- Email: [email protected]
Shrinking, nonshrinking, and expanding diffraction peaks
ANNALS
OF
Shrinking,
PHYSICS:
41, 177-204
Nonshrinking,
(1967)
and
Expanding
M. &I. Department
of Physics,
Brown
Diffraction
Peaks*
ISLAM
clniversity,
Providence,
Rhode
Island
An explanation of the shrinking, nonshrinking, and expanding diffraction peaks at high energy (210 BeV) is given in terms of two kinds of forces: (1) due to the exchange of the r-8, JPa = Of+, T = 0 autibound st,ate or virtual state, and (2) due to the exchange of the J po = l-- vector meson (w + q). It is shown that the energy dependence of the diffraction peaks can essentially come from the vector meson exchange force, because this force falls off more slowly at high energy than the other force. Thus, for PT’P and ?r-p scattering, as the vector meson force does not occur, we have little energy dependence or non-shrinking diffraction peaks. For pp and K+p scattering, the vector meson exchange force can have asymptotic behavior such t,hat it decreases absorption as the energy increases and this behavior then gives shrinking diffraction peaks. For pp and K-p the vector meson exchange force changes sign and as a result, the absorption now increases as the energy increases. This behavior produces expanding diffraction peaks. The absorption (or inelasticity) at high energy has been dynamically related with the input forces, in the framework of an impact parameter formalism. The asymptot,ic behavior, which is found for the vector meson force, is explained in terms of a suitably constructed Mandelstam double spectral function. This explanation also indicates how the compositeness of the vector meson comes about. On the basis of lluitarity argument, it is pointed out that the vector meson cannot behave as an elementary particle. The incident and the outgoing particles are treated as spinless. I. INTRODUCTION
At present no satisfactory explanation of the behavior of elastic diffraction peaks at high energy ( 2 10 BeV) exists. Experimentally, it has been observed that the diffraction peaks for elastic pp and Kfp scattering shrink as the energy increases, those for ~‘p and r-p scattering show little energy dependence, and the diffraction peaks for flp and K-p expand with increase in energy (1). Originally, the experimental study of the diffraction peaks was motivated by the prediction of the Regge pole theory. However, at present, it is well established that the simple version of the Regge pole theory does not work. One needs a large number of Regge poles and correspondingly a considerable number of parameters * Supported
by the U. S. Atomic
Energy
Commission. 177
Copyright
0 1967 by Awdemic
Press
Inc.
178
ISLAM
to fit the experimental data (2, 3). As a result, the Regge pole fits of the diffraction peaks have become physically much less meaningful. In this paper a new explanation of the behavior of diffraction peaks is proposed. The physical basis of this approach is that absorption can be dynamically related with the input forces, under certain simple approximations, and the energy dependence of absorption at high energy is then connected with the asymptotic behavior of these forces. Since the forward diffraction peaks essentially arise from absorption, their high energy behavior is thus governed by the asymptotic behavior of the input forces. The way absorption at high energy can be dynamically related with the forces is formulated in Section II of this paper, in the framework of an impact parameter expansion. The corresponding case, for the partial wave expansion, has been previously developed (4, 5) and applied to pp and 7r*p scattering (5, 6’). These applications have indicated that physically meaningful absorption can indeed be calculated from input forces. The energy dependence of the absorption arising from the asymptotic behavior of the forces is considered in Section III. In Section IV, the energy dependence of the diffraction peaks is related with that of the absorption. In Section V, an explanation of the asymptotic behavior of the vector meson force is given. Finally, in Section VI some concluding remarks are made. II.
IMPACT
PARAMETER
EXPANSION
AND
DYNAMICAL
INELASTICITY
Two important characteristics of high energy elastic scattering are (i) the prominent forward diffraction peak, and (ii) the small wide angle scattering. These features indicate that an impact parameter expansion, rather then a partial wave expansion, is more suitable to describe high energy elastic scattering (7). Recently, several authors have considered rigorous formulation of the impact parameter method (8-10) and we shall follow their approach. We define the impact parameter expansion of the invariant scattering amplitude A( s, t) by the relation A(s,
t) = S lm 5 dEJ0(t sin 0/2)A(s,
0
(1)
where s = square of c.m. energy, t = -22k2( 1 - cos 0), and ?r > 0 2 0. Equation ( 1) can be inverted and gives
A(%.$I= 2 !,I sin
0/2 d(sin
e/a)J,(
21cb sin 0/2)A(s,
cos
e).
(2)
The impact parameter b is defined by putting 4 = 2lcb in the above relations. The partial wave expansion of the amplitude A( s, t) is
A(s,t) = l$
(21 + lP’l(cose)At(s),
(3)
DIFFRACTION
179
PEAKS
where Al(S)
= [e2i*~(s) -
and p = k/W; 12is the cm. momentum in Eq. (2), we get’,
1]/2ip
(4)
and W is t,he cm. energy. Inserting
(3)
(5) Equation (5) gives the impact parameter amplitude in terms of the partial wave amplitudes. It also indicates t’hat A (s, b) is an even function of b. From (4) and ( 5)) using the relation
we find A (s, b) can be written
as
A( s, b) = [~is(s3b) - 1],‘22$,
(6)
where e2i8(s,b)
E 2g
(21 + l)ezi*~(“‘Jzl+l(21cb)/2~b.
(7)
Equations (4) and (6) show a close similarity between the partial wave amplitude and the impact parameter amplitude. The unitarit’y restriction on A i( s) is very simple (namely, Im 61(s) > 0), while that for A( s, 6) is quite complicated (9). However, at high energy the unitarity restriction on A (s, 6) becomes simple and is given by Im 6( s, b) z Im 61(s) > 0, where b = (I + 4$)/k.
Equation Al(s)
(8) follows
(8)
from the inverse relation
= 212f= A(s, b)Jzz+1(2kb)
db
(9)
0
and the approximation
t’hat for large h?, Eq. (9) gives (8) Al(s)
= A(s,
b) + 0(1/L?)
(b = I + 3$/k).
Incidentally, (10) also exhibits the correct interpretation parameter. Comparing the equation for the impact parameter A(s,b) with
that of the partial wave amplitude = ?,$ /“I
&OS
0) Pi
(COS
(10)
of b as an impact amplitude
= 211 sin e/2 rl( sin 0/2)Jo( 2kb sin 0/2)A (s, cos 6)
Al(s)
of (5)
o)A(S, COSe),
180
ISLAM
we notice that the analytic properties of A (s, b) for fixed b in the complex s-plane should be the same as those of Al(s) ; that is, assuming Mandelstam representation for A ( s, t), A (s, b) will have right-hand cuts coming from unitarity cuts in the s-channel and left-hand cuts from unitarity cuts in t-channel and u-channel. A ( s, b) will have an additional singularity at 1s 1 = 00. This singularity arises because Jo( 2kb sin e/2), which is an integral function of k2, has a singularity at Ik2/ = w. We now consider the problem of obtaining the absorption Im 6( s, b) from input forces. The corresponding case for the partial wave amplitude has been studied before (4,5). First, we define a function 0( s, b) by the relation
(11) sT is the value of s at the elastic threshold. That Im 6( s, b) for s + m is a well defined quantity follows from Eq. (8). Next, we construct a new amplitude a( s, b) by writing 1 + 2ipa(s, b) = eC2ie(s,b)[l + 2&4(s, From
(la),
b)].
(12)
we get, a( s, b) = [e2ia(8ub)-
1]/2ip
(13)
where a(s, b) = qs, b) - qs, 6).
(14)
Since for s on the positive real axis Im 6( s, 6) = Im 0( s, b), CY(s, b) is real in the physical region. From ( 13) it then follows that the new amplitude a( s, b)obeys elastic unitarity, i.e., Im u(s, b) = p ] u(s, b)]‘, throughout the physicalregion. Equation ( 12) shows that a( s, b) has the same analytic properties as A (s, b) in the s-plane. If we write O(s, b) = A(s, b) + i Im 6(s, b) then
Also, A(s, b) = Re6(s,
b) - (Y(s, 6).
06)
If now from physical considerations, we have some knowledge of Re 6( s, b) and 01(s, b), then we can try to invert Eq. ( 15) and express Im 6( s, b) as an integral over A( s, b) . This would provide us with a dynamical scheme for obtaining ab-
DIFFRACTION
181
PEAKS
sorption. The inversion of Eq. (15) can indeed be done exactly by using the analytic properties of 8( s, b) and some boundary conditions. These boundary conditions are somewhat different from the corresponding partial wave case and are discussed in the appendix. The final result is Im 6(s, b) = -
k(s -TSrj1’2 p [ m [Re 6(s’, 0) - LY(~, b)] da’. k’(s’ - s-ry’ys - s) ST
(17)
Now, to calculate Im 6(s, b), we make the following approximations: (i) Above a sufhciently large energy sA , the impact parameter amplitude A(s, b) for b < R is completely absorbed, i.e., Re 6(s, b) = 0 for s > sA and b < R, where R is the radius of interaction. (ii) Above the energy SA the amplitude a( s, b), which obeys elast’ic unitarity throughout the physica region, approaches its Born term. From the second assumption, it follows that since t’he Born term is real, a( s, b) is essentially real for large s. However,
- 11_ u(s, b) = [e2ia(s3b)
(Y(s,
Zip
P
bj I i i2(8, ___P 6) .“,
so that (Y(s, b) should be small at high energy. Thus, we get for
(18)
s > SA,
P
where aB( s, b) is the Born approximation of a( s, b). Using approximations (i) and (ii), we therefore obtain from ( 17) Im 6(s, b) = -k(s
:sT)l”
aA [Re 6(s’, b) - a.(~‘, b)] ds’ k,(s, - sT)1’2(d - s) ST
j”
+ k(s -$li2
p [SA
=-
p(&?(i, b) ds’ k’(s’ - sTy2(s’ - s)
k(s - sT)1’2 YA[Re 6(s’, b) - (Y(s’, bjl CZS sST k’(d - sT)1’2(d - s) P
(19)
sT)1’2 + k(s -?i-
(s > sa> For large s (>>a,) the first term on the right-hand side of ( 19) behaves as a constant. Thus, the energy dependence of Im 6(s, b) can essentially come from the second term depending on the asymptotic behavior of the Born term a”( s, b). This we study in the next’ section. As the Born term uB( s, b) represents the input
182
ISLAM
forces, SO Eq. (19) shows how the absorption with these forces at high energy. III.
INPUT
FORCES
AKD
ENERGY
Im 6( s, b) is dynamically
DEPENDENCE
OF
THE
related
ABSORPTION
We consider input forces arising from the exchange of the 7r-r antibound state or virtual state (11) and from the exchange of the T = 0 vector meson.The vector meson exchange represents the combined force coming from the exchange of w and cpmesons.It has been pointed out that the force arising from the exchange of the T-R antibound state is similar to that due to the exchange of a scalar particle of mass ,Z2nl, , but with a residue opposite in sign (11). Thus, the invariant Born term due to the exchange of the antibound state can be represented as -YO/(,WO* - t) with 70 > 0 and m o E %a, in the physical region of the s-channel. To calculate the corresponding Born term of the impact parameter amplitude, we go back to Eq. (2) and obtain AB(s, b) = $$ jzk A dAJo(bA) szz -
0
(-t
=
A”)
O2
(20)
for large k*. Ko( z) is the modified Bessel function of second kind. As an approximation, we take AB( s, b) = aB( s, b). This approximation is equivalent to saying that when we calculate forces, we ignore the inelastic scattering in the physical region; A( s, b) and a( s, 6) then coincide. Thus, the Born term of a( s, b) due to the antibound exchange is UOB
= - $
Ko( bmo)
(70 > 0)
(21)
for large k*. We next want to calculate the force or the Born term due to the exchange of the vector meson. The simplest way to do this is to consider the Feynman diagram and use perturbation theory. Assuming the interaction between the spinless nucleon and the vector meson to be given by H NNv = igNNVCPN ‘~,~NV~ and denoting by s the square of c.m. energy, we find from Fig. 1 the invariant amplitude for pp scattering to be
A;p(s, t> =
2 YNNV
u--s
8ap=-t
&NV 4lr
s
f
mv2/2
-
my2 - t
2mNy2
+g&.
(22)
DIFFRACTIOX
FIG.
1. Exchange
183
PEAKS
of the elementary
vector
meson
The second term corresponds to a very short range force that contributes only to the s-wave in the s-channel and we neglect this force. The negative sign in front of the first term in (22) indicates that the vector meson exchange force is repulsive for pp scattering. The invariant amplitude due to vet tor meson exchange in ISP scattering is now obtained by interchanging s and u and this gives
Worth noticing is that the long range part of the force is now attractive. For K’p and K-p scattering, assuming the interaction between K-meson and the vector meson to be
we obtain Born terms similar to Eqs. (22) and (23) with giNV replaced by sign of gNNV and gKKv should be gNN VgKK v and 2mN2 by ??r#’ + 3%‘. The relative positive, so that the long range vector meson exchange force is repulsive (attractive) for K’p (K-p) scattering (1%). We can now calculate the Born term of the impact parameter amplitude due to vector meson exchange by making the same approximation as before. The final result is
(24) for large k2, where the negative sign is for particle scattering and positive sign is for antiparticle scattering; yV = &NV/4?r for pp and pp; TV = gNNVgKKV/4r for K+p and K-p. Notice that. yv is always positive. The s factor in Eq. (24) occurs because of the spin of the vector meson. In calculating the vector meson exchange contribution by perturbation theory, we
184
ISLAM
have treated it as an elementary particle and the factor s indicates this. The Born terms (22) and (23) blow up like s at high energy. In reality, we believe the vector meson behaves not like an elementary particle but like a composite particle, so that its Born term is damped at high energy. Originally, the Regge pole theory provided a method for the damping mechanism (IS, 14), but the validity of this theory has not so far been established ( 1) . For the moment, we shall take a phenomenological attitude and consider that for a composite vector meson, the Born term is obtained by replacing the factor s in Eq. (24) by c(s), where for s --f ~0 a(s)
= csB + !b(s),
0 -I /3 < 1, C > 0, and $(s) obeys a Holder condition near and at s = CQ; also #( cc ) = 0. Later on, in Section V, we shall show by considering the elastic scattering of two equal mass spinless particles, that a suitable Mandelstam double spectral function can be constructed such that in t-channel it corresponds to the exchange of a vector meson and in s-channel it produces the asymptotic behavior postulated above. The reason C is chosen positive is that if the vector meson force is repulsive for the elementary case, then it will remain repulsive for the composite case, even though its energy dependence is damped. The Born term for the composite vector meson exchange is thus taken to be
We now investigate the energy dependence that occurs in Im 6( s, b) due to the forces (21), (24) and (25), using Eq. (19). Let us denote the first term on the right-hand side in (19) by Im 15~(s, b). Im sL( s, b) then primarily represents the contribution that comes from the low energy region. We further define Im 60(s, b) =
aoB(s’, b) ds’ k(s - sTy* O” ‘lP(s’ s - ~)‘/2(S’ - s) n8.4s
(26)
and Im &(.s, b) = ‘(’ The total absorption Im6(s,
- sT)1’2 I, a
s,1,2( avI($A(da’S’
s’
s) .
(27)
b).
(28)
is now given by b) = ImaL(s,
b) + Im&(s,
b) + Im&(s,
As can be seen from (19), for large s Im aL( s, b) behaves as
DIFFRACTION
185
PEAKS
Im &( s, b) = const. + 0( l/s).
(29)
If s is large so that terms of the order of l/s can be neglected, then Im &,(s, b) is essentially a constant. Inserting (21) in (26), we obtain Im &(s, b) = -&0K0(bm0)k(s
- sT)l” m
x-
s1 C
mk(s
as’
l
-
ii s sA s’ys’
a)
-
ST)l’2(S’
-
-
8,)
s1)
I +
s2
mp(s
1
-
s2)
ii
(30) s .,
s’W(S
-
SA
-- 1
l&r sST
$ys~
r
s’l/2(s’
_
dS sT)‘12(s’
-
S)
+
k
Re
I,(S)]
where s1
=
(7?2
+
PL)’
=
ST,
s2 =
(m
-
p)“,
In denotes the mass of the proton and p denotes the mass of the other particle (p = mforppand@p,p = mKforK&pandp = m,fora*p).Fors> ST, Re L(s) = -
’ $“(s - s$”
log (s)“’ + (8 - sT’)l” (8)“’ - (s - sT)1’2
so that for large s Re L(s) = -klog Since k( s - 8~)~‘~= s/2fors--+ Im 6,,(s, b) = --
0
$
.
w,wefindfrom(30),
1 YoKo(bmo) const. - i 4
( )I
log $
(31)
for s large and neglecting terms of the order of l/s. Equation (31) shows that the dominant energy dependence of Im 60(s, b) comes from the term log s/s and is therefore small. Thus, Im 60(s, b) will be practically constant for large s. Another interesting point about (31) is the b-dependence that it exhibits of the absorption Im &( s, b). It is exactly the same as we would obtain for a pure imaginary optical potential of Yukawa type ( 15). However, we have arrived at the result (31) using dynamical concepts and not any phenomenological absorbing potential. We have from Eq. (27) and the Born term (24) for an elementary vector
186
ISLAM
meson Im
6v(s,
b)
=
frvK:Jnl,)
k(S
-
sT)1’2p
SW SA
Tl/z(,,j
_
s’ ds’
sT)l/2Jc~2(s’ -
’
8)
= =F~~~vK&mv)k(S - sg* (32)
8.4
S
--
l&r
ds’
s ST
s”‘Z(
s’
-
s,yys
-
s)
+ iReL(s)
1 .
For large s, Eq. (32) gives Im &(s,
b) =
From (33), we find for s + 00 Im K-p scattering. However, unitarity (Eq. (8) ) . Therefore, we conclude and that use of the Born term (24) Using the Born term (25) of the Im &(s,
b) = =FYvK:$mv)
C
-
s1
mp(s 82
tf
.
(33)
s,liz(s, ~~~)l;$~(
s’ _ s)
ST)~”
l Sl) ii s
cc
SA
00 +
0
6( s, b) = Im &( s, b) is negative for @p and restriction gives Im 6( s, b) > 0 for s large that the vector meson cannot be elementary, to calculate absorption is not appropriate. composite vector meson, we have
k(s - ST)l”P [,
= =~J&vKo(bmv)k(s
x-
flog
fyvKo(hv)
1
mp(s - s2>ii s 8Aiys’
s‘lys’
a(~‘) ds’ - s,)“ys’ - Sl)
(34)
a(~‘) ds’ - Sp(S’ - sz)
where (35) Now, we have assumedfor s + 00
u(s)= cs@ + l)(s)
DIFFRACTION
where 0 5 /3 < 1, #( m ) = 0, and #(s) I es)
PEAKS
obeys the Hiilder
- $(soj I < KI l/s
for large s (K, E > 0). If, further,
187
-
condition
l/so If
the integral
exists (p > 1), then we have (16,17) I(s) = - qy S’S
+ o(t)
(0 < P < 1)
(36)
and
I(sj = _ 1 c(asTip log(&)+o(;). 8-m T S
CP=O) (37)
From (34) and (37) we find that for p = 0, the dominant energy dependence of Im 6”( s, b) is given by log s/s. That is, the same as t’hat for Im 60( s, b). This is not surprising, since in this case, asymptotically the vector meson Born term behaves like bhat for the 7r-r antibound state. However, if the vector meson Born t,erm does not fall off as fast, i.e., if 0 < fl < 1, then from (34) and (36), Im Ev(s, b) = 7 $ yV K. (bn~~)
[
const. -
4c cot np g-8
1
neglecting 0( l/s) terms. Comparing (3X) with (31 j, we find for 0 < fi < 1 hhe dominant energy dependence of Im 6(s, b) comes from the vector meson exchange. Physically this is understandable, since the vector meson exchange force falls off more slowly compared to the other force at very large energy. Further, from (38) we get, for s large (39) Thus, if 0 < /3 < 45, then aIms(s,b) 8-S
is -ve
for pp and K’p
(absorption
scattering
(40a)
decreases as energy increases)
and dIm6(s,b) &S
. 1s +ve for pp and 25-p scattering. (absorption
increases as energy increases)
(4Ob)
188
ISLAM
We now investigate diffraction peaks. IV.
ENERGY
the implication DEPENDENCE
of the above results OF
THE
on the behavior
DIFFRACTION
of
PEAKS
The imaginary part of the scattering amplitude comes essentially from absorption within an interaction radius R. The interaction radius R can depend on energy. For collisions with impact parameter b < R(s), we consider the scattering to be completely absorptive. From Eq. (1) then, Im A(s, t) 75 21c2j”” b fZbJo(bl/--t)
ImA(s,b)
0
= kW 1” b dbJo(b 4-t)
0
=
ICW lR b db[l
- ?(S,b)l
[I
(41)
sR b” cZb[l -
- &s, b)] + y
ds, b)l +- - -
0
when the momentum transfer is small; here q( s, b) = ew21ms(s*b). From (41)) using the optical theorem,
we have UT(S) = 4n s R b db[l 0
-7
(s, b)l
(43)
where uT( s) is the total cross section. From (41)) we also get a Im A(s, t) ’ at
=- kW R b” db[l - ~(s 7b)l * 4 s0
t=0
(44)
Next, let us define a quantity
7(s) = a Im A(s,t) at
I
/
t-0
Im A(s, 0).
(45)
From (42) and (45), then,
R 7(s) = g I b” UT(S)-0
db[l
- q(s, 611.
(46)
From (43) we get
dcds) = ___ ds
a rm;;s’
b, b db + 4?rR [I
-
~(s,
b)]
+$.
(47)
DIFFRACTION
Similarly,
&iS) -=-,I ds
189
PEAKS
from (46)) after some algebra we obtain
27r R (R”(48)
If we are in an energy region where the total cross section does not vary appreciably, then the last term in (48) can be dropped and we find,
if
a Im Hs, b) is -ve , then
CHS) ~ ds
d Im 6(s, b) . 1s +ve,
fhid -TCQ-
as
if
(ii)
then
l3S
We now relate T(S) with the inverse of the diffraction amplitude is predominantly imaginary, then
au -= at
P du --=k2 dn
k&z
j A(s,
t) I2 M
&
> (). ’
width.
(49b) If the scattering
[In1 Ais, t>l’
(50)
and
a Im at A(s, t) .
aTcr ,LImA(s,t)‘= k2W2
z
From (50) and (51) it follows, t=o
(51)
using the optical theorem (42),
2 E Im A(s, 0)
a Im A(s, t) at t=o = 27(s).
(52)
Again, if au/at = Aeat, bhen (53) so that a = 2~( s), where a is defined as the inverse of the diffraction width. From our statements (49a, b) and (40a, b) it now follows that (i) da/ds > 0 for pp and K+p scattering, i.e., the diffraction peak shrinks; (ii) da/ds < 0 for fjp and K-p scattering, i.e., the diffraction peak expands. In the case of ?r*p scattering, a Im 6(s, b)/& is small as the vector meson exchange force does not occur. Equation (48) then shows that there will be very little energy dependence of a. A few remarks are worthwhile making here: 1. From Eq. (47) it follows that if du,/ds = 0, i.e., uT is constant, then 4&[1
- D(s, b)] g
= -8a
6” ~(s, b) a lmass(s’ ‘) b &.
(55)
190
ISLAM
Thus, if a Im 6(s, b)/ds is -ve, then dR/cls > 0, i.e., in the case of shrinking peaks the radius of interaction increases with energy. On the other hand, if d Im 6( s, b)/& is +ve, then dR/cls < 0, i.e., in the case of the expanding diffraction peaks, the radius of interaction decreases. These results are in agreement with optical model, where a and R are related by a = R2/4. 3. From Eq. (39) of Section III, we note that d Im 6( s, b)/& decreases in magnitude as s increases. If now the radius R tends to a constant value for large s, then from (47) and (48) it follows that the total cross section and the diffraction width tend to become constants asymptotically. V. ASYMPTOTIC
BEHAVIOR
OF
THE
VECTOR
MESON
FORCE
We have found that if the vector meson exchange is given by ~a( s)/(mv2 - t) with U(S) behaving as Cs” (0 < /3 < $5) for s -+ 00, then we can understand the shrinking and the expanding of diffraction peaks. We now want to see how such behavior of the vector meson contribution can occur. For this purpose, let us study the scattering of two equal mass spinless neutral particles and consider the diagram of Fig. 2. The invariant scattering amplitude corresponding to this diagram is dominated by a sharp p-wave resonance in the t-channel and we want’ to investigate the asymptotic behavior of this amplitude in s-channel, If we denote this invariant amplitude by A (s, t) and the corresponding double spectral function by p( s, t), then
For the moment, we postpone the question of subtraction that may be needed in (56). The absorptive parts in s- and t-channels are given by
A,(s, t) = ; j p(,,T)td"
FIG.
2. Exchange
of the
composite
vector
meson
DIFFRACTION
191
PEAKS
and (58) and in terms of these the invariant
amplitude
is
A(& t) = f / L!!!y
(59)
=- 1 - Ah’, t) ds’ 7r J d-s .
(60)
Using elastic unitarity in t-channel, the double spectral by integrals over the absorptive parts in s-channel:
function
p( s, t) is given
where the integration is carried over the region K(z, z’, 2”) = .z* + z” + zN2 2ZZ’XN - 1 > 0; here 4,(x, t) = A,(s, t) andz = 1 + 2s/(t - 4~‘). Besides (61), we have from (57) PCS, t) = ImA,(z,
(62)
t).
If any asymptotic behavior for A,(x, t) when z + 00 is assumed, then this behavior should be such that Eqs. (61) and (62) are consistent. This problem has been investigated by Gribov (18)) who argues that the asymptotic behavior of 9,(z, t) should be A,(&
t) -
x”F,(log
2)
(63)
where F,(log z) decreases faster than l/log z asymptotically tion t. In our problem, we shall assume that A&x, t) = z”F( log z)f(t)
for
x+
and can be a funcm,
(64)
where F(log x) decreases faster than l/log x asymptotically and v is a constant. In Appendix II, we shall verify that this asymptotic behavior is consistent with Eqs. ( 61) and (62). The consistency requirement gives us an equation for f(t) , namely,
Imf(t) We now f(t) :
2 2:
make the following f’(t)
p(t)
1 f(t)
p(t)
I*,
phenomenological =
=
(
t
assumption
rdt, t, -
t -
i&t
-p
-
4p.’
l/2
>* about the function
(66)
192
ISLAM
As we shall see later, t, will correspond to the position of the p-wave resonance and the parameter y will be connected with its width. y will be considered small corresponding to a narrow resonance. With the form (66) for j( tj, Eq. (65) is automatically satisfied for t near t, . The absorptive part in s-channel and the double spectral function corresponding to it are now given by
A& 1) =
&t,z”F(log f, -
t -
i&t
2) -
(67)
4~’
and p(s
7
tj
= r”d~l/t-4,2Z”F(log (6 - t)” + yyt
2)
(68)
- 4/.&Z)
for z asymptotic and t near t, . To seewhether the above equations are consistent with Eq. (57)) we insert (68) in (57) and obtain A,(s, t) = y+
j
yZ/t’ - 4/.&‘F(log z’) dt’ [(b - t’j” + yyt’ - 4/.42j](t’ - t)
(69)
where z’ = 1 + [2s/( t’ - 4~‘)]. For y small the integrand in (69) is essentially a d-function, so that z’“F( log z’j can be replaced by zrYF( log z,.), where X~ = 1 + [as/(&. - 46)]. Also, the lower limit of integration can be taken as 4~~ since s is very large. Thus, we get m y&’ - 4,~~dt A&, t) = r&&‘F(log z,> 1 7r s49 [(& - t’)” + y2(t’ - 4$jJ(t’ - t) (70) r~&‘F(log zr) = t, - t - i&twhich is Eq. (67). Let us now find the absorptive part in t-channel. Writing Eq. (58) as (71) where z’ = 1 + [2s’/(t - 4~‘)], we note that for a double spectral function like (68) the unsubtracted form (71) cannot be used, since p(s, tj behaves as x” for z + CQ(v > 0). A subtraction is needed, and for this purpose let us use an s-wave subtracted dispersion relation’ At(t, zj = A;(t)
+ A 1 P(s’, t) [-& n-
1 It is worth mentioning here that the antibound an s-wave subtraction (see appendix of ref. 11).
state
- Qdz’j] contribution
dzz’. is connected
(72) with
DIFFRACTION
193
PEAKS
For large x’, the quant,ity inside the square bracket does not go like 0( l/x’) but like 0( l/x”), so that the integral in (72) will now converge, if we use the double spectral function (68) with v < 1. Dispersion relation (72) indicates that the asymptotic behavior of the double spectral function we have constructed is such t)hat no knowledge of the s-wave part is obtained. Our primary interest is in the absorpt’ive part coming from p( s, t) given by (6s)) so that in (72) we are mainly concerned with the second term. This can be written as
r”~tr~cq7
A,(4 4 = (& - t)” + y2(t - 4$) xb? where
m x(z)
= J a s
z’“F(log
Z’)
z’-
1
- Qo&)]
(73)
dz’.
(74)
To see whether the absorptive part (73) can indeed be dominated by the p-wave amplitude for -1 5 z 5 1, let us examine the partial wave projection of the function x( 2) : Fl
xz = $4
I-1
cZzPz(x)x~x)
=- 1 m dz’z’“F(log 7 I
(75)
(1 2 1) (76)
.z’)&&Z’).
For x’ large, the integrand in (76) behaves as l/~‘~+l-” (0 < v < 1) ; consequently, if 1 > 1 the integrand is strongly suppressed. Thus, we arrive at the possibility that 1x1=1 ( >> 1xi>1 1, i.e., among all the partial waves occurring in the absorptive part (73), the only important one is 1 = 1. We can, therefore, say that the p( s, t) we have constructed gives a sharp p-wave resonance in the physical t-channel. Let us next go to the s-channel (t < 0) and examine the invariant amplitude A (s, t) arising from the absorptive part A t( t, x) given by (73). Since we are now in the crossed channel, a zero-width approximation (i.e., the limit y = 0) is appropriate in the t-channel. From (73) and (59)) it then follows
A(s,t) = rlAxc.4 t,--t *
(
x,=1+-------
t, -
2s
4p2> To obtain t,he high energy behavior of A (s, t), we have to find the asymptotic behavior of x(2,.) for x,. ---) 00. From Eq. (74), using the expansion of &o( z’) in l/z’, - x’“F(log x’) dz’ x(2,) = ; + O(l). (78) 21 z’(z’ - z,)
I‘
Here the lower limit z1 is a large but finite value and is introduced
for the con-
194
ISLAM
venience of the later discussion. The asymptotic behavior of x(x,.) is, of course, independent of xi . Using the variable {’ = (x’ - 221)/ z’, we transform the range of integration in (78) from (zl , co ) to ( -1, 1). Dropping the 0( 1) term and writing ~(2~) = @( cr), we have
(79) where
p(f) = z”F(lOg 2’)
(SOI
and
Equation
(SO) shows that for {’ near 1
, dr’) = (p”“:,,,.
(81)
We expect F( log x’) to be a continuous function of z’ when z’ --f w ; so we can assume cpl( {‘) will obey a Holder condition near and at the end {’ = 1. Further F( log z’) vanishes at z’ = 00, so that cpl( 1) = 0. The behavior of the principal value of the integral (79) for cr -+ 1 will now be given by (19)
(Y2) where vg < v and +I({) obeys a Holder condition near and at the end { = 1. We can replace +I( cr) by its limiting value (a,( 1). Going back to the original variable x,. , we get from (S2) Re x(x,)
-
const. 2:’
for
xT ---f 00.
From (77), we then obtain (83) ~(~0 < v < 1). where u(s) -CsYofors--+ This is precisely the asymptotic behavior we postulated exchange term. VI.
CONCLUDING
for the vector meson
REMARKS
We have found that the energy dependence of absorption or inelasticity Im 6(s, b) can essentially come from the vector meson exchange force. The
DIFFRACTION
195
PEAKS
reason is this force asymptotically falls off less rapidly than the other long range force, namely, that due to the exchange of the ~‘-7r antibound state ( 11) . It has been found that the vector meson cannot behave as an elementary particle; in other words, its Born amplitude cannot increase as s asymptotically. If it does, then unitarity is violated in JP and K-p scattering.’ We have examined the phenomenological behavior where the factor s in the Born term is replaced by C(S) and g(s) behaves asymptot8ically as Cs” (/3 < 1). For 0 < /3 < 35, we find d Im 6( s, b)/ds is -ve for pp and k’+~j and a Im 6(s, b)/& is +ve for flp and K-p. The change of sign occurs because the vector meson force changes sign when we go from particle to ant,iparticle scattering. The fact that absorption decreases with energy for particle scattering (d Im 6( s, b)/ds < 0) and increases for antipart,icle scattering (8 Im 6( s, b)/ds > 0) gives a shrinking peak for the former and expanding peak for the latter. In the case of r*p scattering, the vector meson exchange force is absent and as such, the energy dependence of the diff raction peaks is small. Thus, we have arrived at a simple explanation of the shrinking, nonshrinking, and expanding diffraction peaks. We have explained the phenomenological asymptotic behavior of the vector meson force in terms of a suit,ably constructed Mandelstam double spectral function. This function, given by (6s)) is such that it produces a sharp p-wave resonance in the physical region of t-channel (Fig. 2). It also produces higher partial waves, but its asymptotic behavior in s shows that bhe higher part’ial waves will be strongly suppressed. In the physical region of s-channel, the real part, of the scattering amplitude coming from this double spectral function has now t’he asymptotic behavior s’ with /3 < 1. Thus, we have a composite vector meson coming from t,he double spectral function (6s). An important result’ that comes from the present investigation can be seen if we consider the part of Im 6( s, b) that arises from the ?T-R antibound state exchange. For large s this is given by Im &(s, b) = const. Ko(b~r~). This is exactly the result’ one will arrive at using an optical model with a purely imaginary Yukawa potential (15). However, here we have arrived at this result using t,he dynamical or S-matrix met’hod. Many authors have recognized the usefulness of the optical model in describing high energy scattering (15, %$-24). It is, therefore, reassuring to find that the optical model results can be obtained using the S-matrix approach and making simple dynamical approximations. Further we note that the validity of the present theory is no longer restricted to small angles. Our model is capable of giving a real part of the scattering amplitude. The p That by other
the vector meson does not behave authors on the basis of unitarity
as an elementary (,%I, 21).
particle
has also been
argued
196
ISLAM
imaginary part of the amplitude, mainly due to absorption, comes from impact parameters b < R. For b > R, we can assume the impact parameter amplitudes are real and essentially equal to their Born amplitudes. Therefore, we can have a real part given by ReA(s,t) ~22k 2 JLrn b dbJo(b+t)AB(s, b). (84) In the forward
direction,
using (20) and (25), we get m Re A(s, 0) z -70 b dbKo(bmo) =l= yv cs@ - b dbKo( bmv) sR sR (85) = --yo to &(Rmo)
=F yv Cs” ;-
V
&(Rmv).
For x large, &(x) z ( ?r/2z)1’2eC”; therefore, Kl( RWLO)>> &( Rmv), as mv 37120. In the 10 BeV lab energy region, we can expect the first term in (85) to dominate and produce a repulsive real part (5, 6). But, asymptotically, when s -+ ~0 the second term should dominate and give us Re A( s, 0) -+ - 00 for pp and K+p and Re A( s, 0) -+ + m for $p and K-p scattering. For ap scattering, as w or cpexchange is absent, the last term does not occur. However, in this case, the asymptotic behavior of Re A (s, 0) can be controlled by other forces which we have not taken into account. Notice that, since fl in (85) lies in the range 0 < 0 < 55, the diffraction picture Re A (s, O)/Im A (s, 0) + 0 as s --) ~0 holds N SUM). Jin and MacDowell(25) have shown, using phase representa(ImA(s,O) tion, that for an asymmetric amplitude, the real part of the forward scattering amplitude (t = 0) in one of the channels (s or U) becomes negative and tends to - to asymptotically. Our considerations above indicate how this result can be understood in simple physical terms.3 Experimentally it is found that the antiparticle cross section da/at is larger than the particle cross section au/at for t = 0; but at small values of / t 1, da/at > at/at. This change of sign of &?/at - au/at or cross-over effect (2, S) can be easily explained on the basis of our discussion in Section IV. For j t ) small, we are in the diffraction region and the imaginary part of the amplitude dominates, so that .N
a5 aa 0~ Im B(s, t) - Im A(s, t>. z-z From Eqs. (41),
(43), and (46), we have ImA(s,t)
3 Many authors have high energy (26-32).
investigated
= ‘2 the real
+ ‘F part
of the forward
scattering
amplitudes
at
DIFFRACTION
where
3~ = (diffraction Im il(s,
width)-‘.
t) - ImA(s,
197
PEAKS
Therefore,
2) = ~{[~-l]-,t,[~i-T]).
Since C~ > uT and ? > 7, the quantities inside the square brackets are positive. We then expect’ a change of sign of [Im 6( s, t) - Im A( s, t)] at small values of 1 t 1, which explains the cross-over effect. In conclusion, we will like to suggest a few applications of the present approach. One of these will be the investigation of high energy large momentum transfer pp and 7r*p scattering, where experimental results are now available (33,34). The large angle scattering shows strong energy dependence. It will be interesting to set whether this dependence is conncc%ed with the vector meson force, which is also energy dependent and, being a short range force, can be important at large momentum transfers. Another application will be to the one meson exchange model with absorptive caorrection (35-38). This model has been quite successful in piou exchange case, but has failed when vector exchange reactions are involved. It, will be worthwhile to examine how the predictions of this model are changed, when the vector meson is treated not as an elementary but as a composite particle and its exchange amplitude is modified in a way similar to what has been done in t’he present’ paper. APPENDIX
I
In this appendix, we shall consider t#hc deduction of Eq. (17). The mathematical steps involved are the same as in the corresponding partial wave case (4)) but the boundary conditions are somewhat different now. So we shall briefly go through the derivation. First, we define functions
then, @(St) + a( se) = 3g( s)
(Al.l)
and @(s+) - qs-) The problem is to find purpose, the infinite cut cut { = fT = -l/s+ to l-plane if the functions
= 2ih(s).
(ST S s < m)
(A1.2)
h(s) assuming g(s) to be a known function. For this along the real axis in s-plane is transformed to a finite < = 0 by introducing a new variable 1 = -l/s. In the corresponding to +(s), g(s), and h(s) are denoted by
198
ISLAM
primed ones, i.e., +(s)
= a’(c)
etc., then the above two equations become
d(t+)
+ d(1L)
= 2g’(t)
(a’@+) - d(L) The homogeneous
equation
= 2&‘(t).
corresponding
to (A1.3)
x(L+> + xct-> and the fundamental
solutions
of this equation
can be written
(A1.4)
is
are
C2(3-J:
CA-(r - hP, (A1.3)
(rT S t 5 0)
= 0
x(T) = Cl(q, Equation
(A1.3)
c4Hs - !w’“.
as
cP’(t+)= $+qt-)
+ 2gl(t)
and its general solution is given by (AM) and
To determine the particular have to use certain boundary for s * sT
solution appropriate to our physical problem we conditions. To this purpose, we first note that A(s, 6) = AM(S)
(A1.7)
= a + ipa’
where a is the s-wave scattering length and we have neglected t,erms of the order of k2. Also, we have assumed, there are no inelastic channels open at threshold. Now, 2&3(s,b) e = 1 + 2ipA(s, b); therefore,
from (A1.7)
we get e
--4Id(s,b)
= 1 + 4p2a2 for
s + sT .
This gives Im6(s,b)
= it
Aa2
,()
ST
for
sjs
T.
4
(A1.8)
4 No/e added in proof: Similar but more careful consideration indicates Im 6(s, b)/k ishes fast,er than k. This, of course, does not change the boundary condition.
van-
DIFFRACTION
Xext we consider the case when Instead of (A1.7), we now have
A(s, b) = AL,,(S)
199
PEAKS
inelastic
channels
are open at threshold.
= a + ib + $(a + ib)’
(A1.9)
for s -+ sT and neglecting k2 t’erms; a + ib is the complex s-wave scattering From (A1.9) -4Im6(s,b)
e
=
( 1 - 2pb)2 + 4p2a2
=
1 - 4&
length.
so that Im 6(s, b) b = ~ k 6 For the fundamental we find that
solutions
~2 = c2[[/({
for
s + ST.
(A1.lO)
- {T)]1’2 and x3 = c3[{({ - j+T)]‘)l-1’2,
However, h’(P) = h(s) = Im 6(s, b)/k is finite when { -+ lT as indicated by Eqs. (A1.8) and (A1.lO). Therefore, solutions x2 and x3 are not appropriate for our problem. On the other hand, for the fundamental solution x1 = CI[(< - ~T)/d1’2 and x4 = cdl(3. - (T)]““, we find (19) h’(t)
a (( - {T)1’2-ao
for
< + [T
(a0 < %I.
If we take 010= 0, then (A1.S) is satisfied. If we choose 010sufficiently close to the value +2/, then the phomomenological behavior indicated by (A1.lO) can also be satisfied. Thus, the solutions ~1 and x4 are only relevant for our problem. For X( [) = xl(T), h’( <) m 1/<1’2 when C --, 0; on the other hand, for x( {) = o(o < 35 ) when [ + 0. Now, h’( {) = Im 6(s, b)/k and in X4(b) m-1 cx i- 1’2--oro( writing down Eq. ( 11) for 8( s, b), we have considered that Im 6( s, b)/k --+ 0 when s -+ ~0. Therefore, h’( <) -+ 0 when { + 0. Thus, the fundamental solution x4 is only acceptable. Now, for s --$ 0
e(s, b)_1s 16
m Im 6(s’, b) C~S’= const lr ST k’s’
Therefore, a’({)
= O(s, b)/k
= const.
when
{ -+ 00.
(Al.ll)
From Eq. (A1.5) with x( {) = x4({), we find that for r + Q) , the first term on the right-hand side behaves like constant; however, the second term behaves like cl+’ for j 2 0. This behavior violates (Al.11). Therefore, we must have P,(l) = 0.
200
ISLAM
Thus, the solution we finally obtain is
sO IT
2g’(t)
dt
Ltct - $+‘T)l”2(t- i-1
(Al.12)
and
g’(t)dt r-rP2(t- i-1*
(A1.13)
Transforming to the variable s, we get
e(s, b)=(s- sTp2 sk
ai
Im 6(s, b) =k
(s -YZ
8T
A(s’, b) ds’ k’(s’ - s,)“Z(S’ - s) ’
and A(& b) ds’ p SW k’(s’ - ST)l’2(S’ - s) ST
APPENDIX
II
In this appendix we want to seewhether the asymptotic behavior of the s-channel absorptive part, A,(z, t) = x’F(log z)f(t)
(Y > 0)
(A2.1)
is consistent with the equations
and p(s,
t)
= z”F(logx)
Imf(t).
(A2.3)
The region of integration in (A2.2) is shown in Fig. 3: For z and zN fixed, the range of integration for x’ is xo 5 z’ < .zzn - 2/(x2 - l)(~“~ - 1) and so forth. Examining the integrand of (A2.2)) we find that it becomes large-(i) when x’ and zNhave values near the boundary line K = 0, and (ii) becauseof the assumed asymptotic behavior (A2.1)) when one of them is very large (correspondingly, the other one would be small). Let us begin with a very large value of z and define a parameter X such that if x = x/X, then 1 << 2 << 2. We can now consider that the double spectral function is given by
DIFFRACTION
201
PEAKS
(A2.4) CIXnA,*(XN,
t)
*a zw-&=i)
A&‘,
Cl2
.s zcz-&cl,
(2,’
t>
z’)l/z(xb’
-
-
~‘)‘/Z
where
&
= 2.2’ f
&
= XX” f 1/(9
FIG.
&2*
3. Integration
-
l)(z’2 -
l)(P
region
1) 5s x(x’ f -
of the
1) w 2(/
double
&‘,
-
f dp=T).
spectral
function
1)
(A2.5) (A2.6)
202
ISLAM
The regions of integrations in (A2.4) are shown in Fig. 3 by the shaded area. For the integration over zn in the first term and that over x’ in the second term of (A2.4)) we can use the asymptotic formula for the absorptive part, since the integrations are on large values of the variables. Further, we may assume that the logarithmic terms F(log z”) and F(log z’) in these integrations can be replaced by F( log x), because x is very large. Thus, we obtain
SW-~)
dz’ zlv
../ zb-d/22--1)
(2,
-
~‘)‘/Z
(Zb’
-
2’)1/2
The inner integrations in (A2.7) can be easily done in the following Let q = X: - 4x2 - 1, qa,b = 2: ,b/z = zn f dzN2 - 1. Then,
way.
(A2.S)
where w = vb/qa . NOW, q/7]b = (x” $- dzNa - 1)/(x + dx2 - 1) << 1, because x0 $ zn 5 x/X and x/X << x. Therefore, the lower limit of integration in (A2.8) can be taken to be zero and we obtain (99)
From Eq. (A2.7)
we now get
PCS, 0 = -1 t - 4p2 1’2 &-) x%(log 7r
x) f*(t)
“’ x’F(log
/“” cZz’As(x’, t)&v(z’) 20 z) f(t)
s”” &“A,*(/, 20
(A2.10) t>&y(z”).
For large x’, Qy(x’) - I/X”+’ and A,(z’, t) - z”F(log x’) so that the integrals in (A2.10) will converge for z --+ 00, if F( log x’) vanishes faster than l/log x’.
DIFFRACTION
203
PEAKS
This, of course, we have already as sumed. For x asymptotic, written as5 PCS, t)
= x”F(lw .w*uMt)
p( s, t) can now be
+ fu>P*u)l
(A2.11)
where (A2.12) Thus, (A2.11)
and (A2.3)
are consistent
Imf(t)
if
= f*(Mt)
We may expect the main contribution large values of x’, so that
that is, cp(t) E cp( t)f( t) where p(t) tion (A2.13) now becomes Im.f(t)
of the integral
= [(t - 4~‘)/t]l’~
in (A2.12)
to come from
and c is a constant. Equa-
E 2cp(t)lf(t)l!
If the constant factor 2c is absorbed in f(t), (A2.13) can be expressed as Imf(t)
(A2.13)
+ f(t)cp*(t).
then the consistency condition iA2.15)
22 p(t)lf(t)12.
ACKNOWLEDGMENT The Feldman
author wishes to thank for his interest.
RECEIVED:
Dr.
Y. S. Jin
for
stimulating
conversations
and
Prof.
D.
June 3,1966 REFERENCES
1. S. J. LINDENB~UM, Review Lecture, Oxford International Conference Particles (1965). 2. R. J. N. PHILLIPS .IND W. RIRIT~, Phys. Rev. 139, B1336 (1965). 3. T. 0. BINFORD AND B. R. DES.41, Phys. Rev. 138, B1167 (1965). 4. M. M. ISLAM AND KYUNGSIH KING, Phys. Rev. 139,B973 (1965). 5. M. M. ISL.ZM, Phys. Rev. 141, 1524 (1966). 6. M. M. ISLAM, Phys. Rev., 147, 1144 (1966). 7. R. BLMVKENBECLER.\ND M. L. GOLDBEKGER, Phys. Rev. 126,766 (1962). 8. W. N. C~TTINGHAM AND R. F. PEIERLS, Phys. Rev. 137, B147 (1965). 9. T. ADSCHI BND T. KOT.\NI, SUppt. P?Og?. Theoret. Phys. (Commemoration (1965). 5 Eyuation uses somewhat
(A2.11) is exactly different method
the same as that obtained by Gribov to evaluate t,he integrals asymptotically.
(18).
on
Elementary
Issue),
316
However,
he
204
ISLAM
10. 11. 12. IS.
E. M. J. S. 14. G. 1.5. R. 16’. F. 17. J. 18. V. 19. N.
PREDAZZI, Ann. Phys. (N. Y.) 36,228 (1966). M. ISLAM, Phys. Rev., 147, 1148 (1966). J. SAKURAI, Ann. Phys. (N. Y.) 11, 1 (1960). C. FRAUTSCHI, M. GELL-MANN, AND F. ZACHARIASEN, Phys. Rev. 126,2204 (1962). F. CHEW, S. C. FRAUTSCHI, AND S. MANDELSTAM, Phys. Rev. 126,121X? (1962). SERBER, Rev. Mod. Phys. 36, 649 (1964); Phys. Rev. Letters 10, 357 (1963). G. TRICOMI, “Integral Equations,” p. 182. Interscience, New York, 1957. HAMILTON AND W. S. WOOLCOCK, Rev. Mod. Phys. 36,737 (1963). N. GRIBOV, NucZ. Phys. 22, 249 (1961). I. MUSKHELISHVILI, “Singular Integral Equations,” Chap. 4. Noordhoof, Groningen, Holland, 1953. 20. P. G. 0. FREUND AND R. OEHME, Phys. Rev. Letters 10,199, 315(E) (1963). 81. K. YAMAMOTO, Phys. Letters 6,355 (1963). 22. Y. NAMBU AND M. SUGAWARA, Phys. Rev. Letters 10, 304 (1963); Phys. Rev. 132, 2724 (1963). 23. N. BYERS AND C. N. YANG, Phys. Rev. 142,976 (1966). 2.4. R. C. ARNOLD, ‘LAn Optical Potential for High Energy Physics: Theory and Applications,” Preprint, Argonne National Laboratory (1966). 25. Y. S. JIN AND S. W. MACDOWELL, Phys. Rev. 138, B1279 (1966). 26. P. SODING, Phys. Letters 8,285 (1964). 27. I. I. LEVINTOV AND G. M. ADELSON-VELSKY, Phys. Letters 13,185 (1964). 28. B. LAUTRUP, P. M$LLER NIELSEN, AND P. OLESEN, Phys. Rev. 140, B984 (1965). 29. G. H~~HLER, G. EBEL, AND J. GIESECKE, 2. Physik 180,430 (1964). 30. V. S. BARASHENKOV AND V. I. DEDJU, Nucl. Phys. 64,636 (1965). 31. H. BANERJEE, Phys. Rev. 140, B989 (1965). 3%. A. BIALAS AND E. BIALAS, Nuovo Cimento 37,1686 (1965). 33. G. COCCONI, V. T. COCCONI, A. D. KRISCH, J. OREAR, R. RUBINSTEIN, D. B. SNARL, B. T. ULRICH, W. F. BAKER, E. W. JENKINS, AND A. L. READ, Phys. Rev. 138, B165 (1965). 34. J. OREAR, R. RUBINSTEIN, D. B. SCARL, D. H. WHITE, A. D. KRISCH, W. R. FRISKEN, A. L. READ, AND H. RUDERMAN, Phys. Rev. Letters 16,309 (1965). 35. J. D. JACKSON, Rev. Mod. Phys. 37,484 (1966). 36. L. DURAND III AND Y. T. CHIU, Phys. Rev. 139, B646 (1965). 37. J. D. JACKSON, J. T. DONOHUE, K. GOTTFRIED, R. KEYSER, AND B. E. Y. SVENSSON, Phys. Rev. 139, B428 (1965). 38. V. BARGER AND M. EBEL, Phys. Rev. 138, B1148 (1965). 39. “Tables of Integral Transform,” A. Erdelyi, ed., Vol. I, p. 310. McGraw-Hill, New York,
1954.
OF
Shrinking,
PHYSICS:
41, 177-204
Nonshrinking,
(1967)
and
Expanding
M. &I. Department
of Physics,
Brown
Diffraction
Peaks*
ISLAM
clniversity,
Providence,
Rhode
Island
An explanation of the shrinking, nonshrinking, and expanding diffraction peaks at high energy (210 BeV) is given in terms of two kinds of forces: (1) due to the exchange of the r-8, JPa = Of+, T = 0 autibound st,ate or virtual state, and (2) due to the exchange of the J po = l-- vector meson (w + q). It is shown that the energy dependence of the diffraction peaks can essentially come from the vector meson exchange force, because this force falls off more slowly at high energy than the other force. Thus, for PT’P and ?r-p scattering, as the vector meson force does not occur, we have little energy dependence or non-shrinking diffraction peaks. For pp and K+p scattering, the vector meson exchange force can have asymptotic behavior such t,hat it decreases absorption as the energy increases and this behavior then gives shrinking diffraction peaks. For pp and K-p the vector meson exchange force changes sign and as a result, the absorption now increases as the energy increases. This behavior produces expanding diffraction peaks. The absorption (or inelasticity) at high energy has been dynamically related with the input forces, in the framework of an impact parameter formalism. The asymptot,ic behavior, which is found for the vector meson force, is explained in terms of a suitably constructed Mandelstam double spectral function. This explanation also indicates how the compositeness of the vector meson comes about. On the basis of lluitarity argument, it is pointed out that the vector meson cannot behave as an elementary particle. The incident and the outgoing particles are treated as spinless. I. INTRODUCTION
At present no satisfactory explanation of the behavior of elastic diffraction peaks at high energy ( 2 10 BeV) exists. Experimentally, it has been observed that the diffraction peaks for elastic pp and Kfp scattering shrink as the energy increases, those for ~‘p and r-p scattering show little energy dependence, and the diffraction peaks for flp and K-p expand with increase in energy (1). Originally, the experimental study of the diffraction peaks was motivated by the prediction of the Regge pole theory. However, at present, it is well established that the simple version of the Regge pole theory does not work. One needs a large number of Regge poles and correspondingly a considerable number of parameters * Supported
by the U. S. Atomic
Energy
Commission. 177
Copyright
0 1967 by Awdemic
Press
Inc.
178
ISLAM
to fit the experimental data (2, 3). As a result, the Regge pole fits of the diffraction peaks have become physically much less meaningful. In this paper a new explanation of the behavior of diffraction peaks is proposed. The physical basis of this approach is that absorption can be dynamically related with the input forces, under certain simple approximations, and the energy dependence of absorption at high energy is then connected with the asymptotic behavior of these forces. Since the forward diffraction peaks essentially arise from absorption, their high energy behavior is thus governed by the asymptotic behavior of the input forces. The way absorption at high energy can be dynamically related with the forces is formulated in Section II of this paper, in the framework of an impact parameter expansion. The corresponding case, for the partial wave expansion, has been previously developed (4, 5) and applied to pp and 7r*p scattering (5, 6’). These applications have indicated that physically meaningful absorption can indeed be calculated from input forces. The energy dependence of the absorption arising from the asymptotic behavior of the forces is considered in Section III. In Section IV, the energy dependence of the diffraction peaks is related with that of the absorption. In Section V, an explanation of the asymptotic behavior of the vector meson force is given. Finally, in Section VI some concluding remarks are made. II.
IMPACT
PARAMETER
EXPANSION
AND
DYNAMICAL
INELASTICITY
Two important characteristics of high energy elastic scattering are (i) the prominent forward diffraction peak, and (ii) the small wide angle scattering. These features indicate that an impact parameter expansion, rather then a partial wave expansion, is more suitable to describe high energy elastic scattering (7). Recently, several authors have considered rigorous formulation of the impact parameter method (8-10) and we shall follow their approach. We define the impact parameter expansion of the invariant scattering amplitude A( s, t) by the relation A(s,
t) = S lm 5 dEJ0(t sin 0/2)A(s,
0
(1)
where s = square of c.m. energy, t = -22k2( 1 - cos 0), and ?r > 0 2 0. Equation ( 1) can be inverted and gives
A(%.$I= 2 !,I sin
0/2 d(sin
e/a)J,(
21cb sin 0/2)A(s,
cos
e).
(2)
The impact parameter b is defined by putting 4 = 2lcb in the above relations. The partial wave expansion of the amplitude A( s, t) is
A(s,t) = l$
(21 + lP’l(cose)At(s),
(3)
DIFFRACTION
179
PEAKS
where Al(S)
= [e2i*~(s) -
and p = k/W; 12is the cm. momentum in Eq. (2), we get’,
1]/2ip
(4)
and W is t,he cm. energy. Inserting
(3)
(5) Equation (5) gives the impact parameter amplitude in terms of the partial wave amplitudes. It also indicates t’hat A (s, b) is an even function of b. From (4) and ( 5)) using the relation
we find A (s, b) can be written
as
A( s, b) = [~is(s3b) - 1],‘22$,
(6)
where e2i8(s,b)
E 2g
(21 + l)ezi*~(“‘Jzl+l(21cb)/2~b.
(7)
Equations (4) and (6) show a close similarity between the partial wave amplitude and the impact parameter amplitude. The unitarit’y restriction on A i( s) is very simple (namely, Im 61(s) > 0), while that for A( s, 6) is quite complicated (9). However, at high energy the unitarity restriction on A (s, 6) becomes simple and is given by Im 6( s, b) z Im 61(s) > 0, where b = (I + 4$)/k.
Equation Al(s)
(8) follows
(8)
from the inverse relation
= 212f= A(s, b)Jzz+1(2kb)
db
(9)
0
and the approximation
t’hat for large h?, Eq. (9) gives (8) Al(s)
= A(s,
b) + 0(1/L?)
(b = I + 3$/k).
Incidentally, (10) also exhibits the correct interpretation parameter. Comparing the equation for the impact parameter A(s,b) with
that of the partial wave amplitude = ?,$ /“I
&OS
0) Pi
(COS
(10)
of b as an impact amplitude
= 211 sin e/2 rl( sin 0/2)Jo( 2kb sin 0/2)A (s, cos 6)
Al(s)
of (5)
o)A(S, COSe),
180
ISLAM
we notice that the analytic properties of A (s, b) for fixed b in the complex s-plane should be the same as those of Al(s) ; that is, assuming Mandelstam representation for A ( s, t), A (s, b) will have right-hand cuts coming from unitarity cuts in the s-channel and left-hand cuts from unitarity cuts in t-channel and u-channel. A ( s, b) will have an additional singularity at 1s 1 = 00. This singularity arises because Jo( 2kb sin e/2), which is an integral function of k2, has a singularity at Ik2/ = w. We now consider the problem of obtaining the absorption Im 6( s, b) from input forces. The corresponding case for the partial wave amplitude has been studied before (4,5). First, we define a function 0( s, b) by the relation
(11) sT is the value of s at the elastic threshold. That Im 6( s, b) for s + m is a well defined quantity follows from Eq. (8). Next, we construct a new amplitude a( s, b) by writing 1 + 2ipa(s, b) = eC2ie(s,b)[l + 2&4(s, From
(la),
b)].
(12)
we get, a( s, b) = [e2ia(8ub)-
1]/2ip
(13)
where a(s, b) = qs, b) - qs, 6).
(14)
Since for s on the positive real axis Im 6( s, 6) = Im 0( s, b), CY(s, b) is real in the physical region. From ( 13) it then follows that the new amplitude a( s, b)obeys elastic unitarity, i.e., Im u(s, b) = p ] u(s, b)]‘, throughout the physicalregion. Equation ( 12) shows that a( s, b) has the same analytic properties as A (s, b) in the s-plane. If we write O(s, b) = A(s, b) + i Im 6(s, b) then
Also, A(s, b) = Re6(s,
b) - (Y(s, 6).
06)
If now from physical considerations, we have some knowledge of Re 6( s, b) and 01(s, b), then we can try to invert Eq. ( 15) and express Im 6( s, b) as an integral over A( s, b) . This would provide us with a dynamical scheme for obtaining ab-
DIFFRACTION
181
PEAKS
sorption. The inversion of Eq. (15) can indeed be done exactly by using the analytic properties of 8( s, b) and some boundary conditions. These boundary conditions are somewhat different from the corresponding partial wave case and are discussed in the appendix. The final result is Im 6(s, b) = -
k(s -TSrj1’2 p [ m [Re 6(s’, 0) - LY(~, b)] da’. k’(s’ - s-ry’ys - s) ST
(17)
Now, to calculate Im 6(s, b), we make the following approximations: (i) Above a sufhciently large energy sA , the impact parameter amplitude A(s, b) for b < R is completely absorbed, i.e., Re 6(s, b) = 0 for s > sA and b < R, where R is the radius of interaction. (ii) Above the energy SA the amplitude a( s, b), which obeys elast’ic unitarity throughout the physica region, approaches its Born term. From the second assumption, it follows that since t’he Born term is real, a( s, b) is essentially real for large s. However,
- 11_ u(s, b) = [e2ia(s3b)
(Y(s,
Zip
P
bj I i i2(8, ___P 6) .“,
so that (Y(s, b) should be small at high energy. Thus, we get for
(18)
s > SA,
P
where aB( s, b) is the Born approximation of a( s, b). Using approximations (i) and (ii), we therefore obtain from ( 17) Im 6(s, b) = -k(s
:sT)l”
aA [Re 6(s’, b) - a.(~‘, b)] ds’ k,(s, - sT)1’2(d - s) ST
j”
+ k(s -$li2
p [SA
=-
p(&?(i, b) ds’ k’(s’ - sTy2(s’ - s)
k(s - sT)1’2 YA[Re 6(s’, b) - (Y(s’, bjl CZS sST k’(d - sT)1’2(d - s) P
(19)
sT)1’2 + k(s -?i-
(s > sa> For large s (>>a,) the first term on the right-hand side of ( 19) behaves as a constant. Thus, the energy dependence of Im 6(s, b) can essentially come from the second term depending on the asymptotic behavior of the Born term a”( s, b). This we study in the next’ section. As the Born term uB( s, b) represents the input
182
ISLAM
forces, SO Eq. (19) shows how the absorption with these forces at high energy. III.
INPUT
FORCES
AKD
ENERGY
Im 6( s, b) is dynamically
DEPENDENCE
OF
THE
related
ABSORPTION
We consider input forces arising from the exchange of the 7r-r antibound state or virtual state (11) and from the exchange of the T = 0 vector meson.The vector meson exchange represents the combined force coming from the exchange of w and cpmesons.It has been pointed out that the force arising from the exchange of the T-R antibound state is similar to that due to the exchange of a scalar particle of mass ,Z2nl, , but with a residue opposite in sign (11). Thus, the invariant Born term due to the exchange of the antibound state can be represented as -YO/(,WO* - t) with 70 > 0 and m o E %a, in the physical region of the s-channel. To calculate the corresponding Born term of the impact parameter amplitude, we go back to Eq. (2) and obtain AB(s, b) = $$ jzk A dAJo(bA) szz -
0
(-t
=
A”)
O2
(20)
for large k*. Ko( z) is the modified Bessel function of second kind. As an approximation, we take AB( s, b) = aB( s, b). This approximation is equivalent to saying that when we calculate forces, we ignore the inelastic scattering in the physical region; A( s, b) and a( s, 6) then coincide. Thus, the Born term of a( s, b) due to the antibound exchange is UOB
= - $
Ko( bmo)
(70 > 0)
(21)
for large k*. We next want to calculate the force or the Born term due to the exchange of the vector meson. The simplest way to do this is to consider the Feynman diagram and use perturbation theory. Assuming the interaction between the spinless nucleon and the vector meson to be given by H NNv = igNNVCPN ‘~,~NV~ and denoting by s the square of c.m. energy, we find from Fig. 1 the invariant amplitude for pp scattering to be
A;p(s, t> =
2 YNNV
u--s
8ap=-t
&NV 4lr
s
f
mv2/2
-
my2 - t
2mNy2
+g&.
(22)
DIFFRACTIOX
FIG.
1. Exchange
183
PEAKS
of the elementary
vector
meson
The second term corresponds to a very short range force that contributes only to the s-wave in the s-channel and we neglect this force. The negative sign in front of the first term in (22) indicates that the vector meson exchange force is repulsive for pp scattering. The invariant amplitude due to vet tor meson exchange in ISP scattering is now obtained by interchanging s and u and this gives
Worth noticing is that the long range part of the force is now attractive. For K’p and K-p scattering, assuming the interaction between K-meson and the vector meson to be
we obtain Born terms similar to Eqs. (22) and (23) with giNV replaced by sign of gNNV and gKKv should be gNN VgKK v and 2mN2 by ??r#’ + 3%‘. The relative positive, so that the long range vector meson exchange force is repulsive (attractive) for K’p (K-p) scattering (1%). We can now calculate the Born term of the impact parameter amplitude due to vector meson exchange by making the same approximation as before. The final result is
(24) for large k2, where the negative sign is for particle scattering and positive sign is for antiparticle scattering; yV = &NV/4?r for pp and pp; TV = gNNVgKKV/4r for K+p and K-p. Notice that. yv is always positive. The s factor in Eq. (24) occurs because of the spin of the vector meson. In calculating the vector meson exchange contribution by perturbation theory, we
184
ISLAM
have treated it as an elementary particle and the factor s indicates this. The Born terms (22) and (23) blow up like s at high energy. In reality, we believe the vector meson behaves not like an elementary particle but like a composite particle, so that its Born term is damped at high energy. Originally, the Regge pole theory provided a method for the damping mechanism (IS, 14), but the validity of this theory has not so far been established ( 1) . For the moment, we shall take a phenomenological attitude and consider that for a composite vector meson, the Born term is obtained by replacing the factor s in Eq. (24) by c(s), where for s --f ~0 a(s)
= csB + !b(s),
0 -I /3 < 1, C > 0, and $(s) obeys a Holder condition near and at s = CQ; also #( cc ) = 0. Later on, in Section V, we shall show by considering the elastic scattering of two equal mass spinless particles, that a suitable Mandelstam double spectral function can be constructed such that in t-channel it corresponds to the exchange of a vector meson and in s-channel it produces the asymptotic behavior postulated above. The reason C is chosen positive is that if the vector meson force is repulsive for the elementary case, then it will remain repulsive for the composite case, even though its energy dependence is damped. The Born term for the composite vector meson exchange is thus taken to be
We now investigate the energy dependence that occurs in Im 6( s, b) due to the forces (21), (24) and (25), using Eq. (19). Let us denote the first term on the right-hand side in (19) by Im 15~(s, b). Im sL( s, b) then primarily represents the contribution that comes from the low energy region. We further define Im 60(s, b) =
aoB(s’, b) ds’ k(s - sTy* O” ‘lP(s’ s - ~)‘/2(S’ - s) n8.4s
(26)
and Im &(.s, b) = ‘(’ The total absorption Im6(s,
- sT)1’2 I, a
s,1,2( avI($A(da’S’
s’
s) .
(27)
b).
(28)
is now given by b) = ImaL(s,
b) + Im&(s,
b) + Im&(s,
As can be seen from (19), for large s Im aL( s, b) behaves as
DIFFRACTION
185
PEAKS
Im &( s, b) = const. + 0( l/s).
(29)
If s is large so that terms of the order of l/s can be neglected, then Im &,(s, b) is essentially a constant. Inserting (21) in (26), we obtain Im &(s, b) = -&0K0(bm0)k(s
- sT)l” m
x-
s1 C
mk(s
as’
l
-
ii s sA s’ys’
a)
-
ST)l’2(S’
-
-
8,)
s1)
I +
s2
mp(s
1
-
s2)
ii
(30) s .,
s’W(S
-
SA
-- 1
l&r sST
$ys~
r
s’l/2(s’
_
dS sT)‘12(s’
-
S)
+
k
Re
I,(S)]
where s1
=
(7?2
+
PL)’
=
ST,
s2 =
(m
-
p)“,
In denotes the mass of the proton and p denotes the mass of the other particle (p = mforppand@p,p = mKforK&pandp = m,fora*p).Fors> ST, Re L(s) = -
’ $“(s - s$”
log (s)“’ + (8 - sT’)l” (8)“’ - (s - sT)1’2
so that for large s Re L(s) = -klog Since k( s - 8~)~‘~= s/2fors--+ Im 6,,(s, b) = --
0
$
.
w,wefindfrom(30),
1 YoKo(bmo) const. - i 4
( )I
log $
(31)
for s large and neglecting terms of the order of l/s. Equation (31) shows that the dominant energy dependence of Im 60(s, b) comes from the term log s/s and is therefore small. Thus, Im 60(s, b) will be practically constant for large s. Another interesting point about (31) is the b-dependence that it exhibits of the absorption Im &( s, b). It is exactly the same as we would obtain for a pure imaginary optical potential of Yukawa type ( 15). However, we have arrived at the result (31) using dynamical concepts and not any phenomenological absorbing potential. We have from Eq. (27) and the Born term (24) for an elementary vector
186
ISLAM
meson Im
6v(s,
b)
=
frvK:Jnl,)
k(S
-
sT)1’2p
SW SA
Tl/z(,,j
_
s’ ds’
sT)l/2Jc~2(s’ -
’
8)
= =F~~~vK&mv)k(S - sg* (32)
8.4
S
--
l&r
ds’
s ST
s”‘Z(
s’
-
s,yys
-
s)
+ iReL(s)
1 .
For large s, Eq. (32) gives Im &(s,
b) =
From (33), we find for s + 00 Im K-p scattering. However, unitarity (Eq. (8) ) . Therefore, we conclude and that use of the Born term (24) Using the Born term (25) of the Im &(s,
b) = =FYvK:$mv)
C
-
s1
mp(s 82
tf
.
(33)
s,liz(s, ~~~)l;$~(
s’ _ s)
ST)~”
l Sl) ii s
cc
SA
00 +
0
6( s, b) = Im &( s, b) is negative for @p and restriction gives Im 6( s, b) > 0 for s large that the vector meson cannot be elementary, to calculate absorption is not appropriate. composite vector meson, we have
k(s - ST)l”P [,
= =~J&vKo(bmv)k(s
x-
flog
fyvKo(hv)
1
mp(s - s2>ii s 8Aiys’
s‘lys’
a(~‘) ds’ - s,)“ys’ - Sl)
(34)
a(~‘) ds’ - Sp(S’ - sz)
where (35) Now, we have assumedfor s + 00
u(s)= cs@ + l)(s)
DIFFRACTION
where 0 5 /3 < 1, #( m ) = 0, and #(s) I es)
PEAKS
obeys the Hiilder
- $(soj I < KI l/s
for large s (K, E > 0). If, further,
187
-
condition
l/so If
the integral
exists (p > 1), then we have (16,17) I(s) = - qy S’S
+ o(t)
(0 < P < 1)
(36)
and
I(sj = _ 1 c(asTip log(&)+o(;). 8-m T S
CP=O) (37)
From (34) and (37) we find that for p = 0, the dominant energy dependence of Im 6”( s, b) is given by log s/s. That is, the same as t’hat for Im 60( s, b). This is not surprising, since in this case, asymptotically the vector meson Born term behaves like bhat for the 7r-r antibound state. However, if the vector meson Born t,erm does not fall off as fast, i.e., if 0 < fl < 1, then from (34) and (36), Im Ev(s, b) = 7 $ yV K. (bn~~)
[
const. -
4c cot np g-8
1
neglecting 0( l/s) terms. Comparing (3X) with (31 j, we find for 0 < fi < 1 hhe dominant energy dependence of Im 6(s, b) comes from the vector meson exchange. Physically this is understandable, since the vector meson exchange force falls off more slowly compared to the other force at very large energy. Further, from (38) we get, for s large (39) Thus, if 0 < /3 < 45, then aIms(s,b) 8-S
is -ve
for pp and K’p
(absorption
scattering
(40a)
decreases as energy increases)
and dIm6(s,b) &S
. 1s +ve for pp and 25-p scattering. (absorption
increases as energy increases)
(4Ob)
188
ISLAM
We now investigate diffraction peaks. IV.
ENERGY
the implication DEPENDENCE
of the above results OF
THE
on the behavior
DIFFRACTION
of
PEAKS
The imaginary part of the scattering amplitude comes essentially from absorption within an interaction radius R. The interaction radius R can depend on energy. For collisions with impact parameter b < R(s), we consider the scattering to be completely absorptive. From Eq. (1) then, Im A(s, t) 75 21c2j”” b fZbJo(bl/--t)
ImA(s,b)
0
= kW 1” b dbJo(b 4-t)
0
=
ICW lR b db[l
- ?(S,b)l
[I
(41)
sR b” cZb[l -
- &s, b)] + y
ds, b)l +- - -
0
when the momentum transfer is small; here q( s, b) = ew21ms(s*b). From (41)) using the optical theorem,
we have UT(S) = 4n s R b db[l 0
-7
(s, b)l
(43)
where uT( s) is the total cross section. From (41)) we also get a Im A(s, t) ’ at
=- kW R b” db[l - ~(s 7b)l * 4 s0
t=0
(44)
Next, let us define a quantity
7(s) = a Im A(s,t) at
I
/
t-0
Im A(s, 0).
(45)
From (42) and (45), then,
R 7(s) = g I b” UT(S)-0
db[l
- q(s, 611.
(46)
From (43) we get
dcds) = ___ ds
a rm;;s’
b, b db + 4?rR [I
-
~(s,
b)]
+$.
(47)
DIFFRACTION
Similarly,
&iS) -=-,I ds
189
PEAKS
from (46)) after some algebra we obtain
27r R (R”(48)
If we are in an energy region where the total cross section does not vary appreciably, then the last term in (48) can be dropped and we find,
if
a Im Hs, b) is -ve , then
CHS) ~ ds
d Im 6(s, b) . 1s +ve,
fhid -TCQ-
as
if
(ii)
then
l3S
We now relate T(S) with the inverse of the diffraction amplitude is predominantly imaginary, then
au -= at
P du --=k2 dn
k&z
j A(s,
t) I2 M
&
> (). ’
width.
(49b) If the scattering
[In1 Ais, t>l’
(50)
and
a Im at A(s, t) .
aTcr ,LImA(s,t)‘= k2W2
z
From (50) and (51) it follows, t=o
(51)
using the optical theorem (42),
2 E Im A(s, 0)
a Im A(s, t) at t=o = 27(s).
(52)
Again, if au/at = Aeat, bhen (53) so that a = 2~( s), where a is defined as the inverse of the diffraction width. From our statements (49a, b) and (40a, b) it now follows that (i) da/ds > 0 for pp and K+p scattering, i.e., the diffraction peak shrinks; (ii) da/ds < 0 for fjp and K-p scattering, i.e., the diffraction peak expands. In the case of ?r*p scattering, a Im 6(s, b)/& is small as the vector meson exchange force does not occur. Equation (48) then shows that there will be very little energy dependence of a. A few remarks are worthwhile making here: 1. From Eq. (47) it follows that if du,/ds = 0, i.e., uT is constant, then 4&[1
- D(s, b)] g
= -8a
6” ~(s, b) a lmass(s’ ‘) b &.
(55)
190
ISLAM
Thus, if a Im 6(s, b)/ds is -ve, then dR/cls > 0, i.e., in the case of shrinking peaks the radius of interaction increases with energy. On the other hand, if d Im 6( s, b)/& is +ve, then dR/cls < 0, i.e., in the case of the expanding diffraction peaks, the radius of interaction decreases. These results are in agreement with optical model, where a and R are related by a = R2/4. 3. From Eq. (39) of Section III, we note that d Im 6( s, b)/& decreases in magnitude as s increases. If now the radius R tends to a constant value for large s, then from (47) and (48) it follows that the total cross section and the diffraction width tend to become constants asymptotically. V. ASYMPTOTIC
BEHAVIOR
OF
THE
VECTOR
MESON
FORCE
We have found that if the vector meson exchange is given by ~a( s)/(mv2 - t) with U(S) behaving as Cs” (0 < /3 < $5) for s -+ 00, then we can understand the shrinking and the expanding of diffraction peaks. We now want to see how such behavior of the vector meson contribution can occur. For this purpose, let us study the scattering of two equal mass spinless neutral particles and consider the diagram of Fig. 2. The invariant scattering amplitude corresponding to this diagram is dominated by a sharp p-wave resonance in the t-channel and we want’ to investigate the asymptotic behavior of this amplitude in s-channel, If we denote this invariant amplitude by A (s, t) and the corresponding double spectral function by p( s, t), then
For the moment, we postpone the question of subtraction that may be needed in (56). The absorptive parts in s- and t-channels are given by
A,(s, t) = ; j p(,,T)td"
FIG.
2. Exchange
of the
composite
vector
meson
DIFFRACTION
191
PEAKS
and (58) and in terms of these the invariant
amplitude
is
A(& t) = f / L!!!y
(59)
=- 1 - Ah’, t) ds’ 7r J d-s .
(60)
Using elastic unitarity in t-channel, the double spectral by integrals over the absorptive parts in s-channel:
function
p( s, t) is given
where the integration is carried over the region K(z, z’, 2”) = .z* + z” + zN2 2ZZ’XN - 1 > 0; here 4,(x, t) = A,(s, t) andz = 1 + 2s/(t - 4~‘). Besides (61), we have from (57) PCS, t) = ImA,(z,
(62)
t).
If any asymptotic behavior for A,(x, t) when z + 00 is assumed, then this behavior should be such that Eqs. (61) and (62) are consistent. This problem has been investigated by Gribov (18)) who argues that the asymptotic behavior of 9,(z, t) should be A,(&
t) -
x”F,(log
2)
(63)
where F,(log z) decreases faster than l/log z asymptotically tion t. In our problem, we shall assume that A&x, t) = z”F( log z)f(t)
for
x+
and can be a funcm,
(64)
where F(log x) decreases faster than l/log x asymptotically and v is a constant. In Appendix II, we shall verify that this asymptotic behavior is consistent with Eqs. ( 61) and (62). The consistency requirement gives us an equation for f(t) , namely,
Imf(t) We now f(t) :
2 2:
make the following f’(t)
p(t)
1 f(t)
p(t)
I*,
phenomenological =
=
(
t
assumption
rdt, t, -
t -
i&t
-p
-
4p.’
l/2
>* about the function
(66)
192
ISLAM
As we shall see later, t, will correspond to the position of the p-wave resonance and the parameter y will be connected with its width. y will be considered small corresponding to a narrow resonance. With the form (66) for j( tj, Eq. (65) is automatically satisfied for t near t, . The absorptive part in s-channel and the double spectral function corresponding to it are now given by
A& 1) =
&t,z”F(log f, -
t -
i&t
2) -
(67)
4~’
and p(s
7
tj
= r”d~l/t-4,2Z”F(log (6 - t)” + yyt
2)
(68)
- 4/.&Z)
for z asymptotic and t near t, . To seewhether the above equations are consistent with Eq. (57)) we insert (68) in (57) and obtain A,(s, t) = y+
j
yZ/t’ - 4/.&‘F(log z’) dt’ [(b - t’j” + yyt’ - 4/.42j](t’ - t)
(69)
where z’ = 1 + [2s/( t’ - 4~‘)]. For y small the integrand in (69) is essentially a d-function, so that z’“F( log z’j can be replaced by zrYF( log z,.), where X~ = 1 + [as/(&. - 46)]. Also, the lower limit of integration can be taken as 4~~ since s is very large. Thus, we get m y&’ - 4,~~dt A&, t) = r&&‘F(log z,> 1 7r s49 [(& - t’)” + y2(t’ - 4$jJ(t’ - t) (70) r~&‘F(log zr) = t, - t - i&twhich is Eq. (67). Let us now find the absorptive part in t-channel. Writing Eq. (58) as (71) where z’ = 1 + [2s’/(t - 4~‘)], we note that for a double spectral function like (68) the unsubtracted form (71) cannot be used, since p(s, tj behaves as x” for z + CQ(v > 0). A subtraction is needed, and for this purpose let us use an s-wave subtracted dispersion relation’ At(t, zj = A;(t)
+ A 1 P(s’, t) [-& n-
1 It is worth mentioning here that the antibound an s-wave subtraction (see appendix of ref. 11).
state
- Qdz’j] contribution
dzz’. is connected
(72) with
DIFFRACTION
193
PEAKS
For large x’, the quant,ity inside the square bracket does not go like 0( l/x’) but like 0( l/x”), so that the integral in (72) will now converge, if we use the double spectral function (68) with v < 1. Dispersion relation (72) indicates that the asymptotic behavior of the double spectral function we have constructed is such t)hat no knowledge of the s-wave part is obtained. Our primary interest is in the absorpt’ive part coming from p( s, t) given by (6s)) so that in (72) we are mainly concerned with the second term. This can be written as
r”~tr~cq7
A,(4 4 = (& - t)” + y2(t - 4$) xb? where
m x(z)
= J a s
z’“F(log
Z’)
z’-
1
- Qo&)]
(73)
dz’.
(74)
To see whether the absorptive part (73) can indeed be dominated by the p-wave amplitude for -1 5 z 5 1, let us examine the partial wave projection of the function x( 2) : Fl
xz = $4
I-1
cZzPz(x)x~x)
=- 1 m dz’z’“F(log 7 I
(75)
(1 2 1) (76)
.z’)&&Z’).
For x’ large, the integrand in (76) behaves as l/~‘~+l-” (0 < v < 1) ; consequently, if 1 > 1 the integrand is strongly suppressed. Thus, we arrive at the possibility that 1x1=1 ( >> 1xi>1 1, i.e., among all the partial waves occurring in the absorptive part (73), the only important one is 1 = 1. We can, therefore, say that the p( s, t) we have constructed gives a sharp p-wave resonance in the physical t-channel. Let us next go to the s-channel (t < 0) and examine the invariant amplitude A (s, t) arising from the absorptive part A t( t, x) given by (73). Since we are now in the crossed channel, a zero-width approximation (i.e., the limit y = 0) is appropriate in the t-channel. From (73) and (59)) it then follows
A(s,t) = rlAxc.4 t,--t *
(
x,=1+-------
t, -
2s
4p2> To obtain t,he high energy behavior of A (s, t), we have to find the asymptotic behavior of x(2,.) for x,. ---) 00. From Eq. (74), using the expansion of &o( z’) in l/z’, - x’“F(log x’) dz’ x(2,) = ; + O(l). (78) 21 z’(z’ - z,)
I‘
Here the lower limit z1 is a large but finite value and is introduced
for the con-
194
ISLAM
venience of the later discussion. The asymptotic behavior of x(x,.) is, of course, independent of xi . Using the variable {’ = (x’ - 221)/ z’, we transform the range of integration in (78) from (zl , co ) to ( -1, 1). Dropping the 0( 1) term and writing ~(2~) = @( cr), we have
(79) where
p(f) = z”F(lOg 2’)
(SOI
and
Equation
(SO) shows that for {’ near 1
, dr’) = (p”“:,,,.
(81)
We expect F( log x’) to be a continuous function of z’ when z’ --f w ; so we can assume cpl( {‘) will obey a Holder condition near and at the end {’ = 1. Further F( log z’) vanishes at z’ = 00, so that cpl( 1) = 0. The behavior of the principal value of the integral (79) for cr -+ 1 will now be given by (19)
(Y2) where vg < v and +I({) obeys a Holder condition near and at the end { = 1. We can replace +I( cr) by its limiting value (a,( 1). Going back to the original variable x,. , we get from (S2) Re x(x,)
-
const. 2:’
for
xT ---f 00.
From (77), we then obtain (83) ~(~0 < v < 1). where u(s) -CsYofors--+ This is precisely the asymptotic behavior we postulated exchange term. VI.
CONCLUDING
for the vector meson
REMARKS
We have found that the energy dependence of absorption or inelasticity Im 6(s, b) can essentially come from the vector meson exchange force. The
DIFFRACTION
195
PEAKS
reason is this force asymptotically falls off less rapidly than the other long range force, namely, that due to the exchange of the ~‘-7r antibound state ( 11) . It has been found that the vector meson cannot behave as an elementary particle; in other words, its Born amplitude cannot increase as s asymptotically. If it does, then unitarity is violated in JP and K-p scattering.’ We have examined the phenomenological behavior where the factor s in the Born term is replaced by C(S) and g(s) behaves asymptot8ically as Cs” (/3 < 1). For 0 < /3 < 35, we find d Im 6( s, b)/ds is -ve for pp and k’+~j and a Im 6(s, b)/& is +ve for flp and K-p. The change of sign occurs because the vector meson force changes sign when we go from particle to ant,iparticle scattering. The fact that absorption decreases with energy for particle scattering (d Im 6( s, b)/ds < 0) and increases for antipart,icle scattering (8 Im 6( s, b)/ds > 0) gives a shrinking peak for the former and expanding peak for the latter. In the case of r*p scattering, the vector meson exchange force is absent and as such, the energy dependence of the diff raction peaks is small. Thus, we have arrived at a simple explanation of the shrinking, nonshrinking, and expanding diffraction peaks. We have explained the phenomenological asymptotic behavior of the vector meson force in terms of a suit,ably constructed Mandelstam double spectral function. This function, given by (6s)) is such that it produces a sharp p-wave resonance in the physical region of t-channel (Fig. 2). It also produces higher partial waves, but its asymptotic behavior in s shows that bhe higher part’ial waves will be strongly suppressed. In the physical region of s-channel, the real part, of the scattering amplitude coming from this double spectral function has now t’he asymptotic behavior s’ with /3 < 1. Thus, we have a composite vector meson coming from t,he double spectral function (6s). An important result’ that comes from the present investigation can be seen if we consider the part of Im 6( s, b) that arises from the ?T-R antibound state exchange. For large s this is given by Im &(s, b) = const. Ko(b~r~). This is exactly the result’ one will arrive at using an optical model with a purely imaginary Yukawa potential (15). However, here we have arrived at this result using t,he dynamical or S-matrix met’hod. Many authors have recognized the usefulness of the optical model in describing high energy scattering (15, %$-24). It is, therefore, reassuring to find that the optical model results can be obtained using the S-matrix approach and making simple dynamical approximations. Further we note that the validity of the present theory is no longer restricted to small angles. Our model is capable of giving a real part of the scattering amplitude. The p That by other
the vector meson does not behave authors on the basis of unitarity
as an elementary (,%I, 21).
particle
has also been
argued
196
ISLAM
imaginary part of the amplitude, mainly due to absorption, comes from impact parameters b < R. For b > R, we can assume the impact parameter amplitudes are real and essentially equal to their Born amplitudes. Therefore, we can have a real part given by ReA(s,t) ~22k 2 JLrn b dbJo(b+t)AB(s, b). (84) In the forward
direction,
using (20) and (25), we get m Re A(s, 0) z -70 b dbKo(bmo) =l= yv cs@ - b dbKo( bmv) sR sR (85) = --yo to &(Rmo)
=F yv Cs” ;-
V
&(Rmv).
For x large, &(x) z ( ?r/2z)1’2eC”; therefore, Kl( RWLO)>> &( Rmv), as mv 37120. In the 10 BeV lab energy region, we can expect the first term in (85) to dominate and produce a repulsive real part (5, 6). But, asymptotically, when s -+ ~0 the second term should dominate and give us Re A( s, 0) -+ - 00 for pp and K+p and Re A( s, 0) -+ + m for $p and K-p scattering. For ap scattering, as w or cpexchange is absent, the last term does not occur. However, in this case, the asymptotic behavior of Re A (s, 0) can be controlled by other forces which we have not taken into account. Notice that, since fl in (85) lies in the range 0 < 0 < 55, the diffraction picture Re A (s, O)/Im A (s, 0) + 0 as s --) ~0 holds N SUM). Jin and MacDowell(25) have shown, using phase representa(ImA(s,O) tion, that for an asymmetric amplitude, the real part of the forward scattering amplitude (t = 0) in one of the channels (s or U) becomes negative and tends to - to asymptotically. Our considerations above indicate how this result can be understood in simple physical terms.3 Experimentally it is found that the antiparticle cross section da/at is larger than the particle cross section au/at for t = 0; but at small values of / t 1, da/at > at/at. This change of sign of &?/at - au/at or cross-over effect (2, S) can be easily explained on the basis of our discussion in Section IV. For j t ) small, we are in the diffraction region and the imaginary part of the amplitude dominates, so that .N
a5 aa 0~ Im B(s, t) - Im A(s, t>. z-z From Eqs. (41),
(43), and (46), we have ImA(s,t)
3 Many authors have high energy (26-32).
investigated
= ‘2 the real
+ ‘F part
of the forward
scattering
amplitudes
at
DIFFRACTION
where
3~ = (diffraction Im il(s,
width)-‘.
t) - ImA(s,
197
PEAKS
Therefore,
2) = ~{[~-l]-,t,[~i-T]).
Since C~ > uT and ? > 7, the quantities inside the square brackets are positive. We then expect’ a change of sign of [Im 6( s, t) - Im A( s, t)] at small values of 1 t 1, which explains the cross-over effect. In conclusion, we will like to suggest a few applications of the present approach. One of these will be the investigation of high energy large momentum transfer pp and 7r*p scattering, where experimental results are now available (33,34). The large angle scattering shows strong energy dependence. It will be interesting to set whether this dependence is conncc%ed with the vector meson force, which is also energy dependent and, being a short range force, can be important at large momentum transfers. Another application will be to the one meson exchange model with absorptive caorrection (35-38). This model has been quite successful in piou exchange case, but has failed when vector exchange reactions are involved. It, will be worthwhile to examine how the predictions of this model are changed, when the vector meson is treated not as an elementary but as a composite particle and its exchange amplitude is modified in a way similar to what has been done in t’he present’ paper. APPENDIX
I
In this appendix, we shall consider t#hc deduction of Eq. (17). The mathematical steps involved are the same as in the corresponding partial wave case (4)) but the boundary conditions are somewhat different now. So we shall briefly go through the derivation. First, we define functions
then, @(St) + a( se) = 3g( s)
(Al.l)
and @(s+) - qs-) The problem is to find purpose, the infinite cut cut { = fT = -l/s+ to l-plane if the functions
= 2ih(s).
(ST S s < m)
(A1.2)
h(s) assuming g(s) to be a known function. For this along the real axis in s-plane is transformed to a finite < = 0 by introducing a new variable 1 = -l/s. In the corresponding to +(s), g(s), and h(s) are denoted by
198
ISLAM
primed ones, i.e., +(s)
= a’(c)
etc., then the above two equations become
d(t+)
+ d(1L)
= 2g’(t)
(a’@+) - d(L) The homogeneous
equation
= 2&‘(t).
corresponding
to (A1.3)
x(L+> + xct-> and the fundamental
solutions
of this equation
can be written
(A1.4)
is
are
C2(3-J:
CA-(r - hP, (A1.3)
(rT S t 5 0)
= 0
x(T) = Cl(q, Equation
(A1.3)
c4Hs - !w’“.
as
cP’(t+)= $+qt-)
+ 2gl(t)
and its general solution is given by (AM) and
To determine the particular have to use certain boundary for s * sT
solution appropriate to our physical problem we conditions. To this purpose, we first note that A(s, 6) = AM(S)
(A1.7)
= a + ipa’
where a is the s-wave scattering length and we have neglected t,erms of the order of k2. Also, we have assumed, there are no inelastic channels open at threshold. Now, 2&3(s,b) e = 1 + 2ipA(s, b); therefore,
from (A1.7)
we get e
--4Id(s,b)
= 1 + 4p2a2 for
s + sT .
This gives Im6(s,b)
= it
Aa2
,()
ST
for
sjs
T.
4
(A1.8)
4 No/e added in proof: Similar but more careful consideration indicates Im 6(s, b)/k ishes fast,er than k. This, of course, does not change the boundary condition.
van-
DIFFRACTION
Xext we consider the case when Instead of (A1.7), we now have
A(s, b) = AL,,(S)
199
PEAKS
inelastic
channels
are open at threshold.
= a + ib + $(a + ib)’
(A1.9)
for s -+ sT and neglecting k2 t’erms; a + ib is the complex s-wave scattering From (A1.9) -4Im6(s,b)
e
=
( 1 - 2pb)2 + 4p2a2
=
1 - 4&
length.
so that Im 6(s, b) b = ~ k 6 For the fundamental we find that
solutions
~2 = c2[[/({
for
s + ST.
(A1.lO)
- {T)]1’2 and x3 = c3[{({ - j+T)]‘)l-1’2,
However, h’(P) = h(s) = Im 6(s, b)/k is finite when { -+ lT as indicated by Eqs. (A1.8) and (A1.lO). Therefore, solutions x2 and x3 are not appropriate for our problem. On the other hand, for the fundamental solution x1 = CI[(< - ~T)/d1’2 and x4 = cdl(3. - (T)]““, we find (19) h’(t)
a (( - {T)1’2-ao
for
< + [T
(a0 < %I.
If we take 010= 0, then (A1.S) is satisfied. If we choose 010sufficiently close to the value +2/, then the phomomenological behavior indicated by (A1.lO) can also be satisfied. Thus, the solutions ~1 and x4 are only relevant for our problem. For X( [) = xl(T), h’( <) m 1/<1’2 when C --, 0; on the other hand, for x( {) = o(o < 35 ) when [ + 0. Now, h’( {) = Im 6(s, b)/k and in X4(b) m-1 cx i- 1’2--oro( writing down Eq. ( 11) for 8( s, b), we have considered that Im 6( s, b)/k --+ 0 when s -+ ~0. Therefore, h’( <) -+ 0 when { + 0. Thus, the fundamental solution x4 is only acceptable. Now, for s --$ 0
e(s, b)_1s 16
m Im 6(s’, b) C~S’= const lr ST k’s’
Therefore, a’({)
= O(s, b)/k
= const.
when
{ -+ 00.
(Al.ll)
From Eq. (A1.5) with x( {) = x4({), we find that for r + Q) , the first term on the right-hand side behaves like constant; however, the second term behaves like cl+’ for j 2 0. This behavior violates (Al.11). Therefore, we must have P,(l) = 0.
200
ISLAM
Thus, the solution we finally obtain is
sO IT
2g’(t)
dt
Ltct - $+‘T)l”2(t- i-1
(Al.12)
and
g’(t)dt r-rP2(t- i-1*
(A1.13)
Transforming to the variable s, we get
e(s, b)=(s- sTp2 sk
ai
Im 6(s, b) =k
(s -YZ
8T
A(s’, b) ds’ k’(s’ - s,)“Z(S’ - s) ’
and A(& b) ds’ p SW k’(s’ - ST)l’2(S’ - s) ST
APPENDIX
II
In this appendix we want to seewhether the asymptotic behavior of the s-channel absorptive part, A,(z, t) = x’F(log z)f(t)
(Y > 0)
(A2.1)
is consistent with the equations
and p(s,
t)
= z”F(logx)
Imf(t).
(A2.3)
The region of integration in (A2.2) is shown in Fig. 3: For z and zN fixed, the range of integration for x’ is xo 5 z’ < .zzn - 2/(x2 - l)(~“~ - 1) and so forth. Examining the integrand of (A2.2)) we find that it becomes large-(i) when x’ and zNhave values near the boundary line K = 0, and (ii) becauseof the assumed asymptotic behavior (A2.1)) when one of them is very large (correspondingly, the other one would be small). Let us begin with a very large value of z and define a parameter X such that if x = x/X, then 1 << 2 << 2. We can now consider that the double spectral function is given by
DIFFRACTION
201
PEAKS
(A2.4) CIXnA,*(XN,
t)
*a zw-&=i)
A&‘,
Cl2
.s zcz-&cl,
(2,’
t>
z’)l/z(xb’
-
-
~‘)‘/Z
where
&
= 2.2’ f
&
= XX” f 1/(9
FIG.
&2*
3. Integration
-
l)(z’2 -
l)(P
region
1) 5s x(x’ f -
of the
1) w 2(/
double
&‘,
-
f dp=T).
spectral
function
1)
(A2.5) (A2.6)
202
ISLAM
The regions of integrations in (A2.4) are shown in Fig. 3 by the shaded area. For the integration over zn in the first term and that over x’ in the second term of (A2.4)) we can use the asymptotic formula for the absorptive part, since the integrations are on large values of the variables. Further, we may assume that the logarithmic terms F(log z”) and F(log z’) in these integrations can be replaced by F( log x), because x is very large. Thus, we obtain
SW-~)
dz’ zlv
../ zb-d/22--1)
(2,
-
~‘)‘/Z
(Zb’
-
2’)1/2
The inner integrations in (A2.7) can be easily done in the following Let q = X: - 4x2 - 1, qa,b = 2: ,b/z = zn f dzN2 - 1. Then,
way.
(A2.S)
where w = vb/qa . NOW, q/7]b = (x” $- dzNa - 1)/(x + dx2 - 1) << 1, because x0 $ zn 5 x/X and x/X << x. Therefore, the lower limit of integration in (A2.8) can be taken to be zero and we obtain (99)
From Eq. (A2.7)
we now get
PCS, 0 = -1 t - 4p2 1’2 &-) x%(log 7r
x) f*(t)
“’ x’F(log
/“” cZz’As(x’, t)&v(z’) 20 z) f(t)
s”” &“A,*(/, 20
(A2.10) t>&y(z”).
For large x’, Qy(x’) - I/X”+’ and A,(z’, t) - z”F(log x’) so that the integrals in (A2.10) will converge for z --+ 00, if F( log x’) vanishes faster than l/log x’.
DIFFRACTION
203
PEAKS
This, of course, we have already as sumed. For x asymptotic, written as5 PCS, t)
= x”F(lw .w*uMt)
p( s, t) can now be
+ fu>P*u)l
(A2.11)
where (A2.12) Thus, (A2.11)
and (A2.3)
are consistent
Imf(t)
if
= f*(Mt)
We may expect the main contribution large values of x’, so that
that is, cp(t) E cp( t)f( t) where p(t) tion (A2.13) now becomes Im.f(t)
of the integral
= [(t - 4~‘)/t]l’~
in (A2.12)
to come from
and c is a constant. Equa-
E 2cp(t)lf(t)l!
If the constant factor 2c is absorbed in f(t), (A2.13) can be expressed as Imf(t)
(A2.13)
+ f(t)cp*(t).
then the consistency condition iA2.15)
22 p(t)lf(t)12.
ACKNOWLEDGMENT The Feldman
author wishes to thank for his interest.
RECEIVED:
Dr.
Y. S. Jin
for
stimulating
conversations
and
Prof.
D.
June 3,1966 REFERENCES
1. S. J. LINDENB~UM, Review Lecture, Oxford International Conference Particles (1965). 2. R. J. N. PHILLIPS .IND W. RIRIT~, Phys. Rev. 139, B1336 (1965). 3. T. 0. BINFORD AND B. R. DES.41, Phys. Rev. 138, B1167 (1965). 4. M. M. ISLAM AND KYUNGSIH KING, Phys. Rev. 139,B973 (1965). 5. M. M. ISL.ZM, Phys. Rev. 141, 1524 (1966). 6. M. M. ISLAM, Phys. Rev., 147, 1144 (1966). 7. R. BLMVKENBECLER.\ND M. L. GOLDBEKGER, Phys. Rev. 126,766 (1962). 8. W. N. C~TTINGHAM AND R. F. PEIERLS, Phys. Rev. 137, B147 (1965). 9. T. ADSCHI BND T. KOT.\NI, SUppt. P?Og?. Theoret. Phys. (Commemoration (1965). 5 Eyuation uses somewhat
(A2.11) is exactly different method
the same as that obtained by Gribov to evaluate t,he integrals asymptotically.
(18).
on
Elementary
Issue),
316
However,
he
204
ISLAM
10. 11. 12. IS.
E. M. J. S. 14. G. 1.5. R. 16’. F. 17. J. 18. V. 19. N.
PREDAZZI, Ann. Phys. (N. Y.) 36,228 (1966). M. ISLAM, Phys. Rev., 147, 1148 (1966). J. SAKURAI, Ann. Phys. (N. Y.) 11, 1 (1960). C. FRAUTSCHI, M. GELL-MANN, AND F. ZACHARIASEN, Phys. Rev. 126,2204 (1962). F. CHEW, S. C. FRAUTSCHI, AND S. MANDELSTAM, Phys. Rev. 126,121X? (1962). SERBER, Rev. Mod. Phys. 36, 649 (1964); Phys. Rev. Letters 10, 357 (1963). G. TRICOMI, “Integral Equations,” p. 182. Interscience, New York, 1957. HAMILTON AND W. S. WOOLCOCK, Rev. Mod. Phys. 36,737 (1963). N. GRIBOV, NucZ. Phys. 22, 249 (1961). I. MUSKHELISHVILI, “Singular Integral Equations,” Chap. 4. Noordhoof, Groningen, Holland, 1953. 20. P. G. 0. FREUND AND R. OEHME, Phys. Rev. Letters 10,199, 315(E) (1963). 81. K. YAMAMOTO, Phys. Letters 6,355 (1963). 22. Y. NAMBU AND M. SUGAWARA, Phys. Rev. Letters 10, 304 (1963); Phys. Rev. 132, 2724 (1963). 23. N. BYERS AND C. N. YANG, Phys. Rev. 142,976 (1966). 2.4. R. C. ARNOLD, ‘LAn Optical Potential for High Energy Physics: Theory and Applications,” Preprint, Argonne National Laboratory (1966). 25. Y. S. JIN AND S. W. MACDOWELL, Phys. Rev. 138, B1279 (1966). 26. P. SODING, Phys. Letters 8,285 (1964). 27. I. I. LEVINTOV AND G. M. ADELSON-VELSKY, Phys. Letters 13,185 (1964). 28. B. LAUTRUP, P. M$LLER NIELSEN, AND P. OLESEN, Phys. Rev. 140, B984 (1965). 29. G. H~~HLER, G. EBEL, AND J. GIESECKE, 2. Physik 180,430 (1964). 30. V. S. BARASHENKOV AND V. I. DEDJU, Nucl. Phys. 64,636 (1965). 31. H. BANERJEE, Phys. Rev. 140, B989 (1965). 3%. A. BIALAS AND E. BIALAS, Nuovo Cimento 37,1686 (1965). 33. G. COCCONI, V. T. COCCONI, A. D. KRISCH, J. OREAR, R. RUBINSTEIN, D. B. SNARL, B. T. ULRICH, W. F. BAKER, E. W. JENKINS, AND A. L. READ, Phys. Rev. 138, B165 (1965). 34. J. OREAR, R. RUBINSTEIN, D. B. SCARL, D. H. WHITE, A. D. KRISCH, W. R. FRISKEN, A. L. READ, AND H. RUDERMAN, Phys. Rev. Letters 16,309 (1965). 35. J. D. JACKSON, Rev. Mod. Phys. 37,484 (1966). 36. L. DURAND III AND Y. T. CHIU, Phys. Rev. 139, B646 (1965). 37. J. D. JACKSON, J. T. DONOHUE, K. GOTTFRIED, R. KEYSER, AND B. E. Y. SVENSSON, Phys. Rev. 139, B428 (1965). 38. V. BARGER AND M. EBEL, Phys. Rev. 138, B1148 (1965). 39. “Tables of Integral Transform,” A. Erdelyi, ed., Vol. I, p. 310. McGraw-Hill, New York,
1954.