Axial anomaly and BABAR data Y.N. Klopot 1 , A.G. Oganesian 2 , O.V. - PowerPoint PPT Presentation

Introduction Anomaly Sum Rule Transition form factors of mesons Quark-hadron duality Corrections interplay and ex Axial anomaly and BABAR data Y.N. Klopot 1 , A.G. Oganesian 2 , O.V. Teryaev 1 1 Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Russia 2 Institute of Theoretical and Experimental Physics, Moscow, Russia International Workshop "Bogoliubov readings" JINR Dubna

Introduction Anomaly Sum Rule Transition form factors of mesons Quark-hadron duality Corrections interplay and ex Outline Introduction Anomaly Sum Rule Transition form factors of mesons Quark-hadron duality Corrections interplay and experimental data

Introduction Anomaly Sum Rule Transition form factors of mesons Quark-hadron duality Corrections interplay and ex • The last data of BaBar collaboration [Phys. Rev. D 80, 052002 (2009)] show unexpectedly large values of photon-pion transition form factor at large Q 2 (violation of QCD factorization!)

Introduction Anomaly Sum Rule Transition form factors of mesons Quark-hadron duality Corrections interplay and ex • [Dolgov, Zakharov, Nucl. Phys. B27,1971] • Dispersive representation of anomaly: exact sun rules (ASR). Opportunity to find high precision results and relations. Pion width [ Ioffe, Oganesian, Phys.Lett. B647,2007] for real photons case with very high accuracy. • !! ASR is valid for virtual photons also [Horejsi, Teryaev, Z.Phys.C65, 1995] It will be shown that ASR is actually saturated only by infinite number of resonances- anomaly reveals itself as a collective effect of meson spectrum. • Allows: ... and transition formfactor pion->2 γ Exactness af ASR -significant • The data CELLO, CLEO,(consistent to LCSR with usual DA) • but BaBar collaboration [Phys. Rev. D 80, 052002 (2009)] show unexpectedly large values of photon-pion transition form factor at large Q 2 (violation of QCD factorization!) It is impossible to explain BaBar data on F πγ ( Q 2 ) at large Q 2 by use of usual form of pion distribution [Khodjamirian 0909.2154, Mikhailov, Stefanis, Nucl. Phys. B821 2009]

Introduction Anomaly Sum Rule Transition form factors of mesons Quark-hadron duality Corrections interplay and ex • [Dolgov, Zakharov, Nucl. Phys. B27,1971] • Dispersive representation of anomaly: exact sun rules (ASR). Opportunity to find high precision results and relations. Pion width [ Ioffe, Oganesian, Phys.Lett. B647,2007] for real photons case with very high accuracy. • !! ASR is valid for virtual photons also [Horejsi, Teryaev, Z.Phys.C65, 1995] It will be shown that ASR is actually saturated only by infinite number of resonances- anomaly reveals itself as a collective effect of meson spectrum. • Allows: ... and transition formfactor pion->2 γ Exactness af ASR -significant • The data CELLO, CLEO,(consistent to LCSR with usual DA) • but BaBar collaboration [Phys. Rev. D 80, 052002 (2009)] show unexpectedly large values of photon-pion transition form factor at large Q 2 (violation of QCD factorization!) It is impossible to explain BaBar data on F πγ ( Q 2 ) at large Q 2 by use of usual form of pion distribution [Khodjamirian 0909.2154, Mikhailov, Stefanis, Nucl. Phys. B821 2009]

Introduction Anomaly Sum Rule Transition form factors of mesons Quark-hadron duality Corrections interplay and ex • There were suggested several models explaining this contradiction [Radyushkin Phys.Rev.D80:094009(2009),Dorokhov1003.4693,Mikhailov&Pimikov&Stefa 1006.2936] [Mikhailov, Pimikov, Stefanis 1006.2936 ] give some arguments against this enhancement. • Let us find out what can be learned about the meson transition form factors behaviour at large Q 2 in QCD beyond the factorization from the anomaly sum rule. • Our (non-perturbative) QCD method does not imply QCD factorization and is valid even if QCD factorization is broken. By use of the axial anomaly (in the dispersive approach) we can get the exact relations between possible corrections to lower states and continuum • We will show that even small continuum corrections provides a possibility of relatively large corrections to the lower states.

Introduction Anomaly Sum Rule Transition form factors of mesons Quark-hadron duality Corrections interplay and ex • There were suggested several models explaining this contradiction [Radyushkin Phys.Rev.D80:094009(2009),Dorokhov1003.4693,Mikhailov&Pimikov&Stefa 1006.2936] [Mikhailov, Pimikov, Stefanis 1006.2936 ] give some arguments against this enhancement. • Let us find out what can be learned about the meson transition form factors behaviour at large Q 2 in QCD beyond the factorization from the anomaly sum rule. • Our (non-perturbative) QCD method does not imply QCD factorization and is valid even if QCD factorization is broken. By use of the axial anomaly (in the dispersive approach) we can get the exact relations between possible corrections to lower states and continuum • We will show that even small continuum corrections provides a possibility of relatively large corrections to the lower states.

Introduction Anomaly Sum Rule Transition form factors of mesons Quark-hadron duality Corrections interplay and ex • There were suggested several models explaining this contradiction [Radyushkin Phys.Rev.D80:094009(2009),Dorokhov1003.4693,Mikhailov&Pimikov&Stefa 1006.2936] [Mikhailov, Pimikov, Stefanis 1006.2936 ] give some arguments against this enhancement. • Let us find out what can be learned about the meson transition form factors behaviour at large Q 2 in QCD beyond the factorization from the anomaly sum rule. • Our (non-perturbative) QCD method does not imply QCD factorization and is valid even if QCD factorization is broken. By use of the axial anomaly (in the dispersive approach) we can get the exact relations between possible corrections to lower states and continuum • We will show that even small continuum corrections provides a possibility of relatively large corrections to the lower states.

Introduction Anomaly Sum Rule Transition form factors of mesons Quark-hadron duality Corrections interplay and ex Anomaly sum rule The VVA amplitude � d 4 xd 4 ye ( ikx + iqy ) � 0 | T { J 5 T αµν ( k , q ) = α ( 0 ) J µ ( x ) J ν ( y ) }| 0 � , (1) u γ 5 γ α u − ¯ where J 5 α = (¯ d γ 5 γ α d ) √ � 0 | J 5 α ( 0 ) | π 0 ( p ) � = i 2 p α f π , (2)

Introduction Anomaly Sum Rule Transition form factors of mesons Quark-hadron duality Corrections interplay and ex The VVA triangle graph amplitude can be presented as a tensor decomposition F 1 ε αµνρ k ρ + F 2 ε αµνρ q ρ T αµν ( k , q ) = + F 3 q ν ε αµρσ k ρ q σ + F 4 q ν ε αµρσ k ρ q σ (3) + F 5 k µ ε ανρσ k ρ q σ + F 6 q µ ε ανρσ k ρ q σ We consider the following case: k 2 = 0 , Q 2 = − q 2 (4)

Introduction Anomaly Sum Rule Transition form factors of mesons Quark-hadron duality Corrections interplay and ex Dispersive approach to axial anomaly leads to [Hořejší1985,Hořejší&Teryaev1995]: � ∞ 4 m 2 A 3 ( s ; Q 2 , m 2 ) ds = 1 (5) 2 π A 3 ≡ Im ( F 3 − F 6 ) / 2 (in the paper F 3 = − F 6 is obtained from direct calculation perturbative calculation) • Holds for any Q 2 and any m 2 . • Does not have α s corrections. • Does not have nonperturbative corrections (’t Hooft’s principle).

Introduction Anomaly Sum Rule Transition form factors of mesons Quark-hadron duality Corrections interplay and ex Transition form factors of mesons The form factor π 0 → γ ∗ γ is defined from the matrix element: � d 4 xe ikx � π 0 ( q ) | T { J µ ( x ) J ν ( 0 ) }| 0 � = ǫ µνρσ k ρ q σ F πγγ , (6) The VVA amplitude � d 4 xd 4 ye ( ikx + iqy ) � 0 | T { J 5 T αµν ( k , q ) = α ( 0 ) J µ ( x ) J ν ( y ) }| 0 � , (7) u γ 5 γ α u − ¯ where J 5 α = (¯ d γ 5 γ α d ) √ � 0 | J 5 α ( 0 ) | π 0 ( p ) � = i 2 p α f π , (8)

Introduction Anomaly Sum Rule Transition form factors of mesons Quark-hadron duality Corrections interplay and ex Three-point correlation function T αµν ( k , q ) has pion and higher states contributions: √ 2 f π i p α k ρ q σ ǫ µνρσ F πγγ + ( higher states ) . T αµν ( k , q ) = (9) p 2 − m 2 π Using the kinematical identities δ αβ ǫ σµντ − δ ασ ǫ βµντ + δ αµ ǫ βσντ − δ αν ǫ βσµν + δ ατ ǫ βσµν = 0 , (10) we can single out the pion contribution to 1 2 ( F 3 − F 6 ) (imaginary part is taken w.r.t. p 2 ): √ 1 2 f π π F πγ ( Q 2 ) δ ( s − m 2 2 Im ( F 3 − F 6 ) = π ) , (11)

Introduction Anomaly Sum Rule Transition form factors of mesons Quark-hadron duality Corrections interplay and ex • Q 2 = 0 : pion contribution saturates ASR: 1 F πγγ ( 0 ) = √ (12) 2 π 2 f π 2 • Q 2 � = 0 : Factorization approach to pQCD [Lepage&Brodsky1980]: √ � 1 � x , Q 2 � 2 f π dx ϕ π F πγγ ( Q 2 ) = + O ( 1 / Q 4 ) , (13) 3 Q 2 x 0 Asymptote at large Q 2 : [Efremov&Radyushkin1980] ϕ asymp ( x ) = 6 x ( 1 − x ) π √ 2 f π F asymp ( Q 2 ) = + O ( 1 / Q 4 ) . (14) πγγ Q 2