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

Low-Energy Pion-Photon Reactions and Chiral Symmetry N. Kaiser International Conference HADRON 2011: Munich, 13. June 2011 Tests of chiral perturbation theory via low-energy π − γ reactions COMPASS@CERN: Primakoff effect to extract π − γ cross sections Pion Compton scattering in ChPT: electric/magnetic polarizabilities Radiative corrections to π − γ → π − γ , isospin-breaking correction Neutral and charged pion-pair production: π − γ → π − π 0 π 0 & π + π − π − Total cross sections and 2 π invariant mass spectra at one-loop order Radiative corrections to π − γ → π − π 0 π 0 (simpler case) N. Kaiser Low-energy pion-photon reactions and chiral symmetry

Introduction: some ChPT highlights Pions π ± 0 : Goldstone bosons of spontaneous chiral symmetry breaking Their low-energy dynamics: systematically (and accurately) calculable in Chiral Perturbation Theory ( = loop-expansion with effective Lagrangian) 2-loop prediction for I = 0 ππ -scattering length: a 0 m π = 0 . 220 ± 0 . 005 confirmed by NA48/2@CERN: K + → π + π − e + ν e ( π + π − mass distribut.) qq | 0 � is large, linear term dominates Implications: quark condensate � 0 | ¯ quark mass expansion of m 2 π : m 2 π f 2 qq | 0 � m q + O ( m 2 q ln m q ) π = −� 0 | ¯ DIRAC@CERN: Pionium lifetime τ pred = ( 2 . 9 ± 0 . 1 ) · 10 − 15 sec Γ(( π + π − ) atom → π 0 π 0 ) = 2 π ( a 0 − a 2 ) 2 + . . . 9 α 3 p cm m 2 Cusp effect in 2 π 0 mass spectrum of K + → π + π 0 π 0 at π + π − threshold: ( a 0 − a 2 ) m π = 0 . 257 ± 0 . 006 , ChPT: ( a 0 − a 2 ) m π = ( 0 . 265 ± 0 . 005 ) Electromagnetic processes with pions allow for further tests of ChPT Pion polarizability difference (2-loops): α π − β π = ( 5 . 7 ± 1 . 0 ) · 10 − 4 fm 3 , experimental determinations from Serpukhov and Mainz in conflict with it N. Kaiser Low-energy pion-photon reactions and chiral symmetry

Introduction: Primakoff effect Primakoff effect: Z Scattering of high energy pions in nuclear Coulomb field (high Z) allows to extract cross sections for π − γ reactions (equivalent-photon method) Q 2 − Q 2 d σ Z 2 α Q min = s − m 2 min σ π − γ ( s ) , π ds dQ 2 = π ( s − m 2 Q 4 2 E beam π ) s = ( π − γ invariant mass) 2 , Q → 0 momentum transfer by virtual photon isolate Coulomb peak from strong interaction background COMPASS@CERN: (E18@TUM, S. Paul, J. Friedrich,...) π -Compton scattering π − γ → π − γ : electric and magnetic polarizabilities π 0 -production π − γ → π − π 0 : test QCD chiral anomaly, F γ 3 π = e / ( 4 π 2 f 3 π ) pion-pair product. π − γ → 3 π : √ s > 1GeV meson spectroscopy, exotics, high statistics allows to continue event rates even down to threshold N. Kaiser Low-energy pion-photon reactions and chiral symmetry

Pion Compton-scattering in ChPT Pion Compton-scattering: π − ( p 1 ) + γ ( k 1 , ǫ 1 ) → π − ( p 2 ) + γ ( k 2 , ǫ 2 ) T-matrix in center-of-mass frame in Coulomb gauge ǫ 0 1 , 2 = 0: k 1 2 � � �� T πγ = 8 πα ǫ 2 A ( s , t ) + � ǫ 1 · � k 2 � ǫ 2 · � A ( s , t ) + B ( s , t ) − � ǫ 1 · � t Mandelstam variables: s = ( p 1 + k 1 ) 2 , t = ( k 1 − k 2 ) 2 Differential cross section: d σ d Ω cm = α 2 � 2 + �� � A ( s , t ) � A ( s , t ) + ( 1 + z ) B ( s , t ) � 2 � � � � 2 s t = ( s − m 2 π ) 2 ( z − 1 ) / 2 s with z = cos θ cm , scattering angle Tree diagrams: s − m 2 A ( s , t ) ( tree ) = 1 , B ( s , t ) ( tree ) = π m 2 π − s − t 9 8 7 _ γ -> π _ γ ) σ tot ( π 6 5 σ [ µ b] 4 3 2 1 0 1 1.5 2 2.5 3 3.5 1/2 [m π ] s N. Kaiser Low-energy pion-photon reactions and chiral symmetry

Pion Compton-scattering in ChPT Pion-loop diagrams (photon scattering off the pion’s “pion cloud”): √ − t 4 m 2 π − t + − t � 1 � � A ( s , t ) ( loop ) = ∼ t 2 > 0 2 − 2 m 2 π ln 2 ( 4 π f π ) 2 2 m π with f π = 92 . 4 MeV, expression corresponds to isospin limit: m π 0 = m π Electric/magnetic polarizabilities = low-energy const. with α π + β π = 0 A ( s , t ) ( pola ) = − β π m π t α π m π (¯ ℓ 6 − ¯ < 0 , α π − β π = ℓ 5 ) 24 π 2 f 2 2 α N. Kaiser Low-energy pion-photon reactions and chiral symmetry

Pion Compton-scattering in ChPT 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 Current-algebra relation: � 0 | A µ V ν | π � ≃ f π � π | V µ V ν | π � plus corrections One-loop ”prediction”: α π = − β π ≃ 3 . 0 · 10 − 4 fm 3 σ tot ( s ) insensitive to pion’s low-energy structure Small effect on backward angular distributions of d σ/ d Ω cm 1 0.25 - - - - structureless pion, Born terms only 0.95 0.2 d σ /d Ω cm [ µ b] 0.15 1/2 = 2m π s d σ / d σ 0 0.9 -4 fm 3 α π = - β π = 3.10 0.1 1/2 = 3m π s 0.85 -4 fm 3 Effect of pion polarizabilities: α π = - β π = 3.10 0.05 1/2 = 4m π s 0 0.8 -1 -0.8 -0.6 -0.4 -0.2 0 1 1.5 2 2.5 3 3.5 4 z=cos θ cm 1/2 [m π ] s Pion-loop compensates partly reduction of d σ/ d Ω cm by polarizabilities Effect of pion polarizabilities on π -Compton cross section: less than 20 % 2-loop corrections to d σ/ d Ω cm are very small (Gasser, Ivanov) N. Kaiser Low-energy pion-photon reactions and chiral symmetry

Pion polarizabilities in ChPT Gasser et al., NPB745, 84 (2006): Pion polarizabilities to 2 loops Analytical expression in terms of low-energy constants ¯ ℓ j : α (¯ ℓ 6 − ¯ α m π ln m π ℓ 5 ) c r + 8 ℓ 6 + 65 � � � ℓ 2 − ¯ ¯ ℓ 1 + ¯ ℓ 5 − ¯ α π − β π = π m π + 24 π 2 f 2 ( 4 π f π ) 4 m ρ 3 12 3 + 4 ¯ ¯ � 53 π 2 + 4 ℓ 3 ℓ 4 ℓ 5 ) − 187 − 41 �� 9 (¯ ℓ 1 + ¯ 3 (¯ ℓ 6 − ¯ ℓ 2 ) − 81 + 48 324 ℓ j from ππ data, c r ≃ 0 via resonance saturation Improved values of ¯ 2-loop prediction including realistic estimate of theoretical errors: α π − β π = ( 5 . 7 ± 1 . 0 ) · 10 − 4 fm 3 , α π + β π = ( 0 . 16 ± 0 . 1 ) · 10 − 4 fm 3 Good reasons to believe that chiral prediction is stable against higher order corrections: ChPT at 2-loop order works very well for γγ → π 0 π 0 Existing expt. determinations α π − β π = ( 15 . 6 ± 7 . 8 ) · 10 − 4 fm 3 from Serpukhov (via Primakoff) and α π − β π = ( 11 . 6 ± 3 . 4 ) · 10 − 4 fm 3 from Mainz (via γ p → γπ + n ) violate chiral low-energy theorem by a factor 2! � d ω ω 2 σ πγ 1 α π + β π = abs ( ω ) agrees with results from dispersion sum rules 2 π 2 N. Kaiser Low-energy pion-photon reactions and chiral symmetry

Radiative corrections to pion Compton scattering Pion-structure effects small: necessary to include radiative corr. of O ( α ) Start with structureless pion: extensive calculation in 1-loop scalar QED Advantage of Coulomb gauge: all s -channel pole diagrams vanish Class: I II Class: III Class: IV Class: VII Class: VIII Class: V Class: VI IX X XI Dimensional regularization to treat both ultraviolet divergencies ( d < 4) and infrared divergencies ( d > 4 ) : 2 ( γ E − ln 4 π ) + ln m π d − 4 + 1 1 ξ = µ Alternative: introduce regulator photon mass m γ , ξ IR = ln ( m π / m γ ) N. Kaiser Low-energy pion-photon reactions and chiral symmetry

Radiative corrections to pion Compton scattering Infrared-finite after inclusion of soft photon bremsstrahlung: d σ/ d Ω cm · δ soft � 2 m 2 d d − 1 l π − t m 2 m 2 � � δ soft = α µ 4 − d π π ( 2 π ) d − 2 l 0 p 1 · l p 2 · l − ( p 1 · l ) 2 − ( p 2 · l ) 2 � l % , | <λ | Evaluated in dim. regularization: ξ IR from photon loops gets canceled, radiative correction depends on a small energy resolution scale λ : �� t − 8 � t + � t ln m π s + 1 4 ˆ 4 − ˆ − ˆ 2 λ + ˆ δ real = α � s s − 1 ln ˆ 2 + ln ˆ π � ˆ t 2 − 4 ˆ t 2 √ � 1 / 2 s + 1 )(ˆ t − 2 ) s + 1 + W � (ˆ ln ˆ dx √ √ + W [ 1 − ˆ tx ( 1 − x )] s + 1 − W ˆ 0 s = s / m 2 t = t / m 2 π and W = (ˆ s − 1 ) 2 + 4 ˆ s ˆ tx ( 1 − x ) π , ˆ where ˆ Terms beyond ln ( m π / 2 λ ) specific for evaluation in center-of-mass frame Idealized experiment with undetected soft photons filling in momentum space a small sphere of radius λ in the center-of-mass frame Further experiment-specific soft/hard γ -radiation can be accounted for N. Kaiser Low-energy pion-photon reactions and chiral symmetry

Radiative corrections to pion Compton scattering 1 Results: - + γ --> π - + γ pion Compton scattering: π 0.5 λ = 5 MeV 0 radiative correction [%] -0.5 -1 -1.5 1/2 = 2m π s -2 1/2 = 3m π s -2.5 1/2 = 4m π s 1/2 = 5m π -3 s -3.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 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 a factor of � 10 In long wavelength limit k 1 , k 2 → 0: all strong and radiative corrections vanish, pure Thomson amplitude T ( 0 ) π − γ = − 8 πα� ǫ 1 · � ǫ 2 survives N. Kaiser Low-energy pion-photon reactions and chiral symmetry