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Anomalous Z decays and g−2 of the electron
Volume 143B, number 1, 2, 3
ANOMALOUS
PHYSICS LETTERS
Z DECAYS
AND
g -
9 August 1984
2 OF T H E E L E C T R O N
Mahiko S U Z U K I Department of Physics and Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720, USA Received 19 April 1984 Revised manuscript received 30 April 1984
If the leptonic Z decays with an energetic photon are interpreted as cascade decays through a new particle, a stringent constraint is imposed on the couplings by the measured value of the magnetic moment of the electron. In models with a fermion E, where Z ---,eE(~E) ~ e~'/, the constraint is avoided if E R and E L are either both singlets or both members of a doublet.
Observation of wide angle photons in leptonic Z decays has attracted much attention [1,2]. It is quite conceivable that they resulted from a tail of ordinary physics accidentally enhanced by meager statistics. Alternatively, this observation may be the first hint of new physics beyond the standard electroweak gauge theory, in particular, compositeness of gauge bosons, quarks, and leptons at an energy scale above 100 GeV. In this note, assuming that these anomalous events are a consequence of new physics, we explore what conditions are imposed on theoretical interpretations when we require that the g - 2 factor of the electron be consistent with measurement [3]. The relation between compositeness and g - 2 was discussed by Brodsky and DreU in their pioneering paper [4]. Our analysis addresses to more specific possibilities that are implied by t h e anomalous Z decay experiment. Since the anomalous Z decays Z ~ ¢+d-3' occupy one quarter of the leptonic decays, it is most likely that the final state ¢+d-3' is produced through a heavy particle in cascade decays rather than in primary decays. Pursuing along this line, we come to two possibilities: Z ---~e+ + E - ---, e+ + e - + y ( - o e - + E+ ~ e - + e+ +~,), Z~y+X°~+e++e
-,
(1) (2)
where E and X ° are new particles. One can imagine E as an excited electron and X ° as a 1S0 bound state of composite models in contrast to 3S 1 states which may be assigned to gauge bosons. The two cases are studied separately. The excited electron E . The Z ~ E+ •-y decay occurs through the diagram of fig. la. The anomalous magnetic moment is produced through the diagram of fig. lb. We assume for simplicity that the spin of E is 1 / 2 instead of 3/2. The relevant couplings are defined by the lagrangian with the SU(2) × U(1) symmetry L i n t = g ( E L--O t ~ v e R ) 0 /xB v + K , ( ~ , ~ ) R O p . p ( ; ) L -0 / ~ B p _
+ I¢"(N, E)RO~/r( ; )g • 0~W ~,
(3)
where W, and B~ are the SU(2) and U(1) gauge bosons, respectively. It is assumed in (3) that the mass of E transforms like a component of a doublet, just as the electron mass. We have x' = r " = 0 if E R and E L are both singlets, and x = 0 if both E g and E L belong to doublets ,1. C P invariance requires that r, r', and ~" be real (up to a common phase). We assume C P invariance and choose them to be real. Otherwise, they would generate an electric dipole moment, in general. *] Models cannot be built with the assignment that E R is a singlet and E L is a member of a doublet, since all the couplings in (3) would be forbidden.
0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
237
Volume 143B, number 1, 2, 3
PHYSICS LETTERS
¥ Z
E"
E
9 August 1984
where K ( A , m, M ) is a quantity of O(1) defined by E
K ( A , m, M ) = l o g ( A 2 / m 2)
M4)/(m 2-
+ [ ( m 4 + 3m2M 2 -
(a)
/b)
Fig. 1. (a) Decay through E. A corresponding diagram with e ± and E - replaced by e n: and E + , respectively, also contributes. (b) Anomalous magnetic moment through E. Z may be replaced by 7.
T h e observed Z---, d + d - y decay rate r relative to the Z ~ [+ d - decay rate r = r ( z --+ e+ e - v ) / r ( z --, e + e - )
(4)
constrains the couphngs, since E ---, e~, is presumably the only significant decay m o d e of E. F r o m the calculated decay rates
X M 2 log(
m 2 / M 2)
--(2m4+
3m2M2 + M4)/2(m2- M2) 2,
(10)
for A >> m and M. The p a r a m e t e r A is a mass scale of a form factor F ( k 2) = A Z / ( A 2 - k 2 ) ,
(11)
which has been inserted in the lepton vertices. It indicates a scale of compositeness of fermions and ensures convergence of F e y n m a n loop integrals. We expect A to be m u c h larger than m z and M. Strictly speaking, all the decay rate formulas are also subject to form factor corrections, but they are negligibly small as far as A >> m z and M. We estimate A# with M = 50 GeV. The result is At* = 1.7 x 10 -8, [ l o g ( A 2 / m
r (Z - , eF. + ~S)
M 2 ) 3]
z) + 0.3]
~'- 0.75#'
K2 + ( x ' + K" cot 0w) 2
= (24~r)-m[x = + ( x ' + x " cot Ow): ] x s i n 2 Ow(1 -
r ( Z -+ e~) =
2
3
sin 2 O w ) ] m z ,
This n u m b e r must be within the limit of possible discrepancy between experiment and the Q E D prediction [5],
(6)
- 4 × lO-'°(e/2me)
where M is the mass of E, we obtain for M = 50 MeV
Ka+(K'+K" cotOw)Z=4.8r(e/mz) z.
(7)
M)M,
< 4.6 X 1 0 - 3 ( 1 / r ) .
K2 + (~¢' + x '' cot 0~ ) 2 (14)
(98)
The spinless boson X °. The decay process occurs through the diagram of fig. 2a. The diagrams of
#v = ( 1 6 * r z ) - a e x ( K " - K" tan Ow) X cos 2 0w K ( A , 0,
K(K' - 0 . 1 5 # ' )
(9a)
/~z ----( 1 6 ~ r 2 ) - ' e K ( K' + K" cot 8w)
M)M,
(13)
This is a stringent condition for r = 1 / 8 - 1 / 4 . T h e condition K ' - 0 . 1 5 # ' = 0 means that the Z and ~ loop contributions accidentally cancel each other in A F. A simple way to satisfy (14) is to forbid either of ( # , # ' ) and ~ by assigning E R and E L both to singlets or both to doublets.
(8)
X sin z Ow K ( A , m z ,
< A F < 0.
F o r A = 1 TeV and sin 2 0w = 0.21, we obtain 0 < -
A straightforward calculation leads us to a f o r m u l a for the correction to the magnetic m o m e n t a t , = J*z +
(12)
(5) (G/24v12*rz) X[1 +(1-4
238
× I¢(-~m~ ).
MZ/rn~)2(1 + 2M2/m2z)m3z,
Volume 143B, number 1, 2, 3
PHYSICS L E T T E R S
1'
¥
9 August 1984
or
~/=~'=d=l
and
~=~'=c=0.
(16b)
The observed Z - ~ d + d - y decay rate fixes the couplings through the relation (4) together with e
0
o
+
0
0
r ( z - , d + d - v ) = B ( X ° -+ e + d - ) r ( z --, v x ° ) , (17) la/
tbt
Fig. 2. (a) Decay through X °. (b) Anomalous magnetic m o m e n t through Z - X °. Corresponding diagrams with Z replaced by 3' can also contribute. The form factors are inserted at the vertices of the electron.
fig. 2b and the diagrams which are obtained from fig. 2b by replacement of Z with y contribute to g - 2. Models of this type have a common problem with the SU(2) × U(1) symmetry. The SU(2) x U(1) symmetry requires that X ° be either a singlet or a member of a triplet since it couples to Z and y. However, the coupling of X ° to lepton pairs must not be proportional to lepton masses in order for Z--+ e + e - y and # + # - 7 not to be dominated by Z - - ' r + * - y . The scalar coupling ERdLxO is allowed only when X ° is a member of a doublet, just like a Higgs particle. It is inevitable that the SU(2) × U(1) symmetry is broken in the d + d - X ° couplings. We assume for simplicity that the leptonic couplings of X ° are universal. Peccei [6] argued that X 0 is more likely to be a singlet than a component of a triplet. Since the basic feature of our conclusion remains the same even if X ° is assumed to be a member of a triplet, we present here the results for a singlet X °. The relevant couplings are defined by Lin t = f W . . " (~W ~'~ + TIqg~)X °
id-rs) dX °,
×sin220w(f-f
(15)
~=~'=c=1
and
r/=rf=d=O,
(16a)
) ( r n z2- - M 2 ) 3.
Denoting B ( X ° ~ d + d - ) by B, we obtain with M = 50 GeV (f_f,)2
= 3.4(r/B)(e/mz)2.
(19)
The correction to the magnetic moment is given by
at, = #z + #~,
(20)
#z = ( 1 6 r r 2 ) - l g h ( f - f
") sin Ow(1 - 4 sin 2 Ow)
x R ( A , mz, M ) ,
(21a)
#v = (4rr2)-leh( f sin2 Ow + f ' cos 2 Ow) × / ~ ( A , 0, M ) ,
(21b)
where / ~ ( A , m, M )
= [rn2/(rn 2 - M2)] l o g ( A 2 / m 2) -
[ M Z / ( m 2 - M2)] l o g ( A 2 / M 2 ) ,
(22)
for A >> m and M. Demanding the condition (13) on A# of (20), we obtain
') < 2.4 × l O - 7 ( B / r ) ,
(23)
where
F(f , f') = (f + 3.2f')z/(f where W~ and B~ are dual tensors, ffP invariance requires that either ~ = ~' = c = 0 or ~/= 7/' = d = 0. Without loss of generality, we choose the phase convention
p 2
(18)
(h2/4~r)F(f,f
+ f ' B ~ (~'B ~ + n'Ba~)X °
+hY',?(c -
r ( Z -~ v X ° ) = (24~rm3) -1
- f ' ) 2,
(24)
and A = 1 TeV, M = 50 GeV, and sin 2 0w = 0.21 have been used. (23) is a tight bound even for the SU(2) × U(1) breaking coupling. The X 0 _~ d+ d - decay mode must compete with the X ° - , y y mode in order for Z ~ d + d - y to be 239
Volume 143B, number 1, 2, 3
PHYSICS LETTERS
observable. The X ° -~ 77 decay rate is given by
r(x°
77)
= ( 4 ¢ r ) - l ( f sin 2 0w + f ' cos 2 Ow)2M3,
(25)
while the X ° ~ g + g - decay rate is given by F(X ° ~+~-)=(h2/8rr)M. Therefore, the ratio of F(X ° --+ # + g - ) to F(X ° ---,7"t) is bounded by (23) as
( r(x ° < 1.6 ×
e+ e - ) / r ( x ° -, v7)} F(f,
f')G(f, f')
lO-5(B/r) 2,
(26)
where
G(f,
f ' ) = ( / sin2 0w + f ' cos 2 O w ) 2 / ( f - f ' ) 2 . (27)
The inequality (27) may be expressed in a slightly different form,
B ( X ° -+ 77) B ( X ° -+ e+e -)
> 6 × 104rZF(f,f')G(f, f').
(28)
Even with r = 1/8, the right-hand side of (28) is equal to - 103FG, where F and G are both roughly of order of unity unless there is a substantial cancellation. Since the left-hand side of (28) cannot be larger than an order of a few percent, it means that ( f + 3 . 2 f ' ) ( f sin 2 0w + f ' cos 2 0w)= 0 must hold in our sign convention given in (16a) and (16b). Such a stringent condition has arisen from the following physical requirement: The large decay rate observed for Z - ~ ~ + ~ - 7 needs both fast Z--~ 7X ° decay and a substantial branching ratio into X ° --* ~+~-. To suppress g - 2, the X -~ g+ ~- coupling must be small. However, fast Z v X ° decay requires, through the S U ( 2 ) x U ( 1 )
240
9 August 1984
symmetry, fast X ° ~ y y decay, in general. Then, fast X ° ~ T y decay dominates over X°--~t°+t*-, making Z ~ t*+~ - 7 invisible. We have found that the coupling parameters of X ° must satisfy a stringent constraint, in order to be consistent with g - 2 of the electron. The possibility that X ° is a spin-one boson may be worth considering. For a spin-one boson, leptonic couplings can be SU(2) × U(1) symmetric, the 77 decay is forbidden, and helicity conservation of the electron prevents a large g - 2 correction. The difficulty in this case is how a spin-one X ° can possibly fit in composite models where there are already spin-one bound states, namely, gauge bosons. It is yet to be seen whether composite models compatible with the magnetic moment of the electron can be actually built in a natural manner [7]. This work was supported in part by the National Science Foundation under research grant No. PHY-81-18547 and in part by the Director, Office of Energy Research, Office of High Energy Physics and Nuclear Physics, Division of High Energy Physics of the US Department of Energy under Contract DE-AC03-76SF00098.
References [1] [2] [3] [4] [5]
UA1 Collab., G. Arisson et al., Phys. Lett. 126B (1983) 398. UA2 Collab., P. Bagnaia et al., Phys. Lett. 129B (1983) 130. P.B. Schwinberg et al., Phys. Rev. Lett. 47 (1981) 1679. S.J. Brodsky and S.D. DreU, Phys. R¢v. D22 (1981) 1573. T. Kinoshita and W.B. Linquist, Phys. Rev. Lett. 47 (1980) 2236. [6] R.D. Peccei, Phys. Lett. 136B (1984) 121. [7] H. Harari and N. Seiberg, Phys. Lett. 98B (1981) 269; O.W. Greenberg and J. Sucher, Phys. Lett. 99B (1981) 339; L. Abbott and E. Fahri, Phys. Lett. 101B (1981) 69; H. Fritz~h and G. Mandelbaum, Phys. Lett. 102B (1981) 319.
ANOMALOUS
PHYSICS LETTERS
Z DECAYS
AND
g -
9 August 1984
2 OF T H E E L E C T R O N
Mahiko S U Z U K I Department of Physics and Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720, USA Received 19 April 1984 Revised manuscript received 30 April 1984
If the leptonic Z decays with an energetic photon are interpreted as cascade decays through a new particle, a stringent constraint is imposed on the couplings by the measured value of the magnetic moment of the electron. In models with a fermion E, where Z ---,eE(~E) ~ e~'/, the constraint is avoided if E R and E L are either both singlets or both members of a doublet.
Observation of wide angle photons in leptonic Z decays has attracted much attention [1,2]. It is quite conceivable that they resulted from a tail of ordinary physics accidentally enhanced by meager statistics. Alternatively, this observation may be the first hint of new physics beyond the standard electroweak gauge theory, in particular, compositeness of gauge bosons, quarks, and leptons at an energy scale above 100 GeV. In this note, assuming that these anomalous events are a consequence of new physics, we explore what conditions are imposed on theoretical interpretations when we require that the g - 2 factor of the electron be consistent with measurement [3]. The relation between compositeness and g - 2 was discussed by Brodsky and DreU in their pioneering paper [4]. Our analysis addresses to more specific possibilities that are implied by t h e anomalous Z decay experiment. Since the anomalous Z decays Z ~ ¢+d-3' occupy one quarter of the leptonic decays, it is most likely that the final state ¢+d-3' is produced through a heavy particle in cascade decays rather than in primary decays. Pursuing along this line, we come to two possibilities: Z ---~e+ + E - ---, e+ + e - + y ( - o e - + E+ ~ e - + e+ +~,), Z~y+X°~+e++e
-,
(1) (2)
where E and X ° are new particles. One can imagine E as an excited electron and X ° as a 1S0 bound state of composite models in contrast to 3S 1 states which may be assigned to gauge bosons. The two cases are studied separately. The excited electron E . The Z ~ E+ •-y decay occurs through the diagram of fig. la. The anomalous magnetic moment is produced through the diagram of fig. lb. We assume for simplicity that the spin of E is 1 / 2 instead of 3/2. The relevant couplings are defined by the lagrangian with the SU(2) × U(1) symmetry L i n t = g ( E L--O t ~ v e R ) 0 /xB v + K , ( ~ , ~ ) R O p . p ( ; ) L -0 / ~ B p _
+ I¢"(N, E)RO~/r( ; )g • 0~W ~,
(3)
where W, and B~ are the SU(2) and U(1) gauge bosons, respectively. It is assumed in (3) that the mass of E transforms like a component of a doublet, just as the electron mass. We have x' = r " = 0 if E R and E L are both singlets, and x = 0 if both E g and E L belong to doublets ,1. C P invariance requires that r, r', and ~" be real (up to a common phase). We assume C P invariance and choose them to be real. Otherwise, they would generate an electric dipole moment, in general. *] Models cannot be built with the assignment that E R is a singlet and E L is a member of a doublet, since all the couplings in (3) would be forbidden.
0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
237
Volume 143B, number 1, 2, 3
PHYSICS LETTERS
¥ Z
E"
E
9 August 1984
where K ( A , m, M ) is a quantity of O(1) defined by E
K ( A , m, M ) = l o g ( A 2 / m 2)
M4)/(m 2-
+ [ ( m 4 + 3m2M 2 -
(a)
/b)
Fig. 1. (a) Decay through E. A corresponding diagram with e ± and E - replaced by e n: and E + , respectively, also contributes. (b) Anomalous magnetic moment through E. Z may be replaced by 7.
T h e observed Z---, d + d - y decay rate r relative to the Z ~ [+ d - decay rate r = r ( z --+ e+ e - v ) / r ( z --, e + e - )
(4)
constrains the couphngs, since E ---, e~, is presumably the only significant decay m o d e of E. F r o m the calculated decay rates
X M 2 log(
m 2 / M 2)
--(2m4+
3m2M2 + M4)/2(m2- M2) 2,
(10)
for A >> m and M. The p a r a m e t e r A is a mass scale of a form factor F ( k 2) = A Z / ( A 2 - k 2 ) ,
(11)
which has been inserted in the lepton vertices. It indicates a scale of compositeness of fermions and ensures convergence of F e y n m a n loop integrals. We expect A to be m u c h larger than m z and M. Strictly speaking, all the decay rate formulas are also subject to form factor corrections, but they are negligibly small as far as A >> m z and M. We estimate A# with M = 50 GeV. The result is At* = 1.7 x 10 -8, [ l o g ( A 2 / m
r (Z - , eF. + ~S)
M 2 ) 3]
z) + 0.3]
~'- 0.75#'
K2 + ( x ' + K" cot 0w) 2
= (24~r)-m[x = + ( x ' + x " cot Ow): ] x s i n 2 Ow(1 -
r ( Z -+ e~) =
2
3
sin 2 O w ) ] m z ,
This n u m b e r must be within the limit of possible discrepancy between experiment and the Q E D prediction [5],
(6)
- 4 × lO-'°(e/2me)
where M is the mass of E, we obtain for M = 50 MeV
Ka+(K'+K" cotOw)Z=4.8r(e/mz) z.
(7)
M)M,
< 4.6 X 1 0 - 3 ( 1 / r ) .
K2 + (~¢' + x '' cot 0~ ) 2 (14)
(98)
The spinless boson X °. The decay process occurs through the diagram of fig. 2a. The diagrams of
#v = ( 1 6 * r z ) - a e x ( K " - K" tan Ow) X cos 2 0w K ( A , 0,
K(K' - 0 . 1 5 # ' )
(9a)
/~z ----( 1 6 ~ r 2 ) - ' e K ( K' + K" cot 8w)
M)M,
(13)
This is a stringent condition for r = 1 / 8 - 1 / 4 . T h e condition K ' - 0 . 1 5 # ' = 0 means that the Z and ~ loop contributions accidentally cancel each other in A F. A simple way to satisfy (14) is to forbid either of ( # , # ' ) and ~ by assigning E R and E L both to singlets or both to doublets.
(8)
X sin z Ow K ( A , m z ,
< A F < 0.
F o r A = 1 TeV and sin 2 0w = 0.21, we obtain 0 < -
A straightforward calculation leads us to a f o r m u l a for the correction to the magnetic m o m e n t a t , = J*z +
(12)
(5) (G/24v12*rz) X[1 +(1-4
238
× I¢(-~m~ ).
MZ/rn~)2(1 + 2M2/m2z)m3z,
Volume 143B, number 1, 2, 3
PHYSICS L E T T E R S
1'
¥
9 August 1984
or
~/=~'=d=l
and
~=~'=c=0.
(16b)
The observed Z - ~ d + d - y decay rate fixes the couplings through the relation (4) together with e
0
o
+
0
0
r ( z - , d + d - v ) = B ( X ° -+ e + d - ) r ( z --, v x ° ) , (17) la/
tbt
Fig. 2. (a) Decay through X °. (b) Anomalous magnetic m o m e n t through Z - X °. Corresponding diagrams with Z replaced by 3' can also contribute. The form factors are inserted at the vertices of the electron.
fig. 2b and the diagrams which are obtained from fig. 2b by replacement of Z with y contribute to g - 2. Models of this type have a common problem with the SU(2) × U(1) symmetry. The SU(2) x U(1) symmetry requires that X ° be either a singlet or a member of a triplet since it couples to Z and y. However, the coupling of X ° to lepton pairs must not be proportional to lepton masses in order for Z--+ e + e - y and # + # - 7 not to be dominated by Z - - ' r + * - y . The scalar coupling ERdLxO is allowed only when X ° is a member of a doublet, just like a Higgs particle. It is inevitable that the SU(2) × U(1) symmetry is broken in the d + d - X ° couplings. We assume for simplicity that the leptonic couplings of X ° are universal. Peccei [6] argued that X 0 is more likely to be a singlet than a component of a triplet. Since the basic feature of our conclusion remains the same even if X ° is assumed to be a member of a triplet, we present here the results for a singlet X °. The relevant couplings are defined by Lin t = f W . . " (~W ~'~ + TIqg~)X °
id-rs) dX °,
×sin220w(f-f
(15)
~=~'=c=1
and
r/=rf=d=O,
(16a)
) ( r n z2- - M 2 ) 3.
Denoting B ( X ° ~ d + d - ) by B, we obtain with M = 50 GeV (f_f,)2
= 3.4(r/B)(e/mz)2.
(19)
The correction to the magnetic moment is given by
at, = #z + #~,
(20)
#z = ( 1 6 r r 2 ) - l g h ( f - f
") sin Ow(1 - 4 sin 2 Ow)
x R ( A , mz, M ) ,
(21a)
#v = (4rr2)-leh( f sin2 Ow + f ' cos 2 Ow) × / ~ ( A , 0, M ) ,
(21b)
where / ~ ( A , m, M )
= [rn2/(rn 2 - M2)] l o g ( A 2 / m 2) -
[ M Z / ( m 2 - M2)] l o g ( A 2 / M 2 ) ,
(22)
for A >> m and M. Demanding the condition (13) on A# of (20), we obtain
') < 2.4 × l O - 7 ( B / r ) ,
(23)
where
F(f , f') = (f + 3.2f')z/(f where W~ and B~ are dual tensors, ffP invariance requires that either ~ = ~' = c = 0 or ~/= 7/' = d = 0. Without loss of generality, we choose the phase convention
p 2
(18)
(h2/4~r)F(f,f
+ f ' B ~ (~'B ~ + n'Ba~)X °
+hY',?(c -
r ( Z -~ v X ° ) = (24~rm3) -1
- f ' ) 2,
(24)
and A = 1 TeV, M = 50 GeV, and sin 2 0w = 0.21 have been used. (23) is a tight bound even for the SU(2) × U(1) breaking coupling. The X 0 _~ d+ d - decay mode must compete with the X ° - , y y mode in order for Z ~ d + d - y to be 239
Volume 143B, number 1, 2, 3
PHYSICS LETTERS
observable. The X ° -~ 77 decay rate is given by
r(x°
77)
= ( 4 ¢ r ) - l ( f sin 2 0w + f ' cos 2 Ow)2M3,
(25)
while the X ° ~ g + g - decay rate is given by F(X ° ~+~-)=(h2/8rr)M. Therefore, the ratio of F(X ° --+ # + g - ) to F(X ° ---,7"t) is bounded by (23) as
( r(x ° < 1.6 ×
e+ e - ) / r ( x ° -, v7)} F(f,
f')G(f, f')
lO-5(B/r) 2,
(26)
where
G(f,
f ' ) = ( / sin2 0w + f ' cos 2 O w ) 2 / ( f - f ' ) 2 . (27)
The inequality (27) may be expressed in a slightly different form,
B ( X ° -+ 77) B ( X ° -+ e+e -)
> 6 × 104rZF(f,f')G(f, f').
(28)
Even with r = 1/8, the right-hand side of (28) is equal to - 103FG, where F and G are both roughly of order of unity unless there is a substantial cancellation. Since the left-hand side of (28) cannot be larger than an order of a few percent, it means that ( f + 3 . 2 f ' ) ( f sin 2 0w + f ' cos 2 0w)= 0 must hold in our sign convention given in (16a) and (16b). Such a stringent condition has arisen from the following physical requirement: The large decay rate observed for Z - ~ ~ + ~ - 7 needs both fast Z--~ 7X ° decay and a substantial branching ratio into X ° --* ~+~-. To suppress g - 2, the X -~ g+ ~- coupling must be small. However, fast Z v X ° decay requires, through the S U ( 2 ) x U ( 1 )
240
9 August 1984
symmetry, fast X ° ~ y y decay, in general. Then, fast X ° ~ T y decay dominates over X°--~t°+t*-, making Z ~ t*+~ - 7 invisible. We have found that the coupling parameters of X ° must satisfy a stringent constraint, in order to be consistent with g - 2 of the electron. The possibility that X ° is a spin-one boson may be worth considering. For a spin-one boson, leptonic couplings can be SU(2) × U(1) symmetric, the 77 decay is forbidden, and helicity conservation of the electron prevents a large g - 2 correction. The difficulty in this case is how a spin-one X ° can possibly fit in composite models where there are already spin-one bound states, namely, gauge bosons. It is yet to be seen whether composite models compatible with the magnetic moment of the electron can be actually built in a natural manner [7]. This work was supported in part by the National Science Foundation under research grant No. PHY-81-18547 and in part by the Director, Office of Energy Research, Office of High Energy Physics and Nuclear Physics, Division of High Energy Physics of the US Department of Energy under Contract DE-AC03-76SF00098.
References [1] [2] [3] [4] [5]
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