Large Mixing Angles From Many Right-Handed Neutrinos
IIPMU11-0195
Large Mixing Angles From Many Right-Handed Neutrinos
Brian Feldstein a and William Klemm a,ba Institute for the Physics and Mathematics of the Universe,University of Tokyo, Kashiwa, 277-8568, Japan b Department of Physics and Astronomy,Uppsala University, Box 516, SE-751 20 Uppsala, Sweden
Abstract
A beautiful understanding of the smallness of the neutrino masses may be ob-tained via the seesaw mechanism, whereby one takes advantage of the key qualitativedistinction between the neutrinos and the other fermions: right-handed neutrinosare gauge singlets, and may therefore have large Majorana masses. The standardseesaw mechanism, however, does not address the apparent lack of hierarchy in theneutrino masses compared to the quarks and charged leptons, nor the large leptonicmixing angles compared to the small angles of the CKM matrix. In this paper,we will show that the singlet nature of the right-handed neutrinos may be takenadvantage of in one further way in order to solve these remaining problems: Unlikeparticles with gauge interactions, whose numbers are constrained by anomaly can-cellation, the number of gauge singlet particles is essentially undetermined. If largenumbers of gauge singlet fermions are present at high energies – as is suggested, forexample, by various string constructions – then the effective low energy neutrinomass matrix may be determined as a sum over many distinct Yukawa couplings, withthe largest ones being the most important. This can reduce hierarchy, and lead tolarge mixing angles. Assuming a statistical distribution of fundamental parameters,we will show that this scenario leads to a good fit to low energy phenomenology,with only a few qualitative assumptions guided by the known quark and leptonmasses. The scenario leads to predictions of a normal hierarchy for the neutrinomasses, and a value for the | m ee | mass matrix element of about 1 − a r X i v : . [ h e p - ph ] F e b Introduction
The quark and lepton masses and mixing angles provide one of the fundamental mysteriesof the standard model of particle physics. They hint at a potentially deep underlyingstructure, while also seeming sufficiently random so as to defy straightforward explanation.There are a number of qualitative features of these parameters which one would like tounderstand:
1. Why are the neutrino masses roughly ten orders of magnitude smaller than themasses of the quarks and charged leptons?2. Why do the quark and charged lepton masses have significant hierarchies of aboutfive orders of magnitude?3. Why is the CKM matrix approximately equal to the identity when the up and downquarks are both ordered by mass?4. Why are the mixing angles in the lepton sector fundamentally different from thosein the quark sector, with two angles close to maximal?Of these questions, it is the first which has lent itself most easily to explanation. Thekey observation is that right-handed neutrinos, unlike any of the other standard modelfermions, have no known gauge interactions. As a result, no symmetry forbids them fromobtaining Majorana masses, which may be many orders of magnitude larger than theweak scale. After integrating out the right-handed neutrinos from the theory, one obtainseffective operators of the form ( HL ) /M R , where H is the Higgs field, L is a lepton doublet,and M R is the right-handed neutrino mass scale. This yields light Majorana neutrinos withmasses of order v /M R , with v = 174 GeV being the Higgs vacuum expectation value, andwith a possible additional suppression from Yukawa couplings [1]. Appropriately smallmasses are then obtained for M of order 10 − GeV, perhaps related to the scaleof grand unification. This picture is so simple that it is now taken almost for granted,although it should be kept in mind that experimental confirmation is still lacking. The strong CP problem may or may not have a solution related to the other flavor mysteries, andwe will not concern ourselves with it in this paper. We may rely, for example, on a separate solutioninvolving an invisible axion with a large Peccei-Quinn symmetry breaking scale. we do not know how many of them there are . In-deed, constructions in string theory often produce large numbers of singlet fermions withMajorana masses close to the GUT scale after the compactification of extra dimensions[6, 7, 8]. In the presence of a large number of singlets, the low energy neutrino mass matrix willbe realized as a sum over many distinct hierarchical numbers with the largest Yukawacouplings dominating. This generically results in a washed out hierarchy, and potentiallylarge mixing angles. It follows that a random hierarchical scan of Yukawa couplings is infact a good fit for the observed masses and mixing angles of the standard model, so longas the possibility for a large number of right-handed neutrinos is taken into account.The reason that many singlet states often arise with masses close to the GUT scalein string constructions is actually reasonably generic: After compactifying d extra dimen-sions in a string model, the resulting Kaluza Klein mass scale is related to the effectivefour-dimensional Planck scale M Pl through the relation M KK ∼ M Pl (cid:18) M KK M s (cid:19) d +1 , (1)where M s is the string scale. In order for the compactification geometry to be reasonablydescribed by a classical gravity picture, it is required that M KK be at least parametri-cally smaller than M s . From the above relation, and considering the 6 available extradimensions of string theory, we thus see that it may be reasonable to expect the KaluzaKlein scale to be roughly a few orders of magnitude smaller than M Pl . Note that it is For more general field theory discussions, see also [9, 10]. It is not necessarily required, however, that the compactification of the extra dimensions have a welldefined classical gravity description. This is perhaps the main caveat to the present argument. − GeV. In any case, the point then is that the KK scale is a naturalscale for the appearance of a large number of gauge singlets which may then serve asright-handed neutrinos; these singlets might be born out of KK towers themselves, as wasthe case considered in [7], or they might be associated with moduli fields, stabilized bymasses close to M KK due to fluxes in the compactified dimensions [8]. For our purposes,the main point is that the number of relevant singlets with masses close to the neededseesaw scale may easily number in the tens or hundreds. Note that we will not attempt toconstruct an explicit top-down model for Yukawa couplings in this paper, but will simplytake a phenomenological point of view based on the known properties of the standardmodel masses and mixing angles. We will leave top-down model building as a possiblesubject for future work.The outline of our paper is as follows: In section 2 we will describe our framework inmore detail, discussing the various qualitative features we require, including the propertiesof the high energy Majorana mass matrix. In section 3, we will give an example of aparticular universal Yukawa coupling distribution which gives a good phenomenologicalfit for the charged fermion masses and mixings, and show how our mechanism then alsoleads to good phenomenology in the neutrino sector. Additionally, we will show that oursetup leads to the predictions of a normal hierarchy for the neutrino masses, and a valueof | m ee | – the neutrino mass parameter relevant for neutrinoless double beta decay – ofbetween about 1 and 6 meV. We will summarize and discuss future directions in section4. The reason that many right-handed neutrinos, which we shall denote generically as ν (cid:48) R s ,can wash out hierarchy in the low energy neutrino mass matrix m is simple. This matrix In this latter case note that the singlet masses are actually given by ∼ M /M s , with a furthersmall suppression below the KK scale. m ij = v (cid:88) lk Y ik M − kl Y jl , (2)where, with N right-handed neutrinos, Y is the 3 × N Yukawa matrix, M is the N × N Majorana mass matrix, and v is the Higgs vacuum expectation value. Taking the typicallargest Yukawa coupling size to be of order one, for example, we can see that, withsufficiently many ν R ’s, each matrix element in m will generically obtain some number oflarge contributions. Even if the original Yukawas were hierarchical, the m ij ’s then comeout to be roughly of the same order, with the hierarchy having been lost (note that,assuming arbitrary signs or phases for each term in the sum, one obtains an enhancementto the overall scale of m by a factor which scales as √ N ).The mechanism we are discussing here is fairly general, and a specific structure ordistribution for the Yukawa couplings and Majorana masses is not necessary. We do,however require the following conditions to be satisfied: • There must be a reasonably well defined upper-bound to the neutrino Yukawa cou-plings. Such an upper-bound could be set by perturbativity or by some other fun-damental physics, and should be reasonably independent of which neutrino field agiven Yukawa coupling is associated with. • Enough right-handed neutrinos must be present so that each left-handed neutrino isexpected to have > ∼ ν R ’s, which will give the dominant contribution to the seesaw.Amongst the standard model quarks and charged leptons, we have one order oneYukawa coupling amongst 27 (associated with the top quark), and so with this asa guide, we anticipate that we may require roughly > ∼
30 right handed-neutrinos forour mechanism to work. • We require a Majorana mass matrix which mixes together the right-handed neutrinofields. One simple way this might be realized is if the physics determining theMajorana masses is distinct from that determining the Yukawa couplings (in which6ase the unitary matrix which diagonalizes the Majorana masses can be taken to bedistributed according to the Haar measure, as will be discussed in section 3), butessentially any sufficiently non-diagonal structure should be viable.The need for this last condition can be seen as follows: Suppose that the Majorana massmatrix were diagonal. In that case there would be a tendency for the diagonal elementsof the low energy mass matrix to be larger than the off-diagonal elements, resulting insmall mixing angles. Let us compare for example the 1 , , m ∼ (cid:88) k M − kk Y k , (3) m ∼ (cid:88) k M − kk Y k Y k . (4)For any given Yukawa coupling y , there is in general a much higher chance that y willbe near its upper bound, than the chance that two separate couplings y and y will bothbe large and lead to a large product y y . It is thus clear from equations 3 and 4 that adiagonal Majorana mass matrix will tend to give poor phenomenology in our scenario. It should be clear that we certainly do not require that all of the Yukawa couplingsof the quarks and leptons share a universal distribution. We will, however, concentrateon such distributions in this work, since in this way we may check that it is actually themany ν R ’s which are resulting in large mixing angles, rather than any fundamental lackof hierarchy assumed for the neutrino Yukawa couplings. Note that for this reason, thereis no intrinsic obstacle to realizing our scenario in a supersymmetric context (in whichtan β would modify the hypothetical quark and charged lepton distributions somewhat),or even in the context of a grand unified theory.Note that a wave-function overlap picture (see e.g. [7, 8, 11, 12]) for obtaining hierarchyin the Yukawa couplings would not work very well for our scenario, since in such casesthe couplings for a given field tend to all be correlated in size. This is the key manner inwhich our scenario with many right-handed neutrinos differs from those already appearingin the literature. Constructions with compactified extra-dimensions may still be used Note that, even with a Majorana mass matrix with substantial off-diagonal components, squaredYukawa couplings will still only contribute to the diagonal elements of m ij . With many right-handedneutrinos, however, the large number of terms (of order N ) which are products of separate Yukawacouplings actually give the dominant contribution.
7o motivate the presence of the many ν R states near the GUT scale, but we require,for example, extra-dimensional wave function profiles to be fairly flat, with the Yukawahierarchy generated by fundamental physics in some other manner.In the next section we will show our mechanism at work quantitatively in a specificexample: We will take a Yukawa coupling distribution fit to the known properties ofthe quarks and charged leptons, and work out the phenomenological consequences as afunction of the number of ν R ’s. We will also show that our scenario (as well as anyscenario with a roughly anarchical low energy neutrino mass matrix) leads to a predictionof | m ee | ∼ − Fitting to the charged fermion masses and CKM angles
Without knowledge of the high-scale theory it is of course impossible to know the fun-damental distribution of the Yukawa couplings. However, existing data can serve as aguide; the best fitting probability distribution for the charged fermions has been exam-ined in some detail in [2], which we use as a starting point for our study. For powerlaw distributions with minimum and maximum cutoffs, they found that the quark andcharged lepton masses are best fit by a distribution that is very close to scale invariant, ρ ( m ) ∼ /m . However, we expect that the Yukawa couplings, not the masses, are thefundamental parameters of the underlying theory. Because of mixing effects, the form ofthe Yukawa probability distribution is not directly transferred to the masses. Consideringdistributions of the form ρ ( y ) ∼ y δ (5)at the GUT scale within the standard model, the authors of [2] found that values of δ just below 1 . . × − ≤ y ≤ / ∼ ρ ( y ) ∼ /y . , while it was found that reasonable masses and smallCKM mixing were obtained, in some cases the observed CKM off-diagonal elements were larger than the mean predicted values, lying on tails of their distributions. This can beseen from the plot of the | V us | distribution in Figure 1. (cid:200) V us (cid:200) P r ob a b ilit y D e n s it y C u m u l a ti v e ∆ (cid:61) ∆ (cid:61) ∆ (cid:61) ∆ (cid:61) Figure 1: | V us | for δ = 1 . √
10 ofGUT scale values. The vertical red line represents the experimental value of | V us | . Themaximum possible value is 1 / √ δ = 1 . − .
3. Verticaldotted lines indicate the median values of the distributions, and horizontal dotted linesindicate the fraction of matrices with | V us | less than the experimental value.To ensure that we consider a distribution which gives reasonable quark mixings, wehave generated a large number of up and down quark mass matrices, by scanning accordingto equation 5 for various values of δ and assigning each element a random phase. Wethen consider those which give a mass eigenvalue for each quark which lies within an9edian % below exp. δ V us V ub V cb V us V ub V cb Exp. .2252 .00389 .0406
Table 1: Median values of CKM element magnitudes for different values of δ along withtheir measured values [15]. The rightmost columns indicate the percent of matrices whichgive a given CKM element smaller in magnitude than the measured value. Restrictionson the mass eigenvalues are as in Figure 1.order of magnitude of the measured value, i.e. in the range ( m q / √ , m q √ Finally,we diagonalize the mass matrices with eigenvalues ordered by mass and determine thecorresponding CKM matrices, eliminating any with an incorrect generation structure,as discussed above. As we see in Figure 1, there is a sharp peak at small mixing, soto get a measure of how consistent each scenario is with the data, we characterize eachdistribution by its median as well as the probability of finding a value smaller than theexperimentally observed value. We will use the 68% and 95% levels of the latter to define1 σ and 2 σ ranges, as in [2]. This is illustrated for | V us | in Figure 1 (inset), which showssome preference for smaller values of δ – smaller δ leads to larger mixing angles. To get anoverall measure of the CKM matrix, we consider the three off-diagonal elements shown inTable 1. The measured values of | V us | , | V ub | , and | V cb | , all show improvement of agreementas δ is decreased, and fall within our 1 σ criterion around δ = 1 .
1. We use this, alongwith the mass only best fit of δ ∼ . ± . δ = 1 . Here and throughout this paper we use charged fermion mass eigenvalues evaluated at the GUT scalein the standard model [13], though [2] demonstrated that the best fit for the distribution feels little effectfrom the running. The experimental values for the CKM elements we cite do not reflect any running,but this effect is small, no more than ∼ V us [14]. he Majorana distribution For the high energy Majorana mass matrix, for which we have no guidance from data, wetake a simple ansatz in accord with the discussion of section 2; we suppose that the physicsdetermining the Majorana mass matrix is independent from that determining the Yukawacouplings. This assumption then implies that the unitary matrix U which diagonalizes M = U DU T may be taken as distributed according to the Haar measure [4]. Haardistributed unitary matrices are easily generated via previously established algorithms[16]. Note that we must further impose a distribution on the mass eigenvalues appearingin the diagonal matrix D ; for simplicity we choose the case of a linear distribution betweenzero and M R , the Majorana mass scale, which we fix by requiring that we match the valuefor the larger measured neutrino mass difference, ∆ m . Probability for large PMNS mixing angles
Given our chosen statistical distributions, we next generate a large number of PMNSmatrices for various numbers of right-handed neutrino species between N = 2 and N =100. We first generate a 3 × N Dirac mass matrix M D = vY , where the entries of theYukawa matrix Y scan between 1 . × − and 1 with probability density ρ ( y ) ∼ y − . andhave random phases. We then generate a Haar distributed N × N Majorana mass matrixwith linearly scanning eigenvalues, to form a neutrino mass matrix m ≡ M D M − M TD = U ν D ν U Tν , where D ν is a diagonal matrix containing the neutrino mass eigenvalues, and U ν is unitary. We generate 3 × M l with the same distributionas the Dirac matrices for neutrinos. To more appropriately model the true charged leptonmass matrix, we only take matrices which agree to within a factor of √
10 of the GUTscale charged lepton masses, as described previously for the quarks. We diagonalize M l = U l D l V † l , from which we can generate the PMNS matrix U P MNS = U † l U ν . Thiscan be parameterized, with unphysical phases rotated away, in the standard fashion with The Haar measure µ on a group G satisfies µ ( gE ) = µ ( E ) for every g ∈ G , and is uniquely definedfor G = U ( N ). θ ij ) and three CP phases ( δ, α , α ), U P MNS = 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 (6) × diag(1 , e i α , e i α ) , (7)where s ij = sin( θ ij ) and c ij = sin( θ ij ). Experimentally, there are two large mixing angles,with a best fit (assuming a normal hierarchy) of sin ( θ ) BF = 0 . +0 . − . and maximalmixing for sin ( θ ) BF = 0 . +0 . − . (1 σ )[17]. The third mixing angle is known to be small,with a best fit of sin ( θ ) BF = 0 . +0 . − . at 1 σ [17], and there is growing evidence for anonzero mixing [18, 19]. F r ac ti ono f M a t r i ce s Figure 2: Fraction of matrices with sin (2 θ ) largest ≥ .
98 and sin (2 θ ) next largest ≥ . Note that in this paper we ignore running effects due to the neutrino Yukawa couplings. Even atand above the seesaw scale these are expected to be small for the distributions we consider, since wetake all of our Yukawa couplings to be smaller than 1, with only a handful approaching this upperbound. If the bound on the Yukawas were taken larger, then renormalization group running could haveimportant, albeit model dependent effects. For example, with low energy supersymmetry, depending onthe superpartner spectrum, it is possible for running at the seesaw scale to contribute to slepton masssplittings and flavor changing neutral currents [9]. If only the standard model were assumed valid upto energies above the seesaw scale, then it is possible to obtain large radiative corrections to the Higgsquartic coupling and thereby large, experimentally excluded, values for the Higgs mass. Note that thislatter conclusion would be avoided in the presence of high scale supersymmetry. (2 θ ) = 1, is a special point, we look for cases which have at least as much mixingas the 1 σ experimental bounds, requiring that one angle satisfies sin (2 θ ) ≥ .
98 andanother satisfies sin (2 θ ) ≥ .
84. The results are shown in Figure 2, from which we seea clear indication that as the number of right-handed neutrinos increases, so too doesthe likelihood of obtaining large mixing angles – as expected for the reasons laid out inSection 2.This effect is further illustrated in Figure 3, where we see the shift to largermixing angles as N increases. N (cid:61) N (cid:61) N (cid:61) N (cid:61) (cid:72) Θ (cid:76) Figure 3: Distribution of mixing angles. The three different bands represent the largest,middle, and smallest sin (2 θ ). Other parameters
While the absolute masses of the neutrinos are not well measured, oscillation experimentsgive us a good measure of their mass squared differences, with a best fit of ∆ m =7 . +0 . − . × − eV and ∆ m = 2 . +0 . − . × − eV (assuming a normal hierarchy, withcomparable values for an inverted hierarchy) [17]. To see if our construction accommodatesthis small but non-trivial hierarchy, and to determine whether there is a preference for anormal or inverted structure, in Figure 4 we consider the ratio of neutrino mass squareddifferences, which we plot as log ∆ m / ∆ m . Here we label the masses such that13 > m > m , so that this quantity is positive for a normal hierarchy and negativefor an inverted one. Observed masses give a value of about ± .
5. We see that forlarge N , the masses are much less hierarchical, and easily accommodate the observedvalues. Furthermore, we see an overwhelming preference for the normal hierarchy, whichin particular justifies our use of the associated mass and mixing angle measurements inlater parts of this section. (cid:45) (cid:73) (cid:68) m (cid:145) (cid:68) m (cid:77) P r ob a b ilit y D e n s it y N (cid:61) N (cid:61) N (cid:61) N (cid:61) Figure 4: Ratio of mass squared differences log ∆ m / ∆ m for N = 3, 10, 30 and100. Here we choose the convention m > m > m , so that positive(negative) valuescorrespond to a normal (inverted) hierarchy.Having seen that the mixing angles and mass splittings observed in nature are in-creasingly typical as N increases, we wish to look at other properties of viable matricesproduced within our framework. To select cases close to reality, we consider only matriceswhich satisfy: 0 . ≤ sin ( θ ) ≤ .
35; 0 . ≤ sin ( θ ) ≤ .
61; 29 . ≤ ∆ m / ∆ m ≤ .
6; and 0 . ≤ sin ( θ ) ≤ . σ bounds [17]. In Figure5, we show the distribution of sin( θ ), subject to the large angle and mass constraints,and find that there is some tension with the best fit, which at 2 σ corresponds to about Note that for an inverted hierarchy, our labeling is non-standard. The reason our scenario strongly prefers a normal versus an inverted hierarchy is that the reasonablylarge observed ratio of solar and atmospheric mass squared differences necessitates that either the heaviest(normal hierarchy) or the lightest (inverted hierarchy) of the neutrinos is a mild outlier. Having theheaviest neutrino as the outlier in our scenario is much more probable, since this requires fewer outlyingelements in our typically degenerate mass matrix. . ≤ sin( θ ) ≤ .
17. On the other hand, after fitting successfully the other parametersof the neutrino mass matrix, the probability of obtaining one mild outlier is not too small.This can be seen on the right side of Figure 5, which gives the fraction of matrices at orbelow a given value of sin( θ ) - there is significant variation across the experimentallypreferred region (in grey), reaching around 10% near the upper 2 σ bound. Our scenarioclearly prefers a nonzero value for sin( θ ), for which there is growing evidence, and asignificant range of allowed values would be unexceptional. The situation for θ in ourscenario is ultimately similar to that of neutrino anarchy, for which a global statisticalanalysis was performed and found good agreement with the data [5]. Performing a similaranalysis for the present case is beyond the scope of this work, but we would expect tofind similar results. (cid:72) Θ (cid:76) P r ob a b ilit y D e n s it y (cid:72) Θ (cid:76) C u m u l a ti v e P r ob a b ilit y Figure 5: Left: sin( θ ) distributions among matrices satisfying 2 σ large angle and massconstraints described in the text, for N = 30 (blue, solid) and N = 100 (red, dashed).Right: The corresponding cumulative probability distribution functions. The grey shadedarea indicates the 2 σ best fit region.We next consider the Majorana mass scale, which we determine by requiring that themass splitting ∆ m matches its experimental value. We plot the resulting distribution inFigure 6, where we see that the distribution rises and sharpens at large N , with typicalvalues around 10 − GeV, suggestive of the framework discussed in Section 2.Neutrinoless double beta decay provides us with an experimental probe of Majoranainteractions of neutrinos, and the figure of merit for such experiments is given by m ee , the15 (cid:72) M R (cid:144) GeV (cid:76) P r ob a b ilit y D e n s it y Figure 6: Majorana mass scale distribution among matrices satisfying 2 σ angle and massconstraints described in the text, for N = 30 (blue, solid) and N = 100 (red, dashed).upper left entry of the neutrino mass matrix in the diagonal charged lepton basis, m ee = (cid:88) i U i m i , (8)where U is the PMNS matrix and m i are the masses of the light neutrinos. We showthe prediction for m ee in Figure 7, again using only matrices lying sufficiently close toobservation, and with the overall scale set by the true value of ∆ m . The 68% centralregions of the distributions are . − . N = 30 and . − . N = 100, which seems typical for theories in which the low energy neutrino mass matrix iscomposed of relatively degenerate numbers – for comparison, we also show the distributionwe get by applying the same procedure to matrices with random O (1) parameters, i.e.anarchical matrices. Unfortunately, the predicted values for m ee are beyond the reachof current experiments. On a 10 year or longer timescale, it is possible that neutrinolessdouble beta decay searches, such as a future iteration of EXO [20], or observations oflarge scale structure [21, 22] may reach the required sensitivity. On shorter timescales, Here we generate anarchical matrices with basis-independent scanning following the prescription of [4]:We generate a complex 3 × M D , and complex symmetric 3 × M , eachwith entries scanning between 0 and 1 with random phases, subject to the basis-independent boundariesTr( M † M ), Tr( M † D M D ) ≤
1. We then form our light neutrino mass matrix, m = Λ M D M − M TD , where Λis chosen to obtain the correct value of ∆ m . Because of the basis independence, we are free to choosethe charged lepton matrix to be diagonal.
16t is interesting to note that our framework could be falsified if neutrinoless double betadecay were to be observed with a large measured value of m ee . (cid:200) m ee (cid:200)(cid:72) eV (cid:76) P r ob a b ilit y D e n s it y (cid:72) e V (cid:45) (cid:76) Figure 7: | m ee | distribution among matrices satisfying 2 σ angle and mass constraintsdescribed in the text, for N = 30 (blue, solid) and N = 100 (red, dashed). Also shown isthe anarchical case (yellow, dot-dashed) under the same constraints.Finally, we have done a similar analysis for the distributions of CP phases; we havefound that their distributions are relatively uniform, not providing us with any particularprediction. Other distributions
To get a sense of how much our results depend on our particular choice of probabilitydistribution, in Table 2 we look at the frequency of obtaining two large angles for afew different scenarios, which we take for the purpose of illustration rather than anyparticular physical motivation. We again require one angle with sin (2 θ ) ≥ .
98 andanother with sin (2 θ ) ≥ .
84. Table 2 also lists the percentage of N = 100 matriceswhich satisfy 0 . ≤ sin ( θ ) ≤ .
028 (2 σ bound), after having first satisfied the 2 σ bounds on sin ( θ ), sin ( θ ), and the mass splitting ratio, described above. Case I is thedistribution ( δ = 1 .
1) we have already considered in detail, and is listed for reference.Case II demonstrates the importance of the second condition listed in Section 2–that enough right-handed neutrinos must be present so that each left-handed neutrino17 N = 3 N = 30 N = 100 N = 200 P ( θ ) CommentsI 1.1 2.5 10.1 17.0 19.0 6II 1.3 2.5 3.2 6.7 10.8 11III 0.9 5.3 17.6 20.2 20.9 5IV 0.9 3.7 16.3 19.7 20.5 5 No minimum cutoffV 1.1 3.1 6.8 15.4 18.4 7 Hierarchical MajoranaTable 2: Percent of events with sin (2 θ ) largest ≥ .
98 and sin (2 θ ) next largest ≥ .
84 fordifferent distributions and numbers of right-handed neutrinos, as described in the text.Also shown (denoted P ( θ )): for matrices satisfying 2 σ constraints for θ , θ , andthe squared mass difference ratio, the percentage which also satisfy the 2 σ bounds on θ ,with N = 100.is expected to have > ∼ δ = 1 . N = 30for δ = 1 .
1, the rise in likelihood of large angles does not occur until much larger N in thiscase. Because the steeper distribution favors smaller mixing angles for a given number ofright-handed neutrinos, this case performs somewhat better than the others for the θ distribution.In case III, we see a distribution logarithmically skewed towards larger couplings ( δ =0 . N effect much more dramatic, becoming noticeable for smallervalues of N . Additionally, because this distribution is well behaved all the way to zero,it is not necessary to impose a lower bound on the Yukawa couplings for this value of δ ,so we remove this requirement for case IV. Because only values close to the top of thedistribution are important at large N , this only has the small impact of removing a bit ofprobability from the top. The change is more apparent at small values of N , where thelower parts of the distribution can be relevant in viable matrices.Finally, we turn to the choice of distribution for the Majorana mass matrix, whichwe have previously been taking to be anarchical and basis independent. In case V wetake a very different approach, choosing the same hierarchical distribution ( δ = 1 .
1) andcutoff structure for both Dirac and Majorana mass matrix elements. While this appears18o suppress washout effects somewhat, the impact is fairly weak, as the hierarchy islargely lost in the inversion of the Majorana matrix; only a small relative suppression ofprobability remains at N = 100.From these scenarios we see that while the quantitative behavior varies somewhat, thequalitative effect of increased probability of large mixing angles from many right-handedneutrinos applies broadly in the class of distributions we have laid out. Furthermore,we see that while there is some tension with a small value of sin( θ ), all cases have asignificant amount of probability lying in the experimentally favored region. In this work we have shown that the origin of the unique flavor properties of the neutrinos– small masses, and large mixing angles – may both have their origin in the seesawmechanism. Even if the neutrino Yukawa couplings have large hierarchies, as are observedfor the charged fermions, the hierarchical structure may generically become washed out ifthe seesaw involves a large number of right handed neutrinos, numbering perhaps in thetens or hundreds. We have given an explicit example showing the mechanism at work,with a statistical distribution of Yukawa couplings of a roughly scale invariant form, fitto the properties of the quarks and charged leptons. After integrating out the right-handed neutrinos, the probability of obtaining a mixing angle as large as the observednear-maximal value of θ , along with a second angle as large as the observed value of θ was found to be about 20%. The mechanism is fairly general, and may work with avariety of Yukawa structures, so long as various conditions are satisfied as described in thetext. General predictions of the framework are a normal hierarchy for neutrino masses,and a neutrino mass matrix element | m ee | of about 1 − If the largest neutrino Yukawacouplings are of order one, as in the case of the quarks, then the seesaw scale we require isfairly large – of order 10 − GeV. It then follows that washout processes will destroyany lepton asymmetry produced by the decays of the right-handed neutrinos. As a result,it may be necessary to move away from the thermal leptogenesis paradigm, and constructa model involving, for example, out of equilibrium inflaton decays to right-handed neu-trinos at temperatures much below the seesaw scale [26, 24]. One other possible futuredirection would be to consider the impact of anthropic selection effects on the types ofYukawa coupling distributions that we have been working with. One amusing possibility,somewhat orthogonal to the direction we have been pursuing here, is that the fundamen-tal Yukawa distributions for the charged fermions may actually be reasonably degenerate,but with strong anthropic selection effects constraining the first generation quarks andcharged leptons [27, 28], and indeed, perhaps even the top quark [29], to be outliers. If thiswere the case, degenerate neutrino Yukawas might in fact be more representative of thefundamental Yukawa distributions than the hierarchical charged fermion ones. It mightbe interesting to see if this picture could be made to work at a quantitative level usingsome relatively degenerate ansatz for the Yukawa distributions, and putting anthropicconstraints on various charged fermions masses.
Acknowledgments
We would like to thank Hitoshi Murayama, Taizan Watari and particularly TsutomuYanagida for useful discussions and input. This work was supported by the World PremierInternational Center Initiative (WPI Program), MEXT, Japan.
References [1] P. Minkowski, Phys. Lett. B67, 421 (1977); see also: M. Gell-Mann, P. Ramond, andR. Slansky in Supergravity, p. 315, edited by F. Nieuwenhuizen and D. Friedman,North Holland, Amsterdam, 1979; T. Yanagida, Proc. of the Workshop on Unified For some previous work on leptogenesis with many right handed neutrinos, see [25]. , 113002 (2006) [arXiv:hep-ph/0511219].[3] L. J. Hall, H. Murayama, N. Weiner, Phys. Rev. Lett. , 2572-2575 (2000). [hep-ph/9911341].[4] N. Haba, H. Murayama, Phys. Rev. D63 , 053010 (2001). [hep-ph/0009174].[5] A. de Gouvea, H. Murayama, Phys. Lett.
B573 , 94-100 (2003). [hep-ph/0301050].[6] W. Buchmuller, K. Hamaguchi, O. Lebedev, S. Ramos-Sanchez and M. Ratz, Phys.Rev. Lett. , 021601 (2007) [hep-ph/0703078 [HEP-PH]].[7] V. Bouchard, J. J. Heckman, J. Seo and C. Vafa, JHEP , 061 (2010)[arXiv:0904.1419 [hep-ph]].[8] R. Tatar, Y. Tsuchiya and T. Watari, Nucl. Phys. B , 1 (2009) [arXiv:0905.2289[hep-th]].[9] J. R. Ellis and O. Lebedev, Phys. Lett. B , 411 (2007) [arXiv:0707.3419 [hep-ph]].[10] J. Schechter and J. W. F. Valle, Phys. Rev. D , 2227 (1980).[11] L. J. Hall, M. P. Salem and T. Watari, Phys. Rev. Lett. , 141801 (2008)[arXiv:0707.3444 [hep-ph]].[12] L. J. Hall, M. P. Salem and T. Watari, Phys. Rev. D , 093001 (2007)[arXiv:0707.3446 [hep-ph]].[13] C. R. Das and M. K. Parida, Eur. Phys. J. C , 121 (2001) [hep-ph/0010004].[14] C. Balzereit, T. Mannel and B. Plumper, Eur. Phys. J. C , 197 (1999) [arXiv:hep-ph/9810350].[15] K. Nakamura et al. [Particle Data Group Collaboration], J. Phys. G G37 , 075021(2010).[16] Alan Edelman and N. Raj Rao (2005). Random matrix theory. Acta Numerica, 14,pp 233-297 doi:10.1017/S0962492904000236.2117] T. Schwetz, M. Tortola, J. W. F. Valle, New J. Phys. , 063004 (2011).[arXiv:1103.0734 [hep-ph]]. Addendum: T. Schwetz, M. Tortola, J. W. F. Valle, NewJ. Phys. , 109401 (2011). [arXiv:1108.1376 [hep-ph]].[18] M. Khabibullin, for the T2K Collaboration, [arXiv:1111.0183 [hep-ex]].[19] Talk given by H. De Kerreckt at 6th International Workshop on Low EnergyNeutrino Physics, LowNu2011, 9-12 November, 2011, Seoul National University,http://workshop.kias.re.kr/lownu11/[20] K. O’Sullivan [EXO Collaboration], J. Phys. Conf. Ser. , 052056 (2008).[21] J. Lesgourgues and S. Pastor, Phys. Rept. , 307 (2006) [astro-ph/0603494].[22] J. R. Pritchard and E. Pierpaoli, Phys. Rev. D , 065009 (2008) [arXiv:0805.1920[astro-ph]].[23] M. Fukugita and T. Yanagida, Phys. Lett. B , 45 (1986).[24] W. Buchmuller, R. D. Peccei and T. Yanagida, Ann. Rev. Nucl. Part. Sci. , 311(2005) [hep-ph/0502169].[25] M. -T. Eisele, Phys. Rev. D , 043510 (2008) [arXiv:0706.0200 [hep-ph]].[26] G. Lazarides and Q. Shafi, Phys. Lett. B , 305 (1991).[27] C. J. Hogan, Rev. Mod. Phys. , 1149 (2000) [astro-ph/9909295].[28] L. J. Hall and Y. Nomura, Phys. Rev. D , 035001 (2008) [arXiv:0712.2454 [hep-ph]].[29] B. Feldstein, L. J. Hall and T. Watari, Phys. Rev. D74