Exclusive Processes in Position Space and the Pion Distribution - PowerPoint PPT Presentation

Motivation Example: LO Conformal OPE State of the Art Outlook Exclusive Processes in Position Space and the Pion Distribution Amplitude V. M. Braun University of Regensburg based on V. Braun and D. Müller, arXiv:0709.1348 [hep-ph] Southampton, 12 October 2007 V. M. Braun Exclusive Processes in Position Space and the Pion Distribution Amplitude

Motivation Example: LO Conformal OPE State of the Art Outlook Conformal Transformations ds 2 = g µν ( x ) dx µ dx ν conserve the interval 1 g ′ µν ( x ′ ) = ω ( x ) g µν ( x ) only change the scale of the metric 2 ⇒ preserve the angles and leave the light-cone invariant Translations Rotations and Lorentz boosts ′ µ = λ x µ x µ → x Dilatation (global scale transformation) ′ µ = x µ / x 2 x µ → x Inversion V. M. Braun Exclusive Processes in Position Space and the Pion Distribution Amplitude

Motivation Example: LO Conformal OPE State of the Art Outlook Conformal Transformations ds 2 = g µν ( x ) dx µ dx ν conserve the interval 1 g ′ µν ( x ′ ) = ω ( x ) g µν ( x ) only change the scale of the metric 2 ⇒ preserve the angles and leave the light-cone invariant Translations Rotations and Lorentz boosts ′ µ = λ x µ x µ → x Dilatation (global scale transformation) ′ µ = x µ / x 2 x µ → x Inversion Special conformal transformation x µ + a µ x 2 x µ → x ′ µ = 1 + 2 a · x + a 2 x 2 = inversion, translation x µ → x µ + a µ , inversion V. M. Braun Exclusive Processes in Position Space and the Pion Distribution Amplitude

Motivation Example: LO Conformal OPE State of the Art Outlook Conformal Anomaly Consider YM theory with a hard cutoff M , integrate out the fields with frequencies above µ S eff = − 1 Z „ 1 β 0 « ˆ d 4 x 16 π 2 ln M 2 /µ 2 G a µν G a µν ˜ ( g (0) ) 2 − slow ( x ) + . . . 4 Under the scale transformation x µ → λ x µ , A µ ( x ) → λ A µ ( λ x ) , ψ ( x ) → λ 3 / 2 ψ ( λ x ) and µ → µ/λ with the fixed cutoff 1 Z µν G a µν ( x ) d 4 x G a δ S = − 32 π 2 β 0 ln λ which implies D ( x ) = − β 0 ∂ µ J µ µν G a µν ( x ) 32 π 2 G a Restoring the standard field definition = β ( g ) ∂ µ J µ D ( x ) = g µν Θ QCD EOM µν G a µν ( x ) 2 g G a µν ( x ) V. M. Braun Exclusive Processes in Position Space and the Pion Distribution Amplitude

Motivation Example: LO Conformal OPE State of the Art Outlook Outline Motivation 1 Example: LO 2 Conformal OPE 3 State of the Art 4 Outlook 5 V. M. Braun Exclusive Processes in Position Space and the Pion Distribution Amplitude

Motivation Example: LO Conformal OPE State of the Art Outlook Towards virtual reality . . . Preparation of states: • Accelerator–based experiments: — scattering of particles with fixed momentum — know what was ’before’, learn what becomes ’after’ the interaction — foundation of QM — observation of quantum phenomena with classical tools • Lattice–based experiments: — correlation functions of sources with fixed position — learn what happens in process of the interaction calls for change of philosophy: coordinate space–based phenomenology emphasize on space–time picture V. M. Braun Exclusive Processes in Position Space and the Pion Distribution Amplitude

Motivation Example: LO Conformal OPE State of the Art Outlook Hadron Distribution Amplitides Feynman: How to transfer a large momentum Q to a hadron? ’77–’80: The answer depends on the underlying field theory. In QCD, find a hadron in a rare configuration with all its consitutuents being at a small transverse separation ∼ 1 / Q E.g. pion distribution amplitude (DA) Z 1 n 2 = 0 � 0 | ¯ d ( 0 ) � n γ 5 u ( n ) | π + ( p ) � = if π ( pn ) du e − iunp φ π ( u , µ ) , 0 DAs enter QCD description of many hard processes: elastic and transition form factors γ ∗ ππ , γ ∗ πγ , γ ∗ πρ , . . . ν , B → K ∗ ℓ + ℓ − , B → ργ , . . . weak decays B → πℓ ¯ hard diffraction γ ∗ p → ρ p , . . . etc. but are known very poorly exclusive channels have small partial cross sections one does not measure DAs directly, but rather some convolution integrals for available momentum transfers, most of the processes are affected by so-called soft contributions etc. V. M. Braun Exclusive Processes in Position Space and the Pion Distribution Amplitude

Motivation Example: LO Conformal OPE State of the Art Outlook Example: Pion Distribution Amplitide based on: 1.4 • QCD sum rule and lattice calculations of 1.2 Z 1 1 ↔ du ( 2 u − 1 ) 2 φ π ( u ) ∼ � 0 | ¯ D + ) 2 γ + γ 5 | π ( p ) � φ ( u ) q ( 0.8 0 0.6 • Measurements of γ ∗ γ → π 0 form factor for 0.4 Q 2 ∼ 1 . 5 − 9 GeV 2 (CLEO) 0.2 Z 1 du 0 0.2 0.4 0.6 0.8 1 ∼ 1 − u φ π ( u ) u 0 How to move forward? ⋄ Lattice calculations of higher moments difficult because the loss of O(4) symmetry becomes more and more punishing; also need very high statistics ⋄ QCD analysis of γ ∗ γ → π 0 is somewhat model–dependent; mostly sensitive to a single overlap integral nonlocal operators that resemble DA [ Aglietti et al. ’98 ] 1 coordinate–space analogue of the γ ∗ γ ∗ → π 0 with two virtual photons, or similar 2 V. M. Braun Exclusive Processes in Position Space and the Pion Distribution Amplitude

Motivation Example: LO Conformal OPE State of the Art Outlook Demonstration: Correlation function of two EM currents x ρ p σ ν ( − x ) } π 0 ( p ) � = − i � 0 | T { j em µ ( x ) j em 4 π 2 x 4 T ( p · x , x 2 ) T µν = 3 f π ǫ µνρσ in leading order (LO) Z 1 1 du e i ( 2 u − 1 ) p · x φ π ( u , µ ∼ 1 / | x | ) T ( p · x , x 2 ) = 2 0 use standard decomposition X ∞ φ n ( µ ) C 3 / 2 φ π ( u , µ ) = 6 u ¯ ( 2 u − 1 ) u n n = 0 the Fourier transform yields Bessel functions Z 1 √ u e i ρ ( 2 u − 1 ) C 3 / 2 2 π ( n + 1 )( n + 2 ) i n ρ − 3 / 2 J n + 3 / 2 ( ρ ) du u ¯ ( 2 u − 1 ) = 8 n 0 obtain partial wave expansion ∞ 3 X T ( p · x , x 2 ) = φ n ( µ ) F n ( p · x ) x 2 → ’scale’ 4 n = 0 p · x → ’distance’ i n √ 2 π ( n + 1 )( n + 2 ) ρ − 3 / 2 J n + 3 / 2 ( ρ ) F n ( ρ ) = 2 V. M. Braun Exclusive Processes in Position Space and the Pion Distribution Amplitude

Motivation Example: LO Conformal OPE State of the Art Outlook Realistic models ? 1.4 1.2 1 φ ( u ) 0.8 φ π ( u , µ 0 ) = 0.6 h 1 + φ 2 C 3 / 2 ( 2 u − 1 ) + φ 4 C 3 / 2 i 0.4 = 6 u ¯ u ( 2 u − 1 ) n 4 0.2 0 0.2 0.4 0.6 0.8 1 u φ 2 ( 2 GeV ) = 0 . 201 ± 0 . 113 Braun et al., PRD74(2006)074501 φ 2 ( 2 GeV ) = 0 . 233 ± 0 . 088 C. Sachrajda, LATTICE-2007 blue φ 2 = 0, φ 4 = 0 asymptotic DA red φ 2 = 0 . 25, φ 4 = − 0 . 1 BMS model green BMS model φ 2 = 0 . 25, φ 4 = + 0 . 1 ⋄ same color identification used in all Figures throughout this talk V. M. Braun Exclusive Processes in Position Space and the Pion Distribution Amplitude

Motivation Example: LO Conformal OPE State of the Art Outlook LO results 0.8 0.5 4 0.4 2 0.6 0 T ( ρ, x 2 ) 0.3 i n F n ( ρ ) 0.4 0.2 0.1 0.2 0 0 -0.1 0 2 4 6 8 10 0 2 4 6 8 10 ρ ρ compare: φ π ( u ) = δ ( u − 1 / 2 ) T ( ρ ) = 1 / 2 φ π ( u ) = 1 T ( ρ ) = 1 / 2 cos ρ diffraction: p 2x V. M. Braun Exclusive Processes in Position Space and the Pion Distribution Amplitude

Motivation Example: LO Conformal OPE State of the Art Outlook Separation of variables O(3) rotational Angular vs. radial ⇒ In Quantum Mechanics: dependence symmetry " # − � 2 ♥ 2 m ∆ + V ( | r | ) Ψ = E Ψ ⇒ Ψ( � r ) = R ( r ) Y lm ( θ, φ ) Y lm ( θ, φ ) are eigenfunctions of L 2 Y lm = l ( l + 1 ) Y lm , [ H , L 2 ] = 0 . SL(2,R) conformal Longitudinal vs. transverse ⇒ In Quantum Chromodynamics: dependence symmetry Migdal ‘77 Brodsky,Frishman,Lepage,Sachrajda ‘80 Makeenko ‘81 Ohrndorf ‘82 V. M. Braun Exclusive Processes in Position Space and the Pion Distribution Amplitude

Motivation Example: LO Conformal OPE State of the Art Outlook Collinear conformal transformations Special conformal transformation x − x − → x ′ − = 1 + 2 ax − P = Pz translations x − → x ′ − = x − + c dilatations x − → x ′ − = λ x − form the so-called collinear subgroup SL ( 2 , R ) a α + b α → α ′ = c α + d , ad − bc = 1 „ a α + b « Φ( α ) → Φ ′ ( α ) ( c α + d ) − 2 j Φ = c α + d 1 = √ ( p 0 + p z ) → ∞ p + 2 where Φ( x ) → Φ( x − ) = Φ( α n − ) is the quantum 1 field with scaling dimension ℓ and spin projection s p − = √ ( p 0 − p z ) → 0 2 “living” on the light-ray px → p + x − Conformal spin: j = ( l + s ) / 2 V. M. Braun Exclusive Processes in Position Space and the Pion Distribution Amplitude