Dirac Neutrino Mass Matrix and its Link to Freeze-in Dark Matter
aa r X i v : . [ h e p - ph ] F e b UCRHEP-T609Feb 2021
Dirac Neutrino Mass Matrix andits Link to Freeze-in Dark Matter
Ernest Ma
Physics and Astronomy Department,University of California, Riverside, California 92521, USA
Abstract
Using a mechanism which allows naturally small Dirac neutrino masses and itslinkage to a dark gauge U (1) D symmetry, a realistic Dirac neutrino mass matrix isderived from S . The dark sector naturally contains a fermion singlet having a smallseesaw mass. It is thus a good candidate for freeze-in dark matter from the decay ofthe U (1) D Higgs boson. ntroduction : It has been shown recently [1] that naturally small Dirac neutrino massesmay be linked to a dark U (1) D gauge symmetry. One specific model is studied here withthe inclusion of an S family symmetry, so that a realistic Dirac neutrino mass matrix isobtained. The dark sector consists of four singlet Majorana fermions. Its structure allowsone to be the lightest from a seesaw mechanism akin to that used in canonical Majorananeutrino mass. It is thus very suitable as freeze-in dark matter which owes its relic abundancefrom the decay of the U (1) D Higgs boson.The simple mechanism in question was first pointed out in 2001 [2]. Consider two Higgsdooublets Φ = ( φ + , φ ) and η = ( η + , η ), where η is distinguished from the standard-model(SM) Φ by a symmetry to be decided. Whereas Φ has the usual µ <
0, the corresponding m for η is positive and large. The aforesaid symmetry is assumed to be broken by the softterm µ ′ Φ † η + H.c.
The spontaneous breaking of the SU (2) L × U (1) Y gauge symmetry ofthe SM then results in the usual vacuum expectation h φ i = v , but h η i = v ′ is now givenby − µ ′ v/m , which is suppressed by the small µ ′ and large m .For neutrino mass, if ν R is chosen to transform in the same way as η , but not the otherSM particles, then it pairs up with ν L to form a Dirac fermion with mass proportional to thesmall v ′ . If the symmetry chosen also forbids ν R to have a Majorana mass, then the neutrinois a Dirac fermion with a naturally small mass. This idea of achieving a small v ′ /v ratio isakin to that of the so-called Type II seesaw, as classified in Ref. [3] and explained in Ref. [4].It is also easily generalized [5] and applicable to light quarks and charged leptons [6].In this paper, following Ref. [1] which incorporates an anomaly-free U (1) D gauge sym-metry to distinguish ν R from the other SM particles, a specific model of two massive Diracneutrinos is proposed. With the implementation of an S discrete family symmetry, a real-istic Dirac neutrino mass matrix is obtained. The natural occurrence of light freeze-in darkmatter is also discussed. 2 utline of Model : The particle content of the proposed model is listed in Table 1.fermion/scalar SU (2) L U (1) Y U (1) D S ( ν, e ) , [( ν µ , µ ) , ( ν τ , τ )] 2 − / ′ , e c , [ τ c , µ c ] 1 1 0 1 ′ , ν c , − , ′ ψ , , , , ζ , = ( φ +1 , , φ , ) 2 1/2 0 1 , ′ η , , , = ( η +1 , , , , η , , , ) 2 1/2 4 1 , ′ , χ χ χ U (1) D and S symmetries.The U (1) D gauge symmetry is anomaly-free becaue1 + 1 + 1 − − , − −
64 + 125 = 0 , (1)which is Solution (C) of Ref. [1]. It is based on the observation [7, 8, 9, 10] that ( − , − , − , − , −
4) as B − L charges for gauge B − L symmetry.The Higgs potential consists of six doublets Φ , , η , , , (which are necessary to enforcethe forms of charged-lepton and Dirac neutrino mass matrices to be discussed) and threesinglets χ , , (which are necessary for masses of the dark fermions and the link between theΦ and η doublets). Their quadratic terms are such that Φ , , χ , have negative µ , , , , but η , , , , χ have large positive m , , , , . The terms connecting them are f χ η † Φ + f χ η † Φ + f ′ χ χ ∗ η † Φ + f ′ χ χ ∗ η † Φ + µ ′ χ η † Φ + µ ′ χ η † Φ + µ χ η † Φ + µ χ η † Φ + µ χ ( η † + η † )Φ + µ χ ( η † + η † )Φ + f χ ∗ χ + f χ ∗ χ χ ∗ + µ ′ χ ∗ χ χ + µ ′ χ ∗ χ + H.c., (2)3here the µ , µ , µ , µ terms break S softly. Let h φ , i = v , , h η , , , i = v ′ , , , , h χ , , i = u , , , then v , , u , obtain nonzero vacuum expectation values in the usual way atthe breaking scales of SU (2) L × U (1) Y and U (1) D respectively, whereas v ′ , , , , u are smallbecause of the large positive m , , , , . Assuming that the soft breaking of S preserves theinterchange symmetry η ↔ η , so that m = m and v ′ = v ′ , the results are v ′ , ≃ − f , u v , − µ ′ , u v , − µ , u v , m , ,v ′ = v ′ = ≃ − µ u v − µ u v m , u ≃ − f u − f u u − µ ′ u u m . (3) Dirac Neutrino Masses and Mixing : The S representation used is that first proposed inRef. [11] and fully explained in Ref. [12]. Let ( a , a ) and ( b , b ) be doublets under S , then a b + a b ∼ , a b − a b ∼ ′ , ( a b , a b ) ∼ . (4)The structure of the charged-lepton mass matrix is determined thus by the Yukawa terms y ee c ¯ φ , y ( µµ c + τ τ c ) ¯ φ and y ( − µµ c + τ τ c ) ¯ φ , so that the 3 × e, µ, τ )to ( e c , µ c , τ c ) is diagonal with m e = y v , m µ = y v − y v , m τ = y v + y v .There are only two singlet neutrinos ν c , which couple to ( ν e , ν µ , ν τ ) through η , , , .One linear combination of the three neutrinos must then be massless. For calculationalconvenience, ν c may be added, so that the 3 × M ν = a b c − d c d , (5)where the (12) entry comes from v ′ , the (13) entry comes from v ′ , the (22) and (32) entriesare the same because they come from ( µη + τ η ) ν c , whereas the (23) and (33) entries comefrom ( − µη + τ η ) ν c .The neutrino mixing matrix is then obtained by diagonalizing M ν M † ν = | a | + | b | ac ∗ − bd ∗ ac ∗ + bd ∗ a ∗ c − b ∗ d | c | + | d | | c | − | d | a ∗ c + b ∗ d | c | − | d | | c | + | d | . (6)4ssuming that | a | | b | << (2 | d | + | b | )(2 | c | + | a | ), the eigenvalues are m ν = 0 , m ν = 2 | c | + | a | , m ν = 2 | d | + | b | . (7)Let b/d = i √ s /c , then ν = is ν e − √ c ν µ + 1 √ c ν τ . (8)Let a/c = √ c s /c , then ν = s c ν e + 1 √ c − is s ) ν µ + 1 √ c + is s ) ν τ . (9)With these choices, the massless eigenstate is automatically ν = c c ν e + 1 √ − s − ic s ) ν µ + 1 √ − s + ic s ) ν τ . (10)In other words, a completely realistic neutrino mixing scenario dubbed cobimaximal [13]with θ = π/ δ = − π/ m = 2 . × − eV , m = 7 . × − eV , s = 0 . , s = 0 . , (11)the values d = 0 .
035 eV , c = 0 . , b/d = 0 . i, a/c = 0 .
93 (12)are obtained.
Deviation from Cobimaximal Mixing : Using v ′ = v ′ , the correlation of δ = − π/ θ = π/ θ > π/ δ changes numerically with θ . Let M ν = a b ǫ ) c − (1 + ǫ ) d − ǫ ) c (1 − ǫ ) d , (13)5hen M ν M † ν = | a | + | b | (1 + ǫ )( ac ∗ − bd ∗ ) (1 − ǫ )( ac ∗ + bd ∗ )(1 + ǫ )( a ∗ c − b ∗ d ) (1 + ǫ ) ( | c | + | d | ) (1 − ǫ )( | c | − | d | )(1 − ǫ )( a ∗ c + b ∗ d ) (1 − ǫ )( | c | − | d | ) (1 − ǫ ) ( | c | + | d | ) . (14)To first order in ǫ , the mass eigenvalues are the same, with the following changes in themixing parameters: s = − √ ǫ ) , c = 1 √ − ǫ ) , e − iδ = ie iθ ′ , (15) bd = i √ s c e iθ ′ , ca = c − iǫs s √ s c , θ ′ = 2 ǫ | c | | d | s c c s . (16)Using the world average [14] s = 0 . ǫ = 0 . , θ ′ = 8 . × − . (17) Dark Sector : The dark sector consists of four fermion singlets ψ , , ∼ ζ ∼
5. Becauseof the chosen scalars χ ∼ χ ∼ χ ∼
6, there is no connection to the two singletneutrinos ν c , ∼ −
4. Hence ψ , , , ζ may be considered odd under an induced Z symmetrywhich stabilizes the lightest among them as dark matter. The 4 × ζ , ψ , , ) is of the form M ζ,ψ = h ′ u h ′ u h ′ u h ′ u h u h ′ u h u h ′ u h u . (18)Recalling that u << u from Eq. (3), it is clear that ζ gets a very small mass, i.e. m ζ = − h ′ u h u − h ′ u h u − h ′ u h u . (19)Since χ has large and positive m so that u is very small, the breaking of U (1) D ismainly through χ , . The relevant part of the Higgs potential is then V = − µ χ ∗ χ − µ χ ∗ χ + [ µ ′ χ ∗ χ + H.c. ]+ 12 λ ( χ ∗ χ ) + 12 λ ( χ ∗ χ ) + λ ( χ ∗ χ )( χ ∗ χ ) . (20)6et H , = √ Re ( χ , ), then the 2 × H , is M H = λ u λ u u + 2 µ ′ u λ u u + 2 µ ′ u λ u − µ ′ u /u ! . (21)Let H be the lighter, with mixing θ to H . Now H does not couple to ψψ , but H doesand through ψ − ζ mixing to ζ ζ with Yukawa coupling y H = m ζ / √ u . From Eqs. (3) and(19), it is clear that m ζ << u , hence y H is very much suppressed.Consider now the very light Majorana fermion ζ as dark matter. It has gauge interac-tions, but if the reheat temperature of the Universe is much below the mass of the U (1) D gauge boson as well as m H , then it interacts only very feebly through H . Its productionmechanism in the early Universe is freeze-in [15] by H decay before the latter decouplesfrom the thermal bath. The decay rate of H → ζ ζ isΓ H = y H θ m H π √ − r (1 − r ) , (22)where r = m ζ /m H . For r <<
1, the correct relic abundance is obtained for [16] y H θ ∼ − r − / . (23)This translates to m ζ θ ( u m H ) / ∼ × − , (24)which may be satisfied for example with u = 10 GeV, m H = 600 GeV, θ = 0 .
1, and m ζ = 80 MeV. The decoupling temperature for ζ is roughly T ∼ m Z D /m Z ) / = 5 . SU (10) → SU (5) × U (1) χ . Concluding Remarks : The right-handed neutrino ν R has been proposed as the link [1]to dark matter by having it transform under a new U (1) D gauge symmetry. The smallDirac neutrino masses are enforced by a seesaw mechanism proposed in Ref. [2] using newHiggs doublets also transforming under U (1) D . With the help of the non-Abelian discrete7ymmetry S , it is shown how a realistic neutrino mixing matrix may be obtained which isapproximately cobimaximal [13].After the spontaneous breaking of U (1) D , a dark parity remains for four Majoranafermions, the lightest of which has a seesaw mass. It is suitable as freeze-in dark matterwith its relic abundance coming from the decay of the U (1) D Higgs boson.
Acknowledgement : This work was supported in part by the U. S. Department of EnergyGrant No. DE-SC0008541.
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