Corrections to Scaling Neutrino Mixing: Non-zero θ 13 , δ CP and Baryon Asymmetry
aa r X i v : . [ h e p - ph ] M a r Corrections to Scaling Neutrino Mixing: Non-zero θ , δ CP andBaryon Asymmetry Rupam Kalita, ∗ Debasish Borah, † and Mrinal Kumar Das ‡ Department of Physics, Tezpur University, Tezpur-784028, India
Abstract
We study a very specific type of neutrino mass and mixing structure based on the idea ofStrong Scaling Ansatz (SSA) where the ratios of neutrino mass matrix elements belonging to twodifferent columns are equal. There are three such possibilities, all of which are disfavored bythe latest neutrino oscillation data. We focus on the specific scenario which predicts vanishingreactor mixing angle θ and inverted hierarchy with vanishing lightest neutrino mass. Motivatedby several recent attempts to explain non-zero θ by incorporating corrections to a leading orderneutrino mass or mixing matrix giving θ = 0, here we study the origin of non-zero θ as well asleptonic Dirac CP phase δ CP by incorporating two different corrections to scaling neutrino massand mixing: one where type II seesaw acts as a correction to scaling neutrino mass matrix andthe other with charged lepton correction to scaling neutrino mixing. Although scaling neutrinomass matrix originating from type I seesaw predicts inverted hierarchy, the total neutrino massmatrix after type II seesaw correction can give rise to either normal or inverted hierarchy. However,charged lepton corrections do not disturb the inverted hierarchy prediction of scaling neutrino massmatrix. We further discriminate between neutrino hierarchies, different choices of lightest neutrinomass and Dirac CP phase by calculating baryon asymmetry and comparing with the observationsmade by the Planck experiment. PACS numbers: 14.60.Pq, 11.10.Gh, 11.10.Hi ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] . INTRODUCTION Origin of tiny neutrino masses and mixing is one of the most widely studied problemsin modern day particle physics. Since the standard model (SM) of particle physics failsto provide an explanation to neutrino masses and mixing, several well motivated beyondstandard model (BSM) frameworks have been proposed to account for the tiny neutrinomass observed by several neutrino oscillation experiments [1]. More recently, the neutrinooscillation experiments T2K [2], Double ChooZ [3], Daya-Bay [4] and RENO [5] have alsoconfirmed the earlier results and also made the measurement of neutrino parameters moreprecise. The latest global fit values for 3 σ range of neutrino oscillation parameters [6] areshown in table I. Another global fit study [7] reports the 3 σ values as shown in table II. Parameters Normal Hierarchy (NH) Inverted Hierarchy (IH) ∆ m − eV . − .
09 7 . − . | ∆ m | − eV . − .
607 2 . − . θ . − .
344 0 . − . θ . − .
643 0 . − . θ . − . . − . δ CP − π − π TABLE I: Global fit 3 σ values of neutrino oscillation parameters [6]Parameters Normal Hierarchy (NH) Inverted Hierarchy (IH) ∆ m − eV . − .
18 7 . − . | ∆ m | − eV . − .
65 2 . − . θ . − .
375 0 . − . θ . − .
643 0 . − . θ . − . . − . δ CP − π − π TABLE II: Global fit 3 σ values of neutrino oscillation parameters [7] σ range for the Dirac CP phase δ CP is 0 − π , there are two possible best fitvalues of it found in the literature: 306 o (NH), 254 o (IH) [6] and 254 o (NH), 266 o (IH) [7]. Itshould be noted that the neutrino oscillation experiments only determine two mass squareddifferences and hence the lightest neutrino mass is still unknown. Cosmology experimentshowever puts an upper bound on the sum of absolute neutrino masses P i | m i | < .
23 eV [8].Within this bound, he lightest neutrino mass can either be zero or very tiny (compared tothe other two) giving rise to a hierarchical pattern. Or, the lightest neutrino mass can be ofsame order as the other two neutrino masses giving rise to a quasi-degenerate type neutrinomass spectrum.Apart from the issue of lightest neutrino mass and hence the nature of neutrino masshierarchy, the CP violation in the leptonic sector is also not understood very well. Non-zeroCP violation in the leptonic sector can be very significant from cosmology point of viewas it could be the origin of matter-antimatter asymmetry in the Universe. The latest dataavailable from Planck mission constrain the baryon asymmetry [8] as Y B = (8 . ± . × − (1)Leptogenesis is one of the most promising dynamical mechanism of generating this observedbaryon asymmetry in the Universe by generating an asymmetry in the leptonic sector firstwhich later gets converted into baryon asymmetry through B + L violating electroweaksphaleron transitions [9]. As pointed out first by Fukugita and Yanagida [10], the outof equilibrium CP violating decay of heavy Majorana neutrinos provides a natural wayto create the required lepton asymmetry. The most notable feature of this mechanism isthat it connects two of the most widely studied problems in particle physics: the originof neutrino mass and the origin of baryon asymmetry. This idea has been explored withinseveral interesting BSM frameworks [11–13]. Recently such a comparative study was done tounderstand the impact of mass hierarchies, Dirac and Majorana CP phases on the predictionsfor baryon asymmetry in [14] within the framework of left-right symmetric models.Motivated by the quest for understanding the origin of neutrino masses and mixing andits relevance in cosmology, we recently studied several models [15] based on the idea ofgenerating non-zero θ , δ CP and matter-antimatter asymmetry by perturbing generic µ − τ symmetric neutrino mass matrix which can be explained dynamically within generic flavorsymmetry models. In these works, type I seesaw [16] is assumed to give rise to the µ − symmetric neutrino mass matrix with θ = 0 whereas type II seesaw [17] acts as aperturbation in order to generate the non-zero reactor mixing angle θ and also the DiracCP phase δ CP in some cases. In continuation of our earlier works on exploring the underlyingstructure of the neutrino mass matrix, in this work we consider a very specific neutrinomass matrix structure proposed few years back by the authors of [18]. The structure of theneutrino mass matrix is based on the idea of strong scaling Ansatz where certain ratios of theelements of neutrino mass matrix are equal. Out of three such possibilities (to be discussedin the next section), one of them predicts θ = 0 and an inverted hierarchy with vanishinglightest neutrino mass. Such a scaling neutrino mass matrix can also find its origin in specificflavor symmetry models as discussed in [18]. Several phenomenological studies based on theidea of SSA have appeared in [19]. The predictions for neutrino sector similar to the scalingansatz can also be found in models based on the abelian symmetry L e − L µ − L τ [20].Although inverted hierarchy as predicted by SSA can still be viable, vanishing reactormixing angle is no longer acceptable after the discovery of non-zero θ . Generation of non-zero θ in models based on the idea of SSA have appeared recently in [21]. In this work westudy two different possibilities of generating non-zero θ as well as Dirac CP phase δ CP byincorporating corrections to either the neutrino mass matrix or the leptonic mixing matrix,also known as the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. In both the cases weassume the origin of scaling neutrino mass matrix in type I seesaw. The required deviationfrom scaling can either come from a different seesaw mechanism (say, type II seesaw) or fromcharged lepton (CL) correction. The crucial difference between the two different scenario isthat in CL correction, the inverted hierarchy prediction of SSA remains intact whereas withthe combination of two different seesaw mechanism both normal and inverted hierarchies canemerge out of the total neutrino mass matrix. We first numerically fit the scaling neutrinomass matrix (from type I seesaw) with neutrino data on two mass squared differences andtwo angles θ , θ (as θ = 0). Then we derive the necessary perturbation to scalingneutrino mixing by comparing with the full neutrino oscillation data including non-zero θ . We further constrain the perturbation by demanding successful production of baryonasymmetry through the mechanism of leptogenesis.This paper is organized as follows. In section II, we briefly discuss the idea of scalingneutrino mass and mixing. In section III, we study the possible deviation from scaling withtype II seesaw as well as charged lepton corrections. In section IV, we briefly outline the4dea of leptogenesis and in section V we discuss our numerical analysis. We finally concludein section VI. II. STRONG SCALING ANSATZ
According to SSA, ratios of certain elements of the neutrino mass matrix are equal. Thestability of such a structure is also guaranteed by the fact that it is not affected by therenormalization group evolution (RGE) equations. Therefore, the scaling which is presentin the neutrino mass matrix at seesaw scale is also remains valid at the weak scale as werun the neutrino parameters from seesaw to weak scale under RGE. We denote the neutrinomass matrix and the leptonic mixing matrix U PMNS as M ν = m ee m eµ m eτ m µe m µµ m µτ m τe m τµ m ττ U PMNS = U e U e U e U µ U µ U µ U τ U τ U τ As noted by the authors of [18], there are three different types of SSA which can be writtenas m eµ m eτ = m µµ m µτ = m τµ m ττ = S (2) m ee m eτ = m µe m µτ = m τe m ττ = S ′ (3) m ee m eµ = m µe m µµ = m τe m τµ = S ′′ (4)Using (2), we can write the neutrino mass matrix as M ν = A B BS B D
DSBS DS DS , (5)Similarly, for the other two cases (3), (4) one can write down the neutrino mass matrix as M ν = A B AS ′ B D BS ′ AS ′ BS ′ AS ′ , (6)5nd M ν = A AS ′′ B AS ′′ AS ′′ BS ′′ B BS ′′ D , (7)One interesting property of the first scaling mass matrix (5) is that it has one of its eigenvalue m zero (rank 2 matrix) and diagonalization of this matrix gives U e = 0. Thus, it givesrise to inverted hierarchy of neutrino mass with θ = 0. Although such a scenario is nowruled out after the discovery of non-zero θ , there still exists the possibility of generatingnon-zero θ by adding perturbations to the scaling neutrino mass and mixing, given the factthat θ is still small compared to the other two mixing angles. However, diagonalizationof the second scaling matrix (6) gives U µ = 0 or U µ = 0 depending on the hierarchy ofneutrino masses. Similarly, diagonalization of the third scaling matrix (7) gives U τ = 0or U τ = 0. The predictions of both the scaling mass matrices obtained using (3) and(4) are not phenomenologically viable. Even if we assume the validity of these two scalingmass matrices at tree level, they will require large corrections in order to generate the correctmixing matrix. Leaving these to future studies, here we focus on the possibility of generatingnon-zero U e and hence non-zero θ by incorporating different corrections to leading orderscaling neutrino mass matrix given by (5). TABLE III: M RR (in GeV) for Type II seesaw correction with δ CP = π MODEL IH m = 0 . eV . × . × − . × i . × − . × i . × − . × i . × − . × i − . × − . × i . × − . × i − . × − . × i . × + 3 . × i . × . × − . × i . × − . × i . × − . × i . × − . × i − . × − . × i . × − . × i − . × − . × i . × + 9 . × i . × . × − . × i . × − . × i . × − . × i . × − . × i − . × − . × i . × − . × i − . × − . × i . × + 3 . × i ABLE IV: M RR (in GeV) for Type II seesaw correction with δ CP = π MODEL IH m = 10 − eV . × . × − . × i . × − . × i . × − . × i . × − . × i − . × − . × i . × − . × i − . × − . × i . × + 5 . × i . × . × − . × i . × − . × i . × − . × i . × − . × i − . × − . × i . × − . × i − . × − . × i . × + 1 . × i . × . × − . × i . × − . × i . × − . × i . × − . × i − . × − . × i . × − . × i − . × − . × i . × + 5 . × i TABLE V: M RR (in GeV) for Type II seesaw correction with δ CP = π MODEL NH m = 0 . eV . × . × − . × i . × − . × i . × − . × i . × − . × i . × − . × i . × − . × i . × − . × i . × + 3 . × i . × . × − . × i . × − . × i . × − . × i . × − . × i . × − . × i . × − . × i . × − . × i . × + 1 . × i . × . × − . × i . × − . × i . × − . × i . × − .i . × − . i . × − . × i . × − . i . × + 364514 . i III. DEVIATIONS FROM SCALING
As discussed in the previous section, the neutrino mass matrices based on the idea of SSAdo not give rise to the correct neutrino mixing pattern. Therefore, the scaling neutrino mass7
ABLE VI: M RR (in GeV) for Type II seesaw correction with δ CP = π MODEL NH m = 10 − eV − . × . × − . × i . × − . × i . × − . × i . × − . × i . × − . × i . × − . × i . × − . × i . × + 5 . × i − . × . × − . × i . × − . × i . × − . × i . × − . × i . × − . × i . × − . × i . × − . × i . × + 1 . × i − . × . × − . × i . × − . × i . × − . × i . × − . × i . × − . × i . × − . × i . × − . × i . × + 5 . × i or mixing matrix has to be corrected in order to have agreement with neutrino oscillationdata. Here we consider two different sources of such corrections to SSA which not only cangive rise to correct neutrino mixing but also have different predictions for neutrino masshierarchy, leptonic Dirac CP phase as well as baryon asymmetry. A. Deviation from Scaling with Type II Seesaw
Type II seesaw mechanism is the extension of the standard model with a scalar field ∆ L which transforms like a triplet under SU (2) L and has U (1) Y charge twice that of lepton dou-blets. Such a choice of gauge structure allows an additional Yukawa term in the Lagrangiangiven by f ij (cid:0) ℓ TiL
C iσ ∆ L ℓ jL (cid:1) . The triplet can be represented as∆ L = δ + L / √ δ ++ L δ L − δ + L / √ The scalar Lagrangian of the standard model also gets modified after the inclusion of thistriplet. Apart from the bilinear and quartic coupling terms of the triplet, there is onetrilinear term as well involving the triplet and the standard model Higgs doublet. From theminimization of the scalar potential, the neutral component of the triplet is found to acquire8
ABLE VII: m LR (in GeV) for Type II seesaw correction with δ CP = π a = b IH m = 0 . eV . − . i . − . i . − . i . − . i . − . i − . − . i . − . i − . − . i − . − . i . − . i . − . i . − . i . − . i − . − . i − . − . i . − . i − . − . i − . − . i . − . i . − . i . − . i . − . i − . − . i − . − . i . − . i − . − . i − . − . i a vacuum expectation value (vev) given by h δ L i = v L = µ ∆ H h φ i M (8)where φ = v is the neutral component of the electroweak Higgs doublet with vev approxi-mately 10 GeV. The trilinear coupling term µ ∆ H and the mass term of the triplet M ∆ canbe taken to be of same order. Thus, M ∆ has to be as high as 10 GeV to give rise to tinyneutrino masses without any fine-tuning of the dimensionless couplings f ij . In the presenceof both type I and type II seesaw the neutrino mass can be written as M ν = m II + m I (9)where m II = f v L is the type II seesaw contribution and m I = m LR M − RR m TLR is the type I seesaw term with m LR , M RR being Dirac and Majorana neutrino mass matrices respectively.We assume the type I seesaw to give rise to scaling neutrino mass matrix. We then introducethe type II seesaw term as a correction to the scaling neutrino mass matrix and constrainthe type II seesaw term from the requirement of generating correct value of θ as well as9 ABLE VIII: m LR (in GeV) for Type II seesaw correction with δ CP = π a = b IH m = 10 − eV . . i . . i . . i . . i . − . i . − . i . . i . − . i . − . i . . i . . i . . i . . i . − . i . − . i . . i . − . i . − . i . . i . . i . . i . . i . − . i . − . i . . i . − . i . − . i baryon asymmetry. One interesting property of scaling is that type I seesaw can give rise toscaling neutrino mass matrix irrespective of the right handed Majorana mass matrix M RR ,if Dirac neutrino mass matrix m LR obeys scaling. As we discuss in section V, this propertyof scaling allows us to derive the type II seesaw correction as well as the Dirac neutrino massmatrix. B. Deviation from Scaling with Charged Lepton Correction
The scaling neutrino mass matrix we discuss here, given by equation (5) predicts m = 0and θ = 0. In the previous subsection, type II seesaw correction to scaling neutrino masswas discussed which not only can result in non-zero θ but also can give rise to non-zero m .Since, an inverted hierarchical neutrino mass pattern with m = 0 is still allowed by neutrinooscillation data, one can generate non-zero θ by incorporating corrections to the leptonicmixing matrix only without affecting the scaling neutrino mass matrix. The PMNS leptonicmixing matrix is related to the diagonalizing matrices of neutrino and charged lepton mass10 ABLE IX: m LR (in GeV) for Type II seesaw correction with δ CP = π b = d IH m = 0 . eV − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i matrices U ν , U l respectively, as U PMNS = U † l U ν (10)The PMNS mixing matrix can be parametrized as U PMNS = c c s c s e − iδ − s c − c s s e iδ c c − s s s e iδ s c s s − c c s e iδ − c s − s c s e iδ c c (11)where c ij = cos θ ij , s ij = sin θ ij and δ is the Dirac CP phase. If U ν originates from scalingneutrino mass matrix given by type I seesaw, then for diagonal charged lepton mass matrix,both the reactor mixing angle θ and the leptonic Dirac CP phase δ vanish. However, anon-trivial charged lepton mixing matrix U l can result in correct leptonic mixing matrix U PMNS even if U ν predicts θ = 0. As discussed in the section on numerical analysis V,we constrain the charged lepton mass matrix by demanding the generation of correct θ required by neutrino oscillation data and also the correct value of δ CP in order to producecorrect baryon asymmetry. 11 ABLE X: m LR (in GeV) for Type II seesaw correction with δ CP = π b = d IH m = 10 − eV − . − . i . − . i . − . i . − . i . − . i . − . i . − . i . − . i . − . i − . − . i . − . i . − . i . − . i . − . i . − . i . − . i . − . i . − . i . . i − . . i − . . i − . . i − . . i − . . i − . . i − . . i − . . i IV. LEPTOGENESIS
As mentioned earlier, leptogenesis is the mechanism where a non-zero lepton asymmetryis generated by out of equilibrium, CP violating decay of a heavy particle which later getsconverted into baryon asymmetry through electroweak sphaleron transitions. In a modelwith both type I and type II seesaw mechanisms at work, such lepton asymmetry can begenerated either by the decay of the right handed neutrinos or the heavy scalar triplet.For simplicity, here we consider only the right handed neutrino decay as the source oflepton asymmetry. One can justify this assumption in those models where type I seesawis dominating and type II seesaw is sub-leading giving rise to a Higgs triplet heavier thanthe lightest right handed neutrino. The lepton asymmetry from the decay of right handedneutrino into leptons and Higgs scalar in a model with only type I seesaw is given by ǫ N k = X i Γ( N k → L i + H ∗ ) − Γ( N k → ¯ L i + H )Γ( N k → L i + H ∗ ) + Γ( N k → ¯ L i + H ) (12)In a hierarchical pattern of heavy right handed neutrinos, it is sufficient to consider thedecay of the lightest right handed neutrino N . Following the notations of [12], the lepton12 ABLE XI: m LR (in GeV) for Type II seesaw correction with δ LR = π a = b NH m = 0 . eV . − . i . − . i . − . i . − . i − . − . i − . − . i . − . i − . − . i − . − . i . − . i . − . i . − . i . − . i − . − . i − . − . i . − . i − . − . i − . − . i . − . i . − . i . − . i . − . i − . − . i − . − . i . − . i − . − . i − . − . i asymmetry arising from the decay of N in the presence of type I seesaw only can be writtenas ǫ α = 18 πv m † LR m LR ) X j =2 , Im[( m ∗ LR ) α ( m † LR m LR ) j ( m LR ) αj ] g ( x j )+ 18 πv m † LR m LR ) X j =2 , Im[( m ∗ LR ) α ( m † LR m LR ) j ( m LR ) αj ] 11 − x j (13)where v = 174 GeV is the vev of the Higgs doublet responsible for breaking the electroweaksymmetry, g ( x ) = √ x (cid:18) − x − (1 + x )ln 1 + xx (cid:19) and x j = M j /M . The second term in the expression for ǫ α above vanishes when summedover all the flavors α = e, µ, τ . The sum over all flavors can be written as ǫ = 18 πv m † LR m LR ) X j =2 , Im[( m † LR m LR ) j ] g ( x j ) (14)13 ABLE XII: m LR (in GeV) for Type II seesaw correction with δ CP = π a = b NH m = 10 − eV . − . i . − . i . − . i . − . i − . . i − . . i . − . i − . . i − . . i . − . i . − . i . − . i . − . i − . . i − . . i . − . i − . . i − . . i . − . i . − . i . − . i . − . i − . . i − . . i . − . i − . . i − . . i From the lepton asymmetry ǫ given by the expression above, the corresponding baryonasymmetry can be obtained by Y B = cκ ǫg ∗ (15)through sphaleron processes [9] at electroweak phase transition. Here the factor c is measureof the fraction of lepton asymmetry being converted into baryon asymmetry and is approxi-mately equal to − .
55. On the other hand, κ is the dilution factor due to wash-out processwhich erase the asymmetry generated and can be parametrized as [22] − κ ≃ √ . K exp[ − / (3(0 . K ) . )] , for K ≥ ≃ . K (ln K ) . , for 10 ≤ K ≤ ≃ √ K + 9 , for 0 ≤ K ≤ . (16)where K is given as K = Γ H ( T = M ) = ( m † LR m LR ) M πv M P l . √ g ∗ M ABLE XIII: m LR (in GeV) for Type II seesaw correction with δ CP = π b = d NH m = 0 . eV − . − . i . − . i − . − . i . − . i . − . i − . − . i − . − . i − . − . i − . − . i . . i . . i . . i . . i . . i . . i . . i . . i . . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i Here Γ is the decay width of N and H ( T = M ) is the Hubble constant at temperature T = M . The factor g ∗ is the effective number of relativistic degrees of freedom at temperature T = M and is approximately 110.We note that the lepton asymmetry shown in equation (14) is obtained by summingover all the flavors α = e, µ, τ . A non-vanishing lepton asymmetry is generated only whenthe right handed neutrino decay is out of equilibrium. Otherwise both the forward andthe backward processes will happen at the same rate resulting in a vanishing asymmetry.Departure from equilibrium can be estimated by comparing the interaction rate with theexpansion rate of the Universe. At very high temperatures ( T ≥ GeV) all charged leptonflavors are out of equilibrium and hence all of them behave similarly resulting in the oneflavor regime. However at temperatures
T < GeV (
T < GeV), interactions involvingtau (muon) Yukawa couplings enter equilibrium and flavor effects become important [23].Taking these flavor effects into account, the final baryon asymmetry is given by Y flavorB = − g ∗ [ ǫ η (cid:18) m (cid:19) + ǫ τ η (cid:18) m τ (cid:19) ]15 ABLE XIV: m LR (in GeV) for Type II seesaw correction with δ CP = π b = d NH m = 10 − eV . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i TABLE XV: Values of δ CP giving correct Y B for inverted hierarchy with Type II seesaw correctionto scaling Model δ CP (radian) for a=b δ CP (radian) for b=d m = 0 .
065 eV 0 . − . , . − . . − . , . − . m = 10 − eV – 2 . − . , . − . m = 0 .
065 eV 3 . − . , . − . . − . , . − . m = 10 − eV – 0 . − . , . − . m = 0 .
065 eV – – m = 10 − eV – – Y flavorB = − g ∗ [ ǫ e η (cid:18) m e (cid:19) + ǫ µ η (cid:18) m µ (cid:19) + ǫ τ η (cid:18) m τ (cid:19) ]where ǫ = ǫ e + ǫ µ , ˜ m = ˜ m e + ˜ m µ , ˜ m α = ( m ∗ LR ) α ( m LR ) α M . The function η is given by η ( ˜ m α ) = "(cid:18) ˜ m α . × − eV (cid:19) − + (cid:18) . × − eV˜ m α (cid:19) − . − In the presence of an additional scalar triplet, the right handed neutrino can also decay16 Y B δ m =0.065 eV, a=b -0.2-0.15-0.1-0.05 0 0.05 0.1 0.15 0.2 0 1 2 3 4 5 6 7 Y B δ m =10 -6 eV, a=b-6-5-4-3-2-1 0 1 2 3 0 1 2 3 4 5 6 7 Y B δ m =0.065 eV, b=d -4-3-2-1 0 1 2 3 4 0 1 2 3 4 5 6 7 Y B δ m =10 -6 eV, b=d FIG. 1: Baryon asymmetry in one flavor regime as a function of δ CP for inverted hierarchy withtype II seesaw correction to scalingTABLE XVI: Values of δ CP giving correct Y B for normal hierarchy with Type II seesaw correctionto scaling Model δ CP (radian) for a=b δ CP (radian) for b=d m = 0 .
07 eV 3 . − . , . − . . − . , . − . m = 10 − eV 0 . − . , . − . m = 0 .
07 eV 0 . − . , . − . . − . , . − . m = 10 − eV – –3 Flavor m = 0 .
07 eV – – m = 10 − eV – – through a virtual triplet. The contribution of this diagram to lepton asymmetry can beestimated as [24] ǫ α ∆1 = − M πv P j =2 , Im[( m LR ) j ( m LR ) α ( M II ∗ ν ) jα ] P j =2 , | ( m LR ) j | (17)We use these expressions to calculate the baryon asymmetry in our numerical analysis section17 Y B δ m =0.065 eV, a=b -0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 0.04 0 1 2 3 4 5 6 7 Y B δ m =10 -6 eV, a=b-20-15-10-5 0 5 10 15 20 0 1 2 3 4 5 6 7 Y B δ m =0.065 eV, b=d -4-3-2-1 0 1 2 3 4 0 1 2 3 4 5 6 7 Y B δ m =10 -6 eV, b=d FIG. 2: Baryon asymmetry in two flavor regime as a function of δ CP for inverted hierarchy withtype II seesaw correction to scaling discussed below. V. NUMERICAL ANALYSIS
We first diagonalize the scaling neutrino mass matrix (5) and find its eigenvalues m = 12 S (cid:16) D + AS + DS − p A S − ADS (1 + S ) + D (1 + S ) + 4 B S (1 + S ) (cid:17) m = 12 S (cid:16) D + AS + DS + p A S − ADS (1 + S ) + D (1 + S ) + 4 B S (1 + S ) (cid:17) m = 0We numerically evaluate the four parameters A, B, D, S by equating m , m to two neutrinomass squared differences ∆ m , ∆ m and two non-zero mixing angles to θ , θ .Now, in the case of type II seesaw correction to scaling neutrino mixing, we assume thecharged leptons mass matrix to be diagonal so that U PMNS = U ν . Therefore, we can write(9) as U PMNS .m diag ν .U T PMNS = m II + m I (18)18 Y B δ m =0.065 eV, a=b -0.001-0.0008-0.0006-0.0004-0.0002 0 0.0002 0.0004 0.0006 0.0008 0.001 0 1 2 3 4 5 6 7 Y B δ m =10 -6 eV, a=b-0.006-0.004-0.002 0 0.002 0.004 0.006 0 1 2 3 4 5 6 7 Y B δ m =0.065 eV, b=d -0.0015-0.001-0.0005 0 0.0005 0.001 0.0015 0 1 2 3 4 5 6 7 Y B δ m =10 -6 eV, b=d FIG. 3: Baryon asymmetry ratio in three flavor regime as a function of δ CP for inverted hierarchywith type II seesaw correction to scaling where m diag ν is the diagonal neutrino mass matrix given by m diag ν =diag( m , p m + ∆ m , p m + ∆ m ) for normal hierarchy and m diag ν =diag( p m + ∆ m − ∆ m , p m + ∆ m , m ) for inverted hierarchy. The type I see-saw mass matrix m I always gives inverted hierarchy whereas m diag ν can give either normalor inverted hierarchy depending on the type II seesaw contribution m II . In the minimalextension of the standard model with type I and type II seesaw mechanisms, the type Iseesaw term depends upon m LR and M RR whereas type II seesaw depends upon the vev ofthe neutral component of Higgs triplet. Since m LR , M RR as well as the type II seesaw termcan be chosen by hand, such a framework is difficult to constrain due to too many numberof free parameters. However, in a specific class of models called left right symmetric models(LRSM) [25], the type II seesaw term is directly proportional to M RR thereby decreasingthe number of free parameters compared to the minimal extension. Another reason forchoosing the framework of LRSM is that here we can find the right handed Majorana massmatrix M RR from the type II seesaw perturbation. However, for a given Dirac neutrinomass matrix m LR , one can not find M RR from the type I seesaw formula alone as the inverse19 Y B δ m =0.07 eV, a=b -6-4-2 0 2 4 6 0 1 2 3 4 5 6 7 Y B δ m =10 -6 eV, a=b-10-8-6-4-2 0 2 4 0 1 2 3 4 5 6 7 Y B δ m =0.07 eV, b=d -0.3-0.2-0.1 0 0.1 0.2 0.3 0 1 2 3 4 5 6 7 Y B δ m =10 -6 eV, b=d FIG. 4: Baryon asymmetry in one flavor regime as a function of δ CP for normal hierarchy withtype II seesaw correction to scaling of type I seesaw mass matrix does not exist due to its scaling property ( m = 0). In LRSMwe can write equation (18) as U PMNS .m diag ν .U T PMNS = γ (cid:18) M W v R (cid:19) M RR + m I (19)where γ is a dimensionless parameter, M W is the W boson mass and v R is the scale of leftright symmetry breaking. Since m I has been numerically evaluated as the leading orderscaling neutrino mass matrix, type II contribution can now be evaluated as a function ofleptonic Dirac CP phase δ CP and the lightest neutrino mass m (NH), m (IH), the twounknowns on the left hand side of the above equation. It should be noted that, we haveomitted the Majorana phases in this discussion. After determining the type II seesaw termand hence M RR , we use it in the type I seesaw term to find out the Dirac neutrino massmatrix m LR . Here we use the already mentioned special property of scaling neutrino massmatrix originating from type I seesaw: if Dirac neutrino mass matrix m LR obeys scaling,then m I obeys scaling irrespective of the structure of M RR . Therefore, we use the scaling20 Y B δ m =0.07 eV, a=b -0.15-0.1-0.05 0 0.05 0.1 0.15 0 1 2 3 4 5 6 7 Y B δ m =10 -6 eV, a=b-15-10-5 0 5 10 15 0 1 2 3 4 5 6 7 Y B δ m =0.07 eV, b=d -0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 0.04 0 1 2 3 4 5 6 7 Y B δ m =10 -6 eV, b=d FIG. 5: Baryon asymmetry in two flavor regime as a function of δ CP for normal hierarchy withtype II seesaw correction to scaling Dirac neutrino mass matrix given by m LR = a b bc b d dcbc dc dc We use the above m LR and the already derived right handed Majorana mass matrix M RR in the type I seesaw formula and equate it to the scaling neutrino mass matrix evalutednumerically earlier.We use two different choices of lightest neutrino mass in order to show the effect ofhierarchy. For inverted hierarchy we take m = 0 .
065 eV, 10 − eV and for normal hierar-chy we take m = 0 .
07 eV, 10 − eV. After choosing the lightest neutrino mass, the onlyundetermined parameters in the equation (19) are δ CP , γ and v R . Choosing generic orderone coupling γ , one can now write M RR in terms of δ CP and v R . We choose the left rightsymmetry breaking scale v R in a way which keeps the lightest right handed neutrino in theappropriate flavor regime of leptogenesis. After we find M RR in terms of δ CP , we use it in21 Y B δ m =0.07 eV, a=b -0.0004-0.0003-0.0002-0.0001 0 0.0001 0.0002 0.0003 0.0004 0 1 2 3 4 5 6 7 Y B δ m =10 -6 eV, a=b-0.006-0.004-0.002 0 0.002 0.004 0.006 0 1 2 3 4 5 6 7 Y B δ m =0.07 eV, b=d -0.0002-0.00015-0.0001-5e-05 0 5e-05 0.0001 0.00015 0.0002 0 1 2 3 4 5 6 7 Y B δ m =10 -6 eV, b=d FIG. 6: Baryon asymmetry in three flavor regime as a function of δ CP for normal hierarchy withtype II seesaw correction to scaling the type I seesaw formula with the m LR obeying scaling as shown above. To simplify thenumerical calculation further, we assume equality between some parameters in m LR : a = b and b = d . The other choice a = d does not give us any solution. We also do not assumeequality of the parameter c with others as c can be found independently of a, b, d when weequate the type I seesaw formula with the numerically evaluated type I seesaw mass matrixof scaling type. The numerical form of the right handed neutrino mass matrix M RR forall the cases discussed are shown in table III, IV, V and VI. Similarly, the Dirac neutrinomass matrices are listed in table VII, VIII, IX, X, XI, XII, XIII and XIV. Although theyare, in general, complicated functions of δ CP , we have used a specific value of δ CP = π/ δ CP continuously and show the variation of Baryon asymmetry in figure 1, 2, 3, 4, 5 and6. It should be noted that the type II seesaw corrections to scaling have been consideredwithin the framework of LRSM where SU (2) R gauge interactions can give rise to sizeablewash-out effects erasing the asymmetry produced. As noted in [26], such wash-out effectscan be neglected by choosing a high value of v R such that M /v R < − is satisfied.22 ABLE XVII: Charge lepton diagonalizing matrix for δ CP = π U l . − . i . . i − . . i − . . i . − . i − . − . i . . i . . i . . i TABLE XVIII: M RR (in GeV) with charged lepton correction to scalingModel M RR (GeV)1 Flavor × ×
00 0 3 × × ×
00 0 3 × × ×
00 0 1 × In the second mechanism we adopt to give correction to the scaling neutrino mixing, wedo not add corrections to the neutrino mass matrix originating from type I seesaw, but in-corporate corrections to the neutrino mixing matrix originating from charged lepton mixing.Diagonalizing the scaling neutrino mass matrix from type I seesaw matrix gives U ν whichis related to the leptonic mixing matrix U PMNS through the charged lepton diagonalizingmatrix U l . We first numerically evaluate U ν by using the best fit values of neutrino masssquared differences and two mixing angles θ , θ . We also substitute the best fit values of23 ABLE XIX: m LR (in GeV) for δ CP = π with charged lepton correction to scaling NH m = 10 − eV − . . i . − . i − . − . i . − . i − . . i . . i − . − . i . . i − . . i − . . i . − . i − . − . i . − . i − . . i . . i . − . i − . . i − . . i . − . i − . . i . . i − . . i . − . i − . − . i − . . i . − . i . − . i TABLE XX: Values of δ CP giving correct Y B with charge lepton correction to scaling Model δ CP (radian) . − . , . − . . − . , . − . . − . , . − . neutrino mixing angles in PMNS mixing matrix (11) and then compute the charged leptondiagonalizing matrix as U l = U ν U † PMNS
We keep the Dirac CP phase δ CP as free parameter so that U l is a function of it. Thenumerical form of U l for δ CP = π/ Y B δ Y B δ Y B δ FIG. 7: Baryon asymmetry in one, two and three flavor regimes as a function of δ CP with chargedlepton correction to scaling as m LR = U l .m LR .U † l (20)where m LR is the scaling type Dirac neutrino mass matrix we choose earlier. We choosea diagonal form of M RR while keeping the lightest right handed neutrino mass M in theappropriate flavor regime of leptogenesis and varying the heavier right handed neutrinomasses M , M between M and the grand unified theory scale M GUT ∼ GeV. Forsuch a choice of M RR , we numerically evaluate the parameters in m LR by equating the typeI seesaw term ( m LR ) M − RR ( m LR ) T to the numerically fitted type I scaling neutrino massmatrix. The numerical values of M RR which give baryon asymmetry closest to the observedin each flavor regime are shown in table XVIII. The numerical form of Dirac neutrino massmatrices in each flavor regime for δ CP = π/ δ CP are shown in figure 7. The values of Dirac CP phasewhich give rise to correct baryon asymmetry are listed in table XX.25 I. RESULTS AND CONCLUSION
We have studied a specific type of neutrino mass matrix based on the idea of strongscaling ansatz where the ratio of neutrino mass matrix elements belonging to two differentcolumns are equal. Out of three such possibilities, we focus on a particular scaling neutrinomass matrix which predicts zero values of reactor mixing angle θ . This choice was moti-vated by several recent works where the leading order neutrino mass matrix obeying certainsymmetries predict θ = 0 and suitable corrections to the neutrino mass matrix or leptonicmixing matrix give rise to small but non-zero values of θ . In this work, we have assumedtype I seesaw to give rise to scaling neutrino mass matrix which (in the diagonal chargedlepton basis) gives θ = 0 and inverted hiearchical mass pattern with m = 0. Then weconsider two different possible corrections to scaling: one with type II seesaw which givesrise to deviations from both θ = 0 and m = 0, the other with charged lepton correctionwhich gives non-zero θ while keeping m = 0. We also assume both the corrections togive rise to non-trivial Dirac CP phase δ CP as well. In both the cases, we first numeri-cally evaluate the type I seesaw scaling neutrino mass matrix by using the best fit values ofneutrino mass squared differences and two mixing angles: solar and atmospheric. We thencalculate the necessary corrections to scaling neutrino mass and mixing by keeping the DiracCP phase as free parameter. We further constrain the Dirac CP phase by calculating thebaryon asymmetry through the mechanism of leptogenesis and comparing with the observedBaryon asymmetry. The important results we have obtained in the case of type II seesawcorrection to scaling can be summarized as: • Type II seesaw correction to scaling neutrino mass matrix with θ = 0 , m = 0 canresult in both normal as well as inverted hierarchy with non-zero θ as well as non-trivial Dirac CP phase δ CP . • For inverted hierarchy with a = b that is m LR (11) = m LR (12), correct values of baryonasymmetry is obtained through the mechanism of leptogenesis only when the lightestneutrino mass m is of same order as the heavier ones m , m . • For inverted hierarchy with b = d that is m LR (12) = m LR (22), both large and mildhierarchy among neutrino masses give rise to correct baryon asymmetry through lep-togenesis. 26 For normal hierarchy with m LR (11) = m LR (12) , both large and mild hierarchy amongneutrino masses can give rise to correct baryon asymmetry in the one flavor regime.In the two flavor regime however, the lightest neutrino mass m should be of sameorder as m , m to give correct baryon asymmetry. • For normal hierarchy with m LR (12) = m LR (22), the lightest neutrino mass m shouldbe of same order as m , m to give correct baryon asymmetry in both one and twoflavor regimes. • Observed baryon asymmetry can not be generated in the three flavor regime of lepto-genesis in this framework.Similarly, the important results in the case of charged lepton correction to scaling are: • Charged lepton correction to scaling neutrino mixing predicts only inverted hierarchywith m = 0, but gives rise to correct values of θ and non-trivial δ CP . • Correct baryon asymmetry can be obtained through leptogenesis for one, two andthree flavor regimes if δ CP is restricted to certain range of values.Since the Dirac CP phase is restricted in all these cases discussed, from the demand of pro-ducing the correct baryon asymmetry, future determination of δ CP should be able to shedsome light on these scenarios. Future experiments may however, measure a different valueof δ CP than the ones which give correct baryon asymmetry through the mechanism of lep-togenesis in the models we have studied here. This will by no means rule out the neutrinomass models based on strong scaling ansatz we discuss, but will only hint at a differentsource of baryon asymmetry than the one discussed in our work. Similarly, determination ofneutrino mass hierarchy in neutrino oscillation experiments will further constrain the modelsand only charged lepton correction to scaling may not be sufficient to reproduce the correctneutrino data if inverted hierarchy gets disfavored by experiments. From theoretical pointof view, such scaling neutrino mass matrix can find a dynamical origin within discrete flavorsymmetry models as pointed out by [18]. Since scaling is not affected by renormalizationgroup running, additional physics are required in order to produce correct low energy neu-trino oscillation data. Undisturbed by such running effects, scaling can be valid all the wayfrom grand unified theory scale down to the TeV scale, where new physics affects like Higgs27riplet in type II seesaw can give rise to the necessary correction to scaling neutrino massmatrix. Although we have studied only one particular type of scaling neutrino mass matrixgiving θ = 0 , m = 0, the other two possibile scaling mass matrices could also give riseto correct neutrino phenomenology if suitable corrections are incorporated, which is left forour future studies. VII. ACKNOWLEDGEMENT
The work of M. K. Das is partially supported by the grant no. 42-790/2013(SR) fromUniversity Grants Commission, Government of India. [1] S. Fukuda et al. (Super-Kamiokande), Phys. Rev. Lett. , 5656 (2001), hep-ex/0103033; Q.R. Ahmad et al. (SNO), Phys. Rev. Lett. , 011301 (2002), nucl-ex/0204008; Phys. Rev.Lett. , 011302 (2002), nucl-ex/0204009; J. N. Bahcall and C. Pena-Garay, New J. Phys. ,63 (2004), hep-ph/0404061; K. Nakamura et al., J. Phys. G37 , 075021 (2010).[2] K. Abe et al. [T2K Collaboration], Phys. Rev. Lett. , 041801 (2011), [arXiv:1106.2822[hep-ex]].[3] Y. Abe et al., Phys. Rev. Lett. , 131801 (2012), [arXiv:1112.6353 [hep-ex]].[4] F. P. An et al. [DAYA-BAY Collaboration], Phys. Rev. Lett. , 171803 (2012),[arXiv:1203.1669 [hep-ex]].[5] J. K. Ahn et al. [RENO Collaboration], Phys. Rev. Lett. , 191802 (2012),[arXiv:1204.0626][hep-ex]].[6] M. C. Gonzalez-Garcia, M. Maltoni and T. Schwetz, JHEP , 052 (2014)[arXiv:1409.5493[hep-ph]].[7] D. V. Forero, M. Tortola and J. W. F. Valle, Phys. Rev.
D90 , 093006(2014)[arXiv:1405.7540[hep-ph]].[8] P. A. R. Ade et al. [Planck Collaboration], Astron. Astrophys. , A16 (2014).[9] V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov, Phys. Lett.
B155 , 36 (1985).[10] M. Fukugita and T. Yanagida, Phys. Lett.
B174 , 45 (1986).[11] J. Ellis, S. Lola and D. V. Nanopoulos, Phys. Lett.
B452 , 87 (1999); G. Lazarides and . D. Vlachos, Phys. Lett. B459 , 482 (1999); M. S. Berger and B. Brahmachari, Phys. Rev.
D60 , 073009 (1999); M. S. Berger, Phys. Rev.
D62 , 013007 (2000); W. Buchm¨uller andM. Plumacher, Int. J. Mod. Phys.
A15 , 5047 (2000); K. Kang, S. K. Kang and U. Sarkar,Phys. Lett.
B486 , 391 (2000); H. Goldberg, Phys. Lett.
B474 , 389 (2000); R. Jeannerot,S. Khalil and G. Lazarides, Phys. Lett.
B506 , 344 (2001); D. Falcone and F. Tramontano,Phys. Rev.
D63 , 073007 (2001); D. Falcone and F. Tramontano, Phys. Lett.
B506 , 1 (2001);H. B. Nielsen and Y. Takanishi, Phys. Lett.
B507 , 241 (2001); E. Nezri and J. Orloff, JHEP , 020 (2003).[12] A. S. Joshipura, E. A. Paschos and W. Rodejohann, Nucl. Phys.
B611 , 227 (2001).[13] S. Davidson, E. Nardi and Y. Nir, Phys. Rept. , 105 (2008).[14] D. Borah and M. K. Das, Phys. Rev.
D90 , 015006 (2014).[15] D. Borah, Nucl. Phys.
B876 , 575 (2013); D. Borah, S. Patra and P. Pritimita, Nucl. Phys.
B881 , 444 (2014); D. Borah, Int. J. Mod. Phys.
A29 , 1450108 (2014); M. Borah, D. Bo-rah, M. K. Das and S. Patra, Phys. Rev.
D90 , 095020 (2014); R. Kalita and D. Borah,arXiv:1410.8437.[16] P. Minkowski, Phys. Lett.
B67 , 421 (1977); M. Gell-Mann, P. Ramond, and R. Slansky(1980), print-80-0576 (CERN); T. Yanagida (1979), in Proceedings of the Workshop on theBaryon Number of the Universe and Unified Theories, Tsukuba, Japan, 13-14 Feb 1979; R.N. Mohapatra and G. Senjanovic, Phys. Rev. Lett , 912 (1980); J. Schechter and J. W.F. Valle, Phys. Rev. D22 , 2227 (1980).[17] R. N. Mohapatra and G. Senjanovic, Phys. Rev.
D23 , 165 (1981); G. Lazarides, Q. Shafi andC Wetterich, Nucl. Phys.
B181 , 287 (1981); C. Wetterich, Nucl. Phys.
B187 , 343 (1981);J. Schechter and J. W. F. Valle, Phys. Rev.
D25 , 774 (1982); B. Brahmachari and R. N.Mohapatra, Phys. Rev.
D58 , 015001 (1998); R. N. Mohapatra, Nucl. Phys. Proc. suppl. ,257 (2005); S. Antusch and S. F. King, Phys. Lett.
B597 , (2), 199 (2004).[18] R. N. Mohapatra and W. Rodejohann, Phys. Lett.
B 644 , 59 (2007).[19] A. Blum, R. N. Mohapatra and W. Rodejohann, Phys. Rev.
D76 , 053003(2007); A. S. Jo-shipura, W. Rodejohann, Phys. Lett.
B678 , 276 (2009); M. Obara, arXiv:0712.2628; M.Chakraborty, H. Z. Devi, A. Ghosal, arXiv:1410.3276; W. Rodejohann, Prog. Part. Nucl.Phys. , 321 (2010).[20] W. Grimus, L. Lavoura, J. Phys. G31 , 683 (2005); W. Grimus, L. Lavoura, Phys. Rev
D62 , D62 , 093011 (2000).[21] B. Adhikary, M. Chakrabarty, A. Ghosal, Phys. Rev.
D86 , 013015 (2012); S. Verma , Phys.Lett.
B714 , 92 (2012); M. Yasue, Phys. Rev.
D86 , 116011 (2012); M. Yasue, arXiv:1209.1866;S. Dev, R. R. Gautam, L. Singh, Phys. Rev
D89 , 013006 (2014).[22] E. W. Kolb and M. S. Turner, The Early Universe, Frontiers in Physics, Vol. 69, Addison-Wesley, Redwood, 1990; A. Pilaftsis, Int. J. Mod. Phys. , 1811 (1999); E. A. Paschos andM. Flanz, Phys. Rev. D58 , 113009 (1998).[23] R. Barbieri, P. Creminelli, A. Strumia and N. Tetradis, Nucl. Phys.
B575 , 61 (2000); A.Abada, S. Davidson, F. -X. Josse-Michaux, M. Losada and A. Riotto, JCAP , 004 (2006);E. Nardi, Y. Nir, E. Roulet and J. Racker, JHEP , 164 (2006); A. Abada, S. Davidson,A. Ibarra, F. -X. Josse-Michaux, M. Losada and A. Riotto, JHEP , 010 (2006); P. S. B.Dev, P. Millington, A. Pilaftsis and D. Teresi, Nucl. Phys.
B886 , 569 (2014).[24] G. Lazarides and Q. Shafi, Phys. Rev.
D58 , 071702 (1998); T. Hambye and G. Senjanovic,Phys. Lett.
B582 , 73 (2004).[25] J. C. Pati and A. Salam, Phys. Rev.
D10 , 275 (1974); R. N. Mohapatra and J. C. Pati, Phys.Rev.
D11 , 2558 (1975); G. Senjanovic and R. N. Mohapatra, Phys. Rev.
D12 , 1502 (1975);R. N. Mohapatra and R. E. Marshak, Phys. Rev. Lett. , 1316 (1980); N. G. Deshpande,J. F. Gunion, B. Kayser and F. I. Olness, Phys. Rev. D44 , 837 (1991).[26] N. Cosme, JHEP , 027 (2004)., 027 (2004).