SLAC-FUB-1260
Baryon-Antibaryon
II.
Institut
Bootstrap
fiir
Model of the Meson Spectrum
G. Schierholz +>
Theoretische Physik
der Universitzt,
Hamburg, Germany
+) Present
address:SIAC,
P.O. Box 4349, Stanford,
California
94305, USA.
Abstract:
In this
work we present
meson spectrum,
we understand
of the nearest
potential.
degenerate.
Any contributions
and only contains
an integral
equation
intermediate
We are looking
coupling
input
bound state
with
constants
values.
and F/D ratios
it
Practically,
solution
singlets
and octets
physical
masses.
consistent
with
experiment
that
Instead,
a solution
singlet-octet
mixing.
are
is exotic
the bootstrap
which is
is chosen to
contributions
such as two-meson
of the bootstrap
system in which
as mesons, are
Furthermore,
have, to a certain
extent,
is required
constants
and other
that
the bootstrap
being decoupled
scalar,
ideas.
Possible
against
approximately.
their
A
and vector
average
turn out to be
origins
small perturbations
of the symmetry breaking
their
system consists
pseudoscalar
and F/D ratios
the output
to agree with
of SU(3) breaking
The spontaneous breakdown of SLJ(3) is ruled
the dynamics is stable
singulari-
it
equation
The potential
with masses being in good agreement with
are then investigated.
fact
governing
to be interpreted
is found comprising
The coupling
for
meson multiplets.
the remainder
of only a few multiplets,
self-consistent
solution
multiplets,
the input
Starting
Reasons why they must be small are discussed.
a self-consistent
to coincide
equation
Inelastic
effects.
are neglected.
for
baryon-antibaryon
forced
The dynamical
the
an SU(3) symmetric
are neglected
in the baryon c.m. momentum.
states
into
analog of the Lippmann-Schwinger
take account of relativistic
for
the baryon and meson multiplets
from the u-channel
the deuteron.
system is the relativistic
from the t-channel
and transformed
stage we assume that
model which,
of the quark model.
are constructed
meson multiplets
At this
bootstrap
to be an alternative
the forces
from the baryon octets,
ties
a baryon-antibaryon
out by the
of the input
masses.
is given in terms of bootstrapped
2
1.
Introduction
Various
dynamical
great
models have been proposed,
understanding
of the meson spectrum.
success in classifying
the hadrons,
of the quark system, however,
has been made recently
of the quark model,
assumption
without
In another
class
underlying
states,
i.e.
particle
that
to give the main attraction
the dynamical
a baryon-antibaryon
origin
BB underlying
the nearest
since it
is exotic
It
higher
Some earlier
that
ideas
the mesons may have
to form a
quarks is that we already
and only contains
two-
The main advantage of starting
a potential
as mesons, then the set of input
from
have some
scattering
[lO,ll].
If
singularities
(the u-channel
the deuteron)
are
[8] have been strengthened
They state
from nucleon-nucleon
into
channels
This leads to the
taken to be given by the t-channel
mesons transformed
interpreted
strap
are likewise
if
which can be considered
of with
knowledge of the Bi interaction
the BB forces
[9].
from two-meson
of a yet unspecified
of the meson spectrum
structure
instead
171.
bound states
to the quark model.
states
feature
the dynamics 14-61. In
to the starting-point.
(BB)-like
alternative
settling
can only be achieved
the mesons are mainly
some progress
as long as quarks have not been found.
principle
in a work by S. Wagner and the author
possible
although
of the mesons, is an ad hoc
justification
self-consistency
giving
The dynamics
hand, the most attractive
the nonet structure
a simple bootstrap
years,
The model, which has met with
understood
On the other
system in contradiction
concerning
eight
of models the meson spectrum is constructed
and installed
conclusion
in the last
is the quark model [I].
is not clearly
[2,3].
dynamical
these calculations
include'd
mostly
of
can be neglected
and the BB bound states
and output
particles
are
form a boot-
system which may serve as a guide to the dynamics of the meson spectrum.
is the aim of this
to the physical
paper to develop a BB bootstrap
model directing
meson spectrum.
The symmetry of the system should emerge from the bootstrap
Several
bootstrap
arguments have been given that
condition
SU(2) proves
crossing
inspection
function.
matrix
does not have real
of the bootstrap
In the following
system (at least
There is no solution
(I being the isospin)
equation
eigenvalues.
restricting
the
In case of the B, bootstrap
only involving
which is related
we assume that
in an approximative
condition.
the symmetry group able to fulfil
is one of the groups SU(n) [12].
to be too small.
I = 1 multiplets
attention
I = 0 and
to the fact
This can easily
that
the
be verified
by
to the pole terms of the wave
SU(3) couples
the mesons to the Bi
sense) and then see if
this
is confirmed.
3
The SLJ(3) representations
irreducible
decomposition
8x8=1+8
S
The subscripts
S and
Any practical
truncation
+8
involved
A
+Io+i?i+27
.
A denote symmetric
and antisymmetric
of the BE bootstrap
of the number of multiplets
system now requires
The unitarity
This would be the case if
channels,
were dynamically
decoupled.
situation
is the reciprocal
bootstrap
octets
(made up of the partial
SU(3) representations).
look simple.
are given by the
of the two baryon octets:
exploitation
irreducible
in the BB interaction
higher
equations
multiplets
a drastic
waves and the
must, furthermore,
and, e.g.
An almost classical
inelastic
example of such a
octet
of the (l/2>+
respectively.
and the (3/2)+
decuplet
are involved can be
[131. The fact that in the BE case only a few multiplets
At short distances there is some evidence of an
expected from the following.
almost universal
BE interaction
Since the vector
exchange is known to be dominant we have calculated
state
gives
low-lying
role,
this
isosinglet-exchange
wo exchange *) in Ref.191.
spectrum for pure
Table 1 demonstrating
[14] which favours
that
for
this
choice
The results
of the coupling
dominance.
the BE bound
are shown in
constant
wo exchange
BB bound states which are already very similar
to the physical
+*I
Bearing in mind that the vector exchange plays a dominant
meson spectrum.
3S - 3DI
1
obvious that
all
W(3)
however,
higher
indicates,
from the bootstrap
3
PO partial
waves (lying
and
isosinglet
multiplets
suffices
point
of view,
lowest)
to shift
A smallattractive
the exotics
only the 'So,
It
become involved.
exchange cannot be completely
are degenerate.
that
(belonging
true
since,
octet
is now
in this
case,
contribution,
to the 10, Eand
27 plet)
to
mass.
component of the w meson. The coupling constant is
+) The w is the singlet
gw.
(v>2o/4Tr = 15.7 (gCT) = 0) which is adjusted to give mout =min =784 MeV.
UO
uo
wo
Note that Ref.[9]
deals with the nucleon-antinucleon
system. However, for
isosinglet
exchange the results
W)As far as the scalar
clear.
include
also hold for
the BE case.
mesons are concerned the experimental
Even though the ~(410) has been omitted
the
a
in Table 1 since
pole on the second sheet with
real
there
part
is strong
situation
from the tables
evidence
- 400 MeV [16].
for
is not
[15] we
a complex
(JP)cn
output
mesons
mass
lT(140)
K(500)
(O--I+
ISO
415
n(550)
n' (960)
P (765)
K*(890)
(l-3-
3S1 - 3D
‘I
784
w (784)
+(1020)
Ipl
(1+>-
1250
B(1235)
6 (970)
TA(1020)
co+>+
3P0
K
(725)
920
KN(-1200)
cl (410)
E (700)
S=(1060)
Al(1070)
o+>+
3P 1
1200
KA(1240-1400)
A2(1300)
3
p2 - 3P2
1550
f (1260)
v+>+
f'(1514)
1
D2
Table 1: Bound state
masses (in MeV) for
compared to the physical
contains
single
isosinglet
meson spectrum.
a second bound state
SU(3) multiplets
1810
1~~(1640)
W)+
are completely
vector
exchange
The 3S1-3 D, partial
at 1530 MeV which is not quoted.
degenerate
in this
case.
wave
The
5
It
is now apparent
that
SU(3) is broken in nature.
amount of SU(3) symmetry survives
a good algebra
to start
is not clearly
understood
It
[17-191.
well
is,
there
We believe
is much room for
In Section
has been discussed
evident
that
our understanding
self-consistent
certain
we have some insight
the Bz bootstrap
symmetry breaking
review
is presented
Outline
B stands for
the BB channel.
from certain
transformations
of the model and discuss
since
discussed
our basic
Here a
bootstrap.
and compared to experiment.
Here
In Section
and an almost quantitain Section
Finally,
[20].
M indicates
This type of equation
properties
of the bootstrap
The discussion
5 we add
on the input
vector
and scalar
recorded
standard
The baryon and meson octets
the baryon octet
the SU(3) multiplet,
is favoured
state
function
forces
and
by arguments
under Lorentz
can be restricted
singlets
and octets
waves
the Lagrangians
[lo].
equation
since, as
1So, 3S1-3D1 and 3Po respectively)
Starting
from
the other multiplets
are actually
decoupled.
to the partial
out later,
be written
answer these questions
be based on the dynamical
to the exchange of pseudoscalar,
(belonging
the symmetry breaking
of the Model
employed in Refs.[9,10].
arising
papers
remarks.
model will
in Appendix A the construction
mass is identified
in the product
form
with
is
of why SU(3) works so
into
of the meson spectrum is given.
Our bootstrap
already
it
effects.
SU(3) symmetry breakingmechanismsare
description
in various
3 is devoted to the SU(3) symmetric
solution
some concluding
turns
will
mechanism so that
of the symmetry breaking
it
2 we give a brief
Section
assumptions.
2.
that
origin
although
can only be improved if
process.
tive
however,
the symmetry breaking
The dynamical
with.
But a considerable
of the potential
are assumed to be degenerate
the nucleon mass.
The potential
is
and
can
4
6
where the factor
CM carries the SU(3) crossing coefficients
and v'p?J,.,t4,c')
e/l (1
NB
the dynamical part.
The t-channel multiplets
are denoted by NS
represents
NB
\
(l)x
M
1
3
1
8A - 8A
1
5 (14avciT -
$ (14a2 - 10a + 5)
1
10
I
1
27
(l-a>2
-- 10
3
- y
(l-a)2
-- 10
3 (~v~T - aV-"T + '>
B is a subscript
presented
--
(2 a2+2a-1)
SU(3) crossing
3PO collectively.
distinguishing
+ 5)
5bV+aT)
- y
1
Table 2:
+ 5)
+2G (aV+aT-2aVcxT)
I
lo I I
(14cl”c.xT-5bV+aT)
+ 2(cxV+aT) - 2
4C(VC1T
I
0 +41/7 a(l-a>
I I
8s-8A
‘,
where
4
loo" + 5)
2+4a-2
4cY
I
- 3D
s1
mixed coupling'
3
s1 - D1 pure V or T coupling
1 i$(14a2 I I
8s - 8s
(8)3
Wx
M
cN .
B
coefficients
labeling
between different
couplings.
in Table 2 and the dynamical
aV-aT
+
1)
3Sl - 3D1 and
'so,
of the multiplets
The crossing
part
-
: ("VCIT + aV+aT - 1)
X denotes
the spin-parity
(ypT
of the potential
and
coefficients
are
is given in
Appendix A.
Table 2 indicates
multiplets
effect
so that,
of octet
by drawing
that
in this
isosinglet
exchange contributes
case, they would be completely
exchange on the singlets
the singlet
crossing
coefficient
and octets
the same to all
degenerate.
The
can best be demonstrated
and the eigenvalues
of the octet
7
crossing
matrix
couplings
for
for
values
cy one would only
then expect two low-lying
bound state
exchange proves to be small as compared to the singlet
spread between the singlet
and the lowest
c1 = 0 and c1 = 1.
increases
going to
low-lying
bound state
octet
with
In order
an effective
It was originally
parametrized
derived
from a Khuri
to have the "right"left-hand
to the picture
also present
off-shell
[lo]
that
the vector
generalization
wave function
where it
t0 = 0
(when it
(Ref.[lO])
is unphysical.
of the potential
The cut is
and if
the simple
also appears in the bound state
In order
to remove this
unphysical
The form factor
The trajectory
= -0.4
for pseudoscalar
a(t)
for vector
and scalar
is parametrized
intermediate
baryon exchange.
This choice
states
is consistent
neglected
diagram with
less
two-meson intermediate
two-pion
intermediate
high-energy
contributions
which are coupled
Part of these contributions
which contribute
with
inelastic
exchange for
built
states
such as, e.g.
included
up by multiple
(an extensive
is develop
in the one-
the u-channel)
to theextrapolatedpole
states
conceptions.
to the BE system by means of
are already
the mesons to be BB bound states
are those with
in the form
exchange and
meson exchange terms (and in the deuteron
The diagrams,
to the potential,
= 0.6 + 0.9t
exchange.
understand
contributions
+ 0.9t
So far we have tacidly
two-pion
(2.3)
is taken to be the same for all
Regge-like.
a(t)
cut
and
=
P(C,a’.,
i.e.
being
This corresponds
N = 2 (3) pions.
is taken on shell),,
is used it
octet
(see Appendix A).
s I 4m2 - N2u2.
of
from the
0.4 5 Cleff < 1.
representation
cut
The
cx = 0.5 and
attractive
the form factor
mesons consist
in the form factor
of
if
exchange.
to remove the exotics
F/D ratio
Some remarks have to be made concerning
octets
has a minimum at
spectrum one would need an overall
exchange contribution
we set
the unmixed
a=a
to
degenerate
the octet
for
The octets are
V =a T in the mixed case).
c1 = 1 and are far spread in the case of c( = 0. For smaller
(corresponding
of
This has been done in Fig.1
0 < cx 5 1.
since we
meson exchange.
terms in this
sense,
study of the box
in Ref.[21]).
They mainly
8
enter
into
.forces
of
the double spectral
r =. 0.1 fermi
On the other
contribute
function
part
(to be compared to
hand, the diagrams with
larger
much to the one-meson-exchange
to the double spectral
function
is frozen
In order
because of the high thresholds.
inelastic
giving rise to very short range
r - 1.4 fermi for pion exchange).
numbers of intermediate
mesons
amplitudes,
and that part belonging
at the expected bound state energies
to estimate
the effect
of the
channels
on the mass of the bound state and on the wave function we
BE t-f & where M stands for meson. The
study the case of two coupled channels
Hamiltonian
is taken to be
(‘1
hOB + VB
VBM
(2.4)
i.e.
the forces
bation
theory
these forces.
first
We assume that low-order perturin the % channel are ignored.
can be employed for vi;) because of the short-range nature of
Suppose there
order perturbation
function
is a low-lying
theory
in the meson channel
bound state
gives a contribution
(for
simplicity
in the BE channel,
to the bound state
the SU(3) specification
then
wave
is
dropped)
(2.5)
where we have set
($,,$,)
= 1.
A mass shift
at first
occurs in second order
yielding
(2.6)
It
threshold.
We will
mB lying above the two-meson production
The imaginary part of (2.6) represents the width
case first.
is complex for
discuss
this
/7=
and the real
(2.7)
part
(2.8)
9
contributes
a correction
constructed
to give the correct
the widths
are small
in a SU(3)-symmetric
even smaller
integral
comes from
if
propagator
occurs
the higher
mass MM channels
the major contribution
closed
is suppressed relative
no proof
we may assume that
equations
this
which are not well
effects
is constrained
channels
Aldrovani
because the
hand, the part
to give the physical
of
to the double spectral
Although
this
is
decoupled
from
on the wave function.
and Caser [22] have recently
be studied
value
may, however,
are approximately
considerably
mass,but
value
wave function
threshold).
Note,
bound states
shown that
inelastic
they make a number of assumptions
founded and do not hold true
should first
show that
The partial
may change in the case of loosely
for deeply bound BB states.
in the case of the p where the dynamics
width.
the mesons as two meson resonances
already
to the principle
E = mB belonging
but can operate
channels may lower the bound state
Inelastic
experimental
On the other
mi/(inelastic
inelastic
situation
(in the BB channel).
to
at
Since
Re m(2) should be small and
here)
to the second power there.
function
that
than its
amount to the norm of the wave function
-
VA:) has to be
double counting.
the two-meson decay of the bound state
a large
however,
would be smaller
E = mB as is expected.
contribute
the eigenvalue
and to exclude
world which we discuss
which describes
(2.51,
width
(the p width
than the width
(2.8)
The
to the mass of the bound state.
The various
(mentioned
attempts
to understand
in the introduction),
the two-meson channel is almost negligible
however,
as far
as the output
meson masses are concerned.
3.
SU(3) Symmetric Bootstrap
The self-consistency
the coupling
Multiplets
not included
in the input
coupling
for
equation
a review
of the F/D ratio
One way of normalizing
(if
in this
model extends
any) of the multiplets
may occur as bound states
and F/D ratios
(which have a pole at
result
involved.
at higher
energy
of the wave function
see Ref.[23])
and g CT),g 0)
so that
from the residues
is,
the residue
however,
not determined
in the Bethe-Salpeter
only permits
(in the case of the 3S1-3D1 partial
the wave function
(being
form factor
of the
Ek = mB> making use of
(the same problem is met with
the electromagnetic
to the mass,
to the input.
constants
The normalization
by the eigenvalue
calculating
only a little
wave functions
Eq.(A,l-A.6).
tion
inherent
and the F/D ratio
The output
equation;
requirement
constant
such as to contribute
bound state
Solution
intuitive
a calculawave).
for our aim) is by
at zero momentum transfer
which
10
gives
the total
finite
q2 [24].
factor
It
This is unambiguous although
is obvious
can be explained
discussion
that
charge.
of the wave function)
a large
even if
state.
For this
applies
to the bound states
they do not effect
The normalization
mesons (if
method is completely
Table 1 leads us to suppose that
are essentially
is shown in Table 3 (for
attractive
and repulsive
respectively
ISO
1
sO
3S1 - 3Dl
3p0
to the lowest
(repulsive)
T
+
+
+
+
partial
refers
V
+
Signs of the pseudoscalar,
tions
waves playing
vector
waves.
and V(T) denotes vebtor
condition
two mesons.
($,,$,>
= 1 and
an active
part
3P
0' Their effect on
where + and - stand for
to the dynamical
3S1 - 3D1
+
of the bound
masses are taken).
to setting
references)
(this
(and the norm
bound states.
the partial
later
the form
the diagrams like
to decay into
'So, 3S1 - 3D1 and
the output
that
the normalization
equivalent
if
From the preceding
the physical
be taken over by the neutral
in the bootstrap
is only relevant
the position
which are not allowed
octet
is not the case for
amount to the form factor
reason we can only assume that
These are the pseudoscalar
Table 3:
procedure
channels we know, however,
may contribute
can, therefore,
this
by the diagram shown in Fig.2a.
on the inelastic
in Fig.2b
that
this
part
only).
3P
0
+
+
+
and scalar
+
meson exchange contribu-
Here + (-) means attractive
(tensor)
coupling.
11
On the input
calculation
the others
octet
side V(T) means pure vector
we only take these partial
later.
We then start
mesons taking
derived
their
in Ref.[lO]
removing the exotics
from the input
the exotics
The
condition
(in reality
because the output
one already
we believe
considering
fact
that
nucleon
left
the pseudoscalar
a limit
to the coupling
physics
should stay within
lying
with
constants
this
pseudoscalar,
values
bootstrap
vector
the physical
can be normalized
described
and vector
are,
be discussed
refinements
recorded
solution
mesons.
value)
however,
by the
by a single
fixed),
constants
however,
sets
to have to do with
program exists.
now (the bootstrap
in the input
are
but the input
Experiment,
to this
since
by only
meson coupling
in Table 4 refer
exhibits
and scalar
choice
This is supported
= 1).
A solution
limit.
this
need not be varied
and any model which claims
some smaller
The experimental
The (partial)
with
ones in any scale.
is given in Table 4 and will
be made).
well
(IJJ, $,)
and the F/D ratios
(g(T)/g(V)
stopped here although
constant
bound states
The scalar
loop diagram [25].
F/D ratios
constants
stage
decay can be quantiiatively
TI' -t yy
and
(we shall come back to
constants x> are subject to the
coupling
(i.e.
singlet
channels which may justify
early
agrees fairly
may not exceed the output
result
at this
the r coupling
the diagram in Fig.2a
the
open
values
that
in the exotic
TI and rlh
low-lying
cormnent on
masses 1151 and the coupling
For admissable
values.
In our bootstrap
account and shall
from the corresponding
does not lead to deeply bound states
bootstrap
coupling.
waves into
average physical
as initial
later).
(tensor)
The
program was
parameters
had to
to Refs.[26,27].
the nonet structure
of the lowThe masses are in good agreement
the average physical
seem to be too low.
to the
meson masses except for the 11; and the
w. which
3PO
In the case of the
partial
wave we assign the singlet
o meson and the octet
constants
W>the octet
and F/D ratios
coupling
to
6 ,
{K,
KN} and {E, S*].
are concerned we are faced with
constants
are labeled
by their
As far as the coupling
the following
situation:
I = 1, Y = 0 members.
12
‘SO
3S -3D
1
1
I
I
3
pO
I
1
8
1
8
1
8
m.
In
580
400
590
905
460
860
m
out
580
400
590
905
460
860
m
exp
-900
-400
-880
-850
400-1100
400-1100
(g2/4n)in
10.0
14.4
v 7.5
T 5.0
0.7
5.0
7.0
0.5
(g2/wo;,
12.4
15.7
?
?
?
?
14.4
vT 5.-15.
7
k2/Wexp
(gW,g(V))
0. - 15.
0.5-I
4. -15..o
o.-15.
-
_
0.82
2. 7
-
-
0.7
2. 6
-
-
_
?
2.4-5.0
-
a.
In
0.15
-
clout
0.15
-
(g(T)/g(V))
in
out
(g (T)/gW))exp
,
o.-15.
v 1.
-I
cf.
ew
Table 4: N(3)
symmetric
stands for vector
0.2-0.4
I
I
bootstrap
(tensor)
T 0.5
-
v 0.97
_
0.6
0.6
T 0.43
?
-
solution
coupling.
compared with
experiment.
V(T)
13
(w) ‘so:
The r coupling
constant
out somewhat lower than that
is consistent
with
the situation
is complicated
(6) 3S1 - 3Dl:
vector
reasonable
derived
experiment.
fixed.
For the
p
is close
overall
strength
of the singlet
coupling
the ratios
to the nucleon
known and
is in agreement with
of the tensor
In addition,
and octet
is not well
mixing.
The F/D ratio
compared to experiment.
The F/D ratio turns
bootstrap
[28], but it
constant
no - r,:
of the vector
[29].
reproduced.
11: coupling
because of
conservation
it
well
from the reciprocal
The
The F/D ratio
current
is fairly
form factor
couplings
turns
coupling
g(T),gW)
isoseems
are
1301. The
prediction
out to be nearly
the
same as was found in low-energy
nucleon-nucleon
scattering
[lOI if the w.
(W
assigned to the w . Moreover, as far as g
is concerned, we are in
P
agreement with the universal
vector coupling hypothesis
[31] predicting
gCVJ2/4Tr a 0.5 - 0.7.
w-4 mixing 1321 is assumed (we shall
If ordinary
P
back to this problem later),
we obtain the physical coupling constants
gg
2
/4n = 0,
gi$
that
the known fact
2
(Y13Po: The coupling
energy nucleon-nucleon
/4n = 0 and gL$2/46 = 13.7,
glfiii2/4n
the I$ is decoupled from the nucleons.
constants
are consistent
interaction
also in the fact
that
gg
known from other
investigations.
[Ill,
with
= 9.7.
waves.
what is known from low-
not only in the order of magnitude but
is small in comparison
to
ga'
The F/D ratio
gives
rise
to a second octet
in all
is not
output
three partial
Their masses are quoted in Table 5.
and vector
situation
solution
come
This reproduces
3S1 - 3D1 and
If
$, is normalized to one for
3p0 the corresponding
coupling constants are larger by a factor of 2-5 than the input ones.
Our bootstrap
be
octet
this
is supported
In the case of the pseudoscalar
by experiment although the experimental
is by no means settled.
1
sO
mass
900
experimental
H(990) c
indications
E(1422)
Table 5: Bound state
3Sl
-
3Dl
1200
3
pO
1500
P(l540)
K*(1270)
masses (in MeV) of the second octet
not included
in the input.
14
The second pseudoscalar
have been included
octet
turns
out to be rather
in the bootstrap.
However, the output
g2/4rr = 5(a = 1) which already proves that
factor of at least 15 than the m exchange.
In the exotic
at 1400 MeV.
channels
some indications
other
partial
[15] that
this
contribution
high and not excluded
a
I = 3/2,
waves not included
coupling
constant
is smaller
by experiment.
Y = 1 meson exists
in the bootstrap
contain
interpretation
of the
is
by a
appears in the 3S1-3 D1 partial
only a 27-plet
This is fairly
low and should actually
wave
There are
at 1200 MeV.
The
bound states above
1300 MeV as shown in Table 6. The tensor singlet
and octet
( 3P2- 3F2) is in
3
good agreement with experiment.
The axial vector bound states
( P 1) are
compatible with the D and KA mesons but are far too high compared to the A,.
This would support
the Deck-effect
recurrence
'(T , the B meson, is also fairly
of the
is taken into
1300
1500
1530
1320
B(1235)
A1(1070)
f (1260)
D (1285)
f'(1514)
KA(1240 - 1400)
A2(1300)
experiment
Table 6: Bound state
involved
the width
dominant one.
constants
indicate
In addition,
and vector
octet
and octets
parametrization
masses (in MeV) for
this
that
partial
the isosinglet
waves not
cancel
for
our choice
exchange is by far
is improved by the fact
exchange contribution
of the vector
that
some higher
in the bootstrap.
The coupling
not believe
(if
I
1570
octet
mass
singlets
reproduced
The first
account).
1
p1
scalar
well
A, [33].
the main part
the pseudo-
to the
'50 and 3S1 - 3D1
as a consequence of the special
exchange (see Table 3).
of the form factor
that
For this
parameters
reason we do
is crucial.
Small
the
15
changes of the form factor
and can be cancelled
constants
parameters
by simply
readjusting
which are not subject
and scalar
model one could argue that
(in the mass-coupling
different
4.
solution
the isosinglet
the vector
dependence of the masses on these coupling
curve
effect
and scalar
to the self-consistency
Since the scale of the vector
by our bootstrap
merely
constant
constants
solutions
constants,
plane)
singlet
coupling
is left
open
condition.
coupling
further
exchange
exist.
however,
The
gives
of small extension
a closed
so that
a much
is not expected.
SU(3) Breaking
So far we have assumed SU(3) invariance
splittings
indicate,
symmetry breaking
however,
pattern
SU(3) symmetric
hand, there
,In our model the binding
are many reasons for believing
of SU(3) symmetry breaking
interaction
itself
of SU(3) and the dynamics it
carefully
[341.
is a result
energy
that
of the properties
This means that
the
of the
our understanding
governs could be advanced if
We are now going to discuss
examined.
the
they are of the order of 25 - 30% which cannot be
On the other
neglected.
special
to that
and that
(1. - 1.4 GeV) sets the scale of the SU(3) viola-
multiplets
According
tions.
way.
The mass
sense.
SU(3) is broken in nature
that
emerges in a definite
of the bound state
in an ideal
the deviations
possible
origins
be
of SU(3)
symmetry breaking.
Ina bootstrap
to unstable
breaking
model the symmetry breaking
Some general
dynamics.
have been pointed
first
arises
if
exhibits
like
discuss
the conjecture
the bootstrap
the full
6mBD8(<D8> = &
and first
{I(I+l)-
order perturbation
of this
tional
6mB D82mB/(t-mi)2
type.
at
The first
t = 0.
""V = 160 MeV, the scalar
of SU(3)
But in order
to favour
calculations.
We
broken SU(3) [17].
mass splitting
This
solution
but
The mass splittings
are then expected to go mainly
12
T Y -1)) since only octets are involved
In the case of the pseudoscalar
exchange vanishes
of spontaneously
leading
theory
one has to make detailed
theory
a solution
to
[35].
system has a self-supporting
symmetry.
direction
of the bootstrap
out in the literature
one mechanism more than the other
shall
features
chooses that
octet
should apply.
order perturbation
being of the order
octet
We have attempted
it
on the output
to construct
is propor-
(6mB D8/mB) . (v/binding
energy).
is even less because the pseudoscalar
For the physical
mass splittings
has not been taken into
( C;mpS= 450 MeV,
account because of the
16
very small contribution)
'couplings)
state
first
gives only a minute
The reason for
masses.
exchange term is strongly
Second, the vector
vector
order perturbation
octet
octet
correction
this
to the
is threefold.
suppressed by the fact
is not far
contributions
theory (taking SU(3) symmetric
1So and
3Sl - 3Dl bound
split
have opposite
that
the pseudoscalar
it
vanishes
at
t = 0.
and third,
the pseudoscalar and
1
signs in the So and 3S1 - 3D1 partial
waves (for
our parametrization)
This rules
out spontaneous breakdown of SU(3). There would also have been no
room for
a (massless)
symmetry breaking,
so that
First,
Goldstone
however,
started
by an extraneous
becomes apparent,
e.g.
pursue the idea that
"driving
vector
in the
symmetry breaking
the right
w-$I mixing
that
cancelled
interactions.
The form of the
case the SU(3) violation
mechanism [361.
Henceforth,
term" is given by singlet-octet
the dominant role.
To begin with
is
Such an interaction
interaction.
spontaneous
(see Table 3).
direction.
symmetry breaking
the "driving
term" must play
nonet mixing.
into
SU(3)noninvariant
In view of the fact
alone.
boson in strong
points
In any but the spontaneous
they are partly
we shall
mixing
effects
SU(3) breaking
is dropped out the
For the present
we only consider
we introduce
w-4 mixingby
hand through
the
interaction
(3.')
If
this
is included
violating
contribution
shown in Fig.3.
perturbation
in the potential
recorded
We now evaluate
theory.
(a>
(I,Y)
# co,01 :
(8)
(I,Y)
= co,01 :
It
(A.7)
there
in Appendix B.
the LO-$mixing
causes an output
arises
It
an explicit
corresponds
SU(3)
to the diagrams
mechanism in second order
mass correction
(3.3)
17
The wave functions
are normalized to
however, is independent of the norm.
and o = 6 means I = I3 = Y = 0.
M'
($J, , $i) = 6M,M . Note that the result,
Apart from the singlet-octet
mixing the
mass shift
component of an octet
where
u
denotes
(I,
transforms
13, Y)
like
by the Gell-Mann-Okubo
collectively
the eighth
The strength
mass formula.
IS1 - 3
D,
as described
of the symmetry violating
Experimental
Mesons
'SO
just
output
Mass
IT (140)
145
K (500)
n (550)
525
505
n' (960)
600
P (765)
760
K*(890)
975
w (784)
520
1120
$(‘020)
600
995
3
pO
a (410)
300
1300
Table 7: Bound state
mixing
part
is given by
subject
mass spectrum
(in MeV) arising
compared to the physical
ml8 .
to the bootstrap
Since the
condition.
from bootstrapped
w-+
meson spectrum.
w-4 mixing
is self-generating,
In second order we have
ml8
is
18
which corresponds
nonet mixing
to the diagrams in Fig.4.
proves to be small for physical
is negligible
that
Evaluating
mixing.
the self-consistency
Eq.(3.5)
we obtain
in which case the experimental
the physical
than in the symmetry breaking
too low singlet
nonet and
mass.
for
Ops = -23'
the experimental
formula
however, we will
Our calculation
phenomenological
the bootstrap
symmetric
principle,
symmetry breaking
The fact
mixing
that
The strength
SU(3) are that
relatively
only supports
origin
connected
from.
small lies
small octet
The magnitude
to calculate
apart
of the mass
from the scalar
octet,
The deviations
in the SU(3) symmetric
from
interaction
They are caused by the somewhat
mechanism.
for the vector
% = 26.2'
This has to be compared with
nonet.
derived
Ops = -23'
giving
rise
from the linear
The second octet
to a further
mass shift
against
the w-4 mixing
mass
does,
which,
that
w-0 mixing.
mechanism is
That means IA-$ mixing
1181.
solution
emerges from
We now have a fully
system and an asymmetric
is too small as to yield
arises
to w-$
one.
which is stable
The bootstrap
against
small
way the symmetry of the system reduces to SU(2), but the
pattern
decoupling
forces
of hypercharge-one
of SU(3) breaking
to the dominant role
of the SU(3) breaking
the dynamics it
This allows
is not settled.
of bein isolated
the characteristic
is closely
experiment
than fundamental.
In this
perturbations.
applies
the dynamics is unstable
of the bootstrap
however,
already
with
by self-consistency
system instead
solution
is,
here.
shows that
rather
condition
ov = 39O, ops = -Ho).
the singlet
is determined
ml8
It
waves.
the pseudoscalar
not discuss
constant.
angles turn out to be
mass formula:
also mix with
coupling
partial
ov = 35O and
values
(quadratic
of course,
Since
The mixing
Scalar mixing
leads to the mass spectrum shown in Table 7
situation
meson spectrum have its
side the pseudoscalar
parameters.
ml8 = 240 MeV.
It
the SU(3) breaking unambiguously.
13
for the
so' % - 3D1 and 3p0
shifts
agrees almost quantitatively
rather
mixing
because of the very small octet
expected
therefore,
On the input
follows
from
played by the vector
depends on the symmetric
couplings.
w-o
exchange.
solution
One of the main reasons why the deviations
in the isosinglet-exchange
channels.
and
from
dominance corresponding
to
19
5. Conclusions
Our model is a first
underlying
The results
states.
explained
as BB composite
possibility
of attributing
The good features
attempt
to treat
indicate
particles.
the meson spectrum starting
that
the physical
In view of this
the fundamental
of our model voting
entities
for
from BB
mesons can be
one should consider
"quarks"
a Bfi underlying
the
to the baryons.
structure
can be
summarized as follows:
1.
The dynamical
consistent
concept and the solution
with what is known from low-energy
other models being supported
seriousness
2.
by experiment.
the low-lying
meson spectrum is well
interaction
This gives an impression
and
of the
only arise
3.
The SU(3) breaking
We have always a second octet
In the quark model a further
nonet
for very high quark mass [2,38].
mechanism is guided by the dynamics of the SU(3) symmetric
solution.
The form of the symmetry breaking
dynamical
concept.
with
of
of the quark model) here is a
dominance.
of small changes of the parameters.
will
The nonet structure
reproduced.
mesons (being one of the pillars
consequence of isosinglet-exchange
consistent
nucleon-nucleon
model is
of the model.
The physical
despite
of our Bi bootstrap
therefore
Our model has an SU(3) breaking
is a critical
bootstrap
test
solution
of the
which is
experiment.
Acknowledgement
It
is a pleasure
to thank G. Kramer for
wish to thank S. Wagner for his cooperation
stimulating
in the early
discussions.
I also
stage of this
work.
20
Appendix A
In order
we write
and octets
to establish
both the dynamics and define
down the Lagrangians
interacting
with
for
the scalar,
the coupling
pseudoscalar
and vector
constants
singlets
the baryon octets:
Singlet
(a) scalar
(B) pseudoscalar
(A.2)
Octet
(a) scalar
(-4.4)
(S) pseudoscalar
(A.51
Note that
the coupling
constant)
differs
Following
symmetric
apart
constant
by a factor
Ref.[lO]
of the vector
of 2 from that
the Lagrangians
under SU(3) transformations.
from the form factor
given by
octet
(i.e.
the pNN coupling
of the compilation
(A.1 - A.6) give rise
The dynamical
part
[26].
to a Bfi potential,
of the potential
is
and the partial
wave expansion
employed in Ref.[37].
(a) scalar
:
O=g,I=l
(R) pseudoscalar
:
0 = gc&
The@and$are
I = 1
(y) vector
where
Ak=(iIE,-Q),
the vector
Z&'-
exchange contribution
J'\
)
. Compared to the baryon-baryon
only changes signs.
interaction
The form factor
is taken
to be of the form [lOI
(A.81
where
J B denotes the spin of the exchanged particle.
.
22
Appendix B
The SU(3) violating
part
of the potential
corresponding
to Fig.3 arises
This changes
propagators.
is taken into account in the vector-meson
I
for thenoninvariant
part into
the pole-term
tmy"
if
v'
and
to the singlet
which
The noninvariant
contribution
to the potential
the other to the octet.
are
The crossing coefficients
in the form (2.2).
we call
v w9 can be written
At the same time,
the vertices
(c) singlet-octet
transition
c
(8) octet-octet
become mixed,
= (2
one belonging
fl-oc>) &r-0( )
03.2)
transition
(B.3)
where
c1 refers
contribution
the vector
refering
substituted
to the vector
and tensor
coupling respectively.
There is no
The dynamical part is given by
transition.
to the singlet-singlet
exchange contribution
to the singlet
by (B.1).
and octet
with
0
!J'+ and Ov,couplings respectively
In case of the octet-octet
transition
(see Appendix A)
and the propagator
we write
References
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G. F. Chew, Phys. Rev. Lett. 2, 233 (1962);
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R. Aldrovani
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N. Nakanishi,
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A. L. Licht
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S. L. Adler,
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J. J. Sakurai,
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C. W. Akerlof,
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J. J. Sakurai,
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S. Okubo, Phys. Lett.
[33]
G. W. Brandenburg, A. E. Brenner, M. L. Ioffredo,
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D. H. Sharp, Recent Development in Particle
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[35]
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[36]
J. J. Sakurai,
[19]
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and A. Pagnamenta, Phys. Rev. E,
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Phys. Rev. 177, 2426 (1969)
Introduction
to Unitary
Ann. Phys. -'11
2,
Symmetry (Wiley,
17,
1021 (1966)
165 (1963)
references
cited
Phys. -B7, 432 (1968)
A. Pagnamenta, Nuovo Cim. -53, 30 (1968).
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Phys. Rev. 132, 434 (1963)
Nucl.
New York,
1 (1960)
Phys. Rev. Lett.
and further
137-j G. Schierholz,
[38]
Theor.
Phys. B39, 306 (1972)
p.14
Figure
Fig.1
Captions
Octet-singlet
octet-octet
Fig.2
Diagrams for
crossing
crossing
coefficient
matrix
(1) and eigenvalues
of the
(gl and g2).
the electromagnetic
form factor
and the normalization
condition:
(a) nucleon
loop and (b) meson loop contribution.
Fig.3
Diagrams arising
from w-I$ mixing
Fig.4
Self-generating
w-$I mixing
to be included
in the potential,
diagrams in second order.
3
-0
--
-- -4
N---A\
/
/
i
(b)
Fig. 2
\
b- --
.
W