The anomaly triangle and muon g 2 SANTI PERIS San Francisco State - PowerPoint PPT Presentation

The anomaly triangle and muon g − 2 SANTI PERIS San Francisco State U. and U. Autonoma de Barcelona The anomaly triangle and muon g − 2 – p.1/8

2-loop EW contribution to g − 2 Kukhto et al. ’92 SP, Perrottet, de Rafael ’95 Czarnecki, Krause, Marciano ’95, ’96 Knecht, SP, Perrottet, de Rafael ’02 µ µ Czarnecki, Marciano,Vainshtein ’03 Z γ � � M 2 m 2 � ∞ G µ g − 2 w L ( Q 2 ) + Z + Q 2 w T ( Q 2 ) ∝ α µ dQ 2 Z µ √ f m 2 2 8 π 2 π M 2 2 � �� � w L =2 w T =2 Nc γ ( one − loop, m f =0) Q 2 g − 2 2 | e,u,d ∼ 2 × 10 − 11 Q 2 = − q 2 , “Gluon-irreducible” quark triangle � w L ( Q 2 ) q ν ǫ µραβ q α k β + W µνρ ( q, k ) = T (3) Q 2 f f w T ( Q 2 ) k σ � q 2 ǫ µνρσ + q ν ǫ µρλσ q λ + q µ ǫ ρνλσ q λ � � + O ( k 2 ) � � ⇒ � T (3) • w L ( Q 2 ) related to the chiral anomaly = Q 2 w L = 0 . ν,e,u,d f f • w T ( Q 2 ) not but... The anomaly triangle and muon g − 2 – p.2/8

Theorem Vainshtein ’02 Knecht, SP, Perrottet, de Rafael, ’03 In the massless limit, to all orders in α s : w L ( Q 2 ) = 2 w T ( Q 2 ) and, since anomaly does not get renormalized: w L = 2 N c Q 2 exact! (Adler,Bardeen ’69; Witten ’83) ⇒ neither does 2 w T = 2 N c Q 2 , to all orders in α s . = � ℓ = ν,e ℓ L γ µ T (3) ℓ L + � Using L (3) q = u,d q L γ µ T (3) q L , etc...in SU (2) L × U (1) Y : µ � d 4 xd 4 y e iqx ( y − x ) λ ǫ µνρλ � � �� Q 2 � � L (3) µ ( x ) V ( Y ) ( y ) R ( Y ) w L ( Q 2 ) − 2 w T ( Q 2 ) quarks ∝ T (0) ν ρ � �� � =0 , (Pert . Theory) • i.e., w L − 2 w T has no pert. contributions in α s , it is like, e.g., � LR � = � V V − AA � . The anomaly triangle and muon g − 2 – p.3/8

Non-perturbative effects w L ( Q 2 ) = 2 N c 1) Adler-Bardeen-Witten : (exact for all Q !) Q 2 2) However, for w T ( Q 2 ) : • Large Q 2 : + ( const. ) α s χ � ψψ � 2 2 w T ( Q 2 ) ≈ 2 N c � � + O (1 /Q 8 ) 1 + NO α s Q 2 Q 6 χ = Π V T (0) Magnetic susceptibility, � ψψ � , very poorly known. • Small Q 2 : 2 w T ( Q 2 ) ≈ ( const. ) C ( p 6 ) C ( p 6 ) + O ( Q 2 ) ∼ 1 /M 2 , (unknown) 22 22 Hadron Chiral Pert. Theory, L eff (parity-odd): (Ebertshauser, Fearing, Scherer ’01; Bijnens, Girlanda, Talavera ’02, Kampf, Moussallam ’09) � u µ � �� L O ( p 6 ) = C ( p 6 ) ∇ γ f γν + , f αβ ǫ µναβ Tr + ... ; SU ( N F ) L × SU ( N F ) R → SU ( N F ) V 22 + � �� � π,η,... The anomaly triangle and muon g − 2 – p.4/8

Non-perturbative effects (II) Very roughly, Anomaly Q 2 ���� 2 w T ( Q 2 ) Q 2 ∼ 2 N c Q 2 + Λ 2 Hadron i.e. 2Nc 2 2w Q T The anomaly triangle and muon g − 2 – p.5/8

Conjectures w L ( Q 2 ) − 2 w T ( Q 2 ) = − 2 Nc π Π LR ( Q 2 ) • Conjecture 1: (Son-Yamamoto ’10) f 2 in wide class of “AdS/QCD” models (chiral limit, N c → ∞ ) (not without caveats, e.g. OPE is exponential; wrong chiral limit in pert. theory) ( Knecht, SP, de Rafael ’11) Chiral log’s respect this relation in SU (2) × SU (2) × U (1) ( m u,d = 0 , m s � = 0 ) (Gorsky, Kopnin,Krikun, Vainshtein ’12) c ( p 6 ) ℓ ( p 4 ) ( µ ) = − N c − 64 π 2 ( µ ) 13 f 2 5 π However, they don’t in SU (3) × SU (3) ( m u,d,s = 0 ) (Knecht, SP , ’12 (unpublished)) ?? C ( p 6 ) L ( p 4 ) N c 128 π 2 ( µ ) � = − ( µ ) 22 10 f 2 π (Magnetic susceptibility) ?? N c π ∼ − 9 GeV − 2 • Conjecture 2: χ = − 4 π 2 f 2 (Vainshtein ’02) : Other results: χ ∼ − 3 GeV − 2 , sum rules, VMD, (Ioffe, Fadin, Lipatov ’10; Balitsky et al. ’85; Belyaev et al. ’84; Ball et al. ’02) The anomaly triangle and muon g − 2 – p.6/8

Another Perturbative Surprise Up to now, special kinematic configuration in � V V A � . Jegerlehner,Tarasov ’06 However, it has been found at two loops for arbitrary momenta that : W µνρ ( q, k ) = W µνρ ( q, k ) | one − loop (1 + O ( α s ) ) � �� � =0 !! i.e., no renormalization, not just for the anomaly, but for the whole triangle ! Given the non-trivial momentum dependence, can this be just a coincidence ? could this be true to all orders in α s ? The anomaly triangle and muon g − 2 – p.7/8

Summary • VVA triangle is a very interesting theoretical laboratory for QCD • Even though most results obtained in chiral limit: can lattice help/check ? • The LbL ← → � V V A � connection: ( Melnikov, Vainshtein ’04; Prades, de Rafael, Vainshtein ’09) k 1 ≈ k 2 ≫ k 3 q q 0 k 0 1 γ γ γ H 5 k 3 k k 3 2 The anomaly triangle and muon g − 2 – p.8/8