Nuclear Physics B196 (1982) 378-393
@ North-Holland
Publishing Company
RADIATIVE
Lampros
DECAYS:
ARNELLOS,
William
W’-+ P*+y
AND
Z”+ P”+y
J. MARCIANO
and Zohreh
PARSA
Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60201, USA
Received
18 August
1981
We present a general analysis of the intermediate
vector boson decays W* + Pi + y and
Z”+ PO+ y (where P denotes any pseudoscalar
meson). Strong interaction effects are parametrized
by vector and axial-vector
form factors which we estimate
using quantum
chromodynamics,
bound-state
analyses and triangle anomaly calculations.
In the case of ordinary light pseudoscalars
we find that these exclusive radiative decays are highly suppressed,
e.g. T(W* + ~*r)/r(W-+
pi;,) = 3 x 10e8 and T(Z’ + ~‘r)/r(Z+
CL/.?)= 9 x 10ml’. We also examine decays in which P is a
very heavy bound-state
meson or pseudo-Goldstone
boson (such as a technion).
For the latter
case the suppression
is much less; so those decays may be observable
at very high luminosity
accelerators
such as ISABELLE
and LEP.
1. Introduction
The coming generation
of accelerators will investigate high-energy
regimes where
production
of the intermediate
vector bosom (IVBs) W* and Z” is anticipated.
In
the “standard”
SU(2)r x U(1) Weinberg-Salam
(WS) model [l] these weak interaction mediators are predicted to have masses [2-4]
38.5 GeV
mw=y-83.0*2.4GeV,
sm Ow(mw)
(l.la)
77.1 GeV
mZ=sin
where we have employed
-93.8*2.0GeV,
2&(mw)
the experimental
sin’ i&(mw)
average
(l.lb)
value [2-41
= 0.215 *0.012
(1.2)
for the renormalized
weak mixing angle (defined by modified minimal subtraction
radiative
with p = mw). The results in eqs. (1.1) and (1.2) include O(a) electroweak
corrections
and hence represent precise testable predictions
[2]. If the W’ and Z”
are found to have masses in the range given by eq. (l.l), it would represent
an
important success for the standard model.
Assuming that the W’ and Z” will be discovered in the near future and copiously
produced by high luminosity
pp, pp, and e+e- colliding beam facilities [5-71, one
can anticipate precise measurements
of their basic properties.
For that reason, we
would like to know as much as possible about the theoretically
predicted decay
378
L. Arnellos et al. / W* -, P+ + y,
modes
of these particles.
jets have been thoroughly
Lowest
analyzed
have been computed
[8,9].
examined [lo]. For example,
order
Z” + P” + y
decay rates into lepton
[8]. Even radiative
379
pairs and hadronic
corrections
to those processes
Some higher order induced decays
the decays [ll, 121 Z”+~o+~Lt+~-
(where 4’ is a Higgs scalar) have been suggested
latter case, one finds [12] for rni <<rn$
have also been
and Z’++‘+y
as ways of finding
w”+~o+Y)~8x10_5
the 4’. In the
(1.3)
r(zo + /.Lfi)
Although quite rare, we expect that such a decay rate would be observable
at high
luminosity
e+e- colliders capable of producing about 1 Z’/sec on resonance.
If this
radiative decay is observed,
it will provide a precise determination
of rn+ via a
measurement
of the photon’s energy. Somewhat more speculative
(but very interesting) is the prospect of discovering
entirely new types of particles such as supersymmetric fermions [13], technions [14], new exotic mesons [ 15], etc. in the decay
products of the IVBs.
Given the above motivations,
we have studied various decay modes of the W’
and Z”. In this paper we present a general analysis of the exclusive radiative decays
(1.4a)
w*+p*+y,
(1.4b)
ZO+PO+y,
where
P generically
denotes
any pseudoscalar
meson.
If observable,
such decays
would yield useful information
regarding strong interaction
dynamics. Furthermore,
detection of the two-body decay W’+ P’+ y combined with a measurement
of the
photon’s energy could provide a precise determination
of mw*. Finally, significant
deviations
from the standard model’s predictions
for these radiative decays may
signal new unanticipated
physics. The presentation
of our results is organized
as
follows: In sect. 2 we present a general analysis of the decays in eq. (1.4), parametrizing strong interaction
effects in terms of hadronic form factors. We discuss in sect.
3 how to estimate these form factors for light pseudoscalars
such as r+, K’, D’,
F’, rr’, 77, 7’ etc. In sect. 4 we examine
the case of very heavy bound-state
pseudoscalars,
employing
the analysis of Guberina,
Kuhn, Peccei and Ruckl [16].
Then, in sect. 5 we comment on the case of pseudo-Goldstone
bosons arising from
chiral symmetry
breaking of exotic quarks or so-called techniquarks.
Finally, in
sect. 6 we conclude with some remarks on our findings.
2. General analysis
2.1. w*+p*+*
We begin by considering
the decay of a W- boson into a pseudoscalar
meson Pwith momentum
P and a photon with momentum
k. (The case W’ + P’+ y follows
* This possibility
was pointed
out to us by G. Kane (private
communication).
L. ArneNos et al. / W*+ P’+
380
I
y, Z”+ PO+ y
P-
P-\
i
d
b,\
Y
k
;P+k
I
I
P+k;
‘W-
‘W-
(al
(b)
Fig. 1. Diagrams contributing to the decay W- + Pm+ y.
simply by charge conjugation.)
The total amplitude for this process is described by
the Feynman
diagrams in fig. 1. The diagram in fig. la illustrates
the W- pole
contribution.
To evaluate it we employ the WS model’s couplings
-ie[(2P+k),g,,
-(P+2k)&,*
-(P-
kLgJ 3
(2.la)
for the yWW vertex and
-- k
(2.lb)
2J2
for the coupling of the W- boson to the weak charged current JT. (The normalization
is such that g2/8m k = J’ 2GF, where Gr= 1.16632 x lo-’ GeV2
is the Fermi
constant.) In that way we find for the amplitude in fig. la
x w&x
- (P + 2kL&LA- 2&&--(~)IJpW+
where c@(k) and E A(P + k) are the polarization
In obtaining
E”(P+k)(P+k)*
Only the axial-vector
such that
part of JT’
contributes
(P-(P)IJr+
where
fP is the charged
pseudoscalar
(2.2)
vectors of the y and W- respectively.
eq. (2.2) we made use of the orthogonality
&“(k)k,
uwv,
conditions
= 0)
(2.3a)
=o.
(2.3b)
to the matrix
element
in eq. (2.2),
(O)lO>= $rJ’p (KM),
decay
constant
(2.4)
and
(KM) stands
for the
381
L. Arnellos et al. / W+ + P* + y, Z” -, P” + y
Kobayashi-Maskawa
decay considered
parametrization
quark mixing matrix element appropriate
for the particular
[17]. (That is (KM) is a generalization
of the Cabibbo
angle
to a six-flavor
M
=
_
a
theory
[S].) Combining
eqs. (2.4) and (2.2) we find
egfp(KM)eW(k)eA(P+k)g,,
.
(2.5)
2JZ
Turning
to the amplitude
in fig. lb, one has by LSZ reduction
Mb=
-egs~(k)E*(P+k)ri;*,
(2.6a)
2J2
(P-(P)IW’:WY+ KVIIO) ,
(2.6b)
(sum over all fermions with charge Qf, where Q, = $1 is
where J’: = Cr Q&J,~~
the electromagnetic
current. Writing out the most general expression for the Lorentz
tensor
TWA,one has
F&A= (KM)
f,(2P+k),(P+k),Fp(k2)
-H1g,*
-&P&h
s-m;
- H3P,PA - H4kFPh - H,k,k,
+ iH6ew&apP
1, (2.7)
tensor (~~~2~ = +l), Hi = Hi(k2, s), (i = 1,2 . . . 6),
where &,A0p is the antisymmetric
s = (P+ k)2 and Fp(k2) is the electromagnetic
form factor of the P- normalized
to
Fp(0) = 1. Our notation in eq. (2.7) has been taken from studies of radiative pion
decay Y+eF,y
where one encounters
the same Hi form factors [18] (actually the
two sets of form factors are related by complex conjugation).
The term proportional
to H6 is due to the weak charged current’s vector component,
all other terms in
eq. (2.7) stem from the axial-vector
current. Using the orthogonality
conditions
in
eq. (2.3), one finds for real W- and y that Ha, H,, and the pseudoscalar
pole term
in eq. (2.7) do not contribute
to the decay W-+P
+ y. Furthermore,
requiring
that the total amplitude
M,+Ms
be gauge invariant
(i.e. it must vanish under the
replacement
E&(k)+ k”), we find for-real photons (k2 = 0), the conditions
[18]
(2.8a)
H3(0, s) = 0,
HI(O, s) =fp-P
Thus, the total gauge-invariant
M,+Mb=
-4KM)
2JZ
amplitude
- kHz(0, s) .
is given by
Ep(k)EA(P+k)WdO,
+ iHe(O, s).s,AUpkaPP}.
(Of course,
(2.8b)
in this decay s = m$.) The axial-vector
s)(P.
kg,* -Z’,k,)
(2.9)
form factor H2(0, s) and vector
382
L. Arnellos et al. / W’+
form
factor
&(O,
s) incorporate
P+ + y, ZO+ PO+ y
all strong
interaction
effects.
Since
we have
explicitly extracted
the W- and P pole contributions
in our analysis, the final
amplitude in eq. (2.9) depends only on the structure of the strong interactions.
The
structure-dependent
form factors H2 and Hc have been the subject of considerable
theoretical
and experimental
interest because of their role in radiative pion and
kaon decays [19] and the connection
between H6(0, 0) and the Adler-Bell-Jackiw
anomaly [20]. However, previous investigations
involved the small-s region, while
we are interested in s = m&.
To clearly distinguish
different final-state
pseudoscalars
and emphasize
axial-vector
and vector origins, we now adopt a new form factor notation:
Hz(O, s) +
their
Ap-(s) ,
(2.10a)
.
(2.10b)
He(O, s) + VP+)
In terms of Ap-(s) and VP-(s) the final amplitude
for W- + P- + y is given by
M(w-~P~r)=-Eg(KM)FL((k)E*(P+k)
2JZ
x [V’ . kg@*-P,k,)A,~(m~)+iE,h~pk*PPV,-(m~)].
(2.11)
[For W’+P’r
the sign of Ap-(mw) in eq. (2.11) should be changed.] Squaring
the amplitude in eq. (2.1 l), averaging over the initial polarization
states of the W ,
summing over y polarizations
and carrying out the phase space integrations,
we
find for the decay rate of the W- at rest
T(W’+ P’y) = ~iI(M,‘(,V,~(~~)12+~A,~(m~)l’)m~(
Comparing
with the WS model’s
predicted
pficI leptonic
1-$3.
(2.12)
rate [8, 211
(2.13)
one obtains
r(w*+P’y)
r(W-+/.LcJ
1
= ~cY~~KM]~(]Vp-( n~&),~+,Ap-(m&)~~)rn&( 1 -$)3,
(2.14)
where LY= e2/47r = l/137. To numerically
estimate the branching ratio in eq. (2.14)
one needs to know the values of VP-(mk) and Ap-(WI&), a problem we subsequently
address beginning in sect. 3.
2.2.
ZO+PO+y
The radiative decay of the Z” boson into a pseudoscalar
meson, PO, and a photon
is simpler to analyze. In that case because the Z” is neutral, there is no analog of
383
L. Arnellos er al. / W* + P’ f y, Z0 + P0 + y
fig.
la. In addition,
assuming
that the P” is even under
the vector
part of the weak neutral
current
contributes
Extracting
a factor -ig/4 cos 13~for the Z” coupling
charge
to the weak neutral
sin2 Ow) ,
J? = 4(J? -JF’
conjugation,
to the Z” analog
only
of fig. lb.
current
J:,
(2.15a)
(2.15b)
(where
T3f = weak isospin
of fermion
M(ZO+ PO,y) =
f, e.g. T3” = $), we find
(2.16a)
-eg
EV#(P+k)F;*,
4 cos lvw
(P”(P)ITIJ”:(x)J: (O)llO)
(2.16b)
(2.16)
= icWAapkaPPV&s)
where
Vp(s) is the weak neutral
we obtain
current’s
vector from factor.
From this amplitude
the decay rate
(2.17)
e
W
Comparing
with the Z” + pii partial
width (in the WS model)
[8]
e (1-4sin28w+8sin40~),
(2.18)
W
one finds
T(Z” + POy)
nz”+
Pfi)
(Y7r
=Q
/ Vp4m~)12m~
(l-4sin20w+8sin40w)
(2.19)
Again we come to a point where numerical
estimates
require a knowledge
of
Vpo(mg). The remainder
of this paper is devoted to evaluating
such form factors
for specific cases.
3. Light pseudoscalar
mesons
We first consider the case in which P is an ordinary relatively light pseudoscalar
with mE/rn& <<1. For that situation the large-s behavior of the vector form factors
VP-(s) and V+(s) of sect. 2 can be obtained from QCD calculations
by Brodsky
and Lepage [22]. These authors actually computed the F,,~v..,X(~) form factor with
one photon “y” off-mass-shell
at k2 = s, and found*
F,~),.‘,~. (s) -+(l+o(&)),
s-00
l
Brodsky and Lepage
for this difference.
used the neutral
pion decay constant.
(3.1)
We have modified
their result to account
384
L. Amellos
et al. / W* + P* + y, Z” + P” + y
where A ~0.4 GeV is the QCD mass scale. For s very large =mf
or m$, the
corrections
in eq. (3.1) should be small and the first term then represents
a very
good approximation
(provided of course rn$ <<s). Adjusting
the coefficient in eq.
(3.1) to account
for the appropriate
couplings
V,-(s) -2,
s-03 s
in our problem,
(P- = r-,
we find
K-, D-, F-, etc.),
(3.2)
for the charged current vector form factor. The neutral weak current case is
somewhat more complicated
because of its dependence
on sin’ Bw and the mixing
between states. For a pure qq onium bound-state
pseudoscalar
it is given by
Vpo(s) -2fP31Q,](1 -4)Q,lsin2
s-m s
13,))
(3.3)
where Q, is the charge of the constituent
quark (the 3 is a color factor). For more
complicated
low-mass mesons such as the ~TO,77,n’, etc. one must include mixing
effects. Considering
the r” as a &dd - au) quark state one finds
-J?f
V$p(S) :(l-4sin2&).
s-m
s
Similarly, using
(957.57 MeV),
the quark
model
(3.4)
description
[23] of the ~(548.8
MeV)
and
(3Sa)
q=n8~os~p+~1sint9p,
q’= n1 cos BP-n8
77’
sin @p,
(3.5b)
+ dd - 2s~) ,
(3.5c)
where
ns = -&iu
ql=&iu+dd+Ss),
with 13~3: -lo”,
we find (taking f,, = f,, = fir)
-JTf
~{cos
V,(s) s+* J3s
hf
--Z{4JZ
V,,(s) s+m J%
8p(~ - 4 sin2 8,) -
cos ep(i -
AS a check on the Brodsky-Lepage
is equivalent
(3.5d)
to the dipole
2 sin’
4J2sin
ep(l-
2 sin2 0,))
,
e,) + sin ep(i - 4 sin2 e,)} .
calculation,
(3.6)
(3.7)
we note that asymptotically
it
approximation
fP
V,-(s) = 2
mv--s’
where mv is the mass of a vector meson that couples to (and presumably
(3.8)
dominates)
L. Arnellos
the Py channel.
et al. 1 W' + P’ + y,
the pion form factor
-$ .
anomaly
(3.9)
by the p meson
effects.) The result in eq. (3.9) is in good numerical
Jackiw triangle
385
For the case P- = 7~ , eq. (3.8) gives
V,-(O) =
(We dominate
Z” -, p” + y
pole and neglect
agreement
finite width
with the Adler-Bell-
value [20, 231
1
V,-(O) = -
4fJT2.
Indeed,
14%.
Since
position
the Z”+
taking
f,, = 0.132
GeV, m. = 0.77 GeV, one finds that they agree to within
the Z”+Po+
y decay rate depends only on Vpo(mc), we are now in a
to evaluate the branching
ratio in eq. (2.19). Considering
first the case of
nay decay mode, we find by combining eqs. (3.4) and (2.19) (with s = m$)
T(ZO+7r0y) ffnf2,
r(Z’+pfi)
Using
(1 - 4 sin2 Sw)2
fii = 0.132 GeV, mz = 93.8 GeV, sin2 8 w=O.215
T(ZO+ Troy)=9x
nzO-* l-L@)
nz” + w) =
nz”+ l-4)
nz” + rl’Y)=
nz”+ PLCL)
7
1
somewhat
larger
and (Y= l/137,
lo-lo,
much too small to observe. Carrying out a similar
Eqs. (3.6) and (3.7) we obtain (for &= -10’)
Although
(3.11a)
=2m:(1-4sin20w+8sin48w)’
x
1o-9
x
1o-7
we find*
(3.11b)
analysis
for the 77 and 7’ using
,
(3.12)
(3.13)
than the 7r”y decay rate, the ny and n’y decay rates
also seem to be too small to measure. For P” a ‘So pseudoscalar
of a qq onium,
the quantity fp in eq. (3.3) should be determined
by a bound-state
analysis. Leaving
fp arbitrary we find (using sin* 8w = 0.215)
r(zO+‘so+y)
QZO+PCL)
T(ZO + ‘so + y)
nz”
* A preliminary
report
+ PLCL)
= 6 x 10-8(fis,JfsJ2,for Q, = $,
= 4 x 10-“(MfTr)”
of the result in eq. (3.11b)
9
(3.14a)
forQ,=-5.
in ref. [9] was in error by a factor
(3.14b)
&.
386
L. Arnellos el al. / W’+
P’ + y, Z”+ P” + y
If for some reason, fis,/f_ is very large, these radiative decays could be considerably
enhanced.
We examine this situation further in sect. 4.
In the case of radiative
W* decays, we must estimate
before a numerical evaluation of the rate for W’+
temporarily
we introduce the parameter
Y&S) = A&)/
then using the Brodsky-Lepage
the axial-vector
form factor
P*y can be completed.
However,
(3.15)
VP-(S) ;
for V,-(s)
[22] calculation
[see eq. (3.2)] we find
from eq. (2.14)
(3.16)
What is the value of ]rr-(mf)]?
Unfortunately,
a detailed QCD analysis has not
been carried out for the case of axial-vector
currents. However, because the weak
charged current is purely left-handed,
we expect the asymptotic behavior
Iw(s)l z
To motivate
this expectation
further,
1.
we consider
(3.17)
the dipole approximation
(3.18)
where mA, is the mass of the A1 meson. This current algebra result [24] when
taken with the dipole approximation
for the vector form factor in eq. (3.8) suggests
(3.19)
which has the asymptotic
limit ]ysI&)l+
and find (using the Weinberg relationship
IrAu
This value is in good agreement
pion decay* [19]
1. As a test of eq. (3.19) we take
mfi, = 2mi)
= 4.
with the experimental
s= 0
(3.20a)
result obtained
from radiative
,y$,m(O)= 0.44 f 0.12 .
(3.20b)
Hence, we find additional
justification
for using Iypm(rn&)j= 1; although
detailed QCD analysis should be carried out.
clearly
a
Taking mw = 83 GeV, cy = l/137, f,, = 0.132 GeV, fK= 1.2f,, Iyp-(m&)1 = 1 and
the phenomenological
values [17] for the appropriate
(KM) elements, we find from
l
In actual fact the photon
is a second experimental
spectrum in radiative pion decay depends
solution r,-(O) = -2.36zt0.12.
quadratically
on y,-(O);
so there
L.Arnellosetal./
387
W*+Pt+y,Zo+po+y
eq. (3.16)
(3.21)
(3.22)
T(W* + D’y)
r(wT(W’+
,
(3.23)
.
(3.24)
= 1 x 10-9(Mf?r)2
+ #ufi&J
F’y)
T(W_+cc.v,)
Unless f~ >>f= (which seems unlikely)
small to measure.
= 2 x lo-“(f&J2
all of these branching
4. Very heavy hound-state
ratios appear
to be too
mesons
In sect. 3 we found that the radiative
decays of the W’ and Z” into light
pseudoscalars
were suppressed by a factor (fp/m)’ where m = mw or mZ. Physically,
this suppression
factor is due to the small part of the three-body
(qlq2y) phase
space in which one constrains
the system by requiring
that the qlq2 form a
pseudoscalar
bound state. For very heavy quarks, we expect less suppression
since
the initial q1q2 pair have a smaller relative velocity and are therefore
somewhat
more likely to bind.
Since the top quark has not been observed at the highest PETRA energies, one
has the bound mt> 18 GeV. Hence bound states such as ti may have masses on
the order of imw. In addition there may be a fourth generation
of very massive
fermions [21] which could give rise to new very massive pseudoscalar
mesons. In
this section we address the following question: What are the radiative decay rates
of the W’ and Z” into very massive bound-state
pseudoscalars?
For the above problem, our general decay rate formulas in eqs. (2.14) and (2.19)
are still valid; however, we can no longer use the asymptotic form factor calculations
of Brodsky and Lepage to evaluate them. (That is because mE/m& is no longer
negligible and we do not know the value of fp appropriate
for heavy quark systems.)
One possibility
is to use a bound-state
wave function approach to estimate the
vector and axial-vector
form factors. Such a program has already been carried out
of the radiative
by Guberina,
Kuhn, Peccei and Riickl [16] in their examination
decay Z”-, y+ heavy quark bound state. Using their results for the ‘So onium
(which corresponds
to the pseudoscalar
case) one has
, vpo(m2z)12
21to.72GeV)mP
IQ,12(1 -4lQ,l
(mg-m2p)2
sin2
Bw)*,
(4.1)
L. Arnellos et al. / W* + P’ f y, Z0 + Pa + y
388
where Q, is the charge (+$ or -4) of the constituent
quark in the qq onium bound
state. Assuming
approximately
the same bound-state
dynamics
for a charged
pseudoscalar
heavy meson we find
(0.02 GeV)mp
lVP-b&)1*
Using
the estimates
=
(&:,
_
42
(4.2)
.
in eqs. (4.1) and (4.2) and assuming
IYP(&)]2=
1, [KM+
1
we find from eqs. (2.19) and (2.14) (with sin2 I% = 0.215)
T(ZO-+POy)
for Q,=$,
(4.3a)
for&=--$,
(4.3b)
T(ZO + CLCZ)
T(W’
-9 P* y)
r(w-
+ /_LLa,>
These ratios are maximized
(4.4)
by m&/m:
and rn;-/m&
= 4 which corresponds
to
(4.5a)
Qq=-4,
=2x1o-6,
(4.5b)
(4.6)
The branching ratios in eqs. (4.5) and (4.6) are considerably
larger than those found
for light pseudoscalar
mesons. Unfortunately,
even at high luminosity
accelerators
such as ISABELLE
and LEP where one expects between
107-10’ W’ and Z”
bosons to be produced during a year of running, these branching ratios imply only
about l-10 radiative decays per year. Such an event rate is probably too small to
measure; it will be masked by competing background
radiation [ 161.
5. Pseudo-Goldstone
bosons
Dynamical
symmetry
breaking
schemes such as technicolor
[25] (also called
hypercolor
in the literature)
and exotic quark models 11151 introduce
new very
strongly interacting fermions which through their condensates
break the local gauge
symmetry and provide masses for the W’ and Z” bosons. Chiral symmetry breaking
in this new quark sector gives rise to a plethora of pseudoscalar
mesons which have
very interesting phenomenological
implications.
For example, the so-called minimal
extended technicolor
model [26, 271 contains eight distinct techniquarks
U”, D”, E, N ,
a = 1,2,3,
(5.1)
L. Arnellos et al. J W* +
where U and D form ordinary
These techniquarks
SU(3), color triplets
have standard
x
SU(2),
389
P* + y, Z”+ PO+ y
while E and N are color singlets.
U(1) quantum
numbers
(i.e. they form
nf, = 4 left-handed
isodoublets
and 8 right-handed
isosinglets)
and each species
comes in N,, technicolors.
Technicolor
forces are the source of their superstrong
condensate
bindings. The techniquarks
in this model have an SU(8),_ x SU(8)R x
U(1) chiral symmetry which is dynamically
broken by their condensation
to SU(8) x
U(1). This breakdown
of chiral symmetry
gives rise to 63 would be Goldstone
bosons. Three of these become the longitudinal
components
of the W’ and Z”,
thereby endowing them with mass. The remaining 60 pseudoscalar
technions acquire
masses as a result of their higher order corrections
and are therefore
called
pseudo-Goldstone
bosons (PGBs). From our point of view, the most interesting
of
the PGBs are four relatively light color singlets which are thought to have masses
in the vicinity of 10 GeV. (There is some uncertainty
regarding their actual mass
values; for our considerations
we need only assume that they are much lighter than
the W’ and Z’.) The other 56 PGBs are very heavy (see ref. [27]) and carry ordinary
color; we will not discuss them in this paper. In terms of the techniquark
states in
eq. (5.1), the four light technions are given by [27]
pi = (P’)’ = (U”D” - 3NE)/J12N,,
P3 = [UaUa -D”D”
,
- 3(NN - EE)]/J24N,,
P” = [U”U” + D”D” - 3(NN + EE)]/dm
where
a = 1,2,3
the color index is summed
(5.2a)
,
,
(5.2b)
’
(5.2c)
over. (The P” state in eq. (5.2~) differs
by the factor of 3 in the brackets from the corresponding
state given in ref. [27].)
The technicolor
index of- these fields is implicitly summed over to form technicolor
singlets; hence the l/dN,,
normalization
factor. In general P3 and P” may mix;
however, we ignore such effects. Our primary concern is to illustrate the approximate
magnitude
of the decay rates for Z” + P” or P3 + y and W” + P’ + y that one should
expect in technicolor models. We do not take the specific model under consideration
very seriously; however, some of its basic properties are rather general features of
such theories and may turn out to be correct.
To determine
the rates for Z”+ P” or P3 + y and W’+ P*+ y in the extended
technicolor model described above, we must compute the induced technion-photonW’ or Z” effective coupling (i.e. we need to know the vector and axial-vector
form
factors introduced
in sect. 2). The vector form factor is obtained from techniquark
triangle diagrams.
quark-antitechniquark
anomaly condition)
To compute
coupling
that loop effect, we employ the technion-techni(determined
by PCAC and the Adler-Bell-Jackiw
-
h&‘Jcdf~,
(5.3)
where
350 GeV
fp=
Jnf,
= 175 GeV
(5.4)
390
is the charged technion
L.Arnellosetal.
decay constant
/ W*+P*+y,Z’+P’+y
and nf, = 4 for the model under consideration.
(The numerical
value of fr is determined
by the requirement
that mw and mZ be
83 and 93.8 GeV, respectively.)
The techniquark
mass m4 in eq. (5.3) is a constituent
mass -0.5-l
TeV. Because the constituent
quark mass is the largest mass in the
triangle loop calculation we are considering,
we can drop all dependence
on external
masses. In that case the evaluation
of the triangle diagram becomes trivial. Using
the basic coupling in eq. (5.3) and the normalized
technion states in eq. (5.2), we
find that the vector form factors appropriate
for the decays Z” + P” or P3 + y and
W*+P*+
y are
(5.5a)
( Vp$m$)12 =
N$ (1 - 4 sin* ew)*
6rr4f;
IVdm$)l’ = 42
(5.5b)
7
N:f
(5.5c)
127l fP
Similar results have been previously obtained by Ali and Beg* and Ellis, Gaillard,
Nanopoulos
and Sikivie**.
Taking sin* 19~ = 0.215, NC, = 4, and fp = 175 GeV, we find from the general decay
rate formula in eq. (2.19)
(5.6a)
(5.6b)
As noted by Ali and Beg, the decay Z”+ P3 + y is suppressed
because of the
(l-4
sin* ow)* factor in eq. (5.5b). [The same suppression
occurs for Z”+ 7r”+ y,
see eq. (3.11a).] The rate for Z”+ PO+ y is somewhat larger; but still probably too
small to measure.
However,
since the decay rates are proportional
to N$ and
ratios
increase approximately
like nr?, one can expect somewhat larger branching
in a more realistic model where NC, and nf, are presumably
greater than 4. An
order of magnitude increase in the Z” + P” + y rate would render it just about equal
to the radiative Higgs decay rate in eq. (1.3). Therefore,
we believe that the decay
Z” + P” + y may be observable
at a high luminosity
e+e- machine such as LEP. At
the Z” resonance
the signal for such a decay will be a monoenergetic
photon with
E, = (m$ -m$)/2mz.
* Ali and BBg [28] estimated the rate for Z” + any PGB + y in the framework
** Our results are a factor of 4 larger than those in ref. 1291.
of a different
model.
391
L. Arnellos ef al. / W’ + P* + y, Z” + P” + y
For the decay W’ + P’ + y, there is potentially an additional
contribution
the axial-vector
form factor A,-(IX&).
From eqs. (2.13), (5.5c), we find
T(W’ + P* + y)
T(W_+
What
is the value
is induced
/.&)
of Iyr-(m&)l
by SU(8) symmetry
= 2 x l&l
for this PGB?
.
+ Iyp&z&)l’)
We expect
from
(5.7)
I-yP-(m&)12<< 1, since y
breaking
effects. (It should certainly be smaller than
the usual current algebra result Iyrm(0)l = rnE/rni, = $ and thus fairly unimportant.)
Since this branching
ratio also grows like N: and n; (approximately),
2 x 10m5
should probably be considered
a lower bound on the branching
ratio in eq. (5.7).
Therefore,
it seems that the rate for W’ + P* + y will be large enough to measure
at a high luminosity facility such as ISABELLE.
6. Conclusion
We have presented
a general analysis of the radiative decays W’ + P’ + y and
Z” + P” + y. The results are applicable to any pseudoscalar
meson. For some of the
specific cases considered,
our numerical estimates of their relative branching ratios
are summarized
below
T(W’+
7T++ y)/r(w-
+ /C) = 3 x lop8
r(W’-+
K* + y)/r(W-
+ p;) = 2
X10p9
rtw'+D*+ y)jr(Wp+K+5
rcw'+ F'+ y)/r(W-+
X lop9
ordinary
light pseudoscalars,
CL;) = 1 X lo-'
r~w~-,~so+y~~r~w~~cL~~I-i~iO-~~heavy bound state,
r(w*+ P*+ y)/r(w-+cL+2xi0P)
PGB technion;
r(zO-J+ y)/r(z"+P~)- 9x1~-‘o
r(zO+q+y)/r(zO+p+
7x1~r3
ordinary
light pseudoscalars,
r(zO+77~+y)/r(zO+pfi)==
1~10-7 I
r(zO+l~~+~)/r(zO+~fi)= 3xl~-6j
heavy bound
state,
r(zO+ p3+ y)/r(zO+Wfi)=2Xi0-6
PGB technions.
r(zO+~O+~)/r(z~+~fi)=7~10-6 I
For ordinary light pseudoscalar
mesons such as the r*, r”, 7, etc. these two-body
radiative decays are highly suppressed and most certainly unobservable.
In the case
of very heavy bound-state
pseudoscalars
with masses -50 GeV, the branching
ratios increase significantly;
but the event rate for such decays is probably still too
L. Arnellos et al. / W*+ P’+
392
small to distinguish
from background.
Finally,
y, Z” + P” + y
our examination
of the PGB radiative
decays in the simplest extended technicolor
model suggests that a year’s running
at ISABELLE
(producing
-6 x lo7 W’ bo sons) will give rise to at least 100 W’+
P*+ y events (assuming, of course, that PGB technions
actually exist). We also
emphasize that in a bigger more realistic technicolor
model, the branching
ratios
T(W* -, P*+ y)/r(W-+
~Lyll) and T(Z”+ PO+ r)/r(Z’+
CL&) may turn out to be
significantly
larger than our estimates since they grow approximately
like N$n;.
An order of magnitude
increase in Z” + P” + y would make its rate comparable
to
the radiative
Higgs decay Z”+ &JO+y and presumably
detectable
at LEP or
ISABELLE.
To observe any of these radiative decays will require good photon
detectors which allow one to trigger on the very energetic photon emitted.
Of all the two-body radiative decays of the W’ and Z” considered
in this paper,
only those containing
PGBs seem to have a viable chance of being detected.
Fortunately,
those modes are also the most interesting and certainly worth searching
for.
We thank M.A.B. Beg, G. Kane, A. Mueller,
and F. Paige for comments
and conversations
related to this work. W.J.M. and Z.P. thank the members of
Brookhaven
National Laboratory
for their hospitality during the completion
of this
paper.
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