Leading asymmetries in two-hadron production in ee annihilation at the Z pole

We present the leading unpolarized and single spin asymmetries in inclusive two-hadron production in electron-positron annihilation at the Z pole. The azimuthal dependence in the unpolarized differential cross section of almost back-to-back Ž . hadrons is a leading cos 2f asymmetry, which arises solely due to the intrinsic transverse momenta of the quarks. An extensive discussion on how to measure this asymmetry and the accompanying time-reversal odd fragmentation functions is given. A simple estimate indicates that the asymmetry could be of the order of a percent. q 1998 Elsevier Science B.V. PACS: 13.65.q i; 13.85.Ni; 13.87.Fh; 13.88.qe Recently, we have presented the results of the complete tree-level calculation of inclusive two-hadron w x production in electron-positron annihilation via one photon up to subleading order in 1rQ 1 , where the scale Q 2 2 ' Ž . Ž . is defined by the timelike photon momentum q with Q 'q and given by Q s s . The quantity Q had to be much larger than characteristic hadronic scales, but – being interested in effects at subleading order – we considered energies only well below the threshold for the production of Z bosons. In this article we extend those results to electron-positron annihilation into a Z boson, such that the results can be used to analyze LEP-I data. We will neglect contributions from photon exchange and g-Z interference Ž .0 terms, which are known to be numerically irrelevant on the Z pole. Only leading order 1rQ effects are discussed, since for QRM the power corrections of order 1rQ are expected to be completely negligible. Z Ž .0 Furthermore, we will focus on tree level, i.e., order a . A rich structure nevertheless arises when taking into s account the intrinsic transverse momentum of the quarks and, possibly, polarization of the detected hadrons in the final state. By accounting for intrinsic transverse momentum effects we extend the results in the analysis of w x Chen et al. 2 , where no azimuthal asymmetries arising from transverse momenta have been considered. w x For details of the calculation and the formalism we refer to 1 . We shortly repeat the essentials. We consider y q Ž y X q the process e qe TMhadrons, where the two leptons with momentum l for the e and l for the e , . X 2 2 respectively annihilate into a Z boson with momentum qs lq l , which is timelike with q 'Q . Denoting 0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. Ž . PII S0370-2693 98 00136-1 ( ) D. Boer et al.rPhysics Letters B 424 1998 143–151 144 Ž . 2 the momentum of the two outgoing hadrons by P h s 1, 2 we use invariants z s 2 P PqrQ . We will h h h Ž consider the case where the two hadronic momenta P and P do not belong to the same jet i.e., P PP is of 1 2 1 2 2 . order Q . In principle, the momenta can also be considered as the jet momenta themselves, but then effects due to intrinsic transverse momentum will be absent. We will treat the production of hadrons of which the spin Ž . 2 states are characterized by a spin vector S h s 1, 2 , satisfying P PS s0 and y1FS F0. In this way we h h h h can treat the case of unpolarized final states or final state hadrons with spin-0 and spin-1r2. In the present Ž . article we will disregard the polarization of hadron two summation over spins . The final states which have to be identified and analyzed for the effects we discuss are simpler than the ones investigated by Artru and Collins w x 3 , who proposed to measure azimuthal correlations in four-hadron production. Ž . The cross section including a factor 1r2 from averaging over incoming polarizations for two-particle inclusive ee annihilation is given by P 0 P 0 ds Že qey. a 2 1 2 w mn s L W , 1 Ž . mn 3 3 2 2 2 2 2 2 d P d P 1 2 4 Q yM qG M Q Ž . ž / Z Z Z 2 Ž 2 2 . Ž with a se r 16p sin u cos u and the helicity-conserving lepton tensor neglecting the lepton masses and w W W . polarization is given by X X X X l2 l2 2 l l r s L l ,l s g qg 2 l l q2 l l yQ g y 2 g g 2 i e l l , 2 Ž . Ž . Ž . Ž . mn V A m n n m mn V A mnrs where g l , g l denote the vector and axial-vector couplings of the Z boson to the leptons, respectively, and the V A hadron tensor is given by dPX 4 W q ;P S ;P S s d qyP yP yP H P ;P S ;P S , 3 Ž . Ž . Ž . Ž . H mn 1 1 2 2 X 1 2 mn X 1 1 2 2 3 0 2p 2 P Ž . X with 2 < < :2 < < : H P ;P S ;P S s 0 J 0 P ;P S ;P S P ;P S ;P S J 0 0 , 4 Ž . Ž . Ž . Ž . mn X 1 1 2 2 m X 1 1 2 2 X 1 1 2 2 n where a summation over spins of the unobserved out-state X is understood. In order to expand the lepton and hadron tensors in terms of independent Lorentz structures, it is convenient ˆ to work with vectors orthogonal to q. A normalized timelike vector t is defined by the boson momentum q and ̃m m 2 m Ž . a normalized spacelike vector z is defined by P s P y PPqrq q for one of the outgoing momenta, say ˆ P , 2 q m m t̂ ' , 5 Ž . Q Q P m q m 2 m m ̃ z ' P s 2 y . 6 Ž . ˆ 2 P Pq z Q Q 2 2 ˆ Vectors orthogonal to t and z are obtained with help of the tensors ˆ mn mn ˆˆ m n g 'g y t t qz z , 7 Ž . ˆ ˆ H 1 mn mnrs mnrs ˆ e 'ye t z s e P q . 8 Ž . ˆ H r s 2 r s P Pq Ž . 2 Since we have chosen hadron two to define the longitudinal direction, the momentum of hadron one can be used ( ) D. Boer et al.rPhysics Letters B 424 1998 143–151 145 Ž . Fig. 1. Kinematics of the annihilation process in the lepton center of mass frame for a back-to-back jet situation. P P is the momentum 1 2 Ž . of a fast hadron in jet one two . ˆ m m m < < ˆ to express the directions orthogonal to t and z. We define the normalized vector h s P r P , with P ˆ 1H 1H 1H mn mn ˆ Ž . s g P , and the second orthogonal direction is given by e h see Fig. 1 . We use boldface vectors to H 1n H n denote the two-dimensional Euclidean part of a four-vector, such that P PP syP PP . 1H 1H 1H 1H In the calculation of the hadron tensor it will be convenient to define lightlike directions using the hadronic Ž . Ž 2 . or jet momenta. The momenta can then be parametrized remember that P PP is of order Q using 1 2 dimensionless lightlike vectors n and n satisfying n sn s0 and n Pn s 1, q y q y q y z Q 1 m m P ' n , 9 Ž . 1 y '2 z Q 2 m m P ' n , 10 Ž . 2 q '2 Q Q m m m m q ' n q n qq , 11 Ž . y q T ' ' 2 2 with q 'yQ . We have neglected hadron mass terms and considering the case of two back-to-back jets we T T have Q <Q. We will use the notation pspPn for a generic momentum p. As momentum P defines T . 2 the vector z , ˆ m m ˆ m P syz q syz Q h . 12 Ž . 1H 1 T 1 T Vectors transverse to n and n one obtains using the tensors q y g mn ' g mn yn m n 4 , 13 Ž . T q y e mn ' e mnrs n n , 14 Ž . T qr ys where the brackets around the indices indicate symmetrization. The lightlike directions can easily be expressed ˆ in t, z and a perpendicular vector, ˆ 1 m m m ˆ w x n s t qz , 15 Ž . ˆ q '2 1 QT m m m m ˆ ˆ n s t yz q2 h , 16 Ž . ˆ y ' Q 2 showing that the differences between g mn and g mn are of order 1rQ. We will see however that taking H T transverse momentum into account does not automatically lead to suppression. ( ) D. Boer et al.rPhysics Letters B 424 1998 143–151 146 To leading order the expression for the hadron tensor, including quarks and antiquarks, is qlyq mn y q 2 2 2 m n q y W s3 dp dk d p d k d p qk yq Tr D p V D k V q , 17 Ž . Ž . Ž . Ž . Ž . H p k T T T T T ž / mln where V m sg g qg g g m is the Z boson-quark vertex. We have omitted flavor indices and summation. The V A 5 w x correlation functions D and D are given by 4 : 1 4 i kP x 2 < < :2 < < : D k s d x e 0 c x P ,S ; X P ,S ; X c 0 0 , 18 Ž . Ž . Ž . Ž . Ý H j i j i 1 1 1 1 4 2p Ž . X 1 4 yi pP x 2 < < :2 < < : D p s d x e 0 c 0 P ,S ; X P ,S ; X c x 0 19 Ž . Ž . Ž . Ž . Ý H i j j 2 2 2 2 i 4 2p Ž . X Ž . Ž . and the quark momentum k and similarly for p and the polarization vector S from now on we omit S are 1 2 decomposed as follows: 1 kf P qk , 20 Ž . 1 T z l1 S f P qS . 21 Ž . 1 1 1T M1 Ž . To leading order in 1rQ one has that zsz . The partly integrated correlation function D is parametrized as: 1 m n r s 1 M Pu e g P k S Pu g 1 1 mnrs 1 T 1T 1 5 q H dk D P ,S ;k s D qD yG Ž . H 1 1 1 1T 1 s y 2 1⁄2 y y z P M M M k sP rz , k 1 1 1 1 1 T is g S m P n is g k P n s k P n mn 5 1T 1 mn 5 T 1 mn T 1 H H yH yH qH , 22 Ž . 1T 1 s 1 2 2 5 M M M 1 1 1 Ž H. where the shorthand notation G and similarly for H stands for the combination 1 s 1 s k PS Ž . T 1T G z ,k sl G qG . 23 Ž . Ž . 1 s T 1 1 L 1T M1 We parametrize the antiquark correlation function D in the same way, except that the distribution functions are overlined and the obvious replacements of momenta are done. Ž . The functions D , . . . in Eq. 22 and G ,G , . . . in G , . . . are called fragmentation functions. One wants 1 1 L 1T 1 s to express the fragmentation functions in terms of the hadron momentum, hence, the arguments of the Ž . y y fragmentation functions are chosen to be the lightcone momentum fraction zsP rk of the produced 1 hadron with respect to the fragmenting quark and k X 'yzk , which is the transverse momentum of the hadron T T in a frame where the quark has no transverse momentum. In order to switch from quark to hadron transverse momentum a Lorentz transformation leaving k and P unchanged needs to be performed. The fragmentation 1 functions are real and in fact, depend on z and k X 2 only. T Ž iy . We note that after integration over k several functions disappear. In the case of Tr D is g a specific T 5 Ž 2 2 . H combination remains, namely H ' H q k r2 M H . 1 1T T 1 1T 2 X Ž X . The choice of factors in the definition of fragmentation functions is such that Hdz d k D z,k sN , where T 1 T h N is the number of produced hadrons. h Note that the decay probability for an unpolarized quark with non-zero transverse momentum can lead to a transverse polarization in the production of spin-1r2 particles. This polarization is orthogonal to the quark transverse momentum and the probability is given by the function D . In the same way, oppositely 1T ( ) D. Boer et al.rPhysics Letters B 424 1998 143–151 147 transversely polarized quarks with non-zero transverse momentum can produce unpolarized hadrons or spinless particles, with different probabilities. In other words: there can be a preference for one or the other transverse Ž . polarization direction of the quark aligned or opposite relative to its transverse momentum to fragment into an unpolarized hadron. This difference is described by the function H H . It is the one appearing in the so-called 1 w x Collins effect 5 , which predicts a single transverse spin asymmetry in for instance semi-inclusive DIS, and arises due to intrinsic transverse momentum. The functions D and H H are what are generally called ‘time-reversal odd’ functions. For a discussion on 1T 1 w x w x the meaning of this, we refer to Ref. 1 and earlier Refs. 6 ; here we only remark that it does not signal a violation of time-reversal invariance of the theory, but rather the presence of final state interactions. The cross sections are obtained from the hadron tensor after contraction with the lepton tensor 1 mn l2 l2 2 2 mn m n m n mn ˆ ˆ L s g qg Q y 1y2 yq2 y g q4 y 1yy z z y4 y 1yy l l q g Ž . Ž . Ž . ˆ ˆ Ž . V A H H H H ž / 2 m n 4 l l 2 mn r w m n x ˆ ˆ ( ( y2 1y2 y y 1yy z l y 2 g g Q qi 1y2 y e y2 i y 1yy l e z , Ž . Ž . Ž . Ž . ˆ ˆ Ž . H V A H H r H 24 Ž . 4 w x where mn indicates symmetrization of indices, mn indicates antisymmetrization. The fraction y is defined y y Ž . to be ysP P lrP Pqf l rq , which in the lepton center of mass frame equals ys 1qcosu r2, where u 2 2 2 2 is the angle of hadron two with respect to the momentum of the incoming leptons. ˆm Azimuthal angles inside the perpendicular plane are defined with respect to l , defined to be the normalized H m m ˆ Ž . ( perpendicular part of the lepton momentum l, l s l r Q y 1yy : Ž . H H ˆ < < l Pa sy a cosf , 25 Ž . H H H a mn ˆ < < e l a s a sinf , 26 Ž . H H m H n H a for a generic vector a. The vector and axial couplings to the Z boson are given by: g j sT j y2 Q j sinu , 27 Ž . V 3 W g j sT j , 28 Ž . A 3 j j Ž j j where Q denotes the charge and T the weak isospin of particle j i.e., T sq1r2 for jsu and T sy1r2 3 3 3 y . for jse ,d,s . Combinations of the couplings occurring frequently in the formulas are c j s g j2 qg j2 , Ž . 1 V A c j s g j2 yg j2 , js l or u ,d ,s Ž . 2 V A c j s2 g j g j . 29 Ž . 3 V A As well, we will use the following kinematical factors: 1 cm 2 2 A y s yyqy s 1qcos u r4, Ž . Ž . 2 ž / 2 cm 2 B y sy 1yy s sin u r4, Ž . Ž . 2 cm C y s 1y2 y s ycosu . 30 Ž . Ž . Ž . 2 ( ) D. Boer et al.rPhysics Letters B 424 1998 143–151 148 We obtain in leading order in 1rQ and a the following expression for the cross section in case of unpolarized s Ž . or spinless final state hadrons: q y 2 2 ds e e TMh h X 3 a Q 1 Ž . a 1 2 w 2 2 l a l a a s z z c c A y y c c C y F D D Ž . Ž . Ý 1 1 2 1 1 3 3 1 2 2 ž / 2 2 2 2 1⁄2 2 dV dz dz d q 1 2 T Q yM qG M Ž . a,a Z Z Z Ha H a H H1 1 l a ˆ ˆ qcos 2f c c B y F 2 hPk hPp yk Pp , 31 Ž . Ž . Ž . ž / 1 1 2 T T T T 5 M M 1 2 l l ˆm where dV s 2 dy df and f gives the orientation of l . We use the convolution notation H a a a 2 2 2 a 2 2 2 2 F D D ' d k d p d p qk yq D z , z k D z , z p . 32 Ž . Ž . Ž . Ž . H T T T T T 1 1 T 2 2 T ˆ Ž . The angle f is the azimuthal angle of h see Fig. 1 . In order to deconvolute these expressions we can define 1 weighted cross sections df l ds eeTMh h X Ž . 1 2 2 2 : W s d q W , 33 Ž . H A T 2 2p dV dz dz d q 1 2 T Ž . where W s W Q ,f ,f ,f ,f . The subscript A denotes the polarization in the final state for hadron one, T 1 2 S S 1 2 Ž . Ž . with as possibilities unpolarized O, including the case of summation over spin , longitudinally polarized L or Ž . transversely polarized T . We postpone the discussion of the additional structures and information accessible by measuring the polarization of one of the final state hadrons to the end of this letter. Even without determining polarization of a final state hadron a subtle test of our understanding of spin transfer mechanisms in perturbative QCD can be done. The information on the production of a transversely Ž . polarized quark-antiquark pair, which subsequently fragment into unpolarized or spinless hadrons with Ž . probabilities depending on the orientation of the anti quark’s spin vector relative to its transverse momentum, is Ž . 1 contained in the cos 2f azimuthal asymmetry . To access this information we utilize the weighted cross 1 sections 3 a 2 Q 1 a w l a l a a 2 : 1 s c c A y y c c C y D z D z , 34 Ž . Ž . Ž . Ž . Ž . Ý O 1 1 1 3 3 1 1 2 2 ž / 2 2 2 2 2 Q yM qG M a,a Ž . Z Z Z Q 3 a 2 Q Ž . T w H 1 a l a H Ž1.a cos 2f s c c B y H z H z , 35 Ž . Ž . Ž . Ž . Ž . Ý 1 1 1 2 1 1 2 2 ¦ ; 2 2 2 2 4M M 1 2 Q yM qG M a,a Ž . O Z Z Z where the k -moments for a generic fragmentation function F are defined by T n 2 kT Žn. 2 2 2 2 F z sz d k F z , z k . 36 Ž . Ž . Ž . H i i T i i T 2 ž / 2 Mi Ž . We now like to focus on the weighted cross section defined in Eq. 35 and discuss its possible measurement. Ž . In order to be able to observe the cos 2f dependence one must look at two jet events in unpolarized electron-positron scattering. In each jet one identifies a fast hadron with momentum fractions z and z 1 2 Ž . respectively. One of the hadrons say two together with the leptons determines the lepton scattering plane as is 1 Ž . w x 2 This asymmetry is not to be confused with the cos 2f asymmetry found by Berger 7 , which is 1rQ suppressed. ( ) D. Boer et al.rPhysics Letters B 424 1998 143–151 149 indicated in Fig. 1. In the lepton CM system hadron two determines the z-direction with respect to which the ˆ Ž . azimuthal angles are measured. One needs in particular the azimuthal angle f of the other hadron one as well 1 < < Ž . as its transverse momentum P , which determines Q s P rz . The cos 2f angular dependence then can 1H T 1H 1 Ž . be analyzed by calculating the weighted cross section of Eq. 35 . For an order of magnitude estimate, we consider the situation of the produced hadrons being a p and a p. q q y y uTMp dTMp dTMp uTMp Ž . Ž . Ž Ž . Ž . . Furthermore, we assume D z sD z D z sD z , respectively and neglect unfa1 1 1 1 dTMpŽ . vored fragmentation functions like D z etc.; and similar for the time-reversal odd functions. The equalities for the D functions seem quite safe on grounds of isospin and charge conjugation, the same 1 assumptions might be non-trivial for the H H functions. As a consequence of these assumptions the fragmenta1 tion functions can be taken outside the flavor summation, and we obtain Q H H Ž1. z H H Ž1. z Ž . Ž . T 1 1 1 2 2 : cos 2f sF y 1 , 37 Ž . Ž . Ž . O 1 ¦ ; 4M M D z D z Ž . Ž . 1 2 1 1 1 2 O where c c B y Ž . Ý 1 2 asu ,d F y s . 38 Ž . Ž . 1 e a e a c c A y y c c C y Ž . Ž . Ý 1 1 3 3 ž / 2 asu ,d Ž 2 w x. This factor is shown in in Fig. 2 as a function of the center of mass angle u we use sin u s0.2315 8 . At 2 W an angle close to 90 o we observe the largest effect. In order to get an estimate of the true asymmetry at the level Ž . 2 of count rates, one should compare Eq. 35 with the weighted cross section Q r4M M . To estimate the 2 : T 1 2 O w x ratio of those two quantities, we use as argued in Ref. 9 , for the ratio of the fragmentation functions H Ž1.Ž . Ž . Ž . H z rD z sO 1 , although this is likely an optimistic estimate. From the average transverse momen1 1 1 1 Ž .2 w x tum squared of produced pions in one jet, for which we take 0.5 GeVrc 10 , one obtains an estimate for the average transverse momentum of pions in jet one with respect to a given pion in jet two. This leads at 2 2 2 : z sz s1r2 to Q r4M f50 1 and consequently to an estimate at the percent level for the ratio 2 : 1 2 T p O O Qr4M 2 cos 2f r Qr4M 2 . Such an azimuthal dependence in the unpolarized cross section, 2 : 2 : Ž . Ž . T p 1 T p O O however, may be detectable in present-day electron-positron scattering experiments. The situation where hadron two is taken to be a jet, which in this back-to-back jet situation is equivalent to Ž . Ž . analyzing the azimuthal structure of hadrons inside a jet, is obtained by considering D z sd 1yz and 2 2 1 H Ž . 2 : H z s0. This gives the familiar result for Hdz 1 and it gives zero for the z -integrated cos2f 2 2 2 O 1 azimuthal asymmetry. Ž . Fig. 2. Factor defined in Eq. 38 depending on the center of mass angle u . 2 ( ) D. Boer et al.rPhysics Letters B 424 1998 143–151 150 Table 1 Weighted cross sections for S /0,S s0 1 2 y1 2 2 2 2 2 2 2 2 : W A W P 3 a Q r Q y M q G M Ž . Ž . A w Z Z Z a 1 l a l a a Ž Ž . Ž .. Ž . Ž . 1 L y l Ý c c A y y c c C y G z D z 1 a,a 1 3 3 1 1 L 1 1 2 2 Ž . H 1 a 2 l a H Ž1.a Ž . Ž . Ž . Ž . Ž . Q r4M M sin 2f L l Ý c c B y H z H z T 1 2 1 1 a,a 1 2 1 L 1 1 2 a 1 l a l a H Ž1.a Ž . Ž . < < Ž Ž . Ž .. Ž . Ž . Q rM sin f yf T S Ý c c A y y c c C y D z D z T 1 1 S 1T a,a 1 1 3 3 1T 1 1 2 2 1 Ž . H 1 a l a a Ž . Ž . < < Ž . Ž . Ž . Q rM sin f qf T y S Ý c c B y H z H z T 2 1 S 1T a,a 1 2 1 1 1 2 1 Ž . H 1 a 3 2 l a H Ž2.a Ž . Ž . < < Ž . Ž . Ž . Q r6M M sin 3f yf T y S Ý c c B y H z H z T 1 2 1 S 1T a,a 1 2 1T 1 1 2 1 a 1 l a l a Ž1.a Ž . Ž . < < Ž Ž . Ž .. Ž . Ž . Q rM cos f yf T S Ý c c A y y c c C y G z D z T 1 1 S 1T a,a 1 3 3 1 1T 1 1 2 2 1 Ž . The experimental determination of the polarization of one of the final state hadrons offers further opportunities to reveal the hadronic structure in terms of spin-dependent fragmentation functions. We assume in Ž . the following that the spin vector of hadron one, i.e. S , is known reconstructed , having in mind the example 1 of a produced L and its self-analyzing properties. We observe a rich structure of angular dependences due to polarization. Again, weighted cross sections are the appropriate means to separate out specific functions. For instance, the weighted cross section Q 3 a 2 Q Ž . T w H 1 a l a a < < sin f qf s S c c B y H z H z 39 Ž . Ž . Ž . Ž . Ž . Ý 1 1 S 1T 1 2 1 1 2 2 ¦ ; 1 2 2 2 2 M2 Q yM qG M a,a Ž . T Z Z Z w x picks out the term which is the closest analogue to the Collins effect 5 in semi-inclusive lepton-hadron w x Ž . scattering 11 . We note that a confirmation of the cos 2f asymmetry, also implies a confirmation of the Collins effect. A complete list of weighted cross sections at leading order is given in Table 1. In conclusion, we have presented the leading asymmetries in inclusive two-hadron production in electronpositron annihilation at the Z pole. We have investigated unpolarized and single spin asymmetries. We included w x the effects of intrinsic transverse momentum and in this sense our results are an extension of those of Ref. 2 . Ž . The azimuthal dependence in the unpolarized differential cross section is a cos 2f asymmetry, which arises solely due to the intrinsic transverse momenta of the quarks. An extensive discussion on how to measure this asymmetry and the accompanying time-reversal odd fragmentation functions is given. A simple estimate indicates that the asymmetry could be at the percent level, hence it can perhaps be observed in present-day electron-positron scattering experiments. In confirming the existence of this asymmetry one also confirms the Collins effect, without the need of a polarization measurement. w x Note added: In a preliminary study 12 a similar correlation in back-to-back jets was already experimentally H H investigated. We find that it involves moments of the functions H and H , different from the ones in our 1 1 correlation. In this study no significant result was found using the 1991 to 1994 LEP data. In this analysis three momenta in the final state need to be determined, namely besides two hadron momenta also the jet axis, and X Ž X . hence there are two azimuthal angles, f and f , yielding a cos fqf asymmetry. We like to thank Daniel van Dierendonck and Niels Kjaer for useful discussions on the experimental aspects. ̈ Furthermore, we thank Thierry Gousset and Oleg Teryaev for several discussions. This work is part of the Ž . research program of the foundation for Fundamental Research of Matter FOM , the National Organization for Ž . Scientific Research NWO and the TMR program ERB FMRX-CT96-0008. ( ) D. Boer et al.rPhysics Letters B 424 1998 143–151 151