Quark Rearrangement Model for Nucleon-Antinucleon Annihilation at Low Energies Faculty 0

16 Pages·2003·0.6 MB·English

937
Progress of Theoretical Physics, Vol. 69, No.3, March 1983
Quark Rearrangement Model
for Nucleon-Antinucleon Annihilation at Low Energies
Masahiro MARUYAMA
Faculty 0/ Engineering Science D
o
w
Osaka University, Toyonaka 560 n
lo
a
d
e
(Received September 16, 1982) d
fro
m
Nucleon·antinucleon annihilation process at low energies has been explained by the quark
h
rearrangement model with parameters ga, where ga describes the strength for a quark·antiquark ttp
opfa itrh eto m ceosmonb ianne di nisto i na thmee soornd e(rJ ro, fl Jt, hPe, mwe).s onT mhiass sp.a raItm ies tsehro hwans sthtraotn tgh edseep epnrdoepnecrtei eosn o trhieg ikniantde s://a
c
from the overlapping of quark wave functions of hadrons. The relation to the potential model a
d
for NN interaction is also discussed. e
m
ic
.o
§ 1. Introduction up
.c
o
m
In a previous paper 1,1) we have proposed a new model for the mechanism of /p
nucleon-antinucleon annihilation at low energies, where the annihilation process tp/a
has been considered to occur through rearrangement of three quarks in nucleon rtic
le
and three antiquarks in antinucleon into three mesons which then decay into a -a
b
number of pions. In this model we have introduced an adjustable parameter ga stra
for each meson which describes the strength for a quark-antiquark (qij) pair to ct/6
combine into a meson. The remarkable feature in this parameter is that its 9/3
value is strongly dependent on the kind of the meson, a, and approximately in the /93
7
order of the meson mass, /1
8
7
1
8
(1) 0
6
b
y
which has been conjectured to be due to the effect of the 5U(6) symmetry g
u
e
breaking on the combining processes. s
t o
The purpose of the present paper is to clarify the origin of the feature of ga n 1
7
by the use of quark wave functions. Although the annihilation mechanism is not N
o
well known, it may be supposed that the strength for combination of qij should v
e
m
be proportional to the overlap squared of the spatial wave functions of initial and b
e
final states. Therefore the strong dependence on the kind of meson and the order r 2
0
would be explained from the properties of the wave functions. 18
In § 2, the quark rearrangement model proposed in paper I is reviewed and
the present approach to the strength for qij combination is described, where we
introduce quark wave functions. We construct the internal quark wave func
tions of hadrons by means of the non-relativistic spring-junction model in § 3.
938 M. Maruyama
Comparisons with the results in paper I and with experimental data are given in
§ 4. Section 5 contains conclusions. Here we also discuss the relation of the
results to the imaginary potential in NN potential models.
§ 2. A quark rearrangement model for NN annihilation
Let us review the quark rearrangement model proposed in the previous paper D
o
w
1.1 The basic idea about the quark rearrangement process for NN annihilation n
) lo
into mesons at low energies is shown in Fig. 1: Three mesons (a, /3, r), or three ad
e
qanijt ipqauiarsr,k sa rien canretiantuecdl eboyn t, haen dre tchoemn bdiencaatyio inn toof pthiornese aqnuda rpkhso tion nnsu acclceoonrd ainngd ttoh rtheee d from
decay modes of the mesons.2) We have assumed the following: http
(i) The relative angular momentum between q and ij is zero, so that the s://a
three mesons (a, /3, r) are any three of 7[, 7J, (jJ and p. *) ca
d
(ii) Since the annihilation is considered at rest or at very low energies, em
orbital angular momentum between nucleon and antinucleon is zero. ic.o
u
(iii) Spins and isospins of quarks and antiquarks do not change during the p
.c
reaction. om
The probability with which the initial NN system with the isospin I and the spin /ptp
5 annihilates into three mesons (a, /3, r) has been written as follows: /artic
le
P(I, 5, a, /3, r)= b(I, 5 )C(I, 5, a, /3, r )gagpg7 V(a, /3, r; ECM=2M), (2) -ab
s
where C(I, 5, a, /3, r) is the overlap squared of the 5U(6) spin-isospin wave tra
c
functions of the initial NN state and the final three meson state. b(I, 5) denotes t/69
/3
the spin and isospin dependent parameter to represent the contribution from the /9
3
initial NN interaction. ga is the parameter which describes the strength for a qij 7/1
pair to combine into the a meson. V(a, /3, r; ECM=2M) represents the phase 87
1
space volume of the final three meson state at the NN threshold, 80
6
b
y
V(a, /3, r; ECM)= !dmldm2dmsla(ml)!p(m2)!7(m3) gu
e
s
t o
n
1
7
N
o
v
e
m
b
e
r 2
0
1
8
N N
Fig.1. The quark diagram of NN annihilation into three mesons (a, /3, y).
*) Here it is also assumed that "fJ and (j) do not involve the strange quarks (ideal mixing).
Quark Rearrangement Model 939
(3)
where pi, Ei and mi are momentum, energy and mass of the i-th meson re
spectively, and Ei is given by
(4) D
o
w
n
is the total energy in the center of mass frame, given at rest by lo
ECM a
d
e
d
(5) fro
m
wacictho utnhte, wnuec lheaovne minatsrso dMuc. edT toh et amkea stsh ed idsterciabyu tiwoind tfhusn cotfi opn a/an(dm ()J). m/e,s,(omns) ianntdo https
/~ ( m) are taken to be ://ac
a
d
/,,(m)=o(m- m,,), (6 ) em
ic
.o
/~(m)= o(m-m~), (7) u
p
.c
o
where m" = 134.96 and 139.57 MeV for 7[0 and 7[± respectively and m~ = 548.8 MeV. m
/p
/p( m) and /w( m) have been determined by the experimental data phenom tp
/a
enologically (see the dotted curves in Figs. 2 and 3 ).1) rtic
Fitting the experimental data of pj5 annihilation at rest, we have obtained le
-a
the values of parameters, ga and b(I, 5), bs
tra
c
(g,,=O.1 fixed), b(O, 1)=174.9±27 , t/6
9
/3
g~=O.74±O.1 , b(1, 1)=173.5±17 , /9
3
7
/1
8
7
1
8
0
6
b
y
g
u
e
s
t o
n
1
7
N
o
" v
/ e
/ m
b
e
200 400 600 800 1000 m (MeV) 400 600 1000 m (MeV) r 201
8
Fig. 2. The mass distribution function !p(m). Fig. 3. The mass distribution function !w(m).
The solid curve indicates !p( m) in the pres· See the caption of Fig. 2.
ent approach, which should be compared
with the dotted curve describing !p( m) in
model I.1
)
940 M. Maruyama
gp = 1.66±0.1 , b(O, 0)=430.5±100,
g",= 1.26±0.2 , b(l, 0)=145.5±50. (8)
In this results we find that ga is strongly dependent on the kind of the meson, a,
and approximately in the order of the meson mass. It is considered to reflect the
difference among the meson wave functions owing to SU(6) symmetry breaking. D
o
In the present paper, we will clarify the relation between the differences w
n
among ga and the spatial wave functions of nucleons and mesons. The interac· loa
d
tion in the rearrangement process shown in Fig. 1 is not well known. Here let us e
d
suppose that nucleon (antinucleon) has a junction (antijunction) and the annihila· fro
m
tion process is induced through the pair annihilation of the junction and the h
antijunction. Other possibilities, for example, the case without junctions would ttps
be investigated in the forthcoming papers. We assume, as in paper I, that the ://a
c
a
process shown in Fig. 1 dominates NN annihilation into mesons, although several d
e
m
other processes are allowed for NN annihilation in the framework of the quark ic
.o
rearrangement model. 3),4) This assumption is supported by the previous result as up
well as some other considerations.3 Instead of Eq. (2) we propose as the .c
),5) o
m
probability according to the present approach as follows: /p
tp
/a
rtic
le
-a
b
s
tra
c
t/6
9
/3
/9
3
7
/1
8
x Jr dI 3r l , d3r z , .. , d3r s ' 'llT:'r f * ( rl", rz , "', rs , ; a, /3 ,r ) 718
0
6
b
y
g
u
e
s
(9 ) t o
n
1
where r;(i=l, 2, 3) is the spatial coordinate of the i·th quark in nucleon, r;(i=4, 7 N
5, 6) is that of the i·th antiquark in antinucleon, and rA r j ) is that of the junction ove
3 m
(antijunction). Voo (rj- r j )o3(rl-r/)'''o3(rs- rs') denotes the interaction for b
e
the annihilation, where 03( rj- r j ) term means that the annihilation occurs only r 2
0
when the junction and the antijunction spatially overlap each other. Iff; (rl, "', 18
rs, rj, Tj;J, 5) and Ifff(rl, "', rs; a,/3, r) represent the quark wave function of
the initial NN system with the isospin J and the spin 5 and that of the final three
meson system respectively. By the use of the internal wave function of a meson,
Xa, and that of nucleon, ¢N, they can be written as follows:
Quark Rearrangement Model 941
cxexp(i(PN+PR)R)f])NR(X,I, S)¢N(r1, r2, r3, r;)¢R(r4, r5, rs, rJ),
(10 )
D
o
w
n
where R and X are the center of mass coordinate and the relative coordinate of loa
d
e
nucleon and antinucleon, and f]) NR(X, I, 5) represents the relative wave function d
between nucleon and antinucleon. Here the meson wave functions are assumed fro
m
to be approximated by the plane waves. Let us consider the annihilation at rest h
ttp
or at very low energies. In this case the relative angular momentum· between s
nucleon and antinucleon can be restricted to zero, so that f]) NR(X, I, 5) could be ://ac
a
d
approximated to be independent of X, e
m
ic
(12) .ou
p
.c
o
Our present model has these four adjustable parameters b(I, 5) in all provided m
/p
that the internal quark wave functions, Xa and ¢N, are prepared. tp
/a
"In the present approach fp( m) and fw( m) are replaced with new ones, as rtic
shown by the solid curves in Figs. 2 and 3. The main points changed are le
-a
(i) fp(m) and fw(m) at large m are lowered to be zero above m=1200 MeV, bs
(ij) fp( m) is made higher and fw( m) is lowered at small m (m < 700 MeV) trac
slightly, t/69
/3
which would be within the ambiguity for the derivation of fa( m) from the /9
3
experimental data. This replacement does not change the results qualitatively, 7/1
8
but is necessary to the quantitative agreement. 7
1
8
0
6
§ 3. Internal wave functions of nucleons and mesons by
g
u
e
s
This section concerns construction of internal wave functions of hadrons in t o
n
terms of non-relativistic spring-junction modelS) based on the string-junction 1
7
picture of hadrons. It is assumed that the confining interaction is harmonic N
o
v
oscil1ator potential with a spring constant x and mass splitting of hadrons e
m
foorri gninuaclteeos nf riosm o fo nthee g lfuoormn exchange potential based on QCD.7) The Hamiltonian ber 20
1
8
H= - 2~q (P p p /)- 2~ P /
12+ 22+
+ ~ x{(r1- r;)2+(r2- r;)2+(r3- rd}
942 M. Maruyama
+b 23 - mC2q3 2 - md237 \f2 +l3i S2 S3 )
(13 )
D
o
w
n
where ri and r] are the coordinates of the i-th quark and the junction re loa
d
spectively, and e
d
fro
=1p m
b .. (14) h
l.J I TOI ' ttp
s
://a
c
(15) a
d
e
m
ic
.o
(16 ) u
p
.c
o
m
(17) /p
tp
/a
as represents the coupling constant of the strong interaction, and mq and f1 are rtic
masses of quark and junction, respectively. le
-a
We construct the eigenfunction of the total Hamiltonian, using the var bs
iational method. Consider the case of the Hamiltonian without the one gluon trac
exchange potential, and as is well known the solution of the ground state in the t/69
/3
center of mass rest frame has the Gaussian form, /9
3
7
/1
8
7
1
8
0
6
b
(18) y g
u
e
s
where t o
n
1
t=rl-r2, (19) 7 N
o
v
e
m
(20) b
e
r 2
0
S= r]-T1( rl + r2+ r3), (21) 18
R=3 1+ {mq(rl+r2+r3)+f1r]}. (22)
mq f1
A trial function is defined as
Quark Rearrangement Model 943
_( 3Cs / mqflX )3/4( Ct )3/4( 2Cq )3/4
rPN(rl, r2, r3, rJ)- -;ry 3mq+fl 2i JmqX ~JmqX
xexp[ - ~ !3::'tflCsS2- ~ JmqxCqq2- ~JmqxCteJ
x (5U(6) spin-isospin wave function) (23 )
D
with variational parameters Cs, Cq and Ct. The root mean square radius given o
w
n
by this wave function is lo
a
d
e
d
(24 ) fro
m
h
where Cq= Ct due to the spherical symmetry. Choosing the relevant mass mq ttp
s
and fl, and fitting experimental values of the mass difference between proton and ://a
c
,1 (1232) and the proton charge radius, we obtain the values of the spring constant a
d
e
x and the coupling constant as. Taking the proton charge radius to be m
ic
.o
R=0.75~0.85 fm, (25) up
.c
o
we find a number of parameter sets m
/p
tp
X= 160~300 MeV fm-2 , (26) /artic
as=1.2~1.5 , (27) le-a
b
s
and for proton tra
c
t/6
(28) 9/3
/9
3
(29) 7
/1
8
7
in a rather wide range of mq and fl, 1
8
0
6
mq=90~150 MeV, (30) by
g
u
fl=100~500 MeV. (31) es
t o
n
The value of quark mass is rather small, compared with ones in the other models, 1
7
for example, the Isgur-Karl modelS) (mq=330 MeV). This is because in our N
o
model the wave functions are taken to be eigenstates of the total Hamiltonian ve
m
with one gluon exchange potential (OGEP) and to reproduce the proton charge b
e
radius (0.75 fm), while in their model the OGEP is treated as perturbation and the r 2
0
1
proton radius is 0.6 fm. In our model kinetic energies of quarks in proton are 8
larger than those in ,1 (1232), so that the energy difference by the OGEP between
proton and ,1(1232) needs to be larger than in the Isgur-Karl model. This leads
to lighter quark mass on account of the term proportional to mq2 in the OGEP.
Moreover, since the radius is proportional to (mqx )-1/4, the larger value of the
944 M. Maruyama
proton radius leads to the smaller quark mass. We also investigated the case in
which the OGEP is treated as perturbation. Taking the proton radius to be 0.75
fm as Eq. (40), we find that the result is in agreement with that in Table I within
10%, independent of mq. Therefore our result is consistent with that in other
models. Our quark mass value is different also from that obtained by a naive
evaluation by use of proton magnetic moment (mq~300 MeV). However it
would be necessary to consider the corrections from meson cloud contribution D
o
w
and the relativistic effect in this evaluation. Therefore we do not take the above n
lo
fact seriously. ad
e
N ow consider the case of meson with the Hamiltonian d fro
m
h
ttp
s
://a
c
(32) ad
e
m
ic
where b', c' and d' are defined in equations similar to Eqs. (14)~ (16). The .o
u
p
internal wave function has the form, .c
o
m
/p
tp
/a
rtic
x (5U(6) spin-isospin wave function), (33) le-a
b
s
where tra
c
t/6
(34) 9
/3
/9
The experimental value of the hyperfine splitting among J[, Tj, p and (J) mesons 37
/1
cannot be, however, reproduced in this model. As the internal wave functions of 8
7
1
mesons do not affect the final results so much, Ca is determined by means of the 8
0
6
following way without getting the solution by the use of the variational method: b
y
g
(i) The values of mq, x and as in Eqs. (26), (27) and (30) are used. ue
s
(ii) The radius of J[ m1e sRon iics }re produced, t on 1
7
v~<r_->{-3 -- 1/2 N
~8 -Cn mqX ov
e
m
=0.6~0.8 fm. (35) be
r 2
0
In addition 1
8
(36)
since meson is more deeply bound.
J[
(iii) The difference between the expectation values of the Hamiltonian for
J[
Quark Rearrangement Model 945
meson and p meson, Llm"p, is taken to be the experimental value of the
mass splitting (~640 MeV): CIT cannot be taken to be a large value, for
the expectation value for meson remarkably decreases with CIT, or for
1[
C=1.0
(37)
Then the possible value of mq is fairly restricted. D
o
w
n
§ 4. Results loa
d
e
d
We compare the present results by means of Eq. (9) with the ones by the fro
m
previous model I using Eq. (2) at first. The results of the present approach have h
ttp
the ambiguity of a factor of two or three according to the choice of the pa· s
rameters. Hereafter let us take a set of parameters to be ://ac
a
d
e
mq=100 MeV, (38 ) m
ic
.o
f.i=150MeV (39 ) up
.c
o
and m
/p
tp
!<r2>=0.75 fm (40) /a
rtic
le
for proton, and then we get -a
b
s
x= 166.5 MeV ·fm-2 , (41 ) trac
t/6
as=1.35 , (42) 9/3
/9
3
and for proton 7
/1
8
7
Cs=0.82 , (43) 18
0
6
Ct=Cq=2.80. (44) by
g
u
e
Consider the following ratio of each three meson channel to pO 1[0 1[0 channel: st o
n
Q(I , 5 ,a, (3 ,r )-- PU,b 5U, ,aS, )(3 , r) .. P(I, 1b, (pIO,,1 1)[ 0, 1[0) , (45) 17 N
o
v
e
where QU, 5, a, (3, r) does not involve bU, 5); that is to say, it has no pa m
b
e
rameters provided that the wave functions are given. The results of numerical r 2
0
calculation are shown in Table I. The fifth column is the result of the previous 1
8
model I with ga given in Eq. (8), which should be compared with the fourth column
describing the case in which all ga have the same value, or only CU, 5, a, (3, r)
and V(a, (3, r; E are taken into account in Eq. (2). The last two columns
CM)
represent the results of the present approach with the use of the hadron wave
946 M. Maruyama
Table I. Q(I, S, a, /:3, Y) defined in Eq. (45).
C(ISa/:3y)
Model I This model
a/:3y IG, S C(ISa/:3y) x (phase (1 ) (A) (B)
space)
7[07[07[0 1-,0 81 8.8 0.5 0.3 0.3
7[07[+7[- 1-,0 54 5.8 0.4 0.2 0.2
7}07[07[0 0+,0 27 1.7 0.7 0.4 0.4 D
o
7}0 7[+ 7[- 0+, 0 54 3.3 1.5 0.7 0.7 w
n
p07[07[0 1 +, 1 27 1.0 1.0 1.0 1.0 loa
p07[+7[- 1+,1 150 5.5 5.5 5.6 5.6 de
d
P+7[-7[° 1 +, 1 6 0.2 0.2 0.2 0.2 fro
P-7[+7[° 1 +, 1 6 0.2 0.2 0.2 0.2 m
7}07}07[0 1-,0 27 0.7 2.4 1.3 1.3 http
Pp+077}}0°JJ[0[- 00--,, 11 66 00..0077 00..55 00..55 00..55 s://ac
a
P-7}°J[+ 0-, 1 6 0.07 0.5 0.5 0.5 de
pOp0J[0 1-,0 81 0.3 4.7 5.5 5.6 mic
pOP+J[- 0-, 1 108 0.4 6.2 7.3 7.5 .o
u
1-,0 18 0.06 1.0 1.2 1.2 p
.c
pOP-J[+ 0-, 1 108 0.4 6.2 7.3 7.5 o
m
1-,0 18 0.06 1.0 1.2 1.2 /p
P+P-7[° 0-, 1 108 0.4 6.3 7.3 7.4 tp
1-,0 450 1.6 26.2 30.4 31.0 /a
olJ[°J[° 0-, 1 75 2.6 1.9 2.2 2.2 rtic
le
olJ[+ J[- 0-, 1 150 5.0 3.8 4.5 4.5 -a
b
WwOOpP0+7J[0[- 01 ++,, 01 10188 00..035 03..67 50..08 05..91 strac
0+,0 18 0.05 0.6 0.8 0.8 t/6
9
wOp-J[+ 1+,1 108 0.3 3.7 5.0 5.1 /3
0+,0 18 0.05 0.6 0.8 0.8 /9
3
wOw°J[° 1-, 1 225 0.5 4.5 7.1 7.2 7/1
7}07}07}0 0+,0 81 0.4 8.7 7.9 8.4 87
1
p07}07}0 1 +, 1 75 0.05 2.8 2.4 2.5 8
0
pOp07}0 0+, 0 225 0.03 3.6 2.6 2.8 6 b
p+p-7}0 1 +, 1 108 0.01 1.7 1.3 1.3 y g
0+,0 450 0.06 7.2 5.3 5.5 u
e
pOpopo 1 +, 1 135 0.004 1.0 0.6 0.6 st o
pOp+p- 01 ++, ,01 332748 00..0010 9 22..47 11..75 11..86 n 17
W°7}°7}° 0-, 1 27 0.009 0.4 0.2 0.2 No
WOp07}0 1-,0 18 0.001 0.08 0.07 0.07 ve
m
wOpopo 0-, 1 189 0.003 0.5 0.2 0.3 b
e
wOp+ p- 0-, 1 378 0.005 1.1 0.5 0.5 r 2
1-,0 324 0.004 0.9 0.4 0.4 0
1
8
wOw°7}° O+, 0 81 0.003 0.2 0.03 0.03
wOwopo 1 +, 1 189 0.001 0.1 0.04 0.04
wOwowo 0-,1 135 0.0004 0.04 0.002 0.002
W°7}°J[° 1 +,1 6 0.06 0.3 0.4 0.4