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Higher Fock states and power counting in exclusive charmonium decays

The role of higher Fock-state contributions to exclusive charmonium decays will be discussed. It will be argued that the J/Psi (Psi') decays into baryon-antibaryon pairs are dominated by the valence Fock-state contributions. P-wave charmonium decays, on the other hand, receive strong contributions from the c cbar g Fock states since the valence Fock-state contributions are suppressed in these reactions. Numerical results for the decay widths of J/Psi (Psi') into baryon-antibaryon and chi(J) into pairs of pseudoscalar mesons will be also presented and compared to data.


Introduction
Exclusive charmonium decays have been investigated within perturbative QCD by many authors, e.g. [1]. It has been argued that the dominant dynamical mechanism is cc annihilation into the minimal number of gluons allowed by colour conservation and charge conjugation, and subsequent creation of light quark-antiquark pairs forming the final state hadrons (cc → ng * → m(qq)). The dominance of annihilation through gluons is most strikingly reflected in the narrow widths of charmonium decays into hadronic channels in a mass region where strong decays typically have widths of hundreds of MeV. Since the c and the c quarks only annihilate if their mutual distance is less than about 1/m c (where m c is the c-quark mass) and since the average virtuality of the gluons is of the order of 1 − 2 GeV 2 one may indeed expect perturbative QCD to be at work although corrections are presumably substantial (m c is not very large). In hard exclusive reactions higher Fock-state contributions are usually suppressed by inverse powers of the hard scale, Q, appearing in the process (Q = 2m c for exclusive charmonium decays), as compared to the valence Fock-state contribution. Hence, higher Fock-state contributions are expected to be negligible in most cases. Charmonium decays are particularly interesting because the valence Fock-state contributions are often suppressed for one or the other reason. In such a case higher Fock-state contributions or other peculiar contributions such as power corrections or small components of the hadronic wave functions may become important. Examples of charmonium decays with suppressed valence Fock-state contributions are the hadronic helicity non-conserving processes like J/ψ → ρπ or η c → BB, the radiative J/ψ decays into light pseudoscalar mesons and , the main topic of this report, the decays of the P -wave charmonium states, χ cJ .
Recently, the importance of higher Fock states in understanding the production and the inclusive decays of charmonium has been pointed out [2]. As has been shown the long-distance matrix elements can there be organized into a hierarchy according to their scaling with v, the typically velocity of the c quark in the charmonium. One may apply the velocity expansion to exclusive charmonium decays as well [3]. The Fock-state expansions of the charmonia start as In [3] it is argued that the amplitude for a two-body decay of a charmonium state satisfies a factorization formula which separates the scale m c from the lower momentum scales. The decay amplitude is then expressed as a convolution of a hard scattering amplitude that involves the scale m c , an initial state factor (charmonium wave function) that involves scales of order m c v and lower, and a final state factor (light hadron wave function) that involves only the scale Λ QCD . If m c was asymptotically large, the dominant terms in the factorization formula would involve the minimal number of partons in the hard scattering. Terms involving additional partons in the initial state are suppressed by powers of v while terms involving additional partons in the final state are suppressed by powers of Λ QCD /m c . The relative strength of various contributions to specific decay processes can easily be estimated.
Combining the fact that a hard scattering amplitude involving a P -wave cc pair is of order v, with the Fock-state expansion (1), one finds for the amplitudes of χ cJ decays into, say, a pair of pseudoscalar mesons the behaviour where a and b are constants and α sof t s comes from the coupling of the Fock-state gluon to the hard process. For the decay reaction J/ψ → BB, on the other hand, one has Thus, one sees that in the case of the χ cJ the colour-octet contributions are not suppressed by powers of either v or 1/m c as compared to the contributions from the valence Fock states [3]. Hence, the colour-octet contributions have to be included for a consistent analysis of P -wave charmonium decays. The situation is different for J/ψ decays into baryon-antibaryon pairs: Higher Fock state contributions first start at O(v 2 ). Moreover, there is no obvious enhancement of the corresponding hard scattering amplitudes, they appear with at least the same power of α s as the valence Fock state contributions. Thus, despite of the fact that m c is not very large and v not small (v 2 ≃ 0.3), it seems reasonable to expect small higher Fock-state contributions to the baryonic decays of the J/ψ.
where x (′) , b (′) stand for sets of momentum fractions and transverse separations characterizing the hadron h (′) . Each Fock state provides such an amplitude (marked by the upper index c) 2 . Ψ h denotes the transverse configuration space light-cone wave function of a light hadron. f (c) is the charmonium decay constant. Because the annihilation radius is substantially smaller than the charmonium radius this is, to a reasonable approximation, the only information on the charmonium wave function required.T

(c)
H is the Fourier transform of the hard scattering amplitude to be calculated from a set of Feynman graphs relevant to the considered process. t represents a set of renormalization scales at which the α s appearing inT The b space form of the Sudakov factor has been calculated in next-to-leading-log approximation by Botts and Sterman [4]. As mentioned before, exclusive charmonium decays have been analysed several times before, e.g. [1]. New refined analyses are however necessary for several reasons: in previous analyses the standard hard scattering approach has been used with the running of α s and the evolution of the wave function ignored. In the case of the χ cJ also the colour-octet contributions have been disregarded. Another very important disadvantage of some of the previous analyses is the use of light hadron distribution amplitudes (DAs), representing wave functions integrated over transverse momenta, that are strongly concentrated in the end-point regions. Despite of their frequent use, such DAs were always subject to severe criticism. In the case of the pion, they lead to a clear contradiction to the recent CLEO data [6] on the πγ transition form factor, F πγ . As detailed analyses unveiled, the F πγ data require a DA that is close to the asymptotic form ∝ x(1 − x) [7,8]. Therefore, these end-point region concentrated DAs should be discarded for the pion and perhaps also for other hadrons like the nucleon [9].
4 Results for J/ψ (ψ ′ ) → BB According to the statements put forward in Sects. 1 and 2, higher Fock-state contributions are neglected in this case. From the permutation symmetry between the two u quarks and from the requirement that the three quarks have to be coupled in an isospin 1/2 state it follows that there is only one independent scalar wave function in the case of the nucleon if the 3-component of the orbital angular momentum is assumed to be zero. Since SU(3) F is a good symmetry, only mildly broken by quark mass effects, one may also assume that the other octet baryons are described by one scalar wave function. It is parameterized as  [10]. The three-gluon contributions are evaluated with m c = 1.5 GeV, Λ QCD = 210 MeV and a B 10 = 0.85 GeV −1 . The widths for the decuplet baryon channels are given for one charge state.
in the transverse momentum space. The set of indices 123 refers to the quark configuration u + u − d + ; the wave functions for other quark configurations are to be obtained from (5) by permutations of the indices. The transverse momentum dependent part Ω is parameterized as a Gaussian in k 2 ⊥i /x i (i = 1, 2, 3). The transverse size parameter, a B 8 , appearing in that Gaussian, as well as the octet-baryon decay constant, f B 8 , are assumed to have the same value for all octet baryons. In [9] these two parameters as well as the nucleon DA have been determined from an analysis of the nucleon form factors and valence quark structure functions at large momentum transfer (a B 8 = 0.75 GeV −1 ; f B 8 = 6.64 × 10 −3 GeV 2 at a scale of reference µ 0 = 1 GeV). The DA has been found to have the simple form It behaves similar to the asymptotic form, only the position of the maximum is shifted slightly. Calculating the hard scattering amplitude from the Feynman graphs for the elementary process cc → 3g * → 3(qq) and working out the convolution (4), one obtains the widths for the J/ψ decays into BB pairs listed and compared to experimental data in Table 1. As can be seen from that table a rather good agreement with the data is obtained. In addition to the three-gluon contribution there is also an isospin-violating electromagnetic one generated by the subprocess cc → γ * → 3(qq). According to [10] this contribution seems to be small. The extension of the perturbative approach to the baryonic decays of the ψ ′ is now a straightforward matter. One simply has to rescale the J/ψ widths by the ratio of the corresponding electronic widths where ρ p.s. (z) = √ 1 − 4z 2 is the phase space factor. In contrast to previous calculations the ψ ′ and the J/ψ widths do not scale as (M J/ψ /M ψ ′ ) 8 since the hard scattering amplitude is evaluated with 2m c instead with the bound-state mass. Results for the baryonic decay widths of the ψ ′ are presented in Table 2. Again good agreement between theory and experiment is found with the exception of the Σ 0 Σ 0 channel. 5 Results for χ cJ → P P , J = 0, 2 Let me first discuss the color-singlet contribution to the ππ channel since this is an essentially parameter-free contribution. The pion wave function is parameterized analogue to (5) with Φ π = Φ AS = 6x (1 − x). The pion decay constant, f π , is 131 MeV and the transverse size parameter is given by a π = 1/( √ 8πf π which automatically satisfies the π 0 → γγ contraint 76.8 55.0 54.6 33.9 32.1 24.4 Γ exp [12] 76 ± 14 26 ± 14 58 ± 12 23 ± 9 25 ± 8 16 ± 8 [11] 53 ± 15   [3]. derived in [13]. This form of the pion wave function is strongly contrained by the πγ transition form factor [7,8]. The information on the χ cJ wave function needed is the singlet decay constant which is related to the derivative of the non-relativistic χ cJ wave function at the origin In [3] R ′ P (0) is chosen to be 0.22 GeV 5/2 and, as before, m c = 1.5 GeV. The hard scattering amplitude is now computed from the Feynman graphs for the elementary process cc → 2g * → 2(qq) and convoluted with the hadronic wave functions according to (4). Evaluating the decays width from the singlet amplitudes, one finds, not surprisingly, values which are too small as compared to experiment [11]: Γ(χ c0(2) → π + π − ) = 8.22 (0.41) keV. Of course, the colour-octet contribution (see (2)) is lacking and one has to estimate it as well and to add it coherently to the colour-singlet amplitude. For the calculation of the colour-octet amplitude an octet wave function is required. This introduces new degrees of freedom and, for a first estimate, it is advisable to simplify as much as possible in order to keep the number of new free parameters as small as possible. It is important to realize that in the |cc 8 g Fock state not only the cc pair is in a colour-octet state, but also the three particles, c, c and g, are in an S-state. Hence orbital angular momenta are not involved and the transverse degrees of freedom can be integrated over. Therefore, one only has to operate with a DA Φ  for each J. It seems reasonable in the spirit of the non-relativistic expansion to approximate the ccg DA by δ(z 3 − z) where z ≃ ǫ/(2m c ) ≃ v 2 /2. ǫ is the binding energy of the charmonium and z 3 refers to the momentum fraction the gluon carries. Analogously to the colour-singlet case, the only information on the colour-octet wave function that then enters the final results is the octet decay constant f (8) χc 2 . A fit of the unknown decay constant f (8) χc to the experimental data on the decay widths provides the value f (8) χc = 1.46 × 10 −3 GeV 2 which appears to be reasonably large [3]. The resulting decay widths are presented in Table 3. With regard to the large experimental uncertainties fair agreement with experiment is obtained. This analysis can be easily extended to other P P channels, see [3].
Exclusive charmonium decays constitute an interesting laboratory for investigating power correstions and higher Fock-state contributions. In particular one can show that in the decays of the χ cJ the contributions from the next-higher charmonium Fock state, ccg, are not suppressed by powers of m c or v as compared to the cc Fock state and therefore have to be included for a consistent analysis of these decays. For J/ψ (ψ ′ ) decays into BB pairs the situation is different: Higher Fock-state contributions are suppressed by powers of 1/m c and v. Indeed, as an explicit analysis reveals, with plausible baryon wave functions a reasonable description of the baryonic J/ψ (ψ ′ ) decay widths can be obtained alone from the cc Fock state while for χ cJ decays into pairs of pseudoscalar mesons agreement with experiment can only be obtained when the color-octet contribution is taken into account.