PHYSICAL REVIEW 0
14, NUMBER
VOLUME
Asymmetry
in
e+ e~K++ K
nnd
9
q~n+y+
1
NOVEMBER 1976
y
M. P. Gokhale, S. H. Patil, and S. D. Rindani
of Physics, Indian Institute of Technology, Bombay 400 076, India
(Received 19 January 1976; revised manuscript received 23 May 1976)
Department
y+y~K++
The final-state interaction for the ofF-shell-photon process
K is discussed in terms of an
"intermediate representation" where one of the photons is on shell. This allows the evaluation of the
K from the two-photon intermediate
absorptive part and the soft-photon contribution for e+
state. The asymmetry due to the interference between the one-photon and the two-photon amphtudes for this
process, near the f'-resonance position, is found to be quite large, but goes through zero. Generalization of
the analysis of the final-state interaction in the s-wave state to SU{3) allows us to predict a decay width
I {q~n. + y+ y)
05 keV, which is in good agreement with the experimental number.
e~K++
-0,
I. INTRODUCTION
The analysis of ee annihilation is important for
testing the validity of quantum electrodynamics as
well as for the observation of small but significant
effects due to weak neutral currents' ' and higherorder radiative corrections. ' ' In particular, the
asymmetry' ' resulting from the interference between the charge-conjugation C = —1 hadron states
produced by the one-photon intermediate state and
the C =+ 1 hadron states produced by the two-photon intermediate state is of considerable interest
because of its enhancement due to
(i) the infrared effects and
(ii) the final-state interaction near the C =+ 1 po-
sitions.
These two effects may produce a large asymmetry,
of the order of 20%%uq, near the 0=+1 resonance positions, in contrast to the generally expected
asymmetry of the order of the fine- structure constant n, i.e. , about 1%%uo.
In a recent note, ' which we will denote as (I), we
discussed the results of an analysis of the asymmetry due to the interference between one-photon
and two-photon amplitudes for
e+e-m'+n
8+ 8 ~77
+ 71' + 7T
(2)
J
in the region of the C =+ 1 resonances with
we found that the asymmetry near
the f-meson resonance position for process (1),
and near the A. „A., meson resonance positions for
process (2), is about 15-20%%ua, but goes through
zero. These are, indeed, large effects and should
=0, 1, 2. There
be easily observed experimentally.
In the first part of our present note, we analyze
the asymmetry due to the interference between onephoton and two-photon amplitudes for the process
e(p, ) + e(p, ) —R'(a, ) + Z-(u, )
(3)
14
in the region of the f'-meson resonance position.
Though the techniques we have used for the evaluation of the two-photon amplitude here are similar
to those used in (I) for processes (1) and (2), there
are some major differences and improvements.
These are as follows:
(i) The R-meson mass is not negligible, unlike
the p-meson mass which is quite small and could
be neglected at the energies under consideration.
(ii) In (I), the final-state interaction for the softphoton amplitude was introduced for processes (1)
and (2) in analogy with the treatment for the onshell photon amplitude. While this is plausible, it
does not follow logically, and requires a detailed
treatment in terms of the off-shell invariant amplitudes for a rigorous justification. Here we have
analyzed the final-state interaction in terms of an
intermediate representation for invariant amplitudes, with only one photon off-shell. Use of such
a representation is valid for the discussion of both
the absorptive part of the amplitude as well as the
soft-photon contribution to the amplitude.
(iii) We have calculated the radiative corrections
leading to the K'K state with charge conjugation
C =+ 1 as well as C = —1. However, the radiative
corrections leading to the hadron state with C = —1,
which were neglected in (I), are calculated only in
the soft-photon limit. Any improvement over this
would require a very sophisticated treatment of the
strong interaction.
The main result of the analysis is that the asymmetry in the forward and backward events for process (3), near the f'-meson resonance position, is
of the order of 35%, but goes to zero within about
half a width from the resonance.
In the second part of our note, we consider the
decay rate for the process
g~7t +p+p
In some of the earlier attempts' to evaluate this
decay, one used either the vector-dominance mod2250
ASYMMETRY IN
14
e+ e ~ E'+ E
AND
q
~
n
+
y+
2251
y
el or an effective Hamiltonian. These models lead
to a rather small rate for this decay, though the
latter model, which is cutoff-dependent, can give
a larger rate. Here we assume that the two photons are in an s-wave state. We relate the amplitude for process (4) to that for y+y-R" +K in the
I = 1 state by crossing and SU(3) symmetry, which
is then evaluated by saturating it with the 5 meson.
We predict a partial width
I'(q-m +y+y) =4. 8x10 ' keV,
which is in good agreement with the average experimental value' of 8. 15&& 10 ' keV, though our
prediction tends to support some of the larger" exvalues.
perimental
(0)
e(p, )+e(p2) K'(k, )+K (k~)
With the assumption of vector-meson dominance
and &u-$ mixing, the one-photon amplitude for proII. ASYMMETRY
IN
cess (3) is
T, = e'
mp
m p ' —s-~I' Pm P
m
-KK .
J
with
1
FIG. 2. The diagrams for (a) vacuum polarization,
and (b), (c) vertex corrections for the process ee
evaluation of this asymmetry, we will need to calculate
(i) the two-photon amplitude corresponding to
Fig. 1, including the I = 0, = 2 term for the K'K
state with the final-state interaction, and the background amplitude,
(ii) vacuum polarization and vertex corrections
due to the electromagnetic interaction correspond-
v(p, ) ( $, —P, ) u(p, ),
2
(b)
my cos 9
mg' —s-zI'~m~
sin'8~
(7)
where s =p', p = (p, +p, ), m, m&, and m„are the
masses of the p, p, and u& mesons, respectively,
I'@, and I' are the corresponding widths, and
8„ is the cu-P mixing angle. The region of major
interest for the analysis of the asymmetry in this
process is around the position of the f' resonance
with I = 0 and 2 = 2, which is produced through the
two-photon intermediate state. Therefore, for the
I",
Fig. 2, and
(iii) the bremsstrahlung contributions, corresponding to process (3), accompanied by the emission of a soft photon, as shown in Fig. 3. In the
soft-photon approximation, diagrams with the zryy
point interaction do not contribute to (ii) or (iii).
We begin with a calculation of the two-photon
amplitude. In the following calculations, the electron mass is taken to be small compared to s' '.
The two-photon amplitude
Taco -Photon amplitude.
corresponding to Fig. 1 is
ing to
/
/
/
(o)
FIG. 1. Diagram for the two-photon amplitude for the
process ee K+K .
(b)
(c)
FIG. 3. The diagrams for the bremsstrahling
es associated with ee K+E
process-
P. GOKHALE, S. H. PATIL,
M.
2252
ie
T2
~
s4
(21)')
dq, dq2
J
q1
2
q2
x (P
where Q =-,'(k, —k, ) and B„B„andB, are the invariant amplitudes. The helicity amplitudes" are
related to these invariant amplitudes by
P)
™
(ql +q2
2
1
)y„yp(P,
)B„„,
1
where m is the electron mass and
the off-shell photon process
(8)
= —q1'q 2B1 —2 &1 I'(q1
1
I
B„,describes
+q, 'f(l,
T1
K" (k, ) + K (k, ).
y (q, ) +y„(q,)
S. D. RINDANI
AND
The general structure of B„„is quite formidable
with five invariant amplitudes. ' However, there
are two parts of T2 which are important for the
discussion of asymmetry in K R production for
process (3). One is the absorptive part of 7'„resulting from on-shell photons, which gives the
dominant contribution to asymmetry at the C =+ 1
resonance positions. The other is the soft-photon
contribution, which is important because of the enhancement due to infrared effects. Both of these
contributions can be evaluated in terms of what we
call an "intermediate representation" of B„„,
where one of the two photons is on its mass shell,
say ql 0 This representation has the form
1
= 2q1'q2~
)&
q
q 2g»
1
—q.
(&2)„„=(q2'Q)'z,
,q
+ s'~'(Q, ~k, ( sine')B, ],
~
ql
Qq2)&q2
q2 Q)(q11
&
) (0
) =-
(q2), I&, =2(s
= j2I = (s
14&I
(0,
l
I
1
*, 0),
o
q2O)
(2k. -q1). (-2k1+q. ).
Ql
are
helicity amplitudes
~k, ~'(
sin'0')
)
)
)
—2
+
Ql
, =e'g(e, ') (2, ('(sis'0')
(
.
—4u')'~', q,
—q2')/2s'~', )(), is the
kaon mass, and 8' is the scattering angle in the
center-of-mass (c.m. ) frame. Here we have used
the representations
«y„(q1) andy,
=(s+q, ')/2s'~'
exchange diagrams and the sea-gull diagram,
which we designate by I3~„, is
2
1
&',
~
of T describe the helicities
where the subscripts
for the photon polarization vectors, where the subscripts denote the helicities.
The Born approximation for B„„due tothe kaon
&
(2»-q1). (-2k2+q2).
r, , = —e'g(q2')
cos g ' pe
sin'0
1
e.(q2)=(q2') "(-1421 o
q2'Qq22Q(
+q2 q1'Qk2p
and the corresponding
1
(10)
2
2.
(sin 8 )B2
(12)
—q1'q2Q„Q.
ql q2Q))q2
' cos'8') B,
1&& &
.
+q2'QQ)(q1(&
(f3)2
I
f
21~
0
21|1&
$, (co«')B„
/
1/2
~
s,(e,
(f 1)))
q, sin'8'+
1
1
71
e
1
slee
For incorporating the effect of the
Ql
(14)
91 02 )
0
f'
2
0
)
meson, one first obtains the
8=2 projections
of these amplitudes:
ASYMMETRY IN
14
T, , (J = 2) = —,'(3 cos'8' —1)
T, , (J=2) =
'TY2
sin'8'
" -1
~ K'+
d cos8(3 cos'8 —1)TB
d cos8( sin'8)
T, ,(J=2) ='~ sinG' cos8'
e + e
AND q
K
~
go +
y+
2253
y
„
(15)
TB,
d cosG( sinG cosG) TB,
These integrals are carried out, and then, using relations (11), one obtains the corresponding
amplitudes
B„B„andB,.
invariant
%e define
B,(J=2)=-a, (s)+a, (s) cos'8',
B,(J = 2) = a, (s),
B,(J = 2) =- a, (s) cosG',
for which the Born approximations
15e'g(q, ')s
2( )
B( )
,
30e'g(q, ')s
2)2(~
(s
4 2)
30e g(q2 )8
(s —q. ')'I&,
I
a„a„a„a,are
(4g' —s/2)
10
Oe'g(q2'),
for
2p'(p.
s(-', q, '+ 3p, ')
„s
r
2fk
f2
5s
8fk, f'
'+ 4 f%, f'/3)
(~zp'q2'+ sq2'+ 2s fk, f'+
2p'
1/2fk
s'/'+2fk,
f3
p'(s+ p')
(s —q, ')'
f
a, (s) =
~,j.
a, (s),
I
' 2)2 a, (s),
(s
2
s'/'+
2 f%,
s- ~ and
f
f
s'/'+2fk,
f
for the amplitude for KK -yy scattering with the
KK intermediate state, in the Logunov-Tavkhelidze
The effects of the final-state interacequation.
tion in both these approaches are rather similar
though there are some relatively minor differences
"
( )]I~
( )
1
t' Drr(s')
i
The Born approximations
as 1/s for
3sp')
2fg
—q2')'
i/2 a, (s).
fg xl s
(s
a2(s) =
f
between them.
In the 8-matrix approach, one writes" dispersion relations for [a&(s)]I=oDr&(s) where [a,. (s)] =
are the 1 =0 projections of a, (s) and D«(s) is the
D function for the I=0, J= 2 partial-wave KK
scattering amplitude. Approximating the left-hand
discontinuity by its. Born approximation, we get
il
(s —q,',')'
1/2
2 fk,
2s' 'fR f'
The analyticity properties of these Born terms indicate that the amplitudes which are free from
kinematic singularities, and which are suitable for
writing partial-wave dispersion relations, are
a, (s) =
s'/'+
for these amplitudes go
do not contain undesirable
threshold factors.
The introduction of the final-state interaction in
the I =0, J=2 state is considered through two approaches. In the approach which is discussed
here, one writes dispersion relations for the Smatrix elements and uses the N/D representation
for a suitable partial-wave scattering amplitude
for KK scattering. In the other approach, which is
discussed in the Appendix, we analyze the solutions
Im[a; (s')]
s —s
since the a,. (s') are quite steep near
may set Dr+(s') =Dry(0) and obtain
Now
[a (s)]I=0
D ICE(
-(0
DrB(s)
[aB(s)]I&
J
s'= 0,
(2o)
the I = 0, = 2 partial-wave KK
scattering amplitude by the f'-meson resonance,
these expressions lead to relations between
[B,(J= 2)]I~ and their Born approximations
[BB( 2) ]t=o
On saturating
J
we
S.
M. P. GOKHALE,
2254
(21)
1
+—
= J3
—
--
2
PQfl—
—
---
—
2 rn&i' —s —im&&I'&
—1
xP (t, )„,B, (v=2),
'.
.
abs& =
d gj.d 0 ~ P
2
x~(P. )~.
(22)
9'z
C2 ~ 0'i
1
)
~
rP
Cxo
&
9'2
&2o
my
t
2,
«(P. ) 2(
,
F
[(4)) vB;(&=2)+(t, )„,B', (g=2)],
)
(23)
(IV). Finally, carrying out the in-
i-
where B; (J =1) are obtained from (16) using the Born approximations
tegrals, we get
0. 6 n m&, 'cosa
abs T2=e'v(P, ) (|I{, t{{,) u(P, ) —
s 2myl yl
s
myt
-
for s=m&, , where n is the fine-structure
cess (3).
Soft-photon contribution.
2 'te
( 2)lp
(2){)4s
constant,
~ 2(@&
~ )
&
(p, ) r. P')~)) u(p,
where sp stands for soft-photon contribution.
{r,)~=e'
' V{y, ){k', -g, )u{)t,)
~
6t
'
(24)
is the scattering angle in the c.m. frame for pro-
in the soft-photon
) B',
.
.
limit is given by
...
myI
'-
—,
The evaluation
!n
(
s t +2~k~~cos8 -+
099 cos08
x ln
s't' —2 k, cos8 0.
~
and
The leading contribution
d~ qj
s
———
' In
' +—
1/2
T
'.
m
m
——1+
3
(26)
(26)
~y'
we include only the electron-loop
(27)
[Fig. 2(b)] has been calculated before
s~/
— ln
'., ln —
+—
'
a
)
.
The electron vertex correction
Tv(ee) =
F-- 1 p ( t, )„„B',. (z = 2)
of the integrals gives the result
where my is a small mass assigned to the photon.
Vacuum polarization meed vertex corrections. For vacuum polarization,
diagram [Fig. 2(a)] which gives'
T =
l4
where the (t,. )„, are given in (10), Bs„ is given by
(13), and B{s(J= 2) are obtained from (16) using the
Born approximations (17). This B „may be substituted in (8) and the integrals carried out. In the
integrations, we will be primarily interested in
the contributions to the integrals from two regions
which are important for the discussion of asymmetry in process (3). These are the absorptive
part and the soft-photon contribution.
Absorptive past. The absorptive part of the twophoton amplitude T2 is generally smaller than the
soft-photon contribution to (8), but gives the dominant interference term at the resonance position
s=mf, ', since the resonant soft-photon term is out
of phase with the one-photon term (6) at s =n)z, '.
The resonant absorptive term is obtained by using
Cutkosky rules'4
where m&, is the mass of the f'-meson resonance
and I'&& is its width. We assume that the finalstate interaction in other states, i.e. , 0 =2, I =1,
andt2, I =0 or 1, is negligible, so that the amplitudes in those states may be represented by
their Born approximations.
We then have for process (9)
J3
S. D. RINDANI
H. PATIL, AND
s'
—
m
m
— + 1 —21n
x/a
——
m
ln
m
mr
and
is'
T, .
(28)
The kaon vertex correction is complicated by the strong interaction. However, in the soft-photon limit, the
KK intermediate state dominates. The amplitude corresponding to Fig. 2(c) is
Tv(KK) =
2s,
ln
'
—-- -1 I, +I,
tBy
T, ,
(29)
ASYMMETRY IN
e+ e ~ K'+ E
AND
~
go +
T
.
q
y+
2255
y
where
2
Ia =2
1—
S
ln
[S(S —4 2]1/
S12 —(S
4+2)12
—
4g2)l/2
+(S
S
. ~(1+F2)ln[x2j(1-x2)]
x'
1
b
—[(s —4p, 2)js]
"o
limit
In the soft-photon
.=
T (KK)
my~o
this gives
—
1n
w
",
2p
my
'.
1+
s —2p,
[s(s —4g )]
The differential
The asymmeh'y.
tions (Fig. 3), is given by
Sl/2
/
ln
(S
4"2)1/2
,
sl/+(s —4')1/
.
cross section to process (3), along
(30)
with the bremsstrahlung
correc-
I
d cose
g coso
aT
g~T 1+b S+2(T2) 2+ Ted+Tv(88)+Tv(KK)
gj Vg)
~
+5 (ee)+
5
(KK)+5 (eK),
(3&)
where
~n'
de,
sin'e, s -
4p.
'
'/'
(32)
processes. In the soft-photon approximation,
are the corrections from the bremsstrahlung
and where 5
these terms are
5'(ee)=
)+(~ —2ln
(ln
)1n(
(33)
)
as given in Ref. 1, where ~E is the maximum energy of the soft photon emitted, and
6'(KN
=
2~ my
—
ln
zE
1+
w
5a(eg
=
s —2 p. 2
[s(s —4p')]'/'
ln
s'/' —(s —4p, 2)'/2
s' '+(s —4y. ')'/'
s'/'+2~kl(' cos8
4u
—
+0.9 eosg
ln —Re ln, /,
nay
PPZ~t
—
—
Vlyt
—8
g
(35)
Wf&I y&
I
The asymmetry in the K K distribution is due to
the presence of the "C=+1" terms, abs T„(T2)„,
and 5 (eK). lt is conveniently represented' in
terms of an asymmetry parameter
&-B
B
where I' and B are the forward and backward
events, respectively. The expected behavior of
the asymmetry parameter near the f'-resonance
position is shown in Fig. 4. The plot is for m&i
= 1516 MeV and I"& = 40 MeV.
Discussion. The asymmetry in the K'K distribution for the process ee- K K is predicted
to be quite large, about 35% near the f'-resonance
position, but goes through zero a little below the
resonance, around s X/2 = m/t —1/i j4. The asymmetry in this case is larger than in the case of
7t'7t production, ' but is sharper in distribution.
This is to be expected, since the f' meson has a
smaller width than the meson. At the resonance,
i. e. , s'/'=m i, the asymmetry isA =0.28. Our
numbers for the asymmetry at the resonances,
f
m'm
and K"K production, are in good
agreement with the estimates' based on the quark
model. They agree with the prediction' for m'w
production based on the universal coupling of the
meson but differ from the prediction for K'K
production based on the universal coupling of the
f' meson. We regard our results as a support for
both for
f
0.40.3
0.2-
0.1-
A~
—
F-B
FW
0
i440
&460
&480
-0.1-
-0.2
-03.
-04.
-O.S
FIG. 4. The
function of s~
as~~etryA
in ee
E+E' plotted as a
P. GOKHALE, S. H. PATIL,
M.
2256
f
the universal coupling" of the meson but a nonuniversal coupling for the
meson, which would
be plausible if were primarily a singlet.
A detailed observation of the asymmetry is important for understanding the properties of and
and the radiative corrections to processes ee
f'
f
f'
f
K'K,
m
S. D. RINDANI
AND
14
s7, (s) is free from kinematic singularities, and
D3(s) is the D function for pseudoscalar-pseudoscalar scattering in the =0, symmetric 8 state.
Approximating the left-hand discontinuity of v2(s)
we get
by its Born approximation,
J
s D, (s )Im22(s )
s —s
1
m
8
8
7r
t
III. DECAY RATE FOR q~no+y+y
The decay process q
-
- n'+ y+ y is related
(41)
to
~„(q,) +r.(q, ) ~'(&,) +q(~, )
(37)
crossing. Furthermore, since the phase space
for the decay is quite small, it is possible to assume that the two photons a, re in the J =0, s-wave
state.
The representation (10) is valid for the process
(37) also, except that B, and B, do not contribute
to the =0 amplitude. We decompose B, into the
various SU(3) (8&&8) states and write for the J'=0
by
J
amplitude
8
279
where 33 and 7» are contributions to (37) from the
symmetric 8 and the 27 states of SU(3), respecis related to the =0 projection of
tively. The
the helicity amplitude T» by
J
8,
1
B, = ——
S
/
I
T, , dcos8.
Born approximation
- K+K
—
it is represented as a pole at
for integration ove r s' & 0. Furthe rmore,
we note that there should be a zero in D, (s) at
s=m6 ', where m& is the mass of the 5 resonance
and we require that D, (s) —constant for s
These features together determine the form of
D, (s) uniquely,
s
sp
&0
—,(
—
—+
k, 'sin'6
&
S3/2(S
dcos8
e i12
—4 i1 ) 1/2
~
"),
(42)
where we expect sp to be of the order of 2mt, ".
We assume that the final-state interaction in the
27 state is negligible so that 7»= 7», so that we
have
1
msD, (s)
„s'
" s'D, (s') Im 7, (s')
—s
(40)
Sl
(S —4i12)1/2
S 1/2 + (S —4 i1 2) 1/2
2
where the factor W3/5 comes from the SU(3) projections, and p, is some average pseudoscalar
octet mass, taken to be approximately the kaon
mass. The final-state interaction is introduced by
writing dispersion relations for s3.3(s)D2(s), where
dS —73 S),
(43)
(39)
has the quantum numbers J~ =0', and which decays
mainly into mg. For introducing the final-state
interaction, we first obtain the Born approximation
for 7„which may be obtained from, say y+y
73 =
jf function, i.e.,
1
for process (37) is
zero, i.e. , B, =O, though the 7; themselves are
not zero. Therefore, it is the final-state interaction which will give a nonzero amplitude for (37).
One expects a dominant final-state interaction
for the process (37) in the symmetric 8 state,
since there is the 5 resonance' which presumably
Now the
For the integration over the left cut, i.e. , s'&0,
we represent the D function as a pole at s' = sp & 0.
This is similar" to what is frequently done for the
D, (s)=1
B„.=(e, e.g„, -~.„q,.)B„
j.
d,
where we have used the fact that v, (s) +3»(s) =0.
The integral here is evaluated numerically and the
B» in (38) with the B, thus obtained is used as the
matrix element for the decay of g- m'+y+y. We
obtain a numerical value of
I (q mo+y+y) =4. 8x10 ' keV
(44)
for s, =2m~' and a somewhat larger value for sp
& 2m, '. This number should be compared with the
experimental value' of 8. 1& 10 ' keV. Actually,
the experimental value quoted here is the average
value. Some experiments,
however, give a considerably larger value.
We note that in one of the earlier attempts, Oppo
and Oneda' assumed that the m' and one of the photons in the decay q-m'+y+y are in 1 state saturated by the vector mesons, thereby introducing a
considerable angular-momentum
barrier, which
gave a very small value for the decay rate, viz. ,
F„„p»=0.44 eV. On the other hand, in the effective-Hamiltonian model of Okubo and Sakita'
one got
p» = 8 eV, though this number is cutoff-dependent.
Therefore, with reference to the
average experimental value for the decay width,
our result (44) should be regarded as quite satis-
"
I'„,
ASYMMETRY IN
14
e+ e ~ K'+
factory and our model for the decay g- w +y+y
should be regarded as essentially correct.
K
AND
q
v, (v", v',
~
+
x
s) =
X
The final-state interaction for the process may
be introduced through an integral equation of the
Bethe-Salpeter type. Here we will consider the
I ogunov- Tavkhelidze equation, which is an approximation to the Bethe-Salpeter equation. This
equation for the partial-wave amplitude for the
process y+y-K+K has the form
"
') =7', (s, v')+
.
—
f
dv
II
vll
y/2
0
xT, (s, v")v,. (v", v',
s),
(Al)
where T, (s, v'} is the I =0, J=2 partial-wave projection of a suitable helicity amplitude for y+y
-K+K, Ts(s, v') is the Born approximation for
T, (s, v'}, v&(v", v', s} is the "potential" or the Born
approximation for the I =0, J =2 partial-wave
amplitude for KK scattering, i.e. , the Born approximation for [(v+g')/v]'~'e' 2sin6„and v =s/4
We take v' =v and assume T, (s, v")=T, (s, v)
to obtain
T, (s, v) =T~s(s,
v)/D(s),
—~"
1
D(s}=1 —
1(
i~
Jo
„,
„—
dv"
v —v
v
v"
+i(,
~
(A2)
v, (v", v", s).
This result is qualitatively similar to the result
(20) in that the final-state interaction multiplies
the Born. term by a "resonant factor. " For the
analysis of this effect, we assume that the denominator produces a resonance corresponding to the
' meson with the observed width. Therefore, if
we expand the denominator at the point s =s„=m& '
—iF& m&. , we get
f
D(s) =[(s —m~. ')+ il'q
m~
dD(s)
] ds
s=s
.
(AS)
The determination of dD(s)/ds ~, , requires a
dynamical model for v, (v", v', s).
A simple model for v, (v", v', s) is to assume that
it is generated by the exchanges of particles with
=1, 2, 3. We evaluate the numerator of the pole
terms at the pole positions, so that the potentials
are asymptotically well behaved, and obtain the
off -shell potential
J
~
J. Godine
and A. Hajokey, Phys. Rev. D 6, 3301 (1972);
V. K. Cung, A. Mann, and E. A. Paschos, Phys. Lett.
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2257
y
g f„(s)
APPENDIX
1',. (s,
y+
m ll 2+vI+vlf
2
(A4)
2(v vn)lip
where the summation is over particles with spins
1, 2, and S. The functions f„(s) for exchanges of
4 =1, 2, 3 particles are
f, (s) =SC»y, (—,'m, '—lP ) J', (I + m' —4p2
I
2s
(
f, (s) = 5C„y, (-,'m, '—u') &, 1+
I
f, (s) = 7C„y,(-,'m, '—v'(P, (I
2
+.
4
~'
2,
A5
2s
m 32 —4p2
respectively, where Col are the crossing matrix
elements for the exchange of a particle with isospin I, for KK scattering in the isospin-zero
state:
(A6)
The values of y'„are related to the coupling constants of these particles with the KK system and
are related to the experimental widths' of these
particles. For the 4=1 case, we have actually
two exchanges, one with I =1 corresponding to the
p exchange, and another with I =0 corresponding
to the exchange of the unmixed co, meson. . Together
they give an effective contribution of C„y, =1.4.
For the J =2 exchange, we have the f' meson which
gives C»y, =2.0/m' where m =1 GeV. The widths
of the = 3 resonances are not so well established
and we adjust it to produce the required zero in
D(s) at s =2. S (GeV)'. This gives us a value of
C,Iy, =1.1/m4 corresponding to the partial width
I'(g- ((w) =150 MeV. This determines the
v, (v", v", s) needed for the calculation of D(s} in
(A2). The value of dD(s)/ds~, , obtained from
this D(s) is approximately -1.S/mz ', so that
J
T, (s,
v)=,
0 ' 75m ~
.
—s —zmyi
Ts(s, v),
(Av)
which is similar in form to the result (21), but is
smaller by a factor of 0.75. Even. this can be
traced to our taking a rather large width for
I
w((} so that one may conclude that there is
consistency between the two approaches.
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