PROTON SPIN-SPIN ASYMMETRIES FOR LARGE ANGLE ELECTRON-PROTON ELASTIC SCATTERING

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PROTON SPIN-SPIN ASYMMETRIES FOR LARGE ANGLE ELECTRON-PROTON ELASTIC

SCATTERING

M. Anselmino

To cite this version:

M. Anselmino. PROTON SPIN-SPIN ASYMMETRIES FOR LARGE ANGLE ELECTRON-

PROTON ELASTIC SCATTERING. Journal de Physique Colloques, 1985, 46 (C2), pp.C2-195-C2-

200. �10.1051/jphyscol:1985218�. �jpa-00224529�

PROTON SPIN-SPIN ASYMMETRIES FOR LARGE ANGLE ELECTRON-PROTON ELASTIC SCATTERING

M. Anselmi.no

Department of Physios, Indiana University, Bloomington, Indiana 47405, U.S.A.

Résumé : Nous considérons un modèle que nous avons développé précédemment pour dé- crire les asymétries spin-spin dans la diffusion p-p à grands angles et nous l'ap- pliquons pour prédire les asymétries spin-spin du proton dans la diffusion e-p.

Nos résultats diffèrent de ceux attendus dans le schéma de la QCD perturbàtive. La mesure des asymétries de spin fournirait une mesure directe des deux facteurs de forme élastiques du proton et un test clair des différents mécanismes. Hous commen- tons d'autres effets de spin.

Abstract - We consider a model previously developed to describe spin-spin asym- metries in proton-proton large angle elastic scattering and apply it to predict proton spin-spin asymmetries in electron-proton elastic scattering. Our results differ from those commonly expected in a perturbative QCD scheme. The measure- ment of the spin asymmetries would provide a direct measurement of the two proton

elastic form factors and a clean further test of different mechanisms. We comment on some other st>in efferta

I am going to report on the results of some work recently done in collaboration with E. Predazzi.W

(2)-(9)

A few years ago many theoretical models tried to explain some striking experimental d a t a ^ " / on the spin-spin asymmetry AJJJJ, in large angle elastic proton-proton scattering, in terms of quark-quark hard scattering.

In particular, application of perturbative QCD to large P^ exclusive reactions was widely discussed. The most general, QCD inspired, scheme is that of Ref. (11).

It gives qualitatively correct predictions for the Q dependence of the hadronic 2 form factors and the fixed angle elastic cross-sections. It also leads to the helicity sum rule according to which in any large Q^ exclusive reaction A&+CD which can be described in its scheme, the sum of the initial helicities must equal

(in the limit in which we neglect the masses of the quarks compared to their energies) the sum of the final ones:

(1) V X B = V V

The helicity sum rule has immediate consequences for many spin observables. In particular it requires the proton-proton double helicity flip amplitude

*„ = <++|*|—> and the single helicity flip one 4_ = <++|*|+-> to be zero. This implies that the longitudinal spin-spin asymmetry A ^ L must be related to the transverse one Ajq

, at 8 = -= , by

(2) 2 VlKL^ = !'

It also implies (12)

that the large angle proton polarization must be zero.

Recent measurements of both ATT at 6 = -r and the proton polarization at large

(14)

P , contradict the previous conclusions . Moreover, also the preliminary re- On leave of absence from Istituto di Fisica Teorica della Universita Torino, Italy.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985218

C2-196 JOURNAL DE PHYSIQUE

s u l t s on t h e measurement of t h e h e l i c i t y d e n s i t y m a t r i x element p, , of t h e p

t

v e c t o r meson produced a t l a r g e PT i n t h e process n p

ptp(15), seem t o c o n t r a d i c t the scheme of Ref. (11): i n f a c t t h e i n d i c a t i o n of a l a r g e v a l u e of pl-l a g a i n v i o l a t e s t h e h e l i c i t y sum r u l e s which f o r c e s p l - l t o be zero.

These experimental r e s u l t s seem t o be a n i n d i c a t i o n t h a t , a t l e a s t when l a r g e PT v a l u e s a r e involved, t h e simple way of computing t h e hadron h e l i c i t i e s by summing t h e h e l i c i t i e s of t h e quarks, which, through t h e h e l i c i t y conserving elementary hard s c a t t e r i n g , a r e always p a r a l l e l t o t h e hadron d i r e c t i o n , may n o t b e s a t i s - f a c t o r y . The recombination of t h e s c a t t e r e d quarks o r , i n o t h e r words, t h e r o l e of t h e hadronic wave f u n c t i o n , may b e important i n giving s p i n e f f e c t s o t h e r than those given by t h e QCD elementary quark i n t e r a c t i o n s .

I n t h e model of Ref. (9) I described t h e spin-spin a s m e t r i e s i n p-p l a r g e a n g l e e l a s t i c s c a t t e r i n g e s s e n t i a l l y t a k i n g i n t o account, when computing t h e proton h e l i c i t i e s i n terms of t h e quark ones, t h e f a c t t h a t i n such r e a c t i o n s we always have some p a r t o n i c c o n f i g u r a t i o n s w i t h one of t h e quarks a t a l a r g e a n g l e t o t h e o t h e r quarks. This implies t h a t we cannot simply sum t h e quark h e l i c i t i e s i n order t o g e t t h e proton ones, b u t we have t o p r o p e r l y compute t h e quark and proton s p i n p r o j e c t i o n s i n d i f f e r e n t d i r e c t i o n s .

This model seems t o b e t h e only one which d e s c r i b e s w e l l t h e experimental r e s u l t s on ANN and A ~ ~ ( ~ ~ ) ; though I have n o t com uted i t s p r e d i c t i o n f o r t h e h e l i c i t y

1-1 5

d e n s i t y m a t r i x p (p+) i n t h e process

p+pfp(15), t h e r e is no need i n i t s scheme f o r p t o be zero; t h e v a l u e of t h e p r o t o n p o l a r i z a t i o n a t l a r g e PT, i n s t e a d , i s

1-1

p r e d i c t e d t o be z e r o , and I w i l l make some comments on t h a t a t t h e end of my t a l k . Since t h e main e f f e c t s of t h e model of Ref. (9) come from t h e d i f f e r e n t s p i n r e - combination mechanism r a t h e r than t h e quark elementary i n t e r a c t i o n s , it seems i m - p o r t a n t t o f u r t h e r t e s t i t s v a l i d i t y by a p p l y i n g i t t o another r e a c t i o n .

I n Ref. (i) we used t h e model of Ref. (9) t o d e s c r i b e t h e proton spin-spin asym- m e t r i e s i n l a r g e a n g l e electron-proton e l a s t i c i n t e r a c t i o n s . Again, we found some q u a l i t a t i v e r e s u l t s which d i f f e r from t h e common e x p e c t a t i o n s , t h u s o f f e r i n g a c l e a r p o s s i b i l i t y of f u r t h e r t e s t i n g t h e main i d e a of t h e model.

The h e l i c i t y amplitudes f o r t h e pe- + pe- e l a s t i c s c a t t e r i n g i n t h e Center of Mass system (C.M., s e e F i g . 1 f o r t h e d e f i n i t i o n of t h e kinematics) a r e given by

where X,v(Xf ,

denote r e s p e c t i v e l y t h e i n i t i a l ( f i n a l ) h e l i c i t i e s of t h e proton and t h e e l e c t r o n . As u s u a l , r ^is t h e most g e n e r a l form of t h e proton v e r t e x allowed by p a r i t y conservation'and time r e v e r s a l i n v a r i a n c e

where

i s t h e anomalous magnetic moment o f t h e proton measured i n Bohr magnetons,

K

= 1.79, and m i s t h e proton mass.

From eqs. (3) and (4) we have, n e g l e c t i n g t h e e l e c t r o n and p r o t o n masses compared t o t h e i r C.M. e n e r g i e s , t h e 3 independent amplitudes:

- 4 ~ ~

= - M*;* = M--;--

{ ^{M+-;+- = M-+;-+ - - - -4E2 42 q 2 2F1 (1 + c o s 8) F 1}

= -M - = -

M,t;++ = M--;+- +-;-- - -"~;-t -4E2 q2 2E s i n 0 C 2m F 2

l o n g i t u d i n a l l y p o l a r i z e d protons ( h e l i c i t y s t a t e s ) we have, w i t h obvious n o t a t i o n s

For protons w i t h s p i n p r o j e c t i o n +-- 1 along t h e y d i r e c t i o n ( t h a t i s perpendicular t o t h e s c a t t e r i n g plane) we have: 2

and f o r protons p o l a r i z e d i n t h e s c a t t e r i n g p l a n e perpendicularly t o t h e i r d i r e c t i o n o f motion:

The denominator of eqs. (6)-(8) i s p r o p o r t i o n a l t o t h e unpolarized cross-section, 1 aa2 F

du/dq2 =

Xv;XIvl I M ~ , ~ ~ Notice a l s o t h a t , by proper combination of t h e spin-spin asymmetries, we have a p o s s i b l e way of measuring s e p a r a t e l y t h e proton fbrm f a c t o r s F, and F?.

-

I n t h e scheme of Ref. (11) t h e proton h e l i c i t y f l i p amplitude M*:-+ is f o r c e d t o

.-3 .

be zero by t h e h e l i c i t y sum r u l e (1) ( i n t h e l i m i t i n which we n e g l e c t t h e quark masses compared t o t h e i r e n e r g i e s ) . I n such a c a s e w e have, from eqs. (5)- (8), t h e s i m p l e l a r g e ~2 p r e d i c t i o n s :

4(1+ cos 6 ) .

4+(1+ cos 0 )

The measurement of t h e spin-spin asymmetries (6)-(8) would then c o n s t i t u t e a f u r t h e r c r u c i a l t e s t of t h e Brodsky-Lepage mechanism.

The main p o i n t s of t h e model of Ref. (9) can be seen from Figure 2. During t h e high energy Center of Nass i n t e r a c t i o n between t h e proton and t h e e l e c t r o n , one quark of t h e p r o t o n i s f r e e l y s c a t t e r e d by t h e e l e c t r o n v i a one photon exchange;

t h e f i n a l proton c o n f i g u r a t i o n i s one i n which t h e s c a t t e r e d quark i s a t a very l a r g e angle a and h a s a high t r a n s v e r s e momentum w i t h r e s p e c t t o t h e o t h e r s p e c t a t o r quarks.

The p-e s c a t t e r i n g amplitude corresponding t o t h e c o n f i g u r a t i o n of Figure 2 can be w r i t t e n , i n terms o f t h e elementary amplitude M and t h e proton wave f u n c t i o n JI, a s (dropping f o r t h e moment a l l t h e s p i n i n d i c e s )

where t h e sum and i n t e r a t i o n a r e over a l l t h e allowed quark and g n e m a t i c a l con- f i g u r a t i o n s and where( 8 )

c o s . 8

(l-X)

(11) cos ^cc

4 ^l+(l-X)' - ^{2(1-X) c o s 8}

X i s t h e f r a c t i o n of l o n g i t u d i n a l momentum c a r r i e d by q l ; t h e f r a c t i o n of l o n g i t u d i n a l momentum and t h e t r a n s v e r s e momentum, w i t h r e s p e c t t o t h e f i n a l proton p ' , c a r r i e d by q: a r e given by

X '

= 1. - ^(l-X) cos e

2 ) [ ⁼

^p

^{s i n 8.}

C2-198 JOURNAL DE PHYSIQUE

One has t o s t r e s s a t t h i s p o i n t t h a t , f o r a l l models l i k e o u r s , t h e v a l i d i t y of u s i n g t h e p a r t o n model p i c t u r e t o d e s c r i b e e x c l u s i v e r e a c t i o n s i n terms of t h e elementary s c a t t e r i n g of two f r e e quarks ( o r one quark and one ' e l e c t r o n ) , taken t o be on t h e i r mass-shell, i s f a r from being c l e a r ; t h e non p e r t u r b a t i v e confine- ment i n t o t h e f i n a l hadron obviously p r e v e n t s t h e s c a t t e r e d quark (q') from being f r e e and on i t s mass s h e l l ( u n l e s s we c o n s i d e r t h e l i m i t ^X + ~ ( ~ 6 ) ) . 1

We do n o t know how much t h i s approximation could a f f e c t o u r r e s u l t s : t h a t would r e q u i r e a complete knowledge of t h e r e l a t i v i s t i c proton wave f u n c t i o n i n terms of quark masses, s p i n s and r e l a t i v e motion. However, we t h i n k t h a t , d e s p i t e t h e i n e v i t a b l e approximations, our model i s c o r r e c t i n assuming, i n a l l t h e s e l a r g e a n g l e e x c l u s i v e r e a c t i o n s , t h e c o n t r i b u t i o n from p a r t o n c o n f i g u r a t i o n s c o n t a i n i n g quarks w i t h a l a r g e i n t r i n s i c t r a n s v e r s e momentum; t h i s i s e s s e n t i a l l y what makes t h e d i f f e r e n c e w i t h t h e o t h e r models.

A s we d i d i n Ref. (9) we consider h e r e o n l y t h e s p i n a n a l y s i s of t h e p-e process;

t h e f u l l computation of t h e energy and PT dependence of t h e e l a s t i c cross-section would r e q u i r e t h e complete knowledge of t h e proton wave f u n c t i o n . Though we could t r y t o give some e s t i m a t e s by using phenomenological kT and

dependence i n

$(X, G), we p r e f e r t o d i s c u s s h e r e o n l y t h e spin-spin asyrnmetries; when computing t h e r a t i o s giving such asymmetries, i n f a c t , we can expect t h e energy dependence to c a n c e l o u t .

Referring a g a i n t o t h e case of Fig. 2, i t i s convenient t o compute t h e s c a t t e r i n g amplitudes i n two d i f f e r e n t steps. F i r s t we describe t h e f i n a l proton spin s t a t e by q u a n t i z i n g it a l o n g t h e z d i r e c t i o n ( c a n o n i c a l s p i n s t a t e ) . We assume t h e s p i n s t a t e of t h e p r o t o n i n terms of t h e quarks t o be d e s c r i b e d by SU(6) wave f u n c t i o n s , w i t h t h e proton and t h e quark s p i n s quantized along t h e z d i r e c t i o n . The

s p e c t a t o r quark s v i n s t a t e s thus c o i n c i d e w i t h h e l i c i t y s t a t e s whereas f o r t h e s c a t t e r e d quark q l i t s canonical s p i n s t a t e i s a mixture of I+> and I-> h e l i c i t y s t a t e s :

The second s t e p is t h a t of r o t a t i n g t h e amplitudes obtained a f t e r t h e f i r s t s t e p t o h e l i c i t y amplitudes; taking i n t o account not only t h e configuration of F i g . 2 , b u t a l s o i t s "timed reversed" one, i n which t h e i n i t i a l proton has quarks with large i n t r i n s i c t r a n s v e r s e momentum, w h i l e t h e f i n a l one c o n t a i n s o n l y p a r a l l e l quarks, we g e t :

a-8

= M -.-- ⁽⁶⁾ - 2 c o s --M,+;-+(a) ₂ a-8

' M-+;4(e) cos 2 M+_;+-(a)

= -M*;-+(%) = M--;+-(%) =

a-B

- s i n T(M,;,(a) + M+-;&(")) where a l l amplitudes a r e h e l i c i t y ones.

I n eqs. (14) we dropped t h e i n t e g r a t i o n over

and t h e p r o t o n wave f u n c t i o n s ( s e e eq. (10)); t h i s simple form corresponds t o a & l i k e choice o f q ( x ) , $ ( X ) =

= 6(x-X,), which i s what we s h a l l u s e i n o r d e r t o g i v e some numerical e s t i m a t e s f o r t h e spin-spin asymmetries. The common f a c t o r $ ( X ' (8,x,), k ' ~ ( ~ , x , ) ) , which we d i d n o t w r i t e i n eqs. ( 1 4 ) , c a n c e l s o u t i n t h e r a t i o s g i v i n g t h e s p i n asym-

m e t r i e s , b u t would c o n t r i b u t e t o t h e c r o s s - s e c t i o n and its energy dependence.

Notice t h a t i n o u r scheme, due t o our s p i n recombination mechanism, t h e proton

s i n g l e h e l i c i t y f l i p amplitudes a r e n o t zero (except i n t h e l i m i t

+ 1, a + B),

b u t they a r e of t h e same o r d e r a s t h e non h e l i c i t y f l i p ones; t h i s l e a d s t o

(q2) (see e q s . (5)),

2 2

(15) F2(9 " Fl(q 1,

r a t h e r than

(16) ~ ~ ( - q ~ ) ^{Fl (P} 2 .

a s i n Ref. (11).

As we a l r e a d y s a i d , though we could t r y t o f i t t h e l a r g e q2 experimental be- haviour of t h e form f a c t o r s and t h e c r o s s - s e c t i o n s by using phenomenological de- p e n d e n c e ~ i n $(x,ST), we p r e f e r , a t t h i s s t a g e , t o g i v e p r e d i c t i o n s f o r t h e spin- s p i n asyrmnetries.

I n o r d e r t o do so we need t o know t h e elementary q-e s c a t t e r i n g amplitudes. I n t h e q-e c e n t e r of mass system (c,m,) t h e y a r e given by

(17) 1<;,--1+ cos a t 1 - cos a '

The elementary c.m. s c a t t e r i n g a n g l e a ' i s r e l a t e d t o t h e s c a t t e r i n g a n g l e a i n t h e p-e C.M. system by

c o s a - ^{8 X - l}

(18) cos a ' =

1 - B c o s c r ' ' B = =

so t h a t we have

(x+l) - (X-l) cos a

M++;++(a)- *(l-cos a)

P + ; + ( a ) - 1 + c o s a x ( 1 - cos a )

where cos a i s given i n terms of 6 _and X by eq. (11).

We a r e now i n t h e p o s i t i o n t o give some q u a l i t a t i v e ' p r e d i c t i o n s f o r t h e spin-spin asymmetries; we assume f o r s i m p l i c i t y a 6-shaped dependence f o r $(X), $(X) =

= 6 (X-X,) , w i t h

= 1 . From eqs. (14), (19) and (11) (evaluated a t

= -) 1 3 we can then g e t numericat v a l u e s f o r t h e spin-spin asymmetries (6) -(8).

I n Figs. 3-5 we p l o t r e s p e c t i v e l y kL, DSS and t h e d i f f e r e n c e DT-Dss a s f u n c t i o n s of B ( s o l i d l i n e s ) ; we a l s o p l o t f o r comparison (dashe l i n e s ) t h e corresponding p r e d i c t i o n s of t h e Brodsky-Lepage scheme, eq. ( 9 ) .

There is a l a r g e d i f f e r e n c e between t h e two s e t s o f p r e d i c t i o n s ; in p a r t i c u l a r i n our model we have some c l e a r q u a l i t a t i v e consequences such a s kL#l and DNNfDSS. The measurement of such q u a n t i t i e s would t h e r e f o r e c o n s t i t u t e a f u r t h e r c r u c i a l t e s t of o u r mechanism and, i n g e n e r a l , a way of b e t t e r uiiderstanding t h e r o l e of t h e p r o t o n wave f u n c t i o n and of non p e r t u r b a t i v e e f f e c t s i n l a r g e a n g l e e x c l u s i v e r e a c t i o n s .

E x i s t i n g experimental data''') on t h e p r o t o n electromagnetic e l a s t i c form f a c t o r s do n o t a l l o w y e t a c l e a n s e p a r a t i o n between F1 and F2 (or GE and GM); a s we s a i d b e f o r e , t h e measurement of t h e proton spin-spin asymmetries would give a p o s s i b l e way of measuring s e p a r a t e l y F1 and F2.

Let me conclude w i t h some comments on t h e r e c e n t measurement of a l a r g e v a l u e o f t h e p o l a r i z a t i o n i n t h e h i h PT e l a s t i c s c a t t e r i n g of p r o t o n s , w i t h EL^^. ⁼

28 &"/c, a t Brookhaven. (11)

C2-200 JOURNAL DE PHYSIQUE

None of t h e e x i s t i n g models which t r y t o d e s c r i b e p-p e l a s t i c s c a t t e r i n g i n pure terms of q-q s c a t t e r i n g ( 2 ) - ( g ) , (11), can e x p l a i n such a l a r g e v a l u e ; i n a l l of them t h e p o l a r i z a t i o n should be zero.

One has t o n o t i c e , however, t h a t f o r protons w i t h a l a b o r a t o r y momentum of 28 GeV/c

m m

t h e C.M. r a t i o i s still r a t h e r l a r g e , E== 0.26. This might mean t h a t t h e l a r g e p o l a r i z a t i o n is a mass e f f e c t ( 1 8 ) , which should go away a t higher energy.

Again, mass e f f e c t s a r e hidden i n t h e unknown proton wave f u n c t i o n and they a r e normally neglected i n t h e quark models; i n my model(9) I have taken i n t o account some wave f u n c t i o n e f f e c t , i n t h e way of computing t h e proton h e l i c i t y i n terms of t h e quark ones, b u t I do n o t t a k e i n t o account any mass e f f e c t .

I f mass e f f e c t s a r e important i n giving a l a r g e p o l a r i z a t i o n , they might be a l s o important when com u t i n double s p i n e f f e c t s , l i k e ANN and ALL, measured a t a n even lower energy ( I 0 ) 113).

Even i f one can hope t h a t mass e f f e c t s do n o t a f f e c t much double s p i n e f f e c t s ( i n QCD o r QED we have l a r g e double s p i n e f f e c t s b u t no s i n g l e s p i n e f f e c t , even w i t h massless quarks) such l a r g e v a l u e of t h e p o l a r i z a t i o n might mean i n s t e a d

t h a t we a r e n o t i n a n energy r e g i o n l a r g e enough a s t o s a f e l y apply t h e p a r t o n model p i c t u r e when d e a l i n g w i t h s p i n e f f e c t s ; t h e wave f u n c t i o n e f f e c t s , i n o t h e r words, could s t i l l b e s o s t r o n g a s t o overwhelmingly i n f l u e n c e , independently o f t h e underlying elementary process, t h e f i n a l hadronic p r o p e r t i e s .

References.

( 1 ) ANSELMINO M, and PREDAZZI E., Indiana U n i v e r s i t y p r e p r i n t , submitted t o Z e i t . Physik C .

( 2) CHEN C.K., Phys. Rev. L e t t . 41, (1978) 1440.

( 3) FARRAR G.R., GOTTLIEB S., SIVERS D., THOMAS G.M., Phys. Rev. D g , (1979) 202.

( 4) BRODSKY S . J . , CARLSON C.E., LIPKIN M., Phys. Rev. ^D%, (1979) 2278.

( 5) ^PREPARATA G . and SOFFER J., Phys. L e t t . E, (1979) 304.

( 6) WALTERS G.F., Phys. Rev. L e t t . 45, (1980) 776.

( 7) SZWED J., Phys. Lett. z, (1980) 485 and Phys. Rev. m, ⁽¹⁹⁸²⁾ 735.

( 8) AVILEZ C . , COCHO G., MORENO M., Phys. Rev. D 2 , (1981) 634.

( 9) ANSELMINO M . , Z. Phys. C g , (1982) 63; AIP Conference Proceddings, 95,

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(10) CRABB D.G., e t a l , Phys. Rev. L e t t . G, (1978) 1257; CROSBIE E.A. e t a l . , Phys. Rev. D G , (1981) 600.

(11) LEPAGE G.?. and BRODSKY S.J., Phys. Rev. D z , (1980) 2157; BRODSKY S . J . , HUANG T . , LEPAGE G.P., SLAC-PUB-2868, published i n Springer T r a c t s i n Modern Physics, Vol. 100, ed. FRIES D. and ZEITHITS B. (1982); BRODSKY S.J.

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(13) AUER I.P. e t a l . , Phys. Rev. L e t t . 2, (1984) 808.

(14) Talk by RAYMOND _{R .} a t t h i s conference.

(15) Talk by BUNCE J . a t t h i s conference.

(16) SZWED J . , Nucl. Phys. m, ^{(1983) 53} and Acta Physica Polonica G, (1983) 239.

(17) KIRK P.N. e t a l . Phys. Rev. DE, (1973) 63; ATWOOD W.A., SLAC-Rep. (1980) 224.

(18) EFREMOV A.V. and TERYAEV 0 .V., Sov. J . Nucl. Phys. 36, (1982) 140.