Active-Sterile Neutrino Oscillations and Leptogenesis

Home , Leptogenesis

Journal of Modern Physics, 2021, 12, 1248-1266 https://www.scirp.org/journal/jmp ISSN Online: 2153-120X ISSN Print: 2153-1196

Bruce Hoeneisen

Universidad San Francisco de Quito, Quito, Ecuador

How to cite this paper: Hoeneisen, B. Abstract (2021) Active-Sterile Neutrino Oscillations and Leptogenesis. Journal of Modern Phys- We study coherent active-sterile neutrino oscillations as a possible source of ics, 12, 1248-1266. leptogenesis. To this end, we add 3 gauge invariant Weyl_R neutrinos to the https://doi.org/10.4236/jmp.2021.129077 Standard Model with both Dirac and Majorana type mass terms. We find that

the measured active neutrino masses and mixings, and successful baryogene- Received: May 18, 2021 Accepted: July 9, 2021 sis via leptogenesis, may be achieved with fine-tuning, if at least one of the Published: July 12, 2021 sterile neutrinos has a mass in the approximate range 0.14 to 1.1 GeV.

Copyright © 2021 by author(s) and Keywords Scientific Research Publishing Inc. This work is licensed under the Creative Leptogenesis, Majorana Neutrino, Matter-Antimatter Asymmetry Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access 1. Introduction and Overview We present a study of coherent active-sterile neutrino oscillations as a possible source of baryogenesis via leptogenesis. Consider a reaction of the form ±± eW→→ν i eW that produces a lepton asymmetry that is partially con- verted to a baryon asymmetry before the sphaleron freeze-out temperature

Tsph =131.7 ± 2.3 GeV [1]. The present baryon-to-photon ratio of the universe − is measured to be η =±×(6.12 0.04) 10 10 [2]. Let us consider a bench-mark

scenario with all numbers calculated at a reference temperature Tsph . We define

δ ≡−nn−+ nn −+ + δ δ the electron asymmetry e ( ee) ( ee) , and similarly for µ , τ

δ n − T and B . e is the number density of electrons. The asymmetry at sph from neutrino oscillations required to obtain η is

δδδle≡++=−µτ δ(37 12) δB [3]. At Tsph the age of =−(37 12)η ( 2 3)⋅( 385 × 22( 43 × 8)) ≈− 3.1 × 10−8 −11 the universe is tu =1.4 × 10 s , and the time between collisions of active neu- ++ trinos in the reaction ννeeee→ is −22 t=1σ nc− ≈× 7 10 s ≈ 1 0.001 GeV σ c ( e ) ( ) , where is the cross-section. We

DOI: 10.4236/jmp.2021.129077 Jul. 12, 2021 1248 Journal of Modern Physics

B. Hoeneisen

note that at Tsph neutrinos are of short wavelength relative to tc , i.e.

tTc  2π sph . Observed neutrino oscillations require that at least two neutrino eigenstates have mass. To this end, we add at least n′ = 2 gauge singlet Weyl_R neutrinos

ν R to the Standard Model. To obtain lepton number violation, we assume the neutrinos are of the Majorana type, i.e. we add both Dirac and Majorana mass terms to the Lagrangian [4]. −+  ± Let us consider the reaction eW→→ν i eW , with neutrino mass eigens-

tates ν i oscillating coherently during time tc . The condition for coherent os-

cillations is that ν i has mass  6 GeV . (The physics described in this overview will be developed in the following Sections.) The cross-section for the lepton number violating reaction is reduced relative to the lepton conserving reaction 2 by a factor mmij(2 E ) due to polarization miss-match, where mi is the neutrino eigenstate mass, and E is the neutrino energy in the laboratory frame. One mechanism to obtain CP violation is to have two interfering amplitudes with different “strong” phases and different “weak” phases [5]. A “strong” phase (the name is borrowed from B-physics) is a phase that does not change sign under CP-conjugation. A “weak” phase changes sign under CP-conjugation. Here, the “weak” phases are the CP-violating phases in the weak mixing matrix U. The “strong” phases are the propagation phases of the interfering ultra-relativistic 22 neutrinos, 22Xij=( m i − mL j ) ( E) , with energy ET≈ 2.8 sph , and Lt= c . To obtain a sizable CP violation asymmetry, the relative propagation phase differ-

ence 2X ij between two neutrinos in time tc should be of order π 2 or less. 22 This requires two neutrinos to satisfy mmij− 1.1 GeV . There are cosmological constraints, mainly from Big Bang Nucleosynthesis

(BBN), that require the mass of sterile neutrinos to be ms  0.14 GeV . Thus, the interesting mass range for sterile neutrinos contributing to leptogenesis is approximately 0.14 GeV to 1.1 GeV. From the following studies, we conclude that nature may have added, to the Standard Model, two or more gauge singlet Weyl_R Majorana neutrinos, with fine tuned parameters, as the source of neutrino masses and mixing, and successful baryogenesis via leptogenesis. This scenario is not new, yet is not men- tioned in several leading leptogenesis reviews. Here, we emphasize analytic solutions, and an understanding of several delicate issues related to Majorana neutrinos, lepton number violation, CP-violation, polarization miss-match, and coherence. In the following Sections, we develop, step-by-step, the physics behind the preceding comments.

2. Dirac Neutrinos

In the following sections we consider a neutrino experiment with a source at the origin of coordinates, and a detector at a distance zL= . We assume

Lp 2π z , so the neutrinos are almost on mass-shell. pz is the neutrino momentum. At first let us consider a single neutrino flavor, and the reaction

DOI: 10.4236/jmp.2021.129077 1249 Journal of Modern Physics

B. Hoeneisen

−+ −+ eW→→ν e eW .

Before electroweak symmetry breaking (EWSB) at TEWSB ≈±159 1 GeV [1],

the neutrino field ν L is massless, carries the 2-dimensional “Weyl_L” representation of the proper Lorentz group, and satisfies the wave equation

iσνµµ∂=L 0, (1)

where σ 0≡ 1 22× , σσ00≡ , σσkk≡− , and σ k are the Pauli matrices [4]. Summation over repeated indices is understood. k = 1, 2, 3 , and µ = 0,1,2,3 .

Multiplying on the left by −∂iσνν, obtains the Klein-Gordon wave equation of a massless field:

µν µ ην∂∂≡∂∂=νµLL µ ν0, (2)

µν where η =diag( 1, −−− 1, 1, 1) is the metric.

After EWSB the Higgs boson acquires a vacuum expectation value vh [4]. N The field ν L forward scatters on vh with amplitude Yvh 2 becoming a N ν R (Y is a Yukawa coupling [4]), that forward scatters on vh with ampli- N* tude Yvh 2 becoming a ν L , etc. The field ν R transforms as “Weyl_R”

[4]. These scatterings are forward because vh does not depend on the space-time coordinates xµ . These scatterings are described by the Dirac equation,

iσνµµ∂=RL mi ν,, σνµµ ∂= LR m ν (3)

N with mYv= h 2 . In this way the field ν R is created (arguably) after EWSB,

on a time scale 1/m, and the fields ν L and ν R couple together forming a 4-dimensional field ψ that carries the reducible Dirac = Weyl_L ⊕ Weyl_R representation of the proper Lorentz group. The solution of (3) proportional to

exp(−+iEt ipz z) , in a Weyl basis, is [4] Ep− ξ z 1

ν Lu Ep+ z ξ2 ψ uz≡ =exp(−+iEt ip z), (4) ν Ru Ep+ z ξ1  Ep− z ξ2  corresponding to a particle of mass m, and momentum p= pezz with 22 pz =+− Em. This is the “stepping stone” mechanism of mass generation [6].

Alternatively, consider (3): the ν L creates ν R on a time scale 1/m, which in  turn creates ν L , etc. Solutions for other p can be obtained with Lorentz

transformations. ξ1 and ξ2 are complex numbers that define the polarization of the neutrino (to be discussed in Section 5).

The solution of (3) proportional to exp(iEt− ipz z) is Ep− η z 1

ν Lv Ep+ zη2 ψ vz≡=exp(iEt− ip z) . (5) ν Rv −+Epzη1  −−Epzη2

The charge conjugate of ψ v is [4]

DOI: 10.4236/jmp.2021.129077 1250 Journal of Modern Physics

B. Hoeneisen

Ep− η* z 2 c * * c (ν ) −iσν −+Epη 2* Rv 2 Rv z 1 (ψvv) =−=i γψ =  = exp(−+iEt ipz z). (6) c iσν* + η* (ν Lv ) 2 Lv Epz 2 * −−Epzη1

* * Note that −iσν2 Rv transforms as Weyl_L, while iσν2 Lv transforms as Weyl_R † † T † * [4]. ννLR, ννRL, νRR σν2 , and νRR σν2 are scalars with respect to the proper µ Lorentz group. The Dirac Equations (3) can be summarized as (imγψ∂−µ ) =0 . We work in the Weyl basis with γ matrices

0500σ00k σσk − 0  γγ= ,=  ,. γ=  (7) σσ000− k 00  σ 

5 5 The Weyl_L and Weyl_R projectors are γγL ≡−(12) , and γγR ≡+(12) . ± For example, the Weyl_L component of ψ is γψL . Note that W and Z only

“see” the Weyl_L fields γψLu or ψγvR. Neutrinos may, or may not, have a conserved U (1) charge q such as lepton number. In quantum field theory, the fields are interpreted as follows:

 ψ u creates a particle with charge +q, and spin angular momentum compo- 1 1 nent s = + with amplitude Ep− ξ , and s = − with amplitude z 2 z 1 z 2 Ep+ z ξ2 ; † 0  ψuu≡ ψγ annihilates this particle; † 0 1  ψvv≡ ψγ creates an antiparticle with charge -q, and spin sz = + with 1 2 amplitude Ep+ η* , and s = − with amplitude Ep− η* ; z 2 z 2 z 1  ψ v annihilates this antiparticle. This interpretation is needed to avoid unstable particles with negative energy. 1 These particles and antiparticles have mass m, spin , positive energy E, and 22 2 momentum pezz=+− E mez. Note that antiparticles have the opposite charge of the corresponding particle.

Let us now consider two neutrino flavors, ν e and ν µ . The field ν Le may N forward scatter on vh with amplitude Yvee h 2 becoming a ν Re , which may E* forward scatter on vh with amplitude Yvehµ 2 becoming a ν Lµ , etc. As a result, two mass eigenstates acquire masses:

ψ11=cos θψe + sin θψ µ , with massm , (8)

ψ22=−+sin θψe cos θψ µ , with massm . (9)

For simplicity, we have suppressed the sub-indices u for neutrinos, or v for an- −+ ti-neutrinos. For example, the interaction eW →ν Le producing a weak 2212 (12) ( ) state has ψ (00) = , ν (00) = , and νν(00) + ( ) is normalized µ Re Le Le to 1. An observation at distance L obtains eW−+ with probability 22 (12) ( ) −+ PLee∝+ψψ Le ( ) Le ( L) , or µ W with probability

22 (12) ( ) PLeLµµ∝+ψψ( ) L µ( L) , where

222 PPeµµ=−=1ee 4cosθθ sin sin( Xe ) , (10)

DOI: 10.4236/jmp.2021.129077 1251 Journal of Modern Physics

B. Hoeneisen

2 2 22 with Xeeµµ≡∆ mL(4 E) , and ∆meeµµ ≡− mm. This is the phenomenon of neutrino oscillations.

3. Majorana Neutrinos

If neutrinos have no additive conserved charge (such as lepton number), it is possible to add Majorana type mass terms to (3): c cc iσνµµ∂=Ru mM ν Lu +( νRv ) ,, i σµµ ∂( νLv ) = m( νRv ) (11)

cc** * iσµµ∂( νRv ) = m( νLv ) + Mi νRu ,. σνµµ ∂=Lu m ν Ru (12)

cc** * iσµµ∂( νRu ) = m( νLu ) + Mi νRv ,, σνµµ ∂=Lv m ν Rv (13)

c cc iσνµµ∂=Rv mM ν Lv +( νRu ) ,. i σµµ ∂( νLu ) = m( νRu ) (14)

Here, with one generation, the masses can be made real by re-phasing the c * c * fields. The charge conjugate fields are (νLu ) ≡ i σν2 Lu , (νLv ) ≡ i σν2 Lv , c * c * (νRu ) ≡−i σν2 Ru , and (νRv ) ≡−i σν2 Rv . Majorana mass terms for fields ν L are not added, at tree level, because such terms are not gauge invariant. Note that c the Majorana mass terms link ν Ru with (ν Rv ) , etc. Then, a created ν Lu may N forward scatter on vh (with amplitude Yvh 2 ) becoming a ν Ru , that may forward scatter on M * (whatever it is, e.g. a dimension 5 operator containing c vh ) becoming a (ν Rv ) , that may forward scatter on vh (with amplitude N c Yvh 2 ) becoming a (ν Lv ) , etc, see Figure 1. Equations (13) and (14) are the charge conjugate of Equations (11) and (12), respectively.

Note that before EWSB, the fields ν Lu and ν Lv are in statistical equilibrium due to their interactions with the gauge bosons W µ and B. From (11) to (14) c c we conclude that after EWSB, the fields ν Lu , ν Ru , (ν Rv ) , (ν Lv ) , and their conjugates, become linked together so that the Majorana character of neutrinos may emerge dynamically after EWSB on a time scale 1/m (for the case of interest Mm ).

Figure 1. Graphical representation of (11), (12), (13), and (14), corresponding, respectively, to the four rows of arrows. Weyl_L and Weyl_R fields forward scatter on m and M. −+ −+ The lepton number conserving reactions are eW→→ν Lu eW and

+− c +− eW →→(ν Lv ) eW . The lepton number violating reactions are

−+ cc+− eW →↔↔ννLu Ru( ν Rv ) ↔( νLv ) →eW , and

+− cc −+ c eW →(νLv ) ↔( νRv ) ↔↔→ ννRu Lu eW . Ultra-relativistic ν Lu and (ν Lv ) have a polarization miss-match, see (4), (6) and Section 5.

DOI: 10.4236/jmp.2021.129077 1252 Journal of Modern Physics

B. Hoeneisen

Equations (11) to (14) are linear and homogeneous, and their general solution is a superposition of mass eigenstates. Each term of (11) transforms as Weyl_L,

and is proportional to exp(−+iEt ipz z) , and so they may be mixed. Equations (11) can be re-written as c ν iσνµµ∂ ( Lv ) 0 m Lu = c . (15) σν∂  i µµRu mM(ν Rv ) Equations (12) can be re-written as σν∂ * c i µµLu 0 m (ν ) = Lv . (16) σν∂ c ** i µµ( Rv ) mMν Ru Both (15) and (16) can be diagonalized simultaneously with a unitary matrix U and its complex-conjugate to obtain the equation in the mass eigenstate basis:

iσνµµ∂ a m 0 ν  = aa, (17) σν∂    i µµs 0 mss ν  ν Lu ν a †  ν≡c ≡UU ννmass,, mass ≡≡ ν (18)  ν (ν Rv ) s

* ma 0 † 0 m UU= . (19) 0 ms mM The unitary matrix U that satisfies (15), (17), and (19), with real and positive

ma , ms , and M, is eiiαα cosθθi e sin U = , (20) isinθθ cos 2 where mm=exp{ − i(α + π 2)} , tan θ = mmas, and tan( 2θ ) = 2 mM. The eigenvalues are

112222   m=−++ MM4, m m =+++ MM4. m (21) as22  

From here on we take the Majorana masses Mmi Dj , so θ  1. Then ν a

is an “active” neutrino that is mostly ν Lu , while ν s is a “sterile” neutrino that c is mostly (ν Rv ) . According to (18), the fields evolve as follows:

22 † ννztU,= diag exp −+ iEtiEmzU − 0,0 . ( ) { ( i )} ( ) (22)

−+ Consider a source that produces neutrinos in a weak state, e.g. eW →ν e . At 2212 c (12) ( ) the source, (ν Rv (00)) = and ννLu (00) + Lu ( ) is normalized to 1. 2 22  Define ∆≡m mmsa −. We obtain 2212 ∆m2 =ν(12) + ν( ) =−=− 222θθ PLLaa Lu ( ) Lu ( ) 1 Pas 1 4cos sin sinL . (23)  4E

The lepton violating reaction has probability Paa that differs from Paa by polarization miss-match factors (to be discussed in Section 5):

2 1 2 ∆m =2θ +− 2 θ222 θθ Paa 2 ( ma cos ms sin) 4 mma s cos sin sinL . (24) 2E 4E

DOI: 10.4236/jmp.2021.129077 1253 Journal of Modern Physics

B. Hoeneisen

The interpretation of these equations is discussed in Section 4. = Let us generalize to ng 3 generations of weak SU (2)L doublets, and n′ = 3 gauge singlet Weyl_R neutrinos. We introduce the notation c  ν (ν1Rv ) ν eLu  Lu  cc ν≡≡,, νν( ν) ≡( ν) . (25) ν c Luµ Lu Rv 2Rv ( Rv )  c ντ Lu ν ( 3Rv )

These fields are related to the mass eigenstates as follows:

†   νmass =UU νν,. = νmass (26)

† † The (nngg+×′′) ( nn +) weak mixing matrix U is unitary: UU = 1, UU= 1. The generalization of (15) is   c T ν iσν∂ ( ) 0 m Lu µµ Lv = D . (27) σν∂   c i µµRu mMD (ν Rv ) N mDh= Yv 2 is the complex nn′× g Dirac mass matrix, and M is the symmetric nn′′× Majorana mass matrix chosen real. The symmetric mass matrix is diagonalized as follows: 0 mT UT D U= diag mm , , , m . (28) ( 12 nng + ′ ) mMD

The masses mi of the mass eigenstates are chosen real and positive. The fields  ν evolve as (22). Then, for ultra-relativistic neutrinos, the probability of a lep- −+ −+ ton conserving event, e.g. eW→→νµi W , is [2]

nng + ′ =δ − ** 2 Pαβ αβ 4∑ Re UUUαi β i α j U β j sin Xij ij< (29) nng + ′ + ** 2∑ Im UUUαβαi i j U β j sin( 2 Xij ) , ij<

≡+PPαβCPC αβ CPV , (30)

22 where Xij=( m i − mL j ) (4 E) . The sums in (29) are over mass eigenstates ij,

with masses mi and m j that cannot be discriminated in the experiment, and are coherent, see Section 6. Under CP conjugation, UU→ * . The first two

terms on the right hand side of (29), denoted PαβCPC , are CP conserving, while

the last term, denoted PαβCPV , may be CP violating. Note that to obtain CP violation, at least one physical phase in U is needed, in addition to the propagation

phase 2X ij (that requires mmij≠ ). Since U is unitary, ∑∑αβPPαβ =αβ = 1 . −+ +− The probability to observe a lepton violating event, e.g. eW→→νµi W , is 2  nngg++′′nn 1  = ** − ** 2 Pαβ 2  ∑∑mUUiiiαβ 4 mmij Re  UUUαβα ii j U β jsin X ij 2E  i ij<  (31) ′ nng +  + ** 2∑ mmi j Im  Uαβαi U i U j U β j sin( 2 Xij ) , ij< 

DOI: 10.4236/jmp.2021.129077 1254 Journal of Modern Physics

B. Hoeneisen

≡+PPαβCPC αβ CPV  1, (32)

where we have included the polarization miss-match factors discussed in Section +− −+ 5. The probability Pαβ for the CP-conjugate event, e.g. eW→→νµi W , is * obtained by UU→ . Note that the first two terms, denoted Pαβ CPC , violate lepton number but conserve the CP symmetry. Note that the last term, denoted

Pαβ CPV , is lepton number violating and may be CP violating, and is the source of the leptogenesis studied in this article. Considering the width of the neutrino

energy distribution, strong CP violation requires 2X ij  π 2 . The last two terms in (29) and (31) are due to coherent interference of the mass eigenstates.

Note that for L → 0 , Pαα is proportional to the square of the “effective 2 mass” ∑i mUiiα . Neutrino-less double beta decay experiments constrain the effective mass of electrons to be less than 0.165 eV [2]. Equations (29) and (31) assume neutrinos are nearly on mass shell, i.e.

Lp λ = 2π z , and that the neutrino mean energy Em i . The sums in these equations only include neutrinos that are coherent, see Section 6.

4. Interpretation

In a neutrino oscillation experiment, most neutrinos traverse the detector with-

out interacting. In the limit L → 0 , Pαβ→ δ αβ . The probabilities Pαβ are de- + fined as the number of ν β lWβ → counts at the far detector, divided by the + number of νααlW→ counts at the near detector, corrected for acceptance

and detector efficiencies. If the efficiency of the detector for ν β is negligible,

Pαβ may still be “measured” as a disappearance in the sum of other channels.

For the probability Pαβ , the detector efficiency factor due to polarization miss-match has already been included. The interpretation of the preceding equations needs an understanding of the entire experiment. In particular we need to consider polarization miss-match (Section 5), and coherence (Section 6). If at the source the neutrino mass is sufficiently uncertain, then a weak state is produced, i.e. a coherent superposition of mass eigenstates. If at the source the neutrino mass is sufficiently well deter- mined, then a mass eigenstate is produced. Even if the neutrino mass eigenstates are produced coherently, they may lose coherence before being detected, either in transit, and/or at the detector. If this is the case, then an “observation” has been made, and we need to pass from amplitudes to probabilities, i.e. interference terms are lost. For simplicity, we consider a single generation, i.e. (15) to (24). If production is coherent, and the mass eigenstates have become incoherent, then the proba- 2 2 bility for ν a is cos θ , and the probability for ν s is sin θ . In the case of coherent production and incoherent detection, e.g. the neutrino mass is meas-

ured at the detector with a resolution sufficient to discriminate between ma

and ms , the combined probability to have a ν a and a lepton conserving event −+ −+ 4 eW→ eW is cos θ , the combined probability to have a ν s and a lepton

DOI: 10.4236/jmp.2021.129077 1255 Journal of Modern Physics

B. Hoeneisen

4 conserving event is sin θ , the combined probability to have a ν a and a lepton −+ +− 42 violating event eW→ eW is mmascosθ ( 2 E ) , and the combined 42 probability to have a ν s and a lepton violating event is mmassinθ ( 2 E ) . 2 The factor mmas(2 E ) is due to polarization miss-match (Section 5) that re- duces the cross-section by this factor, and applies to a real, i.e. finite extent, un-

polarized detector. We assume mEa  1 and mEs  1. E is the energy of the neutrino in the laboratory frame, i.e. the detector. The case of interest to the leptogenesis scenario studied in this article is coherent production and coherent detection, since interference is needed for CP violation. If production is coherent, and the mass eigenstates remain coherent at

detection, then Paa of (23) is the probability to observe the lepton conserving event eW−+→ eW −+. In any practical neutrino oscillation experiment, with a

finite detector, Paa of (24) is the probability to observe the lepton violating event eW−+→ eW +−.

5. Polarization Miss-Match

++ + Consider the decay We→ ν e in the rest frame of W . The neutrino ν e + carries the field ν Lu , see (4). Let Θ be the angle between the W spin and

the ν e momentum. Then ξ1 =cos( Θ 2) and ξ2 =isin( Θ 2) [4]. Therefore, + the amplitude for W to create a right-handed (i.e. helicity +1/2) ν e is

∝Θcos( 2) Ep −z , and the amplitude to create a left-handed ν e is

∝Θisin( 2) Ep +z . Helicity is the projection of the spin angular momentum  in the direction of the momentum, i.e. sp⋅=ˆ (12)σ ⋅ pˆ . Note that most ul-

tra-relativistic ν e are left-handed. −− − Consider the CP-conjugate decay We→ ν e in the rest frame of W . The c anti-neutrino ν e carries the field (ν Lv ) , see (6). Let Θ be the angle between − * the W spin and the ν e momentum. Then η2 =cos( Θ 2) and * − −=η1 isin( Θ 2) . Therefore, the amplitude for W to create a right-handed

(i.e. helicity +1/2) ν e is ∝Θcos( 2) Ep +z , and the amplitude to create a

left-handed ν e is ∝Θisin( 2) Ep −z . Note that most ultra-relativistic ν e are right-handed. Note that for ultra-relativistic Majorana neutrinos we can still distinguish neutrinos (lepton number ≈+1 and helicity ≈−1/2) from anti-neutrinos (lepton number ≈−1 and helicity ≈+1/2), since lepton number is conserved to a high degree of accuracy, see Section 10 for a numerical example. ++ Consider the lepton-conserving sequence of events We→ ν e followed by ++ νµµ → W . We assume Em e and Em µ . The probability, after averag- 2 Θ Θ′ ∝ 2 ing over cos and cos , is ∑i aEi , where ai is the amplitude corresponding to mass eigenstate i, and Θ′ is the angle with respect to the final + ++ W . Consider the lepton-violating sequence of events We→ ν e followed by −− νµµ → W . The probability, after averaging over cos Θ and cos Θ′ , is 2 ∝ ∑i amii 2 . The probability for the lepton conserving events (29) is norma-

DOI: 10.4236/jmp.2021.129077 1256 Journal of Modern Physics

B. Hoeneisen

2 = lized to ∑i ai 1, so the relative polarization miss-match factor for lepton vi- 2 2 olating events is ∑i amii (2 E ) . This factor has been included in (24) and

(31). We note that in the limit mEi → 0 , the experimental distinction between Dirac and Majorana neutrinos fades away, and lepton violation vanishes.

6. Coherence

The sums in (29) and (31) only include coherent neutrinos. To obtain coherent

oscillations between two neutrinos of masses mi and m j it is necessary that

the energy uncertainty σ E of the produced and detected neutrinos be suffi- 2 2 22 ciently large, i.e. σ E  ∆m 8 , where ∆≡m mmji − [7]. If this condition is met, what is created is a flavor eigenstate, i.e. a coherent superposition of mass eigenstates, and oscillations remain coherent while the wave packets of the two components overlap. The overlap ceases after the “coherence time” [7] 2E 2 tcoh = 22σ t , (33) ∆m2

where σ t is the Gaussian wave packet duration. The combined coherence factor after time ∆t is [7] εσ= −∆2 2 ⋅ −∆ 22 coh expm( 8E ) expttcoh . (34)

In the present application we take, arguably, σ tW≈Γ1 , σ EW≈Γ , and

∆≈ttc , where tc is the mean time for a neutrino to collide with a charged lep-

ton. Consider the reference temperature Tsph = 131.7 GeV , and ET≈ 2.8 sph . ++ The cross-section σ for eeννee→ is given by (50.25) of [2]. We obtain −22 t=1 nc− σ ≈× 7 10 s = 1 0.001 GeV n − c ( e ) ( ) , where e is the electron number

density at Tsph . Consider an active-sterile neutrino oscillation with 2 22 2 22 ∆=−≈msa mm s a m s . Requiring coherent oscillations, ∆msa(81σ E )  obtains

the bound ms  6 GeV . For ET≈ 2.8 sph , ∆=ttc = tcoh corresponds to

ms = 19 GeV . Therefore, for ms  6 GeV , oscillations remain coherent for 12 22 10tc . We verify also that tpc  2π , with pEm=( − ) , so the neutrinos of interest are nearly on mass shell.

7. Asymmetry Build-Up

So far we have been studying a neutrino oscillation experiment with baseline L. In this Section we apply the results to the universe when it has the reference

temperature Tsph = 131.7 GeV . Consider a single ν e . The ν e lifetime is tc . ±± The probability that the interaction ν eeW→ occurs in the time interval −ttc from t to tt+ d is dP= ed ttc . The lepton number violating and CP vi-

olating asymmetry, Ptαβ CPV ( ) , is proportional to t for the case of interest

2X ij  π 2 . The mean of Ptαβ CPV ( ) is then Ptαβ CPV ( c ) .

Consider the contribution of the channel αβ+ αβ to δl . Let nβ ,i be the − −− comoving number density of lβ at time ti , where leβ = ,,µτ. Then, at

time ti+1 = tt ic + ,

DOI: 10.4236/jmp.2021.129077 1257 Journal of Modern Physics

B. Hoeneisen

nβ,1i+ =++ n β , i nP α , i αβ nP α , ii αβ ,, (35) nβ,1ii+ =++ n β , nPαα,,iiαβ nP αβ ,i.

Taking the difference of these two equations, and dividing by

nβ,1i+ + nββ,1ii+ ≈+≈+ nnβ,i, nn αα,, ii, obtains δδ≈+ − +δ − β,ii+ 1 β ,(PP αβ CPV αβ CPV ) α,i( PP αβ CPC αβ CPC ). (36)

Summing over α and β obtains

δδli,1+ ≈− li , ∑∑PPαβ CPV − δα ,i αβ CPC . (37) αβ αβ

The last term is the “wash-out” term that tends to restore the equilibrium value

δl = 0 . Note that δl decreases by ∑αβ Pαβ CPV in time tc until it reaches either

tu δlMAX = − ∑ Pαβ CPV , (38) tc αβ or until wash-out sets in at

δδlWO ∑∑Pαβ CPC ≡=−α PPαβ CPC ∑αβ CPV . (39) αβ αβ αβ

We note that Pαβ CPV and Pαβ CPC are proportional to Lt= c if 2X ij  π 2 , so,

in this case of interest, δlMAX and δlWO are independent of tc .

8. Constraints from Cosmology

Constraints from Big Bang Nucleosynthesis (BBN), Baryon Acoustic Oscillations (BAU), and direct searches, limit the mass of sterile neutrinos to be greater than 0.14 GeV [8], so the interesting sterile neutrino mass range, for the leptogenesis scenario being considered, is approximately 0.14 GeV to 1.1 GeV. The lifetimes −5 of these neutrinos range from approximately 10 s at ms = 1 GeV , to 0.1 s for

ms = 0.13 GeV [8]. Big Bang Nuleosynthesis and cosmic microwave background (CMB) measurements do not allow one additional ultra-relativistic degree of freedom at

TBBN ≈ 1 MeV [2]. For the Standard Model, the equivalent number of neutrinos +0.36 (for BBN) is Nν = 3.045 [2]. The Planck CMB result gives Nν = 2.92−0.37 at 95% confidence [2]. So an extra stable neutrino, that was once in statistical equili-

brium with the Standard Model sector, needs to decouple at TT>≈C 0.14 GeV ,

where TC is the confinement-deconfinement temperature. Such an extra neu-

trino contributes ≤0.12 to Nν . Therefore, sterile neutrinos either 1) never reached statistical equilibrium with the Standard Molel sector, or 2) reached sta-

tistical equilibrium, but decoupled at TT> C and hence are sufficiently cooler

than active neutrinos at TBBN [2], or 3) the sterile neutrino mass is mTs > BBN

and these neutrinos decayed before T reached TBBN .

As an example, for ma = 0.005 eV , and ms in the range of interest 0.14 GeV to 1.1 GeV, we find that sterile neutrinos never reach statistical equilibrium with the Standard Model sector, or reach equilibrium but decouple at T > 0.14 GeV , and hence do not affect BBN.

DOI: 10.4236/jmp.2021.129077 1258 Journal of Modern Physics

B. Hoeneisen

9. Leptogenesis with ng = 1 and n′ = 2

Let us study the simplest case with lepton number violation and CP violation.

We take ng = 1 generations of active neutrinos and n′ = 2 gauge singlet

Weyl_R neutrinos. For mMa  1 and mMa  2 , the unitary mixing matrix U 2 2 from (28), to order mMDi i , is |mm ||22* | m m * −−DD12 − D 1 D 2 i1 22 22MM12MM12  2 * mD1 mD1 mmDD12 Uiα ≈ 1,−− (40) i M 2M 2 2MM 1 1 12 2 m mm* m i D2 −−DD121 D2 2 M2 2MM12 2M 2

and the mass eigenstates are 22 mmDD12 ma =,+ (41) MM12

2 m mM=1, + D1 (42) s112 M1 2 mD2 mM=1. + (43) s222 M 2

The active neutrino mass ma , and sterile neutrino masses M1 and M 2 are

real and positive. The Dirac terms mD1 and mD2 are complex. 22 Let us write (38) for the present case ng = 1, n′ = 2 . To order mMa we obtain

*2 nn−+− −t  m δ =ee = u 2 D1 ea3 mM1 Im  nn−++  M ee2⋅( 2.8Tsph )  1 (44) *2 2 *2  2mD2 22 mmDD12  +mMa 2Im  +−(MM12)Im   . M2 MM12 

This equation assumes the approximation sin( 2XXij) ≈ 2 ij valid for

2X ij  π 2 , corresponding to masses 1.1 GeV . The three terms correspond

to interference of neutrinos ννas↔ 1 , ννas↔ 2 , and ννss12↔ , respectively. 22 We find that all terms of order mMai cancel. In conclusion, if both sterile neutrinos have masses in the approximate range 0.14 to 1.1 leptogenesis is negligible.

Let us consider the case 0.14M1 1.1 GeV , and M 2  6 GeV so that

ν s2 is incoherent. In this case we keep only the first term in (44). Leptogenesis

can be successful if we are able to find mD1 and mD2 that satisfy (41) and (44) −8 (omitting the terms with M 2 ) with the required δδle==−×3.1 10 . Note that (41) and (44) are under-constrained: they have multiple solutions. We therefore use the Casas-Ibarra procedure [9], and write the solution to (41) in the form

DOI: 10.4236/jmp.2021.129077 1259 Journal of Modern Physics

B. Hoeneisen

T mDi= MR i i m a , where R is any orthogonal, i.e. RR= 1, nn′× g matrix. In the present case

R1 R = . (45) R2

Substituting in (44) we obtain −tmM22 δ = ua 1 *2 e 3 Im(R1 ) . (46) 2⋅( 2.8Tsph )

As an example, we take M1 = 1 GeV , M 2 = 50 GeV , and ma = 0.015 eV . −14 Also tu =1( 4.6 × 10 GeV) , and Tsph = 131.7 GeV . We obtain *2 8 Im(R1 ) = 6.4 × 10 . A solution then needs fine tuning, e.g.

−πii4π 42 R12= e, K R= e K 1 − iK , (47)

8 *2 where K =6.4 × 10 . Since we have chosen Re(R1 ) = 0 , wash-out remains negligible. Equation (44) can be generalized to n′ > 2 by inspection.

10. Leptogenesis with ng = 3 and n′ = 3

Without loss of generality we work in a basis that diagonalizes the nngg× charged lepton mass matrix, and the nn′′× Majorana mass matrix M. The

(nngg+×′′) ( nn +) weak mixing matrix U, defined in (28), to lowest order in

mMD , has the form [2]

1 † *1−− 1 † *1− 1− mMD M mDl V mMD V h 2 U ≈ . (48)  −1 1 −−1 † *1 −−MmVDl 1 MmmMD D V h 2

To the present order of approximation, we take Vh = 1 . The 3 × 3 PMNS weak

mixing matrix of active neutrinos Vl depends on three angles and one Dirac

CP-violating phase δCP , that have been measured, and two Majorana

CP-violating phases η1 and η2 (with the notation of [2]) that have not been

measured. Unphysical phases of Vl , that can be canceled by re-phasing fields, have already been fixed. We take the central measured values of these parameters for normal (NO) or inverse (IO) neutrino mass ordering from the first col- umn of Table 14.7 of [2]. The two active neutrino mass-squared differences are

also obtained from this Table. η1 and η2 are free parameters until measured. Neutrino oscillation experiments show that at least 2 neutrino eigenstates have mass, so (arguably) at least n′ = 2 gauge singlet Weyl_R neutrinos need to be added to the Standard Model. For the case n′ = 2 , the lightest active neutrino mass is zero. For the case n′ = 3 , the lightest active neutrino mass is a free parameter until measured. The diagonal mass matrix of active neutrinos, obtained from (28) and (48), is [2]

DOI: 10.4236/jmp.2021.129077 1260 Journal of Modern Physics

B. Hoeneisen

l TT− 1 m≡diag( m123 , m , m) ≈− Vl mM D mV Dl. (49)

Successful leptogenesis is possible if we are able to solve (49), (38), and (39) −8 with the needed lepton asymmetry δδδle=++=−×µτ δ 3.1 10 . We focus on the case n′ = 3 . The problem is under-constrained, so again we follow the Ca-

sas-Ibarra procedure [9]. From (49), with the notation M≡ diag( MM12 , ,) , T −−11l we obtain RR= 1 with R≡ i M mVDl m . Finally,

Nlvh † mDl= Y = − i MR mV , (50) 2 where R is any orthogonal, i.e. RRT = 1, (51)

nn′× g matrix. Equation (49) is satisfied by (50). To satisfy (38), the matrix R needs to be complex. Successful leptogenesis requires fine tuning of the unknown parameters. As a proof of principle we present the following example: normal neutrino mass or-

dering, m1 = 0.001 eV , M1 = 1.04 GeV , and M 2 = 1.06 GeV . We also set

η1 = 0 , η2 = 0 , and M 3 = 60 GeV , and note that the results depend negligibly on these last three parameters (for the texture of the matrix R chosen below). At −−11 14 TT= sph , the age of the universe is tu =×=×1.4 10 s 1( 4.6 10 GeV) , and −22 tc =×=7 10 s 1( 0.001 GeV) . Equation (51) has many solutions. Successful

leptogenesis needs RRαβii 1 and RRαβii imaginary to high accuracy, as in −8 (47). To obtain a solution that satisfies δl =−×3.1 10 we choose a particular

texture of R (that makes the results insensitive to M 3 ), and obtain:  −−eiπ4 e−− i π 4 0 − eiiππ44 − e0 −iiππ44 1 iπ4− iπ 4 RK=−e e 0+− e e 0, (52) 4K 1 0 00 00  K with K = 52540 . The probabilities for a “neutrino oscillation experiment” with

Lt= c , from terms in (29) and (31) are respectively: −−2 19 P −− =−×−×1 3.4 10 4.3 10 , ee

−−32 12 −19 P −+ =×+×+×6.5 10 1.1 10 2.0 10 , ee

−−4 15 P −−=×6.2 10 −× 8.2 10 , e µ

−−−32 12 19 P −+=×6.3 10 +× 1.2 10 −× 1.7 10 , e µ

−−4 15 P −− =×5.1 10 −× 8.7 10 , e τ

−−−32 13 19 P −+ =×5.3 10 +× 9.5 10 +× 4.2 10 , e τ

−−4 15 P −− =×6.2 10 +× 8.2 10 , µ e

−−−32 12 19 P −+ =×6.3 10 +× 1.2 10 −× 1.7 10 , µ e

DOI: 10.4236/jmp.2021.129077 1261 Journal of Modern Physics

B. Hoeneisen

=−×−×−−2 19 Pµµ−− 1 3.6 10 8.7 10 , =×+×+×−−32 12 −18 Pµµ−+ 8.0 10 1.2 10 2.0 10 , =×−−4 −×15 Pµτ−− 5.4 10 2.0 10 , =×+×−×−−−32 12 18 Pµτ−+ 7.1 10 1.0 10 1.2 10 ,

−−4 15 P −− =×5.1 10 +× 8.7 10 , τ e

−−−32 13 19 P −+ =×5.3 10 +× 9.5 10 +× 4.2 10 , τ e =×−−4 +×15 Pτµ−− 5.4 10 2.0 10 , =×+×−×−−−32 12 18 Pτµ−+ 7.1 10 1.0 10 1.2 10 , =−×+−2 Pττ−− 1 2.9 10 0.0, = ×−32 +× −−13 +×18 Pττ−+ 6.3 10 8.2 10 1.3 10 . (53)

We note that the lepton number violating reactions are suppressed with respect to the lepton conserving ones, and the CP violating terms are suppressed with

respect to the CP conserving ones. We note that the terms Pαβ CPC are of order −12 −18 10 and positive, while the terms Pαβ CPV are of order 10 and can be positive or negative.

P −+ From the first term in ee we obtain the effective mass of neutrino-less double beta decay experiments for this example: −32 6.5× 10 ×⋅ 2 2.8 ⋅Tsph = 0.00013 eV . The current limit is 0.165 eV [2]. Asymmetries per channel are presented in Table 1 (from (38) and (39) without the sums over α and β ). Note that, in this example, we do not reach sa-

turation due to wash-out, i.e. δδllMAX< WO in each channel. Summing the −8 asymmetries in the last row of Table 1 obtains δδll=MAX =−×3.1 10 as required. In conclusion, successful baryogenesis via leptogenesis may be achieved with the fine-tuning shown in (52), plus the fine tuning of the masses (so that the positive and negative terms in the last sum of (39) cancel to one part in 102).

Table 1. Lepton number asymmetries per channel δlMAX (upper numbers), and wash-out

asymmetries δlWO (in parenthesis) for individual channels. δl is the least of δlMAX

and δlWO .

α δe δ µ δτ

−4.0 × 10−9 3.4 × 10−9 −8.6 × 10−9 e+ (−1.8 × 10−7) (1.4 × 10−7) (−4.4 × 10−7)

3.4 × 10−9 −4.0 × 10−8 2.5 × 10−8 µ + (1.4 × 10−7) (−1.6 × 10−6) (1.2 × 10−6)

−8.6 × 10−9 2.5 × 10−8 −2.7 × 10−8 τ + (−4.4 × 10−7) (1.2 × 10−6) (−1.6 × 10−6)

−9 −8 −8 Sum of δ β −9.2 × 10 −1.2 × 10 −1.0 × 10

DOI: 10.4236/jmp.2021.129077 1262 Journal of Modern Physics

B. Hoeneisen

Several tests with modifications of this example follow: −8  Setting δCP = 0 , or η1 = 0.785 , or η2 = 0.785 , obtains δlMAX =−×3.1 10

as before, so the CP-violating phases in the PMNS matrix Vl contribute negligibly to leptogenesis (in this scenario of active-sterile neutrino oscillations). Leptogenesis is mostly due to the CP-violating asymmetries in R, or

equivalently mD . −34  Choosing a real R obtains δlMAX =1.4 × 10 , so, again, the Dirac phase δCP

of Vl contributes negligibly to leptogenesis.

 Setting δCP = ηη12 = = 0 and a real R obtains δl = 0 , as expected. This cross-check is satisfied for 0, 1, 2 or 3 coherent sterile neutrinos.

 Setting M 2 = 50 GeV , to test a case with one coherent sterile neutrino, ob- −8 −10 tains δlMAX =−×3.0 10 and δlWO =−×2.8 10 . Note that wash-out dominates.

 Setting M1 = 1 GeV , M 2 = 0.5 GeV and M 3 = 0.2 GeV , to test a case with −9 −10 three coherent neutrinos, obtains δlMAX =8.8 × 10 and δlWO =1.6 × 10 . Note that the signs are now wrong, and wash-out dominates.  Results for inverse neutrino mass ordering are similar. However, we were −8 unable to reach successful leptogenesis, i.e. δl =−×3.1 10 .

11. Sterile Neutrino Dark Matter?

Detailed dark matter properties have recently been obtained by fitting spiral galaxy rotation curves, and, independently, by fitting galaxy stellar mass distributions [10]. These measurements imply that dark matter was in thermal and dif- fusive equilibrium with the Standard Model sector in the early universe, and decoupled (from the Standard Model sector and from self-annihilation) at a temperature T  0.2 GeV . If dark matter particles are fermions, the measurements +37 obtain their mass mh = 107−20 eV [10]. This mass is disfavored by the Tre- maine-Gunn limit (that applies to fermion dark matter) [11], which however needs revision [12] [13] [14] [15]. Fermion dark matter is also disfavored, relative to boson dark matter, by spiral galaxy rotation curves and by galaxy stellar mass distributions with a significance of 3.5σ [10].

Nevertheless, let us see if sterile neutrinos of mass mh could have reached statistical equilibrium with the Standard Model sector by the time of the con-

finement-deconfinement temperature TC ≈ 0.14 GeV . This is the minimum temperature at which dark matter in equilibrium with the Standard Model sector can decouple without spoiling the agreement with Big-Bang Nucleosynthesis. At −5 this temperature the age of the universe is tu =1.7 × 10 s , and the neutrino life- −10 time is tc =2.9 × 10 s . The number of baryons per electron is −9 η ⋅(2 3) ⋅( 205 × 22( 43 ×=× 8)) 5.3 10 . The number Pas of sterile neutrinos of

mass mh that need to be produced per electron is −9 5.3× 10 ⋅(mmph) ⋅Ω( CDM Ωb) =0.25 . (We use the standard notation in cos-

mology [2].) For nng =′ = 1, i.e. from (23), we obtain Pas = 0.02 (for

ma = 0.05 eV ), insufficient to produce the observed density of dark matter (as reported in [16]).

DOI: 10.4236/jmp.2021.129077 1263 Journal of Modern Physics

B. Hoeneisen

Let us consider nng =′ = 3. The example of Section 10 has the matrix R with

a texture that makes the results insensitive to M 3 . We may set M 3 = 107 eV with no significant change in the results reported in Section 10. For that fine-tuned example we obtain from (29), before subtracting reverse conversions,

PPe66=1.2,µτ = 30.4, P 6 = 24.1. (54)

The sub-index “6” stands for ν 6 with mass Mm3 = h . Reverse conversion lim- its these numbers to the statistical equilibrium value 1. In conclusion, sufficient sterile neutrino dark matter production is possible. Such dark matter is disfavored by observations but not ruled out.

12. Sterile Neutrino Search?

Consider a neutrino experiment that reconstructs the detected neutrinos in all-charged final states with the capability to discriminate a sterile neutrino mass from the active neutrino masses. In this case there is no interference, and the

probability to detect the sterile neutrino ν i , relative to the probability to detect any neutrino in the αα channel is

2 Ps * = UUααii, (55) Pa

with no sum implied. For nng =′ = 1, i.e. from (20), we obtain 4 2 PPsa=sin θ = ( ma m s) , which is experimentally hopeless. For ng = 1 and 4 2 n′ = 2 , i.e. from (40), we obtain PPsa= R11( ma M) , which is very interest- 17 2 −5 ing! For the example in Section 9 we obtain PPsa≈×4 10 ( ma M1 ) ≈×9 10 ,

which is less hopeless. For nng =′ = 3, and the example in Section 10, we obtain −5 PP4 µµ =9 × 10 , where “4” stands for ν 4 of mass M1 . This is the maximum for all channels, and is experimentally challenging. A search for sterile neutrinos in the approximate mass range 0.14 GeV to 2.0 GeV may be considered. A study has been presented in [17]. In conclusion, the same factor K ≈×5 104 that makes the model fine-tuned, enters to the fourth power, and may allow the model to be tested experimentally!

13. Conclusions

We have studied coherent active-sterile neutrino oscillations as a possible source of leptogenesis. To this end, we add n′ gauge invariant Weyl_R neutrinos to

the Standard Model with both Dirac mD and Majorana M mass terms. We find that for n′ = 3 we can obtain the measured active neutrino masses and mixings, and successful baryogenesis via leptogenesis, with, however, the fine tuning described in Sections 9 and 10, see (47) and (52). The Dirac CP-violating phase

δCP , and Majorana CP-violating phases η1 and η2 of the 3 × 3 PMNS weak

mixing matrix Vl contribute negligibly to this scenario of leptogenesis. The

major contribution comes from the phases of the Dirac mass matrix mD that links the Weyl_L and Weyl_R neutrinos. The Dirac nature of charged particles and the possible Majorana nature of neutrinos, emerge dynamically after elec-

DOI: 10.4236/jmp.2021.129077 1264 Journal of Modern Physics

B. Hoeneisen

troweak symmetry breaking, i.e. just before sphaleron freeze-out. The possible Majorana nature of neutrinos allows lepton number violation, with, however, a 2 cross-section reduced by a factor mmij(2 E ) due to polarization miss-match. This penalty renders lepton number violation beyond the reach of current laboratory experiments. CP-violation is the result of coherent interference of neutrinos with two clashing phases: a phase from the weak mixing matrix U (mainly

from the Dirac mass matrix mD ), and a phase 2X ij from neutrino propaga-

tion. The interference is coherent if the sterile neutrino mass ms is less than

approximately 6 GeV. The condition 2X ij  π 2 , for strong CP violation, im-

plies ms 1.1 GeV . Constraints from Big Bang Nucleosynthesis require

ms  0.14 . We find that at least one of the sterile neutrinos needs to have a mass in the approximate range 0.14 to 1.1 GeV to obtain successful leptogenesis. With n′ = 3 , we may include in the model sterile neutrino dark matter with +37 the measured mass mh = 107−20 eV [10]. However, such dark matter is disfavored by observations (but not ruled out [10]). The present scenario of leptogenesis requires a fine tuning parameter K ≈×5 104 , and a pattern of R such as (52). Why should nature select such a pattern (reminiscent of patterns in chemistry and biology)? It is interesting to note that with this fine tuning parameter K, the neutrino Yukawa coupling mag- nitudes become comparable to the ones of charged leptons and quarks. It is also interesting to note that K may bring sterile neutrino search within experimental reach. The scenario studied in this article is similar to the model νMSM [8], where calculations have been carried out numerically in full detail. The search for ste-

rile neutrinos with ms  2 GeV with sufficient sensitivity may be possible in a dedicated experiment [17].

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

References [1] DOnofrio, M., Rummukainen, K. and Tranberg, A. (2014) Physical Review Letters, 113, Article ID: 141602. https://doi.org/10.1103/PhysRevLett.113.141602 [2] Zyla, P.A., et al. (2020) Progress of Theoretical and Experimental Physics, 2020, 083C01. https://doi.org/10.1093/ptep/ptaa104 [3] Davidson, S., Enrico Nardi, E. and Nir, Y. (2008) Physics Reports, 466, 105. https://doi.org/10.1016/j.physrep.2008.06.002 [4] Schwartz, M.D. (2014) Quantum Field Theory and the Standard Model. Cambridge University Press, Cambridge. https://doi.org/10.1017/9781139540940 [5] Branco, G.C., Lavoura, L. and Silva, J.P. (1999) CP Violation. Claredon Press, Ox- ford. [6] Hoeneisen, B. (2006) Trying to Understand Mass. arxiv:0609080. [7] Akhmedov, E. (2019) Quantum Mechanics Aspects and Subtleties of Neutrino Os-

DOI: 10.4236/jmp.2021.129077 1265 Journal of Modern Physics

B. Hoeneisen

cillations. International Conference on History of the Neutrino: 1930-2018, Paris, 5-7 September 2018. arxiv:1901.05232. [8] Canetti, L., Drewes, M., Frossard, T. and Shaposhnikov, M. (2012) Physical Review D, 87, Article ID: 093006. https://doi.org/10.1103/PhysRevD.87.093006 [9] Casas, J.A. and Ibarra, A. (2001) Nuclear Physics B, 618, 171-204. https://doi.org/10.1016/S0550-3213(01)00475-8 [10] Hoeneisen, B. (2020) International Journal of Astronomy and Astrophysics, 10, 203-223. https://doi.org/10.4236/ijaa.2020.103011 [11] Tremaine, S. and Gunn, J.E. (1979) Physical Review Letters, 42, 407-410. https://doi.org/10.1103/PhysRevLett.42.407 [12] Hammer, F., Yang, Y.B., Arenou, F., Puech, M., Flores, H. and Babusiaux, C. (2019) The Astrophysical Journal, 883, 171. https://doi.org/10.3847/1538-4357/ab36b6 [13] Hammer, F., Yang, Y., Arenou, F., Wang, J., Li, H., Bonifacio, P. and Babusiaux, C. (2020) The Astrophysical Journal, 892, 3. https://doi.org/10.3847/1538-4357/ab77be [14] Yang, Y., Hammer, F., Fouquet, S., Flores, H., Puech, M., Pawlowski, M.S. and Kroupa, P. (2014) MNRAS, 442, Article ID: 24192433. https://doi.org/10.1093/mnras/stu931 [15] Hammer, F., Yang, Y.B., Arenou, F., Babusiaux, C., Puech, M. and Flores, H. (2018) ApJ, 860, 76. https://doi.org/10.3847/1538-4357/aac3da [16] Hoeneisen, B. (2021) International Journal of Astronomy and Astrophysics, 11, 59-72. https://doi.org/10.4236/ijaa.2021.111004 [17] Gninenko, S.N., Gorbunov, D.S. and Shaposhnikov, M.E. (2012) Advances in High Energy Physics, 2012, Article ID: 718259. https://doi.org/10.1155/2012/718259

DOI: 10.4236/jmp.2021.129077 1266 Journal of Modern Physics