MMass matrix parametrization for pseudo-Diracneutrinos
A. GorinNational Research Nuclear University MEPHI(Moscow Engineering Physics Institute),115409 Moscow, RussiaINR RAS(Institute for Nuclear Research of the Russian Academy of Sciences),108840, Troitsk, Russia,e-mail [email protected] 18, 2019
Abstract
An overview of pseudo-Dirac neutrino framework is given startingfrom general spinor phenomenology. The framework is then testedby simulation of oscillations for T2K experiment parameters. Twopossible derivations [7] and [8] of oscillation parameters are indicatedto have the same result.
Keywords: neutrino oscillations, sterile neutrinos, pseudo-Dirac neutrinos,neutrino oscillation experiments
Massive neutrinos directly indicate presence of physics beyond the Standardmodel (BSM). Precise measurements of neutrino oscillations provide thepossibility to probe various BSM theories.Since the absolute values of neutrino masses are currently beyond directmeasurements various experiments are focused on the standard neutrinomodel ( ν SM) oscillation parameters – square mass differences ∆ m and δ -phase.Some experiments however reported the existence of anomalies in ex-perimental data. These anomalies can find explanation in theories withadditional neutrino interactions, most notably the sterile neutrinos.1 a r X i v : . [ h e p - ph ] N ov ecently a number of short-baseline reactor experiments declared anobservation of sterile neutrinos with the significance of 3 σ . However theobservations are not entirely compatible to each other. The matter is underinvestigation in the ongoing STEREO, PROSPECT, SoLid and Neutrino-4experiments. Experimental evidences suggesting sterile neutrino with mass ∼ ( − ) ν µ disappearance [2] and LSND&MiniBooNE ( − ) ν µ → ( − ) ν e appearance [3, 4]. There are two ways to approach this problem.First possibility is to consider 3+1 non-unitary mixing scenario [5]. Itcan be used to explain short-baseline disappearance experiments howeverthe anomalies observed in LSND and MiniBooNE experiments [6] remainunexplained.Second possibility is addressing to more than one sterile neutrino. 3+2scenario can be studied in general framework of 3 active and 3 sterile neu-trino. Here we are probing the pseudo-Dirac scenario with 3 active and 3sterile neutrinos.In Section 2 we will describe how pseudo-Dirac neutrinos naturally arisewhen the neutrino is a composition of Dirac and Majorana spinors.In Section 3 we will show that pseudo-Dirac neutrinos can be effectivelydescribed by three parameters. Then the mass matrix can be effectivelydiagonalized which we show using two different approaches. Then we willplot the oscillation probability for pseudo-Dirac scenario against pure Diracneutrinos for the setup of T2K experiment.In Section 4 we will discuss what can be further done to address theproblem of streile neutrinos and neutrino mass generation. Lagrangian mass term for two spinors χ and η has the form L mass = 12 (cid:0) χ η (cid:1) M (cid:18) χη (cid:19) (1)where mass is given by M = (cid:18) A MM B (cid:19) and
M, A, B are 2x2 matrices.For the most general free field case we can write down “Weyl-Majorana-Dirac equation” iσ µ ∂ µ ψ L − η D,R m D,R ψ R − η L m L ( iσ ) ψ ∗ L = 0 i ¯ σ µ ∂ µ ψ R − η D,L m D,L ψ L − η R m R ( iσ ) ψ ∗ R = 0 (2)2ith non-negative mass terms m and phase terms η = e iϕ from unitarygroup U (1). Defining ˜ m = ηm and ψ R = (cid:18) ψ + iψ ψ + iψ (cid:19) , ψ L = (cid:18) ψ + iψ ψ + iψ (cid:19) this equation can be transformed into the form [1]: (cid:3) Φ + ˆ M Φ = 0 (3)where Φ = ( ψ ..ψ ) T Now let us illustrate only the simple case m D,L = m D,R = m D . For thiscase general spinor mass matrix is positive semi-definite Hermitian matrixof the form ˆ M = M R A M R − A − B M L B M L (4)where M R = (cid:18) ν + m R − ν ν ν + m R (cid:19) , M L = (cid:18) ν + m L − ν ν ν + m L (cid:19) , B = (cid:18) µ µ µ − µ (cid:19) , A = (cid:18) k − k (cid:19) and ˜ m D ˜ m L + ˜ m ∗ D ˜ m R = k (cid:62) m ∗ D ˜ m L + ˜ m D ˜ m R = µ + iµ ˜ m D = ν + iν This matrix has fourdoubly degenerate eigenvalues. Considering real and positive m R and m D and complex m L we are down to just two eigenvalues.Now consider χ and η in 1 to be the left- and right-handed neutrinofields ν L and ν R . We can work with two Majorana neutrinos if we stipulate ν R = ν (cid:48) CL . Then M = (cid:18) m L m D m D m R (cid:19) There are three commonly knownspecial cases for the values of the elements of this matrix.First case is m L = m R . In this scenario we have a pair of eigenval-ues m D ± m L and mixing angle between ν L and ν R is given by tan θ = m D m R − m L = π . No active-sterile oscillations are realized in this case.Second case is m L = m R = 0. In this scenario we have a pure Diracneutrino.Last case is m L , m R (cid:28) m D . This scenario is referred to as pseudo-Diraccase.In general, neutrino can have Majorana and Dirac parts L D + Mmass = L Dmass + L Lmass + L Rmass (5)and Dirac neutrino can be represented as two Majorana neutrinos. Left-handed neutrinos are concerned active while right-handed are sterile i.e.they are singlets under SU (2) L × U (1) Y .For the Pseudo-Dirac neutrino the symmetry of mass matrix is notthe symmetry of the weak interaction. It is easy to obtain Pseudo-Dirac3eutrino decomposition ψ ± L = 1 √ (cid:18) η ± iη (cid:19) = 1 √ N L ± iN L ) → √ N L ± e iϕ N L ) ψ ± R = 1 √ (cid:18) − iσ ( η ∗ ± iη ∗ )0 (cid:19) = 1 √ N C L ± iN C L ) → √ N C L ± e iϕ N C L )(6)for a pair of almost degenerate mass Majorana neutrino with opposite CPsign and lepton number not being conserved in higher order weak interac-tion.Because of the small value of mass matrix distortions the mixing anglebetween two Majorana neutrinos is ∼ π . For chirality preserving processes it is suffice to diagonalize M † M . Wewill now consider two possibilities – M and M diagonalization and showthat in the leading order they provide the same result for pseudo-Diracneutrinos.In general, 6x6 mass matrix diagonalization gives 15 mixing angles, mul-tiple violating CP phases and 6 eigenvalues. Under Pseudo-Dirac assump-tion this can be approximated by ordinary 3x3 Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [7]. M † M (cid:39) (cid:18) m † D m D m ∗ L m TD + m † D m ∗ R m ∗ D m L + m R m D m ∗ D m TD (cid:19) (7)consider bi-unitary transformation U † R m D U L = diag ( m , m , m ) = m then V = (cid:18) U L U ∗ R (cid:19) and V † ( M † M ) V = (cid:18) m U † L m † L U ∗ L m + mU † R m ∗ R U ∗ R mU TL m L U L + U TR m R U R m m (cid:19) (8)If we completely ignore off-diagonal parts then it is just Dirac scenariowith doubly-degenerate eigenvalues. Otherwise in the first order approxi-mation each pair takes the form (cid:18) m i (cid:15) ∗ i m i (cid:15) i m i m i (cid:19) ν iS = √ ( ν iL + e iϕ i ν iR ) ν iA = i √ ( ν iL − e iϕ i ν iR ) such that e iϕ i = (cid:15) i | (cid:15) i | for decomposition 6 and mass eigen-values given by m iS,A = m i ± (cid:15) i m i .Another method for diagonalization M itself is completely removingleft-handed Majorana spinor part of the Dirac one – mass matrix takes theform M = (cid:18) m (cid:48) D m (cid:48) D M s (cid:19) In [8] it is shown that the appropriate diagonalizingtransformation is given in form V = 1 √ (cid:18) U † U (cid:19) (cid:18) δ − δ † (cid:19) (9)where U diagonalizes m (cid:48) D and δ = U ( (cid:15)/ ε ), ε T = − ε and M s = 2 (cid:15)m D − εm D + m D ε . This produces M = V † mV where m = (cid:18) m D (1 + (cid:15) ) 00 − m D (1 − (cid:15) ) (cid:19) Now m in the leading order have the eigenvalues m i ± (cid:15) (cid:48) i m i which arethe same as in the previous case. With these eigenvalues we can write down the oscillation probability interms of ordinary PMNS matrix. Assume that mass eigenvalues splittingfor pseudo-Dirac neutrino is given by m iS,A = m i ± (cid:15) i m i . Using the resultsfrom [7] it is easy to model ν µ → ν e oscillation probability which is P ( ν α → ν β ) = 14 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) j =1 U βj ( e i m jS E t + e i m jA E t ) U ∗ αj (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (10)To illustrate potentially observable differences between Dirac and pseudo-Dirac scenario we will simulate oscillations for T2K experiment parameters: • L = 295 km and E ≤ • δ = − π and sin θ = 0 . sin θ = 0 . sin θ = 0 . • ∆ m = 7 . · − eV ∆ m = 2 . · − eV . • normal mass hierarchy.This allows us to probe the impact of small Majorana additives. Pleasealso note that energy spectrum now depends on the absolute mass of neu-trino because of the splitting. First we will model the situation where (cid:15) i = 0 .
1, Fig. 1. 5lease note that neutrino beam in T2K experiment has energy distri-bution with maximum at 0 . . ÷ ν µ → ν e oscillation probability comparedto pure Dirac scenario for T2K experiment parameters and naive assump-tions for pseudo-Dirac mass eigenvalues.Let us illustrate the difference in energy spectrum for more realistic (cid:15) i parameters. In Fig. 2 we have taken m = 0 .
01 eV, (cid:15) = 2 . · − , (cid:15) =4 . · − and (cid:15) = 5 . · − proportional to mass squares differences.Figure 2: Pseudo-Dirac neutrino ν µ → ν e oscillation probability comparedto pure Dirac scenario for T2K experiment parameters for more realisticmass splittings. 6 Discussion and Conclusion
Now we are in the situation where combined experimental data from atmo-spheric, reactor and accelerator neutrino experiments is in good agreementwith 3 active neutrino model for the first three oscillation peaks. Upcom-ing experiments can provide more experimental data thus clarifying thesituation.Long-baseline experiments can provide precise values of ν SM oscilla-tion parameters and provide enough data to determine the neutrino masshierarchy.Short-baseline experiments can either improve their statistics and cancelout all anomalies or successfully approve that the ν SM needs expansion.Using precise β -decay and K-capture measurements it would be ar-guably possible to measure neutrino masses directly or at least put a con-straints on them. ββ and 0 νββ observations as well as atmospheric, solar, galactic andextra-galactic neutrino experiments are important for probing different neu-trino mass generation mechanisms.It is also important to consider theoretical models for processes in earlyUniverse – the constraints from these models are generally less strict thanfrom direct observations but still helpful either for a cross-checking or forlimiting the potential of exotic mass generation and mixing models.Here we presented the derivation of pseudo-Dirac neutrino from generalspinor formalism.For the parameters of T2K experiment the probability of ν µ → ν e os-cillation was modeled. The current setup of the experiment however is notsensitive to differences in Dirac and pseudo-Dirac oscillations.It was shown that in the leading order approximation PD neutrino canbe effectively described by three (cid:15) parameters of mass splitting – it is validfor M and M diagonalization.There are questions arising naturally in the context of neutrino massgeneration mechanism.First question is whether it is suffice to consider pseudo-Dirac neutrinoto fit observations or general framework is needed? This question will beaddressed by the future observations.Second question is about the compatibility of particular mass generationmechanism with pseudo-Dirac scenario in particular and it’s rigidity topossible observational data as a whole. Which mechanisms are the bestcandidates, Yukawa coupling or multiple scalar fields (like in Zee model) ormaybe even geometric models of mass generation?7 cknowledgements I am grateful to M.Yu. Khlopov for an invitation to XXII Bled Workshopand to the organizing committee of the Workshop for an opportunity tomake a talk via internet.
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