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PHYSICAL REVIEW D 1 APRIL 1996VOLUME 53, NUMBER 7
Heavy quark fragmentation functions for D-wave quarkonium and charmedb-flavored mesons
Kingman Cheung*Center for Particle Physics, University of Texas at Austin, Austin, Texas 78712
Tzu Chiang Yuan†
Davis Institute for High Energy Physics, University of California at Davis, Davis, California 95616~Received 2 October 1995!
At the large transverse momentum region, the production of heavy-heavy bound states such as charmonium,bottomonium, andbc mesons in high energye1e2 and hadronic collisions is dominated by parton fragmen-tation. We calculate the heavy quark fragmentation functions into theD-wave quarkonium andbc mesons toleading order in the strong coupling constant and in the nonrelativistic expansion. In thebc meson case, oneset of itsD-wave states is expected to lie below the open flavor threshold. The total fragmentation probabilityfor a b antiquark to split into theD-wavebc mesons is about 231025, which implies that only 2% of the totalpseudoscalar ground stateBc comes from the cascades of these orbitally excited states.
PACS number~s!: 13.87.Fh, 12.38.Bx 13.85.Ni
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I. INTRODUCTION
The production of heavy-quark–antiquark bound stasuch as charmonium, bottomonium, and the yet undiscovecharmedb-flavored mesons at high energye1e2 and had-ronic machines can provide very interesting tests of perbative QCD. In the charmonium case, our traditional wisdo@1# has taken the following scenario: Most of theJ/c comespredominately from either the radiative decay of tP-wavexcJ states produced by the lowest order gluon fusimechanism or the weak decays ofB mesons, whereas thc8 should be produced almost entirely fromB-meson de-cays. Similarly, most of theY should be produced from theradiative xbJ decay. Only until recently could this naivthinking be confronted by the wealth of high energy expemental data collected at the Fermilab Tevatron. During1992–1993 run, the Collider Detector at Fermilab~CDF! re-corded surprisingly large production rates of the promJ/c, c8, andxcJ which are orders of magnitude above ththeoretical lowest order predictions@2#. In 1993 @3#, it wasrealized that fragmentation can play an important rolequarkonium production at the large transverse momen(pT) region at the Tevatron. Parton fragmentation inquarkonium is formally a higher order process but nevertless it can be enhanced at the sufficiently large transvemomentum region compared with the usual gluon fusprocess@3#. A simple explanation of the enhancement is ththe fragmenting partons are produced at high energieswith small invariant masses which can lead to the enhanment factor of powers of (pT /mc)
2 compared with the gluonfusion mechanism.
Another important theoretical development is the factization formalism of Bodwin, Braaten, and Lepage@4# forthe inclusive decay and production of heavy quarkoniu
* Electronic address~internet!: [email protected]†Electronic address~internet!: [email protected]
530556-2821/96/53~7!/3591~13!/$10.00
tesred
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This new approach, which is based on nonrelativistic QCD~NRQCD!, allows us to go beyond the conventional color-singlet model@5# by considering a double series expansion ofthe decay or production rates in terms of the strong couplinconstant and the relative velocity of the heavy quarks insidthe bound state. This double series expansion leads us narally to include the higher Fock states of the quarkoniumsome of which have the heavy quark pair in the color-octestate. We will discuss more about this factorization formal-ism in the next section.
Most of the fragmentation functions for quarkonium thatare relevant to phenomenology have now been calculateGluon fragmentation functions were calculated in Ref.@3#for theS wave, Refs.@6,7# for theP wave, and Ref.@8# forthe 1D2 state; heavy quark fragmentation functions were calculated in Refs.@9,10# for theS wave and Refs.@11–13# forthe P wave. Phenomenological applications of these fragmentation functions have been performed by a number ogroups@14,15#. The current scenario of charmonium produc-tion at high transverse momentum is as follows. Among alfragmentation contributions that are relevant toJ/c produc-tion, the dominant ones come from~i! gluon fragmentationinto a color-octet 3S1 cc state followed by a doubleE1transition intoJ/c and~ii ! gluon fragmentation intoxcJ fol-lowed by the radiative decayxcJ→J/c1g. The formermechanism is the most dominant one forc8 production be-cause of the absence ofxcJ(2P) states below theDD thresh-old. The current experimental status with this new fragmentation insight has now been reviewed by many people ovarious occasions. We refer the readers to Refs.@14–17# formore details.
The charmed-flavoredbc mesons are expected to be ob-served soon at the Tevatron and preliminary limits on theproduction rate already exist@18#. Parton fragmentation hasbeen applied phenomenologically to the production ofbcmesons as well at the Tevatron@19,20#. The heavy quarkfragmentation functions forb*→(bc) were calculated inRef. @21# for theSwave and in Refs.@11,12# for theP wave.
3591 © 1996 The American Physical Society
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3592 53KINGMAN CHEUNG AND TZU CHIANG YUAN
The validity of the parton fragmentation has been questioin some recent exactO(as
4) calculations @22#. However,these calculations obtained different results and led to cflicting conclusions among themselves. It was only until trecent appearance of the results of Ref.@23# that this contro-versy can be settled and the validity of the fragmentatapproximation can be established in the production of (bc)mesons at the Tevatron. One can, therefore, use the fragmtation approximation with confidence in calculating the prduction cross sections and transverse momentum spectthe (bc) mesons, including the leading logarithmic corretions, induced gluon fragmentation contribution@20#, andcontributions from all orbitally excited states.
The purpose of this paper is to extend the previous calations of heavy quark fragmentation functions to tD-wave case. TheD-wave orbitally excited states are of interest phenomenologically. In the (bc) case, one set of theD-wave states is predicted by potential models to lie beltheBD threshold. These states, once produced, will cascinto the pseudoscalar ground stateBc via pion and/or photonemissions and thereby contribute to the inclusive productrate of Bc . The excitedD-wave charmonium resonancehave also been suggested@24# to resolve thec8 surplus prob-lem observed at the CDF. In Sec. II, we briefly review tfactorization model of Bodwin, Braaten, and Lepage@4# thatcan consistently factor out the long-distance physics ashort-distance perturbative factors for the inclusive prodtion and decay rates of heavy quarkonium. In Sec. III,present the calculation of theD-wave fragmentation func-tions for both the unequal and equal mass cases withinspirit of the factorization model. We discuss our resultsSec. IV and conclude in Sec. V.
FIG. 1. Feynman diagram forQ* (q)→QQ(P,1 or 8)1Q(p8).
ed
on-he
on
en-o-a ofc-
cu-e-
owade
ions
e
ndc-e
thein
II. FACTORIZATION MODEL
According to the factorization formalism@4#, the frag-mentation function for a heavy quarkQ to split into aquarkonium stateX with longitudinal momentum fractionzcan be written as
DQ→X~z,m!5(n
dn~z,m!^O nX&, ~1!
where O n are local four-fermion operators defined inNRQCD. The short-distance coefficientsdn(z,m) are inde-pendent of the quarkonium stateX. For a fragmentation scalem of order of the heavy quark massmQ , the coefficientsdn(z,m) can be calculated using perturbation theory in astrong coupling constantas(2mQ). The relative size of thematrix elementsO n
X& for a given stateX can be estimated byhow they scale withmQ and with the typical relative velocityv of the heavy quarks inside the quarkonium. Thus, the fac-torization formula, Eq.~1!, is a double expansion inas andv. To determine the relative importance of the various termsin this formula, one should take into account both the scalingin v of the matrix elementsO n
X& and the order ofas in thecoefficientsdn(z,m). The leading term inv of this formulacorresponds to the popular color-singlet model@5#. However,keeping only the leading term in this double expansion cansometimes lead to incomplete or even inconsistent resultsdue to the presence of infrared divergences@4,25#.
One important feature in the factorization model is thatthe quarkonium stateX is no longer considered as solely aQQ pair but rather a superposition of Fock states. For ex-ample, the spin-triplet3DJ quarkonium states, denoted bydJQ , have the following Fock state expansion:
FIG. 2. Another Feynman diagram forQ* (q)→QQ(P,8)1Q(p8).
udJQ&5O~1!uQQ~3DJ ,1!&1O~v !uQQ~3PJ8,8!g&1O~v2!uQQ~3S1 ,8 or 1!gg&1•••, ~2!
t
n
where the notation1 and8 refers, respectively, to the color-singlet and color-octet states of theQQ pair. In the aboveFock-state expansion there are also otherO(v2) states, e.g.,uQQ(3DJ ,8 or 1)gg&, but their production will be furthersuppressed by powers ofv. Heavy quark fragmentation intoD-wave quarkoniumdJ
Q can be deduced from the leadingFeynman diagrams ofQ*→(QQ)Q as shown in Figs. 1 and2. In Fig. 1, theQQ pair can be either in the color-single3DJ state, in the color-octet3PJ8, or in the color-octet orcolor-singlet 3S1 states, whereas in Fig. 2 it can only be ithe color-octet3S1 state. The short-distance factors deduce
dfrom these diagrams are all of orderas2 . Furthermore, all
these diagrams are also of the same order inv. According tothe NRQCD @4#, the relevant local four-fermion operatorsdescribing these three Fock states areO 1(
3DJ), O 8(3PJ8),
andO 8(3S1) which have relative scalings likev4, v2, and
v0, respectively. However, theQQ pair in the color-octetP-wave andS-wave states can evolve nonperturbatively intothe physicalD-wave quarkonium statedJ
Q by emitting oneand two soft gluon~s!, respectively. Emitting a soft gluoncosts a factor ofv at the amplitude level in NRQCD andhence a factor ofv2 in the probability. Thus, with a nonrel-
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53 3593HEAVY QUARK FRAGMENTATION FUNCTIONS FORD-WAVE . . .
ativistic normalization convention in the state, all three m
trix elements^O 1d
J
Q
(3DJ)&, ^O 8d
J
Q
(3PJ8)&, and ^O 8d
J
Q
(3S1)&scale likemQ
3 v7. As a result, both Figs. 1 and 2 contributethe fragmentation functions of orderas
2v7 and should all betaken into account for a consistent calculation. With apppriate normalization, the color-singlet matrix elemen
^O 1d
J
Q
(3DJ)& as well as ^O 1d Q
(1D2)& for the spin-singletD-wave statedQ can be related to the nonrelativistic radiaD-wave wave functionRD9 (0), which is the spin average othe spin-singlet and spin-tripletRD9 (0)’s, by
^O 1d Q
~1D2!&'75Nc
4puRD9 ~0!u2, ~3!
^O 1d
J
Q
~3DJ!&'15~2J11!Nc
4puRD9 ~0!u2. ~4!
Potential models can be used to determine the value ofwave function so that the NRQCD matrix elements for tcolor-singlet contributions are fixed. Unfortunately, potentmodels cannot determine the color-octet matrix elemesince dynamical gluons are involved. However, from our eperience of theP-wave quarkonium case@11#, we do notexpect the color-octet component to play a major role in theavy quark fragmentation. We note that this is in sharp ctrast with the gluon fragmentation in which a gluon can frament into the color-octet 3S1 state via the processg*→QQ. This process is of orderas and is at least onepower ofas less than the leading color-singlet term in thgluon fragmentation. We also note that the color-singlet cotribution of heavy quark fragmentation from Fig. 1 is freeinfrared divergences. On the other hand, infrared divergenwill show up in the gluon fragmentation function into thspin-tripletD-wave quarkonium. It is necessary to includthe color-octet contributions in order to achieve finite asensible perturbative answers for the gluon fragmentatfunctions into spin-tripletD-wave states. In what follows wewill restrict ourselves to the color-singlet contribution in thheavy quark fragmentation intoD-wave quarkonium. If thecolor-octet matrix elements can be determined in the futuit is straightforward to include their contributions since thecorresponding short-distance coefficients can be extraceasily from the previousS-wave andP-wave calculations bymodifying the color factors.
The factorization model for the quarkonium system cbe extended to the unequal mass case such as the (bc) me-son system. In what follows, we will denote the spin-sing
a-
o
o-ts
l
theealntsx-
hen-g-
en-fces
edion
e
re,irted
n
et
and the spin-tripletD-wave (bc) meson states bydbc anddJbc , respectively. In this case, Fig. 2 is absent for the coloroctet contribution.
III. HEAVY QUARK FRAGMENTATION FUNCTIONSINTO D-WAVE HEAVY-HEAVY MESONS
The general covariant formalism for calculating the production and decay rates ofS-wave and P-wave heavyquarkonium in the nonrelativistic expansion was developesome time ago@26#. It is straightforward to extend it to thecase of unequal mass and higher orbital excitation. We shapresent a covariant formalism for the production of aD-wave meson (Qq) by some unspecified short-distanceprocesses. LetmQ andmq be the masses of the two quarkswith mQ.mq , and introduce the mass parametersr5mq /(mQ1mq) and r5mQ /(mQ1mq)512r . In theleading nonrelativistic approximation, the massM of the me-son is simplymQ1mq . The amplitude for producing thebound state (Qq) in a state with momentumP, total angularmomentumJ, total orbital angular momentumL, and totalspinS is given by
A~P!5 (LZ ,SZ
E d3k
~2p!3CLLZ
~k!^LLZ ;SSZuJJZ&M~P,k!,
~5!
where
M~P,k!5O GGSSZ~P,k!. ~6!
O G represents the short-distance interaction producing thQ andq in a specific relative orbital angular momentum, andin general is a product of Dirac matrices. The heavy quarkQ and q have momentumrP1k and r P2k, respectively,where 2k is the small relative momentum of the quarks in-side the bound state.CLLZ
(k… is the Bethe-Salpeter wavefunction in the momentum space and is assumed to beslow-varying smooth function ofk. SometimesM(P,k) is atrace by which the on-shell spinorsv(rP1k,s) andu( r P2k,s) are brought together in the right order.GSSZ
(P,k), up to second order ink, is given by
FIG. 3. Feynman diagram forQ* (q)→Qq(P,1)1q(p8).
GSSZ~P,k!5AmQ1mq
2mQmq(s,s
K 12 s; 12 sUSSZL v~rP1k,s!u~ r P2k,s!
'AmQ1mqS rP” 1k”2mq
2mqD S g5
2e/~P,SZ!D S r P” 2k”1mQ
2mQD , ~7!
where in the middle parenthesesg5 is for S5SZ50 ~spin singlet! and2e/(P,SZ) is for S51 ~spin triplet!.Sincek/M is a small quantity, we can expandM(P,k) aroundk50 in a Taylor expansion:
d
nd-
s for
3594 53KINGMAN CHEUNG AND TZU CHIANG YUAN
M~P,k!5M~P,0!1ka
]M~P,k!
]kaUk50
11
2kakb
]2M~P,k!
]kakbUk50
1•••, ~8!
where the first, second, and third terms correspond to quantum numbersL50, 1, and 2 of the orbital angular momentum, anso forth. Thus for theS-wave (L50), P-wave (L51), andD-wave (L52) states, the amplitudeA(P) will depend on theradial wave functionsRS(0), RP8
(0), andRD9 (0) through the relations
E d3k
~2p!3C00~k!5
RS~0!
A4p,
E d3k
~2p!3C1LZ
~k!ka52 iA 3
4pRP8 ~0!ea~P,LZ!,
E d3k
~2p!3C2LZ
~k!kakb5A15
8pRD9 ~0!eab~P,LZ!, ~9!
whereea is the polarization vector for the spin-1 particle, andeab is the totally symmetric, traceless, and transverse secorank polarization tensor for the spin-2 particle. In theD-wave case the amplitude becomes
A~P!51
2A15
8peab~P,JZ!RD9 ~0!
]2M~k!
]ka]kbUk50
, ~10!
for the spin-singlet case, whereJZ5LZ , and
A~P!51
2A15
8pRD9 ~0!Pabr
J ~P,JZ!]2Mr~k!
]ka]kbUk50
, ~11!
where
PabrJ ~P,JZ!5 (
LZ ,SZeab~P,LZ!er~P,SZ!^2LZ ;1SZuJJZ&, ~12!
for the spin-triplet case. Using the appropriate Clebsch-Gordan coefficients, we have@27#
PabrJ51~P,JZ!52A 3
20 @ 23P aber~P,JZ!2P areb~P,JZ!2P brea~P,JZ!#, ~13!
PabrJ52~P,JZ!5
i
MA6@eas~P,JZ!etbrs8P
tgss81ebs~P,JZ!etars8Ptgss8#, ~14!
PabrJ53~P,JZ!5eabr~P,JZ!, ~15!
where
P ab52gab1PaPb
M2 ~16!
and eabr(P,JZ) is the totally symmetric, traceless, and transverse spin-3 polarization tensor. The polarization sumJ51,2, and 3 are given by the familiar expressions@27#
(JZ521
1
ea~P,JZ!eb* ~P,JZ!5P ab , ~17!
(JZ522
2
eab~P,JZ!ers* ~P,JZ!51
2~P arP bs1P asP br!2
1
3P abP rs , ~18!
53 3595HEAVY QUARK FRAGMENTATION FUNCTIONS FORD-WAVE . . .
(JZ523
3
eabg~P,JZ!ersh* ~P,JZ!51
6~P arP bsP gh1P arP bhP gs1P asP brP gh1P asP bhP gr1P ahP bsP gr1P ahP brP gs!
21
15~P abP ghP rs1P abP gsP rh1P abP grP sh1P agP bhP rs1P agP bsP rh
1P agP brP sh1P bgP ahP rs1P bgP asP rh1P bgP arP sh!. ~19!
y
After writing down the formalism for the general production of D-wave mesons, we shall specify the productimechanism. The specific production ofD-wave (Qq) me-sons that we are considering is by the fragmentation oheavy quarkQ:
Q* ~q!→Qq~P,1!1q~p8!, ~20!
where the off-shellQ* is produced by a high energy sourcG. The fragmentation function is then given by the formu@9,21#
D~z!51
16p2E dsuS s2M2
z2r 2M2
12z D limq0→`
(uA~P!u2
(uA0u2,
~21!
where (uA0u25NcTr(GGq” ) is the tree-level amplitudesquared to create an on-shellQ quark with the same threemomentumq, and (uA(P)u2 is the amplitude squared foproducing (Qq)1q from the same sourceG. The generalprocedures to extract the fragmentation function have bdescribed in detail in Refs.@9,21# and we shall be brief inwhat follows.
A. Unequal mass case
We shall first derive the fragmentation functions forheavy quarkQ into an unequal massD-wave (Qq) meson.The fragmentation ofQ* into aQq meson is given by the
-on
f a
ela
-r
een
a
process in Eq.~20!, of which the leading Feynman diagramis given in Fig. 3. In reality, the above fragmentation processapplies tob*→dbc1c (b*→dJ
bc1c) and also its chargeconjugateb*→dbc1 c (b*→dJ
bc1 c). The amplitudes thatare presented in the following are forQ*→(Qq)1q, whilethe amplitudes for the conjugate process can be obtained bcomplex conjugation. However, the final expressions for thefragmentation functions are valid for bothQ*→(Qq)1qandQ*→(Qq)1q fragmentation processes.
The amplitude A(P) for producing the spin-singletD-wave state is given in Eq.~10! with
]2M
]ka]kbUk50
given by
]2M
]ka]kbUk50
5Ni jDFu~p8!Vabg5G, ~22!
where
Ni j5gs2CF
d i jA3
1
4r r MAMwith CF5
4
3, ~23!
Vab5V1ab1~122r !V2
ab , ~24!
and
entation
V1ab5
32r
r 2MDF
3qaqb~P” 12M !~q”2 r M !116r
r 2MDF
2AFqaqbn” ~P” 2M !1
2
r 2DFAFn” ~qagb1qbga!P”
14r
rMDFAF
2~qanb1qbna!n” ~P” 2M !11
rAF2n” ~nagb1nbga!P” 14r MAF
3nanbn” ~P” 2M !, ~25!
V2ab52
4
r 2MDF
2~gaqb1gbqa!~q”2 r M !22
r 2MDFAFn” ~gaqb1gbqa!2
1
rMAF
2n” ~ganb1gbna!, ~26!
with
DF5~s2 r 2M2!21, ~27!
AF5@n•~q2 r P!#21. ~28!
In the above formulas,s5q2 and we have chosen the axial gauge associated with the four-vectornm5(1,0,0,21). In thisgauge, Fig. 3 is the only leading Feynman diagram where factorization between the short-distance process and fragmbecomes manifest. We square the amplitudeA(P) and sum over colors and helicities ofq and (Qq), and put them into Eq.~21!. The heavy quark fragmentation function for the1D2 state thus obtained is given by
3596 53KINGMAN CHEUNG AND TZU CHIANG YUAN
DQ→Qq~1D2!~z!5as2~2rM !uRD9 ~0!u2
324pM7
1
r 6r 2z~12z!2
~12 r z!10$60160~2718r24r 2!z15~2672600r1580r 22160r 3164r 4!z2
110~22551825r21130r 21596r 3288r 4264r 5!z31~3225213050r122585r 2216872r 315036r 4
2576r 511152r 6!z424r ~69022535r14200r 222279r 31968r 42524r 52160r 6!z51 r 2~154524890r
18525r 222176r 313076r 42320r 51320r 6!z622r 3~2552660r11475r 21174r 31740r 41120r 5!z7
15r 4~15230r1107r 2156r 3180r 4!z8%. ~29!
For the spin-triplet case, the amplitudeA(P) is given in Eq.~11! with
]2Mr
]ka]kbUk50
given by
]2Mr~k!
]ka]kbUk50
5Ni jDFu~p8!VabrG, ~30!
where
Vabr5V1abr1~122r !V2
abr ~31!
and
V1abr5
4
rDFF ~gbgra1gagrb!2
1
rDF$@qa~gbgr22r grb!1qb~gagr22r gra!#P” 12M ~qagrb1qbgra!%
18r
rM2DF
2qaqbgrG~q”1 r M !11
rAFn” F2~gagrb1gbgra!1AF$na@~gbgr22rgrb!P” 12rMgrb#
1nb@~gagr22rgra!P” 12rMgra#%12
rDF$qa@~gbgr22rgrb!P” 12rMgrb#1qb@~gagr22rgra!P” 12rMgra#%G
24r MAFF 4r 2qaqbDF21
1
rDFAF~qanb1qbna!1AF
2nanbGn” ~P” 2M !gr, ~32!
V2abr5
M
rAFn” F2r DF~qagb1qbga!1AF~nagb1nbga!Ggr. ~33!
We then square the amplitude and sum over the colors and helicities ofq and (Qq) using the polarization sum formulas in Eqs.~17!–~19!. The fragmentation functions thus obtained are
DQ→Qq~3D1!~z!5as2~2rM !uRD9 ~0!u2
3240pM7
1
r 6r 2z~12z!2
~12 r z!10$60~114r !2260~114r !~7136r228r 2!z15~26112652r16128r 2
212576r 317360r 4!z2110~223722589r28704r 2127328r 3228608r 4111360r 5!z31~2775131350r
1153935r 22637872r 31897816r 42611056r 51183552r 6!z424r ~54016585r148660r 22162629r 3
1199928r 42118084r 5128400r 6!z51 r 2~1095113830r1137935r 22376916r 31385696r 42192320r 5
136800r 6!z622r 3~16512100r126395r 2255836r 3148360r 4226720r 513360r 6!z7
15r 4~91114r11717r 222424r 312476r 421872r 51192r 6!z8%, ~34!
rmThe
53 3597HEAVY QUARK FRAGMENTATION FUNCTIONS FORD-WAVE . . .
DQ→Qq~3D2!~z!5as2~2rM !uRD9 ~0!u2
648pM7
1
r 6r 2z~12z!2
~12 r z!10$1801180~2714r !z15~7772852r1464r 2264r 31128r 4!z2
110~269311087r21126r 21624r 32112r 42128r 5!z31~7875215650r123295r 2222564r 319112r 4
1128r 512304r 6!z424r ~147022005r14740r 224613r 312296r 42488r 52320r 6!z51 r 2~283522050r
112035r 228332r 315952r 42960r 51640r 6!z622r 3~405240r12565r 22602r 311000r 42320r 5!z7
15r 4~21110r1205r 2152r 3184r 4!z8%, ~35!
DQ→Qq~3D3!~z!52as
2~2rM !uRD9 ~0!u2
405pM7
1
r 6z~12z!2
~12 r z!10$90190~2712r !z15~3992174r1200r 2!z2110~23781193r2437r 2
170r 3!z31~472522900r18390r 221918r 312268r 4!z412~2199511700r24930r 211617r 322387r 4
1350r 5!z51~220522830r17940r 225194r 314914r 42930r 511000r 6!z622r ~3602325r11680r 2
2749r 311540r 41295r 5190r 6!z715r 2~21214r1133r 2256r 31189r 4218r 5118r 6!z8%. ~36!
All the above fragmentation functions are valid for the fragmentation ofb→dbc anddJbc , as well as forb→dbc anddJ
bc, whereJ51,2,3.
For a given principal quantum numbern the 1D2 and3D2 states constructed in theLS coupling scheme are mixed in
general to form the physical states 21 and 218 defined as
u218&5cosuu1D2&2sinuu3D2&,
u21&5sinuu1D2&1cosuu3D2&, ~37!
whereu is the mixing angle. Thus, the fragmentation functions into the physical states 21 and 218 are
DQ→Qq~218!~z!5cos2uDQ→Qq~1D2!~z!1sin2uDQ→Qq~3D2!~z!2sinucosuDmix~z!,
DQ→Qq~21!~z!5sin2uDQ→Qq~1D2!~z!1cos2uDQ→Qq~3D2!~z!1sinucosuDmix~z!. ~38!
Therefore, we also need to calculate the fragmentation functionDmix(z). We note that in the case of charmonium it is thecharge-conjugation quantum numberC that prevents the mixing ofcc(1D2) and cc(3D2) states, becausecc(1D2) hasC511 while cc(3D2) hasC521. In the unequal mass case the meson does not have this quantum numberC to prevent themixing. The mixing fragmentation function is obtained by calculating the interference teA(1D2)A* (
3D2)1A* (1D2)A(3D2) and sum over the colors and helicities of the particles in the final state as before.
mixing fragmentation function thus obtained is
Dmix~z!525as
2~2rM !uRD9 ~0!u2
54A6pM7
1
r 6r 2z~12z!2
~12 r z!8$12112~2514r22r 2!z1~1292198r1156r 2164r 3!z2
14~239185r276r 218r 3236r 4!z312~572156r1133r 22130r 31198r 4148r 5!z4
24r ~12227r14r 2251r 322r 4214r 5!z51 r 2~9216r211r 2262r 324r 4!z6%. ~39!
The fragmentation probabilities for these states can be obtained by integrating the fragmentation functions overz:
E0
1
DQ→Qq~1D2!~z!dz510N
21r 7r 10@ r ~79313712r142160r 21123220r 31253315r 41126622r 519778r 611180r 7280r 8!
1315r ~11140r1280r 21580r 31701r 41152r 5116r 6!logr #, ~40!
E0
1
DQ→Qq~3D1!~z!dz5N
7r 7r 10@ r ~2091457r128224r 21479388r 311136191r 42358043r 52430066r 62113308r 7
112528r 8!1105r ~5160r11772r 218644r 314759r 425632r 522108r 62368r 7164r 8!logr #,
~41!
3598 53KINGMAN CHEUNG AND TZU CHIANG YUAN
E0
1
DQ→Qq~3D2!~z!dz55N
21r 7r 10@ r ~11272700r167880r 21320552r 31393317r 41191990r 51118742r 614712r 72160r 8!
1315r ~9144r1604r 211260r 31879r 41532r 51156r 6!logr #, ~42!
E0
1
DQ→Qq~3D3!~z!dz58N
21r 7r 10@ r ~219114828r1100786r 21160027r 31288799r 41428008r 5263854r 6187913r 7
114422r 8!1630r ~15120r1343r 21126r 31861r 41112r 5163r 6178r 716r 8!logr #, ~43!
E0
1
Dmix~z!dz5210A6N7r 7r 10
@ r ~1531358r13214r 2213796r 3215021r 417176r 519696r 61240r 7!1105r ~5110r220r 2
2180r 3237r 41110r 5136r 6!logr #, ~44!
whereN5as2(2rM )uRD
9 (0)u2/(3240pM7). We note that the running scale used in the strong coupling constantas is set at2rM , which is the minimal virtuality of the exchanged gluon@9#.
B. Equal mass case
Heavy quark fragmentation functions for theD-wave quarkonium can be obtained simply by settingr51/2 andu50 in theprevious formulas for the unequal mass case. For convenience we also present their explicit formulas in the following:
DQ→dQ~z!58as
2~2mQ!uRD9 ~0!u2
81pmQ7
z~12z!2
~22z!10~3840215360z130720z2237120z3135328z4
229344z5118344z625848z71775z8!, ~45!
DQ→d1Q~z!5
8as2~2mQ!uRD9 ~0!u2
405pmQ7
z~12z!2
~22z!10~172802103680z1321120z22551840z31546744z4
2314752z51112402z6224594z712915z8!, ~46!
DQ→d2Q~z!5
16as2~2mQ!uRD9 ~0!u2
81pmQ7
z~12z!2
~22z!10~2880214400z137360z2258240z3158604z4238372z5
116517z624014z71445z8!, ~47!
DQ→d3Q~z!5
8as2~2mQ!uRD9 ~0!u2
405pmQ7
z~12z!2
~22z!10~11520269120z1231680z22488960z31675136z4
2592288z51309688z6280736z718285z8!. ~48!
We can also obtain the fragmentation probabilities by integrating overz:
E0
1
dzDQ→dQ~z!58as
2~2mQ!uRD9 ~0!u2
81pmQ7 S 77667721
253355 ln2D , ~49!
E0
1
dzDQ→d1Q~z!5
2as2~2mQ!uRD9 ~0!u2
405pmQ7 ~5471272789300 ln2!, ~50!
E0
1
dzDQ→d2Q~z!5
16as2~2mQ!uRD9 ~0!u2
81pmQ7 S 2889265168
224810 ln2D , ~51!
E0
1
dzDQ→d3Q~z!5
8as2~2mQ!uRD9 ~0!u2
405pmQ7 S 518243221
2356025 ln2D . ~52!
ek
.
y
e
53 3599HEAVY QUARK FRAGMENTATION FUNCTIONS FORD-WAVE . . .
IV. DISCUSSION
We have seen that the covariant formalism used to callate theD-wave fragmentation functions is quite cumbesome and tedious. The complicated results are thereneeded to be cross-checked by other means. To be surehave recalculated theD-wave fragmentation functions forthe equal mass case from scratch and we did end up withsame results as what we have in the above forr51/2. Twoadditional nontrivial checks are performed by using tBraaten-Levin spin-counting rule@28# and the heavy quarkspin-counting rule which we will discuss in turns.
A. Braaten-Levin spin-counting rule
The fragmentation functionDi→H(z) for a partoni split-ting into a hadronH is related to the distribution functionf i /H(x) of finding the partoni inside the hadronH by theanalytic continuation@29,30#
f i /H~x!5xDi→HS 1xD . ~53!
The results of perturbative fragmentation functions allowto study the perturbative tail of the distribution functionsthe heavy quark inside the heavy mesons as well. Fromexplicit calculations, we see thatf i /H(x) has a pole located ax5 r . The pole is cut off by nonperturbative effects relatedthe formation of bound states of the (Qq) pair. This pole isof order 6, 8, and 10 forS-, P-, andD-wave states, respectively. In the general case ofL waves we expect this pole tobe of order 612L. Therefore, we can expandf (x) as aLaurent series:
f ~x!5an~r !
~x2 r !n1
an21~r !
~x2 r !n211 less singular terms,
~54!
with n5612L for the generalL waves.The Braaten-Levin rule states that the leadin
r -dependent coefficientsan(r ) satisfy the simple spin count-ing. Specifically, we have
a6~1S0!:a6~
3S1!51:3 for S waves, ~55!
a8~1P1!:a8~
3P0!:a8~3P1!:a8~
3P2!
53:1:3:5 forP waves, ~56!
a10~1D2!:a10~
3D1!:a10~3D2!:a10~
3D3!
55:3:5:7 forD waves, ~57!
and so on.The applicability of these counting rules has been dem
strated for theS- and P-wave cases in Ref.@11#. For theD-wave case we can expand the distribution functiowhich are obtained from theD-wave fragmentation functionsby Eq. ~53!, aroundx5 r , and obtain the first two terms inthe Laurent series:
cu-r-forewe
the
he
usofourtto
-
g
on-
ns,
f Q/dQq~x!530720N~r r !2
~x2 r !101320N~576r2275!r r
~x2 r !91•••,
~58!
f Q/d1Qq~x!5
18432N~r r !2
~x2 r !101192N~576r2275!r r
~x2 r !91•••,
~59!
f Q/d2Qq~x!5
30720N~r r !2
~x2 r !101320N~576r2275!r r
~x2 r !91•••,
~60!
f Q/d3Qq~x!5
43008N~r r !2
~x2 r !101448N~576r2275!r r
~x2 r !91•••,
~61!
whereN5as2uRD
9 (0)u2/(3240pM7). Therefore, the leadingcoefficients a10’s indeed satisfy the spin-counting ratio5:3:5:7.Actually, we found that not only the first term butalso the second term in the Laurent series off (x) obeys theBraaten-Levin spin-counting rules, while the third term doesnot. We conjecture that the Braaten-Levin spin-counting rulecan be applied to the first two coefficients in the expansion~54! for all L-wave (L 5 0, 1, 2, . . . )cases.
B. Heavy-light limit
Hadrons containing a single heavy quark exhibit heavyquark symmetry in the limitmQ /LQCD→`. In the limit ofmQ /LQCD→`, both the heavy quark spinSWQ and the totalspin JW of a heavy hadron containing a single heavy quarkQ become good quantum numbers. This implies that in thspectroscopy of the hadron containing a single heavy quarQ, the angular momentum of the light degrees of freedomJW l5JW2SWQ is also a good quantum number. We refer collec-tively to all the degrees of freedom in the heavy-light hadronother than the heavy quark as the light degrees of freedomFor heavy-light (Qq) mesons,JW l5SW q1LW whereSW q is thespin of the light quarkq andLW is the orbital angular momen-tum. Thus, hadronic states can be labeled simultaneously bthe eigenvaluesj and j l of the total spinJW and the angularmomentum of the light degrees of freedomJW l , respectively.In general@31#, the spectrum of hadrons containing a singleheavy quark has, for eachj l , a degenerate doublet with totalspins j15 j l11/2 and j25 j l21/2. ~For the case ofj l50,the total spin must be 1/2.) ForD-wave heavy-light mesons,j l can either be 3/2 or 5/2. Thus (j2 , j1)5(1,2) and~2,3!for j l 5 3/2 and 5/2, respectively. As a result, we expect tohave two distinct doublets (3D1,2
18) and (21,3D3) in thelimit of mQ /mq→`, i.e., r→0. In this limit, the mixingcoefficients in Eq.~37! can be determined by the Clebsch-Gordan coefficients in the tensor product of a spin-1/2 statand a spin-2 state with the result
u218&5A2
5u1D2&1A3
5u3D2&, ~62!
r
3600 53KINGMAN CHEUNG AND TZU CHIANG YUAN
u21&52A3
5u1D2&1A2
5u3D2&; ~63!
i.e., we are transforming the states1D2 and3D2 in the LS
coupling scheme to the states 21 and 218 in the j j couplingscheme.
In their discussions of the heavy quark fragmentatiofunctions within the context of heavy quark effective theorJaffe and Randall@32# showed that fragmentation functionscan have a 1/mQ expansion if expanded in terms of a mornatural variable
y5~1/z!2 r
r, ~64!
ny,
e
rather than the usual fragmentation variablez, and the heavyquark mass expansion is given as a power series inr :
D~y!51
ra~y!1b~y!1O~r !, ~65!
wherea(y),b(y), etc., are functions of the variabley. Theleading terma(y) is constrained by the heavy quark spin-flavor symmetry while all the higher order terms containspin-flavor symmetry-breaking effects. One can recast ouresults for theD-wave fragmentation functions derived inSec. III in the above form, by carefully expanding the pow-ers ofr and (12 r z). The leading 1/r terms of the fragmen-tation functions are then given by
DQ→dQq~y!→10N8~y21!2
ry10~307222656y12056y22912y31660y42120y5175y6!,
DQ→d1Qq~y!→
3N8~y21!2
ry10~614425312y11952y221824y311020y42120y5115y6!,
DQ→d2Qq~y!→
5N8~y21!2
ry10~614425312y13392y221824y311220y42200y51105y6!,
DQ→d3Qq~y!→
112N8~y21!2
ry10~3842332y1347y22114y3195y4220y5115y6!,
Dmix~y!→250A6N8~y21!2
ry10~144y2120y428y519y6!,
DQ→218~y!→5N8~y21!2
ry10~614425312y11952y221824y311020y42120y5115y6!,
DQ→21~y!→80N8~y21!2
ry10~3842332y1347y22114y3195y4220y5115y6!,
d-
ofch-eosg,
of
whereN85as2uRD
9 (0)u2/@3240p(rM )7#. Thus, at leading or-der of 1/mQ , we obtain the spin-counting ratios
DQ→d1Qq~y!
DQ→218~y!→
3
5, ~66!
DQ→21~y!
DQ→d3Qq~y!
→5
7, ~67!
as expected from heavy quark spin symmetry.
V. CONCLUSIONS
The detection of theD-wave orbitally excited states ofheavy-heavy mesons is much more difficult than theground states. So far, none ofD-wave quarkonium has beenidentified. The basic reasons are~i! the production rates are
ir
small, ~ii ! small branching ratios in subsequent decays, an~iii ! the very small efficiencies in subsequent levels of identification. From the potential model calculations,D-wavecharmonium is likely to be above theDD threshold andhence it will decay predominantly into a pair ofD mesons.In principle,D-wave quarkonium can be identified partiallyby its pure leptonic decay~e.g., 3D1→m1m2), or by itssubsequent decays into the lower-lying states. It is becausethe subsequent levels of decays that substantially the braning ratios and efficiencies are reduced in identifying thesD-wave bound states. However, both the branching ratiand efficiencies are uncertain at this stage. In the followinwe shall estimate the production rates ofD-wave charmo-nium, bottomonium, and (bc) mesons based upon the frag-mentation functions obtained in Secs. III and IV.
The values ofuRD9 (0)u2 for the (cc), (bb), and (bc) sys-
tems were summarized nicely in Ref.@33# using differentpotential models. We choose the QCD-motivated potential
53 3601HEAVY QUARK FRAGMENTATION FUNCTIONS FORD-WAVE . . .
Buchmuller and Tye@34#, and the corresponding values ouRD9 (0)u
2 for the (cc), (bb), and (bc) systems are 0.015,0.637, and 0.055 GeV7, respectively. These values are fothe first set of theD-wave states. We note that the values ouRD9 (0)u
2 obtained using the Buchmu¨ller-Tye potential arethe smallest ones of the four potential models studied in R@33#.
In the following analysis, we will adopt the valuesmc51.5 GeV andmb54.9 GeV that were used in the fit ofthe Buchmu¨ller-Tye potential to the bound state spectra@33#.Using as(2mc)'0.253, uRD9 (0)u
250.055 GeV7, r5mc /(mb1mc)50.234, and ignoring the possibly small mixingeffects for the (bc) system, we show theD-wave fragmen-tation functions for (bc) mesons in Fig. 4. The fragmentationprobabilities for ab antiquark intoD-wave (bc) mesons canbe obtained from Eqs.~40!–~43!:
E0
1
Db→dbc~z!dz56.731026, ~68!
E0
1
Db→d1bc~z!dz51.731026, ~69!
E0
1
Db→d2bc~z!dz56.531026, ~70!
E0
1
Db→d3bc~z!dz58.531026. ~71!
Therefore, the total probability is about 2.331025. We cancompare this value with the corresponding probabilitiesthe S-wave andP-wave states. Using the sameas and thevalues of uRS(0)u2 and uRP8 (0)u
2 calculated with the sameBuchmuller-Tye potential, the total fragmentation probabilities of the 1S and 1P (bc) states are about 131023 and1.731024, respectively @11#. These total fragmentationprobabilities show a similar suppression factor of orderv2 in
FIG. 4. The fragmentation functions forb to fragment intoD-wave (bc) mesons at the heavy quark mass scale.
f
rf
ef.
of
-
going from 1S to 1P states and from 1P to 1D states, whichare in accordance with the velocity counting rules inNRQCD @4#.
Next, for D-wave charmonium we useas(2mc)'0.253and uRD9 (0)u
250.015 GeV7. The D-wave fragmentationfunctions for charmonium are shown in Fig. 5, and the frag-mentation probabilities are given by
E0
1
Dc→dc~z!dz53.131026, ~72!
E0
1
Dc→d1c~z!dz52.331026, ~73!
E0
1
Dc→d2c~z!dz53.631026, ~74!
E0
1
Dc→d3c~z!dz51.731026. ~75!
The total probability is about 1.131025, to be comparedwith the probabilities ofP(c→J/c)'1.831024 @9# andP(c→hc ,xc)'831025 @11# obtained using the same in-puts. With our inputs, the gluon fragmentation probabilityinto 1D2 charmonium@8# is about 1.231026.
For bottomonium, we use as(2mb)'0.174 anduRD
9 (0)u250.637 GeV7. The shape of the fragmentationfunctions forD-wave bottomonium is identical to those ofcharmonium in Fig. 5, but the overall normalization is afactor of 200 smaller. The fragmentation probabilities aregiven by
E0
1
Db→db~z!dz51.631028, ~76!
E0
1
Db→d1b~z!dz51.231028, ~77!
FIG. 5. The fragmentation functions forc to fragment intoD-wave charmonium at the heavy quark mass scale.
,
e
-
-
n
eel-nis.e
t-
3602 53KINGMAN CHEUNG AND TZU CHIANG YUAN
E0
1
Db→d2b~z!dz51.831028, ~78!
E0
1
Db→d3b~z!dz58.531029. ~79!
The total probability forD-wave bottomonium is very small,;5.431028. This is to be compared with the probabilitieof P(b→Y)'231025 @9# andP(b→hb ,xb)'2.531026.We observe that the relative probabilities for theS-wave toP-wave quarkonium and for theP-wave toD-wave quarko-nium are roughly suppressed by a similar factor ofv2 as inthe bc case.
At the Tevatron theb-quark production cross section is oorder 10mb with pT*6 GeV, which implies about 109 bquarks with an accumulated luminosity of 100 pb21. Usingthe fragmentation probabilities calculated above, we expabout 104 D-wave (bc) mesons, while there are only abou50D-wave bottomonium. At largepT , charm quark produc-tion is very similar to bottom quark production. Thereforethe c-quark production cross section is also of order 10mband there should be about 104 D-wave charmonium for thesame luminosity. Although the production rates ofD-wave(bc) mesons and charmonium are rather substantial,small branching ratios and efficiencies of identificationsubsequent levels of decays render their detections extremdifficult. For example, the combined branching ratiB„1D2(cc)→1P1g,
1P1→J/cp0,J/c→m1m2… has been
estimated to be about 1024 @8#. Even after the installation ofthe Main Injector when the yearly luminosity can be boosteto 1–3 fb21, the chance of detecting theD-wave (bc) me-sons or charmonium is still quite slim. The future CERN
s
f
ectt
,
theinelyo
d
Large Hadron Collider~LHC! can produce 1012–1013 bbpairs andcc pairs per year running. It can then produceabout 107–108 D-wave (bc) mesons and charmonium. Aftertaking into account the branching ratios and efficienciesthere should be a handful number of theseD-wave states thatcan be identified. Nevertheless, detection ofD-wave botto-monium does not seem to be feasible even at the LHC. Thproduction ofbb andcc pairs at the CERNe1e2 collidersLEP, LEP II, and other higher energye1e2 or m1m2 col-liders is much smaller than at the hadronic colliders. Therefore, it is much more difficult to identifyD-wave quarko-nium or (bc) mesons at thee1e2 or m1m2 facilities.
In conclusion, we have computed the heavy quark fragmentation functions intoD-wave heavy-heavy mesons con-taining two heavy quarks to leading order in strong couplingconstant and to leading order in the nonrelativistic expansiowithin the framework of factorization model of Bodwin,Braaten, and Lepage. The color-singlet contributions can bexpressed in terms of the NRQCD matrix elements which arrelated to the second derivative of the nonrelativistic radiaD-wave wave function. The color-octet contributions are expected to be quite small in the heavy quark fragmentatioand therefore we have ignored them in the present analysThe fragmentation functions obtained in this work should buseful in the production of theD-wave charmonium, botto-monium, and (bc) mesons at the future Large Hadronic Col-lider.
ACKNOWLEDGMENTS
This work was supported in part by the U.S. Departmenof Energy under Grants Nos. DE-FG03-93ER40757 and DEFG03-91ER40674.
@1# R. Baier and R. Ru¨ckl, Z. Phys. C19, 251 ~1983!; E.W.N.Glover, A.D. Martin, and W.J. Stirling,ibid. 38, 473 ~1988!.
@2# CDF Collaboration, F. Abeet al., Phys. Rev. Lett.69, 3704~1992!; 71, 2537~1993!.
@3# E. Braaten and T.C. Yuan, Phys. Rev. Lett.71, 1673~1993!.@4# G.T. Bodwin, E. Braaten, and G.P. Lepage, Phys. Rev. D51,
1125 ~1995!.@5# See for example G.A. Schuler, Report No. CERN-TH.7170/9
hep-ph/9403387~unpublished!, and references therein.@6# E. Braaten and T.C. Yuan, Phys. Rev. D50, 3176~1994!.@7# J.P. Ma, Phys. Lett. B332, 398~1994!; Nucl. Phys.B447, 405
~1995!.@8# P. Cho and M. Wise, Phys. Rev. D51, 3352~1995!.@9# E. Braaten, K. Cheung, and T.C. Yuan, Phys. Rev. D48, 4230
~1993!.@10# C.-H. Chang and Y.-Q. Chen, Phys. Lett. B284, 127 ~1992!;
Phys. Rev. D46, 3845~1992!; 50, 6013~E! ~1994!.@11# T.C. Yuan, Phys. Rev. D50, 5664~1994!.@12# Y.-Q. Chen, Phys. Rev. D48, 5181 ~1994!; 50, 6013~E!
~1995!.@13# J.P. Ma, Phys. Rev. D53, 1185~1996!.@14# E. Braaten, M.A. Doncheski, S. Fleming, and M.L. Mangan
Phys. Lett. B333, 548 ~1994!; M. Cacciari and M. Greco,
4,
o,
Phys. Rev. Lett.73, 1586 ~1994!; D.P. Roy and K. Sridhar,Phys. Lett. B339, 141 ~1994!.
@15# E. Braaten and S. Fleming, Phys. Rev. Lett.74, 3327~1995!;M. Cacciari, M. Greco, M.L. Mangano, and A. Petrelli, Phys.Lett. B 356, 553 ~1995!; P. Cho and A.K. Leibovich, Phys.Rev. D53, 150 ~1996!.
@16# M. Mangano, presented at the Xth Topical Workshop onppCollisions, Batavia, Illinois, 1995~unpublished!, Report No.hep-ph/9507353~unpublished!; CDF Collaboration, G. Baueribid.; D0 Collaboration, K. Bazizi~unpublished!.
@17# E. Braaten, presented at the Symposium on Particle Theoryand Phenomenology at Iowa State University, 1995, North-western Report No. NUHEP-TH-95-11, hep-ph/9509210~un-published!.
@18# CDF Collaboration, F. Abeet al., FERMILAB-CONF-95-202-E, 1995~unpublished!.
@19# K. Cheung, Phys. Rev. Lett.71, 3413~1993!.@20# K. Cheung and T.C. Yuan, Phys. Lett. B325, 481~1994!; Phys.
Rev. D53, 1232~1996!.@21# E. Braaten, K. Cheung, and T.C. Yuan, Phys. Rev. D48, R5049
~1993!.@22# C.-H. Chang and Y.-Q. Chen, Phys. Rev. D48, 4086 ~1993!;
C.-H. Chang, Y.-Q. Chen, G.-P. Han, and H.-T. Jiang, Phys.
53 3603HEAVY QUARK FRAGMENTATION FUNCTIONS FORD-WAVE . . .
Lett. B 364, 78 ~1995!; S.R. Slabospitsky, Phys. At. Nucl.58,988 ~1995!; A.V. Berezhnoy, A.K. Likhoded, and M.V. Shev-lyagin, ibid. 58, 672 ~1995!; A.V. Berezhnoy, A.K. Likhoded,and O.P. Yushchenko, Serpukhov Report No. IFVE-95-59hep-ph/9504302~unpublished!; M. Masetti and F. Sartogo,Phys. Lett. B357, 659 ~1995!.
@23# K. Kolodziej, A. Leike, and R. Ruckl, Phys. Lett. B355, 337~1995!.
@24# P. Cho and M. Wise, Phys. Lett. B346, 129 ~1995!.@25# G.T. Bodwin, E. Braaten, and G.P. Lepage, Phys. Rev. D46,
1914 ~1992!.@26# B. Guberina, J.H. Ku¨hn, R.D. Peccei, and R. Ruckl, Nucl.
Phys.B174, 317 ~1980!.
,
@27# L. Bergstrom, H. Grotch, and R.W. Robinett, Phys. Rev. D43,2157 ~1991!.
@28# E. Braaten~private communication!.@29# S.D. Drell, P. Levy, and T.M. Yan, Phys. Rev.187, 2159
~1969!.@30# L.N. Lipatov, Sov. J. Nucl. Phys.20, 94 ~1975!; A.P. Buhvos-
tov, L.N. Lipatov, and N.P. Popov,ibid. 20, 287 ~1975!; V.N.Gribov and L.N. Lipatov,ibid. 15, 438~1972!; 15, 675~1972!.
@31# N. Isgur and M. Wise, Phys. Rev. Lett.66, 1130~1991!.@32# R.L. Jaffe and L. Randall, Nucl. Phys.B412, 79 ~1994!.@33# E. Eichten and C. Quigg, Phys. Rev. D52, 1726~1995!.@34# W. Buchmuller and S. Tye, Phys. Rev. D24, 132 ~1981!.
