Babu - 1988 - Modes of Calculable Majorana Neutrino Masses

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Volume 203, number 1,2

PHYSICS LETTERS B

24 March 1988

MODEL OF CALCULABLE MAJ ORANA NEUTRINO MASSES

KS. BABU

Department of Physics and Ast ronomy , Uni versit y of Rochester, Rochester, NY 14627, USA

Received 30 December 1987

We present an economical model of generating finite and calculable Majorana neutrino masses. The model has a singly

charged and a doubly charged singlet scalar which supply Majorana masses to the neutrinos at the two-loop level. One of the

neutrinos is predicted to be massless in the model.

If the neutrinos have mass, they may be either Dirac or Majorana particles. In the standard model, the neutri-

nos are massless since right-handed neutrinos are absent (so that Dirac mass is not possible) and since lepton

number is exactly conserved (so that Majorana mass is not possible). Any attempt to generate non-zero neutrino

masses has to violate one of the above two assumptions. Since right-handed neutrinos are not essential in the

standard model, it is often more economical to supply Majorana masses to the left-handed neutrinos. For ex-

ample, if a triplet of Higgs particles carrying two units of hypercharge is introduced into the standard model

[ 1 ] *, the neutrinos can acquire Majorana masses. The extreme smallness of their masses compared to the

charged lepton masses may be explained by the smallness of the triplet to doublet vacuum expectation value

ratio

[

31. Another example is the model of Zee

[

41 wherein the neutrinos acquire finite and calculable masses

at the one-looop level. The model requires a second doublet and a charged singlet of scalar particles and predicts

interesting mixing patterns

[ 51.

In this letter we present a new model of calculable Majorana neutrino masses

and mixing which is more economical than the models mentioned above in the sense that the number of new

particles needed is minimal. Finite neutrino masses arise at the two-loop level and are naturally small. A novel

feature of the model is that one of the neutrinos is predicted to be massless.

We extend the standard model to include two charged singlet Higgs fields - a singly charged h+ and a doubly

charged k++. Right-handed neutrinos are not introduced. The interaction of the charged scalars with the fer-

mions is given by

Lf~=_&(~ _C&L)~ijh+ +hnb(laRClbR)k++

+h.c.,

(1)

where wL stands for the left-handed lepton doublet and lR for the right-handed charged lepton singlet. (ab) and

(ij) are the generation and SU( 2) indices, respectively, and C is the charge conjugation matrix. The coupling

fab is antisymmetric (,&,= -fbn) by virtue of the Clebsch factor tij, whereas

bob

s symmetric

(hab= hba) h +

and

k+ +

do not interact with the quarks. Eq. (1) when expanded for the case of three generations reads

=UpGE~L

-~eL>+fer(&L-~eL)+&r(~TL-~pL)]h+

+[heeece,+hKurUC~~+hrr~~R+2he~eC~R+2hereC~R+2h~rlUCrR]k++

+h.c.

(2)

We shall assume_& and

hab

o be real for simplicity. For later convenience we define a matrix Rsuch that Eaob

qhab,with q= 1 for a= b and rl= 2 for a# b. Note that eq. (1) conserves lepton number and by itself cannot be

responsible for neutrino mass generation.

* For a recent review see ref. [21.

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PHYSICS LETTERS B

24 March 1988

Fig. 1. Two-loop diagram which generates finite Majorana neu-

trino masses.

Consider the self-interaction of the scalar particles h +,

k + +

and the usual doublet += ( @+ O)T described by

the Higgs potential

V=~~ +~+~u22h+h-+~L:k++k--+I,(~+d)2+I2(h+h-)2+;l~(k++k--)2+114(9+~)(h+h-)

+n,(~+~)(k++k--)+l,(h+h-)(k++k--)+~(h+h+k--+h-h-k++).

The physical Higgs particles are Re go, h + and k + + with masses given by

(3)

ms=4A,v2, rnz =pU: +&v2,

ml =,u: +&v,

(4)

where v= (@O>.@* and Im @are as usual, the would-be Goldstone bosons eaten up by the

W

and 2. Note

that there is no mixing among 9 and h+.

The last term in eq. (3) violates lepton number by two units and is responsible for neutrino mass generation

at the two-loop level via the diagram shown in fig. 1. There is no infinity associated with the diagram and hence

the neutrino masses are calculable in the model. Fig. 1 (and its conjugate) leads to an induced Majorana mass

term

Y

maSS - faMx9

where we have defined self-conjugate fields

Xn=V=+(va)C.

The mass matrix is given by

M~b=8~~,FcdrncmdcdCt db

where m,, md are the charged lepton masses and

(5)

(6)

(7)

(8)

It is cumbersome to evaluate the convergent integrals of eq. (8) exactly; we simply note that if the region of

integration is limited to k* , q2 e m , it takes a simple form:

Icd

ji (mi:m~)

mz:mi,

mqmyy2)

m:ln(m:y2)]

x [m~ln(m~~2) -m21n(m~~2)].

(9)

If the cut-off parameter

m2

is set equal to

rni

and the lepton masses are neglected in comparison with the

charged scalar masses,

(10)

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24 March 1988

We shall use this approximate expression for our numerical estimates.

From the structure of the mass matrix, eq. (7)) it follows that one of the neutrinos is massless in the three

generation model. This can be seen by noting that Mob in (7) may be written as A&=

uKfl)ab,

where

K

is a

matrix defined by

Kcd=

8dm,m&

(c, d not summed).

(11)

Hence Det M= 1Detf] Det K= 0, since a 3 X 3 antisymmetric matrix has zero determinant #2.This is in clear

contrast with the Zee model which also utilizes antisymmetric Yukawa couplings but predicts all the neutrinos

to be massive. The crucial difference is that in the present model the charged scalar h+ does not mix with the

charged scalar from the doublet, whereas in the Zee model such mixing is precisely the one responsible for

neutrino masses.

Before we analyze the mass matrix for the mass-mixing pattern, we shall constrain the various parameters of

the model from existing experimental data. The scalars h + and k ++ have to be heavier than about 20 GeV, or

else they should have been observed at e+e- collision experiments at PETRA. The exchange of h + contributes

to p decay [4,6]. While the angular and energy distributions are undistorted, the total rate is corrected by a

factor

1 +

4fiFm /g2

rni)*

.

Allowing for a deviation of lo16 rom e-F universality, this requires

j /rnz 5

8

X

10 - GeV -. Similar constraints on falm f and fZ,lm i can be obtained by demanding that the deviation from

e-p-t universality in T decay be less than a few percent.

Both h+ and k++

exchange contribute to the (g- 2) of the leptons. The correction is given by

6(g,-2)=[(2m1)2/48x2][Cfft),~/m~+(j;Rt)lllm:].

From 6(g,-2) < IO-O and 6(g,-2) < lo- [ 71, we obtain

Cff)eelmf+(~~t)ee/m:65x10-2 GeV2, Cff),lm~+(~~~)~/m~~l~10-~ GeV-2.

~1 nd e y decay proceed at the one-loop level by

h +

and

k + +

exchange at a rate given by

(12)

(13)

r(pL--,ey) =

am:

(14)

FromBr(~-+ey)


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Volume 203 number 1 2

PHYSICS LETTERS B

24 March 1988

A=lh*

w b.WU-M4[%U-~M.

The differential cross section is

(18)

+ a h& (I+cose)2 .s

2~41~ l+cose+r

(19)

where 4 is the center of mass energy, 8 the angle between e- and p-, r=2m, ls, c,.,=cos B,, s,=sin 8,. The

first term in (19) is the standard model y and Z contribution [ O], the second term is the k+ +-y and k+ +-Z

interference and the last term is the pure k++

contribution. The charge asymmetry defined as

A

FB= & -NB)I(NF+NB), where & and NB are the number of events in the forward and backward hemi-

spheres (forward is defined by t9c 90 ) is given by (assuming

h /4na e

1)

A

F

N C2+(2h& ,/4na){l+rZ ln[r(2+r)l(l+r)*])

C, +(2hze/4xa){2-2r+r * I n[(r +2)/r]}

where

[lo]

CIZ1, c*=Ls

4c2s2 s-m* *

w

z

(20)

(21)

Agreement with the average PETRA measurements [ 111 A,= ( 17.8 _+ .9)% and A,,= ( - 15.5 f 2.7)% at

$= 44 GeV can be obtained if we choose h& .l4na, h& /471a, 6 10 -*, corresponding to mk - 50-l 00 GeV. The

total cross section e+e--+l+l- is not sensitive to the k++

effects with such small values of the couplings.

Having constrained the various parameters of the model we now turn to the neutrino mass matrix, eq. (IO).

From the discussions above, it is clear that the off-diagonal elements of h are more severely constrained than the

diagonal elements. We shall assume h,b

0 for a # b. The mass matrix (in the limit of neglecting electron mass)

is then given by

(22)

where

(23)

Note that if rn: is also neglected compared to m f , the tau neutrino and one of Y, or v,, are massless. Hence v7 is

not the heaviest neutrino in the model.

The eigenvalues of M are

(24)

Choosing p- 200 GeV, which is the highest scale in the theory, andhb Nh, = 1O-* (which is consistent with all

the constraints derived earlier) and mk= m,, N 100 GeV, as a typical example, we see that r12- 3 X 10m5 eV,

& - 1 x 1O-* eV. It is interesting to note that these masses are in the right range to explain the solar neutrino

puzzle by the MSW mechanism [ 121.

The matrix M can be diagonalized by an orthogonal transformation

OTMO=M&ag,

(25)

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Volume 203, number I,2

PHYSICS LETTERS B

24 March 1988

with

Cl

SlC3

o=

-SIC2

CIc2c3 S2S3

SlS2 -C1S2c3 C2S3

where ci=cos ei, si=sin ei and

(26)

(27)

Thus we see that except for e3, the mixing angles are not necessarily small. The model predicts large oscillations

among the different neutrino flavors. As a consequence, no definite conclusion can be drawn as to which one of

the neutrinos is massless in the model.

In conclusion, we have presented an economical model of Majorana neutrino mass generation. The masses

are finite and naturally small, since they arise via two-loop diagrams. A unique prediction of the model is the

masslessness of one of the neutrinos.

I wish to thank Dr. X.-G. He, Professor V.S. Mathur and Professor R.N. Mohapatra for discussions. This

work is supported in part by the US Department of Energy Contract No. DE-ACOZER13065.

Note added

The k+ + exchange also leads to muonium-antimuonium oscillation [ 141 (pe--p-e+) with a strength given

by G N

h, h,lm:.

With reasonable values of the couplings, G N 1O- GF in our model which may be tested in the

near future. I thank Professor R.N. Mohapatra for pointing this out to me.

References

[

]

T.P. Cheng and L.F. Li, Phys. Rev. D 22 (1980) 2860;

L. Maiani, CERN Report No. TH2846 (1980).

[2] S.M. Bilenky and S.T. Petcov, Rev. Mod. Phys. 59 (1987) 671.

[3] G.B. Gelmini and M. Roncadelli, Phys. Lett. B 99 (1981) 411;

H. Georgi, S.L. Glashow and S. Nussinov, Nucl. Phys. B 193 (1981) 297.

[4] A. Zee, Phys. Lett. B 93 (1980) 389; Phys. Lett. B 161 (1985) 141; University ofwashington preprint 40048-19 (1985).

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S.T. Petcov, Phys. Lett. B 115 (1982) 401.

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[ 91 Particle Data Group, M. Aguilar-Benitez et al., Review of particle properties, Phys. Lett. B 170 (1986) 1.

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[ 111 See e.g. B. Naroska, in: Progress in electroweak interactions, ed. J. Tran Thanh Van (Editions Frontieres, Dreux, 1986) p. 83, and

references therein.

[ 121 L. Wolfenstein, Phys. Rev. D 17 (1978) 2369;

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to be published.

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Babu - 1988 - Modes of Calculable Majorana Neutrino Masses

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