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JPsi Production from Electromagnetic Fragmentation in Z decay


FERMILAB-PUB-94/074-T NUHEP-TH-94-6 March 1994

J/ψ Production from Electromagnetic Fragmentation

arXiv:hep-ph/9403396v1 30 Mar 1994

in Z 0 decay
Sean Fleming Department of Physics and Astronomy Northwestern University,Evanston, IL 60208 and Fermi National Accelerator Laboratory P.O. Box 500, Batavia, IL 60510

Abstract The rate for Z 0 → J/ψ + ?+ ?? is suprisingly large with about one event for every million Z 0 decays. The reason for this is that there is a fragmentation contribution
2 2 that is not suppressed by a factor of Mψ /MZ . In the fragmentation limit MZ → ∞

with Eψ /MZ ?xed, the di?erential decay rate for Z 0 → J/ψ + ?+ ?? factors into electromagnetic decay rates and universal fragmentation functions. The fragmentation functions for lepton fragmentation and photon fragmentation into J/ψ are calculated to lowest order in α. The fragmentation approximation to the rate is shown to match the full calculation for Eψ greater than about 3Mψ .

Introduction Fragmentation is the decay of a high transverse momentum parton into a collinear hadron. The di?erential cross section for the inclusive production of such a hadron in e+ e? annihilation factors into di?erential cross sections d? for the production of large transverse σ momentum partons and fragmentation functions D(z) [1]. The fragmentation function gives the probability for the splitting of a parton into the hadron with momentum fraction z. These functions are independent of the subprocess that creates the fragmenting particle, and can be evolved to any scale via the Altarelli-Parisi evolution equations. It has recently been shown that it is possible to calculate the fragmentation functions for heavy quarkonium states using perturbative quantum chromodynamics (QCD) [2]. Fragmentation functions for several of these states have been calculated explicitly [2, 3, 4, 5, 6]. In particular the paper of Braaten, Chueng and Yuan on charm quark fragmentation [3] is most relevant to the work presented here. Their analysis focuses on decays of the Z 0 into hadrons showing that charmonium production is dominated by charm quark fragmentation. In an analagous manner it is shown here that decays of the Z 0 into charmonium plus electromagnetic particles (leptons and photons) are dominated by electromagnetic fragmentation. The process Z 0 → ψ + ?+ ?? , which is of order α2 , where α is the electromagnetic coupling constant, has a branching ratio of 7.5 × 10?7. This is an order of magnitude larger than the order-α process Z 0 → ψγ [7], which has a branching ratio of 5.2 × 10?8 . That makes Z 0 → ψ + ?+ ?? the dominant ψ production mechanism in electromagnetic Z 0 decays. The unexpectedly large rate can be explained by a fragmentation contribution which is not
2 2 suppressed by a factor of Mψ /MZ [2]. In this paper, it is shown that in the fragmentation

limit MZ → ∞ with Eψ /MZ ?xed, the rate for Z 0 → ψ + ?+ ?? factors into subprocess rates for electromagnetic decays and individual fragmentation functions . At leading order in α there is a contribution from the fragmentation function D?→ψ for a lepton to split into a ψ, and a contribution from the fragmentation function Dγ→ψ for a photon to split into a ψ. The fragmentation functions are calculated in a manner that is independent of the hard process that produces the fragmenting parton, and the fragmentation calculation is shown to match the full calculation for Eψ greater than about 3Mψ .

1

The decay rate for Z 0 → ψ + ?+ ?? The decay rate for Z 0 → ψ + ?+ ?? was calculated in a model-independent way in Ref. [8, 9] using the Feynman diagrams in ?gure 1a. The diagrams in ?gure 1b also contribute to this process at the same order in the electromagnetic coupling α. Fortunately they can be neglected. In order to understand why only the diagrams of ?gure 1a need to be considered it is necessary to understand why the rate for Z 0 → ψγ is smaller than the rate for Z 0 → ψ+?+ ?? . rate is suppressed by a power of α compared to the former rate. However upon closer examination this expectation turns out to be untrue. To see why compare the diagrams that contribute to the decay rates Γ(Z 0 → ψ + ?+ ?? ) ?gure 1 and Γ(Z 0 → ψγ) ?gure 2. It is important to note that the lepton propagator in the diagrams of ?gure 2 is always of
2 order 1/MZ . In contrast there is a substantial region of phase space where both the photon 2 propagator and the lepton propagator in the diagrams of ?gure 1a are of order 1/Mψ . Thus 2 2 these diagrams will be enhanced by a factor of MZ /Mψ , compared to the diagrams of ?gure 2 2 2. The factor of MZ /Mψ is large enough to overwhelm the extra power of α making the

Naively one would expect Γ(Z 0 → ψγ) to be larger than Γ(Z 0 → ψ + ?+ ?? ) since the latter

diagrams of ?gure 1a more important than the diagrams of ?gure 2. The lepton propagator
2 of the diagrams in ?gure 1b is always of order 1/MZ , so these diagrams are suppressed by a

factor of α compared to the diagrams in ?gure 2, and may be neglected. The remaining diagrams that contribute to Γ(Z 0 → ψ + ?+ ?? ) at the same order in α are obtained from the diagrams of ?gure 1 by replacing the photon propagator with a
2 Z-boson. Then the boson propagator in the diagrams of ?gure 1a is always of order 1/MZ . 2 Similarly the lepton propagator in the diagrams of ?gure 1b is also always of order 1/MZ .

Thus these diagrams are suppressed by a factor of α compared to the diagrams of ?gure 2 and can be neglected. Keeping only the diagrams in ?gure 1a, and neglecting the lepton mass, the result of

2

the calculation of Γ(Z 0 → ψ + ?+ ?? ) is
2 Γ = 4α2 gψ Γ(Z 0 → ?+ ?? ) 1+λ √ 2 λ

dy


?



y 2 ?4λ y 2 ?4λ

dw g(y, w, λ)

g(y, w, λ) =

1 2

(y ? 2)2 + w 2 y 2 ? w2

? 2λ

1 2w 2 ? (y 2 ? w 2 )(1 ? y) + 2λ2 2 2 ? w 2 )2 (y y ? w2

(1)

2 2 2 2 where λ = Mψ /MZ , y = 2P · Z/MZ and w = 2(p+ ? p? ) · Z/MZ . Here P , p? , p+ , and Z

are the 4-momenta of the ψ, ?? , ?+ , and Z 0 . The parameter gψ can be determined from the electronic width Γe+ e? of the ψ to be
2 gψ =

3 Γe + e ? . 4π α2 Mψ

(2)

2 Using Γe+ e? = 5.4 keV the photon-to-ψ coupling is gψ = 0.008. Integrating the function

g(y, w, λ) over w yields the full di?erential decay rate dΓ = dEψ
2 8α2gψ

Γ(Z 0 → ?+ ?? ) MZ

y + yL (y ? 1)2 + 1 3 ? 2y 2 log + λ + λ2 ? 2yL , (3) y y y y ? yL

where yL =



y 2 ? 4λ. In the center of mass frame y = 2Eψ /MZ . Later on these results will

be compared to the fragmentation calculation in the fragmentation limit MZ → ∞ with Eψ /MZ ?xed. In this limit Eq. (3) reduces to
2 0 + ? y2 dΓ 2 Γ(Z → ? ? ) (y ? 1) + 1 = 8α2 gψ log ? 2y . dEψ MZ y λ

(4)

The Fragmentation Contribution to Z 0 Decay In reference [3] the general form of the fragmentation contribution for the production of a ψ of energy Eψ in Z 0 decays is given as dΓ(Z 0 → ψ(Eψ ) + X) = dz dΓ(Z 0 → i(Eψ /z) + X, ?2 ) Di→ψ (z, ?2 ), 3 (5)

i

where the sum is over partons and z is the longitudinal momentum fraction of the ψ relative to the fragmenting parton. All of the dependence on the ψ energy Eψ has been factored into the subprocess decay rate Γ and all of the dependence on the ψ mass Mψ has been factored into the fragmentation function Di→ψ (z, ?2 ). A factorization scale ? has to be introduced in order to maintain this factored form in all orders of perturbation theory. This general form was developed in the context of QCD, but it applies equally to QED, where the only partons are leptons and photons. This simpli?es the general electromagnetic fragmentation contribution to dΓ (Z 0 → ψ(Eψ ) + X) = dEψ 2
1 0

dz

dE? dEγ

dΓ 0 (Z → ?? (E? ) + X, ?2 ) D?→ψ (z, ?2 ) δ(Eψ ? zE? ) dE? dΓ (Z 0 → γ(Eγ ) + X, ?2 ) Dγ→ψ (z, ?2 ) δ(Eψ ? zEγ ), dEγ (6)

+

1 0

dz

where X are electromagnetic ?nal states, and E? and Eγ are the lepton and photon energies. The factor of 2 accounts for the fragmentation contribution from both the ?? and the ?+ . Large logarithms of Eψ /? in the subprocess decay rate Γ can be avoided by choosing ? on the order of Eψ . The large logarithms of order Eψ /Mψ which then appear in the fragmentation functions can be summed up by solving the Altarelli-Parisi evolution equation. In the electromagnetic case the evolution of the fragmentation functions is of order α and may be neglected. It is easy to count the order of α for the leading order fragmentation contributions to Eq. (6). The subprocess rate for Z 0 → ?? + X is of order 1, while the subprocess rate be shown to be of order α2 , while the fragmentation function for a photon to split into a ψ for Z 0 → γ + X is of order α. The fragmentation function for a lepton to split into a ψ will

will be shown to be of order α. Therefore both fragmentation processes will contribute to Eq. (6) at leading order in α. At lowest order in α it is possible to simplify things. The energy distribution for the

4

subprocess Z 0 → ?? (E? ) + X at lowest order is MZ dΓ = Γ(Z 0 → ?+ ?? ) δ(E? ? ). dE? 2 Furthermore the photon fragmentation function at lowest order can be written as Dγ→ψ (z, ?2 ) = Pγ→ψ δ(z ? 1) (8) (7)

where the function Pγ→ψ is the probability for a photon to split into a ψ, and z is the longitudinal momentum fraction of the ψ relative to the γ. These simpli?cations reduce the fragmentation contribution to the energy distribution Eq. (6) at leading order in α to: dΓ (Z 0 → ψ(Eψ ) + ?+ ?? ) = dEψ 2Eψ 2 dΓ 4 Γ(Z 0 → ?+ ?? ) D?→ψ ( ,? ) + (Z 0 → γ(Eψ ) + ?+ ?? , ?2 ) Pγ→ψ . MZ MZ dEψ (9)

The physical interpretation of the ?rst term on the right hand side of Eq. (9) is that the Z 0 decays into two leptons, with energies MZ /2 on a distance scale of order 1/MZ . Subsequently one of the leptons decays into a collinear lepton and ψ on a distance scale of order 1/Mψ . The physical interpretation of the second term is that the Z 0 decays into two leptons and a photon with energy Eψ on a distance scale of order 1/MZ , and the photon fragments into a ψ on a distance scale of order 1/Mψ . Note that at this order the only dependence on the factorization scale is in D?→ψ and in the subprocess rate for Z 0 → γ + ?+ ?? . Given the general form of the fragmentation contribution at lowest order in α in Eq. (9), it is only necessary to calculate the fragmentation function D?→ψ (z, ?2 ), the fragmentation probability Pγ→ψ , and the subprocess rate for the Z 0 → γ + ?+ ?? . Photon Fragmentation The fragmentation function Dγ→ψ (z, ?2 ) for a photon to split into a ψ can be calculated in a manner that is independent of the process that produces the fragmenting photon. The Feynman diagram in ?gure 3a represent such a process at lowest order in α. An unknown vertex, represented by the circle, radiates a photon which fragments into a ψ. The 5

fragmentation probability Pγ→ψ can be isolated by dividing the cross section σ1 , for the production of a ψ with energy Eψ , by the cross section σ0 , for the production of a real photon of energy Eγ = Eψ , in the limit Eψ ? Mψ where fragmentation dominates. The general form of the photon production cross section σ0 is σ0 = 1 F lux [dk][dpout ] (2π)4 δ 4 (pin ? k ? pout ) |A0 |2 (10)

where pin is the sum of incoming 4-momenta, k is the photon 4-momentum, and pout is the sum of the remaining outgoing 4-momenta. Here [dk] = d3 k/(16π 3 k0 ) is the Lorentzinvariant phase space for the photon and [dpout ] is the Lorentz invariant phase space for the remaining outgoing particles, F lux denotes the incoming particle ?ux for which no explicit expression is needed since it will cancel the same factor when σ1 is divided by σ0 . The amplitude A0 for the process can be calculated from the Feynman diagram in ?gure 3b A0 = Γ? ?? , (11)

where Γ? is a vertex factor for the production of the photon, for which the explicit form is not needed. Squaring and summing over ?nal spins gives |A0 |2 = ?Γ? Γ? . ? The general form of the γ → ψ cross section σ1 is σ1 = 1 F lux [dP ][dpout ] (2π)4 δ 4 (pin ? P ? pout ) |A1 |2 (13) (12)

where P is the ψ 4-momentum. F lux, pin , and pout are the same as described after Eq. (10). The amplitude A1 can be calculated from the Feynman diagram in ?gure 3a A1 = gψ e ? Γ (?g?ν ) tr[ ?γ ν ], 4 (14)

where ? is the ψ polarization vector, and Γ? is the vertex factor for the production of a virtual photon with invariant mass Mψ . Squaring the amplitude and summing over ?nal spins gives
2 |A1 |2 = ?4πgψ α Γ? Γ? ?

(15)

6

The vertex factor Γ? in Eq. (14) only di?ers from the vertex factor in Eq. (11) in one respect,
2 it is evaluated at the point k 2 = Mψ instead of k 2 = 0. Aside from the vertex factor the

ψ mass enters the cross section σ1 in the phase space integral. The contribution to photon fragmentation will come from a region of phase space where Eψ ? Mψ so that the ψ mass can be neglected. Then the cross section σ1 can be written as
2 σ1 = 4πgψ α σ0 .

(16)

The fragmentation probability Pγ→ψ can be read o? from Eq.(16):
2 Pγ→ψ = 4πgψ α.

(17)

Evaluating this numerically gives a fragmentation probability of Pγ→ψ = 7 × 10?4 . Photon Subprocess The energy distribution of the subprocess Z 0 → γ + ?+ ?? is calculated next. At

this point it is necessary to decide what part of the Z 0 → ψ + ?+ ?? phase space is to be identi?ed as photon fragmentation, and what part is interpreted as lepton fragmentation. In Eq. (9), the dependence on the factorization scale ? cancels between D?→ψ and dΓ/dEψ . Thus by changing ? some of the lepton fragmentation contribution can be moved into the photon fragmentation term and vice versa. There is therefore no clear crossover from lepton fragmentation to photon fragmentation, which makes it necessary to make an arbitrary choice on the appropriate phase space cuto?. In this paper a cuto? on the invariant mass of the ? ? ψ system, where ? is the fragmenting lepton, is introduced. The contribution to the di?erential decay rate from negative lepton fragmentation is considered to come from the region of phase space where s < ?2 . Here s is the invariant mass of the ?? ? ψ system. Similarly the contribution from positive lepton fragmentation is considered to come from the region s′ < ?2 where s′ is the invariant mass of the ?+ ?ψ system. The photon fragmentation contribution is interpreted as coming from the remaining region. In the calculation of the photon subprocess the invariant-mass cuto?s translate into a limit on the phase space of the photon energy distribution.

7

The decay rate for Z 0 → γ + ?+ ?? is Γ(Z 0 → γ + ?+ ?? ) = 1 2MZ [dk][dp+ ][dp? ] (2π)4 δ 4 (Z ? k ? p+ ? p? ) 1 3 |A|2, (18)

where k, p+ , p? and Z are the 4-momenta of the photon, ?+ , ?? , and Z 0 . The amplitude can be calculated from the diagrams in ?gure 4. Averaging over initial spins and summing over ?nal spins reduces the square of the amplitude to 1 3 |A|2 =
2 2 2 gw e2 2 2 (y ? 2) + w (CV + CA ) 6 cos2 θw y 2 ? w2

(19)

2 2 where y = 2k · Z/MZ , and w = 2(p+ ? p? ) · Z/MZ , CV = ?1 + 4 sin2 θw and CA = 1, gw

is the weak coupling constant, and θw is the weak mixing angle. Note that aside from the normalization this is the same as the integrand of Eq. (1) in the limit Mψ → 0. Simplifying the phase space integral, Eq. (18) reduces to MZ Γ(Z → γ + ? ? ) = 32(2π)3
0 + ? 1
2 2?2 /MZ 2 (y?2?2 /MZ ) 2 ?(y?2?2 /MZ )

dy

dw

1 3

|A|2 ,

(20)

where the limits on the invariant masses s and s′ translate into the limits on w. Integrating over w gives the photon energy distribution in the decay Z 0 → γ + ?+ ?? : dΓ (Z 0 → γ(Eγ ) + ?+ ?? , ?2 ) = dEγ 2α Γ(Z 0 → ?+ ?? ) π MZ
2 (y ? 1)2 + 1 y ? ?2 /MZ log 2 y ?2 /MZ

? y + 2

?2 2 MZ

θ(y ?

2?2 ),(21) 2 MZ

where y = 2Eγ /MZ . It is possible to simplify the expression for the energy distribution by taking the limit ? ? MZ . In this limit Eq. (21) simpli?es to dΓ 2α Γ(Z 0 → ?+ ?? ) (Z 0 → γ(Eγ ) + ?+ ?? , ?2) = dEγ π MZ yM 2 (y ? 1)2 + 1 log 2Z ? y . y ? (22)

2 The price that is paid for this simpli?cation is that smooth threshold behavior at y = 2?2 /MZ

is lost, and the di?erential decay rate becomes negative at su?ciently small values of y. Since 8

the fragmentation approximation breaks down in the threshold region anyway nothing is lost by making this simpli?cation. Lepton Fragmentation The calculation of the fragmentation function D?→ψ (z, ?2 ) for a lepton to split into a ψ parallels that of the photon fragmentation function. At leading order in α the process is symbolically represented by the diagram in ?gure 5a. The fragmentation probability is obtained by dividing σ1 , the cross section for the production of ψ + ?? , by σ0 , the cross section for lepton production shown in the diagram of ?gure 5b, in the limit Eψ ? Mψ where fragmentation dominates. The general form of the cross section σ0 for lepton production is σ0 = 1 F lux [dq][dpout ] (2π)4 δ 4 (pin ? q ? pout ) |A0 |2 , (23)

where q, pin , and pout are the 4-momenta of the ?? , the incoming particles, and all other outgoing particles. Just as in the photon fragmentation calculation, [dq] and [dpout ] are the Lorentz invariant phase space for the lepton and the remaining outgoing particles, and F lux represents the incoming particle ?ux (which will cancel with the same quantity in the cross section σ1 ). The square of the amplitude A0 calculated from the Feynman diagram in ?gure 4b for ?? production, averaged over initial spins and summed over ?nal spins, is ? |A0 |2 = tr[ q ΓΓ] (24)

where the Dirac matrix Γ is the matrix element for the production of a real lepton of momentum q for which the explicit form is not needed. The lepton mass m? has been neglected since its 4-momentum q is taken to be large compared to m? . The general form of the cross section σ1 for the production of a lepton that subsequently fragments into a ψ is 1 F lux [dP ][dp? ][dpout ] (2π)4 δ 4 (pin ? P ? p? ? pout ) |A1 |2 , (25)

where P is the ψ 4-momentum, and everything else is as described after Eq. (23). The next step is to write the phase space in an iterated form, by introducing integrals over q = P + p? 9

the virtual lepton momentum, and s = q 2 the invariant mass of the ψ ? ?? system. Then the phase space expression becomes [dP ][dp? ] (2π)4 δ 4 (pin ? P ? p? ? pout ) = ds 2π [dq] (2π)4 δ 4 (pin ? q ? pout ) [dP ][dp? ] (2π)4 δ 4 (q ? P ? p? ) . (26)

The contribution that corresponds to the fragmentation of the lepton in the diagram of ?gure 5a comes from the region of phase space in which the ψ ? ?? system has large momentum q

2 compared to the ψ mass and small invariant mass s = q 2 of order Mψ . In a frame in which

the virtual lepton has a 4-momentum q = (q0 , 0, 0, q3), the longitudinal momentum fraction of the ψ relative to the ψ ??? system is z = (P0 + P3 )/(q0 + q3 ) and its transverse momentum is P⊥ = (P1 , P2 ). Expressing the phase space in terms of these variables and integrating over the 4-momentum p? and over P⊥ , the 2-body phase space reduces to [3] [dP ][dp? ] (2π)4 δ 4 (q ? P ? p? ) = 1 8π
1 0

dz θ s ?

2 Mψ . z

(27)

The lepton mass has been set to zero. An upper limit on the integral over the invariant mass s is introduced by requiring s < ?2 , as discussed earlier. The calculation of the lepton framentation function is simplest to do in the axial gauge, because only the diagram of ?gure 5a, where the lepton is produced and splits into a collinear lepton and ψ, needs to be considered. Other diagrams where both the lepton and ψ are produced separately from the vertex Γ are suppressed in this gauge. If the calculation were done in some other gauge these diagrams would need to be considered, but the resulting expression could be manipulated into the form below using Ward identities. The 4-vector N associated with axial gauge is chosen to be N = (1, 0, 0, ?1). The amplitude A1 calculated in this gauge can be reduced to A1 = e2 gψ ?β (P )? 1 P αN β + P β N α . (?(p? ) γα q Γ) g αβ ? u s P ·N

(28)

where ? is the ψ polarization vector, and Γ is the Dirac matrix element for the production 10

of a virtual lepton with an invariant mass s on the order of the ψ mass. The explicit form of Γ is not needed.
? The ψ 4-momentum can be written as P ? = zq ? + P⊥ + (P0 q3 ? q0 P3 )/(q0 + q3 )N ? . In ? the fragmentation region P⊥ = (0, P⊥ , 0), and P0 q3 ? q0 P3 are of order Mψ while the virtual

lepton momentum q is large compared to Mψ so P ? zq. Note that in the fragmentation
2 region both s = q 2 and P · q are of order Mψ . Using these approximations and keeping only

the leading order terms in q/Mψ simpli?es the square of the amplitude to
2 |A1 |2 = 32π 2 α2 gψ 2 Mψ (z ? 1)2 + 1 1 ? ? 2 tr[ q ΓΓ]. z s s

(29)

Integrating over s up to the scale ?2 the lepton fragmentation probability is obtained by dividing the cross section σ1 by the cross section σ0 . The di?erences between σ1 and σ0 are
2 due to the fact that q 2 ? Mψ in σ1 , while q 2 = 0 in σ0 . These di?erences are on the order of 2 2 Mψ /Eψ and in the fragmentation limit where Eψ ? Mψ they can be neglected. The result

is σ1 = σ0
?2 0

ds

1 0

dz θ(s ?

2 M2 Mψ (z ? 1)2 + 1 1 2 ) 2α2 gψ ? 2ψ . z z s s

(30)

from this it is possible to extract the lepton fragmentation function
2 2 2α2 gψ 2 Mψ z?2 (z ? 1)2 + 1 log 2 ? z + 2 z Mψ ? 2 θ(z?2 ? Mψ )

D?→ψ (z, ? ) =

(31)

Note that the lepton fragmentation function is zero for values of z at which the production of a ψ is kinematically forbidden. Taking the limit ? ? Mψ simpli?es Eq. (31) to
2 D?→ψ (z, ?2 ) ≈ 2α2 gψ

(z ? 1)2 + 1 z?2 log 2 ? z . z Mψ

(32)

Just as in the calculation of the photon subprocess there is a price to be paid for this
2 simpli?cation. Eq. (32) does not have the correct threshold behavior at z = Mψ /?2 , and

it becomes negative for su?ciently small z. The fragmentation function Eq. (32) is shown at the scales ? = 3Mψ and ? = 6Mψ in ?gure 6. Note that there is a dramatic dependence 11

on the arbitrary factorization scale ?. At the scale ? = 6Mψ the fragmentation function is much larger and peaks at a lower z value than at the scale ? = 3Mψ . Comparison with full calculation The fragmentation contribution to the di?erential decay rate for Z 0 → ψ + ?+ ?? is given by inserting Pγ→ψ from Eq. (17), dΓ/dEψ from Eq. (22), and D?→ψ from Eq. (32) into the factorization formula Eq. (9):
0 + ? 2 2 y 2MZ dΓ 2 Γ(Z → ? ? ) (y ? 1) + 1 (Z 0 → ψ(Eψ ) + ?+ ?? ) = 8α2gψ ? 2y . log 2 dEψ MZ y Mψ

(33)

Note that the ?-dependence cancels exactly. It is now possible to verify that this agrees with the full calculation in the fragmentation limit Eq. (4). Figure 7 compares the energy distribution of the full calculation Eq. (3) and the fragmentation calculation Eq. (33). It is clear from the graph that the fragmentation approximation breaks down for su?ciently small Eψ . For Eψ = 3Mψ the di?erence between the two curves is less than 1%, while for Eψ = 2Mψ the di?erence is 5%. In practice there will often be a minimum energy below which detectors do not register particles. If this minimum energy is large enough then it is clear that the fragmentation approximation will give a result very close to the full calculation. Figure 8 shows the energy distribution in the fragmentation limit separated into the lepton fragmentation contribution, the ?rst term on the right hand side of Eq. (9), and the photon fragmentation contribution, the last term on the right hand side of Eq. (9). The contributions are shown at ? = 3Mψ and ? = 6Mψ . The relative contribution of the two production mechanisms depends dramatically on the factorization scale ?, though the total, photon fragmentation plus lepton fragmentation is independent of ?. At the scale ? = 3Mψ the contribution of the lepton fragmentation mechanism is negligable compared to the contribution from the photon fragmentation mechanism, while at the scale ? = 6Mψ both contributions are of the same order. The analysis carried out in this paper applies equally to other heavy quark antiquark states such as the ψ ′ and the Υ. Unfortunately the framentation contribution to 12

Υ production is an order of magnitude smaller than the fragmentation contribution to ψ
2 production. This is because the Υ-photon coupling gΥ = 6 × 10?4 is so much smaller than 2 the ψ-photon coupling gψ = 8 × 10?3 , and the Υ mass is larger than the ψ mass.

Conclusion The process Z 0 → ψ + ?+ ?? has been studied in the fragmentation limit MZ → ∞ keeping Eψ /MZ ?xed. In this limit the decay rate factors into the subprocess rates Γ(Z 0 → ?+ ?? ) and Γ(Z 0 → γ + ?+ ?? ), convoluted with the electromagnetic fragmentation functions D?→ψ and Dγ→ψ . The fragmentation function D?→ψ (z, ?) for a lepton to split into ψ, the fragmentation probability Pγ→ψ for a photon to split into ψ, and the subprocess Γ(Z 0 → γ+?+ ?? ) were calculated at lowest order in α. The lepton fragmentation function was de?ned by imposing a cuto? on the invariant mass s of the lepton and ψ in the ?nal state. This cuto? translated into limits on the phase space of the subprocess Γ(Z 0 → γ + ?+ ?? ). It was then explicitly shown that the ?-dependence of the lepton fragmentation function canceled the ?-dependence of the subprocess Γ(Z 0 → γ + ?+ ?? ). Comparison between the fragmentation calculation and the full calculation shows that the fragmentation approximation is accurate to within 5% at Eψ = 2Mψ , and becomes more accurate for ψ energies greater than this. This work is supported in part by the U.S. Department of Energy, Division of High Energy Physics, under Grant DE-FG02-91-ER40684. I wish to thank E. Braaten for many helpful discussions, and his in?nite patience. I also wish to thank the Fermilab theory group for their hospitality.

13

References
[1] J.C. Collins and G. Sterman, Nucl. Phys. B185 172 (1981); J.C. Collins and D.E. Soper, Nucl. Phys. B194, 445 (1982); G. Curci, W Furmanski and R. Petronzio, Nucl. Phys. B175 (1980) 27. [2] E. Braaten and T.C. Yuan, Phys. Rev. Lett. 71, 1673 (1993). [3] E. Braaten, K. Cheung and T.C. Yuan, Phys. Rev. D48, 4230 (1993). [4] E. Braaten, K. Cheung and T.C. Yuan, Phys. Rev. D48, R5049 (1993). [5] A.F. Falk, M. Luke, M.J. Savage, and M.B. Wise, Phys. Lett. B312, 486 (1993) and Phys. Rev. D49, 555 (1994). [6] C.-H. Chang and Y.-Q. Chen, Phys. Lett. B284, (1991) 142; Phys. Rev. D46 3845 (1992); Y.-Q. Chen, Phys. Rev. D48, 5158 (1993). [7] B.Guberina, J.H. K¨ hn, R.D.Peccei, and R. R¨ ckl, Nucl. Phys. B174, 317 (1980). u u [8] S. Fleming, Phys. Rev. D48, R1914 (1993). [9] L. Bergstr¨m and R.W. Robinett, Phys. Lett. B245, 249 (1990). o [10] J.H. K¨ hn and H. Schneider, Phys. Rev. D24, 2996 (1981) and Z. Phys. C11, 263 u (1981). [11] K. Hagiwara, A.D. Martin and W.J. Stirling, Phys. Lett. B267, 527 (1991).

Figure Captions
1. The Feynman diagrams for Z 0 → ψ + ?+ ?? at leading order in α. a) The two diagrams from which the fragmentation contribution can be isolated, and b) the two diagrams that may be neglected. 2. The two Feynman diagrams for Z 0 → ψγ at leading order in α. 14

3. Feynaman diagrams for a) ψ production by photon fragmentation, b) photon production. The shaded circle represents some general vertex that radiates the photon. 4. The two Feynman diagrams for Z 0 → γ + ?+ ?? at leading order in α. 5. Feynman diagrams for a) ψ production by lepton fragmentation, b) lepton production. The shaded circle represents some general vertex that radiates the lepton. 6. Lepton fragmentation function for ? = 3Mψ (solid) and ? = 6Mψ (dashes). 7. Energy distribution: the full calculation (solid), the fragmentation calculation (dashes). 8. Photon fragmentation (PF) and lepton fragmentation (LF) contributions to the energy distribution for ? = 3Mψ (solid) and ? = 6Mψ (dashes).

15

This figure "fig1-1.png" is available in "png" format from: http://arXiv.org/ps/hep-ph/9403396v1

This figure "fig2-1.png" is available in "png" format from: http://arXiv.org/ps/hep-ph/9403396v1

This figure "fig1-2.png" is available in "png" format from: http://arXiv.org/ps/hep-ph/9403396v1

This figure "fig2-2.png" is available in "png" format from: http://arXiv.org/ps/hep-ph/9403396v1

This figure "fig1-3.png" is available in "png" format from: http://arXiv.org/ps/hep-ph/9403396v1



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