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 (π0, η, η′) pole contributions to aHLbL

µ

q3, λ k, ρ q1, µ q2, ν

=

k, ρ q1, µ q2, ν q3, λ P q1, µ q2, ν q3, λ k, ρ P P q3, λ q2, ν q1, µ k, ρ

+ ...

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 (π0, η, η′) pole contributions to aHLbL

µ

• ff-shellness in χPT?

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

Our aim: pseudoscalar poles in HLbL

• We want the pseudoscalar (π0, η, η′) pole contributions to aHLbL

µ

q3, λ k, ρ q1, µ q2, ν

=

k, ρ q1, µ q2, ν q3, λ P q1, µ q2, ν q3, λ k, ρ P P q3, λ q2, ν q1, µ k, ρ

+ ...

unambiguosly identified (see eg. tomorrow’s talks)

• Commonly off-shellness is coined for high-energy link

q3, λ k, ρ q1, µ q2, ν

⇒

q3, λ k, ρ

j5σ

4 (q1−q2)2ǫµνρσ(q1 − q2)ρ

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

λ, q3 ν, q2 µ, q1 σ, k λ, q3 ν, q2 µ, q1 σ, k λ, q3 ν, q2 µ, q1 σ, k

⇒

σ, k λ, q3 ν, q2 µ, q1

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

The pseudoscalar poles in brief

• It amounts to calculate

λ, q3 ν, q2 µ, q1 σ, k λ, q3 ν, q2 µ, q1 σ, k λ, q3 ν, q2 µ, q1 σ, k

⇒

σ, k λ, q3 ν, q2 µ, q1

• Result expressed as weighted integral over space-like on-shell form factors

aHLbL;P

ℓ

= −2π 3 α π 3 ∞ dQ1dQ2 +1

−1

dt

• 1 − t2Q3

1Q3 2

× FPγ∗γ∗(Q2

1, Q2 3)FPγ∗γ(Q2 2, 0)I1(Q1, Q2, t)

Q2

2 + m2 P

+ FPγ∗γ∗(Q2

1, Q2 2)FPγ∗γ(Q2 3, 0)I2(Q1, Q2, t)

Q2

3 + m2 P

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

The pseudoscalar poles in brief

• It amounts to calculate

λ, q3 ν, q2 µ, q1 σ, k λ, q3 ν, q2 µ, q1 σ, k λ, q3 ν, q2 µ, q1 σ, k

⇒

σ, k λ, q3 ν, q2 µ, q1

• Result expressed as weighted integral over space-like on-shell form factors

aHLbL;P

ℓ

= α π 3 ∞ dQ1dQ2 +1

−1

dt(w1F1 + w2F2)

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

The pseudoscalar poles in brief

• It amounts to calculate

λ, q3 ν, q2 µ, q1 σ, k λ, q3 ν, q2 µ, q1 σ, k λ, q3 ν, q2 µ, q1 σ, k

⇒

σ, k λ, q3 ν, q2 µ, q1

• 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

λ, q3 ν, q2 µ, q1 σ, k λ, q3 ν, q2 µ, q1 σ, k λ, q3 ν, q2 µ, q1 σ, k

⇒

σ, k λ, q3 ν, q2 µ, q1

• 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

λ, q3 ν, q2 µ, q1 σ, k λ, q3 ν, q2 µ, q1 σ, k λ, q3 ν, q2 µ, q1 σ, k

⇒

σ, k λ, q3 ν, q2 µ, q1

• 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

λ, q3 ν, q2 µ, q1 σ, k λ, q3 ν, q2 µ, q1 σ, k λ, q3 ν, q2 µ, q1 σ, k

⇒

σ, k λ, q3 ν, q2 µ, q1

• 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

λ, q3 ν, q2 µ, q1 σ, k λ, q3 ν, q2 µ, q1 σ, k λ, q3 ν, q2 µ, q1 σ, k

⇒

σ, k λ, q3 ν, q2 µ, q1

• 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

λ, q3 ν, q2 µ, q1 σ, k λ, q3 ν, q2 µ, q1 σ, k λ, q3 ν, q2 µ, q1 σ, k

⇒

σ, k λ, q3 ν, q2 µ, q1

• 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

λ, q3 ν, q2 µ, q1 σ, k λ, q3 ν, q2 µ, q1 σ, k λ, q3 ν, q2 µ, q1 σ, k

⇒

σ, k λ, q3 ν, q2 µ, q1

• 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

λ, q3 ν, q2 µ, q1 σ, k λ, q3 ν, q2 µ, q1 σ, k λ, q3 ν, q2 µ, q1 σ, k

⇒

σ, k λ, q3 ν, q2 µ, q1

• 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

λ, q3 ν, q2 µ, q1 σ, k λ, q3 ν, q2 µ, q1 σ, k λ, q3 ν, q2 µ, q1 σ, k

⇒

σ, k λ, q3 ν, q2 µ, q1

• 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: Fπγγ∗(q2) = Fπγγ(1 + bPq2 + ...) ✗ 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?

Laurent exp.: Fπγγ∗(q2) =

• n=−1

cn(q2 − M2)n ✗ next pole(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?

PAs: Fπγγ∗(q2) = PN

M = QN(q2)

RM(q2) = Fπγγ

• 1 + bPq2 + ... + O(qN+M+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?

PAs: Fπγγ∗(q2) = PN

M = QN(q2)

RM(q2) = Fπγγ

• 1 + bPq2 + ... + O(qN+M+1)
• Convergence not only to meromorphic but Stieltjes → beyond large-Nc

Convergence for sequences: PN

1 , PN N , PN N+1, ... analytic related

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

Padé approximants: singly virtual

• How to approximate (not model) non-perturbative hadronic functions?

PAs: Fπγγ∗(q2) = PN

M = QN(q2)

RM(q2) = Fπγγ

• 1 + bPq2 + ... + O(qN+M+1)
• Convergence not only to meromorphic but Stieltjes → beyond large-Nc

Convergence for sequences: PN

1 , PN N , PN N+1, ... analytic related

• Mixed q2 = 0 and q2 = ∞ expansions compatible → pQCD constraints

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

Padé approximants: singly virtual

• How to approximate (not model) non-perturbative hadronic functions?

PAs: Fπγγ∗(q2) = PN

M = QN(q2)

RM(q2) = Fπγγ

• 1 + bPq2 + ... + O(qN+M+1)
• Convergence not only to meromorphic but Stieltjes → beyond large-Nc

Convergence for sequences: PN

1 , PN N , PN N+1, ... analytic related

• Mixed q2 = 0 and q2 = ∞ expansions compatible → pQCD constraints
• Convergence to HVP or sophisticated FF DR analysis in EPJ C73, 2668

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

Padé approximants: singly virtual

• How to approximate (not model) non-perturbative hadronic functions?

PAs: Fπγγ∗(q2) = PN

M = QN(q2)

RM(q2) = Fπγγ

• 1 + bPq2 + ... + O(qN+M+1)
• Convergence not only to meromorphic but Stieltjes → beyond large-Nc

Convergence for sequences: PN

1 , PN N , PN N+1, ... analytic related

• Mixed q2 = 0 and q2 = ∞ expansions compatible → pQCD constraints
• Convergence to HVP or sophisticated FF DR analysis in EPJ C73, 2668

A powerful (non-trivial) case: both real and imaginary parts in loops ⇒ +

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

Padé approximants: singly virtual

• How to approximate (not model) non-perturbative hadronic functions?

PAs: Fπγγ∗(q2) = PN

M = QN(q2)

RM(q2) = Fπγγ

• 1 + bPq2 + ... + O(qN+M+1)
• Convergence not only to meromorphic but Stieltjes → beyond large-Nc

Convergence for sequences: PN

1 , PN N , PN N+1, ... analytic related

• Mixed q2 = 0 and q2 = ∞ expansions compatible → pQCD constraints
• Convergence to HVP or sophisticated FF DR analysis in EPJ C73, 2668

A powerful (non-trivial) case: both real and imaginary parts in loops ⇒ +

• Cannot overemphasize: spectroscopy is theory-forbiden!

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

Padé approximants: singly virtual

• How to approximate (not model) non-perturbative hadronic functions?

PAs: Fπγγ∗(q2) = PN

M = QN(q2)

RM(q2) = Fπγγ

• 1 + bPq2 + ... + O(qN+M+1)
• Convergence not only to meromorphic but Stieltjes → beyond large-Nc

Convergence for sequences: PN

1 , PN N , PN N+1, ... analytic related

• Mixed q2 = 0 and q2 = ∞ expansions compatible → pQCD constraints
• Convergence to HVP or sophisticated FF DR analysis in EPJ C73, 2668

A powerful (non-trivial) case: both real and imaginary parts in loops ⇒ +

• Cannot overemphasize: spectroscopy is theory-forbiden!
• Obtain the derivatives from data (later)

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

Padé approximants: double virtual

• Commonly referred to as Canterbury approximants

Fπγ∗γ∗(q2

1, q2 2) = C N M = QN(q2 1, q2 2)

RM(q2

1, q2 2);

QN(RM) = N(M)

i,j

cQ(R)

n,m q2i 1 q2j 2

Again, match derivatives to get cQ,R

ij

’s

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

Padé approximants: double virtual

• Commonly referred to as Canterbury approximants

Fπγ∗γ∗(q2

1, q2 2) = C N M = QN(q2 1, q2 2)

RM(q2

1, q2 2);

QN(RM) = N(M)

i,j

cQ(R)

n,m q2i 1 q2j 2

Again, match derivatives to get cQ,R

ij

’s

• Again nice convergence properties, as for instance, pQCD models

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

Padé approximants: double virtual

• Commonly referred to as Canterbury approximants

Fπγ∗γ∗(q2

1, q2 2) = C N M = QN(q2 1, q2 2)

RM(q2

1, q2 2);

QN(RM) = N(M)

i,j

cQ(R)

n,m q2i 1 q2j 2

Again, match derivatives to get cQ,R

ij

’s

• Again nice convergence properties, as for instance, pQCD models

F log

Pγ∗γ∗(Q2 1, Q2 2) = FPγγM2

Q2

1 − Q2 2

ln 1 + Q2

1/M2

1 + Q2

2/M2

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

Padé approximants: double virtual

• Commonly referred to as Canterbury approximants

Fπγ∗γ∗(q2

1, q2 2) = C N M = QN(q2 1, q2 2)

RM(q2

1, q2 2);

QN(RM) = N(M)

i,j

cQ(R)

n,m q2i 1 q2j 2

Again, match derivatives to get cQ,R

ij

’s

• Again nice convergence properties, as for instance, pQCD models

F log

Pγ∗γ∗(Q2 1, Q2 2) = FPγγM2

1 dx 1 xQ2

1 + (1 − x)Q2 2 + M2

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

Padé approximants: double virtual

• Commonly referred to as Canterbury approximants

Fπγ∗γ∗(q2

1, q2 2) = C N M = QN(q2 1, q2 2)

RM(q2

1, q2 2);

QN(RM) = N(M)

i,j

cQ(R)

n,m q2i 1 q2j 2

Again, match derivatives to get cQ,R

ij

’s

• Again nice convergence properties, as for instance, pQCD models

F pQCD;as

Pγ∗γ∗

(Q2

1, Q2 2) = 2F a P tr Q2λa

1 dx 6x(1 − x) xQ2

1 + (1 − x)Q2 2

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

Padé approximants: double virtual

• Commonly referred to as Canterbury approximants

Fπγ∗γ∗(q2

1, q2 2) = C N M = QN(q2 1, q2 2)

RM(q2

1, q2 2);

QN(RM) = N(M)

i,j

cQ(R)

n,m q2i 1 q2j 2

Again, match derivatives to get cQ,R

ij

’s

• Again nice convergence properties, as for instance, pQCD models

F pQCD;as

Pγ∗γ∗

(Q2

1, Q2 2) = 2F a P tr Q2λa

1 dx 6x(1 − x) xQ2

1 + (1 − x)Q2 2

• Simple example with appropriate power-like behavior

C 0

1 =

1 1 + cR

0,1(q2 1 + q2 2) + cR 1,1q2 1q2 2

→ 1 1 + cR

0,1(q2 1 + q2 2)

Again, benefits of not doing spectroscopy!

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

Inputs: data fitting

• Inputs: sequences data fitting (not just fitting models)
• 0.20

0.15 0.10 0.05 0.00 0.15 0.10 0.05 0.00

P2

2Q2

P1

7Q2 0. 10. 20. 30. 40. 0.1 0. 0.1 0.2

Q2 GeV2 Q2FΗΓΓQ2 GeV

• P11

P21 P31 P41 P51 P61 P71 0.50 0.55 0.60 0.65

bΗ

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

Inputs: data fitting

• Inputs: sequences data fitting (not just fitting models)

Reduced PN

N+1 :

P0;fit

1

→ bP , ... , P1;fit

2

✗ Data Set PN

1 :

P0;fit

1

→ bP , ... , P4;fit

1

→bP

cP , P5;fit 1

✗

• Let’s see impact on aHLbL;η

µ

(fact) Fit P0

1

14.5 P1

2

—

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

Inputs: data fitting

• Inputs: sequences data fitting (not just fitting models)

Reduced PN

N+1 :

P0;der

1

→ bP , ... , P1;der

2

✓ Data Set PN

1 :

P0;fit

1

→ bP , ... , P4;fit

1

→bP

cP , P5;fit 1

✗

• Let’s see impact on aHLbL;η

µ

(fact) Fit Der P0

1

14.5 13.2 P1

2

— 13.3

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

Inputs: data fitting

• Inputs: sequences data fitting (not just fitting models)

New larger PN

N+1 :

P0;fit

1

→ bP , ... , P1;fit

2

, P2;fit

3

✗ Data Set PN

1 :

P0;fit

1

→ bP , ... , P7;fit

1

→bP

cP , P8;fit 1

✗

• Let’s see impact on aHLbL;η

µ

(fact) Fit Der New Fit P0

1

14.5 13.2 14.0 P1

2

— 13.3 13.4

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

Inputs: data fitting

• Inputs: sequences data fitting (not just fitting models)

New larger PN

N+1 :

P0;der

1

→ bP , ... , P1;der

2

✓ Data Set PN

1 :

P0;fit

1

→ bP , ... , P7;fit

1

→bP

cP , P8;fit 1

✗

• Let’s see impact on aHLbL;η

µ

(fact) Fit Der New Fit New Der P0

1

14.5 13.2 14.0 13.1 P1

2

— 13.3 13.4 13.3

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

Inputs: data fitting

• Inputs: sequences data fitting (not just fitting models)

New larger PN

N+1 :

P0;der

1

→ bP , ... , P1;der

2

✓ Data Set PN

1 :

P0;fit

1

→ bP , ... , P7;fit

1

→bP

cP , P8;fit 1

✗

• Let’s see impact on aHLbL;η

µ

(fact) Fit Der New Fit New Der P0

1

14.5 13.2 14.0 13.1 P1

2

— 13.3 13.4 13.3 Advantage of PAs vs. resonance fits: interrelation, convergence, systematics

Padé approach to pseudoscalar poles in HLbL Application to HLbL and results

Section 3 Application to HLbL and results

Padé approach to pseudoscalar poles in HLbL Application to HLbL and results

Reconstructing FPγ∗γ∗(Q2

1, Q2 2)

• Simplest approach: C 0

1 (Q2 1, Q2 2)

C 0

1 (Q2 1, Q2 2) =

FPγγ(0, 0) 1 + bP(Q2

1 + Q2 2)

Padé approach to pseudoscalar poles in HLbL Application to HLbL and results

Reconstructing FPγ∗γ∗(Q2

1, Q2 2)

• Simplest approach: C 0

1 (Q2 1, Q2 2)

C 0

1 (Q2 1, Q2 2) =

FPγγ(0, 0) 1 + bP(Q2

1 + Q2 2)

• Next element: C 1

2 (Q2 1, Q2 2)

C 1

2 (Q2 1, Q2 2) =

FPγγ(0, 0)(1 + α1(Q2

1 + Q2 2) + α1,1Q2 1Q2 2)

1 + β1(Q2

1 + Q2 2) + β2(Q4 1 + Q4 2) + β1,1Q2 1Q2 2 + β2,1Q2 1Q2 2(Q2 1 + Q2 2) + β2,2Q4 1Q4 2

Padé approach to pseudoscalar poles in HLbL Application to HLbL and results

Reconstructing FPγ∗γ∗(Q2

1, Q2 2)

• Simplest approach: C 0

1 (Q2 1, Q2 2)

C 0

1 (Q2 1, Q2 2) =

FPγγ(0, 0) 1 + bP(Q2

1 + Q2 2)

• Next element: C 1

2 (Q2 1, Q2 2)

C 1

2 (Q2 1, Q2 2) =

FPγγ(0, 0)(1 + α1(Q2

1 + Q2 2) + α1,1Q2 1Q2 2)

1 + β1(Q2

1 + Q2 2) + β2(Q4 1 + Q4 2) + β1,1Q2 1Q2 2 + β2,1Q2 1Q2 2(Q2 1 + Q2 2) + β2,2Q4 1Q4 2

Reconstruction 1.Reduce to Padé Approximants FPγγ(0, 0), α1, β1, β2 → from PAs

Padé approach to pseudoscalar poles in HLbL Application to HLbL and results

Reconstructing FPγ∗γ∗(Q2

1, Q2 2)

• Simplest approach: C 0

1 (Q2 1, Q2 2)

C 0

1 (Q2 1, Q2 2) =

FPγγ(0, 0) 1 + bP(Q2

1 + Q2 2)

• Next element: C 1

2 (Q2 1, Q2 2)

C 1

2 (Q2 1, Q2 2) =

FPγγ(0, 0)(1 + α1(Q2

1 + Q2 2) + α1,1Q2 1Q2 2)

1 + β1(Q2

1 + Q2 2) + β2(Q4 1 + Q4 2) + β1,1Q2 1Q2 2 + β2,1Q2 1Q2 2(Q2 1 + Q2 2) + β2,2Q4 1Q4 2

Reconstruction 1.Reduce to Padé Approximants FPγγ(0, 0), α1, β1, β2 → from PAs

Padé approach to pseudoscalar poles in HLbL Application to HLbL and results

Reconstructing FPγ∗γ∗(Q2

1, Q2 2)

• Simplest approach: C 0

1 (Q2 1, Q2 2)

C 0

1 (Q2 1, Q2 2) =

FPγγ(0, 0) 1 + bP(Q2

1 + Q2 2)

• Next element: C 1

2 (Q2 1, Q2 2)

C 1

2 (Q2 1, Q2 2) =

FPγγ(0, 0)(1 + α1(Q2

1 + Q2 2) + α1,1Q2 1Q2 2)

1 + β1(Q2

1 + Q2 2) + β2(Q4 1 + Q4 2) + β1,1Q2 1Q2 2 + β2,1Q2 1Q2 2(Q2 1 + Q2 2) + β2,2Q4 1Q4 2

Reconstruction 1.Reduce to Padé Approximants 2.Reproduce the OPE behavior (high energies) Fπγ∗γ∗ = 1 3Q2 (2Fπ)

• 1 − 8

9 δ2 Q2 + O(αs(Q2))

• ⇒ β2,2 = 0, α1,1, β2,1

Padé approach to pseudoscalar poles in HLbL Application to HLbL and results

Reconstructing FPγ∗γ∗(Q2

1, Q2 2)

• Simplest approach: C 0

1 (Q2 1, Q2 2)

C 0

1 (Q2 1, Q2 2) =

FPγγ(0, 0) 1 + bP(Q2

1 + Q2 2)

• Next element: C 1

2 (Q2 1, Q2 2)

C 1

2 (Q2 1, Q2 2) =

FPγγ(0, 0)(1 + α1(Q2

1 + Q2 2) + α1,1Q2 1Q2 2)

1 + β1(Q2

1 + Q2 2) + β2(Q4 1 + Q4 2) + β1,1Q2 1Q2 2 + β2,1Q2 1Q2 2(Q2 1 + Q2 2)

Reconstruction 1.Reduce to Padé Approximants 2.Reproduce the OPE behavior (high energies) 3.Reproduce the low energies (aP;1,1)

Padé approach to pseudoscalar poles in HLbL Application to HLbL and results

Reconstructing FPγ∗γ∗(Q2

1, Q2 2)

• Simplest approach: C 0

1 (Q2 1, Q2 2)

C 0

1 (Q2 1, Q2 2) =

FPγγ(0, 0) 1 + bP(Q2

1 + Q2 2)

• Next element: C 1

2 (Q2 1, Q2 2)

C 1

2 (Q2 1, Q2 2) =

FPγγ(0, 0)(1 + α1(Q2

1 + Q2 2) + α1,1Q2 1Q2 2)

1 + β1(Q2

1 + Q2 2) + β2(Q4 1 + Q4 2) + β1,1Q2 1Q2 2 + β2,1Q2 1Q2 2(Q2 1 + Q2 2)

Reconstruction 1.Reduce to Padé Approximants 2.Reproduce the OPE behavior (high energies) 3.Reproduce the low energies (aP;1,1) Be generous: all configurations with no poles ⇒ amin

P;1,1 < aP;1,1 < amax P;1,1

Padé approach to pseudoscalar poles in HLbL Application to HLbL and results

Reconstructing FPγ∗γ∗(Q2

1, Q2 2)

• Simplest approach: C 0

1 (Q2 1, Q2 2)

C 0

1 (Q2 1, Q2 2) =

FPγγ(0, 0) 1 + bP(Q2

1 + Q2 2)

• Next element: C 1

2 (Q2 1, Q2 2)

C 1

2 (Q2 1, Q2 2) =

FPγγ(0, 0)(1 + α1(Q2

1 + Q2 2) + α1,1Q2 1Q2 2)

1 + β1(Q2

1 + Q2 2) + β2(Q4 1 + Q4 2) + β1,1Q2 1Q2 2 + β2,1Q2 1Q2 2(Q2 1 + Q2 2)

Reconstruction 1.Reduce to Padé Approximants 2.Reproduce the OPE behavior (high energies) 3.Reproduce the low energies (amin

P;1,1 < aP;1,1 < amax P;1,1)

Padé approach to pseudoscalar poles in HLbL Application to HLbL and results

Pseudoscalar-pole contribution: Final results

—C 0

1 (Q2 1, Q2 2)—

aHLbL;P

µ

× 1011 OPE (aP;1,1 = 2b2

P)

Fact (aP;1,1 = b2

P)

π0 65.3(1.4)F(2.4)bπ[2.8]t 54.3(1.5)F(2.2)bπ[2.5]t η 17.1(0.6)F(0.2)bη[0.6]t 13.0(0.4)F(0.2)bη[0.5]t η′ 16.0(0.5)F(0.3)bη′ [0.6]t 12.0(0.4)F(0.3)bη′ [0.5]t Total 98.4[2.9]t 79.3[2.6]t

Padé approach to pseudoscalar poles in HLbL Application to HLbL and results

Pseudoscalar-pole contribution: Final results

—C 0

1 (Q2 1, Q2 2)—

aHLbL;P

µ

× 1011 OPE (aP;1,1 = 2b2

P)

Fact (aP;1,1 = b2

P)

π0 65.3(1.4)F(2.4)bπ[2.8]t 54.3(1.5)F(2.2)bπ[2.5]t η 17.1(0.6)F(0.2)bη[0.6]t 13.0(0.4)F(0.2)bη[0.5]t η′ 16.0(0.5)F(0.3)bη′ [0.6]t 12.0(0.4)F(0.3)bη′ [0.5]t Total 98.4[2.9]t 79.3[2.6]t

—C 1

2 (Q2 1, Q2 2)—

aHLbL;P

µ

× 1011 amin

P;1,1

amax

P;1,1

π0 64.1(1.3)L(0)δ[1.3]t 63.0(1.1)L(0.5)δ[1.2]t η 16.3(0.8)L(0)δ[0.8]t 16.2(0.8)L(0.6)δ[1.0]t η′ 14.7(0.7)L(0)δ[0.7]t 14.3(0.5)L(0.5)δ[0.7]t Total 95.1[1.7]t 93.5[1.7]t

Padé approach to pseudoscalar poles in HLbL Application to HLbL and results

Pseudoscalar-pole contribution: Final results

—C 0

1 (Q2 1, Q2 2)—

aHLbL;P

µ

× 1011 OPE (aP;1,1 = 2b2

P)

Fact (aP;1,1 = b2

P)

π0 65.3(1.4)F(2.4)bπ[2.8]t 54.3(1.5)F(2.2)bπ[2.5]t η 17.1(0.6)F(0.2)bη[0.6]t 13.0(0.4)F(0.2)bη[0.5]t η′ 16.0(0.5)F(0.3)bη′ [0.6]t 12.0(0.4)F(0.3)bη′ [0.5]t Total 98.4[2.9]t 79.3[2.6]t

—C 1

2 (Q2 1, Q2 2)—

aHLbL;P

µ

× 1011 amin

P;1,1

amax

P;1,1

π0 64.1(1.3)L(0)δ[1.3]t{1.2}sys 63.0(1.1)L(0.5)δ[1.2]t{2.3}sys η 16.3(0.8)L(0)δ[0.8]t 16.2(0.8)L(0.6)δ[1.0]t η′ 14.7(0.7)L(0)δ[0.7]t 14.3(0.5)L(0.5)δ[0.7]t Total 95.1[1.7]t 93.5[1.7]t

Padé approach to pseudoscalar poles in HLbL Application to HLbL and results

Pseudoscalar-pole contribution: Final results

—C 0

1 (Q2 1, Q2 2)—

aHLbL;P

µ

× 1011 OPE (aP;1,1 = 2b2

P)

Fact (aP;1,1 = b2

P)

π0 65.3(1.4)F(2.4)bπ[2.8]t 54.3(1.5)F(2.2)bπ[2.5]t η 17.1(0.6)F(0.2)bη[0.6]t 13.0(0.4)F(0.2)bη[0.5]t η′ 16.0(0.5)F(0.3)bη′ [0.6]t 12.0(0.4)F(0.3)bη′ [0.5]t Total 98.4[2.9]t 79.3[2.6]t

—C 1

2 (Q2 1, Q2 2)—

aHLbL;P

µ

× 1011 amin

P;1,1

amax

P;1,1

π0 64.1(1.3)L(0)δ[1.3]t{1.2}sys 63.0(1.1)L(0.5)δ[1.2]t{2.3}sys η 16.3(0.8)L(0)δ[0.8]t{0.8}sys 16.2(0.8)L(0.6)δ[1.0]t{0.9}sys η′ 14.7(0.7)L(0)δ[0.7]t{1.3}sys 14.3(0.5)L(0.5)δ[0.7]t{1.7}sys Total 95.1[1.7]t{3.3}sys 93.5[1.7]t{4.9}sys

Padé approach to pseudoscalar poles in HLbL Application to HLbL and results

Pseudoscalar-pole contribution: Final results

—C 0

1 (Q2 1, Q2 2)—

aHLbL;P

µ

× 1011 OPE (aP;1,1 = 2b2

P)

Fact (aP;1,1 = b2

P)

π0 65.3(1.4)F(2.4)bπ[2.8]t 54.3(1.5)F(2.2)bπ[2.5]t η 17.1(0.6)F(0.2)bη[0.6]t 13.0(0.4)F(0.2)bη[0.5]t η′ 16.0(0.5)F(0.3)bη′ [0.6]t 12.0(0.4)F(0.3)bη′ [0.5]t Total 98.4[2.9]t 79.3[2.6]t

—C 1

2 (Q2 1, Q2 2)—

aHLbL;P

µ

× 1011 amin

P;1,1

amax

P;1,1

π0 64.1(1.3)L(0)δ[1.3]t{1.2}sys 63.0(1.1)L(0.5)δ[1.2]t{2.3}sys η 16.3(0.8)L(0)δ[0.8]t{0.8}sys 16.2(0.8)L(0.6)δ[1.0]t{0.9}sys η′ 14.7(0.7)L(0)δ[0.7]t{1.3}sys 14.3(0.5)L(0.5)δ[0.7]t{1.7}sys Total 95.1[1.7]t{3.3}sys 93.5[1.7]t{4.9}sys

—Final Result (combining errors just for clarity)

aπ,η,η′

µ

= (63.6(2.7) + 16.3(1.3) + 14.5(1.8))×10−11 = 94.3(5.3)×10−11

Padé approach to pseudoscalar poles in HLbL Application to HLbL and results

Pseudoscalar-pole contribution: Final results

—C 0

1 (Q2 1, Q2 2)—

aHLbL;P

µ

× 1011 OPE (aP;1,1 = 2b2

P)

Fact (aP;1,1 = b2

P)

π0 65.3(1.4)F(2.4)bπ[2.8]t 54.3(1.5)F(2.2)bπ[2.5]t η 17.1(0.6)F(0.2)bη[0.6]t 13.0(0.4)F(0.2)bη[0.5]t η′ 16.0(0.5)F(0.3)bη′ [0.6]t 12.0(0.4)F(0.3)bη′ [0.5]t Total 98.4[2.9]t 79.3[2.6]t

—C 1

2 (Q2 1, Q2 2)—

aHLbL;P

µ

× 1011 amin

P;1,1

amax

P;1,1

π0 64.1(1.3)L(0)δ[1.3]t{1.2}sys 63.0(1.1)L(0.5)δ[1.2]t{2.3}sys η 16.3(0.8)L(0)δ[0.8]t{0.8}sys 16.2(0.8)L(0.6)δ[1.0]t{0.9}sys η′ 14.7(0.7)L(0)δ[0.7]t{1.3}sys 14.3(0.5)L(0.5)δ[0.7]t{1.7}sys Total 95.1[1.7]t{3.3}sys 93.5[1.7]t{4.9}sys

—Final Result (combining errors just for clarity)

aπ,η,η′

µ

= (63.6(2.7) + 16.3(1.3) + 14.5(1.8))×10−11 = 94.3(5.3)×10−11

• Low-energy emphasis but high-energies too
• η and η′ fulfill high-energies (5 × 10−11 effect: 1/3 of exp error)
• Systematic from sequence results

Padé approach to pseudoscalar poles in HLbL Application to HLbL and results

Summary

• Padé approximants to reconstruct form factors
• Full use of data and theory in a systematic approach; not modelling
• New value aHLbL;π,η,η′

µ

= 94.3(5.3) × 10−11 including systematics

• OPE for all the pseudoscalars implemented
• Bypass η − η′ mixing (output): non-trivial if fully theory-driven approach

Related projects

• Radiative corrections for P → ¯

ℓℓ¯ ℓ′ℓ′: Phys.Rev. D97 (2018) 056010

• In contact with H. Czyz for e+e− → e+e−P

Padé approach to pseudoscalar poles in HLbL Backup

Section 4 Backup

Padé approach to pseudoscalar poles in HLbL Backup

FPγ∗γ∗(Q2

1, Q2 2) The two planes: boundaries for the aP;1,1 region

Padé approach to pseudoscalar poles in HLbL Backup

FPγ∗γ∗(Q2

1, Q2 2) The two planes: boundaries for the aP;1,1 region

Padé approach to pseudoscalar poles in HLbL Backup

Seeing is believing: toy models and systematics

—aπ

µ: Regge Model— F Regge

π0γ∗γ∗(Q2 1, Q2 2) = aFPγγ Q2

1−Q2 2

• ψ(0)
• M2+Q2

1 a

• −ψ(0)
• M2+Q2

2 a

• ψ(1)
• M2

a

• C 0

1

C 1

2

C 2

3

C 3

4

LE 55.2 59.7 60.4 60.6 OPE 65.7 60.8 60.7 60.7 FitOPE 66.3 62.7 61.1 60.8 Exact 60.7

—aπ

µ: Logarithmic Model—

F log

π0γ∗γ∗(Q2 1, Q2 2) = FPγγM2 Q2

1−Q2 2 ln

1+Q2

1/M2

1+Q2

2/M2

• C 0

1

C 1

2

C 2

3

C 3

4

LE 56.7 64.4 66.1 66.8 OPE 65.7 67.3 67.5 67.6 FitOPE 79.6 71.9 69.3 68.4 Exact 67.6

• P. Masjuan & P. Sanchez Phys.Rev. D95, 054026 (2017)

Padé approach to pseudoscalar poles in HLbL Backup

Seeing is believing: toy models and systematics

—aπ

µ: Regge Model— F Regge

π0γ∗γ∗(Q2 1, Q2 2) = aFPγγ Q2

1−Q2 2

• ψ(0)
• M2+Q2

1 a

• −ψ(0)
• M2+Q2

2 a

• ψ(1)
• M2

a

• C 0

1

C 1

2

C 2

3

C 3

4

LE 55.2 59.7 60.4 60.6 OPE 65.7 60.8 60.7 60.7 FitOPE 66.3 62.7 61.1 60.8 Exact 60.7

—aπ

µ: Logarithmic Model—

F log

π0γ∗γ∗(Q2 1, Q2 2) = FPγγM2 Q2

1−Q2 2 ln

1+Q2

1/M2

1+Q2

2/M2

• C 0

1

C 1

2

C 2

3

C 3

4

LE 56.7 64.4 66.1 66.8 OPE 65.7 67.3 67.5 67.6 FitOPE 79.6 71.9 69.3 68.4 Exact 67.6

• The convergence result is excellent!
• The OPE choice seems the best → high energy matters
• Still, low energies provide a good performance
• Error ∼ difference among elements → Systematics!
• P. Masjuan & P. Sanchez Phys.Rev. D95, 054026 (2017)