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The diffractive background to the Veneziano model of ππ scattering
17.B.llNuclear
Physics
B17 (1970) 467-4’71. North-Holland
Publ. Comp.,
Amsterdam
THE DIFFRACTIVE BACKGROUND TO THE VENEZIANO MODEL OFnn SCATTERING P. BROOKER and J. C. TAYLOR Department
of Theoretical Physics, Oxford, England
Received (Revised
Oxford
University,
19 March 1969 19 August 1969)
Abstract: A term which represents the exchange of a flat Pomeron, but which is otherwise as structureless as possible, is added to the Veneziano model amplitude for 7fr scattering. The corrections in the small s, t, u-region are certainly greater than 7% and probably more like 50%. The correction to the Adler-Weisberger sum rule, however, is only of the order of 3%.
1. INTRODUCTION It has been suggested [l] that, whereas other Regge trajectories exhibit some sort of duality with resonances, the Pomeron is to be regarded as a separate, non-dual background. In this spirit, Lovelace’s [2] Veneziano model [3] for 71’11 scattering leaves out the Pomeron. It is therefore important to try to determine how far the Pomeron background would alter the conclusions of the model about, amongst other things, the Adler condition [4] and other current-algebra constraints [5]. It is the purpose of this note to investigate these questions. We construct a very simple and naive model of the Pomeron background, which is just added to the Lovelace-Veneziano amplitude. The exceptional flatness of the P-trajectory and the connection with the inelastic channels (and presumably Regge-cut effects) are incorporated in the model, which is described in sect. 2, where the assumptions made are listed. In sect. 3, we deduce a lower limit on the Pomeron/Veneziano ratio of ‘ITo at the point t = v = 0; but we point out that this ratio is much more likely to be around 50%. In contrast we show in sect. 4 that the Pomeron contribution to the Adler-Weisberger sum rule should only be a few percent. Our approach may be contrasted with that of Wong [6], who treats the Pomeron on the same footing as other Regge trajectories, and who also has the (possibly) disagreeable feature of Z= 2 trajectories, albeit non-leading. 2. THE MODEL AND ASSUMPTIONS Let a, b, c and d be the isospin indices of the four pions, channels be
and let the
468
P. BROOKER and J. C. TAYLOR s,
ab~cd;
t,
u,
ac ~bd;
ad~bc.
We write the amplitude as
A(s; t, U)6ab 6cd + A ( t ; u, S)Sac 6 b d + A ( u ; s, t)6ad 5bc ,
(1)
where
A(s; t, ~,) = ½[V(s, 0 + V(s, u) - V(t, u)] + P(s; t, u)
(2)
Here P is the Pomeron background and
v ( s , t) : - ~ v
r (1- ~(s)} r (1- ~(t)} r {1- ~(s) - ~(t)}
'
(a)
w h e r e a i s t h e c o m m o n p, fo t r a j e c t o r y ( a s s u m e d a l s o to b e a s s o c i a t e d w i t h the P'). F o r a p ~ v e r t e x g p ~ e • ( q l + q 2 ) , f V and gp~n a r e r e l a t e d by 2
f v = 2gp~
.
(4)
A b o u t t h e P o m e r o n t e r m P we m a k e t h e a s s u m p t i o n s (i) A d d i t i v i t y , a s a l r e a d y d i s p l a y e d in eq. (2). (ii) A s s ~ +~o f o r s m a l l t
P(t; u, s) ~ i f p f ( t ) v
,
(5)
w h e r e v = ½ ( s - u) a n d f ( 0 ) = 1. H e r e we h a v e t a k e n a c o m p l e t e l y f l a t P o m e r o n t r a j e c t o r y w h i c h i s c o n s i s t e n t w i t h m a n y R e g g e f i t s of 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 . F r o m s u c h a fit ( s o l u t i o n 1 of R a r i t a et al. [7]) we t a k e t i p = 6.3 G e V - 2 .
(6)
T h e v a l i d i t y of t h i s n u m b e r d e p e n d s on f a c t o r i z a t i o n f o r t h e P o m e r o n , w h i c h m a y not b e c o r r e c t f o r a R e g g e - c u t p h e n o m e n o n ; but i t s o r d e r of m a g n i t u d e i s p r o b a b l y c o r r e c t in a n y c a s e . (iii) T h e a s y m p t o t i c eq. (5) i s s u p p o s e d to be v a l i d up to t e r m s w h i c h t e n d to z e r o a s v ~ 0o. T h e r e a s o n f o r t h i s i s t h e f o l l o w i n g . In a R e g g e - o r T o i l e r - p o l e m o d e l f o r e q u a l - m a s s s c a t t e r i n g , t h e c o r r e c t i o n s to t h e l e a d i n g t e r m va a r e of o r d e r v a - 2 ; we m i g h t e x p e c t t h e s a m e of eq. (5) if t h e P o m e r o n i s a n y t h i n g l i k e a R e g g e p o l e . T h e r e f o r e a n y c o r r e c t i o n to eq. (5) of o r d e r u n i t y w o u l d h a v e t o c o m e f r o m a n o t h e r t r a j e c t o r y b u t by h y p o t h e s i s , a l l o t h e r s a r e i n c l u d e d in t h e V e n e z i a n o t e r m . (iv) T h e P o m e r o n c r o s s s e c t i o n
a p : [ s ( s - 4 m 2 ) ] -½ I m P ( 0 ; s , 4 m 2 - s) ,
(7)
i s a s s u m e d to i n c r e a s e m o n o t o n i c a l l y to i t s l i m i t i n g v a l u e , flip, a s v i n creases. The only reason for this assumption is the notion that the Pomeron should be as featureless as possible. It w i l l b e s e e n t h a t t h e l a t t e r two a s s u m p t i o n s , a t l e a s t , a r e r a t h e r a r b i t r a r y b u t t h e y s e e m t h e s i m p l e s t s e t of a s s u m p t i o n s to t r y out.
DIFFRACTIVE BACKGROUND
469
3. THE CORRECTIONS AT LOW ENERGY We want to i n v e s t i g a t e how P c o m p a r e s with V in eq. (2) at low e n e r g i e s . We note that the s i m p l e s t choice of P c o n s i s t e n t with our a s s u m p t i o n s a m o u n t s to putting ~ p = tip, w h e r e ~ p is defined by r e l a t i o n (7). T h i s , in f a c t , c o n s t i t u t e s an u p p e r bound on (rp; so we c o n s t r u c t the d i f f e r e n c e
g(v) = i[s(s - 4m27r)]½ ~ p - P(0; s, 4m2 - s) .
(8)
By a s s u m p t i o n s (ii) and (iii), g(v) -~ 0 as v ~ o% and so s a t i s f i e s the unsubtracted dispersion relation 2 2f:2 v' I m g ( v ' ) d r ' .
g ( v ) : -~
(9)
v,2 - v2
Also, b e c a u s e of a s s u m p t i o n (iv), I m g ( v ' ) ~ 0 and t h e r e f o r e
g(v) >1 0 ,
for
0 < v < 2m 2 . 7r
(10)
T h i s inequality is m o s t powerful at v = 0, w h e r e we get, using eq. (8),
IP(O;2m27r,2m2r)l >~ 2 r n 2 ~ p = 12.6m 2.~
(11)
Eq. (11) is to be c o m p a r e d with the value of the Veneziano p a r t of eq. (2) at the s a m e point. E q s . (3) and (4) give ½[V(2m2,0) + V(0,2m2) - V ( 2 m 2 , 2 m 2 ) ] =
2~g2
m2a'(0)~ +O(m4) ,
(12)
w h e r e we have u s e d L o v e l a c e ' s [2] condition a(m 2) = ½. Adopting the v a l u e s Fn = 125 GeV, g ~ = 31 and a'(0) = 0.88 GeV -1, eq. (12) has the value 170rn~. T h u s the l i m i t on P in eq. (11) is only about 7% of the Veneziano part. H o w e v e r , the l o w e r l i m i t in eq. (11), coming as it does f r o m the c a s e of constant crp, is m o s t unlikely to be attained. F o r if the P o m e r o n amplitude is connected with inelastic channels, we would only expect a p to take on its full value above the r e l e v a n t inelastic t h r e s h o l d s . A m o r e r e a l i s t i c model, c o n s i s t e n t with the a s s u m p t i o n s of sect. 2, m i g h t be
P(t; s, u) = t3pf(t)[(s - M2)(u - M2)]½ ,
(13)
w h e r e M is a p a r a m e t e r r e p r e s e n t i n g s o m e s o r t of effective value f o r the t h r e s h o l d f o r the dominant inelastic p r o c e s s e s . The equality in eq. (11) would be a c h i e v e d for M = 2rnTr. A m o r e r e a l i s t i c , but still c o n s e r v a t i v e e s t i m a t e might be M = 4rn~ giving
or about 50% of the Veneziano t e r m .
470
P . B R O O K E R a n d J . C. T A Y L O R
F r o m this point of view, it is somewhat unexpected that V s a t i s f i e s Adl e r ' s condition by itself. However we have not actually c o n s i d e r e d the A d l e r point, since we do not know how to take r e l a t i o n (11) o f f - m a s s - s h e l l .
4. THE A D L E R - W E I S B E R G E R SUM RULE T h e r e is one c u r r e n t - a l g e b r a c o n s t r a i n t , the A d l e r - W e i s b e r g e r l o w - e n e r g y - l i m i t t h e o r e m [8] which t u r n s out to be insensitive to a P o m e r o n b a c k g r o u n d of the f o r m of r e l a t i o n (13). We m u s t f i r s t say s o m e t h i n g about the function f(t) in r e l a t i o n (13). This m u s t not spoil the Regge behaviour in the t - c h a n n e l . An e x p r e s s i o n such as f(t)
= e x p [,\ t 4 - ~ ( t ° - t) 4]
(15)
o 1
has been p r o p o s e d in a slightly different context [9]. Here (t o - t ~ is taken to be r e a l and positive for t < 0. The p a r a m e t e r s X and t o >/4rn~ m u s t be c h o s e n c o n s i s t e n t l y with the shape of the diffraction peak. Having noted the existence of suitable functions f(t) all that we shall r e q u i r e below is f ' ( 0 ) , a s s u m i n g that higher d e r i v a t i v e s a r e not large. F r o m ref. [6], we set f ' ( 0 ) £ 1 GeV "2 .
(16)
The A d l e r - W e i s b e r g e r l o w - e n e r g y limit is [8] lim
[Y(s,0) _
Y(u,O)+P(s;O,u)-P(u;O,s)]
2 l = 2a~ 2 = ~150 Lll6JGeV- ,
(17)
V ---~ 0
where the upper value c o m e s f r o m the G o l d b e r g e r - T r e i m a n value for ~ and the l o w e r value f r o m the p i o n - d e c a y rate. The Veneziano contribution to the left-hand side [10] is ~fiVCZ'(0) = 170 GeV -2 .
(18)
The P o m e r o n contribution, using r e l a t i o n (13), is -tip[½
+M2f ' (0) + O(m 2 / M 2 ) ]
•
(19)
Using (16) and taking M ~ 4rn~, we expect relation (19) to have the o r d e r of magnitude tip ~ 6 GeV -2, and thus to be only a few p e r c e n t of r e l a t i o n (18). We a r e grateful to the r e f e r e e for c o r r e c t i n g an i m p o r t a n t n u m e r i c a l e r r o r , and for other helpful c r i t i c i s m . One of us (P. B.) would like to thank the Science R e s e a r c h Council for a R e s e a r c h Studentship.
DIFFRACTIVE BACKGROUND
471
REFERENCES [1] P . G . O . F r e u n d , Phys. Rev. Letters 20 (1968) 235; H . H a r a r i , Phys. Rev. Letters 20 (1968) 1395. [2] C.Lovelace, Phys. Letters 28B (1968) 265. [3] G.Veneziano, Nuovo Cimento 57A (1968) 190. [4] S . L . A d l e r , Phys. Rev. 139 (1965> ~ c~38. [5] S.Weinberg, Phys. Rev. Letters 17 (1966) 616. [6] D.Y.Wong, Phys. Rev. 181 (1969) 1900. [ 7 ] W . R a r i t a , R . J . R i d d e l l , C . B . C h i u a n d R . J . N . P h i l l i p s , Phys. Rev. 165 (1968) 1615. [8] S . L . A d l e r and R . F . D a s h e n , Current algebra (W.A.Benjamin, New York, 1968). [9] R.Jengo, Phys. Letters 28B (1968) 262. [10] P . B r o o k e r , Nucl. Phys. B10 (1969) 59; H.Osborn, Nuovo Cimento Letters i (1969) 513.
Physics
B17 (1970) 467-4’71. North-Holland
Publ. Comp.,
Amsterdam
THE DIFFRACTIVE BACKGROUND TO THE VENEZIANO MODEL OFnn SCATTERING P. BROOKER and J. C. TAYLOR Department
of Theoretical Physics, Oxford, England
Received (Revised
Oxford
University,
19 March 1969 19 August 1969)
Abstract: A term which represents the exchange of a flat Pomeron, but which is otherwise as structureless as possible, is added to the Veneziano model amplitude for 7fr scattering. The corrections in the small s, t, u-region are certainly greater than 7% and probably more like 50%. The correction to the Adler-Weisberger sum rule, however, is only of the order of 3%.
1. INTRODUCTION It has been suggested [l] that, whereas other Regge trajectories exhibit some sort of duality with resonances, the Pomeron is to be regarded as a separate, non-dual background. In this spirit, Lovelace’s [2] Veneziano model [3] for 71’11 scattering leaves out the Pomeron. It is therefore important to try to determine how far the Pomeron background would alter the conclusions of the model about, amongst other things, the Adler condition [4] and other current-algebra constraints [5]. It is the purpose of this note to investigate these questions. We construct a very simple and naive model of the Pomeron background, which is just added to the Lovelace-Veneziano amplitude. The exceptional flatness of the P-trajectory and the connection with the inelastic channels (and presumably Regge-cut effects) are incorporated in the model, which is described in sect. 2, where the assumptions made are listed. In sect. 3, we deduce a lower limit on the Pomeron/Veneziano ratio of ‘ITo at the point t = v = 0; but we point out that this ratio is much more likely to be around 50%. In contrast we show in sect. 4 that the Pomeron contribution to the Adler-Weisberger sum rule should only be a few percent. Our approach may be contrasted with that of Wong [6], who treats the Pomeron on the same footing as other Regge trajectories, and who also has the (possibly) disagreeable feature of Z= 2 trajectories, albeit non-leading. 2. THE MODEL AND ASSUMPTIONS Let a, b, c and d be the isospin indices of the four pions, channels be
and let the
468
P. BROOKER and J. C. TAYLOR s,
ab~cd;
t,
u,
ac ~bd;
ad~bc.
We write the amplitude as
A(s; t, U)6ab 6cd + A ( t ; u, S)Sac 6 b d + A ( u ; s, t)6ad 5bc ,
(1)
where
A(s; t, ~,) = ½[V(s, 0 + V(s, u) - V(t, u)] + P(s; t, u)
(2)
Here P is the Pomeron background and
v ( s , t) : - ~ v
r (1- ~(s)} r (1- ~(t)} r {1- ~(s) - ~(t)}
'
(a)
w h e r e a i s t h e c o m m o n p, fo t r a j e c t o r y ( a s s u m e d a l s o to b e a s s o c i a t e d w i t h the P'). F o r a p ~ v e r t e x g p ~ e • ( q l + q 2 ) , f V and gp~n a r e r e l a t e d by 2
f v = 2gp~
.
(4)
A b o u t t h e P o m e r o n t e r m P we m a k e t h e a s s u m p t i o n s (i) A d d i t i v i t y , a s a l r e a d y d i s p l a y e d in eq. (2). (ii) A s s ~ +~o f o r s m a l l t
P(t; u, s) ~ i f p f ( t ) v
,
(5)
w h e r e v = ½ ( s - u) a n d f ( 0 ) = 1. H e r e we h a v e t a k e n a c o m p l e t e l y f l a t P o m e r o n t r a j e c t o r y w h i c h i s c o n s i s t e n t w i t h m a n y R e g g e f i t s of 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 . F r o m s u c h a fit ( s o l u t i o n 1 of R a r i t a et al. [7]) we t a k e t i p = 6.3 G e V - 2 .
(6)
T h e v a l i d i t y of t h i s n u m b e r d e p e n d s on f a c t o r i z a t i o n f o r t h e P o m e r o n , w h i c h m a y not b e c o r r e c t f o r a R e g g e - c u t p h e n o m e n o n ; but i t s o r d e r of m a g n i t u d e i s p r o b a b l y c o r r e c t in a n y c a s e . (iii) T h e a s y m p t o t i c eq. (5) i s s u p p o s e d to be v a l i d up to t e r m s w h i c h t e n d to z e r o a s v ~ 0o. T h e r e a s o n f o r t h i s i s t h e f o l l o w i n g . In a R e g g e - o r T o i l e r - p o l e m o d e l f o r e q u a l - m a s s s c a t t e r i n g , t h e c o r r e c t i o n s to t h e l e a d i n g t e r m va a r e of o r d e r v a - 2 ; we m i g h t e x p e c t t h e s a m e of eq. (5) if t h e P o m e r o n i s a n y t h i n g l i k e a R e g g e p o l e . T h e r e f o r e a n y c o r r e c t i o n to eq. (5) of o r d e r u n i t y w o u l d h a v e t o c o m e f r o m a n o t h e r t r a j e c t o r y b u t by h y p o t h e s i s , a l l o t h e r s a r e i n c l u d e d in t h e V e n e z i a n o t e r m . (iv) T h e P o m e r o n c r o s s s e c t i o n
a p : [ s ( s - 4 m 2 ) ] -½ I m P ( 0 ; s , 4 m 2 - s) ,
(7)
i s a s s u m e d to i n c r e a s e m o n o t o n i c a l l y to i t s l i m i t i n g v a l u e , flip, a s v i n creases. The only reason for this assumption is the notion that the Pomeron should be as featureless as possible. It w i l l b e s e e n t h a t t h e l a t t e r two a s s u m p t i o n s , a t l e a s t , a r e r a t h e r a r b i t r a r y b u t t h e y s e e m t h e s i m p l e s t s e t of a s s u m p t i o n s to t r y out.
DIFFRACTIVE BACKGROUND
469
3. THE CORRECTIONS AT LOW ENERGY We want to i n v e s t i g a t e how P c o m p a r e s with V in eq. (2) at low e n e r g i e s . We note that the s i m p l e s t choice of P c o n s i s t e n t with our a s s u m p t i o n s a m o u n t s to putting ~ p = tip, w h e r e ~ p is defined by r e l a t i o n (7). T h i s , in f a c t , c o n s t i t u t e s an u p p e r bound on (rp; so we c o n s t r u c t the d i f f e r e n c e
g(v) = i[s(s - 4m27r)]½ ~ p - P(0; s, 4m2 - s) .
(8)
By a s s u m p t i o n s (ii) and (iii), g(v) -~ 0 as v ~ o% and so s a t i s f i e s the unsubtracted dispersion relation 2 2f:2 v' I m g ( v ' ) d r ' .
g ( v ) : -~
(9)
v,2 - v2
Also, b e c a u s e of a s s u m p t i o n (iv), I m g ( v ' ) ~ 0 and t h e r e f o r e
g(v) >1 0 ,
for
0 < v < 2m 2 . 7r
(10)
T h i s inequality is m o s t powerful at v = 0, w h e r e we get, using eq. (8),
IP(O;2m27r,2m2r)l >~ 2 r n 2 ~ p = 12.6m 2.~
(11)
Eq. (11) is to be c o m p a r e d with the value of the Veneziano p a r t of eq. (2) at the s a m e point. E q s . (3) and (4) give ½[V(2m2,0) + V(0,2m2) - V ( 2 m 2 , 2 m 2 ) ] =
2~g2
m2a'(0)~ +O(m4) ,
(12)
w h e r e we have u s e d L o v e l a c e ' s [2] condition a(m 2) = ½. Adopting the v a l u e s Fn = 125 GeV, g ~ = 31 and a'(0) = 0.88 GeV -1, eq. (12) has the value 170rn~. T h u s the l i m i t on P in eq. (11) is only about 7% of the Veneziano part. H o w e v e r , the l o w e r l i m i t in eq. (11), coming as it does f r o m the c a s e of constant crp, is m o s t unlikely to be attained. F o r if the P o m e r o n amplitude is connected with inelastic channels, we would only expect a p to take on its full value above the r e l e v a n t inelastic t h r e s h o l d s . A m o r e r e a l i s t i c model, c o n s i s t e n t with the a s s u m p t i o n s of sect. 2, m i g h t be
P(t; s, u) = t3pf(t)[(s - M2)(u - M2)]½ ,
(13)
w h e r e M is a p a r a m e t e r r e p r e s e n t i n g s o m e s o r t of effective value f o r the t h r e s h o l d f o r the dominant inelastic p r o c e s s e s . The equality in eq. (11) would be a c h i e v e d for M = 2rnTr. A m o r e r e a l i s t i c , but still c o n s e r v a t i v e e s t i m a t e might be M = 4rn~ giving
or about 50% of the Veneziano t e r m .
470
P . B R O O K E R a n d J . C. T A Y L O R
F r o m this point of view, it is somewhat unexpected that V s a t i s f i e s Adl e r ' s condition by itself. However we have not actually c o n s i d e r e d the A d l e r point, since we do not know how to take r e l a t i o n (11) o f f - m a s s - s h e l l .
4. THE A D L E R - W E I S B E R G E R SUM RULE T h e r e is one c u r r e n t - a l g e b r a c o n s t r a i n t , the A d l e r - W e i s b e r g e r l o w - e n e r g y - l i m i t t h e o r e m [8] which t u r n s out to be insensitive to a P o m e r o n b a c k g r o u n d of the f o r m of r e l a t i o n (13). We m u s t f i r s t say s o m e t h i n g about the function f(t) in r e l a t i o n (13). This m u s t not spoil the Regge behaviour in the t - c h a n n e l . An e x p r e s s i o n such as f(t)
= e x p [,\ t 4 - ~ ( t ° - t) 4]
(15)
o 1
has been p r o p o s e d in a slightly different context [9]. Here (t o - t ~ is taken to be r e a l and positive for t < 0. The p a r a m e t e r s X and t o >/4rn~ m u s t be c h o s e n c o n s i s t e n t l y with the shape of the diffraction peak. Having noted the existence of suitable functions f(t) all that we shall r e q u i r e below is f ' ( 0 ) , a s s u m i n g that higher d e r i v a t i v e s a r e not large. F r o m ref. [6], we set f ' ( 0 ) £ 1 GeV "2 .
(16)
The A d l e r - W e i s b e r g e r l o w - e n e r g y limit is [8] lim
[Y(s,0) _
Y(u,O)+P(s;O,u)-P(u;O,s)]
2 l = 2a~ 2 = ~150 Lll6JGeV- ,
(17)
V ---~ 0
where the upper value c o m e s f r o m the G o l d b e r g e r - T r e i m a n value for ~ and the l o w e r value f r o m the p i o n - d e c a y rate. The Veneziano contribution to the left-hand side [10] is ~fiVCZ'(0) = 170 GeV -2 .
(18)
The P o m e r o n contribution, using r e l a t i o n (13), is -tip[½
+M2f ' (0) + O(m 2 / M 2 ) ]
•
(19)
Using (16) and taking M ~ 4rn~, we expect relation (19) to have the o r d e r of magnitude tip ~ 6 GeV -2, and thus to be only a few p e r c e n t of r e l a t i o n (18). We a r e grateful to the r e f e r e e for c o r r e c t i n g an i m p o r t a n t n u m e r i c a l e r r o r , and for other helpful c r i t i c i s m . One of us (P. B.) would like to thank the Science R e s e a r c h Council for a R e s e a r c h Studentship.
DIFFRACTIVE BACKGROUND
471
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