Comment on astrophysical consequences of a neutrinophilic 2HDM
aa r X i v : . [ h e p - ph ] J u l MPP-2011-70
Comment on astrophysical consequences of a neutrinophilic 2HDM
Shun Zhou ∗ Max-Planck-Institut f¨ur Physik (Werner-Heisenberg-Institut), F¨ohringer Ring 6, 80805 M¨unchen, Germany (Dated: November 22, 2018)Several authors have pointed out that the scalar-mediated interaction of neutrinos in a neutrio-philic two-Higgs-doublet model ( ν y i < ∼ . × − by supernova neutrino observation, and further constrained to be y i < ∼ . × − by precision measurements of acoustic peaks of the cosmic microwave background. Based on theenergy-loss argument for supernova cores, we derive a slightly more restrictive bound y i < ∼ . × − .Therefore, the ν PACS numbers: 12.60.Fr, 14.60.Lm
The ν ϕ , which acquires asmall vacuum expectation value (vev) v ϕ = 0 . M ν = Y ν v ϕ with Y ν being neutrinoYukawa coupling matrix. A salient feature of the ν Y ν can be of order one even for sub-eV neutrinomasses [1–3]. In the flavor basis where the charged-leptonYukawa coupling matrix is diagonal, one can further take Y ν to be Hermitian by rotating right-handed neutrinofields in the flavor space. Hence the Yukawa interactionof neutrinos can be written as − L Y = τ X α,β = e ( Y ν ) αβ ν α ν β η = X i =1 y i ν i ν i η , (1)where V † Y ν V = Diag { y , y , y } with V being the neu-trino mixing matrix, which relates neutrino mass eigen-states ν i to flavor eigenstates ν α , and m i = y i v ϕ (for i = 1 , ,
3) are neutrino masses. Here η is a scalar bosonarising from the neutral component of ϕ , and its mass isnaturally around the vev of ϕ , i.e., m η ≈ v ϕ = 0 . y i ∼ O (1) have beendiscussed in Refs. [1–3], however, the restrictive boundson neutrino Yukawa couplings are unfortunately missed.In fact, stringent bounds on the neutrino-Majoron in-teraction have been obtained in the literature by assum-ing a pseudoscalar coupling iy i ν i γ ν i χ with χ being apseudoscalar boson [4, 5]. One can show that thosebounds apply as well to the scalar case in the relativis-tic limit, where small neutrino masses can be neglected.However, it should be noticed that the lepton-number-violating processes are forbidden in the ν ∗ Electronic address: [email protected] larger than the supernova distance, i.e., λ − ν e D < ∼ D = 51 . ν e + η → ¯ ν e + η , ¯ ν e + ν e → η + η and ¯ ν e + ν α → ν e + ¯ ν α for α = e, µ, τ . After removing the lepton-number-violatingcontributions from the Majoron model [4], one can obtaina restrictive bound on neutrino Yukawa couplings y i < ∼ . × − . (2)If η bosons decay rapidly into neutrinos and are absentin the cosmic background, the bound will be weaker buton the same order of magnitude. Given v ϕ = 0 . y i = m i /v ϕ , thus heavier neutrino mass eigen-states interact more strongly with the scalar boson. Inthe case of normal mass hierarchy, the bound in Eq. (2)may be slightly relaxed to y i < ∼ − , because electronantineutrino possesses a small fraction of the heaviestmass eigenstate[6]. For the inverted mass hierarchy ornearly degenerate neutrino mass spectrum, however, thebound in Eq. (2) is still applicable.As indicated by precision measurements of the acous-tic peaks of the cosmic microwave background, neutri-nos should be freely streaming around the time of pho-ton decoupling T γ, dec = 0 .
256 eV in order to avoidthe acoustic oscillations of the neutrino-scalar fluid [5].At this moment, the neutrino temperature is T ν, dec =(4 / / T γ, dec = 0 .
183 eV. For the relevant two-bodyscattering processes ν i + η → ν i + η , ν i + ¯ ν i → η + η and ν i + ¯ ν i → ν j + ¯ ν j via η -exchange, we can sim-ply estimate the scattering rate as Γ ν ≈ y i T ν, dec up tosome numerical factors. The free-streaming argumentrequires Γ ν to be smaller than the cosmic expansionrate H γ, dec = 100 km s − Mpc − (Ω M h ) / ( z dec + 1) / with Ω M h = 0 .
134 being the cosmic matter density and z dec = 1088 the redshift at photon decoupling. Henceone can derive a more restrictive bound y i < ∼ . × − [5]. Taking account of the existing bound on Yukawacouplings in Eq. (2), one should increase v ϕ by three or-ders of magnitude to guarantee sub-eV neutrino masses.Therefore, the mass of η is expected to be m η ≈ v ϕ =100 eV, which is much larger than neutrino temperatureat the time of photon decoupling. As a consequence, η bosons have already decayed into neutrinos, and therelevant process ν i + ¯ ν i → ν j + ¯ ν j is mediated by a vir-tual η boson [6]. The scattering rate is modified to beΓ ν ≈ y i T ν, dec /m η ≈ y i T ν, dec /v ϕ = y i T ν, dec /m i , thusthe true bound from the free-streaming argument is y i < ∼ . × − , (3)for m i ∼ . y i , the neglected numerical factors are indeedunimportant for the bound in Eq. (3).An important point is that η is massive enough to de-cay into neutrino-antineutrino pairs η → ν i + ¯ ν i . Thelifetime in its rest frame is τ η = (3 y i m η / π ) − ≈ . × − s, where y i = 10 − and m η ≈ v ϕ = 1 keV havebeen taken. Although η bosons can be copiously pro-duced in the supernova core, one may expect that theywill decay soon after production and thus cannot causeexcessive energy losses. However, since the temperaturein the cooling phase is sufficiently high T = 30 MeV, thelifetime of thermal η bosons should be lengthened by aLorentz factor E/m η ≈ . Consequently, the relativis-tic η bosons before decaying may have traveled a distance l η ≈ . × cm, which is larger than the core radius R = 10 km. But η bosons cannot freely propagate in theneutrino background, their mean free path can be esti-mated as λ η = ( y i T ) − ≈ . × cm that is comparableto the mean free path of neutrinos in the case of standardneutral-current interaction. Thus η bosons behave like anew species of neutrinos and accelerate the energy trans-fer, which leads to the reduction of the cooling time orthe duration of supernova neutrino burst. There are twopossibilities to avoid the contradiction with the neutrinoobservation of Supernova 1987A [7]: (i) to increase thecoupling ( y i > − ) such that λ η becomes much smallerand the energy transfer by η bosons is negligible; (ii) todecrease the coupling ( y i < − ) such that η bosonshave never been trapped and thermalized in the core.The first possibility is already excluded by the bound inEq. (3), while the second one is subject to the constraintfrom standard energy-loss arguments.The production of η bosons will be efficient via thebremsstrahlung process ν i + N → ν i + N + η becauseof the high nucleon density. The emission rate shouldbe proportional to the thermal average of νN scatteringrate σn ν n B where σ is the νN collision cross section, n ν and n B are respectively the neutrino and baryon number densities. The energy of emitted η bosons is of order T .Put all together, we can get the volume emission rate Q η ≈ y i G T n B ≈ . y i × erg cm − s − , (4)where σ = G E , T = 30 MeV and n B = ρ/m N with ρ = 3 . × g cm − have been assumed. Note thatall the neutrinos have been taken to be relativistic andnon-degenerate, which is an excellent approximation for ν µ and ν τ . For degenerate ν e , there will be a blockingfactor that suppresses the scattering rate, which has beenneglected in Eq. (4) for an order-of-magnitude estimate.The energy loss should be small so as not to shorten theneutrino burst, so we require the volume emission rate tobe Q η < ∼ . × erg cm − s − and then obtain y i < ∼ . × − . (5)As a matter of fact, there are additional contributions tothe production of η bosons, such as ν i + e − → ν i + e − + η and ν i + ¯ ν i → η + η . Hence the bound may be slightlystronger if all the contributions are included. One canthen verify that the mean free path λ η ∼ cm is muchlarger than the core radius, which is consistent with theprerequisite that η bosons can escape from the core andcarry away energies.Based on the above discussions, one may take y i =10 − and figure out the cross section of ν i + ¯ ν i → ν j + ¯ ν j via η -exchange, and that via Z -exchange. It is straight-forward to get σ η ∼ y i /E for the former case, while σ Z ∼ G E for the latter, where E is the neutrino en-ergy. In the neutrinosphere with a typical temperature T = 10 MeV, we further obtain σ η /σ Z ≈ y i /G E ≈ − for y i = 10 − and E = 3 T = 30 MeV. Therefore,the scalar-mediated neutrino interaction is too weak toequilibrate supernova neutrinos of different species in theneutrinosphere and thus cannot wash out the collectiveeffects in supernova neutrino oscillations [1].In conclusion, if the astrophysical bounds on the scalar-mediated neutrino interaction are taken into account,the motivation for the ν The author would like to thank Georg Raffelt for valu-able comments and suggestions, and Marc Sher for manyinteresting discussions. This work was supported by theAlexander von Humboldt Foundation. [1] M. Sher and C. Triola, Phys. Rev. D. , 117702 (2011).[2] F. Wang, W. Wang and J.M. Yang, Europhys. Lett. ,388 (2006).[3] S. Gabriel and S. Nandi, Phys. Lett. B , 141 (2007).[4] E.W. Kolb and M.S. Turner, Phys. Rev. D , 2895(1987). [5] S. Hannestad and G.G. Raffelt, Phys. Rev. D , 103514(2005).[6] M. Sher, private communication.[7] G.G. Raffelt, Phys. Rept.198