Angular dependences in electroweak semi-inclusive leptoproduction

We present the leading order unpolarized and polarized cross sections in electroweak semi-inclusive deep inelastic leptoproduction. The azimuthal dependences in the cross section differential in the transverse momentum of the vector boson arise due to intrinsic transverse momenta of the quarks. However, the presented asymmetries are not suppressed by inverse powers of the hard scale. We discuss the different opportunities to measure specific asymmetries as offered by neutral compared to charged current processes and point out the optimal kinematical regions. The present Ž . and proposed future HERA collider experiments would be most suitable for measuring some of the asymmetries discussed here, especially in case of L production. q 2000 Elsevier Science B.V. All rights reserved. PACS: 13.85.Ni; 13.87.Fh; 13.88.qe w x In this article we extend results of Refs. 1–4 on semi-inclusive deep inelastic leptoproduction cross sections to include contributions from Z-boson exchange and g–Z interference terms in neutral current processes as well as contributions from W-boson exchange in charged current processes. The azimuthal dependences in the cross section differential in the transverse momentum of the vector boson arise due to intrinsic Ž .0 transverse momenta of the quarks. Only leading order 1rQ effects are discussed, w x since higher twist contributions like those of Refs. 5–9 are expected to be negligible at energies for which electroweak contributions are relevant. Also, we will focus on tree Ž .0 level, i.e. order a . A rich structure nevertheless arises when taking into account the s polarization of the initial or final state particles. At the end of this article we will discuss the experimental opportunities to study specific terms in the cross sections. w x For details of the calculation and the formalism we refer to Refs. 1–3 . We shortly repeat the essentials. It is convenient to use the hadron momenta in the process 0550-3213r00r$ see front matter q 2000 Elsevier Science B.V. All rights reserved. Ž . PII: S0550-3213 99 00586-6 ( ) D. Boer et al.rNuclear Physics B 564 2000 471–485 472 l HTM l X hX to define two lightlike vectors n and n , satisfying n Pn s1. These q y q y vectors then define the lightcone components of a vector as p'pPn and we use the . w y q x component notation ps p , p , p . Up to mass terms the momentum P of the target T Ž . Ž . hadron H is along n , the momentum P of the outgoing hadron h is along n . We q h y assume here that we are discussing current quark fragmentation, for which one requires PPP ;Q, where q syQ is the momentum transfer squared. In leading order in h 1rQ the process factorizes into a product of a hard perturbative partonic subprocess and two soft non-perturbative parts, which describe the distribution of quarks in the target and the final fragmentation of the struck quark into hadrons, respectively. The neutral current cross section for unpolarized and polarized electroweak semi-inclusive lepton–hadron scattering is given by ds l HTM l hX a 2 zy Ž . i j mn s L 2 M W x . 1 Ž . Ý mn i j i j 2 4 dx dz dy df d q 4Q l T ij 2 Ž . Ž . Ž . y y We use invariants xsQ r 2 PPq , zsPPP rPPq and ys PPq r PP l fq rl h Ž . l being the momentum of the beam lepton . The cross section is differential in dx, dz, 2 w x dy, df and in d q where q sqqx PyP rzs 0,0,q . The indices i, j can be g l T T h T for the photon or Z for the Z-boson. The relative propagator factors x are given by i j 1 Q 2 x s1, x sx s , x s x . 2 Ž . Ž . gg g Z Zg ZZ g Z 2 2 2 sin 2u Q qM Ž . W Z Here we note that in this process the scale Q is defined by the spacelike vector boson Ž 2 2 . momentum q with Q 'yq , hence the width of the Z-boson plays a negligible role and we have taken it to be zero. Also, we will consider the cross section differential in w x the transverse momentum of the vector boson, but the factorized expression 10 that we < < 2 2 will consider will require q <Q , to insure that one is sensitive to the region of T intrinsic transverse momenta. Ž . The lepton tensor neglecting the lepton masses is given by X X X X i j i j 2 i j r s L l ,l sC 2 l l q2 l l yQ g qD 2 ie l l , 3 Ž . Ž . mn m n n m mn mnrs where the incoming lepton has momentum l and the back-scattered lepton momentum l. We have defined C s1, C Z sC Zg se g l yg l l , Ž . V A e C ZZ s g l2 qg l2 y 2 g l g l l , 4 Ž . Ž . Ž . V A V A e D sl , D Z sD se g l l yg l , Ž . e V e A D s g l2 qg l2 l y 2 g l g l , 5 Ž . Ž . Ž . V A e V A where e denotes the coupling of the photon to the leptons in units of the positron charge; g l , g l denote the vector and axial-vector couplings of the Z-boson to the V A leptons, respectively and l is the helicity of the incoming lepton. e ( ) D. Boer et al.rNuclear Physics B 564 2000 471–485 473 To leading order the expression for the hadron tensor, including quarks and antiquarks, is mn y q 2 2 2 m n 2 M W s dp dk d p d k d p qq yk Tr F p V D k V Ž . Ž . Ž . q y Ž . H i j T T T T T i j p ,k qlyq q , 6 Ž . ž / mln where p and k represent the quark momentum before and after the interaction with the vector boson. The vertices V m can be either the photon vertex V m seg m or the Z-boson i g vertex V m sg g qg g g . The vector and axial-vector couplings to the Z boson are Z V A 5 given by: g k sT k y2 esinu , 7 Ž . V 3 W g k sT k , 8 Ž . A 3 k k Ž k where e denotes the charge and T the weak isospin of particle k i.e. T sq1r2 for 3 3 q q k y y . kse ,m ,n ,u and T sy1r2 for kse ,m ,n ,d,s . We have omitted flavor indices 3 and summation. The correlation functions F and D comprise information on the hadronic structure of the target in terms of quark degrees of freedom and on the quark Ž hadronization process, respectively. They are given by path-ordered exponentials are . suppressed : d x i pP x2 < < : F P ,S ; p s e P ,S c 0 c x P ,S , 9 Ž . Ž . Ž . Ž . H m n n m 4 2p Ž . d x ikP x2 < < :2 < < : D P ,S ;k s e 0 c x P ,S ; X P ,S ; X c 0 0 . Ž . Ž . Ž . ÝH m n h h m h h h h n 4 2p Ž . X 10 Ž . Ž . Using Lorentz invariance, hermiticity, and parity invariance, the partly integrated correlation function y F x , p ' dp F P ,S ; p Ž . Ž . H T q q p sx P , pT sF ŽO . x , p qF Ž L. x , p qF ŽT . x , p , 11 Ž . Ž . Ž . Ž . T T T Žwith indices O, L,T indicating the polarization of the target: unpolarized, longitudinally . and transversely polarized, respectively is parametrized in terms of distribution functions as M Pu s p P n mn T ŽO . 2 H 2 F x , p s f x , p qqh x , p , Ž . Ž . Ž . T 1 T 1 T q 2 1⁄2 5 2 P M M M Pug is g p P n 5 mn 5 T Ž L. 2 H 2 F x , p s yl g x , p yl h x , p , Ž . Ž . Ž . T 1 L T 1 L T q 2 1⁄2 5 2 P M M ( ) D. Boer et al.rNuclear Physics B 564 2000 471–485 474 M P n p S p PS Pug T T T T 5 ŽT . H 2 m 2 F x , p s f x , p e g y g x , p Ž . Ž . Ž . T 1T T mnrs 1T T q 2 1⁄2 2 P M M M is g S P n p PS is g p P n mn 5 T T T mn 5 T 2 H 2 yh x , p y h x , p , Ž . Ž . 1T T 1T T 2 5 M M M 12 Ž . with M being the target hadron mass. We only consider polarization of spin-1r2 hadrons, represented by lsM SrP the lightcone helicity and S the transverse spin T 2 aŽ 2 . of the target hadron. The normalization is chosen by requiring that Hdx d p f x, p s T 1 T n, where n is the number of valence quarks with flavor a. Time reversal invariance is expected to eliminate the T-odd functions f H and h , especially in the case of 1T 1 semi-inclusive deep inelastic scattering due to the absence of initial state interactions, cf. w x Refs. 3,10,11 . Nevertheless, there is a possibility that they might be effectively w x generated by a gluonic background field, cf. e.g. Ref. 12 , hence we keep these functions for completeness. Ž . The partly integrated correlation function 1 q D z ,k ' dk D P ,S ;k Ž . Ž . H T h h y y z k sP rz , k h T sDŽO . z ,k qDŽ L. z ,k qDŽT . z ,k , 13 Ž . Ž . Ž . Ž . T T T Ž . now O, L,T indicating the polarization of the observed final state hadron is parametrized in terms of fragmentation functions as M Pu s k P n h h mn T h ŽO . 2 2 H 2 2 D z ,k s D z , z k qH z , z k , Ž . Ž . Ž . T 1 T 1 T y 2 1⁄2 5 P M M h h h M Pu g is g k P n h h 5 mn 5 T h Ž L. 2 2 H 2 2 D z ,k s yl G z , z k yl H z , z k , Ž . Ž . Ž . T h 1 L T h 1 L T y 2 1⁄2 5 P M M h h h M e g P k S h mnrs h T hT ŽT . H 2 2 D z ,k s D z , z k Ž . Ž . T 1T T y 2 1⁄2 P M h h k PS Pu g is g S m P n Ž . T hT h 5 mn 5 hT h 2 2 2 2 y G z , z k yH z , z k Ž . Ž . 1T T 1T T M M M h h h k PS is g k P n Ž . T hT mn 5 T h H 2 2 y H z , z k , 14 Ž . Ž . 1T T 2 5 M M h h with M the mass, l sM SrP the lightcone helicity, and S the transverse spin h h h h h hT of the produced spin-1r2 hadron. The choice of factors in the definition of fragmenta2 2 aŽ 2 2 . tion functions is such that Ý Hdz z d k D z, z k sN , where N is the total a T 1 T h h number of produced hadrons. The fragmentation functions H H and D are called 1 1T T-odd functions, which in contrast to the T-odd distribution functions are expected to be present, since they are not forbidden by time reversal invariance due to the presence of w x final state interactions, cf. Ref. 10 . ( ) D. Boer et al.rNuclear Physics B 564 2000 471–485 475 We would like to emphasize that the distribution and fragmentation functions as defined above parametrize the soft non-perturbative parts of the scattering process. At present a complete calculation of these non-perturbative objects from first principles, as for instance by a lattice calculation, is not available. Estimates can be made using QCD Ž . Ž . sum rule or model calculations. Apart from the known functions f x , g x and 1 1 Ž . Ž D z , which are functions integrated over the transverse momentum there exists also 1 Ž . w x. marginal information on G z 13 , recently first experimental hints on the size of 1 H w x some asymmetries concerning the so-called Collins function H 10 have been 1 w x presented 14 . None of the other functions has been experimentally accessed. Therefore, many of the observables that we will discuss contain unknown functions, which however are not just parametrizations of ignorance, but represent essential information on the structure of hadrons as can be probed in hard scattering processes and on the hadronizaw x tion process. For the interpretation of the various functions we refer to Refs. 1–3 . In processes with at least two hadrons one needs to be careful with the notion of Ž transÕerse. In the definition of distribution and fragmentation functions which one . wants to be independent of the specifics of the process , transverse momentum components denoted by a subscript T are defined with respect to the momenta P and P such h that P s0 and P s0, respectively. Consequently vectors are decomposed in plus, T hT minus and transverse components defined by the lightlike vectors n and n , conq y structed from P and P , and the transverse projector g mn 'g mn yn m n 4 . h T q y Ž . For defining the process dependent azimuthal angles with respect to the scattering plane, on the other hand, it is more convenient to use a frame where the virtual boson and the target, i.e. the momenta q and P, are collinear. We indicate transverse Ž components in the latter frame with the subscript H and call them perpendicular . henceforth . Thus depending on the choice of frame the covariantly defined vector q m sq m qx P m yP rz can be the transverse component of q with respect to the T h collinear hadrons, or it is up to a factor 1rz the component of P perpendicular to the h scattering plane, q syP rz. The kinematics in the frame where q and P are T hH collinear can be expressed by the set of normalized vectors 2 x ̃ t̂' P , 15 Ž . Q z'yqrQ, 16 Ž . ˆ ĥ'yq rQ sy qqx PyP rz rQ , 17 Ž . Ž . T T h T 2 2 ̃ 2 Ž . where Q syq and P'Py PPq qrq , such that: T T 1 m m m ˆ w x n s t qz , 18 Ž . ˆ q '2 1 QT m m m m ˆ ˆ n s t yz q2 h . 19 Ž . ˆ y ' Q 2 The lepton momentum reads ' 2yy Q 1yy m m m m ˆ ˆ l sQ t y z qQ l , 20 Ž . ˆ H 2 y 2 y ( ) D. Boer et al.rNuclear Physics B 564 2000 471–485 476 Fig. 1. Kinematics for one-particle inclusive leptoproduction. The lepton scattering plane is determined by the momenta l, l and P. The azimuthal out-of-plane angle f of the produced hadron is indicated. and we define the tensors mn mn ˆˆ m n g 'g y t t qz z , 21 Ž . ˆ ˆ H mn mnrs ˆ e 'ye t z . 22 Ž . ˆ H r s The cross sections are obtained from the hadron tensor after contraction with the lepton Ž . tensor Eq. 3 , which to leading order in 1rQ is 2 2 Q Q 1 mn mn m n mn mn ˆ ˆ L sC y2 A y g q4B y l l q g yD i C y e , Ž . Ž . Ž . ž / i j i j H H H H i j H 2 2 2 y y 23 Ž . expressed in terms of the functions 1 2 A y s 1yyq y , 24 Ž . Ž . Ž . 2 B y s 1yy , 25 Ž . Ž . Ž . C y sy 2yy . 26 Ž . Ž . Ž . ˆm Azimuthal angles inside the perpendicular plane are defined with respect to l , defined H to be the normalized perpendicular part of the incoming lepton momentum l, ˆ < < l Pa sy a cosf , 27 Ž . H H H a mn ˆ < < e l a s a sinf , 28 Ž . H H m H n H a for a generic vector a. In the cross sections we will encounter dependences on the three azimuthal angles f, f and f of the produced hadron momentum, its spin vector, and S Sh Ž . of the target hadron spin vector, respectively cf. Fig. 1 . We would like to note that at leading order the spin vector S is identical to the spin vector perpendicular to z, i.e. ˆ T Ž . Ž . Ž . S , and also lsM SPz r PPz and analogously for S and l . This does not ˆ ˆ H hT h w x hold at order 1rQ, cf. Ref. 15 . In order to present our results on cross sections in a compact form we define the following combinations of the couplings and Z-boson propagators a gg 2 g Z a ZZ a K y sA y C e x q2C e g x qC c x Ž . Ž . 1 a gg a V g Z 1 ZZ C y Ž . g Z a ZZ a y 2 D e g x qD c x , 29 Ž . a A g Z 3 ZZ 2 ( ) D. Boer et al.rNuclear Physics B 564 2000 471–485 477 a g Z a ZZ a K y syA y 2C e g x qC c x Ž . Ž . 2 a A g Z 3 ZZ C y Ž . gg 2 g Z a ZZ a q D e x q2 D e g x qD c x , 30 Ž . a gg a V g Z 1 ZZ 2 a gg 2 g Z a ZZ a K y syB y C e x q2C e g x qC c x , 31 Ž . Ž . Ž . 3 a gg a V g Z 2 ZZ which contain the combinations of the Z-boson-to-quark couplings c s g a2 qg a2 , 32 Ž . Ž . 1 V A c s g a2 yg a2 , 33 Ž . Ž . 2 V A c s2 g a g a . 34 Ž . 3 V A Ž . Furthermore, we use the convolution notation where w denotes a weight function 2 2 2 a 2 F w p ,k fD ' d p d k d p qq yk w p ,k f x , p Ž . Ž . Ž . Ž . H T T T T T T T T T T =D z , z 2 k 2 . 35 Ž . Ž . T We find for the leading order unpolarized cross section, taking into account both photon and Z-boson contributions, ds l HTM l hX 2 a 2 x z 2 s Ž . a w x s K y F f D Ž . Ý 1 1 1 2 4 1⁄2 dx dz dy df d q Q l T a,a H H h H 1 1 a ˆ ˆ qK y cos 2f F 2 hPp hPk yp Pk , Ž . Ž . ž / 3 T T T T 5 M M 1 2 36 Ž . Ž . where the sum runs over all quark and anti-quark flavors. Perturbative QCD corrections to the term proportional to f D , which is independent of the azimuthal angle at 1 1 Ž . Ž . tree level, will also produce a cos 2f term, next to a cos f term and the time-reversal Ž . Ž . odd sin f and sin 2f terms, but all these will be suppressed by inverse powers of the w x hard scale 5–9,16,17 . In order to differentiate between the perturbatively generated and Ž . the above given cos 2f asymmetry, one could for instance apply a transverse momenw x tum cut-off to exclude the contributions from intrinsic transverse momentum 18 or one Ž . can study the analogous charged current exchange process, since the cos 2f term of Ž . Ž . Eq. 36 will then be absent K s0 as we will observe below. 3 In case the target hadron is polarized the cross section is found to be TM X 2 2 ds l HTM l hX 2a xz s Ž . a w x s lK y F g D Ž . Ý 2 1 1 2 4 1⁄2 dx dz dy df d q Q l T a,a H H h H 1 L 1 a ˆ ˆ ylK y sin 2f F 2 hPp hPk yp Pk Ž . Ž . ž / 3 T T T T MMh ( ) D. Boer et al.rNuclear Physics B 564 2000 471–485 478 H f D 1T 1 a ˆ < < q S K y sin fyf F hPp Ž . Ž . T 1 S T M g D 1T 1 a ˆ < < q S K y cos fyf F hPp Ž . Ž . T 2 S T M H h H 1 1 a ˆ < < y S K y sin fqf F hPk Ž . Ž . T 3 S T Mh 2 a ˆ ˆ < < y S K y sin 3fyf F 4 hPp hPk Ž . Ž . Ž . T 3 S T T ž H H h H 1T 1 2 ˆ ˆ y2 hPp p Pk yp hPk , 37 Ž . T T T T T / 2 5 2 M Mh where we have not included the unpolarized cross section terms again, i.e. parts which cancel from differences of cross sections with reversed polarization are suppressed here and in the following. The cross section for an unpolarized target, but with spin Õector of the final state ( ) spin-1r2 hadron being determined is TM X 2 2 ds l HTM l hX 2a xz s Ž . a w x s l K y F f G Ž . Ý h 2 1 1 2 4 1⁄2 dx dz dy df d q Q l T a,a H H h H 1 1 L a ˆ ˆ ql K y sin 2f F 2 hPp hPk yp Pk Ž . Ž . ž / h 3 T T T T MMh H f D 1 1T a ˆ < < y S K y sin fyf F hPk Ž . Ž . hT 1 S T h Mh f G 1 1T a ˆ < < q S K y cos fyf F hPk Ž . Ž . hT 2 S T h Mh H h H 1 1 a ˆ < < q S K y sin fqf F hPp Ž . Ž . hT 3 S T h M 2 a ˆ ˆ < < q S K y sin 3fyf F 4 hPk hPp Ž . Ž . Ž . hT 3 S T T ž h H H h H 1 1T 2 ˆ ˆ y2 hPk k Pp yk hPp . 38 Ž . T T T T T / 2 5 2 M Mh Ž . Ž . Eqs. 37 and 38 show some similarity; the latter is obtained from the former by the set of replacements distribution functionsl fragmentation functions, MlM , klp, h 4 lTMl , S TMS , f TMf together with an additional minus sign for each time-reh T hT S Sh versal odd function f H ,h , D , H H , where the replacement of the distribution func1T 1 1T 1 ( ) D. Boer et al.rNuclear Physics B 564 2000 471–485 479 tions by the fragmentation functions means that f , g,h functions are replaced by D,G, H Ž . functions, respectively and vice versa . Finally, the leading order double polarized cross section is found to be TM TM X 2 2 a a ds l HTM l hX 2 a x z s K y K y Ž . Ž . Ž . 1 3 w x s ll F g G qll Ý h 1 1 h 2 4 1⁄2 2 2 dx dz dy df d q Q l T a,a = H H h H 1 L 1 L ˆ ˆ cos 2f F 2 hPp hPk yp Pk Ž . ž / T T T T MMh g G 1 1T a ˆ < < ql S K y cos fyf F hPk Ž . Ž . hT 1 S T h Mh H g D 1 1T a ˆ < < yl S K y sin fyf F hPk Ž . Ž . hT 2 S T h Mh H h H 1 L 1 a ˆ < < ql S K y cos fqf F hPp Ž . Ž . hT 3 S T h M 2 a ˆ ˆ < < ql S K y cos 3fyf F 4 hPp hPk Ž . Ž . Ž . hT 3 S T T ž h H H h H 1 L 1T 2 ˆ ˆ y2 hPk p Pk yhPp k T T T T T / 2 2 MMh K a y Ž . 1 < < < < q S S cos 2fyf yf Ž . T hT S Sh 2 = H H a f D qg G K y Ž . 1T 1T 1T 1T 1 ˆ ˆ < < < < F hPp hPk y S S T T T hT MM 2 h = H H f D 1T 1T cos fyf cos fyf F p Pk Ž . Ž . S S T T h MMh K a y Ž . 1 < < < < q S S sin fyf sin fyf Ž . Ž . T hT S Sh 2 = g G 1T 1T a < < < < F p Pk q S S K y Ž . T T T hT 2 MMh = H f G 1T 1T ˆ ˆ sin 2fyf yf F hPp hPk Ž . S S T T h MMh < < < < a y S S K y cos fyf sin fyf Ž . Ž . Ž . T hT 2 S Sh ( ) D. Boer et al.rNuclear Physics B 564 2000 471–485 480 = H a f G K y Ž . 1T 1T 3 < < < < F p Pk q S S cos f qf Ž . T T T hT S Sh MM 2 h = < < < < a w x F h H q S S K y cos 2fqf yf Ž . Ž . 1 1 T hT 3 S Sh = H a h H K y Ž . 2 1 1T 3 2 ˆ < < < < F 2 hPk yk q S S Ž . T T T hT ž / 2 2 2 Mh = 2 2 ˆ ˆ cos 4fyf yf F 8 hPp hPk Ž . Ž . Ž . S S T T ž h 2 2 ˆ ˆ ˆ y4 hPp hPk p Pk y2 hPk p Ž . T T T T T T H H h H 2 1l2 1T 1T 2 2 2 ˆ y2 hPp k qp k q , 39 Ž . Ž . T T T T / 2 2 ž / plk 5 4M Mh where the additional term indicated by the parentheses in the last line stands for the set of replacements distribution functionsl fragmentation functions, MlM , klp, h 4 lll , S lS , f lf together with an additional minus sign for each time-reh T hT S Sh versal odd function f H ,h , D , H H . 1T 1 1T 1 Ž . Ž . We remark that the appearance of the couplings in the cross sections 36 – 39 shows a clear pattern. All convolutions with chiral-odd distribution functions h ,h ,h ,h 1 1 L 1T 1T and fragmentation functions H H , H H , H , H H couple with K . The chiral-even sector 1 1 L 1T 1T 3 a a Ž . Ž . involves the couplings K and K : In Eqs. 36 and 39 the convolutions with an even 1 2 Ž . a Ž a. odd number of time-reversal odd functions couple with K K ; in the single 1 2 Ž . Ž . polarized cross sections 37 and 38 the situation is reversed, i.e. convolutions with an Ž . a Ž a. even odd number of time-reversal odd functions couple with K K . 2 1 To illustrate the kinematical regions where contributions from Z-boson exchange and Ž interference terms become important for experimental data on azimuthal asymmetries at 2 w x. relatively low values of Q , cf. Refs. 19,20 , we plot the values of the couplings aŽ 2 . K y,Q as appearing in the neutral current cross sections of scattering with electrons i or negatively charged muons, for different fixed values of y over the range 1(Q ( 10 GeV . Fig. 2 shows the couplings for terms when the struck quark is u-like, and Fig. 3 when it is a d-like quark. In both figures the two linear combinations aŽ 2 . aŽ 2 . aŽ 2 . aŽ 2 K y,Q ,l sq1 qK y,Q ,l sy1 and K y,Q ,l sq1 yK y,Q ,l s i e i e i e i e . y1 are plotted as occurring in scattering processes with unpolarized and polarized lepton beams, respectively. In case one scatters with positrons or positively charged muons one has to change the sign of l , hence, the resulting plots for unpolarized e positron beams are identical to the electron scattering case and for polarized beams the given curves for the couplings simply change sign. 1 In this study we consider only tree level, hence, we do not include logarithmic Q behavior due to the perturbative running of the strong and electroweak coupling constants. But this will not affect the relative magnitude of the couplings K a as a function of Q. i ( ) D. Boer et al.rNuclear Physics B 564 2000 471–485 481 uŽ 2 . Fig. 2. The couplings K y,Q for three different fixed values of y: ys0.4 solid line, ys0.6 dashed line, i Ž . and ys0.8 dotted line. On the left: sum of couplings for l sq1 and l sy1 unpolarized beam . On the e e Ž . right: difference of couplings for l sq1 and l sy1 polarized beam . e e Deviations from the Q independent behavior at low Q indicate where interference terms are important. Generally, effects of the weak interaction start to show up above Q ;300 GeV 2 and become significant for Q R10 GeV . It can be read off from the figures that asymmetries involving K a and K a are best measured with an unpolarized 1 3 beam at lower values of y. In particular for asymmetries proportional to the K a 3 couplings the y dependence is sizable. On the other hand, asymmetries involving the K a 2 couplings are best measured with a polarized beam, preferably at high values of y Ž . although the y-dependence here is rather moderate . Moreover, from the r.h.s. of the figures one sees that K a gets an enhancement by a factor 2 for u-like quarks and a 2 dŽ 2 . Fig. 3. The couplings K y,Q for three different fixed values of y: ys0.4 solid line, ys0.6 dashed line, i Ž . and ys0.8 dotted line. On the left: sum of couplings for l sq1 and l sy1 unpolarized beam . On the e e Ž . right: difference of couplings for l sq1 and l sy1 polarized beam . e e ( ) D. Boer et al.rNuclear Physics B 564 2000 471–485 482 2 Ž a factor 4 for d-like quarks at very high Q the same holds for the coupling K for 1 . unpolarized lepton scattering . a Ž . Looking at the definition of K in Eq. 29 reveals that azimuthal asymmetries 2 involving these particular couplings, which are typical polarization measurements, can be determined even in experiments with an unpolarized beam. The axial vector coupling of the Z-boson provides the necessary g structure needed to render the involved traces 5 in the calculation non-vanishing, which in the case of a polarized beam is provided by the helicity terms in the lepton tensor. The observation that polarization of quarks can be tested with unpolarized beams using electroweak interference effects was made in the w x context of electron-positron annihilation 21 . The l.h.s. plots in Figs. 2 and 3, respectively, indicate that azimuthal asymmetries involving K a couplings are accessible with 2 unpolarized lepton beams for Q R10 GeV . The strength of the K a couplings has a 2 small y dependence. Many of the azimuthal asymmetries that arise in the above given cross sections are difficult to measure. Nevertheless, we have presented the complete expressions, since Ž . different terms could be accessed in different future experiments. First of all, in order to go beyond the photon contributions one needs a relatively high energy experiment. On the other hand, at higher energies effects due to intrinsic transverse momentum are w x expected to be less important 10 , although not power-suppressed. One way around the problem of having to go to very high energies is by studying semi-inclusive deep inelastic leptoproduction involving neutrinos. One can either investigate the case of a Ž neutrino beam scattering off a hadron for instance in an experiment like NOMAD at . CERN or scattering with an electron or muon beam off a hadron with a neutrino as Ž . produced lepton for instance COMPASS at CERN or ZEUS and H1 at DESY . In both cases no interference with photons occurs. On the other hand, in the case of a produced neutrino a new problem is that one cannot define a lepton scattering plane as given in Ž . Fig. 1 one does not observe l , hence azimuthal angles cannot be defined, unless one w x can reconstruct the direction of the neutrino by the momentum imbalance 17 . One could also define azimuthal angles with respect to a transverse polarization vector of one initial or final state hadron, but this still limits the number of accessible asymmetries severely and makes the analysis much more difficult. In order to arrive at the expressions for the cross sections of such charged current processes, one can take e s0, g l sg l s1, c s0 and replace a V A 2 2 2 1 Q x TMx s , 40 Ž . ZZ W W 2 2 2 ž / 8sin u Q qM W W in the above given couplings K . In addition, one replaces c s"c s1, depending on i 1 3 a Ž a2 a2 . a Ž a2 a2 . the chirality of the quark, since c s g qg r2 and c s g yg r2. Hence, 1 R L 3 L R 2 A high p azimuthal asymmetry in charged current semi-inclusive deep inelastic scattering arising at T w x w x order a has been studied in Ref. 18 . In Ref. 22 the same asymmetry for the neutral current process was s studied, also at low p , taking into account purely photon exchange. The mechanism to include intrinsic T w x transverse momentum is the one of Ref. 5 and is therefore considered to be of higher twist. ( ) D. Boer et al.rNuclear Physics B 564 2000 471–485 483 for a left-handed quark one finds c sc s1 and for a right-handed quark one finds 1 3 c syc s1. This results in 1 3 C y Ž . 2 ab ab < < K y syK y s4 1yl x V A y " Ž . Ž . Ž . Ž . 1 2 e W W ab ž / 2 1 for a left-handed quark 2 < < s4 1yl x V = , Ž . 2 e W W ab 1⁄2 1yy for a right-handed quark Ž . 41 Ž . K ab y s0, 42 Ž . Ž . 3 where a,b are the incoming and outgoing quark flavor indices, respectively, and Vab stands for the appropriate CKM matrix element. Needless to say, the sum over flavors in Ž the cross section expressions will now only run over the appropriate flavors u-like or . Ž . d-like . We have neglected the lepton masses, hence helicity states l s"1 are equal e Ž . Ž . to the chirality states RrL . The term 1yl reflects the fact that the incoming lepton e Ž y y . e ,m ,n can only be left-handed in charged exchange and the incoming anti-lepton q q Ž . Ž e ,m ,n only right-handed l for the incoming lepton is replaced by yl for an e e . y incoming anti-lepton . For example, for the elementary process n dTMe u, one finds the 2 2 du y ud 2 < < < < Ž . coupling K s8 x V and for n uTMe d one finds K s8 x V 1yy . 1 W W ud 1 W W ud The y independence of K du is explained by the fact that the total spin of a left-handed 1 lepton and a left-handed quark is Js0 such that the partonic scattering becomes ud 2 Ž . isotropic in the c.m. lepton scattering angle; the coupling K has a 1yy depen1 dence characteristic of the Js1 total spin of a left-handed neutrino and a right-handed Ž w x. anti-quark see for instance Ref. 23 . Note that these two elementary processes will be accompanied by distribution and fragmentation functions that in general are completely different in magnitude. The same holds for the difference between neutrinorelectron w x versus anti-neutrinorpositron scattering, cf. e.g. Ref. 16 . One also has to be aware that for the charged current cross-sections no averaging over initial lepton polarizations has to be performed. The present HERA experiments ZEUS and H1 could access asymmetries for which it is important to take into account the Z boson. However, since in these experiments no Ž initial polarization is present although longitudinal lepton beam polarization will . become disposable also for the collider experiments in the near future , one way to access some of the interesting asymmetries is by focusing on L production, since the spin vector of the L can be determined from its subsequent decay. From the enhancement of the coupling K at high Q as apparent on the r.h.s. of Figs. 2 and 3, one 2 concludes that it seems quite promising to measure the helicity fragmentation function G by studying for instance L production using a polarized lepton beam on an 1 unpolarized target at very high Q. This option was investigated in for instance Refs. w x Ž . 13,24,25 . In this case one is sensitive to the first term in Eq. 38 , which does not involve a weight factor. Hence, one can integrate over the transverse momentum of the vector boson, thereby deconvoluting the distribution and fragmentation function. The Ž . resulting cross section is proportional to the integrated function G z multiplied with 1 Ž . the well-known function f x . As mentioned above, from the l.h.s. of the same figures 1 ( ) D. Boer et al.rNuclear Physics B 564 2000 471–485 484 one sees that azimuthal asymmetries involving K a couplings are also accessible with 2 unpolarized lepton beams for Q R10 GeV , thus allowing for a measurement of G1 with unpolarized lepton and proton beams. This is the semi-inclusive deep inelastic TM q y w x scattering analog of the proposed measurement of G in e e TML X 26 . One could 1 Ž . also exploit charged current exchange by using a neutrino or anti-neutrino beam like TM y w x for instance in n pTMm L X 25,27 , but we would like to emphasize that one can use the neutral current exchange process without the need for lepton beam polarization, i.e. TM X l HTM l L X. w x A polarized proton beam, for instance at the proposed polarized HERA collider 28 , would give even more opportunities to measure the different asymmetries presented Ž . Ž . w x here, i.e. in principle those given in Eqs. 37 and 39 , cf. also Ref. 29 . A remark that is relevant here is that measuring the transversity functions h and H via the 1 1 Ž . Ž . cos f qf term in Eq. 39 cannot be done in the charged current exchange case S Sh Ž y ≠ ≠ . abŽ . Ž e.g. via e p TMnL X , since K y s0 a similar observation was made in Ref. 3 w x ≠ ≠ . 30 concerning the process p p TMWX . The same holds for any other chiral-odd function. But they can of course be accessed in the neutral current processes and at very high energies lepton beam polarization can even be of assistance. In conclusion, we have presented the leading order unpolarized and polarized cross sections in electroweak semi-inclusive deep inelastic leptoproduction. We have discussed the present and future possibilities for experimental investigation of some of the asymmetries presented here. In particular, the opportunities offered by neutral and Ž charged current processes were contrasted and the optimal kinematic regions in y and 2 . Q for which one might expect certain asymmetries to be measurable were studied. We have observed that one can measure the helicity fragmentation function G by L 1 production in the neutral current exchange process with both lepton and proton beams unpolarized. Also, we have noted that the transversity distribution and fragmentation functions cannot be measured in charged current exchange semi-inclusive leptoproduction.