Pion Form Factor Research Articles

Gravitational form factors of pion from top-down holographic QCD

The gravitational form factors (GFFs) of pions are calculated from a top-down holographic quantum chromodynamics (QCD) approach with momentum transfer dependence for the first time. It is important because the GFFs of hadrons have information on the internal stress distribution that may provide insight into the mechanism of how QCD forms hadrons. The forward limit value of this GFFs, i.e., the D-term, was also obtained. Furthermore, in this approach, we observe the so-called glueball dominance, in which pions have gravitational interactions via infinite glueball spectra. Published by the American Physical Society 2024

Timelike meson form factors beyond the elastic region from lattice QCD

We present a calculation of the vector-isovector timelike form factors of the pion and the kaon using lattice quantum chromodynamics. We calculate two-point correlation functions with mπ∼280 MeV, extracting both the finite-volume spectrum and matrix elements for these states created from the vacuum by a vector current. After determining the coupled-channel ππ,KK¯ scattering amplitudes, we perform the necessary correction for the significant finite-volume effects present in the current matrix elements, leading to the timelike form factors. We find these to be dominated by the presence of the ρ resonance, and we extract its decay constant by an analytic continuation of the amplitudes to the resonance pole. In addition, the spacelike pion form factor is determined on the same lattice configurations, and a dispersive parametrization is used to simultaneously describe the spacelike and elastic timelike regions. Published by the American Physical Society 2024

QCD Predictions for Meson Electromagnetic Form Factors at High Momenta: Testing Factorization in Exclusive Processes.

We report the first lattice QCD computation of pion and kaon electromagnetic form factors, F_{M}(Q^{2}), at large momentum transfer up to 10 and 28 GeV^{2}, respectively. Utilizing physical masses and two fine lattices, we achieve good agreement with JLab experimental results at Q^{2}≲4 GeV^{2}. For Q^{2}≳4 GeV^{2}, our results provide abinitio QCD benchmarks for the forthcoming experiments at JLab 12GeV and future electron-ion colliders. We also test the QCD collinear factorization framework utilizing our high-Q^{2} form factors at next-to-next-to-leading order in perturbation theory, which relates the form factors to the leading Fock-state meson distribution amplitudes. Comparisons with independent lattice QCD calculations using the same framework demonstrate, within estimated uncertainties, the universality of these nonperturbative quantities.

Off-shell pion properties: Electromagnetic form factors and light-front wave functions

The off-shell pion electromagnetic form factors are explored with corresponding off-shell light-front wave functions modeled by constituent quark and antiquark. We apply the Mandelstam approach for the microscopic computation of the form factors relating the model parameters with the pion decay constant and charge radius. Analyzing the existing data on the cross sections for the Sullivan process, H1(e,e′,π+)n [H. P. Blok (Jefferson Lab Fπ Collaboration), Charged pion form factor between Q2=0.60, and 2.45 GeV2. I. Measurements of the cross section for the H1(e,e′π+)n reaction, .], we extract the off-shell pion form factor using the relation derived from the generalized Ward-Takahashi identity for the pion electromagnetic current. They are compared with our previous results [H.-M. Choi , Pion off-shell electromagnetic form factors: Data extraction and model analysis, .] from exactly solvable manifestly covariant model of a (3+1)-dimensional fermion field theory. We find that the adopted constituent quark model reproduces the extracted off-shell form factor F1(Q2,t) from the experimental data within a few percent difference and matches well with our previous theoretical simulation which exhibits a variation of about 10% for the extracted off-shell pion form factor g(Q2,t). We also identify the pion valence parton distribution function (PDF) and transverse momentum distribution (TMD) in terms of the light-front wave function and discuss their off-shell properties. Published by the American Physical Society 2024

Pion gravitational form factors in the QCD instanton vacuum. I

The pion form factors of the QCD energy-momentum tensor (EMT) are studied in the instanton liquid model (ILM) of the QCD vacuum. In this approach, the breaking of conformal symmetry is encoded in the form of stronger-than-Poisson fluctuations in the number of instantons. For the trace of the EMT, it is shown that the gluonic trace anomaly term contributes half the pion mass, with the other half coming from the quark-mass-dependent σ term. The Q2 dependence of the form factors is governed by glueball and scalar meson exchanges. For the EMT, the spin-0 (trace) and spin-2 (traceless rank-2 tensor) form factors are computed at next-to-leading order in the instanton density using effective quark operators. Relations between the gluon and quark contributions to the EMT form factors are derived. The form factors are also expressed in terms of the pion light-front wave functions in the ILM. The results at the low resolution scale of the inverse instanton size are evolved to higher scales using the renormalization group equation. The ILM results compare well with those of recent lattice QCD calculations. Published by the American Physical Society 2024

Nuclear medium meson structures from the Schwinger proper-time Nambu–Jona-Lasinio model

In this paper, we report the results of our study on the nuclear medium modifications of the meson electromagnetic form factors in the framework of the Nambu–Jona-Lasinio model with the help of the Schwinger proper-time regularization scheme to tame the loop divergence and simulate the effect of QCD confinement. In our current approach, the meson structure and nuclear medium are constructed in the same Nambu–Jona-Lasinio model at the quark level. We examine the free-space and in-medium charge radii of kaon and pion, the spacelike elastic electromagnetic form factors of kaon and pion, and their quark-sector form factors, which reflect their internal structures. By comparing our result to experimental data, we found that the free-space elastic electromagnetic form factors of the mesons are consistent with the data, while the in-medium elastic electromagnetic form factors of the mesons are found to decrease as the nuclear matter density increases, leading to a rise of meson charge radius, which is consistent with the prediction of other theoretical calculations. We also predict the axial nucleon coupling constant gA in nuclear medium computed via the Goldberger-Treiman relation, which is crucial for searching the neutrinoless double beta decay (0νββ). Published by the American Physical Society 2024

Consistency of the pion form factor and unpolarized transverse momentum dependent parton distributions beyond leading twist in the light-front quark model

We investigate the interplay among the pion’s form factor, transverse momentum dependent distributions (TMDs), and parton distribution functions (PDFs) extending our light-front quark model (LFQM) computation based on the Bakamjian-Thomas construction for the two-point function [1,2] to the three-point and four-point functions. Ensuring the four-momentum conservation at the meson-quark vertex from the Bakamjian-Thomas construction, the meson mass is taken consistently as the corresponding invariant meson mass both in the matrix element and the Lorentz factor in our LFQM computation. We achieve the current-component independence in the physical observables such as the pion form factor and delve into the derivation of unpolarized TMDs and PDFs associated with the forward matrix element. We address the challenges posed by twist-4 TMDs and exhibit the fulfillment of the sum rule. Effectively, our LFQM successfully handles the light-front zero modes and offers insights for broader three-point and four-point functions and related observables. Published by the American Physical Society 2024

Onset of scaling violation in pion and kaon elastic electromagnetic form factors

Using a symmetry-preserving truncation of the quantum field equations describing hadron properties, parameter-free predictions are delivered for pion and kaon elastic electromagnetic form factors, FP=π,K, thereby unifying them with kindred results for nucleon elastic electromagnetic form factors. Regarding positive-charge states, the analysis stresses that the presence of scaling violations in QCD entails that Q2FP(Q2) should exhibit a single maximum on Q2>0. Locating such a maximum is both necessary and sufficient to establish the existence of scaling violations. The study predicts that, for charged π, K mesons, the Q2FP(Q2) maximum lies in the neighbourhood Q2≃5 GeV2. Foreseeable experiments will test these predictions and, providing their Q2 reach meets expectations, potentially also provide details on the momentum dependence of meson form factor scaling violation.

Measurement of the Pion Form Factor with CMD-3 Detector and Its Implication to the Hadronic Contribution to Muon (g-2).

The cross section of the process e^{+}e^{-}→π^{+}π^{-} has been measured in the center-of-mass energy range from 0.32 to 1.2GeV with the CMD-3 detector at the electron-positron collider VEPP-2000. The measurement is based on an integrated luminosity of about 88 pb^{-1}, of which 62 pb^{-1} represent a complete dataset collected by CMD-3 at center-of-mass energies below 1GeV. In the dominant region near the ρ resonance a systematic uncertainty of 0.7% was achieved. The implications of the presented results for the evaluation of the hadronic contribution to the anomalous magnetic moment of the muon are discussed.

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.

QCD parameters and SM-high precisions from e+e−→ Hadrons: Summary

In this talk, I summarize the results obtained recently in Ref. [1] using the PDG 22 compilation of the e+e−→ Hadrons ⊕ the recent CMD3 data for the pion form factor. Using the gluon condensate 〈αsG2〉=(6.49±0.35)×10−2 GeV4 from heavy quark sum rules, the extracted QCD four-quark and dimension eight condensate condensates values are: ραs〈ψ¯ψ〉2=(5.98±0.64)×10−4 GeV6 and d8=(4.3±3.0)×10−2 GeV8 from the ratio R10 of Laplace sum rules to order αs4. Inversely using these estimated values of the condensates, we obtain from R10: 〈αsG2〉=(6.12±0.61)×10−2 GeV4 which leads to the average (6.40±0.30)×10−2 GeV4. Using the lowest τ-like decay moment, the mean result of Fixed Order (FO) and Contour Improved (CI) PT series within the standard OPE is: αs(Mτ)=0.3385(50)(136)syst [resp. 0.3262(37)(78)syst] to order αs4 [resp. αs5] leading to αs(MZ)=0.1207(17)(3) [resp. 0.1193(11)(3)], while the sum of the non-perturbative contribution at Mτ is: δNPV(Mτ)=(2.3±0.2)×10−2. Using the same data, one also obtains the LO hadronic vacuum polarization to the muon and τ anomalous magnetic moments: aμ|l.ohvp=(7036.5±38.9)×10−11,aτ|l.ohvp=(3494.8±24.7)×10−9 which leads to: Δaμ≡aμexp−aμth=(143±42th±22exp)×10−11 indicating about 3σ discrepancy between the SM predictions and experiment. One also finds: α(5)(MZ)|had=(2766.3±4.5)×10−5.

Yukawa potential under weak magnetic field

Weak magnetic field induced corrections for the Yukawa potential due to one pion exchange between two constituent quarks (nucleons) are presented. For that, the constant magnetic field effect on the pion propagator and on the pion form factor are taken into account. An effective gluon propagator parametrized with an effective gluon mass (Mg∼0.5 GeV) is considered. In the limit of magnetic field that is weak with respect to the constituent quark mass and pion mass, analytical and semianalytical expressions can be obtained. Different types of contributions are found, isotropic or anisotropic, dependent on the pion mass and also on the constituent quark and effective gluon masses. Overall the corrections are of the order of 2% to 5% of the Yukawa potential at distances close to 2 fm, and they decrease slower than the Yukawa potential. The anisotropic corrections are considerably smaller than the isotropic components. A sizable splitting between results due to the magnetic field dependent neutral or charged pion mass is found. Published by the American Physical Society 2024

Small Angle Initial State Radiation Analysis of the Pion Form Factor at BESIII

The time-like pion form factor (FF), i.e. the cross section of the process e+e− → π+π−, is presently a matter of intense discussion within the particle physics community. The e+e− → π+π− process is the largest contributor to the hadronic vacuum polarization (HVP) correction in the Standard Model prediction of the muon anomalous magnetic moment (aμ). Recent lattice QCD evaluations of the HVP, as well as a new measurement of the pion FF by the CMD-3 collaboration, have hinted at possible shortcomings in previous measurements, thus preventing an unambiguous interpretation of the latest experimental value of aμ. An intense effort is ongoing to provide new experimental insights into the pion FF, and the BESIII collaboration aims to contribute with a new measurement of world-leading precision. In this work, preliminary studies towards such a measurement are presented.

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.

Pion valence-quark generalized parton distribution at physical pion mass

We present the first lattice-QCD x-dependent pion valence-quark generalized parton distribution (GPD) calculated directly at physical pion mass using the Large-Momentum Effective Theory (LaMET) with next-to-next-to-leading order perturbative matching correction. We use clover fermions for the valence action on 2+1+1 flavors of highly improved staggered quarks (HISQ), generated by MILC Collaboration, with lattice spacing a≈0.09 fm and box size L≈5.5 fm; the pion two-point measurements number up to O(106) with boost momentum 1.73 GeV. The pion valence distribution is renormalized in hybrid scheme with Wilson-line mass subtraction at large distances in coordinate space, followed by a procedure to match it to the MS‾ scheme. We focus on the zero-skewness limit, where the GPD has a probability-density interpretation in the longitudinal Bjorken x and the transverse impact-parameter distributions. We take the integral of our GPD functions to generate leading moment so that we can make comparisons with past lattice-QCD and experimental determinations of the pion form factors and found consistent agreement among them. We predict the higher GPD moments and reveal x-dependent tomography of the pion for the first time using lattice QCD.

QCD parameters and SM-high precisions from e+e−→ Hadrons

Using the PDG 22 compilation of the e+e−→ Hadrons ⊕ the recent CMD3 data for the pion form factor and the value of gluon condensate 〈αsG2〉 from heavy quarkonia, we extract the value of the four-quark condensate: ραs〈ψ¯ψ〉2=(5.98±0.64)×10−4 GeV6 and the dimension eight condensate: d8=(4.3±3.0)×10−2 GeV8 from the ratio R10 of Laplace sum rules to order αs4. We show the inconsistency in using at the same time the standard SVZ value of the gluon and the vacuum saturation of the four-quark condensates. Using the previous values of the four-quark and d8 condensates, we re-extract 〈αsG2〉 from R10 to be: (6.12±0.61)×10−2 GeV4 in perfect agreement with the one from heavy quarkonia. We also use the lowest τ-like decay moment Rτee to extract the value of the QCD coupling. We obtain to O(αs4) the mean of FO (fixed order) and CI (contour improved) PT series within the standard OPE: αs(Mτ)=0.3385(50)(136)syst where the last error is an added conservative systematic from the distance of the mean to the FO and CI central values. Assuming that the αs-coefficients of the PT series grow geometrically as observed in the calculated case, we obtain to O(αs5): αs(Mτ)=0.3262(37)(78)syst. These results lead to: αs(MZ)=0.1207(17)(3) [resp. 0.1193(11)(3)] to order αs4 [resp. αs5]. The corresponding value of the sum of the non-perturbative contribution at Mτ is: δNPV(Mτ)=(2.3±0.2)×10−2. Reciprocally, using αs(Mτ), 〈αsG2〉 and d8 as inputs, we test the stability of the value of the four-quark condensate obtained from the lowest τ-like moment. We complete our analysis by updating our previous determinations of the lowest order hadronic vacuum polarization contributions to the lepton anomalies and to α(MZ2). We obtain in Table 3: aμ|l.ohvp=(7036.5±38.9)×10−11, aτ|l.ohvp=(3494.8±24.7)×10−9 and α(5)(MZ)|had=(2766.3±4.5)×10−5. This new value of aμ leads to: Δaμ≡aμexp−aμth=(142±42th±41exp)×10−11 which reduces the tension between the SM prediction and experiment.

Hadron spectra and pion form factor in dynamical holographic QCD model with anomalous 5D mass of scalar field* *Supported in part by the National Natural Science Foundation of China (NSFC) (12235016, 12221005, 11725523, 11735007, 12275108), the Strategic Priority Research Program of Chinese Academy of Sciences (XDB34030000, XDPB15), the start-up funding from University of Chinese Academy of Sciences (UCAS), and the

The simplest version of the dynamical holographic QCD model is described by adding the KKSS model action on a dilaton-graviton coupled background, in which the AdS metric is deformed by the gluon condensation and further deformed by the chiral condensation. In this framework, both the chiral symmetry breaking and linear confinement can be realized. The light-flavor hadron spectra and the pion form factor were investigated, but it was difficult to reconcile the light-flavor hadron spectra and pion form factor. By considering the anomalous 5-dimension mass correction of the scalar field from QCD running coupling, it is found that the light flavor hadron spectra and pion form factor can be described well simultaneously. In particular, the ground state and lower excitation states of the scalar, pseudo scalar, and axial vector meson spectra are improved. However, the vector meson spectra are not sensitive to the anomalous 5-dimension mass correction of the scalar field.

Glimpse into the pion gravitational form factor

We provide a novel approach to calculate the gravitational form factor of pion under the ladder approximation of the Bethe-Salpeter equation, with contact interactions. Central to this approach is a symmetry-preserving treatment of the dressed $\ensuremath{\pi}\ensuremath{\pi}$ amplitude, which shows explicitly the contributions from intrinsic quarks and bound states, the latter being necessary to produce the $D$-term of pion in the soft-pion limit. The approach we provide in this work can be applied to many processes of physical significance.

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.