Exploratory studies for the position-space approach to hadronic light-by-light scattering in the muon g 2

Transcription

1 Exploratory studies for the position-space approach to hadronic light-by-light scattering in the muon g 2 Nils Asmussen in Collaboration with Antoine Gérardin, Jeremy Green, Harvey Meyer, Andreas Nyffeler Institut für Kernphysik, Johannes Gutenberg-Universität Mainz June 23, 2017 Nils Asmussen (KPH, JGU Mainz) Position-space approach to HLbL in g 2 June 23, / 18

2 References I Hadronic Light-by-Light Contribution to the Muon Anomalous Magnetic Moment on the Lattice, Talk by NA at the DPG meeting, Heidelberg, March 24, 2015 II Lattice 2015 proceedings of J. Green et al arxiv: III Lattice 2016 proceedings of NA et al arxiv: IV A. Nyffeler, talk at Workshop (J-PARC, Japan, Nov. 2016) V H. B. Meyer, talk at first workshop of the muon g 2 Theory Initiative (Q-Center, IL, 6 June 2017) Nils Asmussen (KPH, JGU Mainz) Position-space approach to HLbL in g 2 June 23, / 18

3 HLbL contribution to g 2 gyromagnetic moment: µ = g e 2m S anomalous magnetic moment: a µ = g to 4 standard deviations discrepancy between aµ exp and aµ theo new physics? Nils Asmussen (KPH, JGU Mainz) Position-space approach to HLbL in g 2 June 23, / 18

4 HLbL contribution to g 2 gyromagnetic moment: µ = g e 2m S anomalous magnetic moment: a µ = g to 4 standard deviations discrepancy between aµ exp and aµ theo new physics? reduce uncertainties Experiment Theory for HLbL J-PARC phenomenology lattice QCD Fermilab model uncertainties model independant estimates for dominant contribution Blum et al 15,..., 17 (π 0, η, η ; ππ) (talks earlier this session) using experimental input our group Colangelo et al 14,..., 17 Pauk and Vanderhaeghen 14 Nils Asmussen (KPH, JGU Mainz) Position-space approach to HLbL in g 2 June 23, / 18

5 Lattice QCD: Current Approach to HLbL two independent developments by Blum et al our group (I-V) similarities get directly F 2 (q 2 = 0) no cancellation of an O(α 2 ) term position space perturbative treatment of the QED part QED part computed in infinite volume in continuum Lorentz covariance is manifest in our approach no power law effects in the volume (an important motivation for this work) Nils Asmussen (KPH, JGU Mainz) Position-space approach to HLbL in g 2 June 23, / 18

6 Euclidean position-space approach to a HLbL µ z x y 0 master formula (I-V) [ aµ HLbL = me6 d 4 y d 4 x 3 }{{}}{{} =2π 2 y 3 d y i Π ρ;µνλσ (x, y) = L [ρ,σ];µνλ (x, y) }{{} QED = 4π π 0 dβ sin2 (β) 0 d x x 3 i Π ] ρ;µνλσ (x, y). }{{} QCD d 4 z z ρ j µ (x) j ν (y) j σ (z) j λ (0). L [ρ,σ];µνλ (x, y) computed in the continuum & infinite-volume no finite-volume effects from the photons & affordable way (1d integral) to sample the integrand for the fully connected contribution. Nils Asmussen (KPH, JGU Mainz) Position-space approach to HLbL in g 2 June 23, / 18

7 Perturbation theory in Euclidean position-space Scalar propagators: Fermion propagator: d 4 p S(x) (2π) 4 G 0 (x) = 1 4π 2 x 2, G m(x) = m 4π 2 x K 1(m x ). ip µ γ µ + m [ p 2 + m 2 e ipx = m2 K 2 (m x ) ] 4π 2 γ µ x µ + K 1 (m x ), x x U n (z) = Chebyshev polynomials of the second kind: U 0 (z) = 1, U 1 (z) = 2z, U n+1 (z) = 2zU n (z) U n 1 (z) (n 1), Key property: orthogonal basis on S 3 ; if ˆɛ is a unit vector U n (ˆɛ ˆx) U m (ˆɛ ŷ) = δ nm ˆɛ n + 1 U n(ˆx ŷ). Nils Asmussen (KPH, JGU Mainz) Position-space approach to HLbL in g 2 June 23, / 18

8 Sketch of the derivation ˆF 2(0) = i Tr {[γρ, γτ ] ( ip/ + m)γρτ (p, p)( ip/ + m)}, 48m Γ ρσ(p, p) = e x 6 K µνλ (x 1, x 2, p) Π ρ;µνλσ (x 1, x 2), 1,x 2 K µνλ (x 1, x 2, p) = γ µ(ip/ + / (x1) m)γ ν(ip/ + / (x1) + / (x2) m)γ λ I(ˆɛ, x 1, x 2), I(ˆɛ, x, y) = q 2 k 2 (q + k) 2 (p q) 2 + m 2 (p q k) 2 + m 2 e i(qx+ky). q,k With p = imˆɛ. From [III]. Diagrammatic representation of I(ˆɛ, x, y): Nils Asmussen (KPH, JGU Mainz) Position-space approach to HLbL in g 2 June 23, / 18

9 The scalar function I(ˆɛ, x, y) where J(ˆɛ, y) s(x, u) = v δ (x, u) = I(ˆɛ, x, y) = G 0 (u y) J(ˆɛ, u) J(ˆɛ, x u), S(x, y) = V δ (x, y) = T βδ (x, y) = u u G 0 (y u) e mˆɛ u G m (u) = n 0 u u u G 0 (u y)s(x, u), G 0 (u y)v δ (x, u), G 0 (u y)t βδ (x, u), J(ˆɛ, u)j(ˆɛ, x u) = ˆɛ ɛ δ J(ˆɛ, u)j(ˆɛ, x u) ˆɛ ( 1 n=0 =... ) z n (y 2 ) U n (ˆɛ ŷ), (IR regulated) z n (u 2 )z n ((x u) 2 ) U n(û x u), n + 1 Nils Asmussen (KPH, JGU Mainz) Position-space approach to HLbL in g 2 June 23, / 18

10 Explicit form of the QED kernel L [ρ,σ];µνλ (x, y) = A=I,II,III Gδ[ρσ]µανβλ A T (A) αβδ (x, y), with e.g. Gδ[ρσ]µανβλ I 1 } {(γ 8 Tr δ [γ ρ, γ σ ] + 2(δ δσ γ ρ δ δρ γ σ ) )γ µ γ α γ ν γ β γ λ, T (I) αβδ T (II) αβδ (x, y) = (x) α (x, y) = m (x) α ( (x) β + (y) β )V δ(x, y), ( T βδ (x, y) + 1 ) 4 δ βδs(x, y) T (III) αβδ (x, y) = m( (x) β + (y) β (T ) αδ (x, y) δ αδs(x, y) ), S(x, y) = ḡ (0) ( x, ˆx ŷ, y ), V δ (x, y) = x δ ḡ (1) ( x, ˆx ŷ, y ) + y δ ḡ (2) ( x, ˆx ŷ, y ), T αβ (x, y) = (x αx β x2 4 δ αβ) l (1) + (y αy β y 2 4 δ αβ) l (2) + (x αy β + y αx β x y 2 δ αβ) l (3). The QED kernel L [ρ,σ];µνλ (x, y) is parametrized by six weight functions. Nils Asmussen (KPH, JGU Mainz) Position-space approach to HLbL in g 2 June 23, / 18

11 Example: Form Factor g (2) where g (2) (x 2, x y, y 2 1 π ) = 8πy 2 x sin 3 du u 2 dφ 1 β 0 0 { ( y 2 + u 2 ) } log χ 2 sin β + cos β cos φ 1 2 u y sin φ 1 n=0 [ { U n zn ( u )z n+1 ( x u ) x u cos φ 1 n ( u cos φ 1 x ) U ] n+1 n + 2 [ + z n+1 ( u )z n ( x u ) ( u cos φ 1 x ) U ] n n x u cos φ U n+1 } 1 n + 2 x y = x y cos β, x u = χ = y 2 + u 2 2 u y cos(β φ) y 2 + u 2 2 u y cos(β + φ), x 2 + u 2 2 x u cos φ 1 U n = U n ( x cos φ1 u u x z n =linear combination of products of two modified Bessel functions. From [I]. Reminder: V δ (x, y) = x δ ḡ (1) ( x, ˆx ŷ, y ) + y δ ḡ (2) ( x, ˆx ŷ, y ). Nils Asmussen (KPH, JGU Mainz) Position-space approach to HLbL in g 2 June 23, / 18 )

12 Complete set of weight functions: x dependence m 2 µg (0) ( x, y = fm, cos β) l (1) ( x, y = fm, cos β) x fm cos β = cos β = cos β = x fm cos β = cos β = cos β = mµg (1) ( x, y = fm, cos β) l (2) ( x, y = fm, cos β) x fm cos β = cos β = cos β = x fm cos β = cos β = cos β = mµg (2) ( x, y = fm, cos β) l (3) ( x, y = fm, cos β) x fm cos β = cos β = cos β = x fm cos β = cos β = cos β = ḡ (0) ( x, ˆx ŷ, y ) contains an arbitrary additive constant (due to the IR divergence in I (ˆɛ, x, y)), which does not contribute to L [ρ,σ];µνλ (x, y). Nils Asmussen (KPH, JGU Mainz) Position-space approach to HLbL in g 2 June 23, / 18

13 The π 0 pole contribution Assume a vector-meson-dominance transition form factor (parameters: m V, m π and overall normalization) F( q 2 1, q 2 2) = c (q m2 V )(q2 2 + m2 V ), c = N cm 4 V 12π 2 F π. i Π c 2 { ( ρ;µνλσ (x, y) = mv 2 (m2 V ɛ µναβ ɛ σλργ + ) K π (x, y) m2 π) x α y β x γ y γ } +ɛ µλαβ ɛ νσγρ K π (y x, y) + ɛ µσαρ ɛ νλβγ K π (x, x y). y γ x γ where K π (x, y) ( ) d 4 u G mπ (u) G mv (u) G mv (x u)g mv (y u) = K π (y, x). Nils Asmussen (KPH, JGU Mainz) Position-space approach to HLbL in g 2 June 23, / 18

14 Contribution of the π 0 to a HLbL µ m π = 300 MeV m π = 600 MeV m π = 900 MeV a Hlbl µ ( y max ) m π = 300 MeV m π = 600 MeV m π = 900 MeV d d y ahlbl µ y max /fm y max /fm Dashed line = result from momentum-space integration Contribution is perhaps surprisingly long-range. Nils Asmussen (KPH, JGU Mainz) Position-space approach to HLbL in g 2 June 23, / 18

15 The lepton loop: fully analytic result for i Π ρ;µνλσ (x, y) i Π ρ;µνλσ (x, y) = (1) Π ρ;µνλσ (x, y) + Π (1) ρ;νλµσ (y x, x) + x ρ Π (r,1) νλµσ (y x, x) + Π (1) ρ;λνµσ ( x, y x) + x ρ Π (r,1) λνµσ ( x, y x). Π (r,1) (x, y) µνλσ ( m ) 8 [ ( xα)(x y) β K 2 (m x )K 2 (m x y ) = 2 2π x 2 x y 2 l γδ (y) Tr{γαγµγ β γν γγ γσ γ δ γ λ } + K 1 (m x )K 1 (m x y ) p( y ) Tr{γµγν γσ γ λ } x x y + ( xα)(x y) β K 2 (m x )K 2 (m x y ) x 2 x y 2 p( y ) Tr{γαγµγ β γν γσ γ λ } + ( xα) K 2 (m x )K 1 (m x y ) x 2 qγ (y) Tr{γαγµγν γγ γσ γ λ } x y + (x y) β K 1 (m x )K 2 (m x y ) x x y 2 qγ (y) Tr{γµγ β γν γγ γσ γ λ } + ( xα) K 2 (m x )K 1 (m x y ) x 2 q δ (y) Tr{γαγµγν γσ γ δ γ λ } x y + (x y) β K 1 (m x )K 2 (m x y ) x x y 2 q δ (y) Tr{γµγ β γν γσ γ δ γ λ } + K 1 (m x )K 1 (m x y ) ] l γδ (y) Tr{γµγν γγ γσ γ δ γ λ } x x y Nils Asmussen (KPH, JGU Mainz) Position-space approach to HLbL in g 2 June 23, / 18

16 The lepton loop (continued) Π (1) (x, y) ρ;µνλσ = ( 2 m ) 8 [ ( xα)(x y) β K 2 (m x )K 2 (m x y ) 2π x 2 x y 2 f ρδγ (y) Tr{γαγµγ β γν γγ γσ γ δ γ λ } + K 1 (m x )K 1 (m x y ) f ρδγ (y) Tr{γµγν γγ γσ γ δ γ λ } x x y + K 1 (m x )K 1 (m x y ) gρ(y) Tr{γµγν γσ γ λ } x x y + ( xα)(x y) β K 2 (m x )K 2 (m x y ) x 2 x y 2 gρ(y) Tr{γαγµγ β γν γσ γ λ } + ( xα) K 2 (m x )K 1 (m x y ) x 2 hργ (y) Tr{γαγµγν γγ γσ γ λ } x y + (x y) β K 1 (m x )K 2 (m x y ) x x y 2 hργ (y) Tr{γµγ β γν γγ γσ γ λ } + ( xα) K 2 (m x )K 1 (m x y ) x 2 ˆf ρδ (y) Tr{γαγµγν γσ γ δ γ λ } x y + (x y) β K 1 (m x )K 2 (m x y ) ] x x y 2 ˆf ρδ (y) Tr{γµγ β γν γσ γ δ γ λ } l γδ (y) = 2π2 K 1 (m y ) (ŷγ m 2 ŷ δ K 2 (m y ) δ γδ ), hργ (y) = π2 ( ) m y m 3 ŷγ ŷρ m y K 1 (m y ) δγρk 0 (m y ), ˆf ρδ (y) = π2 m 3 f ρδγ (y) = π2 m 3 { } ŷρŷ δ m y K 1 (m y ) + δ ρδ K 0 (m y ) qγ (y) = 2π2 m 2 ŷγ K 1 (m y ), { ŷγ ŷ δ ŷρ m y K 2 (m y ) + (δ ρδ ŷγ δγρŷ δ δ γδ ŷρ) K 1 (m y ) }, p( y ) = 2π2 m 2 K 0 (m y ). Nils Asmussen (KPH, JGU Mainz) Position-space approach to HLbL in g 2 June 23, / 18

17 Lepton loop integrand contribution to a HLbL µ Integrand of lepton loop contribution to a µ HLbL m l = m µ m l =2 m µ 10 8 f( y ) m µ y Nils Asmussen (KPH, JGU Mainz) Position-space approach to HLbL in g 2 June 23, / 18

18 Lepton loop integrand contribution to aµ HLbL (zoom) Integrand of lepton loop contribution to a µ HLbL 10 8 f( y ) m µ y numerically compatible with f ( y ) m µ y log 2 (m µ y ) for small y analytic result reproduced at the percent level Nils Asmussen (KPH, JGU Mainz) Position-space approach to HLbL in g 2 June 23, / 18

19 Conclusion The covariant position-space method looks like a promising approach. We plan to make the QED kernel publicly available. Tests of the QED kernel: π 0 pole, lepton loop. The analytic result for the lepton loop is reproduced at the percent level The π 0 contribution is very long-range, but with the π 0 contribution calculated, we hope to be able to correct for the finite-size effects on this contribution, by computing the transition form factor on the same ensemble (Gérardin 16). A parallel activity: analysis of the eight forward light-by-light scattering amplitudes, constraining the resonance transition form factors (with V. Pascalutsa; next talk by A. Gérardin). Nils Asmussen (KPH, JGU Mainz) Position-space approach to HLbL in g 2 June 23, / 18