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
Physics and Astronomy Department,University of California, Riverside, California 92521, USA
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  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 . 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.  and explained in Ref. .It is also easily generalized  and applicable to light quarks and charged leptons .In this paper, following Ref.  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. . 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.  and fully explained in Ref. . 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 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  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  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  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 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.  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 .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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