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Home , Sterile neutrino

PRL-TH/95-11

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

QUASI GOLDSTONE AS A STERILE

Eung Jin Chun Anjan S. Joshipura and INTERNATIONAL ATOMIC ENERGY Alexei Yu. Smirnov AGENCY

UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL MIRAMARE-TRIESTE ORGANIZATION

IC/95/164 PRL-TH/95-11 1 Introduction International Atomic Energy Agency and All the experimentally known transform uon-trivially under the gauge group 677(3) United Nations Educational Scientific and Cultural Organization SU(2) x f/(l) of the (SM). However there are experimental hints in tiu> INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS neutrino sector which suggest the existence of SU(S) x SU[2) x (7(1) - singlet fermions mixing appreciably with the known . These hints come from (a) the deficits m the solar [1] and atmospheric [2] neutrino fluxes (b) possible need of significant hot. component [3] in tin- QUASI GOLDSTONE FERMION AS A STERILE NEUTRINO dark of the universe and (c) some indication of v, - vl:. oscillations in the laboratory [4], These hints can be reconciled with each other if there exists a fourth very light (< (9(eV)) Eung Jin Chun neutrino mixed with some of the known neutrinos preferably with the out-. The fourth International Center for Theoretical Physics, Trieste, Italy, neutrino is required to be sterile in view of the strong bounds on number of neutrino flavours Anja.li S. .loshipura coming both from the LEP experiment as well as from the primordial nucleosynthesis [5]. Theoretical Physics Group, Physical Research Laboratory, The existence of very light sterile neutrino demands theoretical justification since unlike a. Ahmedabad, 380 009, India the active neutrinos, the mass of the sterile state is not. protected by the gauge symmetry of and the SM and hence could be very large. Usually the sterile neutrino is conskiercd on the saint- Alexei Yu. Smirnov footing as the active neutrinos and some ad hoc symmetry is introduced to keep this neutrino International Center for Theoretical Physics, Trieste, Italy light. Recently there arc several attempts to construct models for sterile neutrinos which have and Institute for Nuclear Research, Russian Academy of Sciences, the origin beyond the usual structure [6, 7, 8], In particular in Ref. [6] we suggested 117312 Moscow, Russian Federation. a possibility that (SUSY) may be responsible for both the existence and tin- lightness of the sterile fermions. ABSTRACT One could consider three different ways in which supersymmetry can keep sterile states The existence of sterile neutrino is hinted by simultaneous explanation of diverse neutrino very light. anomalies. We suggest that the quasi Goldstone fermions (QGF) arising in supersymmetric (1) Combination of supersymmetry and the (continuous) R symmetry present in many Hiipcr- theory as a result, of spontaneous breaking of global symmetry like the Peccei-Quinn symmetry symmetric models may not allow a mass term for the light sterile state. or the symmetry can play a role of the sterile neutrino. The smallness of mass (2) Spontaneous breakdown of some other global symmetry in supersymmetric theory can lead : of QGF (ms ~ 10~* - 10 eV) can be related to the specific choice of superpotential or to massless fermions which form the of the Goldstone . Kahler potential (e.g., no-scale kinetic terms for certain superfields). Mixing of QGF with (3) The spontaneous breakdown of the global supersymmetry itself would give rise to a massless neutrinos implies the R-parity violation. It can proceed via the coupling of QGF with the fermion, the goldstino. Higgs supermultiplets or directly with the lepton doublet. A model which accounts for the The mechanism (1) and its phcnomenological consequences were discussed in Ref. ((>]. solar and atmospheric anomalies and the is presented. Mechanism (3) though appealing is not favoured phenomenologieally in view of the difficulties in building realistic models based on the spontaneously broken global SUf'Y. We discuss in this paper implications of the mechanism (2) concentrating for definiteness on the simplest MIRAMARE • TRIESTE case of a global U (!)<-,•. July 1995 The spontaneously broken global symmetries are required for reasons unrelated to the •"Wp.lli!!

existence of ljf;;ht sterile states. The most interestim;, example:- being spontaneously broken In liic supersynunetnc limit ihe fermiouic component of the Goldstone is masslcss. lepton number .symmetry [9] and the Peceei-Quinri (PQ) syimnetry imposed [10] to :-

In this section and subsequently, we will consider the following general superpotent.ial

SUSY is broken in this example if f'f / f.j. For a minimum with the F-terms: /<„ = Fn, = 0, W = + Wm U) the Goldstone fermion in Eq. (3) remains massless at the tree level in spite of the SUSY where W is assumed to be invariant, under some global symmetry U{\)c;. As we outlined breakdown. As we noticed before this version has phenomenological problems and further on in the introduction, this symmetry may be identified with the PQ symmetry, lepton number we will concentrate on possibilities related to supergravity. symmetry or combination thereof. The first term in Bq. (1) refers to the superpot.ential of the The mass of the QGF in supergravity theory is typically of the order of mass minimal superaymmetric standard model (MSSM). The second term contains .5(7(3) x SU(2) x m :t/v (= WSVSY} [IS, 17, 18]. For instance, the superpot.ential in Eq. (2) leads to ms ~ mA/i C/(l) singlet superfields which arc responsible for the breakdown of U[l)c- The minimal choice when generic soft terms of SUSY breakdown are allowed [16]. Howerver, the mass m,s can be for Wg is much smaller lor specific choices of 1) the superpotential and/or 2) soft. SUSY breaking terms. Ws = \(U

2 of Bq. (1) describes mixing of the singlet fields with the superfields of the MSSM. \(

3 is shown [)7] to generate the tree level mass where we have omitted the generation indices. The first, term in E(j. (8) produces the Dirac masses of neutrinos, whereas the second one gives the Majorana masses of R.H neutrino com- 10 KI ponents. The scale fG ~ 1O - lO^GeV generates M ~ 10 - 10" GeV required by the as in the global < aso if the minimal kinetic terms of the fields are assumed. For commonly HDM and atmospheric neutrinos. 10 2 accepted value of the PQ symmetry breaking .scale. fa = frQ = 10 - 10' GeV, one gets (i) Suppose that only o,rr',y superfields belong to the Z sector, whereas all other HH- :i from Eq. (5) mlS- ~ (10 - 10 ) eV. On the other hand, the value of ms in Eq. (4) desired for perfields have minimal kinetic terms: NJh,L 6 C. Then SUSY breaking induces the soft lfl explanation of the solar neutrino deficit requires fa ~ 10 GeV which can be related to the term M ~ - grand unification scale. To identify f<; with fPq, one should overcome the cosmological bound r N 12 fa fi-Q < 1O GeV. The bound can br removed by nxion mixing with some other Goldstone which generates the mass of QGF in one loop (Fig. 1): boson in their kmctic terms [19] or by field driven to small values in inflationary period [20]. In this case however, the nxioii cannot play the role of . 2). Another possibility to get very light S is based on the idea of no-scale supergravity [21]. This mechanism is similar to that of the mass generation by coupling of S with heavy 3 The Kahler potential and the super-potential < an be arranged in sneh a way that supcrsym- |18, 22). For AN ~ O(mm) and {M/fG) ~ JO" , m,v is in the keV range. metry breaking is communicated to the singlet S via a set of interactions. As the result, the (ii) [/fit us suppose that not only a,a',y but also N have the no-scale kinetic terms, hi mass ot S appears in one, two or even three loops. this case A^- = 0 at. tree level, but. non-zero A^ will be generated in one loop (see Fig. 2) Let us consider the following Kahler potential: by the soft breaking term related to usual LNH?. A^m^LNH'j, and by the quartic coupling aNL'H^ which follows from |WNP term of the supersymmetric scalar T - zuz;,) + c,c;, (6) potential. As the result one has D 1 where T is the moduli field appearing in the underlying superstring theory. Za and Q are the 1 I rn \ matter superfields which have the no-scale kinetic term (Z-sector) and the minimal kinetic term (C -sector) respectively. The corresponding scalar potential at the scale reads, Correspondingly, ms appears in two loops (Fig. 2). Combining Eqs. (10) and (11) we get. the estimation of m.v: V = h.c.} + \Wa (7) m<; ^ ———TTTT—;. . m,, . (12) where m<> = O(m yj). The tree-level masses of the fermionic components of the fields Z are n : a Here m,, = (m )"'IM. For the HDM mass scale ra, ~ 3 eV, A,, ~ v2 ~ 100 GeV and determined by the global supersymmetrk results. Therefore, if the singlet fields triggering 12 fa ^ 10 GeV it follows from Eq. (12) that ms ~ 3 • 10'•' eV can be achieved if the mass of U(\)a breaking are in the Z sector, the QGF will be massless at tree level [18]. The QGF RH component is M - 10H GeV. will acquire the mass through the interactions with fields C\ having minimal kinetic terms, In this version of model the left, and right neutrino components have different kinetic terms and consequently, usual soft SUSY breaking terms. Moreover, 5 (or

V, Here n2 and 7;i,^ arc the SU{2) fine structure constant and gaugino mass respectively- For where Mp is the Planck scale mass. In this case, u = A ^7 is naturally about the weak scale. 12 :1 mu ~ 3 eV, m,/) ~ v2 ^ 100 GeV, and fG - 10 GeV, one gets from Eq. (13) ms ~ 3 • 10" Since fpQ ~ (Q »f the nxionic can occur only in the presence of either explicit or spontaneous violation of the R parity field S. The interaction in Eq. (14) results in the following coupling conventionally imposed in the MSSM [25]. Indeed, the Higgs field which breaks U(1)G may WUS ' (IK) belong either to R even or odd superfield depending upon the nature of the U(\)c,- If it belongs £' to R even (i.e. Higgs like) superfield then the corresponding QGF is R odd and its mixing with , S, h\, ii%): 1. PQ symmetry. The supersymmetric theories with Peccei-Quinn symmetry may contain 0 0 0 a term 0 m°s CjlV COS qw sin (•20) \HiH2a, (14) 0 citvsinf] 0 0 with 2 is the weak scale, tan/V = v-2/Vy antl v\$ are the VEV's of H]2- I" 10 1J2 (a) ~ fpQ would be large ~ 10 — 10 GeV. Since this VEV generates the parameter matrix (20) we have included also the direct axino mass /«"• that, can be generated by the H = \{ff) of the MSSM through the interaction (14), one would need to fine tune A in order to mechanisms of section 2. We have neglected the contribution from the interactions with the understand the smallness of JJ.. The coupling of axionic supermultiplet, S to Higgs superfield in Eq. (20). In general gauginos mix with through v\.2. This mixing will is then given by not change the qualitative results which follow from Eq. (20). Moreover, the mixing can be (15) small if the gaugino mass is chosen much larger than the fi-parameter. Gauginos will also mix JPQ ] The smallness of j.i can be understood if a couples to Higgs through non-renormalizable with neutrinos through the VEV of sneutrino field which may arise due to t ie presence of the t term [11] coupling in Eq. (19) and soft SUSY breaking terms. This mixing generates [26] neutrino mass J 2 of order j (i;) /rai/2 (9 is the SU{2) coupling constant). For 7H1/2 > 100GeV and (£>) < 10 AH,H2^-, (16) keV, this contribution is much smaller than m" ~ 10"a eV which can result from the radiative corrections. Block diagoualizatioii of the matrix (20) leads to the following effective mass matrix for coupling between QGF and neutrino. Note that a is similar to the RH neutrino miiip

/pQ!«,,y^I<4.l0»GoV. (22) where fL denotes the scale associated with the spontaneous breaking of lhe lepton number

In this case, however, cannot provide the cold dark matter of the Universe. Note symmetry and c.t is a parameter of order unity. The mass matrix generated by Fq. (2(i) is that the lightest supersymmetric cannot be cold dark matter either because of their 0 cttv sin p/fL (27) instability due to the /?-parity violation or due to their decay into the lighter axino. For c,€v sin p/fL «'"• 1(l fi'Q > 10 GeV the QGF mass generated via /x-term is too small for the MSW solution. For a and the desired vt — S mixing can be obtained for e ~ 0.1 MeV und /;, ~ l()' GeV. 5 frq ~ 10" GeV, m$ SB 10~ eV is in the region of "just-so" solution of the solar neutrino Let us give an example of models which leads to the mixing term of Ecj. (20). Consider the 12 problem. The axiom can however serve as cold dark matter provided /PQ ~ 10 GeV. In this U{1)L charge assignments (1,-1,-3) for the fields (tr, IT', L) respectively. All other fields are case, the seesaw contribution to ms is very small and one needs a non-vanishing mass raj. taken neutral. The relevant part for the U{l)a invariant superpotcntial is given IUS follows: If m^ is the dominant contribution to the mass of S, ms — m%, one obtains from Bq. (21) W = \{aa' - fl)y (2K) for the v — S mixing Ml' c fusin/3 tan eus (23) where the first term breaks the lepton symmetry and generates supermultiplet. of

2 Eq. (3). The second term in Eq. (28) generates the effective interaction displayed in Eq. (2U1 Then the desired value, taxi6U3 ~ (2-6) • 10~ eV (4), can be obtained if the R parity breaking with c = ^ and e ~ ~£ftfl- Thus specific choice for the lepton charges allows one to convlal e parameter f. equals t 12 6 to the scale fL. In particular, for Sc ~ 0.1 and fL ~ 10 GeV, one has f ~ 1 MeV. f = :—-r— « (2 — 6) • 10" ——- . (24) c v sin p sm (i )t 3. PQ as the lepton number symmetry. If both Higgs and leptons transform non-trivially For JPQ ~ 1012 GeV one has t ~ 0.1 MeV. In general, the appropriate range of t is (10~3 - under the U(l)a symmetry then the latter can play a dual role of the PQ symmetry and the 10) MeV. It can be generated as a radiative correction: e ~ h2m / /16?r2. Alternatively, e 3 2 lepton number symmetry as in Ref. [27]. In this case one can correlate the origin of e and may arise through the coupling of the product LH-i to some fields carrying non zero lepton )j, to the same symmetry breaking scale JFQ. The neutrino coupling to QGF is given by the number. In this case the required smallness of c may be understood in analogy with that of combination of Eqs. (19) and (26): //-parameter.

Wm (29) 2. Lepton number symmetry. Let us identify U{\)a with the Septon number symmetry. alns = Unlike in the previous case, it is possible now to couple the QGF directly to neutrino through the term This WmiIiny generates the following effective mass matrix for v and 5 which is the conibinat ion hbHia - \*%>) of Eq. (21) and Eq. (27):

This is analogous to Eq. (14) but now the scalar component of a is R odd and its VEV breaks 0 {c, -rv)(v*hul/fpQ ,

R parity. Electroweak symmetry breaking v2 ^ 0 leads through the term (25) to the direct

10 According to Eq, (30) tilt \> — S mixing angle 9IIS is determined by • The superpotential (33) leads to the mass matrix in (i/(i. vT, N,,, NT) basis: 0 0 m» (I v2sm20lf ' ^ PQ 0 0 0 ml? The (7-charge prescription {-1,-1, 1,-1,-2) for (Hi, H2, o,

U(\)o invariant superpotcntiak \ 0 in!? MllT MT ) 1 2 l W - Xiao - ff,Q)y +-^-HtH,a + -%LH2o . (32) The above mass matrix gives rise to pseudo-Dirac neutrino with a common mans v Mr Mf, It gives the terms displayed in Eq. (29) with r, = -j$*cp = \/2. JnDM ^ —T7 - 1->'>J

4 Model This mass can be in the eV range as required for the solution of the dark matter problem

by taking the values m£ - 0.1 GeV, mi' - 50 GeV and M,lT ~ Hf GeV. The muss Let us put together the basic ingredients discussed in section 2 and 3 into a model which splitting is given by 2 LI simultaneously explains the solar, atmospheric and the dark matter problems. In principle AmAm2 /m(ml\ I MrM \ the sterile state, like axino, could mix with any of the neutrinos but the possibility of the vr. — S 1 2 mixing which solves the solar neutrino problem seems most preferred phenomenologically. The Taking {jj? -) ~ 1, one reproduces both mixing and Am' required to explain the atmo- required range of the v,, — S mixing and 5 mass is given in Eq. (4). The alternative possibility spheric anomaly. of Up — S mixing accounting for the atmospheric neutrino deficit conflicts with the cosmological The charge prescription, G(Ne) = 0, permits the bare mass term MNeNc or the non-rcnorrnalizable bound coming from the nucleosynthesis. (i 18 term hNeNe(Ta'/MP which will produce Me ~ 10 - 10 GeV. The Dirac mass term is 12 Let, us consider the model with U(l)r; = U(\)PQ broken at }PQ ~ 10 GeV in which the li J generated by high-order non-renormalizable term: hLtNeH2(T /MP, and therefore, m' ~ mass of QGF is generated in two or three loops via the interaction with the RH neutrino me{fpQ/Mp)3 is negligibly small. components (S) and the mixing is induced by the Le-eoupling described by the superpotential One can get more symmetric or regular charge prescription introducing more singlet fields (32). To suppress the mixing of S with v^^ and to get pscudo-Dirac structure for v^ — vT or a horizontal symmetry in addition to U(l)a- system (needed to explain simultaneously the HDM and the atmospheric neutrino problem), we The model presented above does not contain any mixing between v(, and i/l:T. Such mixing suggest that U( \)c, is generation dependent '. Consider, for example, the following prescription can be induced, for example, by adding new Higgs field which could generate a Dirac mans of U(\)c, charges: term meTu,,NT. This give rise to the ve — i/(1 mixing angle dr)1 ~ ^ being in the range of H, H-i u a' L L L N iV N e p T e M r sensitivity of KARMEN and LSND [4] for m.rT ~ 30 MeV, m,, - GeV [6], -I -1 1 -1 -2 -1/2 3/2 0 3/2 -1/2 Tliis choice gives rise to the desired phenomenological results. Specifically, 5 Conclusions • The mixing angle (31) following from the superpotential (32) can fall in the required Simultaneous explanation of different neutrino anomalies hints to the existence of sterile neu- range (4) if f - 1 MeV and f - 1012 GeV. PQ trino. We have considered a possibility that the sterile neutrino is the quasi Goldstone fermion, • The above assignments lead to the following superpotential in the fi — T sector: which appears as the result of spontaneous breaking of a global U(l)c, symmetry in super-

symmetry theory. This global U(\)G symmetry can be identified with the PQ symmetry, the c»=j»,f " fPQ fi'Q lepton number symmetry or the horizontal symmetry. :| These couplings generate the axino mass vn% in the MSW range as discussed in section 2. The mass of QGF generated by SUSY breaking can be as small as 1(T eV so that i/t->5 can introduce for this an additional horizontal symmetry, suggesting that U(l)c is generation blind, resonance conversion solves the solar neutrino problem. In the supergravity theories such a

11 12 smallness of ms is related to special forms of superpotential and the scale of U(l)c breaking [3] J. R. Primack, I Holtzman, A. Klypin, and D. 0. Caldwcll, Phys. Rev. Lett. 74. 2161) 16 fc: ~ 10 GeV or to no-scale kinetic terms for certain superfields, In the last case, m.s is (1995): and references therein. generated in two or three loops. [4] LSND collaboration, LA-UR-95-1238 (micl-ex/9504002). The mixing of QGF with the neutrinos implies spontaneous or explicit violation of the R parity. QGF can mix with neutrino via interaction with Higgs multiplets (in the ca.se of PQ [5] K. EiKivist, ,1. Maalanipi and V. B. Semikoz, preprint. HU-TFT-95/28 (liq>-pli/(jr>0521l>); symmetry) or directly via coupling with the combination LH2 (in the case of lepton number '"•)!• previous works see, X. Shi, D. N. Schramm and B. D. Fields, Phys. Rev. D 48. 25M symmetry). (1903); and references therein. The r/(l)c-symmet.ry being generation dependent can simultaneously explain i.ht: domi- [6) E. J. Chuti, A. S. Joshipura and A. Yu. Smirnov. preprint. IC/95/7G, PRL-TH/95-7 (hep- nance of QGF coupling with electron neutrino and pseudo-Dirac structure of vti — vT system ph/9505275). needed to explain the atmospheric neutrino problem and HDM. The PQ breaking scale JI

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Fig. 1; One-loop diagram for the QGF mass. The solid lines are fermions and t.ht; dotted

lines axe bosons. AN is the soft parameter of NNa.

15 16 Fig. 2: Two-loop diagram for the QGF mass. AD is the soft parameter of LNH2. Fig. 3: Three-loop diagram for the QGF mass. The cross with m,^ denotes gaugino mass insertion.

17 18 Stampato in proprio nella tipografia del Centro Internazionale di Fisica Teorica