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The isospin mixing effect in p p annihilation at rest into π+ π− and KK †
Nuclear Physics North-Holland
A516 (1990) 643-661
THE ISOSPIN
MIXING
EFFECT
INTO w+m-
IN $p ANNIHILATION
AT REST
AND K&
S. FURUI’ institut ftir Theoretische Kernphys~k,
U~~versit~t Bonn, Nussalfee,
G. STROBEL Institut fiir Theoretische Physik,
and Amand
de ~ysi~ue
~~origue,
FAESSLER
Universitiit Tiibingen, Auf der Morgenstelle Fed. Rep. Germany R. VINH
Division
14-16, 5300 Bonn, Fed. Rep. Germany
14, 7400 Tiibingen,
MAU
IPN 91404 Orsay, France and LPTPE, 75252 Paris, France
Uni~rsif~
Pierre et Marie Curie,
Received 9 December 1988 (Revised 2 April 1990) Abstract: The isospin-mixing effect in pp annihilation at rest into r+Y and KK is studied. We compare the results obtained with a pp optical potential and a meson-exchange interaction which is derived either from the Bonn or from the Paris potential. The annihilation potential is derived from the constituent quark model. We adopt the annihilation model A2, i.e. we assume that the two mesons are created by annihilation of two, and creation of one, qq pairs with the quantum numbers of the vacuum. The annihilation into three mesons is described by the rearrangement model R3. The isospin-mixing effect is sensitive to the choice of the meson-exchange potential. The combination of the Paris potential and the quark-model annihilation potential reproduces the ratios of rr+rrand KK production fairly well when the ss pair creation is suppressed as compared to a uii or a dd pair creation by a factor of about 0.3 in the amplitude. In the ‘SD, channel the model predicts substantial KK production from the D-wave. For the &C production in the P-wave, the model yields comparable contributions from ‘PO and 3P, waves. Predictions of the model on several other two-meson decay modes are also presented and compared with those of other quark models.
1. Introduction
The branching ratios of pp annihilation into two mesons are possible sources of information on the mechanism of qq pair creation and annihilation in hadrons. In a previous work ‘), we observed that the experimental data of pp annihilation in flight into two mesons can be reproduced in the annihiIation model A2 with 3Po vertices better than in the rearrangement models R2 or the rearrangement model 52 with 3S1 vertices. In the annihilation model A2 (see fig. l), we assume that the two qq pairs annihilate and a qq pair is created, while in the rearrangement model R2 and 52 (see fig. l), a qq pair is annihilated and the remaining two quarks and i
Supported
by the Deutsche Forschungsgemeinschaft School of Science and Engineering,
’ Present address: 0375-9474/90/%03.50
@ 1990 - Elsevier
Science
Publishers
and Alexander von Humboldt Teikyo University, Utsunomiya B.V. (North-Holland)
Stiftung. 320, Japan.
644
S. Furui ef al. / Isospin mixing e#eci
two antiquarks rearrange into two mesons. The main reason of the deficiency of the rearrangement model is the strong momentum mismatch between the initial pp and the final meson states. Another reason is the stronger selection rule in the R2 model which forbids the S-wave pp annihilation into two s-wave mesons. In a simple Born approximation, we also studied the branching ratios of pp annihilation at rest into mesons I,‘) and observed that the A2 model reproduces the experimental branching ratios better than the R2 model. Recently the ASTERIX collaboration of the LEAR group 3, published branching ratios of the pp annihilation at rest into n+7~- and KK in a hydrogen gas target and compared with data from a liquid hydrogen target “). In liquid hydrogen the annihilation from S-waves dominates due to Stark mixing and K+K- production is about 5 of the m+n- production, while in the gas target the annihilation from the P-wave dominates and the K+K- production is about & of the T+V- production at atmospheric pressure. In order to make a detailed analysis of pp annihilation at rest into mesons, it is necessary to take into account the isospin-mixing effect in the NR atomic state. Kaufmann and Pilkuhn “) showed that protonium in a P-wave annihilates preferentially in one of the two possible isospin states, namely in the 3P, state, isospin-1 dominates, while in the 'P, , 3P0 and 3Pz states, isospin-0 dominates. The effect was studied with the Dover-Richard model which consists of a phenomenological isospin-independent annihilation potential of the Woods-Saxon type and of the G-parity transformed Paris NN potential “). In the Paris Nfi potential, the annihilation part is derived from the baryon-exchange model which is isospin dependent. The Paris group solved the Schrodinger equation and studied isospin-mixing effects 7). In the iSo channel, the Dover-Richard model predicts that the annihilation occurs preferentially in the isospin-0 state *), while the Paris model predicts that it occurs preferentially in the isospin-1 state. The discrepancy suggests that the isospin mixing depends strongly on the annihilation potential. Thus it is important to check whether the branching ratios of the pp annihilation at rest can be reproduced by one of the models consistently. Oades et al. ‘) calculated the branching ratios of pp annihilation at rest into ~+cTand K’K- using an amplitude given by the quark model lo) or by analytical continuation of the helicity amplitude Nti+ ~~6 [ref. “‘)I. The annihilation into two pseudoscalar mesons occurs mainly from 3S,, 3P, and 3P2states. The phenomenological helicity amplitudes of NR-+ rr+rr- extrapolated to the corresponding kinematical region shows that the branching ratios of rtm- for the 3P, and the 3P0 states are comparable. In the quark model of ref. lo), the decay into ~~6 is described by the rearrangement model with one-gluon exchange 52 (see fig. 1) and by a model with two-gluon exchange A2S (see fig. l), whiie the decay into K+K- is only given by A2S (see fig. 1). This model explains the ratio of KfK- to ~+?r- production but since the rearrangement model forbids the decay into V+T- from the 3P2 channel the decay from the 3P0 channel becomes too large as compared to the 3P2 channel.
S. Furui et aL / Isospin mixing eflect
This
is in disagreement
mechanism
with the results
of NR annihilation
and it is necessary also be derived The Tiibingen
of ref. *) and
into two mesons
to study other mechanisms
from the quark group ‘) derived
model
64.5
experimental
adopted
data ‘). The
in ref. lo) is not unique
I*). The annihilation
potential
should
in the constituent
quark
consistently.
the annihilation
potential
model with the assumption that the intermediate state is given by two- or three-meson states out of 9 s-wave mesons (7r+, 7r”, 7r-, p+, p”, p-, w, 77 and 7’) and 16 p-wave mesons (B+, B”, B-, a’, 6’, 6-, A:, A:, A;, A:, A;, A;, E, D, f and H). Channels including more than one p-wave meson are ignored. The T-matrix is calculated from the Lippmann-Schwinger equation and the annihilation amplitude gives the branching ratios of the pp annihilation into mesons in flight. We adopt the same approach in the analysis of the pp annihilation at rest. Since the branching ratios of pp annihilation into two mesons are well reproduced by the A2 model, we neglect the contribution of the R2 diagram (see fig. 1). The suppression of K+K- production as compared to the 7r+?r- production can be explained by the suppression of the SS pair creation as compared to the uii or dd pair creation ‘32). In the annihilation model A2, the decay into two pseudoscalar mesons occur from both 3Po and 3P2 channels and the form factors of the decay processes from the two channels are
L 17
~ 123
84
654
A2
15
24
~ 123
654
R2
17
123
A3
15
~ 123
R
C 09104
654
2’4
52
654
lb
~ 123
25
34
654
R3
17
123
84
654
A2S
Fig. 1. Schematic diagrams of the quark model for N6l annihilation into two or three mesons. A2 and A3 are the annihilation diagrams, and R3 and R2 are the rearrangement diagrams. S2 is the rearrangement diagram with one-giuon exchange and A2S is the annihilation diagram with two-gluon exchange which are used in ref. r”).
S. Furui et al. / Isospin mixing effect
646
different.
In the Born approximation
without
initial-state
interaction,
we observed
that the branching ratio of the 3P, channel is larger than that of the 3P, channel ‘). In the analysis of branching ratios, it is important to take into account all the open channels
since a specific
decay
mode becomes
effectively
decay modes that rob the flux out of this channel
suppressed
13). We consider
if there
are
the annihilation
into three mesons in the rearrangement model R3, although there are indications that the annihilation model A3 dominates over R3 ‘*r4). The annihilation potential that we use in this paper is the combination of R3 and A2 as in ref. ‘). Since the K+K- system is not an eigenstate of G-parity, a coherent sum of isospin-0 and isospin-1 amplitudes contributes to K+K- production. The relative phase of the two amplitudes depends on the initial-state interaction and consequently the isospin mixing is crucial for the K”K- and K°Ko production. We study the initial-state interaction by comparing results obtained with the one-boson exchange potential of the Bonn group with results obtained with the meson-exchange potential of the Paris group. We combine the meson-exchange potential with the above annihilation potential and solve the Schrodinger equation. Using the wave function obtained in this way, we calculate the branching ratios and compare with experiment. Unlike the calculations of ref. ‘) we ignore the Nd f tiA and Ad channels which give only about a 10% correction to the annihilation cross section, but contrary to that work, we include the Coulomb interaction between the proton and antiproton. The plan of this paper is as follows. In sect. 2 we calculate the wave function of the pp atom in momentum space. In sect. 3 we derive the partial decay widths of pp annihilation into nr+F and KK. Numerical results are shown in sect. 4. In sect. 5 we present predictions of the A2 model on several other two-meson decay modes and compare with those of other quark models. In sect. 6 the main conclusions and a discussion of the most important results will be given. 2. The pp atomic wave function in momentum space In order to solve the Schrbdinger equation in momentum space for the NR system we take the states pp and iin as a basis and consider the following coupled equations;
Im I 0
(Vdk k’) + K&(k
k’) + ViAk, WMp,(k’) d3k’
00
+
(K&k,
W+ ViAk k’)Mdk’)
d3k’=EICr,,(k), (2.14
0
$
&(k)+
k’) + V:im(k, k’)hhi(k’)
d3k’
co
+
I
0
where m is the reduced
(Vim&
k’)+ Kn,(k k’)Mp,(W d3k’=&hi(k),
mass of the NR system,
k is the relative
momentum
@lb) between
S. Furui et al. / Isospin mixing effect
N and isoscalar
N in the centre-of-mass isovector
system,
independent
meson-exchange
V,,
VhEX and
Vi&
are the Coulomb,
respectively. The annihilation potential in V&) where Vi,, is the annihilation potential V,‘,, is given by i( c’,,* the isospin-I channel. In the Dover-Richard model, the annihilation potential is isospin
and
641
potential,
and so Vi,,, is zero. Sm is the proton-neutron
mass difference.
For the meson-exchange potential VhEX, we consider the G-parity transformation of the one-boson-exchange potential of the Bonn group (HM3) 15), and the r, w and two-pion-exchange potential of the Paris group “). In the case of the Bonn potential, we approximate the coupling potential between the pp and the nii channel and the p-exchange. In the case of the Paris potential, V&x by the n-exchange Since the meson-exchange potential loses its validity are defined as for V,‘,,. V&x at short range or at high momentum we regularize the meson-exchange potential with a cut-off factor
(2.2) potential and A is the cut-off where V,,,(k, k’) ,, is the original meson-exchange momentum. We choose A = 1 GeV and n = 10. This cut-off makes the too strong short-range attraction of the meson-exchange potential weaker. In the Paris potential 6), the short-range part is cut off in configuration but not in momentum space. We adopt the simple momentum space cut-off since it allows to express the potential in a simple analytical form. It makes comparison with the other published results difficult but indicates the dependence of the physical quantities of the pp atoms on the cut-off procedures. The solution of the coupled equation (2.1) in momentum space faces a problem due to the singularity of the Coulomb potential. Kwon and Tabakin i6) applied the regularization the eigenvalues mic singularity,
prescription
of Land6 and published
a computer
code for calculating
and eigenfunctions. Although the Coulomb potential has a logarithit is multiplied in the integral equation by the wave function and
the integrand can be calculated properly. We adopt the same technique and solve eq. (2.1) for each partial wave by using the matrix inversion method which was used by the Hannover group in the calculation of the NN bound-state problem 17). We first check
the program
for the pure
Coulomb
interaction
and next with the
meson-exchange potentials V,,, of the Bonn group and the Paris group, and the V,,, of the Dover-Richard model. The results are compared with those of Richard and Sainio “) where the Dover-Richard model in the coordinate space was used. Due to the different procedure of the short-range cut-off, we cannot expect complete agreement of our result for the combination of the Paris potential and the DoverRichard annihilation potential with the results of Richard and Sainio, but we can check qualitative agreement. The combination of the Bonn potential and the DoverRichard annihilation potential allows to see from the results the dependence of the decay width and the energy shifts on the meson-exchange potential. Finally we
S. Furui et al./ Isospin mixingeffeci
648
replace V,,, given by the optical potential by our calculation in the quark model. The derivation of the optical potential in the quark model is shown in ref. ‘). We ignore the strange meson production
in the construction
but we treat this small contribution tively. After the partial
(about
wave expansion,
eq. (2.1) reduces
m(v’CL(k,k’)SLr+vtirELXL.(k,
$2(k)+
of the annihilation
5% of the annihilation
potential
at rest) perturba-
to
k’)+v;,:(k,k’)6,,J&$(k’)kf2dk’
I0 al
+
I
(v$&‘(
k, k’) + v;$
(k, k’))c#$(k’)k’2
dk’= E&$(k),
(2.3a)
0
m
$k%Xk)+
(2&n + vL:$(k,
k’)+
v;;:(k,
k’)S,,,)+$(k’)k’2
dk’
I 0
+
k, k’) + v;,f,f (k, k’)S,,.)&$,(k’)kf2
(vj$:(
where for the spin-independent V(k,
k’) =
Y*Lw(k’),
1 vLTL’(k, k’) Y&k) LML’M’
vk( k, k’) = -
3.
(2.3b)
potential,
vL.~ =_ 2 ~
Here QL(x) is the Legendre potential, the J-dependence
dk’= E&,(k),
I
m o r2 drL(kr)
(e'/
V(r)_hJk’r)
rkk’) QL( k2 + kf2/2kk’)
function of the second kind. is treated as in ref. 15).
, . For the spin-dependent
Decay widths of pp annihilation into m+mr- and KK
The decay widths of an atomic
state with angular
momentum
J and energy
E has
the form ‘*)
:r= r
F
I
d~i(E)I({pi}lQ~~~I~(E))12
3
(3.1)
where dp,(E) is the phase space of the final state, pi are the momenta of the final particles, QHP is the transition operator between the final mesonic state (in Q-space) and the initial baryon-antibaryon state (in P-space) ‘) and F is the distortion operator on the initial state. The total width of the system can be derived from the energy eigenvalue and also from the equation
z-r
I
k2 dk kf2 dk’ &‘LT(k)*lmv~~~(k,
k’)+‘LT(k’),
(3.2)
649
S. Furui et al. / Zsospin mixing effect
of angular momentum J, orbital angular where rPJLT is the N6I wave function momentum L and isospin T. The isospin-0 and isospin-1 wave functions are linear combinations
of pp and iin wave functions, ,
#&W
$JLo(k) =&#&k)+
+JL’(k)=&6$(k)-$~(k)].
(3.3)
from the sum Since the annihilation potential z)if;T in our model is constructed of different meson channels, the width can be written as the sum of contributions from these channels
=,v[dPiC&)j
k2 dk kf2 dk’ ~JLT(k)*(kIPHQ/{pi}) (3.4)
x({pi)lQffPlk’)4’LT(k’). The KfKK-K+
state and K°Ko state are not eigenstates
of G-parity.
We describe
the
as &-K+(P)
=4&(p)
+
&&)I,
(3.5)
$M&)
=&MU4
-
&&)I~
(3.6)
and K°Ko as
The partial
decay widths
for K-K+
and K°Ko are
x&+JLo(k’)+c$JL’(k’)], where for the K-K+
(3.7)
the plus sign and for the K°Ko the minus
sign is to be chosen
in the wave functions. fi is the invariant mass of the NR system. The partial wave quark model amplitudes for the meson production (p(QHPlk) in the constituent are derived in ref. ‘). In rr+Y production with the annihilation model A2, the amplitude
for the 33S1 channel
(pIQHP/k)
is* 3’23(3a2+bZ)(a2+b2)
= A,,N;N:/&
a2(3a2+2b2) (3.8)
l The factor & in ref. ‘) is corrected of the parameter hA2.
to I/&
here. The correction
changes
only the normalization
650
S. Furui et al. / Isospin mixing effect
while the 13P0 amplitude (plQHPlk)
is
= h,,NZ,N:/&i
( 3az2;2b’)1’2
3(3a2+2b2) 1_b2(3a2+b*) and the 13P2 amplitude (plQHPjk)
p
2
(ST’*
(-3)
-3ayyib
(3;:;;;‘)’
1
2 (:p +i k2) (GF) ,
(3.9)
is 3’2 b2(3a2+ b2)(a2+ b2)
= h,,N2,N:/&
(3~2’~2b*)~‘~ a2b2 3a2t2b2
(’ 2+ik2) 4p
(5)
(3a2+2b2)2
1
(OS,).
P2k (3.10)
Here a = 3.1 GeV-’ and 6 = 4.1 GeV-’ are the size parameters for the baryons and the mesons, Na = (3a*/ n-)3’* and N, = ( b2/ r)3’4 are the normalizations of the baryon and the s-wave meson wave functions. The description of a n and K in the constituent quark model is controversial. The Regensburg group *‘) argues that the pion is a highly collective state and the qq component is only a fraction of the physical pion. We adopted this picture in the previous work ‘) and multiplied the 7r wave function by a spectroscopic factor v$. It is not clear whether the same spectroscopic factor should be applied to K, since K is more massive than r. In the flux tube model 2’,22) the r is assumed to be pure qq but its size is smaller than that of other mesons. In pp annihilation into mesons, the two approaches give qualitatively the same results, but in some decay processes there appear significant differences “). A study of the meson-baryon interaction and the decay of mesons indicates that a smaller size for the 7~ without a spectroscopic factor less than unity gives a better agreement with experiment 24). But this does not mean that the description of a pion in the nonrelativistic quark model is now solved. Here we assume that the size of a QTand K are the same as that of other mesons and multiply both wave functions n and K with the spectroscopic amplitude 4. With this spectroscopic factor, our effective value for the pion size is the same as in ref. 24). The spin-flavour matrix elements (Os,) for N6I annihilation into ~+6, K+Kand K°Ko are shown in table 1. If there is no isospin mixing in the initial atomic state, the K°Ko production
is suppressed
as compared
to the K+K-
production
TABLE 1 The spin-flavour
matrix
elements
for NN+
T+T-
and Ki( multiplied
by v$$
due
S. Furui et al. / Isospin mixing efict
to the interference KsKL production which indicates
of the isospin-0 from the atomic
a strong isospin
from the atomic
P-state
of the branching
ratios
and isospin-1 S-state is about
mixing
is about of rr+Y
amplitudes.
Experimentally
70% of the K+K-
the
production
r3)
in the S-state. The KsKs + KLKL production
30% of the K+Kand
651
production
KK production
allows
‘). The comparison us to estimate
the
relative strength of the SS pair creation as compared to the uii or the dd pair creation in the NN system. Although in meson decays SU(3) symmetry violation in the qq pair creation is small, there are indications that the SS pair creation is suppressed factor in the amplitude as a. in baryon systems 25,26). We define the suppression One can include the possible difference of the spectroscopic factor for a r and that for K effectively in this parameter. The amplitudes (3.8)-(3.10) have a specific dependence on the momentum p of the meson. At threshold, the relative momentum for the r+Y system is 0.93 GeV/c and that of the K+K- system is 0.80 GeV/c. The amplitude (3.9) for the 3P, channel is proportional to l-(b2(3a2+ b2)/3(3a2+2b2))p2. It becomes zero at the meson centre-of-mass momentum p = 0.49 GeV/c for a = 3.1 GeV-’ and b = 4.1 GeV-’ and at p = 0.70 GeV/c for a = & GeV-’ and b = v’%GeV-‘. This kind of effect occurs also in the meson-baryon coupling in the 3P,, model 24,27)and it has the same origin as the recoil term of Mitra and Ross 28). The 3P0 model predicts the suppression of the two pseudoscalar meson production from the 3P,, pp state. In the 3P2 channel the amplitude (3.10) is proportional to p2 while in the 3S, channel the amplitude (3.8) is proportional to p. Consequently the K production which has a smaller p as compared to r production is more suppressed in the 3P2 channel than in the ‘S1 channel. Experimentally the branching ratio of K+K- production is about 30% of the r+Y production in the S-state and about 6% in the P-state 3*29).
4. Numerical We calculate (i) the energy shifts binding energy, (ii) the half-width antiprotonic atom which, in the limit antinucleon, is described as a pure
results
AE = E - Ecou,, where Ecoul is the Coulomb $, (iii) the nii mixing probability P in the of large distance between the nucleon and the pp and (iv) the ratio of the decay width from
the isospin-1 state and the isospin-0 state T,/T, for S- and P-wave antiprotonic atoms. In the quark model we adjust the strength parameter of the annihilation potential to fit the pw production cross section at plab = 490 MeV/c [ref. ‘)I, since around this energy the experimental error is relatively small. In the present calculation using the Bonn potential, we choose the annihilation strength parameters to be three times larger than the previous in-flight calculation ‘) in order to increase the decay widths. Since we are interested in the dependence on the meson-exchange potential we choose also the same strengths for the Paris potential. To check the program, we calculate the energy eigenvalues for the Paris potential and the annihilation potential of Dover-Richard (Paris + DRl). Although the cut-off
S. Funk et al. / Isospin mixing effect
652
TABLE 2 The energy shift and width of the pp atom in the model of the Dover-Richard 1, Paris-model, the combination of the Paris potential and the quark model (R3 + A2) and the combination of the Bonn potential and the quark model (R3+A2). &r is the imaginary part of the eigenvalue E of the coupled equation (2.3a) and (2.3b). T,C0j is r’rr=‘(‘) defined in eq. (3.2). P(M) is the admixture of nri component in the asymptotically pp atomic state DRl
AE, +r P(nii) r,/ra
‘SD, AEa
p-
P(nii) r,irs
‘PO AE,
+r
P(M) r,/ra
+I A%
yr
P(M) j-,/r,
‘PI
0.54 keV 0.51 keV 4 x 1o-6 0.68
DRl 0.76 keV 0.45 keV 3 x 1o-6 0.76
Paris 0.85 keV 0.82 keV 2.93
Paris 0.71 keV 0.40 keV 0.88
Paris + quark 0.96 keV 0.62 keV 3 x 10-s 14
Paris + quark 0.65 keV 0.71 keV 4 x lo+ 0.04
-0.074 eV 0.056 eV 2 x 1o-9 0.03
-0.072 eV 0.056 eV 0.0001
-0.029 eV 0.012 eV 5 x 1o-‘O 0.09
Paris
Paris + quark
Bonn + quark
0.01 eV 0.006 eV 2 x lo-I0 8.6
0.03 1 eV 0.0036 eV 6 x 1O-6 6.4
Paris + quark
Bonn + quark -0.026 eV 0.0026 eV 7 x 1o-4 3.5
Bonn + quark
DRl 0.036 eV 0.010 eV 5 x lo-lo 9.5
0.032 eV 0.0096 eV 13.8
-0.030 eV 0.014 eV 0.52
0.01 eV 0.005 eV 1 x 1o-‘O 2.6
3PF,
DRl
Paris
Paris + quark
A-%
-0.005 eV 0.015 eV lxlo-‘” 0.63
-0.003 eV 0.014 eV 0.65
$r
P(nii) r,/ra
0.73 keV 0.34 keV 4 x 1om5 2.2
Paris + quark
-0.026 eV 0.013 eV 1 x 1o-‘O 0.96
P(nii) r,/r,
Bonn + quark
Paris
Paris
AEa
-0.31 keV 1.08 keV 3 x 10-s 0.13
DRl
DRl
;r
Bonn + quark
0.01 eV 0.002 eV 6 x lo-‘* 0.47
Bonn-t
quark
11 eV 30 eV 2 x lo-’ 2.4
-0.085 eV 0.018 eV 1 x 1o-5 0.02
S. Furui et al. / Isospin mixing efect
prescription partial
is different
from
waves. Only in the “P,
ref. 18), which, With the optical comparing
we expect,
channel
we found
qualitative
width
in the quark model, obtained
agreement
a weak repulsion
is due to the cut-off dependence
potential
the decay
ref. i8), we obtained
653
contrary
of the spin-spin
we check the numerical
from the energy
eigenvalue
in most to the force.
stability
by
and the sum of
the partial decay widths. Table 2 shows our results for the Paris potential (Paris + quark) and for the Bonn potential (Bonn + quark) both supplemented by an annihilation potential calculated in the quark model. Here the Paris potential is cut-off in momentum space and so is not exactly equivalent to the original. For comparison we show the results of Richard and Sainio ‘*) who use the Paris potential and the Dover-Richard potential calculated in the configuration space (DRl) and then the results of the Paris group who use the annihilation potential derived from the baryon-exchange model ‘) (Paris). The average energy shift and half width for the ‘So state and the 3SD, state in the Paris + quark model is AE -$T = (0.73 - i0.69) keV. The experimental value is (0.70 f 0.15 - iO.80 f 0.20) keV (ref. 30)). Recent measurements using the cyclotrontrap yield AE -$ir = (0.685 f 0.034 - i0.533 f 0.068) keV for singlet S and AE - $ir = (0.845 f 0.043 - iO.35410.080) keV for the triplet S (ref. “)). The results with the Bonn potential differ appreciably from those of the Paris potential in the 3Po channel. In this channel the bound state due to strong interaction and the Coulomb bound state are accidentally degenerate and one gets a large width. This feature remains when changing the annihilation potential from the Dover-Richard model to the quark model. The main difference comes from the fact that in the Bonn potential the scalar component in the J = 0, L = 1 channel remains finite in the limit of small momentum k and k’, while in the Paris potential the component in the L = 1 channel becomes 0 in the same limit. Thus the Bonn potential is more attractive than that of the Paris potential in the NR i3Po channel. The differences in the NN channel do not show up in the NN channel, since there the u-exchange and w-exchange largely cancel each other. The differences in the results of the NR annihilation could be due to the annihilation potential of the R3 diagram which is of long range and concentrates the wave function to smaller distances. In fact in the ISo channel the Bonn+ quark model shows an attraction which is in contradiction to the other models and experiment. By choosing the size parameter of the nucleon to be smaller by 10% the sign of the energy shift changes 32). The KK creation probability depends on the strength of the ss pair creation. In the hadronization process of e+e- at high energy, one observed that the SS pair creation in the flux tube is suppressed as compared to the uii or the dd pair creation due to the higher constituent mass. The argument is analogous to the spontaneous e+e- pair creation in the strong electric field 25). We expect the same effect in the NR annihilation. The ratio of the Kl? production and rTTcr- production in the model of Bonn + quark and Paris + quark without the suppression factor for the SS pair creation are shown in table 3. The experimental branching ratios of rtnT- and
S. Furui et al. / Isospin mixing effect
654
TABLE 3a The ratio of K+K- production and ~T+P- production from the pp atom in the quark model (R3 + A2) with the Paris potential 3SD,
3P0
3P,
K+K-Jr+/ KOKO/n+?TKK/rr+Y
5.0 2.8 7.8
0.66 0.23 0.9
0.8 0.02 0.8
CY
0.28
0.30
KK allow to extract the suppression factor (Y for the SS pair creation as compared to the uii or the dd pair creation. We choose (Y so that in the 3SD1 channel the KK production becomes 60% of the rr+7~- production and that the average value for the 3P0 and 3P2 channels is 8%. In the 3SD, channel the KK is mainly produced from the D-wave. Since in the D-wave K°Ko production is favoured, the ratio of KLKs/K+Kbecomes close to experiment. In the P-wave the ratio of (K,K,+ K,K,)/ K+K- depends on the meson-exchange potential. The Paris potential suggests a suppression factor LY~0.3 for both 3SD, and P-waves and the ratio K°Ko/K+Kis 56% for 3SD,- and 30% for P-waves (table 3). Experimentally KsKL production is about 70% of the K+K- production for a liquid hydrogen target “) and KsKs+ KLKL production is about 30% of the K+K- production for a gas target 29). (Y= 0.3 follows also from the AA annihilation into n+rand K+K- in the region of the 5(2220) resonance 26). The partial widths of rr+C,
K+K- and K°Ko for the Paris + quark model is shown
in table 4. The S-wave rr+C production is small due to the unfavoured isospin mixing. The isospin mixing, however, depends on the meson-exchange potential. In the Bonn+
quark
model
the partial
width
of rrtnT-
in this channel
is twice as
large as that in the Parisfquark. As discussed in ref. 43), we find a strong tensor coupling between the S- and D-states in the 3SD, channel. The annihilation from D-wave is comparable to that from the S-wave. In the P-wave the rr+Y widths in
TABLE 3b The ratio of K+K- production and z-‘v- production from the pp atom in the quark model (R3 + A2) with the Bonn potential
K+K-/T+~KF/n+VKK/V+?Y a
%D,
3P0
3P,
1.6 2.2 3.8
2.5 4.4 6.9
0.2 0.4 0.6
0.40
0.22
S. Furui et al. / Isospin mixing effect
655
TABLET
The width of Kl? production and ~+r-
3SD, 3P0 3P,
1.4 keV 25 meV 4.6 meV
production from the pp atom in the quark model (R3+A2) with the Paris potential
0.45 eV 2.2 meV 0.07 meV
2.24~1~= 0.19 eV 1.4u2=0.12meV 0.06aZ= 0.005 meV
1.27cz2=0.11 eV 0.51u’= 0.043 meV O.O016a*= 0.00013 meV
the Paris+quark model are similar to those of the phenomenological helicity amplitudes calculated by Oades et al. ‘). The branching ratio of rTT+r- from the S-wave is 0.7 x lop3 in the Bonn+ quark model and 0.25 x 1O-3 in the Paris+quark model, which are smaller than the experimental value (3.19+0.20) x 10m3 [ref. ‘)I. In the P-wave, the experimental value is 3, (4.81 f 0.49) x 10p3. The theoretical value depends on the way of weighting the contribution of the different atomic states but in the model of Paris+quark we find it is about 20 x 10m3. The origin of the disagreement in the branching ratio can be attributed to the use of R3 for the three-meson production which yields the annihilation potential with a longer range than the A2 model. We expect the combination of A2+A3 to give better agreement 14,33). 5. Comparison with other models of NN annihilation into two mesons NN decay into two mesons has been discussed by several groups models ‘,12,23,34-36).They disagree on whether the rearrangement annihilation model A2 dominates. In this section we compare the annihilation into two mesons other than rrr and KK in the A2 quark
rearrangement
model
R2 in the following
decay
channels:
by using different model R2 or the prediction of NN model and in the pn and pn’, “7
and Z-V’, ~6 and rB, rf and .rrA2 and give a comment on the decay correspondence to the SS pair creation, we consider also the production %-iTw.
5.1. pp+pv
into rp. In of ~4 and
AND pi’
Experimental branching ratios of pp annihilation to pv and pi’ differ from each other among different measurements. We quote BR( pp + pv) = (0.22 f 0.17, 0.64 f 0.14, 0.97*0.16)x lo-* and BR( pp+pprl’) = (0.11*0.06,0.14*0.08)x lo-* [ref. 37)], which indicates BR( pp + pn)/BR( pp + pv,~) = 2 -9. Since the R2 model does not allow these decays from S-wave pp Green and Liu 36) calculated the amplitudes of the two-step process 33S1 NN + B n, v’( I = 0) + pq T,I’( I = 1). They found that the pq’ production is too large as compared to the pq production. In the A2 model, and choosing the same meson size as Green and Liu 24), i.e. a = 0.6 fm and the mixing
S. Furui et al. / Isospin mixing e#eci
656
angle 0,, = - 10” and the size of a nucleon
b = 0.56 fm = 2.83 GeV-‘,
we find the ratio
BR( pp + pn)/BR( pp + ~77’) = 2.74. The difference between the A2 and the R2 comes mainly from the strength of the damping of the form factor.
5.2.
plZ+“q AND ~7’
In the effective chiral lagrangian model, the flavour singlet component of an 77 meson does not couple to Nr;[ in the lowest order of the l/N, expansion 38). In this model, the ratio of branching ratios BR(pp-+ rr”r]‘)/BR(pp+ ~‘7) is given by tan2 or, = 0.038, ignoring a small correction from the phase space. Experimentally the ratio is (0.5OkO.19) x 10p3/(0.46*0.13) x 10p3, or the n and 7’ production are comparable. Due to the spin-parity selection rules, nn and ~7’ productions from the S-wave pp are forbidden and they are produced from 3Po and 3P2 states. In the A2 model, the n and 17’ production are comparable since the damping of the amplitude for the high relative momentum is not so significant. In the R2 model, however, the ~7 and ~7’ are produced only from the 3Po state and the rn production is suppressed as compared to the rrn’ production due to the high relative momentum.
5.3.
pp+&
AND
Experimental nB) = (0.7+0.1,
vB
branching ratios of pp annihilation to TB and rr8 are BR(13S1Nf + 1.96*0.27)+ 10e2 and BR(“S,NR+ rr6) = (0.69*0.12) x 10-2. In
the R2 model the quotient of the branching ratio BR( pp + nB)/BR( pp + rrS) without initial state interaction and with the statistical weight for the triplet and the singlet state, becomes 104 [ref. ““)I. This large ratio comes from the ratio of the ClebschGordan coefficient squared (50 x 3/6) and the momentum dependence of the form factor. In the A2 model the corresponding Clebsch-Gordan coefficient squared is (4.167 x 3/4.5) and the momentum dependence of the form factor is weak. With the same parameters as in sect. 5.1, we obtain this ratio as 3.4 which is close to experiment.
5.4. pp+rrf
AND
VA,
Experimental data of pp annihilation to nf and nA2 are BR( pp+ rf) = (0.24+ 0.07) x 10d2 and BR( pp+ r*Az) = (3.78 * 0.87, 5.71 f 0.85, 2.83 * 0.32) x 10-2. Since the relative angular momentum between the two mesons are Z= 2, the processes are forbidden in the R2 model. Green and Liu 36) calculated the amplitude for the two-step processes 31SoNR +~~(I=O)+~f(l=2)and”S,N~+rr~(l=O)+ vA,(I = 2). They found that the amplitudes are too small. In the A2 model the processes are allowed and the rA2 production occurs also from the 33SD, state. Without initial state interaction and with the same size parameters as in sect. 5.1,
S. Fumi et al. f Isospin mixing effect
we obtain production
BR( pp + rf)/BR( from the S-wave
pp+ NR
r&A:)
657
= 0.45. The A3 model predicts
is forbidden3’).
The experiment
that the rrf
with a liquid
hydrogen target which is dominated by the S-wave pp annihilation does not show a peak coming from annihilation into wf. The pn annihilation in flight extracted from pd reaction
shows a relatively
large rf production
from the S-wave 40), but a
recent experiment with a hydrogen gas target indicates pp annihilation small in the S-wave but relatively large in the P-wave 41).
into nf is
Experimentally, S-wave NN annihilation into mp is dominated by the 33S, channel although the “So channel is allowed by the spin-parity selection rule. In the case of P-wave, the extraction of the 11P1/33P, ratio from the experimental data has changed from time to time. At the Durham conference, dominance of the “P, channel was suggested 4), but the analysis was corrected at the Mainz conference and dominance of the j3P, channel was reported 42). Improvement of the statistics revealed that there is an additional rrr resonance of 1565 MeV which is expected to have the quantum numbers 2+. The resonance which is called AX(1565) and r” are produced from the 3P, or 3P2 NR state and the branching ratio of rrp from 3P, and 3P2 states turned out to be smaller than that of ‘P, state. The reduction in the 3P,,2 can be understood by the dynamical selection rule 13). The angular momentum dependence of the s-p production from the pp atom has been a puzzle. Based on the pn experiment 40) and using the A2 model, Dover et al. 13) explained the smallness of np production from the ‘So as a result of the large rrf decay mode in the same channel. A more refined calculation of the A2 model shows, however, the rp production from the ISo channel is not so suppressed as compared
to the rrf production ‘743). Th e enhancement of the ‘SrNR over *SONN can be explained by the strong tensor coupling of the 3D,NN channel which yields substantial rp production 43). The ratio of the contribution 33P,,2NR over *‘P,NR predicted in the A2 model agrees with the results given in ref. 42), but does not agree with the new analysis of the data with higher statistics 41). The quotient of the branching ratios BR(31S,NR -+ prr)/BR(31SoN~+ frr) in ref. 43) is much larger than that calculated in ref. 13). There are two sources for the discrepancy. the nucleon
First, in ref. 13) the meson size was assumed to be much smaller than size which is an improper approximation. As a consequence the damping
of the form factor in the model of ref. 13) turned out to be too strong. Second, the spin-flavour matrix element squared for the rp production in the ref. 13) is too small by a factor i relative to the rf production. The authors of ref. 43) claimed that the A3 model does not solve the pv puzzle. There, the philosophy is that one of the diagrams is the leading one and determines the annihilation cross section. For the P-wave pm puzzle the authors excluded contributions from the A3 diagrams. One could invoke a modified A2 model
658
S. Fumi et al. / Isospin mixing effect
in which the p in the final state decays into two pions. processes
are included
over the A2 model and r,)
in different
and incorporated
in the generalized
is that the resonance channels
A3 model. energy
In ref. 39) the corresponding
An advantage
and width
of the A3 model
of two pions
(e.g. mp
such as vp, rrf and rrAX can be taken from experiment
in the coherent
sum of the amplitudes.
An analysis
of the angular
distribution of the Dalitz plot in the generalized A3 model shows that the extraction of the branching ratio “P,(7r~)/~~P,(rrp) depends strongly on the details of the model like the pion radius and a clear separation of the 3P, and 3P2 contribution is not easy 39,4’). Thus it is extremely difficult to extract the above ratio from the data.
5.6.
pD+m+ AND mo
The production of 4 from the NN annihilation provides information on the SS pair creation and the reaction mechanism. The comparison with w production will in addition supply information on the mixing angle of the SU(3) flavour singlet with the octet. Experimentally the branching ratios of pp annihilation at rest into ~4 and ~MI are BR(pp+ rr4) =(0.3OiO.15)~ lop3 and BR(pp+ rra)= (5.2 f 0.5) x 10e3 [ref. “)I. In the standard theory of flavour mixing, one expects the quotient of the branching ratios is BR( pp + rr4)/BR( pp + rm) = tan’ (Bv- 0,) = (4.2 - 8.6) x 10p3, which is much smaller than the experimental data. Here we used Bv = 35.3” and the phenomenological value of 13~= 36” - 39” [ref. “)I. The ratio from the pn annihilation is even larger than that from the pp annihilation. Ellis et al. 45) claimed that it is due to the large SS content in a proton or antiproton. Another interpretation of this enhancement of ~4 production is the contribution of a qsqs resonance or C(1480) with quantum numbers 33S1 [ref. ““)I. We interpret a 4 as a KK resonance and consider the pp annihilation into KKr. The K and K form a 4 by the final-state interaction. In this model, an SS pair creation is necessary and the OZI rule is not violated. The ratio of rr+ and TMJproduction is related to the ratio of the SS to uii or dd creation. An analysis of this process in the A3 model will be discussed
in a separate
paper 39).
6. Summary and conclusions quark model We studied the pp annihilation into nTr+r- and KK in the constituent and assumed that the annihilation amplitude is given by the quark pair creation model or the 3P0 model. We observed that the model predicts the relative strength of the 3P2 state and the 3P0 state consistent with the phenomenological helicity amplitudes of NR + rTr+np. It is different from the quark rearrangement model which finds the 3P0 contribution to be very large. We compared the A2 model, the R2 model and other models for NR annihilation into two mesons in several channels and observed that the A2 model gives the best agreement with the experimental
S. Furui ef al. / Isospin
data.
The Mainz
mesons
group
by SU(3) invariant
calculated amplitudes
branching
mixing egeeect
ratios
and claimed
659
of pp annihilation
the impo~ance
into two
of the rearrange-
ment model in the rrtrTT- production 35). In this calculation, flavour matrix elements are assumed to be independent of the spin angular momentum part. in our quark model with SU(6) wave functions and 3Po vertices, the coupling of spin and flavour is crucial and the selection rules are quite different from that of the SU(3) model. Furthermore, the differences in the form factors of the rearrangement model and the annihilation model should be taken into account when discussing the differences in the two models. We compare the Bonn and the Paris potential for the NN interaction. The Bonn + quark model gives a too large shift and width in the ‘P, channel. It could be due to our specific annihilation potential, especially that of R3 which has relatively long range and makes the wave function sensitive to the intermediate and short-range part of the meson-exchange potential. We find a node in the 3P0 channel wave function which enhances the short-range part of the wave function. The Paris + quark model predicts the ratio of Kl? production to rrfC production and the ratio of K’l?’ and K’K- consistent with experiment when the suppression factor (Y for the SS pair creation compared to UU or dd pair creation is about 0.3. In the 3SD1 channel the KE production from the D-wave turned out to be dominant. The 3Po model predicts that in the ‘S, channel K'l?' production is suppressed but in the 3D, channel it is not suppressed as compared to K+K- production. We took the rea~angement model R3 for the three-meson annihilation and the annihilation model A2 for the two-meson annihilation. The analysis of the branching ratios in flight ‘) and at rest in the Born approximation i4) suggest that the annihilation into three mesons may be dominated by the annihilation model A3. It is clear that the present result must be further improved. A more involved calculation with the combination of A3 and A2 will be done in the future. We have chosen a relatively large size for the pion and used the same spectroscopic factor 4 for 7r and for K mesons. Our effective value for the pion size by using the above spectroscopic factor is 0.35 fm. The flux tube model suggests a size of a pion which is slightly smaller. The size of mesons modify the angular distribution, especially
in the Dalitz plot of the TGT~ decay 39). It will change the branching
ratios
to rp and wf meson states. The analysis of the Dalitz plot of the m~m and the KKrr decay from NR and also from mesons will provide a more realistic description of the r and K wave functions. The sensitivity to the meson size comes from the interference of different quasi-two-body decay modes and is rather specific to three-meson decays. In the r+n. and K+K- angular distribution of the pp annihilation, the initial- and the final-state interactions are more important than the meson size. To conclude we remark that the ratio of 7r+r- to Kl? production is sensitive to the nature of the NN interaction and thus detailed comparison with the experimental data will provide information on the initial NN wave function, on the annihilation mechanism, as well as the structure of the mesons.
S. Furui et al. / Isospin mixing effect
660
The authors
thank
Dr. Maruyama
for helpful
discussions
and
Professor
C.B.
Dover for clarifying problems in the rp puzzle. A. Faessler for his kind hospitality at the Institut
R.VM. is very grateful to Professor fiir Theoretische Physik, Tubingen,
and to the Alexander
for its support.
to Professor during
von Humboldt
M.G. Huber
summer
Foundation
for his interest
1989, Professor
on the subject
and the hospitality
A.M. Green for the hospitality
in Helsinki and Professor E. Klempt for very useful ment. S.F. is grateful to KEK National Laboratory permission to use computers there.
SF. is very grateful
and helpful
in Bonn discussion
information about the experifor High Energy Physics for
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A516 (1990) 643-661
THE ISOSPIN
MIXING
EFFECT
INTO w+m-
IN $p ANNIHILATION
AT REST
AND K&
S. FURUI’ institut ftir Theoretische Kernphys~k,
U~~versit~t Bonn, Nussalfee,
G. STROBEL Institut fiir Theoretische Physik,
and Amand
de ~ysi~ue
~~origue,
FAESSLER
Universitiit Tiibingen, Auf der Morgenstelle Fed. Rep. Germany R. VINH
Division
14-16, 5300 Bonn, Fed. Rep. Germany
14, 7400 Tiibingen,
MAU
IPN 91404 Orsay, France and LPTPE, 75252 Paris, France
Uni~rsif~
Pierre et Marie Curie,
Received 9 December 1988 (Revised 2 April 1990) Abstract: The isospin-mixing effect in pp annihilation at rest into r+Y and KK is studied. We compare the results obtained with a pp optical potential and a meson-exchange interaction which is derived either from the Bonn or from the Paris potential. The annihilation potential is derived from the constituent quark model. We adopt the annihilation model A2, i.e. we assume that the two mesons are created by annihilation of two, and creation of one, qq pairs with the quantum numbers of the vacuum. The annihilation into three mesons is described by the rearrangement model R3. The isospin-mixing effect is sensitive to the choice of the meson-exchange potential. The combination of the Paris potential and the quark-model annihilation potential reproduces the ratios of rr+rrand KK production fairly well when the ss pair creation is suppressed as compared to a uii or a dd pair creation by a factor of about 0.3 in the amplitude. In the ‘SD, channel the model predicts substantial KK production from the D-wave. For the &C production in the P-wave, the model yields comparable contributions from ‘PO and 3P, waves. Predictions of the model on several other two-meson decay modes are also presented and compared with those of other quark models.
1. Introduction
The branching ratios of pp annihilation into two mesons are possible sources of information on the mechanism of qq pair creation and annihilation in hadrons. In a previous work ‘), we observed that the experimental data of pp annihilation in flight into two mesons can be reproduced in the annihiIation model A2 with 3Po vertices better than in the rearrangement models R2 or the rearrangement model 52 with 3S1 vertices. In the annihilation model A2 (see fig. l), we assume that the two qq pairs annihilate and a qq pair is created, while in the rearrangement model R2 and 52 (see fig. l), a qq pair is annihilated and the remaining two quarks and i
Supported
by the Deutsche Forschungsgemeinschaft School of Science and Engineering,
’ Present address: 0375-9474/90/%03.50
@ 1990 - Elsevier
Science
Publishers
and Alexander von Humboldt Teikyo University, Utsunomiya B.V. (North-Holland)
Stiftung. 320, Japan.
644
S. Furui ef al. / Isospin mixing e#eci
two antiquarks rearrange into two mesons. The main reason of the deficiency of the rearrangement model is the strong momentum mismatch between the initial pp and the final meson states. Another reason is the stronger selection rule in the R2 model which forbids the S-wave pp annihilation into two s-wave mesons. In a simple Born approximation, we also studied the branching ratios of pp annihilation at rest into mesons I,‘) and observed that the A2 model reproduces the experimental branching ratios better than the R2 model. Recently the ASTERIX collaboration of the LEAR group 3, published branching ratios of the pp annihilation at rest into n+7~- and KK in a hydrogen gas target and compared with data from a liquid hydrogen target “). In liquid hydrogen the annihilation from S-waves dominates due to Stark mixing and K+K- production is about 5 of the m+n- production, while in the gas target the annihilation from the P-wave dominates and the K+K- production is about & of the T+V- production at atmospheric pressure. In order to make a detailed analysis of pp annihilation at rest into mesons, it is necessary to take into account the isospin-mixing effect in the NR atomic state. Kaufmann and Pilkuhn “) showed that protonium in a P-wave annihilates preferentially in one of the two possible isospin states, namely in the 3P, state, isospin-1 dominates, while in the 'P, , 3P0 and 3Pz states, isospin-0 dominates. The effect was studied with the Dover-Richard model which consists of a phenomenological isospin-independent annihilation potential of the Woods-Saxon type and of the G-parity transformed Paris NN potential “). In the Paris Nfi potential, the annihilation part is derived from the baryon-exchange model which is isospin dependent. The Paris group solved the Schrodinger equation and studied isospin-mixing effects 7). In the iSo channel, the Dover-Richard model predicts that the annihilation occurs preferentially in the isospin-0 state *), while the Paris model predicts that it occurs preferentially in the isospin-1 state. The discrepancy suggests that the isospin mixing depends strongly on the annihilation potential. Thus it is important to check whether the branching ratios of the pp annihilation at rest can be reproduced by one of the models consistently. Oades et al. ‘) calculated the branching ratios of pp annihilation at rest into ~+cTand K’K- using an amplitude given by the quark model lo) or by analytical continuation of the helicity amplitude Nti+ ~~6 [ref. “‘)I. The annihilation into two pseudoscalar mesons occurs mainly from 3S,, 3P, and 3P2states. The phenomenological helicity amplitudes of NR-+ rr+rr- extrapolated to the corresponding kinematical region shows that the branching ratios of rtm- for the 3P, and the 3P0 states are comparable. In the quark model of ref. lo), the decay into ~~6 is described by the rearrangement model with one-gluon exchange 52 (see fig. 1) and by a model with two-gluon exchange A2S (see fig. l), whiie the decay into K+K- is only given by A2S (see fig. 1). This model explains the ratio of KfK- to ~+?r- production but since the rearrangement model forbids the decay into V+T- from the 3P2 channel the decay from the 3P0 channel becomes too large as compared to the 3P2 channel.
S. Furui et aL / Isospin mixing eflect
This
is in disagreement
mechanism
with the results
of NR annihilation
and it is necessary also be derived The Tiibingen
of ref. *) and
into two mesons
to study other mechanisms
from the quark group ‘) derived
model
64.5
experimental
adopted
data ‘). The
in ref. lo) is not unique
I*). The annihilation
potential
should
in the constituent
quark
consistently.
the annihilation
potential
model with the assumption that the intermediate state is given by two- or three-meson states out of 9 s-wave mesons (7r+, 7r”, 7r-, p+, p”, p-, w, 77 and 7’) and 16 p-wave mesons (B+, B”, B-, a’, 6’, 6-, A:, A:, A;, A:, A;, A;, E, D, f and H). Channels including more than one p-wave meson are ignored. The T-matrix is calculated from the Lippmann-Schwinger equation and the annihilation amplitude gives the branching ratios of the pp annihilation into mesons in flight. We adopt the same approach in the analysis of the pp annihilation at rest. Since the branching ratios of pp annihilation into two mesons are well reproduced by the A2 model, we neglect the contribution of the R2 diagram (see fig. 1). The suppression of K+K- production as compared to the 7r+?r- production can be explained by the suppression of the SS pair creation as compared to the uii or dd pair creation ‘32). In the annihilation model A2, the decay into two pseudoscalar mesons occur from both 3Po and 3P2 channels and the form factors of the decay processes from the two channels are
L 17
~ 123
84
654
A2
15
24
~ 123
654
R2
17
123
A3
15
~ 123
R
C 09104
654
2’4
52
654
lb
~ 123
25
34
654
R3
17
123
84
654
A2S
Fig. 1. Schematic diagrams of the quark model for N6l annihilation into two or three mesons. A2 and A3 are the annihilation diagrams, and R3 and R2 are the rearrangement diagrams. S2 is the rearrangement diagram with one-giuon exchange and A2S is the annihilation diagram with two-gluon exchange which are used in ref. r”).
S. Furui et al. / Isospin mixing effect
646
different.
In the Born approximation
without
initial-state
interaction,
we observed
that the branching ratio of the 3P, channel is larger than that of the 3P, channel ‘). In the analysis of branching ratios, it is important to take into account all the open channels
since a specific
decay
mode becomes
effectively
decay modes that rob the flux out of this channel
suppressed
13). We consider
if there
are
the annihilation
into three mesons in the rearrangement model R3, although there are indications that the annihilation model A3 dominates over R3 ‘*r4). The annihilation potential that we use in this paper is the combination of R3 and A2 as in ref. ‘). Since the K+K- system is not an eigenstate of G-parity, a coherent sum of isospin-0 and isospin-1 amplitudes contributes to K+K- production. The relative phase of the two amplitudes depends on the initial-state interaction and consequently the isospin mixing is crucial for the K”K- and K°Ko production. We study the initial-state interaction by comparing results obtained with the one-boson exchange potential of the Bonn group with results obtained with the meson-exchange potential of the Paris group. We combine the meson-exchange potential with the above annihilation potential and solve the Schrodinger equation. Using the wave function obtained in this way, we calculate the branching ratios and compare with experiment. Unlike the calculations of ref. ‘) we ignore the Nd f tiA and Ad channels which give only about a 10% correction to the annihilation cross section, but contrary to that work, we include the Coulomb interaction between the proton and antiproton. The plan of this paper is as follows. In sect. 2 we calculate the wave function of the pp atom in momentum space. In sect. 3 we derive the partial decay widths of pp annihilation into nr+F and KK. Numerical results are shown in sect. 4. In sect. 5 we present predictions of the A2 model on several other two-meson decay modes and compare with those of other quark models. In sect. 6 the main conclusions and a discussion of the most important results will be given. 2. The pp atomic wave function in momentum space In order to solve the Schrbdinger equation in momentum space for the NR system we take the states pp and iin as a basis and consider the following coupled equations;
Im I 0
(Vdk k’) + K&(k
k’) + ViAk, WMp,(k’) d3k’
00
+
(K&k,
W+ ViAk k’)Mdk’)
d3k’=EICr,,(k), (2.14
0
$
&(k)+
k’) + V:im(k, k’)hhi(k’)
d3k’
co
+
I
0
where m is the reduced
(Vim&
k’)+ Kn,(k k’)Mp,(W d3k’=&hi(k),
mass of the NR system,
k is the relative
momentum
@lb) between
S. Furui et al. / Isospin mixing effect
N and isoscalar
N in the centre-of-mass isovector
system,
independent
meson-exchange
V,,
VhEX and
Vi&
are the Coulomb,
respectively. The annihilation potential in V&) where Vi,, is the annihilation potential V,‘,, is given by i( c’,,* the isospin-I channel. In the Dover-Richard model, the annihilation potential is isospin
and
641
potential,
and so Vi,,, is zero. Sm is the proton-neutron
mass difference.
For the meson-exchange potential VhEX, we consider the G-parity transformation of the one-boson-exchange potential of the Bonn group (HM3) 15), and the r, w and two-pion-exchange potential of the Paris group “). In the case of the Bonn potential, we approximate the coupling potential between the pp and the nii channel and the p-exchange. In the case of the Paris potential, V&x by the n-exchange Since the meson-exchange potential loses its validity are defined as for V,‘,,. V&x at short range or at high momentum we regularize the meson-exchange potential with a cut-off factor
(2.2) potential and A is the cut-off where V,,,(k, k’) ,, is the original meson-exchange momentum. We choose A = 1 GeV and n = 10. This cut-off makes the too strong short-range attraction of the meson-exchange potential weaker. In the Paris potential 6), the short-range part is cut off in configuration but not in momentum space. We adopt the simple momentum space cut-off since it allows to express the potential in a simple analytical form. It makes comparison with the other published results difficult but indicates the dependence of the physical quantities of the pp atoms on the cut-off procedures. The solution of the coupled equation (2.1) in momentum space faces a problem due to the singularity of the Coulomb potential. Kwon and Tabakin i6) applied the regularization the eigenvalues mic singularity,
prescription
of Land6 and published
a computer
code for calculating
and eigenfunctions. Although the Coulomb potential has a logarithit is multiplied in the integral equation by the wave function and
the integrand can be calculated properly. We adopt the same technique and solve eq. (2.1) for each partial wave by using the matrix inversion method which was used by the Hannover group in the calculation of the NN bound-state problem 17). We first check
the program
for the pure
Coulomb
interaction
and next with the
meson-exchange potentials V,,, of the Bonn group and the Paris group, and the V,,, of the Dover-Richard model. The results are compared with those of Richard and Sainio “) where the Dover-Richard model in the coordinate space was used. Due to the different procedure of the short-range cut-off, we cannot expect complete agreement of our result for the combination of the Paris potential and the DoverRichard annihilation potential with the results of Richard and Sainio, but we can check qualitative agreement. The combination of the Bonn potential and the DoverRichard annihilation potential allows to see from the results the dependence of the decay width and the energy shifts on the meson-exchange potential. Finally we
S. Furui et al./ Isospin mixingeffeci
648
replace V,,, given by the optical potential by our calculation in the quark model. The derivation of the optical potential in the quark model is shown in ref. ‘). We ignore the strange meson production
in the construction
but we treat this small contribution tively. After the partial
(about
wave expansion,
eq. (2.1) reduces
m(v’CL(k,k’)SLr+vtirELXL.(k,
$2(k)+
of the annihilation
5% of the annihilation
potential
at rest) perturba-
to
k’)+v;,:(k,k’)6,,J&$(k’)kf2dk’
I0 al
+
I
(v$&‘(
k, k’) + v;$
(k, k’))c#$(k’)k’2
dk’= E&$(k),
(2.3a)
0
m
$k%Xk)+
(2&n + vL:$(k,
k’)+
v;;:(k,
k’)S,,,)+$(k’)k’2
dk’
I 0
+
k, k’) + v;,f,f (k, k’)S,,.)&$,(k’)kf2
(vj$:(
where for the spin-independent V(k,
k’) =
Y*Lw(k’),
1 vLTL’(k, k’) Y&k) LML’M’
vk( k, k’) = -
3.
(2.3b)
potential,
vL.~ =_ 2 ~
Here QL(x) is the Legendre potential, the J-dependence
dk’= E&,(k),
I
m o r2 drL(kr)
(e'/
V(r)_hJk’r)
rkk’) QL( k2 + kf2/2kk’)
function of the second kind. is treated as in ref. 15).
, . For the spin-dependent
Decay widths of pp annihilation into m+mr- and KK
The decay widths of an atomic
state with angular
momentum
J and energy
E has
the form ‘*)
:r= r
F
I
d~i(E)I({pi}lQ~~~I~(E))12
3
(3.1)
where dp,(E) is the phase space of the final state, pi are the momenta of the final particles, QHP is the transition operator between the final mesonic state (in Q-space) and the initial baryon-antibaryon state (in P-space) ‘) and F is the distortion operator on the initial state. The total width of the system can be derived from the energy eigenvalue and also from the equation
z-r
I
k2 dk kf2 dk’ &‘LT(k)*lmv~~~(k,
k’)+‘LT(k’),
(3.2)
649
S. Furui et al. / Zsospin mixing effect
of angular momentum J, orbital angular where rPJLT is the N6I wave function momentum L and isospin T. The isospin-0 and isospin-1 wave functions are linear combinations
of pp and iin wave functions, ,
#&W
$JLo(k) =&#&k)+
+JL’(k)=&6$(k)-$~(k)].
(3.3)
from the sum Since the annihilation potential z)if;T in our model is constructed of different meson channels, the width can be written as the sum of contributions from these channels
=,v[dPiC&)j
k2 dk kf2 dk’ ~JLT(k)*(kIPHQ/{pi}) (3.4)
x({pi)lQffPlk’)4’LT(k’). The KfKK-K+
state and K°Ko state are not eigenstates
of G-parity.
We describe
the
as &-K+(P)
=4&(p)
+
&&)I,
(3.5)
$M&)
=&MU4
-
&&)I~
(3.6)
and K°Ko as
The partial
decay widths
for K-K+
and K°Ko are
x&+JLo(k’)+c$JL’(k’)], where for the K-K+
(3.7)
the plus sign and for the K°Ko the minus
sign is to be chosen
in the wave functions. fi is the invariant mass of the NR system. The partial wave quark model amplitudes for the meson production (p(QHPlk) in the constituent are derived in ref. ‘). In rr+Y production with the annihilation model A2, the amplitude
for the 33S1 channel
(pIQHP/k)
is* 3’23(3a2+bZ)(a2+b2)
= A,,N;N:/&
a2(3a2+2b2) (3.8)
l The factor & in ref. ‘) is corrected of the parameter hA2.
to I/&
here. The correction
changes
only the normalization
650
S. Furui et al. / Isospin mixing effect
while the 13P0 amplitude (plQHPlk)
is
= h,,NZ,N:/&i
( 3az2;2b’)1’2
3(3a2+2b2) 1_b2(3a2+b*) and the 13P2 amplitude (plQHPjk)
p
2
(ST’*
(-3)
-3ayyib
(3;:;;;‘)’
1
2 (:p +i k2) (GF) ,
(3.9)
is 3’2 b2(3a2+ b2)(a2+ b2)
= h,,N2,N:/&
(3~2’~2b*)~‘~ a2b2 3a2t2b2
(’ 2+ik2) 4p
(5)
(3a2+2b2)2
1
(OS,).
P2k (3.10)
Here a = 3.1 GeV-’ and 6 = 4.1 GeV-’ are the size parameters for the baryons and the mesons, Na = (3a*/ n-)3’* and N, = ( b2/ r)3’4 are the normalizations of the baryon and the s-wave meson wave functions. The description of a n and K in the constituent quark model is controversial. The Regensburg group *‘) argues that the pion is a highly collective state and the qq component is only a fraction of the physical pion. We adopted this picture in the previous work ‘) and multiplied the 7r wave function by a spectroscopic factor v$. It is not clear whether the same spectroscopic factor should be applied to K, since K is more massive than r. In the flux tube model 2’,22) the r is assumed to be pure qq but its size is smaller than that of other mesons. In pp annihilation into mesons, the two approaches give qualitatively the same results, but in some decay processes there appear significant differences “). A study of the meson-baryon interaction and the decay of mesons indicates that a smaller size for the 7~ without a spectroscopic factor less than unity gives a better agreement with experiment 24). But this does not mean that the description of a pion in the nonrelativistic quark model is now solved. Here we assume that the size of a QTand K are the same as that of other mesons and multiply both wave functions n and K with the spectroscopic amplitude 4. With this spectroscopic factor, our effective value for the pion size is the same as in ref. 24). The spin-flavour matrix elements (Os,) for N6I annihilation into ~+6, K+Kand K°Ko are shown in table 1. If there is no isospin mixing in the initial atomic state, the K°Ko production
is suppressed
as compared
to the K+K-
production
TABLE 1 The spin-flavour
matrix
elements
for NN+
T+T-
and Ki( multiplied
by v$$
due
S. Furui et al. / Isospin mixing efict
to the interference KsKL production which indicates
of the isospin-0 from the atomic
a strong isospin
from the atomic
P-state
of the branching
ratios
and isospin-1 S-state is about
mixing
is about of rr+Y
amplitudes.
Experimentally
70% of the K+K-
the
production
r3)
in the S-state. The KsKs + KLKL production
30% of the K+Kand
651
production
KK production
allows
‘). The comparison us to estimate
the
relative strength of the SS pair creation as compared to the uii or the dd pair creation in the NN system. Although in meson decays SU(3) symmetry violation in the qq pair creation is small, there are indications that the SS pair creation is suppressed factor in the amplitude as a. in baryon systems 25,26). We define the suppression One can include the possible difference of the spectroscopic factor for a r and that for K effectively in this parameter. The amplitudes (3.8)-(3.10) have a specific dependence on the momentum p of the meson. At threshold, the relative momentum for the r+Y system is 0.93 GeV/c and that of the K+K- system is 0.80 GeV/c. The amplitude (3.9) for the 3P, channel is proportional to l-(b2(3a2+ b2)/3(3a2+2b2))p2. It becomes zero at the meson centre-of-mass momentum p = 0.49 GeV/c for a = 3.1 GeV-’ and b = 4.1 GeV-’ and at p = 0.70 GeV/c for a = & GeV-’ and b = v’%GeV-‘. This kind of effect occurs also in the meson-baryon coupling in the 3P,, model 24,27)and it has the same origin as the recoil term of Mitra and Ross 28). The 3P0 model predicts the suppression of the two pseudoscalar meson production from the 3P,, pp state. In the 3P2 channel the amplitude (3.10) is proportional to p2 while in the 3S, channel the amplitude (3.8) is proportional to p. Consequently the K production which has a smaller p as compared to r production is more suppressed in the 3P2 channel than in the ‘S1 channel. Experimentally the branching ratio of K+K- production is about 30% of the r+Y production in the S-state and about 6% in the P-state 3*29).
4. Numerical We calculate (i) the energy shifts binding energy, (ii) the half-width antiprotonic atom which, in the limit antinucleon, is described as a pure
results
AE = E - Ecou,, where Ecoul is the Coulomb $, (iii) the nii mixing probability P in the of large distance between the nucleon and the pp and (iv) the ratio of the decay width from
the isospin-1 state and the isospin-0 state T,/T, for S- and P-wave antiprotonic atoms. In the quark model we adjust the strength parameter of the annihilation potential to fit the pw production cross section at plab = 490 MeV/c [ref. ‘)I, since around this energy the experimental error is relatively small. In the present calculation using the Bonn potential, we choose the annihilation strength parameters to be three times larger than the previous in-flight calculation ‘) in order to increase the decay widths. Since we are interested in the dependence on the meson-exchange potential we choose also the same strengths for the Paris potential. To check the program, we calculate the energy eigenvalues for the Paris potential and the annihilation potential of Dover-Richard (Paris + DRl). Although the cut-off
S. Funk et al. / Isospin mixing effect
652
TABLE 2 The energy shift and width of the pp atom in the model of the Dover-Richard 1, Paris-model, the combination of the Paris potential and the quark model (R3 + A2) and the combination of the Bonn potential and the quark model (R3+A2). &r is the imaginary part of the eigenvalue E of the coupled equation (2.3a) and (2.3b). T,C0j is r’rr=‘(‘) defined in eq. (3.2). P(M) is the admixture of nri component in the asymptotically pp atomic state DRl
AE, +r P(nii) r,/ra
‘SD, AEa
p-
P(nii) r,irs
‘PO AE,
+r
P(M) r,/ra
+I A%
yr
P(M) j-,/r,
‘PI
0.54 keV 0.51 keV 4 x 1o-6 0.68
DRl 0.76 keV 0.45 keV 3 x 1o-6 0.76
Paris 0.85 keV 0.82 keV 2.93
Paris 0.71 keV 0.40 keV 0.88
Paris + quark 0.96 keV 0.62 keV 3 x 10-s 14
Paris + quark 0.65 keV 0.71 keV 4 x lo+ 0.04
-0.074 eV 0.056 eV 2 x 1o-9 0.03
-0.072 eV 0.056 eV 0.0001
-0.029 eV 0.012 eV 5 x 1o-‘O 0.09
Paris
Paris + quark
Bonn + quark
0.01 eV 0.006 eV 2 x lo-I0 8.6
0.03 1 eV 0.0036 eV 6 x 1O-6 6.4
Paris + quark
Bonn + quark -0.026 eV 0.0026 eV 7 x 1o-4 3.5
Bonn + quark
DRl 0.036 eV 0.010 eV 5 x lo-lo 9.5
0.032 eV 0.0096 eV 13.8
-0.030 eV 0.014 eV 0.52
0.01 eV 0.005 eV 1 x 1o-‘O 2.6
3PF,
DRl
Paris
Paris + quark
A-%
-0.005 eV 0.015 eV lxlo-‘” 0.63
-0.003 eV 0.014 eV 0.65
$r
P(nii) r,/ra
0.73 keV 0.34 keV 4 x 1om5 2.2
Paris + quark
-0.026 eV 0.013 eV 1 x 1o-‘O 0.96
P(nii) r,/r,
Bonn + quark
Paris
Paris
AEa
-0.31 keV 1.08 keV 3 x 10-s 0.13
DRl
DRl
;r
Bonn + quark
0.01 eV 0.002 eV 6 x lo-‘* 0.47
Bonn-t
quark
11 eV 30 eV 2 x lo-’ 2.4
-0.085 eV 0.018 eV 1 x 1o-5 0.02
S. Furui et al. / Isospin mixing efect
prescription partial
is different
from
waves. Only in the “P,
ref. 18), which, With the optical comparing
we expect,
channel
we found
qualitative
width
in the quark model, obtained
agreement
a weak repulsion
is due to the cut-off dependence
potential
the decay
ref. i8), we obtained
653
contrary
of the spin-spin
we check the numerical
from the energy
eigenvalue
in most to the force.
stability
by
and the sum of
the partial decay widths. Table 2 shows our results for the Paris potential (Paris + quark) and for the Bonn potential (Bonn + quark) both supplemented by an annihilation potential calculated in the quark model. Here the Paris potential is cut-off in momentum space and so is not exactly equivalent to the original. For comparison we show the results of Richard and Sainio ‘*) who use the Paris potential and the Dover-Richard potential calculated in the configuration space (DRl) and then the results of the Paris group who use the annihilation potential derived from the baryon-exchange model ‘) (Paris). The average energy shift and half width for the ‘So state and the 3SD, state in the Paris + quark model is AE -$T = (0.73 - i0.69) keV. The experimental value is (0.70 f 0.15 - iO.80 f 0.20) keV (ref. 30)). Recent measurements using the cyclotrontrap yield AE -$ir = (0.685 f 0.034 - i0.533 f 0.068) keV for singlet S and AE - $ir = (0.845 f 0.043 - iO.35410.080) keV for the triplet S (ref. “)). The results with the Bonn potential differ appreciably from those of the Paris potential in the 3Po channel. In this channel the bound state due to strong interaction and the Coulomb bound state are accidentally degenerate and one gets a large width. This feature remains when changing the annihilation potential from the Dover-Richard model to the quark model. The main difference comes from the fact that in the Bonn potential the scalar component in the J = 0, L = 1 channel remains finite in the limit of small momentum k and k’, while in the Paris potential the component in the L = 1 channel becomes 0 in the same limit. Thus the Bonn potential is more attractive than that of the Paris potential in the NR i3Po channel. The differences in the NN channel do not show up in the NN channel, since there the u-exchange and w-exchange largely cancel each other. The differences in the results of the NR annihilation could be due to the annihilation potential of the R3 diagram which is of long range and concentrates the wave function to smaller distances. In fact in the ISo channel the Bonn+ quark model shows an attraction which is in contradiction to the other models and experiment. By choosing the size parameter of the nucleon to be smaller by 10% the sign of the energy shift changes 32). The KK creation probability depends on the strength of the ss pair creation. In the hadronization process of e+e- at high energy, one observed that the SS pair creation in the flux tube is suppressed as compared to the uii or the dd pair creation due to the higher constituent mass. The argument is analogous to the spontaneous e+e- pair creation in the strong electric field 25). We expect the same effect in the NR annihilation. The ratio of the Kl? production and rTTcr- production in the model of Bonn + quark and Paris + quark without the suppression factor for the SS pair creation are shown in table 3. The experimental branching ratios of rtnT- and
S. Furui et al. / Isospin mixing effect
654
TABLE 3a The ratio of K+K- production and ~T+P- production from the pp atom in the quark model (R3 + A2) with the Paris potential 3SD,
3P0
3P,
K+K-Jr+/ KOKO/n+?TKK/rr+Y
5.0 2.8 7.8
0.66 0.23 0.9
0.8 0.02 0.8
CY
0.28
0.30
KK allow to extract the suppression factor (Y for the SS pair creation as compared to the uii or the dd pair creation. We choose (Y so that in the 3SD1 channel the KK production becomes 60% of the rr+7~- production and that the average value for the 3P0 and 3P2 channels is 8%. In the 3SD, channel the KK is mainly produced from the D-wave. Since in the D-wave K°Ko production is favoured, the ratio of KLKs/K+Kbecomes close to experiment. In the P-wave the ratio of (K,K,+ K,K,)/ K+K- depends on the meson-exchange potential. The Paris potential suggests a suppression factor LY~0.3 for both 3SD, and P-waves and the ratio K°Ko/K+Kis 56% for 3SD,- and 30% for P-waves (table 3). Experimentally KsKL production is about 70% of the K+K- production for a liquid hydrogen target “) and KsKs+ KLKL production is about 30% of the K+K- production for a gas target 29). (Y= 0.3 follows also from the AA annihilation into n+rand K+K- in the region of the 5(2220) resonance 26). The partial widths of rr+C,
K+K- and K°Ko for the Paris + quark model is shown
in table 4. The S-wave rr+C production is small due to the unfavoured isospin mixing. The isospin mixing, however, depends on the meson-exchange potential. In the Bonn+
quark
model
the partial
width
of rrtnT-
in this channel
is twice as
large as that in the Parisfquark. As discussed in ref. 43), we find a strong tensor coupling between the S- and D-states in the 3SD, channel. The annihilation from D-wave is comparable to that from the S-wave. In the P-wave the rr+Y widths in
TABLE 3b The ratio of K+K- production and z-‘v- production from the pp atom in the quark model (R3 + A2) with the Bonn potential
K+K-/T+~KF/n+VKK/V+?Y a
%D,
3P0
3P,
1.6 2.2 3.8
2.5 4.4 6.9
0.2 0.4 0.6
0.40
0.22
S. Furui et al. / Isospin mixing effect
655
TABLET
The width of Kl? production and ~+r-
3SD, 3P0 3P,
1.4 keV 25 meV 4.6 meV
production from the pp atom in the quark model (R3+A2) with the Paris potential
0.45 eV 2.2 meV 0.07 meV
2.24~1~= 0.19 eV 1.4u2=0.12meV 0.06aZ= 0.005 meV
1.27cz2=0.11 eV 0.51u’= 0.043 meV O.O016a*= 0.00013 meV
the Paris+quark model are similar to those of the phenomenological helicity amplitudes calculated by Oades et al. ‘). The branching ratio of rTT+r- from the S-wave is 0.7 x lop3 in the Bonn+ quark model and 0.25 x 1O-3 in the Paris+quark model, which are smaller than the experimental value (3.19+0.20) x 10m3 [ref. ‘)I. In the P-wave, the experimental value is 3, (4.81 f 0.49) x 10p3. The theoretical value depends on the way of weighting the contribution of the different atomic states but in the model of Paris+quark we find it is about 20 x 10m3. The origin of the disagreement in the branching ratio can be attributed to the use of R3 for the three-meson production which yields the annihilation potential with a longer range than the A2 model. We expect the combination of A2+A3 to give better agreement 14,33). 5. Comparison with other models of NN annihilation into two mesons NN decay into two mesons has been discussed by several groups models ‘,12,23,34-36).They disagree on whether the rearrangement annihilation model A2 dominates. In this section we compare the annihilation into two mesons other than rrr and KK in the A2 quark
rearrangement
model
R2 in the following
decay
channels:
by using different model R2 or the prediction of NN model and in the pn and pn’, “7
and Z-V’, ~6 and rB, rf and .rrA2 and give a comment on the decay correspondence to the SS pair creation, we consider also the production %-iTw.
5.1. pp+pv
into rp. In of ~4 and
AND pi’
Experimental branching ratios of pp annihilation to pv and pi’ differ from each other among different measurements. We quote BR( pp + pv) = (0.22 f 0.17, 0.64 f 0.14, 0.97*0.16)x lo-* and BR( pp+pprl’) = (0.11*0.06,0.14*0.08)x lo-* [ref. 37)], which indicates BR( pp + pn)/BR( pp + pv,~) = 2 -9. Since the R2 model does not allow these decays from S-wave pp Green and Liu 36) calculated the amplitudes of the two-step process 33S1 NN + B n, v’( I = 0) + pq T,I’( I = 1). They found that the pq’ production is too large as compared to the pq production. In the A2 model, and choosing the same meson size as Green and Liu 24), i.e. a = 0.6 fm and the mixing
S. Furui et al. / Isospin mixing e#eci
656
angle 0,, = - 10” and the size of a nucleon
b = 0.56 fm = 2.83 GeV-‘,
we find the ratio
BR( pp + pn)/BR( pp + ~77’) = 2.74. The difference between the A2 and the R2 comes mainly from the strength of the damping of the form factor.
5.2.
plZ+“q AND ~7’
In the effective chiral lagrangian model, the flavour singlet component of an 77 meson does not couple to Nr;[ in the lowest order of the l/N, expansion 38). In this model, the ratio of branching ratios BR(pp-+ rr”r]‘)/BR(pp+ ~‘7) is given by tan2 or, = 0.038, ignoring a small correction from the phase space. Experimentally the ratio is (0.5OkO.19) x 10p3/(0.46*0.13) x 10p3, or the n and 7’ production are comparable. Due to the spin-parity selection rules, nn and ~7’ productions from the S-wave pp are forbidden and they are produced from 3Po and 3P2 states. In the A2 model, the n and 17’ production are comparable since the damping of the amplitude for the high relative momentum is not so significant. In the R2 model, however, the ~7 and ~7’ are produced only from the 3Po state and the rn production is suppressed as compared to the rrn’ production due to the high relative momentum.
5.3.
pp+&
AND
Experimental nB) = (0.7+0.1,
vB
branching ratios of pp annihilation to TB and rr8 are BR(13S1Nf + 1.96*0.27)+ 10e2 and BR(“S,NR+ rr6) = (0.69*0.12) x 10-2. In
the R2 model the quotient of the branching ratio BR( pp + nB)/BR( pp + rrS) without initial state interaction and with the statistical weight for the triplet and the singlet state, becomes 104 [ref. ““)I. This large ratio comes from the ratio of the ClebschGordan coefficient squared (50 x 3/6) and the momentum dependence of the form factor. In the A2 model the corresponding Clebsch-Gordan coefficient squared is (4.167 x 3/4.5) and the momentum dependence of the form factor is weak. With the same parameters as in sect. 5.1, we obtain this ratio as 3.4 which is close to experiment.
5.4. pp+rrf
AND
VA,
Experimental data of pp annihilation to nf and nA2 are BR( pp+ rf) = (0.24+ 0.07) x 10d2 and BR( pp+ r*Az) = (3.78 * 0.87, 5.71 f 0.85, 2.83 * 0.32) x 10-2. Since the relative angular momentum between the two mesons are Z= 2, the processes are forbidden in the R2 model. Green and Liu 36) calculated the amplitude for the two-step processes 31SoNR +~~(I=O)+~f(l=2)and”S,N~+rr~(l=O)+ vA,(I = 2). They found that the amplitudes are too small. In the A2 model the processes are allowed and the rA2 production occurs also from the 33SD, state. Without initial state interaction and with the same size parameters as in sect. 5.1,
S. Fumi et al. f Isospin mixing effect
we obtain production
BR( pp + rf)/BR( from the S-wave
pp+ NR
r&A:)
657
= 0.45. The A3 model predicts
is forbidden3’).
The experiment
that the rrf
with a liquid
hydrogen target which is dominated by the S-wave pp annihilation does not show a peak coming from annihilation into wf. The pn annihilation in flight extracted from pd reaction
shows a relatively
large rf production
from the S-wave 40), but a
recent experiment with a hydrogen gas target indicates pp annihilation small in the S-wave but relatively large in the P-wave 41).
into nf is
Experimentally, S-wave NN annihilation into mp is dominated by the 33S, channel although the “So channel is allowed by the spin-parity selection rule. In the case of P-wave, the extraction of the 11P1/33P, ratio from the experimental data has changed from time to time. At the Durham conference, dominance of the “P, channel was suggested 4), but the analysis was corrected at the Mainz conference and dominance of the j3P, channel was reported 42). Improvement of the statistics revealed that there is an additional rrr resonance of 1565 MeV which is expected to have the quantum numbers 2+. The resonance which is called AX(1565) and r” are produced from the 3P, or 3P2 NR state and the branching ratio of rrp from 3P, and 3P2 states turned out to be smaller than that of ‘P, state. The reduction in the 3P,,2 can be understood by the dynamical selection rule 13). The angular momentum dependence of the s-p production from the pp atom has been a puzzle. Based on the pn experiment 40) and using the A2 model, Dover et al. 13) explained the smallness of np production from the ‘So as a result of the large rrf decay mode in the same channel. A more refined calculation of the A2 model shows, however, the rp production from the ISo channel is not so suppressed as compared
to the rrf production ‘743). Th e enhancement of the ‘SrNR over *SONN can be explained by the strong tensor coupling of the 3D,NN channel which yields substantial rp production 43). The ratio of the contribution 33P,,2NR over *‘P,NR predicted in the A2 model agrees with the results given in ref. 42), but does not agree with the new analysis of the data with higher statistics 41). The quotient of the branching ratios BR(31S,NR -+ prr)/BR(31SoN~+ frr) in ref. 43) is much larger than that calculated in ref. 13). There are two sources for the discrepancy. the nucleon
First, in ref. 13) the meson size was assumed to be much smaller than size which is an improper approximation. As a consequence the damping
of the form factor in the model of ref. 13) turned out to be too strong. Second, the spin-flavour matrix element squared for the rp production in the ref. 13) is too small by a factor i relative to the rf production. The authors of ref. 43) claimed that the A3 model does not solve the pv puzzle. There, the philosophy is that one of the diagrams is the leading one and determines the annihilation cross section. For the P-wave pm puzzle the authors excluded contributions from the A3 diagrams. One could invoke a modified A2 model
658
S. Fumi et al. / Isospin mixing effect
in which the p in the final state decays into two pions. processes
are included
over the A2 model and r,)
in different
and incorporated
in the generalized
is that the resonance channels
A3 model. energy
In ref. 39) the corresponding
An advantage
and width
of the A3 model
of two pions
(e.g. mp
such as vp, rrf and rrAX can be taken from experiment
in the coherent
sum of the amplitudes.
An analysis
of the angular
distribution of the Dalitz plot in the generalized A3 model shows that the extraction of the branching ratio “P,(7r~)/~~P,(rrp) depends strongly on the details of the model like the pion radius and a clear separation of the 3P, and 3P2 contribution is not easy 39,4’). Thus it is extremely difficult to extract the above ratio from the data.
5.6.
pD+m+ AND mo
The production of 4 from the NN annihilation provides information on the SS pair creation and the reaction mechanism. The comparison with w production will in addition supply information on the mixing angle of the SU(3) flavour singlet with the octet. Experimentally the branching ratios of pp annihilation at rest into ~4 and ~MI are BR(pp+ rr4) =(0.3OiO.15)~ lop3 and BR(pp+ rra)= (5.2 f 0.5) x 10e3 [ref. “)I. In the standard theory of flavour mixing, one expects the quotient of the branching ratios is BR( pp + rr4)/BR( pp + rm) = tan’ (Bv- 0,) = (4.2 - 8.6) x 10p3, which is much smaller than the experimental data. Here we used Bv = 35.3” and the phenomenological value of 13~= 36” - 39” [ref. “)I. The ratio from the pn annihilation is even larger than that from the pp annihilation. Ellis et al. 45) claimed that it is due to the large SS content in a proton or antiproton. Another interpretation of this enhancement of ~4 production is the contribution of a qsqs resonance or C(1480) with quantum numbers 33S1 [ref. ““)I. We interpret a 4 as a KK resonance and consider the pp annihilation into KKr. The K and K form a 4 by the final-state interaction. In this model, an SS pair creation is necessary and the OZI rule is not violated. The ratio of rr+ and TMJproduction is related to the ratio of the SS to uii or dd creation. An analysis of this process in the A3 model will be discussed
in a separate
paper 39).
6. Summary and conclusions quark model We studied the pp annihilation into nTr+r- and KK in the constituent and assumed that the annihilation amplitude is given by the quark pair creation model or the 3P0 model. We observed that the model predicts the relative strength of the 3P2 state and the 3P0 state consistent with the phenomenological helicity amplitudes of NR + rTr+np. It is different from the quark rearrangement model which finds the 3P0 contribution to be very large. We compared the A2 model, the R2 model and other models for NR annihilation into two mesons in several channels and observed that the A2 model gives the best agreement with the experimental
S. Furui ef al. / Isospin
data.
The Mainz
mesons
group
by SU(3) invariant
calculated amplitudes
branching
mixing egeeect
ratios
and claimed
659
of pp annihilation
the impo~ance
into two
of the rearrange-
ment model in the rrtrTT- production 35). In this calculation, flavour matrix elements are assumed to be independent of the spin angular momentum part. in our quark model with SU(6) wave functions and 3Po vertices, the coupling of spin and flavour is crucial and the selection rules are quite different from that of the SU(3) model. Furthermore, the differences in the form factors of the rearrangement model and the annihilation model should be taken into account when discussing the differences in the two models. We compare the Bonn and the Paris potential for the NN interaction. The Bonn + quark model gives a too large shift and width in the ‘P, channel. It could be due to our specific annihilation potential, especially that of R3 which has relatively long range and makes the wave function sensitive to the intermediate and short-range part of the meson-exchange potential. We find a node in the 3P0 channel wave function which enhances the short-range part of the wave function. The Paris + quark model predicts the ratio of Kl? production to rrfC production and the ratio of K’l?’ and K’K- consistent with experiment when the suppression factor (Y for the SS pair creation compared to UU or dd pair creation is about 0.3. In the 3SD1 channel the KE production from the D-wave turned out to be dominant. The 3Po model predicts that in the ‘S, channel K'l?' production is suppressed but in the 3D, channel it is not suppressed as compared to K+K- production. We took the rea~angement model R3 for the three-meson annihilation and the annihilation model A2 for the two-meson annihilation. The analysis of the branching ratios in flight ‘) and at rest in the Born approximation i4) suggest that the annihilation into three mesons may be dominated by the annihilation model A3. It is clear that the present result must be further improved. A more involved calculation with the combination of A3 and A2 will be done in the future. We have chosen a relatively large size for the pion and used the same spectroscopic factor 4 for 7r and for K mesons. Our effective value for the pion size by using the above spectroscopic factor is 0.35 fm. The flux tube model suggests a size of a pion which is slightly smaller. The size of mesons modify the angular distribution, especially
in the Dalitz plot of the TGT~ decay 39). It will change the branching
ratios
to rp and wf meson states. The analysis of the Dalitz plot of the m~m and the KKrr decay from NR and also from mesons will provide a more realistic description of the r and K wave functions. The sensitivity to the meson size comes from the interference of different quasi-two-body decay modes and is rather specific to three-meson decays. In the r+n. and K+K- angular distribution of the pp annihilation, the initial- and the final-state interactions are more important than the meson size. To conclude we remark that the ratio of 7r+r- to Kl? production is sensitive to the nature of the NN interaction and thus detailed comparison with the experimental data will provide information on the initial NN wave function, on the annihilation mechanism, as well as the structure of the mesons.
S. Furui et al. / Isospin mixing effect
660
The authors
thank
Dr. Maruyama
for helpful
discussions
and
Professor
C.B.
Dover for clarifying problems in the rp puzzle. A. Faessler for his kind hospitality at the Institut
R.VM. is very grateful to Professor fiir Theoretische Physik, Tubingen,
and to the Alexander
for its support.
to Professor during
von Humboldt
M.G. Huber
summer
Foundation
for his interest
1989, Professor
on the subject
and the hospitality
A.M. Green for the hospitality
in Helsinki and Professor E. Klempt for very useful ment. S.F. is grateful to KEK National Laboratory permission to use computers there.
SF. is very grateful
and helpful
in Bonn discussion
information about the experifor High Energy Physics for
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