A bootstrap calculation for the mass of the ππ (T = 0, l = 0) antibound state

V o l u m e II, n u m b e r 4

PHYSICS

LETTERS

15 August 1964

A BOOTSTRAP

CALCULATION FOR THE MASS OF THE lrTr(T=O, / = 0 ) A N T I B O U N D STATE S. CIULLI and GR. G H I K A Institute f o r Atomic Physics, Bucharest Received 1O July 1964

In the l a s t four y e a r s m a n y p a p e r s 1,2) have been published r e g a r d i n g the e x i s t e n c e or n o n e x i s t e n c e of a p a r t i c l e r e l a t e d to the e n h a n c e m e n t of the ~r~ ~ ~ (l =0, T=O) c r o s s section at zero k i n e t i c e n e r g y (near the e l a s t i c t h r e s h o l d ) , o b s e r v e d in the A b a s h i a n - B o o t h - C r o w e e x p e r i m e n t (A, B, C p a r t i c l e ) . T h r e e m o n t h s ago, Atkinson 3) showed that t h e r e i s a s t r o n g indication that an antibound state will o c c u r in the (l =0, T =0) p a r t i a l wave of the ~rlr s c a t t e r i n g , j u s t u n d e r the e l a s t i c t h r e s h o l d (s =4) *. The effect of such an antibound state for the ~r c r o s s section could be s i m i l a r to that produced by a stable 7r~ bound state. N e v e r t h e l e s s such a " p a r t i c l e " would not be detectable e x p e r i m e n t a l l y , the antibound state being a pole on the second Riemarm sheet of the amplitude, In o r d e r to e s t i m a t e the p o s i t i o n of the antibound s t a t e , Atkinson a p p r o x i m a t e d the behaviour of the p a r t i a l wave amplitude between s = 0 and s = 4 by a constant. By a n a l y t i c a l continuation 4) a pole is found on the r e a l axis of the second Riemarm sheet, close to the e l a s t i c threshold. T h i s p o s s i b l e i n t e r p r e t a t i o n of the A B C - " p a r t i c l e " i s v e r y a t t r a c t i v e . A l e s s crude e s t i m a t i u n b e c o m e s however n e c e s s a r y in o r d e r to compute m o r e a c c u r a t e l y the position of this second R i e m a n n sheet pole. It i s the p u r p o s e of the p r e s e n t p a p e r to p e r f o r m a bootstrap calculation of the m a s s of this antibound state. A complete b o o t s t r a p c a l c u l a t i o n would yield a l l the r e s o n a n c e m a s s e s and coupling c o n s t a n t s (scatt e r i n g lengths). F o r such a bootstrap calculation one m u s t take p r o p e r l y into account the left hand cut. produced by the c r o s s e d r e a c t i o n ** , but in the p r e s e n t work we shall use e x p e r i m e n t a l v a l u e s for T = 0 s c a t t e r i n g length, r e s t r i c t i n g o u r s e l v e s to the Chew pole a p p r o x i m a t i o n for the left hand cut. By the N / D method

ds' D(s) = 0(4) - s - 4 f ~ N ( s ' ) T 4 (s'-4)(s'-s)

(1)

we get in the pole a p p r o x i m a t i o n (N(s) = 1/(S+So) :s o > 0)

T=O

=

1 =

w h e r e D(4) and s o a r e two independent p a r a m e t e r s D(4) i s d e t e r m i n e d by the value oi the amplitude in point n u m b e r 1 (see fig. 1): D(4) = 1/(aO(so+4)) (a ° = s c a t t e r i n g length for T = 0) .

(3)

If an antibound state e x i s t s , its p o s i t i o n m 2 i s c o m p l e t e l y d e t e r m i n e d by s o. Indeed, the f o r m of the a m plitude on i t s second R i e m a n n sheet for 0 < s < 4 i s 4): A°(s)

(4)

:

1-

'

* In the following weshall use natural units with the mass of the ~ mesonequal to one. As usual,

s

=-

(p1+~02)2 =

4(v+l), t - (pl+p3) 2 = -2v(1-cos 0), u ~- (pl+P4)2 = -2~(l+oos O), ~ = ]p2 I. ** This problem will be discussed in the last part of this letter. A m o r e detailed version of the present w o r k will appear in Rev. Phys. Acad. Rep. Populaire Roumaine.

336

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PHYSICS

LETTERS

15 August 1964

having a pole at s = m 2 w h e r e m (antibound state m a s s ) i s given by 3): ½~/m2(4-m 2 ) = A ° ( m 2 ) ,

0
(4') T=0 Now, to obtain a "feed=back" equation for m , we equate the value of A} 0 (s) f r o m eq, (2) at s = 0, to that obtained by i n t e g r a t i o n of AT=O(s, t) on the line connecting p o i n t s lq~ 2 and 3 (the solid line in fig. 1). T h i s e n a b l e s u s to take advantage of the fact that on the s o l i d line AT=O{s ,t), can be w r i t t e n as a d i s p e r s i o n i n t e g r a l over the i m a g i n a r y p a r t of the f o r w a r d (backward) amplitude of the c r o s s e d p r o cesses.

/C//< ///,, //,:z//?/: / 1~

/

\

/ 8tmhd ,q/em#,,~

.~,,/ ~g,=/ (t-e)

Fig. 1 Although the pole a p p r o x i m a t i o n is a somewhat rough approach, having r e q u e s t e d for A°o(S) to have c o r r e c t v a l u e s at s = 0 and s = 4 it i s expected that eq. (2) would be a good a p p r o x i m a t i o n in the i n t e r v a l 0 < m 2 < 4. We feel this p r o c e d u r e " m o r e s t a b l e " than that which u s e s 2) the s y m m e t r y r e l a t i o n s of the d e r i v a t i v e s of A T in a single point (s =$, 4 t = $). 4 The e s t i m a t i o n of the e r r o r in the position of the a n t i bound pole (see l a t e r ) c o n f i r m s this s t a t e m e n t . Hence, at s = 0 we have Im A ° ( a ° ; u t ) { So 2_ 4 S o ' f ~ + J ~ o } -1 1 +lf ^ [1 ~ u - 4

Cso

)ao

l.

3~ 4

+

I:

t '(t '-t)

*-3

7r

4

-1

d cos

w _4

4 (u'-4)(u'-u) du' + ~ 4

dr' +

(u'=4)(u'=u)

du' +'~-~J

t'(t'-t)

(u'=4)(u'-u)

t'(t'-t)

;">='I

It'

du'

(5)

'

w h e r e , u = 2 ( 1 + c o s Oi). The sign of the T = 1 coefficient i m p l i e s that the f o r w a r d s c a t t e r i n g of the t h i r d c h a n n e l was defined along s = 0. T h e r e f o r e Im A T ( u , , s=0) = ~ l (2/+1) Im A T l (u'). Only S and P waves w e r e r e t a i n e d , higher waves tending r a p i d l y to z e r o n e a r the e l a s t i c t h r e s h o l d . In fact, even the P wave (T = 1) c o n t r i b u t i o n to the r i g h t hand side of eq. (5) i s r a t h e r s m a l l * . The T = 1 wave was c a l c u l a t e d f r o m the known e x p e r i m e n t a l p a r a m e t e r s of the p m e s o n . One could w r i t e for the p m e s o n amplitude a B r e i t - W i g n e r e x p r e s s i o n s a t i s f y i n g u n i t a r i t y . 2

-P- ]Vol + 2id-u-Im ~ooo "

(6)

* For the T = 0, 1 and 2 contribution we found respectively 0.489, 0.015, 0.802 for a ° = 2.5; 0.267, 0.016 and 0.401 for a ° = 1. 33q

Volume 11, number 4

PHYSICS LETTERS

15 August 1964

T h i s e x p r e s s i o n was then c o r r e c t e d to take p r o p e r l y into account the a c t u a l t h r e s h o l d behaviour 3) 1'

A i(v=O) = O,

AI

(u=O)~O.05.

(7)

Fojc t h i s p u r p o s e eq. (6) was w r i t t e n in a N / D f o r m and a CDD pole was added to D j u s t at u = 0, ( A ~ = N / ( D + ( ~ / / u ) ) ) in o r d e r to fit eq. (7). Hence, the m e s o n c o n t r i b u t i o n b e c o m e s

~(v-Ti) A ~ ( u ) = 1 . 6 5 v - 7.476 - (20/v)+ i ~/v

(8)

c o r r e s p o n d i n g to a T = 1 c r o s s section 48~y 2 ~P - (1.65u 2 - 7.476u- 20)2+u 3 r e a c h i n g i t s m a x i m u m at sp = 29.26 and having ~ = ½~/spr = 2.325 (i. e. Mp = 755 MeV, r = 120 MeV). The T = 2 t e r m would give the m o s t i m p o r t a n t c o n t r i b u t i o n b e c a u s e of its l a r g e isotopic spin factor. F o r A 2 we take a 1/D approximation. Now, owing to the lack of e x p e r i m e n t a l data on the T = 2 s c a t t e r ing length a 2, in o r d e r to d e t e r m i n e it, we shall use the well known r e l a t i o n AO(s, t) = 5 A 2 ( s , t) in the point s = 2, t = 0 (which i s not too f a r f r o m the m a x i m u m s y m m e t r y point s = -~, t = 4 where the above r e l a t i o n a c t u a l l y holds). =

+ ~ 4j

( s ' - 4 ) ( s ' - 2)

s'(s'-2)

+~4 j

+-~a

s'(s'-2)

--~

(s'-4)(s'-2)

ds' - ~4 j

s'(s'-2)

ds' = 0 .

(9)

Eq. (5) and (9) a r e to be solved together. The e x p e r i m e n t a l value of the T = 0 s c a t t e r i n g length being not known exactly *, the e q u a t i o ~ w e r e solved for a set of v a l u e s of a ° r a n g i n g f r o m 1 to 2.5. The c o r r e s p o n d i n g r e s u l t s a r e listed ~ t a b l e 1 together with the m a s s m of the T = 0, l = 0 antibound state (ABC " p a r t i c l e " ) given by eq./(4'). Table 1 a2

1

1.25

1.50

1.75

2

2.25

2.50

a2

0.26

0.32

0.38

0.43

0.47

0.52

0.56

So(S~) mABC 0VIeV)

57(24)

37(11)

29(9.5)

24(8.5)

21(7.5)

19(7)

17(6.5)

230±2.8

243+3.7

251±2.5

257±1.9

261±1.5

264±1.2

266±0.9

t were obtained using conditton • (The errors of mABC and the values (s~) a 2 = ~2 a o ins tead of the above listed values of a 2. The elastic threshold is 279.16~MeV).

To e s t i m a t e the e r r o r s , we c o m p a r e d the above l i s t e d a 2 v a l u e s to those obtained by u s i n g the A ° = ~ A 2 condition in s = 4, t = 0, which is two t i m e s f a r t h e r f r o m the m a x i m u m s y m m e t r y point than s = 2, t = 0. The obtained d i s c r e p a n c i e s exceed c o n s i d e r a b l y those produced by n e g l e c t i n g the P - w a v e t e r m , and r e p r e s e n t t h e r e f o r e an i n d i c a t i o n for the o r d e r of magnitude of the e r r o r s connected with the h i t h e r t o neglected higher waves. The e r r o r s of s o a r e g r e a t , but those affecting the value of the ABC m a s s , a r e i n s i g n i f i c a n t (ranging f r o m 0.9 to 3.7 M e V : ) , owing to its e x t r e m e neighbourhood to the elastic threshold. Of c o u r s e , although the e r r o r s of the antibound state m a s s a r e p r o b a b l y v e r y low, e x p r e s s i o n (2) does not r e p r e s e n t a f a i r a p p r o x i m a t i o n for the s c a t t e r i n g amplitude at a l l p o s s i b l e v a l u e s of s. We see two d i f f e r e n t p o s s i b i l i t i e s to i m p r o v e the r e s u l t s , taking p r o p e r l y into account the left hand cut. F i r s t l y one Can u s e for the a m p l i t u d e (and i t s cosine d e r i v a t i v e ) a d i s p e r s i o n r e l a t i o n at t = 0, where the c r o s s i n g t r a n s f o r m a t i o n can be p e r f o r m e d simply. Some t r o u b l e would be p r o d u c e d by u n i t a r i t y (which r e l a t e s r e a l and i m a g i n a r y p a r t s of the f o r w a r d amplitude d e r i v a t i v e s ) b e c a u s e the c o s i n e power * A probable value for a ° is 1.50. 338

Volume 11, number 4

PHYSICS LETTERS

15 August 1964

s e r i e s a r o u n d t = 0 would not in g e n e r a l converge over the whole p h y s i c a l r e g i o n 0 >t > 4 - s . T h i s point can be solved by expanding the amplitude into p o w e r s of the function w(s, t) which m a p s the cosine s i n g u l a r i t i e s on the b o u n d a r y of the unit c i r c l e 5,6). The second p o s s i b l e way i s to u s e L e g e n d r e e x p a n s i o n s only in the s - c h a n n e l , and w s e r i e s in the c r o s s e d channels. T h i s l a t e r ones will p e r m i t a n a l y t i c continuation of the amplitude in the dashed r e g i o n of fig. 1, in o r d e r to calculate the left-hand cut of the p a r t i a l waves. H one i s o l a t e s the m a r g i n a l s q u a r e root s i n g u l a r i t y of the s p e c t r a l function, the c o n f o r m a l mapping s e r i e s w a r e s t i l l c o n v e r g e n t even in the s p e c t r a l function r e g i o n 7) and the a n a l y t i c a l continuation i s t h e r e f o r e quite p o s s i b l e . T h i s second approach t a k e s advantage of the N/D method, which i s e s p e c i a l l y suitable for studying the behaviour of the p a r t i a l waves on the second R i e m a n n sheet, where r e s o n a n t and antibound s t a t e s m a y occur. We wish to e x p r e s s our gratitude to Dr. Atkinson for i n f o r m i n g us in advance about his work. The a u t h o r s a r e indebted to t h e i r collegue Dr. M. Micu for an i l l u m i n a t i n g d i s c u s s i o n . It is a p l e a s u r e to acknowledge our thanks to Prof. J. Hulubei and Prof. S. T z i t z e i c a for t h e i r i n t e r e s t in t h i s work.

References 1) A.Abashian, N.E.Booth and K.M. Crowe, Phys. Rev. Letters 5 (1960) 258; Phys. Rev. Letters 7 (1961) 35; Phys. R o y . 132 (1963) 2314. 2) B.R.Desai, Phys. Rev. Letters 6 (1961) 497. 3) D.Atkinson, Physics Letters 9 (1964) 69. 4) R. Oehme, Phys. Rev. 121 (1961) 1840. 5) J. Fischer and S.Ciulli, Zh. Eksperim. i Teor. Fiz. 41 (1961) 256; Soviet P h y s i c s - J E T P 14 (1962) 185. 6) D.Atkinson, Physics Letters 7 (1963) 65. 7) I. Ciulli, S. Ciulli and J. Fischer, Nuovo Cimento 23 (1962) 1129. * * * * *

PION-NUCLEON AND

SCATTERING THE POSSIBLE B. H. B R A N S D E N ,

IN T H E R A N G E 3 0 0 - 7 0 0 Pll RESONANCE *

MeV

P. J. O ' D O N N E L L

Department of Physics, University of Durham, England and R. G. MOORHOUSE

Rutherford High Energy Laboratory, Chilton, England Received 8 July 1964

T h e r e i s a continuing need for a c c u r a t e p a r a m e t r i c a n a l y s e s of p i o n - n u c l e o n s c a t t e r i n g data to d e t e r m i n e the b e h a v i e u r of individual p a r t i a l waves and c o r r e s p o n d i n g phase shifts and the p a r a m e t e r s of r e s o n a n t states. In this note the r e s u l t s of an e n e r g y dependent a n a l y s i s of ~+ a n d ~s c a t t e r i n g by n u c l e o n s in the e n e r g y r a n g e 300700 MeV i s p r e s e n t e d . Ideally one would a t t e m p t to fit the whole of the e x i s t i n g e x p e r i m e n t a l data at all e n e r g i e s s i m u l t a n e o u s l y but, with one exception 1,2), the data have been a n a l y s e d by i n * Work supported in part by the National Institute for Research in Nuclear Science and by the University of Durham.

dependent phase shift a n a l y s e s at each energy. In the work of Feld and Roper 1) and Roper 2) the e n e r g y dependence of the pion n u c l e o n p a r t i a l waves i s i n t r o d u c e d e i t h e r by e x p r e s s i n g them in t e r m s of B r e i t - W i g n e r f o r m s o r , as a p p r o p r i a t e , expanding the phase shifts in a power s e r i e s in energy. An a l t e r n a t i v e method of p a r a m e t r i s a t i o n suggested by the a n a l y t i c p r o p e r t i e s of the p a r t i a l wave a m p l i t u d e s for pion nucleon s c a t t e r i n g i s explored in the p r e s e n t investigation. The p a r t i a l wave a m p l i t u d e s , f/,±, a r e defined by fz± = (e 2 iSl±- 1)/2iq

(1) 339