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Determination of Regge-pole residue functions from fixed-t dispersion relations
-_~
Nuclear Physics B9 (1969) 324-330. North-Holland Pub1. Comp., Amsterdam
D E T E R M I N A T I O N OF REGGE-POLE RESIDUE F U N C T I O N S FROM FIXED-t DISPERSION RELATIONS J. BAACKE Institut ]'ur Theoretische Kernphysik, Universitttt Karlsruhe
Received 28 October 1968
Abstract: Fixed-t dispersion relations are used to derive a simple method to d e t e r mine (or test) Regge-pole residue functions from low-energy data. The method is applied to yN charge exchange scattering.
i. INTRODUCTION
V a r i o u s a u t h o r s [1-3] have d e r i v e d f i n i t e - e n e r g y sum r u l e s (FESR) to o b t a i n i n f o r m a t i o n on R e g g e - p o l e p a r a m e t e r s f r o m l o w - e n e r g y data. T h e s e sum r u l e s may be l o o k e d upon at a s a check of c o n s i s t e n c y b e t w e e n r e a l and i m a g i n a r y p a r t of the R e g g e - p o l e c o n t r i b u t i o n s by f i x e d - t d i s p e r s i o n r e l a tions (FTDR) [3]. H o w e v e r it is e v i d e n t that this c o n s i s t e n c y can a l w a y s be o b t a i n e d if the "finite e n e r g y " i s c h o s e n high enough and the i m a g i n a r y p a r t is taken f r o m R e g g e - p o l e exchange above the h i g h - e n e r g y l i m i t of p h a s e shift a n a l y s i s . This can be s e e n f r o m the r e s u l t s of ref. [4]. On the o t h e r hand in the ~N c h a r g e exchange s c a t t e r i n g the a m p l i t u d e s have not y e t r e a c h e d a s i m p l e Regge b e h a v i o u r even at K l a b = 11 G e V / c and p o s s i b l y a p p r e c i a b l e c o r r e c t i o n s have to be taken into account. We avoid p a r t of this difficulty by the r e s t r i c t e d a s s u m p t i o n , that only the i m a g i n a r y p a r t of the a m p l i t u d e shows Regge b e h a v i o u r above the e n e r g i e s of the p h a s e - s h i f t a n a l y s i s . This a s s u m p t i o n a l l o w s f o r c o r r e c t i o n s in the r e a l p a r t of the s c a t t e r i n g a m p l i t u d e which would o c c u r e.g. in m o d i f i e d Regge r e p r e s e n t a tions o r in the i n t e r f e r e n c e m o d e l . We then d e t e r m i n e the R e g g e - p o l e p a r a m e t e r s f r o m the r e q u i r e m e n t , that the h i g h - e n e r g y c o n t r i b u t i o n should p r o v i d e c o n s i s t e n c y between the p h a s e - s h i f t a n a l y s i s and FTDR. This m e t h od has the a d v a n t a g e that the p h a s e - s h i f t a n a l y s i s is c h e c k e d s i m u l t a n e o u s l y ; so we can d e c i d e c r i t i c a l l y upon the r e l i a b i l i t y of o u r r e s u l t s . In the f o l l o w ing we w r i t e the f o r m u l a s d i r e c t l y f o r 7rN c h a r g e exchange w h e r e the method is a p p l i e d h e r e . 2. DETERMINATION O F R E G G E - P O L E RESIDUE FUNCTIONS FROM F T D R We w r i t e the F T D R f o r the usual 7rN c h a r g e exchange a m p l i t u d e in the form
RESIDUE FUNCTIONS
'(-)(v,
ReA
t) -
v
F B ( v , t) +
t
1 - 4---~
2v ~-
f
325
oo dv, i m A ,(_)(v,,t ) v'2 _ v2
v°
d r ' v' l m B ( - ) ( v ', t)
2 T Re B(-)(v, t) -- FB(V , t) + ~ v
v ;2 - v 2
'
(la)
(lb)
'
o w i t h t h e f o l l o w i n g n o t a t i o n : M and # t h e n u c l e o n a n d p i o n m a s s , r e s p e c t i v e l y , v = w + (t/4M), w h e r e w i s t h e t o t a l p i o n l a b e n e r g y , vo = ~ + (t/4M), F B = [1 - ( t / 2 p 2 ) ] 8 ~ f 2 / ( v 2 - v 2 ) , w h e r e v B = - ( ~ 2 / 2 M ) + (t/4M) a n d f 2 = 0.081 i s the p i o n n u c l e o n c o u p l i n g c o n s t a n t , T h e i m a g i n a r y p a r t of the a m p l i t u d e s i s s u p p o s e d to b e g i v e n by t h e e x c h a n g e o f a s i n g l e R e g g e p o l e ( p - t r a j e c t o r y ) a b o v e w = ~ [P = ~ + ( t / 4 M ) ] a n d h a s the f o r m
imA,(-) Regge
b (t)
-
+
1 - t-L 4M2
...
a
½K(a) ( h )
(2a)
i m B(_) ab (t) ½ K ( a ) ( M ) a - 1 Regge M '
(2b)
with
/¢(~)
~ 2 2+a r(~ + a) =-~ r(i+a) '
(3)
and b±(l) a s d e f i n e d in r e f . [6]. W i t h t h i s a s s u m p t i o n e q s . ( l a , b ) m a y b e w r i t t e n f o r v < ~' in t h e f o r m P
~A,(-) = ReA,(-)(v,t)
v FB(V, t ) t 1 - 4-~
_
K(a)R
=
t
'
2v v° (4a)
'
4M 2
a (t) b_(t) AB(-)
_ Re B(-)(v,t)
_ FB(v,t)
~ _2f P
...
-
(4b)
M
o
where 1
RlX, X, a) = ~-~ f x
oo
X,~
dx'
X' 2 _X-------5 _ 1 :~a-1
zrM
oo
~
- -1 n=O 2n+ l - a
(x/~ 2n '
(x < ~).
(5)
326
J. BAACKE
Then AA '(-) and AB (-) can be c a l c u l a t e d f r o m p h a s e shifts alone. F r o m eqs. (4a,b) we obtain i m m e d i a t e l y t 1 - 4M2 A A ' ( - ) ( v , t)
b (t)=
a(t)b (t)=
1 v Y
hB(-)(v,t).
(6a)
(6b)
The r i g h t - h a n d s i d e s depend on a(t), but this is known r a t h e r a c c u r a t e l y f r o m h i g h - e n e r g y d a t a [9, 6, 7], f u r t h e r m o r e the a - d e p e n d e n c e is not v e r y sensitive. If o u r a s s u m p t i o n (2a,b) f o r the i m a g i n a r y p a r t is valid and the p h a s e shift solution is c o r r e c t , the r i g h t - h a n d s i d e s of eq. (6a,b) have to be independent of the e n e r g y and one would obtain a d e t e r m i n a t i o n of the r e s i d u e functions b+(t) which is m o r e r e l i a b l e than that f r o m F E S R , s i n c e a r a n g e of v - v a l u e s is u s e d and the c o n s i s t e n c y of the input with the d i s p e r s i o n r e lation has been tested.
3. DISCUSSION O F N U M E R I C A L R E S U L T S F r o m the p h a s e shift a n a l y s i s of L o v e l a c e et al. [8] ~ 4 ' ( - ) and AB (-) w e r e c a l c u l a t e d . Fig. 1 s h o w s the e n e r g y d e p e n d e n c e of the e x p r e s s i o n s (6a,b) f o r b+(t) and a(t) b(t) at s o m e v a l u e s of t. U n f o r t u n a t e l y we o b s e r v e s y s t e m a t i c d e v i a t i o n s f r o m the e x p e c t e d e n e r g y i n d e p e n d e n c e . N e v e r t h e l e s s we c a l c u l a t e d b+(t) and a(t) b(t) f r o m the e n e r g y a v e r a g e of t h e s e r e s u l t s , i g n o r e d h o w e v e r the d a t a below 370 MeV b e c a u s e t h e r e A A ' ( - ) and AB(-) a r e s m a l l c o m p a r e d to the whole a m p l i t u d e s . The r e s u l t s a r e plotted in fig. 2, t o g e t h e r with the bounds f o r [b+(t)] as o b t a i n e d in ref. [7]. The r e s i d u e f u n c t i o n s obtained h e r e a r e not c o m p a t i b l e with t h e s e bounds, which a r e i m p o s e d r i g o r o u s l y by 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 s b e t w e e n 3 and 18 GeV/c (ref. [9]), if we n e g l e c t e r r o r s in a(t). A s i m i l a r r e s u l t has been o b t a i n e d by Dolen et al. [3] f r o m FESR. The v i o l a t i o n of the bounds given in ref. [7] is r a t h e r s t r o n g and we have looked f o r p o s s i b l e e x p l a n a t i o n s : (a) In o u r c a l c u l a t i o n we did not take into a c c o u n t the c o n t r i b u t i o n s of the h i g h e r ~N r e s o n a n c e s (above 2 GeV lab e n e r g y ) . H o w e v e r t h e i r influence is n e g l i g i b l e in the r e g i o n of i n t e r e s t , if we c a l c u l a t e t h e i r c o n t r i b u t i o n f r o m the i n t e r f e r e n c e m o d e l with, e.g., the p a r a m e t e r s of B a r g e r and Cline [10]. The use of the i n t e r f e r e n c e m o d e l is s u r e l y justified if one is i n t e r e s t e d in the o r d e r of m a g n i t u d e of t h e s e c o r r e c t i o n s only. (b) The A n s a t z (2a,b) m a y be i n c o r r e c t and o t h e r c o n t r i b u t i o n s m i g h t have to be taken into a c c o u n t . H o w e v e r the a b s o l u t e value of A A ' ( - ) and AB(-) is too g r e a t g e n e r a l l y , so the high e n e r g y c o n t r i b u t i o n should be e n l a r g e d to obtain a g r e e m e n t . The addition of a s e c o n d R e g g e pole would not
RESIDUE FUNCTIONS
327
,I
t J
500
1000
1500
Trr [MeV]
500.
1000
1500
Trc ~-MeV]
15O0
T.~ [MeV]
b+(t) I.
x
I.
~m~x
•
x
b.,(t) 500
b+(tl :
1000
500 ;
:
:
~
1000 :
:
:
:
;
1500 :
:
:
:
I
:
T~[MeV]
x X
Fig. la. The energy dependence of the expressions (6a,b} for b+(t). The energy T is in MeV, the ordinates are dimensionless. a m e l i o r a t e the s i t u a t i o n b e c a u s e i t w o u l d d e c r e a s e f a s t e r with i n c r e a s i n g e n e r g y a n d the high e n e r g y c o n t r i b u t i o n w o u l d be r e d u c e d . The b e s t one c a n do i s to r e p l a c e the i n t e g r a n d of the d i s p e r s i o n i n t e g r a l at v > g by the b o u n d w h i c h f o l l o w s d i r e c t l y f r o m the e x p e r i m e n t a l 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 , w i t h o u t u s i n g the R e g g e Dole m o d e l o r the a b o v e d e c o m p o s i t i o n into n o - f l i p and flip a m p l i t u d e s . At s o m e t - v a l u e s the r e s u l t i n g i n e q u a l i t y i s v i o l a t e d i n d i c a t i n g t h a t the p h a s e - s h i f t s o l u t i o n h a s to be i m p r o v e d [11]. T h e s a m e c o n c l u s i o n f o l l o w s f r o m a c o m p a r i s o n of the T = ½ f o r w a r d a m p l i t u d e a s c a l c u l a t e d f r o m p h a s e s h i f t s and f r o m the f o r w a r d d i s p e r s i o n r e l a t i o n , u s i n g the m o s t r e c e n t * t o t a l c r o s s s e c t i o n d a t a [11]. N e v e r t h e l e s s we t e n d * The discrepancy is considerably s m a l l e r for a new set of forward amplitudes which was kindly sent to us by Dr. Lovelaee. However the new phase shifts are not yet available.
328
J. BAACKE
oqt)b_(t)
X
X
X
10 5 •
.
.
,
.
.
,
,
I
.
.
.
.
.
.
.
I
.
500 l ,,(t) b_(t)
.
.
.
.
.
.
.
.
I
.
1000
..........................
,
,
.
1500
T
1500
Trt EMeV]
~,
,
10 5
500
'('ib-(tl I
I
1000
. . . . . . . .
x I
I
:
:
500
;
!
~
i
;
1000
I
J
I
•
:
1500
~_
T~ [r~ eV] i
10
,*(t)b_ (t)
5
.. .. .. ..
t~-~5_6 ~ 500
I
,
.. .. .. ..
;000
,I
.. .. .. ..
1500
, I
• -
• ~ -
~
-5
-10
Fig. lb. The energy dependence of the expressions (6a,b) for a(t) b_(t). The energy T~ is in MeV, the ordinates a r e dimensionless. to t a k e o u r r e s u l t s (which a r e e s s e n t i a l l y e q u i v a l e n t to t h o s e of r e f . [3]) a s a c o n f i r m a t i o n of the b e h a v i o u r of b+~t) p r o p o s e d in r e f s . [7, 12]. E s p e c i a l l y the z e r o of b+(t) at t ~ - 0 . 1 5 (GeV/c) z and the s i g n of b_(t) a r e c l e a r l y e x h i b i t e d by the c o r r e s p o n d i n g b e h a v i o u r of A A ' ( - ) and AB(-) r e s p e c t i v e l y .
CONCLUSION T h e f i x e d - t d i s p e r s i o n r e l a t i o n s can be u s e d f o r a d e t e r m i n a t i o n of R e g g e - p o l e p a r a m e t e r s . In c o m p a r i s o n w i t h f i n i t e - e n e r g y s u m r u l e s one h a s the a d v a n t a g e t h a t the d e v i a t i o n s f r o m the c o n s i s t e n c y b e t w e e n the i n p u t and the d i s p e r s i o n r e l a t i o n s h o w i m m e d i a t e l y to w h a t e x t e n t the r e s u l t i s r e l i a b l e . A t p r e s e n t t h e s e d e v i a t i o n s a r e so l a r g e t h a t o n l y q u a l i t a t i v e c o n c l u s i o n s on the r e s i d u e f u n c t i o n s c a n b e d r a w n .
RESIDUE FUNCTIONS
I
; -':
:
:
b, (t)
-
~--
.
'
"
x N
x~ a(tl b_ (t)
/
t0.
~'
5. .1 ' ,;"-'5
I
"
%
-5. % \
\
\
'-10.
x
Fig. 2. Residue functions b+(t} and a(t} b(t} obtained by our method: x: our r e s u l t s , - - : proposal of ref. [7], - - - : bounds obtained in ref. [7]. T h e a u t h o r w o u l d l i k e to t h a n k P r o f e s s o r D r . G. H 6 h l e r f o r h i s i n t e r e s t in t h i s w o r k a n d v a l u a b l e d i s c u s s i o n s , a n d M e s s r s R. S t r a u s z and H. G. S c h l a i l e f o r c o m m u n i c a t i o n of u n p u b l i s h e d r e s u l t s .
NOTE
ADDED
~
PROOF
It w a s p o i n t e d out to t h e a u t h o r b y P r o f e s s o r C. L o v e l a c e t h a t t h e s t a t e m e n t in s e c t . 3(b) i s t r i v i a l l y w r o n g . In f a c t t h e a d d i t i o n of a s e c o n d R e g g e p o l e (p') m a y l e a d to an i n c r e a s e of t h e h i g h e n e r g y c o n t r i b u t i o n s , d e s p i t e of t h e c o n s t r a i n t t h a t t h e 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 h o u l d not b e m o d i f i e d by t h i s p r o c e d u r e n e a r v ~ v o. H o w e v e r t h e l a s t c o n s t r a i n t l i m i t s s t i l l t h e i n c r e a s e to 40% o r l e s s in t h e r e g i o n of i n t e r e s t . T h i s l i m i t i s not a t t a i n e d b y t h e p' p a r a m e t e r s of r e f . [13].
329
330
J. BAACKE
RE FERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9]
[10] [11] [12] [13]
A. Logunov, L.D. Soloviev and A. N. Tavkhelidze, Phys. Letters 24B (1967) 181. K. Igi and S. Matsuda, Phys. Rev. Letters 18 (1967) 625. R. Dolen, D. Horn and C. Schmid, Phys. Rev. 166 (1968) 1768. Lehmann, Nucl. Phys. 29 (1962) 300. G. HShler, J. Baacke and R. Strausz, Phys. Letters 21 (1966) 223, G. HShler, J. Baacke, H. Schlaile and P. Sonderegger, Phys. Letters 20 (1966) 79. G. HShler, J. Baacke and G. Eisenbeisz, Phys. i e t t e r s 22 (1966) 203. C. Lovelace, Proc. Heidelberg Int. Conf. on elementary particles (North-Holland Publ. Comp., Amsterdam, 1968) p.79. A. V. Stirling, P. Sonderegger, J. Kirz, P. Falk-Vairant, O, Guisan, C. Bruneton, P. Borgeaud, M. Yvert, J. P. Guillaud, C. Caverzasio and B. Amblard, Phys. Rev. Letters 14 (1965) 763; C. Bruneton, B. Amblard, P. Borgeua, C. Caverzasio, P. Falk-Vairant, J . P . Guillaud, O. Guisan, J. Kirz, P. Sonderegger, A.V. Stirling and M. Yvert, to be published. V. Barger and D. Cline, Phys. Rev. Letters 16 (1966) 913. G. HShler, R. Strausz and H. Schlaile, private communication. D. D. Reeder and K. V. L. Sarma, University of Wisconsin, preprint (December 1967). V . B a r g e r and R. J. N. Phillips, University of Wisconsin, preprint (August 1968).
Nuclear Physics B9 (1969) 324-330. North-Holland Pub1. Comp., Amsterdam
D E T E R M I N A T I O N OF REGGE-POLE RESIDUE F U N C T I O N S FROM FIXED-t DISPERSION RELATIONS J. BAACKE Institut ]'ur Theoretische Kernphysik, Universitttt Karlsruhe
Received 28 October 1968
Abstract: Fixed-t dispersion relations are used to derive a simple method to d e t e r mine (or test) Regge-pole residue functions from low-energy data. The method is applied to yN charge exchange scattering.
i. INTRODUCTION
V a r i o u s a u t h o r s [1-3] have d e r i v e d f i n i t e - e n e r g y sum r u l e s (FESR) to o b t a i n i n f o r m a t i o n on R e g g e - p o l e p a r a m e t e r s f r o m l o w - e n e r g y data. T h e s e sum r u l e s may be l o o k e d upon at a s a check of c o n s i s t e n c y b e t w e e n r e a l and i m a g i n a r y p a r t of the R e g g e - p o l e c o n t r i b u t i o n s by f i x e d - t d i s p e r s i o n r e l a tions (FTDR) [3]. H o w e v e r it is e v i d e n t that this c o n s i s t e n c y can a l w a y s be o b t a i n e d if the "finite e n e r g y " i s c h o s e n high enough and the i m a g i n a r y p a r t is taken f r o m R e g g e - p o l e exchange above the h i g h - e n e r g y l i m i t of p h a s e shift a n a l y s i s . This can be s e e n f r o m the r e s u l t s of ref. [4]. On the o t h e r hand in the ~N c h a r g e exchange s c a t t e r i n g the a m p l i t u d e s have not y e t r e a c h e d a s i m p l e Regge b e h a v i o u r even at K l a b = 11 G e V / c and p o s s i b l y a p p r e c i a b l e c o r r e c t i o n s have to be taken into account. We avoid p a r t of this difficulty by the r e s t r i c t e d a s s u m p t i o n , that only the i m a g i n a r y p a r t of the a m p l i t u d e shows Regge b e h a v i o u r above the e n e r g i e s of the p h a s e - s h i f t a n a l y s i s . This a s s u m p t i o n a l l o w s f o r c o r r e c t i o n s in the r e a l p a r t of the s c a t t e r i n g a m p l i t u d e which would o c c u r e.g. in m o d i f i e d Regge r e p r e s e n t a tions o r in the i n t e r f e r e n c e m o d e l . We then d e t e r m i n e the R e g g e - p o l e p a r a m e t e r s f r o m the r e q u i r e m e n t , that the h i g h - e n e r g y c o n t r i b u t i o n should p r o v i d e c o n s i s t e n c y between the p h a s e - s h i f t a n a l y s i s and FTDR. This m e t h od has the a d v a n t a g e that the p h a s e - s h i f t a n a l y s i s is c h e c k e d s i m u l t a n e o u s l y ; so we can d e c i d e c r i t i c a l l y upon the r e l i a b i l i t y of o u r r e s u l t s . In the f o l l o w ing we w r i t e the f o r m u l a s d i r e c t l y f o r 7rN c h a r g e exchange w h e r e the method is a p p l i e d h e r e . 2. DETERMINATION O F R E G G E - P O L E RESIDUE FUNCTIONS FROM F T D R We w r i t e the F T D R f o r the usual 7rN c h a r g e exchange a m p l i t u d e in the form
RESIDUE FUNCTIONS
'(-)(v,
ReA
t) -
v
F B ( v , t) +
t
1 - 4---~
2v ~-
f
325
oo dv, i m A ,(_)(v,,t ) v'2 _ v2
v°
d r ' v' l m B ( - ) ( v ', t)
2 T Re B(-)(v, t) -- FB(V , t) + ~ v
v ;2 - v 2
'
(la)
(lb)
'
o w i t h t h e f o l l o w i n g n o t a t i o n : M and # t h e n u c l e o n a n d p i o n m a s s , r e s p e c t i v e l y , v = w + (t/4M), w h e r e w i s t h e t o t a l p i o n l a b e n e r g y , vo = ~ + (t/4M), F B = [1 - ( t / 2 p 2 ) ] 8 ~ f 2 / ( v 2 - v 2 ) , w h e r e v B = - ( ~ 2 / 2 M ) + (t/4M) a n d f 2 = 0.081 i s the p i o n n u c l e o n c o u p l i n g c o n s t a n t , T h e i m a g i n a r y p a r t of the a m p l i t u d e s i s s u p p o s e d to b e g i v e n by t h e e x c h a n g e o f a s i n g l e R e g g e p o l e ( p - t r a j e c t o r y ) a b o v e w = ~ [P = ~ + ( t / 4 M ) ] a n d h a s the f o r m
imA,(-) Regge
b (t)
-
+
1 - t-L 4M2
...
a
½K(a) ( h )
(2a)
i m B(_) ab (t) ½ K ( a ) ( M ) a - 1 Regge M '
(2b)
with
/¢(~)
~ 2 2+a r(~ + a) =-~ r(i+a) '
(3)
and b±(l) a s d e f i n e d in r e f . [6]. W i t h t h i s a s s u m p t i o n e q s . ( l a , b ) m a y b e w r i t t e n f o r v < ~' in t h e f o r m P
~A,(-) = ReA,(-)(v,t)
v FB(V, t ) t 1 - 4-~
_
K(a)R
=
t
'
2v v° (4a)
'
4M 2
a (t) b_(t) AB(-)
_ Re B(-)(v,t)
_ FB(v,t)
~ _2f P
...
-
(4b)
M
o
where 1
RlX, X, a) = ~-~ f x
oo
X,~
dx'
X' 2 _X-------5 _ 1 :~a-1
zrM
oo
~
- -1 n=O 2n+ l - a
(x/~ 2n '
(x < ~).
(5)
326
J. BAACKE
Then AA '(-) and AB (-) can be c a l c u l a t e d f r o m p h a s e shifts alone. F r o m eqs. (4a,b) we obtain i m m e d i a t e l y t 1 - 4M2 A A ' ( - ) ( v , t)
b (t)=
a(t)b (t)=
1 v Y
hB(-)(v,t).
(6a)
(6b)
The r i g h t - h a n d s i d e s depend on a(t), but this is known r a t h e r a c c u r a t e l y f r o m h i g h - e n e r g y d a t a [9, 6, 7], f u r t h e r m o r e the a - d e p e n d e n c e is not v e r y sensitive. If o u r a s s u m p t i o n (2a,b) f o r the i m a g i n a r y p a r t is valid and the p h a s e shift solution is c o r r e c t , the r i g h t - h a n d s i d e s of eq. (6a,b) have to be independent of the e n e r g y and one would obtain a d e t e r m i n a t i o n of the r e s i d u e functions b+(t) which is m o r e r e l i a b l e than that f r o m F E S R , s i n c e a r a n g e of v - v a l u e s is u s e d and the c o n s i s t e n c y of the input with the d i s p e r s i o n r e lation has been tested.
3. DISCUSSION O F N U M E R I C A L R E S U L T S F r o m the p h a s e shift a n a l y s i s of L o v e l a c e et al. [8] ~ 4 ' ( - ) and AB (-) w e r e c a l c u l a t e d . Fig. 1 s h o w s the e n e r g y d e p e n d e n c e of the e x p r e s s i o n s (6a,b) f o r b+(t) and a(t) b(t) at s o m e v a l u e s of t. U n f o r t u n a t e l y we o b s e r v e s y s t e m a t i c d e v i a t i o n s f r o m the e x p e c t e d e n e r g y i n d e p e n d e n c e . N e v e r t h e l e s s we c a l c u l a t e d b+(t) and a(t) b(t) f r o m the e n e r g y a v e r a g e of t h e s e r e s u l t s , i g n o r e d h o w e v e r the d a t a below 370 MeV b e c a u s e t h e r e A A ' ( - ) and AB(-) a r e s m a l l c o m p a r e d to the whole a m p l i t u d e s . The r e s u l t s a r e plotted in fig. 2, t o g e t h e r with the bounds f o r [b+(t)] as o b t a i n e d in ref. [7]. The r e s i d u e f u n c t i o n s obtained h e r e a r e not c o m p a t i b l e with t h e s e bounds, which a r e i m p o s e d r i g o r o u s l y by 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 s b e t w e e n 3 and 18 GeV/c (ref. [9]), if we n e g l e c t e r r o r s in a(t). A s i m i l a r r e s u l t has been o b t a i n e d by Dolen et al. [3] f r o m FESR. The v i o l a t i o n of the bounds given in ref. [7] is r a t h e r s t r o n g and we have looked f o r p o s s i b l e e x p l a n a t i o n s : (a) In o u r c a l c u l a t i o n we did not take into a c c o u n t the c o n t r i b u t i o n s of the h i g h e r ~N r e s o n a n c e s (above 2 GeV lab e n e r g y ) . H o w e v e r t h e i r influence is n e g l i g i b l e in the r e g i o n of i n t e r e s t , if we c a l c u l a t e t h e i r c o n t r i b u t i o n f r o m the i n t e r f e r e n c e m o d e l with, e.g., the p a r a m e t e r s of B a r g e r and Cline [10]. The use of the i n t e r f e r e n c e m o d e l is s u r e l y justified if one is i n t e r e s t e d in the o r d e r of m a g n i t u d e of t h e s e c o r r e c t i o n s only. (b) The A n s a t z (2a,b) m a y be i n c o r r e c t and o t h e r c o n t r i b u t i o n s m i g h t have to be taken into a c c o u n t . H o w e v e r the a b s o l u t e value of A A ' ( - ) and AB(-) is too g r e a t g e n e r a l l y , so the high e n e r g y c o n t r i b u t i o n should be e n l a r g e d to obtain a g r e e m e n t . The addition of a s e c o n d R e g g e pole would not
RESIDUE FUNCTIONS
327
,I
t J
500
1000
1500
Trr [MeV]
500.
1000
1500
Trc ~-MeV]
15O0
T.~ [MeV]
b+(t) I.
x
I.
~m~x
•
x
b.,(t) 500
b+(tl :
1000
500 ;
:
:
~
1000 :
:
:
:
;
1500 :
:
:
:
I
:
T~[MeV]
x X
Fig. la. The energy dependence of the expressions (6a,b} for b+(t). The energy T is in MeV, the ordinates are dimensionless. a m e l i o r a t e the s i t u a t i o n b e c a u s e i t w o u l d d e c r e a s e f a s t e r with i n c r e a s i n g e n e r g y a n d the high e n e r g y c o n t r i b u t i o n w o u l d be r e d u c e d . The b e s t one c a n do i s to r e p l a c e the i n t e g r a n d of the d i s p e r s i o n i n t e g r a l at v > g by the b o u n d w h i c h f o l l o w s d i r e c t l y f r o m the e x p e r i m e n t a l 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 , w i t h o u t u s i n g the R e g g e Dole m o d e l o r the a b o v e d e c o m p o s i t i o n into n o - f l i p and flip a m p l i t u d e s . At s o m e t - v a l u e s the r e s u l t i n g i n e q u a l i t y i s v i o l a t e d i n d i c a t i n g t h a t the p h a s e - s h i f t s o l u t i o n h a s to be i m p r o v e d [11]. T h e s a m e c o n c l u s i o n f o l l o w s f r o m a c o m p a r i s o n of the T = ½ f o r w a r d a m p l i t u d e a s c a l c u l a t e d f r o m p h a s e s h i f t s and f r o m the f o r w a r d d i s p e r s i o n r e l a t i o n , u s i n g the m o s t r e c e n t * t o t a l c r o s s s e c t i o n d a t a [11]. N e v e r t h e l e s s we t e n d * The discrepancy is considerably s m a l l e r for a new set of forward amplitudes which was kindly sent to us by Dr. Lovelaee. However the new phase shifts are not yet available.
328
J. BAACKE
oqt)b_(t)
X
X
X
10 5 •
.
.
,
.
.
,
,
I
.
.
.
.
.
.
.
I
.
500 l ,,(t) b_(t)
.
.
.
.
.
.
.
.
I
.
1000
..........................
,
,
.
1500
T
1500
Trt EMeV]
~,
,
10 5
500
'('ib-(tl I
I
1000
. . . . . . . .
x I
I
:
:
500
;
!
~
i
;
1000
I
J
I
•
:
1500
~_
T~ [r~ eV] i
10
,*(t)b_ (t)
5
.. .. .. ..
t~-~5_6 ~ 500
I
,
.. .. .. ..
;000
,I
.. .. .. ..
1500
, I
• -
• ~ -
~
-5
-10
Fig. lb. The energy dependence of the expressions (6a,b) for a(t) b_(t). The energy T~ is in MeV, the ordinates a r e dimensionless. to t a k e o u r r e s u l t s (which a r e e s s e n t i a l l y e q u i v a l e n t to t h o s e of r e f . [3]) a s a c o n f i r m a t i o n of the b e h a v i o u r of b+~t) p r o p o s e d in r e f s . [7, 12]. E s p e c i a l l y the z e r o of b+(t) at t ~ - 0 . 1 5 (GeV/c) z and the s i g n of b_(t) a r e c l e a r l y e x h i b i t e d by the c o r r e s p o n d i n g b e h a v i o u r of A A ' ( - ) and AB(-) r e s p e c t i v e l y .
CONCLUSION T h e f i x e d - t d i s p e r s i o n r e l a t i o n s can be u s e d f o r a d e t e r m i n a t i o n of R e g g e - p o l e p a r a m e t e r s . In c o m p a r i s o n w i t h f i n i t e - e n e r g y s u m r u l e s one h a s the a d v a n t a g e t h a t the d e v i a t i o n s f r o m the c o n s i s t e n c y b e t w e e n the i n p u t and the d i s p e r s i o n r e l a t i o n s h o w i m m e d i a t e l y to w h a t e x t e n t the r e s u l t i s r e l i a b l e . A t p r e s e n t t h e s e d e v i a t i o n s a r e so l a r g e t h a t o n l y q u a l i t a t i v e c o n c l u s i o n s on the r e s i d u e f u n c t i o n s c a n b e d r a w n .
RESIDUE FUNCTIONS
I
; -':
:
:
b, (t)
-
~--
.
'
"
x N
x~ a(tl b_ (t)
/
t0.
~'
5. .1 ' ,;"-'5
I
"
%
-5. % \
\
\
'-10.
x
Fig. 2. Residue functions b+(t} and a(t} b(t} obtained by our method: x: our r e s u l t s , - - : proposal of ref. [7], - - - : bounds obtained in ref. [7]. T h e a u t h o r w o u l d l i k e to t h a n k P r o f e s s o r D r . G. H 6 h l e r f o r h i s i n t e r e s t in t h i s w o r k a n d v a l u a b l e d i s c u s s i o n s , a n d M e s s r s R. S t r a u s z and H. G. S c h l a i l e f o r c o m m u n i c a t i o n of u n p u b l i s h e d r e s u l t s .
NOTE
ADDED
~
PROOF
It w a s p o i n t e d out to t h e a u t h o r b y P r o f e s s o r C. L o v e l a c e t h a t t h e s t a t e m e n t in s e c t . 3(b) i s t r i v i a l l y w r o n g . In f a c t t h e a d d i t i o n of a s e c o n d R e g g e p o l e (p') m a y l e a d to an i n c r e a s e of t h e h i g h e n e r g y c o n t r i b u t i o n s , d e s p i t e of t h e c o n s t r a i n t t h a t t h e 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 h o u l d not b e m o d i f i e d by t h i s p r o c e d u r e n e a r v ~ v o. H o w e v e r t h e l a s t c o n s t r a i n t l i m i t s s t i l l t h e i n c r e a s e to 40% o r l e s s in t h e r e g i o n of i n t e r e s t . T h i s l i m i t i s not a t t a i n e d b y t h e p' p a r a m e t e r s of r e f . [13].
329
330
J. BAACKE
RE FERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9]
[10] [11] [12] [13]
A. Logunov, L.D. Soloviev and A. N. Tavkhelidze, Phys. Letters 24B (1967) 181. K. Igi and S. Matsuda, Phys. Rev. Letters 18 (1967) 625. R. Dolen, D. Horn and C. Schmid, Phys. Rev. 166 (1968) 1768. Lehmann, Nucl. Phys. 29 (1962) 300. G. HShler, J. Baacke and R. Strausz, Phys. Letters 21 (1966) 223, G. HShler, J. Baacke, H. Schlaile and P. Sonderegger, Phys. Letters 20 (1966) 79. G. HShler, J. Baacke and G. Eisenbeisz, Phys. i e t t e r s 22 (1966) 203. C. Lovelace, Proc. Heidelberg Int. Conf. on elementary particles (North-Holland Publ. Comp., Amsterdam, 1968) p.79. A. V. Stirling, P. Sonderegger, J. Kirz, P. Falk-Vairant, O, Guisan, C. Bruneton, P. Borgeaud, M. Yvert, J. P. Guillaud, C. Caverzasio and B. Amblard, Phys. Rev. Letters 14 (1965) 763; C. Bruneton, B. Amblard, P. Borgeua, C. Caverzasio, P. Falk-Vairant, J . P . Guillaud, O. Guisan, J. Kirz, P. Sonderegger, A.V. Stirling and M. Yvert, to be published. V. Barger and D. Cline, Phys. Rev. Letters 16 (1966) 913. G. HShler, R. Strausz and H. Schlaile, private communication. D. D. Reeder and K. V. L. Sarma, University of Wisconsin, preprint (December 1967). V . B a r g e r and R. J. N. Phillips, University of Wisconsin, preprint (August 1968).