Single quark polarization in quantum chromodynamics subprocesses

PHYSICAL REVIEW Single D VOLUME quark polarization 53, NUMBER in quantum of Phy sics, 1 FEBRUARY chromodynamics W. G. D. Dharmaratna Department 3 1996 subprocesses zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK University of Ruhuna, M&ma, Sti Lanka Gary R. Goldstein Department of Phy sics, Tufts University , M edford, M assachusetts 02155 (Received 29 August 1995) It is well known that the single-polarization asymmetries vanish in QCD with massless quarks. But, heavy quarks with a nonzero mass should be transversely polarized due to the breakdown of the helicity conservation. In this paper we give the exact fourth-order pertwbative QCD predictions for the transverse polarization from all QCD subprocesses, qfq’ -+ q+q’, q+g + q+g, gfg + q+b and q + r$+ q’ + q’, which are significant for heavy quark production, with a description of the method of calculation. The kinematical dependence of the polarization is discussed. Top quark polarization from gluon fusion and quark annihilation processes, which are the important subprocesses at high energies, is estimated and its significance is discussed. PACS number(s): 12.38.Bx, 13.88.+e It is well known that in QCD with massless quarks (and order in a,, the strong fine-structure constant. Those gluons) the quark helicity is conserved at each vertex, and so there can be no single-quark polarization in any per- imaginary parts all involve similar integrations using the Cutkosky rules. The imaginary contiibutions to the he- twbatively licity amplitudes calculated nomenologically scattering significant @recess. This is a phe- statement for any hadronic are then projected out and inserted into the relevant expression for the polarization. The one-loop process in which the relevant energies and transverse mo- calculations menta are large compared to quark “mass” scales. What sets such mass scales is a long-standing question in QCD applications. It is foreshadowed by the distinctions made manipulation software, but are quite well defined. To make the procedure more concrete we will sketch the calculation of the QCD analogue of electron-muon between current and constituent quark masses in preQCD quark and p&on models. Whatever the appropri- scattering, p + q’ + Q + q’. The lowest-order diagram is just one-gluon exchange as shown in Fig. l(a). The M ate definition matrix for that diagram can be written as of the quark mass, whether current, con- are lengthy and are facilitated by symbolic stituent, effective, or running mass, once this parameter is nonzero the strict helicity conservation breaks down. And with the possibility of helicity flip vertices, singlequark polarization can occur in scattering processes. The observation of single-quark polarization, if it were possible in spite of confinement, would signal the importance of quark masses as well as higher-order QCD carrections. The latter is necessitated by the dependence of single-quark polarization on an interference between op- posite helicities, which requires, in turn, imaginary parts of bilinear products of amplitudes. Imaginary or complex amplitudes level. arise from loop diagrams, i.e., beyond the tree While it is clear that direct observation of quark polarization is not possible, there are many indirect measures of the putative quantity, such as the polarization or density matrix elements of leading hadrons [l] or the distribution of hadrons in jet fragmentation [2]. Given the fundamental nature of QCD and quark spin, we have been collecting together QCD perturbation theory predictions for single massive quark’ polarization in some two-body processes. Below we present the results for various flavor combinations in quark-quark scattering, quark-gluon scattering, quark-antiquark annihilation and gluon fusion. The technique for obtaining these results is straightforward. We first calculate the lowest-order Feynman diagrams for the particular process and project out the helicity amplitudes. diagrams that contribute Then we calculate all the an imaginary 0556-2821/96/53(3~/1073(14~/$06.M) part in second 53 (b) FIG. 1. Lowest-order (a) and fourth-order (b) Feynman diagrams for the q+q’ + q+q’ scattering process. In the fourth order only the diagrams which contribute to the polarization are shown. 1073 zyxwvutsrqp 01996 The American Physical Society W. G. D. DHARMARATNA AND GARY R. GOLDSTEIN 1074 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 53 (1) M= where the indices and momenta are indicated in Fig. l(a) and q2 = (pl - p3)‘. The resultant six independent helicity flip amplitudes, *I(++ -+ ++) I a?(++ -+ --) , The f-‘s are given by fi = 8p2 - 4(p2 - E1E2) co2 ; , fi = -4mlmz a$(+- + +-) , a%(+- + -+) , f3 = 4(p2 a%(++ + -+) a% (+- - 3 ++) 2 ’ fs = -4mlEz sin’ i, + E1E2) cd 8 f4 = 4mlmz sin’ - i, fe = -4m& 03si sin i , cm i sin i , where the symbols E1, Ea, p, and 0 defined in the centerof-mass frame are the energy of Q, the energy of q’. the , zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA momentum and the scattering angle, respectively. The next order imaginary contribution is shown in Fig. l(b). There is only one Feynman diagram that contributes to , the imaginary part in the s-channel physical region, the box diagram. The M matrix for that diagram is can be expressed in the form (3) which contains the integration over the internal four-momentum k of the loop, and with S,, a gluon mass cutoff that protects against infrared divergences, which will be taken to zero (since such divergences cancel out of the final physical result). First, the k-dependent terms can be separated as MC&, (XaX’);j(~a~“)~~{4~~ .p2(7” @7v)b - 2[(Y7”$, @ yv) - (7” @ .YvYli# l)]b, - (y ” y T @ w f’~,Jbnp} , (4) where the notation (A @ B) = [f&)A(p,)][~@ &% (p)2)] is used and d% (l; k,; k,k,) (b;bx;bxJ =q (k~- 6,$J((k- q)~- 6,z,J((pl - k)Z - mT)((pz+k)2 - +) (5) . The imaginary part (or the discontinuity across the unitarity cut in the physical s channel) is obtained from the Cutkosky rules by replacing the two direct channel propagators with on-shell 6 functions: zyxwvutsrqponmlkjihgfedcbaZYXWVUT (p1 - ?%;2 - r nt (pz + !$ - m; * 2?ri6((p1 - k)Z - mf)O((pl ==3 27w(p, + !q - k)O) ) - m?J6y(pz - k)O) ) where the 0 functions take the positive-energy parts. This replacement puts the particles corresponding intermediate states on their positive-energy mass shell. Equation (5) becomes 6; b,; b,,),ut = i(274’ / wl;kw~((Pl - kY - rnfyq(Pl (@ - 6L)((k These integrations have been performed for electron-electron - rc)“ )s((Pz + k)Z - m;)e((pz - q)Z - &) to the - k)O) scattering in quantum electrodynamics [3], but in the SINGLE QUARK POLARIZATION IN QUANTUM zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR 53 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 1075 large-s limit. Here these integrals are calculated The integrati&s exactly, up to terms that vanish as S, goes to 0. are handled by using cylindrical k E (ko, kl cos4,kl coordinates sin4, k,) for zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH k in the center-of-mass frame: i.e., and d4k E dkok,.dkld4dk, First the integration over 4 is performed and then one of the 8 functions is used to do the ki integration. The remaining 6 function becomes a constraint on ko (in this particular case ko = 0), so only the integration over k, remains to be done. For the simplest case (b,,,) this takes the form where F = 2k,p and S = (pl + pz)’ is the square of the total center-of-mass energy of the subprocess. The integration over 5 is done analytically neglecting the terms that vanish in the zero-gluon mass limit. Equation Imb== = (5) (6) is &!!$*(!& related to the imaginary parts of the integrals via 2Wb; b,; bp) The resulting imaginary Imb= (5) Imb, = = (b; b,; b&t (5) + y] , (8) parts of the integrals are -+I($) [&l*(s) Imb,,= ($) [$I($)-~] , Imbw= (5) [q&p2ln($)] Integrals with other combinations , sin(g) , of indices all vanish because of the symmetry properties of the integrand. Note that the masses never enter explicitly in any of these results. Masses are imbedded in the kinematic definitions of B,.p, and 4’. Note also that there are only 1 -~ l* ( )I Q2 logarithmic terms in the scalar and vector zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA -q= q2+4p2 ing nonlogarithmic Imb,, = Imb,, terms to the polarization. The next step requires the integrated (9) = (5) [~sinOln(~) expressions, but there are some nonlogarithmic terms in the tensor expressions. These latter will be significant in contribut- ’ -isin@] , terms be folded in with the Dirac matrices and s&nom in Eq. (3). This is performed in the helicity basis by using MACSYMA, and the results, the imaginary parts of the second-order six independent helicity amplitudes (4;) can be written as 4; = Here fa’s are d&ned (10) earlier for the lowest-order p2i + mfmi 91 = 16 gz = -16m1mz In sin g + p’(p? - El&) co? i K diagram and ga’s are > sin’ i ((J&L& - p”) In sin i + p’) 1 sin’ % ~~, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJ , 16 g3 = cos2(0/2) In sin i + p2(p2 + EIE2) [( sin’ i co2 i 1 (11) , 1076 W. G. D. DHARMARATNA g4 = 16mlmz sin’ z (&& 9 + p’) In sin 5 - p2 AND GARY R. GOLDSTEIN 53 , sin(fl/2) p2& + Elm; cosz Q5 = -16m1cos(e/2) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA g6 = -16mzp sin(B/2) cos(6/2) p2& + Ezmf cos2 Each amplitude has the expected dependence on quark masses and angular variables. Note that the infrared divergent term which appears only in Eq. (10) is proportional to the lowest-order amplitudes as it should be for cancellation. Having obtained the lowest-order helicity amplitudes and the imaginary parts of the second-order h&city flip amplitudes, the next step is to calculate the leading contribution to the transverse polarization. The transverse polarization (‘P) of a particle in a scattering process can be defined in terms of cross sections measured with different spin configurations of that particle: namely, where the arrows denote spatial spin (up, down) configurations of the particle with respect to the scattering plane. Here spin up (down) direction is chosen to be parallel (antiparallel) to the vector A = pa x p~/Ip, x FBI. When expressed in terms of helicity amplitudes, Eq. (12) takes the following form for the polarization of particle C of the process considered: Finally, the substitution of helicity amplitudes into the above equation leads to the following relatively simple expression for the polarization of the scattered quark [4]: P= a.(-l/3)mlq2sin(8/2){(& 2p~cos(e/2){2p4cos4(~/2) + 2El)ms ln[sin(6’/2)] + E~p2cos2(O/2)} + [(mi + mf + 2E1Ez)p2 - 2p4] cosz(8/2) + 2p4 + mfm;} where the factor (-l/3) is the SU(3) factor for color triplets scattering on &or triplets aAd a. is the strongcoupling constant, which depends on the logarithm of the momentum transfer in the usual treatment of QCD. Note that the polarization is independent of infrared divergences as expected. Note also that this exact, expression contains a logarithtiic term in the numerator, ln[sin(0/2)], and an additional contribution, - [cos(0/2)], that would not be present in a “leading log” expansion. The general form of the color factor for s - 2~scattering can be written as where C!, = 4/3 and C, = 10/9 for a color triplet source and C, = 3 and C, = 0 for a color octet source. For U(1) or QED the color factor and the coupling constant have to be replaced by {l} and a. Otherwise the formula is the same for electron (e-)-muon (II-) scattering and is the same but with opposite sign for e--p+ scattering. Note that the overall sign of the expression depends on whether it is a particle-particle interaction or particle-antiparticle interaction in the case of QED (in addition, the color factor is also different in the case of QCD). The sign change is a consequence in ordering fermion-antifermion fields. In the limit that the mass ’ (14) (mz) of p+ becomes very large, compared to ml and p, this reduces to the polarization of electrons scattered off a spin-i nucleus (2 = l), the Mott asymmetry [5]: mlp z&(0/2) ’ = 2aEf cos(0/2)[1- ln[sin(0/2)] (pZ/Ef) sin2(Cl/2)] (16) That neglects any form factors, so is relevant for low q’. Hence, Eq. (14), with the indicated group and coupling factors altered so that {‘}a. + -laZ, provides the recoil correction to the Mott asymmetry formula for electronnucleus scattering. Whether or not this explains slight discrepancies between the Mott formula and data [6] for scattering from ,~Au remains to be checked. Having calculated the asymmetry for s - u scattering and following the above arguments, it is easy to show that the transverse polarization of the s quark from s-c scattering can be obtained from Eq. (14) by changing the overall sign and replacing the color factor by {7/6}. Therefore the final result gets the same sign for both processes and is equivalent to a scattering of an s quark on an effective color potential, which is attractive, m the limit m2 is very large. The other three subprocesses can be calculated by following a similar procedure. However, there is a number 53 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA SINGLE QUARK POLARIZATION IN QUANTUM zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQP 1077 of fourth-order Feynman diagrams that contribute to the. gluon fusion, and quark annihilation-are given in the imaginary part of the amplitudes of,each of these subprofollowing sections. : cesses. Particularly, quark-gluon elastic scattering and QUARK+GLUON SCATTERING gluon fusion processes involve not only a large number of F:ynman diagrams but also a large number of additional The lowest-order and the fourth-order Feynman diaintegrations zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA (b,,,,b.,,, etc.) due to the triple-gluon cougrams that contribute to the imaginary amplitudes of pling vertex which makes the calculation more difficult. this subprocess are shown in Figs. 2(a) and 2(b), respecFeynman diagrams that contribute to the imaginary part tively. Adding the contributions from all Feynman diof amplitudes and the resultant expressions for the poagrams the following lengthy expression is obtained for larieation of the other processes-quark-gluon scattering, the polarization of the scattered quark of mass m [4]: where 1 F=2 sin(e/2)JEZ NL = ln - p2 cos2(e/2) 2 a4 ;(P - E)p2 + cos2 ;(4Ep2 N2 = l6 cos6 ;(P - E)p2 + 7 Cm4 ;(E2 N3 = -2p cos i E2 -pzcose -p2)p + 2p sin(S/2)JE2 -p2cosz(B/2) - Ezp + 3E3) _ E2(p + SE) , + 3 cos’ ;(Is~~ - 14~~ + 3E2)E - (47p2 + 28Ep + 45Ez)E sin’ i (9p + E)p - llp2 - 12Ep + 7E* 16p sin’ i - 96 >( D = [4P2cos2, +p(p + 9E)cosB ’ + 4p2 + 9E(E +p)][cos6’(p2 , , > + 2Ep + 4E’)p +p3 co? 0 + cos2 e(E3 - p3 + E=p + 2Ep2) + p3 + 2Ep2 + E=p + Es] , where E, p, and B are the energy of the quark, the momentum, and the scattering angle, respectively. All quantities are defined in the center-of-mass frame of the subprocess. The color factors are not separately shown in the expression as in the previous case because the addition of the contributions from different diagrams with different color factors is done. The delicate cancellation of infrared divergences in the imaginary parts of loop diagrams, that must be obtained in the polarization formula, is a good test of the correctness of such a lengthy calculation. The above expression, the polarization of the scattered quark off a gluon, is distinctly different from the analogous QED expression, the polarization of the scattered electron off a photon-Compton scattering. This is mainly due to the existenw of the triple:gluon coupling vertex in &CD. In the lowest order the g-g-9 vertex produces a t-channel exchange and in the fourth order several new diagrams contribute as shown in Fig. 2. By replacing the non-Abelian color factors appropriately, the corresponding scattered electron polarization in Compton scattering is obtained (41: p = am(E + p cos 0) s&(0/2) pcos(8/2)D’ e E-P 2~ , FN:+N:l*E+p ---N;cos~~G zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON > ( D’ = (1 + cm2 e)[p ‘3 cos e + 2Ep2 + E’p + E3] + p3 sin2 e + 2Ep(p + 2E) cos e , N: = 2P2(P - E) ox* ; + E(4p2 - Ep + 3E2) & N; = (P - E) N ( P cos2 ; - E ; - E2(p + 5~) , cm2 ; - E(p + SE) , > = EP ~0s’ ; + 2(pz + Ep + E2) (18) W. G. D, DHARMARATNA AND GARY R. GOLDSTEIN 1078 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA However, the magnitude of the polarization For example, the largest polarization of electron (mass - 0.5 MeV) off a photon, when 0 N 0.85 rad and p - 1.1 MeV, is is quite small. the scattered which occurs about 0.06%. (a) 53 For p (mass N 105.7 MeV) and 7 (mass - the magnitude of the polarization at about the same as it should be due could be written as a functmn of m/p. tude of the polarization is mainly due 1784 MeV) the peak remains to Eq. (18) which The small magnito the small value (a) zyxwvutsrqponmlkjihgfedcbaZY (b) FIG. 2. Lowest-order (a) and fourth-order (b) Feynman diagrams for the g+g + pfg scattering process. In the fourth order only the diagrams which contribute to the polarization are shown. FIG. 3. Lowest-order (a) and fourth-order (b) Feynman diagrams for the glum fusion, 9 f 9 --t 4 + q, process. In the fourth order only the diagrams which contribute to the polarization are shown. SINGLE QUARK POLARIZATION IN QUANTUM.. 3 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA of the coupling constant (a) in QED. This small, 01 effect would be very difficult to measure. GLUON order FUSION The Feynman diagrams that contribute to the polarization through gluon fusion are shown in Figs. 3(a) (lowest order) and 3(b) (fourth order). Note that the existence of the triple-gluon coupling vertex produces an schannel exchange in the lowest order. In one-loop level several new diagrams contribute to the complex am- 1079 plitude which include radiative correction to the gluon propagator-“vacuum polarization.” Particularly, the radiative corrections to the gluon propagator contribute to the polarization as a result of the interference with lowest order t-channel and u-channel amplitudes. This did not happen in s-g scattering since the va.cuum polarization corrections to the gluon propagator are pure real in the cross channel kinematic region. Only the radiative corrections to the quark propagator contributed to the polarization of the s-quark from s-u scattering. The resulting quark polarization for 9 + 9 + p + Q is given by [7] zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA - k2cos*B) p = o18m(p2 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA (~~+~~)Y++(~~-12’2)Y-+~~ln~+~~+18k3sin2RcosB(C1+Cz) 24k sin OD where D=(9k2cos20+7p2)(k4cos40- 2k2m2sin20- p4), Nl = Qk2Cos30(~= + 2k2) + 6kp cosO(27p2 + llkp - 27k=) + 27kacos0 Nz = kp cos20(llp2 , + 76k2) - 162p2m2 + 33k3p , Ns = P COS0[243m=p zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA CoS4 0 - cos’ O(324m2p - 54k3) + 22kp2 - 243m=p + 164k3] , N4 = Gksin2 0 ~0~0172 cos2 6’(27p3 - 18k2p + k3) + 27(Q6kzp - 24p3) _ 8(22kpz + 45k~)1 , cl = ; + rnf)O(&‘zTni) ) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJ c &=?(2P3p2 y* cp*y4*> =I* zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA This QCD polarization formula has an interesting structure, mainly due to the contribution from diagrams with internal fermion loops. rn; is the mass of the internal fermion and the summation runs over all possible intermediate quark masses. For strange quaik production, as an example, rni takes up, down, and strange quark masses at energies below the threshold for charm production. In other words an extra contribution is added to the polarization at each threshold when the center of mass energy is increased. In the case of pair annihilation, the QED analogue of gluon fusion, the corresponding expression for the polarization of the produced lepton can be obtained by setting the QCD structure constant (f& to zero and by replacing the other color factors by 1. The resulting expression, which is distinctly different from that of &CD, takes the form p=,m(p2- k2cos20) 4p sin OD’ where N+y + + N- Y- + 2 cos0 (p” + 5k2) ln;G + 2kp cosO)] , 1080 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA W. G. D. DHARMARATNA AND GARY R. GOLDSTEIN zyxwvutsrqponmlkjihgfedcbaZYXWVUTS D’ = k4 ax4 0 + 2k=p= cos= 0 - 2k4 cm= 0 - p4 - Zk=p= + 2k4 , N* = +[(p* + 2k2) cos’, The center-of-mass + 3k2 z&6kp COST] variables and Y* are the same as above. QUARK PAIR ANNIHILATION Here, only the cake with different flavors in the initial and in the final states is considered. vacuum polarization diagrams appear in the s-channel exchange and add to the imaginary not contribute to the polarization lowest-order and the fourth-order a.mz ss is Nl + E4kzmf(56X- 24p2k3 sin t’D But, they do in this case, since they are just overall factors to the lowest-order amplitude. The Feynman diagrams that contribute to the polarization are shown in Figs. 4(a) and 4(b), respectively. Polarization of the s quark from uti + p=- Similar to gluon fusion, amplitude. + 16X+) + Nz In (~)+W+)] (21) , where D = 4E4 + 2E2(mf + rnz - E’) sin’ 6 + 2~~~77~~ co? 6 , Nl = -p3k sin’ .9[(4k3 + 72Emz + 36E3)p cm 0 + 40E3k] , NZ = E’p[p’mi: sin’ 0(54p cos 6 + 20k) + k’mf(36p N3 = 4E2pk2m;(18k x* = and p(k) cos B - 5~) , (E2 f kp cos S)= - mfm; p(3kp cos 6 f p2 f 2k=) 2EJ(E2 cos 0 - 40k)] , f kp c0s0)~ - m$n; (E2 f kp cost?)= - mfm; ’ state, and the other symbols are the same as defined in Eq. (21). The experimental verification of this formula is ex- respectively, and E is the energy of the quark (or antiquark) which is the same for both initial and final states. For the QED analogue of quark annihilation, e- tremely difficult because of the small magnitude of the polarization. For example, the highest polarization of the produced p from e- + e+ annihilation is of the or- +e+ der of 0.5%, which is approximately and ml(mz) are the center-of-mass and the mass of the particles momenta in the initial (final) --t p7 + JL+, the polarization of the produced CL- is the same for heav- obtained by neglecting the diagram with the triple-gluon ier fermion (7) production vertex and changing the color factors appropriately higher momentum. The suppression due to the small QED p=- L+mz p2k2 sin6’D N; + 2E4kmf(X- - X,) [4]: except the peak appears at a stant does not OCCUI‘in the polarization massive quark from e-+e+ contribution * coupling con- of the produced ~q. In this case the leading to the polarization can be calculated rela- tively easily, since it involves only one Feynman diagram in the one-loop level. The polarization arises from the -2E2p2mf (E-p )I’ gluon radiative correction to the photon-quark vertex as E+P zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA shown in Fig. 4(c). Hence, the resultant polarization is In zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA (24 proportional to the strong-coupling constant a,. Contributions from all the other diagrams are suppressed by a where factor of ~QED. Ni = p3k sin’ 6’(k2p cos B - 4E3] , The resultant polarization duced quark, in the coordinate for the pro- system considered, has the simple form mzk sin 9 cos 6 N; = 2E2p[m:(E2 - p2 cos* 0) - mf(E2 + k2)] , = -as{4’31 [2k2(l + co9 0) + 4m;] ’ (23) SINGLE QUARK POLARIZATION IN QUANTUM 53 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 1081 where mz and zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA k are the mass and the center-of-mass moquarks, as given in [2], for example, mentum of the quark, respectively. {4/3} is the resultant SU [3] color factor from quark vertices. This has been derived previously [S] in the large-Q’ limit. It is clear from Eq. (23) that the polarization of the quark can be written as a function of 8 and mJk. The polarization mz/k N 0.87. peaks when 0 = Therefore the peak occurs above of their momentum direction, has been suggested ful description of fragmentation less than 5% asymmetry KINEMATICAL can be used to ob- However, Eq. (23) predicts at the best. DEPENDENCE POLARIZATION OF THE the able quantity, assuming that the particles in the jet could remember the direction of the polarization and preferhave a component quantities. x/4 (or 3?r/4) and threshold but well below the scaling region. An interesting way to relate the quark polarization to an observ- entially tain observable The magnitude variables. of the polarization The scattering depends on several angle, 6’, and the momentum or the energy in the center-of-mass frame can be varied in that independently. Hence, we need to study the variation of the polarization with these two variables. Furthermore, IS]. Also, a more careto estimate the polarization from the full QCD formula, of transversely polarized zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA one needs quark masses and the strong-coupling aa. What quark mass-current, constituent, a larger effective mass incorporating constant or perhaps confinement effects at lower energies where the perturbative QCD is partially valid-should be inserted? Spontaneous symmetry breaking of chiral invariance, incorporating instanton effects in the vacuum, have been evoked to give quark propagators mass corrections 191.The effective mass that arises for one quark flavor is momentum dependent, as (a) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA anticipated, and gives expected “constituent” values at low momentum. It is an interesting ization expressions point to mention can be written that the polar- as functions of a., 8, and m;/p, eliminating the other parameters by using the kinematical relations. This shows that the properties of polarization can be discussed for any given quark mass; for a higher mass the same properties will appear at a higher momentum. Hence, for simplicity, the constituent quark masses, 0.3, 0.5, 1.5, and 4.5 GeV/c’ for up (or down), strange, charm, and bottom quarks, respectively, are used when necessary. Also, a constant value, a, = 0.4, is used for the strong-coupling constant which is an overall factor in the polarization. The kinematical dependence of the polarization outgoing strange quark in the center-of-mass of the frame is shown in Figs. 5(a)-5(d) for the four subprocesses. The polarization as a function of 0 is shown for different values (b) of center-of-mass momentum (the region of the momen- tum is chosen to study the polarization at the peak). For large momentum relative to quark masses the polarization still becomes negligible as expected. However, for relatively low momenta there is a considerable structure. The first two subprocesses, s+u and s+g scattering, produce smaller polarization glu0.l * change in the polarization Cc) FIG. 4. Lowest-order (a) and fourth-order (b) Feynman diagrams for the annihilation, r~+ rJ + q’ + rj’, process. In the fourth order only the diagrams which contribute to the polarization are shown. (c) The most significant Feynman diagram for the polarization of massive quark through e++eannihilation. compared with the other two. It should be pointed out that the full expression for the polarization is significantly distinct from that of the leading logarithmic approximation, as an example the sign of the additional from s + u is a consequence contribution from the nonlogarithmic term in the expression. Another interesting thing to study is the quark mass dependence of the polarization. For s + u scattering this is done by expressing the polarization fl, C.T= ma/p, and y = mu/p. as a function of Figure 6(a) shows the 3: and I/ dependence of the polarization at the peak which occurs at 0 N 130’. Although the parameters + and y zyxwvuts W . G. D. DHARM ARATNA AND GARY R. GOLDSTEIN 1082 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA are expressed in terms of strange and up quark masses, the result could be read for the scattering of any two quarks of different flavor. The sign of the polarization is determined by the parameter y. For small values of y the polarization is positive and it peaks when y = 0 and cc N 1.5. For large y, the sign changes and the largest negative polarization (- - 5.5%) occurs when y -+ 03 at z N 0.4, i.e., the polarization gets its largest negative value if the s quark is scattering off a static color source. For a color octet static source the polarization is still negative and it is - -12% at the peak. For the s+g scattering, most of the features, peaks, and the sign changes, are very similar to the previous case. The major difference is that the largest negative polarization occurs at moderate momenta (p - 1600 MeV/c) and the positive peak exists at smaller values of p N 200 MeV/c. Therefore the kinematic regions in which the two different subprocesses change sign and peak do not coincide. However, those regions could overlap when smeared by relevant quark and gluon structure functions. The quark mass dependence of the polarization is easy to study here, since this process involves only one quark. The polarization as a function of the center-of-mass scat- FIG. 5. Polarization 53 tering angle (B) and m/p is shown in Fig. 6(b). Although there are several bumps and dips, the overall magnitude is smaller than in the previous case. Again, the magnitude is largest when 0 N 130°, i.e., in the backward scattering and the sign changes with m/p. The sign at the peak changes from negative to positive when m/p increases from - 0.35 to - 2.0. For larger m/p it becomes negative again and peaks at B N 85O. The quark polarization from gluon fusion is more interesting than the two subprocesses discussed above; the magnitude is comparatively larger and it remains significant for larger values of center-of-mass momenta. The estimation of the s-quark polarization involves not only the strange quark mass but also the other quark masses which could be excited depending on the available energy. The polarization vanishes [Fig. 5(c)] at 0 = s/2 and the sign changes leaving the magnitude the same under B + ?r - 6 as one expects from constraints on helicity amplitudes under identical particle interchange. The quark mass dependence of the polarization at the peak, i.e., at 0 N 60”, is shown in Fig. 6(c). The variation with the final-state quark mass (m) and with the gluon centerof-mass momentum (p) is given. The constituent quark of outgoing s quark in the subprocess center-of-mass frame as a function of the scattering angle with me, = 4.5 G&‘/c’. (a) q+q’ -i q+q’, (b) q+g + q+g, a, = 0.4, m, = md = 0.3 GeVjc=, m, = 0.5 GeVJ c=, m, = 1.5 G&/c’, (c) g + g --t q + q, and (d) q + q + q’ + 4’. SINGLE zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA QUARK POLARIZATION IN QUANTUM 53 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA masses are used for all internal fermion loops and the numerical result is almost the s&me if they are replaced by current quark masses., Note that the magnitude of the highest polarization remains the same while the peak gets broader when m increases. The peak occurs when mfp N 0.3. More interesting is the momentum structure of the polarization. For small momenta, polarization increases rapidly with p until the peak value is reached and then decreases slowly, a slower rate than one would expect fxom a leading logarithmic approximation in The new terms that are added to the polarization at each threshold of pair production, when p increases, are partially responsible for that effect. It is an interesting point to mention, even though an m/E. 1083 individual Feynmati diagram is not a gauge-invariant entity, that the dominant contribution to the polarization comes from one particular diagram, namely from the radiative corrections to the gluon propagator--“vacuum polarization.” This appears in the s-channel exchange and contributes 4o the imaginary parts of amplitudes in gluon fusion, in contrast to the gluon propagator corrections in the t-channel exchange for s + u and s + 9 scattering subprocesses with no contribution to the imaginary amplitudes. That imaginary amplitude contributes to the polarization as a result of the interference with the lowest order t- and u-channel amplitudes. The existence of the diagram with an s-channel intermediate gluon, for the subprocesses considered, is a result of the FIG. 6. Mass dependence of the polarization of the outgoing s quark. AU parameters are the same as in Fig. 5. (a) q+q”q+q’,(b)q+g~q+g,(c)g+g’q+4,and(d)q+~--tq’+6. W. G. D. DHARMARATNA 1084 AND GARY R. GOLDSTEIN non-Abelian character of the theory which makes the polarization large and distinct from the analogous QED expression. The polarization of the s quark produced from the quark annihilation [Fig. 5(d)] has interesting characteristics which are similar to those f?om gluon fusion; Namely, the magnitude is large and the highest polarization occurs at reasonably larger center-of-mass, momenta. Interestingly,, the region in the parametric space, p and 8, at which the polarization peaks is the same for these two subprocesses. This will certainly help to minimize the dilution of the polarization in the addition of the two contributions within the convolution integrals. For the values of parameters chosen the polarization is always positive and it peaks at p N 1.5 and 6’ N 80’. The magnitude is slightly larger than all the other processes. More interesting is the quark mass dependence of the polarization which can be parametrized as a function of z = m,/p and y = mJp, where m, and m. are the initial-state (u-quark) and the final-state (s-quark) quark masses. I and y dependences of the polarization at the peak, i.e., at 0 N SO’, is shown in Fig. 6(d). Note that the kinematical constraint for pair production, S > 4mi, leads to 1 + y2 > 1’. While the magnitude is quite small and the z dependence is insignificant for large y, there is a reasonable structure for small y which is the region that produces heavier quarks from lighter quark annihilation. The largest polarization (10.5%) occtirs at z N 0.5 and smallest possible y. i.e., y = 0 or for massless quarks in the initial state. HEAVY-QUARK POLARIZATION It is interesting to pursue the possibility of heavy quarks, charm, or bottom quarks, becoming polarized in the production processes. It is well known [lo] that the heavy-quark production cross section is dominated by the gluon fusion subprocess, especially at large transverse momenta and small Feynman z, while the annihilation processes give the next significant contribution. This general rule is true as long as the beam or the target does not carry any valence heavy quarks, because, then the heavy-quark contribution to the structure functions is negligible. Therefore, the first two subprocesses, the flavor excitation processes, are not considered for heavyquark production. The polarization of down, strange, charm, and bottom quarks from gluon fusion are compared in Fig. 7(a). Here, the center-of-mass momentum of gluons is chosen (p = 13 GeV/c) to obtain the largest polarization for the bottom quark. Aspcan be seen by comparing with Fig. 5(c) the overall magnitude of the polarization has not increased for heavy quarks, but at a given momentum the heaviest quark acquires the largest polarization. This is always true if the center-of-mass energy is well above the’threshold for pair creation. For heavy-quark production from u + ti annihilation, the result is very similar and is shown in Fig. 7(b). Here, the center-of-mass momentum of the u quark is chosen to be p = 0.9 GeV/c to see the peak of bottom quark polarization. Again, the heaviest quark FIG. 7. Polarization of up, strange, charm, and bottom quarks at the subprocess CM momentum of (a) 13 G&‘/c for glum fusion and (b) 9 GeV/c for annihilation. Other parameters are identical to Fig. 5. gets the largest polarization at a given momentum as in the previous case. In addition, the magnitude at the peak gets larger for heavier quarks. For example, the peak polarization for bottom quarks is about - 10% while that for strange quarks is about 7%. This is due to the fact that for heavier quarks the peak polarization occurs at a higher momentum which lowers the value of the quantity y in Fig. 6(d) moving the kinematical region towards the peak of the polarization. However, once we go to the higher momentum the effects of the decreasing coupling constant have to be taken into account, and that is an overall factor. TOP QUARK POLARIZATION The study of the polarization properties of the top quark,, when its mass is near or above 120 GeV, is more interesting for several reasons and worth giving special consideration. As has been already discussed in detail [11,12] the physics of the top quark has distinctly different features from other quarks, especially in its polarization phenomena. The main interesting feature for the SINGLE QUARK POLARIZATION IN QUANTUM s3 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 1085 zyxwvuts spin physics of the top quark is the rapidity of its decay process. First, the depolarization mechanism &ective for other quarks does not play a role for the heavy top quark and, second, it decays as a free quark with no hadronization or recombination So, any polarization effects due to its rapid decay [ll]. information top quark should be directly products with neither dilution thermore, carried by the produced transferred The Fur- the dominant decay process of the top quark, the electroweak two-body decay t + ideal analyzer of its own polarization. gluon into its decay nor enhancement. transverse fusion polarization and quark bW+, of the top annihilation will be an quark subprocesses from are shown in Figs. S(a) and S(b) for a top quark of IWSS 140 GeV/c’. The constituent quark masses given earlier are used for the other quarks. quark polarization Note that the top from gluon fusion depends on all the other quark masses due to the contribution from the vac- uum polarization diagram. The approximate value of the running coupling constant, a. = 0.15, is used in the estimation of the magnitude. Although the magnitude is small, - 2% from gluon fusion and N 4% from annihilation at the peak with the above value of the a,, it is not iero even in the kinematic region where the perturbative QCD is expected to be valid with no doubt. This is an even more interesting result to verify experimentally, if it is possible, since the predicted polarization arises from the one-loop level but not as a correction to the leadingorder predictions. This is a direct test of the fourth-order prediction from perturbative &CD. By using the polarization formula for gluon fusion, published previously 171, an interesting method to test the top quark polarization Collider (LHC) at the CERN Large Hadron has been suggested in [12]. At the LHC energies, the top quark production is dominated by the gluon fusion subprocess and the next significant contribution comes from the annihilation subprocess [12]. Having obtained the polarization from annihilation processes, one can include both subprocesses’ polarization in the convolution with the quark and gluon distribution functions to obtain the resultant polarization for the appro- priate hadron collider, which is neglected in [12]. Since the top quark has been observed at Fermilab, it FIG. 8. Polarization of the top quark of mass 140 GeV/c’ , (a) from gluon fusion and (b) from annihilation. Other parameters are identical to Fig. 5 except a. = 0.15. incorporated into new physics. It is an interesting point to mention that the large transverse polarization in inclusive hyperon production processes, which involve the strange quark production, observed previously (see references in 171) will not occw here since the top quark is produced at very high energy and it decays very rapidly-the polarization of the top quark is preserved as discussed above. Therefore, may be possible to collect a few hundred top events (if the expected luminosity is reached) before the LHC to study any observation of top quark polarization larger than the predictions given here will be an indication of some new the physics of the top quark. or nonstandard energies the annihilation At the Fermilab Tevatron subprocess plays the dominant model interactions, which could happen at high energies where the top quark is produced. This role in the production of the heavy top quark. Therefore, the polarization of the top quark through the annihilation makes the measurement of the top quark polarization very important and interesting experimentally as well as process is more significant here. theoretically. As a result, the resul- tant polarization of the top quark at Tevatron energies is slightly larger than that of LHC. The significance of the top quark polarization at the Fermilab Tevatron has been discussed already [13]. However, in the discussion of the single-spin asymmetry ing plane, the contribution perpendicular to the scatter- from qg subprocesses, which is given here, should be included in order to obtain the upper bound for observable effects due to the standard theory. Then, any sizable asymmetries zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK Note added in proof. An error in N4 of Eq. (19) was brought to our attention by A. Brandenberg 1141to whom we are grateful. observed can be ACKNOWLEDGMENT This work was supported U.S. Department of Energy. in part by a grant from the W. G. D. DHARMARATNA [l] R. H. Dalitz, G. Ft. Goldstein, and R. Marshall, Phys. Lett. B 215, 783 (1988); G. Ft. Goldstein, in High Energy Spin Physics: Eighth Internnational Symposium, Proceedings, Minneapolis, Minnesota, 1988, edited by K. .I. He&r, AIP Conf. Proc. No. 187 (AIP, New York, 1989). [Z] J. Collins, Nucl. Phys. B396, 161 (1993), and references contained therein. [3] R. N. Cahn and Y. S. Tsai, Phys. Rev. D 2, 870 (1970). [4] W. G. D. Dharmaratna, G. R. Goldstein, and G. A. Ringland, Z. Phys. C 41, 673 (1989). (Note: The sign error has been corrected here.) [5] W. A. Mckinley and H. Feshbach, Phys. Rev. 74, 1759 (1948); R. H. Dali& Proc. R. Sot. London A206, 509 (1951); A. I. Akhieaer and V. B. Berestetskii, Quantum Electrodynamics, 3rd ed. (Nauka, Moscow, 1969). AND GARY R. GOLDSTEIN 53 [6] P. E. Spivak et al., Sov. Phys. JETP 14, 759 (1962). [7] W. G. D. Dharmaratna and G. R. Goldstein, Phys. Rev. D 41, 1731 (1990). [8] G. L. Kane et al., Phys. Rev. Lett. 41, 1889 (1978). [9] D. I. Dyakonov,and V. Yu. Petrov, Nucl. Phys. B272, 457 (1986). [lo] P. D. B. Collins and A. D. Martin, Hadron Interactions (Hilger, London, 1985). [ll] R. H. Dalits and G. R. Goldstein, Phys. Rev. D 45, 1531 (1992). [12] G. L. Kane, G. A. Ladinsky, and C. -P. Yuan, Phys. Rev. D 45, 124 (1992). [13] G. A. Ladinsky, Phys. Rev. D 46, 3789 (1992). [14] W. Berm&her, A. Brandenberg, and P. Uwer, Aachen Report No. PITHA 95/26, 1995 (unpublished).