Dyson-Schwinger equations and the muon g-2


  • Khépani Raya Universidad de Granada
  • Adnan Bashir
  • Ángel S. Miramontes
  • Pablo Roig




Electromagnetic form factors, Bound and unstable states, Hadronic light-by-light contributions, Dyson-Schwinger equations


We present a brief introduction to the Dyson-Schwinger equations (DSEs) approach to hadron and high-energy physics. In particular, how this formalism is applied to calculate the electromagnetic form factors $\gamma^* \gamma^* \to \textbf{P}^0$ and $\gamma^* \textbf{P}^\pm \to \textbf{P}^\pm$ (with $\textbf{P}^\pm$ and $\textbf{P}^0$ charged and neutral ground-state pseudoscalar mesons, respectively) is discussed. Subsequently, the corresponding contributions of those form factors to the muon anomalous magnetic moment ($g-2$) are estimated. We look forward to promoting the DSE approach to address theoretical aspects of the muon $g-2$, highlighting some calculations that could be carried out in the future.


i. MCEs are also expected to play a major role in the description of nucleon and hyperon transition form factors, around Q2≈0 [58-61].

ii. Equation (25) must be adapted to account for the flavour decomposition of the η − η 0 systems [52].

iii. The quark propagator is written as S(p) = −iγ · p σv(p2) + σs(p2), with σs,v(p2) being algebraically related to M(p2) and Z(p2) in Eq. (8).

iv. The MCEs take place in the neighborhood of Q2≈0 [58-61], such that, for increasing Q2, BRL → RL.

C. D. Roberts. On Mass and Matter. AAPPS Bull., 31 (2021) 6.

B. Abi et al., Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm. Phys. Rev. Lett., 126 (2021) 141801.

G. W. Bennett et al., Final Report of the Muon E821 Anomalous Magnetic Moment Measurement at BNL. Phys. Rev. D, 73

(2006) 072003.

T. Aoyama et al., The anomalous magnetic moment of the muon in the Standard Model. Phys. Rept., 887 (2020) 1.

N. Saito, A novel precision measurement of muon g-2 and EDM at J-PARC. AIP Conf. Proc., 1467 (2012) 45.

T. Aoyama, M. Hayakawa, T. Kinoshita, and M. Nio, Complete Tenth-Order QED Contribution to the Muon g − 2. Phys. Rev. Lett., 109 (2012) 111808.

A. Czarnecki, W. J. Marciano, and A. Vainshtein, Refinements in electroweak contributions to the muon anomalous magnetic

moment. Phys. Rev., D 67 (2003) 073006. [Erratum Phys. Rev. D73 (2006) 119901].

C. Gnendiger, D. Stockinger, and H. St ¨ ockinger-Kim, The electroweak contributions to (g − 2)µ after the Higgs boson mass

measurement. Phys. Rev., D 88 (2013) 053005.

T. Aoyama, T. Kinoshita, and M. Nio, Theory of the Anomalous Magnetic Moment of the Electron. Atoms, 7 (2019) 28.

M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang, A new evaluation of the hadronic vacuum polarisation contributions to

the muon anomalous magnetic moment and to α(m2Z). Eur. Phys. J., C 80 (2020) 241. [Erratum Eur. Phys. J. C80 (2020)410].

A. Keshavarzi, D. Nomura, and T. Teubner, The g−2 of charged leptons, α(M2Z ) and the hyperfine splitting of muonium. Phys. Rev., D 101 (2020) 014029.

P. Masjuan and P. Sanchez-Puertas, Pseudoscalar-pole contribution to the (gµ − 2) a rational approach. Phys. Rev., D 95 (2017) 054026.

P. Roig and P. Sanchez-Puertas, Axial-vector exchange contribution to the hadronic light-by-light piece of the muon anomalous magnetic moment. Phys. Rev., D 101 (2020) 074019.

A. Guevara, P. Roig, and J. J. Sanz-Cillero. Pseudoscalar pole light-by-light contributions to the muon (g − 2) in Resonance

Chiral Theory. JHEP, 06 (2018) 160.

P. Roig, A. Guevara, and G. Lopez Castro. ´ V V 0P form factors in resonance chiral theory and the π −η −η0 light-by-light

contribution to the muon g−2. Phys. Rev. D, 89 (2014) 073016.

I. Danilkin, C. Florian Redmer, and M. Vanderhaeghen, The hadronic light-by-light contribution to the muon’s anomalous

magnetic moment. Prog. Part. Nucl. Phys., 107 (2019) 20.

G. Colangelo, M. Hoferichter, M. Procura, and P. Stoffer, Dispersion relation for hadronic light-by-light scattering two-pion contributions. JHEP, 04 (2017) 161.

P. Masjuan, P. Roig, and P. Sanchez-Puertas, The interplay of transverse degrees of freedom and axial-vector mesons with

short-distance constraints in g − 2. J. Phys. G 49 (2022) 015002.

J. A. Miranda and P. Roig. New τ -based evaluation of the hadronic contribution to the vacuum polarization piece of the muon anomalous magnetic moment. Phys. Rev. D, 102 (2020) 114017.

C. D. Roberts and A. G. Williams, Dyson-Schwinger equations and their application to hadronic physics. Prog. Part. Nucl. Phys., 33 (1994) 477.

C. S. Fischer, QCD at finite temperature and chemical potential from Dyson–Schwinger equations. Prog. Part. Nucl. Phys., 105

(2019) 1.

H. Sanchis-Alepuz and R. Williams, Recent developments in bound-state calculations using the Dyson–Schwinger and Bethe–Salpeter equations. Comput. Phys. Commun., 232 (2018) 1.

K. Raya, Z.-F. Cui, L. Chang, J.-M. Morgado, C. D. Roberts, and Jose Rodriguez-Quintero. Revealing pion and kaon structure via generalised parton distributions. 9 2021) .

Z.-Fang Cui, M. Ding, F. Gao, K. Raya, D. Binosi, and L. Chang, Craig D Roberts, Jose Rodr´ıguez-Quintero, and Sebastian M Schmidt. Kaon and pion parton distributions. Eur. Phys. J. C, 80 (2020) 1064.

K. Raya, L. Chang, M. Ding, D. Binosi, and C. D Roberts, Unveiling the structure of pseudoscalar mesons. In 18th International Conference on Hadron Spectroscopy and Structure, (2020) pages 565.

M. Ding, K. Raya, D. Binosi, L. Chang, C. D. Roberts, and S. M Schmidt, Drawing insights from pion parton distributions. Chin. Phys. C, 44 (2020) 031002.

Si-xue Qin, Craig D Roberts, and Sebastian M Schmidt. Spectrum of light- and heavy-baryons. Few Body Syst., 60 (2019) 26.

G. Eichmann, H. Sanchis-Alepuz, R. Williams, R. Alkofer, and C. S. Fischer. Baryons as relativistic three-quark bound states. Prog. Part. Nucl. Phys., 91 (2016) 1.

K. Raya et al., Structure of the neutral pion and its electromagnetic transition form factor. Phys. Rev. D, 93 (2016) 074017.

L. Chang, I. C. Cloet, C. D. Roberts, S. M. Schmidt, and P. C. ¨Tandy. Pion electromagnetic form factor at spacelike momenta.

Phys. Rev. Lett., 111 (2013) 141802.

Lei Chang et al., Imaging dynamical chiral symmetry breaking pion wave function on the light front. Phys. Rev. Lett., 110 (2013) 132001.

G. Eichmann, Nucleon electromagnetic form factors from the covariant Faddeev equation. Phys. Rev. D, 84 (2013) 014014.

A. Miramontes, A. Bashir, K. Raya, and P. Roig, Pion and Kaon box contribution to a HLbLµ . (2021) 12.

K. Raya, A. Bashir, and P. Roig, Contribution of neutral pseudoscalar mesons to a HLbLµ within a Schwinger-Dyson equations approach to QCD. Phys. Rev. D, 101 (2020) 074021.

G. Eichmann, C. S. Fischer, W. Heupel, and R. Williams, Single pseudoscalar meson pole and pion box contributions to the anomalous magnetic moment of the muon. Phys. Lett. B, 797 (2019) 134855. [Erratum Phys. Lett. B 799 (2019) 135029].

G. Eichmann, C. S. Fischer, W. Heupel, and R. Williams, Kaonbox contribution to the anomalous magnetic moment of the muon. Phys. Rev., D 101 (2020) 054015.

G. Eichmann, C. S. Fischer, W. Heupel, and R. Williams, The muon g-2 Dyson-Schwinger status on hadronic light-by-light

scattering. AIP Conf. Proc., 1701 (2016) 040004.

T. Goecke, C. S. Fischer, and R. Williams, Hadronic light-bylight scattering in the muon g-2 a Dyson-Schwinger equation approach. Phys. Rev. D, 83 (2011) 094006. [Erratum Phys.Rev. D 86 (2012) 099901].

S.-x. Qin and C. D. Roberts, Impressions of the Continuum Bound State Problem in QCD. Chin. Phys. Lett., 37 (2020) 121201.

D. Binosi, L. Chang, J. Papavassiliou, S.-X. Qin, and C. D. Roberts, Symmetry preserving truncations of the gap and Bethe-Salpeter equations. Phys. Rev. D, 93 (2016) 096010.

L. Albino, A. Bashir, B. El-Bennich, E. Rojas, F. E. Serna, and Roberto Correa da Silveira. The impact of transverse Slavnov-Taylor identities on dynamical chiral symmetry breaking. JHEP, 11 (2021) 196.

M. Atif Sultan, K. Raya, F. Akram, A. Bashir, and B. Masud. Effect of the quark-gluon vertex on dynamical chiral symmetry

breaking. Phys. Rev. D, 103 (2021) 054036.

L. Albino, A. Bashir, L. X. Gutierrez Guerrero, B. El Bennich, and E. Rojas, Transverse Takahashi Identities and Their Implications for Gauge Independent Dynamical Chiral Symmetry Breaking. Phys. Rev. D, 100 (2019) 054028.

Z. Xing, K. Raya, and L. Chang, Quark anomalous magnetic moment and its effects on the ρ meson properties. Phys. Rev. D,

(2021) 054038.

S.-X. Qin, C. D. Roberts, and S. M. Schmidt, Ward–Green–Takahashi identities and the axial-vector vertex. Phys. Lett. B, 733 (2014) 202.

M. S. Bhagwat, L. Chang, Y.-X. Liu, C. D. Roberts, and P. C. Tandy, Flavour symmetry breaking and meson masses. Phys. Rev. C, 76 (2007) 045203.

A. Bender, Craig D. Roberts, and L. Von Smekal. Goldstone theorem and diquark confinement beyond rainbow ladder approximation. Phys. Lett. B 380 (1996) 7.

Y.-Zhen Xu, D. Binosi, Z.-F. Cui, B.-L. Li, C. D Roberts, ShuSheng Xu, and Hong Shi Zong. Elastic electromagnetic form factors of vector mesons. Phys. Rev. D, 100 (2019) 114038.

F. E. Serna, C. Chen, and B. El-Bennich, Interplay of dynamical and explicit chiral symmetry breaking effects on a quark. Phys. Rev. D, 99 (2019) 094027.

L. Chang, Y.-B. Liu, K. Raya, J. Rodr´ıguez-Quintero, and Y.-B. Yang, Linking continuum and lattice quark mass functions via an effective charge. Phys. Rev. D, 104 (2021) 094509.

Si-xue Qin et al., Interaction model for the gap equation. Phys. Rev. C, 84 (2011) 042202.

M. Ding et al., γ∗γ → η, η0 transition form factors. Phys. Rev. D, 99 (2019) 014014.

P. Maris and P. C. Tandy, The Quark photon vertex and the pion charge radius. Phys. Rev. C, 61 (2000) 045202.

P. Maris and P. C. Tandy, The pi, K+, and K0 electromagnetic form-factors. Phys. Rev. C, 62 (2000) 055204.

P. Maris and P. C. Tandy, Electromagnetic transition formfactors of light mesons. Phys. Rev. C, 65 (2002) 045211.

A. S. Miramontes and H. Sanchis-Alepuz, On the effect of resonances in the quark-photon vertex. Eur. Phys. J. A, 55 (2019) 170.

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.

R. A. Williams, C. R. Ji, and S. R. Cotanch, Kinematically accessible vector meson resonance enhancements in p (K-, e+ e-) Lambda, Sigma0, Lambda (1405. Phys. Rev. C, 48 (1993) 1318.

G. Ramalho and M. T. Peña γ (∗) N→N∗(1520) form factors in the timelike regime. Phys. Rev. D, 95 (2017) 014003.

C. Granados, S. Leupold, and E. Perotti, The electromagnetic Sigma-to-Lambda hyperon transition form factors at low energies. Eur. Phys. J. A, 53 (2017) 117.

K. Raya et al., Dynamical diquarks in the γ(∗)p →N(1535)1/2− transition. Eur. Phys. J. A, 57 (2021) 266.

P. Maris and P. C. Tandy, Bethe-Salpeter study of vector meson masses and decay constants. Phys. Rev. C, 60 (1999) 055214.

G. Eichmann, Ch. S. Fischer, E. Weil, and R. Williams, On the large-Q2 behavior of the pion transition form factor. Phys. Lett.

B, 774 (2017) 425.

M. Knecht and A. Nyffeler. Hadronic light by light corrections to the muon g-2 The Pion pole contribution. Phys. Rev. D, 65

(2002) 073034.

K. Raya, M. Ding, A. Bashir, L. Chang, and C. D. Roberts. Partonic structure of neutral pseudoscalars via two photon transition form factors. Phys. Rev. D 95 (2017) 074014.

Chao Shi et al., Flavour symmetry breaking in the kaon parton distribution amplitude. Phys. Lett. B, 738 (2014) 512.

E. B. Dally et al., DIRECT MEASUREMENT OF THE NEGATIVE KAON FORM-FACTOR. Phys. Rev. Lett., 45 (1980) 232.

S. R. Amendolia et al., A Measurement of the Kaon Charge Radius. Phys. Lett. B, 178 (1986) 435.

S. R. Amendolia et al., A Measurement of the Space - Like Pion Electromagnetic Form-Factor. Nucl. Phys. B, 277 (1986) 168.

H. P. Blok et al., Charged pion form factor between Q2=0.60 and 2.45 GeV2. I. Measurements of the cross section for the

H(e, e0π+)n reaction. Phys. Rev. C, 78 (2008) 045202.

Robert Delbourgo and Peter C. West. A Gauge Covariant Approximation to Quantum Electrodynamics. J. Phys. A 10 (1977) 1049.




How to Cite

Raya K, Bashir A, Miramontes Ángel S, Roig P. Dyson-Schwinger equations and the muon g-2. Supl. Rev. Mex. Fis. [Internet]. 2022 Mar. 31 [cited 2022 Dec. 3];3(2):020709 1-9. Available from: https://rmf.smf.mx/ojs/index.php/rmf-s/article/view/6229