A computation of nucleosynthesis bounds on the masses of long-lived Dirac and
Majorana neutrinos is reviewed. In particular an explicit treatment of the
``differential heating'' of the $\nue$ and $\bnue$ ensembles due to the
residual out-of-equilibrium annihilations of decoupled heavy neutrinos is
included. The effect is found to be considerably weaker than recently reported
by Dolgov et al. For example, the bounds for a Dirac tau neutrino are $\mnt <
0.37$ MeV or $\mnt > 25$ MeV (for $\dNu > 1$), whereas the present laboratory
bound is $\mnt < 23.1$ MeV.Comment: 6 pages, 2 eps-figures. Talk at Neutrino 9

Kainulainen, Kimmo

English

arXiv

A computation of nucleosynthesis bounds on the masses of long-lived Dirac and Majorana neutrinos is reviewed. In particular an explicit treatment of the ``differential heating'' of the $\nue$ and $\bnue$ ensembles due to the residual out-of-equilibrium annihilations of decoupled heavy neutrinos is included. The effect is found to be considerably weaker than recently reported by Dolgov et al. For example, the bounds for a Dirac tau neutrino are $\mnt  25$ MeV (for $\dNu > 1$), whereas the present laboratory bound is $\mnt  25$ MeV (for $\dNu > 1$), whereas the present laboratory bound is $\mnt < 23.1$ MeV

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PRECISE NUCLEOSYNTHESIS LIMITS ON NEUTRINOMASSESKimmo KainulainenCERN, CH-1211, Geneva 23, SwitzerlandA computation of nucleosynthesis bounds on the masses of long-lived Dirac andMajorana neutrinos1 is reviewed. In particular an explicit treatment of the \dier-ential heating" of the e and e ensembles due to the residual out-of-equilibriumannihilations of decoupled heavy neutrinos is included. The eect is found to beconsiderably weaker than originally reported by Dolgov et al. 2. For example,the bounds for a Dirac tau neutrino are m < 0:37 MeV or m > 25 MeV (forN > 1), whereas the present laboratory bound is m < 23:1 MeV3.1 IntroductionNucleosynthesis considerations have proved to be an eective tool in ndinglimits on particle properties such as masses, couplings, lifetimes, neutrino mix-ing parameters and so on. The constraining power of nucleosynthesis sensi-tively depends on the constraints on primordial light element abundances thatmust be inferred from the observational evidence. This is the dicult partof NS considerations (see K.A. Olive, these proceedings). Nevertheless, as faras NS bounds on new physics are concerned, these details can be summarizedby a single parameter: the number of equivalent neutrino degrees of freedomN . The theoretical prediction for the helium abundance on the other hand,with or without new physics, is relatively straightforward and can usually bedone very accurately. Parametrizing the deviation of the prediction from thestandard result in units of N , one obtains a mapping of the nucleosynthesisbound on the space of parameters of the model. This is the situation in par-ticular for long-lived massive neutrinos, and this is what I mean with \precise’nucleosynthesis bounds.Massive annihilating particles aect the helium abundance indirectly byaltering the expansion rate, or by directly altering the rate of reactions holdingneutrons and protons in equilibrium:n+ e$ p + e− n + e+ $ p+ e; (1)during the time Tγ ’ 0:7 MeV when the n=p ratio is freezing out. Also, thechanges in the expansion rate alter the time available for the free neutron decayn! p + e+ e.1For example, additional mass density speeds up the expansion rate relativeto reactions (1), which causes the n=p ratio to freeze out earlier, leaving behindmore neutrons and hence eventually more helium than in the reference case of 3massless neutrinos. Secondly, since O(few) MeV neutrinos freeze out at ratherlow temperatures, their annihilations to e e nal states at temperatures belowthe chemical freeze-out temperature of e’s (Tchem ’ 2:3 MeV) can producesome exess in the e and e number densities (\bulk heating"), leading to anincrease of the overall rate of eqs. (1). As a result equilibrium is maintainedlonger, leading to less neutrons and eventually less helium being produced. Insection 2 I will review a computation that accurately accounts for these twoeects.Moreover, the residual annihilations of already decoupled heavy neutrinosbelow Tkin ’ 1 MeV, when e’s fall from kinetic equilibrium, can cause adeviation from equilibrium in the tail of the e and e spectra2 . Althoughsmall in amplitude, the eect of these neutrinos is boosted by the quadraticdependence on energy of reactions (1); the detailed balance argument relatingthe equilibrium conversion rates n! p and p! n does not apply, and simplybecause there are more protons than neutrons around, the presence of thisdistortion has the tendency of increasing the relative number of neutrons and,eventually, the nal helium abundance. I will compute the contribution toN from this \dierential heating" in section 3. All eects will be includedin the nal results, to be presented in section 4.2 Elements of the ComputationThe relevant momentum-dependent Bolzmann equations for the scalar phase-space distribution functions have the form:Ei(@t − pH@p)fi(p; t) = CE;i(p; t) + CI;i(p; t); (2)where Ei = (p2 + m2i )1=2 and H = (8=3M2Pl)1=2 is the Hubble expansionrate (with MPl the Planck mass and  the total energy density). The index iruns over all particle species in the plasma; each distribution function of eachspecies has an equation like (2), and all of them are coupled through the elasticand inelastic collision terms CE(p; t) and CI(p; t).Since the elastic scattering rates are much higher than the annihilationrates, neutrinos freeze out from the chemical equilibrium remaining in goodkinetic contact with the other particles in the plasma. Their spectra can thenbe described to a very good accuracy by the pseudo-chemical potentials zi(T ),20.00.11.010.0T(MeV)0.00.20.40.60.81.0n/n0nenτne,00110T(MeV)0.02.04.0ρ/ρ 0ρ-ρ+ρaveρeFigure 1: 1a) Electron and tau neutrino number densities with a Majorana  with m = 5MeV, normalized to equilibrium density n0  (3(3)=22)Tγ3. The dashed line for compari-son shows the unperturbed electron neutrino temperature. 1b) Energy densities of −, +and e with a Dirac  with m = 20 MeV, normalized to 0 = 72Tγ4=240. Also shownis the energy density corresponding to  in the helicity averaged approximation.i.e. (Ei pp2 +m2i )f(p; zi)  (eEi=T+zi + 1)−1; T 4 + 2he(Tγ)111=3Tγ ; (3)where he(T ) is related to the entropy stored in the electrons and positrons:se  22heT 3=45. With the introduction of pseudo-chemical potentials theBolzmann equations (2) can be integrated over the momenta. One then has aset of coupled ordinary dierential equations for the pseudo-chemical potentialsof each neutrino species and the photon temperature Tγ1.In g. 1a a particular solution of the equation network with a massmM = 5MeV is shown. The importance of tracking the e density is clearly visible:the number density of e’s remains close to the equilibrium value until T ’ 2:3MeV, after which it freezes from chemical equilibrium. Since annihilations of ’s are still occurring at T < 2 MeV, a slight heating of the electron neutrinodistributions results. In the units N, the corresponding eect is5 N ’−3:6ne. In this example ne ’ 0:1 so that one expects to get N  −0:4,in good agreement with the full nucleosynthesis computation1.In the Dirac case there is the additional complication of dierent helicitystates having dierent interaction strengths at high energies: positive helicitystates will freeze out earlier with larger relic density (and the negative helicitystates later with smaller) than the hypotetical Dirac neutrino with helicity-averaged interaction strength. One sees this eect in g. 1b, which shows theenergy density stored into dierent helicities. These deviations almost exactlycancel each other in the sum, however, so that the total number density, and3hence the induced eect on helium synthesis, is very close to the results foundusing the helicity-averaged approach. Figure 1b also shows how the relativeenergy density stored into tau neutrinos sharply increases at small T .3 Dierential HeatingQualitatively the physics was explained in the introduction; for more detailssee Dolgov et al. 2. Since one is considering a very small amplitude eect inthe electron neutrino distributions, one can to a good accuracy use the lin-earized Bolzmann equation for the deviation. Moreover, because the eectis dominated by large energies, it is an exellent approximation to replace theFermi{Dirac distributions by Bolzmann distributions. Then noting that thel.h.s. of eq. (2) annihilates the equilibrium distribution, and trading the vari-ables (t; p) for x  m=T and y  p=Tγ , one gets the generic equation forthe deviation:Hx@@xf(x; y) = Cann(x; y) −Cel(x; y)f(x; y); (4)where x  m =T . The variables x and x are of course simply related; usingx allowed succinctly including the eect of entropy release in eq. (4). Theannihilation term Cann(x; y) acts as a source and it is peaked around y ’ x.The term Cel(x; y) represents the restoring force of the elastic scatterings. Inmagnitude, Cel  Cann. Annihilation terms are easily computed, and I ndthe forms similar to the ones by Dolgov et al. 2. On the Dirac case, for exampleCDann ’ 9:0 10−3m3n(x)2 − neq(x)2x4py exp[−(x− y)2=y]FD(x; y); (5)where n’s are the number densities normalized as in g. 1 and FD(x; y) is afunction with the asymptotic value of 1 for y ! 0; x ! 1 2. In numericalwork I used exact functions, valid for any x and y, but this is not a signicanteect. The elastic scattering term is given by (neq(0)  1)Cel(x; y) ’ 0:75m5yx5TTγ4 1 + 0:23neq(x) + 0:54TγT4neq(me=Tγ)!:(6)The solution of (4) is straightforward:f(x; y) =Z x0dx0A(x0)Cann(x0; y) exp−Z xx0dx00A(x00)Cel(x00; y); (7)41 10y0.00.10.20.30.40.5y2f(z,y)Dy2f(z,y)My2δf(x,y)Dy2δf(x,y)M1.0 10.0m(MeV)-0.20.00.20.4δ∆N νDiracMajoranaFigure 2: a) The distributions y2f(x; y) for m = 10 MeV, and T = 1 MeV (x = 10) areshown with the corresponding bulk-equilibrium distributions. b) The eect on BBN of thedierential heating in units of equivalent  degrees of freedom.whereA(x) 1xH(x)TγT1 +Tγ3hIdhIdTγ; (8)and hI(Tγ) = 2 +he(Tγ). Equations (4) and (7) dier from those of Dolgov etal. 2 in that I included the dilution due to the entropy release in the electronannihilations, as well as the correct energy dependence in the expansion rateH, both of which tend to weaken the eect. Moreover, the elastic scatteringterm (6) is somewhat larger than that of Dolgov et al., which also weakens theeect. Fig. 2a displays a particular solution of equation (7) and nal resultsfor the eect of dierential heating are shown in gure 2b. The eect is largestfor the Dirac neutrino at around m ’ 10 MeV. However, for m = 25 MeVit is already below 0.25 eective neutrino degrees of freedom, i.e. a fourth ofthat found by Dolgov et al. These results are in very good agreement with thefull numerical solution of the Bolzmann equation by Hannestad and Madsen6in the Majorana case.4 ResultsThe mass bounds can be expressed in terms of t functions of N. Includingall the eects to the electron neutrino distributions discussed above, we ndthat the Majorana masses are bounded bymM <px(0:350 + 0:047px+ 0:591x) (0:15− x)+ (0:079 + 0:576x+ 0:598x2− 0:514x3 + 0:211x4) (x − 0:15)mM > 68:59− 63:83x+ 49:18x2− 33:09x3 + 13:18x4− 2:17x5; (9)5where the units are MeV and I used x  N . One sees that opening up awindow for a stable tau neutrino below the laboratory bound of 23.1 MeV 3would require relaxing the nucleosynthesis bound to N > 1:44. Then theNS bound of N < 1, together with the above laboratory bound rules out along-lived Majorana tau neutrino with mM > 0:95 MeV.In the Dirac case the upper bound on the disallowed region leads to theconstraintmD > 41:88− 28:47x+ 20:44x2− 13:60x3 + 5:49x4− 0:90x5; (10)where the units are again MeV. The eect of dierential heating was for N >1 to increase the bound from 22 MeV to m > 25 MeV, closing the windowbelow the experimental bound of m < 23:1 MeV at 95% CL.The computation of the lower bound on the disallowed region is entirelydierent from that in the preceding cases and completely unchanged by thedierential heating eects. I will only quote the results from Fields et al.: forthe bound N > 1 they give m< 0:31 MeV and m< 0:37 MeV, withTQCD = 100 MeV. These constraints are conservative in the sense that theywould be somewhat stricter if the QCD transition temperature was chosenhigher1.In conclusion, even with the weak constraint of N < 1, nucleosynthesisbound is strong enough to exclude a long-lived ( > 100 sec) Majorana tauneutrino with m > 0:95 MeV and a Dirac tau neutrino with m > 0:37MeV.AcknowledgementsI wish to thank Sacha Dolgov, Sten Hannestad and Jes Madsen for usefuldiscussions during the Neutrino 96 conference in Helsinki.References1. B.D. Fields, K. Kainulainen and K.A. Olive, hep-ph 9512321, CERN-TH/95-335, Astroparticle Physics (in press).2. A.D. Dolgov, S. Pastor and J.W.F. Valle, hep-ph/9602233.3. ALEPH collaboration, Phys. Lett. B 349, 585 (1995); for the update to23.1 MeV, see A. Gregorio, these proceedings.4. K. Enqvist, K. Kainulainen and V. Semikoz, Nucl. Phys. B 374, 392(1992).65. K. Enqvist, K. Kainulainen and M. Thomson, Nucl. Phys. B 373, 498(1992).6. S. Hannestad and J. Madsen, Phys. Rev. Lett. 76, 2848 (1996); erratumhep-ph/9606452.7

Precise Nucleosynthesis Limits on Neutrino Masses

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