Pad approach to pseudoscalar poles in HLbL based on P. Masjuan, PS: - PowerPoint PPT Presentation

Padé approach to pseudoscalar poles in HLbL based on P. Masjuan, PS: Phys.Rev. D95 (2017) Pablo Sanchez-Puertas, Charles University Prague � sanchezp@ipnp.troja.mff.cuni.cz

Padé approach to pseudoscalar poles in HLbL Outline 1. A (very) brief reminder 2. Our proposal: Padé approximants 3. Application to HLbL and results 4. Summary

Padé approach to pseudoscalar poles in HLbL A (very) brief reminder Section 1 A (very) brief reminder

Padé approach to pseudoscalar poles in HLbL A (very) brief reminder Our aim: pseudoscalar poles in HLbL • We want the pseudoscalar ( π 0 , η, η ′ ) pole contributions to a HLbL µ k, ρ q 1 , µ k, ρ q 1 , µ k, ρ q 1 , µ q 1 , µ k, ρ P + ... = P P q 3 , λ q 2 , ν q 2 , ν q 3 , λ q 2 , ν q 2 , ν q 3 , λ q 3 , λ unambiguosly identified (see eg. tomorrow’s talks)

Padé approach to pseudoscalar poles in HLbL A (very) brief reminder Our aim: pseudoscalar poles in HLbL • We want the pseudoscalar ( π 0 , η, η ′ ) pole contributions to a HLbL µ off-shellness in χ PT?

Padé approach to pseudoscalar poles in HLbL A (very) brief reminder Our aim: pseudoscalar poles in HLbL • We want the pseudoscalar ( π 0 , η, η ′ ) pole contributions to a HLbL µ k, ρ q 1 , µ k, ρ q 1 , µ k, ρ q 1 , µ q 1 , µ k, ρ P + ... = P P q 3 , λ q 2 , ν q 2 , ν q 3 , λ q 2 , ν q 2 , ν q 3 , λ q 3 , λ unambiguosly identified (see eg. tomorrow’s talks) • Commonly off-shellness is coined for high-energy link q 1 , µ k, ρ k, ρ ⇒ ( q 1 − q 2 ) 2 ǫ µνρσ ( q 1 − q 2 ) ρ 4 j 5 σ q 3 , λ q 2 , ν q 3 , λ To my point of view one of the next obstacles ahead (tomorrow talks?)

Padé approach to pseudoscalar poles in HLbL A (very) brief reminder The pseudoscalar poles in brief • It amounts to calculate σ, k σ, k σ, k σ, k µ, q 1 ν, q 2 λ, q 3 µ, q 1 λ, q 3 ⇒ ν, q 2 ν, q 2 µ, q 1 λ, q 3 λ, q 3 µ, q 1 ν, q 2

Padé approach to pseudoscalar poles in HLbL A (very) brief reminder The pseudoscalar poles in brief • It amounts to calculate σ, k σ, k σ, k σ, k µ, q 1 ν, q 2 λ, q 3 µ, q 1 λ, q 3 ⇒ ν, q 2 ν, q 2 µ, q 1 λ, q 3 λ, q 3 µ, q 1 ν, q 2 • Result expressed as weighted integral over space-like on-shell form factors � + 1 � 3 � ∞ = − 2 π � α a HLbL ; P � 1 − t 2 Q 3 1 Q 3 dQ 1 dQ 2 dt ℓ 2 3 π 0 − 1 � F P γ ∗ γ ∗ ( Q 2 1 , Q 2 3 ) F P γ ∗ γ ( Q 2 + F P γ ∗ γ ∗ ( Q 2 1 , Q 2 2 ) F P γ ∗ γ ( Q 2 2 , 0 ) I 1 ( Q 1 , Q 2 , t ) 3 , 0 ) I 2 ( Q 1 , Q 2 , t ) � × Q 2 2 + m 2 Q 2 3 + m 2 P P

Padé approach to pseudoscalar poles in HLbL A (very) brief reminder The pseudoscalar poles in brief • It amounts to calculate σ, k σ, k σ, k σ, k µ, q 1 ν, q 2 λ, q 3 µ, q 1 λ, q 3 ⇒ ν, q 2 ν, q 2 µ, q 1 λ, q 3 λ, q 3 µ, q 1 ν, q 2 • Result expressed as weighted integral over space-like on-shell form factors � + 1 � 3 � ∞ � α a HLbL ; P = dQ 1 dQ 2 dt ( w 1 F 1 + w 2 F 2 ) ℓ π 0 − 1

Padé approach to pseudoscalar poles in HLbL A (very) brief reminder The pseudoscalar poles in brief • It amounts to calculate σ, k σ, k σ, k σ, k µ, q 1 ν, q 2 λ, q 3 µ, q 1 λ, q 3 ⇒ ν, q 2 ν, q 2 µ, q 1 λ, q 3 λ, q 3 µ, q 1 ν, q 2 • Result expressed as weighted integral over space-like on-shell form factors notice the peaks at the relevant low energies

Padé approach to pseudoscalar poles in HLbL A (very) brief reminder The pseudoscalar poles in brief • It amounts to calculate σ, k σ, k σ, k σ, k µ, q 1 ν, q 2 λ, q 3 µ, q 1 λ, q 3 ⇒ ν, q 2 ν, q 2 µ, q 1 λ, q 3 λ, q 3 µ, q 1 ν, q 2 • Result expressed as weighted integral over space-like on-shell form factors • Trivial if form factors god-given ... Mathematica not so kind yet!

Padé approach to pseudoscalar poles in HLbL A (very) brief reminder The pseudoscalar poles in brief • It amounts to calculate σ, k σ, k σ, k σ, k µ, q 1 ν, q 2 λ, q 3 µ, q 1 λ, q 3 ⇒ ν, q 2 ν, q 2 µ, q 1 λ, q 3 λ, q 3 µ, q 1 ν, q 2 • Result expressed as weighted integral over space-like on-shell form factors • Trivial if form factors god-given ... Mathematica not so kind yet! • Only ab-initio theoretical: lattice → finite points: interpolation!

Padé approach to pseudoscalar poles in HLbL A (very) brief reminder The pseudoscalar poles in brief • It amounts to calculate σ, k σ, k σ, k σ, k µ, q 1 ν, q 2 λ, q 3 µ, q 1 λ, q 3 ⇒ ν, q 2 ν, q 2 µ, q 1 λ, q 3 λ, q 3 µ, q 1 ν, q 2 • Result expressed as weighted integral over space-like on-shell form factors • Trivial if form factors god-given ... Mathematica not so kind yet! • Only ab-initio theoretical: lattice → finite points: interpolation! • Nature solves QCD for us: experiment → reduced points: extrapolation!

Padé approach to pseudoscalar poles in HLbL A (very) brief reminder The pseudoscalar poles in brief • It amounts to calculate σ, k σ, k σ, k σ, k µ, q 1 ν, q 2 λ, q 3 µ, q 1 λ, q 3 ⇒ ν, q 2 ν, q 2 µ, q 1 λ, q 3 λ, q 3 µ, q 1 ν, q 2 • Result expressed as weighted integral over space-like on-shell form factors • Trivial if form factors god-given ... Mathematica not so kind yet! • Only ab-initio theoretical: lattice → finite points: interpolation! • Nature solves QCD for us: experiment → reduced points: extrapolation! • A natural framework for this hihgly desired! • Framework: avoid model-building (as model-independent as possible)

Padé approach to pseudoscalar poles in HLbL A (very) brief reminder The pseudoscalar poles in brief • It amounts to calculate σ, k σ, k σ, k σ, k µ, q 1 ν, q 2 λ, q 3 µ, q 1 λ, q 3 ⇒ ν, q 2 ν, q 2 µ, q 1 λ, q 3 λ, q 3 µ, q 1 ν, q 2 • Result expressed as weighted integral over space-like on-shell form factors • Trivial if form factors god-given ... Mathematica not so kind yet! • Only ab-initio theoretical: lattice → finite points: interpolation! • Nature solves QCD for us: experiment → reduced points: extrapolation! • A natural framework for this hihgly desired! • Framework: avoid model-building (as model-independent as possible) • Keep track of systematic errors

Padé approach to pseudoscalar poles in HLbL A (very) brief reminder The pseudoscalar poles in brief • It amounts to calculate σ, k σ, k σ, k σ, k µ, q 1 ν, q 2 λ, q 3 µ, q 1 λ, q 3 ⇒ ν, q 2 ν, q 2 µ, q 1 λ, q 3 λ, q 3 µ, q 1 ν, q 2 • Result expressed as weighted integral over space-like on-shell form factors • Trivial if form factors god-given ... Mathematica not so kind yet! • Only ab-initio theoretical: lattice → finite points: interpolation! • Nature solves QCD for us: experiment → reduced points: extrapolation! • A natural framework for this hihgly desired! • Framework: avoid model-building (as model-independent as possible) • Keep track of systematic errors • Emphasize the low-energy region

Padé approach to pseudoscalar poles in HLbL A (very) brief reminder The pseudoscalar poles in brief • It amounts to calculate σ, k σ, k σ, k σ, k µ, q 1 ν, q 2 λ, q 3 µ, q 1 λ, q 3 ⇒ ν, q 2 ν, q 2 µ, q 1 λ, q 3 λ, q 3 µ, q 1 ν, q 2 • Result expressed as weighted integral over space-like on-shell form factors • Trivial if form factors god-given ... Mathematica not so kind yet! • Only ab-initio theoretical: lattice → finite points: interpolation! • Nature solves QCD for us: experiment → reduced points: extrapolation! • A natural framework for this hihgly desired! • Framework: avoid model-building (as model-independent as possible) • Keep track of systematic errors • Emphasize the low-energy region • Incorporate theoretical high-energy constraints

Padé approach to pseudoscalar poles in HLbL A (very) brief reminder The pseudoscalar poles in brief • It amounts to calculate σ, k σ, k σ, k σ, k µ, q 1 ν, q 2 λ, q 3 µ, q 1 λ, q 3 ⇒ ν, q 2 ν, q 2 µ, q 1 λ, q 3 λ, q 3 µ, q 1 ν, q 2 • Result expressed as weighted integral over space-like on-shell form factors • Trivial if form factors god-given ... Mathematica not so kind yet! • Only ab-initio theoretical: lattice → finite points: interpolation! • Nature solves QCD for us: experiment → reduced points: extrapolation! • A natural framework for this hihgly desired! Our Proposal: use of Padé approximants

Padé approach to pseudoscalar poles in HLbL Our proposal: Padé approximants Section 2 Our proposal: Padé approximants

Padé approach to pseudoscalar poles in HLbL Our proposal: Padé approximants Padé approximants: singly virtual • How to approximate (not model) non-perturbative hadronic functions? Taylor series: F πγγ ∗ ( q 2 ) = F πγγ ( 1 + b P q 2 + ... ) ✗ poles(cuts)

Padé approach to pseudoscalar poles in HLbL Our proposal: Padé approximants Padé approximants: singly virtual • How to approximate (not model) non-perturbative hadronic functions? c n ( q 2 − M 2 ) n � Laurent exp.: F πγγ ∗ ( q 2 ) = ✗ next pole(cuts) n = − 1

Padé approach to pseudoscalar poles in HLbL Our proposal: Padé approximants Padé approximants: singly virtual • How to approximate (not model) non-perturbative hadronic functions? M = Q N ( q 2 ) 1 + b P q 2 + ... + O ( q N + M + 1 ) PAs: F πγγ ∗ ( q 2 ) = P N � � R M ( q 2 ) = F πγγ