Chiral Symmetry and Low-Energy Pion-Photon Reactions N. Kaiser (TU - PowerPoint PPT Presentation

Chiral Symmetry and Low-Energy Pion-Photon Reactions N. Kaiser (TU München) Hadron Physics Seminar, GSI Darmstadt, 26.9.2018 Tests of Chiral Perturbation Theory via low-energy π − γ reactions COMPASS@CERN: Primakoff effect to extract π − γ cross sections π -Compton scattering π − γ → π − γ : electric/magnetic polarizabilities Radiative corrections to π -Compton scattering (and µ ± p → µ ± p ) Chiral anomaly test: π − γ → π − π 0 Neutral and charged pion-pair production: π − γ → π − π 0 π 0 , π + π − π − Radiative corrections to π − γ → 3 π Radiative corrections to proton and neutron magnetic moments N. Kaiser (TUM) Chiral symmetry and low-energy pion-photon reactions

Introduction: Primakoff effect COMPASS experiment at CERN (S. Paul, J. Friedrich, B. Ketzer, B. Grube,...) Primakoff effect: Z Scattering high-energy pions in nuclear Coulomb field (charge Z) allows to extract cross sections for π − γ reactions (equivalent-photon method) Q 2 − Q 2 Z 2 α Q min = s − m 2 d σ min π ds dQ 2 = σ π − γ ( s ) , π ( s − m 2 π ) Q 4 2 E beam s = ( π − γ invariant mass) 2 , Q → 0 momentum transfer by virtual photon Isolate Coulomb peak from strong interaction background Different final-states π − γ , π − π 0 , π − π 0 π 0 , π + π − π − allow to test different aspects of chiral dynamics (low-energy QCD) Diffractive pion-scattering: meson spectroscopy and search for exotics N. Kaiser (TUM) Chiral symmetry and low-energy pion-photon reactions

Pion Compton-scattering Structure of pion at low energies: calculated in chiral perturbation theory General form of pion Compton-scattering amplitude in cm frame: 2 � � �� ǫ 1 · � ǫ 2 · � t = ( k 1 − k 2 ) 2 T πγ = 8 πα − � ǫ 1 · � ǫ 2 A ( s , t ) + � k 2 � k 1 A ( s , t ) + B ( s , t ) , t Corresponding differential cross section: = α 2 d σ � | A ( s , t ) | 2 + | A ( s , t ) + ( 1 + cos θ cm ) B ( s , t ) | 2 � d Ω cm 2 s Tree diagrams ( s -channel pole diagram vanishes, ǫ 1 · ( 2 p 1 + k 1 ) = 0): s − m 2 π A ( s , t ) = 1 , B ( s , t ) = m 2 π − s − t One-loop diagrams (finite after mass renormalization): π − t + √− t � t � 1 4 m 2 � ∼ t 2 > 0 2 + 2 m 2 π ln 2 A ( s , t ) = − ( 4 π f π ) 2 2 m π Electric/magnetic polarizabilities = low-energy const. with α π + β π = 0 α π − β π = α (¯ ℓ 6 − ¯ A ( s , t ) = − β π m π t ℓ 5 ) < 0 , 2 α 24 π 2 f 2 π m π Combination ¯ ℓ 6 − ¯ ℓ 5 = 3 . 0 ± 0 . 3 determined via radiative pion decay π + → e + ν e γ , PIBETA@PSI: axial-to-vector coupl. ratio F A / F V = 0 . 44 N. Kaiser (TUM) Chiral symmetry and low-energy pion-photon reactions

Pion polarizability measurement Two-loop prediction of chiral perturbation theory [J. Gasser et al. (’06)] α π − β π = α (¯ ℓ 6 − ¯ ℓ 5 ) α m π c r + 8 ℓ 6 + 65 ln m π � � � ℓ 2 − ¯ ¯ ℓ 1 + ¯ ℓ 5 − ¯ + 24 π 2 f 2 ( 4 π f π ) 4 π m π 3 12 m ρ � 53 π 2 ¯ 3 + 4 ¯ + 4 ℓ 3 ℓ 4 ℓ 5 ) − 187 − 41 �� 9 (¯ ℓ 1 + ¯ 3 (¯ ℓ 6 − ¯ ℓ 2 ) − 81 + 48 324 α π − β π = ( 5 . 7 ± 1 . 0 ) · 10 − 4 fm 3 , α π + β π = 0 . 16 · 10 − 4 fm 3 COMPASS result: α π − β π = ( 4 . 0 ± 1 . 8 ) · 10 − 4 fm 3 [PRL 114, 062002 (’15)] pion beam 1.15 1.10 x γ = E γ / E π in lab, cos θ cm = 1 − 2 x γ s / ( s − m 2 π ) 1.05 R S 1 0.95 Analysis of data includes: 0.90 0.85 chiral pion-loop corrections A ( s , t ) ∼ ln 2 ( . t . ) 0.4 0.5 0.6 0.7 0.8 0.9 muon beam 1.15 radiative corrections [NPA 812, 186 (’08)] 1.10 1.05 m 2 π − m 2 π 0 )ln 2 ( . t . ) isospin-breaking correction ∼ ( R P 1 previous results from Mainz and Serpukhov: 0.95 α π − β π = ( 12 − 16 ) · 10 − 4 fm 3 0.90 0.85 0.4 0.5 0.6 0.7 0.8 0.9 x J N. Kaiser (TUM) Chiral symmetry and low-energy pion-photon reactions

Radiative corrections to pion Compton scattering Pion-structure effects small: necessary to include radiative corr. of O ( α ) Start with structureless pion: extensive exercise in one-loop scalar QED Include leading pion-structure α π − β π in form of γγ -contact vertex F µν F µν Virtual photon loops + soft γ -radiation ( ω < λ ) give infrared finite result 1 1 - + γ --> π - + γ pion Compton scattering: π λ = 5 MeV 0.5 0.5 λ = 5 MeV 0 radiative correction [%] radiative correction [%] 0 -0.5 -0.5 -1 -1.5 -1 1/2 = 2m π s -2 1/2 = 2m π s -1.5 1/2 = 3m π s 1/2 = 3m π s -2.5 1/2 = 4m π s 1/2 = 4m π s -2 1/2 = 5m π -3 s -4 fm 3 α π = - β π = 3.10 -3.5 -2.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 z = cos θ cm z = cos θ cm QED radiative corrections are maximal in backward directions z ≃ − 1 Same kinematical signature as pion polarizability difference α π − β π Suppressed by factor of ∼ 10 Relative size and angular depend. not affected by leading pion-structure N. Kaiser (TUM) Chiral symmetry and low-energy pion-photon reactions

Proton radius from elastic muon-proton scattering COMPASS proposal [S. Paul, J. Friedrich, et al.] : Measure proton charge radius r p = ( 0 . 84 − 0 . 88 ) fm in µ ± p → µ ± p scatter. Generalize Rosenbluth formula to massive muons ( m µ ≥ √− t = Q ) d σ ( 1 γ ) = 4 πα 2 s − ( M p + m µ ) 2 � − 1 � s − ( M p − m µ ) 2 � − 1 � t 2 dt �� ( s + M 2 p − m 2 µ ) 2 µ + t �� � + m 2 4 M 2 p G 2 E ( t ) − t G 2 � � m 2 � G 2 µ − s M ( t ) + t M ( t ) × 4 M 2 p − t 2 Advantage of muons over electrons: much smaller radiative corrections 1 - p -> µ - p -8 full lines: µ + p -> µ + p - p -> e - p dashed lines: µ e E lab = 1 GeV Radiative correction [%] Radiative correction [%] -10 0 E lab = 50 GeV E lab = 100 GeV -12 E lab = 200 GeV -1 -14 -2 -16 infrared cutoff: 50 MeV infrared cutoff: 10 MeV -3 -18 0 0.1 0.2 0.3 0.001 0.01 0.1 2 [GeV 2 ] 2 [GeV 2 ] Q Q Analytical calculation of radiative corrections done for point-like proton: Vertex corrections, vacuum polarization, 2-photon exchange, soft bremsstrahlung N. Kaiser (TUM) Chiral symmetry and low-energy pion-photon reactions

Extracting the chiral anomaly π 0 → 2 γ and γ → 3 π couplings determined by chiral anomaly of QCD Amplitude and cross section for π − ( p 1 ) + γ ( k , ǫ ) → π − ( p 2 ) + π 0 ( p 0 ) : e ǫ µνκλ ǫ µ p ν 1 p κ 2 p λ F 3 π = 9 . 8 GeV − 3 T γ 3 π = 0 M ( s , t ) , 4 π 2 f 3 π � 1 σ tot ( s ) = α ( s − m 2 π )( s − 4 m 2 π ) 3 / 2 dz ( 1 − z 2 ) | M ( s , t ) | 2 ( 4 f π ) 6 π 4 √ s − 1 - -> π - π 0 ) 25 σ tot ( γπ tree + loops + loops + lec + ρ resonance 20 σ [ µ b] 15 10 5 0 2 3 4 5 6 1/2 [m π ] s ρ ( 770 ) -resonance must be included: � s t u � M ( s , t ) ( ρ ) = 1 + 0 . 46 ρ − s − i √ s Γ ρ ( s ) + ρ − t + m 2 m 2 m 2 ρ − u N. Kaiser (TUM) Chiral symmetry and low-energy pion-photon reactions

Extracting the chiral anomaly Dispersive representation of πγ → ππ with p-wave phase shift as input [M. Hoferichter, B. Kubis, D. Sakkas, PRD 86, 116009 (’12)] e u = 3 m 2 M ( s , t ) = F ( s ) + F ( t ) + F ( u ) , π − s − t , 4 π 2 f 3 π ∞ F ( s ) = a + b s + s 2 ds ′ Im F ( s ′ ) � 1 ( s ) e − i δ 1 Im F ( s ) = [ F ( s )+ ˆ F ( s )] sin δ 1 1 ( s ) s ′ 2 ( s ′ − s ) , π 4 m 2 π Relevant subtraction constant C = 3 ( a + b m 2 π ) is fitted to data and matched via the chiral representation to F 3 π m 2 � 2 . 9 − ln m π �� � π C = F 3 π 1 + = 1 . 067 F 3 π ( 4 π f π ) 2 m ρ solid line: C = 9 . 78 GeV − 3 dashed line: C = 12 . 9 GeV − 3 close to threshold, one-photon exchange an important correction: 1 → 1 − 2 e 2 f 2 π / t Good theory waiting for good data N. Kaiser (TUM) Chiral symmetry and low-energy pion-photon reactions

Tree level cross sections for π − γ → 3 π Coulomb gauge ǫ · p 1 = ǫ · k = 0, photon does not couple to incoming π − No γ 4 π vertex at leading order 5 - γ --> 3 π total cross sections: π 4 - π 0 π 0 tree approx. π σ tot [ µ b] 3 - π + π - tree approx. π 2 1 0 3 4 5 6 7 1/2 /m π s Example: total cross section for π − ( p 1 ) + γ ( k , ǫ ) → π − π 0 π 0 � √ s − m π α � π ( µ 2 − m 2 µ 2 − 4 m 2 π ) 2 σ tot ( s ) = d µ 16 π 2 f 4 π ( s − m 2 π ) 3 2 m π π − µ 2 + λ 1 / 2 ( s , µ 2 , m 2 π − µ 2 ) ln s + m 2 � π ) � ( s + m 2 − λ 1 / 2 ( s , µ 2 , m 2 √ s π ) 2 m π ( µ 2 − m 2 π ) / f 2 π is LO chiral ππ -interaction, rest from 3-body phase space How large are next-to-leading order corrections from chiral loops + cts? N. Kaiser (TUM) Chiral symmetry and low-energy pion-photon reactions