Unified dispersive approach to real and virtual photon-photon scattering into two pions

Unified dispersive approach to real and virtual photon-photon scattering into two pions Bachir Moussallam Project started with Diogo Boito arxiv:1305.3143 Photon 2013 *** Paris, May 20-24

Introduction Goal: representations for amplitudes γγ (q 2 ) ππ or γ (q 2 ) γππ ( q 2 < 1 GeV 2 ) Method: combine nonperturbative QCD tools [Unitarity, analyticity, Chiral symmetry, soft photon theorems] Relevancefor muon g 2 HVP contribution from γππ channels[reduced model dependence] Future: 2π contribution in light-by-light amplitude γγ (q 2 2 ) ππ γ (q 2 3 )γ (q 2 ) ([Hoferichter et al.]) 4 Account for ππ re-scattering (exp. progress recently: [NA48/2],[DIRAC]...) 2/24

Experimental aspects q 2 >4m 2 π : e + e γπ 0 π 0 direct relation to γ γπ 0 π 0. Measurements performed (q < 1 GeV): SND:[Phys. Lett. B 537 (2002) 201] CMD-2:[Phys. Lett. B 580 (2004) 119] also KLOE[EPJ C49 (2007) 473] but q=m ϕ e + e γπ + π ISR and FSR amplitudesinterfere. q 1 p 1 k 1 q 2 (A) k 2 (B) p 2 must measure angular distributions 3/24

q 2 <0 : From (in principle) e + e e + e π 0 π 0, e + e π + π with single tagging 4/24

Theory: Unitarity key ingredient to FSI ππ scattering elastic: s< 1 GeV 2 Fermi-Watson theorem. Valid for γγ (q 2 ) ππ? Depends on q 2 q 2 4m 2 π : Im(γγ ππ) J =(γγ ππ) J (ππ ππ) J q 2 >4m 2 π : Im(γγ ππ) J =(γγ ππ) (ππ ππ) J J +(γ ππ)(γππ ππ) J [Creutz,Einhorn PR D1 (1970)2537.] 5/24

Chiral low energy expansion: Valid when q 2, s <<1 GeV 2 π 0 π 0 probes loops, NLO calculations [Bijnens, Cornet NP B296 (1988) 557] (q 2 =0) [Donoghue,Holstein,PR D48 (1993) 137] (q 2 =0) H n = 2(s m2 π ) Ḡ(s,q 2 ) ++ NLO F 2 π H c = s Ḡ(s,q 2 )+( l ++ NLO F 2 6 l 5 ) s q2 + H Born 48π 2 F 2 ++ π π with Ḡ(s,q 2 )= sḡ π (s) q 2 Ḡ π (q 2 ) s q 2 q 2 J π (s) J π (q2 ) s q 2 ImḠ π (z), J π (z) =0 when z>4m 2 π 6/24

Analyticity in QCD Combine unitarity with analyticity[omnès, NC 8 (1958) 316]. PW amplitudes analytic s with two cuts right-handcut:[4m 2 π, ] left-hand cut:[, 0] (usually!) Discontinuity on RHC: disc(γγ ππ)=(γγ ππ) s iε (ππ ππ) s+iε also when q 2 >4m 2. FSI problem solved π Amplitude from LHC from Muskhelishvili equation Appl. to γγ ππ[gourdin, Martin NC 17 (1960) 224] Matching w. ChPT[Morgan, Pennington PL B272 (1991) 134, Donoghue, Holstein PR D48 (1993) 137] 7/24

Phenomenology of LHC Pion pole (Born Amplitude) π + π + π + γ γ γ π π (a) (b) (c) π Resonance poles π 0 π + π ω, ρ 0 ρ + γ π 0 γ π γ π q 2 =0: form-factors 8/24

Born helicity amplitudes: H Born λλ (s,q 2,θ)=F v π (q2 ) H Born λλ (s,q 2,θ) H Born ++ (s,q2,θ)= 2(4m2 π q2 (1 σ 2 π cos2 θ)) (s q 2 )(1 σ 2 π cos2 θ) 2(s 4m 2 H Born + (s,q2 π,θ)= )sin2 θ (s q 2 )(1 σ 2 π cos2 θ) H Born +0 (s,q2,θ)= 2 2q 2 (s 4m 2 π )sinθcosθ s(s q 2 )(1 σ 2 π cos2 θ) Partial-waveJ=0 h Born 0,++ (s,q2 )= 1 s q 2 4m2 π σ π (s) log1+σ π(s) 1 σ π (s) 2q2 Singularities: LHC[,0], pole s=q 2 (soft photon), 9/24

Resonance exchange partial-waves: LHC in case q 2 >4m 2 π Im(s)/m 2 π 4 3 2 1 0-1 -2-3 Omnès applicable? 0 s s + -4-5 0 5 10 15 20 25 Re(s)/m 2 π 10/24

Yes a) Use Källen-Lehmann representation for propagators BW V (t)= 1 π 4m 2 π dt σ(t,m V,γ V ) (t t) b) Limiting prescriptions: q 2 =lim ε 0 q 2 +iε (generalized) LHC does not intersect RHC 0.02 0.01 Im (s)/m 2 π 0-0.01-0.02 0 10 20 30 40 50 Re (s)/m 2 π 11/24

Representation based on 2-subtracted DR H I ++ (s,q2,z)=f v π (q2 ) H I,Born ++ (s,q2,z)+ V=ρ,ω F Vπ(q 2 ) H I,V ++ (s,q2,z) +Ω (s q I 0 (s) 2 )b I (q 2 )+sf v s(j π (q2 ) I,π (s,q 2 ) J I,π (q 2,q 2 )) q 2 Ĵ I,π (q 2 ) s q 2 +s V=ρ,ω F Vπ(q 2 ) sj I,V (s,q 2 ) q 2 J I,V (q 2,q ) 2. Remarks: H I ++ isospin amplitude,hn ++, Hc linear combination ++ First line: tree diagrams Omnès function Ω I 0 (s)=exp 1 π 4m 2 π ds s (s s) δi 0 (s ) 12/24

(Continued) J I,π (s,q 2 ),J I,V (s,q 2 ): integrals of h I,Born 0,h I,V 0, phase-shifts J I,π (s,q 2 )= 1 π J I,V (s,q 2 )= 1 π 4m 2 π 4m 2 π (integrals well defined) also ds sinδ I 0 (s ) h I,π (s ) 2 (s s) Ω I 0 (s ) 0,++ (s,q 2 ) ds sinδ I 0 (s ) h I,V (s ) 2 (s s) Ω I 0 (s ) 0,++ (s,q 2 ) Ĵ I,π (q 2 )= JI,π (s,q 2 ) s Satisfies soft-photon theorem s=q 2 Extra polynomial in s: paramatrize higher energy portionsof cuts. Two unknownfunctions: b I (q 2 ) 13/24

Chiral symmetry constraints π 0 π 0 amplitudeat s=0 H n ++ (0,q2,z)= H n,v ++ (0,q2,z) q 2 b n (q 2 ) V Adlerzero b n (q 2 )=O(m 2 π ) Except if q 2 =0! NLO ChPT: H n ++ (0,q2,z) = 2m2 π Ḡπ NLO F 2 (q 2 ) J π (q 2 ) π q 2 = 96π 2 F 2 1+ q2 15m 2 + π π 14/24

Parametrization of subtraction functions: b n (q 2 )=b n (0) F(q 2 ) +β ρ (GS ρ (q 2 ) 1)+β ω (BW ω (q 2 ) 1) b c (q 2 )=b c (0) +β ρ (GS ρ (q 2 ) 1)+β ω (BW ω (q 2 ) 1) F(q 2 ) from chiral NLO b n (0), b c (0): comp. with chiral amplitude Note: relation to pion polarizabilities α π 0 β π 0= 2α 1 lim s=0 m π s Hn,V ++ (s,0,θ)+bn (0) α π + β π += 2α 1 lim s=0 s Hc,V ++ (s,0,θ)+bc (0) m π β ρ, β ω : experimental inputs 15/24

Comparison with NLO ChPT q 2 = 0.2 GeV 2 H++(s, q 2, z) t=m 2 π 0.6 0.4 0.2 0-0.2 Re(H++) disp. Re(H++) chiral Im(H++) disp. Im(H++) chiral q 2 = 0.2 (GeV) 2-0.4-0.2-0.1 0 0.1 0.2 0.3 s (GeV) 2 q 2 =0.2 GeV 2 H++(s, q 2, z) t=m 2 π Form factor=1 at NLO 0.2 0.1 0-0.1-0.2-0.3-0.4 Re(H++) disp. Re(H++) chiral Im(H++) disp. Im(H++) chiral q 2 = 0.2 (GeV) 2-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 s (GeV) 2 16/24

q 2 =0: comparison w. data[γγ π 0 π 0 ] 0(z < 0.8) (nb) σγγ π 0 π 16 14 12 10 8 6 4 2 α β = 1.9 α β = 1.3 α β = 1.0 Crystal Ball Belle 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 s (GeV) 2 Crystal Ball: [PR D41 (1990) 3324] Belle : [PR D78 (2008) 052004] KLOE-2 [expected] 17/24

Fit of σ e + e γπ 0 π 0(q2 ) data β ρ β ω χ 2 /N dof ref. 0.14±0.12 ( 0.39±0.12)10 1 20.2/27 SND (2002) 0.13±0.15 ( 0.31±0.15)10 1 15.0/21 CMD-2 (2003) 0.05±0.09 ( 0.37±0.09)10 1 38.1/50 Combined 18/24

Fit of σ e + e γπ 0 π 0(q2 ) data (cont.) Cross-section (nb) Cross-section (nb) 0.5 0.4 0.3 0.2 0.1 0 0.5 0.4 0.3 0.2 0.1 Akhmetshin (2003) β ρ, β ω : fitted β ρ = β ω = 0 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Achasov (2002) β ρ, β ω : fitted β ρ = β ω = 0 q (GeV) 0 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 q (GeV) 19/24

Differential cross-sections dσ ds (s,q2 ) (nb GeV 1) dσ e + e π 0 π 0 γ d s 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 q = 0.75 q = m ρ q = m ω q = 0.95 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s (GeV) shape changeswhen q>(m ρ +m π ) no σ meson bump 20/24

Illustrative comp. w. other approaches Consider ρ 0 π 0 π 0 γ (GeV) 1 2 1.5 Dispersive UChPT σ-model 10 4 B ρ π 0 π 0 γ d s 1 0.5 0 0.3 0.4 0.5 0.6 0.7 s (GeV) Chiral Lagr.+V + unitarized pion loop [Palomar,Hirenzaki,Oset NP A707 (2002)161] Chiral Lagr.+V +pion loop +sigma meson[bramon et al. PL B517 (2001) 345] 21/24

Muon(g 2)/2: γππ contribution in HVP a [γππ] μ = 1 q 2 max 4π 3 dq 2 K μ q 2 σ e + e γ γππ(q 2 ) 4m 2 π cross-sections: σ c, σ n α 3 σ(q 2 )= 12(q 2 ) 3 q 2 4m 2 π ds(q 2 s)σ π (s) 1 dz 1 σ c : SeparateBorn H c λλ 2 = H Born λλ +Ĥ c λλ 2 σ c (q 2 )=σ Born (q 2 )+ ˆσ Born (q 2 )+ ˆσ c (q 2 ) H λλ (s,q 2,θ) 2 Define σ Born (e.g. add rad. corr. part σ(e + e π + π )) 22/24

σ Born corresponds to sqed σ Born (q 2 )= πα2 Numerical results: channel cross-section 3q 2 σ3 π (q2 ) F v π (q2 ) 2 α π η(q2 ) γπ + π σ Born 41.9 10 11 γπ + π ˆσ Born (1.31±0.30) 10 11... γπ + π ˆσ c (0.16±0.05) 10 11 γπ 0 π 0 σ n (0.33±0.05) 10 11 Remarks a Born comparableto Δa SM μ μ =±49 10 11 a μ [ˆσ Born ]>0 unlike[dubinskyetal. EPJC40 (2005)41] σ-meson approx: a [γσ] =1.2 10 11 [Narison (2003)], μ =1.5 10 11 [Ahmadov,Kuraev,Volkov (2010)] a μ 23/24

Conclusions Analyticity based treatment of FSI in γγ ππ extended to γγ (q 2 ) Main issue: left-hand cut [pion, resonances] becomes generalized one, but properly defined Good description of experimentaldata e + e γπ 0 π 0 (2 parameters) a μ : contributionsfrom γπ 0 π 0, γπ + π [q<0.95 GeV] Other applications: pion generalized polarizabilities, sigma meson (pole)-γ form factor Extensions possible [ q> 1 GeV ] (coupled-channel MO). Double virtual scattering? 24/24