PHYSICAL REVIE%
VOLUME 14, NUMBER 9
D
Radiative corrections to
ee /urn
l
and ee
NOVEMBER 1976
—&p7rn.
M. P. Gokhale, S. H. Patil, and S. D. Rindani
Department
of Physics, Indian Institute of Technology, Bombay 400 076, India
(Received 26 January
1976)
We calculate the radiative corrections to ee ~Q'~ Qn n, and predict that the ratio of the rate for ee ~ Q'~ gw+m
to that for ee~Q'~Qrr~tr
should be about 1.86. We show that the asymmetry in the n'+n
distribution for
ee~p m+m is quite large, about 15%, but it goes through zero for the m+m invariant mass near m~. On the
other hand, for ee ~Q'~Qrr+rr, the asymmetry is negligibly small, less than 1%.
corrections.
I. INTRODUCTION
The discovery' of the P and the P' particles has
introduced a major new element in the domain of
elementary particles. It has given rise to a very
intense effort toward the understanding of the
properties of these particles and their interactions.
Though a considerable amount of knowledge has
accumulated, it has not, as yet, led to definite
conclusions regarding their role in the scheme of
strong and electromagnetic properties of elemen-
tary particles.
A particularly important process' involving the
g and g' particles is
e(P, ) + e(P. )
- 0'-
),
tr(&, ) + u(&. ) + 0(&.
where the two pions are in a predominantly Sstate. The decay g'- g+ tr+tr, which is a
major decay mode of the f', finds a significant
analogy in the decay p' p+w+n. There are' interesting similarities and differences between
these two decays, which are important for the
understanding of the properties of the
and the ('
wave
(
particles.
In this note, we discuss the radiative corrections
to process (1) and to
e(p, ) + e(p, ) —tr {a,) + tr(u, )+ pe(us) .
The electromagnetic
corrections to these pro-
cesses (1) and (2) are qualitatively similar, but
are quite different quantitatively. The differences can be traced to the difference in the isospin
content of the two systems and in the magnitudes
of the decay widths. We will analyze two aspects
of the processes (1) and (2):
1. The ratio of the rates of m'v production
and tretro production is considered for process (1).
With the generally accepted assignment of isospin
I =0 for both g and g', this ratio is expected to be
2 from pure strong interaction, since the two pions
would then be in an Z = 0 state. Any deviation from
this ratio of 2 would be ascribed to electromagnetic
14
The experimental
value4 for the ratio
rate(ee- (tr'tr )
rate(ee- Ptt'n')
is 1.6 +0.4. This number,
while consistent with a
ratio of 2, could allow for a major deviation from
2. We analyze the electromagnetic corrections to
process (1) in the soft-photon approximation, which
we find to be quite large, and predict a value of
R =1.86. It is expected that improved experimental
numbers for R will tend toward this value. It
should be noted that, since p has isospin I =2, the
two pions in process (2) can be in I =0 or 2 states,
so that the ratio of the m'm production rate to the
ttetto production rate for process (2) can be different from 2, even without the electromagnetic interaction, and we make no prediction for this
ratio for process (2).
2. jn processes (1) and (2) for tr+tt production,
the two pions are in the charge-conjugation C =+1
state if the processes proceed via a single-photon
intermediate state. In that case, there will be no
asymmetry in the angular distribution of m'n
However, electromagnetic corrections allow the
m" z to be in the C =-1 state to a small extent.
We calculate the asymmetry in the angular distribution of tr'tt for processes (1) and (2), resulting from the interference between the amplitudes
with n'm in the C =+1 and C =-1 states. Though
this asymmetry is of the order of the fine-structure constant z, it is considerably enhanced by
the infrared effects and the final-state interaction.
In particular, we show that the asymmetry, as a
function of the azimuthal angle defined later, for
process (2) with the invariant mass of the tt'tt
near the p-meson mass, is quite large, about
but goes
15/q in the soft-photon approximation,
through zero at a position slightly below the resonance. The corresponding asymmetry for process
(1) is much smaller because of the small width of
the g' resonance, and the energy restrictions for
the nm system.
Observation of the above effects will be useful
the properties of g and g', espein understanding
cially with reference to processes (1) and (2).
2259
M. P. GOKHALE,
2260
S. H. PATIL,
We neglect the lepton mass in the following con-
S. D. RINDANI
AND
1(a)] gives
siderations.
2a(m2 —2 p, 2)
H. RATIO R OF PRODUCTION RATES
ee~ Qm'm,
(4)
The contributions to come from the vertex correction [Fig. 1(a)], the wave-function correction
[Fig. 1(b)], the bremsstrahlung corrections
[Fig. 1(c)], and a correction due to the difference
in the charged- and neutral-pion masses, so that
we write
&
—4 p. 2)'/2
nm(m2
Pm'omo
The leading electromagnetic correction to order
a, in the soft-photon approximation, to the value
of R =2 comes from the diagrams in Fig. 1, and is
given by ~, where
5
14
x ln
(m2
4~2)1/2
m —(m'
—4 p, 2)'~
p. is the pion mass, m is the "average"
mass of the n'n system, and m& is a small mass
assigned to the photon. Since the m n distribution
is peaked toward a larger mass of the n'n' system, we obtain a value of ~ = 3.5 p, .
Waue-function correction. The wave-function
corrections [Fig. 1(b)] give
'PRy
7l
contributions
Other electromagnetic corrections of order n to
process (1) are the diagrams where the soft photon
interacts only with the leptons, in which case the
value of R is unchanged. The diagrams where the
internal soft photon interacts with the pions only
once, in which case the pions are in a C=-1 state,
give rise to corrections to 8 of the order of e',
which are neglected.
contributions
to 5. The calculations are similar to those done
previously' for ee- p. p. , K'K .
Vertex correction. The vertex correction [Fig.
We now evaluate the soft-photon
x
j
correction.
The bremsstrahlung
[Fig. 1(c)] lead to
m' —2g2
1- m (m2
4i12)1/2
ln
m + (m' —4 p. 2)'/2~
(m2 4i/2)1/2
m
where 4E is the maximum energy of the soft photon emitted. We take (AF)//i/. =2%.
Mass coxrectln. The correction & due to the
difference in the masses of n' and m' is given by
I'2
2„(m, +) —I'2. 2,„(m,p)
I"i,
(b)
(6)
where
BxenzssA'ahlung
=5y+5~+5~+5
my
~„(m„p)
where I'& 2„,(m) is the decay width for p'- gn/w
with the pions having mass m each. The number
is the fractional difference in the integral of the
weighted phase space, due to the difference in the
masses of m' and w'. To evaluate this, we use an
e model, similar to the one used by Schwinger
et al. , ' in which the g'Pe and ewv couplings are of
the form P'„gf'e and ega, respectively, where P
is the pion field. We have used m, =800 MeV and
l", =400 MeV for the mass and width of c.
Numerical evaluation of the above quantities
gives
~,
~
g
+~, = 2. 3x10-~,
=-4.6x10-',
so that
5=
-6.9 x10-2
R =1.86.
FIG. 1. Diagrams contributing to the change in the
ratio of the rate for ee g' g~+7t to that for ee
—y' 7('m'.
This should be compared with the experimental
value' of 1.6 +0.4. It should be possible to verify
the corrections implied by (11) by a more accurate
RADIATIVE CORRECTIONS TO ee ~ Pen' AND ee ~
14
determination
of the experimental
value for the
ratio R.
The above analysis is applicable to a more general process ee-Xmm, where X is a C =-1, I =0
particle, at a relatively low value of the effective
mass of the mm system. For example, for X=co
or P and the effective mass of mw m =3.5 p, , we
expect the same value of about 1.86 for the ratio
of the decay rates into the charged pions and the
neutral pions. So far, however, there is no experimental information on the value of 8 for these
cases.
III. ASYMMETRY
IN
e(p, )+e+2) n'(k, )+z(]&2)+p (p3)
We consider the asymmetry in the w n distribution mainly for process (2), i.e. , ee- p'w'rr .
Though it is present in process (1) as well, it is
considerably suppressed there, because of the
small decay width for g' and the restriction on the
maximum energy of the n'n system.
The asymmetry in the m n distribution is due
to the interference between the leading amplitude
for which the n'm are in a C =+1 state and the
amplitude with electromagnetic corrections for
which the w'n are in a C =-1 state. We will assume that the m'm in the leading amplitude are in
an S-wave state. This is justified at the energies
under consideration for the process.
The diagrams which contribute to asymmetry, to
order a, are shown in Fig. 2. We write the ampli-
tude for process (2), including the leading term
T» and the terms in the soft-photon approximation
for the diagrams in Fig. 2(a), as
T„=T, „(1+&) .
(12)
Here
(13)
P„s, = (P —k;)',
where s = P', P =P, +
amplitude for y, p&n n,
the cn'n vertex, assuming
duced via the E resonance.
given by the expression
-
—i(e/g)
(2m)'
d'q,
'
q,
gt„„ is the
g is the strength of
that the pions are proThe correction & is
and
2p, eTB
[(p, —q, )' —m'+
ic]
(14)
where m is the lepton mass, and 78, the amplitude
for the on-shell-photon process
y8(q,
)+e(k)- m'(k, )+ v
(k, ),
has a form
g
8
=(k, ak, q, —k, sk, q, )r,
where v is the scalar invariant amplitude.
The region of importance for the asymmetry is
the one near the position of the p-meson resoTherefore, one should calnance, i.e., s, =mp
culate the amplitude 7 with the final-state interaction in the I =1, 4=1 n'n state. W'e introduce
the final-state interaction in terms of the helicity
amplitudes.
The helicity +1 amplitude v, is related to the
invariant amplitude v by
4~2
(a)
pmm
i/2
q, (k, +k, )(sing')r,
(17)
where 6' is the scattering angle for process (15)
in the center-of-mass frame. The Born approximation for v, is given by the diagrams in Fig. 3,
and has the expression
/
(b)
FIG. 2. Diagrams contributing to the asymmetry
distribution for ee pox+7t
in
FIG. 3. Born diagrams for e+y&
x++x
.
/
7T
S. H. PATIL,
M. P. GOKHALE,
2262
s —4p, '
eg
'~'
~s
S. D. RINDANI
1
{
Projecting this amplitude
AND
J =1 state,
onto the
J =1) = — Seg) (s .—» S ')(»,'
k}{
k
we get
for the
2—
'
(», + &.))
(
J =1 part
s.(s, —» S ' )I s'
ln
of T~
»s, —(s. —» S ' )'" I '
(19)
where g is the "coupling" constant for e))')) . The analytic properties of ~s(J=1) suggest that the finalstate interaction may be incorporated by writing dispersion relations in the variable s, for 1/s, [q, (k,
+k, )]'T(J =1)D, (s, ), where D„(s,) is the D function for the I =1, J =1 partial-wave m))'-scattering amplitude. Approximating the left-hand discontinuity of v(J= I) by that of ws(J=1), we get
D
(s, )[q, (k, + k, )] '
(20)
„(s,')'~'(s,' —4 p. ')@'(s,' —s.,)
Now, because of the steep behavior
gral, which leads us to the result
of the integrand
near
s,'=0,
we may set
D, (s,'}=D„(0)inside
the inte-
(21)
J =1 partial-wave
We saturate the
'
~(J=I)=,
mP —S3 —imP
VE p
.
I"P
so that we have
by the p meson,
amplitude
~s(J=1),
(22)
where mp is the mass of the p-meson resonance and ~p is its width. For other angular-momentum
states,
i.e., =3,5, . . etc. , we assume that the final-state interaction is negligible so that the corresponding
With this assumption, the total amplitude
amplitudes may be approximated by their Born approximations.
7 1S
J
.
1
{q, k, )(q, k, )
~
~
3s~
(s, —4p. ')[q, (k, +k, )]'
'
[s,
4p,
(s, —4p,
vs, + (s, —4p. ')'~'
~s, —(s, —4g')'"
')]'"
mp'
mp' —s, —il'pm~
carry out the integrals in (14) with the ~() obtained from (16}with this expression for v, and get, in the
limit of the vanishing lepton mass,
We
p, k, p, k,
12(p, k, P, . k,
~
-p,
~
k, p, k, )
~
In this expression we have negiected a term proportional
to {~k~/ko)', which is (Iuite small in the region of
our interest.
For the discussion of the asymmetry in the m m distribution due to &, we choose the symmetric centerof-mass frame where the z axis is along the direction of k, xk, and the x axis is along the direction of k3.
Defining 8 and (t) as the polar and azimuthal angles of p„we have
do
m'
»s»ss
J»s,
Jgl
i
»sl
e„ is the polarization vector for the p, J,
Jf are the spins of the initial and final parti-
where
and
l*,»sss
(25}
f
cles, respectively, and the limits of integration
for s, are p. '+m '+2k„k„+2(k,[(k, [, and for s,
the limits are (s'' —m )' and 4p, ' with k, =(s '')/
2 and k»=(s —m, ' —s, )/2s, '~'. To (25) we add the
contributions from the bremsstrahlung processes
corresponding to the diagrams in Fig. 2(b). The
effect of this is to replace the my in (24) by b, F.,
where 4E is the maximum energy of the soft photon
emitted. We take 4E =0.02 p. .
The asymmetry in the
distribution is conn
veniently described in terms of the asymmetry parameter A defined as
m'
RADIATIVE CORRECTIONS TO ee ~
14
about
0. 3
Pmm
AND ee
s,''=760
~ pox
2263
MeV, which is slightly below the
resonance.
0.2 .
0.1
0.0
A
—O.
'3 (Ve
650
i.
-02- 0.3
—
FIG. 4. Asy~~etry in e +e p +Vt +7t as a function
of the invariant mass of ~+~, near the p-meson resonance position.
The asymmetry analysis can be extended to
other processes ee —Xw'n where X is a C = -1
particle, and the above results are generally
valid. For example, X can be v, P, or P. However, the analysis for ee- it'- gv'v requires
important modifications because of the narrow
width of the $' resonance. Effectively, we find a
similar expression for & as in (24), except that
the factor ln(p, /b. Z), which would have been there
after including the bremsstrahlung contributions,
is now replaced by In/[LE —(il& )/2]/b, EI. This
suppresses the value of & by at least an order of
magnitude. Also we do not have the enhancement
due to the p-meson resonance, since the m'n can
have a maximum effective mass of about 600 MeV
only, which is below the p-meson resonance
threshold. Overall, the asymmetry for this process is less than 1%.
IV. CONCLUSIONS
f,
f"
Our calculations predict that the ratio of the
(vm'
rate for ee- g'- (n'n to that for eeis 1.86. However, the asymmetry in the distribution of w'w for ee-g'- Pm" m is found to be quite
small, less than 1%. On the other hand, the asymmetry in the n m distribution for ee —p n n is
shown to be quite large, of the order of 15% for
the effective mass of n'~ near the p-resonance
('-
(d'o/ds, dy)dP —f„(d'(x/ds, d&p)dy
'
(d'o/ds, dy)dy+ P" (Fg/ds, dy)dy
(26}
is evaluated numerically with
=770 MeV and I'~ =150 MeV, and is plotted in
Fig. 4 as a function of vs„ for s' ' =1700 MeV. It
has a mild dependence on s and increases slowly
as s increases. In general, the asymmetry is
quite large, about 15 /o, but goes through zero at
This parameter
m&
~J. E. Augustin et al. , Phys. Rev. Lett. 33, 1406 (1974);
J. J. Aubert et al. , ibid. 33, 1404 (1974); G. A. Abrams
et al. , iMd. 33, 1453 (1974).
26. S. Abrams et al. , Phys. Rev. Lett. 34, 1181 (1975).
3R. W. Brown, V. K. Cung, K. O. Mikaelian, and E. A.
Paschos, Phys. Lett. 43B, 403 (1973); M. P. Gokhale,
S. H. Patil, and S. D. Tindani, preceding paper, Phys.
Bev. D 14, 2250 (1976).
position, but goes through zero. This dramatic
behavior of the asymmetry should be experimentally easily observable.
4J. Schwinger, et al. , Phys. Rev. D 12, 2607 (1975);
D. Morgan and M. R. Pennington, ibid. 12, 1283 (1975);
L. S. Brown and R. N. Cahn, Phys. Rev. Lett. 35, 1
(1975).
5E. HQger et al. , Phys. Rev. Lett. 35, 625 (1975).
V. Barger, Nuovo Cimento 32, 127 (1964).
S. H. Patil and S. D. Rindani, Phys. Rev. D 13, 730
(1976).