Jet handedness as a measure of quark and gluon polarization

Physics Letters B 284 (1992) 394-400 North-Holland

PHYSICS LETTERS B

Jet handedness as a measure of quark and gluon polarization A.V. E f r e m o v 1, L. Mankiewicz 2 and N.A. T6rnqvist 3 Theoo, Division, CERN, CH- 121I Geneva 23, S~ itzerland Received 23 March 1992 We introduce a new concept, the handedness of a jet, which can be used to determine the polarization of a high-energy quark or gluon produced in a hard collision. The method can be tested in e+c ~ 2 jets at the Z° peak where the polarization of the emerging quarks is known and correlation between jet handedness and parton polarization could be determined. Subsequently, this correlation could be used in other contexts to determine the parton polarization from a measurement of jet handedness. In particular, the method could be very useful for measuring quark and gluon polarization and the quark "transversity" distribution in the proton.

1. During recent years it has been realized that some q u a n t u m numbers of quarks and gluons, in spite of confinement, can be determined experimentally. In fact, as reviewed in ref. [1 ], their spin and charge can be determined through the detailed studies of the jets which arise when a quark or a gluon is produced with a large momentum. In this letter we discuss a new method which allows also the polarization of a patton to be determined through a similar study. Such information is clearly very important for understanding spin effects in QCD. In particular, it can provide supplementary information which could help to resolve the current "proton spin crisis" [2,3]. It appeared because of the SLAC and EMC measurements [4] of the proton spin-dependent structure function g ~ ( x ) and provoked a lively discussion as to which partons carry the proton helicity. According to a popular hypothesis, there is a large contribution to the first m o m e n t of g~ (x) from the gluon-polarized distribution AG arising through the axial anomaly [ 5 ]. It has been suggested that it could partially cancel the contribution due to polarized

l

Permanent address: Joint Institute for Nuclear Research, JINR, 141 980 Dubna, Moscow Region, Russia. 2 Permanent address: Nicolaus Copernicus Astronomical Centre, Bartycka 18, PL-00716 Warsaw, Poland. 3 Permanent address: Research Institute for High Energy Physics, University of Helsinki, Siltavuorenpenger 20C, SF-00170 Helsinki, Finland. 394

quarks. A number of ideas have been proposed for the direct determination of A G ( x ) , see e.g. ref. [3]. In principle experiments can measure the spin asymmetry in a direct photon production in high-energy collisions of polarized protons [6 ], or the polarization of leptons in a Drell-Yan production from a polarized target [7]. However, such experiments are difficult because of the rather small cross section involved. Therefore, we find that it would be very useful to have at our disposal a technique that would allow a determination of the polarization of a high-momentum parton from an analysis of its jet. In this paper we shall discuss how it should be possible to perform such parton polarization measurement. The method relies on the detailed study of the jet structure. We note that such a possibility was realized as early as 1978 by one of us [8] in the context of the polarization of high-pT A's. After this work was essentially finished we learned about the work of ref. [9], where a similar possibility is suggested. For this purpose we propose to study a new measurable property of jets, which we call their handedness. This observable, like the decay asymmetry in the weak hyperon decay of A -+ ~zp, is proportional to the polarization of the parent parton. Note that as a jet we mean any isolated group of more than two particles that have invariant mass much smaller than their relative momenta with respect to all other registered particles. In the following we explain the idea

0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

Volume 284, number 3,4

PHYSICS LETTERS B

behind the concept of handedness and outline the corresponding algorithm for data analysis. The detailed account of the underlying calculations will be published elsewhere [ 10]. For the weak decay of A ~ gp, parity violation allows for a ~ . k term in a general matrix element o'.k .A/IA~ n p = A -t- B

ItS"

The interference of this term with the parity-conserving one results in the asymmetric distribution of the decay products in the A CMS -x 1 + ~ P . k / I k l . This allows one to determine the polarization P of the initial A once the decay parameter c~ is known from other experiments. On the other hand, parity is conserved in a stronginteraction process, such as a strong decay or jet fragmentation. This fact puts severe constraints on the form of possible invariant amplitudes for such processes. In particular, it follows that in a strong twobody decay it is impossible to get information about the polarization of the decaying state solely from the observation of the m o m e n t u m distribution of the decay products. Therefore, in a strong decay, one is forced to consider at least three particles in the final state. Then a pseudovector n ~ can be constructed from their m o m e n t a and contracted with the polarization pseudovector to form a parity-conserving term in the decay amplitude. As a consequence, one can determine the polarization of the initial particle by observing the asymmetry with respect to the direction of n ~. In the case of a jet, the parent parton fragments into many final hadrons so that a pseudovector n/' can be easily constructed.

25 June 1992 (1)

M = f +igor.n,

where n = k 1 X k2/]k 1 X k 2 ] iS the normal to the decay plane. Here k~ and k2 are the momenta of the two pions ordered in some prescribed way, e.g., according to their charge. For example, k ~could always be taken to be the n + and k2 the n - momenta. If the charges are not measured, one can define kl and k2 such that, for instance, Ikll > LkRI. As is very well known, the matrix element ofeq. ( 1 ) leads to an asymmetric decay distribution with respect to the direction of the polarization of the initial fermion da dr2 x (If12 + ]g[Z)(1 + ( t P . n ) ,

(2)

where (~ is the asymmetry parameter or the analyzing power c~ = 2 I m ( f g * ) / ( [ f [ 2 + Igl2),

(3)

often denoted by ,4 or P . Here it will play a role similar to that of the decay parameter ~t in the weak hyperon decay such as A ~ rip. In a strong three-body decay the value of (~ is determined by non-perturbative strong-interaction dynamics and depends only on the invariantss~ = (k + k l ) 2,s2 = (k + k2) 2 (where k denotes m o m e n t u m of the initial fermion) and on the masses of the particles involved. Once the (~(sl, s2, M 2) is known from some previous experiments, with the known initial polarization, one can reverse the argument and determine P by studying the decay asymmetry with respect to n . In covariant notation the interesting term can be written as

~ P - n = - ~ (S/'nl,), 2. Before we consider the conceptually more difficult problem of a jet polarization, let us discuss a simpler case, with very well known spin kinematics. Consider the decay of a heavy fermion with JP = ? l + into two pions and another J R = 5J + fermion. It is completely analogous to the well-known process nN nN with the incoming pion line reversed. Assuming parity conservation, this process is determined by two independent amplitudes, the spin non-flip ( f ) and the spin flip (g) amplitudes. The matrix element in the rest frame of the heavy fermion has the simple form

.a /72 = - 1 , and S u denotes where th, .x C,,,,a~k.u kl.2 k2, the initial polarization vector, S/'k~ = O, S 2 = _ p 2 . It is useful to express these vectors in a frame where the longitudinal component is oriented along k and the two transverse spatial components are inside (el±) or, respectively, orthogonal ( e 2 ± ) to the production plane of the heavy fermion. In this notation we obtain

S t, =

Ikl , , , ~,_~ ,, E Pl177,,±,,±,-llT~ },

1l/` =-

/711TT,/7±,n±,/711-:-;I,

Ikl Al

l

2

E'~

JVI ]

/7~ ---- - 1 ,

(4) 395

Volume 284, number 3,4 k ~' = ( E , O , O , k ) ,

M 2 = E 2 - k 2.

PHYSICS LETTERS B (4cont'd)

In our three-body example the polarization-dependent term in (2) depends on the initial fermion only through the pseudovector S ~', and on the decay products only through the pseudovector ~,n ~'. From the angular distribution of the decay products one can determine three asymmetries Hll, H~,, and H ~', defined with respect to the planes orthogonal to k, e ± ~, and e±-, respectively. We shall call these asymmetries the longitudinal and transverse components of handedness. The longitudinal handedness Htl is conceptually the simplest, and also the easiest to measure. It is defined in terms of left- and right-handed events, and it is proportional to the longitudinal polarization of the initial fermion: HI1 =

NL(nll > 0) -- NR(nll < 0) N = ~7,- Ptl ,

(5)

where N = NL + NR is the total number of events. Whether the event is left- or right-handed depends on the relative orientation of the three vectors k, k j, and k2, such that nil :x e~jkk~k[k~ is > 0 for NL and < 0 for NR. The particles with m o m e n t a k~ and k2 have to be ordered either by their charges or by the magnitudes of their momenta, in the same way for all events. The difference between left- and right-handed events is similar to the difference between the left and right hand when the same three fingers of each hand point in the direction of m o m e n t a of the decay products. The sign of ?hr depends of course on the convention adopted to distinguish left from right. In eq. (5) the parameter ct(sl,s2, M 2) has been replaced by the average 77, since in practice one will have to perform an average over a phase space of the final states. This average can involve some weighting procedure (whence the subscript W ) . I f a (Sl, s,_, M 2 ) alternates in sign the weighting function W (sl, s2, M 2 ) could in principle even compensate for the sign changes. The W should of course be chosen in such a way that I~w] becomes larger, and one could even imagine that its approximate form could be determined by a fitting procedure to the actual data. Once ~7w is determined in an experiment where the initial polarization PII is known and HII is measured, its value can be used in another experiment to determine PII through the measurement of HII, using exactly the same algorithm for its definition. 396

25 June 1992

For the case of three-body decay the transverse polarization can be measured in a very similar way, using the value of?7,, determined from the longitudinal polarization measurement. It is to be noted that, as will be explained below, in the case of jet fragmentation 77, is in general different in the longitudinal and the transverse directions. One has to replace the direction k by e± ~ or e± 2 in the ordered set of three momenta, or equivalently replace nil in eq. (5) by the transverse components n~_ (a = 1,2). Thus, the two transverse-handedness parameters are defined as H~ = N L ( n ~ > O) - N R ( n ~ < O) N

= ?7,, P2_,

(6)

for a = 1,2. By measuring both H~_ one can determine also the transverse polarization of the initial fermion. So far we have defined handedness in the CMS of the decaying fermion. The three handedness parameters defined in eqs. (5) and (6) define a three-vector H which is proportional to the polarization of the initial fermion. Eqs. (5) and (6) can be written in terms o f the cross section a = %(1 + aS¢'n~) as Hi = f a s i g n ( n / ) Wdg2 = 7 f w P i . f a IW[ d~9

(7)

Instead of sign(ni) one can extract the polarizationdependent term by averaging over the direction of n in the CMS: 3 f a n Wd,Q 3(n),- = &wP. H = 5 falWld~9 = ~

(8)

3. Let us now turn to the fragmentation of a jet. First of all, we notice that contrary to the case of a particle decay the criterion for determining a jet is not unambiguous. There is an ambiguity in choosing the opening angle of the jet; also, there is always a number of soft particles that cannot be identified with either jet [ 11 ]. In QCD the arbitrariness is due to the coloured nature of the parent parton, quark or gluon, and to the fact that the hadrons associated with the jet are colourless. Certainly, the parameter ?Tw will depend on the chosen criterion. Another problem that enters the definition of?Tw is the large number of particles in a jet. Hence, one has a large number of choices of pairs ( i j ) corresponding

Volume 284, number 3,4

PHYSICS LETTERS B

to m o m e n t a kti and k2j. It seems natural to define the handedness in such a way that each individual jet is either left-handed (hjet = + 1 ), or right-handed (hjet = - 1 ) . However, because fragmentation is a probabilistic process, a given choice of pairs can sometimes give wrong information about the polarization of the parent hadron. It is believed in general that the fastest, leading particles are most likely to carry the information about the parent parton, while soft, "wee" hadrons are distributed randomly in phase space. Inclusion o f too many soft hadrons into the analyzed sample can therefore dilute the effect. On the other hand, one should not rely entirely on the information from the leading pair. We therefore believe that it is neccessary to include some of the next-to-leading particles in the determination of the handedness o f the jet. Because of this inherent ambiguity, we believe that the most suitable way to proceed is to select a group of leading particles and form all possible pairs among them, adding the contribution from all the pairs with equal weights. Whether a better correlation between handedness and polarization can be achieved, by weighting different pairs according to a certain algorithm, is left as an open question. Most probably, it can only be answered by an analysis of real data. The precise criterion by which the handedness of an individual jet is defined has still to be proposed. One such criterion could be hjet = sign(Kk -- KR),

(9)

where KL,R are the numbers o f left- and right-handed pairs among the leading group. More precisely, the criterion (9) means that if n~jkl is the normalized vector associated with a particular pair ( j k ) , the longitudinal and transverse handedness of an individual jet is defined as

•

h]et = sign

(z

sign(nljk))

,

pairs jk

and

thus the quantity ~ j k slgn(nljk) ' ) replaces sign(n ~) in eq. (7). Another, perhaps simpler, criterion would be to use an average direction formed from the directions n¢jk) to replace n in eq. (8). A more important difference between parton fragmentation and particle decay comes from the essentially non-local character of the parton fragmentation

25 June 1992

process. This can be understood as being basically due to the quantum mechanical uncertainty principle. Note that in the case of a particle decay we deal with a final state of definite energy and momentum. Consequently, the space-time coordinate of the parent is totally undefined, and the initial-state wave function corresponds to a plane wave. The decay itself is therefore insensitive to the mechanism of preparation or "production" of the initial state. All the information is carried by the free-particle density matrix characterized by m o m e n t u m and polarization vectors. On the other hand, in the quark fragmentation, the e n e r g y - m o m e n t u m of the jet is not precisely defined. Therefore a jet corresponds not to a plane wave, but rather to a wave packet of finite extension. Thus, in general it depends on the short-range subprocess from which it has originated. Its space-time extension is therefore of the order of l/q, where q is the large m o m e n t u m characterizing the initial hard subprocess. To make the discussion more quantitative, let us consider the expression of the cross section for jet "production" and subsequent "decay", written in the factorized form [ 12 ]: d a w / d4 ~d~,,f~(q,~) (jetkl~,(~)q~(O)ljetk). (10) Here db,~ corresponds to the hard patton subprocess characterized by the large m o m e n t u m scale q. The second factor is the matrix element of the bilocal quark operator. It can be decomposed into a set of the fragmentation functions of the quark into a jet of m o m e n t u m k using the operator product expansion (OPE) [13]. In the case of the decay the zero-momentum vacuum state is the only one that can be inserted between quark operators in eq. (10), and therefore the ~ dependence of the matrix element exponentiates to exp (i ~ • k). For the fragmentation, a polynomial factor in ~ can appear in the decomposition of the quark operator products into the Dirac tensor structures. Such a decomposition is quite analogous to the one considered by Jaffe and Ji [14] for the structure functions of the polarized hadron. One needs only to make the substitutions

n~ ~i,,

S l, ~-~ ~ n l , ,

f - ~ D v,

h,--D T

g , ~ D A,

397

Volume 284, number 3,4

PHYSICS LETTERS B

in the expansions of ref. [14], to find the required decomposition. Expanding now the matrix element of eq. (10) in all possible Dirac tensors up to twist-three terms containing n ~, one obtains (k; kl, k2 Iq,, (~)q/~ (0) Ik; kl, k2) 1

f dz d3kl d3k2 , ,I. . . . . . =,

z

k°

k°

I. . . . v

I;"r/~'"~"n ¢ x M u 2

0

+ (75h),I3M(D A + D A)

-(?,si~),~f~(n.

~)MD)

+ (7'ii£),,l~D ~

+(75[k,~l),/s(n.~)M-D2 T +

.}

where the dots stand for terms independent on the vector n u. The mass parameter M is of the order of a hadron mass. We have singled it out to make D . dkl/k°l • dk2/k ° dimensionless, and z is the fraction of the quark m o m e n t u m carried by the jet. We shall assume that the jet carries most of the quark momentum, so that the z dependence of the D functions is close r o d ( 1 - z ) . Now let us show how the expression (11) works in the special case of the previously considered particle decay. Then only the ~-independent terms survive, i.e., (D) + D~ ) for the axial part and D~ for the tensor one. In this case the former gives the main contribution to the longitudinal handedness, and the latter to the transversal one. Note that because the decaying particle is a free fermion on the mass shell k -~ = M -~, the matrix element ( 1 1 ) has to satisfy the Dirac equation. This immediately gives D(+D~

-- D T,

(12)

and consequently (~ll = (~±. In the more general case of a jet fragmentation there is no free-particle Dirac equation for quark, and no relation like (12) arises. In addition, the second axial term also gives a twisttwo contribution to the longitudinal handedness. This changes the longitudinal analyzing power (ql, but not the transverse one a±, which as previously remains determined by the tensor term D~. As a result, the analyzing power 77 will be different for longitudinal and transverse handedness: HII = ~711PII, 398

H~_ = ~:± P2.

(13)

25 June 1992

Note, however, that the twist-two transverse handedness due to D~v will contribute only to the double transverse spin correlation, i.e., either when the jet is produced by a transversly polarized target or beam, or when one measures transverse-handedness correlations of two jets produced simultaneously. Single transverse handedness is of twist-three and is determined by the D v term, which is transverse to the scattering plane, and by the D~ + D A term. In addition to these two terms there will also be other twist-three contributions from quark-gluon correlation functions [ 14,1 5 ] due to the quark interaction with the background gluon field of the jet. The former two terms are connected with the latter ones by a sum rule which follows from the QCD quark equation of motion. These contributions determine the order of magnitude of the transverse handedness as ~ M / q , where q is the large m o m e n t u m in the subprocess. At the same time the quark transverse polarization Pq~ decreases as mq/q. In this case, it seems that ~t± ( x M / m q ) is very large. This conclusion, however, could hardly be considered as completely correct. The quark mass mq depends on the ultraviolet cut-off parameter S . Hence Pq~ is not renormalization-group-invariant, i.e., experimentally not a measurable quantity. On the contrary, handedness is directly measurable experimentally and therefore can be regarded as a more fundamental characteristic of jets. It can completely replace the quark polarization in any physical process. Moreover, it could be non-zero even in the case of vanishing transverse quark polarization. As an example one could consider the single-quark polarization transverse to the scattering plane in e+e - -~ q~ 2 jets. There seems to be no fundamental difficulty in applying the above reasoning to the gluon jets as well. In fact, in the rest frame of the jet, eq. (10) and the decomposition ( 1 1 ) can be written in a two-dimensional form in terms of Pauli matrices a. For the gluon jet one has only to replace a by the matrices Z of spin one. The special structure of gluon jets such as oblateness [16] and tensor polarization will be discussed elsewhere [ 10 ].

4. Finally, we note that the analyzing parameters 7711 and ~7± can be measured in a process where the po-

Volume 284, number 3,4

PHYSICS LETTERS B

larization of the parent quark or gluon is known. One such a process is e+e - annihilation into two jets in the region of the Z peak. It is well known that the produced qq pairs are longitudinally polarized because the interference of axial and vector coupling constants which can be expressed through sin-' Ow = 0.23. p~ = _ ~ q 1 4- cos 2 0 - 2Pc/Pq cos0 ~ 1 + cos-' 0

~q,

2PePq cos 0

where 0 is the quark production angle in the CMS, and _

2gvgA gv + gA

has values of 0.93 for down quarks, 0.67 for up quarks, and 0.16 for electrons. The polarization of the antiquark has the opposite sign. The transverse polarization of quarks is effectively zero for mq << q, but the transverse s p i n - s p i n correlation is not zero. For example, the correlation transverse to the scattering plane equals g2 a - g,~ sin 2 0 g~, + g~, 1 + cos: 0 '

A~,,v ~

(14)

which is - 0 . 7 5 for up quarks and - 0 . 3 5 for down quarks in the region 0 ~ 90 °. The analyzing parameters entering (13) can be expressed through the corresponding fragmentation functions D in the rest frame of the jet (k = 0): 7711 -

2M

~

f

A 2

2

DI (kl , k z , k l " k2)n,

d3kl d3k2 k---F

nx > 0

(15) and --(2) c*± = 2 I

kO . DIT ( k ?~, k ~"~, k l .kz)n= kld3kl ~ d3k2

pair. The problem appears because, if one cannot distinguish between quark and antiquark jets, one would obtain zero result for the handedness due to their opposite polarization. Tagging the quark jets by requiring the presence of a baryon, or by the presence of a leptonic weak decay of a heavy flavour, would suppress the statistics drastically. However, it would allow one to discriminate the handedness for different flavours. Then, one could use the criterion k~ > k2, which may be more sensitive to the polarization of the parent patton. A much better strategy seems to be simply to choose particle 1 as the positive particle from the pair, while particle 2 is negative. In this case in the CP-invariant theory the handedness for an antiquark jet has the same sign as for a quark jet with opposite helicity. With the ordering of pairs according to charges, however, the handedness of an up (u or c) quark jet will be opposite to that of a down (d, s or b) quark jet since their charges are opposite. This will dilute the effect. We note, however, that at the Z peak the production of up quarks is suppressed by the factor Cru

(g.~ + g,~,)u

1 + (1 - 8sin-~ O~) 2

O'd

~ (g?~ + g,~)d

1 4- (1 -- 34 sin20w )2

0.78,

(18)

compared with the down quark production with larger polarization. The resulting dilution coefficient, x =

k~"

25 June 1992

3 - 2 (Pu/Pd) (Cru/Crd) ~ 0.45, 3 + 2au/ad

(19)

seems to be large enough to suggest a reanalysis of the LEP two-jet events looking for jet handedness and attempting to determine the parameters ?7.

(16)

n:>0

We also note that the absolute value of the transverse ?7± could also be determined through the handedness correlations of two jets in the same event: ,'IV'V = NLL 4- ~TRR -- ~¢~LR-- /¥RL = ( ~ ) ) 2 A q V N " N

(17) As far as the measurement of c~ll is concerned, special care is necessary in choosing the vector k~ of a

5. In summary, we have proposed a new method to determine the quark and gluon polarization through a measurement of a new property of jets - their handedness. The main question of how large the corresponding analyzing power is remains open, but clearly only experiment can resolve this problem. Apart from suppression due to helicity conservation in QCD, by order o f (Dlhadron/mje I )2, we do not see any reason why it should be prohibitively small. One possible experiment is a search for jet handedness in e+e annihilation into two jets, in the region of the Z peak. A 399

Volume 284, number 3,4

PHYSICS LETTERS B

positive result w o u l d open the d o o r to a whole series o f o t h e r polarization m e a s u r e m e n t s . O n c e our h a n d e d n e s s concept is p r o v e n useful for parton polarization m e a s u r e m e n t s , an i m p o r t a n t application w o u l d be the direct m e a s u r e m e n t o f the gluon spin structure f u n c t i o n o f the nucleon. This is possible through a d e t e r m i n a t i o n o f the longitudinal h a n d e d n e s s o f jets produced, through the gluon C o m p t o n subprocess, at a longitudinally polarized nucleon target. N o t e that b e a m polarization is not required. A n o t h e r very interesting observable is the quark " t r a n s v e r s i t y " d i s t r i b u t i o n in the nucleon, discussed in refs. [ 14,17]. It a m o u n t s to the d e t e r m i n a tion o f the transverse h a n d e d n e s s o f quark jets produced in deep inelastic scattering off a transversely polarized target [9]. T h e difficult and long-standing p r o b l e m s are the single transverse a s y m m e t r i e s at large p±, such as A polarization or pion p r o d u c t i o n asymm e t r y (for a recent m o d e l e x p l a n a t i o n o f the latter see ref. [ 18 ] ). W e hope that m e a s u r i n g the transverse h a n d e d n e s s o f high-px jets could help to u n d e r s t a n d the nature o f these p h e n o m e n a . We gratefully acknowledge useful discussions with G. Altarelli, J. Collins, P. H o y e r and J. Softer, as well as stimulating questions f r o m m e m b e r s o f the Delphi Collaboration. W e also thank J. Ellis and the T h e o r y D i v i s i o n at C E R N for the hospitality e x t e n d e d to us during our stay there.

Note added. After this paper was accepted for publication we h a v e learned about the article by Nachtm a n n [19], which c o n t a i n s a discussion o f the idea very similar to ours, but o r i e n t e d for different applications.

References [1] See the review by P. M~ittig, Phys. Rep. 177 (1989) 141. [2] H. Rollnik, Proc. 9th Intern. Symp. on High energy spin physics (Bonn, 1990), eds. K.H. Althoff and W. Meyer (Springer, Berlin, 1991) Vol. 1, p. 183. [3] E. Reya, Dortmund preprint DO-TH 91/19 (1991), presented at the Lepton photon high energy physics Conf. ( Geneva, 1991).

400

25 June 1992

[4] EMC Collab., J. Ashman et al., Phys. Lett. B 206 (1988) 364; Nucl. Phys. B 328 (1989) 1: M.J. Alguard et al., Phys. Rev. Left. 37 (1978) 1261: 41 (1978) 70: G. Baum el al., Phys. Rev. Left. 51 (1983) 1135. [5] A.V. Efremov and O.V. Teryaev, JINR report No. E288-287 (1988), unpublished; G. Altarelli and G. Ross, Phys. Left. B 212 (1988) 391; R.D. Carlitz, J.C. Collins and A.M. Mueller, Phys. Lett. B 214 (1988) 229. [6] E. Berger and J. Qiu, Phys. Rev. D 40 (1989) 778, 3128. [7] A.P. Contogouris and S. Papadopoulous, Phys. Lett. B 260 (1991) 204. [8] A.V. Efremov, Soy. J. Nucl. Phys. 28 (1978) 83. [9] B. Carlitz, J. Collins S. Heppelmann, G. Ladinsky, R. Jaffe and X. Ji, Measuring transversity densities in singly polarized hadron-hadron collisions (preliminary version of Penn State preprint PSU/TH/101 (March 1992)). [10] A.V. Efremov, L. Mankiewicz and N. A. Tornqvist, to be published. [ I I ] Y u . L . Dokshitzer, V.A. Khoze, A.H. Mueller and S.I. Troyan, Basics of perturbative QCD (Editions Erontieres, Gif-sur-Yveue, 1991 ). [12]A.V. Efremov and A. Radyushkin, Riv. Nuovo Cimento 3 (1980) 1; for the recent discussion of factorization see J.C. Collins, Penn State preprint PSU/TH/100 (1992). [ 13 ] l.l. Balitsky and V.M. Braun, Nucl.Phys. B 361 ( 1991 ) 93. [14] R.L. Jaffe and X. Ji, Phys. Rev. Lett. D 67 (1991) 552; preprint MIT-CPT-2005 (1991). [ 15 ] A.V. Efremov and D.V. Teryaev, Soy. J. Nucl. Phys. 39 (1964) 962: J. Qiu and G. Sterman, Stony Brook preprint ITP-SB91-8 (1991). [ 16] S.J. Brodsky, T. A. DeGrand and R.F. Schwitters, Phys. Lett. B 79 (1978) 255. [17] J. Ralston and D.E. Soper. Nucl. Phys. B 152 (1979) 109; X. Arlru and M. Mekhfi, Z.Phys. C 45 (1990) 669: J. Collins, in: Proc. Penn State Polarized Collider Workshop (University Park), AlP Conf. Proc. No. 223, eds. J. Collins, S.F. Heppelman and R.W. Robinett (AIP, New York, 1991) p. 364: J. Cortes, B. Pire and J. P. Ralston, in: Proc. Penn Stale Polarized Collider Workshop (University Park), AIP Conf. Proc. No. 223, eds. J. Collins, S.F. Heppelman and R.W. Robinctt (AIP, New York, 1991 ) p. 184. [18]J. Softer and N.A. T6rnqvisl, Phys. Rev. Left. 68 (1992) 907. [19] O. Nachtmann, Nucl. Phys. B 127 (1977) 314.