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NEW NEUTRINO PHYSICS WITHOUT FINE-TUNING

arXiv:hep-ph/9208262v1 31 Aug 1992NEW NEUTRINO PHYSICS WITHOUT FINE-TUNINGC.P. BURGESS∗, JAMES M. CLINEPhysics Department, McGill University, 3600 University Street,Montr´eal, Qu´ebec, Canada, H3A 2T8.andMARKUS LUTYPhysics Division, Lawrence Berkeley Laboratory, 1 Cyclotron Road,Berkeley, California, 94707, USA.August 1992McGill-92/36ABSTRACTWe show how a 17 keV neutrino, the solar neutrino problem, and the at-mospheric muon-neutrino deficit could all be the low-energy residues of thesame pattern of lepton-number breaking at and above the weak scale, with norequirement for fine-tuning a symmetry-breaking scale at lower energies.

Therequired pattern of small neutrino masses turns out to be naturally understoodin this framework in terms of powers of the ratio of two high energy scales.All cosmological and astrophysical constraints are satisfied.1IntroductionThere are several kinds of reported experimental anomalies which suggest that newphysics might be lurking in the neutrino sector. The list of titillating phenomena in-cludes (1) the solar-neutrino problem,[1] (2) the 17 keV neutrino,[2] (3) the atmosphericmuon-neutrino deficit,[3] and (4) the excess high-energy electrons in double-beta decayspectra.

[4] Many of these effects remain controversial, with more experiments under-way to determine which might be real. Until the experimental dust settles, theoristscan help by exploring the implications of these anomalies with the goal of discovering∗Talk given at Beyond the Standard Model III, Carleton University, June 1992.1

which are consistent with one another, potentially being signatures of the same typeof underlying new physics.We argue here that the first three items on this list all point toward a specificform for the neutrino mass matrix which can arise naturally from new physics at highscales. [5] The scenario to which we are led does not include the remaining two items,although these may be separately treated in a similar way.

[6]A key feature of our approach is to demand that our models be “natural” —i.e. that they should involve no new symmetry breaking scales below the weak scale.We do so because if such a large heirarchy of scales is fine-tuned in by hand it isunstable against renormalization.

Of all the many well-understood heirarchies of scaleknown in nature, none have ever been unstable in this way. But all of the ways weknow for stabilizing such a heirarchy (supersymmetry or some form of compositenessfor example) necessarily involve new particles and interactions not far above the lowerof the two scales.

It is therefore hard to imagine explaining such a heirarchy belowthe weak scale without introducing new light particles which should already have beenobserved.2General ConstraintsThere are a number of constraints that must be satisfied by any candidate theoryof the solar neutrino problem and the 17 keV neutrino,[7] and these lead to severalgeneral properties that are shared by virtually all models. Firstly, the failure to observeneutrinoless double beta decay implies the cancellation in this decay of the contributionof any 17 keV neutrino which enjoys a 10% mixing with νe.

This cancellation arises mostnaturally if it is enforced by a conservation law, such as the (approximate) conservationof lepton number. In this case the 17 keV state is (pseudo–) Dirac; i.e.

it consistsof a (nearly) degenerate pair of states which have opposite CP parities. Supernovaand laboratory constraints[8] then further suggest that these two states should bedominantly νµ and ντ.

The combination of lepton numbers whose conservation canforbid double beta decay in this way is Le −Lµ + Lτ.Next, a neutrino solution to the solar neutrino problem requires a fourth neu-trino state, νs, that is approximately degenerate with νe. The failure to observed thisnew neutrino in the Z0 width at LEP implies that it must be sterile — i.e.

it cannotcarry SU(2)W × U(1)Y quantum numbers. If this sterile neutrino were massless itsrenormalizable couplings would enjoy a global symmetry, U(1)s, under which νs trans-forms but all other neutrinos are neutral.

This survives as an approximate symmetryin the full theory since this mass must be very small if it is to participate in a solutionto the solar neutrino problem. The total approximate symmetry group then becomesGν ≡U(1)e−µ+τ × U(1)s.Finally, standard cosmology indicates that a 17 keV neutrino cannot be too long-lived.

Consistency with the age of the universe requires such a neutrino to decay intorelativistic daughters with a lifetime no longer than τ17 ∼1012 sec. Requiring thesehot decay products to not overprolong the radiation-dominated era and so delay theonset of galaxy formation gives a shorter bound[9] of τ17 ∼106 sec, although τ17 ∼107

sec might actually be a good thing. [10] In the models to be considered nucleosynthesisconstraints are easily satisfied since all excess light states decouple early enough to bediluted by the QCD phase transition.Another lifetime constraint applies should the 17 keV neutrino decay into νe’s.Consistency of the length of the neutrino pulse observed from SN1987A with supernovamodels requires such a lifetime to lie outside of the interval 3×104 < τ17 < 2×108 sec.All of these lifetime constraints are accomodated in what follows by the decayof the 17 keV state into a lighter neutrino plus a Goldstone boson (majoron) thatis associated with the spontaneous breaking of the approximately-conserved leptonnumbers, Gν.3ModelsIn order to ensure the naturalness of our models we construct them by writing down allpossible interactions that involve the required low-energy particles and which respectthe gauge and Gν symmetries.

Since all of the neutrino effects being described arise inlow-energy experiments we start by writing down the effective lagrangian as seen at theweak scale. Any physics of still higher scales can only affect these experiments throughthe nonrenormalizable interactions they generate in this effective lagrangian.

Renor-malizable models for this underlying physics are easily constructed once the requiredform for the weak-scale lagrangian is known. [5]The particle content of the weak-scale theory consists of the usual standard-modelcontent supplemented by the sterile neutrino, νs, and two electroweak-singlet complexscalars, φi.

These scalars are required in order to spontaneously break Gν, and theymust be electroweak singlets to avoid having the resulting Goldstone bosons acquire aphenomenologically unacceptable coupling to quarks and charged leptons. The variouskinds of models we consider then differ only in their choices for the transformationproperties for φi under the symmetry Gν.

We present two illustrative examples below:3.1A Vacuum-Oscillation ModelConsider first the case for which the scalar fields transform under Gν as φ1 ∼(1/2, −1/2)and φ2 ∼(−1/2, −1/2). Then the renormalizable lagrangian consists of the Stan-dard Model, a kinetic term for νs, and kinetic and potential terms for φi.

The lowest-dimension gauge- and Gν-invariant operators in the effective lagrangian that can con-tribute to the neutrino mass matrix aredimension 5:geM (LeH)(LµH),gτM (LµH)(LτH);dimension 6:ajM2 (LjH) νsφ22,bM2 (LµH) νsφ21;dimension 7:cM3 νsνsφ21φ22;dimension 9:dµµM5 (LµH)(LµH)(φ1φ∗2)2,djkM5 (LjH)(LkH)(φ∗1φ2)2. (1)

H is the usual electroweak Higgs doublet and the L’s are the standard left-handedlepton doublets, for which subscripts represent generation labels. The labels j and kare restricted to take only the two values e and τ.

Appropriate factors of the heavy massscale M are included to ensure that the remaining coupling constants are dimensionless.Replacing the scalars with their vacuum expectation values: ⟨H⟩= v = 174 GeV,⟨φ1⟩= w1, ⟨φ2⟩= w2, and defining g =qg2e + g2τ, gives a mass matrix of the formm = m17γα1βα2α1ǫ1sǫ2βsηcα2ǫ2cǫ3,(2)with m17 = gv2/M, tan θ17 = s/c = ge/gτ, αj = (ajw22)/(gMv), β = (bw21)/(gMv),γ = (cw21w22)/(gM2v2), and ǫj, η = (dw21w22)/(gM4).Requiring m17 = 17 keV puts the new-physics scale at M = 1 ×107gv ≈2 ×109gGeV. Since the expectation values, v, w1 and w2 are of order the weak scale they aremuch smaller than M. This implies the hierarchy ǫ, η ≪γ ≪α, β.

The spectrumof neutrino masses predicted becomes: (1) a pseudo-Dirac pair of 17 keV states splitby ∆m2h = 4m217βα′2, and (2) a pseudo-Dirac pair of light neutrino states with mℓ=m17α′1, and ∆m2ℓ= m217γα′1. In these expressions the coefficients α′i are defined by:α′1 = cα1 −sα2 and α′2 = cα2 + sα1.There are two majorons in this model, χ1 and χ2, which can be thought of as thephases of the fields φ1 and φ2, respectively.

The lifetime for the decay νh →νℓ+ χi inthis model isτ17 = 16πm3h α′22w22+ β2w21!−1(3)A typical choice for the couplings is given by w1, w2 ∼3v, aj = b = c = gτ = 1and ge = 0.1. In this case α′, β ∼10−6, γ ∼10−12, and the neutrino masses becomemℓ∼0.01 eV, ∆m2ℓ∼10−10 eV2, and ∆m2h ∼10−3 eV2.

There is maximal (45o)mixing within each of these pseudo-Dirac pairs. These numbers also imply a lifetimeof τ17 ∼109 sec.It is striking that the hierarchies mh/v and mℓ/mh, as well as ∆m2h/m2h and∆m2ℓ/m2ℓare all explained here by the largeness of M relative to v, w1 and w2.

∆m2his near the experimental upper limit and in the range required to account for theatmospheric neutrino anomaly, and ∆m2ℓfalls naturally into the correct range for “just-so” vacuum oscillations.These numbers easily satisfy the phenomenological constraints with two provisos.Although they can account for the atmospheric neutrino deficit, they do so with max-imal νµ −ντ oscillations. They therefore require that either the atmospheric neutrinodeficit itself, or the most recent IMB constraints coming from the flux of higher-energyupward-coming muons, must disappear.

The 17 keV lifetime is also long compared tothe structure-formation bound although, as may be seen in the following section, it canbe shortened with alternative choices for the dimensionless couplings.

3.2An MSW ModelA slightly different choice for the quantum numbers for φi produces a model withresonant MSW oscillations. In this case the singlet scalar fields transform under Gν asφ1 ∼(−1/2, −1/2) and φ2 ∼(0, −2/3).

The lowest-dimension gauge- and Gν-invariantoperators that contribute to the neutrino mass matrix are in this casedimension 5:geM (LeH)(LµH),gτM (LµH)(LτH);dimension 6:ajM2 (LjH) νsφ21,cM3 νsνsφ32. (4)Contributions to the remaining terms in the neutrino mass matrix are further sup-pressed relative to these by additional powers of M−1.In this case the mass matrix again takes the form of Eq.

(2) with αj = (ajw21)/(gMv),γ = (cw32)/(gMv2), and β, ǫ, η ≪α, γ. This implies the light neutrino states havemasses mℓ± = (m17/2)qγ2 + 4α′12 ± γ, and ∆m2ℓ= m217γqγ2 + 4α′12.

The mixingangle between the two light states works out to be sin2 2θℓ= 4α′12/(γ2 + 4α′12). The 17keV state is negligibly split in this model.

MSW oscillations of the light states can beaccomodated if we choose g = 1, a1 = 0.2, a2 = 1, and c = 1, for which α′1 = 1 × 10−8,α′2 = 1 × 10−7, γ = 1 × 10−7, ∆m2ℓ= 3 × 10−6 eV2, and sin2 2θℓ= 4 × 10−2. Thesevalues do give trouble with the structure-formation bound, however, since they give alifetime for the 17 keV state of ∼1010 sec.Shorter lifetimes as well as the atmospheric neutrino anomaly can be accomodatedby adding a third electroweak singlet scalar transforming under Gν as φ3 ∼(1/2, −1/2)since this allows the additional dimension 6 operator:bM2 (LµH)νsφ23.

(5)Resolution of the atmospheric neutrino anomaly requires w3 near the weak scale. Forexample, w1 = v/2, w2 = v, w3 = 30v, g = 0.1, a1 = 0.1, a2 = 0.01, b = 1, andc = 0.01, gives the MSW effect, atmospheric neutrino oscillations, and a neutrinolifetime of τ17 = (16πw23)/(m3hβ2) ∼2 × 103 sec.4Experimental ImplicationsThere are several ways in which the class of models we have discussed might be exper-imentally probed.

The most obvious way is to confirm or disprove the 17 keV neutrinoand the solar neutrino problem. Solar neutrino oscillations are also predicted to be intoa sterile component — a prediction that is potentially detectable at SNO.

If the atmo-spheric neutrino persists, it must be explained in this picture using maximal νµ −ντoscillations, which would imply that the IMB results on the upcoming muon flux can-not survive. Since the 17 keV lifetime likes to be long in this scenario, it may haveinteresting applications for galaxy formation.

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[6] C.P. Burgess and J.M.Cline, McGill University preprint McGill-92/22; and inthis volume.

[7] G. Gelmini, S. Nussinov, and R. D. Peccei, UCLA preprint UCLA/91/TEP/15(1991), unpublished; J. Cline and T. Walker, Phys. Rev.

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[8] A. Burrows and R. Gandhi, Phys. Lett.

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