]>NUPHB13326S05503213(15)00089910.1016/j.nuclphysb.2015.03.007The AuthorsHigh Energy Physics – PhenomenologyFig. 1Baryon asymmetry in one flavor regime as a function of δCP for inverted hierarchy with type II seesaw correction to scaling. (See Table 15.)Fig. 2Baryon asymmetry in two flavor regime as a function of δCP for inverted hierarchy with type II seesaw correction to scaling. (See Table 16.)Fig. 3Baryon asymmetry ratio in three flavor regime as a function of δCP for inverted hierarchy with type II seesaw correction to scaling.Fig. 4Baryon asymmetry in one flavor regime as a function of δCP for normal hierarchy with type II seesaw correction to scaling.Fig. 5Baryon asymmetry in two flavor regime as a function of δCP for normal hierarchy with type II seesaw correction to scaling.Fig. 6Baryon asymmetry in three flavor regime as a function of δCP for normal hierarchy with type II seesaw correction to scaling.Fig. 7Baryon asymmetry in one, two and three flavor regimes as a function of δCP with charged lepton correction to scaling.Table 1Global fit 3σ values of neutrino oscillation parameters [6].ParametersNormal hierarchy (NH)Inverted hierarchy (IH)
Δm21210−5 eV27.02–8.097.02–8.09
Δm31210−3 eV22.317–2.6072.307–2.590
sin2θ120.270–0.3440.270–0.344
sin2θ230.382–0.6430.389–0.644
sin2θ130.0186–0.02500.0188–0.0251
δCP0–2π0–2π
Table 2Global fit 3σ values of neutrino oscillation parameters [7].ParametersNormal hierarchy (NH)Inverted hierarchy (IH)
Δm21210−5 eV27.11–8.187.11–8.18
Δm31210−3 eV22.30–2.652.20–2.54
sin2θ120.278–0.3750.278–0.375
sin2θ230.393–0.6430.403–0.640
sin2θ130.0190–0.02620.0193–0.0265
δCP0–2π0–2π
Table 3M40 (in GeV) for type II seesaw correction with δCP=π2.MODELIH m3=0.065 eV
1 Flavor(6.65934×10123.13846×1012−1.38275×1012i3.66288×1012−1.66362×1012i3.13846×1012−1.38275×1012i7.76821×1012−3.14624×109i−5.74326×1011−5.85132×108i3.66288×1012−1.66362×1012i−5.74326×1011−5.85132×108i7.5331×1012+3.14624×109i)
2 Flavor (1.9978×10129.41538×1011−4.14824×1011i1.09886×1012−4.99085×1011i9.41538×1011−4.14824×1011i2.33046×1012−9.43872×108i−1.72298×1011−1.7554×108i1.09886×1012−4.99085×1011i−1.72298×1011−1.7554×108i2.25993×1012+9.43872×108i)
3 Flavor (6.65934×1083.13846×108−1.38275×108i3.66288×108−1.66362×108i3.13846×108−1.38275×108i7.76821×108−3.14624×105i−5.74326×107−5.85132×104i3.66288×108−1.66362×108i−5.74326×107−5.85132×104i7.5331×108+3.14624×105i)
Table 4MRR (in GeV) for type II seesaw correction with δCP=π2.MODELIH m3=10−6 eV
1 Flavor (3.66324×10123.14907×1012−4.63416×1011i3.65406×1012−5.57547×1011i3.14907×1012−4.63416×1011i3.3088×1012−5.22573×109i−2.008×1012−9.71871×108i3.65406×1012−5.57547×1011i−2.008×1012−9.71871×108i2.54042×1012+5.22573×109i)
2 Flavor (1.09897×10129.44722×1011−1.39025×1011i1.09622×1012−1.67264×1011i9.44722×1011−1.39025×1011i9.92639×1011−1.56772×109i−6.02401×1011−2.91561×108i1.09622×1012−1.67264×1011i−6.02401×1011−2.91561×108i7.62125×1011+1.56772×109i)
3 Flavor (3.66324×1083.14907×108−4.63416×107i3.65406×108−5.57547×107i3.14907×108−4.63416×107i3.3088×108−5.22573×105i−2.008×108−9.71871×104i3.65406×108−5.57547×107i−2.008×108−9.71871×104i2.54042×108+5.22573×105i)
Table 5MRR (in GeV) for type II seesaw correction with δCP=π2.MODELNH m1=0.07 eV
1 Flavor (5.53224×10123.14101×1012−1.47308×1012i3.66076×1012−1.7723×1012i3.14101×1012−1.47308×1012i7.93362×1012−3.64514×109i9.77698×1011−6.77917×108i3.66076×1012−1.7723×1012i9.77698×1011−6.77917×108i8.27579×1012+3.64514×109i)
2 Flavor (1.65967×10129.42302×1011−4.41923×1011i1.09823×1012−5.31689×1011i9.42302×1011−4.41923×1011i2.38008×1012−1.09354×109i2.93309×1011−2.03375×108i1.09823×1012−5.31689×1011i2.93309×1011−2.03375×108i2.48274×1012+1.09354×109i)
3 Flavor (5.53224×1083.14101×108−1.47308×108i3.66076×108−1.7723×108i3.14101×108−1.47308×108i7.93362×108−364514.i9.77698×107−67791.7i3.66076×108−1.7723×108i9.77698×107−67791.7i8.27579×108+364514.1i)
Table 6MRR (in GeV) for type II seesaw correction with δCP=π2.MODELNH m1=10−6 eV
1 Flavor (−9.27517×10123.42428×1013−4.92222×1012i3.42531×1013−5.92205×1012i3.42428×1013−4.92222×1012i2.73041×1013−5.91445×1011i2.53726×1013−1.09996×1011i3.42531×1013−5.92205×1012i2.53726×1013−1.09996×1011i3.65267×1013+5.91445×1011i)
2 Flavor (−2.78255×10111.02729×1012−1.47667×1011i1.02759×1012−1.77661×1011i1.02729×1012−1.47667×1011i8.19122×1011−1.77434×1010i7.61178×1011−3.29987×109i1.02759×1012−1.77661×1011i7.61178×1011−3.29987×109i1.0958×1012+1.77434×1010i)
3 Flavor (−9.27517×1073.42428×108−4.92222×107i3.42531×108−5.92205×107i3.42428×108−4.92222×107i2.73041×108−5.91445×106i2.53726×108−1.09996×106i3.42531×108−5.92205×107i2.53726×108−1.09996×106i3.65267×108+5.91445×106i)
Table 7mLR (in GeV) for type II seesaw correction with δCP=π2.a=bIH m3=0.065 eV
1 Flavor(5.99679−0.179954i5.99679−0.179954i7.06046−0.211873i5.99679−0.179954i20.0543−7.83146i−23.6114−9.22054i7.06046−0.211873i−23.6114−9.22054i−27.7994−10.856i)
2 Flavor(3.28458−0.098565i3.28458−0.098565i3.86717−0.116048i3.28458−0.098565i−10.9842−4.28947i−12.9325−5.0503i3.86717−0.116048i−12.9325−5.0503i−15.2264−5.94608i)
3 Flavor(0.0599679−0.00179954i0.0599679−0.00179954i0.0706046−0.00211873i0.0599679−0.00179954i−0.200543−0.0783146i−0.236114−0.0922054i0.0706046−0.00211873i−0.236114−0.0922054i−0.277994−0.10856i)
Table 8mLR (in GeV) for type II seesaw correction with δCP=π2.a=bIH m3=10−6 eV
1 Flavor(6.20838+0.177014i6.20838+0.177014i7.30957+0.208411i6.20838+0.177014i44.2529−72.3072i52.1021−85.1326i7.30957+0.208411i52.1021−85.1326i61.3436−100.233i)
2 Flavor(3.40047+0.0969544i3.40047+0.0969544i4.00362+0.114151i3.40047+0.0969544i24.2383−39.6043i28.5375−46.629i4.00362+0.114151i28.5375−46.629i33.5993−54.8997i)
3 Flavor(0.0620838+0.00177014i0.0620838+0.00177014i0.0730957+0.00208411i0.0620838+0.00177014i0.442529−0.723072i0.521021−0.851326i0.0730957+0.00208411i0.521021−0.851326i0.613436−1.00233i)
Table 9mLR (in GeV) for type II seesaw correction with δCP=π2.b=dIH m3=0.065 eV
1 Flavor(−64.8942−43.2459i−0.112269−3.74126i−0.132183−4.40486i−0.112269−3.74126i−0.112269−3.74126i−0.132183−4.40486i−0.132183−4.40486i−0.132183−4.40486i−0.155628−5.18616i)
2 Flavor(−35.544−23.6868i−0.0614925−2.04917i−0.0723996−2.41264i−0.0614925−2.04917i−0.0614925−2.04917i−0.0723996−2.41264i−0.0723996−2.41264i−0.0723996−2.41264i−0.0852412−2.84058i)
3 Flavor(−0.648942−0.432459i−0.00112269−0.0374126i−0.00132183−0.0440486i−0.00112269−0.0374126i−0.00112269−0.0374126i−0.00132183−0.0440486i−0.00132183−0.0440486i−0.00132183−0.0440486i−0.00155628−0.0518616i)
Table 10mLR (in GeV) for type II seesaw correction with δCP=π2.b=dIH m3=10−6 eV
1 Flavor(−4.28715−25.1506i0.110435−3.87326i0.130023−4.56027i0.110435−3.87326i0.110435−3.87326i0.130023−4.56027i0.130023−4.56027i0.130023−4.56027i0.153085−5.36914i)
2 Flavor(−2.34817−13.7755i0.0604877−2.12147i0.0712165−2.49777i0.0604877−2.12147i0.0604877−2.12147i0.0712165−2.49777i0.0712165−2.49777i0.0712165−2.49777i0.0838483−2.9408i)
3 Flavor(0.0428715+0.251506i−0.00110435+0.0387326i−0.00130023+0.0456027i−0.00110435+0.0387326i−0.00130023+0.0456027i−0.00130023+0.0456027i−0.00130023+0.0456027i−0.00130023+0.0456027i−0.00153085+0.0536914i)
Table 11mLR (in GeV) for type II seesaw correction with δLR=π2.a=bNH m1=0.07 eV
1 Flavor(6.36821−0.52521i6.36821−0.52521i7.49775−0.618368i6.36821−0.52521i−31.387−5.45116i−36.9542−6.41805i7.49775−0.618368i−36.9542−6.41805i−43.5088−7.55643i)
2 Flavor(3.48801−0.28767i3.48801−0.28767i4.10669−0.338694i3.48801−0.28767i−17.1914−2.98572i−20.2406−3.51531i4.10669−0.338694i−20.2406−3.51531i−23.8308−4.13883i)
3 Flavor(0.0636821−0.0052521i0.0636821−0.0052521i0.0749775−0.00618368i0.0636821−0.0052521i−0.31387−0.0545116i−0.369542−0.0641805i0.0749775−0.00618368i−0.369542−0.0641805i−0.435088−0.0755643i)
Table 12mLR (in GeV) for type II seesaw correction with δCP=π2.a=bNH m1=10−6 eV
1 Flavor(16.3451−0.40342i16.3451−0.40342i19.2442−0.474975i16.3451−0.40342i−59.0353+7.20559i−69.5065+8.48366i19.2442−0.474975i−69.5065+8.48366i−81.8351+9.98843i)
2 Flavor(2.83105−0.0698744i2.83105−0.0698744i3.33319−0.0822681i2.83105−0.0698744i−10.2252+1.24804i−12.0389+1.46941i3.33319−0.0822681i−12.0389+1.46941i−14.1742+1.73005i)
3 Flavor(0.0516876−0.00127573i0.0516876−0.00127573i0.0608555−0.001502i0.0516876−0.00127573i−0.186686+0.0227861i−0.219799+0.0268277i0.0608555−0.001502i−0.219799+0.0268277i−0.258785+0.0315862i)
Table 13mLR (in GeV) for type II seesaw correction with δCP=π2.b=dNH m1=0.07 eV
1 Flavor(−30.1242−65.0884i0.327667−3.97298i−0.385786−4.67768i0.327667−3.97298i0.327667−3.97298i−0.385786−4.67768i−0.385786−4.67768i−0.385786−4.67768i−0.454214−5.50737i)
2 Flavor(16.4997+35.6504i0.17947+2.17609i0.211304+2.56207i0.17947+2.17609i0.17947+2.17609i0.211304+2.56207i0.211304+2.56207i0.211304+2.56207i0.248783+3.01651i)
3 Flavor(−0.301242−0.650884i−0.00327667−0.0397298i−0.00385786−0.0467768i−0.00327667−0.0397298i−0.00327667−0.0397298i−0.00385786−0.0467768i−0.00385786−0.0467768i−0.00385786−0.0467768i−0.00454214−0.0550737i)
Table 14mLR (in GeV) for type II seesaw correction with δCP=π2.b=dNH m1=10−6 eV
1 Flavor(141.819−379.178i−0.251684−10.1973i−0.296326−12.006i−0.251684−10.1973i−0.251684−10.1973i−0.296326−12.006i−0.296326−12.006i−0.296326−12.006i−0.348886−14.1356i)
2 Flavor(24.5637−65.6755i−0.043593−1.76623i−0.0513252−2.0795i−0.043593−1.76623i−0.043593−1.76623i−0.0513252−2.0795i−0.0513252−2.0795i−0.0513252−2.0795i−0.0604289−2.44835i)
3 Flavor(0.44847−1.19906i−0.000795896−0.0322467i−0.000937066−0.0379664i−0.000795896−0.0322467i−0.000795896−0.0322467i−0.000937066−0.0379664i−0.000937066−0.0379664i−0.000937066−0.0379664i−0.00110328−0.0447006i)
Table 15Values of δCP giving correct YB for inverted hierarchy with type II seesaw correction to scaling.ModelδCP (radian) for a=bδCP (radian) for b=d
1 Flavorm3=0.065 eV0.0797–0.0848,3.0316–3.03790.6069–0.6270,2.3147–2.3329
m3=10−6 eV–2.9009–2.9135,5.8779–5.9099

2 Flavorm3=0.065 eV3.6612–3.6963,5.9017–5.93060.0025–0.0031,2.6778–2.6797
m3=10−6 eV–0.3537–0.3757,2.8582–2.8739

3 Flavorm3=0.065 eV––
m3=10−6 eV––
Table 16Values of δCP giving correct YB for normal hierarchy with type II seesaw correction to scaling.ModelδCP(radian) for a=bδCP (radian) for b=d
1 Flavorm1=0.07 eV3.1648–3.1660,6.2561–6.25740.3348–0.3518,2.6760–2.6998
m1=10−6 eV0.0659–0.0697,2.9285–2.9411–

2 Flavorm1=0.07 eV0.1017–0.1086,3.0486–3.05420.0245–0.0263,3.0963–3.0988
m1=10−6 eV––

3 Flavorm1=0.07 eV––
m1=10−6 eV––
Table 17Charge lepton diagonalizing matrix for δCP=π2.Ul
(0.389343−0.108352i0.0141579+0.0381397i−0.933412+0.0458869i−0.412832+0.0492513i0.881799−0.0404407i−0.313989−0.0486552i0.925292+0.059714i0.3356+0.0906409i0.228991+0.109052i)
Table 18MRR (in GeV) with charged lepton correction to scaling.ModelMRR(GeV)
1 Flavor(1×10130002×10130003×1013)
2 Flavor(1×10100001×10130003×1016)
3 Flavor(1×1080001×1090001×1015)
Table 19mLR (in GeV) for δCP=π2 with charged lepton correction to scaling.NH m1=10−6 eV
1 Flavor(−18.6984+6.60236i12.7243−1.50446i−8.96417−11.8754i11.7542−1.7535i−8.01219+0.437544i5.72898+3.02685i−4.25822−10.3367i2.93615+2.10775i−2.36918+19.5204i)
2 Flavor(−4.11131+36.0857i4.01777−17.0146i−8.06708−20.2467i0.42871−17.3158i−1.20776+8.25435i4.77669+9.26783i9.83461−18.7204i−5.67658+8.37772i−0.331627+12.7398i)
3 Flavor(0.175984−0.191103i−0.111308+0.0777845i0.0423717+0.169387i−0.0896843+0.0881861i0.057542−0.0366476i−0.0257287−0.074506i−0.0647843+0.116022i0.0369503−0.0433691i0.00475925−0.121585i)
Table 20Values of δCP giving correct YB with charge lepton correction to scaling.ModelδCP (radian)
1 Flavor2.2003–2.2072,5.2552–5.2734
2 Flavor3.2440–3.2452,6.2586–6.2624
3 Flavor4.1010–4.1223,6.0136–6.0670
Corrections to scaling neutrino mixing: Nonzero θ13,δCP and baryon asymmetryRupamKalitarup@tezu.ernet.inDebasishBorah⁎dborah@tezu.ernet.inMrinal KumarDasmkdas@tezu.ernet.inDepartment of Physics, Tezpur University, Tezpur784028, IndiaDepartment of PhysicsTezpur UniversityTezpur784028India⁎Corresponding author.Editor: Tommy OhlssonAbstractWe study a very specific type of neutrino mass and mixing structure based on the idea of Strong Scaling Ansatz (SSA) where the ratios of neutrino mass matrix elements belonging to two different columns are equal. There are three such possibilities, all of which are disfavored by the latest neutrino oscillation data. We focus on the specific scenario which predicts vanishing reactor mixing angle θ13 and inverted hierarchy with vanishing lightest neutrino mass. Motivated by several recent attempts to explain nonzero θ13 by incorporating corrections to a leading order neutrino mass or mixing matrix giving θ13=0, here we study the origin of nonzero θ13 as well as leptonic Dirac CP phase δCP by incorporating two different corrections to scaling neutrino mass and mixing: one, where type II seesaw acts as a correction to scaling neutrino mass matrix and the other, with charged lepton correction to scaling neutrino mixing. Although scaling neutrino mass matrix originating from type I seesaw predicts inverted hierarchy, the total neutrino mass matrix 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 mass matrix. We further discriminate between neutrino hierarchies, different choices of lightest neutrino mass and Dirac CP phase by calculating baryon asymmetry and comparing with the observations made by the Planck experiment.1IntroductionOrigin of tiny neutrino masses and mixing is one of the most widely studied problems in modern day particle physics. Since the standard model (SM) of particle physics fails to provide an explanation to neutrino masses and mixing, several well motivated beyond standard model (BSM) frameworks have been proposed to account for the tiny neutrino mass observed by several neutrino oscillation experiments [1]. More recently, the neutrino oscillation experiments T2K [2], Double ChooZ [3], DayaBay [4] and RENO [5] have confirmed the earlier results and made the measurement of neutrino parameters more precise. The latest global fit values for 3σ range of neutrino oscillation parameters [6] are shown in Table 1. Another global fit study [7] reports the 3σ values as shown in Table 2. Although the 3σ range for the Dirac CP phase δCP is 0–2π, there are two possible best fit values of it found in the literature: 306° (NH), 254° (IH) [6] and 254° (NH), 266° (IH) [7]. It should be noted that the neutrino oscillation experiments only determine two mass squared differences and hence the lightest neutrino mass is still unknown. Cosmology experiments however puts an upper bound on the sum of absolute neutrino masses ∑imi<0.23 eV [8]. Within this bound, he lightest neutrino mass can either be zero or very tiny (compared to the other two) giving rise to a hierarchical pattern. Or, the lightest neutrino mass can be of same order as the other two neutrino masses giving rise to a quasidegenerate type neutrino mass spectrum.Apart from the issue of lightest neutrino mass and hence the nature of neutrino mass hierarchy, the CP violation in the leptonic sector is also not understood very well. Nonzero CP violation in the leptonic sector can be very significant from cosmology point of view as it could be the origin of matter–antimatter asymmetry in the Universe. The latest data available from Planck mission constrain the baryon asymmetry [8] as(1)YB=(8.58±0.22)×10−11 Leptogenesis is one of the most promising dynamical mechanism of generating this observed baryon asymmetry in the Universe by generating an asymmetry in the leptonic sector first which later gets converted into baryon asymmetry through B+L violating electroweak sphaleron transitions [9]. As pointed out first by Fukugita and Yanagida [10], the out of equilibrium CP violating decay of heavy Majorana neutrinos provides a natural way to create the required lepton asymmetry. The most notable feature of this mechanism is that it connects two of the most widely studied problems in particle physics: the origin of neutrino mass and the origin of baryon asymmetry. This idea has been explored within several interesting BSM frameworks [11–13]. Recently such a comparative study was done to understand the impact of mass hierarchies, Dirac and Majorana CP phases on the predictions for 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 and its relevance in cosmology, we recently studied several models [15] based on the idea of generating nonzero θ13, δCP and matter–antimatter asymmetry by perturbing generic μ–τ symmetric neutrino mass matrix which can be explained dynamically within generic flavor symmetry models. In these works, type I seesaw [16] is assumed to give rise to the μ–τ symmetric neutrino mass matrix with θ13=0 whereas type II seesaw [17] acts as a perturbation in order to generate the nonzero reactor mixing angle θ13 and also the Dirac CP phase δCP in some cases. In continuation of our earlier works on exploring the underlying structure of the neutrino mass matrix, in this work we consider a very specific neutrino mass matrix structure proposed few years back by the authors of [18]. The structure of the neutrino mass matrix is based on the idea of strong scaling Ansatz where certain ratios of the elements of neutrino mass matrix are equal. Out of three such possibilities (to be discussed in the next section), one of them predicts θ13=0 and an inverted hierarchy with vanishing lightest neutrino mass. Such a scaling neutrino mass matrix can also find its origin in specific flavor symmetry models as discussed in [18]. Several phenomenological studies based on the idea of SSA have appeared in [19]. The predictions for neutrino sector similar to the scaling ansatz can also be found in models based on the abelian symmetry Le–Lμ–Lτ [20].Although inverted hierarchy as predicted by SSA can still be viable, vanishing reactor mixing angle is no longer acceptable after the discovery of nonzero θ13. Generation of nonzero θ13 in models based on the idea of SSA have appeared recently in [21]. In this work we study two different possibilities of generating nonzero θ13 as well as Dirac CP phase δCP by incorporating 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 we assume the origin of scaling neutrino mass matrix in type I seesaw. The required deviation from scaling can either come from a different seesaw mechanism (say, type II seesaw) or from charged lepton (CL) correction. The crucial difference between the two different scenario is that in CL correction, the inverted hierarchy prediction of SSA remains intact whereas with the combination of two different seesaw mechanism both normal and inverted hierarchies can emerge out of the total neutrino mass matrix. We first numerically fit the scaling neutrino mass matrix (from type I seesaw) with neutrino data on two mass squared differences and two angles θ12,θ23 (as θ13=0). Then we derive the necessary perturbation to scaling neutrino mixing by comparing with the full neutrino oscillation data including nonzero θ13. We further constrain the perturbation by demanding successful production of baryon asymmetry through the mechanism of leptogenesis.This paper is organized as follows. In Section 2, we briefly discuss the idea of scaling neutrino mass and mixing. In Section 3, we study the possible deviation from scaling with type II seesaw as well as charged lepton corrections. In Section 4, we briefly outline the idea of leptogenesis and in Section 5 we discuss our numerical analysis. We finally conclude in Section 6.2Strong scaling ansatzAccording to SSA, ratios of certain elements of the neutrino mass matrix are equal. The stability of such a structure is also guaranteed by the fact that it is not affected by the renormalization group evolution (RGE) equations. Therefore, the scaling which is present in the neutrino mass matrix at seesaw scale is also remains valid at the weak scale as we run the neutrino parameters from seesaw to weak scale under RGE. We denote the neutrino mass matrix and the leptonic mixing matrix UPMNS asMν=(meemeμmeτmμemμμmμτmτemτμmττ)UPMNS=(Ue1Ue2Ue3Uμ1Uμ2Uμ3Uτ1Uτ2Uτ3) As noted by the authors of [18], there are three different types of SSA which can be written as(2)meμmeτ=mμμmμτ=mτμmττ=S(3)meemeτ=mμemμτ=mτemττ=S′(4)meemeμ=mμemμμ=mτemτμ=S″ Using (2), we can write the neutrino mass matrix as(5)Mν=(ABBSBDDSBSDSDS2) Similarly, for the other two cases (3), (4) one can write down the neutrino mass matrix as(6)Mν=(ABAS′BDBS′AS′BS′AS′2) and(7)Mν=(AAS″BAS″AS″2BS″BBS″D) One interesting property of the first scaling mass matrix (5) is that it has one of its eigenvalue m3 zero (rank 2 matrix) and diagonalization of this matrix gives Ue3=0. Thus, it gives rise to inverted hierarchy of neutrino mass with θ13=0. Although such a scenario is now ruled out after the discovery of nonzero θ13, there still exists the possibility of generating nonzero θ13 by adding perturbations to the scaling neutrino mass and mixing, given the fact that θ13 is still small compared to the other two mixing angles. However, diagonalization of the second scaling matrix (6) gives Uμ3=0 or Uμ1=0 depending on the hierarchy of neutrino masses. Similarly, diagonalization of the third scaling matrix (7) gives Uτ3=0 or Uτ1=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 scaling mass matrices at tree level, they will require large corrections in order to generate the correct mixing matrix. Leaving these to future studies, here we focus on the possibility of generating nonzero Ue3 and hence nonzero θ13 by incorporating different corrections to leading order scaling neutrino mass matrix given by (5).3Deviations from scalingAs discussed in the previous section, the neutrino mass matrices based on the idea of SSA do not give rise to the correct neutrino mixing pattern. Therefore, the scaling neutrino mass or mixing matrix has to be corrected in order to have agreement with neutrino oscillation data. Here we consider two different sources of such corrections to SSA which not only can give rise to correct neutrino mixing but also have different predictions for neutrino mass hierarchy, leptonic Dirac CP phase as well as baryon asymmetry.3.1Deviation from scaling with type II seesawType II seesaw mechanism is the extension of the standard model with a scalar field ΔL which transforms like a triplet under SU(2)L and has U(1)Y charge twice that of lepton doublets. Such a choice of gauge structure allows an additional Yukawa term in the Lagrangian given by fij(ℓiLTCiσ2ΔLℓjL). The triplet can be represented asΔL=(δL+/√2δL++δL0−δL+/√2) The scalar Lagrangian of the standard model also gets modified after the inclusion of this triplet. Apart from the bilinear and quartic coupling terms of the triplet, there is one trilinear term as well involving the triplet and the standard model Higgs doublet. From the minimization of the scalar potential, the neutral component of the triplet is found to acquire a vacuum expectation value (vev) given by(8)〈δL0〉=vL=μΔH〈ϕ0〉2MΔ2 where ϕ0=v is the neutral component of the electroweak Higgs doublet with vev approximately 102 GeV. The trilinear coupling term μΔH and the mass term of the triplet MΔ can be taken to be of same order. Thus, MΔ has to be as high as 1014 GeV to give rise to tiny neutrino masses without any finetuning of the dimensionless couplings fij. In the presence of both type I and type II seesaw the neutrino mass can be written as(9)Mν=mII+mI where mII=fvL is the type II seesaw contribution and mI=mLRMRR−1mLRT is the type I see saw term with mLR,MRR being Dirac and Majorana neutrino mass matrices respectively. We assume the type I seesaw to give rise to scaling neutrino mass matrix. We then introduce the type II seesaw term as a correction to the scaling neutrino mass matrix and constrain the type II seesaw term from the requirement of generating correct value of θ13 as well as baryon asymmetry. One interesting property of scaling is that type I seesaw can give rise to scaling neutrino mass matrix irrespective of the right handed Majorana mass matrix MRR, if Dirac neutrino mass matrix mLR obeys scaling. As we discuss in Section 5, this property of scaling allows us to derive the type II seesaw correction as well as the Dirac neutrino mass matrix.3.2Deviation from scaling with charged lepton correctionThe scaling neutrino mass matrix we discuss here, given by Eq. (5) predicts m3=0 and θ13=0. In the previous subsection, type II seesaw correction to scaling neutrino mass was discussed which not only can result in nonzero θ13 but also can give rise to nonzero m3. Since, an inverted hierarchical neutrino mass pattern with m3=0 is still allowed by neutrino oscillation data, one can generate nonzero θ13 by incorporating corrections to the leptonic mixing matrix only without affecting the scaling neutrino mass matrix. The PMNS leptonic mixing matrix is related to the diagonalizing matrices of neutrino and charged lepton mass matrices Uν,Ul respectively, as(10)UPMNS=Ul†Uν The PMNS mixing matrix can be parametrized as(11)UPMNS=(c12c13s12c13s13e−iδ−s12c23−c12s23s13eiδc12c23−s12s23s13eiδs23c13s12s23−c12c23s13eiδ−c12s23−s12c23s13eiδc23c13) where cij=cosθij,sij=sinθij and δ is the Dirac CP phase. If Uν originates from scaling neutrino mass matrix given by type I seesaw, then for diagonal charged lepton mass matrix, both the reactor mixing angle θ13 and the leptonic Dirac CP phase δ vanish. However, a nontrivial charged lepton mixing matrix Ul can result in correct leptonic mixing matrix UPMNS even if Uν predicts θ13=0. As discussed in the section on numerical analysis 5, we constrain the charged lepton mass matrix by demanding the generation of correct θ13 required by neutrino oscillation data and also the correct value of δCP in order to produce correct baryon asymmetry.4LeptogenesisAs mentioned earlier, leptogenesis is the mechanism where a nonzero lepton asymmetry is generated by out of equilibrium, CP violating decay of a heavy particle which later gets converted into baryon asymmetry through electroweak sphaleron transitions. In a model with both type I and type II seesaw mechanisms at work, such lepton asymmetry can be generated 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 of lepton asymmetry. One can justify this assumption in those models where type I seesaw is dominating and type II seesaw is subleading giving rise to a Higgs triplet heavier than the lightest right handed neutrino. The lepton asymmetry from the decay of right handed neutrino into leptons and Higgs scalar in a model with only type I seesaw is given by(12)ϵNk=∑iΓ(Nk→Li+H⁎)−Γ(Nk→Li¯+H)Γ(Nk→Li+H⁎)+Γ(Nk→Li¯+H) In a hierarchical pattern of heavy right handed neutrinos, it is sufficient to consider the decay of the lightest right handed neutrino N1. Following the notations of [12], the lepton asymmetry arising from the decay of N1 in the presence of type I seesaw only can be written as(13)ϵ1α=18πv21(mLR†mLR)11∑j=2,3Im[(mLR⁎)α1(mLR†mLR)1j(mLR)αj]g(xj)+18πv21(mLR†mLR)11∑j=2,3Im[(mLR⁎)α1(mLR†mLR)j1(mLR)αj]11−xj where v=174 GeV is the vev of the Higgs doublet responsible for breaking the electroweak symmetry,g(x)=x(1+11−x−(1+x)ln1+xx) and xj=Mj2/M12. The second term in the expression for ϵ1α above vanishes when summed over all the flavors α=e,μ,τ. The sum over all flavors can be written as(14)ϵ1=18πv21(mLR†mLR)11∑j=2,3Im[(mLR†mLR)1j2]g(xj) From the lepton asymmetry ϵ1 given by the expression above, the corresponding baryon asymmetry can be obtained by(15)YB=cκϵg⁎ through sphaleron processes [9] at electroweak phase transition. Here the factor c is measure of the fraction of lepton asymmetry being converted into baryon asymmetry and is approximately equal to −0.55. On the other hand, κ is the dilution factor due to washout process which erase the asymmetry generated and can be parametrized as [22](16)−κ≃0.1Kexp[−4/(3(0.1K)0.25)],forK≥106−κ≃0.3K(lnK)0.6,for10≤K≤106−κ≃12K2+9,for0≤K≤10 where K is given asK=Γ1H(T=M1)=(mLR†mLR)11M18πv2MPl1.66g⁎M12 Here Γ1 is the decay width of N1 and H(T=M1) is the Hubble constant at temperature T=M1. The factor g⁎ is the effective number of relativistic degrees of freedom at temperature T=M1 and is approximately 110.We note that the lepton asymmetry shown in Eq. (14) is obtained by summing over all the flavors α=e,μ,τ. A nonvanishing lepton asymmetry is generated only when the right handed neutrino decay is out of equilibrium. Otherwise both the forward and the 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 the expansion rate of the Universe. At very high temperatures (T≥1012 GeV) all charged lepton flavors are out of equilibrium and hence all of them behave similarly resulting in the one flavor regime. However at temperatures T<1012 GeV (T<109 GeV), interactions involving tau (muon) Yukawa couplings enter equilibrium and flavor effects become important [23]. Taking these flavor effects into account, the final baryon asymmetry is given byYB2flavor=−1237g⁎[ϵ2η(417589m2˜)+ϵ1τη(390589mτ˜)]YB3flavor=−1237g⁎[ϵ1eη(151179me˜)+ϵ1μη(344537mμ˜)+ϵ1τη(344537mτ˜)] where ϵ2=ϵ1e+ϵ1μ,m2˜=me˜+mμ˜,mα˜=(mLR⁎)α1(mLR)α1M1. The function η is given byη(mα˜)=[(mα˜8.25×10−3 eV)−1+(0.2×10−3 eVmα˜)−1.16]−1 In the presence of an additional scalar triplet, the right handed neutrino can also decay through a virtual triplet. The contribution of this diagram to lepton asymmetry can be estimated as [24](17)ϵΔ1α=−M18πv2∑j=2,3Im[(mLR)1j(mLR)1α(MνII⁎)jα]∑j=2,3(mLR)1j2 We use these expressions to calculate the baryon asymmetry in our numerical analysis section discussed below.5Numerical analysisWe first diagonalize the scaling neutrino mass matrix (5) and find its eigenvaluesm1=12S2(D+AS2+DS2−A2S4−2ADS2(1+S2)+D2(1+S2)2+4B2S2(1+S2))m2=12S2(D+AS2+DS2+A2S4−2ADS2(1+S2)+D2(1+S2)2+4B2S2(1+S2))m3=0 We numerically evaluate the four parameters A,B,D,S by equating m1,m2 to two neutrino mass squared differences Δm212,Δm232 and two nonzero mixing angles to θ12,θ23.Now, in the case of type II seesaw correction to scaling neutrino mixing, we assume the charged leptons mass matrix to be diagonal so that UPMNS=Uν. Therefore, we can write (9) as(18)UPMNS.mνdiag.UPMNST=mII+mI where mνdiag is the diagonal neutrino mass matrix given by mνdiag=diag(m1,m12+Δm212,m12+Δm312) for normal hierarchy and mνdiag=diag(m32+Δm232−Δm212,m32+Δm232,m3) for inverted hierarchy. The type I seesaw mass matrix mI always gives inverted hierarchy whereas mνdiag can give either normal or inverted hierarchy depending on the type II seesaw contribution mII. In the minimal extension of the standard model with type I and type II seesaw mechanisms, the type I seesaw term depends upon mLR and MRR whereas type II seesaw depends upon the vev of the neutral component of Higgs triplet. Since mLR,MRR as well as the type II seesaw term can be chosen by hand, such a framework is difficult to constrain due to too many number of 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 MRR thereby decreasing the number of free parameters compared to the minimal extension. Another reason for choosing the framework of LRSM is that here we can find the right handed Majorana mass matrix MRR from the type II seesaw perturbation. However, for a given Dirac neutrino mass matrix mLR, one cannot find MRR from the type I seesaw formula alone as the inverse of type I seesaw mass matrix does not exist due to its scaling property (m3=0). In LRSM we can write Eq. (18) as(19)UPMNS.mνdiag.UPMNST=γ(MWvR)2MRR+mI where γ is a dimensionless parameter, MW is the W boson mass and vR is the scale of left right symmetry breaking. Since mI has been numerically evaluated as the leading order scaling neutrino mass matrix, type II contribution can now be evaluated as a function of leptonic Dirac CP phase δCP and the lightest neutrino mass m1 (NH), m3 (IH), the two unknowns on the left hand side of the above equation. It should be noted that, we have omitted the Majorana phases in this discussion. After determining the type II seesaw term and hence MRR, we use it in the type I seesaw term to find out the Dirac neutrino mass matrix mLR. Here we use the already mentioned special property of scaling neutrino mass matrix originating from type I seesaw: if Dirac neutrino mass matrix mLR obeys scaling, then mI obeys scaling irrespective of the structure of MRR. Therefore, we use the scaling Dirac neutrino mass matrix given bymLR=(abbcbddcbcdcdc2) We use the above mLR and the already derived right handed Majorana mass matrix MRR in the type I seesaw formula and equate it to the scaling neutrino mass matrix evaluated numerically earlier.We use two different choices of lightest neutrino mass in order to show the effect of hierarchy. For inverted hierarchy we take m3=0.065 eV, 10−6 eV and for normal hierarchy we take m1=0.07 eV, 10−6 eV. After choosing the lightest neutrino mass, the only undetermined parameters in Eq. (19) are δCP,γ and vR. Choosing generic order one coupling γ, one can now write MRR in terms of δCP and vR. We choose the left right symmetry breaking scale vR in a way which keeps the lightest right handed neutrino in the appropriate flavor regime of leptogenesis. After we find MRR in terms of δCP, we use it in the type I seesaw formula with the mLR obeying scaling as shown above. To simplify the numerical calculation further, we assume equality between some parameters in mLR: a=b and b=d. The other choice a=d does not give us any solution. We also do not assume equality of the parameter c with others as c can be found independently of a,b,d when we equate the type I seesaw formula with the numerically evaluated type I seesaw mass matrix of scaling type. The numerical form of the right handed neutrino mass matrix MRR for all the cases discussed are shown in Tables 3, 4, 5 and 6. Similarly, the Dirac neutrino mass matrices are listed in Tables 7, 8, 9, 10, 11, 12, 13 and 14. Although they are, in general, complicated functions of δCP, we have used a specific value of δCP=π/2 to show their compact numerical form. To calculate the baryon asymmetry, however, we vary δCP continuously and show the variation of Baryon asymmetry in Figs. 1, 2, 3, 4, 5 and 6. It should be noted that the type II seesaw corrections to scaling have been considered within the framework of LRSM where SU(2)R gauge interactions can give rise to sizeable washout effects erasing the asymmetry produced. The values of δCP which give observed baryon asymmetry are shown in Tables 15 and 16. As noted in [26], such washout effects can be neglected by choosing a high value of vR such that M1/vR<10−2 is satisfied.In the second mechanism we adopt to give correction to the scaling neutrino mixing, we do not add corrections to the neutrino mass matrix originating from type I seesaw, but incorporate corrections to the neutrino mixing matrix originating from charged lepton mixing. Diagonalizing the scaling neutrino mass matrix from type I seesaw matrix gives Uν which is related to the leptonic mixing matrix UPMNS through the charged lepton diagonalizing matrix Ul. We first numerically evaluate Uν by using the best fit values of neutrino mass squared differences and two mixing angles θ12,θ23. We also substitute the best fit values of neutrino mixing angles in PMNS mixing matrix (11) and then compute the charged lepton diagonalizing matrix asUl=UνUPMNS† We keep the Dirac CP phase δCP as free parameter so that Ul is a function of it. The numerical form of Ul for δCP=π/2 is shown in Table 17. Assuming the same diagonalizing matrix of charged leptons and Dirac neutrino mass matrix (a generic case in grand unified theories operating at high scale), we can write down the modified Dirac neutrino mass matrix as(20)mLR=Ul.mLR0.Ul† where mLR0 is the scaling type Dirac neutrino mass matrix we choose earlier. We choose a diagonal form of MRR while keeping the lightest right handed neutrino mass M1 in the appropriate flavor regime of leptogenesis and varying the heavier right handed neutrino masses M2,M3 between M1 and the grand unified theory scale MGUT∼1016 GeV. For such a choice of MRR, we numerically evaluate the parameters in mLR0 by equating the type I seesaw term (mLR0)MRR−1(mLR0)T to the numerically fitted type I scaling neutrino mass matrix. The numerical values of MRR which give baryon asymmetry closest to the observed in each flavor regime are shown in Table 18. The numerical form of Dirac neutrino mass matrices in each flavor regime for δCP=π/2 are shown in Table 19. The predictions for baryon asymmetry as a function of δCP are shown in Fig. 7. The values of Dirac CP phase which give rise to correct baryon asymmetry are listed in Table 20.6Results and conclusionWe have studied a specific type of neutrino mass matrix based on the idea of strong scaling ansatz where the ratio of neutrino mass matrix elements belonging to two different columns are equal. Out of three such possibilities, we focus on a particular scaling neutrino mass matrix which predicts zero values of reactor mixing angle θ13. This choice was motivated by several recent works where the leading order neutrino mass matrix obeying certain symmetries predict θ13=0 and suitable corrections to the neutrino mass matrix or leptonic mixing matrix give rise to small but nonzero values of θ13. In this work, we have assumed type I seesaw to give rise to scaling neutrino mass matrix which (in the diagonal charged lepton basis) gives θ13=0 and inverted hierarchical mass pattern with m3=0. Then we consider two different possible corrections to scaling: one with type II seesaw which gives rise to deviations from both θ13=0 and m3=0, the other with charged lepton correction which gives nonzero θ13 while keeping m3=0. We also assume both the corrections to give rise to nontrivial Dirac CP phase δCP as well. In both the cases, we first numerically evaluate the type I seesaw scaling neutrino mass matrix by using the best fit values of neutrino mass squared differences and two mixing angles: solar and atmospheric. We then calculate the necessary corrections to scaling neutrino mass and mixing by keeping the Dirac CP phase as free parameter. We further constrain the Dirac CP phase by calculating the baryon asymmetry through the mechanism of leptogenesis and comparing with the observed Baryon asymmetry. The important results we have obtained in the case of type II seesaw correction to scaling can be summarized as:•Type II seesaw correction to scaling neutrino mass matrix with θ13=0,m3=0 can result in both normal as well as inverted hierarchy with nonzero θ13 as well as nontrivial Dirac CP phase δCP.•For inverted hierarchy with a=b that is mLR(11)=mLR(12), correct values of baryon asymmetry is obtained through the mechanism of leptogenesis only when the lightest neutrino mass m3 is of same order as the heavier ones m1,m2.•For inverted hierarchy with b=d that is mLR(12)=mLR(22), both large and mild hierarchy among neutrino masses give rise to correct baryon asymmetry through leptogenesis.•For normal hierarchy with mLR(11)=mLR(12), both large and mild hierarchy among neutrino masses can give rise to correct baryon asymmetry in the one flavor regime. In the two flavor regime however, the lightest neutrino mass m3 should be of same order as m2,m3 to give correct baryon asymmetry.•For normal hierarchy with mLR(12)=mLR(22), the lightest neutrino mass m3 should be of same order as m2,m3 to give correct baryon asymmetry in both one and two flavor regimes.•Observed baryon asymmetry can not be generated in the three flavor regime of leptogenesis 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 hierarchy with m3=0, but gives rise to correct values of θ13 and nontrivial δCP.•Correct baryon asymmetry can be obtained through leptogenesis for one, two and three flavor regimes if δCP is restricted to certain range of values. Since the Dirac CP phase is restricted in all these cases discussed, from the demand of producing the correct baryon asymmetry, future determination of δCP should be able to shed some light on these scenarios. Future experiments may however, measure a different value of δCP than the ones which give correct baryon asymmetry through the mechanism of leptogenesis in the models we have studied here. This will by no means rule out the neutrino mass models based on strong scaling ansatz we discuss, but will only hint at a different source of baryon asymmetry than the one discussed in our work. Similarly, determination of neutrino mass hierarchy in neutrino oscillation experiments will further constrain the models and only charged lepton correction to scaling may not be sufficient to reproduce the correct neutrino data if inverted hierarchy gets disfavored by experiments. From theoretical point of view, such scaling neutrino mass matrix can find a dynamical origin within discrete flavor symmetry models as pointed out by [18]. Since scaling is not affected by renormalization group running, additional physics are required in order to produce correct low energy neutrino oscillation data. Undisturbed by such running effects, scaling can be valid all the way from grand unified theory scale down to the TeV scale, where new physics affects like Higgs triplet in type II seesaw can give rise to the necessary correction to scaling neutrino mass matrix. Although we have studied only one particular type of scaling neutrino mass matrix giving θ13=0,m3=0, the other two possible scaling mass matrices could also give rise to correct neutrino phenomenology if suitable corrections are incorporated, which is left for our future studies.AcknowledgementThe work of M.K. 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