Parametrizing the Neutrino sector of the seesaw extension in tau decays

Transcription

1 Parametrizing the Neutrino sector of the seesaw extension in tau decays,a, homas Gajdosik b, Andrius Juodagalis a and omas Sabonis a a Vilnius Uniersity, Institute of heoretical Physics and Astronomy b Vilnius Uniersity, Physics Faculty Darius.Jurciukonis@cern.ch, homas.gajdosik@cern.ch, Andrius.Juodagalis@cern.ch, omas.sabonis@tfai.u.lt he Standard Model includes neutrinos as massless particles, but neutrino oscillations showed, that neutrinos are not massless. A simple extension of adding gauge singlet fermions to the particle spectrum allows normal Yukawa mass terms for neutrinos. he smallness of the neutrino masses can be well understood within the seesaw mechanism. We analyse two cases of the minimal extension of the standard model when are added one or two right-handed fields to the three left-handed fields. In this model second Higgs doublet is included. We calculate the one-loop radiatie corrections to the mass parameters which produce mass terms for the neutral leptons. In both cases we numerically analyse light neutrino masses as functions of the heay neutrinos masses. Parameters of the model are aried to find light neutrino masses that are compatible with experimental data of solar m and atmospheric m atm neutrino oscillations for normal and inerted hierarchy. 36th International Conference on High Energy Physics 4- July 0 Melbourne, Australia Speaker. c Copyright owned by the authors under the terms of the Creatie Commons Attribution-NonCommercial-ShareAlike Licence.

2 . he model We discus the extension of the Standard Model SM with a second Higgs doublet Φ α,α =, with added right-handed neutrino fields to the three left handed neutrino fields. he Yukawa Lagrangian of the leptons is expressed by L Y = k= Φ k l R Γ k + Φ k ν R k D L + H.c.. in a ector and matrix notation, where Φ k = iτ Φ k. In expression. the l R, ν R, and D L = ν L l L are the ectors of the right-handed charged leptons, of the right-handed neutrino singlets, and of the left-handed lepton doublets, respectiely. he Yukawa coupling matrices Γ k are n L n L, while the k are n R n L. In this model, spontaneous symmetry breaking of the SM gauge group is achieed by the acuum expectation alues Φ k ak =. By a unitary rotation of the Higgs doublets, we 0 k / can achiee Φ 0 ak = / > 0 and Φ 0 ak = 0 with 46 GeV. he charged-lepton mass matrix M l and the Dirac neutrino mass matrix M D are M l = Γ and M D =,. respectiely with assumption that M l = diag m e,m µ,m τ. he mass terms for the neutrinos can be written in a compact form with an n L + n R n L + n R symmetric mass matrix 0 M M ν = D,.3 M D ˆM R where the hat indicates that ˆM R is a diagonal matrix. M ν can be diagonalized as U M ν U = ˆm = diagm,m,...,m nl +n R,.4 where the m i are real and non-negatie. In order to implement the seesaw mechanism [, ] we assume that the elements of M D are of order m D and those of M R are of order m R, with m D m R. hen, the neutrino masses m i with i =,,...,n L are of order m D /m R, while those with i = n L +,...,n L +n R are of order m R. It is useful to decompose the n L +n R n L +n R unitary matrix U as U = U L UR, where the submatrix UL is n L n L +n R and the submatrix U R is n R n L +n R [3, 4]. With these submatrices, the left- and right-handed neutrinos are written as linear superpositions of the n L + n R physical Majorana neutrino fields χ i : ν L = U L P L χ and ν R = U R P R χ, where P L and P R are the projectors of chirality. We can express the couplings of the model in terms of mass eigenfields, where three neutral particles are coupling to neutrinos. he interaction of the Z boson with the neutrinos is gien by L nc ν = g [ Z µ χγ µ P L U L 4c U L P R U L UL ] χ,.5 w where g is the SU gauge coupling constant and c w is the cosine of the Weinberg angle.

3 Full formalism for the scalar sector of the multi-higgs-doublet SM is gien in Ref. [3, 4]. he Yukawa couplings of the Higgs bosons Hb 0 to the neutrinos are gien by L ν Y H 0 = ] Hb [U 0 χ R bu L +UL b UR P L + U L b U R +UR bul P R χ,.6 b with b = k b k k, where b are two-dimensional complex unit ectors. he neutral Goldstone boson G 0 b Z is gien by the ector b Z with b Z = i,0. Once the one-loop corrections are taken into account the neutral fermion mass matrix is gien by M ν δml M = D + δm D M D + δm D ˆM R + δm R δml MD,.7 M D ˆM R where the matrix appearing at tree leel.3 is replaced by the contribution δm L. his correction a symmetric matrix, it dominates among all the sub-matrices of corrections. Neglecting the sub-dominant pieces in.7, one-loop corrections to the neutrino masses originate ia the selfenergy function Σ S L 0 = ΣSZ L 0 + Σ SG0 L 0 + Σ SH0 L 0, where the Σ SZ,G0,H 0 L 0 functions arise from the self-energy Feynman diagrams and are ealuated at zero external momentum squared. In the calculation of the self energies the neutrino couplings to the Z boson as well as the Higgs and Goldstone bosons are determined by eqs..5 and.6. Each diagram contains a diergent piece but when summing up the three contributions the result turns out to be finite. he final expression for one-loop corrections is gien by [5] δm L = b 3π b UR ˆm + 3g 64π mw MDU R ˆm ˆm m H 0 b ln where sum b runs oer all neutral physical Higgses H 0 b.. Case n R = ˆm m H 0 b ˆm ˆm ln m Z m Z U R b U R M D,.8 First we consider the minimal extension of the standard model adding only one right-handed field ν R to the three left-handed fields contained in ν L. We use the parametrization of = md a and = md a with a = and a =. Diagonalization of the symmetric mass matrix M ν.3 in block form is 0 U M ν U = U md a ˆM U = l 0.. m D a ˆM R 0 ˆM h he non zero masses in ˆM l and ˆM h are determined analytically by finding eigenalues of the hermitian matrix M ν M ν. hese eigenalues are the squares of the masses of the neutrinos ˆM l = diag0,0,m l and ˆM h = m h. Solutions m D = m hm l and m R = m h m l m h correspond to the seesaw mechanism. In our analysis we fix m H 0 = 5 GeV but m H 0 and m H 0 3 we generate randomly in the range to 000 GeV. 3

4 We can construct the diagonalization matrix U for the tree leel from two diagonal matrices of phases and three rotation matrices U tree = U φ φ i U α U 3 α U 34 βu i, where the angle β is determined by the masses m l and m h. he alues of φ i and α i can be chosen to coer ariations in M D. For calculation of radiatie corrections we use following set of orthogonal complex ectors: b Z = i,0, b =,0, b = 0,i and b 3 = 0,. Diagonalization of the mass matrix after calculation of one-loop corrections is performed with a unitary matrix U loop = U eg U ϕ ϕ,ϕ,ϕ 3, where U eg is an eigenmatrix of M ν M ν and U ϕ is a phase matrix. he second light neutrino obtains its mass from radiatie corrections. he third light neutrino remains massless. It is possible to estimate masses of the light neutrinos from experimental data of solar and atmospheric neutrino oscillations [6] assuming that the lightest m l3 = 0. Considering the normal ordering of the light neutrinos we receie m l = 5.0 ± 0. 0 GeV and m l = 8.7 ± GeV. Numerical analysis shows that we can reach those alues for a heay singlet with the mass bigger than 830 GeV, see Fig.. mli, GeV m h 830 GeV 0 m l m l m h, GeV Figure : Calculated masses of two light neutrinos as a function of the heay neutrino mass m h. he mass of the third light neutrino is zero, when n R =. Solid lines show the boundaries of allowed neutrino mass ranges when the model parameters are constrained by the experimental data on neutrino oscillations. he purple arrows indicates the alues of m l neutrino mass which don t satisfy allowed experimental neutrino mass ranges. Due to the scale, the band of the allowed m l and m l alues are close to each other accordingly their alues are shown separately in the right plots. 3. Case n R = If we add two singlet fields ν R to the three left-handed fields ν L, the radiatie corrections gie masses to all three light neutrinos. 4

5 Now we parametrize = md a~ md b~ and = md a~ md b~ with ~ a =, b~ =, ~ a = and b~ =. Diagonalizing the symmetric mass matrix Mν.3 in block form we write:! 03 3 md~a md~b M 0 l U =. 3. U Mν U = U md~a M R 0 M h ~ md b 0-0 mli, GeV ml ml 0- ml3 - ml, ml ml mh, GeV mh, GeV Figure : he masses mli of the light neutrinos as functions of the heaiest right-handed neutrino mass mh, for the case nr =. he alue of second heaiest right-handed neutrino mass is fixed at mh = 00 GeV. Plot in the left represent normal hierarchy than the plot in the right represent inerted hierarchy of the light neutrinos. he wide solid lines indicate the place of the most frequent alues of the scatter data. In inerted hierarchy case the nearly degenerate masses ml and ml are shown separately in the lower right plot. 5 he non zero masses in M l and M h are determined by the seesaw mechanism: mdi mhi mli and mri mhi, i =,. Here we use m > m > m3 ordering of masses. he third light neutrino is massless at tree leel. he diagonalization matrix for tree leel Utree = U α, α Ueg βi Uφ φi is composed of a M M U and a diagonal phase matrix, respectiely. rotation matrix, an eigenmatrix of U ν ν For calculation of radiatie corrections we use same set of orthogonal complex ectors bi as in first case. Diagonalization of the mass matrix including the one-loop correction is performed with a unitary matrix Uloop = UegUϕ ϕi, where Ueg is the eigenmatrix of Mν Mν and Uϕ is a phase matrix. In numerical calculations the model parameters as well as the deried masses of the light neutrinos are obtained in seeral steps. First, the diagonal mass matrix for tree leel is constructed. he lightest neutrino is massless, and the masses of other two light neutrinos are estimated from experimental data on solar and atmospheric neutrino oscillations. he masses of the heay neutrinos are input parameters. his diagonal matrix is used to constrain the parameters αi and φi that enter the tree-leel mass matrix Mν and its diagonalization matrix Utree. hen the diagonalization matrix is used to ealuate one-loop corrections to the mass matrix. Diagonalization of the corrected mass matrix yields masses for three light neutrinos. If the calculated mass difference is compatible with the experimental neutrino mass difference, the parameter set is kept. Otherwise, another set of the parameters is generated. Figure illustrate the obtained results. Both normal and inerted neutrino mass orderings are considered.

6 4. Conclusions For the case n R = we can match the differences of the calculated light neutrino masses to m and m atm with the mass of a heay singlet bigger than 830 GeV. Only normal ordering of neutrino masses is possible. In the case n R = we obtain three non anishing masses of light neutrinos for normal and inerted hierarchies. he numerical analysis shows that the alues of light neutrino masses especially of the lightest mass depend on the choice of the heay neutrinos masses. he radiatie corrections generate the lightest neutrino mass and hae a big impact on the second lightest neutrino mass. In future we plan to apply our parametrization to study the τ polarization coming from the decay of a W boson in the data of the CMS experiment at LHC and thus determine restrictions to the parameters of the neutrino sector. Acknowledgements. he authors thank Luis Laoura for aluable discussions and suggestions. his work was supported by European Union Structural Funds project "Postdoctoral Fellowship Implementation in Lithuania". References [] M. Gell-Mann, P. Ramond, and R. Slansky, in Supergraity, Proceedings of the Workshop, Stony Brook, New York, 979, edited by F. an Nieuwenhuizen and D. Freedman North Holland, Amsterdam, 979. [] J. Schechter and J. W. F. Valle, Phys. Re. D [3] W. Grimus and H. Neufeld, Nucl. Phys. B [4] W. Grimus and L. Laoura, Soft lepton-flaor iolation in a multi-higgs-doublet seesaw model, Phys. Re. D [hep-ph/004070]. [5] W. Grimus and L. Laoura, One-loop corrections to the seesaw mechanism in the multi-higgs-doublet Standard Model, Phys. Lett. B [hep-ph/0079]. [6] M. C. Gonzalez-Garcia, M. Maltoni, J. Salado and. Schwetz, Global fit to three neutrino mixing: critical look at present precision, arxi: