Massive complex scalar field in the Kerr-Sen geometry Exact solution of wave equation and H

a rXiv:h ep-ph/968354v117Aug1996HU-TFT-96-34hep-ph/9608354Cosmological abundances of right-handed neutrinos Kari Enqvist 1,Petteri Ker ¨a nen 2,Jukka Maalampi 3and Hannes Uibo 4Department of Physics,P.O.Box 9,FIN-00014University of Helsinki,Finland 17August 1996Abstract We study the equilibration of the right-helicity states of light Dirac neutrinos in the early universe by solving the momentum dependent Boltzmann equa-tions numerically.We show that the main effect is due to electroweak gauge boson poles,which enhance thermalization rates by some three orders of mag-nitude.The right-helicity states of tau neutrinos will be brought in equilibriumindependently of their initial distribution at a temperature above the poles if m ντ>∼10keV.1IntroductionPrimordial nucleosynthesis is a remarkable probe of neutrino properties[1].Althoughrecently the increasing accuracy of the cosmological data,such as observations relatedto the primordial abundances of helium,deuterium and the other light elements,hasemphasized systematic errors inherent to primordial nucleosynthesis analysis,thereremains a great potential for constraining neutrino physics via cosmological observa-tions.To some extent primordial nucleosynthesis could be sensitive even to the Diracvs.Majorana nature of neutrinos,because in the Dirac case the small relic abundanceof the inert right-handed component of Dirac neutrino would also contribute at nucle-osynthesis.Of course,presently one cannot hope to differentiate between the Diracand Majorana nature of neutrinos on cosmological grounds,but in principle this isan interesting problem.In practise,because of the smallness of the electron neutrinomass,from this point of view only the right-handed components ofνµandντcan haveinteresting relic abundances.Naively,the cosmological density of light right-handed neutrinosνR(or rather theright-helicity states of light neutrinos,ν+)is expected to be very small.IfνR’s ever werein equilibrium,they decoupled very early because of their low capability of interacting.Assuming this took place well above the electroweak phase transition temperatureT EW,at the onset of primordial nucleosynthesis the contribution of the right-handedcomponent of a Dirac neutrino is given byρR≃[g∗(1MeV)/g∗(T EW)]4/3ρL≃0.044ρL, where g∗(T)is the effective number of degrees of freedom in thermal equilibrium inthe temperature T andρL is the equilibrium energy density of left-handed neutrino.This assumes that at high temperature we may consider the gas of quarks,leptons,and gauge bosons nearly ideal,which may not be true.A recent lattice simulation[2]of the QCD energy density above the critical temperature has revealed that the actualenergy density is some15%smaller than expected,which might signify the existenceof a condensate at high T.One also assumes that there is no significant entropyproduction either at the electroweak or QCD phase ttice simulationsseem to indicate that this is true for QCD,and the latent heat in the electroweakphase transition is also known to be small[3].The right-helicity states of Dirac neutrinos are not completely inert in the StandardModel[5]but can be produced(and destroyed)in spin-flip transitions induced bythe Dirac mass[4,6]or the neutrino magnetic moment.If the neutrino mass islarge enough,ν+would be produced in collisions below the QCD phase transitiontemperature,resulting effectively in an additional neutrino species at nucleosynthesis.An important source ofν+’s are also the non-equilibrium neutrino scatterings and<∼130keV decays of pions,as was pointed out in[6].This gives rise to the bound mνµ<∼150keV[8],using T QCD=100MeV and assuming that nucleosynthesis and mντallows less than0.3extra neutrino families.This is very restrictive bound for tauneutrinos since,basing on primordial nucleosynthesis arguments,there seems to beno window of opportunity for a sufficiently stable(τν>∼102sec)tau neutrino in the MeV region because of the production of non-equilibrium electron neutrinos inντ¯ντannihilations[9].The actual cosmological density of the right-helicity neutrinos depends not onlyon the production rate near the QCD phase transition,but also on whether the right-helicity neutrinos had a chance to equilibrate at some point during the course of theevolution of the universe.In this paper we wish to point out that at T<∼100GeV there is an enhancement of theν+production rate due to the electroweak gauge boson poles.1As a consequence there is a temperature range in which the production rate can exceed the expansion rate of the universe and the right-helicity neutrinos may be brought into thermal equilibrium with other light particles.This will take place if the neutrino mass is sufficiently large.We show that for the tau neutrino the mass limit is about10keV,and of the same order of magnitude for the muon neutrino.At the temperature T=1MeV the energy density of right-helicity tau neutrinos with a mass of10keV is found to be about6%of the energy density of an ordinary left-helicity neutrinos,i.e.their contribution to the effective number of neutrino species is ∆Nν≃0.06.The contribution of right-helicity neutrinos with a smaller mass depends, apart from the mass,on their initial energy density above the electroweak scale.The paper is organized as follows.In Section2we will list all tree-level spin-flipreactions where right-helicity neutrinos are produced and discuss their relative im-portance.We will also demonstrate the W-pole effect by considering the reactionsu1The question of the cosmic abundance of the right-handed component of Dirac neutrinos waspreviously studied in[10].There the pole effect was not considered.neutrino mass squared.Another source for the spin-flip is neutrino magnetic moment[7],which appears in the Standard Modelfirst at one-loop level and is,therefore,toosmall to be of any significance for our considerations.We shall consider light neutrinos with mass less than0.2MeV at temperaturesbetween1MeV and100GeV.Since each right-helicity neutrino introduces in thematrix element a factor mν/|pν|,we may safely neglect processes in which more than one right-helicity neutrino is involved.All relevant processes are listed(up to crossing)in Table1.There are68purely fermionic2→2processes in which a right-helicity muon ortau neutrino can be produced.In addition,a right-helicity tau neutrino can also beproduced in11lepton and quark three-body decays,and the muon neutrino in anotherset of11three-body decays.Since we are especially interested in interactions occuringaround the poles of weak gauge bosons,we have to consider also processes involvingW±,Z and H.There are16such processes.Finally,there are3two-body decaysof W±,Z and H bosons which are capable producing right-helicity muon and tauneutrinos.However,as we will show below,the processes involving gauge or Higgsbosons can be neglected in comparision with the purely fermionic processes.In what follows,for definiteness,we will consider only processes including right-helicity tau neutrinos.Since we are mainly interested on processes at temperaturesabove the muon and tau lepton masses,we expect that the results below are roughlyvalid also for muon neutrinos.2.2Production ratesLet us consider a2→2scattering a+b→ν++d where one of thefinal state particlesis a right-helicity neutrino.To estimate the relative importance of various processeswe approximate the thermally averaged production rate per oneν+by1Γ+=ΠL(ω,k=0),which we approximate by M2i(T)≃M2i+0.1T2(i=W,Z).(This modification is necessary in t-channel propagators only,since s-channel propagators are not infrared sensitive.)In this approximation external particles or interaction vertices do not receive thermal corrections.Thus the dispersion relations in the thermal distributions in Eq.(1)remain unchanged.Admittedly,this is a simplistic approach, but for our purposes,and for the desired accuracies,this should be sufficient.In fact, in the region of interest the effects due to thermal masses turn out to be very small.In addition to thermal corrections,we must account for the imaginary parts of the gauge boson propagators,or the widths.This is particularly important for the s-channel.Thermal corrections will generate additional imaginary parts both in the s-channel and t-channel,but at T<∼100GeV they may safely be neglected.A technical detail worth pointing out is that for afixed helicity,the spin-flip matrix elements are not Lorentz-invariant since the direction of the spin picks out a preferred frame of reference,as was emphasized in[6].Indeed,a Lorentz boost changes the helicity of the particle,so that sometimes afixed-helicity reaction forbidden in the CM may actually take place in another frame.This means that it is not sufficient to compute matrix elements just in e.g.the CM-frame,but instead one should a use a general frame.The main purpose of this paper is to show that interactions at the weak boson pole may bring the right-helicity neutrinos to thermal equilibrium.To demonstrate this effect,let us consider the the t-channel reactionτ−u→ν+d and its crossed s-channel reaction u√M2)¯uνγµ(1−γ5)uτ¯u dγν(1−γ5)u u,(2)Wwhere q=pτ−pν,V ud is the appropriate CKM matrix element,and as in s-channel, the propagator gives rise to a term of the formM2WR W(q2)=M2W m2u(pν·p d)−m2d(pν·p u) (Kν·pτ)− (pτ·pν)−m2τ m2u(Kν·p d)−m2d(Kν·p u)−m2ν m2u(pτ·p d)−m2d(pτ·p u) ,T qq=16m ,E|p|.(6)In the ultra-relativistic limit Kλcan be approximated asKλ≃2pλfor h=−1,(7)Kλ≃m2production rateΓ+of the s-channel process is seen to exceed in a certain temperature interval the expansion rate of the universe,given by the Hubble parameterH≡˙R3M2P l1/2.(10)Hence,with this reaction alone a20keV right-helicity tau(and muon)neutrino wouldbe brought into thermal equilibrium while universe cools through this stage.Thecomplete analysis described in the next section,which is based on solving the Boltz-mann equation with all the relevant processes included,confirms this expectation.Thepole effects are indeed important for an estimate of the relic density of right-helicityneutrinos.In addition to fermionic2→2scattering processes,at high T a potential sourceforν+’s are2→2processes that involve the gauge or the Higgs bosons in thefinalor initial state(we dub such processes“bosonic”).Let us consider as an examplethe processτ−γ→ν+W−,where the photon couples either to the charged lepton(Compton scattering)or to the W-boson in a three-boson vertex.For comparision,the thermally averaged rate of this process is also displayed in Fig. 1.One can seethat it can be neglected even at temperatures around the pole,because there theproduction rate of the s-channel purely fermionic process is about three orders ofmagnitude higher.We have not considered all possible bosonic processes.It is however very plausiblethat generically among the2→2scattering processes only the purely fermionic pro-cesses are important,and among these,s-channel dominates over t-channel because ofthe pole in the s-channel.In what follows,we will always disregard bosonic processes.The importance of decays is less straightforward to discern.Numerical inspectionreveals that the two-body decays listed in the Table1can be neglected:their contribu-tion to the total rate is∼O(10−11)at T>∼10GeV,at lower temperatures their contri-bution vanishes exponentially.Three-body decays are more important.Wefind thatwhile the total contribution from three-body decays is negligible at higher momenta,|p+|/T>∼3,it is as large as few per cents for|p+|/T∼3at T<∼1GeV,increasing up to∼30%for very small momenta.For such small momenta alsoν+production by t→bτ+ν+,which has its maximal contribution(∼10%)around T∼30GeV,is important.3Evolution of the right-helicity neutrino density 3.1The Boltzmann equationOur goal is to estimate the relic abundance of the right-helicity neutrinosν+as a func-tion of time.Of particular interest is the contribution of the right-helicity neutrinosto the effective number of neutrinos,∆Nν,at the onset of primeval nucleosynthesis.Tofind∆Nνone has to determinefirst the evolution of the phase space distributionof the right-helicity neutrinos f+(|p+|,t)2.Then using a primordial nucleosynthesis code one computes the increase in the primordial4He abundance∆Y p resulting fromthe non-zero energy densities ofν+and240T n↔p,(11) where the subscript n↔p refers to the freeze-out of the reactions which transmute protons and neutrons into each other.The justification of this approximation comes from the fact that Y p is predominantly determined by the density of neutrons just after the n↔p freeze-out.Let us briefly describe the method we have applied for solving the evolution of theright-helicity neutrino density.The evolution of the distribution f+(|p+|,t)is governed by the Boltzmann equation∂∂|p+|f+= ∂f+∂R−|p+|∂H ∂f+R0,(16)where R0is arbitrary butfixed.Then Eq.(14)may be written asR ∂H ∂˜f+∂R coll=(C2→2+C1→3)(1−f+)−(C′2→2+C′1→3)f+,(19)where the coefficients C I represent production and C′I destruction ofν+,with I=2→2,1→3.The explicit expressions for the quantities C2→2and C1→3areC2→2(|p+|,R)= scatt12E+ dΠf dΠg dΠh(2π)4δ(4)(p f−p g−p+−p h)×S|M f↔g+h|2f FD f(1−f FD g)(1−f FD h),(20) where,as before,dΠi≡d3p i/((2π)32E i)and S is a symmetry factor taking into account identical particles in the initial and/orfinal states.Note that these expressions do not include the factors g i representing the number of spin degrees of freedom.According to our convention these factors are included already into the matrix elements squared since we sum over polarization states of all particles except the right-helicity neutrino under consideration.We have also used the well justified assumption that charged leptons,quarks and left-helicity neutrinos have thermal FD distributions.The expressions for C′I can be obtained from(20)by the replacementf FDi↔(1−f FD i).(21) Using(20)and(21)one easilyfinds that the primed and unprimed coefficients are related to each other byC′I=e E+/T C I,(22) where we have assumed that all the particles except the right-helicity neutrino are at a common temperature T and the chemical potentials of all the particles can be neglected.With Eq.(22)we are able to rewrite(19)in the more compact form,(23)∂f+f FD+=[exp(E+/T)+1]−1.where f FD+Let us introduce the total production rates per unit volume:Γ2→2(R)= scatt dΠa dΠb dΠ+dΠd(2π)4δ(4)(p a+p b−p+−p d)×S|M ab↔+d|2f FD a f FD b(1−f FD d),Γ1→3(R)= dec dΠf dΠg dΠ+dΠh(2π)4δ(4)(p f−p g−p+−p h)×S|M f↔g+h|2f FD f(1−f FD g)(1−f FD h).(24) Note that the expressions given above do not include the blocking factors(1−f+). Also,the dependence ofΓI’s on R arise through the distribution functions of ambientparing Eqs.(20)and(24)one easily sees thatΓI(R)= d3p+d(|p+|3/(6π2))ΓI(R).(26) From this expression we see that the quantities C I,which are of the dimension of mass,are the production rates ofν+’s per unit volume per unit interval of|p+|3/(6π2) around|p+|.Eq.(26)suggests also a general method for calculating C I’s.Onefirst generates by a Monte Carlo(MC)method a sample of unweighted events,together with their common weight,such that the total integral over the initial andfinal state phase space yieldsΓI.Having this sample,the evaluation of the derivative in(26)essentially reduces to building the histogram the number of unweighted events vs.|p+|3/(6π2). Although this method is very general(applicable both to scatterings and decays)and with a clear physical meaning,it yields a good accuracy only if one generates the initial sample of weighted events such that there are more weighted events in the regions of the phase space where the integrand inΓI is larger.This requires some knowledge of the behaviour of the integrand and an ability to build a MC event generator with the required distribution of events in the phase space.In the case of2→2scatterings there exists also another method of calculation of C2→2.This method follows directly from(20)and is based on the T-invariance of interactions allowing to rewrite C2→2in a form whereν+is an initial state particle. After this transformation the calculation of the corresponding integral proceeds in a usual way.In order to check our results we have determined C2→2with both methods.3.3ResultsThe total rates C(|p+|,R)=C2→2(|p+|,R)+C1→3(|p+|,R)for the right-helicity tau neutrino production for a number offixed momenta are shown in Fig.2.One can clearly see the effect of the s-channel pole,indicating that the rate is dominated by fermionic s-channel processes.Small momentum states pass through the pole at high temperatures,large at low temperatures.Another feature is the increase of the total rate for very small momentumν+’s at0.2GeV<∼T<∼2GeV.This enhancement is due to both scatterings(u s→ν+τ+)and the decays of the tau lepton.Since at these temperatures the thermally averaged center of mass energy of colliding particles is of the same order of magnitude as the mass of the tau lepton,the neutrinos(andτ’s) are preferentially produced in the small-momentum states,for which the probability of the creation of a right-helicity neutrino(spin-flip)is larger.For similar reasons thecontribution ofτ-decays to the total rate can be as large as∼30%.For right-helicity muon neutrinos one expects the analogous mechanisms to be effective at temperatures below100MeV.Because different momentum states have different interaction rates,this causes a distortion of the momentum distribution ofν+’s relative to the equilibrium distribu-tion.This effect is demonstrated in Fig.3.Here we have parametrised the distribution function ofν+forfixed R,f+(|p+|),introducing a momentum-dependent effective tem-perature T eff(|p+|)through f+(|p+|)=[exp(E+/T eff(|p+|))+1]−1.In Fig.3the ratioT eff/T for a tau neutrino of mass10keV is shown at T=30,3and0.3GeV for twoextreme initial conditions at a high temperature T=100GeV:(i)f+=f−=f FD+, plete equilibrium,and(ii)f+=0.One sees that only the lowest momentum right-helicity tau neutrinos of this mass will come into full equilibrium at the pole: T effdepends on the initial condition even at T=0.3GeV for|p+|/T>∼1.The rise of T effat the small-momentum end of the spectrum is because these states,interact-ing more strongly,are kept in good thermal contact with ambient matter even when higher momentum states are(already)decoupled.This kind of the distortion of the distribution function becomes more prominent as the mass of the neutrino increases. As mentioned before,the regime of free expansion reveals itself in the shape-preserving evolution of the distribution function.Another remarkable feature is the rise of T effat higher momenta clearly seen for the curve corresponding to T=30GeV and the zero initial condition f+=0.This rise is there despite the fact the total production rate C(|p+|,R)is a monotonically(at least for higher momenta)dereasing function of|p+|.The explanation can be found from Eq.(23):if f+/f FD+is much smaller than unity, the destruction ofν+’s can be neglected.The regime of well-out-of-equilibrium pro-duction ofν+is effectively maintained only while both the interaction rate C(|p+|,R)and the ratio f+/f FD+are sufficiently small.These conditions are more easily fulfilled by the higher momentum neutrinos at the beginning of their evolution from the zero initial condition.This out-of-equilibrium production results in the distribution func-tion decreasing with|p+|more slowly than the equilibrium distribution function,or, equivalently,an T effrising with|p+|.In contrast with the above discussed rise of T effat lower momenta,the later feature becomes more apparent as the mass of the neutrino decreses.Wefind that spectral distortion changes the energy desity ofν+’s (which were in equilibrium at temperatures above the weak interaction pole)at∼1 MeV typically by only a few percent.The major difference between the actual energy density and the naive estimateρR≃0.044ρL is due to the relatively late decoupling caused by the pole.Once the initial condition isfixed,we may follow the evolution of the energy density ofν+.In Fig.4we show the evolution of the right-helicity neutrino energy density formντ=1,6and20keV for two different initial conditions.If mντ=1keV,spin-flipinteractions are too weak to affect the evolution,and the resulting relic abundance at nucleosynthesis is,to a high accuracy,what one would naively expect.However,if =20keV,spin-flip interactions at the pole are strong enough to equilibrateν+ mντeven if initially f+=0.The right-helicity neutrino energy density at about nucleosynthesis time in units of left-helicity neutrino energy density is shown in Fig.5.We see that almost full>∼10keV. equilibration ofν+is obtained at some temperature below100GeV if mντWe have assumed here that there is no entropy production at QCD phase transition so that the only effect is the dilution of the right-helicity neutrino densities by the appropriate ratios of the effective degrees of freedom before and after the phase tran-sition.We have computed∆Nνassuming also that below T QCD all interactions can<∼30 be ignored.While this is a very good approximation for smaller masses(mντkeV),it has been found[6,8]that for higher masses out-of-equilibrium scatterings and the decayπ0→ν+References[1]For a recent review,see Subir Sarkar,Oxford preprint OUT-95-16P,hep-ph/9602260.[2]G.Boyd et al.,Phys.Rev.Lett.75(1995)4169.[3]K.Enqvist,J.Ignatius,K.Kajantie and K.Rummukainen,Phys.Rev.D45(1992)3415.[4]G.M.Fuller and R.A.Malaney Phys.Rev.D43(1991)3136;K.Enqvist and H.Uibo,Phys.Lett.B301(1993)376.[5]K.J.F.Gaemers,R.Gandhi and ttimer,Phys.Rev.D40(1989)309.[6]A.D.Dolgov,K.Kainulainen and I.Z.Rothstein,Phys.Rev.51(1995)4129.[7]K.Fujikawa and R.Schrock,Phys.Rev.Lett.45(1980)963.[8]B.D.Fields,K.Kainulainen and K.A.Olive,CERN-TH/95-335,UMN-TH-1417/95,hep-ph/9512321(1995).[9]A.D.Dolgov,S.Pastor and J.W.F.Valle,preprint FTUV/98-07,IFIC/96-08,hep-ph/9602233(1996).[10]S.L.Shapiro,S.A.Teukolsky and I.Wasserman,Phys.Rev.Lett.45(1980)689.[11]D.Buskulic et al.(ALEPH collaboration),Phys.Lett.B349(1995)585;K.As-samagan et al.,Phys.Lett.B335(1994)231.Table1:Processes,up to crossings,which produce a right-helicity tau neutrinoν+.fermionic2→23−body decaysντντ→ν+νττ−→ν+l−j W+→τ+ν+(l−jνe,µ−ντνj→ν+νj,(j=e,µ)τ−→ν+d n Z→ν+u m=d u,s c)ντH→ν+H H→ν+ντf ch→ν+f ch,(f ch=τ−)u mτ+ν+,u m=u)τ−H→ν+W−τ−γ→ν+W−Figure captionsFigure1.Thermally averaged production rates per one right-helicity tau neutrinoν+ of mass20keV for the s-channel process u¯d→ν+τ+,the t-channel processτ−u→ν+d and the bosonic processτ−γ→ν+W−.The Hubble expansion rate is also shown for comparision.Figure 2.The total production rates(dashed curves)C(|p+|,R)of right-helicity tau neutrinos as functions of the temperature T.From top to bottom,the rates are given for˜p+/100GeV=0.33,1,2,4,6,8and10.For comparison the evolution of the Hubble parameter H is also given(solid curve).Figure3.The ratio of the momentum-dependent effective temperature T effof the right-helicity tau neutrinos of mass10keV to the plasma temperature T at T= 30GeV,3GeV and0.3GeV.In each case the solid curve corresponds to the equilibrium initial condition f+=f FD,and the dashed curve to the zero initial condition f+=0+at T=100GeV.Figure4.The evolution ofρ+/ρFD+of tau neutrinos as a function of the temperature =20keV(solid curve),6keV(dot-dashed curve)and1keV(dashed curve). T for mντand f+=0at The evolution is given for two different initial conditions:f+=f FD+T=100GeV.Figure5.The energy density of right-helicity tau neutrinosν+in units of the effective number of two-component neutrino species,∆Nν,as a function of the neutrino mass.∆Nνis calculated at T=3MeV(solid curves),1MeV(dashed curves)and0.3MeV<20keV the two sets of curves differ by initial conditions (dot-dashed curves).For mντat T=100GeV.T (GeV)Figure 110-2310-2210-2110-2010-1910-1810-1710-1610-1510-1410-13R a t e (G e V )0110100T (GeV)Figure 210-3010-2510-2010-1510-10H , C (p ,R ) (G e V )+Hubble parameterproduction rates0.0 2.0 4.06.08.010.0p /T +0.60.81.0T /T eff3 GeV30 GeV0.3 GeVFigure 30110100T (GeV)Figure 40.00.20.40.60.81.0ρ /ρFD ++20 keV6 keV 1 keV1101001000m (keV)ντFigure 50.000.020.040.060.08N ∆νT=3 MeV T=1 MeV T=0.3 MeV。

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