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A light Zeldovich-Konopinski-Mahmoud neutrino with a large magnetic moment
Volume 232, number 2
PHYSICS LETTERS B
A LIGHT ZELDOVICH-KONOPINSKI-MAHMOUD WITH A LARGE MAGNETIC MOMENT
30 November 1989
NEUTRINO
G. ECKER, W. G R I M U S Institut fiir Theoretische Physil~ Universitiit Wien, A-f090 Vienna. Austria
and H. N E U F E L D CERN, CH-1211 Geneva 23, Switzerland
Received 1 September 1989
We propose a non-abelian extension of a Zeldovich-Konopinski-Mahmoud lepton number symmetry which gives rise to a naturally light Dirac neutrino with a magnetic moment of O ( 10-tt//B). The neutrino mass appears first at the two-loop level and is well below the experimental upper bound.
A large magnetic moment/z~-~ ( 1 - 10) × 1 0 - ~ # a ( # n = e / 2 m c is the Bohr m a g n e t o n ) o f a Dirac elec-
tron neutrino provides an elegant solution [ 1 ] o f the solar neutrino problem, accounting especially for the a p p a r e n t anticorrelation between the solar neutrino flux and the n u m b e r o f sun spots [2]. The standard procedure to generate Dirac neutrinos introduces right-handed neutral gauge singlets~L These righth a n d e d sterile neutrinos have a tendency to disturb the conventional neutrino astrophysics and cosmology. In particular, from the observed neutrino burst o f the supernova SN1987A [3] an u p p e r b o u n d ~ < 10-~2#B was derived by several authors [4]. This b o u n d would seem to exclude the suggested explanation o f the solar neutrino puzzle unless the righth a n d e d neutrinos have sufficiently strong non-gauge interactions [ 5 ] to get t r a p p e d in the supernova core. A n o t h e r scenario without right-handed neutral singlets avoiding the stringent supernova bound on /A invokes a large transition magnetic m o m e n t between nearly degenerate M a j o r a n a neutrinos [6]. Here we propose to investigate the synthesis o f the two approaches whcre two degenerate M a j o r a n a neutrinos coalesce into a Dirac neutrino due to a ~ We only consider the low-energygauge group SU (2) × U ( 1)r-
Z e l d o v i c h - K o n o p i n s k i - M a h m o u d ( Z K M ) lepton n u m b e r [ 7 ]. All known astrophysical and cosmological constraints are consistent with/A -~ 1 0 - ~~#n for a Z K M neutrino [ 8 ]. The e c o n o m y in the n u m b e r o f neutral fermionic degrees o f freedom and the absence o f sterile neutrinos are very strong arguments in favour o f the Z K M scenario. Another m o t i v a t i o n is the search for a natural mechanism to combine a large magnetic moment with a small neutrino mass. This requirement has turned out to be very difficult to satisfy for the simple reason that any helicity-changing mechanism giving rise to a l a r g e / ~ will in general also produce a big my. A natural coexistence between big p~ and small my seems to call for an underlying symmetry. O f several proposed models for a large magnetic moment, only a few have p r o v i d e d a natural explanation for the small neutrino mass, both for Dirac [9] and for Majorana [10] neutrinos. To emphasize the basic mechanism for a naturally light Z K M neutrino with a large magnetic m o m e n t , we forget about the gauge structure for the m o m e n t . The main ingredients are a charged scalar field q~+ and two lepton generations, each consisting o f a neutral Weyl field Vie and a massive Dirac f e r m i o n f with charge q f = - 1 ( i = 1,2). The theory is assumed to
0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland)
217
Volume 232, number 2
PHYSICS LETTERS B
30 November 1989
possess a conserved ZKM lepton number LZKM= L , - L 2 (i.e., LZKM (U._, f l ) = --LzKM(V2L, f z ) = 1 ) yielding a Dirac neutrino
S of order 2. The resulting group EL (extended lepton number symmetry) is defined by the following composition rules:
11= Ill L "]- (/12L) c •
g(a)og(fl)=g(a+ fl), g(a+ 21rn)=g(a) ,
(1)
Therefore, the fermions have the following Yukawa couplings with q~+:
S2=g(0) =e,
Sog(a)oS=g(a)-'=g(-a) ,
,-~y =/7(hlLfl c "}-hlRflR) ~ + g ( a ) e U ( 1 )ZKM, neT/. + 0C(hzLf2L -k-h2R.f2R) (J~ + +h.c.
Consider now the one-loop contribution with ¢b + exchange to both/a, (shown in fig. 1 are the actual diagrams for the model ofeq. ( 6 ) ) and to m~ (same diagrams without the photon). Depending on the relative signs of the Yukawa couplings h~L and hm, the amplitudes for f~ and t"2 exchange interfere constructively f o r / ~ and destructively for m~ or vice versa. Our strategy will therefore be to achieve a cancellation between the two contributions to m~ by introducing a symmetry" between the two generations. To this end, we will extend the Z K M lepton number symmetry to a non-abelian group yielding the desired result. The simplest extension of U ( I )ZKM tO a non-abelian group involves a single additional group element
I YL
//
\ flR
flL
flL
~
VR
I
\
Fig. 1. One-loop diagrams for u~ for the model ofeq. (6). There are two additional diagrams where the photon is emitted from the scalar lines. Note that vL---vu. and VR------(V2L)c218
The irreducible representations of this group are either one- or two-dimensional: 1 + : g(ot)--* 1, S ~ I ,
l-:g(a)--,1,
S--*- 1 ,
2 (k)'. g ( a ) ---~(~ ikaOe_ik,~),
(4)
ke~.
The Clebsch-Gordan decomposition of tensor products of irreducible representations is easily worked out. The concrete model we are going to investigate in detail is a two-generation model with the usual gauge group SU (2) X U ( 1 ) y. For the momcnt, the second generation can be either ~t or x. In addition to the lefthanded doublets La. there are three (instead of the usual one) S U ( 2 ) singlets lm, fL, fm with q = - 1 ( Y= - 2 ) for each generation i = 1,2. The Higgs sector consists only of doublets: three doublets ¢b~, ~ , , ~R with the usual hypercharge Y= 1 and an unconventional doublet qbL with Y = - 3 (containing ~L-, q~ff- as componcnt fields). With respect to EL, wc postulate the following representation content:
L2L,/ \I2R.]
P /
(3)
(2)
(])I~(J~L~I +,
V2L/"
',J2R/
(J~ll, ( / ) R ~ I - -
.
(5)
To suppress some unwanted couplings, we introduce an additional Z2 symmetry. Under the action of the generating clement of this Z2, q~i, ~H and I,-Rare multiplied by - 1 , while all other fields remain unchanged. The most general fcrmion mass and Yukawa lagrangian with gauge symmetry SU (2) X U ( 1 ) r and global symmetry ELX Z2 has the form
Volume 232, number 2
PHYSICS LETTERS B
~;. + ~< = - .zrf,, f , . - g, E,~. G~ ¢', - g,, E, LA,, l,~ ¢ ' .
-- gR Ea_A,JjR Cl)R- gL Ea B q ~ c )cO L+ h.c., A=(;-7)'
13=(~
10).
U(l)':f~eiaf,
(6)
~l.t: -+e-i;'~i,u , ¢JSL--)eidc.JSL, ~R ---~e - idqSg ,
y, d ~ .
(8)
which break U( 1 ) in (7) to the originally assumed Z:. On the other hand, the Higgs potential can either be U ( 1 )' invariant or it can break U( 1 )' to Z'2. The global symmetry of our model is therefore EL× Z2 X G' with G' = U ( 1 )' or Z~. The Higgs potential can always be chosen to give < ~ >,.~c= 0 so that G' is an exact symmetry. The corresponding conserved quantum number forbids mixing between f and li on the one hand and between @R, q~Land ~ , ~i, on the other hand. The non-abelian group EL must be spontaneously broken. Consider in particular the element S~EL which is represented on the neutral fermions as S Vl. = VII. ~--~ V2L = V~_,
(9)
and which therefore forbids both a magnetic moment and a charge form factor: S _ - ..... ---~V2LO'jzp( VlI.)C"~ (VlL)COlt~,V2L
and, similarly,
(12)
ilI. ~'i" (//R ) c
which forbids both a neutrino mass term and a charge form factor [ 11 ], but allows for a non-vanishing magnetic moment. However, in contrast to our case the Voloshin symmetry does not commute with SU (2) × U ( 1 ) ). because VR is a gauge singlet. The standard gauge group must therefore be extended to a larger group [9], whereas our model is strictly based on SU(2) × U ( 1 )v. The global symmetry E L X Z 2 × G ' is broken down to U( 1 )zr.+,~× G' by (¢'°)va¢' ( q°, )v~ ¢ 0 ,
(13)
giving masses to the charged leptons [i and including mixing between q~{" and q)~ via (8). The mass eigenstates Oj+ with masses Mj(j'= 1,2) are related to the weak eigenstates by a unitary" two-dimensional mixing matrix r':
clJ~:= VRj~7,
(14)
~ + = VL, CI)+ •
For convenience, we write Yukawa couplings from eq. (6):
down
the
neutrino
- . ~ , (v) = v l g , / m q)~+ +gul, R q ~ +gt(12k)c@i --gtx(/2R)C@~ + g R f R ~ ~. +gtf~L@ +
- - g ~ R ) c t b ~ +gL~L)Cq)E ] +h.c. ,
v= V~L+ (V2L)c .
(15)
Due to the exact global symmetry G', only the oneloop diagrams shown in fig. 1 with internal fermions f and scalars @f give rise to a non-zero #,. Neither the usual charged leptons li nor the scalar fields 41, ~II contribute because they cannot induce the necessary transition v~c---, ( v2~.)% The magnetic moment/z~ of the ZKM neutrino v due to the diagrams of fig. 1 is calculated as
It,, = 1.6X 10-Tflt)Re(gcgR V~. 1 VRI )
ValavV-.~-elLa#), ( /121.)c"~ (iJ2L)Co'u~,//II.
=--f)ILtT/ZV(P2L)C--(P2L)CCTI,~Vl/IL=--~O'UVl I
( 11 )
(7)
Continuous global symmetries are either broken explicitly by the remaining lagrangian or they must remain unbroken to avoid Goldstone bosons. Since the singly charged fields in ~R and (~L will have to mix to produce g ~ 0, the Higgs potential necessarily contains terms of the form (and similar ones) i" ~÷ ~)I (~L ¢~)11~ R
s
v7uv-, - P ' ? y .
The symmetry S must be distinguished from Voloshin's symmetry' [9 ]
Neutrino masses are of course forbidden by the gauge symmetry. Although (6) is the most general lagrangian with global group EL × Z2 it actually exhibits a larger global symmetry E L × U ( 1 ) × U ( 1 )' with U(1 ): liR ~eir/,g,
_
30 November 1989
Mw[f(m~)--f(m~--~21) ] Ftlf (10)
f i x ) = x ( x - 1- I n x ) (l_x)2
(16)
219
Volume 232, number 2
PHYSICS LETTERS B
The monotonically increasing function f ( x ) with f ( 0 ) = 0 , f ( 1 ) = 0 . 5 approaches 1 for x - , ~ . Although / t v ¢ 0 requires non-degenerate scalar fields q)~-, qs~-, eq. (16) can naturally yield/A,-~ 10- ~t PB if mf, M~ and Mz are all O (Mw). Eq. ( 16 ) also exhibits the interesting feature that/~--,0 both for mr-.0 and for mf-~oo, keeping the Mi fixed. The Fermi scale (or "l/w) is the natural scale for a big u~ in this model. However, the real problem is not so much getting a sufficiently big/~, but keeping m~ small at the same time. Since rn~ also requires a transition U~L~ 0'2L)% the only relevant one-loop diagrams are the ones of fig. 1 without the photon, because all other couplings are flavour diagonal. Although ELXZ2 is spontaneously broken, this breaking is not effective for rn~ at the one-loop level. Independently o f the mixing between q~+ and q~', the two amplitudes withf~ and J~ exchange cancel exactly. As a side remark, we note that if the global symmetry G' =Z~, a possible spontaneous breaking via ( ~ ) ~ = ~ 0 would induce rn~ at one loop, the magnitude depending, of course, on The present model is the first example o f a gauge model with the standard gauge group SU (2) X U ( 1 ) r where at the one-loop level a massless Dirac neutrino has a magnetic moment of the order 10-1t/tB. In order to veri~" the stability o f our model, we must estimate the natural order o f magnitude o f m~ which arises at the two-loop level. Once again, the only flavour changing Yukawa coupling ge must be involved. Moreover, the symmetry between the two generations must be broken in the second loop which requires exchange o f the charged leptons l~. Altogether four diagrams contribute, two of which are shown in fig. 2 and there are two similar ones with q)E- exchange. As expected, the sum o f the two diagrams in fig. 2 (and similarly for ~ff - exchange) is finite and proportional to m 22- m z. A standard estimate yields Re(gl.gR V[~ my ~
X
VRI) (m22--m~)mf2
(47~)4
(,14~-M~) {Ig.[ ~ .~2
\-~o
[gEl ~'] + "~/2--,/ '
(17)
where ,~ is some average of M,, -I//2 and -'V/o,M _ _ denote the masses o f q ~° and ~ f f - , respectively. Note that q~o is a mass eigenstate (trivially true for q)£ - ) because q)g does not mix with q~, q)u due to the global 220
30 November 1989
/
VL
[ f'tR
/
Yr.
/
IlL
fir
flL \
V'R
f~d\
v.
,¢
/
f2 c
f2Rc
12Lc
k\
/
I
-
Fig. 2. Two-loop diagrams for rn,. There are two similar diagrams with q)E- (instead of q~R) exchange. ~R
Fig. 3. Liu's diagram [ 11 ] for m~ with magnetic moment form factorp~(q 2) and charge form factor R~(q2). G ' . Comparing eqs. ( 16 ) and ( 17 ), we find m v ~ 0 . 1 cV.
m22 10'%tv m~ #B
(IgRI X j~2[f(x2)_f(x,)] \.'-~o
+ ,-TT--M__]'
(18)
with xi= rn ~/M2,. As a consequence, even for m2 = m~, i.e., for Lz~a,a= L c - L,, the neutrino mass is naturally below the experimental upper bound of 18 eV [ 12 ]. For Lzr, M = L ~ - Lu, my is even expected to be several orders of magnitude lighter than the present bound. In a recent paper [ 11 ], Liu has considered a special three-loop contribution to my shown in fig. 3. The two vertices stand for the magnetic moment form f a c t o r / ~ ( q 2 ) and the charge form factor Rv(q2).
Volume 232, number 2
PHYSICS LETTERS B
T a k i n g / ~ (q2) = / ~ ( 0 ) = p~, employing the s t a n d a r d model result [ 13 ] for Rvo (q2) and cutting the integral & f a t M~v, Liu obtains a natural b o u n d /tv< 10-11/ts(mJeV)
level. The neutrino mass arises at the two-loop level and is well below the present experimental b o u n d irrespective o f whether the conserved Z K M lepton n u m b e r is L ¢ - L ~ or L~-L~.
(19)
by d e m a n d i n g that my is not smaller than the contribution from the diagram in fig. 3. This b o u n d is surprisingly strict and should be confronted with our twoloop estimate ( 18 ) which would lead to a less stringent bound. The solution o f this seeming puzzle is related to the global s y m m e t r y EL which forbids a charge form factor as indicated by eq. (1 1 ). More explicitly, for a Z K M neutrino the charge form factor R,, (q2) is given by Rv(q 2 ) = R~, (q 2 ) - R v 2 ( q 2 ) .
30 November 1989
(20)
The fermions f do not contribute to R~ at one loop for the same reason as for m~. R~, (q2) is therefore given by the usual s t a n d a r d model result [ 13 ]. Compared to it, eq. (20) furnishes a suppression factor ( m 2,- m ~ )/M~v. Consequently, the right-hand side o f Liu's b o u n d ( 1 9 ) has to be multiplied by M~v / (mZ~-rn~) in our case. Thus, the b o u n d becomes trivial irrespective o f whether m2 is rn~ or me. In the present model, the neutrino mass is indeed naturally small. Concerning the magnetic m o m e n t s o f charged leptons, the situation is essentially the same as for a twoHiggs-doublet version o f the standard model. Due once again to the global s y m m e t r y G ' , o f the four scalar doublets in our model only q)x, ~ n are relevant for the magnetic m o m e n t s o f charged leptons while ~R, q~L induce the neutrino magnetic m o m e n t . Therefore, the magnetic m o m e n t s o f charged and neutral leptons are completely i n d e p e n d e n t and there are no reciprocal constraints. We have presented the first example o f an SU ( 2 ) × U ( 1 ) y gauge model where a naturally light Dirac neutrino has a large magnetic m o m e n t o f O ( 1 0 - ' '~tn ). The crucial ingredient is a non-abelian extension o f a Z K M lepton n u m b e r s y m m e t r y which guarantees a vanishing neutrino mass at the one-loop
References
[ 1] M.B. Voloshin, M.I. Vysotskii and L.B. Okun, Sov. J. Nucl. Phys. 44 (1986) 440; M.B. Voloshin and M.I. Vysotskii, Sov. J. Nucl. Phys. 44 (1986) 544; A. Cisneros, Astrophys. Space Sci. 10 ( 1971 ) 87. [2] R. Davis, Proc. "Neutrino" '88 (Boston, June 1988); P. Raychaudburi, Mod. Phys. Lett. A 3 (1988) 1319. [ 3 ] K. Hiram et al., Phys. Rev. Lett. 58 ( 1987 ) 1490; R.M. Bionta et al., Phys. Rev. Lett. 58 (1987) 1494. [4]A. Dar, Princeton preprint IAS-PUB (February 1987), unpublished; I. Goldman et al., Phys. Rev. Lett. 60 ( 1988 ) 1789; J.M. Lattimer and J. Cooperstein, Phys. Rev. Lett. 61 (1988) 23; R. Barbieri and R.N. Mohapatra, Phys. Rev. Lett. 61 (1988) 27; D. Notzold, Phys. Rev. D 38 (1988) 1658. [5] R. 13arbieri,R.N. Mohapatra and T. Yanagida, Phys. Len. 13213 (1989) 69; M.B. Voloshin, JETP Lett. 47 (1988) 501; Phys. Left. B 209 (1988) 360. [ 6 ] C.-S. Lim and W.J. Marciano, Phys. Rev. D 37 ( 1988 ) 1368; E.Kh. Akhmedov, Phys. Lett. 13213 (1988) 64. [7] Ya.B. Zcldovich, Dokl. Akad. Nauk SSSR 86 (1952) 505; E.J. Konopinski and H.M. Mahmoud, Phys. Rev. 92 (1953) 1045. [ 8 ] M. Leurer and J. Liu, Phys. Left. 13219 ( 1989 ) 304. [9] M.B. Voloshin, Sov. J. Nucl. Phys. 48 (1988) 512; R. 13arbieriand R.N. Mohapatra, Phys. Len. 13218 (1989) 225; J. Liu, Phys. Lett. 13225 (1989) 148. [10l K.S. Babu and R.N. Mohapatra, Phys. Rev. Lctt. 63 (1989) 228. [11] J. Liu, University of Michigan preprint UM-TH-88-19. [12] M. Fritschi et al., Phys. Lett. 13 173 (1986) 485. [13] W.A. Bardeen, R. Gastmans and B. Lautrup, Nucl. Phys. B 46 (1972) 319; W.J. Marciano and A. Sirlin, Phys. Rev. D 22 (1980) 2695.
221
PHYSICS LETTERS B
A LIGHT ZELDOVICH-KONOPINSKI-MAHMOUD WITH A LARGE MAGNETIC MOMENT
30 November 1989
NEUTRINO
G. ECKER, W. G R I M U S Institut fiir Theoretische Physil~ Universitiit Wien, A-f090 Vienna. Austria
and H. N E U F E L D CERN, CH-1211 Geneva 23, Switzerland
Received 1 September 1989
We propose a non-abelian extension of a Zeldovich-Konopinski-Mahmoud lepton number symmetry which gives rise to a naturally light Dirac neutrino with a magnetic moment of O ( 10-tt//B). The neutrino mass appears first at the two-loop level and is well below the experimental upper bound.
A large magnetic moment/z~-~ ( 1 - 10) × 1 0 - ~ # a ( # n = e / 2 m c is the Bohr m a g n e t o n ) o f a Dirac elec-
tron neutrino provides an elegant solution [ 1 ] o f the solar neutrino problem, accounting especially for the a p p a r e n t anticorrelation between the solar neutrino flux and the n u m b e r o f sun spots [2]. The standard procedure to generate Dirac neutrinos introduces right-handed neutral gauge singlets~L These righth a n d e d sterile neutrinos have a tendency to disturb the conventional neutrino astrophysics and cosmology. In particular, from the observed neutrino burst o f the supernova SN1987A [3] an u p p e r b o u n d ~ < 10-~2#B was derived by several authors [4]. This b o u n d would seem to exclude the suggested explanation o f the solar neutrino puzzle unless the righth a n d e d neutrinos have sufficiently strong non-gauge interactions [ 5 ] to get t r a p p e d in the supernova core. A n o t h e r scenario without right-handed neutral singlets avoiding the stringent supernova bound on /A invokes a large transition magnetic m o m e n t between nearly degenerate M a j o r a n a neutrinos [6]. Here we propose to investigate the synthesis o f the two approaches whcre two degenerate M a j o r a n a neutrinos coalesce into a Dirac neutrino due to a ~ We only consider the low-energygauge group SU (2) × U ( 1)r-
Z e l d o v i c h - K o n o p i n s k i - M a h m o u d ( Z K M ) lepton n u m b e r [ 7 ]. All known astrophysical and cosmological constraints are consistent with/A -~ 1 0 - ~~#n for a Z K M neutrino [ 8 ]. The e c o n o m y in the n u m b e r o f neutral fermionic degrees o f freedom and the absence o f sterile neutrinos are very strong arguments in favour o f the Z K M scenario. Another m o t i v a t i o n is the search for a natural mechanism to combine a large magnetic moment with a small neutrino mass. This requirement has turned out to be very difficult to satisfy for the simple reason that any helicity-changing mechanism giving rise to a l a r g e / ~ will in general also produce a big my. A natural coexistence between big p~ and small my seems to call for an underlying symmetry. O f several proposed models for a large magnetic moment, only a few have p r o v i d e d a natural explanation for the small neutrino mass, both for Dirac [9] and for Majorana [10] neutrinos. To emphasize the basic mechanism for a naturally light Z K M neutrino with a large magnetic m o m e n t , we forget about the gauge structure for the m o m e n t . The main ingredients are a charged scalar field q~+ and two lepton generations, each consisting o f a neutral Weyl field Vie and a massive Dirac f e r m i o n f with charge q f = - 1 ( i = 1,2). The theory is assumed to
0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland)
217
Volume 232, number 2
PHYSICS LETTERS B
30 November 1989
possess a conserved ZKM lepton number LZKM= L , - L 2 (i.e., LZKM (U._, f l ) = --LzKM(V2L, f z ) = 1 ) yielding a Dirac neutrino
S of order 2. The resulting group EL (extended lepton number symmetry) is defined by the following composition rules:
11= Ill L "]- (/12L) c •
g(a)og(fl)=g(a+ fl), g(a+ 21rn)=g(a) ,
(1)
Therefore, the fermions have the following Yukawa couplings with q~+:
S2=g(0) =e,
Sog(a)oS=g(a)-'=g(-a) ,
,-~y =/7(hlLfl c "}-hlRflR) ~ + g ( a ) e U ( 1 )ZKM, neT/. + 0C(hzLf2L -k-h2R.f2R) (J~ + +h.c.
Consider now the one-loop contribution with ¢b + exchange to both/a, (shown in fig. 1 are the actual diagrams for the model ofeq. ( 6 ) ) and to m~ (same diagrams without the photon). Depending on the relative signs of the Yukawa couplings h~L and hm, the amplitudes for f~ and t"2 exchange interfere constructively f o r / ~ and destructively for m~ or vice versa. Our strategy will therefore be to achieve a cancellation between the two contributions to m~ by introducing a symmetry" between the two generations. To this end, we will extend the Z K M lepton number symmetry to a non-abelian group yielding the desired result. The simplest extension of U ( I )ZKM tO a non-abelian group involves a single additional group element
I YL
//
\ flR
flL
flL
~
VR
I
\
Fig. 1. One-loop diagrams for u~ for the model ofeq. (6). There are two additional diagrams where the photon is emitted from the scalar lines. Note that vL---vu. and VR------(V2L)c218
The irreducible representations of this group are either one- or two-dimensional: 1 + : g(ot)--* 1, S ~ I ,
l-:g(a)--,1,
S--*- 1 ,
2 (k)'. g ( a ) ---~(~ ikaOe_ik,~),
(4)
ke~.
The Clebsch-Gordan decomposition of tensor products of irreducible representations is easily worked out. The concrete model we are going to investigate in detail is a two-generation model with the usual gauge group SU (2) X U ( 1 ) y. For the momcnt, the second generation can be either ~t or x. In addition to the lefthanded doublets La. there are three (instead of the usual one) S U ( 2 ) singlets lm, fL, fm with q = - 1 ( Y= - 2 ) for each generation i = 1,2. The Higgs sector consists only of doublets: three doublets ¢b~, ~ , , ~R with the usual hypercharge Y= 1 and an unconventional doublet qbL with Y = - 3 (containing ~L-, q~ff- as componcnt fields). With respect to EL, wc postulate the following representation content:
L2L,/ \I2R.]
P /
(3)
(2)
(])I~(J~L~I +,
V2L/"
',J2R/
(J~ll, ( / ) R ~ I - -
.
(5)
To suppress some unwanted couplings, we introduce an additional Z2 symmetry. Under the action of the generating clement of this Z2, q~i, ~H and I,-Rare multiplied by - 1 , while all other fields remain unchanged. The most general fcrmion mass and Yukawa lagrangian with gauge symmetry SU (2) X U ( 1 ) r and global symmetry ELX Z2 has the form
Volume 232, number 2
PHYSICS LETTERS B
~;. + ~< = - .zrf,, f , . - g, E,~. G~ ¢', - g,, E, LA,, l,~ ¢ ' .
-- gR Ea_A,JjR Cl)R- gL Ea B q ~ c )cO L+ h.c., A=(;-7)'
13=(~
10).
U(l)':f~eiaf,
(6)
~l.t: -+e-i;'~i,u , ¢JSL--)eidc.JSL, ~R ---~e - idqSg ,
y, d ~ .
(8)
which break U( 1 ) in (7) to the originally assumed Z:. On the other hand, the Higgs potential can either be U ( 1 )' invariant or it can break U( 1 )' to Z'2. The global symmetry of our model is therefore EL× Z2 X G' with G' = U ( 1 )' or Z~. The Higgs potential can always be chosen to give < ~ >,.~c= 0 so that G' is an exact symmetry. The corresponding conserved quantum number forbids mixing between f and li on the one hand and between @R, q~Land ~ , ~i, on the other hand. The non-abelian group EL must be spontaneously broken. Consider in particular the element S~EL which is represented on the neutral fermions as S Vl. = VII. ~--~ V2L = V~_,
(9)
and which therefore forbids both a magnetic moment and a charge form factor: S _ - ..... ---~V2LO'jzp( VlI.)C"~ (VlL)COlt~,V2L
and, similarly,
(12)
ilI. ~'i" (//R ) c
which forbids both a neutrino mass term and a charge form factor [ 11 ], but allows for a non-vanishing magnetic moment. However, in contrast to our case the Voloshin symmetry does not commute with SU (2) × U ( 1 ) ). because VR is a gauge singlet. The standard gauge group must therefore be extended to a larger group [9], whereas our model is strictly based on SU(2) × U ( 1 )v. The global symmetry E L X Z 2 × G ' is broken down to U( 1 )zr.+,~× G' by (¢'°)va¢' ( q°, )v~ ¢ 0 ,
(13)
giving masses to the charged leptons [i and including mixing between q~{" and q)~ via (8). The mass eigenstates Oj+ with masses Mj(j'= 1,2) are related to the weak eigenstates by a unitary" two-dimensional mixing matrix r':
clJ~:= VRj~7,
(14)
~ + = VL, CI)+ •
For convenience, we write Yukawa couplings from eq. (6):
down
the
neutrino
- . ~ , (v) = v l g , / m q)~+ +gul, R q ~ +gt(12k)c@i --gtx(/2R)C@~ + g R f R ~ ~. +gtf~L@ +
- - g ~ R ) c t b ~ +gL~L)Cq)E ] +h.c. ,
v= V~L+ (V2L)c .
(15)
Due to the exact global symmetry G', only the oneloop diagrams shown in fig. 1 with internal fermions f and scalars @f give rise to a non-zero #,. Neither the usual charged leptons li nor the scalar fields 41, ~II contribute because they cannot induce the necessary transition v~c---, ( v2~.)% The magnetic moment/z~ of the ZKM neutrino v due to the diagrams of fig. 1 is calculated as
It,, = 1.6X 10-Tflt)Re(gcgR V~. 1 VRI )
ValavV-.~-elLa#), ( /121.)c"~ (iJ2L)Co'u~,//II.
=--f)ILtT/ZV(P2L)C--(P2L)CCTI,~Vl/IL=--~O'UVl I
( 11 )
(7)
Continuous global symmetries are either broken explicitly by the remaining lagrangian or they must remain unbroken to avoid Goldstone bosons. Since the singly charged fields in ~R and (~L will have to mix to produce g ~ 0, the Higgs potential necessarily contains terms of the form (and similar ones) i" ~÷ ~)I (~L ¢~)11~ R
s
v7uv-, - P ' ? y .
The symmetry S must be distinguished from Voloshin's symmetry' [9 ]
Neutrino masses are of course forbidden by the gauge symmetry. Although (6) is the most general lagrangian with global group EL × Z2 it actually exhibits a larger global symmetry E L × U ( 1 ) × U ( 1 )' with U(1 ): liR ~eir/,g,
_
30 November 1989
Mw[f(m~)--f(m~--~21) ] Ftlf (10)
f i x ) = x ( x - 1- I n x ) (l_x)2
(16)
219
Volume 232, number 2
PHYSICS LETTERS B
The monotonically increasing function f ( x ) with f ( 0 ) = 0 , f ( 1 ) = 0 . 5 approaches 1 for x - , ~ . Although / t v ¢ 0 requires non-degenerate scalar fields q)~-, qs~-, eq. (16) can naturally yield/A,-~ 10- ~t PB if mf, M~ and Mz are all O (Mw). Eq. ( 16 ) also exhibits the interesting feature that/~--,0 both for mr-.0 and for mf-~oo, keeping the Mi fixed. The Fermi scale (or "l/w) is the natural scale for a big u~ in this model. However, the real problem is not so much getting a sufficiently big/~, but keeping m~ small at the same time. Since rn~ also requires a transition U~L~ 0'2L)% the only relevant one-loop diagrams are the ones of fig. 1 without the photon, because all other couplings are flavour diagonal. Although ELXZ2 is spontaneously broken, this breaking is not effective for rn~ at the one-loop level. Independently o f the mixing between q~+ and q~', the two amplitudes withf~ and J~ exchange cancel exactly. As a side remark, we note that if the global symmetry G' =Z~, a possible spontaneous breaking via ( ~ ) ~ = ~ 0 would induce rn~ at one loop, the magnitude depending, of course, on The present model is the first example o f a gauge model with the standard gauge group SU (2) X U ( 1 ) r where at the one-loop level a massless Dirac neutrino has a magnetic moment of the order 10-1t/tB. In order to veri~" the stability o f our model, we must estimate the natural order o f magnitude o f m~ which arises at the two-loop level. Once again, the only flavour changing Yukawa coupling ge must be involved. Moreover, the symmetry between the two generations must be broken in the second loop which requires exchange o f the charged leptons l~. Altogether four diagrams contribute, two of which are shown in fig. 2 and there are two similar ones with q)E- exchange. As expected, the sum o f the two diagrams in fig. 2 (and similarly for ~ff - exchange) is finite and proportional to m 22- m z. A standard estimate yields Re(gl.gR V[~ my ~
X
VRI) (m22--m~)mf2
(47~)4
(,14~-M~) {Ig.[ ~ .~2
\-~o
[gEl ~'] + "~/2--,/ '
(17)
where ,~ is some average of M,, -I//2 and -'V/o,M _ _ denote the masses o f q ~° and ~ f f - , respectively. Note that q~o is a mass eigenstate (trivially true for q)£ - ) because q)g does not mix with q~, q)u due to the global 220
30 November 1989
/
VL
[ f'tR
/
Yr.
/
IlL
fir
flL \
V'R
f~d\
v.
,¢
/
f2 c
f2Rc
12Lc
k\
/
I
-
Fig. 2. Two-loop diagrams for rn,. There are two similar diagrams with q)E- (instead of q~R) exchange. ~R
Fig. 3. Liu's diagram [ 11 ] for m~ with magnetic moment form factorp~(q 2) and charge form factor R~(q2). G ' . Comparing eqs. ( 16 ) and ( 17 ), we find m v ~ 0 . 1 cV.
m22 10'%tv m~ #B
(IgRI X j~2[f(x2)_f(x,)] \.'-~o
+ ,-TT--M__]'
(18)
with xi= rn ~/M2,. As a consequence, even for m2 = m~, i.e., for Lz~a,a= L c - L,, the neutrino mass is naturally below the experimental upper bound of 18 eV [ 12 ]. For Lzr, M = L ~ - Lu, my is even expected to be several orders of magnitude lighter than the present bound. In a recent paper [ 11 ], Liu has considered a special three-loop contribution to my shown in fig. 3. The two vertices stand for the magnetic moment form f a c t o r / ~ ( q 2 ) and the charge form factor Rv(q2).
Volume 232, number 2
PHYSICS LETTERS B
T a k i n g / ~ (q2) = / ~ ( 0 ) = p~, employing the s t a n d a r d model result [ 13 ] for Rvo (q2) and cutting the integral & f a t M~v, Liu obtains a natural b o u n d /tv< 10-11/ts(mJeV)
level. The neutrino mass arises at the two-loop level and is well below the present experimental b o u n d irrespective o f whether the conserved Z K M lepton n u m b e r is L ¢ - L ~ or L~-L~.
(19)
by d e m a n d i n g that my is not smaller than the contribution from the diagram in fig. 3. This b o u n d is surprisingly strict and should be confronted with our twoloop estimate ( 18 ) which would lead to a less stringent bound. The solution o f this seeming puzzle is related to the global s y m m e t r y EL which forbids a charge form factor as indicated by eq. (1 1 ). More explicitly, for a Z K M neutrino the charge form factor R,, (q2) is given by Rv(q 2 ) = R~, (q 2 ) - R v 2 ( q 2 ) .
30 November 1989
(20)
The fermions f do not contribute to R~ at one loop for the same reason as for m~. R~, (q2) is therefore given by the usual s t a n d a r d model result [ 13 ]. Compared to it, eq. (20) furnishes a suppression factor ( m 2,- m ~ )/M~v. Consequently, the right-hand side o f Liu's b o u n d ( 1 9 ) has to be multiplied by M~v / (mZ~-rn~) in our case. Thus, the b o u n d becomes trivial irrespective o f whether m2 is rn~ or me. In the present model, the neutrino mass is indeed naturally small. Concerning the magnetic m o m e n t s o f charged leptons, the situation is essentially the same as for a twoHiggs-doublet version o f the standard model. Due once again to the global s y m m e t r y G ' , o f the four scalar doublets in our model only q)x, ~ n are relevant for the magnetic m o m e n t s o f charged leptons while ~R, q~L induce the neutrino magnetic m o m e n t . Therefore, the magnetic m o m e n t s o f charged and neutral leptons are completely i n d e p e n d e n t and there are no reciprocal constraints. We have presented the first example o f an SU ( 2 ) × U ( 1 ) y gauge model where a naturally light Dirac neutrino has a large magnetic m o m e n t o f O ( 1 0 - ' '~tn ). The crucial ingredient is a non-abelian extension o f a Z K M lepton n u m b e r s y m m e t r y which guarantees a vanishing neutrino mass at the one-loop
References
[ 1] M.B. Voloshin, M.I. Vysotskii and L.B. Okun, Sov. J. Nucl. Phys. 44 (1986) 440; M.B. Voloshin and M.I. Vysotskii, Sov. J. Nucl. Phys. 44 (1986) 544; A. Cisneros, Astrophys. Space Sci. 10 ( 1971 ) 87. [2] R. Davis, Proc. "Neutrino" '88 (Boston, June 1988); P. Raychaudburi, Mod. Phys. Lett. A 3 (1988) 1319. [ 3 ] K. Hiram et al., Phys. Rev. Lett. 58 ( 1987 ) 1490; R.M. Bionta et al., Phys. Rev. Lett. 58 (1987) 1494. [4]A. Dar, Princeton preprint IAS-PUB (February 1987), unpublished; I. Goldman et al., Phys. Rev. Lett. 60 ( 1988 ) 1789; J.M. Lattimer and J. Cooperstein, Phys. Rev. Lett. 61 (1988) 23; R. Barbieri and R.N. Mohapatra, Phys. Rev. Lett. 61 (1988) 27; D. Notzold, Phys. Rev. D 38 (1988) 1658. [5] R. 13arbieri,R.N. Mohapatra and T. Yanagida, Phys. Len. 13213 (1989) 69; M.B. Voloshin, JETP Lett. 47 (1988) 501; Phys. Left. B 209 (1988) 360. [ 6 ] C.-S. Lim and W.J. Marciano, Phys. Rev. D 37 ( 1988 ) 1368; E.Kh. Akhmedov, Phys. Lett. 13213 (1988) 64. [7] Ya.B. Zcldovich, Dokl. Akad. Nauk SSSR 86 (1952) 505; E.J. Konopinski and H.M. Mahmoud, Phys. Rev. 92 (1953) 1045. [ 8 ] M. Leurer and J. Liu, Phys. Left. 13219 ( 1989 ) 304. [9] M.B. Voloshin, Sov. J. Nucl. Phys. 48 (1988) 512; R. 13arbieriand R.N. Mohapatra, Phys. Len. 13218 (1989) 225; J. Liu, Phys. Lett. 13225 (1989) 148. [10l K.S. Babu and R.N. Mohapatra, Phys. Rev. Lctt. 63 (1989) 228. [11] J. Liu, University of Michigan preprint UM-TH-88-19. [12] M. Fritschi et al., Phys. Lett. 13 173 (1986) 485. [13] W.A. Bardeen, R. Gastmans and B. Lautrup, Nucl. Phys. B 46 (1972) 319; W.J. Marciano and A. Sirlin, Phys. Rev. D 22 (1980) 2695.
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