The reactor mixing angle and CP violation with two texture zeros in the light of T2K
aa r X i v : . [ h e p - ph ] J a n UWThPh-2011-31IFIC/11-47
The reactor mixing angle and CP violation with twotexture zeros in the light of T2K
P.O. Ludl ∗ ( a ) , S. Morisi † ( b ) and E. Peinado ‡ ( b ) ( a ) University of Vienna, Faculty of Physics, Boltzmanngasse 5, A–1090 Vienna, Austria ( b ) AHEP Group, Institut de F´ısica Corpuscular – C.S.I.C./Universitat de Val`enciaEdificio Institutos de Paterna, Apt 22085, E–46071 Valencia, Spain
24 January 2012
Abstract
We reconsider the phenomenological implications of two texture zeros in sym-metric neutrino mass matrices in the light of the recent T2K results for the reactorangle and the new global analysis which gives also best fit values for the Dirac CPphase δ . The most important results of the analysis are: Among the viable casesclassified by Frampton et al. only A and A predict θ to be different from zeroat 3 σ . Furthermore these two cases are compatible only with a normal mass spec-trum in the allowed region for the reactor angle. At the best fit value A and A predict 0 . ≥ sin θ ≥ .
012 and 0 . ≤ sin θ ≤ . δ = − δ = 1, re-spectively. The cases B , B , B and B predict nearly maximal CP violation, i.e.cos δ ≈ PACS-numbers: 14.60.-z, 14.60.Pq, 14.60.St, 23.40.Bw.
Recently the T2K Collaboration [1] gave hints for a nonzero reactor angle, and also theresults of the MINOS Collaboration [2] point towards the same direction. The global fits ∗ E-mail: [email protected] † E-mail: morisi@ific.uv.es ‡ E-mail: epeinado@ific.uv.es
1f neutrino oscillation experiments give . ≤ sin θ ≤ .
035 (NS) , . ≤ sin θ ≤ .
039 (IS) , [3]0 . ≤ sin θ ≤ . , [4] (1)and the best fit values are sin θ = 0 .
013 and sin θ = 0 . θ = 0 . , δ = − . π (NS) , (2)sin θ = 0 . , δ = − . π (IS) . (3)A lot of papers have been proposed recently in order to reproduce such a large valueof the reactor mixing angle [5]. Already before the recent T2K data there have beenmodels based on discrete flavor symmetries which predict a large reactor mixing angle—for an incomplete list see Ref. [6], and for a classification of models with flavor symmetriesclassified by their predictions for the reactor angle see [7].Here we reconsider the interesting case of Majorana neutrino mass matrices with twotexture zeros in the basis where the charged lepton mass matrix is diagonal, which hasbeen extensively studied in the past. Our aim is to point out the phenomenologicalimplication of such textures in the light of the T2K results .It was shown in [10, 11] and [8] that, in the basis where the charged lepton massmatrix is diagonal, there are seven types of two texture zeros in symmetric neutrino massmatrices compatible with the experimental data on neutrino oscillations. In this work wewant to analyze the correlation between the CP violating phase δ and the reactor mixingangle θ in the framework of these two texture zeros.Another interesting possibility is to place texture zeros in the inverted neutrino massmatrix–see e.g. [12]. The implications of this type of two texture zeros on the reactormixing angle and CP violation have been studied in [13]. In the basis where the charged lepton mass matrix is diagonal, the lepton mixing matrix U and the Majorana neutrino mass matrix M ν are related via M ν = U ∗ diag( m , m , m ) U † . (4)The standard parameterization [14] for U is given by U = e i ˆ α V e i ˆ σ , (5) Throughout this work the abbreviations NS and IS will stand for normal and inverted neutrino massspectrum, respectively. While we were finishing this work two papers treating the same problem were published in [8, 9]. The parameterization used here is a re-writing of the symmetrical parameterization proposed in [15]. V = c c c s s e − iδ − c s − s s c e iδ c c − s s s e iδ s c s s − c s c e iδ − s c − c s s e iδ c c , (6)ˆ α = diag( α , α , α ) and ˆ σ = diag( σ , σ , σ ). c ij = cos θ ij , s ij = sin θ ij with θ ij ∈ [0 , π/ δ ∈ [0 , π ) is the CP violating phase and the σ i ∈ [0 , π ) are the Majorana phases, whichare not measurable in oscillation experiments. The phases α i are irrelevant for neutrinooscillations and will play no role in our analysis, as we will see in the following. Inserting(5) into (4) we obtain( M ν ) ∗ ij = X k m k U ik U jk = X k m k e iσ k V ik V jk e i ( α i + α j ) . (7)Placing a texture zero in the neutrino mass matrix corresponds to the condition( M ν ) ij = 0 ( ⇔ ( M ν ) ∗ ij = 0) (8)for some indices ( i, j ). Defining µ k := m k e iσ k and dividing by e i ( α i + α j ) we arrive at X k µ k V ik V jk = 0 . (9)The assumption of two texture zeros can thus be described by the two equations X k µ k V ak V bk = 0 , X k µ k V ck V dk = 0 . (10)The viable cases of two texture zeros given in [10] and the corresponding parameters( a, b, c, d ) can be found in table 1.case texture zeros (a,b,c,d)A ( M ν ) ee = ( M ν ) eµ = 0 (1,1,1,2)A ( M ν ) ee = ( M ν ) eτ = 0 (1,1,1,3)B ( M ν ) µµ = ( M ν ) eτ = 0 (2,2,1,3)B ( M ν ) ττ = ( M ν ) eµ = 0 (3,3,1,2)B ( M ν ) µµ = ( M ν ) eµ = 0 (2,2,1,2)B ( M ν ) ττ = ( M ν ) eτ = 0 (3,3,1,3)C ( M ν ) µµ = ( M ν ) ττ = 0 (2,2,3,3)Table 1: The viable cases in the framework of two texture zeros in the Majorana neutrinomass matrix M ν and a diagonal charged-lepton mass matrix M ℓ [10].3 General remarks
The system (10) is equivalent to V a V b V a V b V c V d V c V d ! µ µ ! = − µ V a V b V c V d ! . (11)The set of solutions of this system of linear equations depends on the determinant D abcd := det V a V b V a V b V c V d V c V d ! = V a V b V c V d − V a V b V c V d . (12)For D abcd = 0 we find µ µ ! = − µ D abcd V c V d − V a V b − V c V d V a V b ! V a V b V c V d ! . (13)Since at least two neutrino masses must be nonzero, the above equation implies that thelightest neutrino mass is different from zero. Thus we are allowed to divide by µ andwe can easily calculate r := ∆ m ∆ m = m m − m m − m m = (cid:12)(cid:12)(cid:12) µ µ (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) µ µ (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) µ µ (cid:12)(cid:12)(cid:12) . (14)Inserting (13) into (14) we find an equation which relates the six quantities∆ m , ∆ m , θ , θ , θ and δ. Fixing the mass squared differences and the two mixing angles θ and θ (e.g. to theirbest fit values or their nσ -ranges), we obtain a relation between the reactor mixing angle θ and δ . Note that in this way one can eliminate the unknown absolute neutrino massscale. This approach has been previously used in [16].The main question we have to answer before beginning our analysis is whether thedeterminant D abcd can become zero for the seven different cases within the experimentallimits. The first issue we notice is that all entries of the 2 × − matrix V a V b V a V b V c V d V c V d ! (15)are nonzero (by experiment). Thus D abcd = 0 implies V a V b V c V d = V a V b V c V d . (16) In general a normal (inverted) neutrino mass spectrum allows µ = 0 ( µ = 0). However, one canverify that within the experimental 3 σ -range (13) implies µ = 0 ⇔ µ = 0 for all types of two texturezeros we will study in this work. Thus the lightest neutrino mass must be nonzero. V are of the same order of magnitude, we find (cid:12)(cid:12)(cid:12)(cid:12) V a V b V c V d (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) V a V b V c V d (cid:12)(cid:12)(cid:12)(cid:12) ≃ ⇒ | V a V b | ≃ | V c V d | . (17)From | V i | ≃ (2 / √ , / √ , / √ T (18)one easily finds that the only case allowing (17) is C. Therefore for the cases A to B wecan assume D abcd = 0 and use equation (14) for our analysis.Let us now turn to case C. In [18] it has been shown that for θ = 0 the determinant D becomes zero and the system (10) is therefore singular in this case. Inverselyassuming D = 0, we can proceed as follows. Defining ǫ = s e iδ we find D = ǫ (cid:18)
12 sin(2 θ )sin(2 θ )(1 + ǫ ) − ǫ cos(2 θ )cos(2 θ ) (cid:19) . (19)Thus D can be zero only for ǫ = 0 ortan 2 θ tan 2 θ = 2 ǫ ǫ . (20)For 0 ≤ s ≤ .
05 we find (cid:12)(cid:12)(cid:12)(cid:12) ǫ ǫ (cid:12)(cid:12)(cid:12)(cid:12) = 2 s | s exp(2 iδ ) | ≤ s − s < . . (21)Using the 3 σ − ranges provided in [3] one easily finds that at 3 σ tan 2 θ tan 2 θ > . , (22)which implies that s = 0 is indeed the only possibility for D to become 0 at 3 σ .Since we are not interested in the limit s →
0, we can use (10) and (14) also to analyzecase C.
Analysis of the relation between sin θ and cos δ It turns out that for all texture zeros studied in this work r (see equation (14)) can beexpressed as a rational function of at most cubic polynomials in cos δ , i.e. r = p (cos δ ) q (cos δ ) , (23)where p and q are polynomials of order at most 3. Thus we find r q (cos δ ) − p (cos δ ) = 0 , (24)which is an equation of at most third order in cos δ . Thus the dependence of cos δ on themixing angles can be computed exactly . Note that (24) will in general have more solutionsthan (23), because we have multiplied by q (cos δ ). In fact we have the additional solution q (cos δ ) = p (cos δ ) = 0 , (25)5hich corresponds to the limit ∆ m ij /m → quasi-degenerate neutrino mass spectrum.For the cases A and A (24) is linear in cos δ . B , B and C lead to quadratic equationsand B and B yield cubic equations for cos δ , respectively. We used Mathematica toobtain the coefficients of (24). The well-known formulae for the general solutions ofquadratic and cubic equations were implemented in C -programs which, scanning overthe experimentally allowed ranges for r , θ and θ , allowed us to plot sin θ versuscos δ . sin θ was varied between 0 and 0 .
05 and for all other experimentally accessiblequantities we used the values obtained from the newest global fit [3] including already thenew T2K data [1]. Our numerical analysis consists of the following steps: − Input: sin θ , sin θ , sin θ , r = ∆ m / ∆ m (best fit values or nσ − range ( n =1 , , θ (0 to 0.05) is divided into 600 steps. The ranges of r, sin θ and sin θ are divided into steps of equal length in such a way that thecorresponding 1 σ − ranges are divided into 40 steps. Thus e.g. the computation forthe 1 σ -range alone consists of 600 × = 3 . × individual calculation cycles. − Equation (24) is solved for cos δ (there can be up to three solutions). − Only the real solutions ∈ [ − ,
1] are processed further, the others are discarded. − Now we want to insert the remaining solutions for cos δ into (13) to calculate themass ratios m i m j = (cid:12)(cid:12)(cid:12)(cid:12) µ i µ j (cid:12)(cid:12)(cid:12)(cid:12) . (26)In order to do that we have to calculate e iδ from cos δ . There are two solutions tothis problem, namely e iδ := cos δ ± i √ − cos δ, (27)but since they are complex conjugates of each other the mass ratios (26) do notdepend on the choice of the solution. − Finally the program checks whether the following inequalities are fulfilled. m m < , m m < , m m < ,m m > , m m > , m m < . (28)If they are fulfilled the data point (cos δ , sin θ ) is stored. − Finally, when the whole parameter range has been scanned, all stored data pointsare plotted . In order to create plots of a suitable size (in terms of disk space) we constructed a lattice dividingthe range of sin θ into 600 and the range of cos δ ( −
6t turns out that a simple scan over the allowed nσ -ranges for the parameters asdescribed above works very well for all cases of types A and B. However, for case C (NS)the sizes of the steps chosen in our systematic scan are just too large in order to obtainenough data points to produce good and reliable results. The reason for this issue is thatC (NS) implies θ ≈ ◦ [18], so one would need an enormously high resolution in thescan over the nσ -ranges of sin θ to produce reliable results. Thus we have to analyze C(NS) in a different way. A good method to deal with case C (NS) is to assign the inputparameters sin θ , sin θ , sin θ , r = ∆ m / ∆ m random values in their nσ -ranges,rather than varying them step by step. In this way one obtains a so-called scatter plot .Since also case C (IS) shows some hints of problems using a systematic scan, we also dida scatter plot for this case. The number of random points (sin θ , sin θ , sin θ , r ) weused was 10 for each of the nσ -ranges ( n = 1 , , We will now present the results of our numerical analysis. As already explained we haveproduced plots of sin θ versus cos δ (see figures 1–12). The color code is the same forall plots: • The best fit value for the point (cos δ, sin θ ) according to the global fit [3] isindicated by a black cross. • The best fit values for sin θ as a function of cos δ according to our analysis areindicated by a black line. • The nσ − regions are shown as colored areas in the plots (red=1 σ , green=2 σ andblue=3 σ ).The cases A and A are incompatible with an inverted spectrum if the reactor angle isvaried within the 3 σ -range 0 ≤ s ≤ .
05. Assuming a normal neutrino mass spectrumA and A predict θ to be different from zero at 3 σ . For the best fit values for theobservables, namely θ , θ and r , we predict 0 . ≥ sin θ ≥ .
012 corresponding tothe bounds − ≤ cos δ ≤ and 0 . ≤ sin θ ≤ .
032 corresponding to − ≤ cos δ ≤ .The cases B , B , B and B predict the Dirac CP phase to be close to maximal, i.e.cos δ ≈
0. Note furthermore that the cases B (NS), B (IS), B (NS) and B (IS) areincompatible with θ > ◦ . Therefore, for these cases, the plots do not show the black“best fit line” (the best fit value for sin θ given in [3] is 0 .
52 which corresponds to anatmospheric mixing angle larger than 45 ◦ ). The generic predictions we found for the casesof type B concerning the atmospheric angle are shown in table 2.For case C and an inverted spectrum we do not find a strong correlation betweensin θ and cos δ . However, from figure 12 we can see that the best fit value for the point(cos δ, sin θ ) lies in the 1 σ -region of the plot. For case C and a normal spectrum thereis no correlation between the Dirac CP phase and the reactor angle (see figure 11), but, This is in accordance with the best fit results for sin θ given in [17] for B and B . θ ≤ ◦ θ ≥ ◦ B θ ≥ ◦ θ ≤ ◦ B θ ≤ ◦ θ ≥ ◦ B θ ≥ ◦ θ ≤ ◦ Table 2: Inequalities for the atmospheric mixing angle for the cases of type B.Figure 1: The relation between sin θ and cos δ for case A (normal spectrum). Fordescription of the colors see the text.as was pointed out by Grimus and Lavoura [18], atmospheric neutrino mixing is close tomaximal. As shown in figure 13 there is a correlation between the reactor angle and theatmospheric angle, but the deviation from the maximal value of the atmospheric angle isnegligible. In the light of the recent T2K result which points towards a large reactor mixing angle θ , we reconsidered the interesting case of two texture zeros in the neutrino mass matrix.In particular we studied the correlation between the reactor mixing angle θ and theDirac CP phase δ for the viable cases classified in [10] as A , A , B , B , B , B and C.All of these cases are still compatible with the global fit of the neutrino data at 3 σ , butonly the cases A and A predict the reactor angle to be different from zero at 3 σ . In8igure 2: The relation between sin θ and cos δ for case A (normal spectrum).Figure 3: The relation between sin θ and cos δ for case B (normal spectrum).9igure 4: The relation between sin θ and cos δ for case B (inverted spectrum).Figure 5: The relation between sin θ and cos δ for case B (normal spectrum).10igure 6: The relation between sin θ and cos δ for case B (inverted spectrum).Figure 7: The relation between sin θ and cos δ for case B (normal spectrum).11igure 8: The relation between sin θ and cos δ for case B (inverted spectrum).Figure 9: The relation between sin θ and cos δ for case B (normal spectrum).12igure 10: The relation between sin θ and cos δ for case B (inverted spectrum).Figure 11: The relation between sin θ and cos δ for case C (normal spectrum).13igure 12: The relation between sin θ and cos δ for case C (inverted spectrum). s i n θ sin θ Figure 13: The relation between sin θ and sin θ for case C (NS) (scatter plot for the1 σ -range with 10 random points). 14articular for the case A , asserting all the free parameters their best fit values, predicts0 . ≤ sin θ ≤ .
024 while for the case A assuming the best fit values predicts0 . ≤ sin θ ≤ . Acknowledgments
This work was supported by the Spanish MICINN under grants FPA2008-00319/FPA,FPA2011-22975 and MULTIDARK CSD2009-00064 (Consolider-Ingenio 2010 Programme),by Prometeo/2009/091 (Generalitat Valenciana), by the EU Network grant UNILHCPITN-GA-2009-237920. S. M. is supported by a Juan de la Cierva contract. E. P. issupported by CONACyT (Mexico).
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