92
Progress of Theoretical Physics, Vol. 33, No. 1, January 1965
Classical Boson Wave Excitation in Inelastic Collisions and Shadow
Scatterings at Very High Energies
Mikio
NAMIKI
Department of Applied Physics
and
Science and Engineering Research Laboratory
Waseda Uni·versity, Tokyo
(Received September 21, 1964)
Under the assumption that most of elastic scatterings at very high energies are the
unitary reflection--such as the shadow scattering--of inelastic collisions resulting in the
classical heavy boson wave excitation, some important interrelations between the elastic
scattering and the multiple particle production at very high energies are investigated. Recent
analyses of cosmic ray jets have suggested that direct products of the multiple particle
production are mainly heavy bosons rather than pions, and that observed pions come from
the decay of heavy bosuns. Two extreme cases are considered of the final states of the
boson assembly; (i) no correlation case, and (ii) coherent boson wave case. The real states
would be between the two cases. The boson assembly with no correlations ar~ well described
by a classical boson wave state which may be identified with the eigenstate of the boson
annihilation operator. The coherent wave state, describing the definite phase wave, is defined
by the eigenvector of the phase operator. One of the important results is such that there
hold the relations
m
the no correlation case
m the coherent wave case
between the mean variance oN and the average N of the boson multiplicity, and that the
mean vanance of the boson momentum of~ or of the invariant momentum transfer oil must
depend on the primary energy p like
ok or aLl oc pfl,
!3<
1 in the coherent wave case. The present
where /3= 1 in the no correlation case and
status of experiments is too poor to descriminate sharply between both cases, although the
above results are consistent with experiments. Qualified high energy data above about
1,000 GeV are highly desired.
In the no correlation case the interrelations between the elastic diffraction scattering
and the multiple particle production are formulated in some details, and it is shown that
the results are consistent with the existing experiments. Some brief remarks are given on
the large angle elastic scattering and the anti-shrinkage of the proton-antiproton diffraction
peak. Finally a fundamental assumption on the asymptotic behavior of the elastic scattering
amplitude in the modern S matrix theory is criticized from the viewpoint of the multiple
particle production phenomena. In Appendices some mathematical supplements are given
for the eigenvectors of the boson annihilation operator and their reciprocal vectors, and
the physical meaning of the complex energy variable and the analytic continuation of the
elastic scattering amplitude are discussed.
Classical Boson "\IVave Excitation in Inelastic Collisions
~
l.
93
Introduction
Recent researches suggest us to assume that most of elastic scatterings at
very high energies are nothing but the shadow scatterings of inelastic collisions.
Under the assumption, it is worth-while to investigate the interrelation between
the high energy elastic scatterings and the cosmic ray jet phenomena, the latter
being the typical and dominant phenomena of high energy collisions. An introductory work along the line of thought was given by van Hove, 1l but he has
not considered recoil nucleon effects and heavy boson formation effects together
with rather large values of the invariant momentum transfer squared as indicated
by recent experiments. In this paper we develop the problem both on the
experimental basis of a recent analysis 2 l of cosmic ray jets and on the theoretical
basis of the classical wave aspect of heavy bosons.
Let us first remark on the multiple particle production phenomena, such as
comic ray jets, at very high energies. As was firstly pointed out by the Japanese
ICEF group and Niu,Bl it is found that the mean value of the invariant momentum transfer squared J 2- - -one of the most important parameters to describe
the phenomena 4l--is a constant of the order of nucleon mass squared irrespectively of primary energies. 2l This is against the multi peripheral model, but
prepares the starting point of the linked heavy particle model. In a previous
work 2l the present author and his collaborators have introduced the linked heavy
particle model as one of the simplest models consistent with the above experimental results, and successfully analyzed the recent ICEF data5 l based on the
model. The underlying idea of the model is such that the multiple particle
production mainly comes from the excitation of the inner clouds of colliding
nucleons--such as (J), p or 1VlV-cloud, but not from the excitation of the outer
pion cloud. These observed pions are to be regarded as the decay products of
(J), p or NN-systems directly produced in the nucleon-nucleon collisions at very
high energies. Thus we assert that there are some significant correlations among
the pions due to the formation of such heavy bosons as (J), p or NN-systems.
Now we discuss the rather large values of J 2 shown by the recent experiments. The mean value of J=== V A2 is of the same order as mass m of directly
produced bosons. If J would be much smaller than the boson masses as encountered in the multiperipheral model, there would be no phase correlations
among produced heavy bosons. Because of J:::~tn, however,. we may expect
some kind of phase correlations among them. Nevertheless it is not always
possible to observe the phase correlations, since the phase correlations directly
after the collision may be smeared out with some possible final interactions
among the heavy bosons. · Whether there remain some significant phase correlations among the heavy bosons depends on the amount of the final interactions
among them. The situation may be also understood by the following arguments
giVen by the present author and Iso 6 J in their work on the applicability condi-
94
!Ill. Namiki
tions of the hydrodynamical model : The collision complex being formed directly
after 1VN collision has a thickness ocp- 1 much narrower than the mean boson
de Broglie wave length p··- 112 , so that various waves in it should be in phase.
Consequently, the collision complex at the first stage cannot be regarded as a
drop of fluid, but as an assembly of coherent waves. Of course, one may
anticipate that the strong final interactions can change the collision complex
into those which are composed of various waves with random phases, or probably
a drop of fluid.
By the way it is very difficult to estimate precisely the final interaction
effects on the resultant phase correlations. Here we do not estimate the final
interaction effects in detail, but consider only the two extreme cases; the first
case in which there are no correlations among the heavy bosons, and the second
in which the whole heavy bosons are observed as a coherent wave. General
cases can be treated by means of interpolation between the two extreme cases.
The present paper is concerned mainly with the first case and briefly with the
second. One of our aims is to derive some results serving to discriminate between the above extreme cases through comparison with experiments.
Both the extreme cases are characterized by excitation of a classical (or
semi-classical) boson wave in the nucleon-nucleon collisions at very high energies.
We are now interested in the classical wave excitations to be regarded as those
which reveal the wave behavior or the field aspect of bosons. One may say
that we have not recognized the real need to introduce the quantum field
theory into the elementary particle physics. The fashionable elementary particle
physics of two or a few particle processes seems to be only concerned with a
relativistic quantum mechanics, but never with the quantum field theory. Only
exception comes from the photon, since we can certainly observe the classical
electromagnetic fields. The photon should be described by the quantum field
theory. If some boson has the field nature (or wave nature), the multiple
particle production should be well described by the classical boson wave excitation. In other words, actual observations of the classical boson waves may
answer the question as to whether the future theory will need the quantum
field theory or not. Thus it may be expected that the multiple particle production phenomena will play an important role in a search of the future theory.
In § 2 we present the general relations between elastic scatterings and inelastic collisions. There recoil nucleons are taken into account. In § 3 we are
concerned with the classical boson wave states in which there are no correlations
among bosons. The important conclusion is such that the boson state as given
by van Hove may be understood as a classical boson wave state. Section 4 is
devoted to the detailed discussions on the interrelations between the elastic
diffraction scatterings and the multiple particle productions. In particular, the
energy dependence of the Lt-distribution is investigated in some detail. The
large angle scatterings are briefly discussed too. In § 5 some special remarks
95
Classical Boson 1iVave Excitation in Inelastic Collisions
are given on the proton-antiproton collisions, together with the topics of the
anti-shrinkage. In ~ 6 the coherent boson wave states are investigated and are
compared with the no correlation case. Section 7 is concerned with the concluding remarks. In particular, we make some comments on the fundamental
assumption for the asymptotic behavior of the elastic scattering amplitude in
the modern S-matrix theory, from the viewpoint of the multiple particle production. In Appendix A we discuss some detailed mathematical properties of the
classical boson wave states. In Appendix B we remark on the physical meaning
of the use of the complex energy variable and the analytic continuation in the
modernS-matrix theory. This prepares a preliminary knowledge of discussions
in ~ 7.
~
2.
General relations between elastic scatterings
and inelastic collisions
For the sake of simplicity we write down the theory only in the center-ofmass system, throughout the paper. Consider an elastic scattering of two
nucleons from the state (p, -p) to the state (p', -p'), where P=lpi=Jp'j
and E 7, = Ep' = vj} + M 2- =p, 1_\;f being the nucleon mass. By defining the invariant elastic scattering amplitude T (p', p) or T (p, 0) (0 being the scattering
angle) with
T(p,0)=2rcpEv<P', -p'jRjp, -p),
(2·1)*>
then the unitarity condition is written in the form
(2 ·2)
where
(2 ·3)
Here jinel: p) represents the state vector describing the system after
elastic collision of two nucleons with momenta p and - p. P~" and
respectively, the total energy-momentum operator and its initial value,
W~"=(0,0,0,2Ev). The unitarity condition for 0==0 can be rewritten
where
2
Im T (p, 0)
::=
p
2rc
2
Otot,
*> R stands for the transition matrix.
F(p, 0)
=
p
2rc 6;,,.,,
l
the inW~" are,
that is,
as
(2 ·4)
(2 ·5)
96
1\;f. Namiki
Of course, Cf1ut, C5' 1nel and CJ,,t are, respectively, the total cross section, the total
inelastic cross section and the total elastic cross section.
Now we assume that the elastic scattering at very high energies must be
nothing but the shadow scattering of all inelastic collisions, along the line of
thought as mentioned at the beginning of § 1. Because of the assumption T
becomes purely imaginary, so that we can determine T by solving (2 ·2) for
a given F(p, 0). For small angle scatterings the function F(p, fJ) may be reduced to the simple form
F(jY, fJ) =Fa (p) exp ( -1/2 · T (p) fP).
(2 ·6)
Then F(p, 0) is completely characterized by Fa(P) =F(p, O) and
being functions of the primary energy (or momentum) alone.
Fa (p) is replaced by CJ ineJ as seen in (2 · 5) . Therefore we can obtain
angle scatterings, provided that uinel and
are given. Following
we can write as
r
T(p), both
Notice that
T for small
van Hove/)
(2 ·7)
where t = -jp' -- pj 2 = -j}0 2 •
and Cfinel as follows :
Here
r
and
r =a(T/p
Cft .. t
are represented 1n terms of T
2
(2 ·8)
),
1 ( 1- _Uine~-)
2
2nr
112
]
__ (
a being a constant approximately equal to unity.
1_
1/2
Qi11Cl
2nr
)
'
+1l,
(2 ·9)
J
The existing experiments give
2
us r=10(GeV/c)- or 10/NJ2.
Now the first work is to write r in terms of parameters characterizing the
multiple particle production. The parameters must be embedded in the state
vectors jinel: p) and jinel: p') of (2 ·3). The relative distribution of secondary
particles around p' in I inel : p') has the same form as the distribution around
p in jinel: p). In other words, the vector jinel: p') is produced by means of
a rotation of the vector jinel: p) by the scattering angle fJ, that Is,
jinel: p')=exp(ifJL) jinel: p),
(2 ·10)
where L is the component of the total angular momentum operator perpendicular
to the plane determined by p and p'. Therefore we have only to know the
details of the vector jinel: p) alone. For the sake of simplicity, we make an
additional assumption that only two recoil nucleons are left besides heavy bosons
after the collision. The assumption permits us to write the vector as
1
v2!
(2 ·11)
97
Classical Boson \IVave Excitation in Inelastic Collisions
c(;
where
is the creation operator of a nucleon with momentum q, and ::J(q1, q 2 : p)
stands for the amplitude to describe the final distribution of two recoil nucleons.
Here we have suppressed explicit dependences on spin and isospin. We have
also discarded the possibility of exciting the recoil nucleons. It is not so dif- ·
ficult to take them into account.
The state vector lbosons) exclusively describes produced heavy bosons.
~
3.
Classical boson wave states
In this section we discuss the detailed form. of the boson state lbosons),
under the assumption that there are no correlations among produced heavy
bosons. Let us write the heavy boson operator*) as
¢(x)
=
y1
1
L....J--'--'-'-
1 .:
Y V ": Y
[ akexp(zk·x
·
)
+a~.;+
•
J
exp(-zk·x)
(3 ·1a) **)
2oJk
for the box normalization with volume V, and
¢(x) =-
·_cc1~cc~ 3
v
(2n)
3
\
,)
d
v
1£_
2w~.;
[b(k)exp(ik·x) +b+(k)exp(--ik·x)]
(3 ·1b) **)
for the delta-function normalization, w,, being the energy of the boson with mass
rn. The commutation relations become
}
(3 ·2a)
or
[b(k), !J+(k')]
=c)(J~-k'),
[b(k)' b(k')] = [b+ (k)' !J-i- (k') J :=0,
}
(3 ·2b)
where ak or b (k) stands for the annihilation operator of a boson with momentum k, and a};; or b+ (k) for the creation opertor.
In our scheme the direct products in inelastic collisions should be such
heavy bosons as p, w or low energy NN-system. For the sake of simplicity we
suppose that there are rnany bosons of only one kind after the collision. The
boson is considered to be described by the above operator, whose mass should
be at least of the order of nucleon mass M. The observed pions in the multiple particle production phenomena are to be explained as decay products from
the bosons, and the transverse momentum of each pion should have a magnitude of the order of the pion mass /1.
*) For simplicity we deal with neutral scalar bosons alone.
**> Notice that
a!f=
j
(2n-)3
V
.
b(k),
and
Jvl. 1Vamil<i
98
We start with van f{ove's representation of the boson state
lbosons) =
"
{g/! + .fg/J (!i:) !J+ (lii-) d
3
lii- + .\
j/,;
j/,;
f
j
j'g/! (ffs:l'
y._2) !J+ (1£1) !/'
u~2) d
3
kld:3k2
j/,;
j
+ ··} IO),
(3 ·3)
where IO) stands for the vacuum state, and g/0 l, g/l (k),. .. are c-number functions.
All the integrations are carried out over the j-th subinterval ilkf of the whole
interval ( -- p, p) with respect to each longitudinal component of boson momenta,
and over the whole intervals with respect to the other components. It is repeatedly noticed that p is the magnitude of momentum of an incident nucleon in the
center-of-mass system, and that the theory is thoroughly described in the centerof-mass system. Here it is implicit that the probability of finding bosons with
momentum belonging to any subinterval is necessarily finite, and that the total
number of subintervals is sufficiently large. The length of each subinterval is
of the order of (1/n). Now if the functions g/l (k), g/l (kh k2) , ... are regular,
then rough estimates of the ratios of the second term, the third term,··· to the
first term in the bracket of (3 · 3) come to be of the order (1/ n) , (1/ n) 2 , • • •
for large n, respectively. Hence one easily gets
lbosons) =
{fl
g/l} exp {~
J(Y/
j/,;
1
)
(k) /Y/0 )) !J+ (k) d 3k} IO) +
o( ~ ) '
j
which may as well be rewritten as
(3 ·4)
where C =lim
U" g/ 0!,
n->r-oo j=
L
and cp (k) IS the function defined by lim { (g/l (k) f.q/l):
n-:,.co
j = 1, 2, .. ·, n}. The integration with respect to k in the exponent of (3 · 4) is
carried out over the whole region of boson momentum. The assumption that
the functions g/ 1 ) (k), g/ 2 l (kr, k 2) , • • · are regular means that there are no correlations among bosons. Possible correlations among bosons would make the functions singular. For example, if there exist such correlations as formation of p
or w from two or three pions- -where the operator !J+ is considered to describe
the pion----, then g/l or g/ 3 > becomes singular and contributes the leading term
to lbosons) for large n. In our theory, however, these types of correlations are
to be embedded into the heavy boson operator b+ (k) from the outset. Another
important singularity comes from the correlation originating in the coherent
wave excitation, by which we have so many in-phase boson waves with wave
numbers in a very narrow region around a value. For a while we discarded
the coherent correlation until we return back to the problem in § 6. Therefore
we are concerned only with the case in which there are no correlations among
heavy bosons. Moreover we are only interested in the asymptotic limit as n
99
Classical Boson Wa·pe Excitation in Inelastic Collisions
tends to infinity, so that we can write down the boson state as
lbosons) =I cp) = C exp 0 b+ (lc) cp (k) {ZSk) IO)
(3 ·Sa)
Icp) = C exp (L;ai;;ak) IO),
(3 ·5b)
or
k
where
It is easy to see that I ({J) is the right-eigenvector of the annihilation operator
b(k) or a"' belonging to the eigenvalue ({J(k) or a 1., and that <({JI is the lefteigenvector of the creation operator b+ (lr,) or ai;; belonging to the eigenvalue
({J* (k) or a 1/ . For the proof we have only to show that
exp (- ~ IJ+ (k') ({J (k') d 3k') b (k) exp ( ~ IJ+ (k') cp (k') d 3k')
b (k)
+ cp (k)
(3 ·6a)
and
exp 0 ({J* (k') b (k') cZSk') b+ (k) exp (- f ({J* (k') b (k') d 3k') = b+ (k)
+ ({J* (k),
(3 ·6b)
which yield immediately
b(k)lcp)=({J(k)lcp),
<({Jih-r (k)
c::::
}
(3 ·7)
<cpl cp* (k).
Even though there holds the eigenvalue problem for the boson creation or annihilation operator, it must be noted that the eigenvectors never form an orthogonal set, because the creation or annihilation operator is not hermitian. The
mathematical properties of the eigenvalue problem are elucidated in some details
in Appendix A.
Let us consider the expectation value of an arbitrary operator A in Icp)
defined by
<OlAIO)
------ -<OIO)
----~--
(3 ·8)
If we put A=b or b+, we get
(3 ·9)
which yield
<¢ (x) )rp == cp (x), <[r;> (x) J )rp =
2
where
[cp (x) J2 ,
(3 ·10)
100
cp (x) =
-c-!,,. ___ ~ _da~
V (27!) 3 JV 2uJ~;;
[cp (k) exp (ik ·X)
+ ~0* (k) exp (- ik ·X)].
(3 ·11)
vV e also obtain the formula
(3 ·12)
where the symbol : · · · : stands for the normal product. Equations (3 ·10) and
(3 ·12) permit us to say that the boson state I cp) represents a classical wave
state in which the classical wave cp (x) is excited. The expectation value of
the boson number operator and its square become
(3 ·13)
and
(3 ·14)
respectively.
Thus we get the mean variance
(3 ·15)
The last relation between r'J1V and N shows us that, in the classical wave state
[cp), bosons are subject to the Poisson distribution law and consequently, have no
correlations among themselves. In other words, the boson state I cp) originating
from a single nucleon-nucleon collision is characterized by a rather low boson
density in the phase space. There bosons behave as if they would be independent particles. Therefore the real classical wave cp (x) itself will be visualized through very high energy nucleon collisions caused by a sufficiently high
intensity beam or through possible accumulation of cosmic ray jet data. For
example, we have the accumulated plots of the multiplicity versus the incident
energy for many cosmic ray jet events, which may be used for test of the
characteristic relation, r'JN = V N, of the no correlation cases. See discussions in
§ 6.
Now we can derive the Poisson distribution in a direct way.
pation number state I··· n,,,. · ·) is defined by
The occu-
Making use of Eq. (3 ·16) and
<cpla!~ = <cp[a]~,
(3 ·17)
C* = <cp[O),
we easily obtain
<cp~···nk···
I
)
=
C* --J-[·
k
-
V
1
!
nk.
n,.
(a~.- *) ..
(3 ·18)
Classical Boson T--Vave Excitation in Inelastic Collisions .
101
Hence the probability of finding the boson number state I· .. nk· · ·) in the classical wave state Icp) is written as
(3 ·19)
This is just what we want to derive.
Here we have used the equation
}
(3 ·20)
derived in the following way :
<cpl cp) =I Ci 2<0iexp(J cp* (k) b (k) d 3k )exp(j'b+ (k) cp (k) d 3k) IO)
=
ICI 2<0iexp(Jb+ (k) cp(le:)d 3k )exp(Jcp* (k) {b(k)
2
= ICI exp(Jicp(k)
dk),
2
l
+ cp(k) }d 3k) IO)
3
where (3·6a) has been used in the first step, and bjO)=O and <OJb+=O in the
second step. ·
Similarly we can derive the probability of finding the N boson state defined
by
(3 ·21)
in the state Icp).
With the help of (3 · 7) one gets
(3 ·22)
which yields the probability
W'P
(kh
k2, .. ·,
k;v) =
l<cplkb 1.:2, ···, k;v)l 2
<q;Jcp)
~!
2
2
2
3
Jq;(kl) I2 Jcp(k2) l ···lcp(kN) l exp{- Jlcp(k) l d k}.
(3 ·23)
Thus we find that the state vector linel: p) is completely specified by the
e-n umber function cp and g. It is not so easy to derive the concrete forms of
the functions, because we must be concerned with the detailed dynamics of the
strong interactions. One may use some variational method, for example,
JVi. .l\fami ki
102
0
(inel: pllllinel: p) _
0
q>,q
(inel: plinel: p) - '
(3. 24)
H being the total Hamiltonian of the system. However, most of the methods
lead us to a set of nonlinear equations for cp and
which may not be solvable
except by the perturbation method. Solving the set of equations derived from
(3 · 24) and the ordinary Hamiltonian of a nucleon-meson system by the perturbation method, we can obtain the Lewis-Oppenheimer-Wouthuysen state. 7 > Anyway we do not enter into the troublesome problem of deriving cp and g from
dynamics, but only put the rough forms to be expected for cp and 9: from the
experimental informations.
Once Taketani insisted that the classical field nature of pion is to be observed as a static internucleon potential describing the nuclear force in the most
outer region. In contrast with this static field, our classical boson waves are
dynamical fields. It seems to us that the Taketani's aspect-to establish the
field nature---should be supplemented by the present work.
s-,
~
4.
Multiple particle productions and shadow scatterings
In this section we obtain the explicit form of F(p, ()) and discuss some
significant interrelations between the multiple particle production and the
elastic scattering at very high energies.
Our preliminary work is to get
F 0 (p)=F(p,O) or uind· Inserting (2·11) and (3··5) into (2·3), we get
f1~(p)
=
(,Pi£P)\J
Ep
47CjY
4_39!cZSq2_J;J(ql, q2: p) lz
Eq,Eq2
.J
Xo
4
(qlp+qz'"-l-
j1~'"lcp(k)
2
l
3
d k- Wp),
where we have reserved only leading terms at the large limit of the boson
multiplicity. The delta function means the conservation law of energy-momentum in the inelastic collision resulting in the classical wave excitation. After
simple calculations one gets
Fo(P)=
1
8rc
·qE P·(cpjcp)· \ dQqJ:l(q,-q:p)l 2 ,
pEq
,
(4 ·1)
where the forward-backward symmetry of secondary particles has been assumed
for the sake of simplicity.*> By making use of (2 ·5), the total inelastic cross
section is explictly written as
(4 ·2)
'''> Note that we can write the conservation law of momentum as ~ kiq:>(k) l2d3k=O.
Classical Boson Wave Excitation zn Inelastic Collisions
103
Now we know
together with (3 ·19) and (3 · 23). Therefore it is easy to get the differential
cross sections of the inelastic collision as follows :
_du;"el ~p: q, · · · n~r · · ·)
dQf)
II
19(q, -q: p) 12
.\dQqiS"(q, -q:p)i
2
k
·zv,_It/"•
n~;;!
exp ( -w,.),
(4 ·3)
where
and
dujuel(p :q, k1, k2, ···, kN)
dQq{f8k1d 3k2 · · · d 3kN
lg(q, -q:p)i 2
Jdf2q!g(q,
X
q: p) !2
exp{- J [cp(k) j 2d 3h}.
(4 ·4)
Notice that ( 4 · 3) has the same form as the formula obtained by Lewis,
Oppenheimer and Wouthuysen.7l Indeed we can get their formula if we use
q and ak determined by the perturbation theory for the ordinary nucleon-meson
coupling. As is easily seen from (4·3) and (4·4), the multiple particle production phenomena are completely specified in terms of the recoil nucleon
amplitude q(q, -q: p) and the classical boson wave cp(k). For example, if we
average the total boson energy
or
by the differential cross section (4 ·3) or (4 ·4), respectively, then we immediately obtain the formula
E(b) := Juh·l cp (k) i 2d 3k,
=
iJ<(u)J..)) ,
l
l
(4 ·5)
where the last step must be regarded as the definition of the double bracket,
namely,
M. Namikz
104
'
\u),..icp(k) [2d 3k
((w,J; =
(4 ·6)
,.
Jlc;o(k) [ d k
2
3
Here the double bracket ((oh)) stands for the average of one boson energy.
should be emphasized that
It
(4 ·7)
where the right bracket has been defined by (3 ·8). Averages of other quantities than energy can be calculated in a similar fashion. Throughout the paper
we use the double brackets as averages of one boson quantities. For later uses,
let us define the average inelasticity IJ by the formula
(4 ·8)
by which one can express the conservation law of energy m the form
(4 ·9)
The left member is the recoil nucleon energy.
Now we turn to the calculation of F(p, fJ) for fJ=!=O .
we have
F(p, fJ) =
On account of (2 ·10),
1 · qEp ·(c;o[exp(-ifJLb)[c;o)
8n pEq
X Jd..QqS"* (q, - q : p) exp (- ifJ ...{N) 9 (q, - q: p),
where
(4 ·10)
(4·11)
(4 ·12)
O"\ and (1' 2 are, respectively, the spin matrices of the first and the second recoil
nucleons. With the help of definitions
(4 ·13a)
Classical Boson W a·ue Excitation in Inelastic Collisions
105
\ dQq9"* (q, - q : p) exp (- i0.£N) 9 (q, - q : p)
(exp (
iOJ:v) )g=
·'
fdQql g (q, -q: p)
"
2
1
(4 ·13b)
we can rewrite F(jY, 0) as
2
F(jJ, 0)
=
( 4 ·14)
p CJineJ(exp(-iOLb))"'(exp(-iOJ;v))cr.
2rr
-~
First let us calculate the boson term (exp ( -- iOL{)))"'. Recalling (3 · 6a, b)
and (3 · 7), one may proceed with the calculations irt the following way : Because of the formula
exp (iOLb) b (k) exp (
(4 ·15)
exp (- i01'b) b (k),
iOLb)
the boson term becomes
(Y?Jexp (- iOLb) I9) =
/CI
2
3
3
(0 lexp(JY?*bd k )exp (- iOLb) exp(Jb+ 9d k) IO)
=
ICI 2(0/exp(J 9*exp (- i0.£6 ) bd
=
(Y?I <0) exp( -- J
I<0/
2
3
3
k )exp(Jb+'f?d
3
k) jO)
3
d k )exp(J 9* exp (- i()_[o) 9d k) .
If we use the axial symmetry of secondary particle distribution with respect to
the collision axis along p, we should have
( 4 ·16)
so that
(4 ·17)
It is here noted that any small angle approximation has not been used, and
that ( 4 ·17) may be applied to large angle problems. Now we make the small
angle approximation :
(4 ·18)
where
(4 ·19)
v <<.£
2
means the average angular momentum per heavy boson, because the
axial symmetry of the boson distribution gives us
0 )) ··
(4 ·20)
The meaning of v((.l~ 2)) permits us to define a parameter t;, corresponding to
M. Natniki
106
the classical impact paramete r of one heavy boson, by
(4 ·21)
where V<(e)'> is consider ed to be the average momentu m of one boson.
the average transver se momentu m of one boson, IC, may be defined by
!C =
1 ~
v <(k
2
(4 ·21')
))
v «-D,
Also
2
))
Further this may be replaced with the definitio n of the average emission angle
of bosons rJr, , namely,
(4 ·22)
smce we may put ;c = V «k 2 )) sin (JIJ. Thus we may infer that the average scatter~
ing angle Oo by the elastic scatterin gs is of the order of 1JIJ/ V iJ , that is,
eo=
1 =
Tu
. . 1Ib
v
(4. 23)
N
The small angle approxim ation of the recoil nucleon term (4 ·13b) gives
rise to
( 4. 24)
where
JdQq!~LN:J(q,
-
i{j:
p)
j
2
J'.v=
(4 ·25)
Jd.Qq[51(q, -q:p)l2
Here the axial symmetr y for the recoil nucleon distribut ion was also used,
namely,
Jd.Qq9"* (q, -q: p) ~Lvc;J(q, -q: p) =0.
T,v may be regarded as the inverse square of the average recoil angle 1J.v.
Thus the diffractio n constant r (see (2 · 8)) becomes
Y=
a
jJ2
(T,,+TN) .
(4 ·26)
Althoug h it is difficult to draw a definite conclusi on on the interrela tion between
Tu and Tv, we may presume that the average recoil nucleon angular distribut ion
in the multiple particle producti on is much duller than the nucleon angular
distribut ion of the same energy elastic scatterin g if the average inelastic ity 1J IS
not so small. That is to say, it would be reasonab le to assume that
Classical Boson Wa·ve Excitation in Inelastic Collis;ons
or
107
(4 ·27)
even though (4 ·27) must be justified by future experiments. Up to the present,
this seems to be consistent with experiments. Thus we are led to the formula
(4 ·28)
Inserting the relation
(4 ·29) *l
into ( 4 · 28), one gets
, -~~ 4-r/((P);a
r- !C2N«k»2 .
(4 ·30)
It is noted that the diffraction constant r can be written in terms of the multiple
particle production data. The existing experiments seem to suggest us that
r= (10/ M. 2) irrespective of p at the high energy limit. Now we are going to
examine whether or not the observed properties of r can be reproduced through
(4·29) or (4·30) from the cosmic ray jet data.
(i) Diffraction constant Through (4 ·30) we can see how r depends on the
distribution of secondary particles in the multiple particle production. For the
purpose, let us first consider the two extreme cases. The first case is specified
by Noc p and constant ((k 2)) independent of p. l-Ienee we have roc (1/p). In this
case the incident energy is spent on increasing the boson multiplicity with highest
efficiency. The energy (or momentum) distribution of secondary particles in
the center-of-mass system has a lump in the lower energy region irrespective of
p, and rapidly falls down in the higher energy region. In the second case we
have the constant N and ((k 2))ocp 2 • Hence r is constant. The incident energy
is spent on increasing the speed of each produced boson rather than on increasing the number of bosons. Consequently the main part of the distribution
should shift to the higher energy region with increasing p.
- It is obvious that the first case must be excluded both by cosmic ray data
and elastic scattering data. The second case is also not real, because N is constant. Nevertheless these extreme cases would tell us some outstanding interrelationship between the elastic scattering and the multiple particle production.
In real cases N should gradually increase with increasing p, but N((/< 2)) / f> 2 or
consequently r should be kept constant. Indeed we can find the distribution
satisfying these requirements. To do this, it is temporarily assumed that the
average inelasticity r; and the average transverse momentum of bosons JC are
*l This is obtained from (4 · 8) by the approximations, EP:-:::_p and ~(J)k~=:::::.~k~, to be justified at
very high energies.
NI. Namiki
108
constant at the high energy limit. The existing experiment*l seem· to support
the assumption. In particular, the air shower data guarantee that 'lJ has a non
zero constan~ near unity at the high energy limit.
Suppose that the momentum (or energy) spectrum of emitted bosons is
described by the distribution function
w(k)dk,
where w (k) is proportional to /:z 2 ! cp (k) [2 integrated with respect to angle vanabies. The double brackets are now calculated from w, for example,
k
.\· ~;.;C) (k) dl:z
( 4. 31)
II
k
\
0·'
~~~x(/?) dk
Here the kinematical limit kmax should be equal to p if we adhere to the forwardbackward symmetric case. After some calculations we can easily find that the
first case appears if
w (k)
< constant
1
/;:a
1n the higher energy region, whereas we get the second case if
U'
(k)
1
2: constant ka
3
in the higher energy region. The case a= 3, namely, of w (k) =constant (1/ /:z )
is essentially the same as the first, because this function gives rise to constant
\(k)), «l:z 2))cx::logp and then rex:: (logp/p). The case 2<a<3 is also the same as
the .first. Special attention should be paid to the case a= 2, namely, of
2
w (k) =constant (l/k 2) , because this leads to ((k))cx:: log p, ((k ))ocp and then
rex:: (logp) - l . Strictly speaking only about the high energy limit, we may abandon
the case. However, we may have to reserve the case, because the logarithmic
dependence is really observed as nearly constant in a limited interval. The real
and practical cases come from such distribution functions as
w (k) =constant
1
/.::a
For a=1 we have
'''l
The experimental analysis of heavy boson transverse momenta is in progress.
(4 ·34)
Classical Boson W a1Je Excitation zn Inelastic Collisions
((k))=l~g (~/ Nfo)
N
109
'
( 4. 35)
r=
2r; log (p/Nfo) ,
fJ a
/C2
'
where Mo is a constant to specify the function ·w (lz) in the lower energy regwn.
For 1<a·<-:=2 we obtain
a -1 ( p
2-a Mo
-
2
N=2r;
Putting a
·)1-ap,
p )a-1
\(k))=
a
a-1
(
1t1o
(4 ·36)
'
3/2, we get
r=
2
• '/)
'I
3
/C
2
a.
(4 ·36')
Looking over these results toget.her with the two extreme cases, we can find that,
as a general trend, the larger values of a give rise to the large average multiplicities and the smaller r's. It is so easy to understand the situation. Now the
distribution function (4 ·34) certainly gives us the energy-independent diffraction
constant of the right order of magnitude. However, one must pay attention to
the fact that the results have been derived on the basis of the assumption on r;
and /C. In contrast with the constancy of r;, for example, we may accept the
assumption
(4 ·37)
which yields the Regge pole behavior of the diffraction scattering. The dependence never conflicts with the observed values of pion transverse momenta, because the pion transverse momenta can be explained in terms of the Q-value in
the heavy boson decay.
Here it is remarked that we have so far been concerned with only or very
near the high energy limit. This is the reason why we can use the cosmic
ray data for calculating the diffraction constant of the elastic scattering. One
may say that the approach to the high energy limit, such as the shrinkage of
the diffraction peak, cannot be formulated within the framework of the present
theory. For instance, the real part of the elastic scattering amplitude-·-which
has been omitted in the present--may play an important role in the approach.
Recently, however, Isdl has suggested that the shrinkage does not come from
the real part effect but from the energy-dependence of ((.1~) )) appearing in the
formula of r. If it is true, we may discuss the shrinkage based on the inelastic collision data within the framework of the present theory. At any rate we
should need there not only the high energy limits of ((k)) and ((k 2)), but also
2
110
Namil?.i
their energy-dependence in the lower energy regwn. This would reqmre the
rather full knowledge about the distribution function over the wide energy
region. Inversely speaking, the shrinkage would serve to check the consistency
between the elastic scattering and the inelastic collision at rnoderate energies,
although the present theory may be altered due to the decreasing of the multiplicity.
(ii) Invariant momentmn transfer Next we are interested in some restrictions
for the secondary particle distribution brought through (4 ·30). By the inspection of ( 4 · 30) we are immediately led to the formula for the mean variance
of the momentum (or the energy) of an emitted boson:
(4 ·38)
That is to say, the momentum (or energy) spectrum is getting wider and wider
proportionally to the primary energy in the center-of-mass system. This is one
of the important results of the constant r and the formula ( 4 · 30) .
Let us apply the similar arguments to the important parameter--the invariant
L1 I" 2 • To do this, we start with the definitions
momentum transfer L1 =
v
fA= p-q- 'f:.k,'
f
I
(4 ·39)
L
runs over the forward group of bosons in the center.r
of-mass system. Taking the symmetry of collisions in the center-of-mass system
into account, we easily prove
where the summation
For practical purpose we have only to deal with the distribution of the longi4
tudinal component L/ 11 • (See Hasegawa and Yokoi's paper, l or the previous
paper. 2l) By making use of the approximations
M2
p=Ej) ________ '
2Ep
the longitudinal component L/ 11 becomes
Clas:·;ical Boson vV(rve Excitation in Inelastic Collision .. ,·
It is obvious that the second tenTl tends to zero with increasing E/).
get the average ~ 1 at the high energy limit as follows :
111
Thus we
where we have used ( 4 · 40) and the fact the integration over the forward group
in average is a half of the whole integration. Since the term containing k3_
never modifies the order of magnitude of
we may safely use the formula
i;,,
(4 ·41')
For an illustrative example, let us calculate .d 11 with the distribution function
·w (k) =constant
as suggested by ( 4 · 34) .
1
k+Mo
(4 ·42)
-
After elementary calculations, .d 11 becomes
(4 ·43)
where
tan 0 0 =
M0
m
•
--·
Notice that L1 11 giVen by (4·43) is independent of p.
reduced to
For Mo>m (4·43)
IS
while we get
2m ) = 0 (m)
LJ- 11 ::_-==:: (1/2) l}lll log ( Mo
for Mo~1ll. It is also true that Li; 1 is of the order of m for Mo~m. Thus we
have
of the order of the nucleon mass unless . .!\,10 >m. The restriction Mo~'? m
should be regarded as a requirement for the distribution function 'LV (1~).
Now we turn to the mean variance iJLI 11 defined by
Jr
1
( 4. 44)
From ( 4 · 39), Ll 11
-
Ll 11
ts
written as
112
NI. iVamiki
- (.J,,-- ~,)
+ (~kllf- I:f~,;).
= (qll- i],,)
J
J
For simplicity we restrict ourselves to the forward-backward symmetric case, m
which we can put
(4 ·45) *)
Therefore the mean vanance of J 11 can be expressed In terms of boson momentum alone, that IS,
(O'Jr,) = (J,r- ~,) = 4 (~klif- 'Ekr11)
2
2
f
2
(4 ·46)
•
J
It is easy to see that we have essentially the same formula In more general
cases free from the above restriction.
In no correlation cases it holds that
(~k 1 u) =~k 11 /, so that the mean variance is reduced to
2
f
f
CoJ ) 2~&2«k»2[ 2((k2» -1] .
N((k)/
II
Here we have put ~k 11 >=<k>.
rewritten as
"
By the h~lp of (4·30), the mean variance is
~
oJ 11 -2r;p
j
2
IC....Y
2r/a
-1
(4 ·47)
Thus it is found that the mean variance of the invariant momentum transfer
increases in proportion to the primary energy in the center-of-mass system.
This is consistent with a recent analysis of experiments. 9) Equation ( 4 · 47)
also means that there should hold the inequality
/Czj
2r/a
~
(4 ·48)
/1.
This imposes a very severe condition on the values of IC, 1J and r, although it Is
compatible with ( 4 · 35) or ( 4 · 36) and the existing experiments. The inequality
may not be considered to be a strict condition, because it has been deduced
directly from the assumption of no correlations. In contrast with this, the complete correlation among bosons permits us to put (~k 111) 2 = ~L.:kil!, which yields
f
f
f
(4·49)
Inequality (4·48) no longer holds here.
(Do not insert (4·38) into (4·49).
( 4 · 38) has been derived in no correlation cases, whereas ( 4 · 49) holds on the
*> This means that there is a significant correlation between recoil nucleons and emitted bosons.
The author misunderstood the fact in a preliminary report: M. Namiki, Soryushiron Kenkyu
(mimeographed circular in Japanese) 29 (1964), 399; contributed paper to the International Conference on High Energy Physics held at Dubna, August 1964. Nevertheless the results do not
suffer any serious change except a trivial numerical factor.
Classical Boson 1-Vave Excitation in Inelastic Collisions
113
basis of the correlation.) Correlation cases are discussed in ~ 6 in connection
with the coherent wave excitation.
(iii) Nuclear forces and large angle scatterings at high energies Recent researches10) have shown that, at high energies over 1 Ge V, the hard core of the
nuclear force disappears and the real part of the effective ..tVN potential decreases
to zero with increasing energy, while its imaginary part grows up in the core
region. It turns out that the effective potential would come to be purely imaginary at the high energy limit, and that the amplitude (2 · 7) is responsible for
the potential. Rewriting (2 · 7) in terms of the ordinary scattering amplitude
f(p, fJ), we get
j '( p,
where L1p= p-- p'.
t:l) ~/ ·
II
-- - [
j>
(}to!
4n
exp [ -- 1
2
r ( £Jp 2] ,
A
)
(4 ·50)
The effective potential, defined by
(p' I Vr!flp) =
4
n f(jJ, 0) ,
p
( 4 ·51)
becomes
(4 ·52)
or
(4·53)*)
This is just the high energy limit of the nuclear force. Notice that the radius
ro= Vr =VlO/M of the potential region is nearly equal to the hard core radius.
It is also remarked that ( 4 ·50) or ( 4 ·53) corresponds to the nuclear force
at rather large distances, since it has been derived for scatterings of small angle
deflections or small momentum transfers. Of course, ( 4 ·50) or ( 4 ·53) leads
to the Gaussian PT-distribution as follows :
w (pT) dpT =constant exp (- rp/) pTdp'J'.,
which yields P-T= (1/ V
r ) =1\11/ V 10.
(4 ·54)
This is to be applied to small values of
PT·
Recently the large angle scatterings 12 ) are coming into our interests. It becomes clear that the large angle scatterings are characterized by the exponential
Prdistribution
*'
Veff has a tail of the Gaussian shape.
Otsuki et al. have shown that the hard core of the
nuclear force has also a tail of the Gauss-like shape. However, both come from quite different
origins. The former is a direct result of random opening of various channels at very high energies, whereas the latter--not from random origins--has been observed in rather low energy
.NN scatterings. Recently Machida and the present authorm have shown that the hard core is
nothing but a repulsive force originating from the many fermion nonlocal structure of nucleon
and the exclusion principle.
114
1\I. Namiki
(4 -55)
applicable to the rather larger Pr than in (4·54), and by outstanding decrease
of the large angle cross section with increasing energy.
The veer of ·w (Pr) from the Gaussian to the exponential seems to suggest
us that the PT-process could be understood as some stochastic process. Really
we have some reasons why it is so: The Prprocess may ·be translated into the
0-process described by the evolution operator exp ( ~ £0L), so that there may
hold some analogy to the ordinary time-processes described by the evolution
operator exp (-- itl-1).
We may now discuss the outstanding decrease of the large angle cross
section in the following way : Recall that the equation
(4 ·17)
for the factor of F(j:>, 0) is still valid for large angle scatterings.
The operator
2 3
cosOLa has a contribution to cancel out the term N=Ji9(k)\ d k for small
angle scatterings, but never does for large angle scatterings. Therefore N is
no longer cancelled out but is kept remaining for large angle scatterings, so that
the large angle cross section has naturally exp ( -- 2N) as an important factor,
namely,
It seems to be interesting that the large angle elastic cross section is governed
by the energy-dependence of the multiplicity of the multiple particle production.
For example, according as we have
(4·57a)
1\i =A log (p/ N1 0)
or
(4·57b)
we get the following dependence of the large angle cross section :
dud ).
( dQ . larg•
(. p ) ""~l··oc Mo.
2
A
(4 ·58a)
or
( due~_)
\ dQ .
hrgr·
""~!<>
ocexp(-2B / jJ \.
~ 1\1 0 J
(4·58b)
Further calculations of residual factors should be required for the complete
discussion of the large angle scattering.
Anyway it will become an important problem in strong interaction theories
Classical Boson T1Vave Excitation in Inelastic Collisions
115
to investigate the relationships between the large angle scattering and the
multiple particle production at very high energies, since the former should be
regarded as a unitary reflection of the multiple particle production with large
momentum transfers.
§ 5.
Proton=antiproton collisions
In connection with discussions on the diffraction constant in the last section,
we have mentioned that the approach to the high energy limit-----for example,
the shrinkage of the diffraction peak--requires the full knowledge of inelastic
collisions not only at very high energies but also at moderate energies. Nevertheless it may be worth while to make an attempt to explain qualitatively the
anti-shrinkage of the diffraction peak observed in proton-antiproton elastic scatterings, within the framework of the present theory.
The proton-antiproton scattering is essentially different from the protonproton scattering by the possibility of being able to annihilate the proton-antiproton pair in its intermediate states. Accordingly the function F(p, ()) defined
by (2 · 3) for the proton-antiproton scattering is divided into two parts :
F(jY, B):::::::.-{;-- {trf~~l (I>) exp ( (1/2) r1t)
+ t1i<~~1 (p) exp ( (1/2) r2t)}.
(5 ·1)
The first term corresponds to intermediate states containing the proton-antiproton
pair besides b'osons. O"f~~l (p) is the total cross section of the inelastic collision,
after which the pair are left not annihilated. The second term comes from
those intermediate states which never contain the pair but only bosons. cr};~l (jY)
is the total cross section of the inelastic collision resulting in pair annihilation.
From experiments we know that
r,(l)
v ind
(
P)
~ (2) (- )
-O"inel j:>
(5 ·2)
at relatively low energies, but
·(-jJ )
---~ Q .
crf,;~l (p) 1)-HJO
(~)
O";_,Jel
(5 ·3)
The Pomeranchuk theorem should read that
(jf~~ I (j>) ---')- () pp, inl'l (jJ) ,
(5 ·4)
fJ->00
(5 ·5)
where (]' pp,inel and rPP are, respectively, the total inelastic cross section and the
diffraction constant in the proton-proton case. N or «k2)) of the annihilation
process is larger than that of the non-annihilation process, because bosons in
the former case receive the larger amount of energy than in the latter case.
116
iVamiki
Therefore we find that
(5 ·6)
smce r lS proportional to N((k 2 )).
Now we may rewrite F(jY, 0) for small angles as
2
F(t>, r7) = P [!Jf~~~ (JY) u{~~] (p)] exp ( (1/2) rvi7 t),
2rr
(5 ·7)
where
(5 ·8)
Hence we can see that rP;;- starts with the value
(5 ·9)
at relatively low energies and decreases to the high energy limit
(5 ·10)
where (5 · 2) and (5 · 3) have been used. It 1s sure that the energy-dependence
of r Pil means the anti-shrinkage of the diffraction peak of the proton-antiproton
elastic scattering. However, one should not accept these arguments as quantitative ones.
~
6.
Coherent boson wave excitation
In this section we briefly discuss the coherent boson wave excitation by the
nucleon-nucleon collision. The coherent wave is defined as a definite phase wave.
The coherent wave appears when many bosons occupy, with a high density,
some small region of phase space. Let the region be a very small volume
around the momentum k 0 • An ideal coherent wave state IS represented by
[coherent)= [0)\0)· · ·[O)[Xo)[O)·· · .
(6 ·1)
All [O)'s mean that there are no particles except in the state of the momentum
1;. 0 • The vector I X0) is an eigenvector of the phase operator X belonging to an
eigenvalue X0 , where X is an hermitian operator defined by
a 1, 0 = exp (ix) V n
n IS also hermitian.
mutation relation
As is well known,
[x, n]
=
-i.
X
(6 ·2)
and n satisfy the canonical com-
(6 ·3)
Hence we can conclude that the boson number is completely indefinite in the
state [x 0) , because the probability \(n\X 0) [ 2 of finding a n-boson state in the state
Classical Boson Wave Excitation in Inelastic Collisions
Thus we get N and N 2 as
jX 0) is independent of n, In) being an-boson state.
follows:
n max
I: nl<niXo)l
JV=
1
n;,,ax
2
~
2
n=O
n
rna~'{
I: l<niXo)l
117
=
(1/2) llrnax
,
(6 ·4)
=
(1/3) n;nax '
(6 ·5)
'llmax
2
n=O
7t
·N2=
rna.:c
~
n=O
n2 l<nlxo)l 2
-
n max
I: l<niXo)l
2
1
n?nax
3
nmax.
no~'O
nmu being the maximum boson number permitted by the conservation law of
energy. Here we have left only the leading terms for n111ax~ 1. Then the mean
vanance becomes
1
v3
N.
(6 ·6)
Notice that the proportionality to N is a significant characteristic of the coherent
wave case. To summarize the tJN- N relation, we may say that
oNocviJ
m the no correlation case,
ocN
m the coherent wave case.
(6 ·7)
One can draw, in principle, the o~l"- N relation from the experimental N- Ep
plot, and may guess the final state of the boson assembly produced by the
multiple particle production. An introductory work along the line of thought
was attempted by the present author and his collaborators,~) but any definite
conclusion could not be obtained yet, because the statistics of the existing data
of primary energies above about 1,000 Ge V is too poor to discriminate between
the no correlation case and the coherent wave case. The problem relies upon
the future accumulation of qualified high energy data.
Another important characteristic of the coherent wave case is the energydependence of the mean variance ok or lJL1 11 • We have got the relation
(4 ·49)
in the correlation cases. Now let us explicitly write down the condition of exciting the coherent boson wave: Since it is plausible to consider the coherent
wave directly after emission to occupy a narrow volume as mentioned by Fermi
and Landau/ 3 ) the total intensity of the boson wave is of the order of
l=1J(2p)/(M/;ip) =1J
2f'I.3P2
M
(6 ·8)
1\11. 1Varniki
118
Then the coherent boson wave appears when the con-
/1 being the piOn mass.
dition
(6 ·9)
1s fulfilled, where
bosons, namely,
oQ
stands for the mean variance of the emission solid angle of
(6 ·10)
Also we have
(6 ·11)
and then
«e))=ko2+(r'Jk)2=
4p~r/ [ 1 + fP2 ( opk \)
1'J2
4ij
2
].
(6 ·12)
'
By usmg (6 ·10), (6 ·11) and (6 ·12), the above condition becomes
3
( f1
f! \[ 1 +
\ rr:M1r} )
2
IV 2
4YJ
(
ak )
p
]>(\ okp )\.
2
(6 ·13)
Inequality (6 ·13) tells us that the coherent wave should appear if
Of?= constant ( P J /J; p<('l.
]\![I
'
(6 ·14)
This is just what we want to get. There the constant in front of jY/J might
2
depend logarithmically on jY, but should be much smaller than (tJ. 3 N/TCtc ) . Equation (6 ·14) is to be compared with ( 4 · 38') in the no correlation case.
Thus we conclude that the energy-dependence of oJII is written as
(6 ·15)
vhere (i= 1 in the no correlation case, and !3<1 in the coherent wave case.
As was recently pointed out by the present author and Ohba, 9l the energy-dependence is consistent with recent cosmic ray jet data, although the statistics of the
data is also too poor to detern1ine precisely (3.
y ..
§ 7.
Concluding remarks
In the present paper we have formulated the interrelations between the
elastic scattering and the multiple particle production at very high energies,
after showing that most of high energy collisions would be characterized by the
classical heavy boson wave excitation. It is found that the present scheme seems
to be consistent with the recent data of high energy scattering experiments and
Classical Boson Wave Excitation in Inelastic Collisions
119
cosmic ray jet observations. In particular, the diffraction constant of the elastic
shadow scattering given by the theory comes to be of the right order of magnitude at the high energy limit. The observed r'JN --lv relation and the energydependence of ok or Od11 are not inconsistent with those predicted by the theory,
although the statistics of the data are too poor to draw some definite conclusion
on the final state of the boson assembly. For the last purpose, it is highly
desired to accumulate qualified high energy data above about 1,000 Ge V. Another
important work is to analyze the relations between the large angle elastic scatterings and the multiple particle productions with large momentum transfers, because the former should be a unitary reflection on the latter, too.
Finally it may be of some interests to make comments on the asymptotic
behavior of the elastic scattering amplitude in the complex energy plane, frmn
the viewpoint of the consistency with the multiple particle production phenomena.
In the modern S matrix theory or the dispersion theory, it is indispensable to
introduce the fundamental assumption:
T(E)--+EL,
(7 ·1)
!Ei->oo
where L is an integer. By (7 ·1) we should read that T (E) is a good function
like a polynomial around the infinity. On the other hand, one would expect
that the infinity may become an essentially singular point of T (E) as an accumulated point of branches, since T (E) has an infinite number of branch points,
corresponding to the successive opening of various inelastic channels, distributed on the real axis of the complex energy plane. Therefore the assumption
(7 ·1) would impose some severe restrictions on the theory. If (7 ·1) were
true, all the branch points would be limited in a finite energy region bordered
by some critical energy, or at least, the branch effects would rapidly decrease
outside the energy region. In other words, the future theory might be set up
without a serious ·resort to the channels outside the region.
Another possible understanding may be such that (7 ·1) should not be imposed on T (E) itself, but on its average, f\ (Eo), over a range with the spread
c around Eo. Here it is considered that the average amplitude Ts (Eo) is nothing but the practically observed one by a beam with energy spread s, whereas
T (E) itself corresponds to ideal observations by a 1nonoenergetic beam. The
assumption (7 ·1) should be replaced with
T
8
(Eo)
- - + E/.
I 1'0 1-> oo
(7 ·2)
The energy spread in practice may become wider and wider as the energy increases, so that the analyticity of T s (Eo) will be getting better and better. Consequently, one may say that (7 · 2) makes a reasonable condition on 1's (Eo).
It is no longer to cut off many particle channels as in the previous understanding. According to discussions in Appendix B, we have
M. Namiki
120
(7 ·3)
Equation (7 · 3) means that the value of T at a complex point E =Eo+ is corresponds to a practical observation by a beam with spread c around Eo. Consequently, it turns out, in the last understanding, that the S matrix theory based
on (7 ·1) or (7 · 2) should be regarded as an asymptotic form of the future
comprehensive theory, because the S matrix theory never enters into the large
vicinity of the real axis in the larger side of energy.
Anyway (7 ·1) or (7 · 2) would be concerned with the slowly increasing
dependence of the multiplicity on energy, which may reflect a characteristic of
high energy strong interactions.
Appendix A
Mathematical details of classical boson wave states
For a while we are concerned with one degree-of-freedom case. As shown
in the text, the boson state in no correlation cases is described by the vector
Ia) = Ca exp (a+a) IO),
(A ·1)
which satisfies the eigenvalue equation
(A ·2)
ala)=ala),
Since the annihilation operator a is not hermitian, its eigenvectors la)'s do not
make an orthogonal set. Nonetheless we can set up the reciprocal set of the
eigenvectors with the help of the operator K defined by
(A·3)
(A·4)
Kc (c-number) Kc- 1 = (c-number) *.
(A ·5)
Ka 1s explicitly written clown as follows:
(A·6)
because it 1s easy to prove that
exp(-- (1/2) · (a~a+)')aexp((l/2) ·(a -a+)') =a+,
l
exp ( (1/2) · (a- a+) 2) a+ exp (- (1/2) · (a- a+) 2 ) =a.
Now we can define the reciprocal vector by
!ct)=DaKia)
=DaCa* exp(- (1/2) · (a-a+) 2)exp(a+a*) IO), }
Da being an appropriate normalization constant.
the eigenvalue equation
(A·7)
It is obvious that Iii) satisfies
121
Clas:·;ical Boson Wave Excitation in Inelastic Collisions
a+ Ia'>
/
=a* Ia)
I
(A ·8)
'
From (A· 2) and (A· 8) one can easily derive
(a- a') <aia') = 0,
which yields
<ala')=r)(a, a'),
(A·9)
o
provided that Da is appropriately chosen. Here (a, a') stands for the Kro~
necker delta for the discrete a, the Dirac delta function for the continuous
real a, and the modified delta function for the complex a. To get Dm we have
to calculate
To deal with the last factor, it may be convenient to use a set of canonical
variables jJ and q defined by
p=j ~
q=
(a+a+),
(
l
v
a---a +) ,
2(1)
1
(A ·10)
l
(A ·11)
I
or inversely
a=
1
(p- iwq) ,
V2oJ
1
a+=
(p + iwq) .
V2o)
Since the vacuum state has the representative
<qiO) =
t/
I
~ exp (- (w/2) q 2)
(A ·12)
m the diagonal representation of q, it is easily found that
<ala')=Da*CaCa' exp(a'a)
j
~
J'dq exp{ -oJc/- (1/2) [ --iv2wq+a'-ar}
Hence we get
(A ·13)
(A ·14)
From (A ·1) and (A ·14) it is found that Ia) and Id) have the representatives
lvf. iVamiki
122
(qla)
=Cay
(qla)=
(;~ exp (a 2/2) exp ( ~ (rv/2) q 2 + £aV 2w q),
1
4
V2JZ Ca*
uJ
V rr
exp(~ (a* 2 /2))exp((w/2)q 2 +ia*V2o)q).
I
(A ·15)
Here we have used the formula
(q!exp (a+ a) lq') = exp (a 2/ 4) exp (iaV rv/2 · q) exp ( ~ i (a/V 2w) (8 /aq)) iJ (q- q')
=exp(a 2/4)exp(iaVu)j2·q)o(q--i(a/V 2w) --q').
Now we return to the field case with an infinite degree of freedom, m
which we naturally have
!rn)=ff[aY.)
- ·) y"
~~
k
'
(A ·16)
The set { lcp)} and its reciprocal set {I <P)} satisfy the orthogonal condition and,
if the set {I cp)} is complete, have the closure property as follows :
<<Plcp')=o[cp, cp'J,
~Jcp)(<P'i =
l
1.
(A ·17)
tp
Finally it is also noted that the definite phase state IXo) used m ~ 6 can be expressed as
(A ·18)
Appendix B
Physical ineaning of the comj)lex energy variable zn the modern S matrix
theory
Analytic functions of the complex energy variable are extensively used in
the modern S matrix theory. They have so far come only from the mathematical version. \V e discuss here the physical meaning of the complex energy
variable and the analytic function T (E). The average TE (Eo), used in ~ 7 may
be defined by
rt" (Eo)
= .\ T
(E) Ps (E --Eo) dE,
(B ·1)
where PE (E -- E 0) is a positive function rapidly decreasing outside the energy
region with spread 2 around Eo. Now (B ·1) can be rewritten as
T
8
(E 0)
1
\r·exp Cz"Eot) Ps (t) T (t) dt ,
2TC,
= ,
(B·2)
Classical Boson "VVa've Excitation in Inelastic Collisions
123
where
Pr; (t)
=::
1 (~)
Ps
exp ( -- i~t) d~ ,
T(t) = JTCE)exp( -iEt)dE.
We know that
If one puts
p, (t)
(B ·3)
(B·4)
must vanish with the decay time nearly equal to (1/e).
Pc (~)
=
1
(>O,
(B·5)
then one gets ·
rJs (t)
=
exp (-- et) .
(B·6)
Therefore (B ·1) comes to be the form
T,(Eo) =
1
lexp(-i(Eo+ie)t)T(t)dt.
27r J
(B·7)
The right member of (B · 7) defines an analytic function of the complex energy
variable in its upper half plane, namely,
(B·8)
which is nothing but (7 · 3) used for discussions in ~ 7. Equation (B · 8) permits
us to interpret the analytic function T (z) with a complex energy z as the
scattering amplitude caused by a wave packet with an energy spread nearly
equal to Im z. In other words, T (z) represents some partial informations carried away by a wave packet with a time spread of the order of (Im z) - 1 • In fact,
it is easy to show that
T"(t) =Jexp( -iEat)T8 (Ea)dEo
J
= exp ( --- iEot) T (Eo+ ie) dEo
A
=exp( -et) T(t) .
(B·9)
The analytic continuation to the lower half plane describes the time reversed
processes.
Thus we find that the full informations about scatterings caused by wave
packets with arbitrary time spreads distribute over the complex energy plane,
and that the imaginary part of the energy variable represents the energy spread
of the incident wave packet.
124
M. Namiki
References
L. van Hove, Nuovo Cim. 28 (1963), 798; Rev. Mod. Phys. :36 (1964), 655.
And see, Y. Yamaguchi, Stanford Conference Report, 1962.
2) T. Kobayashi, M. Namiki and I. Ohba, Prog. Theor. Phys. 31 (1964), 840.
T. Kobayashi, M. Namiki, I. Ohba and S. Orito, Prog. Theor. Phys. 32 (1964), 738.
3) G. Fujioka et al., Uchusen Kenkyu (mimeographed circular in Japanese) 8 (1963), 511.
K. Niu, Nuovo Cim. 10 (1958), 994.
4) S. Hasegawa and K. Yokoi, Uchusen Kenkyu (mimeographed circular in Japanese) 7
(1962)' 186.
5) M. Koshiba, The ICEF data book.
6) M. Namiki and C. Iso, Prog. Theor. Phys. 18 (1957), 591.
C. Iso, K. Mori and M. Namiki, Prog. Theor. Phys. 22 (1959), 403.
7) H. W. Lewis, J. R. Oppenheimer and S. A. Wouthuysen, Phys. Rev. 73 (1948), 127.
8) C. Iso, private communication.
9) M. Namiki and I. Ohba, Prog. Theor. Phys. 32 (1964), 866.
10) S. Otsuki, R. Tamagaki and M. Wada, Prog. Theor. Phys. 32 (1964), 220.
11) S. Machicla and M. Namiki, Prog. Theor. Phys. 33 (1964), 125.
12) For example, see G. Cocconi's paper, CERN preprint, March, 1964.
13) E. Fermi, Prog. Theor. Phys. 5 (1950) 570; Phys. Rev. 81 (1951), 683.
L. D. Landau, Izv. Akacl. Nauk SSSR 17 (1953), 51.
S. D. Belen'kij and L. D. Landau, Uspekhi fiz. Nauk SSSR 56 (1955), 309.
1)