Green functions of currents in the odd-intrinsic parity sector of - PowerPoint PPT Presentation

Green functions of currents in the odd-intrinsic parity sector of QCD Tom´ aˇ s Kadav´ y with Karol Kampf and Jiˇ r´ ı Novotn´ y Institute of Particle and Nuclear Physics Faculty of Mathematics and Physics, Charles University in Prague XI th Quark Confinement and the Hadron Spectrum Saint Petersburg, September 8–12, 2014 Supported by the Charles University in Prague, project GA UK no. 700214. September 12, 2014 1 / 26

Motivation: Why Green functions? Based on works: [K. Kampf and J. Novotn´ y ’11 (arXiv: 1104.3137)] [T. Kadav´ y ’13: bachelor thesis] [T. Kadav´ y ’14: diploma thesis (in preparation)] Theoretical objects with important physical interpretation. Calculations and predictions of decay widths, etc. Connections with the current problems of the SM. 2 / 26

Chiral perturbation theory ( χ PT) An effective theory in a low-energy region ( ≤ 1 GeV). The basic building block of χ PT [S. Weinberg ’79], [J. Gasser and H. Leutwyler ’84, ’85]: � � i u ( φ ) = exp √ φ . 2 F The matrix of pseudoscalar meson fields:   2 π 0 + 1 1 π + K + 6 η 8 √ √  2 π 0 +  − 1 1 K 0 π − 6 η 8 φ =  √ √  . 0 − 2 K − K 6 η 8 √ The lowest order Lagrangian: χ = F 2 4 � u µ u µ + χ + � . L (2) 3 / 26

Resonance chiral theory (R χ T) An effective theory for an intermediate region (1 GeV ≤ E ≤ 2 GeV). Massive multiplets of vector V(1 −− ), axial-vector A(1 ++ ), scalar S(0 ++ ) and pseudoscalar P(0 − + ) resonances. A singlet and octet decomposition of the fields ( R = V , A , S , P ): 8 � λ a 1 R = √ R 0 + √ R a . 3 2 a =1 The interaction resonance Lagrangian up to O ( p 4 ) [G. Ecker, J. Gasser, A. Pich and E. de Rafael ’89]: R = F V + � + iG V � V µν [ u µ , u ν ] � + F A L (4) � V µν f µν � A µν f µν √ √ √ − � 2 2 2 2 2 2 + c d � Su µ u µ � + c m � S χ + � + id m � P χ − � + id m 0 � P �� χ − � . N F 4 / 26

Odd-intrinsic parity sector of QCD The leading order of the pure Goldstone-boson part of the odd-intrinsic parity sector starts at O ( p 4 ). The parameters are set entirely by the chiral anomaly. The Wess-Zumino-Witten Lagrangian contributes to the anomalous term and has the form [E. Witten ’83]: � � 1 � L WZW = N C φ ε µναβ . F � − i � W µναβ − � d ξ � σ ξ µ σ ξ ν σ ξ α σ ξ W µναβ � β 48 π 2 0 Resonance saturation at O ( p 6 ): Even sector: [V. Cirigliano, G. Ecker, M. Eidemuller, R. Kaiser, A. Pich and J. Portoles ’06]. Odd sector. 5 / 26

Independent operator basis The antisymmetric tensor formalism. The independent operator basis, constructed using the large N C approximation and relevant in the odd-intrinsic parity sector: i µναβ ε µναβ . i = � O X O X Lagrangians up to O ( p 6 ): � � L ( 6 , odd ) O X i κ X = i . R χ T X i 67 operators in total. Except for the single resonance fields, the basis gives us the following set of the multiple field combinations: X = VV , AA , SA , SV , VA , PA , PV , VVP , VAS , AAP . 6 / 26

Three-point Green functions The amplitudes of physical processes can be computed using LSZ reduction formula from the Green functions, the time ordered products of quantum fields. Definition: � 0 | T [ � O 1 ( p 1 ) � O 2 ( p 2 ) � O 3 (0)] | 0 � = � � d 4 x 2 e i ( p 1 x 1 + p 2 x 2 ) � 0 | T [ O 1 ( x 1 ) O 2 ( x 2 ) O 3 (0)] | 0 � . d 4 x 1 = Operators O = V , A , S , P . Only five nontrivial Green functions in the odd-intrinsic parity sector of QCD. � VVP � , � VAS � , � AAP � , � VVA � and � AAA � . 7 / 26

Three-point Green functions: The calculation The task is to calculate all contributing Feynman diagrams. An example: the Feynman diagram with the inner vertex contributing to the Lagrangian L AA = �{∇ σ A µν , A ασ } u β � κ AA 3 ε µναβ . 3 The result: (Π 2 ) abc µν = ( V 4 ) def ρσγδ ( S χ ) fc ( S 2 ( p )) ad µρσ ( S 2 ( q )) eb γδν 4 iB 0 F 2 A d abc A ) r 2 ( p 2 + q 2 − r 2 ) κ AA 3 p α q β ε µναβ . = − ( p 2 − M 2 A )( q 2 − M 2 8 / 26

Three-point Green functions: The topology A general graph topology of the three-point Green functions in the antisymmetric tensor formalism: Double lines stand for resonances and dash lines for GB (double lines together with dash lines is the sum of both possible contributions). The crossing is implicitly assumed. 9 / 26

� VVP � Green function The most important example in the odd-intrinsic parity sector of QCD. Important phenomenological applications. A tensor structure: (Π VVP ) abc µν ( p , q ) = Π VVP ( p 2 , q 2 , r 2 ) d abc p α q β ε µναβ . OPE constraints dictate for high values of all independent momenta: � 1 � p 2 + q 2 + r 2 Π(( λ p ) 2 , ( λ q ) 2 , ( λ r ) 2 ) VVP = B 0 F 2 + O . 2 λ 4 p 2 q 2 r 2 λ 6 Π R χ T VVP ( p 2 , q 2 , r 2 ): substituing the constraints into Π VVP ( p 2 , q 2 , r 2 ). The isolation of two low-energy constants: C W and C W 22 . 7 10 / 26

� VVP � Green function: Decay π 0 → γγ A formfactor: 2 r 2 F R χ T 3 B 0 F Π R χ T π 0 → γγ ( p 2 , q 2 , r 2 ) = VVP ( p 2 , q 2 , r 2 ) . The Brodsky-Lepage behaviour for large momentum [G. P. Lepage and S. J. Brodsky ’80, ’81]: π ) ∼ − 1 Q 2 →∞ F R χ T π 0 → γγ (0 , − Q 2 , m 2 lim Q 2 . An amplitude: A π 0 → γγ = e 2 F π 0 → γγ (0 , 0 , 0). A decay width ( π 0 ( p ) → γ ( k ) + γ ( l )): � 1 ν ( l ) | 2 . |A π 0 → γγ ε µναβ k α l β ǫ ∗ Γ π 0 → γγ = µ ( k ) ǫ ∗ 32 π m π 0 pol 11 / 26

� VVP � Green function: Decay π 0 → γγ 0.30 � 0.25 � � Q 2 F ΠΓΓ � GeV � 0.20 � � � � � � � ���� � � � � 0.15 �� �� � � � � � �� � � � 0.10 0.05 0.00 0 5 10 15 20 25 30 35 Q 2 � GeV 2 � Figure : CLEO (blue points) and BABAR (green squares) data with fitted function F R χ T π 0 → γγ (0 , − Q 2 , m 2 π ) [B. Aubert et al. (The BABAR Collaboration) ’09]. 12 / 26

� VVP � Green function: Decay π 0 → γγ (updated) CELLO 0,3 CLEO BaBar Belle B-L asymptotic behaviour 0,25 Our fit 2 ) (GeV) 0,2 2 F πγγ∗ (Q 0,15 Q 0,1 0,05 0 0 10 20 30 40 2 (GeV 2 ) Q Figure : Updated data with fitted function F R χ T π 0 → γγ (0 , − Q 2 , m 2 π ), [P. Roig, A. Guevara and G. L. Castro ’14]. 13 / 26

� VVP � Green function: Decay of ρ → πγ The connection ρ + ( q ) → π + ( p ) + γ ( k ) with the previous process: e ( q 2 − M 2 V ) F π 0 → γγ (0 , q 2 , 0) . A ρ + → π + γ = lim 2 F V M V q 2 → M 2 V A decay width: Γ ρ + → π + γ = m 2 ρ − m 2 � ν ( k ) | 2 . π |A ρ + → π + γ ε µναβ p α q β ǫ µ ( p ) ǫ ∗ 48 π m 3 ρ pol with [B. Moussallam ’95] � � � � 2 eF V A ρ + → π + γ � � � ≡ 1 + x . � M V A ρ 0 → γγ R χ T: x = − 0 . 010 ± 0 . 005 and Γ ρ + → π + γ = 67 . 0 ± 2 . 3 keV . 14 / 26

� VVP � Green function: Decays of π (1300) Two channels studied. The amplitudes: √ √ 2 κ PV 3 M 2 V − F V κ VVP π ′ → γγ = e 2 8 2 2 A R χ T 3 F V , M 4 V √ √ 2 κ PV 3 M 2 V − F V κ VVP π ′ → ργ = − e 4 2 A R χ T . M 2 3 M V V Belle collaboration [K. Abe et al. ’06]: Γ π ′ → γγ < 72 eV . An experimental bound on Γ π ′ → γγ can be used to get the estimate: κ VVP ≈ ( − 0 . 57 ± 0 . 13) GeV . 15 / 26

� VVP � Green function: Decays of π (1300) 30 25 � Π ’ �ΡΓ � keV � 20 15 10 5 0 20 40 60 80 100 � Π ’ �ΓΓ � eV � Figure : The connection of decay widths for π (1300) → γγ and π (1300) → ργ . The dashed line denotes the Belle collaboration limit. [K. Abe et al.) ’06]. 16 / 26

� VVP � Green function: The muon g − 2 factor Hadronic contributions: hadronic light-by-light scattering. The main source of theoretical error in the SM. The four point Green function � VVVV � can be simplified into: π ± and K ± loops, π 0 , η, η ′ exchanges: the � VVP � case etc. Using the fully off-shell F R χ T π 0 → γγ ( p 2 , q 2 , r 2 ) formfactor we get: a LbyL ,π 0 = (65 . 8 ± 1 . 2) · 10 − 11 . µ The result based on AdS/QCD conjecture [L. Cappiello, O. Cata and G. D’Ambrosio ’11]: a π 0 µ = (65 . 4 ± 2 . 5) · 10 − 11 . The updated result using Belle data [P. Roig, A. Guevara and G. L. Castro ’14]: a π 0 µ = (66 . 6 ± 2 . 1) · 10 − 11 . 17 / 26

� VAS � Green function A tensor structure: (Π VAS ) abc µν ( p , q ) = Π VAS ( p 2 , q 2 , r 2 ) f abc p α q β ε µναβ . OPE constraints dictate for high values of all independent momenta: � 1 � p 2 − q 2 − r 2 Π(( λ p ) 2 , ( λ q ) 2 , ( λ r ) 2 ) VAS = B 0 F 2 + O . 2 λ 4 p 2 q 2 r 2 λ 6 At low energies, up to O ( p 6 ): Π( p 2 , q 2 , r 2 ) = − 32 B 0 C W 11 . An experiment: decay K + → l + νγ suggests [A. A. Poblaguev et al. ’02], [R. Unterdorfer and H. Pichl ’08]: 11 = (0 . 68 ± 0 . 21) · 10 − 3 GeV − 2 C W and κ VAS = (0 . 61 ± 0 . 40) GeV . 18 / 26