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# Low-energy formulas for neutrino masses with $tan beta$-dependent hierarchy

LOW¨CENERGY FORMULAS FOR NEUTRINO MASSES WITH tan ¦Â-DEPENDENT HIERARCHY
M.K.Parida
High Energy Physics Group, International Centre for Theoretical Physics, I¨C34100 Trieste, Italy

arXiv:hep-ph/9710328v3 9 Nov 1997

and Physics Department, North¨CEastern Hill University, P.O.Box 21,Laitumkhrah Shillong 793 003,India

N.Nimai Singh
Physics Department, Gauhati University, Guwahati 781 014, India (October 13,1997)

Abstract
Using radiative corrections and seesaw mechanism, we derive analytic formulas for neutrino masses at low energies in SUSY uni?ed theories exhibiting a new hierarchial relation,for the ?rst time, among them. The new hierarchy is found to be quite signi?cant especially for smaller values of tan ¦Â. 12.10.Dm, 12.60.Jv

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Uni?cation of gauge and Yukawa couplings in supersymmetric grand uni?ed theories and their predictions have received considerable attention over the last decades [1-4]. In addition to solving the well-known gauge hierarchy problem, SUSY uni?ed theories based upon N = 1 supergravity, or originating from superstrings, provide an elegant method of uni?cation with gravity. Independent of these, the problem of neutrino masses has intrigued experimental as well as theoretical physicists since more than sixty years. If a uni?ed theory, based upon supergravity or superstrings, is to be accepted as the ultimate fundamental theory of nature, fermion masses including those of neutrinos have to emerge as necessary predictions of such a theory. The simplest and the most elegant method of obtaining neutrino masses is through seesaw mechanism [5] leading to the Majorana masses of left¨Chanded neutrinos: m¦Íi = C¦Íi m2 i , i = 1, 2, 3 MN (1)

where ¦Íi (i = 1, 2, 3) = (¦Íe , ¦Í? , ¦Í¦Ó ) and mi (i = 1, 2, 3) = (mu , mc , mtop ). With C¦Íi ? 1, the canonical seesaw formula is valid at the scale (? = MX ) where, due to spontaneous symmetry breaking, a higher gauge symmetry such as SUSY SO(10), or one of its intermediate gauge groups, breaks down to the SUSY standard model and the right¨Chanded Majorana neutrinos acquire mass MN ( assumed degenerate). But the limits on neutrino masses extending over wide range of values, are estimated either from cosmology or experimental measurements like tritium ¦Â-decay, neutrinoless double ¦Â-decay, solar and atmospheric neutrino ?uxes and neutrino oscillations at LSND [6]. Therefore, it is of utmost importance and considerable interest that accurate predictions of the seesaw mechanism are made available at low energies. For the ?rst time, Bludmann, Kennedy and Langacker (BKL) [7] found that C¦Íi ¡¯s deviate drastically from unity due to radiative corrections. Very signi?cant re?nements using semi-analytic and numerical methods, including renormalisation of Yukawa couplings and neutrino-mass operator, have been carried out in SUSY [8] and nonSUSY SO(10) [8-9]. In this work, we derive low-energy formulas for neutrino masses, analytically, in SUSY uni?ed theories demonstrating,for the ?rst time, the existence of a new hierarchial relation among them. We also estimate the impact of such a new hierarchy on neutrino-mass ratio predic2

tions for interesting regions of tan ¦Â corresponding to t ? b ? ¦Ó and b ? ¦Ó Yukawa uni?cation models.In the small tan ¦Â region the mass ratios are found to be very signi?cantly dependent upon tan ¦Â. The derivation of analytic formulas is carried out using the renornalisation group equations (RGEs) at one¨Cloop level for gauge couplings (gi(t) , i = 1, 2, 3), Yukawa couplings (hi (t)) [8,10¨C12], neutrino mass operator (K(t)) [8] and vacuum expectation value (VEV) of u-type Higgs doublet (Vu (t)) [11] in SUSY standard model embedded in GUTs like SO(10) without [3,10¨C12] or with an intermediate gauge symmetry [4], dhf (t) = Ohf (t) , f = u, c, top, b, ¦Ó dt Ohtop = htop 16¦Ð 2
3

(2)

6h2 + h2 ? top b

2 Ci g i i=1

(3)

hb Ohb = 16¦Ð 2

3

6h2 b

+

h2 ¦Ó

+

h2 top

?
3

2 Ci¡ä gi i=1

(4)

Oh¦Ó =

h¦Ó 16¦Ð 2

4h2 + 3h2 ? ¦Ó b

2 Ci¡ä¡ä gi i=1

(5)

Ohu,c =

hu,c 16¦Ð 2

3

3h2 ? top

2 Ci g i i=1

(6)

Ci = (

13 16 , 3, ) 15 3 7 16 , 3, ) 15 3 9 , 3, 0) 5

Ci¡ä = (

Ci¡ä¡ä = (

3 bi gi dgi = , i = 1, 2, 3 dt 16¦Ð 2

bi = (

33 , 1 , ?3 ) 5 3

(7)

dK K = dt 16¦Ð 2

6 2 2 6h2 + 2h2 ? g1 ? 6g2 top ¦Ó 5

(8)

In eqs.(2)¨C(8), i = 1, 2, 3 correspond to the gauge groups U(1)Y , SU(2)L , and SU(3)C of the SUSY standard model assumed to hold for ? ¡Ý mtop = MSU SY . The variable t = ln ? is the natural logarithm of the scale parameter ?. We include scale (t = ln ?) dependence in the Yukawa couplings [8,10-12] as well as the vacuum expectation value [11] in the de?nition of the quark masses, mu,c,top (t) = hu,c,top (t)Vu (t) ¡Ì 2 (9)

Using eq.(9) and evolving K from the higher mass scale ? = MX to ? = ?0 = mtop , the renormalized coe?cients in the seesaw formula are, m¦Íi (t0 ) = C¦Íi (t0 ) m2 (t0 ) i , i = 1, 2, 3 MN
2

(10)

C¦Íi (t0 ) =

htop,c,u (tx ) htop,c,u (t0 )

K(t0 ) K(tx )

(11)

where t0 = ln mtop and tx = ln MX . It is to be noted that, the running VEV[11] cancels out from in eq.(11) yielding the same renormalization as noted in ref.[8]. Now using RGEs (2)¨C(8), we obtain in a straight forward manner, C¦Í¦Ó (t0 ) = R(t0 ) e(6Itop +2Ib ?2I¦Ó ) (12)

C¦Í?,e (t0 ) = R(t0 ) e?2I¦Ó
3
di bi

(13)

R(t0 ) =
i=1

¦Ái (tx ) ¦Ái (t0 )

,

di = 1 16¦Ð 2

?4 ?16 , 0, 15 3
(tx )

(14)

If =

(t0 )

h2 (t) dt , f = top, b, ¦Ó. f 4

(15)

In evolving the masses and gauge couplings to scales below ?0 = mtop , we ignore negligible contribution to C¦Íi between the scales MZ and mtop . Below ? = MZ (tz = ln MZ ), the renormalisation of the quark masses, mc and mu , occur due to the presence of the gauge symmetry SU(3)C ¡Á U(1)em leading to well-known QCD¨CQED rescaling factors [11], ¦Çu ? We then obtain m¦Í? (tc ) ? C¦Í? (tc ) m2 (tc ) c MN (17) mc (tc ) mu (0) , ¦Çc ? . mu (t0 ) mc (t0 ) (16)

m¦Íe (0) ? C¦Íe (0) where C¦Í? (tc ) =

m2 (0) u MN

(18)

R(t0 ) ?2I¦Ó e 2 ¦Çc R(t0 ) ?2I¦Ó e 2 ¦Çu

(19)

C¦Íe (0) =

(20)

In eqs.(16)¨C(20), tc = ln mc and t1 = ln ?1 = 0 for ?1 = 1GeV. The quantities in the R.H.S. of (12) have negligible contributions due to renormalisation e?ects carried down to lower energies, ? = ?1 = 1GeV (t = t1 = 0) although those in (13) change signi?cantly. Combining eqs.(10)¨C(20), we obtain the low energy formulas for the left¨Chanded Majorana neutrinos of three generations, m¦Í¦Ó = R e(6Itop +2Ib ?2I¦Ó ) m2 top MN (21)

m¦Í? =

R ?2I¦Ó m2 c e 2 ¦Çc MN

(22)

m¦Íe

R ?2I¦Ó m2 u = 2e ¦Çu MN 5

(23)

where R = R(t0 ) as de?ned in (14) and the masses occuring on the R.H.S. are the low¨Cenergy values. The formulas (21)¨C(23) predict low energy values of neutrino masses, analytically ,in a large class of SUSY uni?ed theories including SO(10) which can be compared with the experimental values.If the right-handed neutrino masses are nondegenerate,these formulas can be easily generalised by replacing MN by MNi for the ith generation in eqs.(21)-(23).But independent of any assumed value of the degenerate right-handed neutrino mass ,the formulas predict a new hierarchial relation, m¦Í¦Ó : m¦Í? : m¦Íe :: m2 exp(6Itop + 2Ib ) top
2 2 : (m2 /¦Çc ) : (m2 /¦Çu ) c u

(24)

Using the experimentally observed values of the quark masses mu , mc and mtop = 174GeV, the CERN¨CLEP measurements, and ¦Çu and ¦Çc [12], we have evaluated the central values and uncertainties of C¦Íi (i = e, ?, ¦Ó ), for di?erent values of tan ¦Â =
Vu , Vd

the ratio of VEVs of

the u¨Ctype and d¨Ctype Higgs doublets. The details of such analyses and their aapplications to di?erent SO(10) models will be reported elsewhere. It is quite clear that the low-energy predictions on neutrino masses depend crucially on the nature and values of Yukawa couplings between mtop and MX .For example,if the Yukawa couplings increase at faster rate in certain GUTs,which may occur due to appearance of new degrees of freedom at higher mass scales ,the new contributions to integrals over squares of Yukawa couplings , are likely to predict neutrino masses di?erent from conventional SUSY SO(10).However,here we con?ne to numerical results on the relative hierarchy between m¦Í¦Ó and m¦Í? or m¦Íe in conventional SUSY GUTs, which doe not depend upon the arbitrariness on MN , m2 2 (6Itop +2Ib ) m¦Í¦Ó top ¦Ç e = m¦Í? m2 c c m2 2 m¦Í¦Ó = top ¦Çu e(6Itop +2Ib ) m¦Íe m2 u (25)

(26)

It may be recognised that the second and third factors in the R.H.S. of (25)¨C(26) are due to radiative corrections from MX down to low energies, corresponding to the ratios, 6

C¦Í ¦Ó 2 = ¦Çc e(6Itop +2Ib ) C¦Í ? C¦Í ¦Ó 2 = ¦Çu e(6Itop +2Ib ) C¦Í e

(27)

For larger values of tan ¦Â corresponding to t ? b ? ¦Ó Yukawa uni?cation in supergrand desert models, the Yukawa couplings at the GUT scale and hence the values of Ib ,I¦Ó , and Itop change slowly with tan ¦Â. The value of exp [ 6Itop , ] almost remains constant giving rise to C¦Í¦Ó /C¦Íe ? 15 ? 20 and m¦Í¦Ó /m¦Íe ? (2 ? 3) ¡Á 1010 for tan ¦Â = 10 ? 59.As compared to the BKL value[7] of m¦Í¦Ó /mnue ? 9.2 ¡Á 109 ,the enhancement in the hierarchy in this region is only 2-2.5. But as tan ¦Â decreases and approaches the value of 1.7, as in b ? ¦Ó uni?cation models with a super grand desert,htop (MX ) and hence Itop increase with the decrease of tan ¦Â while hb (MX ) ,h¦Ó , Ib and I¦Ó are negligible. Thus,we ?nd that,for smaller values of tan ¦Â ? 1.8,Itop = 0.4 and the ratio (m¦Í¦Ó /m¦Íe ) has a substantial contribution due to the rise of the top quark Yukawa coupling near the uni?cation scale which leads to the central value of C¦Í¦Ó /C¦Íe ? 83.6 and m¦Í¦Ó /m¦Íe ? 1.1 ¡Á 1011 .These may be compared the results of BKL[7] corresponding to C¦Í¦Ó /C¦Íe ? 7.6 and m¦Í¦Ó /m¦Íe ? 9.2 ¡Á 109 .Whereas in the large tan ¦Â region,our enhancement factor is only 2-3,for smaller tan ¦Â=1.8,the enhancement factor ia as large as ? 10. Similar features are also noted in intermediate scale SUSY SO(10) model corresponding to MX = 1013 GeV where the intermediate gauge symmetry breaks down to the standard gauge group giving large mass to MN [4]. When the ratio m¦Í¦Ó /m¦Í? is examined as a function of tan ¦Â, similar type of enhancements are noted. For example, in the supergrand desert type models (MX ? 2 ¡Á 1016 GeV), C¦Í¦Ó /C¦Í? ? 65 and m¦Í¦Ó /m¦Í? ? 1.2 ¡Á 106 for tan ¦Â = 1.8, exhibiting an enhancement of hierarchy by a factor ? 15 over the BKL value[7] of m¦Í¦Ó /m¦Í? ? 8.1 ¡Á 104 GeV.The details of evaluation of neutrino masses and mass ratios including uncertainties,as a function of intermediate scales and tan ¦Â would be reported elsewhere. In summary we have obtained analytic formulas for left¨Chanded Majorana¨C neutrino masses at low energies demonstrating, for the ?rst time ,the existence of a new hierarchy 7

between the neutrino masses which depends quite signi?cantly upon tan ¦Â in the region of smaller values of this parameter. These results hold true in a large class of models including SO(10). We suggest that these formulas be used while estimating neutrino mass predictions in uni?ed theories with a degenerate right¨Chanded Majorana neutrino mass. The formulas can be easily generalised for nondegenerate right-handed neutrino masses. After this work was completed,the contents of ref.[13] was brought to our notice where the renormalization for m¦Í¦Ó has been discussed,but to prove the new hierarchy low energy formulas for all the three neutrino masses ,as demonstrated here,are essential.

ACKNOWLEDGMENTS

One of us (M.K.P) thanks Professor R.N.Mohapatra for useful discussions and to Dr.K.S.Babu for useful comments and bringing ref.[13] to our notice.The authors acknowledge support from the Department of Science and Technology, New Delhi through the research project No. SP/S2/K¨C09/91.

8

REFERENCES
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[11] H.Arason, D.J.Castano, E.J.Piard and P.Ramond, Phys.Rev.D 47, 232 (1993); H.Arason, D.J.Castano, B.Keszthelyi, S.Mikaelian, E.J.Piard, P.Ramond and

B.D.Wright, Phys.Rev.D 46, 3945 (1992). [12] V.Barger, M.S.Berger and P.Ohmann,Phys.Rev.D 47, 1093 (1993). [13] K.S.Babu,Q.Y.Liu, and A.Yu.Smirnov,hep-ph/9707457.

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