New U(1) Gauge Model of Radiative Lepton Masses with Sterile Neutrino and Dark Matter
UUCRHEP-T560December 2015
New U(1) Gauge Model of Radiative Lepton Masseswith Sterile Neutrino and Dark Matter
Rathin Adhikari
Centre for Theoretical Physics, Jamia Millia Islamia (Central University),Jamia Nagar, New Delhi 110025, India
Debasish Borah
Department of Physics, Indian Institute of Technology Guwahati, Assam 781039, India
Ernest Ma
Physics & Astronomy Department and Graduate Division,University of California, Riverside, California 92521, USAHKUST Jockey Club Institute for Advanced Study,Hong Kong University of Science and Technology, Hong Kong, China
Abstract
An anomaly-free U(1) gauge extension of the standard model (SM) is presented.Only one Higgs doublet with a nonzero vacuum expectation is required as in the SM.New fermions and scalars as well as all SM particles transform nontrivially underthis U(1), resulting in a model of three active neutrinos and one sterile neutrino, allacquiring radiative masses. Charged-lepton masses are also radiative as well as themixing between active and sterile neutrinos. At the same time, a residual Z symmetryof the U(1) gauge symmetry remains exact, allowing for the existence of dark matter. a r X i v : . [ h e p - ph ] F e b he notion that neutrino mass is connected to dark matter has motivated a large numberof studies in recent years. The simplest realization is the one-loop “scotogenic” model [1],where the standard model (SM) of quarks and leptons is augmented with a second scalardoublet ( η + , η ) and three neutral singlet fermions N R as shown in Fig. 1. Under an exactly ν νN R η η φ φ Figure 1: One-loop “scotogenic” neutrino mass.conserved discrete Z symmetry, ( η + , η ) and N are odd, allowing thus the existence of darkmatter (DM). Whereas such models are viable phenomenologically, a deeper theoreticalunderstanding of the origin of this connection is clearly desirable.Another important input to this framework is the 2012 discovery of the 125 GeV parti-cle [2, 3] at the Large Hadron Collider (LHC) which looks very much like the one Higgs bosonof the SM. This means that any extension of the SM should aim for a natural explanation ofwhy electroweak symmetry breaking appears to be embodied completely in one Higgs scalardoublet and no more.To these ends, we propose in this paper an anomaly-free U (1) X gauge extension of theSM with three active and one sterile neutrinos. Whereas there exist many studies on lightsterile neutrino masses [4, 5], we consider here for the first time the case where all massesand mixing of active and sterile neutrinos are generated in one loop through dark matter,which is stabilized by a residual Z symmetry of the spontaneously broken U (1) X gaugesymmetry. To maintain the hypothesis of only one electroweak symmetry breaking Higgs2oublet (which couples directly only to quarks in this model), charged-lepton masses are alsoradiatively generated through dark matter.The U (1) X gauge symmetry being considered is a variation of Model (C) of Ref. [6]. Ithas its origin from the observation [7, 8, 9, 10] that replacing the neutral singlet fermion N of the Type I seesaw for neutrino mass with the fermion triplet (Σ + , Σ , Σ − ) of the TypeIII seesaw also results in a possible U(1) gauge extension. The former is the well-known B − L , the latter is the model of Ref. [8], where there is one Σ for each of the three familiesof quarks and leptons. Here we consider a total of only two Σ’s, in which case several N ’sof different U (1) X charges must be added to render the model anomaly-free. Model (C) ofRef. [6] is the first such example with three N ’s. It allows radiative neutrino masses withdark matter [11, 12, 13]. It may also accommodate a sterile neutrino with radiative mass [5],but then dark matter is lost. Here we choose to satisfy the anomaly-free conditions withthree different N ’s. In so doing, we obtain a model with dark matter as well as radiativemasses and mixing for three active and one sterile neutrinos as described below.Under U (1) X , let three families of ( u, d ) L , u R , d R , ( ν, e ) L , e R transform as n , , , , respec-tively. We add two copies of (Σ + , Σ , Σ − ), each transforming as n . As shown in Ref. [6],the conditions for the absence of axial-vector anomalies in the presence of U (1) X determine n , , , in terms of n and n with 3 n + n (cid:54) = 0. To satisfy the (cid:80) U (1) X = 0 condition andthe (cid:80) U (1) X = 0 condition due to the mixed gravitational-gauge anomaly, three neutralsinglet N ’s are added. In Model (C) of Ref. [6], their charges [in units of (3 n + n ) /
8] are(3 , , − − , , + 2 + ( − = − , − , (1)( − + 1 + 5 = − , − , (2)i.e. they give identical contributions to the anomaly-free conditions. However, the latterchoice leads to the new model with particle content given in Table 1. The various scalars3article a a Z ( u, d ) L u R d R ν, l ) L l R -9/4 5/4 +Σ (+ , , − )1 R, R N R -9/4 -3/4 + S R S R φ (+ , η (+ , η (+ , χ χ χ χ +4 ξ (++ , + , U (1) X assignment given by a n + a n where 3 n + n (cid:54) = 0.have been added to allow for all fermions to acquire nonzero masses. An automatic residual Z symmetry is obtained as U (1) X is spontaneously broken by χ , . The three neutral singletfermions are relabelled N R and S R, R .The two heavy fermion triplets obtain masses from the Σ R Σ R ¯ χ interactions, whereas S R, R do so through S R S R ¯ χ and S R S R ¯ χ . The quarks get tree-level masses from¯ u R ( u L φ − d L φ + ) and (¯ u L φ + + ¯ d L φ ) d R . Note that Φ is the only scalar doublet with even Z , corresponding to the one Higgs doublet of the SM, solely responsible for electroweaksymmetry breaking. The three active neutrinos ν L and the one singlet “sterile” neutrino N R R ν L η , Φ † η ¯ χ , χ χ ¯ χ , ¯ S R ν L η , Φ † η χ , and N R S R χ . ν i ν j Σ R Σ R η η χ χ χ χ φ φ Figure 2: One-loop active neutrino mass from Σ. ν i ν j S R S R η η χ χ χ χ φ φ Figure 3: One-loop active neutrino mass from S .Looking at the one-loop diagrams of Figs. 2 and 3, we see that instead of just one extrascalar doublet η in the original scotogenic model [1], we now have two: one to couple to thetwo Σ’s, the other to S R . More importantly, because of the U (1) X assignments of Σ R and S R which come from the anomaly-free conditions, they are odd under the unbroken residual Z allowing the existence of dark matter. At the same time, the neutral singlet fermion N R is even under Z and massless at tree level, so it is suitable as a light sterile neutrino5 R S R χ S R χ χ S R χ S R N R χ χ Figure 4: One-loop sterile neutrino mass from S .once it acquires a radiative mass through S . Thus this anomaly-free U (1) X model naturallyaccommodates three active neutrinos and one sterile neutrino, all of which obtain radiativemasses. Note that the quartic couplings Φ † η ¯ χ χ and Φ † η ¯ χ χ are allowed, which alsocontribute to Figs. 2 and 3 respectively.To evaluate the one-loop diagrams of Figs. 1 to 4, we note first that each is a sum of simplediagrams with one internal fermion line and one internal scalar line. Each contribution isinfinite, but the sum is finite. In Fig. 1, it is given by [1]( M ν ) ij = (cid:88) k h ik h jk M k π [ F ( m R /M k ) − F ( m I /M k )] , (3)where M k ( k = 1 , ,
3) are the three N R Majorana masses, m R is the √ Re ( η ) mass, m I isthe √ Im ( η ) mass, and F ( x ) = x ln x/ ( x − √ Re ( η , ), √ Im ( η , ), √ Re ( χ ), √ Im ( χ ), √ Re ( ξ ), √ Im ( ξ ). Let their mass eigenstates be ζ l with mass m l . There are 4 Majorana fermion fields, spanning Σ R , Σ R , S R , S R . Let theirmass eigenstates be ψ k with mass M k . 6n Fig. 2, let the ¯Σ R ν i η and ¯Σ R ν i η couplings be h (2) i and h (2) i , then its contribution to M ν is given by( M ν ) (2) ij = h (2) i h (2) j π (cid:88) k ( z Σ1 k ) M k (cid:88) l [( y R l ) F ( x lk ) − ( y I l ) F ( x lk )]+ h (2) i h (2) j π (cid:88) k ( z Σ2 k ) M k (cid:88) l [( y R l ) F ( x lk ) − ( y I l ) F ( x lk )] , (4)where Σ R = (cid:80) k z Σ1 k ψ k , Σ R = (cid:80) k z Σ2 k ψ k , √ Re ( η ) = (cid:80) l y R l ζ l , √ Im ( η ) = (cid:80) l y I l ζ l , with (cid:80) k ( z Σ1 k ) = (cid:80) k ( z Σ2 K ) = (cid:80) l ( y R l ) = (cid:80) l ( y I l ) = 1, and x lk = m l /M k . In Fig. 3, let the¯ S R ν i η coupling be h (1) i , then its contribution to M ν is given by( M ν ) (1) ij = h (1) i h (1) j π (cid:88) k ( z S k ) M k (cid:88) l [( y R l ) F ( x lk ) − ( y I l ) F ( x lk )] , (5)where S R = (cid:80) k z S k ψ k , √ Re ( η ) = (cid:80) l y R l ζ l , √ Im ( η ) = (cid:80) l y I l ζ l , with (cid:80) k ( z S k ) = (cid:80) l ( y R l ) = (cid:80) l ( y I l ) = 1. In Fig. 4, let the S R N R χ coupling be h (3)2 , then m N = h (3)2 h (3)2 π (cid:88) k ( z S k ) M k (cid:88) l [( y R l ) F ( x lk ) − ( y I l ) F ( x lk )] , (6)where S R = (cid:80) k z S k ψ k , √ Re ( χ ) = (cid:80) l y R l ζ l , √ Im ( χ ) = (cid:80) l y I l ζ l , with (cid:80) k ( z S k ) = (cid:80) l ( y R l ) = (cid:80) l ( y I l ) = 1.In the above, the three active neutrinos ν , , acquire masses through their couplings tothree dark neutral fermions, i.e. Σ R , Σ R , S R , whereas the one sterile neutrino N acquiresmass through its coupling to S R . However, since S R mixes with S R at tree level, there isalso mixing between ν i and N as shown in Fig. 5, with m νN = h (1) i ( h (3)2 ) ∗ π (cid:88) k z S k z S k M k (cid:88) l [ y R L y R l F ( x lk ) − y I l y I l F ( x lk ] , (7)where (cid:80) k z S k z S k = (cid:80) l y R l y R l = (cid:80) l y I l y I l = 0. Note that the structures of these one-loopformulas are all similar, and there is enough freedom in choosing the various parameters toobtain masses of order 0.1 eV for ν and 1 eV for N , as well as a sizeable ν − N mixing. Note7 i N R S R S R η χ φ χ χ Figure 5: One-loop active-sterile neutrino mixing from S .also that the last term in each case corresponds to the cancellation among several scalarswhich allow the loops to be finite and should be naturally small. In Fig. 1, it is representedby the well-known ( λ / † η ) + H.c. term which splits Re ( η ) and Im ( η ) in mass. Inour case for example, in Eqs. (4) and (5), let h ∼ − , the (cid:80) z M factor ∼ (cid:80) [( y R ) − ( y I ) ] F factor ∼ − (which means that the ¯ χ χ coupling is very small), then m ν ∼ . (cid:80) [( y R ) − ( y I ) ] F factor ∼ − instead, then m N ∼ (cid:80) z z M factor be 100 GeV, and the (cid:80) [ y R y R − y I y I ] F factor ∼ − (whichmeans that the η † Φ ¯ χ χ coupling is very small), then m νN ∼ . m = 0 .
93 eV , | U e | = 0 . , | U µ | = 0 . χ , to ease the long-standing tension between ν e appearance and ν µ disappearance experiments in the ∆ m ∼ few eV range.Our model is also an example of the recently proposed framework [15, 16], where chargedleptons also acquire radiative masses. Indeed, they do so here also through the same fourdark fermions, i.e. Σ R , Σ R , S R , S R , with the addition of a scalar triplet ξ (++ , + , and ascalar singlet χ +4 , as shown in Figs. 6 and 7. There are 4 charged scalars, spanning η +1 , η +2 , ξ + , χ +4 . Let their mass eigenstates be ω + r with mass m r . In Fig. 6, let the l jR Σ R ξ + and8 iL l jR Σ R Σ R η +2 ξ + φ χ χ Figure 6: One-loop charged-lepton mass from Σ. l iL l jR S R S R η +1 χ +4 φ χ χ Figure 7: One-loop charged-lepton mass from S . l jR Σ R ξ + couplings be h ξj and h ξj , then its contribution to M l is( M l ) ( ξ ) ij = h (2) i ( h ξj ) ∗ π (cid:88) k ( z Σ1 k ) M k (cid:88) r y +2 r y ξr F ( x rk )+ h (2) i ( h ξj ) ∗ π (cid:88) k ( z Σ2 k ) M k (cid:88) r y +2 r y ξr F ( x rk ) , (8)where η +2 = (cid:80) r y +2 r ω + r , ξ + = (cid:80) r y ξr ω + r . In Fig. 7, let the l jR S R χ +4 coupling be h χj , then itscontribution to M l is( M l ) ( χ ) ij = h (2) i ( h χj ) ∗ π (cid:88) k z S k z S k M k (cid:88) r y +1 r y χr F ( x rk ) , (9)where η +1 = (cid:80) r y +1 r ω + r , χ +4 = (cid:80) r y χr ω + r . Let the (cid:80) y + yF factor ∼
1, and vary h ξ,χ from 1 to0.1 to 0.001, then m τ , m µ , m e may be obtained. Here we require the η Φ ¯ χ ξ † and η Φ ¯ χ χ ll from the value m l / (246GeV) required by the SM. Detailed analyses [17, 18] have been performed for some specificmodels.The neutral dark scalars ζ l have in general components which are not electroweak singlets( η , , ξ ). As such, they are not good dark-matter candidates because their interactions withthe Z gauge boson would result in too large a cross section for their direct detection inunderground experiments. Hence one of the neutral dark fermions ψ k is a much better DMcandidate. Note that whereas Σ R and Σ R are components of SU (2) L triplets, they do notcouple to Z because they have I = 0. Note also that they mix with S R and S R only in oneloop. The case of Σ as dark matter in the triplet fermion analog of the scotogenic modelwas discussed in Ref. [19]. Here the important change is that Σ R and Σ R have both SU (2) L and U (1) X interactions. On the other hand, suppose the lighter linear combination of S R and S R is dark matter, call it ψ , then only U (1) X is involved. As shown recently in [12, 13],the allowed region of parameter space from dark matter relic abundance, direct detectionand collider constraints corresponds to the s-wave resonance region near m X ≈ m ψ . The U (1) X gauge boson mass m X in our model comes from (cid:104) χ , (cid:105) = u , . If n = n = 1 ischosen in Table 1, then the SM Higgs does not transform under U (1) X and there is no X − Z mixing. In that case, m X = 2 g X ( u + 9 u ) . (10)The U (1) X charges of ( u, d ) L , u R , d R , ( ν, l ) L are all 1, and those of l R , N R , χ , χ are − −
3, 1, 3. These particles are even under the residual Z of U (1) X . The dark sector consistsof fermions Σ R , Σ R , S R , S R , with U (1) X charges 3/2, 3/2, 1/2, 5/2, as well as scalars η , η , χ , χ , ξ , with U (1) X charges − /
2, 1/2, 1/2, − / − / N a warm dark-matter candidate. This may require h (3)2 in Eq. (6) tobe much greater than the corresponding Yukawa couplings in Eqs. (4) and (5) for the activeneutrinos. On the other hand, ν − N mixing has to be much more suppressed in order notto overclose the Universe or conflict with observed X-ray data. According to Ref. [20], thesemay be avoided if the mixing | U i | is less than 10 − . Such a small mixing also makes the keVsterile neutrino long-lived on cosmological time scales. It could also provide an explanationto the recently observed 3.55 keV X-ray line [21] after analysing the data taken by the XMM-Newton X-Ray telescope in the spectrum of 73 galaxy clusters. The same line also appearsin the Chandra observations of the Perseus cluster [22] and the XMM-Newton observationsof the Milky Way Centre [23]. In the absence of any astrophysical interpretation of the linedue to some atomic transitions, the origin of this X-ray line can be explained naturally bysterile neutrino dark matter with mass approximately 7.1 keV decaying into a photon anda standard model neutrino. As reported in Ref. [22], the required mixing angle of sterileneutrino with active neutrino should be of the order of sin θ ≈ − − − in order togive rise to the observed X-Ray line flux. For such a tiny mixing angle, Fig. 5 must bestrongly suppressed, implying thus almost zero χ − η mixing. This in turn will make Fig. 3vanish, thus predicting one nearly massless active neutrino.We have shown in this paper how a stabilizing Z symmetry for dark matter may bederived from a new anomaly-free U (1) X extension of the standard model. Using just the oneHiggs doublet of the SM, we have also shown how three charged leptons and active neutrinosplus a sterile neutrino acquire radiative masses through the dark sector. This explains whythe sterile neutrino mass itself is also small. Apart from the possibilities of long lived sterileneutrino dark matter and cold dark matter separately as discussed above, our model is well-suited for the much more interesting mixed-dark-matter scenario, i.e. the coexistence of both.Such a scenario could be important from the point of view of large structure formation, as11ell as offering proofs in different indirect detection experiments ranging from gamma raysto X-rays. We leave such a complete analysis to future investigations.This work is supported in part by the U. S. Department of Energy under Grant No. de-sc0008541. References [1] E. Ma, Phys. Rev.
D73 , 077301 (2006).[2] ATLAS Collaboration, G. Aad et al. , Phys. Lett.
B716 , 1 (2012).[3] CMS Collaboration, S. Chatrchyan et al. , Phys. Lett.
B716 , 30 (2012).[4] A. Merle and V. Niro, JCAP , 023 (2011); J. Barry, W. Rodejohann and H. Zhang,JHEP , 091 (2011); H. Zhang, Phys. Lett.
B714 , 262 (2012); J. Barry, W. Rodejo-hann and H. Zhang, JCAP , 052 (2012); J. Heeck and H. Zhang, JHEP , 164(2013); P. S. Bhupal Dev and A. Pilaftsis, Phys. Rev.
D87 , 053007 (2013); Y. Zhang,X. Ji and R. N. Mohapatra, JHEP , 104 (2013); M. Frank and L. Selbuz, Phys.Rev.
D88 , 055003 (2013).[5] D. Borah and R. Adhikari, Phys. Lett.
B729 , 143 (2014).[6] R. Adhikari, J. Erler, and E. Ma, Phys. Lett.
B672 , 136 (2009).[7] S. M. Barr, B. Bednarz, and C. Benesh, Phys. Rev.
D34 , 235 (1986).[8] E. Ma, Mod. Phys. Lett.
A17 , 535 (2002).[9] E. Ma and D. P. Roy, Nucl. Phys.
B644 , 290 (2002).[10] S. M. Barr and I. Dorsner, Phys. Rev.
D72 , 015011 (2005).[11] D. Borah and R. Adhikari, Phys. Rev.
D85 , 095002 (2012).[12] D. Borah and A. Dasgupta, Phys. Lett.
B741 , 103 (2015).[13] D. Borah, A. Dasgupta and R. Adhikari, Phys. Rev.
D92 , 075005 (2015).1214] J. Kopp, P. A. N. Machado, M. Maltoni and T. Schwetz, JHEP , 050 (2013).[15] E. Ma, Phys. Rev. Lett. , 091801 (2014).[16] E. Ma, Phys. Rev.
D92 , 051301(R) (2015).[17] S. Fraser and E. Ma, Europhys. Lett. , 11002 (2014).[18] S. Fraser, E. Ma, and M. Zakeri, arXiv:1511.07458.[19] E. Ma and D. Suematsu, Mod. Phys. Lett.
A24 , 583 (2009).[20] K. Abazajian, Phys. Rev.
D73 , 063506 (2006).[21] A. Boyarsky, O. Ruchayskiy, D. Iakubovskyi and J. Franse, Phys. Rev. Lett. , 251301(2014).[22] E. Bulbul,M. Markevitch,A. Foster,R. K. Smith,M. Loeewenstein and S. W. Randall,Astrophys. J. , 13 (2014)[23] A. Boyarsky, J. Franse, D. Iakubovskyi and O. Ruchayskiy, Phys. Rev. Lett.115