Higher Fock States Research Articles

Current in the light-front Bethe-Salpeter formalism. I. Replacement of non-wave-function vertices

We apply the light-front reduction of the Bethe-Salpeter equation to matrix elements of the electromagnetic current between bound states. Using a simple (1+1)-dimensional model to calculate form factors, we focus on two cases. In one case, the interaction is dominated by a term instantaneous in light-cone time. Here effects of higher Fock states are negligible and the form factor can be effectively expressed using non-wave function vertices and crossed interactions. If the interaction is not instantaneous, non-wave function vertices are replaced by contributions from higher Fock states. These higher Fock components arise from the covariant formalism via the energy poles of the Bethe-Salpeter vertex and the electromagnetic vertex. The replacement of non-wave function vertices in time-ordered perturbation theory is a theorem which directly extends to generalized parton distributions, e.g., in (3+1) dimensions.

Foreword

The transverse lattice approach to non-perturbative light-front Hamiltonian QCD is described. Preliminary results on the π - ϱ system are presented, at fixed DLCQ and Tamm-Dancoff cut-offs. A renormalized, approximately Lorentz covariant light-front Hamiltonian is found to leading order of the colour-dielectric expansion, compatible with a massless pion. The π light-front wavefunction is compared with experiment. Exclusive processess agree reasonably well, given the approximations, but inclusive processes, sensitive to higher Fock state structure, still exhibit large cut-off artifacts.

Evading the CKM hierarchy: Intrinsic charm inBdecays

We show that the presence of intrinsic charm in the hadrons' light-cone wave functions, even at a few percent level, provides new, competitive decay mechanisms for B decays which are nominally CKM suppressed. For example, the weak decays of the B-meson to two-body exclusive states consisting of strange plus light hadrons, such as $\stackrel{\ensuremath{\rightarrow}}{B}\ensuremath{\pi}K,$ are expected to be dominated by penguin contributions since the tree-level $\stackrel{\ensuremath{\rightarrow}}{b}\mathrm{su}\ifmmode \bar{u}\else \={u}\fi{}$ decay is CKM suppressed. However, higher Fock states in the B wave function containing charm quark pairs can mediate the decay via a CKM-favored $\stackrel{\ensuremath{\rightarrow}}{b}\mathrm{sc}\overline{c}$ tree-level transition. Such intrinsic charm contributions can be phenomenologically significant. Since they mimic the amplitude structure of ``charming'' penguin contributions, the latter need not be penguin contributions at all.

A Phenomenological Analysis of Higher Fock State Contributions to the χcJ Decays**The project supported in part by National Natural Science Foundation of China

We present a phenomenological analysis of higher Fock state contributions to the χcJ decays by using the recent BES experimental data.It is found that the higher Fock state makes an important contribution to the inclusive and exclusive processes with respect to that from the valence Fock state of the χcJ and some constraints of these contributions are obtained for the χc0 and χc2 states in order to fit the experimental data.

HIGHER-FOCK STATES AND THE COVARIANCE OF THE LIGHT-FRONT QUARK MODEL

The light-front quark model has been successful in describing various meson properties. We apply the model to the meson weak decays and discuss the covariance of the model. We find that the inclusion of the higher-Fock state (i.e. nonvalence or so called Z-graph) contributions is necessary to recover the covariance of the light-front quark model. We present the possibility of the effective calculation of the higher-Fock state contributions to restore the covariance in the light-front model.

Transverse lattice

The transverse lattice approach to non-perturbative light-front Hamiltonian QCD is described. Preliminary results on the π - ϱ system are presented, at fixed DLCQ and Tamm-Dancoff cut-offs. A renormalized, approximately Lorentz covariant light-front Hamiltonian is found to leading order of the colour-dielectric expansion, compatible with a massless pion. The π light-front wavefunction is compared with experiment. Exclusive processess agree reasonably well, given the approximations, but inclusive processes, sensitive to higher Fock state structure, still exhibit large cut-off artifacts.

Colour octet contribution to exclusive P-wave charmonium decay into nucleon-antinucleon

We show that although there is no infrared divergence in exclusive P-wave quarkonium decays, the colour octet contribution is no less important than in the inclusive decay. Results from more proper calculations with only colour singlet contribution are not sufficient to account for the measured partial decay widths and so the next higher Fock state must be included. Using the χ J decay into NN̄ as an example, a scheme is devised to yield estimates of the decay widths using both contributions, the results are comparable with experimental measurements.

Entanglement of Fock-space expansion and covariance in light-front Hamiltonian dynamics

We investigate in a model with scalar ``nucleons'' and mesons the contributions of higher Fock states that are neglected in the ladder approximation of the Lippmann-Schwinger equation. This leads to a breaking of covariance, both in light-front and in instant-form Hamiltonian dynamics. The lowest Fock sector neglected has two mesons in the intermediate state and corresponds to the stretched box. First we show in a simplified example that the contributions of higher Fock states are much smaller on the light-front than in instant-form dynamics. Then we show for a scattering amplitude above threshold that the stretched boxes are small, however necessary to retain covariance. For an off-energy-shell amplitude covariance is not necessarily maintained and this is confirmed by our calculations. Again, the stretched boxes are found to be small.

Rotation average in light cone time-ordered perturbation theory

We present a rotation average of the two-body scattering amplitude in light cone time- (\ensuremath{\tau}-)ordered perturbation theory. Using a rotation average procedure, we show that the contribution of an individual time-ordered diagram can be quantified in a Lorentz invariant way. The number of time-ordered diagrams can also be reduced by half, if the masses of the two bodies are the same. In the numerical example of ${\ensuremath{\varphi}}^{3}$ theory, we find that the higher Fock-state contribution is quite small in light cone quantization.

Influence of higher Fock states in light-cone gauge theories

In the light-cone Fock state expansion of gauge theories, the influence of non-valence states may be significant in precision non-perturbative calculations. In two-dimensional gauge theories, it is shown how these states modify the behaviour of the light-cone wavefunction in significant ways relative to endemic choices of variational ansatz. Similar effects in four-dimensional gauge theories are briefly discussed.

An Explanation for theρ−πPuzzle ofJ/ψandψ′Decays

We propose a new explanation for the long-standing puzzle of the tiny branching fraction of ${\ensuremath{\psi}}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\rho}\ensuremath{\pi}$ relative to that for $J/\ensuremath{\psi}\ensuremath{\rightarrow}\ensuremath{\rho}\ensuremath{\pi}$. In the case of $J/\ensuremath{\psi}$, we argue that this decay is dominated by a higher Fock state in which the $c\overline{c}$ pair is in a color-octet $^{3}S_{1}$ state and decays via the annihilation process $c\overline{c}\ensuremath{\rightarrow}q\overline{q}$. In the case of the ${\ensuremath{\psi}}^{\ensuremath{'}}$, we argue that the probability for the $c\overline{c}$ pair in this higher Fock state to be close enough to annihilate is suppressed by a dynamical effect related to the small energy gap between the mass of the ${\ensuremath{\psi}}^{\ensuremath{'}}$ and the $D\overline{D}$ threshold.

χcJproduction ine+e−annihilation

Inclusive production of $\chi_{c J}$ in $e^+e^-$ annihilation is an excellent probe of the role of higher Fock states in the production of heavy quarkonia. Within the non-relativistic QCD approach, contributions from the short-distance production of colour-octet $c\bar{c}$ pairs are significantly larger than those from colour-singlet production. At the same time, $\chi_{c J}$ production rates are significantly smaller than expected in the colour evaporation approach. Measurements of $\chi_{\c J}$ production at CLEO and future B-factories will thus constitute a major test of theoretical approaches to the production of heavy quarkonia.

Power counting in exclusive quarkonium decays

Recent progress in the factorization of exclusive decays of heavy quarkonia into long-distance and short-distance contributions is presented. The role of high Fock states and colour-octet contributions is outlined.

Variational mass perturbation theory for light-front bound-state equations

We investigate the mesonic light-front bound-state equations of the 't Hooft and Schwinger model in the two-particle, i.e. valence sector, for small fermion mass. We perform a high precision determination of the mass and light-cone wave function of the lowest lying meson by combining fermion mass perturbation theory with a variational approach. All calculations are done entirely in the fermionic representation without using any bosonization scheme. In a step-by-step procedure we enlarge the space of variational parameters. For the first two steps, the results are obtained analytically. Beyond that we use computer algebraic and numerical methods. We achieve good convergence so that the calculation of the meson mass squared can be extended to third order in the fermion mass. Within the numerical treatment we include higher Fock states up to six particles. Our results are consistent with all previous numerical investigations, in particular lattice calculations. For the massive Schwinger model, we find a small discrepancy (less than 2 percent) in comparison with known bosonization results. Possible resolutions of this discrepancy are discussed.

Higher Fock states and power counting in exclusive

Exclusive processes at large momentum transfer $Q$ factor into perturbatively calculable short-distance parts and long-distance hadronic wave functions. Usually, only contributions from the leading Fock states have to be included to leading order in $1/Q$ . We show that for exclusive decays of $P$ -wave quarkonia the contribution from the next-higher Fock state $|{\rm Q}\bar{\rm Q}{\rm g}\rangle$ contributes at the same order in $1/Q$ . We investigate how the constituent gluon attaches to the hard process in order to form colour-singlet final-state hadrons and argue that a single additional long-distance factor is sufficient to parametrize the size of its contribution. Incorporating transverse degrees of freedom and Sudakov factors, our results are perturbatively stable in the sense that soft phase-space contributions are largely suppressed. Explicit calculations yield good agreement with data on $\chi_{{\rm c} J}$ decays into pairs of pions, kaons, and etas. We also comment on $J/\psi$ decays into two pions.

Phenomenological Connection Between Hadronic Wavefunction and Its Radius11The project supported by National Natural Science Foundation of China

A picture for the hadronic structure is discussed based on previous work. The radius contributed from the valence Fock-state wavefunction (i.e., component) is calculated in the light-cone framework. We propose a concept of the characteristic length to represent the contributions of higher Fock states to hadronic radii phenomenologically and calculate its value. Using an equivalence between the “intrinsic” radii of constituent quarks and the radii coming from higher Fock states we put forward a constraint on the hadronic wavefunctions of mesons. It is found that these wavefunctions lead to consistency between theories and experiments for D and B exclusive decays. We also give the radii for K, D and B mesons. The results show that the radius of K meson is in good agreement with experimental data.

Novel nuclear and heavy quark phenomena in QCD

A central focus of QCD studies is the non-perturbative structure of hadron wavefunctions, not only bound-state valence quark distributions, but also the intrinsic gluon and heavy quark distributions associated with higher Fock states. In this talk I discuss how the production of heavy quark systems and the use of nuclear targets can help to identify and separate the various components of hadron wavefunctions.

Hadronic transition amplitudes under null-plane Bethe-Salpeter dynamics.

A two-tier form of the Bethe-Salpeter (BS) framework is employed for the evaluation of a few typical transition amplitudes involving strong and electromagnetic decays of qq\ifmmode\bar\else\textasciimacron\fi{} hadrons with L\ensuremath{\ge}0. The two-tier BS framework, whose rationale has been described in some recent publications, is characterized by (i) a three-dimensional reduction of the BS equation under the null-plane ansatz to make a first contact with mass spectral data and (ii) a reconstructed four-dimensional BS wave function to restore the virtual qq\ifmmode\bar\else\textasciimacron\fi{} effects (higher Fock states) for the evaluation of hadronic transition amplitudes through the lowest-order Feynman (quark-triangle) diagrams. The processes explicitly considered are the pion form factor ${F}_{\ensuremath{\pi}}$(${k}^{2}$), (\ensuremath{\omega},${a}_{2}$)\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\pi}, (\ensuremath{\rho},${f}_{2}$)->\ensuremath{\pi}\ensuremath{\pi}, (${a}_{1}$,${a}_{2}$)\ensuremath{\rightarrow}\ensuremath{\rho}\ensuremath{\pi}, and ${b}_{1}$\ensuremath{\rightarrow}\ensuremath{\omega}\ensuremath{\pi}. These illustrate the different types of coupling structures that are generally encountered in such a composite-hadron model. The agreement with the data on almost all these amplitudes is excellent, using the same basic constants (${\ensuremath{\omega}}_{0}$,${C}_{0}$,${m}_{\mathrm{ud}}$) as employed recently for a successful description of the mass spectra of qq\ifmmode\bar\else\textasciimacron\fi{} and qqq hadrons, as well as of some pionic amplitudes.