].Fig.2collects published results for the inter-face tension(scaled by critical temperature)α/T3cof the SU(2)–Higgs model.In order to makethis useful we had to recalculate the3d resultsto4d conventions(adjusting to a renormalisedgauge coupling0.58characteristic for the4d lat-tice data).This comparison further supports thevalidity of dimensional reduction.3.Wavefunctions and excited statesKnowledge of the mass spectrum of screeningstates in the symmetric phase is important to de-scribe the high T phase by non-perturbative an-alytic models.The wave functions may charac-terise the change between the phases.First stud-ies have been reported recently[7].To study ground state and excited states(andpossibly wave functions)one has to use cross-correlations between operators O i from a com-plete set in a given J P C channel.In the transfermatrix formalism they describe the spectral de-composition(Ψ(n)i= vac|O i|Ψ(n) ,|Ψ(n) beingthe zero momentum energy eigenstates)C ij(t)=∞n=1Ψ(n)iΨ(n)∗je−m n t(3)where C(t)is the connected correlation matrix fordistance t in one of the spatial directions.On the lattice one can use only a truncated setof operators O i(i=1,...,N)which is assumedto allow the extraction of the lowest lying statesby considering the eigenvalue problem for C ij(t).Solving instead the generalised eigenvalue prob-lemjC ij(t)Ψ(n)j=λ(n)(t,t0) j C ij(t0)Ψ(n)j(4)errors related to this truncation can be minimised(t>t0,t0=0,1)[8].The wavefunction of state n in the given oper-ator basis is found(atfixed small distance t>t0)to beΨ(n)i(t)= vac|O i e−Ht|Ψ(n) .(5)The components characterise the contribution ofthe original lattice operators to the(ground orexcited)state in the J P C ing opera-tors of different transverse extension(eventuallyonly a subset of a used operator set withfixedquantum numbers)one can derive their relativecontribution to an optimised wavefunction.Themasses of these states are obtainedfitting the di-agonal elementsµ(n)(t)µ(n)(t)= ijΨ(n)∗i C ij(t)Ψ(n)j(6)to cosh form.Our base in the Higgs-channel(0++)is built bythe operatorsρ2x,S x,µ(l)as well as quadratic Wil-son loops of size l×l.In the W-channel(1−−)we use the operators V b x,µ(l)and in the(2++)-channel S x,µ(l)−S x,ν(l).Here the following no-tation is used(l counts the string length in latticeunits)S x,µ(l)=12tr(τbΦ+x U x,µ...U x+(l−1)µ,µΦx+lµ).As an example we show the squared wave-function for ground state andfirst excitation inthe1−−-channel in the high temperature phasenear to the transition as function of the operator3length l for a 503lattice.The length is mappedto universal g 23units in order to accommodate three βG values atthe line of constant physicsm 23/g 43≈0.002(Figs.3,4)at M ∗H =70GeV.Figure 3.Squared wavefunction |Ψ(1)l |2for 1−−groundstate n =1vs.physical lengthFigure 4.Same as Fig.3for first excited state n =2A good scaling behaviour is observed.The last point (l =25)seems to accumulate contributions from longer operators which do not fit to the lat-tice.First results on the spectrum using these wave-functions are presented in Fig.5showing to whatFigure 5.Masses of ground state and first exci-tation in physical units for different channels asfunction of 1/βG ∝a .extent different masses are already approach-ing the continuum limit.Aspects of different smearing procedures,mixing properties of excited states and results on different Higgs masses will be discussed in a future publication [9].REFERENCES1.M.G¨u rtler, E.M.Ilgenfritz,J.Kripfganz,H.Perlt,and A.Schiller,Nucl.Phys.B483(1997)3832.K.Kajantie,ine,K.Rummukainen,andM.Shaposhnikov,Nucl.Phys.B458(1996)903.G.M¨u nster,Nucl.Phys.B340(1990)5594.K.Kajantie,ine,K.Rummukainen,andM.Shaposhnikov,Nucl.Phys.B466(1996)1895.M.G¨u rtler,E.M.Ilgenfritz,and A.Schiller,hep-lat/9702020,to appear in Z.Phys.C 6.J.Heitger,these proceedings7.O.Philipsen,M.Teper,and H.Wittig,Nucl.Phys.B469(1996)4458.M.L¨u scher and U.Wolff,Nucl.Phys.B339(1990)2229.M.G¨u rtler,E.M.Ilgenfritz,A.Schiller,andC.Strecha,in preparation。