Neutrino mass spectrum from the seesaw extension Darius Jurciukonis, homas Gajdosik, Andrius Juodagalvis and omas Sabonis arxiv:11.691v1 [hep-ph] 31 Dec 01 Vilnius University, Universiteto 3, -01513, Vilnius, ithuania 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 one or two right-handed fields are added to the three left-handed fields. A second Higgs doublet is included in our model. We calculate the one-loop radiative 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 heavy neutrinos masses. Parameters of the model are varied to find light neutrino masses that are compatible with experimental data of solar m and atmospheric m atm neutrino oscillations for normal and inverted hierarchy. PACS numbers: 11.30.Rd, 13.15.+g, 14.60.St 1. he model We extend the Standard Model SM by adding a second Higgs doublet and right-handed neutrino fields. he Yukawa agrangian of the leptons is expressed by Y = k=1 Φ k l R Γ k + Φ k ν R k D + H.c. 1.1 in a vector and matrix notation, where Φ k = iτ Φ k. In expression 1.1 l R, ν R, and D = ν l are the vectors of the right-handed charged leptons, of the right-handed neutrino singlets, and of the left-handed lepton Presented at International Symposium on Multiparticle Dynamics, Kielce 01 1
MultPartDyn Proceedings Final printed on November 16, 018 doublets, respectively, and Φ k, k = 1, are the two Higgs doublets. he Yukawa coupling matrices Γ k are n n, while the k are n R n. In this model, spontaneous symmetry breaking of the SM gauge group is achieved by the vacuum expectation values Φ k vac = 0, v k /, k = 1,. By a unitary rotation of the Higgs doublets we can achieve Φ 0 1 vac = v/ > 0 and Φ 0 vac = 0 with v 46 GeV. he charged-lepton mass matrix M l and the Dirac neutrino mass matrix M D are ˆM l = v Γ 1 and M D = v 1, 1. with the assumption that ˆM l = diag m e, m µ, m τ. he hat indicates that ˆM l is a diagonal matrix. he mass terms for the neutrinos can be written in a compact form with a n + n R n + n R symmetric mass matrix M ν can be diagonalized as 0 M M ν = D M D ˆMR. 1.3 U M ν U = ˆm = diag m 1, m,..., m n +n R, 1.4 where the m i are real and non-negative. In order to implement the seesaw mechanism [1, ] 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 = 1,,..., n are of order m D /m R, while those with i = n + 1,..., n + n R are of order m R. It is useful to decompose the n + n R n + n R unitary matrix U as U = U, UR, where the submatrix U is n n +n R and the submatrix U R is n R n +n R [3, 4]. With these submatrices, the left- and right-handed neutrinos are written as linear superpositions of the n + n R physical Majorana neutrino fields χ i : ν = U P χ and ν R = U R P R χ, where P and P R are the projectors of chirality. It is possible to express the couplings of the model in terms of the mass eigenfields, where three neutral particles namely, the Z boson, the neutral Goldstone boson G 0 and the Higgs bosons Hb 0 couple to neutrinos. he full formalism for the scalar sector of the multi-higgs-doublet SM is given in Refs. [3, 4]. Once the one-loop corrections are taken into account the neutral fermion mass matrix is given by M ν 1 δm = MD + δm D δm M D, 1.5 M D + δm D ˆMR + δm R M D ˆMR
MultPartDyn Proceedings Final printed on November 16, 018 3 where the 0 3 3 matrix appearing at tree level 1.3 is replaced by the contribution δm. his correction is a symmetric matrix, it dominates among all the sub-matrices of corrections. Neglecting the sub-dominant pieces in 1.5, one-loop corrections to the neutrino masses originate via the self-energy function Σ S 0 = ΣSZ 0+Σ SG0 0+Σ SH0 0, where the Σ SZ,G0,H 0 0 functions arise from the self-energy Feynman diagrams and are evaluated at zero external momentum squared. Each diagram contains a divergent piece but when summing up the three contributions the result turns out to be finite. he final expression for one-loop corrections is given by [5] δm = b 1 3π b U R ˆm + 3g 64π m W ˆm m H 0 b m Z 1 1 ln ˆm m H 0 b ˆm MDU R 1 ˆm ˆm 1 ln m Z U R b U R M D, 1.6 with b = k b k k, where b are two-dimensional complex unit vectors and the sum b runs over all neutral physical Higgses H0 b.. Case n R = 1 First we consider the minimal extension of the standard model by adding only one right-handed field ν R to the three left-handed fields contained in ν. For this case we use the parametrization i = m D /v a i, where a 1 = 0, 0, 1 and a = 0, 1, e iφ. Diagonalization of the symmetric mass matrix M ν 1.3 in block form is U M ν U = U 03 3 m D a 1 ˆMl 0 m D a U = 1 ˆM R 0 ˆMh..1 he non zero masses in ˆM l and ˆM h are determined analytically by finding eigenvalues of the hermitian matrix M ν M ν. hese eigenvalues 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.
4 MultPartDyn Proceedings Final printed on November 16, 018 6. 10-11 m l1 5. 10-11 6.0 10-11 ml i, GeV 4. 10-11 3.5 10-11 m l 5.83 10-11 5.65 10-11 10 6 10 6.5 10 7 3.05 10-11.98 10-11 3. 10-11.9 10-11 10 6 10 6.5 10 7.5 10-11 10 4 10 5 10 6 10 7 10 8 10 9 10 10 m h, GeV Fig. 1. Calculated masses of two light neutrinos as a function of the heavy neutrino mass m h. Solid lines show the boundaries of allowed neutrino mass ranges when the model parameters are constrained by the experimental data on neutrino oscillations with θ atm = 45. he allowed values of m l1 and m l form bands, their scattered values are shown separately in the middle plots. he diagonalization matrix U for the tree level is constructed from a rotation matrix and a diagonal matrix of phases U tree = U 34 βu i, where the angle β is determined by the masses m l and m h. For the calculation of radiative corrections we use the following set of orthogonal complex vectors: b Z = i, 0, b 1 = 1, 0, b = 0, i and b 3 = 0, 1. Diagonalization of the mass matrix after calculation of one-loop corrections is performed with a unitary matrix U loop = U egv U ϕ ϕ 1, ϕ, ϕ 3, where U egv is an eigenmatrix of M ν 1 M ν 1 and U ϕ is a phase matrix. he second light neutrino obtains its mass from radiative corrections. he third light neutrino remains massless. he masses of the neutrinos are restricted by experimental data of solar and atmospheric neutrino oscillations [6] and by cosmological observations. he numerical analysis shows that we can reach the allowed neutrino mass ranges for a heavy singlet with the mass close to 10 4 GeV and with the angle of oscillations fixed to θ atm = 45, see Fig. 1. he free parameters m H 0, m H 0 3, and φ are restricted by the parametrization used and by oscillation data. Figure illustrates the allowed values of Higgs masses for different values of the heavy singlet.
MultPartDyn Proceedings Final printed on November 16, 018 5 10000.00 1000 35110.00 13300.00 mh03, GeV 43900.00 800 1.5 106 5.34 106 1.87 107 600 6.58 107.31 108 8.11 108 400.85 109 1.00 1010 00 00 400 600 800 1000 mh, GeV mh0, GeV Fig.. he values of the free parameters mh0 and mh30 as functions of the heaviest right-handed neutrino mass mh, for the case nr = 1. he mass of the SM Higgs boson is fixed to mh10 = 15 GeV and the angle of oscillations is θatm = 45. 3. Case nr = When we add two singlet fields νr to the three left-handed fields ν, the radiative corrections give masses to all three light neutrinos. Now we parametrize i = v md~ai, md1~bi with ~a1 = 1, ~b1 = 1, ~a = 1 and ~b = 1. Diagonalizing the symmetric mass matrix Mν 1.3 in block form we write: 03 3 md~a md1~b M 0 l U = U Mν U = U. 3.1 md~a M R 0 M h ~ md1 b he non zero masses in M l and M h are determined by the seesaw mechanism: mdi mhi mli and mri mhi, i = 1,. Here we use m1 > m > m3 ordering of masses. he third light neutrino is massless at tree level. tree U φ is composed he diagonalization matrix for tree level Utree = Uegv φ i of an eigenmatrix of Mν Mν and a diagonal phase matrix, respectively. For calculation of radiative corrections we use the same set of orthogonal complex vectors bi as in the first case. Diagonalization of the mass matrix including the one-loop corrections is performed with a unitary ma1 1 loop loop trix Uloop = Uegv Uϕ ϕi, where Uegv is the eigenmatrix of Mν Mν and Uϕ is a phase matrix.
6 MultPartDyn Proceedings Final printed on November 16, 018 A broader description of the case n R = and graphical illustrations of the obtained light neutrino mass spectra is given in Ref. [7]. Both normal and inverted neutrino mass orderings are considered. 4. Conclusions For the case n R = 1 we can match the differences of the calculated light neutrino masses to m and m atm with the mass of a heavy singlet close to 10 4 GeV. he parametrization used for this case and restrictions from the neutrino oscillation data limit the values of free parameters. Only normal ordering of neutrino masses is possible. In the case n R = we obtain three non vanishing masses of light neutrinos for normal and inverted hierarchies. he numerical analysis [7] shows that the values of light neutrino masses especially of the lightest mass depend on the choice of the heavy neutrino masses. he radiative corrections generate the lightest neutrino mass and have a big impact on the second lightest neutrino mass. Acknowledgements he authors thank uis avoura for valuable discussions and suggestions. his work was supported by European Union Structural Funds project Postdoctoral Fellowship Implementation in ithuania. REFERENCES [1] M. Gell-Mann, P. Ramond, and R. Slansky, in Supergravity, Proceedings of the Workshop, Stony Brook, New York, 1979, edited by F. van Nieuwenhuizen and D. Freedman North Holland, Amsterdam, 1979. [] J. Schechter and J. W. F. Valle, Phys. Rev. D 1980 7. [3] W. Grimus and H. Neufeld, Nucl. Phys. B 35 1989 18. [4] W. Grimus and. avoura, Phys. Rev. D 66 00 014016 [hep-ph/004070]. [5] W. Grimus and. avoura, Phys. ett. B 546 00 86 [hep-ph/0079]. [6] M. C. Gonzalez-Garcia, M. Maltoni, J. Salvado and. Schwetz, arxiv:109.303. [7] D. Jurčiukonis,. Gajdosik, A. Juodagalvis and. Sabonis, arxiv:11.5370.