Pion Electromagnetic Form Factor Research Articles

Next-to-Next-to-Leading-Order QCD Corrections to Pion Electromagnetic Form Factors

We calculate the next-to-next-to-leading-order (NNLO) QCD radiative correction to the pion electromagnetic form factor with large momentum transfer. We explicitly verify the validity of the collinear factorization to two-loop order for this observable and obtain the respective IR-finite two-loop hard-scattering kernel in the closed form. The NNLO QCD correction turns out to be positive and significant. Incorporating this new ingredient of correction, we then make a comprehensive comparison between the finest theoretical predictions and numerous data for both space- and timelike pion form factors. Our phenomenological analysis provides a strong constraint on the second Gegenbauer moment of the pion light-cone distribution amplitude obtained from recent lattice QCD studies. Published by the American Physical Society 2024

Pion and kaon electromagnetic and gravitational form factors

A unified set of predictions for pion and kaon elastic electromagnetic and gravitational form factors is obtained using a symmetry-preserving truncation of each relevant quantum field equation. A key part of the study is a description of salient aspects of the dressed graviton + quark vertices. The calculations reveal that each meson’s mass radius is smaller than its charge radius, matching available empirical inferences; and meson core pressures are commensurate with those in neutron stars. The analysis described herein paves the way for a direct calculation of nucleon gravitational form factors.

Pseudoscalar current and covariance with the light-front approach

Quantum Field Theory (QFT) is used to describe the physics of particles in terms of their fundamental constituents. The Light-Front Field Theory (LFFT), introduced by Paul Dirac in 1949 [1], is an alternative approach to solve some of the problems that arise in quantum field theory. The LFFT is similar to the Equal Time Quantum Field Theory (EQT), however, some particularities are not, such as the loss of covariance in the light-front. Pion electromagnetic form factor is studied in this work at lower and higher momentum transfer regions to explore the constituent quark models and the differences among these and other models. The electromagnetic current is calculated with both the “plus” and “minus” components in the light-front approach. The results are compared with other models, as well as with experimental data.

Electromagnetic time-like form factors

We present the calculation of charged pion and kaon electromagnetic form factors in the time-like regime using Schwinger-Dyson equations and the Poincare covariant Bethe-Salpeter equations within an SU(2) isospin symmetric limit. To accurately represent the behavior of a time-like photon, we have incorporated non-valence contributions into the system of equations, enabling the decays ρ → ππ and φ → KK. The inclusion of these decay mechanisms is essential for capturing the expected behavior of the electromagnetic pion and kaon form factors. Our results for the form factors reasonably reproduce the experimental data for a time-like photon.

Momentum dependence of mixing in the pion vector form factor and its effect on * *Supported in part by the Fundamental Research Funds for the Central Universities (FRF-BR-19-001A), and the National Natural Science Foundation of China (11975028, 11974043)

The inclusion of the mixing effect is essential for a precise description of the pion electromagnetic form factor in the process, which quantifies the two-pion contribution to the anomalous magnetic moment of muon . In this study, we analyze the momentum dependence of mixing by considering loop contributions at the next-to-leading order in expansion within the framework of resonance chiral theory. We revisit a previous study [Y. H. Chen, D. L. Yao, and H. Q. Zheng, Commun. Theor. Phys. 69, 1 (2018)] and consider the contribution arising from the kaon mass splitting in the kaon loops and latest experimental data. We perform two types of fits (with momentum-independent or momentum-dependent mixing amplitude) to describe and data within the energy region of 600900 MeV and decay width of . Furthermore, we compare their results. Our findings indicate that the momentum-independent and momentum-dependent mixing schemes provide appropriate descriptions of the data. However, the momentum-dependent scheme exhibits greater self-consistency, considering the reasonable imaginary part of the mixing matrix element obtained. Regarding the contribution to the anomalous magnetic moment of the muon, , the results obtained from the fits considering the momentum-dependent mixing amplitude are in good agreement with those obtained without incorporating the momentum dependence of mixing, within the margin of errors. Furthermore, based on the fitted values of the relevant parameters, we observe that the decay width of is predominantly influenced by the mixing effect.

Study pion distribution amplitudes from the electromagnetic form factor by data-driven dispersion relation

We study the pion electromagnetic form factor in the modulus squared dispersion relation, and do the model independent extraction of the chiral mass embedded in the sub-leading twist light-cone distribution amplitude. The motivation of this work is the recent measurement of timelike form factor in the resonant regions, which makes up the piece lacking solid QCD-based calculation. With the perturbative QCD calculation up to next-to-leading-order of strong coupling and twist four level of meson distribution amplitudes, we obtain the chiral mass of pion meson as m_0^pi (1 , textrm{GeV}) = 1.31^{+0.27}_{-0.30} , textrm{GeV}. Due to the chiral enhancement contribution from the sub-leading twist distribution amplitudes in the perturbative QCD calculation, we can not extract the lowest Gegenbauer moments of pion from the study of electromagnetic form factor. This problem could be settled down with the foresee Belle-II and BESIII measurement of the transition form factor especially in the large momentum transfers regions.

Pion observables calculated in Minkowski and Euclidean spaces with Ansätze for quark propagators

We study two quark--propagator meromorphic Ans\"atze that admit clear connection between calculations in Euclidean space and Minkowski spacetime. The connection is established through a modified Wick rotation in momentum space, where the integration contour along the imaginary axis is adequately deformed. The Ans\"atze were previously proposed in the literature and fitted to Euclidean lattice QCD data. The generalized impulse approximation is used to calculate the pion transition form factor and electromagnetic form factor, correcting an earlier result. The pion decay constant and distribution amplitude are also calculated. The latter is used to deduce the asymptotic behavior of the form factors. Such an asymptotic behavior is compared with those obtained directly from the generalized impulse approximation and the causes of differences are pointed out.

The electromagnetic Sigma-to-Lambda transition form factors with coupled-channel effects in the space-like region

Using dispersion theory, the electromagnetic Sigma-to-Lambda transition form factors are expressed as the product of the pion electromagnetic form factor and the Sigma {bar{varLambda }}rightarrow pi pi scattering amplitudes with the latter estimated from SU(3) chiral perturbation theory including the baryon decuplet as explicit degrees of freedom. The contribution of the K{bar{K}} channel is also taken into account and the pi pi –K{bar{K}} coupled-channel effect is included by means of a two-channel Muskhelishvili–Omnès representation. It is found that the electric transition form factor shows a significant shift after the inclusion of the K{bar{K}} channel, while the magnetic transition form factor is only weakly affected. However, the K{bar{K}} effect on the electric form factor is obscured by the undetermined coupling h_A in the three-flavor chiral Lagrangian. The error bands of the Sigma-to-Lambda transition form factors from the uncertainties of the couplings and low-energy constant in three-flavor chiral perturbation theory are estimated by a bootstrap sampling method.

Search for Damped Oscillating Structures from Charged Pion Electromagnetic Form Factor Data

The damped oscillating structures recently revealed by a three parametric formula from the proton “effective” form factor data extracted of the measured total cross section σtotbare(e+e−→pp¯) still seem to have an unknown origin. The conjectures of their direct manifestation of the quark-gluon structure of the proton indicate that they are not specific only of the proton and neutron, but they have to be one’s own, similar to other hadrons. Therefore, the oscillatory structures from the charged pion electromagnetic form factor timelike data, extracted of the process e+e−→π+π− are investigated by using the same procedure as in the case of the proton. The analysis shows the appearance of the oscillating structures in the description of the charged pion electromagnetic form factor timelike data by three parametric formula with a rather large value of χ2/ndf, while the description of the data by the physically well-founded Unitary and Analytic model has not revealed any damped oscillating structures. From the obtained result on the most simple object of strong interactions, one can conclude that damped oscillating structures received from the “effective” proton form factor data are probably generated by a utilization of the improper three parametric formula which does not describe these data with sufficient precision.

Timelike electromagnetic kaon form factor

We compute the electromagnetic charged kaon form factor in the timelike region by employing a Poincar\'e covariant formalism of the Bethe-Salpeter equation to study quark-antiquark bound states in conjunction with the Schwinger-Dyson equation for the quark propagator. Following a recent kindred calculation of the timelike electromagnetic pion form factor, we include the most relevant intermediate composite particles permitted by their quantum numbers in the interaction kernel to allow for a decay mechanism for the resonances involved. This term augments the usual gluon mediated interaction between quarks. For a sufficiently low energy timelike probing photon, the electromagnetic form factor is saturated by the $\ensuremath{\rho}(770)$ and $\ensuremath{\phi}(1020)$ resonances. We assume $SU(2)$ isospin symmetry throughout. Our results for the absolute value squared of the electromagnetic form factor agree qualitatively rather well and quantitatively moderately so with available experimental data.

Pion and nucleon relativistic electromagnetic four-current distributions

The quantum phase-space approach allows one to define relativistic spatial distributions inside a target with arbitrary spin and arbitrary average momentum. We apply this quasiprobabilistic formalism to the whole electromagnetic four-current operator in the case of spin-0 and spin-$\frac{1}{2}$ targets, study in detail the frame dependence of the corresponding spatial distributions, and compare our results with those from the light-front formalism. While former works focused on the charge distributions, we extend here the investigations to the current distributions. We clarify the role played by the Wigner rotation and argue that electromagnetic properties are most naturally understood in terms of Sachs form factors, contrary to what the light-front formalism previously suggested. Finally, we illustrate our results using the pion and nucleon electromagnetic form factors extracted from experimental data.

Relativistic composite-particle theory of the gravitational form factors of the pion: Quantitative results

We use a version of the instant-form relativistic quantum mechanics of composite systems to obtain the gravitational form factors of the pion in a common approach to its electroweak and gravitational properties. In the preceding work [Phys. Rev. D 103, 014029 (2021)], we formulated the mathematical background, presented the principal scheme of calculation and testified the obtained qualitative results to satisfy the general constraints given by the principles of the theory of hadron structure. In the present work, we give the detailed calculation of the gravitational form factors in the large range of momentum transfer, their static limits, and the slopes at zero value, the mean-square mass and mechanical radii of the pion. Now we take into account the gravitational structure of the constituent quarks. We show that the results are almost insensitive to the type of model two-quark wave function in a close analogy to the case of the pion electromagnetic form factor. We present a correct calculation of the form factor $D$ and corresponding matrix element of the energy-momentum tensor, going beyond the scope of the modified impulse approximation. Most of the parameters that we use for the calculation had been fixed even earlier in our works on the pion electromagnetic form factors. The only free parameter is the $D$-term of the constituent quark, which we fix by fitting the result for the slope at zero of the normalized to pion $D$-term form factor $D$ of the pion, to a chosen experimental value.

Timelike behavior of the pion electromagnetic form factor in the functional formalism

The electromagnetic form factor of the pion is calculated within the use of functional formalism. We develop integral representation for the minimal set of Standard Model Green's functions and derive the dispersion relation for the form factor in the two-flavor QCD isospin limit ${m}_{u}={m}_{d}$. We use the dressed quark propagator as obtained from the gap equation in Minkowski space and within the Dyson-Schwinger equations formalism to derive the approximate dispersion relation for the form factor for the first time. We evaluate the form factor for the spacelike as well as for the timelike momentum in the presented formalism. A new Nakanishi-like form of integral representation is proved on the basis of the vector Bethe-Salpeter equation for the quark-photon vector with a ladder-rainbow kernel. The gauge technique turns out to be a part of the entire structure of the vertex. In the analytic approach presented here, it is shown that a large amount of the $\ensuremath{\rho}$-meson peak in the cross section ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ is governed by the gauge invariance of QED/QCD---i.e., by a gauge-technique-constructed quark-photon vertex. This approximation naturally explains the broad shape of the $\ensuremath{\rho}$-meson peak.

Electromagnetic Pion form factor in a deformed background

This work discusses the electromagnetic (EM) pion form factor ($\pi FF$) in a deformed AdS geometry. We consider the conformal dimension of the hadron bulk field defined by the scaling dimension of the $q\,\bar{q}$ operator instead of the twist. We also compute the pion EM radius and compare it with the experimental data, finding a relative error of $2\,\%$.

Kaon electromagnetic form factors in dispersion theory

The electromagnetic form factors of charged and neutral kaons are strongly constrained by their low-energy singularities, in the isovector part from two-pion intermediate states and in the isoscalar contribution in terms of omega and phi residues. The former can be predicted using the respective pi pi rightarrow {{bar{K}}} K partial-wave amplitude and the pion electromagnetic form factor, while the latter parameters need to be determined from electromagnetic reactions involving kaons. We present a global analysis of time- and spacelike data that implements all of these constraints. The results enable manifold applications: kaon charge radii, elastic contributions to the kaon electromagnetic self energies and corrections to Dashen’s theorem, kaon boxes in hadronic light-by-light (HLbL) scattering, and the phi region in hadronic vacuum polarization (HVP). Our main results are: langle r^2rangle _text {c}=0.359(3),text {fm}^2, langle r^2rangle _text {n}=-0.060(4),text {fm}^2 for the charged and neutral radii, epsilon =0.63(40) for the elastic contribution to the violation of Dashen’s theorem, a_mu ^{Ktext {-box}}=-0.48(1)times 10^{-11} for the charged kaon box in HLbL scattering, and a_mu ^text {HVP}[K^+K^-, le 1.05,text {GeV}]=184.5(2.0)times 10^{-11}, a_mu ^text {HVP}[K_SK_L, le 1.05,text {GeV}]=118.3(1.5)times 10^{-11} for the HVP integrals around the phi resonance. The global fit to {{bar{K}}} K gives {{bar{M}}}_phi =1019.479(5),text {MeV}, {bar{ varGamma }}_phi =4.207(8),text {MeV} for the phi resonance parameters including vacuum-polarization effects.

Regularization dependence of pion generalized parton distributions * *Supported by the National Natural Science Foundation of China (11775118)

Pion generalized parton distributions are calculated within the framework of the Nambu–Jona-Lasinio model using different regularization schemes, including the proper time regularization scheme, the three-dimensional (3D) momentum cutoff scheme, the four-dimensional momentum cutoff scheme, and the Pauli-Villars regularization scheme. Furthermore, we check the theoretical constraints of pion generalized parton distributions required by the symmetries of quantum chromodynamics in different regularization schemes. The diagrams of pion parton distribution functions are plotted, in addition, we evaluate the Mellin moments of generalized parton distributions, which are related to the electromagnetic and gravitational form factors of pion. Pion generalized parton distributions are continuous but not differential at , when considering the effect of the contact contribution term, generalized parton distributions become not continuous at in all the four regularization schemes. Generalized parton distributions in impact parameter space are considered, the width distribution of u quark in the pion and the mean-squared are calculated. The light-front transverse-spin distributions are studied when quark polarized in the light-front-transverse direction, the transverse-spin density is no longer symmetric around , the peaks shift to , we compare the average transverse shift and in different regularization schemes. The light-cone energy radius and the light-cone charge radius are also evaluated, we found that in the proper time regularization scheme the values of these quantities were the largest, in the 3D momentum cutoff scheme they were the smallest.

Pion form factor from an AdS deformed background

We consider a bottom-up AdS/QCD model with a conformal exponential deformation ekIz2 on a Lorentz invariant AdS background, where kI stands for the scale kπ that fixes confinement in the pion case and kγ for the kinematical energy scale associated with the virtual photon. In this model we assume the conformal dimension associated with the operator that creates pions at the boundary as Δ=3, as in the original bottom-up AdS/QCD proposals. Regarding the geometric slope related to photon field kγ, we analyze two cases: constant and depending on the transferred momentum q. In these two cases we computed the electromagnetic pion form factor as well as the pion radius. We compare our results with experimental data as well as other theoretical (holographic and non-holographic) models. In particular, for the momentum dependent scale, we find good agreement with the available experimental data as well as non-holographic models.

Light mesons with one dynamical gluon on the light front

We obtain the light meson mass spectroscopy from the light-front quantum chromodynamics (QCD) Hamiltonian, determined for their constituent quark-antiquark and quark-antiquark-gluon Fock components, together with a three-dimensional confinement. The eigenvectors of the light-front effective Hamiltonian provide a good quality description of the pion electromagnetic form factor, decay constant, and the valence quark distribution functions following QCD scale evolution. We also show that the pion's gluon densities can be probed through the pion-nucleus induced J/ψ production data. Our pion parton distribution functions provide excellent agreement with J/ψ production data from widely different experimental conditions.

Elucidating the ρ-meson’s role as intermediate resonance in the time-like electromagnetic pion form factor

Motivated by the planned measurements of the time-like electromagnetic proton form factors an exploratory study of the time-like electromagnetic pion form factor is presented in a formalism which describes mesons as Poincaré-invariant bound states. In the respective quark interaction kernel, beyond the gluon-intermediated interactions for valence-type quarks, non-valence effects are included by allowing the pions to couple back in a self-consistent manner to the quarks. Consequently, the opening of the dominant ρ decay channel, ρ → ππ, and the presence of a multi-particle branch cut, setting in when the two-pion threshold is crossed, are included consistently, first in the correspondingly calculated quark-photon vertex, and then consequently in the time-like electromagnetic pion form factor. The obtained results are in agreement with the available experimental data and provide further evidence for the effcacy of a vector-meson dominance model. Effects of going beyond the here employed isospin-symmetric limit are discussed. Last but not least, an outlook is provided on how to include in Poincaré-covariant Faddeev approaches the effects of intermediate resonances in the calculation of baryon properties.

Pion form factor and charge radius from lattice QCD at the physical point

We present our results on the electromagnetic form factor of pion over a wide range of $Q^2$ using lattice QCD simulations with Wilson-clover valence quarks and HISQ sea quarks. We study the form factor at the physical point with a lattice spacing $a=0.076$ fm. To study the lattice spacing and quark mass effects, we also present results for 300 MeV pion at two different lattice spacings $a=0.04$ and 0.06 fm. The lattice calculations at the physical quark mass appear to agree with the experimental results. Through fits to the form factor, we estimate the charge radius of pion for physical pion mass to be $\langle r_{\pi}^2 \rangle=0.42(2)~{\rm fm}^2$.