/Fermilab /UC, Irvine

Chen, Mu-Chun

Mahanthappa, K.T.

U., /Colorado

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arXiv:hep-ph/0610373v2  19 Nov 2006Relating Leptogenesis to Low Energy CPViolationMu-Chun Chen∗ and K.T. Mahanthappa†∗Theoretical Physics Department, Fermilab, Batavia, IL 60510, USAandDepartment of Physics & Astronomy, University of California, Irvine, CA 92697, USA†Department of Physics, University of Colorado, Boulder, CO 80309-0390, USAAbstract. In the minimal left-right symmetric model with spontaneous CP violation, there are onlytwo intrinsic CP violating phases to account for all CP violation in both the quark and lepton sectors.In addition, the left- and right-handed Majorana mass terms for the neutrinos are proportional to eachother due to the parity in the model. This is thus a very constrained framework, making the existenceof correlations among the CP violation in leptogenesis, neutrino oscillation and neutrinoless doublebeta decay possible.Keywords: CP violation; neutrino oscillation; leptogenesisPACS: 12.10.Dm, 14.60.PqINTRODUCTIONThe evidence of non-zero neutrino masses opens up the possibility that the leptonicCP violation might be responsible, through leptogenesis, for the observed asymmetrybetween matter and anti-matter in the Universe. It is generally difficult, however, to makeconnection between leptogenesis and CP-violating processes at low energies due to thepresence of extra phases and mixing angles in the right-handed neutrino sector. Recentlyattempts have been made to induce spontaneous CP violation (SCPV) from a singlesource. Here we focus on the minimal left-right symmetric model. In this model SCPVcould be due to two intrinsic CP violating phases associated with VEVs of two scalarfields which account for all CP-violating processes observed in Nature; these exhaustsources of CP-violation. As the left-handed (LH) and right-handed (RH) Majorana massmatrices are identical up to an overall mass scale, in this model there exist relationsbetween low energy processes, such as neutrino oscillations, neutrinoless double betadecay and lepton flavor violating charged lepton decay, and leptogenesis which occursat very high energy, as pointed out in Ref. [1].THE MINIMAL LEFT-RIGHT SYMMETRIC MODELIn the minimal left-right symmetric SU(2)L×SU(2)R×U(1)B−L model [2], the left- andright-handed matter fields transform as doublets of SU(2)L and SU(2)R, respectively,and the Higgs sector that breaks the left-right symmetry to the SM gauge group containsa SU(2) bi-doublet Higgs, Φ, a SU(2)L triplet, ∆L, and a SU(2)R triplet, ∆R. The Yukawainteractions responsible for generating the lepton masses are,−Lℓ = Li,R(Pi jΦ+Ri j ˜Φ)L j,L + i fi j(LTi,LC τ2∆LL j,L +LTi,RCτ2∆RL j,R)+h.c. , (1)where the matrices Pi j, Ri j and fi j are real due to the assumption of SCPV. The completeLagrangian of the model is invariant under the unitary transformation, under which thematter fields transform as ψL → ULψL and ψR → URψR where ψL,R are left-handed(right-handed) fermions, and the scalar fields transform according to Φ → URΦU†L ,∆L →U∗L ∆LU†L , and ∆R →U∗R∆RU†R, with the unitary transformations UL and UR beingUL =(eiγL 00 e−iγL), UR =(eiγR 00 e−iγR). (2)Under these unitary transformations, the VEV’s transform as κ → κe−i(γL−γR), κ ′ →κ ′ei(γL−γR), vL → vLe−2iγL and vR → vRe−2iγR . Thus by re-defining the phases of matterfields with the choice of γR = αR/2 and γL = ακ +αR/2 in the unitary matrices UL andUR, we can rotate away two of the complex phases in the VEV’s of the scalar fields andare left with only two genuine CP violating phases, ακ ′ and αL,< Φ >=(κ 00 κ ′eiακ ′), < ∆L >=(0 0vLeiαL 0), < ∆R >=(0 0vR 0). (3)The quark Yukawa interactions give rise to quark masses after the bi-doublet acquiresVEV’sMu = κFi j +κ ′e−iακ ′Gi j, Md = κ ′eiακ ′Fi j +κGi j . (4)Thus the relative phase in the two VEV’s in the SU(2) bi-doublet, ακ ′ , gives rise tothe CP violating phase in the CKM matrix. To obtain realistic quark masses and CKMmatrix elements, it has been shown that the VEV’s of the bi-doublet have to satisfyκ/κ ′ ≃ mt/mb ≫ 1. When the triplets and the bi-doublet acquire VEV’s, we obtain thefollowing mass terms for the leptonsMe = κ ′eiακ ′Pi j +κRi j, MDiracν = κPi j +κ ′e−iακ ′Ri j (5)MRRν = vR fi j, MLLν = vLeiαL fi j . (6)The effective neutrino mass matrix, Meffν , which arises from the Type-II seesaw mecha-nism, is thus given by Meffν = MLLν −MIν . Assuming the charged lepton mass matrix isdiagonal, the Yukawa couplings Ri j can be determined by the charged lepton masses. Inthe limit κ ≫ κ ′, the conventional type-I see-saw term is dominated by the term propor-tional to κ ,MIν = (MDiracν )T (MRRν )−1(MDiracν )≃κ2vRPT f−1P = vLβ PT f−1P . (7)RESULTSIn the left-right symmetric model with the particle content we have, leptogenesis re-ceives contributions both from the decay of the lightest RH neutrino, N1, as well as fromthe decay of the SU(2)L triplet Higgs, ∆L. We consider the SU(2)L triplet Higgs beingheavier than the lightest RH neutrino, M∆L > MR1. For this case, the decay of the light-est RH neutrino dominates. In the SM, the canonical contribution to the lepton numberasymmetry from one-loop diagrams mediated by the Higgs doublet and the charged lep-tons is given by [3] ,εN1 =316pi(MR1v2)·Im(MD(MIν)∗M TD)11(MDM†D)11. (8)Now, there is one additional one-loop diagram mediated by the SU(2)L triplet Higgs. Itcontributes to the decay amplitude of the right-handed neutrino into a doublet Higgsand a charged lepton, which gives an additional contribution to the lepton numberasymmetry [3],ε∆L =316pi(MR1v2)·Im(MD(MIIν)∗M TD)11(MDM†D)11, (9)where MD is the neutrino Dirac mass term in the basis where the RH neutrino Majoranamass term is real and diagonal, MD = ORMD and f diag = OR f OTR . Because there is nophase present in either MD = Pκ or MIν or OR, the quantity MD(MIν)∗M TD is real,leading to a vanishing εN1 . We have checked explicitly that this statement is true forany chosen unitary transformations UL and UR defined in Eq. (2). On the other hand, thecontribution, ε∆L , due to the diagram mediated by the SU(2)R triplet is proportional tosinαL. So, as long as the phase αL is non-zero, the predicted value for ε∆L is finite. Anon-vanishing value for εN1 is generated at the sub-leading order when terms of orderO(κ ′/κ) in MD are included. At the leading order, leptogenesis is generated solely fromthe decay mediated by the SU(2)L triplet Higgs.Assume the neutrino Yukawa coupling Pi j to be proportional to the up quark massmatrix and the matrix fi j having hierarchical elements,Pi j = qmumt0 00 mcmt00 0 1 , fi j = t2 t −tt 1 1−t 1 1 , (10)with q being the proportionality constant and t a small and positive number. CP violationin neutrino oscillation is governed by the leptonic Jarlskog invariant, JCP, which canbe expressed model-independently in terms of the effective neutrino mass matrices asJCP =−Im(H12H23H31)/(∆m221∆m231∆m232) and H ≡Meffν Meff†ν , where ∆m2i j ≡m2i −m2j withmi being the mass eigenvalues of the effective neutrino mass matrix, Meffν .−0.003 −0.002 −0.001 0 0.001 0.002 0.003JCP−0.06−0.04−0.0200.020.040.06∆ε’FIGURE 1. Correlation between lepton number asymmetry and the leptonic Jarlskog invariant.To the leading order in ε the leptonic Jarlskog invariant is given by,JCP ≃−2st2(1− t2)v6L∆m221∆m231∆m232mumtsinαL , (11)and the total amount of lepton number asymmetry, εtotal = ε∆L , is proprtional to ∆ε ′,defined as∆ε ′ = 316pif 01vL·Im(MD(vL f eiαL)∗M TD)11(MDM†D)11=ε∆Lβ ∝ sinαL , (12)where β is a function of order one coupling constants in the scalar potential. Theleptonic Jarlskog J and the amount of leptogenesis ε∆L are both proportional to sinαL.Thus the observation of low energy CP violation then implies a non-vanishing leptonnumber asymmetry. We note that an amount of lepton number asymmetry εtotal ∼ 10−8corresponds to a leptonic Jarlskog invariant |JCP| ∼ 10−5.With an additional broken U(1) symmetry, the seesaw scale can be lowered to becloser to the EW scale. In that case, the phase ακ ′ responsible for CPV in the quarksector can also be relevant to the leptonic CP violating processes. Connections such asthe one between electron EDM and leptogenesis can thus be established [4].REFERENCES1. M.-C. Chen and K. T. Mahanthappa, Phys. Rev. D71, 035001 (2005).2. J. C. Pati and A. Salam, Phys. Rev. D10, 275 (1974); R. N. Mohapatra and J. C. Pati, Phys. Rev. D11,566 (1975).3. A. S. Joshipura, E. A. Paschos and W. Rodejohann, Nucl. Phys. B611, 227 (2001). S. Antusch andS. F. King, Phys. Lett. B 597, 199 (2004).4. M.-C. Chen and K. T. Mahanthappa, hep-ph/0609288.

Relating leptogenesis to low energy CP violation

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Relating leptogenesis to low energy CP violation

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