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Volume 34B, number 3 PHYSICS L E T T E R S 15 February 1971
LOW E N E R G Y D A T A , A N A L Y T I C I T Y AND T H E V I O L A T I O N O F P O M E R A N C H U K ' S T H E O R E M
R . W I T *
CERN - Geneva, Switzerland
Received 10 December 1970
In the case of 7r±p scattering the low energy data do not seem to agree with the predictions following from the recently introduced parametrizations violating the Pomeranchuk theorem.
As rea l i zed by many authors [1-4] the usual forward d ispers ion re la t ion for the c ross ing odd ampl i - tude is not a very powerful tool for analyzing the high energy data. For different (but "reasonable") asymptot ic behaviours of the sca t te r ing ampli tudes the low energy predic t ions a re not d is t inguishable f rom each other. Therefore , it is obvious one should look for fur ther , more r e s t r i c t i ve cons t ra in t s on the poss ib le asymptotic behaviour of the total c ro s s - s e c t i ons .
In this le t ter we der ive an in tegra l r ep resen ta t ion for the c ross ing odd ampli tude valid only if the Pomeranchuk theorem is violated. P r e sen t high and low energy data suggest, however, that this r e l a - tion is unlikely to be satisfied. For the sake of s impl ic i ty we r e s t r i c t our se lves to the case of ~=p scat- ter ing. The analys is given below can be, however, extended to the cases of KN and NN sca t te r ing with some obvious changes.
Our in te res t will be concent ra ted on the high energy behaviour of the c ross ing odd ampli tude** T2(k) = ½ ( T (k) - T+(k)). Its asymptotic p roper t i e s are very often d i scussed in connection with the poss ib le violat ion of Pomeranchuk ' s theorem [cf. ref. [6] and the r e f e rences contained therein], In this case the high energy pa rame t r i za t ion of T2(k) takes usual ly the form [1-4]
T2(k) = y_ co(lnk - B - in/2) + iypeXp (- in /2) co0.5 , (1)
where, f rom the beginning y is a ssumed to be different f rom zero (the bes t fit solut ions yield 7_ smal l and negative).
In what follows we would like to make use of two observat ions : i) if eq. (1) holds, then
k / I T 2 ( k ) t k~oo 1 / lnk 4 0 ; (2)
ii) in the la rge momentum in terva l 1.65 GeV/c --< k < 65 GeV/c the imaginary par t of T2(k) does not change i ts sign [7]. The re su l t s of seve ra l f i ts to the exist ing h igh-energy data (in the region, say, 10 GeV/c-- < k --< 65 GeV/c) extrapolated to higher energ ies do not change this proper ty of Im T2(k). Therefore , we take for granted the inequali ty
cr2(k) >/ 0 (3)
above k = 1.65 GeV/c ~- k N. Let us reca l l the most genera l form of d i spers ion re la t ion for T2(k) [8]
+ + y i k'c°'(k' 2 , /~ ReT2(k ) = R e T 2 ( 0 )_ __2f 2 k 2 2 k 2 d k ' t z I m T 2 ( k ' ) (4)
U k 2 ~2 0 - k 2)
where ~2 = ~2[1 _ (/~/M)2]. This equation follows in fact f rom the Cauchy formula (valid also for com- plex values of k) which can be wr i t ten as
* Present address: Jagellonian University, Cracow, Poland. ** We follow here the notation adopted in ref. [5].
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Volume 34B, number 3 P H Y S I C S L E T T E R S 15 February 1971
T2(z) 2f2 z ReT2(O) ~fk20 dk'2ImT2(k') f~dk'2ImT2(k') k2 . . . . . . , z = . (5)
w + 2 _2 tl k '2co'(k '2 - z) z k'2w'(k '2 -z) z+
2 T h e r e i s no d i f f icu l ty in ca l cu l a t i ng the l e f t - h a n d s ide of th is equa t ion fo r z = k N s ince Im T2(k N) = 0 by def in i t ion and the coupl ing cons tan t as w e l l as the t h r e s h o l d va lue Re T 2 (0) a r e r a t h e r we l l known [9-10].
Now, le t us i n t r o d u c e an a u x i l i a r y funct ion
oo dk '2 Im T2(k')
F(z) = _~2 + ~ k'2w'(k'2 z) ' /z N
w h e r e a2 is chosen in such a way that F(k 2) < 0. In th is c a s e F(z) does not have any z e r o and the ho lo- m o r p h y doma in for F ( z ) and 1 / F ( z ) i s the s ame . Combin ing th i s i n f o r m a t i o n with the a s y m p t o t i c b e h a v - i o r o r F ( z ) [cf. a s s u m p t i o n (2)] one can w r i t e the Cauchy f o r m u l a fo r 1 / F ( z ) in the f o r m
1 l f~ dk'2ImF(k ') - ~ = ~ ~
To s imp l i fy our d i s c u s s i o n we put z = 0, then
(6)
1 1 re° dk,2 I m F ( k ' )
- ~ = ~ 2 N ~ , 2 1 F ( k , ) l 2 (7)
The i n t e g r a l in th is equat ion c o n v e r g e s but p robab ly r a t h e r s lowly. Indeed, the i n t e g r a l b e h a v e s l ike (,Txx lnx) -2 for x --. ~ . T h e r e f o r e , i t m a y v e r y we l l happen that the r e l a t i o n (7) wi l l be m o r e s e n s i t i v e to a p r o p e r cho ice of the p a r a m e t e r s d e s c r i b i n g the h i g h - e n e r g y b e h a v i o u r of T2(k) than, e .g . , to the o r i g i n a l d i s p e r s i o n f o r m u l a (4). Ac tua l ly , one can interpolate v e r y n i ce ly the ex i s t i ng h i g h - e n e r g y e x p e r i m e n t a l po in t s with qu i te d i f f e r en t func t iona l f o r m s of the s c a t t e r i n g a m p l i t u d e s [cf. r e f s . [1-4, 11, 12]] but the extrapolation to the a s y m p t o t i c r e g i o n i s f a r f r o m be ing unique. Thus , r e l a t i o n s s i m i l a r to (7) m a y appea r to be helpful in r e j e c t i n g s o m e (o the rw i se accep t ab l e ) m o d e l s p r o p o s e d r e c e n t l y for the h i g h - e n e r g y a n a l y s i s of the ex i s t ing e x p e r i m e n t a l data.
The n u m e r i c a l a n a l y s i s of eq. (7) can be p e r f o r m e d in the fo l lowing way. F i r s t , we c a l c u l a t e the con t r i bu t ion to the r i g h t - h a n d s ide of eq. (7) c o m i n g f r o m the i n t e r v a l k N ~< k ~< 8 GeV/c. We have t aken the v a l u e s of R e T 2 ( k ) and I m T 2 ( k ) f r o m the compi la t ions*J13] . Then, Re F(k) can be c a l c u l a t e d f r o m the l e f t - h a n d s ide of eq. (5). H igher m o m e n t a w e r e d e s c r i b e d by the a s y m p t o t i c expans ion (1) wi th the p a r a m e t e r s taken f r o m ref . [1] and f r o m s o m e o the r independen t f i t s to the data. So, one knows aga in the b e h a v i o u r of R e F ( k ) f r o m the l e f t - h a n d s ide of eq. (5). C a l c u l a t i o n s w e r e p e r f o r m e d fo r d i f f e r e n t v a l u e s of ~2; the m a i n r e s u l t r e m a i n e d the s a m e : the r a t i o of the r i g h t - h a n d s ide to the l e f t - h a n d s ide of eq. (7) was at most equa l to 0.64. The abso lu t e va lue of the d i f f e r e n c e be tween the l e f t - h a n d s ide and the r i g h t - h a n d s ide depends , of c o u r s e , on the cho ice of uni t s [note that the ampl i t ude T 2 a p p e a r i n g in the f o r m u l a (4) d i f f e r s by a f a c t o r 4~ f r o m that u s e d in re f . [12] and that the t r a n s i t i o n f r o m the s y s t e m of uni t s w h e r e ti = c = GeV = 1 to the na tu r a l s y s t e m of uni t s i n t r o d u c e s ano the r f a c t o r ~ 7.194].
A m o r e d i f f icu l t ques t i on is what e r r o r one a s s i g n s to the n u m e r i c a l r e s u l t g iven above. Eva lua t ing the i n t e g r a l in eq. (7) we o b s e r v e r a t h e r s t r o n g c a n c e l l a t i o n s among the con t r i bu t i ons c o m i n g f r o m the l e f t - h a n d s ide of eq. (5). The i n t e g r a l o v e r the i n t e r v a l 0 --< k ~< 1.65 a p p e a r s to be l a r g e so that the con t r i bu t i ons c o m i n g f r o m k >/ 65 GeV/c in eq. (7) a r e a l m o s t neg l ig ib l e . T h e r e f o r e , the way of ana lyz - ing the h i g h - e n e r g y da ta p r o p o s e d in th is note s e e m s to be s t rong ly c o r r e l a t e d with the low e n e r g y input.
P o s s i b l e dev i a t i ons f r o m the e s t i m a t e d va lue of the c o n s i d e r e d r a t i o 2 w e r e i n v e s t i g a t e d in the fo l low- ing way. As a guid ing r e l a t i o n we have chosen eq. (7) e v a l u a t e d at z = k N (the d i f f e r e n c e be tween a s i m - p le p t e r m con t r ibu t ion and a con t r ibu t ion v io l a t i ng the P o m e r a n c h u k t h e o r e m was a l m o s t not seen) .
* This procedure is not inconsistent with the remarks made in refs. [1-4].
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Volume 34B, number 3 P H Y S I C S L E T T E R S 15 February 1971
Varying the values of the coupling constant and Re T 2 (0) between the numbers given in ref . [10] we have obse rved the following genera l p ic ture : e i ther the guiding re la t ion is fulf i l led and eq. (7) is v io la ted or eq. (7) is a lmos t sa t i s f ied but then eq. (5) is badly viola ted (in the e x t r e m e case of about 200~o deviation). Changing the va lues o f f 2 and Re T2(0) in a range "al lowed" by eq. (5) we have obtained no m o r e than 15% change in the ra t io of the r igh t -hand side to the lef t -hand side of eq. (7). For tunate ly , the value of ReT2(kN), being c lose to a local max imum of ReT2(k) , has a good chance of being well es t imated. It is, however , obvious that the analys is desc r ibed above should be r epea ted for al l ene rg ie s where ReT2(k) was measu red , in full analogy with the usual d i spe r s ion re la t ions (actually, our conclusion is at the moment e s sen t i a l ly a "one point resu l t " ) . Any d i s ag reem en t with the data would indicate a non- violat ion of P o m e r a n c h u k ' s theorem since, e.g., eq. (6) was de r ived under the assumption (2) [the p r e s - ent expe r imen ta l data do not give any indication for a change of sign by 02 (k) at momenta g r e a t e r than 65 GeV/c]. T h e r e f o r e , the re la t ion (6) r e p r e s e n t s in fact a s t ronger cons t ra in t than a s i m i l a r d i spers ion fo rmula [5] de r ived for the c ros s ing even ampli tude where only the non-vanishing asymptot ic l imi t of the ave raged total c r o s s - s e c t i o n ~1 = ½(a_ + %) was assumed.
The ac t ive col labora t ion in the ea r ly s tage of this work by Dr. J. Pi~fzt is highly acknowledged. Many thanks a re due to Dr. C. Schmid for helpful and c r i t i ca l r e m a r k s . Useful d i scuss ions with Dr. J. E l l i s a r e apprecia ted. Dr. R. S t rauss has kindly provided me with his unpublished data compilat ion. I am gra teful to the Theo re t i c a l Study Divis ion for the hospi ta l i ty extended to me at CERN.
References [1] v. Barger and R. J. N. Phillips, Phys. Letters 31B (1970) 643. [2] D. Horn and A. Yahil, Phys. Rev. 1D (1970) 2610. [3] R. Thews, Nuovo Cimento 67 (1970) 437. [4] J. Ellis and P. H. Weisz, private communication. [5] A. D. Martin and R. Wit, Nucl. Phys., to be published. [6] R. J. Eden and G. D. Kaiser, Cavendish Laboratory preprint, HEP 70-9, Cambridge (1970). [7] G. Giaeomelli, P. Pini and S. Stagni, CERN preprints, HERA 69-1 and HERA 69-3 (1969). [8] H. Lehmann, Nucl. Phys. 23 (1962) 300. [9] G. H~hler, H. Schlaile and R. Strauss, Z. Physik 223 (1969) 217.
[10] G. Ebel, H. Pilkuhn and F. Steiner, Nucl. Phys. 17B (1970) l. [11] G. Htihler, F. Steiner and R. Strauss, Z. Physik 233 (1970) 240. [12] V. Barger and R. J. N. Phillips, Phys. Rev. Letters 24 (1970) 291. [13] A. A. Carter, Cavendish Laboratory preprint, HEP 68-1, Cambridge;
R. Strauss, private communication.
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