Anarchy and Neutrino Physics
AAnarchy and Neutrino Physics
Jean-Fran¸cois Fortin i , Nicolas Giasson ii and Luc Marleau iii D´epartement de Physique, de G´enie Physique et d’Optique,Universit´e Laval, Qu´ebec, QC G1V 0A6, Canada
The neutrino sector of a seesaw-extended Standard Model is investigated under the anarchyhypothesis. The previously derived probability density functions for neutrino masses and mixings,which characterize the type I-III seesaw ensemble of N × N complex random matrices, are used toextract information on the relevant physical parameters. For N = 2 and N = 3, the distributionsof the light neutrino masses, as well as the mixing angles and phases, are obtained using numericalintegration methods. A systematic comparison with the much simpler type II seesaw ensemble isalso performed to point out the fundamental differences between the two ensembles. It is foundthat the type I-III seesaw ensemble is better suited to accommodate experimental data. Moreover,the results indicate a strong preference for the mass splitting associated to normal hierarchy.However, since all permutations of the singular values are found to be equally probable for aparticular mass splitting, predictions regarding the hierarchy of the mass spectrum remains out ofreach in the framework of anarchy.February 2017 i [email protected] ii [email protected] iii [email protected] a r X i v : . [ h e p - ph ] J a n . Introduction The neutrino sector of the Standard Model (SM) is quite peculiar. Indeed, although the quark andcharged lepton mass spectra are quite hierarchical, the neutrino spectrum is simple: all neutrinosare massless. Neutrino oscillations [1–3], where neutrinos seemingly change flavor in flight, cannotbe accommodated in the SM due to the masslessness of the neutrinos. Neutrino oscillations thusimply massive neutrino eigenstates and the SM must be extended. Moreover, neutrino oscillationexperimental data suggest that the neutrino spectrum is not hierarchical, with three massive lightneutrinos and a mixing matrix, the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix, exhibitingnear-maximal mixing.The anarchy principle, introduced in [4], was put forward to explain the peculiarities ofthe neutrino sector by postulating a low-energy neutrino mass matrix generated by one of theseesaw mechanisms from randomly-generated high-energy mass matrices with elements distributedfollowing a Gaussian ensemble. In a series of papers [5], several numerical analysis of the anarchyprinciple were performed by randomly generating high-energy mass matrices and computing thecorresponding low-energy neutrino mass matrices.Recently, the low-energy neutrino mass matrix probability density function (pdf) for theanarchy principle was obtained from first principles in [6] following the extensive literature onrandom matrix theory [7]. It was shown that the pdfs associated to type I and type III seesawmechanisms were given by the same complicated integral equation while the pdfs associated to typeII seesaw mechanism was simple. A partial investigation of these seesaw ensembles was completedin [6] but an analysis of the physical case of three light neutrinos was not undertaken. This papercloses the gap by studying the implications of the seesaw ensembles for neutrino physics.The low-energy neutrino mass matrix pdf is known to factorize into a singular value pdf anda group variable pdf for all seesaw mechanisms. The singular value pdf corresponds to the pdffor the light neutrino masses while the group variable pdf corresponds to the pdf for the mixingangles and phases of the PMNS matrix. The singular value pdf for type I-III is given in [6]as a multidimensional integral while the singular value pdf for type II is a simple Gaussian-likedistribution. The group variable pdf for all types of seesaw mechanisms is the Haar measure,which seems to prefer near-maximal mixing. However, as stressed in [8], the mode of a pdf is nota well-defined quantity ( e.g. it is not invariant under change of variables), hence it is preferable tocompare pdfs by comparing their probabilities associated to a particular outcome. This probabilitytest, which is hard to perform by randomly generating low-energy neutrino mass matrices, canhowever be straightforwardly implemented from the analytic pdfs.Moreover, the factorization of the pdfs for the singular values and the group variables impliesthat there is no link between the light neutrino masses and the light neutrino mass eigenstates,forbidding an investigation of the preferred mass hierarchy (normal or inverted). From the analyticresults for the singular value pdfs, it is however possible to determine which mass splitting, i.e. m − ˆ m and ˆ m − ˆ m , is favored.This paper is organized as follows: Section 2 gives the relevant pdfs for the seesaw ensemblesobtained in [6]. In section 3 the pdfs for the complex seesaw ensembles are studied in the N = 2and N = 3 cases. For both cases, a comparison is made between the analytic results and thenumerical results, showing perfect agreement. For the N = 3 case relevant to neutrino physics, athorough investigation of the implications of the seesaw ensembles is completed. For example, it isshown from the probability test that both type I-III and type II seesaw ensembles prefer the masssplitting associated to normal hierarchy with a neutrino energy scale of O (10 − ) eV. Finally, adiscussion and a conclusion are presented in section 4.It is important to note that throughout this paper, the term “analytic results” should beunderstood as results obtained from the analytic pdfs which are numerically integrated while theterm “numerical results” corresponds to results obtained from randomly-generated mass matrices.
2. The Seesaw Ensembles
This section states without proof the relevant quantities that appear in the seesaw ensembles,which were derived from the anarchy principle applied to the SM extended with the type I-IIIseesaw mechanism or with the type II seesaw mechanism. The reader interested in the proofs isreferred to [6].
The pdfs for the dimensionless N × N light neutrino mass matrix ˆ M ν = M ν / ( √ ν ) where Λ ν isthe (naturally small) light neutrino mass scale, can be expressed in terms of the light neutrinomass matrix singular values ˆ m ν,i and the light neutrino mass matrix group variables U ν with thehelp of the decomposition ˆ M ν = U ν D ν U Tν where D ν = diag( ˆ m ν, , · · · , ˆ m ν,N ) and ˆ m ν,i ≥ i .The pdfs are found to factorize into two independent pdfs, one pdf for the singular values and onepdf for the group variables, as in P ν ( ˆ m ν ; U ν ) d ˆ m ν dU ν = P ν ( ˆ m ν ) P ν ( U ν ) d ˆ m ν dU ν . For real ( β = 1) and complex ( β = 2) matrix elements, the pdfs are given respectively by P I-III ν ( ˆ m ν ) d ˆ m ν = C I-III βνN I βN ( ˆ m ν, , · · · , ˆ m ν,N ) (cid:89) ≤ i 1] while all the remaining (unphysical, CP-violatingDirac and Majorana) phases have flat distributions. Thus the preferred value for all mixing angles θ i,i +1 is π/ 4, which corresponds to maximal mixing, while the preferred value for all mixing angles θ i,i +2 is π/ 6. It is important however to notice that most probable values are not invariant underchange of variables, as pointed out in [8].Then, the consequences for the light neutrino masses, which are obtained from the singular valuepdfs (2.1), are not as sharp. Indeed, although the chosen decomposition is here fixed, the singularvalue pdfs (2.1) are invariant under permutations of the singular values. Hence, the light neutrinomasses and the light neutrino mixing angles and phases are completely independent. In otherwords, although each singular value ˆ m ν,i has a corresponding singular vector ( u ν,i ) j = U ν,ji suchthat ˆ M ν u ∗ ν,i = ˆ m ν,i u ν,i , the probability that a particular neutrino mass spectrum occurs is at most1 /N !. For example, the dimensionless neutrino mass spectrum ( ˆ m ν, , · · · , ˆ m ν,N ) = ( µ , · · · , µ N ) isas probable as the spectrum ( ˆ m ν, , · · · , ˆ m ν,N ) = ( µ , µ , µ , · · · , µ N ) or any other permutations. Itis thus possible to fix an ordering for the singular values, 0 ≤ ˆ m min ≤ · · · ≤ ˆ m max , keeping in mindthat the relationship between the light neutrino masses and the light neutrino mixing matrix iscompletely lost. Fixing the ordering implies that the singular value pdfs (2.1) are multiplied by N !.Therefore, by working with a fixed basis as described above, some mixing angle preferredvalues correspond to maximal mixing but the ordering of the light neutrino masses for a givenspectrum is completely free. It is thus clear that a spectrum exhibiting one of the two hierarchypatterns preferred by the data (normal or inverse) is as probable as the same spectrum but withpermuted mass eigenstates. These observations are in the spirit of [8], although with the analyticknowledge of the singular value pdfs (2.1), it is now possible to complete an appropriate statistical5 m ˜ P I - III ν ( ˆ m ) Type I-III seesaw ensemble 0 1 2 3 4 50.01.0 ˆ m ˜ P II ν ( ˆ m ) Type II seesaw ensemble0 2 4 6 8 10 12 140.00.10.2 ˆ m ˜ P I - III ν ( ˆ m ) m ˜ P II ν ( ˆ m ) Fig. 1: Probability density functions for the singular values of the complex seesaw ensembleswith N = 2. The red curve corresponds to the analytic result while the histogram corresponds tonumerical results (with 2 . × dimensionless light neutrino mass matrices generated). The leftand right columns show the pdfs for the type I-III and the type II seesaw mechanisms respectively.The singular values are ordered such that 0 ≤ ˆ m ≤ ˆ m and an extra factor of 2! is introduced tocorrect the singular value pdfs.test to better check the validity of the anarchy principle. As stressed in [8], the most appropriatestatistical test seems to be the probability test which computes the probability that the variablesare in a given volume. By choosing the observed values with their error bars for the volumeof the light neutrino masses and mixings, the probability that one ensemble leads to the SM isobtained. Clearly, since the variables are continuous, the calculated probability is very small forvery precisely-known observed values. One can nevertheless discriminate ensembles, for examplethe type I-III and type II seesaw ensembles, by comparing their respective probabilities, as shownbelow. N = 2As a warm-up exercise, the case N = 2 is studied analytically and compared to randomly-generatedlight neutrino mass matrices. Using the parametrization (2.3) for both unitary matrices [ i.e. theone appearing in (2.2) and the light neutrino mixing matrix], the relevant pdfs (2.1) can be written6 arginal pdfs Mean Median Mode˜ P I-III ν ( ˆ m ) 0 . 59 0 . 39 0 . P I-III ν ( ˆ m ) 5 . 39 3 . 41 1 . P II ν ( ˆ m ) (cid:112) π . ˜ P II ν ( ˆ m ) √ π (cid:0) √ (cid:1) . 48 1 . Table 1: Location parameters for the marginal singular value pdfs of figure 1.as P I-III ν ( ˆ m , ˆ m ) = 2 | ˆ m − ˆ m | ˆ m ˆ m (cid:90) ∞ dx dx (cid:90) π/ dθ (cid:48) ( x − x ) x x sin(2 θ (cid:48) ) I (cid:20) ( x − x ) [sin(2 θ (cid:48) )] m ˆ m (cid:21) × e − ˆ m x θ (cid:48) )]2+ x θ (cid:48) )]2)2+ ˆ m x θ (cid:48) )]2+ x θ (cid:48) )]2)2ˆ m 21 ˆ m − x − x ,P II ν ( ˆ m , ˆ m ) = 4 | ˆ m − ˆ m | ˆ m ˆ m e − ˆ m − ˆ m ,P ν ( θ, φ, ϕ , ϕ ) = 18 π sin(2 θ ) , (3.1)where the subscript ν on the masses was omitted to simplify the equations. The modifiedBessel function of the first kind I ( z ) is generated by the integral over the phase in (2.4). The4-dimensional integral is thus simplified to a 3-dimensional integral. Figures 1 and 2 show acomparison between the analytic results (3.1) and numerical results for a fixed ordering of thesingular values (chosen to be 0 ≤ ˆ m ≤ ˆ m , such that ˆ m ≡ ˆ m min and ˆ m ≡ ˆ m max ). Consequently,the resulting marginal singular value pdfs are obtained by computing the following integrals˜ P Σν ( ˆ m ) = 2! (cid:90) ∞ ˆ m d ˆ m P Σν ( ˆ m , ˆ m ) , ˜ P Σν ( ˆ m ) = 2! (cid:90) ˆ m d ˆ m P Σν ( ˆ m , ˆ m ) , for both ensembles ( i.e. Σ = I-III or II).Although the analytic behavior of the type I-III singular value pdf (3.1) is hard to see intuitivelydue to the integral, it is clear that the pdfs (3.1) are correct as seen in figure 1. The behavior ofthe type I-III singular value pdf at vanishing and large singular values matches the expectationsof [6]. The vanishing of ˜ P I-III ν ( ˆ m ) at ˆ m → β = 2) seesaw ensemble is the first moment (the average singular values). Therefore,a meaningful characterization of the previous pdfs is limited to the first moment, the mode and7 π/ π/ θ ˜ P ν ( θ ) = s i n ( θ ) π π φ ˜ P ν ( φ ) = π π/ π ϕ ˜ P ν ( ϕ ) = π π/ π ϕ ˜ P ν ( ϕ ) = π Fig. 2: Probability density functions for the mixing angle and phases of the complex seesawensembles with N = 2. The red curve corresponds to the analytic result while the histogramcorresponds to numerical results (with 2 . × dimensionless light neutrino mass matricesgenerated). The top and bottom rows show the pdfs for the mixing angle θ , the CP-violatingphase δ , and the unphysical phases ϕ and ϕ (note the range is halved due to the extra freedom ϕ i → ϕ i ± π [6]).the median (their location parameters). Their respective values for each distribution are presentedin table 1.From these results, it can be seen that the average singular values coming from the type I-IIIseesaw ensemble are spread over a wider range than in the type II seesaw ensemble. Moreover,when comparing the mean of a distribution with its respective median, one can quantify theasymmetry of the pdfs presented in figure 1. It turns out that the means are much closer to themedians (and thus the modes) in the type II seesaw ensemble, which leads to more symmetricalpdfs as can already be seen from figure 1.Moving forward, the group variable pdf (3.1) is easier to analyze. First, all phases have flatdistributions as mentioned previously. Moreover, the mixing angle has a non-trivial distributionthat prefers near-maximal mixing. From figure 2 the pdf for the mixing angle and the phasesagree well with the normalized Haar measure. Since these pdfs were studied extensively in theliterature and are easier to analyze, their statistical parameters (mean, median and mode), whichare easily obtained from (3.1), are not presented here.Finally, to provide a global overview of the pdfs with unordered singular values, the densityplots of the singular value pdfs for the type I-III and type II seesaw ensembles are shown in figure8 Fig. 3: Density plots of the singular value pdfs for the complex seesaw ensembles with N = 2.The left and right panels show the density plots of the type I-III [ P I-III ν ( ˆ m , ˆ m )] and type II[ P II ν ( ˆ m , ˆ m )] seesaw ensembles respectively. The plots are split along the symmetry axis so thatthe upper and lower triangles show the analytical and numerical results (with 10 dimensionlesslight neutrino mass matrices generated) respectively. There is no ordering of the singular values.3. The symmetry pattern of the pdfs under the exchange ˆ m ↔ ˆ m is easily seen from figure 3.In fact, the plots are constructed in a way that takes advantage of this symmetry to provide ameaningful comparison between analytical and numerical results in both cases. Indeed, by showingonly half the data for the analytical ( ˆ m < ˆ m ) and numerical ( ˆ m > ˆ m ) results in each plot, itcan be seen that the agreement between the two is once again very good. Moreover, by comparingthe distances between modes in each plot, it is possible to determine which ensemble has a strongerrepulsion between the singular values. It is found that the modes are 1 . P I-III ν ( ˆ m , ˆ m ). A possibleexplanation as to why the singular values are closer together in the type II seesaw ensembleis suggested in expression (3.1). Indeed, one can see that the Vandermonde-like contribution | ˆ m − ˆ m | in P I-III ν ( ˆ m , ˆ m ) (responsible for the spreading of the singular values with reference tothe symmetry axis), is strongly suppressed by a product of masses to the fifth power, which tendsto spread the singular values in a narrow band along the two axes. Eventually, in order to get abetter understanding of the type I-III seesaw ensemble, the density plot could be used to guessan analytical form for P I-III ν ( ˆ m , ˆ m ) by fitting some appropriate functions of the singular valueswith free parameters to be determined. Such density plots are not really conceivable for the case N = 3 as they would require heavy numerical computation and would be rather hard to illustrateproperly. However, the same reasoning and conclusions apply to this case as well.9 ormal Hierarchy Inverted Hierarchy θ ( ◦ ) 33 . +0 . − . . +0 . − . θ ( ◦ ) 42 . +3 . − . . +1 . − . θ ( ◦ ) 8 . +0 . − . . +0 . − . δ ( ◦ ) 306 +39 − +63 − ∆ m (10 − eV ) 7 . +0 . − . . +0 . − . ∆ m (cid:96) (10 − eV ) 2 . +0 . − . − . +0 . − . m (eV) < . < . Table 2: Best-fit values for the SM neutrino physics parameters for the normal and invertedhierarchies. The intervals correspond to ± σ . In the case of ∆ m (cid:96) , (cid:96) = 1 for the normal hierarchyand (cid:96) = 2 for the inverted hierarchy. N = 3 : SM Neutrino Physics With the tools and insights developed in the previous sections, it is now possible to fully analyzethe more interesting case of a seesaw-extended SM.First, recent experimental values of the physical parameters in the neutrino sector are summa-rized in table 2. The mixing angles, the CP-violating Dirac phase and the squared mass differences∆ m ij = m i − m j are extracted from [2]. The upper bound on the mostly-electronic neutrino m is the 1 σ upper bound of [10]. This upper bound on m comes from the study of supernova. It isconservative and quite model-independent. Indeed, it is the weakest bound when compared to thecosmological bound on the sum of the neutrino masses which is somewhat model-dependent orthe neutrinoless double β -decay bound and the direct neutrino mass bound which are intertwinedwith some mixing matrix parameters [11]. These values will be used in the probability test at theend of this section.For neutrino physics, the most convenient parametrization for the unitary group U (3) is ofcourse the PMNS mixing matrix [1] for the light neutrino U ν , U ν = θ ) sin( θ )0 − sin( θ ) cos( θ ) cos( θ ) 0 sin( θ ) e − iδ − sin( θ ) e iδ θ ) × cos( θ ) sin( θ ) 0 − sin( θ ) cos( θ ) 00 0 1 e iα / 00 0 e iα / . (3.2) While numerically computing the results of this paper, an updated fit of the neutrino sector experimental valueswas published in [3]. Since the experimental values did not change much, the analysis presented here will not changesignificantly. arginal pdfs Mean Median Mode˜ P I-III ν ( ˆ m ) 0 . 36 0 . 26 0 . P I-III ν ( ˆ m ) 1 . 93 1 . 65 1 . P I-III ν ( ˆ m ) 9 . 65 6 . 55 4 . P II ν ( ˆ m ) (cid:112) π . √ ˜ P II ν ( ˆ m ) (cid:112) π . √ ˜ P II ν ( ˆ m ) √ π (cid:0) 72 + 45 √ − √ (cid:1) . 96 1 . Table 3: Location parameters for the marginal singular value pdfs of figure 4.The group variable pdf (2.1) is given by P ν ( θ , θ , θ , δ, α , α ) = 12 π sin(2 θ ) sin( θ )[cos( θ )] sin(2 θ ) , (3.3)which is the same as the normalized Haar measure (2.4) obtained from the parametrization (2.3).Hence the complex seesaw ensemble prefers the mixing angles θ and θ around π/ θ around π/ 6. The pdfs for the CP-violating Dirac phase δ and the two CP-violatingMajorana phases α and α are flat. Therefore, any value for the CP-violating phases is equallyprobable in the complex seesaw ensemble. It is important to note that the unphysical phases arenot explicitly included in the PMNS parametrization (3.2).The full expression for the type I-III singular value pdf is quite long (the explicit expression fillsup a few pages) and not enlightening. The form (2.1) with the parametrization (2.3) and N = 3 issufficient for both type I-III and type II. As discussed in section 3.1, these pdfs are invariant underpermutations of the singular values ˆ m , ˆ m and ˆ m . Consequently, the hierarchy of the neutrinomass spectrum of the extended SM cannot be predicted under the anarchy hypothesis. The onlyclaim that can be made is that all hierarchy scenarios (that is to say, every 3! permutations of thethree singular values) are equiprobable for a given mass splitting (again, the term mass splitting isunderstood to be a particular ordering of the two quantities ˆ m − ˆ m and ˆ m − ˆ m ). Tofurther study the mass splitting and the resulting marginal pdfs, it is convenient to introduce aparticular singular value ordering. From now on, the ordering is chosen to be 0 ≤ ˆ m ≤ ˆ m ≤ ˆ m (such that ˆ m ≡ ˆ m min , ˆ m ≡ ˆ m med and ˆ m ≡ ˆ m max ) although it must remain clear that thoseare not straightforwardly related to the experimental neutrino masses (once again, the choice iscompletely arbitrary). Thus, one cannot single out any of the two hierarchy scenarios selected byexperimental data. Nevertheless, by studying the marginal singular value pdfs of the type I-IIIand type II seesaw ensembles, some important and interesting results on the regime of low-energyneutrino physics can be obtained. 11 m ˜ P I - III ν ( ˆ m ) Type I-III seesaw ensemble 0 1 2 3 4 50.00.40.81.2 ˆ m ˜ P II ν ( ˆ m ) Type II seesaw ensemble0 1 2 3 4 50.00.20.4 ˆ m ˜ P I - III ν ( ˆ m ) m ˜ P II ν ( ˆ m ) m ˜ P I - III ν ( ˆ m ) m ˜ P II ν ( ˆ m ) Fig. 4: Probability density functions for the singular values (masses) of the complex seesawensembles with N = 3. The red curve corresponds to the analytic result while the histogramcorresponds to numerical results (with 2 . × dimensionless light neutrino mass matricesgenerated). The left and right columns show the pdfs for the smallest, median and largest singularvalues for the type I-III and the type II seesaw mechanisms respectively. The singular valuesare ordered such that 0 ≤ ˆ m ≤ ˆ m ≤ ˆ m and an extra factor of 3! is introduced to correct thesingular value pdfs.First, the marginal singular value pdfs can be obtained by computing the following integrals,˜ P Σν ( ˆ m ) = 3! (cid:90) ∞ ˆ m d ˆ m (cid:90) ∞ ˆ m d ˆ m P Σν ( ˆ m , ˆ m , ˆ m ) , ˜ P Σν ( ˆ m ) = 3! (cid:90) ∞ ˆ m d ˆ m (cid:90) ˆ m d ˆ m P Σν ( ˆ m , ˆ m , ˆ m ) , ˜ P Σν ( ˆ m ) = 3! (cid:90) ˆ m d ˆ m (cid:90) ˆ m d ˆ m P Σν ( ˆ m , ˆ m , ˆ m ) , for both ensembles ( i.e. Σ = I-III or II). The marginal singular value pdfs are shown in figure 4.12 .0 0.5 1.0 1.5 2.00.03.06.09.012.0 R ˜ P I - III ν ( R ) Type I-III seesaw ensemble 0 1 2 3 40.00.40.81.2 R ˜ P II ν ( R ) Type II seesaw ensemble Fig. 5: Probability density functions for the ratios of the complex seesaw ensembles with N = 3.The red curve corresponds to the analytic result while the histogram corresponds to numericalresults (with 2 . × dimensionless light neutrino mass matrices generated). The left and rightplots show the pdfs for the ratios of the type I-III and the type II seesaw mechanisms respectively.The singular values are ordered such that 0 ≤ ˆ m ≤ ˆ m ≤ ˆ m and an extra factor of 3! isintroduced to correct the singular value pdfs.A first observation that comes to mind when looking at figure 4 is the diversity of the massspectrum obtained with the type I-III seesaw ensemble compared to the type II seesaw ensemble.This can be traced back to the fact that the pdfs ˜ P I-III ν ( ˆ m i ) are much more complex than thesimple Gaussian-like pdfs ˜ P II ν ( ˆ m i ) that arise in the type II seesaw ensemble. For example, there isno internal angular dependence associated to extra variables that needs to be integrated out inthe type II pdf. A second observation worth mentioning is the remarkable agreement betweenanalytical and numerical results. For reasons previously mentioned, analytical results derived fromthe type I-III seesaw ensemble are much more challenging to get than those of the type II seesawensemble. Following heavy numerical computation based on adaptive Monte Carlo integration, the11-dimensional integrals resulting from the marginalization procedure can be obtained for givenvalues of ˆ m , ˆ m or ˆ m . The red curves produced in such a way are in very good agreement withtheir corresponding histograms (generated from a sample of light neutrino mass matrices), whichcan be viewed as a validation of the Monte Carlo integration method (estimated to be accurate toat least three significant figures) used in this case or a numerical check of the singular value pdfsobtained in [6].Once the numerical integration method is carefully tested, the next step is to compute therelevant statistical parameters. The results are presented in table 3. Following the same approachas in the N = 2 case, it is found that the average singular values are once again spread over amuch wider range in the type I-III seesaw ensemble. Moreover, when compared to the N = 2 case,it can be seen that this range expands significantly more in the type I-III seesaw ensemble as N increases. Next, comparing these values with their respective medians, one can conclude that thepdfs are much more symmetric in the type II seesaw ensemble, as can be expected when looking13 π/ π/ θ ˜ P ν ( θ ) = s i n ( θ ) Mixing angles 0 π π δ ˜ P ν ( δ ) = π Phases π/ π/ π/ θ ˜ P ν ( θ ) = c o s ( θ ) s i n ( θ ) π π α ˜ P ν ( α ) = π π/ π/ θ ˜ P ν ( θ ) = s i n ( θ ) π π α ˜ P ν ( α ) = π Fig. 6: Probability density functions for the mixing angles and phases of the complex seesawensembles with N = 3. The red curve corresponds to the analytic result while the histogramcorresponds to numerical results (with 2 . × dimensionless light neutrino mass matricesgenerated). The left and right columns show the mixing angles and the remaining flat phasesrespectively. Since these distributions depend only on the Haar measure of the corresponding Liegroup [ U (3) in this case], there is no distinction between type I-III and type II seesaw mechanisms.at figure 4.Even though determining the hierarchy of the mass spectrum is out of reach in the contextof the seesaw ensembles, the previous results can still be used to help identify which of thetwo possible mass splittings (according to our previous ordering, the mass splittings can bewritten as ∆ ˆ m = ˆ m − ˆ m and ∆ ˆ m = ˆ m − ˆ m so that the two possibility are either∆ ˆ m < ∆ ˆ m or ∆ ˆ m > ∆ ˆ m ) is more likely to occur under the anarchy hypothesis.14y studying the distribution of the ratio R , R = ∆ ˆ m ∆ ˆ m = ˆ m − ˆ m ˆ m − ˆ m , which leads to the marginal pdf ˜ P Σν ( R ) for both ensembles, it becomes clear that the pdf resultingfrom type I-III seesaw ensemble is more likely to reproduce the experimental value of R exp (cid:39) . 03 forthe normal hierarchy (also when compared to R exp (cid:39) . 65 for the inverted hierarchy). Moreover,by integrating these distributions over the range 0 ≤ R ≤ ≤ R ≤ ∞ ), one gets the probabilitythat the mass splitting ∆ ˆ m < ∆ ˆ m (∆ ˆ m > ∆ ˆ m ) is realized. For the type I-III seesawensemble, the probability is 95 . 8% (4 . . 0% (21 . m < ∆ ˆ m reminiscent of the normal hierarchy. Moreover, looking at the very distinct behavior of the twopdfs and comparing the resulting probability, it is possible to state that the type I-III seesawensemble is better suited to generate this particular mass splitting. In the context of the typeI-III seesaw ensemble, this means that the mass differences are way more likely to coincide withthe ones from normal hierarchy, yet the ordering of the masses is still unknown (once again, every3! permutations are equally probable for this particular splitting).Next, considering the group variable pdf for the mixing angles and phases (3.3), similarconclusions as with the case N = 2 can be drawn. Once again, the phases have flat distributionsand two of the three mixing angles prefer near-maximal values as shown in figure 6. Here, theunphysical phases were not considered in the making of figure 6 as they were deemed not interestingfor the present discussion. The numerical data coming from a sample of light neutrino massmatrices is once again consistent with the marginal pdfs obtained from the Haar measure. Theonly non-symmetric pdf for the mixing angles is the one associated to θ . Its mode, located at π/ 6, is in agreement with the results of section 3.1 obtained with the parametrization (2.3) sincethe Haar measure is the same for the PMNS matrix (3.2).To further emphasize the differences between the two ensembles, the probability test is nowused to determine how well the complex seesaw ensembles can generate the observed values ofthe extended SM physical parameters in the neutrino sector. The results of the probability testwill then be compared between the type I-III and type II seesaw ensembles to determine whichensemble is more likely to generate the observed values of physical parameters. The experimentalvalues are given in table 2.With the observed values of table 2, the probabilities (where | det J | is the Jacobian of the15ppropriate hierarchy) are P NH m (Λ ν ) = (cid:90) (7 . . × − ν (7 . − . × − ν d ∆ ˆ m (cid:90) (2 . . × − ν (2 . − . × − ν d ∆ ˆ m × (cid:90) . √ ν d ˆ m | det J | P Σν ( ˆ m , ˆ m , ˆ m ) ,P IH m (Λ ν ) = (cid:90) (7 . . × − ν (7 . − . × − ν d ∆ ˆ m (cid:90) ( − . . × − ν ( − . − . × − ν d ∆ ˆ m × (cid:90) . √ ν d ˆ m | det J | P Σν ( ˆ m , ˆ m , ˆ m ) ,P U = (cid:90) V exp dθ dθ dθ dδdα dα P ν ( θ , θ , θ , δ, α , α ) , (3.4)where NH stands for normal hierarchy while IH stands for inverted hierarchy. The first test isachieved by using only the singular value pdfs for both ensembles (see figure 7). For the typeI-III seesaw ensemble, the 12-dimensional integrals over the experimental volume defined by the1 σ range are obtained using the same Monte Carlo algorithm. At this point, it is necessary tostress that this test does not require any ordering of the singular values. In fact, all permutationsare accounted for in these integrals and there is thus no need for an extra factor of 3! to ensurethe normalisation of the pdfs. Since the only free parameter left in the equations is Λ ν , the ideais to plot the probability as a function of Λ ν over a range were the curves reach a maximum.This allows for a simple comparison of their maximum values (by taking appropriate ratios) todetermine the likelihood of each ensemble to generate the observed values. It is important tonote that beside the fact that the probabilities obtained this way are invariant under a changeof basis, the explicit values of the probabilities are not particularly meaningful. In fact, theyare bound to shrink further and further as the experimental values get more and more precise.However, the ratios are considered to be relevant quantities since they are subject to only smallfluctuations during this process (the order of magnitude should remain the same). The results ofthe probability test are presented in figure 7.First, from within the same ensemble, one can compare the probabilities obtained from thenormal and inverted hierarchy scenarios. The maximum probability values resulting from a scanover Λ ν reveal that the mass splitting in (3.4) are ∼ σ ) than from the one defined by inverted hierarchy inthe type I-III seesaw ensemble. In other words, this means that the type I-III seesaw ensemble isway more likely to generate values for these physical parameters that are contained within theregion allowed by the normal hierarchy data set (rather than the inverted hierarchy data set).For the type II seesaw ensemble, the same tendency is observed but with a much smaller ratiobetween the maximum probability values. Indeed, figure 7 shows that the mass splitting is ∼ .0 0.5 1.0 1.5 2.0 2.5 3.00.00.30.60.91.21.5 Λ ν (10 − eV) P NH m ( Λ ν )( × − ) Type I-III seesaw ensembleΛ max ν = 0 . × − eV P NH m (Λ max ν ) = 1 . × − ν (10 − eV) P NH m ( Λ ν )( × − ) Type II seesaw ensembleΛ max ν = 1 . × − eV P NH m (Λ max ν ) = 6 . × − ν (10 − eV) P I H m ( Λ ν )( × − ) Λ max ν = 1 . × − eV P IH m (Λ max ν ) = 1 . × − ν (10 − eV) P I H m ( Λ ν )( × − ) Λ max ν = 2 . × − eV P IH m (Λ max ν ) = 2 . × − Fig. 7: Probability test for the singular values of the complex seesaw ensembles with N = 3. Theleft and right columns show the probability distribution as a function of Λ ν (in the normal andinverted hierarchy scenarios) for the type I-III and the type II seesaw mechanisms respectively.The singular values are in no particular order for this test.times more likely to originate from the region defined by normal hierarchy. It is then possible toconclude that between the two regions scanned in the probability test, both ensembles naturallylead to preferred values for these physical parameters that lie in the region defined by normalhierarchy. Moreover, this preference is strongly accentuated in the type I-III seesaw ensemble.Second, one can make a comparison between the two ensembles based on the result of figure 7.Since it was shown that one region is actively preferred over the other, it becomes useful tocompare the maximum probability values in the case of normal hierarchy for both ensembles.This time, the conclusions are not as striking as in the previous case but one can state that thetype I-III seesaw ensemble is roughly 2 times better than type II for generating values of theseparameters in this particular region. The results obtained from this probability test are thus inagreement with what was found previously by comparing the pdfs of the ratios R and consequentlyhelp quantify the underlying trends in both ensembles.A final point of interest regarding this particular test concerns the energy scale Λ ν . Again fromfigure 7 one can see that choosing the integration region to be over the accepted experimentalvalues (within the 1 σ confidence level) naturally fixes the energy scale of the models. Each scanshows that the maximum probability values are attained for values of Λ ν which are of the same It is interesting to note that the type II seesaw ensemble is approximately 18 times more probable than the typeI-III seesaw ensemble for the inverted hierarchy. ormal Hierarchy Inverted Hierarchy P U . × − . × − P flat . × − . × − P U /P flat . 031 2 . Table 4: Probability test for the mixing angles and phases of the complex seesaw ensembles with N = 3 and a (trivial) normalized flat distribution. The probability P U and P flat are obtained byintegrating the normalized Haar measure and the flat distribution over the experimental volume V exp defined by the data at the 1 σ confidence level (see table 2).order of magnitude, namely Λ ν ∼ O (10 − ) eV for both ensembles. Since Λ ν , which correspondsto the light neutrino mass scale, takes the general form Λ ν = v / Λ new for each type of seesawmechanisms, with v (cid:39) 246 GeV the usual Higgs vacuum expectation value, a quick estimate ofthe new energy scale Λ new associated to the particle content introduced in the extended SM withtype I, type II or type III seesaw mechanism (right-handed neutrinos singlets, Higgs triplet andfermionic triplets respectively) can be made. Indeed, using the previously-mentioned values, onegets Λ new ∼ O (10 ) GeV for the new energy scale of the extended SM, which is very close tothe energy scale of grand unified theory (GUT). Naturally, Λ new is directly related to the massesof these newly-introduced particles. However, in order to assess their corresponding mass scales,one needs to specify the order of magnitude of the coupling constants arising from each seesawscenario. The usual approach is to set the coupling constants to be of O (1) since there is nofundamental principle or symmetry pattern that require particularly small couplings. This inturn suggests that the new particles introduced in the SM are quite heavy since Λ new becomesessentially their corresponding mass scale. In fact, this result is typical of seesaw mechanismsand is often regarded as a prerequisite (when taking the naturalness argument into considerationto avoid seesaw-induced fine-tuning or hierarchy problems) for these mechanisms to give sensiblepredictions concerning the light neutrino masses. It is however possible to lower the mass scales bysimply postulating smaller coupling constants, somewhat disregarding the naturalness argument.Overall, the results of the probability test are therefore consistent with high-energy phenomenologyof the seesaw-extended SM.The second probability test, with results shown in table 4, concerns the mixing parameters ofthe neutrino sector, namely the mixing angles and phases of table 2. In this case, the analysis ismuch simpler since there is no free parameter with which to scan a particular region and the pdf(the Haar measure) is also a lot less complicated. Since both ensembles have the same pdf for themixing angles and phases, a comparison between the two is not possible. However, it is interestingto see how well these ensembles perform when compared to a trivial normalized flat distribution.Here, the idea is simply to test whether there is any improvement when generating parametervalues from the Haar measure obtained in the seesaw ensembles as opposed to a less interesting18odel where there would be no information or explicit dependence on the angular part in thepdf. By comparing the probabilities that the generated values lie within V exp in both cases andfor the two types of hierarchy, one gets the results of table 4. The first conclusion that can bedrawn from these results is that, up to the level of accuracy acknowledged for this test (rememberthat this analysis remains sensible to the choice of integration volume to some extent), there canbe no distinction between normal or inverted hierarchy. Both regions are thus equally probable.However, when comparing P U with P flat , there is indeed improvement as the seesaw ensembles areessentially ∼ To follow up on our previous work [6] regarding the close resemblance of the type I-III singularvalue pdf at N = 1 and the level density at large N , this section further investigates this connectionby adding the comparison with the pdfs at N = 2 and N = 3 for the type I-III seesaw ensemble.The starting point for an appropriate comparison of these quantities is the correlation function ρ I-III νN ( x ) = N (cid:90) P I-III ν ( x, ˆ m , · · · , ˆ m N ) (cid:89) ≤ i ≤ N d ˆ m i , for N = 2 and N = 3 respectively. Introducing a convenient rescaling of the variable x → √ N ˆ m ν ,the resulting correlation functionsˆ ρ I-III ν ( ˆ m ν ) = √ (cid:90) ∞ d ˆ m P I-III ν ( ˆ m ν , ˆ m ) , ˆ ρ I-III ν ( ˆ m ν ) = √ (cid:90) ∞ d ˆ m (cid:90) ∞ d ˆ m P I-III ν ( ˆ m ν , ˆ m , ˆ m ) , with ˆ ρ (ˆ x ) = ρ ( x ) / √ N , can be compared directly to the large N histogram ( N = 60). The resulting11-dimensional integral for N = 3 is carried out using the previously-mentioned Monte Carloalgorithm.From figure 8, one can see that the agreement between analytical and numerical results becomessurprisingly good as N reaches 3. However, there is a priori no clue as to why the correlationfunctions for finite and small N are able to reproduce with great precision the level density atlarge N since they are independent quantities. This represents the first clear indication that aproper large N analysis would indeed be a good approximation of the physical case N = 3, aswas previously suggested (without proof or evidence) in the literature. Thus, there is no doubtthat this particular behavior should be investigated further since it motivates the search for ananalytical expression (coming from a large N analysis) to better understand the physical case athand. 19 m ν ˆ ρ I - III ν ( ˆ m ν ) Comparison with N = 2 0 1 2 3 4 50.00.51.0 ˆ m ν ˆ ρ I - III ν ( ˆ m ν ) Comparison with N = 3 Fig. 8: Comparison between the correlation function at N = 2 and N = 3 (red curves) and thelevel density at large N (histogram with N = 60) for the type I-III complex seesaw ensemble. Thered curves correspond to the analytic result for these specific values of N while the histogramscorrespond to numerical results (with 10 dimensionless light neutrino mass matrices generated).The left and right panel show the comparison with N = 2 and N = 3 respectively. There is noordering of the singular values. 4. Discussion and Conclusion In this work the statistical implications of the seesaw ensembles, following the anarchy principle,for the physical case of three neutrinos were obtained. It is shown that the analytic pdfs computedin [6] are in perfect agreement with the numerical results of randomly-generated light neutrino massmatrices for the complex seesaw ensembles with N = 2 and N = 3. The repulsion between thesingular values is stronger in the type I-III seesaw ensemble than in the type II seesaw ensemble,and the strength of the difference between the repulsions of type I-III and type II ensemblesincreases as N increases.The loss of correlation between the light neutrino masses and the light neutrino mass eigenstatesforbids an investigation of the favored hierarchy pattern (normal or inverted). However, an analysisof the preferred mass splitting, i.e. the preferred ordering of ˆ m − ˆ m and ˆ m − ˆ m , iscompleted. The probability test implies that for both seesaw ensembles, the preferred mass splittingis the one associated to normal hierarchy, although any permutation of the mass eigenstates isequally likely. However, a comparison between ensembles shows that the type I-III seesaw ensembleis only twice as likely as the type II seesaw ensemble to generate the neutrino sector experimentaldata assuming the preferred normal hierarchy.For all seesaw mechanisms, the preferred neutrino energy scale is of O (10 − ) eV, which leadsto a scale of new physics similar to the GUT scale when the associated coupling constants are oforder one. Smaller coupling constants can partly lower the new physics scale.A comparison of the group variable pdf for all seesaw ensembles (the Haar measure) and a flatdistribution shows that the seesaw ensemble is only twice as likely as the flat distribution to lead20o the neutrino sector experimental data. One thus concludes that the type I-III seesaw ensembleis marginally favored over the other ensembles, predicting the hierarchy of the neutrino sector tobe normal.Finally, a comparison of the complex type I-III seesaw ensemble level density for N = 3 andlarge N shows that the properly-normalized N = 3 level density is well approximated by theproperly-normalized large N level density. Because of the complexity of the analytic N = 3singular value pdf and the link between the large N level density and the physical neutrino sector,it would be interesting to obtain an analytical level density at large N . A step in that directionwas made in [6] following the usual Coulomb gas technique, but it was shown there that theresulting level density is wrong. The authors hope to return to this question in the near future. Acknowledgments The authors would like to thank Patrick Desrosiers for useful discussions on random matrix theory.This work is supported by NSERC. 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