POLARIZATION IN LARGE PT PHOTOPRODUCTION OF VECTOR MESONS

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POLARIZATION IN LARGE PT

PHOTOPRODUCTION OF VECTOR MESONS

P. Kroll

To cite this version:

P. Kroll. POLARIZATION IN LARGE PT PHOTOPRODUCTION OF VECTOR MESONS. Journal de Physique Colloques, 1985, 46 (C2), pp.C2-225-C2-234. �10.1051/jphyscol:1985224�. �jpa-00224535�

JOURNAL DE PHYSIQUE

Colloque C2, supplément au n°2, Tome 46, février 1985 page C2-225

POLARIZATION IN LARGE pT PHOTOPRODUCTION OF VECTOR MESONS

P. Kroll

Physics Department, University of Wuppevtal, Gauss-Strasse, SO, D-5600 Wuppertal 1, F.R.G.

Résumé - Nous discutons le modèle QCD pour la diffusion dure et ses prédictions pour les éléments de matrice densité de spin du méson vecteur (V) dans la réaction à grands pj_YP •* VX. Les ré- sultats peuvent constituer un test du schéma probabiliste sous- jacent au modèle QCD parton et des interactions élémentaires et aussi bien une façon de mieux comprendre la dépendance en spin du processus de fragmentation des quarks et des gluons. Nous dis- cutons aussi la réaction e+e~ -> VX dans la région d'interférence Y-Z.

Abstract - QCD hard scattering model predictions for the vector meson (V) spin density matrix elements in the large pT process YP -»• VX are discussed. These results can be a test of both the underlying probabilistic scheme of the QCD parton model and of the elementary interactions, as well as a way of learning about the spin dependence of the fragmentation process of quarks and gluons. The process e+e~ •*• VX in the Y-Z interference region is also discussed.

1 - INTRODUCTION

I am going to report on an investigation by M. Anselmino and myself /1/

in which we studied the production of vector mesons at large pT. in the interactions of polarized and unpolarized photons with protons. The QCD hard scattering model predictions constitute a rather crucial test of both the overall probabilistic scheme of the QCD hard scattering model, which might not hold when spins are involved, and of the QCD elementary interactions. They can also give a way of learning about the fragmentation of polarized partons into polarized hadrons, thus leading to a better understanding of the hadronization process. More- over, independently of spin effects, the production of p's (or other vector mesons) is interesting in itself since p's are to a large extent primarily produced particles in contrast, e.g., to it's.

The general stucture of the results, i.e. which of the vector meson density matrix elements are zero or non-zero respectively, holds also in the case of photoproduction with polari2ed protons, virtual photo- production or even purely hadronic production of vector mesons at large pT. Out of all these only the case of e+e- annihilation, which offers an opportunity to measure the weak coupling constants of quarks, will be discussed in some detail.

2 - THE CROSS-SECTION FOR Vp -» VX

The cross-section for producing a large pT partice V in the interac- tion of a pointlike Y with a proton is given, in the usual lowest order

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985224

QCD hard scattering scheme, by:

where the sum is over all the kinds of constituents; x is the longi- tudinal fraction of the proton momentum carried by-thg constituent j:

s, t and u are the overall Mandelstam variables. do/dt js the cross- section for the elementary reactions yq + gq and yg + qq respectively.

In terms of the elementary amplitudes given in the table below it reads

where eq is the quark charge in units of e; i?, and d are the elemen- tary Mandelstarn variables; as is the strong coupling (with A=O.aGev/c) and Q~ is choosen as

TABLE

-

- _~p

+ VX (quark masses neglected)

As parton distribution functions we use the parametrization given by the CDHS group /2/, assuming a value of Q~~ = 5 (GeV/c) 2 . As fragmen- tation f-actions into vector mesons (p's, K*'s and $Is) we use a para- metrization similar to that given by Baier et a1 /3/ for pions and kaons. The function D is obtained from a fit to the p0 production data of the EMC group /4/ (see Fig. 1 ) :

photon-parton

.

polarization transfer parameter

6 A

+ g =; = - = .$+a2 d3 h = 1

+ A A

q d1=d2-0

A 2 2 - ²²

d 3 = -

2 = o

A 6 2 - ;2

d 3 = -

6 2 + c2

reaction

Yq-' gq

Y 9 + q:

non-zero helicity amplitudes T++++ = T_--- = r n

= P3-73 T + _ + - = T-+-+

T-++ = - ^{T+-+- =} m

T-++- = - ^{T+--+ =} m

C

4/3

1/2

The mean value of Q2 for the EMC data is 17 GeV 2

.

(E = 0.4)

and correspondingly for the other favoured and non-favoured fragmenta- tion functions. By charge conjugation one gets the fragmentation func- tions for the other vector mesons [ p , p 0 etc.).

Fig. 1:

p0 and no z spectra from the EMC /4/ compared to the fit (4).

5

a5

N 0 1

n z= 0.1 I

OD5

0.01

For the gluon fragmentation functions we take

..

a1

$ ) ^EMC

;

-

.

'..;

-

0 ! 3 9 5 ~ * * . * c j . . .

0. .5

-

as it was done in Ref./3/ for pions and kaons. The gluon fragmentation functions are the least well-known quantities significantly contribut- ing to eq.(l); thus we can only expect our estimates to be correct within a factor of about 2.

1.

The fragmentation function evolve with Q' according to the Altarelli- Parisi equations.

The predictions for the inclusive cross-section at a SPS energy is given in Fig. 2. Depending on the kinematical situation one may also get sizeable contributions from the photon structure functions in addition to that from pointlike photons. The photon structure function consists of two parts the anomalous component and the hadronic-like component. For the anomalous part we use a parametrization given by Nicolaidis /5/ which is based on perturbative QCD (these calculations are afflicted by spurious singularities at x = 0 ) . For the hadronic- like part we make use of a parametrization given by Busenitz and Sullivan /6/ which has been obtained from vector meson dominance and the assumption ql _D (x) = qk(x) .

Fig. 2 ( l e f t ) : The c r o s s - s e c t i o n f o r photoproduction of p0 mesons v e r s u s p0 momentum a t SPS energy f o r s e v e r a l production a n g l e s . The s o l i d l i n e s r e p r e s e n t t h e c o n t r i b u t i o n s from t h e p o i n t - l i k e photon, dashed l i n e s t h o s e from t h e anomalous p a r t of t h e photon s t r u c t u r e f u n c t i o n and t h e dashed-dotted l i n e s t h o s e from t h e h a d r o n i c - l i k e p a r t . Fig. 3 : ( r i g h t ) : The r a t i o s of K*' over p0 and 4 over p0 production under go0 a t SPS energy. For symbols r e f e r t o F i g . 2 .

A s can be seen from Fig. 2 t h e p o i n t - l i k e photon c o n t r i b u t i o n domi- n a t e s a t l a r g e pV. The dominance is more pronounced i n t h e forward hemisphere and l e s s i n t h e backward one. T h i s o f f e r s a way t o g e t r i d of t h e c o n t r i b u t i o n s from t h e photon s t r u c t u r e f u n c t i o n . Another p o s s i b i l i t y i s t o s e l e c t 3 j e t e v e n t s which come from t h e p o i n t - l i k e photon. The extended photon a s w e l l a s o t h e r ~(a:) c o n t r i b u t i o n s pro- duce 4 j e t e v e n t s .

To g i v e an impression about t h e production r a t e f o r o t h e r v e c t o r mesons we p r e s e n t i n Fig. 3 t h e r a t i o s of i n c l u s i v e c r o s s - s e c t i o n f o r K*'/pO and @/po a t SPS e n e r g i e s a g a i n s e p a r a t i n g t h e t h r e e d i f f e r e n t c o n t r i b u t i o n s .

As we a l r e a d y s a i d , we must t a k e a l l t h e s e r e s u l t s with a c e r t a i n c a u t i o n : while w e expect t h e p o i n t - l i k e c o n t r i b u t i o n t o be c o r r e c t w i t h i n a f a c t o r - 2 ( u n c e r t a i n t i e s i n d i s t r i b u t i o n and fragmentation f u n c t i o n s , n e g l e c t of i n t r i n s i c kT and s o o n ) , t h e o t h e r two c o n t r i b u - t i o n s a r e much less r e l i a b l e , i n p a r t i c u l a r t h e anomalous component one, which i s l a r g e and whose c a l c u l a t i o n we know t o have very s e r i o u s problems.

3 - THE VECTOR MESON DENSITY MATRIX

I n t h e framework of t h e hard s c a t t e r i n g model t h e v e c t o r meson cm h e l i c i t y d e n s i t y m a t r i x elements r e a d

where we have restricted ourselves to point-like photons. The exten- sion to extended photons or even hadrons is straightforward.

Considering the case of polarized photons and unpolarized protons, the elementary density matrix for the constituent k is (after averag- ing over colour degrees of freedom)

p(k) is normalized such that Trp(k) = 1. Decomposing tQe photon densi- ty matrix in the usual way

we have the standard form of the elementary density matrix a a

~ ( k ) = pO(k) + C P., P (k) and a similar decomposition for p(V).

Making use of the amplitudes given in the table, one finds as a conse- quence of the nature of the lowest order QCD interactions (helicity conservation, real amplitudes etc.)

A

where the polarization transfer parameters d, are also presented in the table.

The function D in eq.(7) is the fragmentation density matrix which is defined in terms.of parton fragmentation amplitudes by

The diagonal elements of L3 are the usual spin-dependent fragmentation functions DIV = gX; p. The non-diagonal elements have no probabilistic interpretathn; ~ffer&ore, the hard scattering scheme requires that they are negligible. Detailed measurements of p(V) could reveal whether they are indeed negligible or not.

In the limit of negligible transverse momentum p with respect to the jet axis, a case to which we restrict ourselves Twe are interested in primary vector mesons cgrrying a large fraction of the jet momentum) we get help from rotational invariance. It implies that only those fragmentation matrix elements may be non-zero which fulfill the condi- tion

h , - ^{A+ = Xk} - ^{X i} . ⁽¹³⁾

The other matrix elements behave as (PTf/P)n where n depends on the net helicity flip. This property of D has been first noted by Nieves /7/.

4 - WPOLARIZED PHOTONS

Insering (1 1) into (7) w e find for the density matrix in the case of unpolarized photons

As a consequence of lowest order QCD only fragmentation matrix ele- ments with Xk = Xkl appears. For negligible pTf rotational invariance eq.(13), requires XV

-

In order to obtain a numerical estimate of the diagonal elements of pO(V) we need some assumptions about the spin-dependent fragmentation functions appearing in (14). A reasonable choice, corresponding to a kind of statistical fragmentation, seems

In fact, for valence quarks this is in agreement with SU(6) spin wave functions. Eq. ( I 5) , from the fact that

implies for all AV

Insering this into eq.(14) we immediately obtain the simple result for po(V) and the decay distribution

Whereas the diagonality of pO(v) is a very general result the explicit values of its elements depend strongly on the assumption about the fragmentation functions. Instead of (15) one may, for instance, use the statistical picture of Feynman and Field /8/ wherc as distinguished from us it is assumed that if the helicities of the qq pair which frag- ments into a hadron are opposite a pion is formed a fraction f of times and a p the remaining 1-f. If the helicities are the same a p is produced. In this case one obtains

Our case corresponds to f = 1 and Field-Feynman use f = 1 .

A more dynamical fragmentation picture for quarks at least (for-gluons too if fragmentation proceeds via the sequential reaction g + qq +

hadrons) has been advocated for by other authors /9/, /lo/. The most

general coupling of two on-shell quarks to a vector meson is

c(q') (ayp + ^bq ) u (q) . Neglecting transverse momenta one finds a non- vanishing CoupYing only if h = hq + A p . Augustin and Renard /9/ then assume that leading vector mesons are formed by the fast jet quark with a soft antiquark out of the sea. In this case (pV= q > > m v l q t ) longitudinal vector mesons are suppressed, implying

w0 = 3/8rr. sin 2 9 (20) In the higher twist limit /lo/, on the other hand, where it is believed that quark and antiquark forming a meson share about equally the meson momentum (q 2 q: pV/2>> mV), only longitudinal vector mesons remain.

Thus, in this plcture one gets

It is now clear that it would be very interesting to have data on pZo in order to learn about the spin dependent fragmentation functions.

We know only of one experimental result from electroproduction /11/

at an admittedly small value of Q ~ . It gives p0 = 0.41 2 0.08 which indicates that the zero helicity state is popu?:ted at least as much as the other states.

5 - CIRCULARLY POLARIZED PHOTONS

With such photons p 3 (V) is measured. Using eqs. (7) , ⁽¹ 1 ) , ^{(1 3)} , ^{we ob-}

tain for the only non-zero matrix element

where

In Fig. 4 a numerical estimate of l:p is snown at a HERA energy where circularly polarized photons will be available. Unfortunately, the quantity p31 (V) cannot be measured through the decay distribution W

(because oh parity conservation) which, therefore, is predicted to be zero.

6 - LINEARLY POLARIZED PHOTONS

For definiteness let us assume that the polarization vector of the photon points perpendicular ( I ) or parallel ( U ) to the scattering plane. Differences l - ^{If measure p 1 ( V )} . Generalization to other direc- tions of the photon polarizations is trivial. Linearly polarized photons can for instance be obtained at the SPS.

Inspection of the table reveals that only gluon jets can carry a linear polarization (i.e. dl (g) 0) . Again making use of eqs. ( 7 ) , ^{(1 1 )} ,

(131, we obtain for the only non-zero matrix element of _(V)

Here in the'helicity basis appear non-diagonal elements of the frag- mentation matrix. This is not a surprise because they are related to diagonal elements of the fragmentation matrix in the I , / / basis,

I I I

fi ^{: 19.3} GeV , 0.10 -

Fig. 4 ( l e f t ) : The d e n s i t y m a t r i x element p l 3 ( V ) f o r p0 production a t a HERA energy f o r s e v e r a l production a n g l e s .

Fig. 5 ( r i g h t ) : The d e n s i t y matrix element p l - l ( V ) 1 f o r p0 and @ pro- d u c t i o n a t a SPS energy f o r s e v e r a l production a n g l e s .

namely,

,VJ - ^{,VJ =}

,+-

^,+;

9 1 4fl +-

I t seems p l a u s i b l e t h a t t h e l i n e a r p o l a r i z a t i o n of t h e gluon i s some- how t r a n s f e r r e d t o t h e v e c t o r meson. Such fragmentation f u n c t i o n s have been i n t r o d u c e d by Olsen e t a 1 /12/ some time ago.

From (22) we f i n d f o r t h e decay d i s t r i b u t i o n

1 3 1 2 2

W = - p l - l ( V ) s i n 9 cos 8

p l - l (VI i s r e l a t e d t o t h e t r a n s m i t t e d s p i n asymmetry

Assuming f o r a q u a n t i t a t i v e e s t i m a t e of p I m 1 ( V ) 1 i n t h e most o p t i m i s t i c c a s e , t h a t D ~ $ - ^{D? = D~} we o b t a i n t h e r e s u l t s shown i n F i g . 5 f o r t h e c a s e of $0 and 4 prod%ction. The l a t t e r one i s p a r t i c u l a r l y s u i t - a b l e f o r t h i s computation s i n c e t h e 4's a r e mainly produced i n gluon j e t s . The s i z e of (V) depends on t h e assumptions a b o u t t h e fragrnen- t a t i o n f u n c t i o n s . ~ d - k h e h i g h e r t w i s t model, f a r example, one g e t s

PI-1 1 (V) = 0.

The r e a c t i o n e+e- ^{-t VX} w i t h u n p o l a r i z e d e l e c t r o n s and p o s i t r o n s h a s been s t u d i e d w i t h i n t h e framework of t h e hard s c a t t e r i n g model by many a u t h o r s (e.g. r e f s . 7 , 8 and 1 3 ) . To s i m p l i f y m a t t e r s and because of a c t u a l r e l e v a n c e we w i l l r e s t r i c t o u r s e l v e s t o e n e r g i e s s m a l l compared with t h e Z-mass b u t b e s i d e s t h e p u r e l y e l e c t r o m a g n e t i c c o n t r i b u t i o n s we keep t h e y - Z i n t e r f e r e n c e terms.

The d e n s i t y m a t r i x pO(v) i s d i a g o n a l with

p:,(v) = I D'O e 2 / E D' e 2 p y l (V) 0 (V) = C !:D():+ :D e i / P D' e 2 (28) 4+ q q 4 q'

4 9 9 q

Our statistical fragmentation leads then again to the predictions given in eq. ( 18) . ^A measurement of WO provides DV A~ and tests the consistency of the whole picture. The observatio2 k?€ non-negligible non-diagonal elements of po(V) would spoil the picture. For example, coherence effects in the fragmentation (i.e. if the two jets do not fragment independently) lead to a non-vanishing 07-,(V) in addition to the diaqonal elements / I /

where the coherent fragmentation functions are defined by

Rotational invariance requires for non-zero elements in the limit of PT + 0

A - ^A' - ^A- + ^1'

4 9 4 q ⁼ hv - ^{A; (31)}

There is no quantitative estimate of py-, (V) available yet. However, its measurement would be highly interesting.

In the ca e of polar'zed electrons (or positrons) one finds E 4

p 1 (V) = p (V) = 0. p (V) is only zero for the parity conserving electro- magnetic interaction. In the y - Z interference region it may have non- zero diagonal elements

and a similar expression for :p (V) + 3 ^(V)

.

one has a way of measuring both the vector and axial vector coupling constants of the quarks. This would test crucially the standard model of electroweak interactions.

Particular interesting examples are the production of + ' s and Po's.

For $ production on$ may assume all fragmentation functions DO to be zero except D$ = D-

.

c A

s cos 0

A ~- ~^{~ J S Z}=^{. G,'} P I ( + ) = 2~

5

s-mZ itcOs2e m~

In p0 production, with DP = DP ^{= :D = D} and other fragmentation func- tions neglected, one mea#uresu

d

2

2 G: - ^{G: and 2 G:} - ^Gv .

The same information is obtained when both the beams are polarized.

8 - CONCLUSIONS

We have computed the hard scattering model predictions for the vector meson spin density matrix p(v1 in the large pT process yp + VX.

In the limit of negligible transverse momentum with respect to the jet axis all the results become extremely simple in that all or most of the non-diagonal terms of P(V) are forced to vanish: therefore any experi- mental measurement of sizeable non-diagonal terms pXVfv(V) would cast very severe doubts on the validity of the probabilistic hard scatter- ing formalism in dealing with spin effects. We want to stress here again that the structure of p(V) comes out to be the same both for the contributions from point-like photons and for the photon structure function.

The only non-diagonal terms which, even in the hard scattering proba- bilistic scheme, may be different from zero, that is

~ 1 , ~

0%- (V) for polarized photons, have a particularly important signif i- cance: in fact they are strictly related to the special role of the gluon and of the y q -+ gq subprocess and their measurement might help to gain some deeper insight into the nature of the gluon and its hadro- nization process.

The simple measurement of the vector meson density matrix for unpola- rized photons ~O(V), can also give us some very useful information about the fragmentation functions of polarized partons into polarized particles.

REFERENCES

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.

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