Electromagnetic time-like form factors


  • Angel Miramontes Universidad Michoacana de San Nicolás de Hidalgo
  • Adnan Bashir Universidad Michoacana de San Nicolás de Hidalgo




Meson form factors; Schwinger-Dyson equations; Bethe-Salpeter equations


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.


R. R. Akhmetshin et al., High-statistics measurement of the pion form factor in the ρ-meson energy range with the CMD- 2 detector, Phys. Lett. B 648 (2007) 28, https://doi.org/10.1016/j.physletb.2007.01.073

V. M. Aul’chenko et al., Measurement of the pion formfactor in the range 1.04-1.38-GeV with the CMD-2 detector, JETP Lett. 82 (2005) 743, https://doi.org/10.1134/1.2175241

L. Barkov et al., Electromagnetic Pion Form-Factor in the Timelike Region, Nucl. Phys. B 256 (1985) 365, https://doi.org/10.1016/0550-3213(85)90399-2

D. Bisello et al., The Pion Electromagnetic Form-factor in the Timelike Energy Range 1.35≤ √ s ≥2.4-GeV, Phys. Lett. B 220 (1989) 321, https://doi.org/10.1016/0370-2693(89)90060-9

C. N. Brown et al., Coincidence electroproduction of charged pions and the pion form-factor, Phys. Rev. D 8 (1973) 92, https://doi.org/10.1103/PhysRevD.8.92

C. J. Bebek et al., Further measurements of forwardchargedpion electroproduction at large κ 2 , Phys. Rev. D 9 (1974) 1229, https://doi.org/10.1103/PhysRevD.9.1229

T. Horn et al., Determination of the Charged Pion Form Factor at Q ∗∗2 = 1.60 and 2.45 (GeV/c)∗∗2 , Phys. Rev. Lett. 97 (2006) 192001, https://doi.org/10.1103/PhysRevLett.97.192001

C. J. Bebek et al., Measurement of the pion form-factor up to q 2 = 4-GeV2 , Phys. Rev. D 13 (1976) 25, https://doi.org/10.1103/PhysRevD.13.25

S. R. Amendolia et al., A Measurement of the Kaon Charge Radius, Phys. Lett. B 178 (1986) 435, https://doi.org/10.1016/0370-2693(86)91407-3

E. B. Dally et al., Direct measurement of the negative kaon form-factor, Phys. Rev. Lett. 45 (1980) 232, https://doi.org/10.1103/PhysRevLett.45.232

C. J. Shultz, J. J. Dudek, and R. G. Edwards, Excited meson radiative transitions from lattice QCD using variationally optimized operators, Phys. Rev. D 91 (2015) 114501, https://doi.org/10.1103/PhysRevD.91.114501

X. Gao et al., Pion form factor and charge radius from lattice QCD at the physical point, Phys. Rev. D 104 (2021) 114515, https://doi.org/10.1103/PhysRevD.104.114515

P. Maris and P. C. Tandy, The π, K+, and K0 electromagnetic form-factors, Phys. Rev. C 62 (2000) 055204, https://doi.org/10.1103/PhysRevC.62.055204

L. Chang et al., Pion electromagnetic form factor at spacelike momenta, Phys. Rev. Lett. 111 (2013) 141802, https://doi.org/10.1103/PhysRevLett.111.141802

H. J. Kwee and R. F. Lebed, Pion form-factors in holographic QCD, JHEP 01 (2008) 027, https://doi.org/10.1088/1126-6708/2008/01/027

J. Bijnens and P. Talavera, Pion and kaon electromagnetic formfactors, JHEP 03 (2002) 046, https://dx.doi.org/10.1088/1126-6708/2002/03/046

Z. Abidin and P. T. P. Hutauruk, Kaon form factor in holographic QCD, Phys. Rev. D 100 (2019) 054026, https://doi.org/10.1103/PhysRevD.100.054026

C. Burden, C. Roberts, and M. Thomson, Electromagnetic form-factors of charged and neutral kaons, Phys. Lett. B 371 (1996) 163, https://doi.org/10.1016/0370-2693(96)00006-8

X. Wang et al., Pion scalar, vector and tensor form factors from a contact interaction Phys. Rev. D 106 (2022) 054016, https://link.aps.org/doi/10.1103/PhysRevD.106.054016

A. Miramontes et al., Pion and Kaon box contribution to aˆI 1 4HLbL, Phys. Rev. D 105 (2022) 074013, https://doi.org/10.1103/PhysRevD.105.074013

L. Albino et al., Pseudoscalar mesons: Light front wave functions, GPDs, and PDFs, Phys. Rev. D 106 (2022) 034003, https://doi.org/10.1103/PhysRevD.106.034003

R. J. Hernandez-Pinto et al., Electromagnetic Form Factors and Charge Radii of Pseudoscalar and Scalar Mesons: A Comprehensive Contact Interaction Analysis, Phys. Rev. D 107 (2023) 054002, https://link.aps.org/doi/10.1103/PhysRevD.107.054002

D. Stamen et al., Kaon electromagnetic form factors in dispersion theory, Eur. Phys. J. C 82 (2022) 432, https://doi.org/10.1140/epjc/s10052-022-10348-3

D. Melikhov, O. Nachtmann, and T. Paulus, The Pion formfactor at time - like momentum transfers in a dispersion approach (2002). https://doi.org/10.48550/arXiv.hep-ph/0209151

C. Bruch, A. Khodjamirian, and J. H. Kuhn, Modeling the pion and kaon form factors in the timelike region, Eur. Phys. J. C 39 (2005) 41, https://doi.org/10.1140/epjc/s2004-02064-3

X. Feng et al., Timelike pion form factor in lattice QCD, Phys. Rev. D 91 (2015) 054504, https://doi.org/10.1103/PhysRevD.91.054504

A. S. Miramontes, H. Sanchis Alepuz, and R. Alkofer, Elucidating the effect of intermediate resonances in the quark interaction kernel on the timelike electromagnetic pion form factor, Phys. Rev. D 103 (2021) 116006, https://doi.org/10.1103/PhysRevD.103.116006

A. S. Miramontes and A. Bashir, Timelike electromagnetic kaon form factor, Phys. Rev. D 107 (2023) 014016, https://doi.org/10.1103/PhysRevD.107.014016

D. Nicmorus, G. Eichmann, and R. Alkofer, Delta and Omega electromagnetic form factors in a Dyson-Schwinger/BetheSalpeter approach, Phys. Rev. D 82 (2010) 114017, https://doi.org/10.1103/PhysRevD.82.114017

G. Eichmann and C. S. Fischer, Nucleon axial and pseudoscalar form factors from the covariant Faddeev equation, Eur. Phys. J. A 48 (2012) 9, https://doi.org/10.1140/epja/i2012-12009-6

H. Sanchis-Alepuz, R. Alkofer, and C. S. Fischer, Electromagnetic transition form factors of baryons in the space-like momentum region, Eur. Phys. J. A 54 (2018) 41, https://doi.org/10.1140/epja/i2018-12465-x

K. Raya et al., Structure of the neutral pion and its electromagnetic transition form factor, Phys. Rev. D 93 (2016) 074017, https://doi.org/10.1103/PhysRevD.93.074017

I. C. Cloët and C. D. Roberts, Explanation and Prediction of Observables using Continuum Strong QCD, Prog. Part. Nucl. Phys. 77 (2014) 1, https://doi.org/10.1016/j.ppnp.2014.02.001

B. El-Bennich et al., Excited hadrons and the analytical structure of bound-state interaction kernels, Few Body Syst. 57 (2016) 955, https://doi.org/10.1007/s00601-016-1133-x

A. S. Miramontes et al., Electromagnetic and strong isospin breaking in light meson masses, Phys. Lett. B 833 (2022) 137291, https://doi.org/10.1016/j.physletb.2022.137291

G. Eichmann et al., Baryons as relativistic three-quark bound states, Prog. Part. Nucl. Phys. 91 (2016) 1, https://doi.org/10.1016/j.ppnp.2016.07.001

A. Bashir et al., Collective perspective on advances in DysonSchwinger Equation QCD, Commun. Theor. Phys. 58 (2012) 79, https://doi.org/10.1088/0253-6102/58/1/16

A. S. Miramontes and H. Sanchis-Alepuz, On the effect of resonances in the quark-photon vertex, Eur. Phys. J. A 55 (2019) 170, https://doi.org/10.1140/epja/i2019-12847-6

C. S. Fischer, D. Nickel, and J.Wambach, Hadronic unquenching effects in the quark propagator, Phys. Rev. D 76 (2007) 094009, https://doi.org/10.1103/PhysRevD.76.094009

C. S. Fischer, D. Nickel, and R. Williams, On Gribov’s supercriticality picture of quark confinement, Eur. Phys. J. C 60 (2009) 47, https://doi.org/10.1140/epjc/s10052-008-0821-1

H. Sanchis-Alepuz, C. S. Fischer, and S. Kubrak, Pion cloud effects on baryon masses, Phys. Lett. B 733 (2014) 151, https://doi.org/10.1016/j.physletb.2014.04.031

R. Williams, Vector mesons as dynamical resonances in the Bethe-Salpeter framework, Phys. Lett. B 798 (2019) 134943, https://doi.org/10.1016/j.physletb.2019.134943

P. D. Group, Review of Particle Physics, Progress of Theoretical and Experimental Physics 2022 (2022), https://doi.org/10.1093/ptep/ptac097

R. Alkofer, A. S. Miramontes, and H. Sanchis-Alepuz, Elucidating the ρ-meson’s role as intermediate resonance in the time-like electromagnetic pion form factor, EPJWeb Conf. 262 (2022) 01020, https://doi.org/10.1051/epjconf/202226201020

F. V. Ignatov et al., Measurement of the e +e − → π +π − cross section from threshold to 1.2 GeV with the CMD-3 detector (2023), https://doi.org/10.48550/arXiv.2302.08834

E. A. Kozyrev et al., Study of the process e +e − → K+K− in the center-of-mass energy range 1010-1060 MeV with the CMD-3 detector, Phys. Lett. B 779 (2018) 64, https://doi.org/10.1016/j.physletb.2018.01.079




How to Cite

Miramontes A, Bashir A. Electromagnetic time-like form factors. Supl. Rev. Mex. Fis. [Internet]. 2023 Sep. 18 [cited 2024 Jul. 17];4(2):021114 1-6. Available from: https://rmf.smf.mx/ojs/index.php/rmf-s/article/view/7111