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Fermi National Accelerator Laboratory

arXiv:hep-ph/9206256v1 26 Jun 1992FERMILAB–PUB–92/149–TNovember 2018Seesaw Neutrino MassRatios withRadiative CorrectionsD. C. Kennedy*Fermi National Accelerator LaboratoryP.O.

Box 500, Batavia, Illinois 60510ABSTRACTUnlike neutrino masses, the ratios of neutrino masses can be predicted by up-quarkseesaw models using the known quark masses and including radiative corrections,with some restrictive assumptions. The uncertainties in these ratios can be reducedto three: the type of seesaw (quadratic, linear, etc.

), the top quark mass, and theLandau-triviality value of the top quark mass. * Work supported by the U.S. Department of Energy under contract no.

DOE-AC02-76-CHO-3000.

The inconclusive but suggestive results of recent solar and atmospheric neutrino andbeta decay experiments [1] lead to the possibility of neutrino masses, which additionallymay have important application to cosmology, astrophysics and laboratory searches forneutrino oscillations. The most economical model of light neutrinos is the so-called “see-saw” of the grand-unified type, which requires a superheavy right-handed neutrino for eachordinary neutrino and arises naturally in partially or completely unified theories with left-right symmetry, such as SO(10) [2,3,4].

These grand unified seesaw models predict smallbut non-zero Majorana masses for the ordinary neutrinos in terms of the Dirac masses ofthe up-type quarks (u, c, t) and the superheavy right-handed Majorana masses. Thesepredictions are made uncertain, however, by the unknown right-handed masses and by ra-diative corrections.

But the ratios of neutrino masses are more definite in seesaw models,under some neccesary and minimal assumptions (printed below in italics) about the physicsunderlying the seesaw [5]. The uncertainties in the mass ratios can then be narrowed to ahandful.The general tree-level form of the seesaw model mass matrix for three families is: 0mDmTDMN,(1)in the left- and right-handed neutrino basis, where each entry is a 3×3 matrix.

We assumethat the upper left corner is zero, as a non-zero Majorana mass for left-handed ν generallyrequires an SU(2)L Higgs triplet, an unnatural addition to the Standard Model in lightof known electroweak neutral-current properties [6]. The Dirac matrix mD is both anSU(2)L and an SU(2)R doublet.

The symmetric superheavy Majorana mass matrix MNfor the right-handed neutrinos N violates lepton number, but is a Standard Model gaugesinglet.MN must be a remnant of a broken SU(2)R or larger symmetry.Assumingthe eigenvalues of MN are much greater than those of mD, the light neutrinos acquirea symmetric Majorana mass matrix mν = mDM−1N mTD and the superheavy neutrinos a2

mass matrix MN upon block diagonalization of (1). The superheavy neutrinos have massesequal to the eigenvalues (MN1, MN2, MN3) of MN.

The matrix MN can have a variety ofsources [3,5]. In models with tree-level breaking of SU(2)R, the right-handed mass requiresan SU(2)R Higgs triplet — in SO(10) models, a Higgs 126.

In models with minimal Higgscontent (SU(2)L,R singlets and doublets only, as in superstring models), the matrix MNmust arise either from loop effects [3,7] or from non-renormalizable terms, presumablyinduced by gravity [8].Making predictions from the seesaw matrix mν requires additional assumptions. To ob-tain simple scaling dependence of light neutrinos masses on the eigenvalues of mD requiresthe assumptions that the matrix mD can be freely diagonalized and that the intergener-ational mixings in MN are no larger than the ratios of eigenvalues between generations.We then need to know the eigenvalues of mD : here the simple grand-unified seesaw isassumed, so that mD ∝mu, the up-type quark mass matrix.

† For predictiveness, theeigenvalues of MN are assumed proportional to a power p of the eigenvalues of mu. Thep = 0 and p = 1 cases are the “quadratic” and “linear” seesaws, respectively, because ofthe dependence of mν,i on mu,i [4,5].

‡The family kinship of quarks and leptons in order of ascending mass is assumed; adifferent kinship merely requires relabelling the neutrinos appropriately. Forming the ratioof any two light neutrino masses,mν,imν,j=m2u,im2u,j·MpN,jMpN,i,(2)†The Dirac mass matrix for the charged leptons ml is proportional to the down-typequark (d, s, b) mass matrix md in the simplest grand unified seesaw models.

The matrixproportionalities of mD and mu, and ml and md, require that each pair of Dirac massesbe generated by only or mainly one Higgs representation. Otherwise, a specific ansatz ofDirac masses is needed.‡If the eigenvalues of MN increase no more than linearly with the hierarchy of eigenvaluesin mu (p ≤1 for a simple power law), and ml ∝md, then, additionally, the neutrino mixingmatrix is identical to quark CKM mixing matrix, at least at the putative unification scale.For reasonable values of the top quark mass, this equality approximately holds at lowenergies [4].3

we obtain the power-law dependence of the seesaw, with exponent 2 −p. Taking the ratiosof neutrino masses eliminates the overall unknown scale in MN.

However, the form (2)requires radiative corrections to the fermion masses to arrive at predictions. The tree-levelresult (2) is taken to be exact at some scale µ = MX, typically the grand unification scale;the masses mν(µ), mu(µ), and MN(µ) are then run down to low energies and related to thephysical masses to yield radiatively modified seesaw predictions.

The leading logarithmapproximation is sufficient for our purposes and is evaluated here in the MS scheme. As anumber of authors have noted, much of the uncertainty in these corrections cancels out infermion mass ratios, if some general conditions hold about the physics that produces thecorrections [5,9].Corrections to the fermion masses are assumed to come from two sources, Higgs-Yukawacouplings and gauge couplings.

A generalized family symmetry is assumed for the gaugeinteractions, so that, apart from differences in mass thresholds, the gauge corrections are“family-blind”. The mass matrices can then be diagonalized and corrections applied toindividual eigenvalues.

Higgs corrections to the masses are proportional to their underlyingYukawa couplings. For the light ν, these are negligible, as they are for the up-type quarks,except for the top quark.

* For the superheavy N, the eigenvalues MN,i(X) are proportionalto the power p of the eigenvalues mu,i(X).Considering only gauge corrections first, the MS renormalization group equations forthe fermion masses and gauge couplings 1...n... are standard [10]:d ln m(µ)d ln µ=Xnb(n)m · g2n(µ),dg2n(µ)d ln µ = −2bn · g4n(µ),(3)with the general solutionm(µ)/m(µ0) =Yn[gn(µ)/gn(µ0)]−b(n)m /bn. (4)* The large top quark Yukawa coupling also leads to renormalization group corrections tothe first-third and second-third family CKM quark mixings.4

The ν mass ratios at the scale MX are the same as the physical ratios:mν,i(X)/mν,j(X) = mν,i/mν,j. (5)The equality holds because the known and unknown gauge corrections to light neutrinomasses are due to heavy, flavor-blind interactions that begin to run only at the W bosonmass, far above any neutrino mass.

The gauge corrections to the up-type quark mass ratiosare substantial, because they partly arise from QCD and because the quark masses havea large hierarchy in the presence of massless gauge bosons. To evaluate these correctionscompletely requires the assumption that there are no new particles of mass between the Zboson and top quark masses with Standard Model gauge couplings.

The gauge correctionsrequire the top quark mass to logarithmic accuracy, which we take from the best neutral-current data to be mt = 160 GeV [6]. (Powers of the top quark mass are left explicit.

)Apart from differences in mass thresholds, the gauge corrections from QCD, QED andthe hypercharge U(1)Y are the same for all up-type quarks. The weak isospin SU(2)Lcorrections to the quark masses are zero, since these masses are of the Dirac type, mixingleft- and right-handed fields.

Corrections due to new gauge couplings would begin at scalesabove mt and would cancel in the ratios. With κ = 1 GeV and taking mu(κ) = 5 MeV,mc(κ) = 1.35 GeV [10], and mt as free if it occurs as a power,mc(X)/mu(X) = mc(κ)/mu(κ) = 270mt(X)/mc(X) = (1.90)mt/mc(κ) = 140(mt/100GeV)mt(X)/mu(X) = (1.90)mt/mu(κ) = 38000(mt/100GeV).

(6)The top quark mass is defined by mt = mt(mt). Since MN,i(X) ∝mpu,i(X), the gaugecorrections to MN are accounted for in the gauge corrections to mu(X).

Any correctionsto MN due to new gauge interactions either cancel in the ratios or are assumed to be weaklycoupled and thus small.The other set of corrections are due to the fermions’ couplings to the Higgs sector. TheYukawa couplings and fermion masses are simultaneously diagonal.

In the neutrino mass5

ratios, under our assumptions, only the Yukawa coupling to the top quark is important.The renormalization group equation for the top quark mass is modified from (3) tod ln mt(µ)d ln µ=Xnb(n)m · g2n(µ) + bHm · [mt(µ)/MW ]2,(7)where the factor bHm depends on the Higgs sector. The solution to (7) can be written asmt(µ) = f(µ) · mt(µ)0, where mt(µ)0 is the solution to (3).

Taking f(mt) = 1,1 −1f2(X) = 2bHmZ MXmtdµµ · m2t (µ)0M2W. (8)The numerical evaluation of f(X) requires the function mt(µ)0 over the full range fromthe top quark mass to unification.

However, our ignorance of this function and of theHiggs sector can be collapsed into a single number, the Landau-triviality value of the topquark mass, mtL. This is the top quark mass for which, with a fixed MX, the right-handside of (8) is unity and f(X) diverges.

That is, f(µ) diverges before µ reaches MX, if mtexceeds mtL. The triviality value mtL is the upper limit of the top quark mass:1m2tL= 2bHmZ MXmtdµµ · m2t (µ)0M2W m2t,(9)with the presence of the unknown mt as the lower bound inducing only a small logarithmicerror.

(The r.h.s. of (9) contains no powers of the top quark mass.) Thenf2(X) =11 −m2t /m2tL.

(10)For example, in the minimal Standard Model, with MX = MPl ≃1×1019 GeV, mtL ≃760 GeV; in the supersymmetric (SUSY) case, with the same MX, mtL ≃190 GeV. Ofparticular interest because of its successful prediction of the weak mixing angle, the SUSYSU(5) grand unified model yields mtL ≃180 GeV, with MX ≃2×1016 GeV.

The non-SUSY SO(10) model, breaking through an intermediate left-right model, gives mtL ≃380GeV [4,6].6

With the aforementioned assumptions, the final mass ratios for the light neutrinos aremνµ/mνe = (270)2−pmντ /mνµ =1(1 −m2t /m2tL)1−p/2 · [140 · mt/100GeV]2−pmντ /mνe =1(1 −m2t /m2tL)1−p/2 · [38000 · mt/100GeV]2−p. (11)For a given νe or νµ mass, the ντ mass can be sensitive to the top quark mass beyond thenaive seesaw dependence, because of the triviality factor.It would be interesting to check how varying these assumptions changes the neutrinomass ratios.

Unfortunately, most of the assumptions cannot be changed without losingpredictiveness. The flavor-blindness of the gauge interactions is especially crucial.

How-ever, switching to a leptonic seesaw, with mD ∝ml, does lead to predictive neutrino massratios, if the eigenvalues MN,i(X) ∝mpl,i(X) and all neutrinos and charged leptons aresubject only to family-blind, weakly-coupled gauge interactions. Thenmν,i/mν,j = [ml,i/ml,j]2−p(12)is a good approximation.AcknowledgmentThe author wishes to thank Paul Langacker of the University of Pennsylvania for anearlier collaboration on the seesaw [4] on which this letter is based, Miriam Leuer andYosef Nir of the Weizmann Institute of Science for helpful discussions, and Carl Albrightof Fermilab for informative comments.References[1]The most recent results for the solar, atmospheric, and Simpson 17 keV neutrinos maybe found in Proc.

XVth Conf. Neutrino Phys.

& Astrophys. (NEUTRINO ’92), Granada,Spain (June 1992).7

[2]M. Gell-Mann et al., in Supergravity, eds. F. van Nieuwenhuizen, D. Freedman (North-Holland, Amsterdam, 1979) p. 315; T. Yanagida, Prog.

Theo. Phys.

B135 (1978) 66;S. Weinberg, Phys. Rev.

Lett. 43 (1979) 1566.[3]P.

Langacker, Phys. Rep. 72 (1981) 185; in TASI 1990, eds.

M. Cvetic, P. Langacker (WorldScientific, Singapore, 1991) p. 863.[4]S.

Bludman, D. Kennedy, P. Langacker Nucl. Phys.

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D45(1992) 1810.[5]H. Harari, Y. Nir, Nucl.

Phys. B292 (1987) 251.[6]P.

Langacker, M. Luo, Phys. Rev.

D44 (1991) 817; P. Langacker, U. Pennsylvania preprintUPR-0492-T (1992) and private communication.[7]E. Witten, Phys.

Lett. 91B (1980) 81.[8]M.

Cvetic, P. Langacker, U. Pennsylvania preprint UPR-0505-T (1992).[9]A. Buras, J. Ellis, M. Gaillard, D. Nanopoulos, Nucl.

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