Towards a data-driven analysis of hadronic light-by-light scattering

Transcription

1 Towards a data-driven analysis of hadronic light-by-light scattering Bastian Kubis Helmholtz-Institut für Strahlen- und Kernphysik (Theorie) Bethe Center for Theoretical Physics Universität Bonn, Germany PWA 8 / ATHOS 3 Ashburn/VA, April 12th 2015 B. Kubis, Data-driven hadronic LbL p. 1

2 Hadronic light-by-light scattering using dispersion relations An application of amplitude analyses: η γ γ ( ) transition form factor π 0 γ γ ( ) transition form factor Pion pion intermediate states Jülich Bonn Bern B. Kubis, Data-driven hadronic LbL p. 2

3 Hadronic light-by-light scattering using dispersion relations An application of amplitude analyses: η γ γ ( ) transition form factor π 0 γ γ ( ) transition form factor Pion pion intermediate states Jülich Bonn Bern e + e e + e π 0 γπ ππ e + e π 0 γ ω,φ ππγ e + e ππγ ππ ππ Pion transition ( form factor F π 0 γ γ q 2 1,q2 2 ) Pion vector Partial waves for γ γ ππ e + e e + e ππ form factor F V π e + e 3π pion polarizabilities γπ γπ ω,φ 3π ω,φ π 0 γ Colangelo, Hoferichter, BK, Procura, Stoffer 2014 B. Kubis, Data-driven hadronic LbL p. 2

4 Hadronic light-by-light scattering hadronic light-by-light soon to dominate Standard Model uncertainty in (g 2) µ hadrons µ B. Kubis, Data-driven hadronic LbL p. 3

5 Hadronic light-by-light scattering hadronic light-by-light soon to dominate Standard Model uncertainty in (g 2) µ hadrons different contributions estimated (in ): µ π 0, η, η π ±, K ± scalars axials, quarks µ µ µ µ 99±16 19±13 15±7 21±3 how to control hadronic modelling? Jegerlehner, Nyffeler 2009 B. Kubis, Data-driven hadronic LbL p. 3

6 Hadronic light-by-light scattering hadronic light-by-light soon to dominate Standard Model uncertainty in (g 2) µ hadrons different contributions estimated (in ): µ π 0, η, η π ±, K ± scalars axials, quarks µ µ µ µ 99±16 19±13 15±7 21±3 how to control hadronic modelling? Jegerlehner, Nyffeler 2009 dispersive point of view: analytic structure, cuts and poles (on-shell) form factors and scatt. amplitudes from experiment expansion in masses of intermediate states, partial waves B. Kubis, Data-driven hadronic LbL p. 3

7 Pseudoscalar transition form factors and(g 2) µ largest individual HLbL contribution: pseudoscalar pole terms singly / doubly virtual form factors F Pγγ (q 2,0) and F Pγ γ (q2 1,q2 2 ) π 0, η, η B. Kubis, Data-driven hadronic LbL p. 4

8 Pseudoscalar transition form factors and(g 2) µ largest individual HLbL contribution: pseudoscalar pole terms singly / doubly virtual form factors F Pγγ (q 2,0) and F Pγ γ (q2 1,q2 2 ) π 0, η, η normalisation fixed by Wess Zumino Witten anomaly, e.g.: F π0 γγ(0,0) = e2 4π 2 F π F π : pion decay constant measured at 1.5% level PrimEx 2011 B. Kubis, Data-driven hadronic LbL p. 4

9 Pseudoscalar transition form factors and(g 2) µ largest individual HLbL contribution: pseudoscalar pole terms singly / doubly virtual form factors F Pγγ (q 2,0) and F Pγ γ (q2 1,q2 2 ) π 0, η, η normalisation fixed by Wess Zumino Witten anomaly, e.g.: F π0 γγ(0,0) = e2 4π 2 F π F π : pion decay constant measured at 1.5% level PrimEx 2011 qi 2 -dependence: often analysed by vector-meson dominance what can we learn from analyticity and unitarity constraints? what experimental input sharpens these constraints? B. Kubis, Data-driven hadronic LbL p. 4

10 Dispersive analysis ofπ 0 /η γ γ isospin decomposition: F π 0 γ γ (q2 1,q2) 2 = F vs (q1,q 2 2)+F 2 vs (q2,q 2 1) 2 F ηγ γ (q2 1,q2) 2 = F vv (q1,q 2 2)+F 2 ss (q2,q 2 1) 2 B. Kubis, Data-driven hadronic LbL p. 5

11 Dispersive analysis ofπ 0 /η γ γ isospin decomposition: F π 0 γ γ (q2 1,q 2 2) = F vs (q 2 1,q 2 2)+F vs (q 2 2,q 2 1) F ηγ γ (q2 1,q 2 2) = F vv (q 2 1,q 2 2)+F ss (q 2 2,q 2 1) analyze the leading hadronic intermediate states: see also Gorchtein, Guo, Szczepaniak 2012 γ v π + γ ( ) s/v π π 0 /η isovector photon: 2 pions pion vector form factor γπ ππ / η ππγ all determined in terms of pion pion P-wave phase shift B. Kubis, Data-driven hadronic LbL p. 5

12 Dispersive analysis ofπ 0 /η γ γ isospin decomposition: F π 0 γ γ (q2 1,q 2 2) = F vs (q 2 1,q 2 2)+F vs (q 2 2,q 2 1) F ηγ γ (q2 1,q 2 2) = F vv (q 2 1,q 2 2)+F ss (q 2 2,q 2 1) analyze the leading hadronic intermediate states: see also Gorchtein, Guo, Szczepaniak 2012 γ v π + γ ( ) s/v γ s π + π 0 γ ( ) v/s π π 0 /η π π 0 /η isovector photon: 2 pions pion vector form factor γπ ππ / η ππγ all determined in terms of pion pion P-wave phase shift isoscalar photon: 3 pions B. Kubis, Data-driven hadronic LbL p. 5

13 Dispersive analysis ofπ 0 /η γ γ isospin decomposition: F π 0 γ γ (q2 1,q 2 2) = F vs (q 2 1,q 2 2)+F vs (q 2 2,q 2 1) F ηγ γ (q2 1,q 2 2) = F vv (q 2 1,q 2 2)+F ss (q 2 2,q 2 1) analyze the leading hadronic intermediate states: see also Gorchtein, Guo, Szczepaniak 2012 γ v π + γ ( ) s/v γ s ω, φ γ ( ) v/s π π 0 /η π 0 /η isovector photon: 2 pions pion vector form factor γπ ππ / η ππγ all determined in terms of pion pion P-wave phase shift isoscalar photon: 3 pions dominated by narrow ω, φ ω/φ transition form factors; very small for the η B. Kubis, Data-driven hadronic LbL p. 5

14 Warm-up: charged pion form factor disc = 1 2i discfv π (s) = ImFπ V (s) = Fπ V (s) θ(s 4Mπ) sinδ 2 1(s)e 1 iδ1 1 (s) final-state theorem: phase of Fπ V (s) is just δ1 1 (s) Watson 1954 B. Kubis, Data-driven hadronic LbL p. 6

15 Warm-up: charged pion form factor disc = 1 2i discfv π (s) = ImFπ V (s) = Fπ V (s) θ(s 4Mπ) sinδ 2 1(s)e 1 iδ1 1 (s) final-state theorem: phase of Fπ V (s) is just δ1 1 (s) Watson 1954 solution: { s Fπ V (s) = P(s)Ω(s), Ω(s) = exp π 4M 2 π } ds δ1(s 1 ) s (s s) P(s) polynomial, Ω(s) Omnès function Omnès 1958 ππ phase shifts from Roy equations Ananthanarayan et al. 2001, García-Martín et al P(0) = 1 from symmetries (gauge invariance) below 1 GeV: F V π (s) (1+0.1GeV 2 s)ω(s) slope due to inelastic resonances ρ, ρ... Hanhart 2012 B. Kubis, Data-driven hadronic LbL p. 6

16 Final-state universality: η, η π + π γ η ( ) π + π γ driven by the chiral anomaly, π + π in P-wave final-state interactions the same as for vector form factor ansatz: F η( ) ππγ = A P(t) Ω(t), P(t) = 1+α ( ) t, t = M 2 ππ B. Kubis, Data-driven hadronic LbL p. 7

17 Final-state universality: η, η π + π γ η ( ) π + π γ driven by the chiral anomaly, π + π in P-wave final-state interactions the same as for vector form factor ansatz: F η( ) ππγ = A P(t) Ω(t), P(t) = 1+α ( ) t, t = M 2 ππ divide data by pion form factor P(t) Stollenwerk et al P(t) t [GeV 2 ] exp.: α KLOE = (1.52±0.06) GeV 2 cf. KLOE 2013 B. Kubis, Data-driven hadronic LbL p. 7

18 Transition form factorη γ γ F ηγ γ(q 2,0) = 1+ κ ηq 2 96π 2 F 2 π 4M 2 π dsσ(s) 3 P(s) FV π (s) 2 s q 2 + F I=0 ηγ γ(q 2,0) [ VMD] η Hanhart et al γ π + π γ B. Kubis, Data-driven hadronic LbL p. 8

19 Transition form factorη γ γ F ηγ γ(q 2,0) = 1+ κ ηq π 2 F 2 π 4M 2 π dsσ(s) 3 P(s) FV π (s) 2 s q 2 + F I=0 ηγ γ(q 2,0) [ VMD] η Hanhart et al γ π + π γ ) 2 F(M l + l M l + l -2 [GeV 2 ] figures courtesy of C. Hanhart data: NA , A B. Kubis, Data-driven hadronic LbL p. 8

20 Anomalous decayη π + π γ α KLOE = (1.52±0.06) GeV 2 large implausible to explain through ρ, ρ... for large t, expect P(t) const. rather γ η γ γ transition form factor: dispersion integral covers larger energy range η π + π γ B. Kubis, Data-driven hadronic LbL p. 9

21 Anomalous decayη π + π γ α KLOE = (1.52±0.06) GeV 2 large implausible to explain through ρ, ρ... for large t, expect P(t) const. rather γ η γ γ transition form factor: dispersion integral covers larger energy range η π + π γ Intriguing observation: naive continuation of Fππγ η = A(1+αt)Ω(t) has zero at t = 1/α 0.66GeV 2 test this in crossed process γπ π η "left-hand cuts" in πη system? BK, Plenter 2015 B. Kubis, Data-driven hadronic LbL p. 9

22 Primakoff reaction γπ πη can be measured in Primakoff η reaction S-wave forbidden P-wave exotic: J PC = 1 + COMPASS π γ ( ) Z π D-wave a 2 (1320) first resonance B. Kubis, Data-driven hadronic LbL p. 10

23 Primakoff reaction γπ πη can be measured in Primakoff η reaction S-wave forbidden P-wave exotic: J PC = 1 + COMPASS π γ ( ) Z π D-wave a 2 (1320) first resonance include a 2 as left-hand cut in decay couplings fixed from a 2 πη, πγ π + η a 2 π + π γ η a + 2 π π + γ η π a 2 P π π γ compatible with decay data? predictions for γπ πη cross sections and asymmetries [ spares] BK, Plenter 2015 B. Kubis, Data-driven hadronic LbL p. 10

24 Formalism including left-hand cuts π + η a 2 π + π γ η a + 2 π π + γ η π a 2 P π π γ a 2 + rescattering essential to preserve Watson s theorem formally: Fππγ(s,t,u) η = F(t)+G a2 (s,t,u)+g a2 (u,t,s) { } F(t) = Ω(t) A(1+αt)+ t2 dx sinδ(x)ĝ(x) π x 2 Ω(x) (x t) 4M 2 π Ĝ: t-channel P-wave projection of a 2 exchange graphs re-fit subtraction constants A, α B. Kubis, Data-driven hadronic LbL p. 11

25 η, η π + π γ witha P(t) t [GeV 2 ] KLOE 2013 α = 1.52±0.06, χ 2 /ndof = 0.94 B. Kubis, Data-driven hadronic LbL p. 12

26 η, η π + π γ witha P(t) t [GeV 2 ] KLOE 2013 α = 1.52±0.06, χ 2 /ndof = 0.94 α = 1.42±0.06,χ 2 /ndof = 0.90 B. Kubis, Data-driven hadronic LbL p. 12

27 η, η π + π γ witha 2 P(t) t [GeV 2 ] KLOE 2013 α = 1.52±0.06, χ 2 /ndof = 0.94 α = 1.42±0.06,χ 2 /ndof = 0.90 equally good why care? sum rule for η γ γ transition form factor slope reduced by 7 8% cf. Hanhart et al B. Kubis, Data-driven hadronic LbL p. 12

28 η, η π + π γ witha P(t) 1.3 P(t) t [GeV 2 ] KLOE 2013 α = 1.52±0.06, χ 2 /ndof = 0.94 α = 1.42±0.06,χ 2 /ndof = t [GeV 2 ] Crystal Barrel 1997 α = 0.6±0.2, χ 2 /ndof = 1.2 equally good why care? sum rule for η γ γ transition form factor slope reduced by 7 8% cf. Hanhart et al B. Kubis, Data-driven hadronic LbL p. 12

29 η, η π + π γ witha P(t) 1.3 P(t) t [GeV 2 ] KLOE 2013 α = 1.52±0.06, χ 2 /ndof = 0.94 α = 1.42±0.06,χ 2 /ndof = t [GeV 2 ] Crystal Barrel 1997 α = 0.6±0.2, χ 2 /ndof = 1.2 α = 1.4±0.4, χ 2 /ndof = 1.4 equally good why care? sum rule for η γ γ transition form factor slope reduced by 7 8% cf. Hanhart et al α α (large-n c ) better fulfilled including a 2 BK, Plenter 2015 B. Kubis, Data-driven hadronic LbL p. 12

30 Anomalous process γπ ππ γπ ππ fully crossing symmetric: odd partial waves P-waves only (neglecting F- and higher) amplitude decomposed into single-variable functions F(s,t,u) = F(s)+F(t)+F(u) B. Kubis, Data-driven hadronic LbL p. 13

31 Anomalous process γπ ππ γπ ππ fully crossing symmetric: odd partial waves P-waves only (neglecting F- and higher) amplitude decomposed into single-variable functions F(s,t,u) = F(s)+F(t)+F(u) left-hand cut ˆF(s) and right-hand cut F(s) self-consistent: { C1 F(s) = Ω(s) 3 + s π ˆF(s) = M 2 π dz(1 z 2 )F ( t(s,z) ) } ds sinδ1 1(s ) ˆF(s ) s Ω(s ) (s s) F(s) = B. Kubis, Data-driven hadronic LbL p. 13

32 Dispersion relations for γπ ππ low-energy theorem / Wess Zumino Witten anomaly: F(0,0,0) = F 3π = e 4π 2 F 3 π F 3π only verified experimentally to 10% accuracy Serpukhov 1987, Ametller et al B. Kubis, Data-driven hadronic LbL p. 14

33 Dispersion relations for γπ ππ low-energy theorem / Wess Zumino Witten anomaly: F(0,0,0) = F 3π = e 4π 2 F 3 π F 3π only verified experimentally to 10% accuracy Serpukhov 1987, Ametller et al high-accuracy data from Primakoff spectrum COMPASS fit dispersive representation to data, extract F 3π C 1 Hoferichter, BK, Sakkas 2012 counts beam K - ρ preliminary COMPASS 2004 hadron data m π - π 0 [GeV] figure courtesy of T. Nagel 2009 B. Kubis, Data-driven hadronic LbL p. 14

34 Extension to decays: ω/φ 3π same quantum numbers as γπ ππ Niecknig, BK, Schneider 2012; see also talk by I. Danilkin test accuracy on KLOE φ 3π Dalitz plot: events bin number ˆF = 0 once-subtracted twice-subtracted χ 2 /ndof B. Kubis, Data-driven hadronic LbL p. 15

35 Extension to decays: ω/φ 3π same quantum numbers as γπ ππ Niecknig, BK, Schneider 2012; see also talk by I. Danilkin test accuracy on KLOE φ 3π Dalitz plot: events bin number second subtraction improves accuracy Niecknig, BK, Schneider 2012 ω 3π Dalitz plot? CLAS, KLOE, WASA-at-COSY B. Kubis, Data-driven hadronic LbL p. 15

36 Transition form factorω(φ) π 0 l + l disc ω = π + π 0 π 0 π ω ω transition form factor related to pion vector form factor ω 3π decay amplitude B. Kubis, Data-driven hadronic LbL p. 16

37 Transition form factorω(φ) π 0 l + l ω π + ω disc = π 0 π 0 π ω transition form factor related to pion vector form factor ω 3π decay amplitude form factor normalization yields rate Γ(ω π 0 γ) (2nd most important ω decay channel) works at 95% accuracy Schneider, BK, Niecknig 2012 B. Kubis, Data-driven hadronic LbL p. 16

38 Numerical results: ω π 0 µ + µ 9 F ωπ 0(s) NA60 09 NA60 11 Lepton-G VMD Terschlüsen et al. f 1 (s) = aω(s) full dispersive dγ ω π 0 µ + µ /ds [10 6 GeV 1 ] s [GeV] s [GeV] clear enhancement vs. naive vector-meson dominance incompatible with data (from heavy-ion coll.) NA , 2011 more "exclusive" data? CLAS NA60 data potentially in conflict with unitarity bounds Ananthanarayan, Caprini, BK 2014 B. Kubis, Data-driven hadronic LbL p. 17

39 One step further: e + e 3π, e + e π 0 γ e e + ω(φ) π 0 π + π decay amplitude for ω/φ 3π: { F(s) = a ω/φ Ω(s) 1+ s π 4M 2 π M ω/φ F(s)+F(t)+F(u) ds sinδ1(s 1 ) ˆF(s } ) s Ω(s ) (s s) a ω/φ adjusted to reproduce total width ω/φ 3π B. Kubis, Data-driven hadronic LbL p. 18

40 One step further: e + e 3π, e + e π 0 γ e e + π 0 π + π decay amplitude for e + e 3π: M e+ e F(s)+F(t)+F(u) { F(s,q 2 ) = a e + e (q2 )Ω(s) 1+ s ds sinδ1(s 1 ) ˆF(s },q 2 ) π s Ω(s ) (s s) 4M 2 π a e+ e (q2 ) adjusted to reproduce spectrum e + e 3π contains 3π resonances no dispersive prediction B. Kubis, Data-driven hadronic LbL p. 18

41 One step further: e + e 3π, e + e π 0 γ e e + π 0 π + π decay amplitude for e + e 3π: M e+ e F(s)+F(t)+F(u) { F(s,q 2 ) = a e + e (q2 )Ω(s) 1+ s ds sinδ1(s 1 ) ˆF(s },q 2 ) π s Ω(s ) (s s) 4M 2 π a e+ e (q2 ) adjusted to reproduce spectrum e + e 3π parameterisation: a e + e (q2 ) = F 3π 3 +βq2 + q4 π BW(q 2 ) = V=ω,φ c V thr ds ImBW(s ) s 2 (s q 2 ) M 2 V q2 i q 2 Γ V (q 2 ) B. Kubis, Data-driven hadronic LbL p. 18

42 One step further: e + e 3π, e + e π 0 γ e π + π 0 e + π γ decay amplitude for e + e 3π: M e+ e F(s)+F(t)+F(u) { F(s,q 2 ) = a e + e (q2 )Ω(s) 1+ s ds sinδ1(s 1 ) ˆF(s },q 2 ) π s Ω(s ) (s s) 4M 2 π a e+ e (q2 ) adjusted to reproduce spectrum e + e 3π parameterisation: a e + e (q2 ) = F 3π 3 +βq2 + q4 π BW(q 2 ) = V=ω,φ c V thr ds ImBW(s ) s 2 (s q 2 ) M 2 V q2 i q 2 Γ V (q 2 ) fit to e + e 3π data prediction for e + e π 0 γ B. Kubis, Data-driven hadronic LbL p. 18

43 Fit toe + e 3π data 10 3 fit SND+BaBar fit HLMNT SND 10 2 BaBar σe + e 3π [nb] q2 [GeV] Hoferichter, BK, Leupold, Niecknig, Schneider 2014 one subtraction/normalisation at q 2 = 0 fixed by γ 3π fitted: ω, φ residues, linear subtraction β B. Kubis, Data-driven hadronic LbL p. 19

44 Comparison toe + e π 0 γ data σe + e π 0 γ [nb] 10 2 SND CMD q2 [GeV] Hoferichter, BK, Leupold, Niecknig, Schneider 2014 "prediction" no further parameters adjusted data well reproduced B. Kubis, Data-driven hadronic LbL p. 20

45 Pion loop contributions / ππ intermediate states Colangelo, Hoferichter, Procura, Stoffer 2014 [figures courtesy of M. Hoferichter] Decompose light-by-light scattering tensor Π µνλσ into form factor scalar QED part preserves gauge invariance Π FsQED µνλσ = F V π (q 2 1 )FV π (q 2 2 )FV π (q 2 3 ) B. Kubis, Data-driven hadronic LbL p. 21

46 Pion loop contributions / ππ intermediate states Colangelo, Hoferichter, Procura, Stoffer 2014 [figures courtesy of M. Hoferichter] Decompose light-by-light scattering tensor Π µνλσ into form factor scalar QED part preserves gauge invariance Π FsQED µνλσ = F V π (q 2 1 )FV π (q 2 2 )FV π (q 2 3 ) + remainder Π µνλσ expanded in γ γ ππ helicity partial waves organised according to left-hand-cut structure see also Hoferichter, Phillips, Schat 2011; Moussallam 2013 B. Kubis, Data-driven hadronic LbL p. 21

47 Pion loop contributions / ππ intermediate states Colangelo, Hoferichter, Procura, Stoffer 2014 [figures courtesy of M. Hoferichter] Decompose light-by-light scattering tensor Π µνλσ into form factor scalar QED part preserves gauge invariance Π FsQED µνλσ = F V π (q 2 1 )FV π (q 2 2 )FV π (q 2 3 ) + remainder Π µνλσ expanded in γ γ ππ helicity partial waves contains automatically polarisability effects Engel, Patel, Ramsey-Musolf 2012 ππ resonances: f 0 (500), f 2 (1270) can be extended to K K ( f 0 (980)) B. Kubis, Data-driven hadronic LbL p. 21

48 Summary / Outlook e + e e + e π 0 γπ ππ e + e π 0 γ ω,φ ππγ e + e ππγ ππ ππ Pion transition ( form factor F π 0 γ γ q 2 1,q2 2 ) Pion vector Partial waves for γ γ ππ e + e e + e ππ form factor F V π e + e 3π pion polarizabilities γπ γπ ω,φ 3π ω,φ π 0 γ B. Kubis, Data-driven hadronic LbL p. 22

49 Summary / Outlook e + e e + e π 0 γπ ππ e + e π 0 γ ω,φ ππγ e + e ππγ ππ ππ Pion transition ( form factor F π 0 γ γ q 2 1,q2 2 ) Pion vector Partial waves for γ γ ππ e + e e + e ππ form factor F V π e + e 3π pion polarizabilities γπ γπ ω,φ 3π ω,φ π 0 γ Further experimental input for π 0 and η transition form factors: Primakoff reactions γπ ππ, γπ πη ω 3π precision Dalitz plot improved e + e 3π amplitudes via interpolation ω/φ π 0 γ test doubly virtual F π 0 γ γ with precision e + e ηπ + π differential data COMPASS CLAS C.-W. Xiao et al. B. Kubis, Data-driven hadronic LbL p. 22

50 Spares B. Kubis, Data-driven hadronic LbL p. 23

51 Paradigm: hadronic vacuum polarization characteristic scale set by muon mass this is not a perturbative QCD problem! dispersion relations to the rescue: use the optical theorem! µ µ hadrons B. Kubis, Data-driven hadronic LbL p. 24

52 Paradigm: hadronic vacuum polarization characteristic scale set by muon mass this is not a perturbative QCD problem! dispersion relations to the rescue: use the optical theorem! µ µ hadrons Im γ γ γ hadrons hadrons 2 σ tot (e + e hadrons) B. Kubis, Data-driven hadronic LbL p. 24

53 Paradigm: hadronic vacuum polarization characteristic scale set by muon mass this is not a perturbative QCD problem! dispersion relations to the rescue: use the optical theorem! had VP aµ 4M 2 π K(s)σ tot (e + e hadrons) µ µ hadrons K(s): kinematical function, for large s: σ tot (e + e hadrons) 1/s K(s) 1/s, B. Kubis, Data-driven hadronic LbL p. 24

54 Paradigm: hadronic vacuum polarization characteristic scale set by muon mass this is not a perturbative QCD problem! dispersion relations to the rescue: use the optical theorem! had VP aµ 4M 2 π K(s)σ tot (e + e hadrons) µ µ hadrons K(s): kinematical function, for large s: σ tot (e + e hadrons) 1/s K(s) 1/s, had VP more than 75% ofaµ given by energies s 1 GeV 2 Jegerlehner, Nyffeler 2009 ρ, ω 0.0 GeV, Υ ψ 9.5 GeV 3.1 GeV φ, GeV 1.0 GeV B. Kubis, Data-driven hadronic LbL p. 24

55 Total cross sectionγπ πη σ(s) [µb] s [GeV] blue: t-channel dynamics / "ρ" only red: full amplitude t-channel dynamics dominate below s 1GeV uncertainty bands: Γ(η π + π γ), α, a 2 couplings BK, Plenter 2015 B. Kubis, Data-driven hadronic LbL p. 25

56 Differential cross sections γπ πη amplitude zero visible in differential cross sections: 0.20 s = 0.9GeV 0.4 s = 1.0GeV 0.8 s = 1.1GeV dσ/dzs [µb] dσ/dzs [µb] dσ/dzs [µb] z s = cosθ s z s = cosθ s blue: t-channel dynamics / "ρ" only red: full amplitude z s = cosθ s B. Kubis, Data-driven hadronic LbL p. 26

57 Differential cross sections γπ πη amplitude zero visible in differential cross sections: 0.20 s = 0.9GeV 0.4 s = 1.0GeV 0.8 s = 1.1GeV dσ/dzs [µb] dσ/dzs [µb] dσ/dzs [µb] z s = cosθ s z s = cosθ s blue: t-channel dynamics / "ρ" only red: full amplitude z s = cosθ s strong P-D-wave interference can be expressed as forwardbackward asymmetry A FB = σ(cosθ > 0) σ(cosθ < 0) A FB (s) σ total s [GeV] B. Kubis, Data-driven hadronic LbL p. 26

58 π 0 γ (qv 2)γ (qs 2 ) transition form factor q s 2 e π 0 e e + e + π 0 e + e e + e q v 2 e e + π 0 π 0 γγ B. Kubis, Data-driven hadronic LbL p. 27

59 π 0 γ (qv 2)γ (qs 2 ) transition form factor q s 2 e π 0 e e + e + π 0 e + e e + e q v 2 e e + π 0 π 0 γγ e e + F π V π + π 0 π γ T(γπ ππ) B. Kubis, Data-driven hadronic LbL p. 27

60 π 0 γ (qv 2)γ (qs 2 ) transition form factor q s 2 e ω(φ) π 0 M φ 2 φ π 0 e + e e + γ M ω 2 ω π 0 e + e e π 0 e e + e + π 0 e + e e + e q v 2 e e + π 0 π 0 γγ e e + F π V π + π 0 π γ T(γπ ππ) B. Kubis, Data-driven hadronic LbL p. 27

61 Extension to spacelike region; slope continuation to spacelike region: use another dispersion relation F π 0 γ γ(q 2,0) = F πγγ + q2 π 4M 2 π ds ImF π 0 γ γ(s,0) s (s q 2 ) work in progress; high-energy completion of the integral? convergence/uncertainties? { sum rule for slope F π0 γ γ(q 2 q 2 },0) = F πγγ 1+a π Mπ 2 +O(q 4 ) 0 a π = M2 π 0 F πγγ 1 π 4M 2 π ds s 2 ImF π 0 γ γ(s,0) = (30.7±0.6) 10 3 Hoferichter et al compare: a π = (32±4) 10 3 PDG 2014 theory error estimate: ππ phases and cutoff effects (in γ 3π partial waves and [γ 3π] [γ π 0 γ]) only! B. Kubis, Data-driven hadronic LbL p. 28

62 Summary: processes and unitarity relations forπ 0 γ γ γ v process unitarity relations SC 1 SC 2 γv γ s γ s P F π 0 γγ P F 3π σ(γπ ππ) Colangelo, Hoferichter, BK, Procura, Stoffer 2014 γπ ππ γ s ω,φ ω,φ γ v Γ π 0 γ γ v ω 3π, φ 3π ω,φ d P Γ 2 Γ 3π dsdt (ω,φ 3π) γ s γ v γ s γ v σ(e + e π 0 γ) γ s P σ(e + e 3π) σ(γπ ππ) d 2 Γ dsdt (ω,φ 3π) γ 3π γ s F 3π σ(e + e 3π) common theme: resum ππ rescattering B. Kubis, Data-driven hadronic LbL p. 29