In this talk we explore the possibility that the smallness of the observed
neutrino masses is naturally understood in a modified version of the standard
model with N extra generations of fermions and N right-handed neutrinos, in
which light neutrino masses are generated at two loops. We find that with N = 1
it is not possible to fit the observed spectrum of masses and mixings while
with N = 2 it is. Within this extension, we analyse the parameters which are
allowed and the possible phenomenological signals of the model in future
experiments. Contribution to the proceedings of Les Rencontres de Moriond EW
2011, Young Scientist Forum

Aparici, A.

Herrero-Garcia, J.

Rius, N.

Santamaria, A.

English

arXiv

In this talk we explore the possibility that the smallness of the observed neutrino masses is naturally understood in a modified version of the standard model with N extra generations of fermions and N right-handed neutrinos, in which light neutrino masses are generated at two loops. We find that with N = 1 it is not possible to fit the observed spectrum of masses and mixings while with N = 2 it is. Within this extension, we analyse the parameters which are allowed and the possible phenomenological signals of the model in future experiments. Contribution to the proceedings of Les Rencontres de Moriond EW 2011, Young Scientist Forum

Herrero García, Juan Andrés

Rius Dionis, Nuria

Aparici Benages, Alberto

Santamaria, Arcadi

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

arXiv:1106.0415v2  [hep-ph]  3 Jun 2011 CAN NEW GENERATIONS EXPLAIN NEUTRINO MASSES?A. APARICI, J. HERRERO-GARCIA, N. RIUS and A. SANTAMARIADepto.de Fisica Teorica, and IFIC, Universidad de Valencia-CSICEdificio de Institutos de Paterna, Apt. 22085, 46071 Valencia, SpainIn this talk we explore the possibility that the smallness of the observed neutrino masses isnaturally understood in a modified version of the standard model with N extra generationsof fermions and N right-handed neutrinos, in which light neutrino masses are generated attwo loops. We find that with N = 1 it is not possible to fit the observed spectrum of massesand mixings while with N = 2 it is. Within this extension, we analyse the parameters whichare allowed and the possible phenomenological signals of the model in future experiments.Contribution to the proceedings of Les Rencontres de Moriond EW 2011, Young ScientistForum.1 IntroductionNeutrino oscillations require at least two massive neutrinos with large mixing, providing one ofthe strongest evidences of physics beyond the Standard Model (SM). On the other hand, one ofthe most natural extensions of the SM is the addition of extra sequential generations 1.LEP II limits on new generation leptons are: mℓ′ > 100.8 GeV and mν′ > 80.5 (90.3) GeVfor pure Majorana (Dirac) particles. When neutrinos have both Dirac and Majorana masses,the bound on the lightest neutrino is 63GeV. For stable neutrinos LEP I measurement of theinvisible Z width, Γinv, implies mν′ > 39.5 (45)GeV for pure Majorana (Dirac) particles.The new heavy fermions contribute to the electroweak parameters and might spoil the agree-ment of the SM with experiment. Global fits of models with additional generations to the elec-troweak data have been performed and they favour no more than five generations. It should bekept in mind that most of the fits make some simplifying assumptions on the mass spectrumof the new generations and do not consider Majorana neutrino masses for the new generationsor the possibility of breaking dynamically the gauge symmetry via the condensation of the newgenerations’ fermions; all these would give additional contributions to the oblique parametersand will modify the fits. Therefore, in view that we will soon see or exclude new generationsthanks to the LHC, it is wise to approach this possibility with an open mind.In this talk (see 2 for further details and a complete list of references) we focus on howneutrino masses can be naturally generated at two loops by adding extra families and singlets.Recall that right-handed neutrinos do not have gauge charges and are not needed to cancelanomalies, therefore their number is not linked to the number of generations.2 Four generationsWe extend the SM by adding a complete fourth generation and one right-handed neutrino νRwith a Majorana mass term 3. We denote the new charged lepton E and the new neutrino νE .νiνjeαWWeβν4, ν 4¯Figure 1: Two-loop diagram contributing to neutrino masses in the four-generation model.The relevant part of the Lagrangian isLY = −ℓ¯YeeRφ− ℓ¯YννRφ˜− 12νcRmRνR +H.c. , (1)where ℓ are the left-handed lepton SU(2) doublets, eR the right-handed charged leptons andνR the right-handed neutrino. In generation space ℓ and eR are organized as four-componentcolumn vectors. Thus, Ye is a general, 4 × 4 matrix, Yν is a general four-component columnvector with elements yα with α = e, µ, τ, E, and mR is a Majorana mass term.After spontaneous symmetry breaking (SSB) the mass matrix for the neutral leptons attree level is a 5 × 5 Majorana symmetric matrix which has only one right-handed neutrinoMajorana mass term. Therefore, it leads to two massive Majorana and three massless Weylneutrinos. Then it is clear that only the linear combination of left-handed neutrinos ν ′4 ∝yeνe + yµνµ + yτντ + yEνE will pair up with νR to acquire a Dirac mass term. Thus, it isconvenient to pass from the flavour basis (νe, νµ, ντ , νE) to a new one ν′1, ν′2, ν′3, ν′4 where the firstthree states will be massless at tree level and only ν ′4 mixes with νR.After this change of basis, να =∑i Vαiν′i (i = 1, · · · , 4, α = e, µ, τ, E) with Vα4 = yα/√∑β y2β,we are left with a 2 × 2 mass matrix for ν ′4 and νR which leads to two Majorana neutrinos ν4and ν4¯ of masses m4,4¯ =12(√mR2 + 4m2D ∓mR),where mD = v√∑i y2i , with v = 〈φ(0)〉, andtan2 θ = m4/m4¯. If mR ≪ mD, we have m4 ≈ m4¯ and tan θ ≈ 1 (pseudo-Dirac limit) whilewhen mR ≫ mD, m4 ≈ m2D/mR, m4¯ ≈ mR and tan θ ≈ mD/mR (see-saw limit).Since lepton number is broken by the νR Majorana mass term, there is no symmetry whichprevents the tree-level massless neutrinos ν ′1, ν′2, ν′3 from gaining Majorana masses; they aregenerated at two loops by the diagram of Figure 1, and are given byMij = − g4m4WmRm2D∑αVαiVα4m2α∑βVβjVβ4m2βIαβ (2)where the sums run over the charged leptons α, β = e, µ, τ, E while i, j = 1, 2, 3, and the loopintegral Iαβ can be found in2. It is easy to show that the eigenvalues of the light neutrino massmatrix are proportional to m4µ, m4τ , m4E which gives a huge hierarchy between neutrino masses:m2m3≤ 14N2E(mτmE)2(mτm4¯)2≤ 10−7N2E, (3)where we have taken ln(m4¯/m4) ≈ ln(mE/m4¯) ≈ 1 and in the last step we used that mE,m4¯ ≥100GeV. To overcome this huge hierarchy very small values of NE are needed, which wouldimply that the heavy neutrinos are not mainly νE but some combination of the three knownones νe, νµ, ντ ; however this is not possible since it would yield observable effects in a variety ofprocesses, like π → µν, π → eν, τ → eνν... This requires that ye,µ,τ ≈ 10−2yE, so NE ≈ 1.Therefore, the simplest version of the model is unable to accommodate the observed spectrumof neutrino masses and mixings. However, notice that whenever a new generation and a right-handed neutrino with Majorana mass at (or below) the TeV scale are added to the SM, thetwo-loop contribution to neutrino masses is always present and provides an important constraintfor this kind of SM extensions.3 The five-generation modelWe add two generations to the SM and two right-handed neutrinos. We denote the two chargedleptons by E and F and the two right-handed singlets by ν4R and ν5R. The Lagrangian isexactly the same we used for four generations but now ℓ and e are organized as five-componentcolumn vectors while νR is a two-component column vector containing ν4R and ν5R. Thus, Yeis a general, 5× 5 matrix, Yν is a general 5× 2 matrix and mR is now a general symmetric 2× 2matrix. The model, contrary to the four-generation case, has additional sources of CP violationin the leptonic sector, however, for simplicity we take all yα and y′α real.As in the four-generation case, the linear combination ν ′4 ∝∑α yανα only couples to ν4Rand the combination ν ′5 ∝∑α y′ανα only couples to ν5R. Therefore, the tree-level spectrum willcontain three massless neutrinos (the linear combinations orthogonal to ν ′4 and ν′5) and fourheavy Majorana neutrinos. For simplicity, we choose ν ′4 and ν′5 orthogonal to each other, i.e.,∑α yαy′α = 0. We change from the flavour fields νe, νµ, ντ , νE , νF to a new basis ν′1, ν′2, ν′3, ν′4, ν′5where ν ′1, ν′2, ν′3 are massless at tree level, so we are free to choose them in any combination ofthe flavour states as long as they are orthogonal to ν ′4 and ν′5.The model should be compatible with the observed universality of fermion couplings and havesmall rates of lepton flavour violation in the charged sector, which requires ye, yµ, yτ , y′e, y′µ, y′τ ≪yE , yF , y′E , y′F . In addition, it should fit the observed pattern of masses and mixings, for instance,reproducing the tribimaximal (TBM) mixing structure. A successful choice of the Yukawas toobtain normal hierarchy (see 2 for an analysis of inverted hierarchy and more details), i.e,m3 ≈√∣∣△m231∣∣ ≈ 0.05 eV, m2 ≈ √△m221 ≈ 0.01 eV, will be yα = yE(ǫ, ǫ,−ǫ, 1, 0) andy′α = y′F (0, ǫ′, ǫ′, 0, 1) which, keeping only terms up to order ǫ2, leads to the 5× 5 unitary matrixV that passes from one basis to the other, να =∑i Vαiν′i (i = 1, · · · , 5, α = e, µ, τ, E, F ):V ≈√231√3−√32 ǫ2 0 ǫ 0− 1√61√3−√32 ǫ2 1√2− 1√2ǫ′2 ǫ ǫ′1√6− 1√3+√32 ǫ2 1√2− 1√2ǫ′2 −ǫ ǫ′0 −ǫ√3 0 1− 32 ǫ2 00 0 −ǫ′√2 0 1− ǫ′2(4)Assuming that mE,F ≫ m4,4¯,5,5¯ ≫ mW , we find:m2 ≈ 3g42(4π)4m4Wǫ2m24Dm4Rm2E lnmEm4¯(5)m3 ≈ g4(4π)4m4Wǫ′2m25Dm5Rm2F lnmFm5¯, (6)and the required ratio m3/m2 ≈ 5 can be easily accommodated for different combinations ofmasses and mixing parameters ǫ, ǫ′.4 Phenomenological analysis of the modelIn general, the most restrictive experimental bound comes from B(µ→ eγ) < 1.2 × 10−11, andit is translated into ǫ < 0.03. From B(τ → µγ) < 4.4 × 10−8, we obtain |ǫ′2 − ǫ2| < 0.09. Wedisplay in Figure 2 a) B(µ→ eγ) versus the mass of the heavy neutrino m4 in the NH case. WeΕ = 0.025Ε = 0.008Current boundFuture bound100 500200 300150 70010-1410-1310-1210-1110-10Heavy neutrino mass m4 HGeVLBHΜeΓLm4 D Hm5 D) = 100 GeVCurrent boundFuture boundmE HmF) = 100 GeVm4 D Hm5 D) = 1000 GeVmE HmF) = 1000 GeV10-6 10-4 0.01 1 100 10410-610-510-40.0010.01Majorana mass m4 R Hm5 RL HGeVLMixingΕHΕ'LFigure 2: a) Left: B(µ → eγ) against m4 for different values of ǫ. We also display present and future limits.b) Right: Parameter space that predicts the right scale for heavy and light neutrinos (the region between thecurves). We also present the current µ→ eγ bound and the expected µ–e conversion limit.also display present and near future limits. Also, µ–e conversion in nuclei gives information onǫ. We expect it to set much stronger bounds in the future.Violations of universality constrain the model in both hierarchies. For example, from piondecay, we obtain ǫ′ < 0.04. Regarding neutrinoless double beta decay, there are contributions ofthe new heavy neutrinos, however, we obtain the same combination of parameters as in the lightneutrino masses expressions, m4Rǫ2, leading to unobservable effects in 0νββ when the formerare fitted (of course the light neutrino contribution can be observed (in IH) in the future).Also, a very striking effect of new generations is the enhancement of the Higgs-gluon-gluonvertex which arises from a triangle diagram with all quarks running in the loop, by a factor of9 (25) in the presence of a fourth (fifth) generation. We estimate roughly that mH > 300GeVin the case of five generations. However, these limits may be softened in some cases.To summarize the phenomenology of the model we present in figure 2 the allowed regionsin the ǫ −m4R plane which lead to m3 ∼ 0.05 eV varying the charged lepton masses mE (mF )and the Dirac neutrino masses m4D (m5D) between 100 GeV–1 TeV, and imposing the boundon the neutrino mass, m4 > 63GeV. We also plot the present bounds on the mixings ǫ (ǫ′) fromµ→ eγ and future limits from µ–e conversion if expectations are attained.So our conclusion is that with four generations and one singlet the correct spectrum of lightneutrino masses cannot be generated. However, with five generations and two singlets all currentdata can be accomodated in the region of the parameter space between the curves of Figure 2b), which will be probed in the near future.AcknowledgmentsThis work has been partially supported by the grants FPA-2007-60323, FPA-2008-03373, CSD2007-00060, CSD2007- 00042, PROMETEO/2009/116, PROMETEO/2009/128 andMRTN-CT-2006-035482 (FLAVIAnet). A.A. and J.H.-G. are supported by the MICINN under the FPU program.References1. B. Holdom et. al., PMC Phys. A3 (2009) 4, [arXiv:0904.4698]. V. A. Novikov et. al.,Phys. Atom. Nucl. 73 (2010) 636–642, [arXiv:0904.4570]. L. M. Carpenter et. al.,Phys. Rev. D82 (2010) 114019, [arXiv:1005.0628]. G. D. Kribs et. al., Phys. Rev.D76 (2007) 075016, [arXiv:0706.3718]. PDG Coll., J. Phys. G37 (2010) 075021.2. A. Aparici, J. Herrero-Garcia, N. Rius, and A. Santamaria, [arXiv:1104.4068].3. S. T. Petcov et. al., Phys. Lett. B143 (1984) 175. K. S. Babu et. al., Phys. Rev. Lett.61 (1988) 674, Phys. Lett. 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Can new generations explain neutrino masses?

Can new generations explain neutrino masses?

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