L'ETT]~RE AL NUOVO CIMENTO VOL. 4, N. 6 10 Giugno 1972

Symmetry.Point Sum Rules and Reaction ~-p-~ ~n.

M. AHMAD

International Centre /or Theoretical Physics - Trieste

(ricevuto il 16 ~Iarzo 1972)

Symmetry-poin t dispersion sum rules (1) have been proposed for ampli tudes de- scribing reactions with two identical channels (say s and u). These sum rules exploit the fact tha t the fixed u-dispersion relat ions do not respect s-u crossing and are obtained by sett ing these dispersion relations equal to zero at the symmet ry point s = u (for a crossing odd amplitude). Any possible necessity of subtract ions can be el iminated by suitable differentiation of the ampli tudes. LYTH (1) has successfully applied the technique to nov and 7:7: scat ter ing to obta in relations between the masses and coupling constants of the resonances contr ibut ing in these reactions. Recently, the applicat ion of this technique to pion photoproduct ion ampl i tudes has also been considered (3). In this le t ter we apply the symmet ry-po in t dispersion sum rules to the reaction z~-p-+ ~n, and, using the available exper imenta l da ta on the baryon resonances decay widths and information on the A2A~ coupling constants, we determine the ~J~'A ~ coupling constant.

Following LXTH (1), we wri te the following symmet ry point sum rules for the invar iant ampli tudes B and A describing the react ion z:-p-+Bn:

(1)

and

co co

- - f t--~Tdt Im B b ( t ) = f d~ Im Bb(S )

fO $1

(2)

cO r

f dt Imdb( t ) = ~ds [ImA~(s)--s ImAm(s)] . t - - X J s 2

fO $0

Here f i = OA/~u (fixed t), A b means A at u = 0 (backward), X = 2M ~ + m~ + m~ and the pr ime denotes differentiation with respect to u at fixed s. In eq. (1) the contribu- tion of the nucleon pole should be understood. The Born term in the s-channel and the

(1) H. GOLDBERG: Phys. •ev., 171, 1485 (1968); D. H. T~yTH: Lett. Nuovo ~imento, 2, 729 (1969). (t) S. MALLIK and B. I)t.~TA-ROY" Nuol. Phys. , 37 B, 226 (1972).

237

238 ~. A ~ a D

contribution of the A 2 pole in the t-channel are given by

(3) B - ~/2 g~r~og~5"x M S - s

(4)

(5)

4IF 1Gl(s - - u) 1 B - -

M(mn + m~) m ] , - - t '

A = [4MZ{m~ + m~) (u - - s) ~ + non-u-dependent terms m]~-- t"

Hero G1, G~ are the A z ~ coupling constants and F 1 is the A2~n coupling constant (3). We shall saturate the s-channel dispersion integrals in (1) and (2) by the isospin-�89 resonances, i.e. {-(1535), {-(1520), ~+(1860), {-(1670) and ~+(1688). The contr ibu tion of these resonances to the invariant amplitudes can be writ ten as

(6) A = - - l , ]~\ f l ( V ~ ) 3 r ~ l ~ [(5xZ--1)( M ~ w ) W+-- 2x( M + W) W_] F3_(s ' (3- - ) - -s ) - l +

1 + ~ [(Sx ~ - 1)(M+ W) W_-- 2x (~ - -W) W+] r,+(s'(2 +) - - s) -~ +

2 + ~ [3x(M - - W) W+-- (M + W) W_]/~z_(s'(2 --) - - s ) - 1 ~-

2 7 t- ~ [3x(M -~ W) W_-- ( 3 / - W) W+]/~1+(s'(1 ~-) - - 8) -I -~

+ ~ [(~t + w) w_] ro+(s'(o + ) - s ) -1 ,

]/2• 2~ / 3 (7) B = V ~ /~q~q,~ [ (5x ' - - 1) W+-- 6xW_] r~_(s'(3 --) - - s -~) +

3 1 + ~ [(5x * - 1) W_-- 6xW+] Fz+(s'(2 + ) - - s) -1 + ~q,q. [3xW+--W_]F,_(s'(2 --) - - s) -1 +

, } + q--~q'~l [3xW_-- W+] Fl+(s'(1 + ) - - s) -1 + ~ W_ ro+(s'(o +) - s)-i

where ~r is an isospin factor, q and ~' are the initial and final centre-of-mass momenta, s(1 • denotes the mass square of the resonance with J = l ~: �89 and / ~

= % / / , ( ~ , 0 ~ pr,-)F(jW,o_> n~). Other quantities are defined as follows:

(8) w = V ~ , w ~ = V ( ( w + : ~ ) , - m~)((w + : ~ ) ' - ~ ) .

(3) For these coupling constants, we use the definitions of S. R. DEANS and J. W. WOOTEN: Phys. Roy., 185, 1797 (1969).

S Y M M E T R Y - P O I N T SUM R U L E S AND R E A C T I O N ~-p- ->~n 239

S u b s t i t u t i n g t h e pole c o n t r i b u t i o n s (3)-(7) in eqs. (1) and (2), we ge t t h e s u m ru les

(9) 4F~ GI V/2g~'~'~vg~5'~ -t- .M(m= + m,~) M ~

~- [12.0/~0+-- 6 0 . 8 F I + - - 1 3 7 . 8 F 2_ -~- 264.1F~+ + 246.3F~_] (GeV) -~ ,

(lO) M(m~ + m~)

= - - [ - - l l . S Fo+- - 50 .4Fi+ -~ 8 4 3 8 . 4 / ' 2 _ - - 221 .3F~+- - 1743.8Fs_ ] (GeV) -a .

Now t h e r e is ev idence in f a v o u r of t h e h y p o t h e s i s t h a t t h e m e s o n t r a j e c t o r i e s f, r a n d p, A 2 couple to nuc l eons p r e d o m i n a n t l y to s - c h a n n e l he l i c i t y nonf l ip a n d he l i e i t y flip amp l i t udes , r e s p e c t i v e l y (4.5). T h e r e is f u r t h e r e v i d e n c e t h a t t h e on -mass - she l l coupl ing c o n s t a n t s of t h e pa r t i c l e s ly ing on t h e s e t r a j e c t o r i e s possess t h e s a m e p roper - t ies (e), i n d i c a t i n g a s m o o t h b e h a v i o u r of flip to nonf l ip r a t i o w h e n c o n t i n u i n g t f r om fo rwa rd d i r ec t ion to m e s o n mass . I n t h e case of A~, KL~INERT a n d W ~ I s z (s) h a v e i n d e e d shown, b y us ing t h e f o r w a r d a n d b a c k w a r d d i spers ion re l a t ions , t h a t t h e r a t io of he l i e i ty nonf l ip a n d he l i c i ty flip coup l ing c o n s t a n t s comes o u t to be v e r y smal l , i .e.

(11) G++ g 2 + 4gl 0.07 . G+_ g2

U s i n g t h i s va lue of t h e r a t i o in eqs. (9) a n d (10), we ge t t h e s u m ru le

(12) gr:~,~g~5'~ = [ - - 14.3Fo+ ~- 8.6F~+ + 4967 .8F~_- - 292 .4F~+- - 1162.0/"3_ ] (GeV) -~ .

E x a c t S U 3 c o n n e c t s g ~ 2 e , g=:r,~' coup l ing c o n s t a n t s as fol lows:

1 (13) g ~ x x : - - ~ (1 - - 4 ] ) g = ~ x ,

w h e r e ], t h e m i x i n g p a r a m e t e r , is t h e f r a c t i o n of F - t y p e coup l ing (7), i .e. , ] = F / F ~- D . I n t h e s u m ru le (12) we fix t h e s igns of F~• (which d e p e n d on t h e r e l a t i v e s igns of g~,.Opu- a n d g ~ . O ~ coup l ing c o n s t a n t s ) f rom t h e low-ene rgy f i t p a r a m e t e r s of m-p-~ ~n b y DEA~S a n d WOOTEN (3), b u t for t h e a c t u a l va lues of decay w i d t h s we use t h e exper i - m e n t a l da t a . F i r s t we keep in t h e s um ru le (12) on ly t h e r e s o n a n c e s �89 ~-(1520) a n d ~+(1860), for w h i c h t h e ~n p a r t i a l d e c a y w i d t h is r e a s o n a b l y a c c u r a t e l y k n o w n . W i t h t h i s a p p r o x i m a t i o n , a n d us ing t h e l a t e s t d a t a on t h e b a r y o n decay w i d t h s (s),

(') G. HOHLER and :R. STRAUSS: Fortschr. Phys., 232, 205 (1970); F. J. GILMAN, J. PUMPLIN, A. SCHWIM~[ER aRd L. STODOLSKY: Phys. Left., 31 B, 387 (1970). (~) H. HARARI and Y. ZARMI: Phys. Leit., 32 B, 291 (1970); R. ODORICO, A. GARC~A and C. A. GARCIA CANAL: Phys. Left., 32 B, 375 (1970); C. MICHAEL and R. ODORICO: Phys. Left., 84 B, 422 (1971). (e) J. ENGELS and G. HOHLER: Karlsruhe preprint (1970); H. ]~:LEINERT and P. H. WEISZ: Lelt. Nuovo Cimento, 2, 459 (1971). (7) J. J. DE SWART: Rs ~/Iod. Phys., 35, 916 (1963). (s) PARTICLE DATA GROUP: •ev. Mod. Phys., 43, $21 (1971).

240 M. AI~IMAI)

eqs. (12) a n d (13) g ive

(14:) ] = 0.335 ~ 0.015 ,

w h e r e we h a v e u sed ( g ~ / 4 ~ ) = 14.7 a n d t h e e r ror co r responds to t h e u n c e r t a i n t y in t h e decay w id t h s . Th i s va lue of ] is to be c o m p a r e d w i t h t h e v a l u e 0.31 r ecen t ly o b t a i n e d b y BLACKI~ON a n d WALl f rom t h e Venez iano r e p r e s e n t a t i o n of ~-p-->~n (9). H o w e v e r , t h e c u r r e n t a lgebra (10) a n d SUe (11) gives a h i g h e r v a l u e ] = 0.40. As far as t h e {• r e s o n a n c e s a re conce rned , t h e l a t e s t d a t a i n d i c a t e s (s)

0 5 ) F ( ~ * - + ~ ) < 0 . 5 % , F ( t o t )

a n d

(16) /'(2~'*-+ ~'N~)< 1% F(tot)

for 3+(1688)

for ~- (1670) .

I f we use F ( J ~ * - + ~ n ) / F ( t o t ) m 0 . 5 % a n d ~ 1 % for 3 + a n d 3- resonances , respec t ive ly , t h e n t h e sum ru le (*) g ives

] = 0.365 ~ 0.015 ,

w h i c h is closer to t h e c u r r e n t a lgebra a n d S U e value .

T h e a u t h o r is i n d e b t e d to Prof . P . BUDINI for r e a d i n g t he m a n u s c r i p t . H e also wishes to t h a n k Prof . A. SALAM, t he I n t e r n a t i o n a l A t o m i c E n e r g y Agency

a n d U N E S C O for h o s p i t a l i t y a t t h e I n t e r n a t i o n a l C e n t r e for Theo re t i c a l Phys ics , T r i e s t e .

(') M . L . BLACKMON and K. C. WALl: Phys. Rev. D, 2, 258 (1970). (10) B. ~7. LEE: Phi/8. Rc~. J ~ . , 14, 675 (1965). (u) F. G~RSEY, •. PAIS and L. A. RADICATI: Phi/8. Rev. Left., 13, 299 (1964). (*) We use negative signs for 2"8-, F~+ consistent with ref. (8).

by Societ& Italiana di Fisica ProprietA letteraria riservata

Direttore res!oonsabile: GIULIANO TORALDO DI FRANCIA

Stampato in Bologna dalla Tipografla Compositori coi tipi della Tipografla Monograf Questo faseicolo b state licenziato dai torehi il 31-V-1972