Nonlocal Condensates Research Articles

Cross-link relations between π and ρ -meson channels and the QCD vacuum

We discuss cross-link relations between the $\ensuremath{\pi}$ and $\ensuremath{\rho}$-meson channels emerging from two different descriptions of the QCD vacuum: instanton physics and QCD sum rules with nonlocal condensates (NLCs). We derive in both schemes an intriguing linear relation between the $\ensuremath{\pi}$ and the ${\ensuremath{\rho}}^{\ensuremath{\parallel}}$-meson distribution amplitudes in terms of their conformal coefficients and work out the specific impact of the scalar NLC in these two channels. Using a simple model with Gaussian decay of the scalar NLC, we are able to relate it to the moments of the pion nonsinglet parton distribution function measurable in experiment---a highly nontrivial result. The implications for the pion and the ${\ensuremath{\rho}}^{\ensuremath{\parallel}}$-meson distribution amplitudes entailed by the obtained cross-link relations are outlined in terms of two generic scenarios.

Chimera distribution amplitudes for the pion and the longitudinally polarized ρ-meson

Using QCD sum rules with nonlocal condensates, we show that the distribution amplitude of the longitudinally polarized ρ-meson may have a shorttailed platykurtic profile in close analogy to our recently proposed platykurtic distribution amplitude for the pion. Such a chimera distribution de facto amalgamates the broad unimodal profile of the distribution amplitude, obtained with a Dyson–Schwinger equations-based computational scheme, with the suppressed tails characterizing the bimodal distribution amplitudes derived from QCD sum rules with nonlocal condensates. We argue that pattern formation, emerging from the collective synchronization of coupled oscillators, can provide a single theoretical scaffolding to study unimodal and bimodal distribution amplitudes of light mesons without recourse to particular computational schemes and the reasons for them.

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.

Rho Meson Distribution Amplitudes from QCD Sum Rules with Nonlocal Condensates

The leading-twist distribution amplitude for the longitudinal rho-meson was studied using QCD Sum Rules with nonlocal condensates and a spectral density which includes next-to-leading order radiative corrections. The obtained profile is compared with results from standard QCD sum rules, lattice QCD, holographic QCD, a light-front quark model, and the instanton liquid model. Preliminary estimates for the first two moments of the transverse ρ-meson distribution amplitude are also given.

Endpoint behavior of the pion distribution amplitude in QCD sum rules with nonlocal condensates

Starting from the QCD sum rules with nonlocal condensates for the pion distribution amplitude, we derive another sum rule for its derivative and its "integral" derivatives---defined in this work. We use this new sum rule to analyze the fine details of the pion distribution amplitude in the endpoint region $x\sim 0$. The results for endpoint-suppressed and flat-top (or flat-like) pion distribution amplitudes are compared with those we obtained with differential sum rules by employing two different models for the distribution of vacuum-quark virtualities. We determine the range of values of the derivatives of the pion distribution amplitude and show that endpoint-suppressed distribution amplitudes lie within this range, while those with endpoint enhancement---flat-type or CZ-like---yield values outside this range.

Pion form factor in the NLC QCD SR approach

We present results of a calculation of the electromagnetic pion form factor within a framework of QCD Sum Rules with nonlocal condensates and using a perturbative spectral density which includes \mathcal{O}(\alpha_s) contributions.

Pion form factor in QCD sum rules, local duality approach, and [formula omitted] fractional analytic perturbation theory

Using the results on the electromagnetic pion Form Factor (FF) obtained in the $O(\alpha_s)$ QCD sum rules with non-local condensates \cite{BPS09} we determine the effective continuum threshold for the local duality approach. Then we apply it to construct the $O(\alpha_s^2)$ estimation of the pion FF in the framework of the fractional analytic perturbation theory.

Nonlocal condensates and pion form factor

We review a nonlocal-condensate approach and its application in QCD sum rules for the spacelike electromagnetic pion form factor. It is shown that the nonlocality of the condensates is a key point to include nonperturbative contributions to the pion form factor.

Nonlocal quark condensate and photon distribution amplitude

The magnetic susceptibility of the quark condensate and the light-cone photon distribution amplitude of the leading twist is studied in the leading order of the perturbative expansion within the nonlocal condensates framework.

QCD sum rules with nonlocal condensates and the spacelike pion form factor

We present a detailed investigation of the spacelike pion's electromagnetic form factor using three-point QCD sum rules that exclusively involve nonlocal condensates. Our main methodological tools are a spectral density which includes $O({\ensuremath{\alpha}}_{s})$ corrections, a suitably improved Gaussian ansatz to model the distribution of the average momentum of quarks in the QCD vacuum, and a perturbative scheme that avoids Landau singularities. Using this framework, we obtain predictions for the pion form factor together with error estimates originating from intrinsic theoretical uncertainties owing to the perturbative expansion and the nonperturbative method applied. We also discuss our results in comparison with other calculations, in particular, with those following from the AdS/QCD correspondence. We find good agreement of our predictions with measurements in the range of momenta covered by the existing experimental data between 1 and $10\text{ }\text{ }{\mathrm{GeV}}^{2}$.

Virtualities of quark and gluon in QCD vacuum

The non-local vacuum condensates of quantum chromodynamics (QCD) describe the distributions of quarks and gluons in the non-perturbative QCD vacuum state. Physically, this means that vacuum quarks and gluons have a nonzero mean-squared momentum in the vacuum, called virtuality. The quark virtuality is given by the ratio of the local quark-gluon mixed vacuum condensate to the quark local vacuum condensate. The gluon virtuality is expressed by gluon vacuum condensates and four-quark vacuum condensates. We study the two virtualities by solving Dyson-Schwinger Equations and calculating quark and gluon vacuum condensates. Our theoretical results for quark virtuality are in good agreement with many other theoretical model predictions such as QCD sum rules and lattice QCD calculations. Our calculation on gluon virtuality is initial and the results are quite interesting.

PION QUARK STRUCTURE IN QCD

We describe the present status of the pion distribution amplitude as it originated from two sources: (i) a nonperturbative approach, based on QCD sum rules with nonlocal condensates and (ii) a NLO QCD analysis of the CLEO data on Fγγ*π(Q2), supplemented by the recent high-precision lattice calculations of the second moment of the pion distribution amplitude.

Deep inside the pion. Reconciling QCD theory with data *

AbstractRecent developments in the QCD description of the pion structure are reviewed. The CLEO pion‐photon transition data analysis favors a distribution amplitude for the pion that is double‐humped but endpoint‐suppressed. After a short outline of the derivation of this amplitude from QCD sum rules with nonlocal condensates, we present the fully fledged analysis of the CLEO data prefaced by predictions for the Fγρπ form factor and commenting on the inherent theoretical uncertainties due to higher twists and NNLO perturbative corrections. We supplement our discussion by considering within QCD factorization theory, the electromagnetic pion form factor at NLO accuracy on one hand, and diffractive di‐jets production on the other, comparing our predictions with the respective experimental data from JLab and the Fermilab E791 collaboration. In all cases, the agreement is impressive.

Baryons as Fock states of 3,5, ... quarks

Recent developments in the QCD description of the pion structure are reviewed. The CLEO pion-photon transition data analysis favors a distribution amplitude for the pion that is double-humped but endpoint-suppressed. After a short outline of the derivation of this amplitude from QCD sum rules with nonlocal condensates, we present the fully fledged analysis of the CLEO data prefaced by predictions for the Fγρπ form factor and commenting on the inherent theoretical uncertainties due to higher twists and NNLO perturbative corrections. We supplement our discussion by considering within QCD factorization theory, the electromagnetic pion form factor at NLO accuracy on one hand, and diffractive di-jets production on the other, comparing our predictions with the respective experimental data from JLab and the Fermilab E791 collaboration. In all cases, the agreement is impressive.

Publisher's Note: Pion form factor in QCD: From nonlocal condensates to next-to-leading-order analytic perturbation theory [Phys. Rev. D 70, 033014 (2004)

Received 22 September 2004DOI:https://doi.org/10.1103/PhysRevD.70.079906©2004 American Physical Society

Pion form factor in QCD: From nonlocal condensates to next-to-leading-order analytic perturbation theory

We present an investigation of the pion's electromagnetic form factor ${F}_{\ensuremath{\pi}}{(Q}^{2})$ in the spacelike region utilizing two new ingredients: (i) a double-humped, end-point-suppressed pion distribution amplitude derived before via QCD sum rules with nonlocal condensates---found to comply at the $1\ensuremath{\sigma}$ level with the CLEO data on the $\ensuremath{\pi}\ensuremath{\gamma}$ transition---and (ii) analytic perturbation theory at the level of parton amplitudes for hadronic reactions. The computation of ${F}_{\ensuremath{\pi}}{(Q}^{2})$ within this approach is performed at the next to leading order (NLO) of QCD perturbation theory (standard and analytic), including the evolution of the pion distribution amplitude at the same order. We consider the NLO corrections to the form factor in the $\overline{\mathrm{MS}}$ scheme with various renormalization scale settings and also in the ${\ensuremath{\alpha}}_{V}$ scheme. We find that using standard perturbation theory, the size of the NLO corrections is quite sensitive to the adopted renormalization scheme and scale setting. The main results of our analysis are the following: (i) Replacing the QCD coupling and its powers by their analytic images, both dependencies are diminished and the predictions for the pion form factor are quasi-scheme- and scale-setting independent. (ii) The magnitude of the factorized pion form factor, calculated with the aforementioned pion distribution amplitude, is only slightly larger than the result obtained with the asymptotic one in all considered schemes. (iii) Including the soft pion form factor via local duality and ensuring the Ward identity at ${Q}^{2}=0,$ we present predictions that are in remarkably good agreement with the existing experimental data both in trend and magnitude.

CLEO and E791 data: a smoking gun for the pion distribution amplitude?

The CLEO experimental data on the πγ transition are analyzed to next-to-leading order accuracy in QCD perturbation theory using light-cone QCD sum rules. By processing these data along the lines proposed by Schmedding and Yakovlev, and recently revised by us, we obtain new constraints for the Gegenbauer coefficients a2 and a4, as well as for the inverse moment 〈x−1〉π of the pion distribution amplitude (DA). The former determine the pion DA at low momentum scale, the latter is crucial in calculating pion form factors. From the results of our analysis we conclude that the data confirm the end-point suppressed shape of the pion DA we previously obtained with QCD sum rules and nonlocal condensates, while the exclusion of both the asymptotic and the Chernyak–Zhitnitsky DAs is reinforced at the 3σ- and 4σ-level, respectively. The reliability of the main results of our updated CLEO data analysis is demonstrated. Our pion DA is checked against the di-jets data from the E791 experiment, providing credible evidence for our results far more broadly.

Vanishing dynamical quark mass at zero virtuality?

We show that the dynamical quark mass in effective nonlocal models can vanish at zero virtuality of the quark as $M(p^2)\propto p^2$. Our arguments follow from the constrained-instanton model of the QCD vacuum and from QCD sum rules calculations with nonlocal condensates. The discussed models also lead to analyticity of $M(p^2)$ in the vicinity of zero.

QCD-based pion distribution amplitudes confronting experimental data

We use QCD sum rules with non-local condensates to recalculate more accurately the moments and their confidence intervals of the twist-2 pion distribution amplitude including radiative corrections. We are thus able to construct an admissible set of pion distribution amplitudes which define a reliability region in the a 2, a 4 plane of the Gegenbauer polynomial expansion coefficients. We emphasize that models like that of Chernyak and Zhitnitsky, as well as the asymptotic solution, are excluded from this set. We show that the determined a 2, a 4 region strongly overlaps with that extracted from the CLEO data by Schmedding and Yakovlev and that this region is also not far from the results of the first direct measurement of the pion valence quark momentum distribution by the Fermilab E791 Collaboration. Comparisons with recent lattice calculations and instanton-based models are briefly discussed.

New shapes of light-cone distributions of the transversely polarized $\rho$ -mesons

We re-analyze the leading twist light-cone distributions for transversely polarized \rho-, \rho'- and b_1-mesons in the framework of QCD sum rules with nonlocal condensates. Using different kinds of sum rules to obtain reliable predictions, we estimate the 2-, 4-, 6- and 8-th moments for transversely polarized \rho- and \rho'-meson distributions and re-estimate tensor couplings f^T_{\rho,\rho',b_1}. We stress that the results of standard sum rules also support our estimation of the second moment of the transversely polarized \rho-meson distribution. New models for light-cone distributions of these mesons are constructed. Phenomenological consequences from these distributions are briefly discussed. Our results are compared with those found by Ball and Braun (1996), and the latter is shown to be incomplete.