Abstract

A simple model to accomodate light sterile neutrinos naturally with large mixing with the usual neutrinos has been proposed. The standard model gauge group is extended to include an S⁒U⁒(2)Sπ‘†π‘ˆsubscript2𝑆SU(2)_{S} gauge symmetry. Heavy triplet higgs scalars give small masses to the left-handed neutrinos, while a heavy doublet higgs scalar give mixing with the sterile neutrinos of the same order of magnitude. The neutrino mass matrix thus obtained can explain the solar neutrino deficit, the atmospheric neutrino deficit, the LSND data and hot dark matter. Lepton number is violated here through decays of the heavy triplet higgs, which generates the lepton asymmetry of the universe, which in turn generates a baryon asymmetry of the universe.

DESY 98-083

July 1998

Naturally light sterile neutrinos

[0pt] Utpal Sarkar (a,b)π‘Žπ‘{}^{(a,b)} 11E-mail: utpal@prl.ernet.in

(a)π‘Ž{}^{(a)} Theory Group, DESY, Notkestraße 85, 22603 Hamburg, Germany

(b)𝑏{}^{(b)} Theory Group, Physical Research Laboratory, Ahmedabad, 380 009, India 22permanent address


Recently the Super Kamiokande [1]announced a positive evidence of neutrino oscillations. They attribute the Ξ½ΞΌsubscriptπœˆπœ‡\nu_{\mu} deficit in the atmospheric neutrino to a Ξ½ΞΌsubscriptπœˆπœ‡\nu_{\mu} oscillating into a Ξ½a⁒t⁒msubscriptπœˆπ‘Žπ‘‘π‘š\nu_{atm} , where Ξ½a⁒t⁒msubscriptπœˆπ‘Žπ‘‘π‘š\nu_{atm} could be a Ξ½Ο„subscript𝜈𝜏\nu_{\tau} or a sterile neutrinos (which is a singlet under the standard model) with Δ⁒ma⁒t⁒m⁒o⁒s2=mΞ½ΞΌ2-ma⁒t⁒m2∼(0.5-6)Γ—10-3⁒eV2.Ξ”superscriptsubscriptπ‘šπ‘Žπ‘‘π‘šπ‘œπ‘ 2superscriptsubscriptπ‘šsubscriptπœˆπœ‡2superscriptsubscriptπ‘šπ‘Žπ‘‘π‘š2similar-to0.56superscript103superscripteV2\Delta m_{atmos}^{2}=m_{\nu_{\mu}}^{2}-m_{atm}^{2}\sim(0.5-6)\times 10^{-3}{% \rm eV}^{2}. There are also indications of neutrino oscillations in the neutrinos coming from the sun. The solar neutrino deficit can be explained if one considers Ξ½eβ†’Ξ½s⁒o⁒lβ†’subscriptπœˆπ‘’subscriptπœˆπ‘ π‘œπ‘™\nu_{e}\to\nu_{sol} oscillations (where Ξ½s⁒o⁒lsubscriptπœˆπ‘ π‘œπ‘™\nu_{sol} could be Ξ½ΞΌsubscriptπœˆπœ‡\nu_{\mu} or Ξ½Ο„subscript𝜈𝜏\nu_{\tau} or a sterile neutrino) with the mass squared difference [2]Δ⁒ms⁒o⁒l⁒a⁒r2=mΞ½s⁒o⁒l2-mΞ½e2∼(0.3-1.2)Γ—10-5⁒eV2,Ξ”superscriptsubscriptπ‘šπ‘ π‘œπ‘™π‘Žπ‘Ÿ2superscriptsubscriptπ‘šsubscriptπœˆπ‘ π‘œπ‘™2superscriptsubscriptπ‘šsubscriptπœˆπ‘’2similar-to0.31.2superscript105superscripteV2\Delta m_{solar}^{2}=m_{\nu_{sol}}^{2}-m_{\nu_{e}}^{2}\sim(0.3-1.2)\times 10^{% -5}{\rm eV}^{2}, This mass squared difference is for resonant oscillation [3]. If one assumes vacuum oscillation solution of the solar neutrino deficit, then this number will be several orders of magnitude smaller. If we assume a three generation scenario, Ξ½a⁒t⁒msubscriptπœˆπ‘Žπ‘‘π‘š\nu_{atm} is then identified with Ξ½Ο„subscript𝜈𝜏\nu_{\tau} and Ξ½s⁒o⁒lsubscriptπœˆπ‘ π‘œπ‘™\nu_{sol} could be a Ξ½ΞΌsubscriptπœˆπœ‡\nu_{\mu} or a Ξ½Ο„subscript𝜈𝜏\nu_{\tau} . Consider, Ξ½s⁒o⁒l≑ντsubscriptπœˆπ‘ π‘œπ‘™subscript𝜈𝜏\nu_{sol}\equiv\nu_{\tau} , which implies, mΞ½ΞΌ2-mΞ½e2=[mΞ½ΞΌ2-mΞ½Ο„2]+[mΞ½Ο„2-mΞ½e2]=Δ⁒ma⁒t⁒m⁒o⁒s2+Δ⁒ms⁒o⁒l⁒a⁒r2∼10-2⁒eV2.superscriptsubscriptπ‘šsubscriptπœˆπœ‡2superscriptsubscriptπ‘šsubscriptπœˆπ‘’2delimited-[]superscriptsubscriptπ‘šsubscriptπœˆπœ‡2superscriptsubscriptπ‘šsubscript𝜈𝜏2delimited-[]superscriptsubscriptπ‘šsubscript𝜈𝜏2superscriptsubscriptπ‘šsubscriptπœˆπ‘’2Ξ”superscriptsubscriptπ‘šπ‘Žπ‘‘π‘šπ‘œπ‘ 2Ξ”superscriptsubscriptπ‘šπ‘ π‘œπ‘™π‘Žπ‘Ÿ2similar-tosuperscript102superscripteV2m_{\nu_{\mu}}^{2}-m_{\nu_{e}}^{2}=[m_{\nu_{\mu}}^{2}-m_{\nu_{\tau}}^{2}]+[m_{% \nu_{\tau}}^{2}-m_{\nu_{e}}^{2}]=\Delta m_{atmos}^{2}+\Delta m_{solar}^{2}\sim 1% 0^{-2}{\rm eV}^{2}. Then we cannot explain the LSND result [4], which announced a positive evidence of Ξ½ΞΌΒ―β†’Ξ½eΒ―β†’Β―subscriptπœˆπœ‡Β―subscriptπœˆπ‘’\overline{\nu_{\mu}}\to\overline{\nu_{e}} oscillations with the mass squared difference (alternate explanation is not possible [5]) Δ⁒mL⁒S⁒N⁒D2=mΞ½ΞΌ2-mΞ½e2∼(0.2-2)⁒eV2Ξ”superscriptsubscriptπ‘šπΏπ‘†π‘π·2superscriptsubscriptπ‘šsubscriptπœˆπœ‡2superscriptsubscriptπ‘šsubscriptπœˆπ‘’2similar-to0.22superscripteV2\Delta m_{LSND}^{2}=m_{\nu_{\mu}}^{2}-m_{\nu_{e}}^{2}\sim(0.2-2){\rm eV}^{2} . This conclusion is true even if we consider Ξ½s⁒o⁒l≑νμsubscriptπœˆπ‘ π‘œπ‘™subscriptπœˆπœ‡\nu_{sol}\equiv\nu_{\mu} .

As a solution to this problem one can say that either LSND result is wrong or there has to be some other explanation for the solar neutrino deficit. But a more popular solution is to extend the standard model to incorporate a sterile neutrino and explain all these experiments [6]. Since LEP data [7]ruled out any possibility of a fourth S⁒U⁒(2)Lπ‘†π‘ˆsubscript2𝐿SU(2)_{L} doublet left-handed neutrino, this fourth neutrino has to be a sterile neutrino, which does not interact through any of the standard model gauge bosons. Incorporating such light sterile neutrino with large mixing with the other light neutrinos in extensions of the standard model is non-trivial [6]. Recently there is one attempt to extend the radiative neutrino mass generation model by Zee [8]to incorporate a sterile neutrino [9]. However, that model cannot explain the baryon asymmetry of the universe.

In this article, we propose a new scenario with an S⁒U⁒(2)π‘†π‘ˆ2SU(2) symmetry, which can provide a naturally light sterile neutrino with large mixing with the other left-handed neutrinos. The neutrino mass matrix with the sterile neutrinos can now explain the LSND data, the solar neutrino problem, the atmospheric neutrino anomaly and the dark matter problem. The lepton number violation at a very high scale generates a lepton asymmetry of the universe, which then gets converted to the baryon asymmetry of the universe during the electroweak phase transition.

We work in an extension of the standard model which includes a couple of heavy triplet higgs scalars [11, 10], whose couplings violate lepton number explicitly at a very high scale, which in turn gives small neutrino masses naturally. Decays of these triplet higgses generates a lepton asymmetry of the universe [10]. We extend this minimal scenario to include a S⁒U⁒(2)Sπ‘†π‘ˆsubscript2𝑆SU(2)_{S} gauge group so as to extend the standard model gauge group to

𝒒e⁒x⁒t≑S⁒U⁒(3)cΓ—S⁒U⁒(2)LΓ—U⁒(1)YΓ—S⁒U⁒(2)S.subscript𝒒𝑒π‘₯π‘‘π‘†π‘ˆsubscript3π‘π‘†π‘ˆsubscript2πΏπ‘ˆsubscript1π‘Œπ‘†π‘ˆsubscript2𝑆{\cal G}_{ext}\equiv SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}\times SU(2)_{S}.

The S⁒U⁒(2)Sπ‘†π‘ˆsubscript2𝑆SU(2)_{S} symmetry breaks down alongwith the lepton number at very high energy ( M𝑀M ), and the out-of-equilibrium condition for generating baryon asymmetry of the universe determines this scale M𝑀M [13]. Since all representations of the S⁒U⁒(2)π‘†π‘ˆ2SU(2) groups are pseudo-real and anomaly free, there is no additional constraints coming from cancellation of anomaly. This makes this mechanism easy to implement in different scenarios.

The fermion and the scalar content of the standard model, which transformations under S⁒U⁒(3)cΓ—S⁒U⁒(2)LΓ—U⁒(1)Yπ‘†π‘ˆsubscript3π‘π‘†π‘ˆsubscript2πΏπ‘ˆsubscript1π‘ŒSU(3)_{c}\times SU(2)_{L}\times U(1)_{Y} as

qi⁒L≑(3,2,1/6)subscriptπ‘žπ‘–πΏ3216\displaystyle q_{iL}\equiv(3,2,1/6) ui⁒R≑(3,1,2/3)subscript𝑒𝑖𝑅3123\displaystyle u_{iR}\equiv(3,1,2/3) di⁒R≑(3,1,-1/3)subscript𝑑𝑖𝑅3113\displaystyle d_{iR}\equiv(3,1,-1/3)
li⁒L≑(1,2,-1/2)subscript𝑙𝑖𝐿1212\displaystyle l_{iL}\equiv(1,2,-1/2) ei⁒R≑(1,1,-1)subscript𝑒𝑖𝑅111\displaystyle e_{iR}\equiv(1,1,-1) ϕ≑(1,2,1/2)italic-Ο•1212\displaystyle\phi\equiv(1,2,1/2)

are all singlets under the group S⁒U⁒(2)Sπ‘†π‘ˆsubscript2𝑆SU(2)_{S} . i=1,2,3𝑖123i=1,2,3 is the generation index. The two heavy triplet higgs scalars ΞΎa≑(1,3,1),a=1,2formulae-sequencesubscriptπœ‰π‘Ž131π‘Ž12\xi_{a}\equiv(1,3,1),a=1,2 , required to give masses to the left-handed neutrinos are also singlets under S⁒U⁒(2)Sπ‘†π‘ˆsubscript2𝑆SU(2)_{S} . In this mechanism we add a S⁒U⁒(2)Sπ‘†π‘ˆsubscript2𝑆SU(2)_{S} doublet neutral left-handed fermion SLsubscript𝑆𝐿S_{L} and two scalars Ξ·πœ‚\eta and Ο‡πœ’\chi , which transform under 𝒒e⁒x⁒tsubscript𝒒𝑒π‘₯𝑑{\cal G}_{ext} as

SL≑(1,1,0,2)subscript𝑆𝐿1102\displaystyle S_{L}\equiv(1,1,0,2) η≑(1,2,1/2,2)πœ‚12122\displaystyle\eta\equiv(1,2,1/2,2) χ≑(1,1,0,2).πœ’1102\displaystyle\chi\equiv(1,1,0,2).

There are two scales in the theory, the S⁒U⁒(2)Sπ‘†π‘ˆsubscript2𝑆SU(2)_{S} and the lepton number violating scale M𝑀M and the electroweak symmetry breaking scale mWsubscriptπ‘šπ‘Šm_{W} . At a high energy M𝑀M , Ο‡πœ’\chi acquires a vacuum expectation value ( v⁒e⁒v𝑣𝑒𝑣vev ) and breaks S⁒U⁒(2)Sπ‘†π‘ˆsubscript2𝑆SU(2)_{S} . Lepton number is broken explicitly at this scale through the couplings of the scalar triplets. All the new scalars are considered to be very heavy,

Mη∼MΟ‡βˆΌMΞΎa∼M.similar-tosubscriptπ‘€πœ‚subscriptπ‘€πœ’similar-tosubscript𝑀subscriptπœ‰π‘Žsimilar-to𝑀M_{\eta}\sim M_{\chi}\sim M_{\xi_{a}}\sim M.

The fields Ξ·πœ‚\eta and ΞΎasubscriptπœ‰π‘Ž\xi_{a} do not acquire any v⁒e⁒v𝑣𝑒𝑣vev . However, once the standard model higgs doublet Ο•italic-Ο•\phi acquires a v⁒e⁒v𝑣𝑒𝑣vev , these fields ΞΎasubscriptπœ‰π‘Ž\xi_{a} and Ξ·πœ‚\eta acquires a very tiny v⁒e⁒v𝑣𝑒𝑣vev , which in turn gives very small masses and large mixing to the neutrinos.

Consider the most general potential of all the scalars in the model ( ΞΎa,Ξ·,Ο‡,Ο•subscriptπœ‰π‘Žπœ‚πœ’italic-Ο•\xi_{a},\eta,\chi,\phi ). There will be quadratic and quartic couplings of the form,

MH2⁒(H†⁒H),Ξ»H⁒(H†⁒H)⁒(H†⁒H)⁒and⁒λ12′⁒(H1†⁒H1)⁒(H2†⁒H2),superscriptsubscript𝑀𝐻2superscript𝐻†𝐻subscriptπœ†π»superscript𝐻†𝐻superscript𝐻†𝐻andsuperscriptsubscriptπœ†12β€²superscriptsubscript𝐻1†subscript𝐻1superscriptsubscript𝐻2†subscript𝐻2M_{H}^{2}(H^{\dagger}H),\lambda_{H}(H^{\dagger}H)(H^{\dagger}H)~{}~{}{\rm and}% ~{}~{}\lambda_{12}^{\prime}(H_{1}^{\dagger}H_{1})(H_{2}^{\dagger}H_{2}),

where H𝐻H correspond to any one of the scalar fields. In addition, there will be two coupled terms,

V𝑉\displaystyle{V} =\displaystyle= ΞΌa⁒(ΞΎa0⁒ϕ0⁒ϕ0+2⁒ξa-⁒ϕ+⁒ϕ0+ΞΎa--⁒ϕ+⁒ϕ+)subscriptπœ‡π‘Žsuperscriptsubscriptπœ‰π‘Ž0superscriptitalic-Ο•0superscriptitalic-Ο•02subscriptsuperscriptπœ‰π‘Žsuperscriptitalic-Ο•superscriptitalic-Ο•0superscriptsubscriptπœ‰π‘Žabsentsuperscriptitalic-Ο•superscriptitalic-Ο•\displaystyle\mu_{a}(\xi_{a}^{0}\phi^{0}\phi^{0}+\sqrt{2}\xi^{-}_{a}\phi^{+}% \phi^{0}+\xi_{a}^{--}\phi^{+}\phi^{+}) (1)
+\displaystyle+ m⁒[Ο•0⁒(Ξ·+0⁒χ--Ξ·-0⁒χ+)-Ο•+⁒(Ξ·+-⁒χ--Ξ·--⁒χ+)]π‘šdelimited-[]superscriptitalic-Ο•0subscriptsuperscriptπœ‚0subscriptπœ’subscriptsuperscriptπœ‚0subscriptπœ’superscriptitalic-Ο•subscriptsuperscriptπœ‚subscriptπœ’subscriptsuperscriptπœ‚subscriptπœ’\displaystyle m[\phi^{0}(\eta^{0}_{+}\chi_{-}-\eta^{0}_{-}\chi_{+})-\phi^{+}(% \eta^{-}_{+}\chi_{-}-\eta^{-}_{-}\chi_{+})]
+\displaystyle+ m⁒[Ο•0⁒(Ξ·+0⁒χ-*-Ξ·-0⁒χ+*)-Ο•+⁒(Ξ·+-⁒χ-*-Ξ·--⁒χ+*)]+h.c.formulae-sequenceπ‘šdelimited-[]superscriptitalic-Ο•0subscriptsuperscriptπœ‚0subscriptsuperscriptπœ’subscriptsuperscriptπœ‚0subscriptsuperscriptπœ’superscriptitalic-Ο•subscriptsuperscriptπœ‚subscriptsuperscriptπœ’subscriptsuperscriptπœ‚subscriptsuperscriptπœ’β„Žπ‘\displaystyle m[\phi^{0}(\eta^{0}_{+}\chi^{*}_{-}-\eta^{0}_{-}\chi^{*}_{+})-% \phi^{+}(\eta^{-}_{+}\chi^{*}_{-}-\eta^{-}_{-}\chi^{*}_{+})]+h.c.

where Ξ·+-subscriptsuperscriptπœ‚\eta^{-}_{+} represents the component of Ξ·πœ‚\eta with electric charge -11-1 and T3=+1/2subscript𝑇312T_{3}=+1/2 of S⁒U⁒(2)Sπ‘†π‘ˆsubscript2𝑆SU(2)_{S} ; Ο‡+subscriptπœ’\chi_{+} represents the component of Ο‡πœ’\chi with T3=+1/2subscript𝑇312T_{3}=+1/2 of S⁒U⁒(2)Sπ‘†π‘ˆsubscript2𝑆SU(2)_{S} ; and Ο‡+*subscriptsuperscriptπœ’\chi^{*}_{+} is the component of χ†superscriptπœ’β€ \chi^{\dagger} with T3=+1/2subscript𝑇312T_{3}=+1/2 of S⁒U⁒(2)Sπ‘†π‘ˆsubscript2𝑆SU(2)_{S} .

For consistency [10]we require ΞΌasubscriptπœ‡π‘Ž\mu_{a} to be less than but of the order of masses of ΞΎasubscriptπœ‰π‘Ž\xi_{a} , and we choose

μ∼0.1⁒M.similar-toπœ‡0.1𝑀\mu\sim 0.1~{}M.

When the field Ο‡πœ’\chi acquires a v⁒e⁒v𝑣𝑒𝑣vev , a mixing of the fields Ο•italic-Ο•\phi and Ξ·πœ‚\eta of amount m⁒<Ο‡>π‘šexpectationπœ’m<\chi> will be induced. Since Mη∼Msimilar-tosubscriptπ‘€πœ‚π‘€M_{\eta}\sim M and mΟ•βˆΌmWsimilar-tosubscriptπ‘šitalic-Ο•subscriptπ‘šπ‘Šm_{\phi}\sim m_{W} , to protect the electroweak scale we then require

m∼mW.similar-toπ‘šsubscriptπ‘šπ‘Šm\sim m_{W}.

This fixes all the mass parameters in this scenario. We can now proceed to minimize the potential. In ref [10]it was shown that the triplet higgs scalars get a very small v⁒e⁒v𝑣𝑒𝑣vev consistent with the minimisation of the potential. In the present scenario both the higgs triplet ΞΎasubscriptπœ‰π‘Ž\xi_{a} and the new doublet higgs scalar Ξ·πœ‚\eta get a tiny v⁒e⁒v𝑣𝑒𝑣vev on minimisation, without any fine tuning of parameters. We assume that T3=+1/2subscript𝑇312T_{3}=+1/2 component of Ο‡πœ’\chi acquires a v⁒e⁒v𝑣𝑒𝑣vev . But that can induce v⁒e⁒v𝑣𝑒𝑣vev s to both the neutral S⁒U⁒(2)Sπ‘†π‘ˆsubscript2𝑆SU(2)_{S} components Ξ·+0superscriptsubscriptπœ‚0\eta_{+}^{0} and Ξ·-0superscriptsubscriptπœ‚0\eta_{-}^{0} . They are given by,

<ΞΎa>expectationsubscriptπœ‰π‘Ž\displaystyle<\xi_{a}> ≃similar-to-or-equals\displaystyle\simeq -μ⁒<Ο•>2MΞΎa2πœ‡superscriptexpectationitalic-Ο•2superscriptsubscript𝑀subscriptπœ‰π‘Ž2\displaystyle-{\mu<\phi>^{2}\over M_{\xi_{a}}^{2}}
<Ξ·-0>≃-m⁒<Ο•>⁒<Ο‡+>MΞ·2similar-to-or-equalsexpectationsubscriptsuperscriptπœ‚0π‘šexpectationitalic-Ο•expectationsubscriptπœ’superscriptsubscriptπ‘€πœ‚2\displaystyle<\eta^{0}_{-}>\simeq-{m<\phi><\chi_{+}>\over M_{\eta}^{2}} andand\displaystyle{\rm and} <Ξ·+0>≃-m⁒<Ο•>⁒<Ο‡-*>MΞ·2.similar-to-or-equalsexpectationsubscriptsuperscriptπœ‚0π‘šexpectationitalic-Ο•expectationsuperscriptsubscriptπœ’superscriptsubscriptπ‘€πœ‚2\displaystyle<\eta^{0}_{+}>\simeq-{m<\phi><\chi_{-}^{*}>\over M_{\eta}^{2}}. (2)

Since, μ∼MΞΎa∼Mη∼<Ο‡+>∼Msimilar-toπœ‡subscript𝑀subscriptπœ‰π‘Žsimilar-tosubscriptπ‘€πœ‚similar-toexpectationsubscriptπœ’similar-to𝑀\mu\sim M_{\xi_{a}}\sim M_{\eta}\sim<\chi_{+}>\sim M and m∼mΟ•βˆΌ<Ο•>∼mwsimilar-toπ‘šsubscriptπ‘šitalic-Ο•similar-toexpectationitalic-Ο•similar-tosubscriptπ‘šπ‘€m\sim m_{\phi}\sim<\phi>\sim m_{w} , we get,

<ΞΎa>∼<Ξ·-0>∼<Ξ·+0>∼O⁒(mW2M).similar-toexpectationsubscriptπœ‰π‘Žexpectationsuperscriptsubscriptπœ‚0similar-toexpectationsubscriptsuperscriptπœ‚0similar-to𝑂superscriptsubscriptπ‘šπ‘Š2𝑀<\xi_{a}>~{}~{}\sim~{}~{}<\eta_{-}^{0}>~{}~{}\sim~{}~{}<\eta^{0}_{+}>~{}~{}% \sim~{}~{}O\left({m_{W}^{2}\over M}\right). (3)

The v⁒e⁒v𝑣𝑒𝑣vev s of ΞΎasubscriptπœ‰π‘Ž\xi_{a} now give small masses to the left-handed neutrinos and the v⁒e⁒v𝑣𝑒𝑣vev of Ξ·Β±0subscriptsuperscriptπœ‚0plus-or-minus\eta^{0}_{\pm} allows mixing of the S⁒U⁒(2)Lπ‘†π‘ˆsubscript2𝐿SU(2)_{L} singlet neutrinos SLsubscript𝑆𝐿S_{L} with the usual left-handed neutrinos, both of which are now of the same order of magnitude n⁒a⁒t⁒u⁒r⁒a⁒l⁒l⁒yΒ―Β―π‘›π‘Žπ‘‘π‘’π‘Ÿπ‘Žπ‘™π‘™π‘¦\underline{naturally} .

The Yukawa couplings of the leptons are given by,

β„’=fα⁒ie⁒li⁒L¯⁒eα⁒R⁒ϕ+fa⁒i⁒j⁒li⁒L⁒lj⁒L⁒ξa+hi⁒x⁒ϡx⁒y⁒li⁒L⁒SL⁒x⁒ηy+h.c.formulae-sequenceβ„’subscriptsuperscript𝑓𝑒𝛼𝑖¯subscript𝑙𝑖𝐿subscript𝑒𝛼𝑅italic-Ο•subscriptπ‘“π‘Žπ‘–π‘—subscript𝑙𝑖𝐿subscript𝑙𝑗𝐿subscriptπœ‰π‘Žsubscriptβ„Žπ‘–π‘₯subscriptitalic-Ο΅π‘₯𝑦subscript𝑙𝑖𝐿subscript𝑆𝐿π‘₯subscriptπœ‚π‘¦β„Žπ‘{\cal L}=f^{e}_{\alpha i}\overline{l_{iL}}e_{\alpha R}\phi+f_{aij}l_{iL}l_{jL}% \xi_{a}+h_{ix}\epsilon_{xy}l_{iL}S_{Lx}\eta_{y}+h.c. (4)

where x,y=1,2formulae-sequenceπ‘₯𝑦12x,y=1,2 are the S⁒U⁒(2)Sπ‘†π‘ˆsubscript2𝑆SU(2)_{S} indices. The first term contributes to the charged lepton masses, while the second and third terms contributes to the neutrino mass and mixing matrix. In the basis, [Ξ½i⁒L⁒SL⁒x]delimited-[]subscriptπœˆπ‘–πΏsubscript𝑆𝐿π‘₯[\nu_{iL}~{}~{}S_{Lx}] we can now write down the mass matrix as,

β„³Ξ½=(βˆ‘afa⁒i⁒j⁒<ΞΎa>hi⁒x⁒ϡx⁒y⁒<Ξ·y0>hi⁒xT⁒ϡx⁒y⁒<Ξ·y0>0).subscriptβ„³πœˆsubscriptπ‘Žsubscriptπ‘“π‘Žπ‘–π‘—expectationsubscriptπœ‰π‘Žsubscriptβ„Žπ‘–π‘₯subscriptitalic-Ο΅π‘₯𝑦expectationsuperscriptsubscriptπœ‚π‘¦0subscriptsuperscriptβ„Žπ‘‡π‘–π‘₯subscriptitalic-Ο΅π‘₯𝑦expectationsuperscriptsubscriptπœ‚π‘¦00{\cal M}_{\nu}=\pmatrix{\sum_{a}f_{aij}<\xi_{a}>&h_{ix}\epsilon_{xy}<\eta_{y}^% {0}>\cr h^{T}_{ix}\epsilon_{xy}<\eta_{y}^{0}>&0}. (5)

There is no Majorana mass terms for the sterile neutrinos. We shall now discuss how to generate baryon asymmetry of the universe [12, 13]in this scenario and what constraint it gives on the new mass scale M𝑀M and then come back to the neutrino mass matrix.

Lepton number is violated when the scalars ΞΎasubscriptπœ‰π‘Ž\xi_{a} decays

ΞΎaβ†’{lic⁒ljc (L=-2)ϕ⁒ϕ (L=0)β†’subscriptπœ‰π‘Žcasessuperscriptsubscript𝑙𝑖𝑐superscriptsubscript𝑙𝑗𝑐𝐿2italic-Ο•italic-ϕ𝐿0\xi_{a}\rightarrow\left\{\begin{array}[]{l@{\quad}l}l_{i}^{c}l_{j}^{c}\quad&(L% =-2)\\ \phi\phi\quad&(L=0)\end{array}\right. (6)

All other couplings conserve lepton number. By assigning a lepton number -11-1 to SL⁒xsubscript𝑆𝐿π‘₯S_{Lx} , we can ensure conservation of lepton number in the decays of the doublet scalar field Ξ·πœ‚\eta .

We choose the mass matrix of ΞΎasubscriptπœ‰π‘Ž\xi_{a} to be real and diagonal (MΞΎ100MΞΎ2)subscript𝑀subscriptπœ‰100subscript𝑀subscriptπœ‰2\pmatrix{M_{\xi_{1}}&0\cr 0&M_{\xi_{2}}} ; but once the one loop self energy type contributions are included, imaginary phases from ΞΌasubscriptπœ‡π‘Ž\mu_{a} and fa⁒i⁒jsubscriptπ‘“π‘Žπ‘–π‘—f_{aij} makes it complex. The absorptive part of the one loop self-energy type diagram will introduce observable CP violation in the mass matrix [14], which would produce unequal amount of leptons and anti-leptons in the decays of the ΞΎa++superscriptsubscriptπœ‰π‘Žabsent\xi_{a}^{++} and ΞΎa--superscriptsubscriptπœ‰π‘Žabsent\xi_{a}^{--} respectively. This will create a charge asymmetry, which will be compensated by equal and opposite amount of charge asymmetry in the production of Ο•+superscriptitalic-Ο•\phi^{+} and Ο•-superscriptitalic-Ο•\phi^{-} in the decays of ΞΎa++superscriptsubscriptπœ‰π‘Žabsent\xi_{a}^{++} and ΞΎa--superscriptsubscriptπœ‰π‘Žabsent\xi_{a}^{--} , so that the universe remains charge neutral.

The interference of the tree level and the one loop diagram of figure 1 will generate a lepton asymmetry in the decays of ΞΎasubscriptπœ‰π‘Ž\xi_{a} , which is given by,


FigureΒ 1: The decay of ΞΎaβ†’lc⁒lcβ†’subscriptπœ‰π‘Žsuperscript𝑙𝑐superscript𝑙𝑐\xi_{a}\to l^{c}l^{c} at tree level (a) and in one-loop order (b). C⁒P-limit-from𝐢𝑃CP- violation comes from an interference of these diagrams.
Ξ΄a≃I⁒m⁒[ΞΌ1⁒μ2*β’βˆ‘k,lf1⁒k⁒l⁒f2⁒k⁒l*]8⁒π2⁒(MΞΎ12-MΞΎ22)⁒[MΞΎaΓξa].similar-to-or-equalssubscriptπ›Ώπ‘ŽπΌπ‘šdelimited-[]subscriptπœ‡1superscriptsubscriptπœ‡2subscriptπ‘˜π‘™subscript𝑓1π‘˜π‘™superscriptsubscript𝑓2π‘˜π‘™8superscriptπœ‹2superscriptsubscript𝑀subscriptπœ‰12superscriptsubscript𝑀subscriptπœ‰22delimited-[]subscript𝑀subscriptπœ‰π‘ŽsubscriptΞ“subscriptπœ‰π‘Ž\delta_{a}\simeq{{Im\left[\mu_{1}\mu_{2}^{*}\sum_{k,l}f_{1kl}f_{2kl}^{*}\right% ]}\over{8\pi^{2}(M_{\xi_{1}}^{2}-M_{\xi_{2}}^{2})}}\left[{{M_{\xi_{a}}}\over% \Gamma_{\xi_{a}}}\right]. (7)

where, the decay width of these scalars ΞΎasubscriptπœ‰π‘Ž\xi_{a} is given by,

Γξa=18⁒π⁒(|ΞΌ1|2+|ΞΌ2|2MΞΎa+βˆ‘i,j|fa⁒i⁒j|2⁒Ma).subscriptΞ“subscriptπœ‰π‘Ž18πœ‹superscriptsubscriptπœ‡12superscriptsubscriptπœ‡22subscript𝑀subscriptπœ‰π‘Žsubscript𝑖𝑗superscriptsubscriptπ‘“π‘Žπ‘–π‘—2subscriptπ‘€π‘Ž\Gamma_{\xi_{a}}={\frac{1}{8\pi}}\left({\frac{|\mu_{1}|^{2}+|\mu_{2}|^{2}}{M_{% \xi_{a}}}}+\sum_{i,j}|f_{aij}|^{2}M_{a}\right). (8)

These decays should be slower [13]than the expansion rate of the universe ( H𝐻H ), otherwise the lepton asymmetry Ξ΄asubscriptπ›Ώπ‘Ž\delta_{a} in decays of ΞΎasubscriptπœ‰π‘Ž\xi_{a} will be suppressed by an amount K⁒(ln⁒K)0.6𝐾superscriptln𝐾0.6K~{}({\rm ln}~{}K)^{0.6} , where K=ΓξaH𝐾subscriptΞ“subscriptπœ‰π‘Žπ»K={\Gamma_{\xi_{a}}\over H} ; H=1.7⁒g*⁒T2MP⁒l⁒at⁒T=MΞΎa𝐻1.7subscript𝑔superscript𝑇2subscript𝑀𝑃𝑙at𝑇subscript𝑀subscriptπœ‰π‘ŽH=\sqrt{1.7g_{*}}\frac{T^{2}}{M_{Pl}}~{}~{}~{}{\rm at}~{}~{}T=M_{\xi_{a}} ; MP⁒lsubscript𝑀𝑃𝑙M_{Pl} is the Planck scale; and g*subscript𝑔g_{*} is the total number of relativistic degrees of freedom.

We consider MΞΎ2<MΞΎ1subscript𝑀subscriptπœ‰2subscript𝑀subscriptπœ‰1M_{\xi_{2}}<M_{\xi_{1}} , so that when ΞΎ2subscriptπœ‰2\xi_{2} decays, ΞΎ1subscriptπœ‰1\xi_{1} has already decayed away and only the asymmetry Ξ΄2subscript𝛿2\delta_{2} generated in decays of ΞΎ2subscriptπœ‰2\xi_{2} will contribute to the final lepton asymmetry of the universe. The lepton asymmetry thus generated will be the same as the (B-L)𝐡𝐿(B-L) asymmetry of the universe, which will then get converted to a baryon asymmetry during the electroweak phase transition [15]. The final baryon asymmetry of the universe is given by,

nBs∼δ23⁒g*⁒K⁒(ln⁒K)0.6.similar-tosubscript𝑛𝐡𝑠subscript𝛿23subscript𝑔𝐾superscriptln𝐾0.6{n_{B}\over s}\sim{\delta_{2}\over 3g_{*}K({\rm ln}~{}K)^{0.6}}. (9)

To obtain the desired amount of baryon asymmetry of the universe one possibility could be [10], M2=1013subscript𝑀2superscript1013M_{2}=10^{13} GeV, and ΞΌ2=2Γ—1012subscriptπœ‡22superscript1012\mu_{2}=2\times 10^{12} GeV, which gives us mΞ½Ο„=1.2⁒f233subscriptπ‘šsubscript𝜈𝜏1.2subscript𝑓233m_{\nu_{\tau}}=1.2f_{233} eV, assuming that the M1subscript𝑀1M_{1} contribution is negligible. Now let M1=3Γ—1013subscript𝑀13superscript1013M_{1}=3\times 10^{13} GeV, ΞΌ1=1013subscriptπœ‡1superscript1013\mu_{1}=10^{13} GeV, and f1⁒k⁒l∼0.1similar-tosubscript𝑓1π‘˜π‘™0.1f_{1kl}\sim 0.1 , then the decay of ψ2subscriptπœ“2\psi_{2} generates a lepton asymmetry Ξ΄2subscript𝛿2\delta_{2} of about 8Γ—10-48superscript1048\times 10^{-4} if the CP phase is maximum. Using MP⁒l∼1019similar-tosubscript𝑀𝑃𝑙superscript1019M_{Pl}\sim 10^{19} GeV and g*∼102similar-tosubscript𝑔superscript102g_{*}\sim 10^{2} , we find K∼2.4Γ—103similar-to𝐾2.4superscript103K\sim 2.4\times 10^{3} , so that nB/s∼10-10similar-tosubscript𝑛𝐡𝑠superscript1010n_{B}/s\sim 10^{-10} .

Thus, with the heavy mass scale to be of the order of M∼1013-14similar-to𝑀superscript101314M\sim 10^{13-14} GeV it is possible to get the desired amount of baryon asymmetry of the universe and v⁒e⁒v𝑣𝑒𝑣vev s of ΞΎπœ‰\xi and Ξ·πœ‚\eta to be of the order of a few eV. Then with proper value of the Yukawa couplings fa⁒i⁒jsubscriptπ‘“π‘Žπ‘–π‘—f_{aij} and hi⁒xsubscriptβ„Žπ‘–π‘₯h_{ix} we can get a neutrino mass matrix (equation 5) which can explain all the neutrino experiments. All the elements of the mass matrix could be non-zero and are about a few eV or less, except for the Majorana mass term of the sterile neutrinos. One can then have several possible scenarios [6]which can explain all the neutrino experiments. Cosider for example [16]one sterile neutrino ( SL⁒1subscript𝑆𝐿1S_{L1} ) with mass of about (2-3)Γ—10-323superscript103(2-3)\times 10^{-3} eV, which mixes with the Ξ½esubscriptπœˆπ‘’\nu_{e} , while the other sterile neutrino ( SL⁒2subscript𝑆𝐿2S_{L2} ) is lighter. This will satisfy the constraints on the sterile neutrinos from nucleosynthesis. If we choose mΞ½esubscriptπ‘šsubscriptπœˆπ‘’m_{\nu_{e}} to be much lighter than SL⁒1subscript𝑆𝐿1S_{L1} , that satisfies all laboratory constraints on its mass. Then the Ξ½eβ†’Ξ½S1β†’subscriptπœˆπ‘’subscript𝜈subscript𝑆1\nu_{e}\to\nu_{S_{1}} oscillations can explain the solar neutrino deficit. We can further assume that Ξ½eβ†’Ξ½S1β†’subscriptπœˆπ‘’subscript𝜈subscript𝑆1\nu_{e}\to\nu_{S_{1}} oscillations satisfies the resonant oscillation condition, while Ξ½eβ†’Ξ½S2β†’subscriptπœˆπ‘’subscript𝜈subscript𝑆2\nu_{e}\to\nu_{S_{2}} oscillations satisfies the vacuum oscillation condition, so there are both the components.

For a solution of the atmospheric neutrino anomaly we assume the [νμ⁒ντ]delimited-[]subscriptπœˆπœ‡subscript𝜈𝜏[\nu_{\mu}~{}~{}\nu_{\tau}] mass matrix to be of the form,

M=(mama⁒bma⁒bmb)𝑀subscriptπ‘šπ‘Žsubscriptπ‘šπ‘Žπ‘subscriptπ‘šπ‘Žπ‘subscriptπ‘šπ‘M=\pmatrix{m_{a}&m_{ab}\cr m_{ab}&m_{b}} (10)

where, ma⁒b>mb>masubscriptπ‘šπ‘Žπ‘subscriptπ‘šπ‘subscriptπ‘šπ‘Žm_{ab}>m_{b}>m_{a} , so that the two physical states are almost degenerate with masses ma⁒bsubscriptπ‘šπ‘Žπ‘m_{ab} , but the mass squared difference is given approximately by, mΞ½ΞΌ2-mΞ½Ο„2∼ma⁒b⁒mbsimilar-tosuperscriptsubscriptπ‘šsubscriptπœˆπœ‡2superscriptsubscriptπ‘šsubscript𝜈𝜏2subscriptπ‘šπ‘Žπ‘subscriptπ‘šπ‘m_{\nu_{\mu}}^{2}-m_{\nu_{\tau}}^{2}\sim m_{ab}m_{b} . We can then have ma⁒b∼similar-tosubscriptπ‘šπ‘Žπ‘absentm_{ab}\sim eV and mb∼10-2similar-tosubscriptπ‘šπ‘superscript102m_{b}\sim 10^{-2} eV, so that the Ξ½ΞΌsubscriptπœˆπœ‡\nu_{\mu} and Ξ½Ο„subscript𝜈𝜏\nu_{\tau} are almost degenerate with mass about a few eV to be the hot component of the dark matter and the mass squared difference mΞ½ΞΌ2-mΞ½Ο„2∼10-2similar-tosuperscriptsubscriptπ‘šsubscriptπœˆπœ‡2superscriptsubscriptπ‘šsubscript𝜈𝜏2superscript102m_{\nu_{\mu}}^{2}-m_{\nu_{\tau}}^{2}\sim 10^{-2} eV 22{}^{2} and maximal mixing can explain the atmospheric neutrino anomaly. These numbers show the freedom available to the present scenario to explain all the data. In practice, as in the case of quark and charged lepton masses, only future experiments can determine the exact form of the Yukawa couplings fa⁒i⁒jsubscriptπ‘“π‘Žπ‘–π‘—f_{aij} and hi⁒xsubscriptβ„Žπ‘–π‘₯h_{ix} .

To summarise, we propose a simple scenario to accomodate naturally light sterile neutrinos in extensions of the standard model with an S⁒U⁒(2)Sπ‘†π‘ˆsubscript2𝑆SU(2)_{S} symmetry. There are two mass scales in the model, the electroweak scale and the scale of lepton number and S⁒U⁒(2)Sπ‘†π‘ˆsubscript2𝑆SU(2)_{S} violation, which is fixed by the conditions for lepton asymmetry of the universe. The heavy triplet and a doublet acquires very tiny v⁒e⁒v𝑣𝑒𝑣vev , which gives masses and mixing of the left-handed neutrinos and the sterile neutrinos. The low energy mass matrix can then explain the solar neutrino deficit, atmospheric neutrino anomaly, LSND result and the dark matter problem. The decays of the triplets generates a lepton asymmetry of the universe, which gets converted to a baryona symmetry of the universe during the electroweak phase transition.


ACKNOWLEDGEMENT

I would like to thank Prof W. Buchmuller and the Theory Division, DESY, Hamburg for hospitality and acknowledge financial support from the Alexander von Humboldt Foundation.

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