Predicting theta_13 and the Neutrino Mass Scale from Quark Lepton Mass Hierarchies
aa r X i v : . [ h e p - ph ] M a r DESY 11-173December 2011
Predicting θ and the Neutrino Mass Scale fromQuark Lepton Mass Hierarchies W. Buchm¨uller, V. Domcke, and K. Schmitz
Deutsches Elektronen-Synchrotron DESY, 22607 Hamburg, Germany
Abstract
Flavour symmetries of Froggatt-Nielsen type can naturally reconcile the largequark and charged lepton mass hierarchies and the small quark mixing angleswith the observed small neutrino mass hierarchies and their large mixing angles.We point out that such a flavour structure, together with the measured neutrinomass squared differences and mixing angles, strongly constrains yet undeterminedparameters of the neutrino sector. Treating unknown O (1) parameters as randomvariables, we obtain surprisingly accurate predictions for the smallest mixingangle, sin θ = 0 . +0 . − . , the smallest neutrino mass, m = 2 . +1 . − . × − eV,and one Majorana phase, α /π = 1 . +0 . − . . Introduction
It remains a theoretical challenge to explain the observed pattern of quark and lep-ton masses and mixings, in particular the striking differences between the quark sectorand the neutrino sector. Promising elements of a theory of flavour are grand uni-fication (GUT) based on the groups SU(5), SO(10) or E , supersymmetry, the see-saw mechanism and additional flavour symmetries [1]. A successful example is theFroggatt-Nielsen mechanism [2] based on spontaneously broken Abelian symmetries,which parametrizes quark and lepton mass ratios and mixings by powers of a small‘hierarchy parameter’ η . The resulting structure of mass matrices also arises in com-pactifications of higher-dimensional field and string theories, where the parameter η is related to the location of matter fields in the compact dimensions or to vacuumexpectation values of moduli fields (cf. [3]).In this article we consider a Froggatt-Nielsen symmetry which commutes with theGUT group SU(5), and which naturally explains the large ν µ − ν τ mixing [4]. Thissymmetry implies a particular hierarchy pattern in the Majorana mass matrix for thelight neutrinos, m ν ∝ η η ηη η , (1)which can be regarded as a key element for our analysis. The predicted Dirac andMajorana neutrino mass matrices are also consistent with leptogenesis [5]. Despite thesesuccesses, the predictive power of the Froggatt-Nielsen mechanism is rather limited dueto unknown O (1) coefficients in all entries of the mass matrices. For example, theconsidered model [5] can accommodate both a small as well as a large ‘solar’ mixingangle θ [4, 6]. To get an idea of the range of possible predictions for a given flavourstructure, it is instructive to treat the O (1) parameters as random variables [7].In the following we shall employ Monte-Carlo techniques to study quantitatively thedependence of yet undetermined, but soon testable parameters of the neutrino sector onthe unknown O (1) factors of the mass matrices. Using the already measured neutrinomasses and mixings as input, we find surprisingly sharp predictions which indicate alarge value for the smallest mixing angle θ in accordance with recent results fromT2K [8], Minos [9] and Double Chooz [10], a value for the lightest neutrino mass of O (10 − ) eV and one Majorana phase in the mixing matrix peaked at α = π .2 i ∗ ∗ ∗ H u H d SQ i a a a + 1 b c d Table 1: Froggatt-Nielsen charge assignments. From Ref. [5].
As far as orders of magnitude are concerned, the masses of quarks and charged leptonsapproximately satisfy the relations m t : m c : m u ∼ η : η ,m b : m s : m d ∼ m τ : m µ : m e ∼ η : η , (2)with η ≃ /
300 for masses defined at the GUT scale. This mass hierarchy can bereproduced by a simple U(1) flavour symmetry. Grouping the standard model leptonsand quarks into the SU(5) multiplets = ( q L , u cR , e cR ) and ∗ = ( d cR , l L ), the Yukawainteractions take the form L Y = h ( u ) ij i j H u + h ( e ) ij ∗ i j H d + h ( ν ) ij ∗ i j H u + 12 h ( n ) i i i S + c . c . , (3)where = ν cR denote the charge conjugates of right-handed neutrinos and i, j = 1 . . . h ( n ) for the right-handed neutrinoscan always be chosen to be real and diagonal. H u , H d and S are the Higgs fieldsfor electroweak and B − L symmetry breaking, i.e., their vacuum expectation valuesgenerate the Dirac masses of quarks and leptons and the Majorana masses for the right-handed neutrinos, respectively. In this setup, the Yukawa couplings are determined upto complex O (1) factors by assigning U(1) charges to the fermion and Higgs fields inEq. (3), h ij ∼ η Q i + Q j . (4)With the charge assignment given in Tab. 1 the mass relations in Eq. (2) are re-produced. Additionally, perturbativity of the Yukawa couplings and constraints ontan β = h H u i / h H d i require 0 ≤ a ≤
1. 3 asses
From Eq. (3) and Tab. 1 one obtains for the Dirac neutrino mass matrix m D and theMajorana mass matrix of the right-handed neutrinos M , m D v EW sin β = h ( ν ) ij ∼ η a η d +1 η c +1 η b +1 η d η c η b η d η c η b , Mv B − L = h ( n ) ij ∼ η d η c
00 0 η b , (5)with the electroweak and B − L symmetry breaking vacuum expectation values v EW = p h H u i + h H d i and v B − L = h S i , respectively. In the seesaw formula m ν = − m D M m TD , (6)the dependence on the right-handed neutrino charges drops out, and one finds for thelight neutrino mass matrix, m ν ∼ v EW sin βv B − L η a η η ηη η . (7)The charged lepton mass matrix is given by m e v EW cos β = h ( e ) ij ∼ η a η η ηη η η η . (8)Note that the second and third row of the matrix m e have the same hierarchy pattern.This is a consequence of the same flavour charge for the second and third generation ofleptons, which is the origin of the large neutrino mixing. Hence, diagonalizing m e cana priori give a sizable contribution to the mixing in the lepton sector. Mixing
The lepton mass matrices are diagonalized by bi-unitary and unitary transformations,respectively, V TL m e V R = m diag e , U T m ν U = m diag ν , (9)with V † L V L = V † R V R = U † U = . From V L and U one obtains the leptonic mixing matrix U PMNS = V † L U , which is parametrized as [11] U PMNS = c c s c e i α s e i ( α − δ ) − s c − c s s e iδ (cid:0) c c − s s s e iδ (cid:1) e i α s c e i α s s − c c s e iδ (cid:0) − c s − s c s e iδ (cid:1) e i α c c e i α , (10)4ith c ij = cos θ ij and s ij = sin θ ij . Since the light neutrinos are Majorana fermions, allthree phases are physical.In the following we study the impact of the unspecified O (1) factors in the lep-ton mass matrices on the various parameters of the neutrino sector by using a MonteCarlo method, taking present knowledge on neutrino masses and mixings into account.Naively, one might expect large uncertainties in the predictions for the observables ofthe neutrino sector obtained in this setup. For instance, the neutrino mass matrix iscalculated by multiplying three matrices, in which each entry comes with an unspeci-fied O (1) factor, cf. Eq. (6). However, carrying out the analysis described below andcalculating the 68% confidence intervals, we find that in many cases our results aresharply peaked, yielding a higher precision than only an order-of-magnitude estimate. Monte-Carlo study
The unknown O (1) coefficients of the Yukawa matrices h ( e ) , h ( ν ) and h ( n ) are constrainedby the experimental data on neutrino masses and mixings, with the 3 σ confidence rangesgiven by [11]: 2 . × − eV ≤ | ∆ m | ≤ . × − eV , . × − eV ≤ ∆ m ≤ . × − eV , . ≤ sin (2 θ ) ≤ . , . ≤ sin (2 θ ) ≤ . (11)In the following we explicitly do not use the current bound on the smallest mixing angle( θ < .
21 at 3 σ [11]). This allows us to demonstrate that nearly all values we obtainfor θ automatically obey the experimental bound, cf. Fig. 1.In a numerical Monte-Carlo study we generate random numbers to model the 39 realparameters of the three mass matrices. The absolute values are taken to be uniformlydistributed in [10 − / , / ] on a logarithmic scale. The phases in h ( e ) and h ( ν ) are Nine complex O (1) factors in each h ( ν ) and h ( e ) , as well as three real O (1) factors in h ( n ) . Notethat here we are treating the low energy Yukawa couplings as random variables, which are related tothe couplings at higher energy scales via renormalization group equations. However, we expect thatthe effect of this renormalization group running can essentially be absorbed into a redefinition of theeffective scale ¯ v B − L , hence leaving the results presented in the following unchanged. , π ). In the following, we shall refer to thosesets of coefficients which are consistent with the experimental constraints in Eq. (11)as hits.In a preliminary run, we consider the neutrino mixing matrix U , with the effectivescale ¯ v B − L ≡ η − a v B − L / sin β treated as random variable in the interval [10 − / , / ] × GeV. We find that the percentage of hits strongly peaks at ¯ v B − L ≃ × GeV.This is interesting for two reasons. Firstly, it implies that given 0 ≤ a ≤
1, the highseesaw scale lies in the range 3 × GeV . v B − L / sin β . × GeV. Note thatthe upper part of this mass range is close the GUT scale, which is important for recentwork on the connection of leptogenesis, gravitino dark matter and hybrid inflation [12].Secondly, this result allows us to fix the parameter ¯ v B − L in the following computationswithout introducing a significant bias.In the main run, for fixed ¯ v B − L , we include the mixing matrix V L of the chargedleptons to compute the full PMNS matrix. We require the mass ratios of the chargedleptons to fulfill the experimental constraints up to an accuracy of 5% and allow for1 ≤ tan β ≤
60 to achieve the correct normalization of the charged lepton mass spec-trum. Finally, imposing the 3 σ constraints on the two large mixing angles of the fullPMNS matrix, we find parameter sets of O (1) factors which yield mass matrices fulfill-ing the constraints in Eq. (11). Our final results are based on roughly 20 000 such hits.For each hit we calculate the observables in the neutrino sector as well as parametersrelevant for leptogenesis. The resulting distributions are discussed below. Statistical analysis
In our theoretical setup the relative frequency with which we encounter a certain valuefor an observable might indicate the probability that this value is actually realizedwithin the large class of flavour models under study. In the following we shall thereforetreat the distributions for the various observables as probability densities for continuousrandom variables. That is, our predictions for the respective observables represent best-guess estimates according to a probabilistic interpretation of the relative frequencies.For each observable we would like to deduce measures for its central tendency andstatistical dispersion from the respective probability distribution. Unfortunately, it isinfeasible to fit all obtained distributions with one common template distribution. Sucha procedure would lack a clear statistical justification, and it also appears impracticalas the distributions that we obtain differ substantially in their shapes. We thereforechoose a different approach. We consider the median of a distribution as its centre6nd we use the 68 % ‘confidence’ interval around it as a measure for its spread. Ofcourse, this range of the confidence interval is reminiscent of the 1 σ range of a normaldistribution.More precisely, for an observable x with probability density f we will summarizeits central tendency and variability in the following form [13], x = ˆ x ∆ + ∆ − , ∆ ± = x ± − ˆ x . (12)Here, x − and x + denote the 16 %- and 84 %-quantiles with respect to the densityfunction f . The central value ˆ x is the median of f and thus corresponds to its 50 %-quantile. All three values of x can be calculated from the quantile function Q , Q ( p ) = inf { x ∈ [ x min , x max ] : p ≤ F ( x ) } , F ( x ) = Z xx min dt f ( t ) , (13)where F stands for the cumulative distribution function of x . We then have: x − = Q (0 . , ˆ x = Q (0 . , x + = Q (0 . . (14)Intuitively, the intervals from x min to x − , ˆ x , and x + respectively correspond to the x ranges into which 16 %, 50 % or 84 % of all hits fall. This is also illustrated in thehistogram for sin θ in Fig. 1. Moreover, we have included vertical lines into eachplot to indicate the respective positions of x − , ˆ x , and x + .In our case the median is a particularly useful measure of location. First of all, it isresistant against outliers and hence an appropriate statistic for such skewed distribu-tions as we observe them. But more importantly, the average absolute deviation fromthe median is minimal in comparison to any other reference point. The median is thusthe best guess for the outcome of a measurement if one is interested in being as close aspossible to the actual result, irrespective of the sign of the error. On the technical sidethe definition of the median fits nicely together with our method of assessing statisticaldispersion. The 68 % confidence interval as introduced above is just constructed in sucha way that equal numbers of hits lie in the intervals from x − to ˆ x and from ˆ x to x + ,respectively. In this sense, our confidence interval represents a symmetric error withrespect to the median.As a test of the robustness of our results, we checked the dependence of our distri-butions on the precise choice of the experimental error intervals. The results presentedhere proved insensitive to these variations. For definiteness, we therefore stick to the7 % % % % % m e d i a n - - - - Θ r e l a ti v e fr e qu e n c y Θ r e l a ti v e fr e qu e n c y Figure 1: Neutrino mixing angles θ and θ . The vertical lines denote the position of the median (solidline) and the boundaries of the 68% confidence region (dashed lines) of the respective distribution. σ intervals. We also checked the effect of taking the random O (1) factors to be dis-tributed uniformly on a linear instead of a logarithmic scale. Again, the results provedto be robust. Mass hierarchy
An important open question which could help unravel the flavour structure of theneutrino sector is the mass hierarchy. Since the sign of ∆ m is not yet known, wecannot differentiate with current experimental data between a normal hierarchy withone heavy and two light neutrino mass eigenstates and an inverted hierarchy, which hastwo heavy and one light neutrino mass eigenstate. Measuring the Mikheyev-Smirnov-Wolfenstein (MSW) effect of the earth could resolve this ambiguity.With the procedure described above, all hits match the structure of the normalhierarchy and there are no examples with inverted hierarchy. It is however notablethat imposing the structure of the neutrino mass matrix given by Eq. (7) alone doesnot exclude the inverted mass hierarchy. Only additionally imposing the measuredbounds on the mixing angles rejects this possibility. Mixing angles
The mixing in the lepton sector is described by the matrix U PMNS given in Eq. (10).Of the three angles, two are only bounded from one side by experiment: for the largestmixing angle θ there exists a lower bound, whereas the smallest mixing angle θ - - - - - m @ eV D r e l a ti v e fr e qu e n c y m Β @ eV D r e l a ti v e fr e qu e n c y Figure 2: Lightest neutrino mass m and effective neutrino mass in tritium decay m β . Vertical linesand shadings as in Fig. 1. is so far only bounded from above. Recent results from T2K [8], Minos [9] and thepreliminary result of Double Chooz [10] point to a value of θ just below the currentexperimental bound. The respective best fit points, assuming a normal hierarchy, aresin θ = 0 .
11 (T2K), 2 sin θ sin θ = 0 .
041 (MINOS) and sin θ = 0 . . < sin θ < .
28 T2K, 90 % CL , δ CP = 0 , θ sin θ < .
12 MINOS, 90 % CL , δ CP = 0 , (15)0 . < sin θ < .
16 Double Chooz, 68 % CL . With the procedure described above, we find sharp predictions for the smallest andthe largest mixing angle within the current experimental bounds,sin θ = 0 . +0 . − . , sin θ = 0 . +0 . − . ; (16)the corresponding distributions are shown in Fig. 1. These results are quite remarkable:the atmospheric mixing angle points to maximal mixing, while the rather large valuefor θ is consistent with the recent T2K, Minos and Double Chooz results.In our Monte-Carlo study we observe that the dominant contribution to the strongmixing in the lepton sector is primarily due to the neutrino mass matrix m ν . Thenumerical results are not much affected by including the charged lepton mixing matrix V L . The PMNS matrix is thus approximately given by the matrix U which diagonalizesthe light neutrino mass matrix m ν . 9 - - - m ΝΒΒ @ eV D r e l a ti v e fr e qu e n c y (cid:144) Π Π (cid:144) Π Π Α r e l a ti v e fr e qu e n c y Figure 3: Effective mass in neutrinoless double-beta decay m νββ and Majorana phase α . Verticallines and shadings as in Fig. 1. Absolute mass scale
The absolute neutrino mass scale is a crucial ingredient for the study of neutrinolessdouble-beta decay and leptogenesis. Although inaccessible in neutrino oscillation ex-periments, different experimental setups have succeeded in constraining this mass scale.Cosmological observations of the fluctuations in the cosmic microwave background, ofthe density fluctuations in the galaxy distribution and of the Lyman- α forest yield aconstraint for the sum of the light neutrino masses, weighted by the number of spindegrees of freedom per Majorana neutrino, g ν = 2, [11] m tot = X ν g ν m ν . . . (17)The Planck satellite is expected to be sensitive to values of m tot as low as roughly0 . β -spectrum in tritiumdecay experiments. The current bound [11] is m β = X i | ( U P MNS ) ei | m i < . (18)By comparison, the KATRIN experiment, which will start taking data soon, aims atreaching a sensitivity of 0 .
04 eV [15]. Finally, the neutrino mass scale can also beprobed by neutrinoless double-beta decay. The relevant effective mass is m νββ = | X i ( U P MNS ) ei m i | . (19)Here, Ref. [16] claims a value of 0 . − .
56 eV. Dedicated experiments, such asGERDA [17] with a design sensitivity of 0 . − .
20 eV, are on the way. Note that10 - - - - - m Ž @ eV D r e l a ti v e fr e qu e n c y - - - ¶ (cid:144) ¶ max r e l a ti v e fr e qu e n c y Figure 4: Effective neutrino mass of the first generation e m and CP violation parameter ε . Verticallines and shadings as in Fig. 1. m νββ does not only depend on the absolute neutrino mass scale and the mixing angles,but also on the phases ( α − δ ) and α in the PMNS matrix.We find sharp predictions for the neutrino mass parameters discussed above. Thelightest neutrino, ν , is found to be quite light, cf. Fig. 2, m = 2 . +1 . − . × − eV , (20)hence favouring a relatively low neutrino mass scale beyond the reach of current andupcoming experiments. More precisely, we find for the neutrino mass parameters dis-cussed above: m tot = 6 . +0 . − . × − eV , m β = 8 . +3 . − . × − eV , m νββ = 1 . +0 . − . × − eV . (21) CP-violating phases
The small value of the mass parameter measured in neutrinoless double-beta decay, m νββ , is due to the relative minus sign between the m and m terms in Eq. (19),caused by a strong peak of the value for the Majorana phase α at π , α π = 1 . +0 . − . . (22)This is depicted in Fig. 3. An analytic analysis of how this phenomena arises from thestructure of the neutrino mass matrix, cf. Eq. (7), is presented in Appendix A. For theother Majorana phase α and the Dirac phase δ we find no such distinct behaviourbut approximately flat distributions. 11 eptogenesis parameters Finally, leptogenesis [18] links the low energy neutrino physics to the high energy physicsof the early universe. The parameters that capture this connection are the effectiveneutrino mass of the first generation e m and the CP violation parameter ε [19], e m = ( m † D m D ) M , ε = − X j =2 , Im (cid:2) ( h ( ν ) † h ( ν ) ) j (cid:3) π ( h ( ν ) † h ( ν ) ) F (cid:18) M j M (cid:19) , (23)with F ( x ) = √ x (cid:0) ln xx + x − (cid:1) and M j denoting the masses of the heavy neutrinos.Here, e m determines the coupling strength of the lightest of the heavy neutrinos tothe thermal bath and thus controls the significance of wash-out effects. It is boundedfrom below by the lightest neutrino mass m . The absolute value of the CP violationparameter ε is bounded from above by [20] ε max = 38 π | ∆ m | / M v EW sin β ≃ . × − (cid:18) β (cid:19) (cid:18) M GeV (cid:19) . (24)With the procedure described above, we find e m = 4 . +3 . − . × − eV , ε ε max = 0 . +0 . − . , (25)and hence a clear preference for the strong wash-out regime [19]. Notice that theretypically is a hierarchy between e m and m of about one order of magnitude. Therelative frequency of the CP violation parameter ε peaks close to the upper bound ε max , with the majority of the hits lying within one order of magnitude or less below ε max , cf. Fig. 4. This justifies the use of ε max when estimating the produced leptonasymmetry in leptogenesis. Here, in the discussion of ε , we assumed hierarchicalheavy neutrinos, M , ≫ M . Theoretical versus experimental input
The results of this section are obtained by combining two conceptually different inputs,on the one hand the hierarchy structure of the neutrino mass matrix m ν given by Eq. (1)and on the other hand the experimentally measured constraints listed in Eq. (11). Ingeneral, the distributions presented above really arise from the interplay between bothof these ingredients. For example, the hierarchy structure alone does not favour a largesolar mixing angle θ and the ratio ∆ m / ∆ m tends to be too large (cf. [21, 22]).This discrepancy is eased by generating the random coefficients in Eq. (1) via theseesaw mechanism. Imposing the experimental constraints finally singles out the subset12f parameter sets used for the distributions presented above. As another example,consider the smallest mixing angle θ and the lightest neutrino mass eigenstate m . Inthese cases, the hierarchy structure of the neutrino mass matrix automatically impliessmall values, similar to those shown in the distributions above. However, the exactdistributions including the precise position of the peaks only arise after implementingthe experimental constraints. A notable exception to this scheme is the Majorana phase α . Here the peak at α = π is a result of the hierarchy structure of the neutrinomatrix m ν alone, as demonstrated in Appendix A. In summary, we find that starting from a flavour symmetry which accounts for themeasured quark and lepton mass hierarchies and large neutrino mixing, the presentknowledge of neutrino parameters strongly constrains the yet unknown observables, inparticular the smallest mixing angle θ , the smallest neutrino mass m , and the Majo-rana phase α . This statement is based on a Monte-Carlo study: Treating unspecified O (1) parameters of the considered Froggatt-Nielsen model as random variables, theobservables of interest are sharply peaked around certain central values.We expect that these results hold beyond Froggatt-Nielsen flavour models. Anobvious example are extradimensional models which lead to the same type of light neu-trino mass matrix (cf. [23]). On the other hand, quark-lepton mass hierarchies and thepresently known neutrino observables cannot determine the remaining observables ina model-independent way. This is illustrated by the fact that our present knowledgeabout quark and lepton masses and mixings is still consistent with θ ≃ Acknowledgements
The authors thank G. Altarelli, F. Br¨ummer, G. Ross, D. Wark, W. Winter andT. Yanagida for helpful discussions and comments. This work has been supported bythe German Science Foundation (DFG) within the Collaborative Research Center 676“Particles, Strings and the Early Universe”.13
Analytic derivation of the Ma jorana phase α O (1) coefficients in the neutrino mass matrix m ν and thelepton mass matrix m e are randomly distributed. One would thus naively expect thatalso the Majorana phases α and α in the PMNS matrix can take arbitrary values.By contrast, the distribution of values for α that we obtain from our numerical Monte-Carlo study, cf. Fig. 3, clearly features a prominent peak at α = π . In this appendixwe shall demonstrate by means of a simplified example how the structure of the neutrinomass matrix m ν may partly fix the phases of the corresponding mixing matrix U .Consider the following simplified Majorana mass matrix m ν for the light neutrinos, m ν = v η ηe iϕ ηηe iϕ η , v = v ¯ v B − L , (26)where ϕ is an arbitrary complex phase between 0 and 2 π . For simplicity, let us neglectany effects on the mixing matrix U from the diagonalization of m e . That is, we define U such that U T m ν U = diag ( m i ), with m i denoting the eigenvalues of m † ν m ν , m , v = η sin ( ϕ/ h ∓ η (5 + 3 cos ( ϕ )) / i + O (cid:0) η (cid:1) , (27) m v = 4 (cid:0) η (cid:2) − sin ( ϕ/ (cid:3)(cid:1) + O (cid:0) η (cid:1) . Notice that the first two mass eigenvalues are nearly degenerate. This is a consequenceof the particular hierarchy pattern of the matrix m ν which originally stems from theequal flavour charges of the ∗ and ∗ multiplets. The relative sign of the O ( η )contributions to m and m eventually shows up again in entries of U , for instance, U , = ∓ ϕ )) / e iϕ exp (cid:18) − i ∓ z ] (cid:19) + O ( η ) . (28)with z = 1 − cos ( ϕ ) − i sin ( ϕ ) . The phase α = 2 (Arg [ U /U ] mod π ) in the matrix U represents the analog of the Majorana phase α in the PMNS matrix, cf. Eq. (10).According to our explicit results for U and U it is independent of the arbitrary phase ϕ to leading order in η , α ≃ (cid:18) Arg (cid:20) − exp (cid:18) − i z ] + i − z ] (cid:19)(cid:21) mod π (cid:19) = π . (29)14n a similar way we may determine the phase analogous to the Majorana phase α .However, due to the hierarchy between the mass eigenvalues m and m , the first andthird column of the matrix U differ significantly from each other, thus leading to aphase that depends on ϕ at all orders of η .Including corrections to all orders in η and scanning over the phase ϕ numericallyshows that the maximal possible deviation of α from π is, in fact, of order η . Addingmore complex phases to the matrix m ν in Eq. (26) gradually smears out the peakin the distribution of α values. The distribution that is reached in the case of sixdifferent phases is already very similar to the one in Fig. 3. We conclude that despitethe need for corrections the rough picture sketched in this appendix remains valid: Thehierarchy pattern of the neutrino mass matrix directly implies that α tends to beclose to α = π . References [1] For a review and references see, for example,S. Raby, Eur. Phys. J. C , 223 (2009)[2] C. D. Froggatt and H. B. Nielsen, Nucl. Phys. B , 277 (1979)[3] Y. Grossman and M. Neubert, Phys. Lett. B , 361 (2000); T. Gherghettaand A. Pomarol, Nucl. Phys. B , 141 (2000); W. Buchmuller, K. Hamaguchi,O. Lebedev and M. Ratz, Nucl. Phys. B , 149 (2007)[4] J. Sato and T. Yanagida, Phys. Lett. B , 127 (1998); N. Irges, S. Lavignac andP. Ramond, Phys. Rev. D , 035003 (1998)[5] W. Buchmuller and T. Yanagida, Phys. Lett. B , 399 (1999)[6] F. Vissani, JHEP , 025 (1998)[7] L. J. Hall, H. Murayama and N. Weiner, Phys. Rev. Lett. , 2572 (2000); J. Satoand T. Yanagida, Phys. Lett. B , 356 (2000); F. Vissani, Phys. Lett. B ,79 (2001)[8] K. Abe et al. [T2K Collaboration], Phys. Rev. Lett. , 041801 (2011)[9] P. Adamson et al. [MINOS Collaboration], Phys. Rev. Lett. , 181802 (2011).1510] Talk given by H. de Kerret at the Sixth International Workshop on Low EnergyNeutrino Physics (LowNu11) at Seoul, Korea during November 9-12, 2011[11] K. Nakamura et al. [Particle Data Group], J. Phys. G , 075021 (2010)[12] W. Buchmuller, K. Schmitz and G. Vertongen, Phys. Lett. B , 421 (2010);W. Buchmuller, K. Schmitz and G. Vertongen, Nucl. Phys. B , 481 (2011);W. Buchmuller, V. Domcke and K. Schmitz, arXiv:1202.6679 [hep-ph].[13] G. Cowan, “Statistical data analysis,” Oxford, UK: Clarendon (1998) 197 p[14] [Planck Collaboration], ESA-SCI(2005)1 (2006), arXiv:astro-ph/0604069[15] M. Beck [KATRIN Collaboration], J. Phys. Conf. Ser. (2010) 012097[16] H. V. Klapdor-Kleingrothaus, A. Dietz, H. L. Harney, I. V. Krivosheina, Mod.Phys. Lett. A16 , 2409-2420 (2001)[17] G. Meierhofer [GERDA Collaboration], J. Phys. Conf. Ser. (2011) 072011.[18] M. Fukugita, T. Yanagida, Phys. Lett.
B174 , 45 (1986).[19] For reviews containing the relevant formulae see, for example,W. Buchmuller, R. D. Peccei and T. Yanagida, Ann. Rev. Nucl. Part. Sci. , 311(2005); S. Davidson, E. Nardi and Y. Nir, Phys. Rept. , 105 (2008)[20] S. Davidson and A. Ibarra, Phys. Lett. B , 25 (2002); see also K. Hamaguchi,H. Murayama and T. Yanagida, Phys. Rev. D , 043512 (2002)[21] G. Altarelli, F. Feruglio and I. Masina, JHEP , 035 (2003); I. Masina andC. A. Savoy, Phys. Rev. D , 093003 (2005) [hep-ph/0501166].[22] F. Plentinger, G. Seidl and W. Winter, Phys. Rev. D , 113003 (2007);F. Plentinger, G. Seidl and W. Winter, Nucl. Phys. B , 60 (2008)[23] T. Asaka, W. Buchmuller, L. Covi, Phys. Lett. B563 , 209-216 (2003)[24] For a review and references see, for example,G. Altarelli, F. Feruglio, Rev. Mod. Phys. , 2701-2729 (2010); H. Ishimori,T. Kobayashi, H. Ohki, Y. Shimizu, H. Okada, M. Tanimoto, Prog. Theor. Phys.Suppl.183