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Some remarks about a second pomeranchuk trajectory
Volume 5, n u m b e r 4
PHYSICS
LETTERS
15 July 1963
I t i s i n t e r e s t i n g to c o n s i d e r s o m e of t h e t h e o r e b l e c h a m b e r a n d of t h e B e v a t r o n , a n d to o u r s c a n t i c a l i m p l i c a t i o n s of t h e x m e s o n . T h e SU 3 u n i t a r y ning and measuring staff, without whose help this s y m m e t r y s c h e m e p r o p o s e d b y G e l l - M a n n 8) a n d experiment would not have been possible. N e' e m a n O) has been used with considerable success r e c e n t l y in c l a s s i f y ~ g t h e known m e s o n s a n d b a r y o n s into m u l t i p l e t s . It i s i m p o r t a n t to n o t e t h a t no o t h e r known m e s o n s c o u l d f o r m a u n i t a r y m u l t l p l e t w i t h t h e x m e s o n . I t w i l l b e i n t e r e s t i n g to s e e if s u c h Refe~'ences " u n i t a r y p a r t n e r s " e x i s t . T h e r e a r e p r e s e n t l y two p o s s i b l e a s s i g n m e n t s f o r t h e x t h a t do not r e q u i r e 1) G.Alexander et a l . , Phys. Rev. Letters 8 (1962) 447. G. Alexander et a l . , Proc. Intern. Conf. on High Energy any " p a r t n e r s " . T a k e d a h a s p o i n t e d out a t h e o r y of Phys. (CERN, Geneva, 1962), p. 320. t h e w e a k i n t e r a c t i o n s w h i c h i s c o m p a t i b l e with a l l D. H. M i l l e r et a l . , Phys. Letters 5 (1963) 279. p r e s e n t l y k n o w n e x p e r i m e n t a l data, b u t w h i c h a l 2) H.K. Ticho et a l . , Lawrence Radiation Laboratory, l o w s t h e p o s s i b i l i t y of copious p r o d u c t i o n of t h e i n Berkeley (unpublished work). t e r m e d i a t e v e c t o r b o s o n s 11). A l s o N a m b u a n d 3) V. Cook et a l . , Phys. Rev. 123 (1961) 320. 4) L . W . A l v a r e z et a l . , Phys. Rev. Letters 10 (1963) S a k u r a i h a v e c o n s i d e r e d t h e h y p o t h e s i s t h a t the K* 184. (890 MeV) m e s o n s a r e c o u p l e d to s t r a n g e n e s s 5) M. Baqi B~g and P . D e Celles, Phys. Rev. Letters 6 changing currents which are conserved "as exactly (1961) 145, 428 (E); see also C. Chan, Phys. Rev. a s p o s s i b l e " 12). T h i s h y p o t h e s i s s u g g e s t s t h e e x Letters 6 (1961) 383 and W. G.Wagner and D. H. Sharp, i s t e n c e of a Y = + 1, T = ½, J = 0 + m e s o n w h o s e Phys.Rev. 128 (1962) 2899. 6) S. Glashow, private communication. c o u p l i n g to o t h e r s t r o n g l y i n t e r a c t i n g p a r t i c l e s v a n 7) A. Rosenfeld, D. Carmony and R. Van de Walle, Phys. i s h e s in t h e l i m i t of e x a c t u n i t a r y s y m m e t r y . P r e s Rev. Letters 8 (1962) 293. e n t e x p e r i m e n t a l d a t a do n o t a l l o w u s to d r a w a n y 8) M.Gell-Mann, Phys.Rev. 125 (1962)1067; California c o n c l u s i o n s a s to t h e a s s i g n m e n t of t h e x m e s o n to Institute of Technology Report CTSL-20, 1961 (una n y of t h e s e s c h e m e s . Any d i s c u s s i o n of a s s i g n published). 9) Y. Ne'eman, Nuclear Phys. 26 (1961) 222. m e n t s a t t h i s s t a g e m u s t b e c o n s i d e r e d to b e p u r e l y S. L. Glashow and A. H. Rosenfeld, Phys. Rev. Letters IO) speculative. 10 (1963) 192. 11) G.Takeda, Phys.Rev. Letters 10 (1963) 167. W e w i s h to t h a n k P r o f e s s o r L u i s W. A l v a r e z f o r 12) Y. Nambu and J. J. Sakurai, K*(725) and the Strangeness h i s e n c o u r a g e m e n t a n d s u p p o r t in t h i s e x p e r i m e n t . Changing Currents of Unitary Symmetry, Enrico W e a r e i n d e b t e d to t h e o p e r a t o r s of t h e 7 2 - i n c h b u b F e r m i Institute Report 63-26, April 1963 (to be pub* F o r a review of the experimental situation see r e f s . 4, 10). lished).
$$*$$
SOME R E M A R K S
ABOUT
A SECOND
POMERANCHUK
TRAJECTORY
V. I. L E N D Y E L * a n d J. M A T H E W S
California Institute of Technology, Pasadena, California Received 17 June 1963
R e c e n t l y , a t t e n t i o n h a s b e e n d r a w n to t h e p o s s i b i l i t y of a s e c o n d P o m e r a n c h u k t r a j e c t o r y . It w a s p o i n t e d out by I g i 1) t h a t t h e a s s u m p t i o n of no t r a j e c t o r y ~(t) w i t h ~(0) > 0 e x c e p t t h e P o m e r a n c h u k t r a j e c t o r y (~p(0) = 1) l e a d s to a s u m r u l e f o r t h e non-isotopic-spin-flip scattering length which disagrees seriously with the experimental value. Igi f u r t h e r o b s e r v e d 1, 2) t h a t by i n c l u d i n g a s e c o n d v a c u u m t r a j e c t o r y ~ p , (t), w i t h ~ p , (0) ~ 0.5, t h e c o r rect scattering length can be obtained. * On leave of absence from Uzhgorod University, USSR.
286
On t h e o t h e r h a n d , i t h a s b e e n p o i n t e d out 3) t h a t h i g h e n e r g y pp a n d p~ t o t a l c r o s s s e c t i o n d a t a a l s o s e e m to r e q u i r e a P'. T h e n e a r l y c o n s t a n t or__ p and rapidly decreasing a~ have been fitted by S~p a n d W a g n e r 3), who ~ g g e s t O~p,(0) ~. 0.3, w h i c h e s sentially agrees with Igi's result. It w o u l d s e e m of i n t e r e s t to d e t e r m i n e a p , (0) a s a c c u r a t e l y a s p o s s i b l e , u s i n g b o t h of t h e a b o v e a p p r o a c h e s . If o n e a s s u m e s t h a t no c u t s e x i s t b e t w e e n cx = 0 a n d ~ = 1, t h e n t h e two m e t h o d s s h o u l d g i v e t h e s a m e r e s u l t . On t h e o t h e r h a n d , if a d i s c r e p a n c y
Volume 5, number 4
PHYSICS LETTERS
between the two values of ap, (0) should appear, one would have to conclude that either a cut or an additional vacuum trajectory must be taken into account. Therefore we have first applied at least squares analysis to the latestexperimental data on ~±p cross sections f r o m 3 - 20 GeV t1~--4-7!'and pp,. pp c r o s . s sections f r o m 5 - 30 GeV , ). The dam w e r e ilkted by e x p r e s s i o n s of the usual f o r m 2, 10) *:
g2 aB = - ~
XZ 1.5
(1)
05
~w-1
In principle, totalnp cross section data should also be used, but the large uncertainties render them practically useless. The #p and (pp, p~) data were fitted separately. The results are not particularly sensitive to ,~p and ~w; best fits occur for a o ,~ a~ -~ 0.4. Typical c u r v e s of x2/(N-n) (N = n u m b e r of data points, n = number of p a r a m e t e r s fitted) against ap, a r e shown in figs. 1 and 2. As one can see f r o m the f i g u r e s , x 2 has a r a t h e r b r o a d minimum in the region ap, ~ 0.1. Next, we examined Igi's approach to see how p r e c i s e a value of ap, it gives. We made one modification in IgPs f o r m a l i s m , however. Rather than using the specific f o r m p r e d i c t e d by Reggeism f o r the asymptotic behaviour of the f o r w a r d ~rp s c a t t e r i n g amplitude f(+)(v), we adopted the following, m o r e g e n e r a l , a p p r o a c h : A s s u m e the total c r o s s section
(r(v) (= ½ (Cr~+p(v) + cr -p(V)) has the asymptotic f o r m ~(~) ~ ~
N-n I0
Crpp = A4 + A5 yap,-1 _ A6 yap-1 _ A7 va~-I -1+A7
(4M 2 _ # 2 )
; /
2.5
2.0
~ - P = A1 + A2 yap,-1 + A3 yap-1
=A4+A5v~P'-I+A6v'Vp
~2
w i t h g 2 / 4 ~ ~ 15. a(+) is the scattering length (-(0.001 ± 0.004)/~).
~Tr+P = .41 + A2 p a p , - 1 _ A3 pap-1
UPP
15 July 1963
+ g(~) + o(!/~)
I -0 5
-I 0
I 0
I 0.5
ap I
1.0
Fig. 1. Plot of X2/(N-n) for ,~*'pdata from refs. 4-7); N=57, n=3. 2.5
P-C X i I.~
j O.~ -I.0
I -0 5
I 0
l 0.5
ap #
1.0
Fig. 2. Plot of X2/~V-~) for pp, p~ data from refs. 8, 9); N=44, n = 4 . In a c c o r d with the a s s u m e d f o r m s (1), we set
g (v) = A v '~- 1 in (3) and obtain where g(u) is an arbitrary function which vanishes as v -- oo. The optical theorem
I m f (+)(v) _kcr(v) -~
(k = ~ v 2 _ #2) X [e(k) - G~o - Av a- 1]
and the d i s p e r s i o n r e l a t i o n obeyed by f(+)(v) gives for the asymptotic f o r m of f(+)(v) as v -. oo the following :
~2
•
?g(~')d~'
A
~a~r(1
2~2 a
-
(4)
½~)
r(~-~
In deriving (4), we have assumed that
5 being an a r b i t r a r y constant. The sum r u l e which r e s u l t s by proceeding in d i r e c t analogy with Igi's work is G(+)
(1 + D ) = a B + 5 +
a B is the Born contribution:
I);
A# a 2n2a
since this choice for 5 is required if (2) is to become
1 ;dk[cr(k)_V~o_g(v)] 2~r2 o
1 ~o +2~r----~ / dug(v)~-
=-
iucroo Av a ( 1 + e-i~a~
(3)
+0(1) f(+)(V) ~ ~ "-~-Sin~a / * Hereafter we write ap, for av,(0), etc. Also, v refers to totallaboratoryenergies thrSughout,while ~ and M are the pion and nucleon masses.
28'7
Volume 5, number 4
PHYSICS LETTERS
as suggested by Reggeism. If one s e r i o u s l y believed in a cut in the s - p l a n e , one could a p p r o p r i ately m o d i f y g ( v ) and d e r i v e the c o r r e c t e d sum rule f r o m (3). We have c o m p a r e d (4) with the existing experimental data for v a r i o u s values of ~, which is c l e a r l y t o b e identified with (Xp,. The integral in (4) was evaluated numerically-up to 5 GeV; contributions f r o m above this point w e r e a s s u m e d to be negligible. The r e s u l t s a r e basically in a g r e e m e n t with Igi 2). We wish to emphasize one point, however ; we believe the e r r o r s in the values f o r the right side of (4) to be considerably s m a l l e r than those given by Igi. The r e a s o n is that the e r r o r s in the p a r a m e t e r s A and ~=o, as given by our l e a s t s q u a r e s a n a l y s i s , a r e strongly c o r r e l a t e d . When account is taken of this c o r r e l a t i o n , it turns out that n e a r l y all of the uncertainty in the n u m e r i c a l evaluation of (4) c o m e s f r o m the uncertainty in the integral
/ 5 GeV dk (k). o This integral was evaluated using the ~+p and ~r-p total c r o s s sections tabulated by B a r a s h e n k o v and Mattsev 11); the e r r o r was estimated to be 5% of the result. The n u m e r i c a l r e s u l t s for the right side of (4) a r e shown in fig. 3, together with our estimated e r r o r s . Fig. 3 strongly suggests that ,, lies in the r a n g e 0.6 - 0.8.
'1
TI
0 -2 -4 -6 -I0
I -0 5
I 0 =1
I
I 0.5
I0
Fig. 3. Numerical evaluation of the right side of sum rule (4) in GeV- . Note that the experimental value of a(+) (1 + p/M) is 0.01 + 0.03 in these units.
288
15 July 1963
The i n d e t e r m i n a c y of our fits to the h i g h - e n e r g y total c r o s s sections, as shown by the c u r v e s of figs. 1 and 2, does not allow us to establish a disc r e p a n c y between the value of ~ so d e t e r m i n e d , and the value of ~ f r o m the sum rule (4) (fig. 3). N e v e r theless, the data appear to suggest a d i s c r e p a n c y , and we would like to call attention to the usefulness of improving this c o m p a r i s o n , principally by i m proving the h i g h - e n e r g y c r o s s section m e a s u r e m e n t s and by extending them to higher energies. If the d i s c r e p a n c y t u r n s out to be r e a l , t h e r e a r e v a r i o u s possible r e a s o n s . We have a l r e a d y mentioned the possibility of cuts in the complex angular m o m e n t u m plane. Another possibility is the existence of a third vacuum t r a j e c t o r y P " c o r r e s ponding to the ABC two-pion r e s o n a n c e (?.) near 300 MeV. This does not affect the f o r m of our sum rule (4), but it s e r v e s to r e d u c e the value of A significantly when introduced into the l e a s t s q u a r e s analysis. PreLiminary e s t i m a t e s show that ~p,, ~ - 0.3, ~i ~, ~ 0.2 enable (4) to be satisfied. It is interesting t o note that an ABC-type r e s o n a n c e has also been shown to be n e c e s s a r y 12) in an a n a l y s i s of the S~(+), P½(+) and P~(+)IrN s c a t t e r i n g ampLitudes using d i s p e r s i o n r ~ a t i o n s and including TrTr s c a t t e r i n g effects. We should like to thank Dr. H. R u d e r m a n for sending us unpublished n u m e r i c a l total c r o s s s e c tion data f r o m Brookhaven National L a b o r a t o r y . One of us (V.I.L.) would like to e x p r e s s his g r a t itude to the California Institute of Technology f o r the hospitality extended to him during the a c a d e m i c y e a r 1962-1963. The other (J. M. ) would like to acknowledge the support of the Radio C o r p o r a t i o n of A m e r i c a . 1) K. Igl, Phys. Rev. Letters 9 (1962) 76. 2) K. Igi, Phys.Rev. 130 (1963) 820. 3) D.H.Sharp and W.G.Wagner, Phys.Rev., to be published. 4) G. yon Dardel et al., Phys.Rev. Letters 7 (1961) 127. 5) S.J. Llndenbaum et al., Phys.Rev. Letters 7 (1961) 352. 6) G. von Dardel et al., Phys. Rev. Letters 8 (1962) 173. 7) A.N.Diddens et al., Phys.Rev. Letters 10 (1963) 262. 8) S.J. Lindenbaum et al., Phys.Rev. Letters 7 (1961) 185. 9) A.N.Diddens et al., Phys.Rev. Letters 9 (1962) 32. 10) B.M.Udgaonkar, Phys.Rev. Letters 8 (1962) 142. 11) V.S. Barashenkov and V. Maltsev, Fozschzitte der Fizik. 12) P.S. Isaev, V.I. Lendyel and W.A. Meshcheryakov, Dubna preprint E-1189 (1963).
PHYSICS
LETTERS
15 July 1963
I t i s i n t e r e s t i n g to c o n s i d e r s o m e of t h e t h e o r e b l e c h a m b e r a n d of t h e B e v a t r o n , a n d to o u r s c a n t i c a l i m p l i c a t i o n s of t h e x m e s o n . T h e SU 3 u n i t a r y ning and measuring staff, without whose help this s y m m e t r y s c h e m e p r o p o s e d b y G e l l - M a n n 8) a n d experiment would not have been possible. N e' e m a n O) has been used with considerable success r e c e n t l y in c l a s s i f y ~ g t h e known m e s o n s a n d b a r y o n s into m u l t i p l e t s . It i s i m p o r t a n t to n o t e t h a t no o t h e r known m e s o n s c o u l d f o r m a u n i t a r y m u l t l p l e t w i t h t h e x m e s o n . I t w i l l b e i n t e r e s t i n g to s e e if s u c h Refe~'ences " u n i t a r y p a r t n e r s " e x i s t . T h e r e a r e p r e s e n t l y two p o s s i b l e a s s i g n m e n t s f o r t h e x t h a t do not r e q u i r e 1) G.Alexander et a l . , Phys. Rev. Letters 8 (1962) 447. G. Alexander et a l . , Proc. Intern. Conf. on High Energy any " p a r t n e r s " . T a k e d a h a s p o i n t e d out a t h e o r y of Phys. (CERN, Geneva, 1962), p. 320. t h e w e a k i n t e r a c t i o n s w h i c h i s c o m p a t i b l e with a l l D. H. M i l l e r et a l . , Phys. Letters 5 (1963) 279. p r e s e n t l y k n o w n e x p e r i m e n t a l data, b u t w h i c h a l 2) H.K. Ticho et a l . , Lawrence Radiation Laboratory, l o w s t h e p o s s i b i l i t y of copious p r o d u c t i o n of t h e i n Berkeley (unpublished work). t e r m e d i a t e v e c t o r b o s o n s 11). A l s o N a m b u a n d 3) V. Cook et a l . , Phys. Rev. 123 (1961) 320. 4) L . W . A l v a r e z et a l . , Phys. Rev. Letters 10 (1963) S a k u r a i h a v e c o n s i d e r e d t h e h y p o t h e s i s t h a t the K* 184. (890 MeV) m e s o n s a r e c o u p l e d to s t r a n g e n e s s 5) M. Baqi B~g and P . D e Celles, Phys. Rev. Letters 6 changing currents which are conserved "as exactly (1961) 145, 428 (E); see also C. Chan, Phys. Rev. a s p o s s i b l e " 12). T h i s h y p o t h e s i s s u g g e s t s t h e e x Letters 6 (1961) 383 and W. G.Wagner and D. H. Sharp, i s t e n c e of a Y = + 1, T = ½, J = 0 + m e s o n w h o s e Phys.Rev. 128 (1962) 2899. 6) S. Glashow, private communication. c o u p l i n g to o t h e r s t r o n g l y i n t e r a c t i n g p a r t i c l e s v a n 7) A. Rosenfeld, D. Carmony and R. Van de Walle, Phys. i s h e s in t h e l i m i t of e x a c t u n i t a r y s y m m e t r y . P r e s Rev. Letters 8 (1962) 293. e n t e x p e r i m e n t a l d a t a do n o t a l l o w u s to d r a w a n y 8) M.Gell-Mann, Phys.Rev. 125 (1962)1067; California c o n c l u s i o n s a s to t h e a s s i g n m e n t of t h e x m e s o n to Institute of Technology Report CTSL-20, 1961 (una n y of t h e s e s c h e m e s . Any d i s c u s s i o n of a s s i g n published). 9) Y. Ne'eman, Nuclear Phys. 26 (1961) 222. m e n t s a t t h i s s t a g e m u s t b e c o n s i d e r e d to b e p u r e l y S. L. Glashow and A. H. Rosenfeld, Phys. Rev. Letters IO) speculative. 10 (1963) 192. 11) G.Takeda, Phys.Rev. Letters 10 (1963) 167. W e w i s h to t h a n k P r o f e s s o r L u i s W. A l v a r e z f o r 12) Y. Nambu and J. J. Sakurai, K*(725) and the Strangeness h i s e n c o u r a g e m e n t a n d s u p p o r t in t h i s e x p e r i m e n t . Changing Currents of Unitary Symmetry, Enrico W e a r e i n d e b t e d to t h e o p e r a t o r s of t h e 7 2 - i n c h b u b F e r m i Institute Report 63-26, April 1963 (to be pub* F o r a review of the experimental situation see r e f s . 4, 10). lished).
$$*$$
SOME R E M A R K S
ABOUT
A SECOND
POMERANCHUK
TRAJECTORY
V. I. L E N D Y E L * a n d J. M A T H E W S
California Institute of Technology, Pasadena, California Received 17 June 1963
R e c e n t l y , a t t e n t i o n h a s b e e n d r a w n to t h e p o s s i b i l i t y of a s e c o n d P o m e r a n c h u k t r a j e c t o r y . It w a s p o i n t e d out by I g i 1) t h a t t h e a s s u m p t i o n of no t r a j e c t o r y ~(t) w i t h ~(0) > 0 e x c e p t t h e P o m e r a n c h u k t r a j e c t o r y (~p(0) = 1) l e a d s to a s u m r u l e f o r t h e non-isotopic-spin-flip scattering length which disagrees seriously with the experimental value. Igi f u r t h e r o b s e r v e d 1, 2) t h a t by i n c l u d i n g a s e c o n d v a c u u m t r a j e c t o r y ~ p , (t), w i t h ~ p , (0) ~ 0.5, t h e c o r rect scattering length can be obtained. * On leave of absence from Uzhgorod University, USSR.
286
On t h e o t h e r h a n d , i t h a s b e e n p o i n t e d out 3) t h a t h i g h e n e r g y pp a n d p~ t o t a l c r o s s s e c t i o n d a t a a l s o s e e m to r e q u i r e a P'. T h e n e a r l y c o n s t a n t or__ p and rapidly decreasing a~ have been fitted by S~p a n d W a g n e r 3), who ~ g g e s t O~p,(0) ~. 0.3, w h i c h e s sentially agrees with Igi's result. It w o u l d s e e m of i n t e r e s t to d e t e r m i n e a p , (0) a s a c c u r a t e l y a s p o s s i b l e , u s i n g b o t h of t h e a b o v e a p p r o a c h e s . If o n e a s s u m e s t h a t no c u t s e x i s t b e t w e e n cx = 0 a n d ~ = 1, t h e n t h e two m e t h o d s s h o u l d g i v e t h e s a m e r e s u l t . On t h e o t h e r h a n d , if a d i s c r e p a n c y
Volume 5, number 4
PHYSICS LETTERS
between the two values of ap, (0) should appear, one would have to conclude that either a cut or an additional vacuum trajectory must be taken into account. Therefore we have first applied at least squares analysis to the latestexperimental data on ~±p cross sections f r o m 3 - 20 GeV t1~--4-7!'and pp,. pp c r o s . s sections f r o m 5 - 30 GeV , ). The dam w e r e ilkted by e x p r e s s i o n s of the usual f o r m 2, 10) *:
g2 aB = - ~
XZ 1.5
(1)
05
~w-1
In principle, totalnp cross section data should also be used, but the large uncertainties render them practically useless. The #p and (pp, p~) data were fitted separately. The results are not particularly sensitive to ,~p and ~w; best fits occur for a o ,~ a~ -~ 0.4. Typical c u r v e s of x2/(N-n) (N = n u m b e r of data points, n = number of p a r a m e t e r s fitted) against ap, a r e shown in figs. 1 and 2. As one can see f r o m the f i g u r e s , x 2 has a r a t h e r b r o a d minimum in the region ap, ~ 0.1. Next, we examined Igi's approach to see how p r e c i s e a value of ap, it gives. We made one modification in IgPs f o r m a l i s m , however. Rather than using the specific f o r m p r e d i c t e d by Reggeism f o r the asymptotic behaviour of the f o r w a r d ~rp s c a t t e r i n g amplitude f(+)(v), we adopted the following, m o r e g e n e r a l , a p p r o a c h : A s s u m e the total c r o s s section
(r(v) (= ½ (Cr~+p(v) + cr -p(V)) has the asymptotic f o r m ~(~) ~ ~
N-n I0
Crpp = A4 + A5 yap,-1 _ A6 yap-1 _ A7 va~-I -1+A7
(4M 2 _ # 2 )
; /
2.5
2.0
~ - P = A1 + A2 yap,-1 + A3 yap-1
=A4+A5v~P'-I+A6v'Vp
~2
w i t h g 2 / 4 ~ ~ 15. a(+) is the scattering length (-(0.001 ± 0.004)/~).
~Tr+P = .41 + A2 p a p , - 1 _ A3 pap-1
UPP
15 July 1963
+ g(~) + o(!/~)
I -0 5
-I 0
I 0
I 0.5
ap I
1.0
Fig. 1. Plot of X2/(N-n) for ,~*'pdata from refs. 4-7); N=57, n=3. 2.5
P-C X i I.~
j O.~ -I.0
I -0 5
I 0
l 0.5
ap #
1.0
Fig. 2. Plot of X2/~V-~) for pp, p~ data from refs. 8, 9); N=44, n = 4 . In a c c o r d with the a s s u m e d f o r m s (1), we set
g (v) = A v '~- 1 in (3) and obtain where g(u) is an arbitrary function which vanishes as v -- oo. The optical theorem
I m f (+)(v) _kcr(v) -~
(k = ~ v 2 _ #2) X [e(k) - G~o - Av a- 1]
and the d i s p e r s i o n r e l a t i o n obeyed by f(+)(v) gives for the asymptotic f o r m of f(+)(v) as v -. oo the following :
~2
•
?g(~')d~'
A
~a~r(1
2~2 a
-
(4)
½~)
r(~-~
In deriving (4), we have assumed that
5 being an a r b i t r a r y constant. The sum r u l e which r e s u l t s by proceeding in d i r e c t analogy with Igi's work is G(+)
(1 + D ) = a B + 5 +
a B is the Born contribution:
I);
A# a 2n2a
since this choice for 5 is required if (2) is to become
1 ;dk[cr(k)_V~o_g(v)] 2~r2 o
1 ~o +2~r----~ / dug(v)~-
=-
iucroo Av a ( 1 + e-i~a~
(3)
+0(1) f(+)(V) ~ ~ "-~-Sin~a / * Hereafter we write ap, for av,(0), etc. Also, v refers to totallaboratoryenergies thrSughout,while ~ and M are the pion and nucleon masses.
28'7
Volume 5, number 4
PHYSICS LETTERS
as suggested by Reggeism. If one s e r i o u s l y believed in a cut in the s - p l a n e , one could a p p r o p r i ately m o d i f y g ( v ) and d e r i v e the c o r r e c t e d sum rule f r o m (3). We have c o m p a r e d (4) with the existing experimental data for v a r i o u s values of ~, which is c l e a r l y t o b e identified with (Xp,. The integral in (4) was evaluated numerically-up to 5 GeV; contributions f r o m above this point w e r e a s s u m e d to be negligible. The r e s u l t s a r e basically in a g r e e m e n t with Igi 2). We wish to emphasize one point, however ; we believe the e r r o r s in the values f o r the right side of (4) to be considerably s m a l l e r than those given by Igi. The r e a s o n is that the e r r o r s in the p a r a m e t e r s A and ~=o, as given by our l e a s t s q u a r e s a n a l y s i s , a r e strongly c o r r e l a t e d . When account is taken of this c o r r e l a t i o n , it turns out that n e a r l y all of the uncertainty in the n u m e r i c a l evaluation of (4) c o m e s f r o m the uncertainty in the integral
/ 5 GeV dk (k). o This integral was evaluated using the ~+p and ~r-p total c r o s s sections tabulated by B a r a s h e n k o v and Mattsev 11); the e r r o r was estimated to be 5% of the result. The n u m e r i c a l r e s u l t s for the right side of (4) a r e shown in fig. 3, together with our estimated e r r o r s . Fig. 3 strongly suggests that ,, lies in the r a n g e 0.6 - 0.8.
'1
TI
0 -2 -4 -6 -I0
I -0 5
I 0 =1
I
I 0.5
I0
Fig. 3. Numerical evaluation of the right side of sum rule (4) in GeV- . Note that the experimental value of a(+) (1 + p/M) is 0.01 + 0.03 in these units.
288
15 July 1963
The i n d e t e r m i n a c y of our fits to the h i g h - e n e r g y total c r o s s sections, as shown by the c u r v e s of figs. 1 and 2, does not allow us to establish a disc r e p a n c y between the value of ~ so d e t e r m i n e d , and the value of ~ f r o m the sum rule (4) (fig. 3). N e v e r theless, the data appear to suggest a d i s c r e p a n c y , and we would like to call attention to the usefulness of improving this c o m p a r i s o n , principally by i m proving the h i g h - e n e r g y c r o s s section m e a s u r e m e n t s and by extending them to higher energies. If the d i s c r e p a n c y t u r n s out to be r e a l , t h e r e a r e v a r i o u s possible r e a s o n s . We have a l r e a d y mentioned the possibility of cuts in the complex angular m o m e n t u m plane. Another possibility is the existence of a third vacuum t r a j e c t o r y P " c o r r e s ponding to the ABC two-pion r e s o n a n c e (?.) near 300 MeV. This does not affect the f o r m of our sum rule (4), but it s e r v e s to r e d u c e the value of A significantly when introduced into the l e a s t s q u a r e s analysis. PreLiminary e s t i m a t e s show that ~p,, ~ - 0.3, ~i ~, ~ 0.2 enable (4) to be satisfied. It is interesting t o note that an ABC-type r e s o n a n c e has also been shown to be n e c e s s a r y 12) in an a n a l y s i s of the S~(+), P½(+) and P~(+)IrN s c a t t e r i n g ampLitudes using d i s p e r s i o n r ~ a t i o n s and including TrTr s c a t t e r i n g effects. We should like to thank Dr. H. R u d e r m a n for sending us unpublished n u m e r i c a l total c r o s s s e c tion data f r o m Brookhaven National L a b o r a t o r y . One of us (V.I.L.) would like to e x p r e s s his g r a t itude to the California Institute of Technology f o r the hospitality extended to him during the a c a d e m i c y e a r 1962-1963. The other (J. M. ) would like to acknowledge the support of the Radio C o r p o r a t i o n of A m e r i c a . 1) K. Igl, Phys. Rev. Letters 9 (1962) 76. 2) K. Igi, Phys.Rev. 130 (1963) 820. 3) D.H.Sharp and W.G.Wagner, Phys.Rev., to be published. 4) G. yon Dardel et al., Phys.Rev. Letters 7 (1961) 127. 5) S.J. Llndenbaum et al., Phys.Rev. Letters 7 (1961) 352. 6) G. von Dardel et al., Phys. Rev. Letters 8 (1962) 173. 7) A.N.Diddens et al., Phys.Rev. Letters 10 (1963) 262. 8) S.J. Lindenbaum et al., Phys.Rev. Letters 7 (1961) 185. 9) A.N.Diddens et al., Phys.Rev. Letters 9 (1962) 32. 10) B.M.Udgaonkar, Phys.Rev. Letters 8 (1962) 142. 11) V.S. Barashenkov and V. Maltsev, Fozschzitte der Fizik. 12) P.S. Isaev, V.I. Lendyel and W.A. Meshcheryakov, Dubna preprint E-1189 (1963).