Derivation of very tiny masses of neutrinos in the left-right symmetric gauge model

Volume 167B, number 1 PHYSICS LETTERS 30 January 1986

D E R I V A T I O N O F VERY T I N Y M A S S E S O F N E U T R I N O S IN T H E L E F T - R I G H T S Y M M E T R I C G A U G E M O D E L

Tsunehiro K O B A Y A S H I 1

Max-Planck-lnstitut fftr Physik und Astrophysik - Werner Heisenberg-lnstitut fftr Physik - D- 8000 Munich 40, Fed. Rep. Germany

Received 29 October 1985

Very tiny mass values of neutrinos are derived in the left-right symmetric gauge model with spontaneous parity violation. In our scheme the Dirac mass terms of neutrinos vanish in the tree level, but they are induced in higher order and proportional to the charged-lepton mass terms. The relation, m~e = Gom2/mwR, with G O _< 10-13 for the three generations, is derived. We also discuss the model where the mixings among the lepton generations vanish.

Neutrino masses in the lef t - r ight symmetric model with spontaneous parity violation based on the gauge group SU(2)L ® SU(2)R ® U(1)B_ L were discussed by Mohapatra and Senjanovi6 [1,2]. Their model is described in terms of the general Yukawa couplings as

£Y = h l ~L ~b~ £R + h2 J~L ~ £R + h3 qL ~bq qR + h4qL~q qR

+ i h 5 ( £ T c r 2 A L £ L + £TCr2AR£R ) + h.c., (1)

where C is the Dirac charge-conjugation matr ix , '~ = r2~*r2, £L(£R) and qL(qR) are, respectively, the left (right)- handed leptons and quarks, and ~ , q~q, A L and A R are mesons. Those particles are represented with the following representations of the group (SU(2)L , SU(2)R , NB_ L = baryon number - lepton number):

£L(~R) = (2(1), 1(2),--1), qL(qR) = (2(1), 1(2), 1/3),

~b £, ~q = (2, 2, 0), AL(AR) = (3(1), 1(3), --2). (2)

In ref. [1] ~b ~ = ~b q is taken. For spontaneous parity violation the vacuum expectation values are given as

<¢~(q)) = , (AL) = , (AR) = , (3)

0 b~(q) VL VR

where the relations V~t >> ai(o3 ~ b~2(q) >> V2L must be realized. In ref. [1] a 2 >> b 2 and V L = 0 are also assumed. A possibly different so lu t ion I3 ] , represented with

V i > b 2 - a q 2 - b ~ : / : O , VL=a~=0 (4)

is pointed out from the standpoint of a composite model for leptons, quarks and Higgs mesons [4]. In the general scheme such a solution is obtained from the following couplings among the scalar mesons:

1 On leave from Institute of Physics, University of Tsukuba, Ibaraki 305, Japan.

0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North.HoUand Physics Publishing Division)

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=,,o [Tr(":, 'L + '-,,: Tr(" [ + " : "R" :AR>

+/31 Tr(A~AL¢£~b'Qt +AtR aR~b'~1"~b~) + f12 Tr(A~¢£AR~~'f + AtR~b'Qt AL~b'~)

+~eTr(AtLA L +AtAR) ~ Tr(¢atq~')+~/ ~ Tr(cbatcbadp"t~a)+'7,( ~ Tr(e#a'~¢a)) 2 a=£,q a=£,q \a=£,q

+ VI(~b, A) - /a2Tr(A~AL + AtRAR) + ~ ma2 Tr(¢at~a), (5) a=~,q

with the constraints [3] t~ 0 + a l > 0, a I < 0, 7 + 7e > 0,/31 > 0, fll > 1/321 (or - 2 a 13, > (ill - 1/32 I)2),/3e < 0 and /a 2 >> m 2 > 0, and Vi(~b, A) stands for the interaction among ¢ and A which does not contribute to the Higgs potential. It is noted that the couplings of ~q with A are quite different from those of ~ in this model [3,4]. For the choice of the parameters satisfying the above constraints the absolute minimum of the Higgs potential is realized at the points represented with eq. (4) [3]. In the model [4] the Yukawa couplings are written as

d~y = f ~ L O ~ R +fq~lL~bqqR + ih(~TC72AL~L + ~TCr2AR ~R) + h.c. (6)

For simplicity we shall first study the case with one generation of leptons and quarks, and mixings among the generations will be discussed later. From eqs. (4) and (6) we see that in the tree level the Dirac mass terms of fermions are given by

m v = O, m~ =f~b~, mqu =fqaq , mqd =fqbq . (7)

Of course, the right-handed neutrino has a non-zero Majorana mass term h V R . The Dirac mass of the neutrino is induced from the higher order diagram shown in fig. 1, the order of which may be estimated as

mDt, ~ 2(2rc)-4 g4f~(fq)2 b~a q b q/m2 R , (8)

where g is the gauge coupling constant for SU(2)L ® SU(2)R and m w stands for the mass of the right-handed R

weak boson given by roW_ "gWR I. Considering that row_ (~IV R I) may represent the characteristic energy scale N N

of the dynamics in the next sublevel, we use mWR as the cut-off and ignore the log-dependent factor in the evalua- tion. From eq. (7) we can rewrite eq. (8) as

m D ~ 2(2~)-4g 4 m~mqumqd/m2 R. (9)

(lq

J~ I R v VL IL bt R Fig. 1. The lowest order diagram inducing the Dirac mass term of the neutrino.

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Even if we put m s = m r ~ 1.8 GeV, mqu = m t "~ 40 GeV, mqd = m b = 5 GeV, mWR ~ 1 TeV [5 ] and g "~ 1, we have

m D ~ 5 X 10 -2 eV. (10) /1

When the Majorana mass term for the right-handed neutrino is taken into account, we may diagonalize the mass

matrix

0 m D c~v = v , (11)

m D hV R

where the neutrino doublet (v, N) is defined by v = v L and N -~ C(~R)T. Then the mass eigenvalues are given as

m~. = (mD) 2/h V R ~ G O m2 /mWR ,

m R ~ h V R , (12)

where the small value G O is estimated as

G O ~ [2(2rr)-4g 4 mqumqd ] 2lh V R m3WR ~ 10 -13 , (13)

for the choice of rn, = rot, m , . = rn b and h ~ g. The mass m 7 given by eq. (12) is quite similar to that derived i n ref. [1] , but the~gmall valueUGo gives us much smaller values than those predicted in ref. [1]. When we ignore the mixings among the leptons, we have

m 3, ~ 10 -13 eV, for re,

10 -9 eV, for v ,

10 -6 eV, for vr, (14)

for the three-generation model. These values should be compared with those, m v ~ 1.5 eV, rn v ~ 56 keV and rn v ~ 18 MeV, given in ref. [1] . Even if there exist more than three generationseand the heavieUst quark masses arerin the order of m . . . . we have G - ~ 10 -5 which gives m ~ 10 -5 eV These values are so small that observa-

"R u v e "

t ion is difficult at present. Taking account of the mixings among the generations in terms of the couplings of fermions with C-mesons, we

may write the Yukawa interaction as

i 1

where the observed generations are obtained by diagonalizing the mass matrices derived from eq. (15) and the ""---" i i charged weak currents are as usual given by j L(R) = Zi(£[(R)3, UL01) + dL(R)7~ UL(R))" We can estimate the Dirac

mass terms o f neutrinos as

D (16) ( m y ) k n = mo(m~)kn'

where (m~)kn represents the kn component of the charged-lepton mass matrix and the contr ibution of the loops can be factorized into the common factor rn 0. This equation implies that the Dirac mass matrix of the neutrinos is proport ional to the charged-lepton mass matr ix and both matrices can be diagonalized in terms of the same unitary matrix. As far as the Majorana mass terms o f the right-handed neutrinos are concerned, we cannot say anything in general. We may only say that the Majorana mass matrix is diagonalized in terms of the same unitary matrix as derived in the charged-lepton case, if either hi/= hSii or hii = constant X fly is satisfied. In these cases

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all mixings among the lepton generations vanish. Since we have a reason to put hi] = h~ii in the composite model proposed in refs. [3,4] , no mixings among the lepton generations may be expected.

I would like to thank Professor W. Zimmerman and Professor H. Fritzsch for kind hospital i ty in the Max- Planck-Institut for Physik und Astrophysik in Munich.

References

[1 ] R.N. Mohapatra and G. Senjanovid, Phys. Rev. Lett. 44 (1980) 912. [2] R.N. Mohapatra and G. Senjanovid, Phys. Rev. D23 (1981) 165. [3] T. Kobayashi and S. Toldtake, University of Tsukuba preprint UTHEP-144 (1985). [4] T. Kobayashi, Phys. Rev. D31 (1985) 2340. [5] G. BeaR, M. Bander and A. Soni, Phys. Rev. Lett. 48 (1982) 848.

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Derivation of very tiny masses of neutrinos in the left-right symmetric gauge model