The azimuthal dependence of hadrons produced in lepton scattering off a
polarized hadron probes the quark transverse-spin distributions. In the
calculation of the asymmetries, transverse momenta of quarks in the
distribution and fragmentation functions must be incorporated. In addition to
the $\sin (\phi + \phi_S)$ asymmetry for transversely polarized hadrons, known
as the Collins effect, we find $\sin 2\phi$ asymmetries for both transversely
and longitudinally polarized hadrons.Comment: 7 pages RevTe

Artru

Collins

Jaffe

Kotzinian

Levelt

Meng

P.J. Mulders

R.D. Tangerman

Ralston

Soper

Tangerman

Physics Letters B

English

arXiv

Tangerman, R.D.

Mulders, P.J.G.

UvA-DARE

Probing transverse quark polarization in deep-inelastic leptoproduction

International Migration, Integration and Social Cohesion online publications

UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)Probing transverse quark polarization in deep-inelastic leptoproductionTangerman, R.D.; Mulders, P.J.G.DOI10.1016/0370-2693(95)00485-4Publication date1995Published inPhysics Letters BLink to publicationCitation for published version (APA):Tangerman, R. D., & Mulders, P. J. G. (1995). Probing transverse quark polarization in deep-inelastic leptoproduction. Physics Letters B, 352, 129. https://doi.org/10.1016/0370-2693(95)00485-4General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:11 Mar 20238 June 1995 PHYSICS LETTERS B ELSEVIER Physics Letters B 352 (1995) 129-133 Probing transverse quark polarization in deep-inelastic leptoproduction R.D. Tangerman, P.J. Mulders ’ National Institute for Nuclear Physics and High Energy Physics (NIKHEF)), l?O. Box 41882, NL-1009 DB Amsterdam, The Netherlands Received 2 January 1995; revised manuscript received 28 March 1995 Editor: PV. Landshoff Abstract The azimuthal dependence of hadrons produced in lepton scattering off a polarized hadron probes the quark transverse- spin distributions. In the calculation of the asymmetries, transverse momenta of quarks in the distribution and fragmentation functions must be incorporated. In addition to the sin( 4 + 4s) asymmetry for transversely polarized hadrons, known as the Collins effect, we find a sin( 3d, - 4~) asymmetry. Furthermore, we find a sin 24 angular dependence for longitudinally polarized hadrons. In hard scattering processes one has the possibility of measuring specific matrix elements of quark and gluon fields. In leading order they can be readily in- terpreted as quark and gluon densities including also the spin (helicity and transverse spin) densities. In- clusive measurements in deep-inelastic lepton-hadron scattering (DIS) enable the extraction of the unpo- larized distribution fi (x) and one of two spin distri- butions, namely the helicity distribution gl (x). The other (transverse) spin distribution hl (x), cannot be measured in inclusive DIS because of its chiral struc- ture. It has been suggested to measure this in Drell- Yan scattering [ 1 ] or in semi-inclusive measurements in lepton-hadron scattering [ 21. One of the ways in which semi-inclusive measurements could be used in- volves transverse momentum dependence. In Ref. [ 31, Collins shows how the semi-inclusive deep-inelastic process e + pT + e + h + X, where pt denotes a ’ Also at Physics Department, Free University, NL-1081 HV Amsterdam, The Netherlands. transversely polarized hadron (spin vector Sk), en- ables one to probe the quark transverse spin through a leading asymmetry depending on the azimuthal an- gle of the outgoing hadron’s momentum and that of the target hadron’s spin vector, the so-called Collins effect. In Ref. [ 41, however, we found that terms pro- portional to k:(k~ . ST/M2), where k~ is the quark transverse momentum with respect to the hadron, in the relevant matrix element are neither forbidden by the symmetries of QCD, nor suppressed by powers of l/Q. In this letter we present all leading contri- butions that enter when transverse momentum depen- dence is considered, treating also the case where the initial hadron is longitudinally polarized. We consider the process e + H -+ e’ + h + X, where e and C’ are the incoming and scattered leptons (considered massless) with momenta 1 and I’, H is the incoming hadron with momentum P (with P* = M2) andspinvectorS(withP.S=OandS’=-l),his the produced hadron with momentum Ph (with Pl = ME). The cross section for this process can be written 0370-2693/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO370-2693(95)00485-S 130 R.D. Tangerman, P.J. hfulders/Physics Letters B 352 (1995) 129-133 as a product of a leptonic and a hadronic tensor da dxg dy dz d2 Phi (1) where we consider the situation in which the transverse momentum ?$!J~ is assumed to be of O( M2). We have used the scalar variables Q2 p.9 xB=K’ Y=p.l’ P ’ P,, (2) Here q = 1 - 1’ is the momentum of the exchanged virtual photon (we will limit ourselves to electromag- netic interactions), which is spacelike ( -q2 = Q2). We assume that Q2 becomes large as compared to the hadronic scale, say M2, but xn and z remain constant, well away from their endpoints 0 and 1. It is conve- nient to introduce the vector Pp E P@ - (P .q/q2) qp which is orthogonal to q, leading to the timelike unit vector P z 2xn p@/Q (with i2 = 1). Together with the spacelike unit vector 4’” 3 qp/Q (with G2 = -l), we can define tensors in the space orthogonal to P and 9, (3) w = EkLvPO~ * 2xB El pqo = - Q2 Epvp=Ppqp (4) Azimuthal angles will be fixed with respect to the lepton scattering plane. The x-axis is derived from the lepton momentum which can be written as with I: = - I: = Q2( 1 - y) /y”. We define the unit vector in the x-direction to be ?fi E l/;/111). The (unpolarized) lepton tensor Lp” = 21pl” + 21”l’fi - Q2f” can be written as Lc”’ = f$,_; (1 - y + iy’) $1” + (1 - y) (23” -t- &gy) +(l-Y)f(l-iy) (nV+a”P) +(l -Y)?V,]. (6) The four tensor structures are mutually orthogonal. The interesting physics is in the hadron tensor which, in leading order in l/Q, is given by 2MWC”” = 2e2 s d4kd4k’ @(k + q - k’) xTr[@(k)yl*A(k’)y”]. (7) We consider for the moment only the quark contri- bution for one flavor. Furthermore we limit ourselves to the symmetric part because the unpolarized lepton tensor is symmetric. One has on the distribution side (involving the target hadron) [ 51 @ap(k) = d4x J eik.* (23-j4 x (P~l~~(o>o(o,x>~~(~~lP~), (8) while on the fragmentation side (involving the pro- duced hadron) [6] A,p( k’) = c i / $ eik”x X In the first equation a color-summation is understood, in the second a color-average. A link operator G( 0, X) = Pexp [-ig Jo” dspA,( s)] is inserted to make the definitions color gauge-invariant. With an appropriate choice of the path structure in the link operator used in the above definitions and an appropriate choice of gauge, the above expression for the hadronic tensor VV“ corresponds to the Born graph in a diagrammatic expansion [ 41. In order to analyse the hadronic tensor it is con- venient to work in a frame in which the hadrons H and h are collinear (see also [ 71). We can define the rank-two tensors fl = gpy - nyn: - n:n!, (10) ET w = ElLV @pT n”_ , (11) using the null vectors, defined implicitly by Q p=- XBJZ n+ + eJzn-’ ph=++z;$+. (12) (13) R.D. Tangerman, P.J. Mulders/ Physics Letters B 352 (1995) 129-133 131 satisfying n+ ’ II_ = 1. In analogy to the Drell-Yan process [ 41, we define a ‘transverse’ vector as a: E gay, having only transverse components ar in the above-mentioned collinear frames. Note that in gen- eral the photon momentum q has non-zero transverse components qr. Also, the hadron spin vector is de- composed according to s=*en 43M xBM& +- A- Qfi n- + ST, (14) with A the helicity and ST the transverse spin, satisfy- ingh2+$=-S*=l. A ‘perpendicular’ vector we define as a: E $““a 1 TV = a: - (aT . qT/Q2)qp. These have only two non-zero components al in the frames where the hadron and the virtual photon are collinear. So in general the outgoing hadron has non-zero Phi, specifically, Phi x -zq,. The use of ‘M’ means ‘up to corrections of (relative) order l/Q*‘. The target hadron has neither T- nor I-components. In short, a transverse tensor is perpendicular to both P and Ph, whereas a perpendicular tensor is perpendicular to both P and q. The two types are connected by boosts oforder ]%]/Q, so that at leading order they may be freely interchanged. Only at c3( l/Q) the differences become important [ 81. After this kinematical intermezzo, we return to the calculation of the hadronic tensor (7). Assum- ing quark momenta in hadrons to be limited, i.e., in the expressions for CD and A the quantities k*, k f P, k’*, and k’ + Ph, are of hadronic scale, one infers that k+ > k’+ and k- < k’-, so that @(k+q-k’) ~6(k++q+)6(q--k’-) x6*(kT--;-qT)r (15) such that kf/P+ x -4+/P+ z Xg and P;/k’- z PJq- M z. Another consequence of Eq. ( 15) is that one becomes sensitive to the integrals s dk-@ (k) and J dk’+ A (k') . Only specific Dirac projections will contribute in leading order. For the distributions we need the matrix elements (i = 1,2) t /dk-Tr [ia’+yscD(k)] = ktr(X,k$)$. 2 kT * ST hfi.(X9 kc)A + hj$(X, Ii+) g M I M’ (17) where x = kf /Pi x xB. Performing the kT- integration, only the distribution functions fr (x, k;) and h(x, k:) = hIT(x, kg) +(k~/Ztf*) hff(x, kc), contribute to the distributions ft (x) and hr (x), re- spectively. Similarly, the leading parts for the (fragmentation) matrix elements in Eq. (9) for the case that one sums over the polarization of the produced hadron are (i, j = 192) &/dk”Tr [r-A(k’)] = Dl(z,z*kF), & s dk’+ Tr [i& ys A( k’)] (18) &k’, = -$$-+(z.z*k:), (19) where z = P;/k- is the longitudinal momentum fraction of the produced hadron, and pr = -z kb is its transverse momentum with respect to the fragment- ing quark. The normalization of Dt is given by the momentum sum rule Ch Ji dz J d*p, zD1 (z, &) = 1. Hermiticity and parity invariance have been used in deriving the above (real) parametrizations. In- terestingly, a structure corresponding to Hf in the distribution part will be excluded because of time- reversal invariance. The fragmentation functions are not invariant under the time-reversal operation be- cause the states ]Ph, X) are out-states. That Hf can well be nonzero in QCD has been made plausible in Refs. [ 3,9] by applying simple models. The function also appears in the cross section for scattering of polarized leptons off unpolarized hadrons, but in that case it is suppressed by a factor 1 /Q [ 101. Note that we use somewhat different functions as in Ref. [ 31, the connection being zD1 (z, z*kF) = b( z, kg) and E$kt zHf(z, z’kF)/Mh = Ab(z, k&, er), where &?r is a unit vector in the i-direction. By means of Fierz transformation of the two y- matrices in Rq. (7), one finds that the projection D1 in ( 18) selects from the distribution side the function fl (called p in Ref. [ 31). On the other hand, Hf 132 R.D. Tangerman, PJ. Mulders/Physics Letters B 352 (199.5) 129-133 comes with the matrix element in Eq. (17). In the parametrization of this matrix element three functions come in. In Ref. 131 only the function htr (called fr) was considered. We will consider here the addi- tional function hfr for transversely polarized hadrons and hfL for longitudinally polarized hadrons. The rest of the calculation is a matter of inserting the above parametrizations into the (Fierz transformed) trace in Eq. (7). The result is 2M Wr”” = 2e2 s d* krd2pr S2 ( pT + z kT + z qT) f&J:) zDdz,p:) r-&"I h-(X, 6) H;(z,p;) Mh where {,u V} indicates symmetrization of the indices. The way to project the Lorentz indices of the convo- lution variables kr and pi on external momenta was discussed in Appendix A of Ref. [ 41 for the Drell-Yan process. The resulting leading-order hadronic tensor is most conveniently expressed in terms of four structure functions constructed from the perpendicular vectors and tensors defined earlier, wpc”” = -WT< -UT x [ S$ypfp - (EP;SLph&y] + VT [(if Sl) R{L”e”I)Pip + (EYSI,K,) (2iz’lP + g”l”>] _ VL A j&YPK P’ (21) where R = &/I& 1. We define the convolution product (reinstating the sum over quark and antiquark flavors) Z[.fDl = Ce:/ d2 krd2pT a2 (pr + z kr + z qr) (I x fn(xn, k;P’(z& x ES e; d2kr f(l(Xnt k;) a X Da (Z, (Phi - zkT>*) , (22) where the use of transverse two-vectors is most appro- priate because of the definitions of the fragmentation functions. For the second line we used qT M - P,,l/z. The structure functions now read MWT=ZI[~IDII, (23) I ) (24) MVT =--I K 4(kTdi)2(pT*h) -2(k&Z)(kT.pT) Iv (25) 4) MVL -+p&& . i’ h I (26) Note that in MT precisely the combination hl = hlT + ( kg/2M2) hh appears. The dot products can easily be converted into squares kg, p& and &_, using the delta function in the convolution integrals. So the structure functions depend on the variables xn, z, and Pi,. Finally, the leading-order cross section is obtained by contracting Eq. (21) with the leptonic tensor, Eq. (6). do dnn dy dz d2P/,J_ = s[wr(l -y+ ;y”) tuTIsTI (1 - y) sin(d) t 4s) t VTISTl (1 - y) sin(34 - 4s) +VLA(I -y)sin2+], (27) where cos 4 = -P . h and sin Q, = e~,yZ,~,, and like- wise for the azimuth of the spin vector, /S_L 1 cos Cps = -P . S_L and ISJJ sin 4s = ~yvft~.Sl~ and we used (SIJ x I&I. Our analysis shows that the sin( 4 + Cps ) asymmetry in the process e + pT + e + h + R.D. Tangerman, P.J. Mulders/Physics Letters B 352 (1995) 129-133 133 X, found in Ref. [ 31 probes the transverse spin dis- tribution hl (x, k:), which upon integration over k~ gives hl (x) . One must be aware, however, of an addi- tional sin( 34 - q5~) azimuthal dependence for trans- versely polarized hadrons. The corresponding struc- ture function, VT, probes the new distribution func- tion hllT( x, kc). For longitudinally polarized hadrons there is a sin 24 asymmetry multiplying the structure function VL, which probes the distribution function hfL( x, kc). These three distribution functions deter- mine the transverse spin density in a polarized hadron including the dependence on quark transverse mo- menta as given in Eq. ( 17). They are all convoluted with the same fragmentation function Hf (z, &) . Although the structure functions contain compli- cated convolutions in transverse momentum space, they simplify considerably if one assumes a Gaussian transverse momentum dependence. That is, f(x, k?) = f(x, 0) exp( -&kc), D(z,&> = D(z,O) exp(-R$k*) (28) = D(z,O) exp(--Z&+/z*), (29) for a general distribution and fragmentation function, respectively. The structure function YVT of Eq. (23)) for instance, with this Gaussian ansatz becomes mv~=zI[flLhl= R2z;R2 H h x exp I fl(x,O) Dl(z,O). (30) Likewise, the other structure functions reduce to MU= = R’H lPhll I,h, Hf, R;+ R; Mh (31) R:,% Iph113 MVT=-(R$+R;)j 2Z2~2~h*[h:,H:1' (32) where the convolutions have the same simple form as in Eq. (30). Note that we considered only two transverse radii, one for the target and one for the measured hadron, but not one for every distribution separately. On completion of this work, we became aware of Ref. [ 111 treating semi-inclusive deep-inelastic scat- tering in a similar fashion. The author finds similar asymmetries in addition to the sin( 4 + 4s) asymme- try. He assumes Gaussian kT-behavior for the distribu- tion and fragmentation functions throughout the paper. Our expressions for the structure functions, Eqs. (23) - (26)) are valid for any kT-dependence. We acknowledge discussions with J.C. Collins. This work is part of the research program of the foun- dation for Fundamental Research of Matter (FOM) and the National Organization for Scientific Research (NWO). References [I] J.P. Ralston and D.E. Soper, Nucl. Phys. B 152 (1979) 109; X. Artru and M. Mekhfi, Zeit. Phys. 45 (1990) 669; R.L. Jaffe and X. Ji, Phys. Rev. L&t. 67 (1991) 552. [2] X. Artru, AIP Conf. Proc. 223 (1991) 176; X. Artru and M. Mekhfi, Nucl. Phys. A 532 (1991) 351~. [ 31 J.C. Collins, Nucl. Phys. B 396 (1993) 161. [ 41 R.D. Tangeman and P.J. Mulders, preprint NIKHEF-94-PI, hep_ph/9403227, to appear in Phys. Rev. D. [5] D.E. Soper, Phys. Rev. D 63 ( 1977) 1141; Phys. Rev. Len. 43 ( 1979) 1847. [6] J.C. Collins and D.E. Soper, Nucl. Phys. B 194 (1982) 445. [7] R. Meng, F.I. Olness and D.E. Soper, Nucl. Phys. B 371 (1992) 79. [ 81 R.D. Tangemm and PJ. Mulders, preprint NIKHEF-94-P7, hep-ph/9408305. [9] J.C. Collins and G.A. Ladinsky, preprint PSU/TH/ 114, hep- ph/9411444. [ 101 J. Levelt and P.J. Mulders, Phys. Lett. B 338 (1994) 357. [ 111 A. Kotzinian, Yerevan preprint, hep-ph/9412283, to appear in Nucl. Phys. B. MVL=- R; R; lphl I* -Wf,Hf], CR;+ R;)* zMMh (33) 

UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)Probing transverse quark polarization in deep-inelastic leptoproductionTangerman, R.D.; Mulders, P.J.G.DOI10.1016/0370-2693(95)00485-4Publication date1995Published inPhysics Letters BLink to publicationCitation for published version (APA):Tangerman, R. D., & Mulders, P. J. G. (1995). Probing transverse quark polarization in deep-inelastic leptoproduction. Physics Letters B, 352, 129. https://doi.org/10.1016/0370-2693(95)00485-4General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:25 Jul 20228 June 1995 PHYSICS LETTERS B ELSEVIER Physics Letters B 352 (1995) 129-133 Probing transverse quark polarization in deep-inelastic leptoproduction R.D. Tangerman, P.J. Mulders ’ National Institute for Nuclear Physics and High Energy Physics (NIKHEF)), l?O. Box 41882, NL-1009 DB Amsterdam, The Netherlands Received 2 January 1995; revised manuscript received 28 March 1995 Editor: PV. Landshoff Abstract The azimuthal dependence of hadrons produced in lepton scattering off a polarized hadron probes the quark transverse- spin distributions. In the calculation of the asymmetries, transverse momenta of quarks in the distribution and fragmentation functions must be incorporated. In addition to the sin( 4 + 4s) asymmetry for transversely polarized hadrons, known as the Collins effect, we find a sin( 3d, - 4~) asymmetry. Furthermore, we find a sin 24 angular dependence for longitudinally polarized hadrons. In hard scattering processes one has the possibility of measuring specific matrix elements of quark and gluon fields. In leading order they can be readily in- terpreted as quark and gluon densities including also the spin (helicity and transverse spin) densities. In- clusive measurements in deep-inelastic lepton-hadron scattering (DIS) enable the extraction of the unpo- larized distribution fi (x) and one of two spin distri- butions, namely the helicity distribution gl (x). The other (transverse) spin distribution hl (x), cannot be measured in inclusive DIS because of its chiral struc- ture. It has been suggested to measure this in Drell- Yan scattering [ 1 ] or in semi-inclusive measurements in lepton-hadron scattering [ 21. One of the ways in which semi-inclusive measurements could be used in- volves transverse momentum dependence. In Ref. [ 31, Collins shows how the semi-inclusive deep-inelastic process e + pT + e + h + X, where pt denotes a ’ Also at Physics Department, Free University, NL-1081 HV Amsterdam, The Netherlands. transversely polarized hadron (spin vector Sk), en- ables one to probe the quark transverse spin through a leading asymmetry depending on the azimuthal an- gle of the outgoing hadron’s momentum and that of the target hadron’s spin vector, the so-called Collins effect. In Ref. [ 41, however, we found that terms pro- portional to k:(k~ . ST/M2), where k~ is the quark transverse momentum with respect to the hadron, in the relevant matrix element are neither forbidden by the symmetries of QCD, nor suppressed by powers of l/Q. In this letter we present all leading contri- butions that enter when transverse momentum depen- dence is considered, treating also the case where the initial hadron is longitudinally polarized. We consider the process e + H -+ e’ + h + X, where e and C’ are the incoming and scattered leptons (considered massless) with momenta 1 and I’, H is the incoming hadron with momentum P (with P* = M2) andspinvectorS(withP.S=OandS’=-l),his the produced hadron with momentum Ph (with Pl = ME). The cross section for this process can be written 0370-2693/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO370-2693(95)00485-S 130 R.D. Tangerman, P.J. hfulders/Physics Letters B 352 (1995) 129-133 as a product of a leptonic and a hadronic tensor da dxg dy dz d2 Phi (1) where we consider the situation in which the transverse momentum ?$!J~ is assumed to be of O( M2). We have used the scalar variables Q2 p.9 xB=K’ Y=p.l’ P ’ P,, (2) Here q = 1 - 1’ is the momentum of the exchanged virtual photon (we will limit ourselves to electromag- netic interactions), which is spacelike ( -q2 = Q2). We assume that Q2 becomes large as compared to the hadronic scale, say M2, but xn and z remain constant, well away from their endpoints 0 and 1. It is conve- nient to introduce the vector Pp E P@ - (P .q/q2) qp which is orthogonal to q, leading to the timelike unit vector P z 2xn p@/Q (with i2 = 1). Together with the spacelike unit vector 4’” 3 qp/Q (with G2 = -l), we can define tensors in the space orthogonal to P and 9, (3) w = EkLvPO~ * 2xB El pqo = - Q2 Epvp=Ppqp (4) Azimuthal angles will be fixed with respect to the lepton scattering plane. The x-axis is derived from the lepton momentum which can be written as with I: = - I: = Q2( 1 - y) /y”. We define the unit vector in the x-direction to be ?fi E l/;/111). The (unpolarized) lepton tensor Lp” = 21pl” + 21”l’fi - Q2f” can be written as Lc”’ = f$,_; (1 - y + iy’) $1” + (1 - y) (23” -t- &gy) +(l-Y)f(l-iy) (nV+a”P) +(l -Y)?V,]. (6) The four tensor structures are mutually orthogonal. The interesting physics is in the hadron tensor which, in leading order in l/Q, is given by 2MWC”” = 2e2 s d4kd4k’ @(k + q - k’) xTr[@(k)yl*A(k’)y”]. (7) We consider for the moment only the quark contri- bution for one flavor. Furthermore we limit ourselves to the symmetric part because the unpolarized lepton tensor is symmetric. One has on the distribution side (involving the target hadron) [ 51 @ap(k) = d4x J eik.* (23-j4 x (P~l~~(o>o(o,x>~~(~~lP~), (8) while on the fragmentation side (involving the pro- duced hadron) [6] A,p( k’) = c i / $ eik”x X In the first equation a color-summation is understood, in the second a color-average. A link operator G( 0, X) = Pexp [-ig Jo” dspA,( s)] is inserted to make the definitions color gauge-invariant. With an appropriate choice of the path structure in the link operator used in the above definitions and an appropriate choice of gauge, the above expression for the hadronic tensor VV“ corresponds to the Born graph in a diagrammatic expansion [ 41. In order to analyse the hadronic tensor it is con- venient to work in a frame in which the hadrons H and h are collinear (see also [ 71). We can define the rank-two tensors fl = gpy - nyn: - n:n!, (10) ET w = ElLV @pT n”_ , (11) using the null vectors, defined implicitly by Q p=- XBJZ n+ + eJzn-’ ph=++z;$+. (12) (13) R.D. Tangerman, P.J. Mulders/ Physics Letters B 352 (1995) 129-133 131 satisfying n+ ’ II_ = 1. In analogy to the Drell-Yan process [ 41, we define a ‘transverse’ vector as a: E gay, having only transverse components ar in the above-mentioned collinear frames. Note that in gen- eral the photon momentum q has non-zero transverse components qr. Also, the hadron spin vector is de- composed according to s=*en 43M xBM& +- A- Qfi n- + ST, (14) with A the helicity and ST the transverse spin, satisfy- ingh2+$=-S*=l. A ‘perpendicular’ vector we define as a: E $““a 1 TV = a: - (aT . qT/Q2)qp. These have only two non-zero components al in the frames where the hadron and the virtual photon are collinear. So in general the outgoing hadron has non-zero Phi, specifically, Phi x -zq,. The use of ‘M’ means ‘up to corrections of (relative) order l/Q*‘. The target hadron has neither T- nor I-components. In short, a transverse tensor is perpendicular to both P and Ph, whereas a perpendicular tensor is perpendicular to both P and q. The two types are connected by boosts oforder ]%]/Q, so that at leading order they may be freely interchanged. Only at c3( l/Q) the differences become important [ 81. After this kinematical intermezzo, we return to the calculation of the hadronic tensor (7). Assum- ing quark momenta in hadrons to be limited, i.e., in the expressions for CD and A the quantities k*, k f P, k’*, and k’ + Ph, are of hadronic scale, one infers that k+ > k’+ and k- < k’-, so that @(k+q-k’) ~6(k++q+)6(q--k’-) x6*(kT--;-qT)r (15) such that kf/P+ x -4+/P+ z Xg and P;/k’- z PJq- M z. Another consequence of Eq. ( 15) is that one becomes sensitive to the integrals s dk-@ (k) and J dk’+ A (k') . Only specific Dirac projections will contribute in leading order. For the distributions we need the matrix elements (i = 1,2) t /dk-Tr [ia’+yscD(k)] = ktr(X,k$)$. 2 kT * ST hfi.(X9 kc)A + hj$(X, Ii+) g M I M’ (17) where x = kf /Pi x xB. Performing the kT- integration, only the distribution functions fr (x, k;) and h(x, k:) = hIT(x, kg) +(k~/Ztf*) hff(x, kc), contribute to the distributions ft (x) and hr (x), re- spectively. Similarly, the leading parts for the (fragmentation) matrix elements in Eq. (9) for the case that one sums over the polarization of the produced hadron are (i, j = 192) &/dk”Tr [r-A(k’)] = Dl(z,z*kF), & s dk’+ Tr [i& ys A( k’)] (18) &k’, = -$$-+(z.z*k:), (19) where z = P;/k- is the longitudinal momentum fraction of the produced hadron, and pr = -z kb is its transverse momentum with respect to the fragment- ing quark. The normalization of Dt is given by the momentum sum rule Ch Ji dz J d*p, zD1 (z, &) = 1. Hermiticity and parity invariance have been used in deriving the above (real) parametrizations. In- terestingly, a structure corresponding to Hf in the distribution part will be excluded because of time- reversal invariance. The fragmentation functions are not invariant under the time-reversal operation be- cause the states ]Ph, X) are out-states. That Hf can well be nonzero in QCD has been made plausible in Refs. [ 3,9] by applying simple models. The function also appears in the cross section for scattering of polarized leptons off unpolarized hadrons, but in that case it is suppressed by a factor 1 /Q [ 101. Note that we use somewhat different functions as in Ref. [ 31, the connection being zD1 (z, z*kF) = b( z, kg) and E$kt zHf(z, z’kF)/Mh = Ab(z, k&, er), where &?r is a unit vector in the i-direction. By means of Fierz transformation of the two y- matrices in Rq. (7), one finds that the projection D1 in ( 18) selects from the distribution side the function fl (called p in Ref. [ 31). On the other hand, Hf 132 R.D. Tangerman, PJ. Mulders/Physics Letters B 352 (199.5) 129-133 comes with the matrix element in Eq. (17). In the parametrization of this matrix element three functions come in. In Ref. 131 only the function htr (called fr) was considered. We will consider here the addi- tional function hfr for transversely polarized hadrons and hfL for longitudinally polarized hadrons. The rest of the calculation is a matter of inserting the above parametrizations into the (Fierz transformed) trace in Eq. (7). The result is 2M Wr”” = 2e2 s d* krd2pr S2 ( pT + z kT + z qT) f&J:) zDdz,p:) r-&"I h-(X, 6) H;(z,p;) Mh where {,u V} indicates symmetrization of the indices. The way to project the Lorentz indices of the convo- lution variables kr and pi on external momenta was discussed in Appendix A of Ref. [ 41 for the Drell-Yan process. The resulting leading-order hadronic tensor is most conveniently expressed in terms of four structure functions constructed from the perpendicular vectors and tensors defined earlier, wpc”” = -WT< -UT x [ S$ypfp - (EP;SLph&y] + VT [(if Sl) R{L”e”I)Pip + (EYSI,K,) (2iz’lP + g”l”>] _ VL A j&YPK P’ (21) where R = &/I& 1. We define the convolution product (reinstating the sum over quark and antiquark flavors) Z[.fDl = Ce:/ d2 krd2pT a2 (pr + z kr + z qr) (I x fn(xn, k;P’(z& x ES e; d2kr f(l(Xnt k;) a X Da (Z, (Phi - zkT>*) , (22) where the use of transverse two-vectors is most appro- priate because of the definitions of the fragmentation functions. For the second line we used qT M - P,,l/z. The structure functions now read MWT=ZI[~IDII, (23) I ) (24) MVT =--I K 4(kTdi)2(pT*h) -2(k&Z)(kT.pT) Iv (25) 4) MVL -+p&& . i’ h I (26) Note that in MT precisely the combination hl = hlT + ( kg/2M2) hh appears. The dot products can easily be converted into squares kg, p& and &_, using the delta function in the convolution integrals. So the structure functions depend on the variables xn, z, and Pi,. Finally, the leading-order cross section is obtained by contracting Eq. (21) with the leptonic tensor, Eq. (6). do dnn dy dz d2P/,J_ = s[wr(l -y+ ;y”) tuTIsTI (1 - y) sin(d) t 4s) t VTISTl (1 - y) sin(34 - 4s) +VLA(I -y)sin2+], (27) where cos 4 = -P . h and sin Q, = e~,yZ,~,, and like- wise for the azimuth of the spin vector, /S_L 1 cos Cps = -P . S_L and ISJJ sin 4s = ~yvft~.Sl~ and we used (SIJ x I&I. Our analysis shows that the sin( 4 + Cps ) asymmetry in the process e + pT + e + h + R.D. Tangerman, P.J. Mulders/Physics Letters B 352 (1995) 129-133 133 X, found in Ref. [ 31 probes the transverse spin dis- tribution hl (x, k:), which upon integration over k~ gives hl (x) . One must be aware, however, of an addi- tional sin( 34 - q5~) azimuthal dependence for trans- versely polarized hadrons. The corresponding struc- ture function, VT, probes the new distribution func- tion hllT( x, kc). For longitudinally polarized hadrons there is a sin 24 asymmetry multiplying the structure function VL, which probes the distribution function hfL( x, kc). These three distribution functions deter- mine the transverse spin density in a polarized hadron including the dependence on quark transverse mo- menta as given in Eq. ( 17). They are all convoluted with the same fragmentation function Hf (z, &) . Although the structure functions contain compli- cated convolutions in transverse momentum space, they simplify considerably if one assumes a Gaussian transverse momentum dependence. That is, f(x, k?) = f(x, 0) exp( -&kc), D(z,&> = D(z,O) exp(-R$k*) (28) = D(z,O) exp(--Z&+/z*), (29) for a general distribution and fragmentation function, respectively. The structure function YVT of Eq. (23)) for instance, with this Gaussian ansatz becomes mv~=zI[flLhl= R2z;R2 H h x exp I fl(x,O) Dl(z,O). (30) Likewise, the other structure functions reduce to MU= = R’H lPhll I,h, Hf, R;+ R; Mh (31) R:,% Iph113 MVT=-(R$+R;)j 2Z2~2~h*[h:,H:1' (32) where the convolutions have the same simple form as in Eq. (30). Note that we considered only two transverse radii, one for the target and one for the measured hadron, but not one for every distribution separately. On completion of this work, we became aware of Ref. [ 111 treating semi-inclusive deep-inelastic scat- tering in a similar fashion. The author finds similar asymmetries in addition to the sin( 4 + 4s) asymme- try. He assumes Gaussian kT-behavior for the distribu- tion and fragmentation functions throughout the paper. Our expressions for the structure functions, Eqs. (23) - (26)) are valid for any kT-dependence. We acknowledge discussions with J.C. Collins. This work is part of the research program of the foun- dation for Fundamental Research of Matter (FOM) and the National Organization for Scientific Research (NWO). References [I] J.P. Ralston and D.E. Soper, Nucl. Phys. B 152 (1979) 109; X. Artru and M. Mekhfi, Zeit. Phys. 45 (1990) 669; R.L. Jaffe and X. Ji, Phys. Rev. L&t. 67 (1991) 552. [2] X. Artru, AIP Conf. Proc. 223 (1991) 176; X. Artru and M. Mekhfi, Nucl. Phys. A 532 (1991) 351~. [ 31 J.C. Collins, Nucl. Phys. B 396 (1993) 161. [ 41 R.D. Tangeman and P.J. Mulders, preprint NIKHEF-94-PI, hep_ph/9403227, to appear in Phys. Rev. D. [5] D.E. Soper, Phys. Rev. D 63 ( 1977) 1141; Phys. Rev. Len. 43 ( 1979) 1847. [6] J.C. Collins and D.E. Soper, Nucl. Phys. B 194 (1982) 445. [7] R. Meng, F.I. Olness and D.E. Soper, Nucl. Phys. B 371 (1992) 79. [ 81 R.D. Tangemm and PJ. Mulders, preprint NIKHEF-94-P7, hep-ph/9408305. [9] J.C. Collins and G.A. Ladinsky, preprint PSU/TH/ 114, hep- ph/9411444. [ 101 J. Levelt and P.J. Mulders, Phys. Lett. B 338 (1994) 357. [ 111 A. Kotzinian, Yerevan preprint, hep-ph/9412283, to appear in Nucl. Phys. B. MVL=- R; R; lphl I* -Wf,Hf], CR;+ R;)* zMMh (33) 

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Probing transverse quark polarization in deep-inelastic leptoproduction

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