- Email: [email protected]
A model for a large neutrino magnetic transition moment and naturally small mass
Volume 237, number
1
PHYSICS
A MODEL FOR A LARGE NEUTRINO AND NATURALLY SMALL MASS Miriam LEURER
LETTERS
MAGNETIC
B
TRANSITION
8 March
1990
MOMENT
and Neil MARCUS
Department of Physics, Technion, Haifa 32000, Israel Received
5 December
1989
Models that attempt to solve the solar neutrino problem by producing neutrino mass corrections in the keV range, and therefore require fine with a Voloshin-type SU(2) horizontal symmetry that protects the gW( m, -m,) /2Mw = 4 x 1O-“, resulting in masses which are naturally and muon lepton numbers is conserved, so that troubling rare decays tau lepton number conservation. Low energy effects are very difftcult seen in colliders with a clear experimental signature.
a large neutrino magnetic moment, generally also produce tuning. Here, we construct a model for Majorana neutrinos masses. This symmetry is broken only by terms of order of the order of a few eV. The difference between electron like fi+ey and p+3e are forbidden. One may also demand to observe, but the model has physical scalars which can be
Twenty years of measurements by Davis and collaborators have suggested that there may be an anticorrelation between solar-neutrino flux and sunspot activity [ 11. Such an anticorrelation cannot be explained in the framework of the standard solar model and the standard Weinberg-Salam theory. Voloshin, Vysotsky and Okun therefore suggested that nonstandard interactions could induce a large magnetic moment, or transition moment, for the electron neutrino. Such a magnetic (transition) moment would induce a precession of the left-handed electron neutrino into an observable right-handed neutrino or antineutrino [ 21. This can explain the solar neutrino problem, including the anticorrelation, if the magnetic moment is of the order of lo-” ,us. In the case of Majorana neutrinos, such magnetic moments are also consistent with the recent supernova observations [ 3 1. The question left open by the VVO papers is what are the interactions that produce such a large magnetic moment? A model for Dirac neutrinos was proposed in refs. [ 4,5 1, and a model for Majorana neutrinos in ref. [ 6 1, to be referred to as I. However, all such models encounter the same problem: A large magnetic moment is generally associated with a one-loop correction to the neutrino mass in the keV range. Since this is 2-3 orders of magnitude above the experimental upper bound, fine tuning is necessary to suppress the mass down to phenomenologically acceptable values, Voloshin pointed out that this problem could be solved by an SU (2) symmetry that would protect the neutrino masses, but not their magnetic moments [ 71. However, such a symmetry is broken in nature. In the case of Majorana neutrinos, the SU (2 ) is simply a horizontal symmetry between two neutrino flavours. In order to avoid breaking by weak interaction effects, the symmetry must be extended to act between two full leptonic generations, and it is then broken by the mass difference of the charged leptons. If one wishes to implement Voloshin’s SU(2), one must also specify how it is broken. One possibility is to gauge the symmetry, and then break it spontaneously [ 81. However, this leads to difficulties since the symmetry must be broken at very high energies, reintroducing line tuning. In addition, SU(2)n has global anomalies, as there is an odd number of Weyl doublets [ 61. Here we will adopt a different approach: Note that the standard model is almost invariant under a horizontal SU(2)” acting between the first and second leptonic generations. This symmetry is broken only by Y/I-#&
f= -
2
=gw
0370-2693/90/$03.50
mp--m, -4x 2Mw
10-4
)
0 Elsevier Science Publishers
B.V. (North-Holland
)
81
Volume 237, number 1
PHYSICS LETTERS B
8 March 1990
where y, and yp are the Yukawa couplings. In our model we shall require that this SU (2), remains an approximate symmetry, broken explicitly by O(t) terms. Then, the mass of the neutrino will be smaller by a factor t than it is in models without this symmetry. Thus, instead of being of the order of a few keV, the mass will naturally be in the experimentally allowed eV range. Since we are considering Majorana neutrinos, it is important to note that they can have only magnetic transition moments. In that case, it is also necessary to protect the mass difference between the two flavors (in our case, between v, and up). VVO concluded that for these neutrinos to precess in the sun, their mass difference should be very small: Am2 < lo-’ eV* [ 21. In I, it was suggested that the natural way to achieve such a strong bound is to force the mass difference to vanish exactly by conserving N,- N,, the difference between muon and electron lepton numbers. Then, (v,)= and ( v”,)~ are naturally combined into one massive ZKM Dirac neutrino [ 9 ] #I. The U ( 1) symmetry generated by N,, - N, is the diagonal subgroup of SU (2 ) “, and is, of course, conserved in the standard model. We will require that this U ( 1) remains an exact symmetry of our model. As for the other lepton numbers: total lepton number, N, + NP + N,, is clearly broken, since magnetic transition moments of Majorana neutrinos are lepton-number violating quantities. One has the option of conserving or violating T lepton number, N,. The consequences of this choice will be discussed later. The magnetic transition moments will be generated via loops involving charged physical scalars, see fig. 1. The mass correction is given by the same diagram with the photon line removed. The first question to answer is what representations of Higgs fields to choose. Since the magnetic transition moment is proportional to the mass of the charged lepton in the loop, it is preferable that it be the T. Then, since the T is a singlet of SU( 2)H and the external neutrinos are doublets, the charged scalar in the loop must be a doublet of SU (2) H. By similarly following the SU (2)w x U ( 1) y representations in the graph, it was concluded in I that the charged scalar must be a mixture of an SU( 2)w doublet and a singlet. We therefore need a multiplet D in the ( j, t*) + , representation of SU(2)w~SU(2).~U(l).,andamultipletSinthe(O,f*)+~ representation #*. In matrix notation
S=(S_
S,),
D=(;: f(‘;)>
where the subscripts indicate N,- N, charge. S, and Dk have electromagnetic charge + 1, while di are neutral. SU(2)H acts from the right, and SU(2)w acts on D from the left. It is worthwhile to note here that if N, is conserved, Sand D must both carry - 1 unit of this charge. This also follows from the Feynman diagrams of fig. ” Magnetic moments of ZKM neutrinos were discussed in ref. [ lo]. ” This is the simplest SU (2 )H extension of a model previously suggested in ref. [ 5]
Fig. I, The contribution of the Higgs to the magnetic transition moment. Another graph, with the photon coupling to the Higgs, is needed for gauge invariance, but does not contribute to the magnetic moment. The graph without the photon gives the mass correction.
82
Volume 237, number
PHYSICS
1
LETTERS
B
8 March
1: Since the internal lepton is the tau, while the external particles are ve and u,, the positively the loop must carry N,= 1. In addition to Sand D, we need a “standard” Higgs, @,in the ( f , 0) + 1 representation:
1990
charged scalar in
This multiplet plays the usual roles of the Higgs in the standard model - it develops a VEV, breaking the weak gauge group to U( 1 ),, and inducing masses for the quarks and leptons. In addition, @has another role in our model: The lagrangian contains a term SD+@*+ h.c. When @ gets its vacuum expectation value, the charged scalars of the S multiplet mix with the charged components of the D multiplet. As mentioned above, this mixture is necessary for the creation of a neutrino magnetic transition moment. The most general SU ( 2)H invariant Higgs potential for @,D and S can be written as ~H=~,(~+~-~2)2+~2[Tr(DDt)]2+M~Tr(DDt)+~3(SSt)2+M~SSt+;14(~t~-V2)
Tr(DD+)
+ A,(~t~-I/2)SSt+AgTr(DDt)SSt+;1,Tr(DDtDDt)+d,~tDDt~+~gSDtDSt + ilO( VS+@Yr,D)( The SU (2)n breaking,
VSt-Dtir2@*) Np- N, conserving,
(4) corrections
can be written as
Ay;,=E{~~Tr(Do~Dt)+~~Sa,St+I,,[Tr(Da3Dt)]2+~,2(S~,St)2+i,,(~t~-~2)Tr(D~3Dt) + A,,(@+@- 1/2)Sa-,St+L,,
Tr(DD+)
Tr(Do3Dt)+/Z,,
+ ~,,SStSa,St+~,9Tr(Da3Dt)Sa3St+~20~tDo3Dt~+~21S( + L22( IG’+@‘ir,D)o,(
Tr(DDt)Sg3St+L,,SSt 1 -&)D+D(
Tr(Do,D+) l+ioX)St
VSt-Dtiz2~*)+6(~tD+e’*~‘D*ia2)a3(Dt-e-i”i~2Dt~*)},
(5)
where the coefficients of the SU ( 2)H breaking terms have been written with an explicit E factor. The a, are Pauli matrices acting in the SU(2)” space, while is, acts in the SU(2)w space. The parameters in 5% are all real, except for the phase e’“, and the i in the lz2, term. To avoid introducing fine tuning in the Higgs sector, all the dimensionful parameters ( I”, Mi, Mi, ,LL~and pi) are assumed to be of a similar order of magnitude, as are the dimensionless parameters A,. However, 6 is a measure of 7 symmetry breaking, and it can therefore be chosen to be as small as one wishes. 9” is manifestly positive definite for a reasonable choice of parameters. It vanishes at
which values, The charge
is therefore an absolute minimum of the potential. Since D and S do not get any vacuum expectation and @gets is usual VEV, the longitudinal modes of the I+‘+ and Z are @’ and Im @’ respectively. scalar mass matrix in the charged sector splits into two submatrices, depending on whether the N,- N, is equal to or opposite the electromagnetic charge. The matrices are given by
(D;
S*,)
MZ,+A~oV*TE(&+ (
-(;l,oicA22)V2
V2&)
-(A,,Td22)V2 M;+il,oV*Tt(p;+v21*2)
Di >(
s+
>
.
(7)
We denote the mixing angles by PA, and the mass eigenstates by Hk and Ki, where M* ( Hi ) GM* ( Kt ). As can be seen, because of the approximate SU (2) H symmetry, the mixing angles and mass eigenvalues in the + and - sectors differ only by 0 ( c ) corrections. The charged physical scalars Hk and Kk are SU (2 )w doublet-singlet mixtures, and all of them contribute to the neutrino magnetic transition moment and mass. In the neutral sector, Re Go is the standard physical Higgs, and is a mass eigenstate. The extra neutral fields 83
Volume 237, number
1
are d, and d_. Although mass matrix is given by
they are electrically
PHYSICS
LETTERS
B
8 March
neutral,
they are complex,
since they carry NM-N, charge. Their
d-)[(
Cd:
1990
(8)
We denote the mass eigenstates in the + sector by h, and k,. (There are no independent fields h_ and k_ in the - sector, but only h: and k: .) h+ and k, are degenerate up to O(E) corrections. Note that if 6 vanishes and T lepton number is a good symmetry, d, and dt cannot mix, since they have opposite N, charge. TheusualfermionsfallintofourSU(2)wXSU(2)HXU(l) multiplets: ‘P,_ina ($, 1*)-i, YRina (0, i*)_2, TLina(f,0)_,ands,ina(0,0)_2.1notherwords P)~,
YR=(e
T,=
0UTL’ 5
The leptonic
Yukawa potential
is given by
The E and 6 symmetry breaking factors in the lagrangian are again explicitly exhibited. It is reasonable to assume that y, -y,, y, - y, and y, -y,. We have used the remaining phase freedom of our theory to make all the parameters except y, and y, real. Since all the phases in lz;, and Spyare multiplied by e, CP violation in the lepton sector will vanish as e-+0. There are now several points to note about the theory. The @couplings to the leptons (and also to the quarks) are identical to the couplings of the Higgs in the standard model. Everything that is new relative to the standard model is associated with the D and S multiplets. These multiplets always couple to one T or v,, and one first or second generation lepton. Even if 62 0 and N, is not conserved, ( - 1)“” is conserved, since N, is always broken in pairs. Finally, if one associates two units of total lepton number with S, and none with @and D, lepton number is conserved in the Yukawa sector. Then, returning to &,, one sees that 1110and e122 are the lepton-number violating parameters of the model. The magnetic transition moment will therefore depend on A,0 and eAz2 (through the mixing angles p* ), since it is a lepton-number breaking quantity. We now have all the ingredients to calculate the one-loop magnetic transition moment and mass correction. The mass correction is, in fact, the complete mass, since the neutrino cannot have a tree-level mass in our model. To write our results, we introduce the following notation for the Yukawa couplings of the * sectors: y:=y,*y,,
y:=y,sy;,
Then, summing
the contributions
‘=
y:=y,+yg.
(11)
of the four Higgs particles to fig. 1, one obtains
#3
e sin 2/?+ Y!+ Y$ 327~~
(12)
m=&sin2/3+Y!+Y$m,Lf(x’:)-f(x:)]-(+++-),
(13)
where 2 x
H.K *
E
> ’
&[&og(_t)-q
>
f(x)=
&%(_:).
M Since p and m are complex, one should redefine u, and v,, to make m real. Then, the real and imaginary and electric dipole moments respectively. For relativistic particles, the relevant quantity is 1p 1=,/m
84
(14,lS)
g(x)=
parts of @give the magnetic [ 11 ]
Volume 237, number
PHYSICS LETTERS B
1
8 March 1990
The x variables are small, since the masses of the charged physical Higgs must exceed 19 GeV [ 121. The most important feature of eqs. ( 12) and ( 13) is that the + and - sectors add in p and cancel in m. Since all the parameters of the two sectors - masses, mixing angles and Yukawa couplings - differ only by O(E) corrections, this means that 9 m-cm,,
P”2P+
(16)
where ,u+ and m, are the contributions approximate SU(2)H. We now have m
m,
-N
P
P+
_
~m,f(x3-f(x3 e ‘s(x):)--g(x5)
of the + sector alone. Thus we see the expected suppression
N IM:,,
(17)
e
In the last step, we have dropped logarithmic
of m by the
corrections.
Eq. ( 17) can be written as
(18) Therefore, if we generate magnetic transition moments of the desired order of magnitude, we get reasonable neutrino masses for Higgs masses up to 0 ( 100 GeV), depending on the exact parameters of the model. This can be contrasted to the case of models without the Voloshin symmetry, in which the Efactors in eq. ( 17) are absent. Then, the 1.5 eV in eq. ( 18) becomes 4 keV, and fine tuning is required to suppress the mass correction. We can also calculate the magnetic moment and mass corrections of other leptons. The T neutrino cannot have a magnetic moment, since it is a Majorana particle, but it will get a one-loop mass if N, is not conserved. Since the dominant contribution now comes from a .Drunning in the loop, one finds that (19) if, for simplicity, we now assume Y’ N Y2 N Y3. Thus the scale of m (v,) is 1 keV, but it vanishes as N, becomes a good symmetry. Turning to the charged leptons, one may worry that contributions proportional to m, to the magnetic and electric dipole moments of the electron and muon are too large. However, such contributions exist only if 6# 0, and arise from the mixing of d, and d? . Since the mass difference between h, and k+ is O(t), and the mixing is 0 (6)) one finds 6~(e,~)~~E62eia~L,~~2ei(Y4X10-‘5~~.
(20)
This is far below the experimental sensitivity for both the magnetic and electric dipole moments. To study other phenomenological consequences of the model, the most important ingredient is the strength of the Yukawa couplings required to give the desired neutrino magnetic transition moment. This will depend on the masses of the charged scalars. Since they must be heavier than 19 GeV, and the lighter of them should not exceed - 100 GeV, because of the m/p ratio, we shall for definiteness take II~~= 50 GeV and n/l,= 100 GeV. We shall also takep=45”. Then, substitutingpe lo-” pB in eq. (12), one gets Y’Y2N5X
10-4.
(21)
Effective four-Fermi interactions the weak interactions, since Y2 -N_--. M:,
involving
the new Higgs fields will therefore be about 300 times weaker than
1 g’w 300 MZ,
This means that low-energy
(22) deviations
from the standard
model will be very small. One should also note that 85
Volume 237, number
1
PHYSICS
LETTERS
B
8 March
1990
these interactions involve only leptons, and two of them must be third generation leptons. Since this sector is not known very precisely, it is very difficult to observe the new interactions experimentally. For example, the will be 0 (0.05%) at PETRA energies. new contribution to the forward-backward asymmetry in e+e-+~+~Also, while universality in T decay to e and p is unaffected, because of the approximate SU(2)u symmetry, universality between T-+evv and 7+nv will have deviations of the order of 0.5%. Both these corrections are more than an order of magnitude below present experimental errors. One would therefore like to find processes which are forbidden in the standard model. These do exist, since separate lepton numbers are no longer conserved. However, usual rare processes like P-Q, p+eee and neutrinoless double-beta decays are forbidden by N, - N, conservation [ 6 1. The only rare decays in our model involve decays to unusual neutrinos; for example 7-+pv,v,. These are impossible to distinguish from usual decays like ?s--t~v~v,. There are also scattering processes like v,e- +z-V’ which look like neutrino mixing, and, if 620, there is the striking scattering e-p-+‘s-~-. However, none of these processes can be expected to be seen in the near future. In contrast to this murky picture, the model has clear and direct experimental consequences in the Higgs sector. There must be a pair of charged scalars H? with a mass between 20 and about 100 GeV, and a splitting of order 25 MeV. (Their widths are about two orders of magnitude less. ) There is also the Kt pair which is very similar to H,, except that the mass is somewhat larger. Finally, there are the neutral complex scalars h, and k,, which also form a pair with an 0 (e) mass splitting. Their masses are not determined, but are presumably of a similar order of magnitude to those of the charged scalars. The charged scalars decay into a charged lepton and a neutrino, while the neutral ones always decay into two charged leptons. In both cases, one and only one of the leptons is in the third generation. All decay modes that conserve N@- N, are allowed. The new scalars can all be produced via a W or a Z, and the charged ones, of course, can also be produced via a photon. The Wand Z couplings are not diagonal in H-K space, so the Z, for example, can produce an H, and a K: . In an e+e- machine, the first new charged particles to be produced will be H, H: and H_ HT pairs. Since the H+-H_ mass splitting is so small, it will be extremely difficult to resolve them directly. However, one can see that there are, in fact, two different particles because they have different decay modes. The H, decays to ec v or r+ v, while the H_ decays top+ v or T+ v. Therefore, the end reaction is e+e-+l+l+ missing energy, where 1+1- is any pair of charged leptons exceptfor ep. The neutral scalars h, and k, can also be produced via an intermediate Z. If N, is conserved, the signature will be e+e-+T+T-e+eor s+z-,u+,u-, and one can again see that there are two new particles with different decay modes. If N, is broken, the two particles will mix and can no longer be distinguished by their decay modes. In this case, there is also a spectacular e+e-+r-7-e+p+ reaction. The scalars can also be produced in hadron colliders. In that case, one can produce one charged and one neutral scalar, via an intermediate W. If the mass of any of these scalars is below -M,/2, they should appear in Z decays. The recent experiments at SLC, Fermilab and LEP therefore presumably exclude this region. This remaining mass range can be explored at LEP II and the SSC. In conclusion, we have presented a model which gives a magnetic transition moment between the electron and muon neutrinos of the order of 1O- ’ ’ ,uB,and a mass of order of a few eV. The tau neutrino mass must be below a few keV. Because of an N,- N, symmetry, easily detectable rare decays are forbidden, and, since the Yukawa couplings are of the order of lo-‘, low energy phenomenology is essentially unchanged. The model has two closely-spaced pairs of charged scalars, the lightest of which has a mass between 20 and - 100 GeV, and a pair of complex neutral scalars. These particles should be easily seen in high-energy colliders. We would like to thank G. Eilam, D. Granite and M. Gronau for useful discussions. This work was supported in part by the United States-Israel Binational Science Foundation (BSF), the Israeli Academy of Sciences and the ijsterreichische Nationalbank Jubilaumsfondsprojekt Nr. 3485.
86
Volume 237, number 1
PHYSICS LETTERS B
8 March 1990
References [ I] R. Davis, K. Lande, B.T. Cleveland, J. Ullman and J.K. Rowley, in: Proc. 13th Intern. Conf. on Neutrino physics and astrophysics, Neutrino 88, eds. J. Schneps et al. (World Scientific, Singapore), to be published. [2] M.B. Voloshin and M.I. Vysotsky, Yad. Fiz. 44 (1986) 845 [Sov. J. Nucl. Phys. 44 (1986) 5441; L.B. Okun, M.B. Voloshin and MI. Vysotsky, Yad. Fiz. 44 (1986) 677 [Sov. J. Nucl. Phys. 44 (1986) 4401; Zh. Eksp. Teor. Fiz. 91 (1986) 754 [Sov. Phys. JETP 64 (1986) 4461. [3] M. Leurer and J. Liu, Phys. Lett. B 219 (1989) 304. [4] M. Fukugita and T. Yanagida, Phys. Rev. Lett. 58 ( 1987) 1807. [5]K.S.BabuandV.S.Mathur, Phys.Lett. B 196 (1987) 218. [ 6 ] M. Leurer and M. Golden, Fermilab preprint 89/ 175-T. [7] M.B. Voloshin, Yad. Phys. 48 (1988) 804 [Sov. J. Nucl. Phys. 48 (1988)]. [ 81 K.S. Babu and R.N. Mohapatra, Phys. Rev. Lett. 63 (1989) 228. [9] Ya.B. Zeldovich. Dokl. Akad. Nauk SSSR 86 ( 1952) 505; E.J. Konopinski and H.M. Mahmoud, Phys. Rev. 92 ( 1953) 1045. [lo] L. Wolfenstein, Nucl. Phys. B 186 (1981) 147; S.T. Petcov, Phys. Lett. B 1I5 (1982) 401. [ 111 L.B. Okun, Sov. J. Nucl. Phys. 44 ( 1986) 546. [ 121 Particle Data Group, G.P. Yost et al., Review of particle properties, Phys. Lett. B 204 ( 1988 ) 1.
87
1
PHYSICS
A MODEL FOR A LARGE NEUTRINO AND NATURALLY SMALL MASS Miriam LEURER
LETTERS
MAGNETIC
B
TRANSITION
8 March
1990
MOMENT
and Neil MARCUS
Department of Physics, Technion, Haifa 32000, Israel Received
5 December
1989
Models that attempt to solve the solar neutrino problem by producing neutrino mass corrections in the keV range, and therefore require fine with a Voloshin-type SU(2) horizontal symmetry that protects the gW( m, -m,) /2Mw = 4 x 1O-“, resulting in masses which are naturally and muon lepton numbers is conserved, so that troubling rare decays tau lepton number conservation. Low energy effects are very difftcult seen in colliders with a clear experimental signature.
a large neutrino magnetic moment, generally also produce tuning. Here, we construct a model for Majorana neutrinos masses. This symmetry is broken only by terms of order of the order of a few eV. The difference between electron like fi+ey and p+3e are forbidden. One may also demand to observe, but the model has physical scalars which can be
Twenty years of measurements by Davis and collaborators have suggested that there may be an anticorrelation between solar-neutrino flux and sunspot activity [ 11. Such an anticorrelation cannot be explained in the framework of the standard solar model and the standard Weinberg-Salam theory. Voloshin, Vysotsky and Okun therefore suggested that nonstandard interactions could induce a large magnetic moment, or transition moment, for the electron neutrino. Such a magnetic (transition) moment would induce a precession of the left-handed electron neutrino into an observable right-handed neutrino or antineutrino [ 21. This can explain the solar neutrino problem, including the anticorrelation, if the magnetic moment is of the order of lo-” ,us. In the case of Majorana neutrinos, such magnetic moments are also consistent with the recent supernova observations [ 3 1. The question left open by the VVO papers is what are the interactions that produce such a large magnetic moment? A model for Dirac neutrinos was proposed in refs. [ 4,5 1, and a model for Majorana neutrinos in ref. [ 6 1, to be referred to as I. However, all such models encounter the same problem: A large magnetic moment is generally associated with a one-loop correction to the neutrino mass in the keV range. Since this is 2-3 orders of magnitude above the experimental upper bound, fine tuning is necessary to suppress the mass down to phenomenologically acceptable values, Voloshin pointed out that this problem could be solved by an SU (2) symmetry that would protect the neutrino masses, but not their magnetic moments [ 71. However, such a symmetry is broken in nature. In the case of Majorana neutrinos, the SU (2 ) is simply a horizontal symmetry between two neutrino flavours. In order to avoid breaking by weak interaction effects, the symmetry must be extended to act between two full leptonic generations, and it is then broken by the mass difference of the charged leptons. If one wishes to implement Voloshin’s SU(2), one must also specify how it is broken. One possibility is to gauge the symmetry, and then break it spontaneously [ 81. However, this leads to difficulties since the symmetry must be broken at very high energies, reintroducing line tuning. In addition, SU(2)n has global anomalies, as there is an odd number of Weyl doublets [ 61. Here we will adopt a different approach: Note that the standard model is almost invariant under a horizontal SU(2)” acting between the first and second leptonic generations. This symmetry is broken only by Y/I-#&
f= -
2
=gw
0370-2693/90/$03.50
mp--m, -4x 2Mw
10-4
)
0 Elsevier Science Publishers
B.V. (North-Holland
)
81
Volume 237, number 1
PHYSICS LETTERS B
8 March 1990
where y, and yp are the Yukawa couplings. In our model we shall require that this SU (2), remains an approximate symmetry, broken explicitly by O(t) terms. Then, the mass of the neutrino will be smaller by a factor t than it is in models without this symmetry. Thus, instead of being of the order of a few keV, the mass will naturally be in the experimentally allowed eV range. Since we are considering Majorana neutrinos, it is important to note that they can have only magnetic transition moments. In that case, it is also necessary to protect the mass difference between the two flavors (in our case, between v, and up). VVO concluded that for these neutrinos to precess in the sun, their mass difference should be very small: Am2 < lo-’ eV* [ 21. In I, it was suggested that the natural way to achieve such a strong bound is to force the mass difference to vanish exactly by conserving N,- N,, the difference between muon and electron lepton numbers. Then, (v,)= and ( v”,)~ are naturally combined into one massive ZKM Dirac neutrino [ 9 ] #I. The U ( 1) symmetry generated by N,, - N, is the diagonal subgroup of SU (2 ) “, and is, of course, conserved in the standard model. We will require that this U ( 1) remains an exact symmetry of our model. As for the other lepton numbers: total lepton number, N, + NP + N,, is clearly broken, since magnetic transition moments of Majorana neutrinos are lepton-number violating quantities. One has the option of conserving or violating T lepton number, N,. The consequences of this choice will be discussed later. The magnetic transition moments will be generated via loops involving charged physical scalars, see fig. 1. The mass correction is given by the same diagram with the photon line removed. The first question to answer is what representations of Higgs fields to choose. Since the magnetic transition moment is proportional to the mass of the charged lepton in the loop, it is preferable that it be the T. Then, since the T is a singlet of SU( 2)H and the external neutrinos are doublets, the charged scalar in the loop must be a doublet of SU (2) H. By similarly following the SU (2)w x U ( 1) y representations in the graph, it was concluded in I that the charged scalar must be a mixture of an SU( 2)w doublet and a singlet. We therefore need a multiplet D in the ( j, t*) + , representation of SU(2)w~SU(2).~U(l).,andamultipletSinthe(O,f*)+~ representation #*. In matrix notation
S=(S_
S,),
D=(;: f(‘;)>
where the subscripts indicate N,- N, charge. S, and Dk have electromagnetic charge + 1, while di are neutral. SU(2)H acts from the right, and SU(2)w acts on D from the left. It is worthwhile to note here that if N, is conserved, Sand D must both carry - 1 unit of this charge. This also follows from the Feynman diagrams of fig. ” Magnetic moments of ZKM neutrinos were discussed in ref. [ lo]. ” This is the simplest SU (2 )H extension of a model previously suggested in ref. [ 5]
Fig. I, The contribution of the Higgs to the magnetic transition moment. Another graph, with the photon coupling to the Higgs, is needed for gauge invariance, but does not contribute to the magnetic moment. The graph without the photon gives the mass correction.
82
Volume 237, number
PHYSICS
1
LETTERS
B
8 March
1: Since the internal lepton is the tau, while the external particles are ve and u,, the positively the loop must carry N,= 1. In addition to Sand D, we need a “standard” Higgs, @,in the ( f , 0) + 1 representation:
1990
charged scalar in
This multiplet plays the usual roles of the Higgs in the standard model - it develops a VEV, breaking the weak gauge group to U( 1 ),, and inducing masses for the quarks and leptons. In addition, @has another role in our model: The lagrangian contains a term SD+@*+ h.c. When @ gets its vacuum expectation value, the charged scalars of the S multiplet mix with the charged components of the D multiplet. As mentioned above, this mixture is necessary for the creation of a neutrino magnetic transition moment. The most general SU ( 2)H invariant Higgs potential for @,D and S can be written as ~H=~,(~+~-~2)2+~2[Tr(DDt)]2+M~Tr(DDt)+~3(SSt)2+M~SSt+;14(~t~-V2)
Tr(DD+)
+ A,(~t~-I/2)SSt+AgTr(DDt)SSt+;1,Tr(DDtDDt)+d,~tDDt~+~gSDtDSt + ilO( VS+@Yr,D)( The SU (2)n breaking,
VSt-Dtir2@*) Np- N, conserving,
(4) corrections
can be written as
Ay;,=E{~~Tr(Do~Dt)+~~Sa,St+I,,[Tr(Da3Dt)]2+~,2(S~,St)2+i,,(~t~-~2)Tr(D~3Dt) + A,,(@+@- 1/2)Sa-,St+L,,
Tr(DD+)
Tr(Do3Dt)+/Z,,
+ ~,,SStSa,St+~,9Tr(Da3Dt)Sa3St+~20~tDo3Dt~+~21S( + L22( IG’+@‘ir,D)o,(
Tr(DDt)Sg3St+L,,SSt 1 -&)D+D(
Tr(Do,D+) l+ioX)St
VSt-Dtiz2~*)+6(~tD+e’*~‘D*ia2)a3(Dt-e-i”i~2Dt~*)},
(5)
where the coefficients of the SU ( 2)H breaking terms have been written with an explicit E factor. The a, are Pauli matrices acting in the SU(2)” space, while is, acts in the SU(2)w space. The parameters in 5% are all real, except for the phase e’“, and the i in the lz2, term. To avoid introducing fine tuning in the Higgs sector, all the dimensionful parameters ( I”, Mi, Mi, ,LL~and pi) are assumed to be of a similar order of magnitude, as are the dimensionless parameters A,. However, 6 is a measure of 7 symmetry breaking, and it can therefore be chosen to be as small as one wishes. 9” is manifestly positive definite for a reasonable choice of parameters. It vanishes at
which values, The charge
is therefore an absolute minimum of the potential. Since D and S do not get any vacuum expectation and @gets is usual VEV, the longitudinal modes of the I+‘+ and Z are @’ and Im @’ respectively. scalar mass matrix in the charged sector splits into two submatrices, depending on whether the N,- N, is equal to or opposite the electromagnetic charge. The matrices are given by
(D;
S*,)
MZ,+A~oV*TE(&+ (
-(;l,oicA22)V2
V2&)
-(A,,Td22)V2 M;+il,oV*Tt(p;+v21*2)
Di >(
s+
>
.
(7)
We denote the mixing angles by PA, and the mass eigenstates by Hk and Ki, where M* ( Hi ) GM* ( Kt ). As can be seen, because of the approximate SU (2) H symmetry, the mixing angles and mass eigenvalues in the + and - sectors differ only by 0 ( c ) corrections. The charged physical scalars Hk and Kk are SU (2 )w doublet-singlet mixtures, and all of them contribute to the neutrino magnetic transition moment and mass. In the neutral sector, Re Go is the standard physical Higgs, and is a mass eigenstate. The extra neutral fields 83
Volume 237, number
1
are d, and d_. Although mass matrix is given by
they are electrically
PHYSICS
LETTERS
B
8 March
neutral,
they are complex,
since they carry NM-N, charge. Their
d-)[(
Cd:
1990
(8)
We denote the mass eigenstates in the + sector by h, and k,. (There are no independent fields h_ and k_ in the - sector, but only h: and k: .) h+ and k, are degenerate up to O(E) corrections. Note that if 6 vanishes and T lepton number is a good symmetry, d, and dt cannot mix, since they have opposite N, charge. TheusualfermionsfallintofourSU(2)wXSU(2)HXU(l) multiplets: ‘P,_ina ($, 1*)-i, YRina (0, i*)_2, TLina(f,0)_,ands,ina(0,0)_2.1notherwords P)~,
YR=(e
T,=
0UTL’ 5
The leptonic
Yukawa potential
is given by
The E and 6 symmetry breaking factors in the lagrangian are again explicitly exhibited. It is reasonable to assume that y, -y,, y, - y, and y, -y,. We have used the remaining phase freedom of our theory to make all the parameters except y, and y, real. Since all the phases in lz;, and Spyare multiplied by e, CP violation in the lepton sector will vanish as e-+0. There are now several points to note about the theory. The @couplings to the leptons (and also to the quarks) are identical to the couplings of the Higgs in the standard model. Everything that is new relative to the standard model is associated with the D and S multiplets. These multiplets always couple to one T or v,, and one first or second generation lepton. Even if 62 0 and N, is not conserved, ( - 1)“” is conserved, since N, is always broken in pairs. Finally, if one associates two units of total lepton number with S, and none with @and D, lepton number is conserved in the Yukawa sector. Then, returning to &,, one sees that 1110and e122 are the lepton-number violating parameters of the model. The magnetic transition moment will therefore depend on A,0 and eAz2 (through the mixing angles p* ), since it is a lepton-number breaking quantity. We now have all the ingredients to calculate the one-loop magnetic transition moment and mass correction. The mass correction is, in fact, the complete mass, since the neutrino cannot have a tree-level mass in our model. To write our results, we introduce the following notation for the Yukawa couplings of the * sectors: y:=y,*y,,
y:=y,sy;,
Then, summing
the contributions
‘=
y:=y,+yg.
(11)
of the four Higgs particles to fig. 1, one obtains
#3
e sin 2/?+ Y!+ Y$ 327~~
(12)
m=&sin2/3+Y!+Y$m,Lf(x’:)-f(x:)]-(+++-),
(13)
where 2 x
H.K *
E
> ’
&[&og(_t)-q
>
f(x)=
&%(_:).
M Since p and m are complex, one should redefine u, and v,, to make m real. Then, the real and imaginary and electric dipole moments respectively. For relativistic particles, the relevant quantity is 1p 1=,/m
84
(14,lS)
g(x)=
parts of @give the magnetic [ 11 ]
Volume 237, number
PHYSICS LETTERS B
1
8 March 1990
The x variables are small, since the masses of the charged physical Higgs must exceed 19 GeV [ 121. The most important feature of eqs. ( 12) and ( 13) is that the + and - sectors add in p and cancel in m. Since all the parameters of the two sectors - masses, mixing angles and Yukawa couplings - differ only by O(E) corrections, this means that 9 m-cm,,
P”2P+
(16)
where ,u+ and m, are the contributions approximate SU(2)H. We now have m
m,
-N
P
P+
_
~m,f(x3-f(x3 e ‘s(x):)--g(x5)
of the + sector alone. Thus we see the expected suppression
N IM:,,
(17)
e
In the last step, we have dropped logarithmic
of m by the
corrections.
Eq. ( 17) can be written as
(18) Therefore, if we generate magnetic transition moments of the desired order of magnitude, we get reasonable neutrino masses for Higgs masses up to 0 ( 100 GeV), depending on the exact parameters of the model. This can be contrasted to the case of models without the Voloshin symmetry, in which the Efactors in eq. ( 17) are absent. Then, the 1.5 eV in eq. ( 18) becomes 4 keV, and fine tuning is required to suppress the mass correction. We can also calculate the magnetic moment and mass corrections of other leptons. The T neutrino cannot have a magnetic moment, since it is a Majorana particle, but it will get a one-loop mass if N, is not conserved. Since the dominant contribution now comes from a .Drunning in the loop, one finds that (19) if, for simplicity, we now assume Y’ N Y2 N Y3. Thus the scale of m (v,) is 1 keV, but it vanishes as N, becomes a good symmetry. Turning to the charged leptons, one may worry that contributions proportional to m, to the magnetic and electric dipole moments of the electron and muon are too large. However, such contributions exist only if 6# 0, and arise from the mixing of d, and d? . Since the mass difference between h, and k+ is O(t), and the mixing is 0 (6)) one finds 6~(e,~)~~E62eia~L,~~2ei(Y4X10-‘5~~.
(20)
This is far below the experimental sensitivity for both the magnetic and electric dipole moments. To study other phenomenological consequences of the model, the most important ingredient is the strength of the Yukawa couplings required to give the desired neutrino magnetic transition moment. This will depend on the masses of the charged scalars. Since they must be heavier than 19 GeV, and the lighter of them should not exceed - 100 GeV, because of the m/p ratio, we shall for definiteness take II~~= 50 GeV and n/l,= 100 GeV. We shall also takep=45”. Then, substitutingpe lo-” pB in eq. (12), one gets Y’Y2N5X
10-4.
(21)
Effective four-Fermi interactions the weak interactions, since Y2 -N_--. M:,
involving
the new Higgs fields will therefore be about 300 times weaker than
1 g’w 300 MZ,
This means that low-energy
(22) deviations
from the standard
model will be very small. One should also note that 85
Volume 237, number
1
PHYSICS
LETTERS
B
8 March
1990
these interactions involve only leptons, and two of them must be third generation leptons. Since this sector is not known very precisely, it is very difficult to observe the new interactions experimentally. For example, the will be 0 (0.05%) at PETRA energies. new contribution to the forward-backward asymmetry in e+e-+~+~Also, while universality in T decay to e and p is unaffected, because of the approximate SU(2)u symmetry, universality between T-+evv and 7+nv will have deviations of the order of 0.5%. Both these corrections are more than an order of magnitude below present experimental errors. One would therefore like to find processes which are forbidden in the standard model. These do exist, since separate lepton numbers are no longer conserved. However, usual rare processes like P-Q, p+eee and neutrinoless double-beta decays are forbidden by N, - N, conservation [ 6 1. The only rare decays in our model involve decays to unusual neutrinos; for example 7-+pv,v,. These are impossible to distinguish from usual decays like ?s--t~v~v,. There are also scattering processes like v,e- +z-V’ which look like neutrino mixing, and, if 620, there is the striking scattering e-p-+‘s-~-. However, none of these processes can be expected to be seen in the near future. In contrast to this murky picture, the model has clear and direct experimental consequences in the Higgs sector. There must be a pair of charged scalars H? with a mass between 20 and about 100 GeV, and a splitting of order 25 MeV. (Their widths are about two orders of magnitude less. ) There is also the Kt pair which is very similar to H,, except that the mass is somewhat larger. Finally, there are the neutral complex scalars h, and k,, which also form a pair with an 0 (e) mass splitting. Their masses are not determined, but are presumably of a similar order of magnitude to those of the charged scalars. The charged scalars decay into a charged lepton and a neutrino, while the neutral ones always decay into two charged leptons. In both cases, one and only one of the leptons is in the third generation. All decay modes that conserve N@- N, are allowed. The new scalars can all be produced via a W or a Z, and the charged ones, of course, can also be produced via a photon. The Wand Z couplings are not diagonal in H-K space, so the Z, for example, can produce an H, and a K: . In an e+e- machine, the first new charged particles to be produced will be H, H: and H_ HT pairs. Since the H+-H_ mass splitting is so small, it will be extremely difficult to resolve them directly. However, one can see that there are, in fact, two different particles because they have different decay modes. The H, decays to ec v or r+ v, while the H_ decays top+ v or T+ v. Therefore, the end reaction is e+e-+l+l+ missing energy, where 1+1- is any pair of charged leptons exceptfor ep. The neutral scalars h, and k, can also be produced via an intermediate Z. If N, is conserved, the signature will be e+e-+T+T-e+eor s+z-,u+,u-, and one can again see that there are two new particles with different decay modes. If N, is broken, the two particles will mix and can no longer be distinguished by their decay modes. In this case, there is also a spectacular e+e-+r-7-e+p+ reaction. The scalars can also be produced in hadron colliders. In that case, one can produce one charged and one neutral scalar, via an intermediate W. If the mass of any of these scalars is below -M,/2, they should appear in Z decays. The recent experiments at SLC, Fermilab and LEP therefore presumably exclude this region. This remaining mass range can be explored at LEP II and the SSC. In conclusion, we have presented a model which gives a magnetic transition moment between the electron and muon neutrinos of the order of 1O- ’ ’ ,uB,and a mass of order of a few eV. The tau neutrino mass must be below a few keV. Because of an N,- N, symmetry, easily detectable rare decays are forbidden, and, since the Yukawa couplings are of the order of lo-‘, low energy phenomenology is essentially unchanged. The model has two closely-spaced pairs of charged scalars, the lightest of which has a mass between 20 and - 100 GeV, and a pair of complex neutral scalars. These particles should be easily seen in high-energy colliders. We would like to thank G. Eilam, D. Granite and M. Gronau for useful discussions. This work was supported in part by the United States-Israel Binational Science Foundation (BSF), the Israeli Academy of Sciences and the ijsterreichische Nationalbank Jubilaumsfondsprojekt Nr. 3485.
86
Volume 237, number 1
PHYSICS LETTERS B
8 March 1990
References [ I] R. Davis, K. Lande, B.T. Cleveland, J. Ullman and J.K. Rowley, in: Proc. 13th Intern. Conf. on Neutrino physics and astrophysics, Neutrino 88, eds. J. Schneps et al. (World Scientific, Singapore), to be published. [2] M.B. Voloshin and M.I. Vysotsky, Yad. Fiz. 44 (1986) 845 [Sov. J. Nucl. Phys. 44 (1986) 5441; L.B. Okun, M.B. Voloshin and MI. Vysotsky, Yad. Fiz. 44 (1986) 677 [Sov. J. Nucl. Phys. 44 (1986) 4401; Zh. Eksp. Teor. Fiz. 91 (1986) 754 [Sov. Phys. JETP 64 (1986) 4461. [3] M. Leurer and J. Liu, Phys. Lett. B 219 (1989) 304. [4] M. Fukugita and T. Yanagida, Phys. Rev. Lett. 58 ( 1987) 1807. [5]K.S.BabuandV.S.Mathur, Phys.Lett. B 196 (1987) 218. [ 6 ] M. Leurer and M. Golden, Fermilab preprint 89/ 175-T. [7] M.B. Voloshin, Yad. Phys. 48 (1988) 804 [Sov. J. Nucl. Phys. 48 (1988)]. [ 81 K.S. Babu and R.N. Mohapatra, Phys. Rev. Lett. 63 (1989) 228. [9] Ya.B. Zeldovich. Dokl. Akad. Nauk SSSR 86 ( 1952) 505; E.J. Konopinski and H.M. Mahmoud, Phys. Rev. 92 ( 1953) 1045. [lo] L. Wolfenstein, Nucl. Phys. B 186 (1981) 147; S.T. Petcov, Phys. Lett. B 1I5 (1982) 401. [ 111 L.B. Okun, Sov. J. Nucl. Phys. 44 ( 1986) 546. [ 121 Particle Data Group, G.P. Yost et al., Review of particle properties, Phys. Lett. B 204 ( 1988 ) 1.
87