Pion Distribution Amplitude Research Articles

Theoretical Description and Measurement of the Pion–Photon Transition Form Factor

Detailed predictions for the scaled pion-photon transition form factor are given, derived with the method of light-cone sum rules and using pion distribution amplitudes with two and three Gegenbauer coefficients obtained from QCD sum rules with nonlocal condensates. These predictions agree well with all experimental data that are compatible with QCD scaling (and collinear factorization), but disagree with the high-$Q^2$ data of the BaBar Collaboration that grow with the momentum. A good agreement of our predictions with results obtained from AdS/QCD models and Dyson-Schwinger computations is found.

Inclusive gluon production in pion-proton collisions and the principle of maximum conformality renormalization scale

The contribution of the higher-twist mechanism to the large-$p_T$ inclusive gluon production cross section in $\pi p$ collisions is calculated in case of the principle of maximum conformality and Brodsky-Lepage-Mackenzie approaches in the dependence of the pion distribution amplitudes. The higher-twist cross sections obtained in the framework of the principle of maximum conformality and Brodsky-Lepage-Mackenzie approaches, and compared and analyzed in relation to the leading-twist cross sections and each other.

Pion Distribution Amplitude from Lattice QCD

A method is explained through which a pointwise accurate approximation to the pion's valence-quark distribution amplitude (PDA) may be obtained from a limited number of moments. In connection with the single nontrivial moment accessible in contemporary simulations of lattice-regularized QCD, the method yields a PDA that is a broad concave function whose pointwise form agrees with that predicted by Dyson-Schwinger equation analyses of the pion. Under leading-order evolution, the PDA remains broad to energy scales in excess of 100 GeV, a feature which signals persistence of the influence of dynamical chiral symmetry breaking. Consequently, the asymptotic distribution φπ(asy)(x) is a poor approximation to the pion's PDA at all such scales that are either currently accessible or foreseeable in experiments on pion elastic and transition form factors. Thus, related expectations based on φ φπ(asy)(x) should be revised.

Determination of the pion distribution amplitude

Right now, we have not enough knowledge to determine the hadron distribution amplitudes (DAs) which are universal physical quantities in the high energy processes involving hadron for applying pQCD to exclusive processes. Even for the simplest pion, one can't discriminate from different DA models. Inversely, one expects that processes involving pion can in principle provide strong constraints on the pion DA. For example, the pion-photon transition form factor (TFF) can get accurate information of the pion wave function or DA, due to the single pion in this process. However, the data from Belle and BABAR have a big difference on TFF in high $Q^2$ regions, at present, they are helpless for determining the pion DA. At the present paper, we think it is still possible to determine the pion DA as long as we perform a combined analysis of the most existing data of the processes involving pion such as $\pi \to \mu \bar{\nu}$, $\pi^0 \to \gamma \gamma$, $B\to \pi l \nu$, $D \to \pi l \nu$, and etc. Based on the revised light-cone harmonic oscillator model, a convenient DA model has been suggested, whose parameter $B$ which dominates its longitudinal behavior for $\phi_{\pi}(x,\mu^2)$ can be determined in a definite range by those processes. A light-cone sum rule analysis of the semi-leptonic processes $B \to \pi l \nu$ and $D \to \pi l \nu$ leads to a narrow region $B = [0.01,0.14]$, which indicate a slight deviation from the asymptotic DA. Then, one can predict the behavior of the pion-photon TFF in high $Q^2$ regions which can be tested in the future experiments. Following this way it provides the possibility that the pion DA will be determined by the global fit finally.

Minimal Ward—Takahashi Vertices and Light Cone Pion Distribution Amplitudes from the GND Quark Model

The gauge-invariant, nonlocal, dynamical quark model is shown to generate minimal vertices that satisfy the Ward—Takahashi identities and the flat-like form of the light-cone pion distribution amplitudes. Non-flat form amplitudes can be produced only if we take a finite momentum cutoff and include nonzero pion mass corrections or go beyond the minimal vertices. A by-product of our investigation shows that the variable u appearing in light-cone pion distribution amplitudes is just the standard Feynman parameter in the Feynman parameter integrals.

Information on the pion distribution amplitude from the pion-photon transition form factor with the Belle and BABAR data

We perform an analysis of all existing experimental data on the pion-photon transition form factor (TFF). In the calculation, we include the next-to-leading order correction to the valence-quark contribution and estimate the non-valence-quark contribution by a phenomenological model based on the TFF's limiting behavior at both Q2 → 0 and Q2 → ∞. At present, the pion distribution amplitude (DA) is not definitely determined, it is helpful to have a pion DA model that can mimic all the behaviors suggested in the literature. For the purpose, we adopt the conventional model for pion wavefunction/DA whose broadness is dominantly controlled by a single parameter B. We fix the DA parameters by using the CELLO, CLEO, BABAR and Belle data within small Q2 region (Q2 ≤ 15 GeV2), where all the data are consistent with each other. The pion-photon TFF is then extrapolated into a larger Q2 region. It is found that we still need more data at a large Q2 region in order to determine the precise value of B, and we hope that the definite behavior of pion DA can be concluded finally by the consistent data in the coming future.

Can we understand an auxetic pion-photon transition form factor within QCD?

A state-of-the-art analysis of the pion-photon transition form factor is presented based on an improved theoretical calculation that includes the effect of a finite virtuality of the quasireal photon in the method of light-cone sum rules. We carry out a detailed statistical analysis of the existing experimental data using this method and by employing pion distribution amplitudes with up to three Gegenbauer coefficients ${a}_{2}$, ${a}_{4}$, ${a}_{6}$. Allowing for an error range in the coefficient ${a}_{6}\ensuremath{\approx}0$, the theoretical predictions for ${\ensuremath{\gamma}}^{*}\ensuremath{\gamma}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}$ obtained with nonlocal QCD sum rules are found to be in good agreement with all data that support a scaling behavior of the transition form factor at higher ${Q}^{2}$, like those of the Belle Collaboration. The data on ${\ensuremath{\gamma}}^{*}\ensuremath{\gamma}\ensuremath{\rightarrow}\ensuremath{\eta}/{\ensuremath{\eta}}^{\ensuremath{'}}$ from CLEO and BABAR are also reproduced, while there is a strong conflict with the auxetic trend of the BABAR data above $10\text{ }\text{ }{\mathrm{GeV}}^{2}$. The broader implications of these findings are discussed.

Finding a Way to Determine the Pion Distribution Amplitude from the Experimental Data

It is believed that one can extract more accurate information of the pion distribution amplitude from the pion-photon transition form factor (TFF) due to the single pion in this process. However, the BABAR and Belle data of the pion-photon TFF have a big difference for Q2 ∈ [15, 40] GeV2, and at present, the pion DA can not be definitely determined from the pion-photon TFF. It is crucial to find the right pion DA behavior and to determine which data is more reliable. We perform a combined analysis of the most existing data of the processes involving pion by using a general model for the pion wavefunction/DA. Such a DA model can mimic all the existed pion DA behaviors, whose parameters can be fixed by the constraints from the processes π0 → γγ, π → μν, and B → πlμ, etc. Especially, we examine the B → π transition form factors that provides another constraint to the parameter B in our DA model, which results in B ∈ [0.00, 0.29]. This inversely shows that the predicted curve for the pion-photon TFF is between the BABAR and Belle data in the region Q2 ∈ [15, 40] GeV2. It will be tested by coming more accurate data in large Q2 region, and the definite behavior of pion DA can be concluded finally.

The axial anomaly and consistency of the finite-width light-cone local duality sum rule for the form factor of the transition γ* γ* → π0

The pion transition form factor at the photon mass-shell of q21 = q22 = 0 is fixed by the axial anomaly, which may serve as a criterion in checking the consistency of the underlying quantum chromodynamics (QCD). We choose to work in the symmetric kinematical region where two photons have the same virtualities and hope to reproduce the experimental value of from light-cone QCD sum rules instead of direct extrapolation of the QCD expression to such a low-energy scale. The form factor of the transition γ*γ* → π0 is investigated in the framework of the zero- and finite-width light-cone local duality QCD sum rules. The expression of the pion transition form factor at large t is independent of the twist-two and twist-three pion distribution amplitudes; therefore a more accurate conclusion may be reached. The influence of the light quark mass and the ωρπ coupling constant to our sum rules is also considered. The consistency between the axial anomaly, the intermediate hadronic structure and the QCD perturbative dynamics is tested to be true within a reasonable region of input parameters.

Pion-Photon Transition Form Factor and Pion Distribution Amplitude in QCD: Facing the Enigmatic Behavior of the BaBar Data

We present an extended analysis of the data for the pion-photon transition form factor from different experiments, CELLO, CLEO, and BaBar, and discuss various theoretical approaches which try to reason from them. We focus on the divergent behavior of the BaBar data for the pion and those for the $\eta(\eta')$ pseudoscalar mesons and comment on recently proposed explanations for this discrepancy. We argue that it is not possible at present to accommodate these data within the standard QCD framework self-consistently.

THE FORM FACTOR OF ργ*→π0IN THE FINITE-WIDTH LIGHT-CONE LOCAL DUALITY SUM RULE

The form factor of the transition ργ*→π0is investigated by analyzing the γγ*→π0transition in the framework of the light-cone local duality quantum chromodynamics sum rules. Both finite- and zero-width forms for the low-energy physical absorptive part of the γγ*→π0transition are considered. The numerical results for the transition form factor Fρπ(Q2) in the asymptotic form and the CZ one of the pion distribution amplitude are obtained respectively, the difference between the finite- and zero-width form is not big but still distinguishable. It is found that the results for the form factor Fρπ(Q2) from CZ, asymptotic and the flat pion DAs are compatible with the predictions of perturbative QCD. Our results will be helpful to distinguish the shape of the pion DA of twist-two, when the concrete information of the experimental data for the transition ργ*→π0are provided.

Photon-to-pion transition form factor and pion distribution amplitude from holographic QCD

We try to understand the recently observed anomalous behavior of the photon-to-pion transition form factor in the holographic QCD approach. First the holographic description of the anomalous \gamma^*\gamma^*\pi^0 form factor is reviewed and applied to various models. It is illustrated that in describing the anomalous form factor, the holographic approach is asymptotically dual to the perturbative QCD (pQCD) framework, with the pion mode \pi(z)\sim z corresponding to the asymptotic pion distribution amplitude. This indicates some inconsistency in light-front holography, since \pi(z)\sim z would be dual to \varphi(x)\sim \sqrt{x(1-x)} there. After clarifying these subtleties, we employ the relation between the holographic and the perturbative expressions to study possible asymptotic violation of the transition form factor. It is found that if one require that the asymptotic form factor possess a pQCD-like expression, the pion mode can only be ultraviolet-enhanced by logarithmic factors. The minimally deformed pion mode will then be of the form \pi(z)\sim z\ln (z\Lambda)^{-1}. We suppose that this deformation may be due to the coupling of the pion with a nontrivial open string tachyon field, and then the parameter $\Lambda$ will be related to the quark condensate. Interestingly, this pion mode leads immediately to Radyushkin's logarithmic model, which fitted very well the experimental data in the large-Q^2 region. On the other side, the pQCD interpretation with a flat-like pion distribution amplitude, proposed by Radyushkin and Polyakov, fails to possess a holographic expression.

Pion-photon transition form factor using light-cone sum rules: theoretical results, expectations, and a global-data fit

A global fit to the data from different collaborations (CELLO, CLEO, BaBar) on the pion-photon transition form factor is carried out using light-cone sum rules. The analysis includes the next-to-leading QCD radiative corrections and the twist-four contributions, while the main next-to-next-to-leading term and the twist-six contribution are taken into account in the form of theoretical uncertainties. We use the information extracted from the data to investigate the pivotal characteristics of the pion distribution amplitude. This is done by dividing the data into two sets: one containing all data up to 9 GeV$^2$, whereas the other incorporates also the high-$Q^2$ tail of the BaBar data. We find that it is not possible to accommodate into the fit these BaBar data points with the same accuracy and conclude that it is difficult to explain these data in the standard scheme of OCD.

Pion-photon transition: The new QCD frontier

We perform a detailed analysis of all existing data (CELLO, CLEO, BaBar) on the pion-photon transition form factor by means of light-cone sum rules in which we include the NLO QCD radiative corrections and the twist-four contributions. The NNLO radiative correction together with the twist-six contribution are also taken into account in terms of theoretical uncertainties. Keeping only the first two Gegenbauer coefficients $a_2$ and $a_4$, we show that the $1\sigma$ error ellipse of all data up to 9 GeV$^2$ greatly overlaps with the set of pion distribution amplitudes obtained from nonlocal QCD sum rules---within the range of uncertainties due to twist-four. This remains valid also for the projection of the $1\sigma$ error ellipsoid on the $(a_2,a_4)$ plane when including $a_6$. We argue that it is not possible to accommodate the high-$Q^2$ tail of the BaBar data with the same accuracy, despite opposite claims by other authors, and conclude that the BaBar data still pose a challenge to QCD.

Evolved QCD predictions for the meson-photon transition form factors

The QCD evolution of the pion distribution amplitude (DA) $\phi_\pi(x,Q^2)$ is computed for several commonly used models. Our analysis includes the nonperturbative form predicted by light-front holographic QCD, thus combining the nonperturbative bound state dynamics of the pion with the perturbative ERBL evolution of the pion distribution amplitude. We calculate the meson-photon transition form factors for the $\pi^0$, $\eta$ and $\eta^\prime$ using the hard-scattering formalism. We point out that a widely-used approximation of replacing $\phi(x, (1-x) Q)$ with $\phi(x,Q)$ in the calculations will unjustifiably reduce the predictions for the meson-photon transition form factors. It is found that the four models of the pion DA discussed give very different predictions for the $Q^2$ dependence of the meson-photon transition form factors in the region of $Q^2>30$ GeV$^2$. More accurate measurements of these transition form factors at the large $Q^2$ region will be able to distinguish the four models of the pion DA. The rapid growth of the large $Q^2$ data for the pion-photon transition form factor reported by the BaBar Collaboration is difficult to explain within the current framework of QCD. If the BaBar data for the meson-photon transition form factor for the $\pi^0$ is confirmed, it could indicate physics beyond-the-standard model, such as a weakly-coupled elementary $C=+$ axial vector or pseudoscalar $z^0$ in the few GeV domain, an elementary field which would provide the coupling $\gamma^*$ $\gamma$ to $z^0$ to $\pi^0$ at leading twist. Our analysis thus indicates the importance of additional measurements of the pion-photon transition form factor at large $Q^2$.

Instanton effects on the twist-three light-cone distribution amplitudes of pion and light scalar mesons above 1 GeV

By use of single instanton approximation the twist-3 light-cone distribution amplitudes of pion as well as the p-wave mesons, i.e., $f_{0}(1370)$, $K_{0}^{\ast}(1430)$ and $a_{0}(1450)$ are investigated within the framework of QCD moment sum rules with inclusion of instanton effects based on the valence quark model. Results show that there is much change on light-cone distribution amplitudes by the isospin- and chirality-dependent instanton contribution compared with the instanton-free ones. We find the intanton involved twist-3 LCDAs are non-positive-definite within some range of momentum fraction and there are rapid changes at two ends of momentum fraction. To guarantee the convergence of moments by the method in this work a low instanton density should be adopted, for instance $n_{c}=1/2\rm{fm^{-4}}$ is welcomed. Possible ingredients which might have impact on the results are briefly discussed. These light-cone distribution amplitudes may be helpful to analyze exclusive heavy flavored processes.

Lattice results for low moments of light meson distribution amplitudes

As part of the UKQCD and RBC collaborations' N_f=2+1 domain-wall fermion phenomenology programme, we calculate the first two moments of the light-cone distribution amplitudes of the pseudoscalar mesons pion and kaon and the (longitudinally-polarised) vector mesons rho, K-star and phi. We obtain the desired quantities with good precision and are able to discern the expected quark-mass dependence of SU(3)-flavour breaking effects. An important ingredient of the calculation is the nonperturbative renormalisation of lattice operators using the RI'/MOM technique.

Light cone sum rules for theπ0γ*γform factor revisited

We provide a theoretical update of the calculations of the pi0-gamma*-gamma form factor in the LCSR framework, including up to six polynomials in the conformal expansion of the pion distribution amplitude and taking into account twist-six corrections related to the photon emission at large distances. The results are compared with the calculations of the B-> pi l nu decay and pion electromagnetic form factors in the same framework. Our conclusion is that the recent BaBar measurements of the pi0-gamma*-gamma form factor at large momentum transfers are consistent with QCD, although they do suggest that the pion DA may have more structure than usually assumed.

Abelian anomaly and neutral pion production

We show that in fully self-consistent treatments of the pion, namely, its static properties and elastic and transition form factors, the asymptotic limit of the product ${Q}^{2}{G}_{{\ensuremath{\gamma}}^{*}\ensuremath{\gamma}{\ensuremath{\pi}}^{0}}({Q}^{2})$, determined a priori by the interaction employed, is not exceeded at any finite value of spacelike momentum transfer. Furthermore, in such a treatment of a vector-vector contact-interaction one obtains a ${\ensuremath{\gamma}}^{*}\ensuremath{\gamma}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}$ transition form factor that disagrees markedly with all available data. We explain that the contact interaction produces a pion distribution amplitude that is flat and nonvanishing at the endpoints. This amplitude characterizes a pointlike pion bound state. Such a state has the hardest possible form factors (i.e., form factors that become constant at large momentum transfers and hence are in striking disagreement with completed experiments). However, interactions with QCD-like behavior produce soft pions, a valence-quark distribution amplitude that vanishes as $~(1\ensuremath{-}x){}^{2}$ for $x~1$, and results that agree with the bulk of existing data. Our analysis supports a view that the large-${Q}^{2}$ data obtained by the BaBar Collaboration is not an accurate measure of the ${\ensuremath{\gamma}}^{*}\ensuremath{\gamma}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}$ form factor.

The pion transition form factor and the pion distribution amplitude

Recent BaBaR data on the pion transition form factor, whose Q^2 dependence is much steeper then predicted by asymptotic Quantum Chromodynamics (QCD), have caused a renewed interest in its theoretical description. We present here a formalism based on a model independent low energy description and a high energy description based on QCD, which match at a scale Q_0. The high energy description incorporates a flat pion distribution amplitude, phi(x)=1, at the matching scale Q_0 and QCD evolution from Q_0 to Q>Q_0. The flat pion distribution is connected, through soft pion theorems and chiral symmetry, to the pion valance parton distribution at the same low scale Q_0. The procedure leads to a good description of the data, and incorporating additional twist three effects, to an excellent description of the data.