986
Progress of Theoretica l Physics, Vol. 52, No. 3, September 1974
All-An gle Featur es of Urbary on Rearra ngeme nt Amplit udes
·and Two-St ep Structu re of Constr uctive Forces
Yuji IGARASHI, Takeo MATSUOKA and Shoji SAWADA
Departm ent of Physics, Nagoya Universi ty, Nagoya
(Received March 4, 1974)
An urbaryon picture having the two-step structure of the constructiv e forces is
proposed.
The very strong constructiv e force of the first kind works among the urbaryons
q with
large mass and charm charge. By the effect of urbaryon pairs qij in the non-charm
state,
the urbaryons behave as clusters Q whose effective masses are very light. The
rather weak
constructiv e force of the second kind distinguish es hadron states QQ, QQQ
from exotic
ones QQQQ, QQQQQ, etc., and plays an essential role in hadronic reactions.
In the process
with a large mome~tum transfer, the urbaryons Q are rearranged as if they were
independe nt
free particles. On the basis of this picture, it is shown that the factorizabi lity
of urbaryon
rearrangem ent amplitude can be realized.
§ I. Introdu ction
During the last decade, many empirica l facts have been accumula ted, which
indicates the existence of certain countabl e subhadro nic entities referred to
as
'urbaryo n '. Although the urbaryon has not been discover ed directly, many
evidence s have been found which suggest that the urbaryon possesse s a half
spin,
unitary spins (isospin, strangen ess and so on) in addition to mass and baryon
number. Cosmic ray events observed by Niu et al. 1l indicate a possibili ty
that
urbaryon brings further feedom of unitary spin including those of U(3).
At
present we have no logical basis of the argumen t to answer to what extent
the
freedom of unitary spins (referred to as 'the freedom of abscissa' ) exists
or
when the history of dicovery of new kinds of unitary spins U(l)~,U(2) ~
U(3)
~U(4) ~-·· terminat es.
If the heavy mass (mx=2~3 GeV) and short life
(~l0- 18 sec) particle observed by Niu et al. is a baryon having
a new quantum
number, we have a sequence of mass differenc es; mn- mp ~1.3 MeV, mAmN
~175 MeV, mx-mN~2 GeV, .... If a picture of approxim
ate U(4) symmetr y
holds, the masses of constitue nt urbaryon s are expected to be much larger
than
these mass differenc es.
On the other hand, the large angle behavior of the hadronic reactions and
the precocio us scaling phenome na of lepton-ha dron interacti ons suggest that
the
masses of urbaryon s are at most several hundred MeV and interacti ons among
the urbaryon s are consider ed not so strong (less than a few hundred MeV) 2
. J,BJ
These rsults have been derived from several different approach es such as parton
model, urbaryon rearrang ement (UR) model. These models contain by implicati
on
All-Angle Features of Urbaryon Rearrangement Amplitudes
987
the independent constituent picture. Particularly, in view of the validity of
UR model, it seems that the countable nature of constituent urbaryon prescribes
strongly the characteristic features of scattering amplitudes. In fact, on the
basis. of the UR model the following important properties of the scattering
amplitude have been pointed out:
1) Additivity: the amplitude of hadronic reaction Tab~cd (s, t, u) is given by a
sum of UR amplitudes T, (s, t, u) (i =X, H, Z, etc.):
Tab-->cd(s, t, u) =I:; C(ab, cd; i)Ti(s, t, u),
(1·1)
i
2)
wll.ere coefficients C (ab, cd; i) are given by counting the contribution of i-th
type UR amplitude. 4 l
Factorizability: each UR amplitude is specified by UR diagram and factorized
into a product of factors corresponding to urbaryon lines :2l
(1·2)
where n., nt and nu stand for the number of rearranged urbaryon in the s-,
t- and u-channels respectively and specify the UR amplitudes and n. = (nt + nu
- n.) /2, etc., denote the number of created and annihilated urbaryon pair
in the s-channei, etc. Each factor in Eq. (1· 2) is overlapping integral which
shows a Regge-like behavior ip. the small Itl or small Iul region and a simple
power behavior in the large momentum transfer region.
3) Duality: the UR diagram implies the duality in itself"l and gives an unified
understanding about the genera1 features of hadronic reactions from low
to high energy regions such as existence or absence of resonant states, dip
structure, rotating phase. 4),B) The dual properties of the scattering amplitudes
are systematized in terms of the UR amplitudes.
From the standpoint of simple quark models, it is difficult to explain the
above features of hadron scattering. Furthermore it is also hard to understand
the reason why the qq and qqq states are stable and the qq, qqq, etc., are unstable and why the quarks have not been discovered in spite of their small
masses and rather weak binding.
In this paper a picture of urbaryon is proposed which leads to the abovementioned properties of the UR amplitudes. In § 2 the necessity of .two-step
structure of the constructive forces and corresponding urbaryon pictures are
discussed. On the basis of this picture the structure of UR amplitudes are
investigated in § 3, and it is shown there that the factorizability can be realized
by considering the urbaryons to be pseudo-independent particl~s. The final section is devoted to further discussion and comments. ·
§ 2. Two-step structure of the constructive forces
In order to overcome the problems stated in § 1, it is shown that the most
988
Y. Igarashi, T. Matsuoka and S. Sawada
hopeful model of urbaryon is the three-triplet models proposed by Nambu7l and
Hori. 7l Among the models which have other freedom (referred to as 'the freedom
of ordinate') besides the freedom of abscissa, the triplet type models with
nearly exact U(3) symmetry in triality charge have superiority over the other
becaus_e a baryon is subject to the Fermi statistics as the three-urbary on system
in the new U(3) singlet state. In these models if one forgets the freedom,
of ordinate, the mesons and baryons are totally symmetric states with respect
to spin, unitary spin and coordinates of constituent urbaryons. The symmetric
form of the wave function in spin, unitary spin and coordinate parts is consistent
with the present experimenta l observations . Although the simple quark model
with the para-fermi statistic of order three 8l has the same algebraic results as
the model having freedom of ordinate, in the latter model there are importjtht
dynamical implications and possibilities . In the following, on the basis of the
urbaryon model with new U(3) symmetry in triality charge space (denoted as
U(3)'), the two-step structure of constructive ~orces and· urbaryon pictures is
discussed.
The constructive force of the first kind
This is the force which gives rise .to the difference between hadronic states
and the unobserved urbaryon system such as q, qq, qqq; according to zero or
non-zero values of triality charge (U(3)' singlet or not) (see Table I). The
force is considered very strong (super-stron g) but of very short range. It is
caused by the direct triality-curr ent X triality-curr ent Fermi type interaction:
(2·1)
Table I. The two-step structure of constructive forces and the' classification of various systems.
triality charge
(U (3) 1 -multiplets)
U (3) -multiplets
non-zero
3', 3*', 6 1 ,
8 1 , 10',-··
3, 3*, 6,-··
examples
q, q, qq,
(qqq).,, ...
remarks
I
l
M.)>1 GeV
M.~-,-4/3VI
separated by constructive force of the first kind
ordinary
hadrons
zero
1'
mesons
1, 8 or 9
qq+qq§_q+···
baryons
1, 8, 10
qqq+qqqqq+· ··
=QQ
Mq~400
MeV
Mq~-vn
=QQQ
separated by constructive force of the second kind exotic
10*, 27, ...
(QQ) (QQ),
(QQQ) (QQQ),
...
nucleus, exotic matter
having large
hypercharge, isospin,
etc. (Zo, Z,) .
~
states
zero baryon number states such as
JCP=o~-, o+-, 1-+, ...
All-Angle Features of Urbaryon Rearrangement Amplitudes
989
or by the exchange of U(3)' octet or nonet vector bosons b:i (Fujii boson or
gluon) with very heavy mass;
(2·2)
where Greek indices attached to urbaryon field q denote ordinary unitary spin
and J../ (i=l, ···,8) is U(3)' spin matrices. For both cases, any way, the total
energy for the n-body system of urbaryon is expressed by
nMq- {C- (4/3)n} VIf2,
(2· 3)
where C is the second order Casimir operator of U(3)' group and Mq is the
mass of urbaryon q. If the effective binding energy V 1 due to the first kind
constructive force satisfies a relation (8/3) V1 ~Mq, the total energies of the states
belonging to the same U(3)' multiplet are nearly equal irrespectively of n (see
Fig. 1). In other words, no creations of urbaryon pairs with the U(3)' singlet
states give essential effects on the_ energy of the system. · This feature would
be realized under the relativistic treatments required for super-strong forces. A
number of creation of urbaryon pair of U(3)' singlet would occur ceaselessly.
The cloud of urbaryon pairs is also the source of constructive force and spreads
over the range. which is given by the Compton wave length of urbaryon pair
~0.7 x 10- 18 em. Therefore, in spite of the short range of the first kind constructive force, the influence of the force reaches to almost full size of hadrons.
Nevertheless, we have the countable entities as discussed in § 1. Thus in the
U(3)' singlet state, the urbaryons move as clu~ters Q~, whose quantum numbers
are sp~cified by those of original urharyons q~, left after stripping their cloud
o urbaryon q
• antiurbaryon
q
Fig. 1. The total energies of the
states q, qq, qlj, qqq, etc. The
SU (3) I singlet StateS are IDOSt
stable and correspond to hadron states.
990
Y. Igarashi, T. Matsuoka and S. Sawada
in U(3) and U(3)' symmetric ways. For example, a baryon which is specified
by usual U(3) indices a, (3, r is considered as a composite system represented as
IBa.Sr) =
~
a', /1', T'
antisymmetrized
~·
[ Coq~,q~,q~,
+ C1 ~ q~,q~,q~,qg,qg' + ···J
tJ, IJ'.
Q~,Q~,Q~, .
anti~;~:frized
(2·4)
It is the cluster Q that appears as a countable entity-urbary on-in tlie hadronic
phenomena. The effective mass of Q is very light and its size is of order of
hadron's one 0.7 x l'0- 13 em.
The constructive force of the se~ond kind
The constructive force of the :first kind cannot explain the differ~nce between
ordinary hadron states (singlet and octet or nonet states in the U(3) symmetry
for mesons and singlet, octet and decuplet states for baryons) and exotic states
(10*, 27, etc., in the U(3) symmetry) with zero triality charge. If one tries to
explain the difference between hadrons and exotic states from the force of the
:first kind, the four(:five)-body system QQQQ(QQQQ Q) may have room to become
more stable than two (three)-body system QQ (QQQ). Therefore, it is expected
that there exists a constructive force of the second kind that distinguishes hadron
states QQ and QQQ from exotic ones QQQQ, QQQQQ, etc. It has been shown
that the strength of the constructive force between urbaryon Q and antiurbaryon
Q which reproduces the observed Regge recurrence of resonance is represented
by a potential of a breaker-shape with depth about - Vu = 0.1 Ge V and range
a~3.5 Gev- 1 ~0.7 x 10-18 cm. 9l
On the other hand, from the investigation of the hadronic reactions in terms
of UR, the existence or absence of non-exotic resonances in the s-channel is
closely related to the type of UR diagram. Each line in the UR diagram which
expresses the flow of usual unitary spin quantum number, should be regarded
as representative of the urbaryon Q since the U(3)' freedom is frozen in the
UR diagram. Furthermore, it has also been shown that the t-dependence of the
magnitude and phase of the H-type VR amplitudes and the s-dependence of
breaking of the line reversal relation can be reproduced by the potential with
the same depth and range as those used for Regge recurrence.10l These facts
suggest that the constructive force of the second kind shares its origin with the
interactive force of UR. A similar viewpoint has been implied by the concept
X-type
a
~-~>:<! ~-H
s
s
Fig. 2. The X-type and H-type UR interactions for M-B
scattering. Since in the H-type UR diagram the QQ
pair annihilation and creation occurs in the s-channel,
the constructive force of the second kind affects attractively the H-type UR process.
All-Angle Features of Urbaryon Rearrangement Amplitudes
991
of duality. Therefore, it is necessary to clarify the origin of the force with
reference to the duality and UR diagram.
In a UR process, whether the constructive force of the second kind affects
attractively or not depends on the occurrence of the QQ pair annihilation resulting in QQ or QQQ in the s-channel (see Fig. 2). The existence of the pair
annihilation in the s-channeT specifies the property under s-u crossing of the QQ
system propagating in the t-channel. If an urbaryon Q has a half spin and is
subject to the Fermi statistics as usual particles, for the QQ systems with I= Y
= 0, the following six series of spin-parity are allowed:
natural parity
J 0 P =even++,
(2·5)
unnatural parity
JOP
=even-- (except for o--),
J 0 P=even+-,
JOP
=Odd-+.
JOP
=odd++,
(2·6)
If the contribution of the natural parity .series in the t-channel are predominant
as indicated by the leading Regge trajectory, both the contributions from J =even
and J =odd give rise to attractive force between rearranged urbaryon Q and
· antiurbaryon Q in the H~type UR, while in the X- type UR the J =even series
gives an attractive force but the J =odd series a repulsive one and their contributions cancel out. The situation comes from the general structure of the propagator of QQ system within natural parity states. 11l In the low energy region,
however, the cancellation of the lowest QQ state 1-- which causes repulsive
force in the case of the X-type UR is not perfect. As to the unnatural parity
series, the contribution is considered to be small and the correspondence to the
UR diagram has not yet been clarified.' Furthermore, there is a possibility that
most of their contributions from the series in (2 · 6) cancel with each other for
both the H-type and X-type UR.
Taking all the above discussion into account, we suppose that high energy
collisions of hadrons take place as follows {see Fig. 3): Each urbaryon Q (or
Q) in a colliding baryon (meson) with momentum p which is larger than a few
GeV jc, is moving with momentm rvp/3 (~p/2) in the rather flat bottom of very
Fig. 3. The UR in the flat bottom of very
deep potentials Vr which have a range
a;;;;3.5 Gev-• and has an effect only
on the triality charge. Each urbaryon
Q (Q) in a colliding baryon (meson)
with momentum p which is larger than
a few GeV/c, is moving with momentum ~p/3 c~p/2) in the bottom.
992
Y. Igarashi, T. Matsuoka and S Sawada
deep potentials VI which have a range a~3.5 GeV- 1 and has an effect only on
the triality charge (constructiv e force of the ·first kind). The Q's behave as if
they felt only the potential Vn and construct rather weakly bound systems.
When the hadrons collide and the two potentials VI stick together, the Q's are
rearranged without influence of VI from a hadron to anoth~r. In the reaction
with a small momentum transfer, however, the effect of binding due to Vn cannot be neglected. Therefore, c'orrelation among the rearranged urbaryons is
important. On the other hand, ·in the p:rocess with a large momentum transfer,
the urbaryons are rearranged as if they were independent free particles as far
as they compose a non-triality charge system with anothet urbaryons after collision.
§ 3.
All~angle.
behavior of UR amplitndes
As pointed out in the previous paper, 2l the UR amplitudes can be factorized
into pieces correspondi ng to urbaryon line. ·In the large momentum transfer
region, the UR amplitude is expressed in terms of the product of each overlapping integral of urbaryon Q as' if each urbaryon Q rearranged freely; while,
in the small momentum transfer region, the amplitude is reflected by the properties. of binding force among urbaryons Q and the hadron state composed of them.
As a result of the correlation among urbaryons Q rearranged in the crossed
channel, the amplitude exhibits Regge-like behavior. On the basis of the .urbaryon
picture, the UR amplitude for two-body scattering is written as
(3·1)
where Fi 1 (s) stands for the overlapping integral of urbaryon Q rearranged from
hadron with momentum Pi .to that with p 1 ((Pi+P1i=s,t or u) and Tn,nu(t,u)
has a close relation to the degree of correlation among urbaryons Q rearranged
in the t- and u-channels. The normalizatio n g is a coupling constant. The
overlapping integral FiJ (s) is a function of (ki + k 1Y where ki (k 1) is p./2 or
Pi/3 (P1/2 ~r p 1 j3) for urbaryon Q in meson or baryon with momenta Pi (PJ).
The explicit form is given by
(
FiJ (s) = 1
(
~i j
-
-
~i j
(k, + k,y ) - 1
Ao
(ki + k,Y ·) -1
-'--'-'-~'-
-
-'---'----'"-
Ao
(3·2)
C>O)
The universal parameter ,\0 is related with the effective mass of urbaryon Q.
The constant ~£/ represents the overlapping at (k 1 + k1Y = 0 which is distinguishe d
according as i and j are meson or baryon (anti baryon).. From the analyses in
the previous paper, 2l we obtain A.o=0.31(GeVY, ~MM=~BB=1.2 and ~MB'=l.8. The
exponent Tn,nu(t,u) assures Regge-like behavior in both smallJtJ and JuJ regions
All-Angle Features of Urbaryon Rearrangement Amplitudes
and tends to unity for large momentum transfer region.
by theform
and
as
r ,., (t),
993
This situation is realized
is linked to the Regge trajectory in n,-body state of urbaryons
a,., (t)
r,.,(t)~1-a,.,(t)
(3· 4a)
r,.,(t)~n,
(3·4b)
for small Itl region and
for large Itl region. · The function r ,., (t) is considered to express the degree
of the correlation among urbaryons Q rearranged in the t-channel. In view of
these properties of F; 1 (s) and r ,..(t), the UR amplitude (3 ·1) can be expressed as
(3· 5a)
for small Itl region,
(3·5b)
for small Iul region and
( -s)-n, ( - t).-n'.( -u )-n,.
T
-=
(3·5c)
.S
m the large momentum transfer region.
The above characteristic behavior of UR amplitudes leads us to independent particle picture for urbaryon Q receiving large momentum transfer aSi far
as the urbaryons Q are imprisoned in the bottom of the Hartree field for the
singlet state of U(3)', where the freedom of ordinate is frozen. In the following, from the viewpoint of the independent urbaryon picture, we investigate the
structure of UR amplitude and the properties of the UR interaction. For simplicity, we take the MM-7MM scattering
(3·6)
where a, b, c and d are mesons. The center-of-mass and inner coordinates of
the constituent urbaryon Q are expressed as (X;, i;) (i = 1, 2, 3, 4). The X. type UR amplitude for MM scattering is given by
(c,
dl Tla, b)=
t!!}d'X;d4r;)d'x1···d'x,P"* (X., rc, X1, Xa,Pc)
c, d
X
P"* (Xd, rd, x2, X4,pd)K (Xa, xb, x., xd, ra, rb, rc, rd)
X
P"(Xa, ra, Xr, X2,Pa) P"(Xb, rb, Xa, X4,Pb),
(3·7)
994
Y. Igarashi, T. Matsuoka and S. Sawada
where Xa=t(XI +X2), ra=XI-X 2, etc., K(Xa,Xb, Xc,Xa,ra ,rb,rc,ra) is an interaction kernel and 7JI (Xa, r a, xi, x 2, Pa) represents the wave function of hadron
a with incoming momentum Pa as shown in Fig. 4. The H-type and Z-type UR
amplitude s are given by interchang ing of suffices 1, 2, 3 and 4. Since a hadron is
considered to be composed of urbaryons Q by the second kind constructi ve force,
it is assumed that
=
s
d'qa¢(XI, XI, PI)¢* (X2, X2,P2)f(qa ),
(3·8)
where qa =(PI- P2) /2. The function f(qa)
gives the relative momentum distributio n of
urbaryons Q bound in hadron a, When
the relative motion of urbaryons Q in hadron
is neglected compared with that of hadron,
then we have an approxima tion
(3·9)
and this leads to PI =P 2=Pa/2 in Eq. (3 · 8).
The. wave function of urbaryon ¢(XI, xr,PI)
with momentum PI is written as
t-
·a
s
Fig. 4. The X-type UR diagram for M-M
scattering. A number is assigned to
each urbaryon whose center-of-ma ss
and inner coordinates are represented
as (X,, x,) (i=l, 2, 3, 4).
Here, cp (xr, PI) represents the inner wave function of urbaryon Q.
of Eq. (3 ·10), Eq. (3 · 8) is rewritten as
.7J!(Xa, ra, Xr, x2, Pa) =eiPaXa
s
d'qaf(qa) cp ( Xr,
By the use
~a +qa) cp* ( X2, ~a- qa).
(3·11)
Inserting this equation into Eq. (3 · 7) we obtain the following form:
(c,d/T/a, b)
= Sl.LCd'X;d 'r,d'q;ei(p ,X,+q'r':f( q;) )K(Xa, xb, Xc, Xa, ra, rb, rc, ra)
c, d
(3· 12)
where the overlappin g integral F ( (k + k'Y) is .of the form
F((k+k'Y )= Jd'xcp(x,k )cp*(x, -k').
(3·13)
All-Angle Features of Urbaryon Rearrangt:ment Amplitudes
995
Here we introduce new sets of variables
W=t(Xa+Xb+Xe+Xc~),
X=t(Xa.+Xb-Xe -Xa),
(3·14)
Y=t(Xa-Xb+Xe -Xa),
Z=t(Xa-Xb-Xe+Xc~),
and denote their conjugate momenta as
N=HPa+Pb+Pe+ Pd),
P=t(Pa +h -Pe -pd),
(3·15)
Q=t(Pa -P.+Pe -Pc~),
R =HPa -pb-Pe +Pa).
Then, the relations s=P 2 , t=Q 2, u=R 2 are satisfied.
in Eq. (3 · 12) with respect to W, we have
<c, d!T!a, b)
=(j4(N)
Carrying out the integral
s
d4Xd4Yd4ZJ!}d4rid4qieiq,r'f(q.))ei<PX+QY+RZ)
e, c!
(3·16)
Let us suppose that the interaction kernel has a peculiar form
K(X, Y, Z, ra, rb, re,'ra) =rr(X)(j4(Y)o4(Z)K (ra, rb, re, rc~).
(3·17)
Then we have the following UR amplitude:
<c, d!Tia, b)
= 0 (N)
4
Sd~qad4qbd4qed4qaF( (qa, qb, qe, qa)f(qa)J(qb)j(qe)f(qa)
xF( (~ +qa +qer)F( (~ -qa +qar)
xF(( -~ +qb-qer)F((-~ -qb-qa)}
(3·18)
where
K. (qa, qb, qe, qa) =
J
,;Q}d4r,eiq,r')K Cra, rb, re, rc~).
e, c!
(3·19)
996
Y.
lgarashz~
T. Afatsuok a and S. Sawada
If the urbaryon s Q are bound rather weakly, the relative momentu m distribut
ion
of urbaryon s f(q) is restricte d to narrow region. The width of f(q) would
be
in the range of a few hundred MeV /c provided that the effective mass of urbaryon
Q is about 400 MeV as indicated in the previous paper. Therefo1 e, ir:t the large
momentu m transfer region where P, Q, R are larger than a few Ge V / c, the
overlapping integrals in Eq. (3 · 18) can be replaced by their averaged one
unless
K (q,., qb, qc, qd) is consider ably enhanced in the large qi (i =a, b, c,
d) region.
Then we have
<c, dl Tla, b)
=o4 (N)
[F ((~ ) JTF ((~) )J Jd4q~d4qb~qcd4qdK
2
)
(q,., qb, qc, qd)
(3. 20)
'fhis is just the factorize d amplitud e. The same procedur e as shown here
can
be applied to the other types of UR diagram and to the processe s including
baryons. If the overlapp ing integral defined by Eq. (3 · l3) behaves as Eq.
(3 · 2)
in the large momentu m transfer region, Eq. (3 ·.20) leads to Eq. (3 · 5c).
As an
example, ignoring the spin of urbaryon Q and assuming that the inner.
wave
function of urbaryon Q is of the form 12l
(3. 21)
we have the overlapp ing integral
(3·22)
where . mQ stands for the effective mass of urbaryon Q and a represen
ts the
extensio n of urbaryon Q with respect to space-tim e coo~dinates. In this
case
the overlapp ing integral (3 · 22) is due to the effect of Lorentz contracti on.
From
(3 · 2) and (3 '22) we have
Ao =:2mQ2 •
(3· 23)
The results obtained in the previous P<,tper 2l lead to the value
mQ=400 MeV.
(3· 24)
The property of the interacti on kernel (3 ·17) means that the center-of -mass
of hadrons collide with each other in the zero range. Furtherm ore, the factorize
d
amplitud e (3· 5c) is closely connecte d with the independ ent urbaryon picture
in
which urbaryon Q has its effective mass of about 400 MeV and the)nner relative
momentu m in hadron extends only to a few hundred MeV /c. Therefor e,
in the
case that the transferr ed momentu m are sufficien tly large as compare d with
the
inner relative momentu m of urbaryon Q, the UR amplitud e has the factorize
d
All-Angle Features of Urbaryon Rearrangement Amplitudes
997
form (3 · 5c). On the other hand, in the small momentum transfer region the
correlation _among urbaryons Q becomes important and the UR amplitude shows
Regge-like behavior. In this region, the zero range approximation 'of the interaction kernel (3 ·17) is not applicable. In the low energy region, urbaryons Q
co·rrelate strongly with each other in s-, t- and u-channels and the UR amplitude
would have a structure similar to the Veneziano amplitude.
§ 4.
Discussion and remarks
From a viewpoint that the urbaryon Q are rather weakly bound in a hadron,
it has been shQwn that the'UR amplitudes are factorized into overlapping integrals
with respect to the coordinate of each rearranged urbaryon. The comparisons of
the. factorized UR amplitudes with the experimental cross sections of the elastic
pp, pp and K±p scattering made in the previous paper 2l lead to a value about
400 MeV for the urbaryon mass mQ. This value of mQ implies that the binding
energy of hadron is a few hundred MeV. M_oreover, this situation assures the
above• viewpoint that the constructive force of the second kind is rather weak
and that the urbaryon Q can be treated as independent particles in the large
momentum transfer region. The urbaryon Q is realized as a quasi-particle given
by (2 · 4) only in the inside of the very deep _potential (constructive fore~ of the
first kind) for U(3)' singlet state. Therefore, we cannot isolate the urbaryon
Q. On the other hand, the urbaryon q with large mass and triality charge will
be observed in the future. The viewpoint given here is based on the two-step
structure of· the constructive forces. The same situation has been suggested from
investigations . of the deep inelastic lepton-hadron collisions in terms of the parton
model and the light cone al~ebra. 3 >
As shown in the previous paper, 2> the factorized UR amplitude (3·1) can
reproduce the experimental cross sections in both the small and the large momentum transfer regions in a unified way by introducing functions r ,., (t), etc.,
given by Eqs. (3·3) and (3·4). It is an interesting problem to study the
behavior of r ,., (t), etc., around the critical momentum transfer tc"""- 2 (Ge VY,
where the functions transform from (3·4a) to (3·4b). It will supply a useful
clue to clarify the structure of the urbaryon Q as a cluster of many urbaryons
q and q.
In Fig. 5, we summarize the general features of the two-body hadronic reactions in terms of the UR diagrams by taking the MB-~MB pro-cess as an example. In this figure, the interactions due to the exchange of unnatural parity
objects are not yet clarified. In the interactions, the effects 'of the creation of
urbaryon pairs qq, etc., will play an import~mt role and will require the fully
relativistic treatment. As for the natural parity exchanges, almost full correspondence of the· experimental features to the UR diagrams has been obtained
for all energy regions and for all momentum transfer regions available at present.
998
Y. Igarashi, T. Matsuoka and S. Sawada
UR diagram
forward
large momentum
transfer
back'ward
~ )S~§""'l8( 8. l8(
X€:M~~tr,"",))/-1 X X
(5, 2' 3)
exchange
H- type
( 3, 2, 5)
- sOiz ( t)e -ii'CO',itt),
-5 • S -2,ffl-3
J
-
S Clls
(U·) e-:-if[ots(u} J
~~=ll~I~,)X X)!-\(
exchange
z- type
-s"•"'e-t'""'•"'
(3,4,3)
-s -s-•1 tr1Jur• -s"'•<"'e·-ima,<u>
}
'
Fig. 5. The general features of UR amplitudes for M-B scattering. The scattering amplitudes
at high and low energies behave as shown graphically on the right- and left-hands side.
This fact supports the viewpoint that many degrees of freedom of the urbaryon
pairs qlj, etc., are essentially frozen and their effects· have already been contained
within the urbaryon Q. The effective freezing of the freedom of urbaryon pairs
qq is considered to be related with the success of non-relativistic approaches with
U(6) symmetry.
For exotic state the constructive force of the second kind does not work
attractively. However, the existence of the nuclear matter indicates that if there
' exists a long range force due to unnatural parity exchange such as one pion exchange, the exotic system may have resonance or bound state. Therefore, it is
expected that there are many kinds of the exotic matter as the many-body system
of hadrons (not directly constructed from urbaryons Q) with large hypercharge
or isospin, etc. If Z 0 and Z 1 observed in K+p and/or K+n scattering are resonances, they can be regarded as· the evidence for' such exotic systems. In order
to investigate such exotic st~tes, a phenomenological study of so-called exotic
peaks in the forward and backward differential cross sections in terms of r,., (t)
and r,.u (u) will supply a useful clue as well as the careful phase shift analyses
for the exotic states such as KN and NN systems.
Acknowledgement s
The authors would like to thank the members of elementary particle and
nuclear research groups of Nagoya University for valuable discussions.
All-Angle Features of Urbaryon Rearrangement Amplitudes
999
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Note added in proof:
After submitting this paper for publication we. were informed that a view that the "constructive force" (the second kind force in our model) is essentially the same as the "interactive force"
has been proposed also by S. Machida in view of dual realistic quark model with Bethe-Salpeter
equation.