arXiv:1810.04613v4 [physics.gen-ph] 17 Aug 2019 eevd ue21 eie:2 uy21 cetd 6Ju 26 Accepted: / 2019 July 25 Revised: / 2019 June 5 Received: Amsterda of University Physics, Theoretical for Institute Nieuwenhuizen Maria Theodorus neutri Diracian with physics sector particle of Model Standard The Article Symmetry .Introduction 1. ol rz npyiswsaaddt .Kjt n .B McDon B. A. [ and mass” Kajita have neutrinos T. that to show awarded which oscillations was physics in becaus prize BSM involve Noble must sector neutrino the But (BSM). model h Netherlands The hsfrteLreHdo oldr(H)hsntpoue evi produced not has (LHC) Collider Hadron Large the far Thus ASclassifications: PACS ANITA Keywords: ult Earth-traversing of ANITA. interpretation do neutrino neutrinoless feeble sterile very a allows and conservation neutri sterile number the to assigned –1 number proble lepton Oscillation With matter. offered. is f e data of three model o description and neutrino better phases not a standard Dirac do physical the become pair and phases a Majorana undone, two of be Members can mixing signature. their becom Dirac masses a physical the attain is limit, which neutrinos “Diracian” occurs, a Majo states In the sterile case, neutrinos. and pseudo-Dirac active the of In mixing D maximal matrix. a mass by Majorana ones active a the by to coupled neutrinos, sterile or handed Abstract: 2019 , 11 h iial xeddsadr oe fpril hsc c physics particle of model standard extended minimally The tnadmdl trl etio,Mjrn as suoD pseudo mass, Majorana neutrinos, sterile model, Standard -5 doi:10.3390/sym11080994 1-25; , 46.t 46.q 12.60.-i 14.60.Pq, 14.60.St, 1 ]. ,SinePr 0,19 HAmsterdam, XH 1098 904, Park Science m, saewre u nvcu n in and vacuum in out worked are ms ahg nryeet eetdby detected events energy high ra etio aems.Ide,te2015 the Indeed, mass. have neutrinos e o,temdlsilvoae lepton violates still model the nos, y21 ulse:3Ags 2019 August 3 Published: / 2019 ly ec o hsc eodtestandard the beyond physics for dence rcms arxadmutually and matrix mass irac ariedgnrt n the and degenerate pairwise e bebt ea.I supports It decay. beta uble eeal xlddfrsolar for excluded generally www.mdpi.com/journal/symmetry l frtedsoeyo neutrino of discovery the “for ald tams aaeesoccur, parameters mass xtra losa ii.While limit. a as ollows aatrsaesaland small are terms rana claemtal othat so mutually scillate symmetry nan he right three ontains rcneutrinos, irac SN2073-8994 ISSN PNACCESS OPEN no Symmetry 2019, 11 2

The standard neutrino model (SνM) with its three Majorana neutrinos has measured values for the mass-squared differences, the mixing angles θ12, θ23 and θ13 and the weak Dirac phase δ. But the absolute mass scale, the order of the hierarchy, normal or inverted, and the Majorana phases are unknown. There is stress in the fit to the standard solar model[2]; there is a reactor neutrino anomaly [3,4]; MiniBooNE finds 4.5σ evidence for a sterile neutrino[5], while MINOS/MINOS+ does not [6]. At present, there is no definite conclusion about the existence of an eV sterile neutrino [7]. There is also input from cosmology. From the lensing of background galaxies by the large, reasonably relaxed galaxy clusters Abell 1689[8–10] and Abell 1835[11] there is indication for 3 active and 3 sterile neutrinos with common mass of 1.5–1.9 eV, which act as the cluster dark matter. We shall not dwell here into the many questions raised by and counter-evidence to that possibility, but refer to the discussion and cited articles in these references. Be it as it may, the 3+3 case puts forward to consider the minimal extension of the standard model (SM) in the neutrino sector. By default, this accepts all SM physics without extension in the Higgs, gauge, quark and charged lepton sectors. Gauge invariance then forbids the presence of a ‘left handed’ Majorana mass matrix between the left handed active neutrinos, so that there must be a Dirac mass matrix to give them mass. As such a term mixes left and right handed fields, this presupposes the existence of 3 right handed neutrinos, also called sterile, i. e., not involved in elementary particle processes[12]. For that reason, they are allowed to have a mutual ‘right handed’ Majorana mass matrix. In order to make up for half of the cluster dark matter, sterile neutrinos have to be generated in the early cosmos by oscillation of active ones. This is only possible when the Dirac mass matrix is accompanied by a non-trivial right handed Majorana mass matrix. In the pseudo-Dirac limit, the right handed Majorana masses are much smaller than the eigenvalues of the Dirac mass matrix. The maximal mixing of the resulting pseudo-Dirac neutrinos implies that in principle half of the emitted solar neutrinos has become sterile here on Earth, and thus unobservable1; this is ruled out by the standard solar model[2]. Hence the pseudo-Dirac case is often considered to be ruled out. We intend to show, however, that there is a way out of this conundrum, so as to faithfully include neutrino mass in the SM without changing its high energy sector.

While excellent studies such as [12–14] discuss the theory for general number Ns of sterile neutrinos, we shall work out the case Ns = 3 in a nontrivial limit where the 6 Majorana neutrinos combine into 3 Dirac neutrinos so that the maximal mixing is harmless and can be circumvented. We call them Diracian neutrinos, i. e., Dirac neutrinos in a model with both Dirac and Majorana masses. In section 2 we treat the theory and in section 3 we consider various applications. We close with a summary.

2. The Lagrangian for active plus sterile neutrinos

In this section we concentrate on the neutrino sector of the SM. For completeness we present the full Lagrangian in Appendix B.

2.1. Active neutrinos only

1 See section 2.4 for details. Symmetry 2019, 11 3

We start from the SM Lagrangian where the e, µ and τ fields are diagonal in the mass basis. Left handed neutrinos and right handed antineutrinos exist are called “active neutrinos” since they participate in the weak interactions2. Additional neutrinos are not involved in them and called sterile. If only active ones exist, they are Majorana particles. Their mass term involves the quantized left handed fermionic

flavor fields νeL, νµL, ντL, 1 M = νT †(M M ) ν + h.c. (1) LmL 2 αLC L αβ βL α,βX=e,µ,τ where is the charge conjugation matrix, T denotes transposition, Hermitian conjugation, and h.c. C M † Hermitian conjugated terms. ML is called the left handed Majorana mass matrix. In the SM gauge M invariance forces ML to vanish[12]; if it is present, it must originate from high energy BSM, such as M Weinberg’s dimension-5 operator. Considering new physics only in the neutrino sector, we neglect ML .

2.2. The Dirac and Majorana mass matrices

M In absence of ML , the only possibility to give mass to the active neutrinos is by a Dirac mass matrix. Since that involves products of left and right handed fields, this presupposes the existence of N 3 s ≥ sterile neutrinos, that must be right handed and represented by quantized fermionic fields νiR,

Ns D = ν M Dν + ν M D †ν , (2) Lm − αL αi iR iR iα αL α=e,µ,τ i=1 X X   where the Dirac mass matrix M D is a complex 3 N matrix. The sterile fields do not enter the weak × s interactions; they are singlets under the U(1) SU(2) SU(3) gauge groups of the SM and affect Y × L× C neither gauge invariance, anomalies nor renormalization. Hence they preserve its full functioning while accounting for neutrino masses. Moreover, the sterile fields may have a mutual mass term like Eq. (1),

1 Ns M = νT †M M† ν +(ν c )T †M M ν c , (3) LmR 2 iRC R,ij jR iR C R,ij jR i,j=1 X   M c where the right handed Majorana mass matrix MR is symmetric and complex valued, and where νiR is the charge conjugate of νiR,

ν c = ν T = (γ 0)T (ν† )T = γ 0 (ν† )T . (4) iR C iR C iR − C iR While ν is a right handed field, ν c is left handed (see Appendix A for properties of γ and matrices). iR iR C The kinetic term has a common form for all species[12],

Ns = ν i←→∂/ ν + ν i←→∂/ ν , (5) Lk αL αL iR iR α=e,µ,τ i=1 X X

2 Left and right handedness refers to the chirality; see AppendixA . Symmetry 2019, 11 4 where the slash denotes contraction with γ matrices, and the partial derivatives acting as

3 3 −→∂ ←−∂ 1 ∂b ∂a ←→∂/ = γ µ←→∂ , ←→∂ = µ − µ , a ∂b/ aγµ γµb . (6) µ µ 2 ≡ 2 ∂x µ − ∂x µ µ=0 µ=0 X X   2.3. The general mass matrix for 3 sterile neutrinos

Though the number of right handed neutrinos is not fixed in principle, the case Ns = 3 has, if not a practical value[8–11], at least an esthetic one: for each left handed neutrino there is a right handed one, in the way it occurs for charged leptons and quarks. The three families of active left and sterile right handed neutrinos have the flavor 3 vectors3

ν ν =(ν , ν , ν )T , ν ν =(ν , ν , ν )T . (7) fL ≡ aL eL µL τL fR ≡ sR 1R 2R 3R With the combined left handed flavor vector

νT ν c T T νT νc T T NfL =( fL, fR ) =( aL, sR ) , (8) the above mass Lagrangians combine into 1 = NT †M DM N + h.c. (9) Lm 2 fLC fL In general, the mass matrix consists of four 3 3 blocks, × M M M DT DM L (10) M = D M . M MR !

M M As stated, we take ML = 0. In the (“standard”, “pure” or “trivial”) Dirac limit also MR = 0. For M D pseudo-Dirac neutrinos MR will be small with respect to M , or, more precisely, small with respect to the variation in the eigenvalues of M D. Though we consider general M D, we are inspired by the case of galaxy cluster lensing where it has nearly equal eigenvalues with central value 1.5–1.9 eV [8–11]; in M section 3.1 we shall show that the entries of MR then typically lie well below 1 meV.

2.4. Intermezzo: One neutrino family

c T In case of one family the flavor vector is NfL =(νL, νR) . The entries of Eq. (10) are scalars, so

0m ¯ M DM = . (11) m¯ µ¯ ! Its eigenvalues are

1 2 1 2 λ1,2 = µ¯ m¯ + µ¯ . (12) 2 ∓ r 4

3 In our case Ns =3 one may be tempted to denote (ν1R,ν2R,ν3R) as (νeR,νµR,ντR). Symmetry 2019, 11 5

The physical masses are their absolute values[12]. For µ¯ nonnegative, this leads to

2 1 2 1 2 1 2 1 m1 = m¯ + µ¯ µ,¯ m2 = m¯ + µ¯ + µ.¯ (13) r 4 − 2 r 4 2 The corresponding eigenvectors are

1 m¯ 1 m 1 m e(1) = = 2 , e(2) = 1 . (14) 2 2 2 2 2 2 m¯ + m m1 m¯ + m m¯ m¯ + m m¯ 1 − ! 2 − ! 1 ! For small µ¯ thesep are 45◦ rotations, i. e.,p maximal mixing of the active and sterilep basis vectors. Formally we may undo the rotations over 45◦, by considering

e(1) + e(2) 1 e(2) e(1) µ/¯ 4¯m ea˜ = , es˜ = − − , (15) √2 ≈ µ/¯ 4¯m! √2 ≈ 1 ! where the approximations are to first order in µ¯, the pseudo Dirac regime. The first vector, ea˜, has its main weight on the first component, so it is mainly active, which we indicate by the tilde on a. The s˜ a˜ s˜ second one, e , is mainly sterile. But unless µ¯ = 0, the masses m1,2 are different, so that e and e are not eigenvectors and have no physical meaning. In fact, the mass squares have the difference

2 2 2 2 1 2 ∆m21 = m2 m1 = 2¯µ m¯ + µ¯ 2¯µm.¯ (16) − r 4 ≈ An initially active state,

(1) (2) 1 m¯ e + m1e νa(0) = = , (17) 2 2 | i 0! m¯ + m1 with momentum p will at time t have oscillated into p

(1) − iE1t (2) − iE2t m¯ e e + m1e e νa(t) = , (18) 2 2 | i m¯ + m1 p 2 2 where E1,2 = p + m1,2. The occurrence probability is q m¯ 4 + m4 +2¯m2m2 cos ∆Et 1 + cos ∆Et 2 1 1 (19) Paa(t) = νa(0) νa(t) = 2 2 2 , |h | i| (¯m + m1) ≈ 2 where for p m¯ ≫ ∆m2 µ¯m¯ ∆E E E 21 . (20) ≡ 2 − 1 ≈ 2p ≈ p In practice there will not be a pure initial state but some wave packet[12]. For t ~/∆E the cosine ≫ 1 in Eq. (19) will average out, so that the fraction of observable neutrinos is approximately 2 . In plain terms: for t large enough, half of the neutrinos are sterile and thus unobservable. For the solar neutrino problem the one-family approximation happens to work quite well[15] and the detection rates are well established. Hence for the pseudo Dirac model it would mean that twice as many neutrinos should Symmetry 2019, 11 6 be emitted as in the standard solar model. The corresponding doubling of heat generated by nuclear reactions is ruled out by the measurements of the solar luminosity, so the case is rarely discussed. Only in the pure Dirac case, i. e., with Majorana mass µ¯ = 0, the oscillations will not take place, a˜ since m1,2 =m ¯ and ∆E =0. When starting from an initial active state νa(0), it now equals e , and this can be taken as eigenstate. The sterile state will merely be a spectator, “just sitting there and wasting its time”. This can be generalized to three families. If one would follow the Franciscan William of Ockham (Occam’s razor), it would be preferable for active neutrinos to be Majorana rather than Dirac with unobservable right handed partners. The SνM differs from the neutrino sector in the SM by accounting for finite masses of its 3 Majorana neutrinos. Below we discuss a “Diracian” setup in which the sterile fields become physical, namely partly active, and the active fields partly sterile, even though the mass eigenstates have Dirac signature in vacuum.

2.5. Diagonalization of the Dirac mass matrix

M We return to the 3 family case and its total mass matrix Eq. (10) with ML = 0. We notice that any 3 3 unitary matrix U can be decomposed as a product of 5 standard ones, ×

′ DM DM D M D M iη1 iη2 iη3 U = D U , U = U D , U = U1U2U3,D = diag(e , e , e ). (21)

The diagonal matrix DM is called the Majorana phase matrix. Likewise we denote the diagonal phase ′ ′ ′ matrix4 D′ by diag(e iη1 , e iη2 , e iη3 ). The matrix U D is the product of

− iδ 1 0 0 c2 0 s2e c3 s3 0

U1 = 0 c s , U2 = 0 1 0 , U3 = s c 0 , (22)  1 1   − 3 3  0 s c s e iδ 0 c 0 01  − 1 1 − 2 2          where ci = cos θi, si = sin θi where the angles θi are termed in standard notation θ1 = θ23, θ2 = θ13 and

θ3 = θ12. The Dirac phase δ is also called weak CP violation phase. The complex valued Dirac mass matrix M D can be diagonalized by two unitary matrices of the form ′ D M ′ D M (21), viz. UL = DLUL DL and UR = DRUR DR . The result reads

D T † d † d M = UR M UL, M = diag(m ¯ 1, m¯ 2, m¯ 3), (23)

5,6 D D M with the real positive m¯ i . We identify UL with the PMNS mixing matrix U = U1U2U3 and DL with the Majorana matrix DM employed in literature.

′ 4 ′ iη1 M iη1 ′ For U in Eq. (21) only 5 of the ηi and ηi are needed; this can be seen by factoring out e from D and e from D ′ and setting η1 η1 η1. Both sides of Eq. (21) thus involve 9 free parameters. 5 → − We denote Dirac mass eigenvalues by m¯ i to distinguish them from the physical masses mi, the eigenvalues in absolute value of the total mass matrix. 6 While the left hand side of Eq. (23) has 9 complex or 18 real parameters, the right hand side has 9+3+9; but since M d M† M∗ M∗ M is diagonal, the diagonal matrices DL = DL and DR only act as a product. Hence it is allowed to fix DR before M solving DL , see below Eq. (28). The number of parameters available for the diagonalization is then still 18. Symmetry 2019, 11 7

To connect the transformation (23) to M DM , we introduce the 6 6 unitary matrix ×

UL 0 ULR = , (24) 0 UR! and define, using that M d T = M d since it is diagonal,

0 M d T DM (25) = ULR M ULR = d N , M M M !

† ν T ν New active and sterile fields naL = UL fL, nsR = UR fR, merged as

1 2 3 1c 2c 3c T nL =(naL, naL, naL, nsR, nsR, nsR) , (26) express (8) as

νT ν c T T † NfL =( fL, fR ) = ULRnL, nL = ULRNfL. (27)

With these steps the right handed Majorana mass matrix transforms into

N T M M = UR MR UR. (28)

M D Like MR , it is complex symmetric, but since UR was needed to diagonalize M , it will in general not N D M result in a diagonal M . With the decomposition UR = DRUR DR as in Eq. (21), one can, however, M N 6,7 use the phases in DR to make the off-diagonal elements of M real and nonnegative . N We denote the diagonal elements of the Majorana matrix M by µ¯i, that may still be complex, and N the real positive off-diagonal elements by µi. The right handed Majorana mass matrix M then takes the form

µ¯1 µ3 µ2 N M = µ3 µ¯2 µ1 , (29) µ µ µ¯  2 1 3   so that the total mass matrix reads M

0 0 0¯m1 0 0  0 0 0 0¯m2 0  00000¯m =  3 . (30) M m¯ 0 0µ ¯ µ µ   1 1 3 2     0m ¯ 2 0 µ3 µ¯2 µ1     0 0m ¯ µ µ µ¯   3 2 1 3    Except in the pure Dirac limit where µi =µ ¯i =0, the nL are not rotations of mass eigenstates.

7 Actually, for n lepton families there are 1 n(n 1) independent complex valued off-diagonal elements and n Majorana 2 − phases, so making all off-diagonal elements real and nonnegative is possible for n =3 or 2. Symmetry 2019, 11 8

2.6. Diracian limit

For reasons explained above, we wish to achieve pairwise degeneracies in the masses. The standard Dirac limit, just taking µ and µ¯ 0, is a trivial way to achieve this; we shall, however, need finite i i → values for them and design the more subtle “Diracian” limit. To start, we notice that the eigenvalues of the mass matrix (29) follow from det( λI)=0, where M − det( λI)= (31) M − (λ2 m¯ 2)(λ2 m¯ 2)(λ2 m¯ 2) (¯µ +¯µ +¯µ )λ5 (µ2 +µ2 +µ2 µ¯ µ¯ µ¯ µ¯ µ¯ µ¯ )λ4 − 1 − 2 − 3 − 1 2 3 − 1 2 3 − 1 2 − 2 3 − 3 1 +[¯m2(¯µ +¯µ )+¯m2(¯µ +¯µ )+¯m2(¯µ +¯µ )+µ2µ¯ +µ2µ¯ +µ2µ¯ 2µ µ µ µ¯ µ¯ µ¯ ]λ3 1 2 3 2 3 1 3 1 2 1 1 2 2 3 3 − 1 2 3 − 1 2 3 +[¯m2(µ2 µ¯ µ¯ )+¯m2(µ2 µ¯ µ¯ )+¯m2(µ2 µ¯ µ¯ )]λ2 (¯m2m¯ 2µ¯ +¯m2m¯ 2µ¯ +¯m2m¯ 2µ¯ )λ. 1 1 − 2 3 2 2 − 3 1 3 3 − 1 2 − 1 2 3 2 3 1 3 1 2 The criterion to get pairwise degeneracies in the eigenvalues (up to signs), is simply that the odd powers in λ vanish. Let us denote

∆¯ =m ¯ 2 m¯ 2, ∆¯ =m ¯ 2 m¯ 2, ∆¯ =m ¯ 2 m¯ 2, 1 2 − 3 2 3 − 1 3 1 − 2 m¯ 2m¯ 3 m¯ 3m¯ 1 m¯ 1m¯ 2 M¯1 = , M¯ 2 = , M¯ 3 = , (32) m¯ 1 m¯ 2 m¯ 3 and express the µ¯i in a common dimensionless parameter u¯ through

∆¯ i µ¯i = u.¯ (33) M¯ i 2 ¯ ¯ 5 1 The relations i m¯ i ∆i = i ∆i =0 make the coefficients of λ and λ ofEq. (31) vanish, respectively. To condense further notation, we express the µ into dimensionless non-negative parameters u , P P i i

ui µi = ∆¯ 1∆¯ 2∆¯ 3 . (34) | |m¯ ∆¯ q i | i| For normal ordering of the m¯ i (notice that these are Diracp masses, not the physical masses), m¯ 1 < m¯ 2 < m¯ 3 implies ∆¯ 1 < 0, ∆¯ 2 > 0, ∆¯ 3 < 0, hence ∆¯ 1∆¯ 2∆¯ 3 > 0; this is also the case for the inverted ordering m¯ 3 < m¯ 1 < m¯ 2 whence ∆¯ 1 > 0, ∆¯ 2 < 0, ∆¯ 3 < 0. It thus holds that

∆¯ 1∆¯ 2∆¯ 3 µ1µ2µ3 = u1u2u3. (35) m¯ 1m¯ 2m¯ 3 Equating the λ3 coefficient of Eq. (31) to zero requires ¯ ¯ ¯ ¯ 3 2 2 ∆1 2 ∆2 2 ∆3 2 ∆1 2 2 2 u¯ (1 + u )¯u +2u1u2u3 =0, u u1 + u2 + u3 = u1 u2 u3. (36) − ≡ ∆¯ 1 ∆¯ 2 ∆¯ 3 ∆¯ 1 − − | | | | | | | |
This cubic equation has the solutions for n = 1, 0, 1 and positive or negative u2 − i u¯ = − e2πin/3 u1/3 e−2πin/3(1 + u2) u−1/3 , (37) √3 + − + h i u = √D+ i√27u u u ,D = (1+u2)3 27u2u2u2. + 1 2 3 − 1 2 3 We restrict ourselves to real solutions; there is always one. Then the matrix is real valued. All M solutions are real when D> 0, which occurs in particular when the ui are small, i.e., in the pseudo-Dirac Symmetry 2019, 11 9 case. Then there exist the large solutions n = 1 with u¯ 1, which in both cases leads to the ± ≈ ± eigenvalues λ± M for i =1, 2, 3. For small u the n =0 solution has a small u¯ and µ¯ , viz. i ≈ ± i i i ¯ µ1µ2µ3 µ1µ2µ3 2 ¯ ∆i u¯ 2u1u2u3 =2 m¯ 1m¯ 2m¯ 3, µ¯i 2 m¯ i ∆i =2u1u2u3 . (38) ≈ ∆¯ 1∆¯ 2∆¯ 3 ≈ ∆¯ 1∆¯ 2∆¯ 3 Mi The Diracian limit, defined by Eqs. (33), (34) and (37), reduces Eq. (31) to a cubic polynomial in λ2,

(λ2 m¯ 2)(λ2 m¯ 2)(λ2 m¯ 2) (39) − 1 − 2 − 3 (λ2 m¯ 2)(¯u2 u2) (λ2 m¯ 2)(¯u2 u2) (λ2 m¯ 2)(¯u2 u2) +λ2∆¯ ∆¯ ∆¯ − 1 − 1 + − 2 − 2 + − 3 − 3 =0. 1 2 3 2 ¯ 2 ¯ 2 ¯ m¯ 1∆1 m¯ 2∆2 m¯ 3∆3 h i 2 Its analytical roots are intricate, but they are easily calculated numerically. Denoting them as mi , the squares of the physical masses, the eigenvalues of are λ = m and λ = + m > 0 for M 2i−1 − i 2i i i =1, 2, 3. From det = m¯ 2m¯ 2m¯ 2 it holds that m m m =m ¯ m¯ m¯ . The eigenvectors are set by M − 1 2 3 1 2 3 1 2 3 6 e(i) = λ e(i), (i =1, , 6). (40) Mjk k i j ··· Xk=1 and they are real and orthonormal. They can be expressed as

e(ia˜) e(is˜) e(ia˜) + e(is˜) e(2i−1) = − , e(2i) = . (41) √2 √2

(ia˜) (is˜) (ia˜) (is˜) 8 with orthonormal e and e for i =1, 2, 3. For small µi and µ¯i the e and e read to first order

T T (1˜a) µ¯1 µ3 µ2 (1˜s) µ¯1 µ3 µ2 e = 1, 0, 0, , m¯ 1 , m¯ 1 , e = , m¯ 2 , m¯ 3 , 1, 0, 0 , 4¯m1 ∆¯ 3 − ∆¯ 2 − 4¯m1 ∆¯ 3 − ∆¯ 2  T  T (2˜a) µ3 µ¯2 µ1 (2˜s) µ3 µ¯2 µ1 e = 0, 1, 0, m¯ 2 , , m¯ 2 , e = m¯ 1 , , m¯ 3 , 0, 1, 0 , (42) − ∆¯ 3 4¯m2 ∆¯ 1 − ∆¯ 3 −4¯m2 ∆¯ 1  T  T (3˜a) µ2 µ1 µ¯3 (3˜s) µ2 µ1 µ¯3 e = 0, 0, 1, m¯ 3 , m¯ 3 , , e = m¯ 1 , m¯ 2 , , 0, 0, 1 . ∆¯ 2 − ∆¯ 1 4¯m3 ∆¯ 2 − ∆¯ 1 −4¯m3     With the first three components of these vectors relating to active neutrinos and the last three to sterile (ia˜) (is˜) ones, it is seen that for small µi and µ¯i the e are mainly active and the e mainly sterile, which we indicate by the tildes. The e(2i−1) and e(2i) are 45◦ rotations of the e(ia˜) and e(is˜), which is maximal mixing. In the standard Dirac limit, it is customary to work with Dirac states and not with Majorana states. Likewise, in our Diracian limit the mass degeneracies allow the rotations to be circumvented by working with the e(ia˜) and e(is˜) themselves. Indeed, there holds the exact decomposition

6 3 = λ e(i)e(i) T = m e(ia˜)e(is˜) T + e(is˜)e(ia˜) T . (43) M i i i=1 i=1 X X h i These steps allow to retrieve the standard Dirac expressions in the limit where the Majorana masses µi, (ia˜) (is˜) µ¯i vanish, whence the e and e become purely active and purely sterile states, respectively.

8 Notice that the i and i +3 components of e(ia˜) and e(is˜) stem with the one-family case Eq. (15). Symmetry 2019, 11 10

For neutrino oscillation probabilities in vacuum (see subsections 2.4 and 3.2) one needs the eigenvalues m2 and eigenvectors e(ia˜) and e(is˜) of 2, i M 3 2 = m2 e(ia˜)e(ia˜) T + e(is˜)e(is˜) T . (44) M i i=1 X h i In terms of nL defined above Eq. (27) and related there to the flavor states (8), the fields for the mass eigenstates are

6 6 i (ia˜) i (is˜) c νaL˜ = ej njL, νsR˜ = ej njL, (i =1, 2, 3). (45) j=1 j=1 X X i Here j =1, 2, 3 label active fields and j =4, 5, 6 sterile ones. Hence the fields νaL˜ annihilate chiral left i handed, mainly active neutrinos and create similar right handed antineutrinos, while the νsR˜ annihilate chiral right handed, mainly sterile neutrinos and create similar left handed antineutrinos. The mass term of the 6 Majorana fields now takes the form of 3 Dirac terms,

3 3 1 1 T = m νi T †νic +(νic )T †νi + h.c. = m νi T νi νi νi + h.c. Lm 2 i aL˜ C sR˜ sR˜ C aL˜ 2 i aL˜ sR˜ − sR˜ aL˜ i=1 h i i=1   X3 3 X = m νi νi + νi νi = m νiνi, (46) − i sR˜ aL˜ aL˜ sR˜ − i i=1 i=1 X   X because fermion fields anticommute and left and right handed fields are orthogonal. The here introduced Dirac fields,

i i i ν = νaL˜ + νsR˜ , (i =1, 2, 3), (47) combine left and right handed chiral fields, as usual. They are the mass eigenstates. In this basis the Dirac-Majorana neutrino Lagrangian is a sum of Dirac terms,

3 = νi i←→∂/ νi m νiνi . (48) L − i i=1 X   2.7. Charged and neutral current

Neutrinos also enter the currents coupled to the W and Z gauge bosons, which are part of the covariant derivatives in the Lagrangian, see Eq. (110) below. The W boson couples to the charged weak current. On the flavor basis it reads g = W J µ † + W †J µ , J µ =2 ν γ µℓ , J µ † =2 ℓ γ µν , (49) LCC − √ µ CC µ CC CC αL αL CC αL αL 2 2 α=e,µ,τ α=e,µ,τ h i X X with g the weak coupling constant. The neutral weak current reads on the flavor basis g = Z J µ , J µ = ν γ µν , (50) LNC −2 cos θ µ NC NC αL αL w α=e,µ,τ X with θw the weak or Weinberg angle. Symmetry 2019, 11 11

To express these in the mass eigenstates, we define and as matrices consisting of the active A S components of the 6-component eigenvectors e(ia˜) and e(is˜), respectively,

= e(ia˜), = e(is˜), (j, i =1, 2, 3), (51) Aji j Sji j and, likewise, s and s for the sterile components A S s = e(ia˜) , s = e(is˜) , (j, i =1, 2, 3). (52) Aji j+3 Sji j+3

From (42) we read off that for small µi and µ¯i

3 µ¯i µk ji δij, ji δij + εijkm¯ j , (53) A ≈ S ≈ − 4¯mi ∆¯ k=1 k X3 s s µ¯i µk ji δij, ji δij + εijkm¯ i . (54) S ≈ A ≈ 4¯mi ∆¯ k Xk=1 From the orthonormality of the eigenvectors it follows that the real valued 3 3 matrices and satisfy × A S the unitarity relation

T + T =1 , (55) AA SS 3×3 while T + T =1 . A A S S 6 3×3 M M ν ′ D M From Eqs. (7), (8), (21), (22), (24) and (27), and denoting D DL , we have fL = DLU D naL. ′ ≡ As shown below Eq. (50), the diagonal phase matrix DL can be absorbed in the fields. Inverting Eq. (45) leaves Eq. (48) invariant and expresses the flavor eigenstates as superpositions of mass eigenstates ν =(ν , ν c ). In vector notation, and using νc = νT †, one has mL aL˜ sR˜ sR˜ − sR˜ C ν = U DM ( ν + ν c )= A ν + S ν c , fL A aL˜ S sR˜ aL˜ sR˜ ν = (ν T + ν c T )U DM† = ν A† + ν c S†, (56) fL aL˜ A sR˜ S aL˜ sR˜ = (ν T νT † T )U DM† = ν A† νT †S†. aL˜ A − sR˜ C S aL˜ − sR˜ C Here U DM = U DDM , with U D is the standard PMNS matrix, see (21), while DM = diag(e iη1 , e iη2 , e iη3 ) 9 is the Majorana matrix of the 3-neutrino problem; its ηi are Dirac phases now . We also introduced

A = U DM , S = U DM , AA† + SS† =1 . (57) A S 3×3 ν c 1 c 2 c 3 c The sterile field sR =(νsR, νsR, νsR) similar to (56) reads

ν c = D′ U DDM ( sν + sν c). (58) sR R R R A a˜ S s˜ The only current knowledge of the involved matrix elements lies in (54).

9 A word on nomenclature: The Majorana phases in the matrix DM stem from the 3+0 SνM, without sterile neutrinos. While they become physical Dirac phases in the 3+3 DMνSM, there appear no true 3+3 Majorana phases, so we propose to keep this name for them. Hence the DMνSM has 3 physical phases: 1 Dirac and 2 “Majorana”phases. They all appear in the CP-invariance breaking part of the neutrino oscillation probabilities, see Eq. (81). Symmetry 2019, 11 12

The flavor eigenstates can also be written as single sums over mass eigenstates,

6 i ν ν ν 1 2 3 1 c 2 c 3 c T ναL = UαiνmL, fL = U mL, mL =(νaL˜ , νaL˜ , νaL˜ , νsR˜ , νsR˜ , νsR˜ ) , (59) i=1 X with the 3 6 PMNS matrix U having elements × U = A =(U DM ) , U = S =(U DM ) , (UU †) = δ , (α, β =1, 2, 3). (60) α,i αi A αi α,i+3 αi S αi αβ αβ the latter deriving from (55), while U †U = 1 , because U represents the 3 active rows of a unitary 6 6×6 6 6 matrix which also involves s and s. Hence the GIM theorem that J µ has the same form on × A S NC flavor and mass basis, does not hold[12]. Inserted in the currents the relations (56) yield

J µ = 2ν U †γ µℓ = 2(ν A† + ν c S†)γ µℓ = 2(ν T νT † T )U DM†γ µℓ (61a) CC mL L aL˜ sR˜ L aL˜ A − sR˜ C S L J µ † = 2ℓ γ µUν =2ℓ γ µ(Aν + Sνc ) =2ℓ γ µU DM ( ν + νc ), (61b) CC L mL L aL˜ sR˜ L A aL˜ S sR˜ J µ = ν γ µ T ν ν γ µ T ν νT †γ µ T ν ν† γµ† T ν†T . (61c) NC aL˜ A A aL˜ − sR˜ S S sR˜ − sR˜ C S A aL˜ − aL˜ CA S sR˜

2.8. Lepton number for sterile neutrinos

There is an ambiguity in defining the lepton number of the sterile neutrinos. The lepton number of neutrinos is investigated by making the transformation

ν e iLaφν , ν e iLsφν , (62) aL → aL sR → sR

This leaves the kinetic terms invariant and for the standard choice Ls =La =1 also the Dirac mass terms (2). Only the Majorana mass terms (1) and (3) will vary by factors e±2 iφ: they violate lepton number conservation by ∆L = 2. This approach connects the lepton number L =1 of active neutrinos also to ± a sterile neutrinos, hence L = 1 for sterile antineutrinos (charge conjugated sterile ones). This assigns ν¯s − lepton number +1 to the components j = 1, 2, 3 of N of Eq. (8) and n of Eq. (26), but 1 to the fL L − components j = 4, 5, 6. Then the mixing (45), or its reverse (56), (58), enforced by the nonvanishing right handed Majorana mass matrix, makes it impossible to consistently connect a lepton number to the i i particles connected to the mass eigenstates νa˜ and νs˜. The opposite choice L = 1, L = 1 circumvents this problem for general models with active and a s − sterile neutrinos. According to (8), (45) and (56) the lepton number La =1 of νa particles is consistent with L = 1 of a ν particle and L = 1 of ν particles. This choice is henceforward consistent with a˜ a˜ s˜ − s˜ (58). The benefit of this convention is that in pion and neutron decay both channels π− µ +ν ¯ , → aL˜ π− µ + ν , and n p + e +¯ν , n p + e + ν , respectively, satisfy lepton number conservation. → sR˜ → e˜ → s˜ The minor price to pay is that now both the Dirac mass term and the Majorana terms violate lepton number conservation by two units, so that the unsolvable problem of lepton number violation remains unsolved. Indeed, as we shall discuss below, the Majorana mass terms still allow for neutrinoless double β decay, where a nucleus decays by emitting two electrons (or two positrons) but no (anti)neutrinos. Symmetry 2019, 11 13

3. Applications

3.1. Estimates for the Dirac and Majorana masses

For later use we present the eigenvalues up to third order in µ1,2,3 and µ¯1,2,3, viz.

2 2 2 ± µ2 µ3 µ¯1 m¯ 1µ1µ2µ3 λ1 = m¯ 1 1 + + , ± − 2∆¯ 2 2∆¯ 3 2 − ∆¯ 2∆¯ 3  2 2  2 ± µ3 µ1 µ¯2 m¯ 2µ1µ2µ3 λ2 = m¯ 2 1 + + , (63) ± − 2∆¯ 3 2∆¯ 1 2 − ∆¯ 3∆¯ 1  2 2  2 ± µ1 µ2 µ¯3 m¯ 3µ1µ2µ3 λ3 = m¯ 3 1 + + . ± − 2∆¯ 1 2∆¯ 2 2 − ∆¯ 1∆¯ 2   DuetoEq.(38) the last terms cancel, to make the m = λ± pairwise degenerate. Employing the averages i | i | m¯ 2 = 1 (¯m2 +m ¯ 2 +m ¯ 2) and m2 = 1 (m2 + m2 + m2), the mass-squared differences ∆m2 m2 m2 3 1 2 3 3 1 2 3 ij ≡ i − j become approximately,

2 2 2 2 2 2 2 ¯ 2 2µ1 µ2 µ3 2 ¯ 2 2µ2 µ3 µ1 ∆1 ∆m23 = ∆1 +m ¯ , ∆2 ∆m31 = ∆2 +m ¯ , (64a, b) ≡ ∆¯ 1 − ∆¯ 2 − ∆¯ 3 ≡ ∆¯ 2 − ∆¯ 3 − ∆¯ 1  2 2 2    2 ¯ 2 2µ3 µ1 µ2 ∆3 ∆m12 = ∆3 +m ¯ , (64c) ≡ ∆¯ 3 − ∆¯ 1 − ∆¯ 2   2 ¯ 2 −5 2 provided that µi ∆i . It holds that ∆3 = ∆msol = (7.53 0.18)10 eV [16]. Normal ordering ≪ | | − 2 ± −3 2 m1 < m2 < m3 is connected to ∆1 = ∆matm = (2.44 0.06)10 eV [16] and ∆2 = ∆1 ∆3 > 0, − ±2 − − while inverse ordering m3 < m1 < m2 leads to ∆1 = ∆matm and ∆2 < 0. Cluster lensing puts forward a value m¯ 1.5 1.9 eV for the absolute scale of the neutrino masses[8–11]. ∼ − With εijk the Levi-Civita symbol, there hold the exact relations

1 3 1 3 m¯ 2 =m ¯ 2 + ε ∆¯ , m2 = m2 + ε ∆ . (65) i 3 ijk k i 3 ijk k j,kX=1 j,kX=1 Let us investigate Eq. (64) for normal ordering. The effects of the Majorana masses are anticipated 2 to occur at the level of ∆msol. We fix m¯ 2 and express m¯ 1,3 in d1,3 as

m¯ 2 =m ¯ 2 d ∆m2 , m¯ 2 =m ¯ 2 + ∆m2 d ∆m2 , (66) 1 2 − 3 sol 3 2 atm − 1 sol so that

∆¯ = ∆m2 d ∆m2 , ∆¯ = d ∆m2 . (67) − 1 atm − 1 sol − 3 3 sol With m¯ m¯ we set also ≈ 2 2 2 2 µ3m¯ ∆matm u3 ∆msol ∆msol Θ3 = 2 = 2 , µ3 = d3Θ3 , u3 = d3Θ3 2 . (68) d3∆msol ∆msol d3 m¯ ∆matm Symmetry 2019, 11 14

With d and Θ fixed we can determine µ or, equivalently, u . Imposing ∆m2 ∆m2 ≪ m¯ 2, 1,3 3 1,2 1,2 sol ≪ atm we deduce from (64c) and from the difference of (64a) and (64b) that

2 2d1 5(1 d3) 2 ¯ ¯ u1 u1 = − − +Θ3, µ1 = ∆2∆3 , 12d3 | |m¯ 1 q 2 2d1 + 7(1 d3) 2 ¯ ¯ u2 u2 = − Θ3, µ2 = ∆3∆1 , (69) 12d3 − | |m¯ 2 q 2 2 2 2 2 2 1 d3 2 2 2 ∆msol u = u2 u1 u3 = − 2Θ3 d3Θ3 2 . − − d3 − − ∆matm   These are expressions of order unity and exact to leading order in d , d 1 and Θ . Hence the typical 1 3 − 3 scale is µ ∆m2 ∆m2 /m¯ =4.310−4eV2/m¯ and µ 810−5eV2/m¯ . 1,2 ∼ atm sol 3 ∼ In the eigenvectorsp (42) the coefficients µ¯ µ¯ ∆m2 µ¯ ∆m2 1 2 u u u atm , 3 u u u sol , (70) 4¯m ≈ −4¯m ≈ − 1 2 3 2¯m2 4¯m ≈ 1 2 3 2¯m2 are typically rather small. Hence the mixing matrix will essentially involve elements Θ S ± 1,2,3 2 mµ¯ 1,2 ∆msol u3 Θ1,2 = u1,2 2 =0.18u1,2, Θ3 32 . (71) ∆1,2 ≈ s∆matm ≈ d3 | | with Θ3 introduced in (68). If one of the Θi dominates but is still small, it can be seen as a mixing angle. In particular Θ 0.1 is possible, which is relevant for the ANITA events to be discussed below. 3 ∼

3.2. Neutrino oscillations in vacuum

We consider neutrino oscillations in the plane wave approximation. See Ref. [17] for an excellent discussion of its merits. In the notation (59), (60) an initially pure active state vector reads

6 ∗ i ν (0) = U νmL . (72) | αL i αi| i i=1 X It evolves after time t into 6 ν (t) = U ∗ e− iφi νi , (73) | αL i αi | mLi i=1 X with the Lorentz invariant phase at r ct given by | | ≈ 2 2 2 3 Eit p r mi c pct mi c t φi = − · 1+ 1 . (74) ~ ≈ s p2 − ~ ≈ 2~p   This result can be motivated for a wave packet [17]. The amplitude for transition into active state β is

6 3 ν ν (t) = U U ∗ e− iφi = U D U D∗( + )e i(ηk−ηl)− iφi . (75) h βL| αL i βi αi βk αl AkiAli SkiSli i=1 X i,k,lX=1 where we used that φ2i−1 = φ2i. The transition probability after time t may be expressed as two terms,

P DM = ν ν (t) 2 = P D + P M , (76) να→νβ |h βL| αL i| να→νβ να→νβ Symmetry 2019, 11 15 where the first one

3 P D = δ U DU D∗U D∗U D (1 e iφj − iφi ), (77) να→νβ αβ − βi αi βj αj − i,j=1 X represents the standard “Dirac” result and where the Majorana masses add the “Majorana” expression

3 P M = U D U D∗U D∗U D e i(ηk−ηl−ηm+ηn)(e iφj − iφi 1) να→νβ βk αl βm αn − i,j,k,l,m,nX =1 [( + )( + ) δ δ δ δ ]. (78) × AkiAli SkiSli AmjAnj SmjSnj − ki li mj nj The fact that 3 P M 0 reflects oscillation into sterile states. β=1 να→νβ ≤ While the Majorana phases η cancel in P D as usual, they remain present in P M . This occurs P i να→νβ να→νβ DM in the DMνSM because they are upgraded to physical Dirac phases, see footnote 9. Pνα→νβ has the schematic δ-dependence 1 + cos δ + sin δ + cos2δ + sin 2δ, but the cos2δ, sin 2δ terms are turned into cos δ, sin δ terms by taking η η′ = δ + η . This is equivalent to replacing the U D of (21) by 3 → 3 3

c2c3 c2s3 s2 D iδ iδ iδ iδ U˜ = U1U2U3 diag(1, 1, e )= c s s s c e c c s s s e s c e , (79) × − 1 3 − 1 2 3 1 3 − 1 2 3 1 2  s s c s c e iδ s c c s s e iδ c c e iδ  1 3 − 1 2 3 − 1 3 − 1 2 3 1 2    wherein δ enters only in the schematic form 1+e iδ. Absolute Majorana phases have no physical meaning; indeed, the expression (78) involves them only through their differences. The CP violation effect,

∆P CP = P DM P DM = P DM P DM , (80) να↔νβ να→νβ − ν¯α→ν¯β να→νβ − νβ→να can be read off from the above by switching α β. The terms with sin(2η η η ), cos(2η η η ) ↔ i − j − k i − j − k with j = k and j = k from (78) cancel, leaving the dependence on δ, the η and t of the form 6 i 3 3 3 ∆ t ∆P CP (t)= d sin δ + d sin(δ + η η )+ c sin(η η ) sin k . (81) να↔νβ k ijk i − j ijk i − j 2q 6 i>j=1 Xk=1 h i=Xj=1 X i

Choosing η1 =0 this vanishes only for the trivial values δ, η2, η,3 equal to 0 or π. It confirms that in the 9 DMνSM two of the Majorana phases ηi of the SνM are physical Dirac phases .

3.3. Neutrino oscillations in matter

Relativistic neutrinos have energy E =(q2 + m2)1/2 q + m2/2q. Neutrino oscillations in vacuum i i ≈ i are ruled by the Hamiltonian H0 = E, which reads on the flavor basis

6 m2 (H ) U q + i νi νi U † . (82) 0 αβ ≈ αi 2q | mLih mL| iβ i=1 X
The q term leads to qδαβ and can be omitted, as it plays no role for the eigenfunctions. For propagation in matter one adds the matter potential. The charged and neutral currents induce the scalar potentials 1 V = √2G n , V = √2G n , (83) CC F e NC −2 F n Symmetry 2019, 11 16

involving the electron number density n = n − n + and the neutron number density n , and yielding e e − e n

Ve = VCC + VNC , Vµ = Vτ = VNC , (84)

The potential of the active neutrinos is diagonal on the flavor basis, while the sterile ones do not sense any. This results in the total matter potential on the flavor basis

Vm = diag(Ve,Vµ,Vτ , 0, 0, 0). (85)

From Eq. (25) and its real, symmetric nature it follows that

M DM = U ∗ U † , M DM † = U U T . (86) LR M LR LRM LR Hence the m2, the eigenvalues of 2, arise from i M M DM † M DM = U 2 U † . (87) LR M LR The total matter Hamiltonian therefore reads on the flavor basis 1 H = M DM † M DM + V . (88) m 2q m

Let us set =2qU † V U =( a , s ) with s = diag(0, 0, 0) and Vm LR m LR Vm Vm Vm a =2q U † V a U , V a = diag(V ,V ,V ). (89) Vm L m L m e µ τ The factor D′ in the decomposition (21) for U , also drops out from a since V a is diagonal, hence it L L Vm m can be totally omitted. Eq. (88) can be expressed as

2 + 1 M d 2 + a M dM N † m m (90) Hm = ULR m ULR, m = M V = N Vd d 2 N 2 . H H 2q 2q M M M + M ! First solving the eigenmodes of and then going to the flavor basis allows to evaluate the effects Hm of oscillations on the active neutrinos without having knowledge of the undetermined matrix UR. The eigenfunctions do not alter upon subtracting (¯m2/2q)1 from , after which all elements are small. 6×6 Hm Inside matter the Diracian properties are lost, there are just 6 Majorana states with different masses. While the neutral current potential V can be omitted in the limit M N 0, this is not allowed in NC → general. In matter one has real potentials V , α = e, µ, τ. Due to V the matrix a is complex α CC Vm hermitian. The hermitian has 6 different positive eigenvalues but complex valued eigenmodes. Hm Inside the Sun the neutrino transport is dominated by a mostly electron-neutrino mode[15]; in the 3+3 model this is represented by two nearly degenerate, nearly maximally mixed Majorana modes. The resonance condition in the standard solar model now splits up as a condition for each of them. The so-called solar abundance problem stems from the inconsistency between the standard solar model parameterized by the best description of the photosphere and the one parameterized to optimize agreement with helioseismic data sensitive to interior composition[18]. The biggest deviations in the solar composition are of relative order 1% and occur at 0.7R . Our modified resonance conditions ∼ ⊙ offer hope for an improved description of the data. Symmetry 2019, 11 17

3.4. Pion decay

One of the simplest elementary particle reactions is

π− W − µ +¯ν . (91) → → µR It describes a negatively charged π− particle, π− =(du¯), consisting of a down quark (charge e/3) and − an anti-up quark (charge 2e/3), decaying into a W − boson (charge e), which in its turn decays into − − a muon (charge e) and an muon-antineutrino (charge 0). Related reactions are π− e +¯ν , π+ − → e → + + + µ † ℓ µ ν ℓ µ ν νc µ + νµ and π e + νe. In the DMνSM the current JCC =2 Lγ U mL =2 Lγ (A aL˜ + S sR˜ ) → i replaces Eq. (91) by decays with any of the 6 mass eigenstates (νm)R emitted. They can be grouped as

π− W − µ +¯νi , (i =1, 2, 3), π− W − µ + νi−3 (i =4, 5, 6). (92) → → aR˜ → → sR˜ i ic The νaL˜ and the charge conjugated νsR˜ fields have identical chiral structure (up to a phase factor, see Ref. [12], eqs (2.139) vs. (2.356)), differing only by their creation and annihilation operators. Hence all decay channels involve the standard chiral factors, and a new factor, the sum over final neutrino states, 6 U 2 = 3 (U DDM ) 2 + 3 (U DDM ) 2. It equals i=1 | µi| i=1 | A µi| i=1 | S µi| P P UU † = U DDMP( T + T )DM†U D† = δ , (93) αβ AA SS αβ αβ for α = β = µ. (In this equality
we employed Eq. (55)). So charged
pions decay in the DMνSM at the same rate as in the SM. Neutral pion decay does not involve neutrinos, so it is also not modified.

3.5. Neutron decay

A neutron n = (ddu) consists of 2 down quarks and one up quark, and a proton p = (duu) of 1 down and 2 up quarks. Neutron decay n p + e +ν ¯ involves a transition from a down quark to an → e up quark producing a virtual W − boson, which decays into an electron and an electron antineutrino. As coded in the charged current J µ † of (49), it occurs in the DMνSM in two channels, n p + e +¯ν and CC → a˜ n p + e + ν . Both decay channels involve the standard chiral factors, and a new factor, the sum over → s˜ final neutrino states (U DDM ) 2 + (U DDM ) 2. This is the α = β = e element of Eq. i | A ei| i | S ei| (93), so it is equal to unity. Hence the neutron lifetime in the DMνSM stems with the one in the SνM. P P The main decay channel is n p + e +ν ¯ . With our convention L = 1, also the channel → a˜ νs˜ − n p + e + ν conserves the lepton number. The latter occurs at a slower rate due to the small term → s˜ T in (93). We could not convince ourselves that it would be ineffective in beam experiments and SS hence be capable to explain the neutron decay anomaly between beam and bottle measurements[19].

3.6. Muon decay

With the neutron decay going into two channels, muon decay µ e +¯ν + ν goes into four, → eR µL µ e +¯ν + ν , µ e +¯ν +¯ν , µ e + ν + ν , µ e + ν +¯ν , (94) → aR˜ aL˜ → aR˜ sL˜ → sR˜ aL˜ → sR˜ sL˜ with rates of leading schematic order 1, T , T and ( T )2, respectively, adding up to the SM result. SS SS SS Symmetry 2019, 11 18

3.7. Neutrinoless double β-decay

In a simultaneous double neutron decay (double β-decay) the emission of two electrons involves i 2 ic 2 i ic the schematic neutrino terms νaL˜ + νsR˜ + νaL˜ νsR˜ , corresponding to the emission of two mostly active antineutrinos, two mostly sterile neutrinos, or one of each. With L = 1 and L = 1, all three νa˜ νs˜ − channels conserve the lepton number. i 2 In the standard neutrino model also neutrinoless double β-decay is possible. Then only the νaL˜ term occurs, subject to the Majorana condition νi c = νi ; with δ and 0, it yields an aL˜ aL˜ Aij → ij Sij → 3 D 2 2 iηi amplitude proportional to mee = i=1 Uei e mi for small mi. The GERDA search puts a bound m 0.15 0.33 eV [20]. Does this rule out the DMνSM for m 2 eV? Not, as we show now. ee P | | ≤ − i 2 ic 2 ∼ In our situation with Diracian neutrinos neither νaL˜ nor νsR˜ contributes, but neutrinoless double-β c decay does arise from the νaL˜ νsR˜ terms. All spinor terms are again as in the SνM. The only change occurs in the effective mass, which now reads m = [AmST + SmAT ] = [U DDM ( m T + m T )DM U D T ] , m = diag(m , m , m ). (95) ee ee A S S A ee 1 2 3 It involves cancellations, since is nearly asymmetric while and m/m are close to the identity matrix. S A But the cancellations are maximal. From the definitions (51) we can go back to the six eigenvectors e(i) of Eqs. (40) and (41). Recalling that λ = m and λ = m , (i =1, 2, 3), it follows that 2i i 2i−1 − i 6 ( m T + m T ) = λ ei ei = . (96) A S S A jk i j k Mjk i=1 X From (30) it is seen that jk =0 for j, k =1, 2, 3, because we neglected the left handed Majorana mass M M DM M matrix ML . In general mee = Mee =(ML )ee [13]. Hence mee =0 for this leading order diagram. Nevertheless, neutrinoless double β–decay, involving lepton number violation ∆L = 2, is not 3 2 forbidden in the DMνSM. It occurs in the mi /q correction to mi in (95) stemming from the internal propagator m /(q2 m2) = m /q2 + m3/q4 + . But the suppression factor m2/q2 makes its i − i i i ··· i measurement impractical for realistic q MeV GeV. Loop effects may fare better, but are also ∼ − tiny. If a finite mee is established, it points at new high-energy physics. In conclusion, the non-detection of neutrinoless double β-decay is compatible with the DMνSM.

3.8. Small twin-oscillation

If the degeneracy of the solar twin modes is slightly lifted by non-cancellation of the last two terms in each line of Eq. (63), one gets

3 2 + 2 − 2 2¯m µ1µ2µ3 ∆¯m1 (λ1 ) (λ1 ) m¯ µ¯1 . (97) ≡ − ≈ − ∆¯ 2∆¯ 3 From the standard solar model we know that oscillations should not occur underway to Earth[2], so that ∆¯m2 . 10−12 eV2. For the supernova SN1987A at distance of 51.4 kpc the absence of twin-oscillations | 1| even implies that ∆¯m2 . 10−22 eV2; the alternative is that twin-oscillation did take place, and only half | 1| of the emitted neutrinos arriving here on Earth were active and could be detected. The implied doubling of power emitted in neutrinos then requires an adjustment of the SN1987A explosion model. Symmetry 2019, 11 19

2 2 The similarly defined ∆¯m2 and ∆¯m3 may be larger. Either of them may describe the MiniBooNE anomaly [5] (disputed by MINOS [6] and still debated [7]) with ∆m2 0.04 eV2. But a more elegant ≈ approach hereto is to keep the 3 Diracian neutrinos and add a fourth sterile one.

3.9. Sterile neutrino creation in the early Universe

The creation of sterile neutrinos in cosmology is an important process based on loss of coherence in oscillation processes. It is well studied, see e.g. [14], and is important when sterile neutrinos are to make up half of the cluster dark matter[8–11]. + The charged currents (61a,b) allow the creation of sterile neutrinos out of active ones via e + νe + + − → W e +¯νs, and creation of sterile antineutrinos out of active ones in the process e +¯νe W → ± → → e + νs, that is, by four-Fermi processes with virtual W exchange. The first term in Eq. (61c) describes interaction of active neutrinos with Z, the second of sterile ones, and the last two the exchange of active versus sterile neutrinos, and vice versa. In particular the creation of sterile neutrinos out of active ones is possible in two channels via the four-Fermi process ν +¯ν ν +¯ν with the exchange of a virtual Z α α → s s boson. All these processes conserve lepton number. As is seen from the sterile component in the flavor eigenstate (56) or from the charged and neutral currents (61), to achieve the sterile neutrino creation a finite matrix is needed. Hence it does not occur in the standard Dirac limit where both Majorana mass MS M matrices ML and MR vanish.

3.10. Muon g 2 anomaly −

The gyromagnetic factor of the muon is gµ = 2(1+ aµ). Dirac theory yields gµ = 2 and aµ is the anomaly due to quantum effects. The leading term is Schwinger’s famous result, α a = + =0.00116 + . (98) µ 2π ··· ··· a is known up to its 9th digit, but there is a 3.5σ discrepancy between measurement and prediction, µ ∼ aexp aSM = 288(63)(43) 10−11 [16], where the first error is statistical and the second systematic. µ − µ Our interest lies in the contribution of neutrinos, which occurs in a simple triangle diagram with virtual W bosons. Ref. [21] presents the result for an arbitrary number of sterile neutrinos. For neutrino masses well below MW it reads 3+Ns G 5m2 aν = (aν )SM U U ∗ =(aν )SM(UU †) =(aν )SM = F µ = 38910−11. (99) µ µ µi µi µ µµ µ √ 12π2 i=1 2 X † The unitarity relation (60), viz. (UU )αβ = δαβ, is valid even beyond our Ns = 3 case[12]. So the DMνSM reproduces the one-loop outcome of the SM, as well as the dominant two-loop electroweak contributions of Ref. [22].

3.11. ANITA detection of UHE cosmic neutrino events

Scattering of ultra high energy (UHE) cosmic rays on cosmic microwave background photons puts the GZK limit on their maximal energy[23,24] and acts as a source for EeV (1018 eV) (anti)neutrinos via the creation and decay of charged pions[25], as considered in section 3.4. Symmetry 2019, 11 20

The Antarctic Impulsive Transient Antenna (ANITA) is a balloon experiment at the South Pole that detects the radio pulses emitted when UHE cosmic neutrinos interact with the Antarctic ice sheet. In a set of & 30 cosmic ray events, ANITA has discovered an upward going event with energy E 0.6 ∼ EeV [26] and one with 0.56 EeV [27]. Both are consistent with the cascade caused by a τ lepton ∼ created beneath the ice surface10. But the SM connects a relatively large neutrino-nucleon cross section σ G2 m E to an UHE neutrino [29], so that the probability for it to traverse a large path L through ∼ F N the Earth to reach Antartica is P exp( n σL) is small, where n is the effective density of nuclei. The T ∼ − numbers are P 410−6 and 210−8, respectively [26,27]. In Ref. [30] it is pointed out that a sterile T ∼ neutrino with a smaller cross section, viz. σ σ sin2 Θ, with Θ the mixing angle with respect to active → neutrinos, may be involved. To explain the events and relate them to the detections at IceCube, AUGER and Super-Kamiokande, these authors fix Θ at 0.1 [30]. For typical models a sterile neutrino with such a large mixing angle should have been discovered already. In the DMνSM the situation is different, however. It contains the reaction ν τ + W +, s˜ → where the W + is quickly lost locally but the τ escapes and decays while creating a shower. In the approximation where one mixing angle dominates, an emitted electron neutrino will have components of strength sin2 Θ on mostly sterile states. Inside the Earth they are scattered less, and, given that they enter the other side of the Earth, can be measured in the τ-flavor mode with modified probability P˜ sin2 Θ exp( n σL sin2 Θ) and modified flux Φ /Φ sin4 Θ exp( n σL sin2 Θ). The T ∼ − sterile active ∼ − estimates of section 3.1 show that the value Θ = 0.1 is reasonable for the component Θ3 of the mixing matrix , see in particular the expression for Θ in Eq. (71). Hence the DMνSM supports S 3 the sterile-neutrino interpretation of ANITA events. For determining the DMνSM parameters, it seems worth to include UHE data.

4. Summary and outlook

Since the maximal mixing of pseudo Dirac neutrinos runs into observational problems, neutrino mass is often supposed to stem from a high-energy sector beyond the standard model (BSM), for instance by the seesaw mechanism[12,14]. We show that the mixing effects can be suppressed in the Dirac-Majorana neutrino standard model (DMνSM), the minimal extension of the SM with 3 sterile neutrinos (3+3 model) with both a Dirac and a right handed Majorana mass matrix. Indeed, to have the 6 physical masses condense in 3 degenerate pairs poses 3 conditions, which leaves 3 Dirac and 3 Majorana masses free. In this Diracian limit the neutrino mass eigenstates act as Dirac particles like the other fermions in the SM. There is no change in the pion, neutron and muon decay, nor on the muon g 2 problem. − Compared to the general case, less mixing occurs since members of the same Dirac pair undergo no mutual oscillation. For small Majorana masses the left handed mass eigenstate is still mostly active, and the right handed one still mostly sterile. A flavor eigenstate has a component on mass eigenstates with a mostly sterile character. With mixing angles up to 0.2 – 0.3, this allows to explain the ANITA ultra high energy events. Hence for determining the DMνSM parameters, it is natural to include UHE data.

10 See [28] for a possible explanation due to sub-surface reflection in the ice for a downward event. Symmetry 2019, 11 21

In the Diracian limit the model keeps some of its Majorana properties. Neutrino oscillations in matter involve the usual 6 nondegenerate Majorana states. Lepton number is not conserved. Neutrinoless double-β decay remains possible, be it at an impractically small rate. Sterile neutrino generation in the early cosmos is possible at temperatures in the few MeV range. It is interesting to investigate whether processes involving the neutral current can further test the model. They are relevant e.g. in nonresonant sterile neutrino production in the early universe. By connecting lepton number L=1 to (mostly) active neutrinos but L = 1 to (mostly) sterile − neutrinos, neutron decay and double β-decay conserve the lepton number, while lepton number violation is restricted to feeble neutrinoless double-β decay. Should that be observed, it would invalidate our assumption of negligible left handed Majorana mass matrix, and prove the presence of BSM physics in the high energy sector. The SM has 19 parameters while 6 neutrino parameters are established and 2 anticipated11. The DMνSM adds 3 further Majorana masses. In the limit where they vanish, the sterile partners decouple and the standard neutrino model emerges. The extra Majorana masses and Dirac role of the “Majorana” phases may alleviate some of the tensions in solar, reactor and other neutrino problems. From a philosophical point of view, we do not consider the values of the Dirac and Majorana masses and phases as problematic properties in urgent need of an explanation, but rather as further mysteries of the standard model.

Acknowledgements The author is grateful for inspiring lectures by and discussion with his teachers Martinus ‘Tini’ Veltman and Gerardus ‘Gerard’ ’t Hooft.

Appendix A: gamma and charge conjugation matrices The four 4 4 anticommuting γ-matrices were introduced by Dirac. They play a role in the description × of, e.g., the chiral left handed and right handed electron and positron. In the convention of Giunti and Kim[12] the Lorentz indices µ = 0, 1, 2, 3 label the coordinates x µ =(ct, x, y, z). The anticommutation relations read for any representation of the γ matrices,

γ µ,γν =2ηµν , ηµν = diag(1, 1, 1, 1) = η . (100) { } − − − µν The γ 5 matrix has the properties

γ 5 = iγ 0γ1γ 2γ 3, γ µ,γ 5 =0, (γ 5)2 =1. (101) { } Left and right handed chiral projectors are, respectively, 1 1 P = (1 γ 5), P = (1 + γ 5). (102) L 2 − R 2

11 The SM in Eq. (110) has 3 gauge coupling constants; 2 Higgs self couplings; 6 quark masses; 3 charged lepton masses; 3 strong mixing angles and a strong Dirac phase. Parameter 19 is the strong CP angle. The established neutrino parameters are 2 mass-squared differences, 3 weak mixing angles and, to some extent, the weak Dirac phase[16]. The 2 weak Majorana phases can in the SνM only be measured via neutrinoless double β decay, but in the DMνSM in many ways. Symmetry 2019, 11 22

Chiral left handed fields are νL = PLν and chiral right handed ones νR = PRν. The projections are 2 2 orthogonal, viz. PRPL = PLPR =0, while PL = PL and PR = PR. Hermitian conjugation brings γ 0† = γ 0, γ i† = γ i, summarized as − µ† 0 µ 0 ν 5† 5 γ = γ γ γ = γµ = ηµν γ , γ = γ . (103)

The charge conjugation matrix has the properties C † = −1, T = . (104) C C C −C It is defined up to an overall phase factor, which plays no physical role, and connected to transpositions,

γ µT = †γ µ , γ 5T = †γ 5 . (105) −C C C C The Pauli matrices are

1 0 0 1 0 i 1 0 σ0 = , σ1 = , σ2 = − , σ3 = . (106) 0 1 1 0 i 0 0 1 ! ! ! − ! The charge conjugate of any spinor ν has the properties

ν c = ν¯T = γ 0 ν†T , ν c† = νT †γ 0, ν = γ 0 ν cT †, C − C − C − C νT = ν c†γ 0 , ν† = ν cT †γ 0, νT † = †γ 0ν c. (107) − C − C −C For four-component spinors ν and ν (with i = j allowed) the contraction νT †ν = 4 ν † ν is i j i C j k,l=1 ikCkl jl a nonvanishing scalar, since the fermion fields ν anticommute and is antisymmetric. The relation i,j C P (νT †ν )† = ν† νT † = ν cT †ν c, (108) i C j j C i j C i assures that the Majorana mass Lagrangian (3) is hermitean. In the chiral representation the γ and matrices have the 2 2 blocks C × 0 σ0 0 σi iσ2 0 γ0 = − , γi = , = − , σ0 0 σ 0 C 0 iσ2 − ! − i ! ! 0 0 5 σ 0 0 0 σ 0 γ = , PL = , PR = . (109) 0 σ0 0 σ0 0 0 − ! ! ! Appendix B: The standard model with sterile neutrinos For completeness we present the Lagrangian of the standard model with Dirac-Majorana neutrinos in a compact form (leaving out the strong CP violating term), 1 1 1 = BµνB AµνA GµνG + D Φ†D µΦ µ2Φ†Φ λ(Φ†Φ)2 L −4 µν − 4 µν − 4 µν µ − − U U D D ℓ ℓ + QL iD/QL + qR iD/qR + qR iD/qR + LL iD/LL + R iD/ R Q ΦY DqD qDY D†Φ†Q Q Φ˜Y U qU qU Y U†Φ˜ †Q L ΦY ℓℓ ℓ Y ℓ†Φ†L (110) − L R − R L − L R − R L − L R − R L 1 1 + ν iD/ν L Φ˜Y νν ν Y ν†Φ˜ †L + νT †M M† ν + ν c T †M M ν c . (111) R R − L R − R L 2 R C R R 2 R C R R Symmetry 2019, 11 23

Up to (110) included, this represents the SM itself: the first line contains the U(1)Y , SU(2)L and SU(3)C gauge fields, respectively, and the Higgs kinetic and potential energy. The second line contains the kinetic terms for three families of quarks, charged leptons and active neutrinos; the third line lists the quark couplings to the Higgs field with 3 3 Yukawa matrices Y U,D and the charged lepton couplings × to the Higgs field with Yukawa matrix Y ℓ. Eq. (111) exhibits the kinetic term for 3 sterile neutrinos and the Yukawa couplings between the active leptons, the Higgs field and the sterile neutrinos with a 3 3 × Yukawa matrix Y ν. Eq. (111) also contains the right handed Majorana mass terms of Eq. (3). M The Diracian limit of the main text refers a special form for MR . The quark doublets in Eq. (110) contain the left handed up, down, charm, strange, top and bottom quarks,

Q1L uL cL tL QL = Q , Q1L = , Q2L = , Q3L = . (112)  2L d s b Q L! L! L!  3L   The lepton doublets contain the left handed electron, muon and tau (tau lepton, tauon), and their active neutrinos: the left handed electron, mu and tau neutrino,

L1L νeL νµL ντL LL = L , L1L = , L2L = , L3L = . (113)  2L e µ τ L L ! L ! L !  3L   The right handed quark and lepton singlets are grouped as

uR dR eR ν1R qU qD ℓ ν R = cR  , R = sR , R = µR , R = ν2R . (114) t b τ ν  R   R   R   3R         µ The covariant derivatives Dµ and D/ = γ Dµ in Eq. (110) contain currents from gauge fields, except for D/νR = ∂/νR, since νR is gauge invariant. Indeed, the weak currents JCC from Eq. (49) and JNC from Eq. (50) arise as parts of LL iD/LL. The strong currents from QL iD/QL do not involve the neutrino sector. In the unitary gauge the normal and conjugated Higgs doublets read, respectively,

1 0 †T 1 v + H Φ= , Φ=˜ iσ2 Φ = , (115) √2 v + H! √2 0 ! where v = µ2/λ is the vacuum expectation value and H the dynamical Higgs field. − After spontaneous symmetry breaking the 3 3 Dirac mass matrices for the Down = d,s,b and p × { } Up = u,c,t quarks are M D = vY D/√2 and M U = vY U /√2; for the charged leptons M ℓ = vY ℓ/√2 { } q q and the Dirac mass matrix for the active neutrinos is M ν = vY ν/√2 (it is denoted as M D in the main D ℓ text). From unitary transformations of the fields it follows that the matrices Mq and M can be taken U diagonal with the respective particle masses as entries. Next, the diagonalization of Mq is performed with the CKM mixing matrix and of M D = M ν with the PMNS mixing matrix of Eq. (21). Symmetry 2019, 11 24

References

1. Nobel-Prize-Committee. https://www. nobelprize.org/prizes/physics/2015/summary/, 2015. 2. Vinyoles, N.; Serenelli, A.M.; Villante, F.L.; others. A New Generation of Standard Solar Models. The Astrophysical Journal 2017, 835, 202. 3. Mention, G.; Fechner, M.; Lasserre, T.; others. Reactor antineutrino anomaly. Phys. Rev. D 2011, 83, 073006. 4. Dentler, M.; Hernández-Cabezudo, Á.; Kopp, J.; others. Sterile neutrinos or flux uncertainties? – Status of the reactor anti-neutrino anomaly. Journal of High Energy Physics 2017, 2017, 99. 5. Aguilar-Arevalo, A.; Brown, B.; Bugel, L.; others. Significant excess of electronlike events in the MiniBooNE short-baseline neutrino experiment. Physical Review Letters 2018, 121, 221801. 6. Adamson, P.; Anghel, I.; Aurisano, A.; others. Search for sterile neutrinos in MINOS and MINOS+ using a two-detector fit. Physical Review Letters 2019, 122, 091803. 7. Böser, S.; Buck, C.; Giunti, C. et al. Status of Light Sterile Neutrino Searches. arXiv preprint arXiv:1906.01739 2019. 8. Nieuwenhuizen, T.M. Do non-relativistic neutrinos constitute the dark matter? EPL (Europhysics Letters) 2009, 86, 59001. 9. Nieuwenhuizen, T.M.; Morandi, A. Are observations of the galaxy cluster A1689 consistent with a neutrino dark matter scenario? Monthly Notices of the Royal Astronomical Society 2013, 434, 2679–2683. 10. Nieuwenhuizen, T.M. Dirac neutrino mass from a neutrino dark matter model for the galaxy cluster Abell 1689. J. Phys.: Conf. Ser. 2016, 701, 012022. 11. Nieuwenhuizen, T.M. Subjecting dark matter candidates to the cluster test. arXiv preprint arXiv:1710.01375 2017. 12. Giunti, C.; Kim, C.W. Fundamentals of neutrino physics and astrophysics; Oxford University Press, 2007. 13. Gonzalez-Garcia, M.C.; Maltoni, M. Phenomenology with massive neutrinos. Physics Reports 2008, 460, 1–129. 14. Lesgourgues, J.; Mangano, G.; Miele, G.; Pastor, S. Neutrino Cosmology; Cambridge University Press, 2013. 15. Smirnov, A.Y. Solar neutrinos: Oscillations or No-oscillations? arXiv preprint arXiv:1609.02386 2016. 16. Tanabashi, M.; Hagiwara, K.; Hikasa, K.; others. Review of particle physics. Physical Review D 2018, 98, 030001. 17. Akhmedov, E.K.; Smirnov, A.Y. Paradoxes of neutrino oscillations. Physics of Atomic Nuclei 2009, 72, 1363–1381. 18. Basu, S.; Chaplin, W.J.; Elsworth, Y.; New, R.; Serenelli, A.M. Fresh insights on the structure of the solar core. The Astrophysical Journal 2009, 699, 1403. 19. Wietfeldt, F.E.; Greene, G.L. Colloquium: The neutron lifetime. Reviews of Modern Physics 2011, 83, 1173. Symmetry 2019, 11 25

20. GERDA-Collaboration. Background-free search for neutrinoless double-β decay of 76Ge with GERDA. Nature 2017, 544, 47–52. 21. Abdallah, W.; Awad, A.; Khalil, S.; Okada, H. Muon anomalous magnetic moment and µ eγ → in B-L model with inverse seesaw. The European Physical Journal C 2012, 72, 2108. 22. Kukhto, T.; Kuraev, E.; Schiller, A.; Silagadze, Z. The dominant two-loop electroweak contributions to the anomalous magnetic moment of the muon. Nuclear Physics B 1992, 371, 567–596. 23. Greisen, K. End to the cosmic-ray spectrum? Physical Review Letters 1966, 16, 748. 24. Zatsepin, G.T.; Kuzmin, V.A. Upper limit of the spectrum of cosmic rays. JETP Lett. 1966, 4, 78–80. [Pisma Zh. Eksp. Teor. Fiz.4,114(1966)]. 25. Beresinsky, V.; Zatsepin, G.T. Cosmic rays at ultra high energies (neutrino?). Physics Letters B 1969, 28, 423–424. ISSN 0370-2693. 26. Gorham, P.; Nam, J.; Romero-Wolf, A.; others. Characteristics of four upward-pointing cosmic-ray-like events observed with ANITA. Physical Review Letters 2016, 117, 071101. 27. Gorham, P.; Rotter, B.; Allison, P.; others. Observation of an Unusual Upward-going Cosmic-ray-like Event in the Third Flight of ANITA. Physical Review Letters 2018, 121, 161102. 28. Shoemaker, I.M.; Kusenko, A.; Munneke, P.K.; others. Reflections On the Anomalous ANITA Events: The Antarctic Subsurface as a Possible Explanation. arXiv preprint arXiv:1905.02846 2019. 29. Connolly, A.; Thorne, R.S.; Waters, D. Calculation of high energy neutrino-nucleon cross sections and uncertainties using the Martin-Stirling-Thorne-Watt parton distribution functions and implications for future experiments. Physical Review D 2011, 83, 113009. 30. Cherry, J.F.; Shoemaker, I. A sterile neutrino origin for the upward directed cosmic ray shower detected by ANITA. arXiv preprint arXiv:1802.01611 2018. c 2019 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).