ELSWIER

2 March 1995

Physics Letters B 346 (1995) 129-136

PHYSICS LETTERS B

Spin symmetry predictions for heavy quarkonia alignment

Peter Cho ‘, Mark B. Wise 2 Law&en Laboratory, Cal$wnia Institute oj’Technolcqy. Pasadena, CA 9I125, USA

Received 30 November 1994 Editor: H. Georgi

Abstract

We investigate the implications of spin symmetry for heavy quarkonia production and decay. We first compare spin symmetry predictions for charmonia and bottomonia radiative transitions with data and find they agree quite well. We next use spin symmetry along with nonrelativistic QCD power counting to determine the leading order alignment of P- and D-wave ortho- quarkonia that are produced through gluon fragmentation. Finally, we discuss a mechanism which may resolve the current factor of 30 discrepancy between theoretical predictions and experimental measurements of prompt I)’ production at the Tevatron. This mechanism involves gluon fragmentation to a subleading Fock component in the $I’ wavefunction and yields 100% transversely aligned 9”s to lowest order. Observation of such a large alignment would provide strong support for this resolution to the I)’ problem.

1. Introduction

Spin symmetry has played an important role in the study of heavy hadrons during the past several years. Many of its consequences for mesons and baryons with quark content Q4 and Qqq have been investigated within the

context of the Heavy Quark Effective Theory (HQET) [ 11. The spin and flavor symmetries of QCD ‘that become exact in the IVY --, CQ limit are transparent in this effective theory framework. But even without the HQET apparatus, several aspects of charm and bottom hadron phenomenology may be understood from symmetry considerations

alone. For example, the rates for strong and electromagnetic transitions between states belonging to separate spin doublets are simply related by group theory factors. Moreover, the sums of partial widths for each member of a doublet to decay to states in another doublet are equal. These predictions that follow from spin symmetry in the infinite mass limit appear to be reasonably well satisfied for finite mass hadrons in the charm and bottom sectors

[21. In this letter, we consider the implications of spin symmetry for heavy quarkonia. Since the spins of both the

quark and antiquark inside a Qfj bound state are separately conserved as M, -+ 03, transitions between members of different quarkonia multiplets within the same flavor sector are more tightly constrained than their heavy-light meson analogues. Ratios of decay rates can be predicted and checked against the large body of charmonia and bottomonia data collected over the past two decades. Spin symmetry also constrains Qe bound state formation and

’ Work supported in part by a DuBridge Fellowship and by the U.S. Department of Energy under DOE Grant no. DE-FG03-92-ER40701. ’ Work supported in part by the U.S. Department of Energy under DOE Grant no. DE-FG03-92-ER40701.

0370-2693/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSD!O370-2693(94)01658-5

130 P. Cho, M. B. Wise /Physics Letters B 346 (I 995) 129-I 36

ties in closely with recent perturbative QCD calculations of parton fragmentation to heavy quarkonia [ 3-61. As we

shall see, the union of these ideas yields several interesting predictions which can be experimentally tested at hadron colliders.

Our paper is organized as follows. We first review the general relation that spin symmetry imposes upon heavy

hadron transitions in Section 2. We then test this relation for quarkonia by comparing its predictions for QQ

electromagnetic transitions with experimental data. In Section 3, we use the spin symmetry relation along with nonrelativistic QCD power counting arguments to determine the leading order alignment of P- and D-wave ortho- quarkonia produced via gluon fragmentation. Finally, we discuss the implications of spin symmetry for a fragmen-

tation mechanism which may resolve the current large discrepancy between theory and experiment in prompt r,V

production at the Tevatron.

2. Radiative quarkonia transitions

It is useful to recall the basic symmetry relation which constrains transitions between any initial and final hadronic

states H and H' that contain heavy constituents [ 21. For Q4 mesons and Qqq baryons, we let jb”, jp’ and J(‘)

denote the angular momentum of H ('j's tends to infinity, the separate angular momenta of the light and heavy

degrees of freedom become good quantum numbers. A similar phenomenon occurs for QQ quarkonia. But in this

case, the quantum numberji” represents just the total spin of the QQ pair, whilej$J’ stands for the bound state’s

remaining angular momentum which includes the orbital contribution from the heavy quark and antiquark.

In the infinite mass limit, the amplitude for a general H + H' process that is mediated by an interaction Hamiltonian

Z’i and results in the emission or absorption of some light quant can be decomposed as

= c (J’Ji Ij)j%i .kk )okjpz 1 xl bf_kz >ofbz; jhjhz IJJz >!itii %&, (2.1)

where repeated angular momentum labels are summed. If the reaction proceeds through the Lth multipole of Zi,

the amplitude becomes proportional to the reduced matrix element Gke ]I,!&+) and may be conveniently rewritten

in terms of a 6j-symbol [ 7,8] :

(2.2)

The consequences of spin symmetry follow from this last formula ‘.

We will apply Eq. (2.2) to heavy quarkonia transitions. In order to test its validity, we first investigate charmonia and bottomonia radiative decays. We start with the electric dipole process is3P, + ‘*3S,, + y. In the initial state, the

heavy spin angular momentumj,Z = 0 or j,, = 1 can be combined with the P-wave orbital angular momentum je = 1 contains an S-wave qe or GQ with j> = 0. Eq. (2.2) implies that the decay rate r( Ie3PJ+ ‘t3SJ, + r) a Et, averaged and summed over initial and final polarizations, is independent of J and J’ in the infinite mass limit. For finite mass quarkonia, the rate indirectly depends upon these angular momentum quantum numbers through formally subleading

’ The only 6j-symbols which we will need for applications of (2.2) in this letter are

P. Cho, M.B. Wise/Physics Letters B 346 (1995) 129-136 131

but phenomenologically important splittings between members of the S- and P-wave multiple&. Taking these splittings into account, we find the following radiative partial width ratios in the cc sector 4:

~(X~-‘J~rCI+r):~(x,~~J~rCI+y):~(Xc2-fJ~~+~):~(~,~rlr+Y)

= 0.095 : 0.20 : 0.27 : 0.44 (Theory) , (2.3a)

= 0.092 + 0.041: 0.24 f 0.04 : 0.27 + 0.03 : unmeasured (Experiment) . (2.3b)

Analogous spin symmetry predictions in the bb sector cannot be checked against data, for absolute radiative partial widths have not yet been measured. However, we can compare theoretical and experimental values for the branching

fraction ratio

R

J = nxbJw) + n 1s) + Y)

r(&J(2p) + T(2s) + r> (2.4)

as a function of J:

Ro:R, :R,=3.22:2.55:2.28 (Theory) , (2.5a)

=5.11&4.13:2.47f0.60:2.28+0.47 (Experiment) . (2.5b)

We do not include the corresponding prediction for Qh,( 2P) --) 776( 2s) + 7) lQh,(2P) + r]b( IS) + -y) in Eq. (2.5a) since the parabottomonium phase space factors are uncertain as the Q,( lS), qb(2S) and h,(2P) masses have not been experimentally measured.

We can further test spin symmetry for heavy quarkonia by considering lT3SJ + 1.3PJ, + y electric dipole decays. Inserting jf= 0, jp = 1 and L= 1 into amplitude (2.2), we find r(3S, + 3PJ, + y) a [ (2J’+ 1) /9] E; and

r( ‘S, + ‘Pi + y) a E:. These partial width results yield the following charmonium and bottomonium sector ratios:

~(v+xco+Y):~(v~xrl +Y):r(~‘-,Xc2+Y):r(77:-jh,+Y)

=9.3:7.9:5.4: 1.3 (Theory) , (2.6a)

= 9.3 + 0.8 : 8.7 f 0.8 : 7.8 f 0.8 : unmeasured (Experiment) , (2.6b)

~~~~2~~-,xbo~~~~+Y~:~~~~2~~~xb,~~~~+Y~:~~~~2~~’xb2~~~~+Y~

4.29 : 6.70 : 6.59 (Theory) , (2.7a)

=4.3&1.0:6.7f0.9:6.6&-0.9 (Experiment) , (2.7b)

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

=7.0: 11.3: 12.3 (Theory) , (2.8a)

=5.4+0.6: 11.3f0.6: 11.4+0.8 (Experiment) . (2.8b)

Again we do not include parabottomonium partial widths in Eqs. (2.7a) and (2.8a) since their symmetry breaking

phase space factors are uncertain. Comparing the theoretical and experimental numbers in Eqs. (2.3)-(2.8), we see that the predictions of spin

symmetry agree quite well with the quarkonia data. The discrepancies in a given bb mode are smaller than those in its CI? analogue, as one would expect. Spin symmetry also works best for the lowest lying QQ states where the ratio of the heavy quarks’ kinetic energy to their rest mass is minimal. But the general success of these radiative transition findings bolsters one’s confidence that results for other processes which follow from the amplitude expression in Eq. (2.2) will also work well. We therefore use it to investigate quarkonia production in the following section.

4 We use the experimentally measured parachannonia masses M, =2979MeV[9~,M,=3526MeV(10,11~andM,,=3592MeV[12lto predict the r( h, + q -t y) and r( ~1: + h, + y) entries in Eqs. (2.3a) and (2.6a).

132 P. Cho. M. B. Wise /Physics Letters B 346 (I 99.5) 129-136

(4 (b) Fig. I. Feynman diagrams which mediategluon fragmentation to P-waveXQ, orthoquarkoniaatorder(a)o5[a,(21We)/~]*and(b)u5cr,(2Mn)l

T.

3. Orthoquarkonia alignment

Quarkonia physics involves a number of different scales separated by the velocity u of the heavy quarks inside

Qe bounds states. These scales are set by the quarks’ mass MQ, typical momentum Mou and kinetic energy A40u2. Within the past few years, an effective field theory formalism called Nonrelativistic Quantum Chromodynamics

(NRQCD) has been established to keep track of this quarkonia scale hierarchy [ 131. The effective theory is based upon a double power series expansion in the heavy quark velocity and strong interaction fine structure constant. Its power counting rules allow one to methodically determine the most important contributions to a particular quar-

konium process [ 141. We will utilize these NRQCD rules in conjunction with spin symmetry arguments to investigate quarkonia production via gluon fragmentation. As we shall see, these tools yield phenomenologically

interesting predictions for orthoquarkonia alignment. We first consider g -+ xoJ fragmentation. The Feynman diagrams that mediate this process at lowest order in the

the velocity expansion are illustrated in Figs. la and lb. The graphs contribute to xoJ production through scattering collisions like gg + gg * -+ ggxo> Such fragmentation reactions dominate over lower order parton fusion processes

when the lab grame energy q. of the fragmenting gluon g * is large, but its squared four-momentum q2 is close to the bound state’s squared mass M& = (2Mo) ‘. In this case, g * is transverse up to corrections of order q’/qg.

Consequently, the fragmenting gluon is essentially real so long as its energy q. is significantly greater than M,o> The QQ states that emerge from the shaded ovals in Fig. 1 reside within the first two Fock components of the xQJ

wavefunction

Ixa~)=O~~>lQ~~‘~:‘~l)+O~u~/Q~~3~~8~lg)+... . (3.1)

The angular momentum quantum numbers of the Qe pairs are indicated in spectroscopic notation inside the square brackets, and their color configurations are labeled by singlet or octet superscripts. In Fig. la, the incoming hard gluon with four-momentum q fragments into QG[ 3P$1)] plus an outgoing hard gluon. The rate for this process is proportional to the square of the first derivative of the P-wave bound state’s wavefunction evaluated at the origin ]R{(O) (2,u5. In Fig. lb, the incoming gluon turns into Qo[ 3S$8’] which appears in the next-to-leading Fock component in (3.1). The formation rate for the colored S-wave pair is proportional to the square of its wavefunction JR,(O)]‘-U’.T~~“S$~’ state subsequently converts to a xaJ through a soft gluon coupling which is depicted as a

P. Cho, M. B. Wise/Physics Letters B 346 (1995) 129-136 133

“black box” in Fig. lb. This “black box” may be regarded as mediating the dominant chromoelectric dipole transition %I*’ + ‘Pj’) + g. Although the resulting P-wave state has order unity overlap with the full xaJ wave-

function, the soft gluon emission costs one power of v in the amplitude and two in the rate. The total transition in Fig. lb therefore proceeds at 0( us). Alternatively, the “black box” may be viewed as combining the 3S$8’ QQ

pair with a soft gluon to form the complete 1 QQ[ 3Si8’ ] g) Fock component in Eq. (3.1) which has O(v) overlap with 1 ,yaJ). This interpretation again leads one to conclude that the transition in Fig. lb takes place at 0( v”) [ 61.

Although the Feynman diagrams in Fig. 1 are of the same order in the NRQCD velocity expansion, their dependence upon the short distance fine structure constant is different. The graphs in Fig. la involve two hard

gluons and are suppressed by 42MQ) /7~ relative to the graph in Fig. lb which has only one. This perturbative QCD suppression is partly offset by the numerical values H, = 15 MeV and H1, = 3 MeV for the nonperturbative

matrix elements associated with the color-singlet and octet graphs [ 61. However, prompt xQJ production at high transverse momenta is known to be dominated by gluon fragmentation to QQ[ 3SIR’ ] [ 15-171. Therefore, we will

simply neglect the subleading contributions to g --) xQJ which come from the diagrams in Fig. 1 a ‘. The intermediate 3S$8’ state in Fig. lb has the same quantum numbers as the incoming hard gluon. In particular,

it is transversely aligned at high energies. Spin symmetry fixes the degree to which the final xu, inherits the color- octet state’s alignment through the soft gluon processes 3Si8’ + 3Pj’) +g or 3Sj8’ + g + ‘Pjl). The populations of

all the individual xQJ helicity components may be determined by setting Je = 0, ji = 1 and L = 1 in the general amplitude expression (2.2). After a straightforward computation, we find that the helicity levels are filled according

lo

x*:x$,=o) :x&w=l) :xg2=o’ :X~yl=l):XQZ (lhl=2) ,1.1.1.1.1.6 9’6’6’18’18’18 ’

A few points about this result should be noted. Firstly, if we compare the helicity populations in (3.2) with their unaligned counterparts

(3.3)

we see that gluon fragmentation at high transverse momenta yields a sizable xQJ alignment. Secondly, the ratios in Eq. (3.2) are consistent with the leading color-octet terms in the polarized g + xcJ fragmentation functions which were calculated in Ref. [ 181. The effect of the xcJ alignment upon the angular distribution of photons that result from the electric dipole transition xcJ + J/ I) + y as well as the induced J/ I) alignment were discussed in this previous article. We will not reiterate those findings of Ref. [ 181 here, but we note that they may be simply recovered from spin symmetry arguments. Finally, uncertainty in the numerical value for the nonperturbative matrix element Hk drops out from the ~$1 ratios in (3.2). Therefore, they represent more reliable predictions than corresponding

estimates for absolute g + xaJ (“) fragmentation probabilities which depend upon the presently poorly constrained

Hb parameter. We consider next gluon fragmentation to D-wave quarkonia. In the absence of a well-established nomenclature

convention, we adopt the symbol 6, to refer to L = 2 quarkonia with total angular momentum J. quark model calculations indicate that n = 1 and n = 2 S,, bottomonia lie below the BB threshold. On the other hand, all n = 1 S,, charmonia are predicted to have masses greater than 2Mn but less than M,+M,, [ 19-211 ‘. The l”D, and I’D,

states may decay to Db and are broad. But parity forbids their 11D2 and 13D2 counterparts from decaying to two spinless mesons. The J = 2 charmonia are consequently quite narrow. Gluon fragmentation to 1 ‘D2 paracharmonia has been found to yield a nonnegligible pp + 1 ‘D, +X cross section at the Tevatron [ 221. Fragmentation to

experimentally more accessible 13D2 orthocharmonia starts at the same order in the velocity expansion but one lower order in CU,~( 2M,) / 7~. The 1 ‘D, states should therefore be produced in greater abundance.

5 In order to completely determine the O( [ a,( 2M,) /PI’) contribution to g --* xe, fragmentation, the interference between the graph in Fig. 1 b

and its one-loop corrections must be calculated along with the diagrams in Fig. la. Such interference terms have not been included so far in

g + xnJ fragmentation function computations.

6 Evidence for a 3.836 GeV 13D2 charmonium state has recently been reported in Ref. [ I1 1.

P. Cho. M. B. Wise /Physics Letters B 346 (1995) I29-136

(4 M

Fig. 2. Feynman diagrams which mediate gluon fragmentation to D-wave 8, orthoquarkonia at order (a) u ’ [ a,( 24,) /?r] 3, (b) u’ [ a,( 2Mo) / P] * and (c) ~‘a,( 2Mo) /T. These same diagrams also mediate gluon fragmentation to S-wave I& states at order (a) u 3 [ q( 2Mo) / V] 3, (b) ~‘[q(2Mo)l7r]~and (c) 07a,(2Mp)/7r. Crossed graphs in (a) and (b) are not displayed.

spinless mesons, The J= 2 charmonia are consequently quite narrow. Gluon fragmentation to 1 ID, paracharmonia has been found to yield a nonnegligible pp -+ 1 ‘D, +X cross section at the Tevatron [ 221. Fragmentation to

experimentally more accessible 13D2 orthocharmonia starts at the same order in the velocity expansion but one lower order in a,( 2M,) /T. The I’D, states should therefore be produced in greater abundance.

Gluon fragmentation to So, orthoquarkonia proceeds at 0( u ‘) via Feynman diagrams like those illustrated in

Figs. 2a, 2b and 2c. The states emerging from the shaded ovals in these graphs correspond to the QQ pairs in the

following Fock components of the So, wavefunction:

~S~,)=O(I)~Q~[3D:“])+O(u)~Q~[3P:~~]g)+O(uZ)~Q~[3S~8’lgg)+.... (3.4)

The diagrams in Figs. 2a and 2b are suppressed relative to that in Fig. 2c by [ a,,,J 2MQ) / r] ’ and cu,( 2Mo) /V respectively. So the dominant contribution to g -+ So, fragmentation comes from the Fock component that contains

the S-wave color-octet state. Its subsequent conversion to So, through long distance processes such as

3S$8) -+3Dj’) +g+g and 3S{8’ +g+g+“D$” conserves j,* = 1. We can thus determine the alignment which the

S,, orthoquarkonium inherits from its transverse 3S$8’ progenitor by setting je = 0, ji = 2 and L = 2 in Eq. (2.2).

We find that the individual 6, VI) helicity states are populated according to

~$*=a’ :~$:‘I=” #&;‘a #&I=” #2w=2’ :y&=O’#.&l=1 q3w=2’ 4_y7:‘1=3’

=L.L.L.A._2.L.I.M.II 50’50~30~ 30.30. 7s. 75. 75. 75 . (3.5)

Once So, orthoquarkonia are formed, spin symmetry fixes the leading order transfer of alignment from these d- wave states to S- and P-wave orthoquarkoniadescendants in subsequent strong and electromagnetic decays. Consider for example the decay mode S,, + Jl I& mr in the cF sector. This reaction conserves heavy spin angular momentum since the outgoing pions are soft. So we can again use Eq. (2.2) to compute the alignment which the final J/+ inherits from the initial S,,:

J/I+~“‘=~’ = Br( S,., + J/G mT)(()~~;=o’ + $ll”i =‘I + $li” =2’) ,

J/~‘I”I=“=Br(S,,-,J/rl,rr~)(lS,, (h=O) + ;s!,d”l = 1) + fs:i”l =29 (3.6)

The names of the helicity states in this expression denote fragmentation probabilities. Inserting the J= 2 D-wave results from Eq. (3.5)) we deduce that I& + Jl I) n-n- feeddown produces longitudinal and transverse J/#‘s in the ratio Jl I,!J (ll=o,:J/rc,‘1”1=“=,:3.6,

P. Cho, M. B. Wise/Physics Letters B 346 (1995) 129-136 135

fusion alone underestimates the rate for J/II, production at large transverse momenta by more than a factor of 10. But if fragmentation is included, the disagreement between theory and experiment reduces to roughly a factor of 2 [ 15-171. The remaining discrepancy may be attributed to a number of theoretical uncertainties in perturbative QCD fragmentation function computations. The application of fragmentation ideas of Jf t,b production thus represents a qualitative success.

The situation for prompt $’ production is totally different. The prediction for da (pp + +’ +X) ldp I at high p I underestimates the measured cross section by a factor of 30 even after fragmentation is taken into account [ 231. Recently, gluon fragmentation to some of the n = 2 XL states that lie above the DD threshold has been proposed as a possible significant source of 4”s [ 18,241. This mechanism requires an uncomfortably large radiative branching fraction Br(& + I+!/ + y) = (5-lo)% in order to explain the factor of 30 discrepancy between theory and data. However, a search for the photons that result from the electromagnetic transition along with an analysis of their angular distribution would at least provide a means for testing this proposal.

There is another possible resolution to the I/J’ surplus problem which does not involve any charmonia states above the DL) threshold [ 25 1. Instead, the answer may lie in fragmentation to higher Fock state components within the I/J’ wavefunction

+O(u*) IcF[~D:‘*~) ]gg)+O(?) Icc[‘s~8’]g)+o(u*) IcC[3S$‘*8)]gg)+... . (3.7)

Gluon fragmentation to the cE[ ‘Si” ] pair in theleadingFockcomponent startsat 0( u3[ (r,(2M,) 1~1~) viaFeynman diagrams like the one illustrated in Fig. 2a. Fragmentation to the CE states in all other Fock components of I$‘) takes place at higher order in the velocity expansion. But the heavy charm-anticharm pairs in the subleading Fock components can be formed at lower orders in a,( 2M,) / r. In particular, the transformation shown in Fig. 2c of a hard gluon into a 3S18’ CE pair followed by its conversion into Jr’ via the emission or absorption of two soft gluons occurs at 0( u ‘czs(2M,) / ‘TT) . If we evaluate the relative strengths of the I,V production processes in Figs. 2a and 2c by setting ~~~0.3 and a,(2M,)lr=9.0X lo-*, we find u~[cY,(~M,)/~~]~= 1.2~ 10e4 and u’[o~,(2M,)l ~1 = 1.4 X lo-“. Hence, gluon fragmentation to the color-octet CC pair within the last Fock state of Eq. (3.7) may naively be an order of magnitude more important than that to the color-singlet pair within the first Fock state. If so, its inclusion could resolve the I+!J’ problem 7.

Without knowing the value for the nonperturbative matrix element associated with g + CE[ 3S$8’ ] fragmentation, we cannot draw any definite conclusion regarding this possible i(l’ production mechanism. But spin symmetry provides a powerful means for testing the hypothesis that this proposal accounts for most of the prompt $“s observed at the Tevatron. Recall that the ‘S$” cc pair in Fig. 2c is transverse up to O(M$ /qg ) corrections. This intermediate je = 0 state transforms at leading order in the velocity expansion into a j2 = 0 I+!J’ through the L = 0 emission or absorption of two soft gluons. The outgoing +!J’ in Fig. 2c has exactly the same angular momentum quantum numbers as the incoming hard gluon. Therefore, I,!J”S produced at high transverse momenta via gluon fragmentation to the final Fock state in Eq. (3.7) are 100% transversely aligned!

This striking prediction can be tested by measuring the lepton angular distribution in the 9’ + e+/- decay channel. The decay products are generally distributed according to

(3.8)

where 0 denotes the angle between the lepton three-momentum in the I/J’ rest frame and the t,!~’ three-momentum in the lab frame. The parameter (Y ranges over the interval - 1~ LY Q 1 and equals unity for totally transverse I,!J”s. Subleading corrections to the infinite mass limit prediction will shift the value for CK to less than one. But experimental

’ Fluon fragmentation to the CE pair within the ( CC[ ?3~81 ]gg) Fock component of 1 J/I) also enhances prompt J/4 production. However, its

impact is smaller than that for I/I’ production since high transverse momenta J/$‘s predominantly come from g 4 xel fragmentation.

136 P. Cho, M. B. Wise/Physics Letters B 346 (1995) 129-136

observation of a lepton angular distribution close to dI’/d cos f?a ( 1 + cos2B) would provide strong support for this

resolution to the I,!/ surplus problem.

Acknowledgements

We thank Eric Braaten and Sandip Trivedi for helpful discussions.

References

] I ] For HQET reviews, see M. Wise, New Symmetries of the Strong Interactions, in: Proc. 6th Lake Louise Winter Institute, eds. B.A.

Campbell, A.N. Kamall, P. Kitching and F.C. Khanna (World Scientific, Singapore, 1991);

H. Georgi, Heavy Quark Effective Theory, in: Proc. Theoretical Advanced Study Institute (TASI) 1991, eds. R.K. Ellis, C.T. Hill and

J.D. Lykken (World Scientific, Singapore, 1992);

B. Grinstein, Lectures on Heavy Quark Effective Theory, in: High Energy Phenomenology, Proc. Workshop (Mexico City, July 1991).

eds. R. Heurta and M.A. Perez (World Scientific, Singapore);

M. Neubert, Phys. Rep. 245 ( 1994) 259.

] 2 ] N. Igsur and M.B. Wise, Phys. Rev. I&t. 66 ( 199 I ) I 130.

[3 1 E. Braaten and T.C. Yuan, Phys. Rev. Len. 71 (1993) 1673.

14 I E. Braaten, K. Cheung and T.C. Yuan. Phys. Rev. D 48 ( 1993) 4230.

[S] Y.-Q. Chen, Phys. Rev. D48 (1993) 5181.

[6] E. Braaten and T.C. Yuan, Phys. Rev. D 50 ( 1994) 3176.

171 A. Falk and M.E. Peskin, Phys. Rev. D 49 (1994) 3320.

[ 8 1 A.R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U.P., Princeton, NJ, 1957).

[ 9 I Particle Data1 Group, Review of Particle Properties, Phys. Rev. D 50, Part I ( 1994).

] IO] E760 Collab., T.A. Armstrong et al., Phys. Rev. Lett. 69 (1992) 2337.

[ I1 1 E705 Collab., L. Antoniazzi et al., Phys. Rev. D 50 ( 1994) 4258.

1 121 Crystal Ball Collab., C. Edwards et al., Phys. Rev. Lett. 48 ( 1982) 70.

[ 131 G.T. Bodwin, E. Braaten and G.P. Lepage, ANL-HEP-PR-94-24 ( 1994). unpublished, and references therein.

1 141 G.P. Lepage et al., Phys. Rev. D 46 (1992) 4052.

1 151 E. Braaten, M.A. Doncheski, S. Fleming and M.L. Mangano, Phys. Lett. B 333 (1994) 548.

[ 161 M. Cacciari and M. Greco, Phys. Rev. Lett. 73 ( 1994) 1586.

I 171 D.P. Roy and K. Sridhar. Phys. Lett. B 339 (1994) 141.

[ 181 P. Cho, S. Trivedi and M. Wise, CALT-68-1943 (1994). unpublished.

[ 191 E. Eichten and F. Feinberg, Phys. Rev. D 23 (1981) 2724.

[ 201 W. Buchmiiller. Phys. Lett. B 112 (1982) 479.

[21] S. Godfrey and N. Isgur, Phys. Rev. D 32 ( 1985) 189.

[ 22 1 P. Cho and M. Wise. CALT-68-1954 ( 1994), unpublished.

I23 1 CDF Collab., Fermilab-conf-94/ 136-E ( 1994), unpublished.

1241 F.E. Close, RAL-94-093 ( 1994). unpublished.

12.5 1 E. Braaten. private communication.