Hadronic light-by-light dispersion relations: short-distance - PowerPoint PPT Presentation

Hadronic light-by-light dispersion relations: short-distance constraints Martin Hoferichter Institute for Nuclear Theory University of Washington Second Plenary Workshop of the Muon g − 2 Theory Initiative Mainz, June 19, 2018 G. Colangelo, MH, M. Procura, P . Stoffer, work in progress M. Hoferichter (Institute for Nuclear Theory) HLbL dispersion relations: short-distance constraints Mainz, June 19, 2018 1

Dispersive representation: overview + + + · · · Π π 0 -pole ¯ Π π -box Π µνλσ = + + Π µνλσ + · · · µνλσ µνλσ Organized in terms of on-shell intermediate states Numerics for a π 0 -pole , a ππ,π -pole LHC talk by B.-L. Hoid and a π -box talks by G. Colangelo and P . Stoffer µ µ µ, J = 0 Other pseudoscalar ( η , η ′ ) and two-meson states ( K ¯ K , πη ) to be included along the same lines Here: attacking the ellipsis with short-distance constraints M. Hoferichter (Institute for Nuclear Theory) HLbL dispersion relations: short-distance constraints Mainz, June 19, 2018 2

BTT decomposition: reminder � ∞ � 1 � 2 π 12 α 3 � a HLbL d Σ Σ 3 � T i ( Q 1 , Q 2 , Q 3 )¯ 1 − r 2 = dr r d φ Π i ( Q 1 , Q 2 , Q 3 ) µ 432 π 2 0 0 0 i = 1 54 54 Π µνλσ = � T µνλσ � T µνλσ ˆ ˆ Π i subset of ˆ ¯ Π i = Π i Π i Q i = Q i (Σ , r , φ ) i i i = 1 i = 1 Bardeen–Tung–Tarrach (BTT) decomposition Π i free of kinematic singularities and zeros ֒ → dispersive treatment A lot of the complexity separated into kernel functions T i Dispersion relations for the Π i at small virtualities, but need to account for Asymptotic region : all Q 2 i large Mixed regions : Q 2 3 ≪ Q 2 1 ∼ Q 2 2 etc. ֒ → short-distance constraints M. Hoferichter (Institute for Nuclear Theory) HLbL dispersion relations: short-distance constraints Mainz, June 19, 2018 3

Familiar contributions in BTT form Pion pole F π 0 γ ∗ γ ∗ ( q 2 1 , q 2 2 ) F π 0 γ ∗ γ ∗ ( q 2 F π 0 γ ∗ γ ∗ ( q 2 1 , q 2 3 ) F π 0 γ ∗ γ ∗ ( q 2 3 , 0 ) 2 , 0 ) Π 1 ( q 2 1 , q 2 2 , q 2 Π 2 ( q 2 1 , q 2 2 , q 2 ˆ ˆ 3 ) = 3 ) = q 2 3 − M 2 q 2 2 − M 2 π 0 π 0 Pion loop � 1 � 1 − x 1 Π π -box ( q 2 1 , q 2 2 , q 2 3 ) = F V π ( q 2 1 ) F V π ( q 2 2 ) F V π ( q 2 ˆ 3 ) dx dy I i ( x , y ) i 16 π 2 0 0 8 xy ( 1 − x − y )( 1 − 2 x ) 2 ( 1 − 2 y ) 8 xy ( 1 − 2 x )( 1 − 2 y ) I 1 ( x , y ) = I 7 ( x , y ) = − · · · ∆ 3 ∆ 123 ∆ 23 123 ∆ ijk = M 2 π − xyq 2 i − x ( 1 − x − y ) q 2 j − y ( 1 − x − y ) q 2 ∆ ij = M 2 π − x ( 1 − x ) q 2 i − y ( 1 − y ) q 2 k j BTT decomposition isolates the dynamical content , separates the kinematics ֒ → do the same for the fermion loop M. Hoferichter (Institute for Nuclear Theory) HLbL dispersion relations: short-distance constraints Mainz, June 19, 2018 4

Fermion loop in BTT decomposition Fermion loop � 1 � 1 − x 1 Π f -loop ( q 2 1 , q 2 2 , q 2 3 ) = N c Q 4 ˆ dx dy I i ( x , y ) f i 16 π 2 0 0 64 xy 2 ( 1 − x − y )( 1 − 2 x )( 1 − y ) 16 x ( 1 − x − y ) 16 xy ( 1 − 2 x )( 1 − 2 y ) I 1 ( x , y ) = − − I 7 ( x , y ) = − ∆ 2 ∆ 3 ∆ 132 ∆ 32 132 132 ∆ ijk = m 2 f − xyq 2 i − x ( 1 − x − y ) q 2 j − y ( 1 − x − y ) q 2 ∆ ij = m 2 f − x ( 1 − x ) q 2 i − y ( 1 − y ) q 2 k j Numerical cross checks f e µ τ c b a f -loop [ 10 − 11 ] 26257 ( 3 ) 464 . 97 ( 5 ) 2 . 686 ( 3 ) 3 . 038 ( 3 ) 0 . 018 ( 3 ) µ Jegerlehner, Nyffeler 2009 26253 . 5102 ( 2 ) 464 . 971652 2 . 68556 ( 86 ) Asymptotic expansion, K¨ uhn et al. 2003 3 . 04 0 . 0182 M. Hoferichter (Institute for Nuclear Theory) HLbL dispersion relations: short-distance constraints Mainz, June 19, 2018 5

Asymptotic region and pQCD quark loop Four point function � Π µνλσ ( q 1 , q 2 , q 3 ) = − i d 4 x d 4 y d 4 z e − i ( q 1 · x + q 2 · y + q 3 · z ) � 0 | T { j µ ( x ) j ν ( y ) j λ ( z ) j σ ( 0 ) }| 0 � Q = 1 j µ ( x ) = ¯ ψ ( x ) Q γ µ ψ ( x ) ψ = ( u , d , s ) T � � 3 diag 2 , − 1 , − 1 All q 2 i large: free propagators give the most singular configuration in position space ֒ → pQCD quark loop should be adequate for the asymptotic region 3 ≡ q 2 simple analytic results For q 2 1 = q 2 2 = q 2 √ � �� 4 8 � π Π pQCD ˆ Π pQCD ˆ = − = − 33 − 16 3 Cl 2 · · · 1 9 π 2 q 4 4 243 π 2 q 4 3 For rough estimate, implement step function θ ( Q 1 − Q min ) θ ( Q 2 − Q min ) θ ( Q 3 − Q min ) M. Hoferichter (Institute for Nuclear Theory) HLbL dispersion relations: short-distance constraints Mainz, June 19, 2018 6

Asymptotic region and pQCD quark loop 20 15 × 10 11 10 a pQCD µ 5 0 1 1.5 2 2.5 3 Q min [GeV] For Q min ∼ 2 GeV asymptotic region � 5 × 10 − 11 , but quite sensitive to matching scale M. Hoferichter (Institute for Nuclear Theory) HLbL dispersion relations: short-distance constraints Mainz, June 19, 2018 7

Mixed regions: OPE and triangle amplitude What to do for mixed regions q 2 1 ∼ q 2 2 ≫ q 2 3 ? OPE! Melnikov, Vainshtein 2004 Non-renormalization theorems for VVA triangle (in chiral limit), c.f. a EW µ Czarnecki, Marciano, Vainshtein 2003, Knecht, Peris, Perrottet, de Rafael 2002, 2004, Mondejar, Melnikov 2013 Proposed interpolation between ABJ anomaly and asymptotic behavior F π 0 γ ∗ γ ∗ ( q 2 1 , q 2 F π 0 γ ∗ γ ∗ ( q 2 1 , q 2 2 ) F π 0 γ ∗ γ ∗ ( 0 , 0 ) 3 ) F π 0 γ ∗ γ ∗ ( 0 , 0 ) Π MV ( q 2 1 , q 2 2 , q 2 Π MV ( q 2 1 , q 2 2 , q 2 ˆ ˆ 3 ) = 3 ) = 1 2 q 2 3 − M 2 q 2 2 − M 2 π 0 π 0 Ad-hoc model that disturbs the low-energy properties a π 0 -pole, VMD = 57 . 1 × 10 − 11 → 69 . 8 × 10 − 11 µ a π 0 -pole, disp = 62 . 6 × 10 − 11 → 79 . 9 × 10 − 11 µ → sizable effect, ( 13–17 ) × 10 − 11 for pion pole alone! ֒ Here: revisit OPE in BTT formalism M. Hoferichter (Institute for Nuclear Theory) HLbL dispersion relations: short-distance constraints Mainz, June 19, 2018 8

Mixed regions: OPE and triangle amplitude Starting point: OPE of two vector currents for ( q 1 + q 2 ) 2 ≪ ( q 1 − q 2 ) 2 d 4 x d 4 y e − i ( q 1 · x + q 2 · y ) T � j µ ( x ) j ν ( y ) � = − � � d 4 z e − i ( q 1 + q 2 ) · z 2 i q 2 ǫ µνλσ ˆ q λ j σ i 5 ( z ) + · · · ˆ q = q 1 − q 2 Q = e j µ = ¯ j µ 5 = ¯ ψ Q 2 γ µ γ 5 ψ ψ Q γ µ ψ ˆ 3 diag ( 2 , − 1 , − 1 ) 2 HLbL tensor in terms of VVA correlator W µνλ , valid for q 2 1 ∼ q 2 2 ≫ q 2 3 , q 2 4 Π µνλσ ( q 1 , q 2 , q 3 ) = 8 q α W β � C 2 q 2 ǫ µναβ ˆ λσ ( − q 3 , q 4 ) a ˆ a = 3 , 8 , 0 C a = 1 C 3 = 1 1 2 2 Tr ( Q 2 λ a ) C 8 = √ C 0 = √ 6 6 3 3 6 For BTT projection, need W µνλ ( q 1 , q 2 ) for general kinematics Knecht, Peris, Perrottet, de Rafael 2004 ֒ → one longitudinal and three transversal structures w ± w − w L ( q 2 1 , q 2 2 , ( q 1 + q 2 ) 2 ) T ( q 2 1 , q 2 2 , ( q 1 + q 2 ) 2 ) T ( q 2 1 , q 2 2 , ( q 1 + q 2 ) 2 ) ˜ M. Hoferichter (Institute for Nuclear Theory) HLbL dispersion relations: short-distance constraints Mainz, June 19, 2018 9

VVA non-renormalization theorems Axial anomaly 2 N c w L ( q 2 1 , q 2 2 , ( q 1 + q 2 ) 2 ) = ( q 1 + q 2 ) 2 Transversal structures 0 = ( w + T + w − q 2 1 , q 2 2 , ( q 1 + q 2 ) 2 � − ( w + T + w − ( q 1 + q 2 ) 2 , q 2 2 , q 2 � � � T ) T ) , 1 2 , ( q 1 + q 2 ) 2 � + (˜ w − T + w − T ) � q 2 1 , q 2 w − T + w − T ) � ( q 1 + q 2 ) 2 , q 2 2 , q 2 0 = (˜ � , 1 ( q 1 + q 2 ) 2 , q 2 2 , q 2 = ( w + w − q 2 1 , q 2 2 , ( q 1 + q 2 ) 2 � + ( w + w − ( q 1 + q 2 ) 2 , q 2 2 , q 2 � � T + ˜ � T + ˜ � � w L T ) T ) 1 1 + 2 q 2 · ( q 1 + q 2 ) � − 2 q 1 · q 2 w + � ( q 1 + q 2 ) 2 , q 2 2 , q 2 w − � ( q 1 + q 2 ) 2 , q 2 2 , q 2 � T 1 T 1 q 2 q 2 1 1 Validity: All theorems apply in the chiral limit ֒ → application requires further assumptions such as pion dominance Vainshtein 2003 w L is renormalized neither perturbatively nor non-perturbatively The transversal theorems only hold perturbatively M. Hoferichter (Institute for Nuclear Theory) HLbL dispersion relations: short-distance constraints Mainz, June 19, 2018 10

Mapping onto BTT 2 ≡ q 2 ≫ q 2 For q 2 1 ∼ q 2 3 (other combinations from crossing) Π 1 = 2 ξ ( q 2 ) w L ( q 2 3 , 0 , q 2 ˆ ˆ 3 ) Π { 2 , 3 , 4 , 7 , 8 , 9 , 11 , 13 , 16 , 54 } = 0 3 ) = ξ ( q 2 ) ˆ Π { 5 , 6 } = ξ ( q 2 ) w + w − ( q 2 3 , 0 , q 2 w L ( q 2 3 , 0 , q 2 � T + ˜ � 3 ) T 2 Π { 17 , 39 , 50 , 51 } = ξ ( q 2 ) ξ ( q 2 ) w + w − ( q 2 3 , 0 , q 2 w L ( q 2 3 , 0 , q 2 Π { 10 , 14 } = − ˆ ˆ � T + ˜ � 3 ) = 3 ) T q 1 · q 2 2 q 1 · q 2 1 1 ξ ( q 2 ) = − � C 2 a = − 2 π 2 q 2 18 π 2 q 2 a = 3 , 8 , 0 M. Hoferichter (Institute for Nuclear Theory) HLbL dispersion relations: short-distance constraints Mainz, June 19, 2018 11