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Model of “calculable” Majorana neutrino masses
Volume 203, number 1,2
MODEL OF “CALCULABLE”
PHYSICS LETTERS B
24 March 1988
MAJORANA NEUTRINO MASSES
KS. BABU Department of Physics and Astronomy, University 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 neutrinos 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 example, 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 fermions is given by Lf’~=_&(~$_C&L)~ijh+ +hnb(laRClbR)k++ +h.c.,
(1)
where wL stands for the left-handed lepton doublet and lR for the right-handed charged lepton singlet. (ab) and (ij) are the generation and SU( 2) indices, respectively, and C is the charge conjugation matrix. The coupling fab is antisymmetric (,&,= -fbn) by virtue of the Clebsch factor tij, whereas bobis symmetric ( hab= hba). h + and k+ + do not interact with the quarks. Eq. (1) when expanded for the case of three generations reads %=UpGE~L
-~eL>+fer(&L-~eL)+&r(~TL-~pL)]h+
+[heeece,+hKurUC~~+hrr~~R+2he~eC~R+2hereC~R+2h~rlUCrR]k+++h.c.
(2)
We shall assume_& and habto be real for simplicity. For later convenience we define a matrix Rsuch that Eaob = qhab, with q= 1 for a= b and rl= 2 for a# b. Note that eq. (1) conserves lepton number and by itself cannot be responsible for neutrino mass generation. *’ For a recent review see ref. [ 21.
132
PHYSICS LETTERS B
Volume 203, number 1,2
24 March 1988
Fig. 1. Two-loop diagram which generates finite Majorana neutrino masses.
Consider the self-interaction of the scalar particles h +, k + + and the usual doublet += ( @+, #O)T described by the Higgs potential V=~~$+~+~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++).
(3)
The physical Higgs particles are Re go, h + and k + + with masses given by 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
(5)
where we have defined self-conjugate fields
(6)
Xn=V=‘+(va‘)C. The mass matrix is given by
(7)
M~b=8~~,FicdrncmdIcdCft)db,
where m,, md are the charged lepton masses and
(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 $, it takes a simple form: Icd =
&ji&(mi:m~)(mz:mi, [mqmyy2) -m:ln(m:y2)]
x [m~ln(m~~2)
-m21n(m~~2)].
(9)
If the cut-off parameter m2 is set equal to rni and the lepton masses are neglected in comparison with the charged scalar masses,
(10)
133
PHYSICS LE’ITERS B
Volume 203, number I,2
24 March 1988
We shall use this approximate expression for our numerical estimates. From the structure of the mass matrix, eq. (7)) it follows that one of the neutrinos is massless in the three generation model. This can be seen by noting that Mob in (7) may be written as A&= uKfl)ab, where K is a matrix defined by Kcd= 8&#dm,m&
(c, d not summed).
(11)
Hence Det M= 1Detf] ’ Det K= 0, since a 3 X 3 antisymmetric matrix has zero determinant #2.This is in clear contrast with the Zee model which also utilizes antisymmetric Yukawa couplings but predicts all the neutrinos to be massive. The crucial difference is that in the present model the charged scalar h+ does not mix with the charged scalar from the doublet, whereas in the Zee model such mixing is precisely the one responsible for neutrino masses. Before we analyze the mass matrix for the mass-mixing pattern, we shall constrain the various parameters of the model from existing experimental data. The scalars h + and k ++ have to be heavier than about 20 GeV, or else they should have been observed at e+e- collision experiments at PETRA. The exchange of h + contributes to p decay [4,6]. While the angular and energy distributions are undistorted, the total rate is corrected by a factor (1 + 4fiFm$/g2 rni)* . Allowing for a deviation of lo16from e-F universality, this requires j$/rnz 5 8 X 10 -’ GeV -‘. Similar constraints on falm f and fZ,lm i can be obtained by demanding that the deviation from e-p-t universality in T decay be less than a few percent. Both h+ and k++ exchange contribute to the (g- 2) of the leptons. The correction is given by 6(g,-2)=[(2m1)2/48x2][Cfft),~/m~+(j;Rt)lllm:]. From 6(g,-2)
< IO-‘O and 6(g,-2)
Cff’)eelmf+(~~t)ee/m:65x10-2
(12)
< lo-’ [ 71, we obtain GeV2,
Cff),lm~+(~~~)~/m~~l~10-~
GeV-2.
(13)
~1and e y decay proceed at the one-loop level by h + and k + + exchange at a rate given by r(pL--,ey) = am:
(14)
FromBr(~-+ey)<2x10-10,wehave Cfft),2/m2h+(&/?),2/m$<2x10-8
[8] GeV-2.
(16)
Lemon number changing decays such as +eee, z-+,uph etc. occur at the tree level via k+ + exchange. h + contributes to such decays only at the one-loop level which can be safely neglected. The rate for p-+ 3e is
r(v3d=
1 h$d& m: 8 192n3 m4.
(16)
k
From Br(p+3e) <2x lo-l2 [ 91 we have a stringent constraint hIreh e.?/m~<4xlO-*’
GeVm2.
(17)
Similar constraints from r decay are somewhat weaker: h,,h,lm i, h,h,,Jm i < 6 X 10 -’ GeVm2. The k++ scalar also contributes to high energy electron-positron scattering e+e--+f+l(l=p or r) in the 2 channel. The observed charge asymmetry in the scattering is in good agreement with the standard model Z”-y interference effects [IO] and this should not be upset by the extra k++ contribution. The k++ contribution to the amplitude is (after a Fierz rearrangement) u2The existence of a zero mass neutrino when using antisymmetric Yukawa couplings was first noted in the context of Dirac neutrinos by Mohapatra [ 131.
134
Volume 203, number 1,2
A=lh*2
w
&
PHYSICS LETTERS B
24 March 1988
(18)
b.WU -M4[%U-~M.
The differential cross section is + a! h& (I+cose)2 2~41~ l+cose+r
.s
(19)
where 4 is the center of mass energy, 8 the angle between e- and p-, r=2m,$ls, c,.,=cos B,, s,=sin 8,. The first term in (19) is the standard model y and Z” contribution [ IO], 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 where & and NB are the number of events in the forward and backward hemiA FB= (&-NB)I(NF+NB), spheres (forward is defined by t9c 90” ) is given by (assuming h$/4na e 1) A FB
N C2+(2h&,/4na){l+rZ ln[r(2+r)l(l+r)*]) $C, +(2hze/4xa){2-2r+r* In[(r+2)/r]}
’
(20)
where [lo] CIZ1,
c*=Ls 4c2s2 ww s-m* z *
(21)
Agreement with the average PETRA measurements [ 111 A,= ( - 17.8 _+1.9)% 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+lis 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 N h, = 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) 135
PHYSICS LETTERS B
Volume 203, number I,2
24 March 1988
with Cl o=
SlS2
where
SlC3 CIc2c3
-SIC2
ci=cos
-C1S2c3
ei,
+S2S3
(26)
+C2S3
si=sin ei and (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 (p’e--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 [ 1 ] 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). [ 5 ] L. Wolfenstein, Nucl. Phys. B 175 (1980) 93; S.T. Petcov, Phys. Lett. B 115 (1982) 401. [6] KS Babu andV.S. Mathur, Phys. Lett. B 196 (1987) 218. [7] T. Kinoshita, B. Nizic andY. Okamoto, Phys. Rev. Lett. 52 (1984) 717; T. Kinoshita and W.B. Lindquist, Phys. Rev. Lett. 47 (1981) 1573. [8] W.W. Kinnison et al., Phys. Rev. D 25 (1982) 2846. [ 91 Particle Data Group, M. Aguilar-Benitez et al., Review of particle properties, Phys. Lett. B 170 (1986) 1. [ 101 R. Bundy, Phys. Lett. B 45 (1973) 340; B 55 (1975) 227. [ 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; S.P. Mikheyev and A.Yu. Smimov, Yad. Fiz. 42 (1985) 1441 [Sov. J. Nucl. Phys. 42 (1985) 9131. [ 131 R.N. Mohapatra, Phys. Lett. B 198 (1987) 69. [ 141 R.N. Mohapatra, in: Proc. Eighth Workshop on Grand unification, ed. K Wali (World Scientific, Singapore) and references therein, to be published.
136
MODEL OF “CALCULABLE”
PHYSICS LETTERS B
24 March 1988
MAJORANA NEUTRINO MASSES
KS. BABU Department of Physics and Astronomy, University 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 neutrinos 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 example, 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 fermions 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 bobis 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 habto 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.
132
PHYSICS LETTERS B
Volume 203, number 1,2
24 March 1988
Fig. 1. Two-loop diagram which generates finite Majorana neutrino 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++).
(3)
The physical Higgs particles are Re go, h + and k + + with masses given by 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
(5)
where we have defined self-conjugate fields
(6)
Xn=V=‘+(va‘)C. The mass matrix is given by
(7)
M~b=8~~,FicdrncmdIcdCft)db,
where m,, md are the charged lepton masses and
(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 $, it takes a simple form: Icd =
&ji&(mi:m~)(mz:mi, [mqmyy2) -m:ln(m:y2)]
x [m~ln(m~~2)
-m21n(m~~2)].
(9)
If the cut-off parameter m2 is set equal to rni and the lepton masses are neglected in comparison with the charged scalar masses,
(10)
133
PHYSICS LE’ITERS B
Volume 203, number I,2
24 March 1988
We shall use this approximate expression for our numerical estimates. From the structure of the mass matrix, eq. (7)) it follows that one of the neutrinos is massless in the three generation model. This can be seen by noting that Mob in (7) may be written as A&= uKfl)ab, where K is a matrix defined by Kcd= 8&#dm,m&
(c, d not summed).
(11)
Hence Det M= 1Detf] ’ Det K= 0, since a 3 X 3 antisymmetric matrix has zero determinant #2.This is in clear contrast with the Zee model which also utilizes antisymmetric Yukawa couplings but predicts all the neutrinos to be massive. The crucial difference is that in the present model the charged scalar h+ does not mix with the charged scalar from the doublet, whereas in the Zee model such mixing is precisely the one responsible for neutrino masses. Before we analyze the mass matrix for the mass-mixing pattern, we shall constrain the various parameters of the model from existing experimental data. The scalars h + and k ++ have to be heavier than about 20 GeV, or else they should have been observed at e+e- collision experiments at PETRA. The exchange of h + contributes to p decay [4,6]. While the angular and energy distributions are undistorted, the total rate is corrected by a factor (1 + 4fiFm$/g2 rni)* . Allowing for a deviation of lo16from e-F universality, this requires j$/rnz 5 8 X 10 -’ GeV -‘. Similar constraints on falm f and fZ,lm i can be obtained by demanding that the deviation from e-p-t universality in T decay be less than a few percent. Both h+ and k++ exchange contribute to the (g- 2) of the leptons. The correction is given by 6(g,-2)=[(2m1)2/48x2][Cfft),~/m~+(j;Rt)lllm:]. From 6(g,-2)
< IO-‘O and 6(g,-2)
Cff’)eelmf+(~~t)ee/m:65x10-2
(12)
< lo-’ [ 71, we obtain GeV2,
Cff),lm~+(~~~)~/m~~l~10-~
GeV-2.
(13)
~1and e y decay proceed at the one-loop level by h + and k + + exchange at a rate given by r(pL--,ey) = am:
(14)
FromBr(~-+ey)<2x10-10,wehave Cfft),2/m2h+(&/?),2/m$<2x10-8
[8] GeV-2.
(16)
Lemon number changing decays such as +eee, z-+,uph etc. occur at the tree level via k+ + exchange. h + contributes to such decays only at the one-loop level which can be safely neglected. The rate for p-+ 3e is
r(v3d=
1 h$d& m: 8 192n3 m4.
(16)
k
From Br(p+3e) <2x lo-l2 [ 91 we have a stringent constraint hIreh e.?/m~<4xlO-*’
GeVm2.
(17)
Similar constraints from r decay are somewhat weaker: h,,h,lm i, h,h,,Jm i < 6 X 10 -’ GeVm2. The k++ scalar also contributes to high energy electron-positron scattering e+e--+f+l(l=p or r) in the 2 channel. The observed charge asymmetry in the scattering is in good agreement with the standard model Z”-y interference effects [IO] and this should not be upset by the extra k++ contribution. The k++ contribution to the amplitude is (after a Fierz rearrangement) u2The existence of a zero mass neutrino when using antisymmetric Yukawa couplings was first noted in the context of Dirac neutrinos by Mohapatra [ 131.
134
Volume 203, number 1,2
A=lh*2
w
&
PHYSICS LETTERS B
24 March 1988
(18)
b.WU -M4[%U-~M.
The differential cross section is + a! h& (I+cose)2 2~41~ l+cose+r
.s
(19)
where 4 is the center of mass energy, 8 the angle between e- and p-, r=2m,$ls, c,.,=cos B,, s,=sin 8,. The first term in (19) is the standard model y and Z” contribution [ IO], 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 where & and NB are the number of events in the forward and backward hemiA FB= (&-NB)I(NF+NB), spheres (forward is defined by t9c 90” ) is given by (assuming h$/4na e 1) A FB
N C2+(2h&,/4na){l+rZ ln[r(2+r)l(l+r)*]) $C, +(2hze/4xa){2-2r+r* In[(r+2)/r]}
’
(20)
where [lo] CIZ1,
c*=Ls 4c2s2 ww s-m* z *
(21)
Agreement with the average PETRA measurements [ 111 A,= ( - 17.8 _+1.9)% 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+lis 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 N h, = 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) 135
PHYSICS LETTERS B
Volume 203, number I,2
24 March 1988
with Cl o=
SlS2
where
SlC3 CIc2c3
-SIC2
ci=cos
-C1S2c3
ei,
+S2S3
(26)
+C2S3
si=sin ei and (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 (p’e--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 [ 1 ] 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). [ 5 ] L. Wolfenstein, Nucl. Phys. B 175 (1980) 93; S.T. Petcov, Phys. Lett. B 115 (1982) 401. [6] KS Babu andV.S. Mathur, Phys. Lett. B 196 (1987) 218. [7] T. Kinoshita, B. Nizic andY. Okamoto, Phys. Rev. Lett. 52 (1984) 717; T. Kinoshita and W.B. Lindquist, Phys. Rev. Lett. 47 (1981) 1573. [8] W.W. Kinnison et al., Phys. Rev. D 25 (1982) 2846. [ 91 Particle Data Group, M. Aguilar-Benitez et al., Review of particle properties, Phys. Lett. B 170 (1986) 1. [ 101 R. Bundy, Phys. Lett. B 45 (1973) 340; B 55 (1975) 227. [ 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; S.P. Mikheyev and A.Yu. Smimov, Yad. Fiz. 42 (1985) 1441 [Sov. J. Nucl. Phys. 42 (1985) 9131. [ 131 R.N. Mohapatra, Phys. Lett. B 198 (1987) 69. [ 141 R.N. Mohapatra, in: Proc. Eighth Workshop on Grand unification, ed. K Wali (World Scientific, Singapore) and references therein, to be published.
136