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We study coherent active-sterile neutrino oscillations as a possible source of leptogenesis. To this end, we add 3 gauge invariant Weyl_R neutrinos to the Standard Model with both Dirac and Majorana type mass terms. We find that the measured active neutrino masses and mixings, and successful baryogenesis via leptogenesis, may be achieved with fine-tuning, if at least one of the sterile neutrinos has a mass in the approximate range 0.14 to 1.1 GeV.

We present a study of coherent active-sterile neutrino oscillations as a possible source of baryogenesis via leptogenesis. Consider a reaction of the form e ∓ W ± → ν i → e ± W ∓ that produces a lepton asymmetry that is partially converted to a baryon asymmetry before the sphaleron freeze-out temperature T sph = 131.7 ± 2.3 GeV [

the universe is t u = 1.4 × 10 − 11 s , and the time between collisions of active neutrinos in the reaction ν e e + → ν e e + is t c = 1 / ( σ n e − c ) ≈ 7 × 10 − 22 s ≈ 1 / ( 0.001 GeV ) , where σ is the cross-section. We note that at T sph neutrinos are of short wavelength relative to t c , i.e. t c ≫ 2 π / T 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 [

Let us consider the reaction e − W + → ν i → e ∓ W ± , with neutrino mass eigenstates ν i oscillating coherently during time t c . The condition for coherent oscillations 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 by a factor m i m j / ( 2 E 2 ) due to polarization miss-match, where m i 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 [

There are cosmological constraints, mainly from Big Bang Nucleosynthesis (BBN), that require the mass of sterile neutrinos to be m s ≳ 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 mentioned 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.

In the following sections we consider a neutrino experiment with a source at the origin of coordinates, and a detector at a distance z = L . We assume L ≫ 2 π / p z , so the neutrinos are almost on mass-shell. p z is the neutrino momentum. At first let us consider a single neutrino flavor, and the reaction e − W + → ν e → e − W + .

Before electroweak symmetry breaking (EWSB) at T EWSB ≈ 159 ± 1 GeV [

i σ ¯ μ ∂ μ ν L = 0 , (1)

where σ 0 ≡ 1 2 × 2 , σ ¯ 0 ≡ σ 0 , σ ¯ k ≡ − σ k , and σ k are the Pauli matrices [

η μ ν ∂ ν ∂ μ ν L ≡ ∂ μ ∂ μ ν L = 0 , (2)

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

After EWSB the Higgs boson acquires a vacuum expectation value v h [

i σ μ ∂ μ ν R = m ν L , i σ ¯ μ ∂ μ ν L = m ν R , (3)

with m = | Y N | v 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 ( − i E t + i p z z ) , in a Weyl basis, is [

ψ u ≡ ( ν L u ν R u ) = ( E − p z ξ 1 E + p z ξ 2 E + p z ξ 1 E − p z ξ 2 ) exp ( − i E t + i p z z ) , (4)

corresponding to a particle of mass m, and momentum p → = p z e → z with p z = + E 2 − m 2 . This is the “stepping stone” mechanism of mass generation [

The solution of (3) proportional to exp ( i E t − i p z z ) is

ψ v ≡ ( ν L v ν R v ) = ( E − p z η 1 E + p z η 2 − E + p z η 1 − E − p z η 2 ) exp ( i E t − i p z z ) . (5)

The charge conjugate of ψ v is [

( ψ v ) c = − i γ 2 ψ v * = ( ( ν R v ) c ( ν L v ) c ) = ( − i σ 2 ν R v * i σ 2 ν L v * ) = ( E − p z η 2 * − E + p z η 1 * E + p z η 2 * − E − p z η 1 * ) exp ( − i E t + i p z z ) . (6)

Note that − i σ 2 ν R v * transforms as Weyl_L, while i σ 2 ν L v * transforms as Weyl_R [

γ 0 = ( 0 σ 0 σ 0 0 ) , γ k = ( 0 σ k − σ k 0 ) , γ 5 = ( − σ 0 0 0 σ 0 ) . (7)

The Weyl_L and Weyl_R projectors are γ L ≡ ( 1 − γ 5 ) / 2 , and γ R ≡ ( 1 + γ 5 ) / 2 . For example, the Weyl_L component of ψ is γ L ψ . Note that W^{±} and Z only “see” the Weyl_L fields γ L ψ u or ψ ˜ v γ R . 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 component s z = + 1 2 with amplitude E − p z ξ 1 , and s z = − 1 2 with amplitude E + p z ξ 2 ;

• ψ ˜ u ≡ ψ u † γ 0 annihilates this particle;

• ψ ˜ v ≡ ψ v † γ 0 creates an antiparticle with charge -q, and spin s z = + 1 2 with amplitude E + p z η 2 * , and s z = − 1 2 with amplitude E − p z η 1 * ;

• ψ v annihilates this antiparticle.

This interpretation is needed to avoid unstable particles with negative energy. These particles and antiparticles have mass m, spin 1 2 , positive energy E, and momentum p z e → z = + E 2 − m 2 e → z . 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 L e may forward scatter on v h with amplitude Y e e N v h / 2 becoming a ν R e , which may forward scatter on v h with amplitude Y e μ E * v h / 2 becoming a ν L μ , etc. As a result, two mass eigenstates acquire masses:

ψ 1 = cos θ ψ e + sin θ ψ μ , with mass m 1 , (8)

ψ 2 = − sin θ ψ e + cos θ ψ μ , with mass m 2 . (9)

For simplicity, we have suppressed the sub-indices u for neutrinos, or v for anti-neutrinos. For example, the interaction e − W + → ν L e producing a weak state has ψ μ ( 0 ) = 0 , ν R e ( 0 ) = 0 , and [ | ν L e ( 1 ) ( 0 ) | 2 + | ν L e ( 2 ) ( 0 ) | 2 ] 1 / 2 is normalized to 1. An observation at distance L obtains e − W + with probability P e e ∝ | ψ L e ( 1 ) ( L ) | 2 + | ψ L e ( 2 ) ( L ) | 2 , or μ − W + with probability P e μ ∝ | ψ L μ ( 1 ) ( L ) | 2 + | ψ L μ ( 2 ) ( L ) | 2 , where

P e μ = 1 − P e e = 4 cos 2 θ sin 2 θ sin 2 ( X e μ ) , (10)

with X e μ ≡ Δ m e μ 2 L / ( 4 E ) , and Δ m e μ 2 ≡ m e 2 − m μ 2 . This is the phenomenon of neutrino oscillations.

If neutrinos have no additive conserved charge (such as lepton number), it is possible to add Majorana type mass terms to (3):

i σ μ ∂ μ ν R u = m ν L u + M ( ν R v ) c , i σ μ ∂ μ ( ν L v ) c = m ( ν R v ) c , (11)

i σ ¯ μ ∂ μ ( ν R v ) c = m * ( ν L v ) c + M * ν R u , i σ ¯ μ ∂ μ ν L u = m * ν R u . (12)

i σ ¯ μ ∂ μ ( ν R u ) c = m * ( ν L u ) c + M * ν R v , i σ ¯ μ ∂ μ ν L v = m * ν R v , (13)

i σ μ ∂ μ ν R v = m ν L v + M ( ν R u ) c , i σ μ ∂ μ ( ν L u ) c = m ( ν R u ) c . (14)

Here, with one generation, the masses can be made real by re-phasing the fields. The charge conjugate fields are ( ν L u ) c ≡ i σ 2 ν L u * , ( ν L v ) c ≡ i σ 2 ν L v * , ( ν R u ) c ≡ − i σ 2 ν R u * , and ( ν R v ) c ≡ − i σ 2 ν R v * . Majorana mass terms for fields ν L are not added, at tree level, because such terms are not gauge invariant. Note that the Majorana mass terms link ν R u with ( ν R v ) c , etc. Then, a created ν L u may forward scatter on v h (with amplitude Y N v h / 2 ) becoming a ν R u , that may forward scatter on M * (whatever it is, e.g. a dimension 5 operator containing v h ) becoming a ( ν R v ) c , that may forward scatter on v h (with amplitude Y N v h / 2 ) becoming a ( ν L v ) c , etc, see

Note that before EWSB, the fields ν L u and ν L v are in statistical equilibrium due to their interactions with the gauge bosons W μ and B. From (11) to (14) we conclude that after EWSB, the fields ν L u , ν R u , ( ν R v ) c ,

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

Equations (12) can be re-written as

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:

The unitary matrix U that satisfies (15), (17), and (19), with real and positive

where

From here on we take the Majorana masses

According to (18), the fields evolve as follows:

Consider a source that produces neutrinos in a weak state, e.g.

The lepton violating reaction has probability

The interpretation of these equations is discussed in Section 4.

Let us generalize to

These fields are related to the mass eigenstates as follows:

The

The masses

where

The probability to observe a lepton violating event, e.g.

where we have included the polarization miss-match factors discussed in Section 5. The probability

Note that for

Equations (29) and (31) assume neutrinos are nearly on mass shell, i.e.

In a neutrino oscillation experiment, most neutrinos traverse the detector without interacting. In the limit

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 determined, 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 probability for

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

Consider the decay

Consider the CP-conjugate decay

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

The sums in (29) and (31) only include coherent neutrinos. To obtain coherent oscillations between two neutrinos of masses

where

In the present application we take, arguably,

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

Consider the contribution of the channel

Taking the difference of these two equations, and dividing by

Summing over

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

or until wash-out sets in at

We note that

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 [^{−}^{5} s at

Big Bang Nuleosynthesis and cosmic microwave background (CMB) measurements do not allow one additional ultra-relativistic degree of freedom at

As an example, for

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

and the mass eigenstates are

The active neutrino mass

Let us write (38) for the present case

This equation assumes the approximation

Let us consider the case

Substituting in (44) we obtain

As an example, we take

where

Equation (44) can be generalized to

Without loss of generality we work in a basis that diagonalizes the

To the present order of approximation, we take

The diagonal mass matrix of active neutrinos, obtained from (28) and (48), is [

Successful leptogenesis is possible if we are able to solve (49), (38), and (39) with the needed lepton asymmetry

where R is any orthogonal, i.e.

Successful leptogenesis requires fine tuning of the unknown parameters. As a proof of principle we present the following example: normal neutrino mass ordering,

with

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 ^{−}^{12} and positive, while the terms ^{−}^{18} and can be positive or negative.

From the first term in

Asymmetries per channel are presented in ^{2}).

−4.0 × 10^{−}^{9} | 3.4 × 10^{−}^{9} | −8.6 × 10^{−}^{9} | |

(−1.8 × 10^{−}^{7}) | (1.4 × 10^{−}^{7}) | (−4.4 × 10^{−}^{7}) | |

3.4 × 10^{−}^{9} | −4.0 × 10^{−}^{8} | 2.5 × 10^{−}^{8} | |

(1.4 × 10^{−}^{7}) | (−1.6 × 10^{−}^{6}) | (1.2 × 10^{−}^{6}) | |

−8.6 × 10^{−}^{9} | 2.5 × 10^{−}^{8} | −2.7 × 10^{−}^{8} | |

(−4.4 × 10^{−}^{7}) | (1.2 × 10^{−}^{6}) | (−1.6 × 10^{−}^{6}) | |

Sum of | −9.2 × 10^{−}^{9} | −1.2 × 10^{−}^{8} | −1.0 × 10^{−}^{8} |

Several tests with modifications of this example follow:

• Setting

• Choosing a real R obtains

• Setting

• Setting

• Setting

• Results for inverse neutrino mass ordering are similar. However, we were unable to reach successful leptogenesis, i.e.

Detailed dark matter properties have recently been obtained by fitting spiral galaxy rotation curves, and, independently, by fitting galaxy stellar mass distributions [

Nevertheless, let us see if sterile neutrinos of mass

Let us consider

The sub-index “6” stands for

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

with no sum implied. For

We have studied coherent active-sterile neutrino oscillations as a possible source of leptogenesis. To this end, we add

With

The present scenario of leptogenesis requires a fine tuning parameter

The scenario studied in this article is similar to the model νMSM [

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

Hoeneisen, B. (2021) Active-Sterile Neutrino Oscillations and Leptogenesis. Journal of Modern Physics, 12, 1248-1266. https://doi.org/10.4236/jmp.2021.129077