Transversely polarized parton densities their evolution and their measurement

The transverse spin asymmetry of a quark in a baryon and the linear polarization of a gluon in a vector meson are studied from the t-channel point of view. Using the Altarelli-Parisi approach, they are shown to obey independent evolution equations and to decrease with increasing Q2• We investigate the possibility to measure them at leading twist, to leading order in et. and et.8 and without analyzing the final polarizations. This requires simultaneous polarization of the beam and the target; the observable effect is in the azimuthal distribution of the high PT particle or jet. Assuming a simple (quark+ scalar diquark) model for the baryon, a large asymmetry is expected in pp Drell-Yan collisions, a smaller one in high Pr pp collisions, from the interference term in the scattering of two identical quarks.


Introduction
In the parton model [1][2][3], the relevant quark and gluon densities for polarized beam or target experi ments are the following ones: (i) a(x) = unpolarized density of parton a .a= quark(q) or gluon(G).xis the Bj orken variable; the dependence in Q 2 is understood.
(ii) L1a(x) = a+(x)-a_(x), where a + (x) and a_(x) are respectively the densities of parton a of positive and negative helicity, when the hadron has positive helicity.
(iii) L11 q(x) = qix) -q _�(x), where q + ix) is the density of quarks polarized in the transverse direction ± n, when the hadron spin points in the direction + n.
(iv) L1 2 G(x)=GAx) -Gp(y), where G�(x) (n= x or y) is the density of gluon linearly polarized along n, * Laboratoire associe au Centre National de la Recherche Scienti fique the hadron being a vector meson linearly polarized along x.
We have obviously q(x) = q + (x) + q_(x) = q � (x) + q _� (x) G(x) = G + (x) + G _(x) = Gx(x) + G9(x).Whereas a lot of theoretical and experimental work has been devoted to a(x) and L1a(x), the transverse polarization asymmetries L11q(x) and L12G(x) have not been, up to now, popular topics.A good reason for this is that they are not easy to measure; we shall discuss this question is Sect.4. But, in the case of ,11 q(x), there is also the prejudice that it vanishes in the limit of zero mass and zero transverse momentum of the quark.This is totally unjustified; in the case of electrons, for instance, the smallness of the electron mass does not preclude large transverse polarizations of ultrarelativistic electrons; the effect of such polariz ation is well known in e + ecollisions [4].To show that L11 q(x) is not necessarily vanishing, we present, in Appendix C, a naive covariant parton model where the baryon is composed of a quark and a scalar diquark.It predicts Concerning the evolution of L11 q(x), previous theoretical works have yielded different results [5].As for L1 2 G( x), we have not found any explicit mention of this quantity in the literature.It is involved, however, implicitly in the polarized photon structure function [6].
For the above reasons, we think it worthwhile to derive independently the evolution equations of the transverse asymmetries, using the Altarelli-Parisi ap proach, and to discuss their observable effects at leading twist and to lowest order in et. and et.8•These problems will be more easily treated in the t-channel formalism which we have already introduced to handle spin in multiparton reactions [7].This work will be developed as follows.In the next subsection, we present the t-channel formalism for the single parton distributions.In Sect.3, we derive the evolution equations of the transversely polarized par ton distributions.We discuss the observability of these distributions in Sect.(2.1) is a partial discontinuity of the forward parton-parton amplitude, p a (x) is a density matrix in spin space, related to the hadronic density matrix pA by ( A a l p a (x) I A�) and similarly for p b (y).r a /A(X) is the hadron-parton cut amplitude ; unlike p a (x) it does not depend on the actual hadron polarization.
For each particle-antiparticle pair in the t-channel of Fig. 1 , we define the t-channel helicity state to be simply IA.)®I J:' ), (2.4a) where J:' = -A '. (2.4b) Then we build a basis of particle-antiparticle helicity states I A ) which have definite total helicity J = A.
and definite symmetry t: = ±.For particles of spin s which can only take two possible helicities, + s ors, the I A ) states are: Let us first consider the "t-channel amplitude" f' a !A(x) obtained by crossing r a /A(x) and sandwiching it between I A ) states.Due to rotational and parity invariance, f' a !A(x) conserves J and e, i.e.
(j a = A a -A�= (j A = A A -A�, (2.9) As expected, transverse polarization asymmetries are associated with non-zero total helicity J in the t-channel, i.e., helicity flip in the s-channel.Due to the conservation of J and e, we can already predict that L11 q and L1 2 G satisfy simple (unmixed) evolution equations, contraryly to what happens for Liq and L1G [2].
As we have done for r, we can similarly decompose the matrix densities p and the parton-parton dis continuity Hin the t-channel basis.For this purpose, we prefer to reformulate the t-channel analysis in a more physical way: To each t-channel state (2.lOa) defined in (2.6), we associate the s-channel operator @ (A)= L C;.µIA.><-µ J . (2.lOb) ).,µ The orthogonality of the I A) states yields (2.11) (2.12) Then we decompose p A, p 8 , p"(x) , p b ( y ) , r af A (x) , r b f B ( y ) and H on the s-channel operators l'D(A) [8]: (2.13b) (2.l 3c)

Aa Ab
The inverse formulas are obtained by use of (2.12); for instance The coefficients f'".{A(x), H A .A • are just the t-channel amplitudes taken between the I A) states.In fact we could have introduced the lD (A)'s directly by (2.11), without any reference tot-channel states.This provides an alternative presentation of the formalism.We are now able to rewrite (2.1) and (2.The "unpolarized" quantities are We give below the values of p A/Po+ corresponding to different polarization cases.
3 Evolution of Li1q(x) and Li2G(x) In the physical gauge, the evolution of p"(x , Q 2 ) or, equivalently, of r a f A(x, Q 2 ) is given by the ladder of Fig. 2  where IJC, {J, y are color indices, µ, v, A. are helicities, b = za + kT and c = (1z) a -kT.For L11 q(x) and L1 2 G (x), we have A.=µ= -A.'= -µ'.The correspond ing element L11or2P(z) = < -A., A.I 1T(z)IA., -A.) appears then as an inter f erence term between two V's (at least in the helicity basis).Alternatively, we have: L1 1 P(z) = P q+nq+/ z) -P q_nq+/ z), L11 P(z) = P GxGJz)-P GxG/z). (3. 2) The detailed calculation of the transverse splitting functions is given in Appendix B and C. We give here just the result [11]: � td a �a • f (z) being any test function which is sufficiently regular at the end points.
For the quark transverse asymmetry, the master equation reads where oc.(Q 2 ) = g 2 /4n is the running coupling constant.
In terms of the moments the solution of (3.4) is given by In the leading logarithm approximation, we have and we can write the exponential factor of (3.5) as The moments of L1 1 P(z) are For the gluon transverse asymmetry, we have just to replace L1 1 by L1 2 and q by Gin (3.4-3.7).We have now We note that, for any n,L1 1 Pn and L1 2 Pn are negative.This means that the transverse asymmetries decrease with increasing Q 2 , whereas the longitudinal spin asymmetry [12] is constant for the quark and growing for the gluon.
4 Observabilit y of the transverse as y mmetries While there is no reason to assume that the transverse polarization of, say, a proton is not transmitted to its quarks (for instance, recall (1.2)), it sometimes happens that the hard process is insensitive to such a quark transverse polarization, at least to zeroth order in (mq/Q), (Pr(q)/Q) and to lowest order in oc and oc •.This is the case in deep inelastic lepton scattering, where helicity conservation at the quark photon vertex selects only the fJ =A -A'= 0 components of the quark density matrix, whereas transverse polarization lies in the fJ = ± 1 components.To observe transverse polarization at leading twist and to lowest order in oc and oc., we have therefore to look at other hard processes.
In this work, we shall assume that the polarizations of the final particles are not measured.Since the final angular distribution may depend on the initial polarizations, we specify the polar angle 8 and azimuthal angle <p of the relative momentum inf and write Finally, it has been shown [13] that the Born ampli tudes for the 2--+ 2 processes ( 4.1 c-f ) also conserve the total helicity, in the massless limit, although no simple proof has been given for it.Then, Aa +Ab and A,� + A,� are both equal to the total helicity of the intermediate state f in (2.2), hence <> a + Jb = 0 (to leading order in IX, IX, and m/Q) (4.9)From (4.3-4.9)we can draw the following conclusions: --if we integrate over <p, the only observable asym metries (i.e., O'p01arized =I O'unpolarized) are characterized by (4.10)This is the case for helicity asymmetries (Ja = Jb = 0).By constrast, in qq or qij scattering we have no net transverse polarization effect after <p integration (the null theorem of [ 4b] ).The same situation holds in GG or Gy scattering to leading order in IX and IX,.
We therefore consider experiments where the data are not integrated over <p.Furthermore we assume parity conservation in the hard process [14].

Conclusion
The transversely polarized quark distribution Ll 1 q(x) and the linearly polarized gluon distribution Ll 2 G(x), which correspond to non zero helicities in the t channel, obey simple, uncoupled evolution equations.
Their moments decrease for all n as negative powers of In Q 2 , unlike the helicity asymmetries for n = 1.
Ll1 q(x) and Ll 2 G(x) should a priori exist even in the limit of vanishing parton mass and transverse momentum.For instance, a naive quark + scalar di quark model gave us Ll1q(x) = q+(x); this, combined with the experimental indication [23] that J q+(x)dx::::: J q_(x)dx, would imply J L11 q(x)dx::::: t J q(x)dx.
L11 q(x) and L12 G(x) can be measured at leading twist and to leading order in a and a,, provided both the beam and the target are polarized (transverse spin + transverse spin for L11 q, linear polarization + linear polarization for L12 G).There is no effect in the total cross section but in the azimuthal distribution.Another method is to analyse one final polarization, in which case only one incoming polarization is needed.We did not study this possibility in the present work.
The principle of the measurement of L11q(x) is the same as for the asymmetry parameter aNN in elastic scattering of two spin t particle; from (4.11) we have an e1iective asymmetry parameter given by where x and y are the fractional momenta of colliding partons a and b respectively and aNN is the asymmetry parameter of the hard subprocess (a = q or 1w, b = q or ij).A strong effect is expected in Drell-Yan pp collisions where aNN is of the order of unity.In pp collisions, the scattering of two identical quarks also has a non vanishing aNN• which comes from the interference term, but this effect is only of order l/N 0010,.Nevertheless it should be interesting to detect it, looking at high xT particles or jets, for which the valence + valence mechanism is dominant.The remarkably strong peaking of aNN(O) at 0 = n/2 pre dicted by (4.15) could be tested.The quantity L1 2 G( x) exists only for spin1E 1 projectile or target, in practice for a real or quasi real photon, in the vector dominance model.The linear polarization of both the beam and the target seems extremely difficult.

4 .
Conclusions are presented in Sect. 5. 2 The t-channel spin formalism for parton distributions In a recent work [7], we have set up a quite general and straightforward formalism to handle spin in single-and multiparton scattering, based on the t channel analysis of the helicity amplitude.Consider the hard collision whose unitarity diagram is drawn in Fig. 1.The cross section for producing a particular hard final state f is a= L J dx( A. a lP0 (x) IA.�) J dy( A.bll ( Y ) IA.b) all ).'s .< A�, Ab I H a +b -+ f I A a , A.b), where x =P a /P A , Y = Pb / PB , < A.�, A.b I w +b -+ I I A. a , A.b) �(a, A.�; b, A.bl r+ If) < fl Tla,A.a ;b,A.b),