Pion Wave Function Research Articles

Applicability of perturbative QCD and NLO power corrections for the pion form factor

In many years ago, Isgur and Llewellyn Smith addressed that PQCD is inapplicable to exclusive processes, such as the pion form factor.The main problem is that the asymptotic of PQCD is only about one fourth of the experimental value. We reexamine this PQCD deep problem. By including NLO power corrections to the pion form factor, we may arrive at a perturbative explanation for the data. The key realization is that we need to interprete that the strong interaction coupling constant involved in the PQCD result should be taken as an effective coupling constant under nonperturbative QCD vaccum. This implies that one can equally identify the relevant scale for the effective coupling constant as the factorization scale about 1 GeV. We also find that the average momentum fraction variable locates about 0.5, which is in favor of the asymptotic pion wave function. By employing photon-pion form factor with NLO power corrections to factorize out the nonperturbative effects involved in the effective coupling constant, we can extract an effective running coupling constant, which represents an effective coupling involving in the hard scattering subprocesses. The difference between the effective running coupling constant and the usual perturbative running coupling constant ($\Lambda_{QCD}=0.3$ GeV) is very small for $Q^2> 1 GeV^2$. The effective running coupling constant $\alpha_s/\pi$ is smaller than 0.2 for $Q^2>1 GeV^2$. This directly showes that PQCD is applicable to exclusive processes at energy $Q^2> 1 GeV^2$. In summary, with NLO power corrections, PQCD can completely explain the $Q^2$ spectrum of pion form factor.

Pion light cone wave function in the nonlocal NJL model

We use the simple instanton motivated NJL-type model to calculate the leading twist pion light cone wave function. The model consists in employing the momentum dependent quark mass in the quark loop entering the definition of the wave function. The result is analytical up to a solution of a certain algebraic equation. Various properties including the ${k}_{T}$ dependence of the pion wave function are discussed. The resulting ${k}_{T}$ integrated wave function is not asymptotic and is in agreement with recent analysis of the CLEO data.

Has the E791 experiment measured the pion wave function profile?

The cross section of hard diffractive dissociation of the pion into two jets is calculated. It is obtained that the distribution of longitudinal momenta for jets is not simply proportional to the profile of the pion wave function, but depends on it in a complicated way. In particular, it is shown that, under the conditions of the E791 experiment, the momentum distribution of jets is similar in its shape for the asymptotic and CZ wave functions, and even the ratio of the differential cross sections is not far from unity.

Perturbative QCD factorization ofπγ*→γ(π)andB→γ(π)lν¯

We prove factorization theorem for the processes $\pi\gamma^*\to\gamma$ and $\pi\gamma^*\to\pi$ to leading twist in the covariant gauge by means of the Ward identity. Soft divergences cancel and collinear divergences are grouped into a pion wave function defined by a nonlocal matrix element. The gauge invariance and universality of the pion wave function are confirmed. The proof is then extended to the exclusive $B$ meson decays $B\to\gamma l\bar\nu$ and $B\to\pi l\bar\nu$ in the heavy quark limit. It is shown that a light-cone $B$ meson wave function, though absorbing soft dynamics, can be defined in an appropriate frame. Factorization of the $B\to\pi l\bar\nu$ decay in $k_T$ space, $k_T$ being parton transverse momenta, is briefly discussed. We comment on the extraction of the leading-twist pion wave function from experimental data.

Skewed parton distributions and the scale dependence of the transverse size parameter

We discuss the scale dependence of a skewed parton distribution of the pion obtained from a generalized light-cone wave function overlap formula. Using a simple ansatz for the transverse momentum dependence of the light-cone wave function and restricting ourselves to the case of a zero skewedness parameter, the skewed parton distribution can be expressed through an ordinary parton distribution multiplied by an exponential function. Matching the generalized and ordinary DGLAP evolution equations of the skewed and ordinary parton distributions, respectively, we derive a constraint for the scale dependence of the transverse size parameter, which describes the width of the pion wave function in transverse momentum space. This constraint has implications for the Fock state probability and valence distribution. We apply our results to the pion form factor.

Glue in the light-front pion

It may be possible to approximate the full pion wave function in light-front QCD using only q q and q qg Fock space components. Removing zero modes and using invariant-mass cutoffs that make a constituent approximation possible leads to non-canonical terms in the QCD hamiltonian, and forces us to work in the broken symmetry phase of QCD in which chiral symmetry breaking operators must appear directly in the hamiltonian because the vacuum is trivial. Assuming any candidate interactions are local in the transverse direction, I argue that they are probably relevant operators so that perturbation theory is not modified at high energies. Since light-front chiral symmetry corresponds to quark helicity conservation, we can readily identify candidate chiral symmetry breaking interactions. The only candidate relevant operator is the quark-gluon emission/absorption operator with a quark spin flip. I argue that this operator can only produce the physical π - ϱ mass splitting if the q qg component of the pion is significant.

Analytic coupling and Sudakov effects in exclusive processes: pion and $\gamma ^*\gamma \to \pi ^0$ form factors

We develop and discuss in technical detail an infrared-finite factorization and optimized renormalization scheme for calculating exclusive processes, which enables the inclusion of transverse degrees of freedom without entailing suppression of calculated observables, like form factors. This is achieved by employing an analytic, i.e., infrared stable, running strong coupling $\alpha _{\rm s}(Q^{2})$ which removes the Landau singularity at $Q^{2}=\Lambda_{\rm QCD}^{2}$ by a minimum power-behaved correction. The ensuing contributions to the cusp anomalous dimension - related to the Sudakov form factor - and to the quark anomalous dimension - which controls evolution - lead to an enhancement at high $Q^{2}$ of the hard part of exclusive amplitudes, calculated in perturbative QCD, while simultaneously improving its scaling behavior. The phenomenological implications of this framework are analyzed by applying it to the pion's electromagnetic form factor, including the NLO contribution to the hard-scattering amplitude, and also to the pion-photon transition at LO. For the pion wave function, an improved ansatz of the Brodsky-Huang-Lepage type is employed, which includes an effective (constituent-like) quark mass, $m_{\rm q}=0.33$ GeV. Predictions for both form factors are presented and compared to the experimental data, applying Brodsky-Lepage-Mackenzie commensurate scale setting. We find that the perturbative hart part prevails at momentum transfers above about 20 GeV ${}^{2}$, while at lower $Q^{2}$-values the pion form factor is dominated by Feynman-type contributions. The theoretical prediction for the $\gamma ^{*}\gamma \to \pi ^{0}$ form factor indicates that the true pion distribution amplitude may be somewhat broader than the asymptotic one.

The [formula omitted] ratio and meson–baryon couplings from QCD sum rules II

Using QCD sum rules, we compute the diagonal meson–baryon couplings, πNN , ηNN , πΞΞ , ηΞΞ , πΣΣ and ηΣΣ , from the baryon–baryon correlation function with a meson: i ∫ d 4x e iq·x 〈0|TJ B(x) J ̄ B(0)|M(p)〉 . The calculations are performed to leading order in p μ by considering the two separate Dirac structures, iγ 5γ μp μ and γ 5σ μνq μp ν separately. We first improve the previous sum rule calculations on these Dirac structures for the πNN coupling by including three-particle pion wave functions of twist 4 and then extend the formalism to calculate the other couplings, ηNN , πΞΞ , ηΞΞ , πΣΣ and ηΣΣ . In the SU(3) symmetric limit, we identify the terms responsible for the F/D ratio in the OPE by matching the obtained couplings with their SU(3) relations. Depending on the Dirac structure considered, we find different identifications for the F/D ratio. The couplings including the SU(3) breaking effects are also discussed within our approach.

Mesonic and non-mesonic decay widths of Λ12C

Results are reported on the precise measurement of the lifetime and the weak-decay branching ratios with charged particle( π − and proton) emission from Λ 12C formed via 12C( π +, K +) reaction. The π − mesonic decay width is consistent with the calculations including the pion wave function distortion effect. We observed large contribution of the neutron-stimulated decay widths, Γ( Λ + n → n + n), in the non-mesonic weak decay, which cannot be explained by the current theoretical calculations.

Formula omitted] beyond naive factorization in the heavy quark limit

Nonleptonic decays B→Dπ − or B→D ∗π − are dominated by factorizable contributions. In the heavy quark limit, nonfactorizable contribution arises from strong radiative correction and power corrections in 1/ m b . I calculate the decay rates for B→D (∗)π − at next-to-leading order in strong interaction, including nonfactorizable corrections. The result is expressed in terms of a convolution of the hard scattering amplitude and the pion wave function. The decay amplitudes in this method are independent of the gauge, the renormalization scale, and the renormalization scheme. The effects of the nonfactorizable contribution are discussed and numerical estimates are presented.

Meson–baryon couplings and the F/ D ratio from QCD sum rules

Motivated by the recent work on the calculation of the πNN coupling constant using QCD sum rule beyond the chiral limit, we construct the corresponding sum rules for the couplings, ηNN, πΞΞ, ηΞΞ, πΣΣ and ηΣΣ. In constructing the η-baryon sum rules, we use the second moment of the η wave function, which we obtain from the pion wave function after SU(3) rotation. In the SU(3) symmetric limit, we can identify the term responsible for the F/ D ratio in the OPE, which after the sum rule analysis gives F/ D∼0.2, a factor of three smaller than from other studies. We also present a qualitative analysis including the SU(3) breaking terms.

Pion and photon light-cone wave functions from the instanton vacuum

The leading-twist wave functions of the pion and the photon at a low normalization point are calculated in the effective low--energy theory derived from the instanton vacuum. The pion wave function is found to be close to the asymptotic one, consistent with the recent CLEO measurements. The photon wave function is non-zero at the endpoints. This different behavior is a consequence of the momentum dependence of the dynamical quark mass suggested by the instanton vacuum. We comment on the relation of meson wave functions and off-forward parton distributions in this model.

Combined effects of nuclear Coulomb field, radial flow, and opaqueness on two-pion correlations

Correlations of two like charged pions emitted from a hot and charged spherically expanding nuclear system are investigated. The motion of the pions is described quantum mechanically using the Klein-Gordon equation which includes Coulomb field and pion absorption. Flow modifies the radial distribution of the source function and rescales the pion wave functions. The radii extracted from the correlation functions are calculated in sideward and outward direction as a function of the pair momentum. Comparison with recent measurements at bombarding energies of 1 GeV and 11 GeV per nucleon is made.

πNNandπNN(1535)couplings in QCD

We study the two point correlation function of two nucleon currents sandwiched between the vacuum and the pion state. The light cone QCD sum rules are derived for the $\ensuremath{\pi}\mathrm{NN}$ coupling ${g}_{\ensuremath{\pi}\mathrm{NN}}$ and the $\ensuremath{\pi}\mathrm{NN}(1535)$ coupling ${g}_{\ensuremath{\pi}{\mathrm{NN}}^{*}}.$ The contribution from the excited states and the continuum is subtracted cleanly through the double Borel transform with respect to the two external momenta ${p}_{1}^{2}, {p}_{2}^{2}=(p\ensuremath{-}{q)}^{2}.$ We first improve the original sum rule for ${g}_{\ensuremath{\pi}\mathrm{NN}}$ and determine the value of the pion wave function ${\ensuremath{\varphi}}_{\ensuremath{\pi}}{(u}_{0}=\frac{1}{2})=1.5\ifmmode\pm\else\textpm\fi{}0.2,$ which is a fundamental nonperturbative quantity similar to the quark condensate. Our calculation shows that the $\ensuremath{\pi}\mathrm{NN}(1535)$ coupling is strongly suppressed.

Vector mesons from tensor couplings in the NJL model

We propose to describe vector mesons in a pure quark model such as the one of Nambu and Jona-Lasinio by means of four point tensor interactions, but with vanishing vector and axial couplings. This becomes possible due to tensor-vector mixing at the one quark loop level. As a consequence a bound particle with ρ quantum numbers is generated. This has the advantage of avoiding the A− π mixing which is known to generate a wrong off-shell pion wave function renormalization. In addition, there is no A 1 meson. We make our arguments quantitative by computing Euclidean correlators via dispersion relations and comparing with lattice QCD and random instanton model results.

TWIST-TWO LIGHTCONE PION WAVE FUNCTION

We present an analysis of the existing constraints for the twist-two lightcone pion wave function. We find that existing information on the pion wave function does not exclude the possibility that the pion wave function attains its asymptotic form. New bounds on the parameters of the pion wave function are presented.

π − decay width of [formula omitted

We have measured the branching ratio of the π − mesonic weak decay and the weak decay lifetime of C Λ 12 with the use of the SKS spectrometer at KEK 12 GeV PS. The π − mesonic decay width (Г π − was derived and compared with theoretical calculations. The measured Г π − is consistent with the theoretical calculation including the distortion effect of the pion wave function.

Exclusive QCD processes, quark-hadron duality, and the transition to perturbative QCD

Experiments at CEBAF will scan the intermediate-energy region of the QCD dynamics for the nucleon form factors and for Compton Scattering. These experiments will definitely clarify the role of resummed perturbation theory and of quark-hadron duality (QCD sum rules) in this regime. With this perspective in mind, we review the factorization theorem of perturbative QCD for exclusive processes at intermediate energy scales, which embodies the transverse degrees of freedom of a parton and the Sudakov resummation of the corresponding large logarithms. We concentrate on the pion and proton electromagnetic form factors and on pion Compton scattering. New ingredients, such as the evolution of the pion wave function and the complete two-loop expression of the Sudakov factor, are included. The sensitivity of our predictions to the infrared cutoff for the Sudakov evolution is discussed. We also elaborate on QCD sum rule methods for Compton Scattering, which provide an alternative description of this process. We show that, by comparing the local duality analysis to resummed perturbation theory, it is possible to describe the transition of exclusive processes to perturbative QCD.

Pion light-cone wave functions and light-front quark model

We discuss a relation between the light-front quark model and QCD. We argue that this model can be used for an evaluation of the light-cone wave functions for moderate values of u, but that it is inapplicable for this purpose in the region near the end points u=(0,1). We find additional support for a recent analysis in which it was claimed that the twist-two pion wave function attains its asymptotic form. The asymptotic twist-four two-particle wave function is also in good agreement with the light-front quark model.

TRANSVERSE MOMENTUM DEPENDENCE IN THE PERTURBATIVE CALCULATION OF THE PION FORM FACTOR

Transverse momentum dependence in the perturbative calculation of the pion electromagnetic form factor is re-analyzed. It is found that (a) the intrinsic transverse momentum dependence associated with the pion wave function significantly suppresses perturbative prediction for the pion form factor at the energy region of a few GeV, and (b) the transverse momentum dependence in the fermion propagator of the hard scattering leads to a mild reduction of the perturbative calculation. Our formalism is available for both the intermediate energy region (Q~ a few GeV) and the asymptotic energy region (Q→∞).