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Volume 34B, number 3 PHYSICS LETTERS 15 February 1971 LOW ENERGY DATA, ANALYTICITY AND THE VIOLATION OF POMERANCHUK'S THEOREM R. WIT * 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 realized by many authors [1-4] the usual forward dispersion relation for the crossing odd ampli- tude is not a very powerful tool for analyzing the high energy data. For different (but "reasonable") asymptotic behaviours of the scattering amplitudes the low energy predictions are not distinguishable from each other. Therefore, it is obvious one should look for further, more restrictive constraints on the possible asymptotic behaviour of the total cross-sections. In this letter we derive an integral representation for the crossing odd amplitude valid only if the Pomeranchuk theorem is violated. Present high and low energy data suggest, however, that this rela- tion is unlikely to be satisfied. For the sake of simplicity we restrict ourselves to the case of ~=p scat- tering. The analysis given below can be, however, extended to the cases of KN and NN scattering with some obvious changes. Our interest will be concentrated on the high energy behaviour of the crossing odd amplitude** T2(k) = ½(T (k) - T+(k)). Its asymptotic properties are very often discussed in connection with the possible violation of Pomeranchuk's theorem [cf. ref. [6] and the references contained therein], In this case the high energy parametrization of T2(k) takes usually the form [1-4] T2(k) = y_ co(lnk - B - in/2) + iypeXp (-in/2) co0.5 , (1) where, from the beginning y is assumed to be different from zero (the best fit solutions yield 7_ small and negative). In what follows we would like to make use of two observations: i) if eq. (1) holds, then k/IT2(k) t k~oo 1/lnk 40; (2) ii) in the large momentum interval 1.65 GeV/c --< k < 65 GeV/c the imaginary part of T2(k) does not change its sign [7]. The results of several fits to the existing high-energy data (in the region, say, 10 GeV/c-- < k --< 65 GeV/c) extrapolated to higher energies do not change this property of Im T2(k). Therefore, we take for granted the inequality cr2(k) >/ 0 (3) above k = 1.65 GeV/c ~- k N. Let us recall the most general form of dispersion relation for T2(k) [8] + +y i k'c°'(k' 2 , /~ ReT2(k ) =ReT2(0 )_ __2f 2 k 2 2 k 2 dk'tzImT2(k' ) (4) U k 2 ~2 0 - k 2) where ~2 = ~2[1 _ (/~/M)2]. This equation follows in fact from the Cauchy formula (valid also for com- plex values of k) which can be written as * Present address: Jagellonian University, Cracow, Poland. ** We follow here the notation adopted in ref. [5]. 211

Low energy data, analyticity and the violation of Pomeranchuk's theorem

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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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