The "complementary" Ansatz, Tr$(M_\nu)=0$, where $M_\nu$ is the prediagonal
neutrino mass matrix, seems a plausible approximation for capturing in a
self-contained way some of the content of Grand Unification. We study its
consequences in the form of relations between the neutrino masses and CP
violation phases.Comment: 8 pages, 1 figure, talk at MRST200

Moussa, S.

Nasri, S.

Schechter, J.

English

arXiv

The  complementary  Ansatz, Tr(M_\nu)=0, where M_\nu is the prediagonal neutrino mass matrix, seems a plausible approximation for capturing in a self-contained way some of the content of Grand Unification. We study its consequences in the form of relations between the neutrino masses and CP violation phases

Schechter, Joseph

Nasri, Salah

Moussa, Sherif

Syracuse University Research Facility and Collaborative Environment

Syracuse University SURFACE Physics College of Arts and Sciences 6-22-2004 Study of Leptonic CP Violation Joseph Schechter Syracuse University Salah Nasri University of Maryland - College Park Sherif Moussa Ain Shams University Follow this and additional works at: https://surface.syr.edu/phy  Part of the Physics Commons Recommended Citation Schechter, Joseph; Nasri, Salah; and Moussa, Sherif, "Study of Leptonic CP Violation" (2004). Physics. 275. https://surface.syr.edu/phy/275 This Article is brought to you for free and open access by the College of Arts and Sciences at SURFACE. It has been accepted for inclusion in Physics by an authorized administrator of SURFACE. For more information, please contact surface@syr.edu. arXiv:hep-ph/0406243v1  22 Jun 2004February 2, 2008 0:11 WSPC/Trim Size: 9in x 6in for Proceedings mrst2004talkSTUDY OF LEPTONIC CP VIOLATIONS. NASRIA, J. SCHECHTERB∗AND S. MOUSSAC(A) Department of Physics,University of Maryland,College Park, MD 20742-4111, USA.E-mail: snasri@physics.umd.edu(B) Physics Department,Syracuse University,Syracuse, NY 13244-1130, USA.E-mail: schechte@phy.syr.edu(C) Department of Mathematics,Faculty of Science,Ain Shams University, Egypt.E-mail: sherif@asunet.shams.eun.egThe “complementary” Ansatz, Tr(Mν)=0, where Mν is the prediagonal neutrinomass matrix, seems a plausible approximation for capturing in a self contained waysome of the content of Grand Unification. We study its consequences in the formof relations between the neutrino masses and CP violation phases.1. IntroductionA favorite topic of discussion at the MRST meetings over many years hasbeen the prediction of quark and lepton masses and mixings. Usually, anattempt is made to predict everything at once based on a suitable guess(Ansatz). Here we discuss a “complementary” Ansatz which adds just alittle information to the system. This is 1,2,3 , for the symmetric pre-diagonal neutrino mass matrix, Mν :Tr(Mν) = 0 (1)∗Speaker1February 2, 2008 0:11 WSPC/Trim Size: 9in x 6in for Proceedings mrst2004talk2If, at first, leptonic CP violation is neglected so that Mν is diagonalized bya real orthogonal matrix, Eq. (1) yields simplym1 +m2 +m3 = 0, (2)where the neutrino masses, mi can be taken here to be either positive ornegative. Now a very recent analysis 4 of solar, atmospheric, reactor andaccelerator neutrino oscillation data (but neglecting LSND) gives the bestfit:m22 −m21 = 6.9× 10−5eV 2,|m23 −m22| = 2.6× 10−3eV 2. (3)Together, Eqs. (2) and (3) provide three equations for three unknowns.There are two essentially different types of solutions. Type I is characterizedby |m3| being largest:m1 = 0.0291 eV, m2 = 0.0302 eV, m3 = −0.0593 eV, (4)while type II has |m3| smallest:m1 = 0.0503 eV, m2 = −0.0510 eV, m3 = 0.00068 eV. (5)Here we will, following ref.5 (see this for further references), discuss thesituation when CP violation effects are included and give an application toleptogenesis.2. Plausibility argument for AnsatzSO(10) grand unified theories have the nice feature that they contain acomplete fermion generation in a single irreducible representation of thegroup. There are three Higgs irreducible representations which can directlycontribute to tree level fermion masses via the Yukawa sector: the 10, the120 and the 126. In principle any number of each is allowed. For everyHiggs field there is a 3 ×3 matrix of unknown coupling constants. Thefermion mass matrices are linear combinations of these matrices. Clearly avery large number of different models can be envisioned.We start from the “kinematical” relation:Tr(M−1/3 − rM−1) ∝ Tr(M0,LIGHT ), (6)which holds when any number of 10’s and 120’s are present but only asingle 126. Here the subscript on the mass matrix indicates the electriccharge of the fermion. The quantity r ≈ 3 takes account of running massesFebruary 2, 2008 0:11 WSPC/Trim Size: 9in x 6in for Proceedings mrst2004talk3from the GUT scale to about 1 GeV. Under the same conditions one alsohas:M0,HEAV Y ∝M0,LIGHT . (7)Note that the physical light neutrino mass matrix, Mν is given by the wellknown formula:Mν ≈M0,LIGHT −MTDIRACM−10,HEAV YMDIRAC . (8)The initial assumption we shall make is that the second, “see-saw” term inEq. (8) is small compared to M0,LIGHT .Now, it has been known for a long time that the quark mixing matrix isof the form diag(1, 1, 1) +O(ǫ). Thus it was very surprising when analysisof neutrino oscillation observations showed that the lepton mixing matrix isnot at all close to the unit matrix but rather has large (12) and (23) mixingelements. In a GUT framework, this suggests that a first approximation tothe prediagonal mass matrices might be to take the charged fermion massmatrices to be diagonal while the neutrino mass matrix would presumablydiffer drastically from the diagonal form. As examples we would setM−1 ≈diag(me,mµ,mτ ) and M−1/3 ≈ diag(md,ms,mb). Substituting these intothe left hand side of Eq. (6) shows it to be about (mb − 3mτ), which isabout zero. Hence the right hand side should also be about zero as shouldTr(Mν) in the non-seesaw dominance case. Although this is clearly anapproximation, it seems likely to be close to the physical situation in thesame sense as mb ≈ 3mτ . Of course, the approximation gets better as themixing matrices needed to bi-diagonalize the charged lepton mass matrixget closer to the unit matrix. For simplicity, in what follows we shall alsoapproximate these matrices to equal the unit matrix.3. Parameterized Ansatz equationThe prediagonal neutrino mass matrix may be brought to diagonal form bya transformation:UTMνU = Mˆν = diag(m1,m2,m3), (9)where U is a unitary matrix. Mν is a symmetric but complex matrixwhich has in general 12 real parameters. This equals the sum of the threeparameters from mi and the nine parameters from U . Now the observablelepton mixing matrix, K is given 1 by K = Ω†U , where we have just agreedto approximate the charged lepton diagonalizing matrix factor, Ω to beessentially the unit matrix (or alternatively we could choose to work inFebruary 2, 2008 0:11 WSPC/Trim Size: 9in x 6in for Proceedings mrst2004talk4a basis where the charged leptons are diagonal). Thus we replace U bythe observable matrix K. K is parameterized in a conventional way asK = Kexpω−10 (τ), where ω0(τ) = diag(eiτ1 , eiτ2 , eiτ3) with τ1 + τ2 + τ3 = 0and:Kexp =c12c13 s12c13 s13e−iδ−s12c23 − c12s13s23eiδ c12c23 − s12s13s23eiδ c13s23s12s23 − c12s13c23eiδ −c12s23 − s12s13c23eiδ c13c23 (10)where sij = sinθij and cij = cosθij . As written, the matrix K is parame-terized by three angles and three independent CP violating phases. To getthe most general K three more phases are required; these can be insertedfor example 1 by multiplying K on the left by ω0(σ). However these phasescan always be canceled by rephasing the (diagonal) charged lepton fieldswhich sit to the left of K. Thus even if the phases, σi were included inU there would always be an allowed choice of charged lepton phases whichwould cancel their effect when we get restrictions (as we shall) on the phys-ical K. Taking this into account the Ansatz reads in terms of physicalquantities:Tr(MˆνK−1expK∗expω0(2τ)) = 0. (11)In more detail it is:m1e2iτ1[1− 2i(c12s13)2sinδe−iδ]+m2e2iτ2[1− 2i(s12s13)2sinδe−iδ]+m3e2iτ3[1 + 2i(s13)2sinδeiδ]= 0. (12)In this equation we can choose the diagonal masses m1,m2,m3 to be realpositive. The mixing angles are known from the best fit 4,s212 = 0.30, s223 = 0.50, s213 = 0.003, (13)wherein the first two have about 25 per cent uncertainty while the third isjust known to be small.Together Eqs. (3) and (12) are now seen to provide 4 real equations forthe 6 unknowns: m1,m2,m3, δ, τ1, τ2. Thus further assumptions are neededto make some predictions. We already saw that assuming all the CP phasesto vanish gives three equations for three unknowns. If we assume only the“conventional” CP phase δ not to vanish there are four equations for fourunknowns, with the results described in 5. This case would become trivialin the limit where s213 is assumed to vanish. There is no reason to expectit to vanish exactly but, considering our lack of knowledge, that seems toFebruary 2, 2008 0:11 WSPC/Trim Size: 9in x 6in for Proceedings mrst2004talk5be also an interesting assumption to investigate. According to Eq. (12) ityields the same result as setting δ = 0. Then we have 4 equations for 5unknowns and can get a one parameter family of solutions.4. Family of Majorana phasesIn this case, Eq. (12) takes the formm1e2iτ1 +m2e2iτ2 +m3e2iτ3 = 0, (14)which corresponds, as illustrated in Fig. 1, to a triangle in the complexplane.Figure 1. Vector triangle representing Eq. (14).We proceed to obtain the family of solutions by assuming a value form3,getting m1 and m2 from Eq. (3) and finally by solving for the two interiorangles µ1 and µ2 using trigonometry. Interesting CP violation quantitiesturn out to be:sin[2(τ1 − τ2)] = −sin(µ1 + µ2),sin[2(τ1 − τ3)] = sinµ2,sin[2(τ2 − τ3)] = −sinµ1, (15)The criterion for the existence of CP violation effects is the area of thetriangle being different from zero. We may express the area as:Area =14([(m1 +m2)2 −m23][m23 − (m1 −m2)2])1/2. (16)February 2, 2008 0:11 WSPC/Trim Size: 9in x 6in for Proceedings mrst2004talk6Now we see that the vanishing of the first factor corresponds to the typeI real solution while the vanishing of the second factor corresponds to thetype II real solution. Furthermore, for a solution to exist, the argument ofthe square root should be positive. With the second factor, that establishesthe minimum allowed value ofm3 while with the first factor, that establishesthe minimum value of m3 which allows a type I solution.The table below shows a “panorama” of solutions decreasing fromm3 =0.3 eV, (which is about the highest value compatible with the cosmologybound 6 that the sum of the neutrino masses be less than about 1 eV) tothe lowest value imposed by the model. In the type I solutions m3 is thelargest mass while in the type II solutions m3 is the smallest mass. Foreach value of m3, the values of the model predictions for m1 and m2 as wellas the internal angles µ1 and µ2 are given. The model prediction for theneutrinoless double beta decay quantity |mee| is next shown. Finally, thelast column shows the estimated lepton asymmetries due to the decays ofthe heavy neutrinos. Note that the reversed sign of lepton asymmetry isalso possible.m1,m2,m3 in eV µ1, µ2 rad. |mee| ǫ1, ǫ2, ǫ3I .2955, .2956, .300 1.038, 1.039 .185 .342, .433, .017II .3042, .3043, .300 1.055,1.056 .187 .330, .426, -.0172I .0856, .0860, .100 0.946, 0.952 .058 .138, .060, .00137II .1119, .1122, .100 1.106, 1.111 .065 .194, .088,-.0024I .0305, .0316, .060 0.258, 0.268 .030 .00982, .00422, .00004II .0783, .0787, .060 1.172, 1.187 .043 .094, .041,-.0011I .0291, .0302, .000552, .030 1.96 ×10−6,0.0592715649 0.000574 .84 ×10−6, .71 ×10−7II .0774, .0782, 1.174, 1.188 .042 .047, .020, -.00110.0592715649II .0643, .0648, .040 1.243, 1.268 .033 .052, .023,-.000681II .0541, .0548, .020 1.355. 1.442 .024 .018, .0078,-.000335II .0506, .0512, .005 1.386, 1.658 .021 .0057, .0025,-.0000824II .0503, .0510, .001 0.814, 2.313 .021 .00073, .00031,-.0000122II 0.0503, 0.0510, 0.051361, .021 .0000348, .0000150,0.0006996 3.089536 -0.601 ×10−6February 2, 2008 0:11 WSPC/Trim Size: 9in x 6in for Proceedings mrst2004talk7Notice that when m3 is decreased to about 0.0593 eV, we get to thereal type I case (no CP violation). Below this value of m3 only the type IIsolutions exist. At m3 about 7× 10−4 eV, we get the real type II case andno solutions exist for m3 below this value. In the m3 regions just above thetwo real cases we can evidently tune the CP violation phases continuouslyto be as small as desired.An interesting application of the model is to neutrinoless double betadecay (for example, the decay 76Ge → 76Se + e− + e−). The currentexperimental bound on the amplitude factor 7 is: |mee| < (0.35 → 1.30)eV, where|mee| =∑|mi(Kexp1i)2e−2iτi |. (17)From the table we see that the predicted values of |mee| are typically aboutone order of magnitude below the experimental bound. Furthermore, thepredicted values do not vary drastically for m3 less than about 0.1 eV.5. Estimate for leptogenesisAn intriguing possibility for learning more about leptonic CP violation isthe study of the proposed leptogenesis mechanism for generation of thepresent baryon number asymmetry of the universe. According to thisscheme, the lepton number violating decays of the heavy neutrinos at ahigh temperature (early universe) establish a lepton asymmetry which getsconverted as the universe cools, through a (B+L) violating but (B-L) con-serving “sphaleron” interaction to a baryon asymmetry. References and arough estimate in the present framework are given in ref. 5. According toEq. (7) the heavy neutrino masses are supposed to be proportional to thelight ones here and have the same diagonalizing matrix, U . The effectiveterm for calculating the heavy neutrino decays at very high temperature isLY UKAWA = −∑L¯ihijΦcNˆj +H.c., (18)wherehij ≈M2/3iKexpije−iτj< φ0 > r′. (19)The quantities needed for the calculation are the matrix products (h†h)ij ;it is thus seen that the effect of a diagonal matrix of phases multiplyingKexpij on the left would cancel out. The lepton CP asymmetry ǫi, due tothe decay of the ith heavy neutrino, is defined as the ratio of decay widths:ǫi =Γ(Ni → L+ Φ)− Γ(Ni → L¯+ Φ¯)Γ(Ni → L+ Φ) + Γ(Ni → L¯+ Φ¯). (20)February 2, 2008 0:11 WSPC/Trim Size: 9in x 6in for Proceedings mrst2004talk8In this equation L+Φ stands for all lepton- Higgs pairs of the types e−j +φ+and νj+φ¯0. This is an effect which violates C and CP conservation, in agree-ment with the requirement of Sakharov. The numerical values of the ǫi,which depend on the ratios of heavy neutrino masses rather than their ab-solute values, are displayed in the last column of the table. Notice that Eq.(18) represents the same term which generates the Dirac mass, MDIRACin Eq. (8). Since our motivation for the Ansatz assumes dominance of thenon seesaw term, this feature requires 5 the heavy neutrino mass scale tobe suitably large. This scale plays a role in the estimation of the presentbaryon to photon ratio, ηB of the universe which is obtained by convolutingthe ǫi with factors obtained by solving the Boltzmann evolution equationsfor the (B-L) asymmetry. It turns out 5 that for typical values of the pa-rameter, m3 in the table, ηB is considerably larger than its experimentalvalue 6 of about 6.5 × 10−10. Thus agreement with experiment requirestuning close to the two real type solutions; the correct order of magnitudeis obtained when either m3 ≈ 0.059 eV (type I) or m3 ≈ 0.005 eV (typeII).We thank D. Black, A. H. Fariborz, C. Macesanu , M. Trodden and D.Schechter for their help. The work of S.N is supported by National Sciencefoundation grant No. PHY-0099544. The work of J.S. is supported in partby the U. S. DOE under Contract no. DE-FG-02-85ER 40231.Finally, we are pleased to take this opportunity to express our wishesfor Happy Birthday, Good Health and Continued Important Contributionsto Physics to PAT O’DONNELL and to HARRY LAM.References1. D. Black, A. H. Fariborz, S. Nasri and J. Schechter, Phys. Rev. D62, 073015(2000).2. X.-G. He and A. Zee, Phys. Rev. D68, 037302, (2003).3. W. Rodejohann, Phys. Lett. B579, 127 (2004).4. M. Maltoni, T. Schwetz, M. A. Tortola and J. W. F. Valle, arXiv:hep-ph/0309130.5. S. Nasri, J. Schechter and S. Moussa, arXiv:hep-ph/0402176.6. D. N. Spergel et al, arXiv:astro-ph/0302209; S. Hannestad, arXiv:astro-ph/0303076.7. H. V. Klapdor-Kleingrothaus et al, Eur. Phys. J. A12, 147 (2001).

Study of Leptonic CP Violation

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Study of leptonic CP violation

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