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Straight-line paths approximation for studying high-energy elastic and inelastic hadron collisions in quantum field theory
Volume 33B, number 7
PHYSICS LETTERS
7 December 1970
STRAIGHT-LINE PATHS APPROXIMATION FOR HIGH-ENERGY ELASTIC AND INELASTIC HADRON IN Q U A N T U M FIELD THEORY B. M. BARBASHOV,
S. P. KULESHOV, V. A. MATVEEV, A. N. SISSAKIAN and A, N. TAVKHELIDZE
STUDYING COLLISIONS
V.N. PERVUSHIN,
Joint Institute for Nuclear Research, Dubna, USSR Received 21 August 1970
The problem of the asymptotic behaviour of high-energy elastic and inelastic amplitudes is studied by means of the functional methods of quantum field theory. The straight-line paths approximation (SLPA), making it possible to effectively calculate the functional integrats which arise, is formulated.
In a n u m b e r of r e c e n t p a p e r s [1-11] the p r o b l e m of the validity of the eikonal a p p r o x i m a t i o n for the t w o - p a r t i c l e e l a s t i c s c a t t e r i n g amplitude at high e n e r g i e s was c o n s i d e r e d in the f r a m e w o r k of v a r i o u s m o d e l s of quantum field theory. In t h e s e p a p e r s the a s y m p t o t i c b e h a v i o u r of the sum of the F e y n m a n d i a g r a m s of the l a d d e r type (when all the p o s s i b l e c r o s s i n g s of " m e s o n " lines between two " n u c l e o n s " a r e taken into account) was e s s e n t i a l l y i n v e s t i g a t e d in the l i m i t of high e n e r g i e s and fixed m o m e n t u m t r a n s f e r s . One of the i m p o r t a n t r e s u l t s of these i n v e s t i g a t i o n s is the fact that the p r i n c i p l e l o g a r i t h m i c t e r m s c a n c e l in the sum of the ladder type d i a g r a m s in the a s y m p t o t i c l i m i t S ~ 0% t = fixed. F u r t h e r , the sum of the ladder type d i a g r a m s tends a s y m p t o t i c a l l y to the sum of quasipotential graphs f o r the t w o - p a r t i c l e s c a t t e r i n g amplitude [12-13] and c o in c i d es with the G l a u b e r - t y p e eikonal expansion of the s c a t t e r i n g amplitude at high e n e r g i e s and s m a l l angles. In ref. [1] the functional i n t e g r a t i o n methods in quantum field theory w e r e used in studying this p r o b l e m . As has been shown in these p a p e r s the functional i n t e g r a t i o n methods p r e s e n t an e f f e c t i v e tool for i n v e s t i g a t i n g the a s y m p t o t i c b e h a v i o u r of the s c a t t e r i n g amplitudes. In subsequent p a p e r s [14,15] the functional i n t e g r a t i o n methods have been used for studying the i m p o r t a n t p r o b l e m of r a d i a t i v e c o r r e c t i o n s to the ladder type graphs f o r t w o - p a r t i c l e e l a s t i c s c a t t e r i n g (see a l s o p a p e r s [16,17]) and in i n v e s t i g a t i n g the i n e l a s t i c s c a t t e r i n g p r o c e s s e s . In this l e t t e r we p r e s e n t some r e s u l t s of i n v e s t i g a t i o n of the model of s c a l a r nucleons i n t e r a c t i n g with neu t r al v e c t o r m e s o n s [11]. We find closed analytic e x p r e s s i o n s for the t w o - n u cl eo n e l a s t i c s c a t t e r ing amplitude and f o r the a m p l i t u d e s of i n e l a s t i c p r o c e s s e s of m eso n production in nucleon c o l l i s i o n s . The a s y m p t o t i c b eh a v i o u r of th e s e a m p l i t u d e s in the high energy l i m i t is studied in the f r a m e w o r k of the s t r a i g h t - l i n e paths approximation (SLPA) f o r m u l a t e d below. It is shown that the p r i n c i p a l l o g a r i t h m i c t e r m s in the s c a t t e r i n g a m p l i t u d e s cancel in the a s y m p t o t i c l i m i t S ~ ~o, t - fixed when the d i a g r a m s with nucleon closed loops a r e neglected. It is shown then that the contributions of the r a d i a t i v e c o r r e c t i o n s to the l a d d e r type graphs in the s t r a i g h t - l i n e paths a p p r o x i m a t i o n a r e f a c t o r i z e d and a r e d e t e r m i n e d by the quantity H(t), which depends only upon the s q u a r e of m o m e n t u m t r a n s f e r s . In the r e g i o n [t I << m2 the quantity H(t) has an exponential dependence on t and p r o d u c e s the d i f f r act i o n peak in e l a s t i c s c a t t e r i n g in a c c o r d a n c e with the hypothesis of s m o o t h n e s s of the l oc al quasipotential of two p a r t i c l e s at high e n e r g i e s [18-20]. Such a b eh av i o u r of the e l a s t i c s c a t t e r i n g amplitude was p r e d i c t e d r e c e n t l y in p ap er [21] and c o r r e s p o n d s , in s o m e s e n s e , to the c o h e r e n t i n t e r a c t i o n of v i r t u a l m e s o n s which belong to the ni~cleon clouds. F u r t h e r the d i f f e r e n t i a l c r o s s s e c t i o n of i n e l a s t i c p r o c e s s e s w e r e obtained. Under the r e q u i r e m e n t of " s o f t n e s s " of s e c o n d a r y m e s o n s the P o i s s o n d i s t r i b u t i o n s in the n u m b e r of p a r t i c l e s e m i t t e d in c o l l i s i o n is found. We note that the total d i f f e r e n t i a l c r o s s s e c t i o n s u m m e d o v e r all the s e c o n d a r y m e s o n s may have, g e n e r a l l y speaking, no pronounced d i f f r a c t i o n peak in the r e g i o n tx2 ~ It l <<: m 2. 484
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Such a b e h a v i o u r is due to the c a n c e l l a t i o n under c e r t a i n conditions of two exponential f a c t o r s , which c o r r e s p o n d to the r a d i a t i v e c o r r e c t i o n c o n t r i b u t i o n s in the e l a s t i c s c a t t e r i n g a m p l i t u d e and to the total c o n t r i b u t i o n s f r o m m u l t i - m e s o n production. We would like to point out the analogy of such a r e g u l a r i t y with the a u t o m o d e l b e h a v i o u r [22] of deep i n e l a s t i c p r o c e s s e s of high e n e r g y hadron i n t e r a c t i o n . We choose an i n t e r a c t i o n L a g r a n g i a n of the f o l lowing f o r m Lin t :
g:t,h2(x)i'~h(x)A a
(x): + g2:A2
(x)~/*(x)tp(x):
(1)
w h e r e g is s o m e d i m e n s i o n l e s s coupling constant. The o n e - p a r t i c l e G r e e n function of the quantum field in the e x t e r n a l field {[iOa
+ gA a(x)] 2 - m2 } G(x,yl A )
Aa(x)
s a t i s f i e s the equation
=-64(x-y)
(2)
The f o r m a l solution of eq. (2) for the F o u r i e r t r a n s f o r m of the G r e e n function can be r e p r e s e n t e d by m e a n s of the functional i n t e g r a l ac
G(p,q[A)
: i f dz e x p { i r ( p 2 - m 2 ) } f ( d4y 2 ~ exp
{i(p-q)y}
f[64v~0
o
(3) x e x p { 2 i g f d~ [va(~) +Pa]Aa[y+2p~ +2 0
where
T2 "~2 [54v]r 1 : 6 4 v
~'2
v2(~)d~}/f64v
exp{-if
v(~)dr/]} 0
exp(-if
(4)
v2(~)d~}
~'1 rl is a n o r m a l i z e d volume e l e m e n t of the functional s p a c e of the f o u r - d i m e n s i o n a l function va(zl) d e t e r mined on the i n t e r v a l ~-1 "< ~ ~< T2" The t w o - p a r t i c l e e l a s t i c s c a t t e r i n g a m p l i t u d e is defined a s a vacuum a v e r a g e of the p r o d u c t of two G r e e n functions [11]. Below, for s i m p l i c i t y , we s h a l l n e g l e c t the vacuum p o l a r i z a t i o n e f f e c t s a s well a s the d i a g r a m s with c l o s e d nucleon loops. The p r o p a g a t o r of the f r e e v e c t o r field @aft is d e t e r m i n e d by the e x p r e s s i o n
cba~ = 5aft _ ~:aKfl/~2/(K2_
~2)
(5)
Taking into account eq. (3) and eq. (5) we obtain the following c l o s e d e x p r e s s i o n for the t w o - p a r t i c l e scattering amplitude
if(pl,P2;
qlq2 )
×[2v2(0) +P2
= g2 f d4yexp{ y(p l _ q l ) } @ a f t ( y ) f [54Vl]~-oo[54v2]~-,,o[2Vl(0)
+q2]flf
d• exp ½ig2 fd4K@y~(K)
i=~1j(C)(K;
+Pl +ql]a
pi,qi]vi)j~ ) ( - g ; pi,qilvi)
(6)
0 + 2~exp{iKy}j(yl)(K;
pl,ql]Ul)j~2)(-K; p2,q21v2)]-
i?
6m2d~ 1
--o(3
where
j(~) (g;
pi,qilvi)
: 2i f d~[viy(~)
+PiyO(~) +qiyo(-~)] exp{2iKy[pi~O(~ ) +qiy~O(-4) +f
viT0?)d~]} (7) 0 is a c o n s e r v i n g t r a n s i t i o n c u r r e n t . Obviously, an e x a c t functional i n t e g r a t i o n in e x p r e s s i o n (2.9) is not p o s s i b l e . T h e r e f o r e , we use below the a p p r o x i m a t e method of c a l c u l a t i n g the i n t e g r a l s o v e r v1 and v2 [23,24], c a l l e d by us SLPA. The functional v a r i a b l e s v 1 and v2 f o r m a l l y i n t r o d u c e d in eq. (3) f o r obtaining the solution f o r the G r e e n function, d e s c r i b e the deviation of a p a r t i c l e t r a j e c t o r y f r o m the s t r a i g h t - l i n e paths. In f a c t , if we put v = 0 in f o r m u l a (7) for the t r a n s i t i o n c u r r e n t , we would obtain In eq. (6) the mass renormalization ~00 = m2 + 5m2 has been made. It removes divergences appearing in the integration over the variables 41 and 42 [14]. 485
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the c l a s s i c a l c u r r e n t of the n u c l e o n , m o v i n g with m o m e n t u m p at ~ . 0 and with m o m e n t u m q at ~ < 0. We note. h o w e v e r , that the a p p r o x i m a t i o n u = 0 is known to be i n a p p l i c a b l e at v a l u e s of the p r o p e r t i m e of the p a r t i c l e c l o s e to z e r o , when the p a r t i c l e c l a s s i c a l t r a j e c t o r y c h a n g e s its d i r e c t i o n . In the l a n g u a g e of F e y n m a n g r a p h s the a p p r o x i m a t i o n m e a n s that the q u a d r a t i c ~ - d e p e n d e n c e in the n u c l e o n p r o p a g a t o r is n e g l e c t e d . It can lead, g e n e r a l l y s p e a k i n g , to the a p p e a r e n c e of d i v e r g e n c e s of the i n t e g r a l s o v e r d4K at the u p p e r l i m i t . A b e t t e r a p p r o x i m a t i o n is given by a v e r a g i n g the n u c l e o n c u r r e n t (7) o v e r the f u n c t i o n a l v a r i a b l e u, i.e.
___
( 2pT+K7
_ _ 2 q 7 - K~__] j7(K; p . q ) = - "2Kp+K2~-i£ - 2Kq- K2 - i < /
(8)
F o r this r e a s o n , in s e e k i n g the t w o - p a r t i c l e e l a s t i c s c a t t e r i n g a m p l i t u d e we s h a l l use the SLPA, which c o n s i s t s in s u b s t i t u t i n g in the e x p o n e n t i a l exponent in eq. (6) the c u r r e n t p r o d u c t a v e r a g e d o v e r the f u n c t i o n a l v a r i a b l e s u 1 and u 2 [11]. T h u s . in SLPA the e x p r e s s i o n for the e l a s t i c s c a t t e r i n g a m p l i t u d e t a k e s the f o r m 1 if(pl,P2; q l , q 2 ) = g 2 H ( t ) jd4yexp{iy(p 1 - q l ) } A ( y ; p l , q l ; p 2 , q 2 ) f dXexp{iXX(y; P l , P 2 ; q l , q 2 ) } (9) 0 where A(y; P l , q l ;
P2,q2) = fd4KQ)c~fi(K) [K + /51 + ql]cr [-K + P2 + q2]fi exp {iKy}
X(Y; P l ' P 2 ; q l , q2 )
g2 4 - (2~)
fd4K
exp{iKy} (DOt~(K)j(~)~; i l , q i ) j ~ ) ( - K ;
r.2 .......... ¢K;
H(I)
= exp i
g2
2(2n) 2
.
Di,qi)J
f d4KC-Dc~fl(K)i~,=l j(~)
"=
.
.
.
.
.
(10)
p 2 ,q 2 ;
(11)
.
~:; Pi,qi)
(12)
It is i n t e r e s t i n g to note that the c o n t r i b u t i o n of the r a d i a t i v e c o r r e c t i o n is f a c t o r i z e d in a f o r m of the f a c t o r H(t) d e p e n d i n g only on the s q u a r e of the m o m e n t u m t r a n s f e r t = (Pl - q l )~, as in the s c a l a r field i n t e r a c t i o n m o d e l [14]. The a n a l o g o u s p h e n o m e n a of the f a c t o r i z a t i o n of the r a d i a t i v e c o r r e c t i o n c o n t r i b u t i o n in q u a n t u m e l e c t r o d y n a m i c s was found in the a r t i c I e s [25-27]. In the high e n e r g y l i m i t S -* ~ at fixed m o m e n t u m t r a n s f e r s t l i m i t e d by the c o n d i t i o n [t ( << m 2 the e x p r e s s i o n for the e l a s t i c s c a t t e r i n g a m p l i t u d e has the f o r m 1'
f(s,
t) : i(s
- u) v(t)
exp
(at)
(13)
where v(t) : ½
f d2y±
I~Ko(~Iy.])
exp (iy± A±) (exp l - " g2
a = { g 2 / 3 ( 2 n ) 2 m 2 } { I n ( m 2 / i a2) + ½}
f - 1) "
t ~ _A2
.L
(14)
(15)
and K 0 is the K e l v i n f u n c t i o n of z e r o o r d e r . The a m p l i t u d e s of i n e l a s t i c p r o c e s s e s of m e s o n p r o d u c t i o n in t w o - n u c l e o n c o l l i s i o n s [11] at high e n e r g i e s can be d e t e r m i n e d by m e a n s of a g e n e r a t i n g f u n c t i o n f(pl,P2; ql,q2 IA ext) h a v i n g a m e a n i n g of the s c a t t e r i n g a m p l i t u d e of two n u c l e o n s in the p r e s e n c e of the e x t e r n a l f i e l d A ext. In what follows we will c o n s i d e r the c a s e in which the m o m e n t a of the s e c o n d a r y m e s o n in the c e n t e r of m a s s s y s t e m s a t i s f y the r e q u i r e m e n t of " s o f t n e s s " [15]:
J We note that taking into account the identity of nucleons reads on symmetrization of eq. (13) to terms vanishing in the limit S ~ oo with fixed t. 486
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~s-=
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N
Koi
IS i=1 xi±[ ]Pl±-ql±]~
1;
[P2±-q21J
(16)
w h e r e the m o m e n t a of the initial nucleons a r e chosen along the Z - a x i s . Under this r e q u i r e m e n t the a m p l i t u d e of N - m e s o n p r o d u c t i o n is f a c t o r i z e d and can be w r i t t e n in the following f o r m : N f i n e l (N) = f ( P l , P 2 ;
ql ' q2)
i-~l
.(1)/K gE~(tci) [:oe ' i;. P l , q l ) +
~(2)(,~txz.*, P2 , q2 )] aOg
(17)
w h e r e Eot(g ) is the p o l a r i z a t i o n v e c t o r of a m e s o n with the m o m e n t u m t(. We find a l s o the a s y m p t o t i c e x p r e s s i o n for d i f f e r e n t i a l c r o s s s e c t i o n s of "soft" meson p r o d u c t i o n . when the m e s o n m o m e n t a s a t i s f y eq. (16). It has the f o r m : d(~ )n
1 V2 (l) U'n l(S, l) U'n 2(S, l) 1 ,r12 S -+ m 47r t-fixed<,.m2
(1 {3)
where 1
Wn(S,t ) --.~.. e x p { 2 a t } f
~ d × i - - ~ ij~)(Ki; pl, ql) l 2 ~p i=1 2Koi (2~r)3
(19)v
The i n t e g r a t i o n r e g i o n ~p o v e r s e c o n d a r y m e s o n m o m e n t a is d e t e r m i n e d by the condition n
-t<2p
~
Ki -
(A_ ~
i=1
Ki)2_
where
nl t=A2
;
(20)
i-1
A = (ql-Pl
- ~
n2 gi) = ( q 2 - P 2 -
~
K~)
C o n s i d e r now an a p p r o x i m a t i o n , in which the total momentum of the s e c o n d a r y m e s o n s can be n e g l e c t e d in a c c o r d a n c e with the r e q u i r e m e n t of " s o f t n e s s " (16). In this a p p r o x i m a t i o n eq. (19) t a k e s the f o r m of the P o i s s o n d i s t r i b u t i o n s 1 exp {2at} [n(s,t)] n Wn(S , t) : nT.
(21)
w h e r e the quantity J n ( s , t ) =-(2g~)~ f d ~
Ij(I)(K; pl,ql)]2
(22)
is the a v e r a g e n u m b e r of secondar}, p a r t i c l e s p r o d u c e d in the two-nucleon c o l l i s i o n at S - ' ~ and fixed t. Using eq. (8) for j ~ , we find f o r t t[ .< m 2 that
if(s, t) = - bt
(23)
The p a r a m e t e r b d e p e n d s , in g e n e r a l , on a special__ f o r m of cut-off of the i n t e g r a l in eo. (22) o v e r the m e s o n momentum. In the p a r t i c u l a r c a s e when R~ ~ m 2, 1 >>~2 >> t~2/m2, ln(m2/~t2) >., In (1/ce) 2 w h e r e c~ : RZ/Po we get
b = {2g2/3(2v)2m 2} ( In ( m 2 / p 2) + ½}
(24)
which c o i n c i d e s with the double slope p a r a m e t e r of the d i f f r a c t i o n exponential (15). Notice that the equality 2 a = b is true a l s o in the i n f r a r e d a s y m p t o t i c l i m i t g ~ 0. F o r this c a s e , a f t e r s u m m i n g in eq. (19) o v e r the n u m b e r of s e c o n d a r y m e s o n s , we find that the dependence on the v a r i a b l e t c a n c e l s and the d i f f r a c t i o n p e a k in the total d i f f e r e n t i a l c r o s s s e c t i o n d i s a p p e a r s. $ The i n t e g r a t i o n r e g i o n in eq. (22) is e f f e c t i v e l y l i m i t e d by ] K Z I "-< RZ, ] ~ £ ] "-< R±. 487
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T h i s r e g u l a r i t y w a s m e n t i o n e d in p a p e r [20] and i s in a n a l o g y w i t h the a u t o m o d e l b e h a v i o u r of d e e p i n e l a s t i c p r o c e s s e s of h a d r o n i n t e r a c t i o n s at h i g h e n e r g y [22]. The s t r a i g h t - l i n e p a t h s a p p r o x i m a t i o n , u s e d in t h i s w o r k c o r r e s p o n d s to a p h y s i c a l p i c t u r e in w h i c h c o l l i d i n g h i g h e n e r g y n u c l e o n s at the a c t of i n t e r a c t i o n r e c e i v e a s m a l l r e c o i l c o n n e c t e d w i t h the e m i s s i o n of " s o f t " m e s o n s and r e t a i n t h e i r i n d i v i d u a l i t y . T h e a u t h o r s e x p r e s s t h e i r d e e p g r a t i t u d e to P r o f s . N. N. B o g o l u b o v , D. I. B l o k h i n t s e v and A. A. L o g u n o v for stimulating discussions.
References [1] B. M. Barbashov, S. P. Kuleshov, V. A. Matveev and A. N. Sissakian, P r e p r i n t JINR E2-4692, Dubna {1969); T.M.P. 3 {1970) 342. [2] H . D . I . A b a r b a n e l and C.Itzykson, Phys. Rev. L e t t e r s 23 (1969) 53. [3] M. Levy a n d J . Sucher, Phys. Rev. 186 (1969) 1656. [4] I, V. Andreev, Zh. Eksp. i Teor. Fiz. 58 (1970) 257. [5] V.N. Pervushin, T.M.P. 4 (1970) 28. [6] H. Cheng and T . T . W u , p r e p r i n t MIT {1969). [7] S . J . C h a n g a n d S.K. Ma, Phys. Rev. 188 {1969) 2378. [8] F. Englert, P. Nico[etopoulos, R. Brout and C. Truffin, Nuovo Cimento 64 (1969) 561. [9] B. M. Barbashov and V. V. Nesterenko, p r e p r i n t JINR P2-4900, Dubna (1970). [10] G. Tiktopoulos and S. B. Treiman, p r e p r i n t Princeton University (1970). [11] B M. Barbashov, S~ P. Ku/eshov, V. A. Matveev, V.N. Pervushin, A.N. Sissakian and A. N. Tavkhe[idze, p r e p r i n t JINRE2-5217, Dubna (i970). [12] A. A. Logunov and A N. Tavkhelidze, Nuovo Cimento 29 (1963) 380. [13] V. R. Garsevanishvili, V.A. Matveev, L.A. S[epchenko and A. N. Tavkhe[idze, Phys. L e t t e r s 29B (1969) 191. [14] B. M. Barbashov, S. P. Ku[eshov, V.A.Matveev and A. N. Sissakian, p r e p r i n t JINR E2-4983, Dubna (1970). [15] B. M. Barbashov, S . P . Kuleshov, V. N. Pervushin and A N. Sissakian, p r e p r i n t JINR E2-4955, Dubna (1970). [16] S. J. Chang, p r e p r i n t University of Illinois {1969). [17] Y . P . Y a o , p r e p r i n t University of Michigan (1969). [i8] S. P. Alliluyev, S.S. Gershtein and A. A. Logunov, Phys. L e t t e r s 18 (1965) 195. [19] D.l.B[okhintsev, Nucl. Phys. 31 (1962) 628. [20] O. A. Khrustalev, V I. Savrin and N. Ye. Tyurin, p r e p r i n t JINR E2-4479, Dubna (1969). [21] V. A. Matveev and A. N. Tavkhelidze, p r e p r i n t JINR E2-5141, Dubna {1970). [22] V. A. Matveev, R. M. Muradyan and A.N..Tavkhe[idze, p r e p r i n t JINR P2-4578, Dubna (1969). [23] B . M . B a r b a s h o v , Zh. Eksp. i T e o r . Fiz. 48 (1965) 607. [24] E.S. Fradkin, Nuel. Phys. 76 (1966) 588. [25] D.R. Yennie, S.C. F r a u t s c h i and H. Suura, Ann. Phys. 13 (1961) 379. [26] K . E . E r i c s s o n , Nuovo C i m e n t o l 9 (1961) 1010. [27] L.D. Soloviev, Nuel. Phys. 64 (1965) 657.
488
PHYSICS LETTERS
7 December 1970
STRAIGHT-LINE PATHS APPROXIMATION FOR HIGH-ENERGY ELASTIC AND INELASTIC HADRON IN Q U A N T U M FIELD THEORY B. M. BARBASHOV,
S. P. KULESHOV, V. A. MATVEEV, A. N. SISSAKIAN and A, N. TAVKHELIDZE
STUDYING COLLISIONS
V.N. PERVUSHIN,
Joint Institute for Nuclear Research, Dubna, USSR Received 21 August 1970
The problem of the asymptotic behaviour of high-energy elastic and inelastic amplitudes is studied by means of the functional methods of quantum field theory. The straight-line paths approximation (SLPA), making it possible to effectively calculate the functional integrats which arise, is formulated.
In a n u m b e r of r e c e n t p a p e r s [1-11] the p r o b l e m of the validity of the eikonal a p p r o x i m a t i o n for the t w o - p a r t i c l e e l a s t i c s c a t t e r i n g amplitude at high e n e r g i e s was c o n s i d e r e d in the f r a m e w o r k of v a r i o u s m o d e l s of quantum field theory. In t h e s e p a p e r s the a s y m p t o t i c b e h a v i o u r of the sum of the F e y n m a n d i a g r a m s of the l a d d e r type (when all the p o s s i b l e c r o s s i n g s of " m e s o n " lines between two " n u c l e o n s " a r e taken into account) was e s s e n t i a l l y i n v e s t i g a t e d in the l i m i t of high e n e r g i e s and fixed m o m e n t u m t r a n s f e r s . One of the i m p o r t a n t r e s u l t s of these i n v e s t i g a t i o n s is the fact that the p r i n c i p l e l o g a r i t h m i c t e r m s c a n c e l in the sum of the ladder type d i a g r a m s in the a s y m p t o t i c l i m i t S ~ 0% t = fixed. F u r t h e r , the sum of the ladder type d i a g r a m s tends a s y m p t o t i c a l l y to the sum of quasipotential graphs f o r the t w o - p a r t i c l e s c a t t e r i n g amplitude [12-13] and c o in c i d es with the G l a u b e r - t y p e eikonal expansion of the s c a t t e r i n g amplitude at high e n e r g i e s and s m a l l angles. In ref. [1] the functional i n t e g r a t i o n methods in quantum field theory w e r e used in studying this p r o b l e m . As has been shown in these p a p e r s the functional i n t e g r a t i o n methods p r e s e n t an e f f e c t i v e tool for i n v e s t i g a t i n g the a s y m p t o t i c b e h a v i o u r of the s c a t t e r i n g amplitudes. In subsequent p a p e r s [14,15] the functional i n t e g r a t i o n methods have been used for studying the i m p o r t a n t p r o b l e m of r a d i a t i v e c o r r e c t i o n s to the ladder type graphs f o r t w o - p a r t i c l e e l a s t i c s c a t t e r i n g (see a l s o p a p e r s [16,17]) and in i n v e s t i g a t i n g the i n e l a s t i c s c a t t e r i n g p r o c e s s e s . In this l e t t e r we p r e s e n t some r e s u l t s of i n v e s t i g a t i o n of the model of s c a l a r nucleons i n t e r a c t i n g with neu t r al v e c t o r m e s o n s [11]. We find closed analytic e x p r e s s i o n s for the t w o - n u cl eo n e l a s t i c s c a t t e r ing amplitude and f o r the a m p l i t u d e s of i n e l a s t i c p r o c e s s e s of m eso n production in nucleon c o l l i s i o n s . The a s y m p t o t i c b eh a v i o u r of th e s e a m p l i t u d e s in the high energy l i m i t is studied in the f r a m e w o r k of the s t r a i g h t - l i n e paths approximation (SLPA) f o r m u l a t e d below. It is shown that the p r i n c i p a l l o g a r i t h m i c t e r m s in the s c a t t e r i n g a m p l i t u d e s cancel in the a s y m p t o t i c l i m i t S ~ ~o, t - fixed when the d i a g r a m s with nucleon closed loops a r e neglected. It is shown then that the contributions of the r a d i a t i v e c o r r e c t i o n s to the l a d d e r type graphs in the s t r a i g h t - l i n e paths a p p r o x i m a t i o n a r e f a c t o r i z e d and a r e d e t e r m i n e d by the quantity H(t), which depends only upon the s q u a r e of m o m e n t u m t r a n s f e r s . In the r e g i o n [t I << m2 the quantity H(t) has an exponential dependence on t and p r o d u c e s the d i f f r act i o n peak in e l a s t i c s c a t t e r i n g in a c c o r d a n c e with the hypothesis of s m o o t h n e s s of the l oc al quasipotential of two p a r t i c l e s at high e n e r g i e s [18-20]. Such a b eh av i o u r of the e l a s t i c s c a t t e r i n g amplitude was p r e d i c t e d r e c e n t l y in p ap er [21] and c o r r e s p o n d s , in s o m e s e n s e , to the c o h e r e n t i n t e r a c t i o n of v i r t u a l m e s o n s which belong to the ni~cleon clouds. F u r t h e r the d i f f e r e n t i a l c r o s s s e c t i o n of i n e l a s t i c p r o c e s s e s w e r e obtained. Under the r e q u i r e m e n t of " s o f t n e s s " of s e c o n d a r y m e s o n s the P o i s s o n d i s t r i b u t i o n s in the n u m b e r of p a r t i c l e s e m i t t e d in c o l l i s i o n is found. We note that the total d i f f e r e n t i a l c r o s s s e c t i o n s u m m e d o v e r all the s e c o n d a r y m e s o n s may have, g e n e r a l l y speaking, no pronounced d i f f r a c t i o n peak in the r e g i o n tx2 ~ It l <<: m 2. 484
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7 December 1970
Such a b e h a v i o u r is due to the c a n c e l l a t i o n under c e r t a i n conditions of two exponential f a c t o r s , which c o r r e s p o n d to the r a d i a t i v e c o r r e c t i o n c o n t r i b u t i o n s in the e l a s t i c s c a t t e r i n g a m p l i t u d e and to the total c o n t r i b u t i o n s f r o m m u l t i - m e s o n production. We would like to point out the analogy of such a r e g u l a r i t y with the a u t o m o d e l b e h a v i o u r [22] of deep i n e l a s t i c p r o c e s s e s of high e n e r g y hadron i n t e r a c t i o n . We choose an i n t e r a c t i o n L a g r a n g i a n of the f o l lowing f o r m Lin t :
g:t,h2(x)i'~h(x)A a
(x): + g2:A2
(x)~/*(x)tp(x):
(1)
w h e r e g is s o m e d i m e n s i o n l e s s coupling constant. The o n e - p a r t i c l e G r e e n function of the quantum field in the e x t e r n a l field {[iOa
+ gA a(x)] 2 - m2 } G(x,yl A )
Aa(x)
s a t i s f i e s the equation
=-64(x-y)
(2)
The f o r m a l solution of eq. (2) for the F o u r i e r t r a n s f o r m of the G r e e n function can be r e p r e s e n t e d by m e a n s of the functional i n t e g r a l ac
G(p,q[A)
: i f dz e x p { i r ( p 2 - m 2 ) } f ( d4y 2 ~ exp
{i(p-q)y}
f[64v~0
o
(3) x e x p { 2 i g f d~ [va(~) +Pa]Aa[y+2p~ +2 0
where
T2 "~2 [54v]r 1 : 6 4 v
~'2
v2(~)d~}/f64v
exp{-if
v(~)dr/]} 0
exp(-if
(4)
v2(~)d~}
~'1 rl is a n o r m a l i z e d volume e l e m e n t of the functional s p a c e of the f o u r - d i m e n s i o n a l function va(zl) d e t e r mined on the i n t e r v a l ~-1 "< ~ ~< T2" The t w o - p a r t i c l e e l a s t i c s c a t t e r i n g a m p l i t u d e is defined a s a vacuum a v e r a g e of the p r o d u c t of two G r e e n functions [11]. Below, for s i m p l i c i t y , we s h a l l n e g l e c t the vacuum p o l a r i z a t i o n e f f e c t s a s well a s the d i a g r a m s with c l o s e d nucleon loops. The p r o p a g a t o r of the f r e e v e c t o r field @aft is d e t e r m i n e d by the e x p r e s s i o n
cba~ = 5aft _ ~:aKfl/~2/(K2_
~2)
(5)
Taking into account eq. (3) and eq. (5) we obtain the following c l o s e d e x p r e s s i o n for the t w o - p a r t i c l e scattering amplitude
if(pl,P2;
qlq2 )
×[2v2(0) +P2
= g2 f d4yexp{ y(p l _ q l ) } @ a f t ( y ) f [54Vl]~-oo[54v2]~-,,o[2Vl(0)
+q2]flf
d• exp ½ig2 fd4K@y~(K)
i=~1j(C)(K;
+Pl +ql]a
pi,qi]vi)j~ ) ( - g ; pi,qilvi)
(6)
0 + 2~exp{iKy}j(yl)(K;
pl,ql]Ul)j~2)(-K; p2,q21v2)]-
i?
6m2d~ 1
--o(3
where
j(~) (g;
pi,qilvi)
: 2i f d~[viy(~)
+PiyO(~) +qiyo(-~)] exp{2iKy[pi~O(~ ) +qiy~O(-4) +f
viT0?)d~]} (7) 0 is a c o n s e r v i n g t r a n s i t i o n c u r r e n t . Obviously, an e x a c t functional i n t e g r a t i o n in e x p r e s s i o n (2.9) is not p o s s i b l e . T h e r e f o r e , we use below the a p p r o x i m a t e method of c a l c u l a t i n g the i n t e g r a l s o v e r v1 and v2 [23,24], c a l l e d by us SLPA. The functional v a r i a b l e s v 1 and v2 f o r m a l l y i n t r o d u c e d in eq. (3) f o r obtaining the solution f o r the G r e e n function, d e s c r i b e the deviation of a p a r t i c l e t r a j e c t o r y f r o m the s t r a i g h t - l i n e paths. In f a c t , if we put v = 0 in f o r m u l a (7) for the t r a n s i t i o n c u r r e n t , we would obtain In eq. (6) the mass renormalization ~00 = m2 + 5m2 has been made. It removes divergences appearing in the integration over the variables 41 and 42 [14]. 485
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7 December 1970
the c l a s s i c a l c u r r e n t of the n u c l e o n , m o v i n g with m o m e n t u m p at ~ . 0 and with m o m e n t u m q at ~ < 0. We note. h o w e v e r , that the a p p r o x i m a t i o n u = 0 is known to be i n a p p l i c a b l e at v a l u e s of the p r o p e r t i m e of the p a r t i c l e c l o s e to z e r o , when the p a r t i c l e c l a s s i c a l t r a j e c t o r y c h a n g e s its d i r e c t i o n . In the l a n g u a g e of F e y n m a n g r a p h s the a p p r o x i m a t i o n m e a n s that the q u a d r a t i c ~ - d e p e n d e n c e in the n u c l e o n p r o p a g a t o r is n e g l e c t e d . It can lead, g e n e r a l l y s p e a k i n g , to the a p p e a r e n c e of d i v e r g e n c e s of the i n t e g r a l s o v e r d4K at the u p p e r l i m i t . A b e t t e r a p p r o x i m a t i o n is given by a v e r a g i n g the n u c l e o n c u r r e n t (7) o v e r the f u n c t i o n a l v a r i a b l e u, i.e.
___
( 2pT+K7
_ _ 2 q 7 - K~__] j7(K; p . q ) = - "2Kp+K2~-i£ - 2Kq- K2 - i < /
(8)
F o r this r e a s o n , in s e e k i n g the t w o - p a r t i c l e e l a s t i c s c a t t e r i n g a m p l i t u d e we s h a l l use the SLPA, which c o n s i s t s in s u b s t i t u t i n g in the e x p o n e n t i a l exponent in eq. (6) the c u r r e n t p r o d u c t a v e r a g e d o v e r the f u n c t i o n a l v a r i a b l e s u 1 and u 2 [11]. T h u s . in SLPA the e x p r e s s i o n for the e l a s t i c s c a t t e r i n g a m p l i t u d e t a k e s the f o r m 1 if(pl,P2; q l , q 2 ) = g 2 H ( t ) jd4yexp{iy(p 1 - q l ) } A ( y ; p l , q l ; p 2 , q 2 ) f dXexp{iXX(y; P l , P 2 ; q l , q 2 ) } (9) 0 where A(y; P l , q l ;
P2,q2) = fd4KQ)c~fi(K) [K + /51 + ql]cr [-K + P2 + q2]fi exp {iKy}
X(Y; P l ' P 2 ; q l , q2 )
g2 4 - (2~)
fd4K
exp{iKy} (DOt~(K)j(~)~; i l , q i ) j ~ ) ( - K ;
r.2 .......... ¢K;
H(I)
= exp i
g2
2(2n) 2
.
Di,qi)J
f d4KC-Dc~fl(K)i~,=l j(~)
"=
.
.
.
.
.
(10)
p 2 ,q 2 ;
(11)
.
~:; Pi,qi)
(12)
It is i n t e r e s t i n g to note that the c o n t r i b u t i o n of the r a d i a t i v e c o r r e c t i o n is f a c t o r i z e d in a f o r m of the f a c t o r H(t) d e p e n d i n g only on the s q u a r e of the m o m e n t u m t r a n s f e r t = (Pl - q l )~, as in the s c a l a r field i n t e r a c t i o n m o d e l [14]. The a n a l o g o u s p h e n o m e n a of the f a c t o r i z a t i o n of the r a d i a t i v e c o r r e c t i o n c o n t r i b u t i o n in q u a n t u m e l e c t r o d y n a m i c s was found in the a r t i c I e s [25-27]. In the high e n e r g y l i m i t S -* ~ at fixed m o m e n t u m t r a n s f e r s t l i m i t e d by the c o n d i t i o n [t ( << m 2 the e x p r e s s i o n for the e l a s t i c s c a t t e r i n g a m p l i t u d e has the f o r m 1'
f(s,
t) : i(s
- u) v(t)
exp
(at)
(13)
where v(t) : ½
f d2y±
I~Ko(~Iy.])
exp (iy± A±) (exp l - " g2
a = { g 2 / 3 ( 2 n ) 2 m 2 } { I n ( m 2 / i a2) + ½}
f - 1) "
t ~ _A2
.L
(14)
(15)
and K 0 is the K e l v i n f u n c t i o n of z e r o o r d e r . The a m p l i t u d e s of i n e l a s t i c p r o c e s s e s of m e s o n p r o d u c t i o n in t w o - n u c l e o n c o l l i s i o n s [11] at high e n e r g i e s can be d e t e r m i n e d by m e a n s of a g e n e r a t i n g f u n c t i o n f(pl,P2; ql,q2 IA ext) h a v i n g a m e a n i n g of the s c a t t e r i n g a m p l i t u d e of two n u c l e o n s in the p r e s e n c e of the e x t e r n a l f i e l d A ext. In what follows we will c o n s i d e r the c a s e in which the m o m e n t a of the s e c o n d a r y m e s o n in the c e n t e r of m a s s s y s t e m s a t i s f y the r e q u i r e m e n t of " s o f t n e s s " [15]:
J We note that taking into account the identity of nucleons reads on symmetrization of eq. (13) to terms vanishing in the limit S ~ oo with fixed t. 486
V o l u m e 33B, n u m b e r
7
PHYSICS
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~s-=
LETTERS
7 December
1970
N
Koi
IS i=1 xi±[ ]Pl±-ql±]~
1;
[P2±-q21J
(16)
w h e r e the m o m e n t a of the initial nucleons a r e chosen along the Z - a x i s . Under this r e q u i r e m e n t the a m p l i t u d e of N - m e s o n p r o d u c t i o n is f a c t o r i z e d and can be w r i t t e n in the following f o r m : N f i n e l (N) = f ( P l , P 2 ;
ql ' q2)
i-~l
.(1)/K gE~(tci) [:oe ' i;. P l , q l ) +
~(2)(,~txz.*, P2 , q2 )] aOg
(17)
w h e r e Eot(g ) is the p o l a r i z a t i o n v e c t o r of a m e s o n with the m o m e n t u m t(. We find a l s o the a s y m p t o t i c e x p r e s s i o n for d i f f e r e n t i a l c r o s s s e c t i o n s of "soft" meson p r o d u c t i o n . when the m e s o n m o m e n t a s a t i s f y eq. (16). It has the f o r m : d(~ )n
1 V2 (l) U'n l(S, l) U'n 2(S, l) 1 ,r12 S -+ m 47r t-fixed<,.m2
(1 {3)
where 1
Wn(S,t ) --.~.. e x p { 2 a t } f
~ d × i - - ~ ij~)(Ki; pl, ql) l 2 ~p i=1 2Koi (2~r)3
(19)v
The i n t e g r a t i o n r e g i o n ~p o v e r s e c o n d a r y m e s o n m o m e n t a is d e t e r m i n e d by the condition n
-t<2p
~
Ki -
(A_ ~
i=1
Ki)2_
where
nl t=A2
;
(20)
i-1
A = (ql-Pl
- ~
n2 gi) = ( q 2 - P 2 -
~
K~)
C o n s i d e r now an a p p r o x i m a t i o n , in which the total momentum of the s e c o n d a r y m e s o n s can be n e g l e c t e d in a c c o r d a n c e with the r e q u i r e m e n t of " s o f t n e s s " (16). In this a p p r o x i m a t i o n eq. (19) t a k e s the f o r m of the P o i s s o n d i s t r i b u t i o n s 1 exp {2at} [n(s,t)] n Wn(S , t) : nT.
(21)
w h e r e the quantity J n ( s , t ) =-(2g~)~ f d ~
Ij(I)(K; pl,ql)]2
(22)
is the a v e r a g e n u m b e r of secondar}, p a r t i c l e s p r o d u c e d in the two-nucleon c o l l i s i o n at S - ' ~ and fixed t. Using eq. (8) for j ~ , we find f o r t t[ .< m 2 that
if(s, t) = - bt
(23)
The p a r a m e t e r b d e p e n d s , in g e n e r a l , on a special__ f o r m of cut-off of the i n t e g r a l in eo. (22) o v e r the m e s o n momentum. In the p a r t i c u l a r c a s e when R~ ~ m 2, 1 >>~2 >> t~2/m2, ln(m2/~t2) >., In (1/ce) 2 w h e r e c~ : RZ/Po we get
b = {2g2/3(2v)2m 2} ( In ( m 2 / p 2) + ½}
(24)
which c o i n c i d e s with the double slope p a r a m e t e r of the d i f f r a c t i o n exponential (15). Notice that the equality 2 a = b is true a l s o in the i n f r a r e d a s y m p t o t i c l i m i t g ~ 0. F o r this c a s e , a f t e r s u m m i n g in eq. (19) o v e r the n u m b e r of s e c o n d a r y m e s o n s , we find that the dependence on the v a r i a b l e t c a n c e l s and the d i f f r a c t i o n p e a k in the total d i f f e r e n t i a l c r o s s s e c t i o n d i s a p p e a r s. $ The i n t e g r a t i o n r e g i o n in eq. (22) is e f f e c t i v e l y l i m i t e d by ] K Z I "-< RZ, ] ~ £ ] "-< R±. 487
Volume 33B, number 7
PHYSICS
LETTERS
7 December 1970
T h i s r e g u l a r i t y w a s m e n t i o n e d in p a p e r [20] and i s in a n a l o g y w i t h the a u t o m o d e l b e h a v i o u r of d e e p i n e l a s t i c p r o c e s s e s of h a d r o n i n t e r a c t i o n s at h i g h e n e r g y [22]. The s t r a i g h t - l i n e p a t h s a p p r o x i m a t i o n , u s e d in t h i s w o r k c o r r e s p o n d s to a p h y s i c a l p i c t u r e in w h i c h c o l l i d i n g h i g h e n e r g y n u c l e o n s at the a c t of i n t e r a c t i o n r e c e i v e a s m a l l r e c o i l c o n n e c t e d w i t h the e m i s s i o n of " s o f t " m e s o n s and r e t a i n t h e i r i n d i v i d u a l i t y . T h e a u t h o r s e x p r e s s t h e i r d e e p g r a t i t u d e to P r o f s . N. N. B o g o l u b o v , D. I. B l o k h i n t s e v and A. A. L o g u n o v for stimulating discussions.
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