Transition form factors and HLBL from Dyson-Schwinger equations Christian S. Fischer Justus Liebig Universität Gießen June 2018 Together with Gernot Eichmann, Esther Weil, Richard Williams 1 Christian S. Fischer (University of Gießen) Hadronic light-by-light from DSEs / 20
Overview F(Q,Q’) 1.(Transition-) form factors DSE TFF LMD+V Q 2.Hadronic light by light 2 Christian S. Fischer (University of Gießen) Hadronic light-by-light from DSEs / 20
Nonperturbative QCD: Complementary approach Quarks and gluons Dyson-Schwinger Equations Lattice simulations Physical quark masses Ab initio Full momentum dependencies Gauge invariant Multi-scale problems feasible Hadrons Effective theories and models ( χ PT, chiral models,...) Dispersive approach Physical degrees of freedom 3 Christian S. Fischer (University of Gießen) Hadronic light-by-light from DSEs / 20
DSE-approach: truncations [ S ( p )] − 1 = [ − ip / + M ( p 2 )] /Z f ( p 2 ) Rainbow-Ladder (RL): Beyond the rainbow (BRL): backup slides Williams, CF, Heupel, PRD 93 (2016) 034026 CF, Williams, PRL 103 (2009) 122001 4 Christian S. Fischer (University of Gießen) Hadronic light-by-light from DSEs / 20
(Transition-) Form factors and decays Pion form factor: Pion transition FF Q 2 π π π 0 → e + e − e + e − π 0 → e + e − γ 5 Christian S. Fischer (University of Gießen) Hadronic light-by-light from DSEs / 20
(Transition-) Form factors and decays Pion form factor: Pion transition FF Q 2 π π π 0 → e + e − e + e − π 0 → e + e − γ 5 Christian S. Fischer (University of Gießen) Hadronic light-by-light from DSEs / 20
(Transition-) Form factors and decays Pion form factor: Pion transition FF Q 2 π π π 0 → e + e − e + e − π 0 → e + e − γ 5 Christian S. Fischer (University of Gießen) Hadronic light-by-light from DSEs / 20
(Transition-) Form factors and decays Pion form factor: Pion transition FF Q 2 π π π 0 → e + e − e + e − π 0 → e + e − γ 5 Christian S. Fischer (University of Gießen) Hadronic light-by-light from DSEs / 20
Quark mass and pion BSE π [ S ( p )] − 1 = [ − ip / + M ( p 2 )] /Z f ( p 2 ) Dynamical mass: M ≈ 350 MeV 0.4 Lattice: quenched Lattice: unquenched (N f =2+1) h ¯ DSE: quenched ΨΨ i ⇡ (250 MeV) 3 0.3 DSE: unquenched (N f =2) M(p) [GeV] GMOR 0.2 0.1 0.0 0.0 1.0 2.0 3.0 4.0 p [GeV] CF, Nickel, Williams, EPJ C 60 (2009) 47 6 Christian S. Fischer (University of Gießen) Hadronic light-by-light from DSEs / 20
Quark mass and pion BSE π [ S ( p )] − 1 = [ − ip / + M ( p 2 )] /Z f ( p 2 ) Dynamical mass: M ≈ 350 MeV 0.4 Lattice: quenched Lattice: unquenched (N f =2+1) h ¯ DSE: quenched ΨΨ i ⇡ (250 MeV) 3 0.3 DSE: unquenched (N f =2) M(p) [GeV] GMOR NJL 0.2 0.1 0.0 0.0 1.0 2.0 3.0 4.0 p [GeV] CF, Nickel, Williams, EPJ C 60 (2009) 47 6 Christian S. Fischer (University of Gießen) Hadronic light-by-light from DSEs / 20
Quark-photon vertex and dynamical vector mesons Basis: k { γ µ , Q µ , k µ } ⊗ { 1 , Q / , k / , Q / k / } Q 12 elements X λ i L µ X τ i T µ Γ µ ( k, Q ) = Γ µ BC ( k, Q ) + Γ µ T ( k, Q ) = i + i i =1 , 4 i =1 , 8 gauge part transverse part ‘Ball-Chiu’ vector-mesons Ball and Chiu, PRD 22 (1980) 2542. WTI: Q µ Γ µ ( k, Q ) = S − 1 ( k + Q/ 2) − S − 1 ( k − Q/ 2) ✓ EM gauge invariance satisfied Vector mesons: dynamically generated ✓ 7 Christian S. Fischer (University of Gießen) Hadronic light-by-light from DSEs / 20
Pion form factor 2 ( ) F Q not timelike: spacelike: ¯ e + e − → X ¯ + accessible e e N N e e − X → e − X N e N − X → − − → ’’ charge, ’ magnetic moment,... radius 2 0 2 Q 4 M − Pion form factor: Q 2 π π Krassnigg, Schladming 2011; Maris, Tandy NPPS 161, 2006 Vector meson poles dynamically generated! : backup slides ρ → ππ Williams, arXiv:1804.11161 8 Christian S. Fischer (University of Gießen) Hadronic light-by-light from DSEs / 20
Pion form factor 2 ( ) F Q not timelike: spacelike: ¯ e + e − → X ¯ + accessible e e N N e e − X → e − X N e N − X → − − → ’’ charge, ’ magnetic moment,... radius 2 0 2 Q 4 M − Pion form factor: Q 2 excluded region π π Krassnigg, Schladming 2011; Maris, Tandy NPPS 161, 2006 Vector meson poles dynamically generated! : backup slides ρ → ππ Williams, arXiv:1804.11161 8 Christian S. Fischer (University of Gießen) Hadronic light-by-light from DSEs / 20
Pion transition form factor Q 2 Q 0 2 η + = Q 2 + Q 0 2 2 Maris, Tandy, Phys. Rev. C 65 045211 (2002) Eichmann, CF, Weil and Williams, PLB 774 (2017) 425 9 Christian S. Fischer (University of Gießen) Hadronic light-by-light from DSEs / 20
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Scalar + axial vector transition from factors 11 Christian S. Fischer (University of Gießen) Hadronic light-by-light from DSEs / 20
Rare pion decay ◆ 2 s B ( π 0 → e + e − ) ✓ m α em 1 − 4 m 2 | A ( − m 2 π / 4) | 2 = 2 B ( π 0 → γγ ) π m π m 2 π Usual: dispersive approach DSE: direct calculation same result as everybody else discrepancy with exp. remains Weil, Eichmann, CF and Williams, PRD96 (2017) no.1, 014021 12 Christian S. Fischer (University of Gießen) Hadronic light-by-light from DSEs / 20
More rare decays π 0 → e + e − γ π 0 → e + e − e + e − Weil, Eichmann, CF and Williams, Phys.Rev. D96 (2017) no.1, 014021 13 Christian S. Fischer (University of Gießen) Hadronic light-by-light from DSEs / 20
Overview F(Q,Q’) 1.(Transition-) form factors DSE TFF LMD+V Q 2.Hadronic light by light 14 Christian S. Fischer (University of Gießen) Hadronic light-by-light from DSEs / 20
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