Towards a Dispersive Analysis of the π 0 Transition Form Factor Bastian Kubis Helmholtz-Institut f¨ ur Strahlen- und Kernphysik (Theorie) Bethe Center for Theoretical Physics Universit¨ at Bonn, Germany “Hadronic Probes of Fundamental Symmetries” March 8th, 2014 B. Kubis, Towards a Dispersive Analysis of the π 0 Transition Form Factor – p. 1
Outline Introduction • Meson transition form factors and ( g − 2) µ • π 0 → γ ∗ γ ∗ : dispersive ingredients Dispersion relations . . . • . . . for the anomalous process γπ → ππ • . . . for vector meson decays ω/φ → 3 π Transition form factors • From hadronic decays to transition form factors: ω/φ → π 0 γ ∗ • Towards the π 0 transition form factor Summary / Outlook B. Kubis, Towards a Dispersive Analysis of the π 0 Transition Form Factor – p. 2
Meson transition form factors and ( g − 2) µ Czerwi´ nski et al., arXiv:1207.6556 [hep-ph] • leading and next-to-leading hadronic effects in ( g − 2) µ : had had − → hadronic light-by-light soon dominant uncertainty B. Kubis, Towards a Dispersive Analysis of the π 0 Transition Form Factor – p. 3
Meson transition form factors and ( g − 2) µ Czerwi´ nski et al., arXiv:1207.6556 [hep-ph] • leading and next-to-leading hadronic effects in ( g − 2) µ : had had − → hadronic light-by-light soon dominant uncertainty • important contribution: pseudoscalar pole terms singly / doubly virtual form factors π 0 , η, η ′ F P γγ ∗ ( q 2 , 0) and F P γ ∗ γ ∗ ( q 2 1 , q 2 2 ) B. Kubis, Towards a Dispersive Analysis of the π 0 Transition Form Factor – p. 3
Meson transition form factors and ( g − 2) µ Czerwi´ nski et al., arXiv:1207.6556 [hep-ph] • leading and next-to-leading hadronic effects in ( g − 2) µ : had had − → hadronic light-by-light soon dominant uncertainty • important contribution: pseudoscalar pole terms singly / doubly virtual form factors π 0 , η, η ′ F P γγ ∗ ( q 2 , 0) and F P γ ∗ γ ∗ ( q 2 1 , q 2 2 ) • for specific virtualities: linked to vector-meson conversion decays ω ) measurable in ω → π 0 ℓ + ℓ − etc. → e.g. F π 0 γ ∗ γ ∗ ( q 2 1 , M 2 − B. Kubis, Towards a Dispersive Analysis of the π 0 Transition Form Factor – p. 3
Dispersive analysis of π 0 → γ ∗ γ ∗ • isospin decomposition: F π 0 γ ∗ γ ∗ ( q 2 1 , q 2 2 ) = F vs ( q 2 1 , q 2 2 ) + F vs ( q 2 2 , q 2 1 ) B. Kubis, Towards a Dispersive Analysis of the π 0 Transition Form Factor – p. 4
Dispersive analysis of π 0 → γ ∗ γ ∗ • isospin decomposition: F π 0 γ ∗ γ ∗ ( q 2 1 , q 2 2 ) = F vs ( q 2 1 , q 2 2 ) + F vs ( q 2 2 , q 2 1 ) • analyze the leading hadronic intermediate states: see also Gorchtein, Guo, Szczepaniak 2012 γ ( ∗ ) π + s γ ∗ v π − π 0 ⊲ isovector photon: 2 pions pion vector form factor ∝ × γπ → ππ all determined in terms of pion–pion P-wave phase shift + Wess–Zumino–Witten anomaly for normalisation B. Kubis, Towards a Dispersive Analysis of the π 0 Transition Form Factor – p. 4
Dispersive analysis of π 0 → γ ∗ γ ∗ • isospin decomposition: F π 0 γ ∗ γ ∗ ( q 2 1 , q 2 2 ) = F vs ( q 2 1 , q 2 2 ) + F vs ( q 2 2 , q 2 1 ) • analyze the leading hadronic intermediate states: see also Gorchtein, Guo, Szczepaniak 2012 γ ( ∗ ) γ ( ∗ ) π + π + s v π 0 γ ∗ γ ∗ v s π − π − π 0 π 0 ⊲ isovector photon: 2 pions ∝ pion vector form factor × γπ → ππ all determined in terms of pion–pion P-wave phase shift + Wess–Zumino–Witten anomaly for normalisation ⊲ isoscalar photon: 3 pions B. Kubis, Towards a Dispersive Analysis of the π 0 Transition Form Factor – p. 4
Dispersive analysis of π 0 → γ ∗ γ ∗ • isospin decomposition: F π 0 γ ∗ γ ∗ ( q 2 1 , q 2 2 ) = F vs ( q 2 1 , q 2 2 ) + F vs ( q 2 2 , q 2 1 ) • analyze the leading hadronic intermediate states: see also Gorchtein, Guo, Szczepaniak 2012 γ ( ∗ ) γ ( ∗ ) π + s v ω, φ γ ∗ γ ∗ v s π − π 0 π 0 ⊲ isovector photon: 2 pions pion vector form factor ∝ × γπ → ππ all determined in terms of pion–pion P-wave phase shift + Wess–Zumino–Witten anomaly for normalisation ⊲ isoscalar photon: 3 pions dominated by narrow resonances ω, φ B. Kubis, Towards a Dispersive Analysis of the π 0 Transition Form Factor – p. 4
π 0 → γ ∗ ( q 2 v ) γ ∗ ( q 2 s ) transition form factor 2 q s e − π 0 e − e + e + π 0 → e + e − e + e − 2 q v π 0 → γγ e − π 0 e + B. Kubis, Towards a Dispersive Analysis of the π 0 Transition Form Factor – p. 5
π 0 → γ ∗ ( q 2 v ) γ ∗ ( q 2 s ) transition form factor 2 q s e − π 0 e − e + e + π 0 → e + e − e + e − 2 q v e − π + π 0 π 0 → γγ e − π 0 e + π − e + γ ∝ F π V × T ( γπ → ππ ) B. Kubis, Towards a Dispersive Analysis of the π 0 Transition Form Factor – p. 5
π 0 → γ ∗ ( q 2 v ) γ ∗ ( q 2 s ) transition form factor 2 q s φ → π 0 e + e − 2 e − π 0 M φ ω ( φ ) ω → π 0 e + e − 2 M ω e + γ e − π 0 e − e + e + π 0 → e + e − e + e − 2 q v e − π + π 0 π 0 → γγ e − π 0 e + π − e + γ ∝ F π V × T ( γπ → ππ ) B. Kubis, Towards a Dispersive Analysis of the π 0 Transition Form Factor – p. 5
γπ → ππ and the Wess–Zumino–Witten anomaly • controls low-energy processes of odd intrinsic parity e 2 • π 0 decay π 0 → γγ : F π 0 γγ = 4 π 2 F π F π : pion decay constant − → measured at 1.5% level PrimEx 2011 B. Kubis, Towards a Dispersive Analysis of the π 0 Transition Form Factor – p. 6
γπ → ππ and the Wess–Zumino–Witten anomaly • controls low-energy processes of odd intrinsic parity e 2 • π 0 decay π 0 → γγ : F π 0 γγ = 4 π 2 F π F π : pion decay constant − → measured at 1.5% level PrimEx 2011 e = (9 . 78 ± 0 . 05) GeV − 3 • γπ → ππ at zero energy: F 3 π = 4 π 2 F 3 π how well can we test this low-energy theorem? B. Kubis, Towards a Dispersive Analysis of the π 0 Transition Form Factor – p. 6
γπ → ππ and the Wess–Zumino–Witten anomaly • controls low-energy processes of odd intrinsic parity e 2 • π 0 decay π 0 → γγ : F π 0 γγ = 4 π 2 F π F π : pion decay constant − → measured at 1.5% level PrimEx 2011 e = (9 . 78 ± 0 . 05) GeV − 3 • γπ → ππ at zero energy: F 3 π = 4 π 2 F 3 π how well can we test this low-energy theorem? π − e − → π − e − π 0 Primakoff reaction π − π − π − π − π 0 π 0 γ ∗ γ ∗ Z e − e − F 3 π = (10 . 7 ± 1 . 2) GeV − 3 F 3 π = (9 . 6 ± 1 . 1) GeV − 3 Serpukhov 1987, Ametller et al. 2001 Giller et al. 2005 − → F 3 π tested only at 10% level B. Kubis, Towards a Dispersive Analysis of the π 0 Transition Form Factor – p. 6
Chiral anomaly: Primakoff measurement • previous analyses based on ⊲ data in threshold region only Serpukhov 1987 ⊲ chiral perturbation theory for extraction B. Kubis, Towards a Dispersive Analysis of the π 0 Transition Form Factor – p. 7
Chiral anomaly: Primakoff measurement • previous analyses based on ⊲ data in threshold region only Serpukhov 1987 ⊲ chiral perturbation theory for extraction counts • Primakoff measurement COMPASS 2004 hadron data 700 of whole spectrum COMPASS, work in progress 600 500 • idea: use dispersion - beam K ρ - relations to exploit all 400 data below 1 GeV for 300 y r a n i anomaly extraction m i 200 l e r p • effect of ρ resonance 100 included model- 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 independently via ππ m [GeV] π π - 0 P-wave phase shift figure courtesy of T. Nagel 2009 B. Kubis, Towards a Dispersive Analysis of the π 0 Transition Form Factor – p. 7
Warm-up: pion form factor from dispersion relations • just two particles in final state: form factor; from unitarity: = disc 1 2 i disc F I ( s ) = Im F I ( s ) = F I ( s ) × θ ( s − 4 M 2 π ) × sin δ I ( s ) e − iδ I ( s ) − → final-state theorem: phase of F I ( s ) is just δ I ( s ) Watson 1954 B. Kubis, Towards a Dispersive Analysis of the π 0 Transition Form Factor – p. 8
Warm-up: pion form factor from dispersion relations • just two particles in final state: form factor; from unitarity: = disc 1 2 i disc F I ( s ) = Im F I ( s ) = F I ( s ) × θ ( s − 4 M 2 π ) × sin δ I ( s ) e − iδ I ( s ) − → final-state theorem: phase of F I ( s ) is just δ I ( s ) Watson 1954 • solution to this homogeneous integral equation known: � s � ∞ � δ I ( s ′ ) ds ′ F I ( s ) = P I ( s )Ω I ( s ) , Ω I ( s ) = exp s ′ ( s ′ − s ) π 4 M 2 π P I ( s ) polynomial, Ω I ( s ) Omnès function Omnès 1958 • today: high-accuracy ππ phase shifts available Ananthanarayan et al. 2001, García-Martín et al. 2011 • constrain P I ( s ) using symmetries (normalisation at s = 0 etc.) B. Kubis, Towards a Dispersive Analysis of the π 0 Transition Form Factor – p. 8
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