LETTERE AL NUOVO CIMENTO

VOL. 15, N. 6

7 F e b b r a i o 1976

Predictions of the Spin Parameter D in the Inclusive Process pp~ ~ p+X. K. HIDAK~

Department o] Physics, University o] Tokyo - Tokyo, J a p a n ( r i c e v u t o il 28 N o v e m b r e 1975)

T h e r e h a v e b e e n a lot of discussions o n t h e ~: e x c h a n g e in i n c l u s i v e r e a c t i o n s (~) since t h e i m p o r t a n c e of t h e ~ e x c h a n g e in p p - ~ p X w a s s u g g e s t e d b y BISHARI (z). J u s t as in t h e case of t w o - b o d y s c a t t e r i n g , p o l a r i z a t i o n m e a s u r e m e n t s a r e crucial in t e s t i n g a n d r e f i n i n g t h e o r e t i c a l m o d e l s of i n c l u s i v e r e a c t i o n s . T h e m e a s u r e m e n t of t h e s p i n d e p o l a r i z a t i o n p a r a m e t e r D e n a b l e s u s to s e p a r a t e t h e u n n a t u r a l - p a r i t y e x c h a n g e a , f r o m t h e i n v a r i a n t i n c l u s i v e cross-section a in t h e process p p ~ p X (*). S i n c e a m a j o r p a r t of a~ is e x p e c t e d to c o m e f r o m t h e g e x c h a n g e for s m a l l m o m e n t u m trar~sfer, t h e D m e a s u r e m e n t is a good m e a n s t o m e a s u r e t h e m a g n i t u d e of t h e ~ e x c h a n g e c o n t r i b u t i o n . I n t h i s n o t e we discuss t h e use of t h e d a t a on t h e D - p a r a m e t e r in t h e process pp4-~ p~X t o t e s t t h e v a l i d i t y of t h e e s t i m a t i o n of t h e ~ e x c h a n g e m a d e b y BISHARI (z) a c c o r d i n g to w h i c h t h e = e x c h a n g e t a k e s a s i g n i f i c a n t p a r t i n t h e i n v a r i a n t cross-section a. W e t r y to m a k e a p r e d i c t i o n of t h e D - p a r a m e t e r in t h e n o r m a l R e g g e ( N R ) region ( i n c l u d i n g t h e t r i p l e - R e g g e (TR) r e g i o n ) in t h e f r a m e w o r k of M u e l l e r - R e g g e m o d e l . I n t h i s pred i c t i o n we a s s u m e t h a t t h e = e x c h a n g e d o m i n a t e s t h e u n n a t u r a l - p a r i t y e x c h a n g e a , a n d use t h e e s t i m a t i o n of t h e 7: c o n t r i b u t i o n c a l c u l a t e d b y BISHARI w i t h t h e ChewL o w e x t r a p o l a t i o n t e c h n i q u e (2) a n d t h e e x i s t i n g d a t a o n a. F u t u r e D m e a s u r e m e n t s will give u s a t e s t of t h e v a l i d i t y of t h e e s t i m a t i o n of t h e = e x c h a n g e . W e c o n s i d e r a n i n c l u s i v e process b - { - a ~ c-~ X . T h e s q u a r e of t h e a m p l i t u d e _F~r t, M s) is r e l a t e d t o t h e d i s c o n t i n u i t y of t h e f o r w a r d 3 t o 3 a m p l i t u d e Aab~_~-c(s, t, M 2) t h r o u g h t h e g e n e r a l i z e d o p t i c a l t h e o r e m of MVELLF.R. H e r e ).o, 2b a n d 2r are t h e c o m p o n e n t s of t h e i r s p i n s (i.e. s - c h a n n e l helicity, t r a n s v e r s i t y , etc.) a n d we define i n v a r i a n t s s ~ - - ( p a - { - p b ) 2 , t ~ = - - ( p a - - p ~ ) 2 a n d M ~ - - ( p ~ - p b - - p c ) z. I n

(l) K. HIDAKA and A. HOSOYA: Left. Nuovo Cimento, 5, 856 (1972); Prog. Theor. Phys., 50, 1666 (1973); K. ISH/KAWA, T. KASAHARA and S. MIYAKE: Lett. Nuovo Cimento, 6, 263 (1973); T. KASAHARA and T. AKH~: Pro#. Theor. Phys., 51, 1473 (1974); K. HIDAKA: Pro#. Theor. Phys., ,~3, 175 (1975); E. GOTSMAN:Phys. Roy. D, 9, 1575 (1974); R. D. FIELD and G. C. Fox: Nucl. Phys., 80 B, 367 (1974). (I) M. BISHARI: Phys. Letl., 38 B, 510 (1972). (*) For parity-conserving inclusive reactions of the type unpolarized+spln 89 we have a total of 8 observables; the invariant cross-section a and 7 spin observables (P, P', R, R', A, A', and D) (a). Among those only the D measurement enables us to separate the unnatural-parity exchange from o'. (s) M. G. DONCEL and A. M~NDEZ: Phys. Left., 41 B, 83 (1972). 195

196

K. I~II)AKA

particular defining the invariant cross-scetion d(~

(1)

~(s, t, ~ )

-~ s d t ~

(s, t, Mz),

we have (2)

a(s, t, M ~) =

1

16~ts(2s=-4-1)(2sb+ 1) ~

~ iF~';a~

~tal~t

t, M2)Iz,

which by the optical theorem is equal to (3)

a(s, t, M 2) =

1

16zts(2s~+ i)(2s~+ 1)

/'2

discr,

\%^t^ll~tS'

t,

where s~ and Sb arc the spins of particles a and b, respectively. Equation (3) yields, in the region (large s/MZ), the usual Regge formulae (see fig. la)) (4)

1

a(s, t, M z) -

9~ , ( t ) ~ ( t ) s~,(l)+~,l') ~ Im a~>_+s~i(.Mz, 0 ,

where a ( M Z, t) is referred to as the forward reggeon-partiele scattering amplitude, and fl~o~;(t) is the coupling of the Regge pole ~(t) to the a~-channel. ~(t) is the usual Regge-signature factor. Consider the case where a and c are spin-89+ baryons. The invariant cross-section (4) can be written ~(s, 1, M l) = ~ ( s , t, M ~) 4- %(s, l, M 2) ,

(5)

where a~ (or a,) is the invariant cross-section for the case where Regge poles i and j have natural (or unnatural) parity (4). (Terms of the form i = natural, j = unnatural vanish in (4) by the parity conservation (*)).

t o.)

b)

F i g . 1. - a) R e g g e d i a g r a m w h i c h c o n t r i b u t e s to t h e i n v a r i a n t cross-section ~ i n t h e N R region a n d b) Reggo d i a g r a m w h i c h c o n t r i b u t e s to t h e p o l a r i z e d cross-section D~ i n t h e N R region.

(') R. D . FIELD: CALT p r e p r i n t , CALT-68-459 (1974). N (*) P a r i t y eon.servation i m p l i e s t h a t for n a t u r a l ( u n n a t u r a l ) - p a r i t y Reg~e e x c h a n g e ~++(D = fl--(D, ,B~_(t) = - fl_N§ ( ~ + ( t ) = ~__(t), ~._(~)= ~_+(t)), w h e r e ~ r e f e r s to h e l i e i t y .

PI~J~DICTIONS OF TIIE SPIN PA.~AMETEtr D IN TH]~ INCLUSIVE PROCESS pp,--> p}X

197

F o r the derivation of the Rcgge f o r m u l a of the D - p a r a m c t e r we follow the salne procedure as FIELD (~). W e define the D - p a r a m e t e r by

(6) with

(7) ,AaAahc

X

X

where ~(~ ) implies t h e spin of a and c p o i n t i n g u p w a r d (downward) r e l a t i v e to the transv e r s i t y axis ~ - - / ~ x ~ r (normal to t h e p r o d u c t i o n plane). I n the case w h e r e a a n d c are spin-~ + baryons we have, by p a r i t y consideration,

D(s, t, M2)a(s, t, M 2) -= a~(s. t, M2)--a~(s, t, M 2)

(8)

in t h e N R region (large s/M 2) (see fig. lb)). H e r e aN and a~ are t h e n a t u r a l and u n n a t u r a l p a r i t y i n v a r i a n t cross-sections defined in (5). Combining eqs. (5) a n d (8) we get in t h e N R region (including t h e T R region)

(9)

D -- (aN-- a,~)/(aN ~- at,) = 1 --- 2a,,/a .

T h u s t h e degree of d e p a r t u r e of D from § 1 measures t h e r e l a t i v e m a g n i t u d e of a, to a. N o w we consider the process ppt-~ p~X. I n this case the u n n a t u r a l p a r t a u is f o r m e d m a i n l y by r~ and A~ exchanges. T h e interference t e r m of t h e ~ a n d A 1 is negligible for small t because ~ = ( t ) ~ r e a l and ~ A , ( t ) ~ p u r e i m a g i n a r y . I n t h e small-t region we expect t h e T: d o m i n a n c e in a~ due to the nearness of t h e 7: pole to our physical region. The r, contribution has been e s t i m a t e d by BISHA_~I (~) w i t h t h e C h e w - L o w extrapolation. U n d e r the a s s u m p t i o n of t h e rc d o m i n a n c e of au we can predict t h e D - p a r a m e t e r from (9), using the existing d a t a on a. N o r m a l i z i n g t h e residue fl~(t) at the ~ pole we h a v e as the 7: c o n t r i b u t i o n

(1o)

( a,--

da )~ Sara.s/~ r

G2

....

--t

18~2ff'~r. (t)-I

-- (4~t)' (~-~ra~Tk~-~ /

tot

2

a~%(M ) e x p [ a ( M 2 ) ( t - - m ~ ) ] .

tot 2 where G is the ppT: coupling c o n s t a n t (G2/4~ = 15) and a~%(M ) is the on-the-mass-shell t o t a l cross-section of t h e ~0p collision at t h e i n v a r i a n t energy M. W e t a k e t h e r~ trajectory as a n ( t ) = t - - m ~ . T h e factor exp [a(t--m~)] takes a c c o u n t of t h e off-shell effect of t h e e x c h a n g e d m which comes b o t h f r o m ppa~ v e r t e x and from a~p t o t a l cross-section. I n general t h e off-shell p a r a m e t e r a depends on M L F o r the elastic scattering ppt-+ p~p we h a v e in the same w a y

D ~l- 1 - - 2 a uel/ a

(11)

el

and (da)

(12)

(4'~-: ~

(;* ,,.,o~

lO~s ~

(--t) 2

~st~~(t)

-( t - m~)~ \So/

exp [b(t-- m~)]

w i t h s o == i (GeV)L W e t a k e t h e off-shell p a r a m e t c r s a and b w h i c h are suggested f r o m t h e observed t-slopc of t h e i n v a r i a n t cross-section a and of the unpolarized elastic cross-

198

K. HIDAKA

s e c t i o n a +1, r e s p e c t i v e l y (5.6). I n fig. 2 a n d 3 we s h o w s o m e p r e d i c t i o n of t h e D - p a r a m e t e r o b t a i n e d in t h i s w a y . I n t h e s a m e figures we also p r e s e n t t h e p r e d i c t i o n s for o t h e r choices of t h e off-shell p a r a m e t e r s , n a m e l y a ~ 0, 3 (GeV) -2 a n d b ~ 0, 3 (GeV) -z. T h e D - p a r a m e t e r does n o t d e p e n d on t h e choice of t h e off-shell p a r a m e t e r s so m u c h i n t h e small-t region. T h e D - p a r a m e t e r in t h e p r o c e s s p p t - ~ p t X can be m e a s u r e d , for e x a m p l e , b y t h e p r e s e n t a p p a r a t u s w i t h a p o l a r i z e d p r o t o n t a r g e t a n d a c a r b o n

1.0

o

I

i

i

=

i

|

,

I

i

,

D 0.5

--0.5

,v,

~)

I

,l

1400

t

I

1500

1690

"

--1.0

I

I

I

I

I

I

|

I

1.0

,

,

i

i

,

i

,

+

,

I

I

D 0.5

b) ~1236 I

0

I

14001500 I

l

1690 1

I

--~

1.0 D 0.5

c) o I

i 121'36 i

110

1400

1500

t

!

1.2

1.4 M(GeV)

I

1690 i

1.6

I

I

1.8

i

I

2.0

F i g . 2. - L o w - m i s s i n g - m a s s p r e d i c t i o n s of D p l o t t e d a g a i n s t m i s s i n g m a s s M w i t h t y p i c a l e r r o r b a r s a t t = - 0 . 0 4 4 (GeV/c) = a n d PlLb = 6.2 ( a ) ) , 15.1 (b)) a n d 29.7 (c)) (GcV/c). D a s h e d - d o t t e d , d a s h e d a n d s o l i d c u r v e c o r r e s p o n d t o t h e c h o i c e of t h e off-shell p a r a m e t e r a ~ 0 . 3 a n d a(M =) (GeV/c) -=, r e s p e c t i v e l y , where a ( M ~) means the parameter suggested from the observed t-slope of the invariant cross~

section ~. Unfilled circles show the predictions of the elastic ])-parameter which have negligibly small uncertainties (the throe choices of the parameter b give nearly the same prediction). We have used the data on the invariant cross-section # of ref. (').

{s) R . M . EDELSTEIN, R . A . CARRIGAN I t . , N. C. H r E ~+, T. J . McMAHoN, I. NADET.WAZ~T,E. W . ANDERSON, E . J . BLESER, G. B. COLLINS, T. FUJII, J . MENES a n d F . TURKOT: PhyS. - ~ v . D , 5, 1073 (1972). (i) J . V. ~T.r.ARy, A . N. DIDDEN8, R . W . DOBINSON, A . KLOVNING, J . L r l ~ , L. S. ROCHF~TER, ~ . SCHL~PMANN, A . M. WE~[ERELL, U. AMALDI, R . BIANCASTELLI, C. BOBIO a n d G. MAq~rHIAE" NUCl. Phz/s., 52 B, 316 (1973).

PREDICTIONS

OF

THE

SPIN PA.RAM~T]~R

D

~.~ T H ~

INCLUSIVE

PROCESS

pp~--~-p~X

199

p o l a r i i n e t e r for a recoil p r o t o n a t A N L . A n a g r e e m e n t of o u r p r e d i c t i o n w i t h f u t u r e d a t a s h o u l d b e c o n s i d e r e d as a s t r o n g s u p p o r t of t h e v a l i d i t y of t h e e s t i m a t i o n of t h e e x c h a n g e b y BlSHARX a c c o r d i n g to w h i c h t h e ~: e x c h a n g e t a k e s a n i m p o r t a n t p a r t in a (*). I f t h e r e is a s i g n i f i c a n t difference of o u r p r e d i c t i o n f r o m f u t u r e m e a s u r e m e n t , i t s h o u l d 1.o

i

a)

D

~

i

~)

0.5 _

0

_

_

J

t

0.85

!

X

0.90

0.85

X

0.90

Fig. 3. - High-missing-mass predictions of D plotted against x (Feynman's scale parameter) with typical error bars at a) t = --0.16 (GeV/c)t, s = 108 (GeV) ~ and b) P~L = 0.063 (GeV/c)~, Pl~b = 24 (GeV/e). Here p• i s the transverse momentum of the outgoing proton. The three curves mean the s a m e a s i n fig. 2. We have used the data on the invariant cross-section a of ref. (').

b e r e g a r d e d e i t h e r as a n i n d i c a t i o n of t h e i n v a l i d i t y of t h e e s t i m a t i o n of t h e = c o n t r i b u t i o n or m o r e p r o b a b l y as t h e effect of t h e A 1 e x c h a n g e . T h e s u b s t a n t i a l e x i s t e n c e of t h e A1 c o n t r i b u t i o n w o u l d necessarily m a k e t h e D - p a r a m e t e r lower t h a n o u r pred i c t i o n b e c a u s e of t h e a b s e n c e of t h e i n t e r f e r e n c e t e r m b e t w e e n r: a n d A 1 (see eq. (9)). I n t h e s a m e w a y we c a n also m a k e p r e d i c t i o n s of t h e D - p a r a m e t e r in t h e processes ~ p ~ - ~ p~X a n d K + p t ~ p t X . F o r t h i s w e h a v e o n l y t o r e p l a c e a ~ b y a ~ a n d a ~ in eq. (10). T h e v a l u e s of t h e a s y m p t o t i c on-shell cross-sections a r ~ a n d a r ~ c a n b e e s t i m a t e d w i t h t h e f a c t o r i z a t i o n a r ~ ~ a2~/a~,r a n d an~r~ a~caTrjv/a~,q~.~. B e f o r e closing t h i s a r t i c l e we w o u l d like to p o i n t o u t s o m e g e n e r a l f e a t u r e s of t h e D - p a r a m e t e r in pp~-~ p~X w h i c h do n o t d e p e n d o n t h e a s s u m p t i o n of t h e =rr domin a n c e in a~. i) Since az~ a n d a= are e x p e c t e d t o scale in t h e T R r e g i o n (s/M 2 large a n d M I large) a c c o r d i n g to t h e u s u a l T R a r g u m e n t , D scales as s i n c r e a s e s w i t h fixed x a n d t. ii) T h e r a t i o a~,/a~ b e h a v e s like sz(a,(t)a~(t))~s -~ w i t h i n c r e a s i n g s a n d fixed M 2 a n d t. so D b e c o m e s -~ 1 i n d e p e n d e n t l y of M ~ a n d t in t h i s limit. H e r e ~ a n d ~, mean natural- and unnatural-parity trajectory, respectively. iii) T h e r a t i o (~=/aN b e h a v e s like ( l - - - x ) ~ ( a ~ ( t ) - a = ( ~ ) ) ~ ( l - - x ) 2 i n t h e T R so D increases as x c o m e s closer t o x = t-1 w i t h fixed s a n d t.

region.

iv) B y t h e isospin a n d t h e a b s e n c e of t h e i n t e r f e r e n c e t e r m b e t w e e n ~ a n d Ax we h a v e a r e l a t i o n {13)

2aP(I--D p)-

a~

~)

in t h e T R region. H e r e a A~ a n d D ff are c o n c e r n e d w i t h t h e process p v ~ t - ~ p t X (or equivalently pp~j~tX) w h e r e ~V is a n u c l e o n . (') K . ABE, T. DELILLO, B. ROBINSON, F. SANNES, J . CARR, J . KEYNE a n d I. SIOTIS: P h y s . Rev. Left., 3 1 , 1527 (1973); J . V . ALLABY, A . N . DIDDENS, R . ~V. DOBLN'SON, A. KLOVNING, J . LITT, L . S. ROCHFA~R, At. SC]KL~PMANN, A..-'t~. WETHERELL, U. AMALDI, R . BIKNCASTELLI, C. BOSIO and. G. MATrHIAE: r e p o r t a t t h e Fourth International Con]erence on High-Energy Collisions ( O x f o r d , 1972). <*) W e c a n p e r f o r m t h i s t e s t a t a n y p o i n t of (s, t), p r o v i d e d we m e a s u r e b o t h D a n d a a t t h e p o i n t (s, t).