CHPT RESULTS FOR HVP AND A NEW EVALUATION OF THE PION LOOP CONTRIBUTION

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ChPT for HVP and a

new evaluation of the pion loop contribution Johan Bijnens

Introduction HLbL HVP Conclusions

CHPT RESULTS FOR HVP AND A NEW EVALUATION OF THE PION LOOP

CONTRIBUTION

Johan Bijnens

Lund University

bijnens@thep.lu.se http://thep.lu.se/~bijnens http://thep.lu.se/~bijnens/chpt/

http://thep.lu.se/~bijnens/chiron/

Towards high precision muon g-2/EDM measurement at J-PARC 28-29 November 2016

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ChPT for HVP and a

Introduction To ChPT or not to ChPT HLbL HVP Conclusions

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Why do we do this?

The muon a_µ= g − 2

2 will be measured more precisely

J-PARC Fermilab

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ChPT for HVP and a

Hadronic contributions

HVP HLbL

The blobs are hadronic contributions

I will present some results on both HVP and HLbL There are higher order contributions of both types

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ChPT for HVP and a

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This talk

HLbL (models)

J. Bijnens and J. Relefors, “Pion light-by-light contributions to the muon g − 2,” JHEP 1609 (2016) 113 [arXiv:1608.01454 [hep-ph]].

Disconnected contributions are (expected to be) large A new evalution of the pion loop

HVP (Chiral perturbation theory (ChPT))

J. Bijnens and J. Relefors, “Connected, Disconnected and Strange Quark Contributions to HVP,” JHEP 1611 (2016) 086

[arXiv:1609.01573 [hep-lat]].

An estimate of the disconnected and strange quark contributions

Finite volume corrections

J. Bijnens and J. Relefors, to be published

All can be found in the PhD thesis of Johan Relefors (with some smaller changes)

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ChPT for HVP and a

To ChPT or not to ChPT

ChPT = Effective field theory describing the lowest order pseudo-scalar representation

or the (pseudo) Goldstone bosons from spontaneous breaking of chiral symmetry.

Describes pions, kaons and etas at low-energies

It’s an effective field theory: new parameters or LECs at each new order

Recent review of LECs:

JB, Ecker,Ann.Rev.Nucl.Part.Sci. 64 (2014) 149 [arXiv:1405.6488]

a_µ is a very low-energy quantity, why notjust calculate it in ChPT?

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ChPT for HVP and a

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To ChPT or not to ChPT

HVP HLbL

Fill the blobs with pions and kaons Lowest order for both HVP and HLbL:

pure pion loop (or scalar QED):well defined answer NLO: the blob is nicely finite

but notafterthe muon/photon integrations

Needs a counterterm (NLO LEC)that is the muon g − 2

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ChPT for HVP and a

To ChPT or not to ChPT

So need more than ChPT Experiment

Dispersion relations lattice QCD

Models

ChPT can be used to put constraints, help understanding results and estimate not evaluated parts,. . .

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ChPT for HVP and a

Introduction

HLbL Disconnected/

connected π-loop HVP Conclusions

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HLbL: the main object to calculate

p₁^ν

p₂^α q^ρ

p₃^β

p₅ p₄

p

′

p

Muon line and photons: well known The blob: fill in with hadrons/QCD Trouble: low and high energy very mixed

Double counting needs to be avoided: hadron exchanges versus quarks

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ChPT for HVP and a

Introduction

HLbL Disconnected/

A separation proposal: a start

E. de Rafael, “Hadronic contributions to the muon g-2 and low-energy QCD,”

Phys. Lett. B322 (1994) 239-246. [hep-ph/9311316].

Use ChPT p counting and large N_c p⁴, order 1: pion-loop

p⁸, order N_c: quark-loop and heavier meson exchanges p⁶, order Nc: pion exchange

Does not fully solve the problem

only short-distance part of quark-loop is really p⁸ but it’s a start

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ChPT for HVP and a

Introduction

HLbL Disconnected/

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A separation proposal: a start

E. de Rafael, “Hadronic contributions to the muon g-2 and low-energy QCD,”

Phys. Lett. B322 (1994) 239-246. [hep-ph/9311316].

Use ChPT p counting and large Nc

p⁴, order 1: pion-loop

p⁸, order N_c: quark-loop and heavier meson exchanges p⁶, order Nc: pion exchange

Implemented by two groups in the 1990s:

Hayakawa, Kinoshita, Sanda: meson models, pion loop using hidden local symmetry, quark-loop with VMD, calculation in Minkowski space (HKS)

JB, Pallante, Prades: Try using as much as possible a

consistent model-approach, ENJL, calculation in Euclidean space (BPP)

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ChPT for HVP and a

Introduction

HLbL Disconnected/

Papers: BPP and HKS

JB, E. Pallante and J. Prades

“Comment on the pion pole part of the light-by-light contribution to the muon g-2,” Nucl. Phys. B 626 (2002) 410

[arXiv:hep-ph/0112255].

“Analysis of the Hadronic Light-by-Light Contributions to the Muon g − 2,” Nucl. Phys. B 474 (1996) 379 [arXiv:hep-ph/9511388].

“Hadronic light by light contributions to the muon g-2 in the large Nc limit,” Phys. Rev. Lett. 75 (1995) 1447 [Erratum-ibid. 75 (1995) 3781] [arXiv:hep-ph/9505251].

Hayakawa, Kinoshita, (Sanda)

“Pseudoscalar pole terms in the hadronic light by light scattering contribution to muon g - 2,” Phys. Rev. D57 (1998) 465-477.

[hep-ph/9708227], Erratum-ibid.D66 (2002) 019902[hep-ph/0112102].

“Hadronic light by light scattering contribution to muon g-2,” Phys.

Rev. D54 (1996) 3137-3153. [hep-ph/9601310].

“Hadronic light by light scattering effect on muon g-2,” Phys. Rev.

Lett. 75 (1995) 790-793. [hep-ph/9503463].

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ChPT for HVP and a

Introduction

HLbL Disconnected/

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Some main observations

The largest constribution is π⁰ (and η, η⁰) exchange For the pseudo-scalar exchange most evaluations are in reasonable agreement

That will be used for an estimate of disconnected/connected

The pion loop can be sizable but a large difference between the two evaluations

For the pure pion loop part, even larger numbers have been proposed byEngel, Ramsey-Musolf

Discussed in my second part

Another approach is the dispersive by Colangelo et al.

There are other contributions but the sum is smaller than the leading pseudo-scalar exchange

I interpret present HLbL lattice results as saying that no major contributions have been missed in the model estimates (but error and actual value might still change)

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ChPT for HVP and a

Introduction

HLbL Disconnected/

Disconnected/Connected

Connected Disconnected

gray=lots of quarks/gluons

Disconnected Estimate the full result with pseudo-scalar exchange Connected diagrams only:

the gluon exchanges responsible for U(1)A breaking are not included at all

η⁰ becomes light, mainly (¯uu + ¯d d )/√

2 (πη) and has the same mass as the pion

Or the two-light states are πu(¯uu) and πd (¯d d )

η becomes mainly ¯ss and much heavier than the pion (and thus small contribution)

Assume that couplings are not affected (not too bad experimentally)

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ChPT for HVP and a

Introduction

HLbL Disconnected/

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Disconnected/Connected

π_u, π_d π⁰, πη

Two flavour case only: up and down quarks (three flavour not more difficult, just more numbers)

Meson couplings to two-photons is via quark-loop Look at charge factors for Connected

As “quark-loop”: q_u⁴+ q_d⁴=¹⁷₈₁ As π_u, π_d: q_u²q²_u+ q²_dq²_d= ¹⁷₈₁ As π⁰, π_η: _q2

u−q²_d

√2

2

+_q2 u+q_d²

√2

2

= ₁₆₂⁹ +₁₆₂²⁵ =¹⁷₈₁ Include U(1)_A breaking: π_η heavy

π⁰: _q2 u√−q²_d

2

²

= ₁₆₂⁹

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ChPT for HVP and a

Introduction

HLbL Disconnected/

Disconnected/Connected

So in this limit:

Two-flavour case

U(1)A breaking makes πη infinitely heavy Full result dominated by pseudo-scalar exchange U(1)A breaking does not affect couplings

Connected: ₁₆₂³⁴ Disconnected:−₁₆₂²⁵ Sum: ₁₆₂⁹

All assumptions get corrections but final conclusion stays The disconnected contribution is expected to be large and of opposite sign with significant cancellations Argument used to go from large-Nc to π⁰, η, η⁰ in

JB, Pallante, Prades, Nucl. Phys. B 474 (1996) 379 [arXiv:hep-ph/9511388]

This form: JB, Relefors, JHEP 1609 (2016) 113 [arXiv:1608.01454]

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ChPT for HVP and a

Introduction

HLbL Disconnected/

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Disconnected/Connected

So in this limit:

Two-flavour case

U(1)A breaking makes πη infinitely heavy Full result dominated by pseudo-scalar exchange U(1)A breaking does not affect couplings

Connected: ₁₆₂³⁴ Disconnected:−₁₆₂²⁵ Sum: ₁₆₂⁹

All assumptions get corrections but final conclusion stays The disconnected contribution is expected to be large and of opposite sign with significant cancellations Argument used to go from large-Nc to π⁰, η, η⁰ in

JB, Pallante, Prades, Nucl. Phys. B 474 (1996) 379 [arXiv:hep-ph/9511388]

This form: JB, Relefors, JHEP 1609 (2016) 113 [arXiv:1608.01454]

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ChPT for HVP and a

Introduction

HLbL Disconnected/

General properties

Π^ρναβ(p₁, p₂, p₃) =

p3

p₂ p₁

q

Actually we really need δΠ^ρναβ(p1, p2, p3) δp_3λ

p3=0

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ChPT for HVP and a

Introduction

HLbL Disconnected/

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General properties

Π^ρναβ(p1, p2, p3):

In general 138 Lorentz structures (but only 28 contribute to g − 2)

Using q_ρΠ^ρναβ = p_1νΠ^ρναβ = p_2αΠ^ρναβ = p_3βΠ^ρναβ = 0 43 gauge invariant structures

Bose symmetry relates some of them

All depend on p₁², p₂² and q², but before derivative and p3 → 0 also p²₃, p1· p₂, p1· p₃

Actually 2 less but singular basisFischer et al.

Compare HVP: one function, one variable

General calculation from experiment: how difficult:

Colangelo

In four photon measurement: lepton contribution

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ChPT for HVP and a

Introduction

HLbL Disconnected/

General properties

R _d⁴_p₁

(2π)⁴

R _d⁴_p₂

(2π)⁴ plus loops inside the hadronic part 8 dimensional integral, three trivial,

5 remain: p₁², p₂², p1· p₂, p1· p_µ, p2· p_µ Rotate to Euclidean space:

Easier separation of long and short-distance Artefacts (confinement) in models smeared out.

More recent: can do two more using Gegenbauer techniquesKnecht-Nyffeler,

Jegerlehner-Nyffeler,JB–Zahiri-Abyaneh–Relefors P₁², P₂² and Q² remain

studya^X_µ =R dl_P₁dl_P₂a^XLL_µ =R dl_P₁dl_P₂dl_Qa^XLLQµ

l_P = ln (P/GeV ), to see where the contributions are Study the dependence on the cut-off for the photons

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ChPT for HVP and a

Introduction

HLbL Disconnected/

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General properties

R _d⁴_p₁

(2π)⁴

R _d⁴_p₂

(2π)⁴ plus loops inside the hadronic part 8 dimensional integral, three trivial,

5 remain: p₁², p₂², p1· p₂, p1· p_µ, p2· p_µ Rotate to Euclidean space:

Easier separation of long and short-distance Artefacts (confinement) in models smeared out.

More recent: can do two more using Gegenbauer techniquesKnecht-Nyffeler,

Jegerlehner-Nyffeler,JB–Zahiri-Abyaneh–Relefors P₁², P₂² and Q² remain

studya^X_µ =R dl_P₁dl_P₂a^XLL_µ =R dl_P₁dl_P₂dl_Qa^XLLQµ

l_P = ln (P/GeV ), to see where the contributions are Study the dependence on the cut-off for the photons

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ChPT for HVP and a

Introduction

HLbL Disconnected/

π-loop

A bare π-loop (sQED) give about −4 · 10⁻¹⁰ The ππγ^∗ vertex is always done using VMD ππγ^∗γ^∗ vertex two choices:

Hidden local symmetry model: only one γ has VMD Full VMD

Both are chirally symmetric

The HLS model used has problems with π⁺-π⁰mass difference (due to not having an a1)

Final numbers quite different: −0.45 and −1.9 (×10⁻¹⁰) For BPP stopped at 1 GeV but within 10% of higher Λ

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ChPT for HVP and a

Introduction

HLbL Disconnected/

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π loop: Bare vs VMD

0.1

1

10

0.1 1

10 0

5e-11 1e-10 1.5e-10 2e-10

-a_µ^LLQ

π loop

VMD bare

P₁ = P₂

Q -a_µ^LLQ

plotted a_µ^LLQ for P₁ = P₂ aµ=R dl_P₁dl_P₂dl_Qa^LLQ_µ lQ = log(Q/1 GeV)

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ChPT for HVP and a

Introduction

HLbL Disconnected/

π loop: VMD vs HLS

0.1 1

0.1 1 10 10

-4e-11 -2e-11 0 2e-11 4e-11 6e-11 8e-11 1e-10

-a_µ^LLQ

π loop

VMD HLS a=2

P₁ = P₂ Q

-a_µ^LLQ

Usual HLS, a = 2

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ChPT for HVP and a

Introduction

HLbL Disconnected/

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π loop: VMD vs HLS

0.1 1

0.1 1 10 10

0 2e-11 4e-11 6e-11 8e-11 1e-10 1.2e-10

-a_µ^LLQ

π loop

VMD HLS a=1

P₁ = P₂ Q

-a_µ^LLQ

HLS with a = 1, satisfies more short-distance constraints

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ChPT for HVP and a

Introduction

HLbL Disconnected/

π loop

ππγ^∗γ^∗ for q₁² = q²₂ has a short-distance constraint from the OPE as well.

HLS does not satisfy it

full VMD does: so probably better estimate

Ramsey-Musolf suggested to do pure ChPT for the π loop

K. T. Engel and M. J. Ramsey-Musolf, Phys. Lett. B 738 (2014) 123 [arXiv:1309.2225 [hep-ph]].

Polarizability (L9+ L10) up to 10%, charge radius 30% at low energies, more at higher

Both HLS and VMD have charge radius effect but not polarizability

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ChPT for HVP and a

Introduction

HLbL Disconnected/

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π loop

ππγ^∗γ^∗ for q₁² = q²₂ has a short-distance constraint from the OPE as well.

HLS does not satisfy it

full VMD does: so probably better estimate

Ramsey-Musolf suggested to do pure ChPT for the π loop

K. T. Engel and M. J. Ramsey-Musolf, Phys. Lett. B 738 (2014) 123 [arXiv:1309.2225 [hep-ph]].

Polarizability (L9+ L10) up to 10%, charge radius 30% at low energies, more at higher

Both HLS and VMD have charge radius effect but not polarizability

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ChPT for HVP and a

Introduction

HLbL Disconnected/

π loop: L

₉

, L

₁₀

ChPT for muon g − 2 at order p⁶ is not powercounting finite so no prediction for a_µ exists.

But can be used to study the low momentum end of the integral over P₁, P₂, Q

The four-photon amplitude is finite still at two-loop order (counterterms start at order p⁸)

Add L₉ and L₁₀ vertices to the bare pion loop:

JB, Relefors, Zahiri-Abyaneh, 1208.3548,1208.2554,1308.2575,1510.05796 JB, Relefors, JHEP 1609 (2016) 113 [arXiv:1608.01454 [hep-ph]].

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Introduction

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π loop: VMD vs charge radius

0.1 0.2 0.1 0.4

0.2

0.4 -4e-11

-2e-11 0 2e-11 4e-11 6e-11 8e-11 1e-10

-a_µ^LLQ

π loop VMD

L₉=-L₁₀

P₁ = P₂ Q

-a_µ^LLQ

low scale, charge radius effect well reproduced

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ChPT for HVP and a

Introduction

HLbL Disconnected/

π loop: VMD vs L

₉

and L

₁₀

0.1 0.2 0.1 0.4

0.2

0.4 0

2e-11 4e-11 6e-11 8e-11 1e-10 1.2e-10

-a_µ^LLQ

π loop

VMD L₁₀,L₉

P₁ = P₂ Q

-a_µ^LLQ

L₉+ L₁₀6= 0 gives an enhancement of 10-15%

To do it fully need to get a model: include a1

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ChPT for HVP and a

Introduction

HLbL Disconnected/

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Include a

₁

L9+ L10 effect is from a₁

But to get gauge invariance correctly need

a1 a1

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ChPT for HVP and a

Introduction

HLbL Disconnected/

Include a

₁

Consistency problem: full a1-loop?

Treat a1 and ρ classical and π quantum: there must be a π that closes the loop

Argument: integrate out ρ and a₁ classically, then do pion loops with the resulting Lagrangian

To avoid problems: representation without a₁-π mixing Check for curiosity what happens if we add a₁-loop

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ChPT for HVP and a

Introduction

HLbL Disconnected/

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Include a

₁

Use antisymmetric vector representation for a1 and ρ Fields Aµν, Vµν (nonets)

Kinetic terms: −¹₂ D

∇^λVλµ∇_νV^νµ−^M₂²^VVµνV^µν E

−¹₂D

∇^λAλµ∇_νA^νµ− ^M₂^A²AµνA^µν E Terms that give contributions to the L^r_i:

FV

2√

2hf_+µνV^µνi +^iG^√^V

2 hV^µνuµuνi + ^F^A

2√

2hf_−µνA^µνi L9= ^F_2M^V^G2^V

V

, L10= − ^F

2 V

4M_V² + ^F

2 A

4M_A²

Weinberg sum rules: (Chiral limit) F_V² = F_A²+ F_π² F_V²M_V² = F_A²M_A² VMD for ππγ: FVGV = F_π²

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ChPT for HVP and a

Introduction

HLbL Disconnected/

V

_µν

only

Π^ρναβ(p1, p2, p3) is not finite (but was also not finite for HLS) But δΠ^ρναβ(p₁, p₂, p₃)

δp3λ

p3=0

also not finite (but was finite for HLS)

Derivative one finite for GV = FV/2

Surprise: g − 2 identical to HLS with a = ^F_F^V²2 π

Yes I know, different representations are identical BUT they do differ in higher order terms and even in what is higher order

Same comments as for HLS numerics

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ChPT for HVP and a

Introduction

HLbL Disconnected/

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V

_µν

only

Π^ρναβ(p1, p2, p3) is not finite (but was also not finite for HLS) But δΠ^ρναβ(p₁, p₂, p₃)

δp3λ

p3=0

also not finite (but was finite for HLS)

Derivative one finite for GV = FV/2

Surprise: g − 2 identical to HLS with a = ^F_F^V²2 π

Yes I know, different representations are identical BUT they do differ in higher order terms and even in what is higher order

Same comments as for HLS numerics

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ChPT for HVP and a

Introduction

HLbL Disconnected/

V

_µν

and A

_µν

Add a1

Calculate a lot δΠ^ρναβ(p1, p2, p3)

δp_3λ

p3=0

finite for:

G_V = F_V = 0 and F_A²= −2F_π²

If adding full a₁-loop G_V = F_V = 0 and F_A²= −F_π²

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V

_µν

and A

_µν

Start by adding ρa₁π vertices

λ₁h[V^µν, A_µν] χ−i +λ₂h[V^µν, A_να] h_µ^νi

+λ3hi [∇^µVµν, Aνα] uαi +λ₄hi [∇_αVµν, Aαν] u^µi +λ5hi [∇^αVµν, Aµν] uαi +λ₆hi [V^µν, Aµν] f−α

νi +λ₇hiV_µνA^µρA^ν_ρi

All lowest dimensional vertices of their respective type Not all independent, there are three relations

Follow from the constraints on Vµν and Aµν (thanks to Stefan Leupold)

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ChPT for HVP and a

Introduction

HLbL Disconnected/

V

_µν

and A

_µν

: big disappointment

Work a whole lot δΠ^ρναβ(p₁, p₂, p₃)

δp3λ

p3=0

not obviously finite Work a lot more

Prove that δΠ^ρναβ(p1, p2, p3) δp_3λ

p3=0

finite, only same solutions as before

Try the combination that show up in g − 2 only Work a lot

Again, only same solutions as before

Small loophole left: after the integration for g − 2 could be finite but many funny functions of mπ, mµ, MV and MA

show up.

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a

₁

-loop: cases with good L

₉

and L

₁₀

0.1

1

10 0.1

1

10 -2e-10

-1.5e-10 -1e-10 -5e-11 0 5e-11 1e-10 1.5e-10 2e-10

-a_µ^LLQ

π loop

bare Weinberg no a₁-loop

P₁ = P₂

Q -a_µ^LLQ

Add FV, GV and FA

Fix values by Weinberg sum rules and VMD in γ^∗ππ no a₁-loop

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ChPT for HVP and a

Introduction

HLbL Disconnected/

a

₁

-loop: cases with good L

₉

and L

₁₀

0.1

1

10 0.1

1

10 -2e-10

-1.5e-10 -1e-10 -5e-11 0 5e-11 1e-10 1.5e-10 2e-10

-a_µ^LLQ

π loop

bare Weinberg with a₁-loop

P₁ = P₂

Q -a_µ^LLQ

Add F_V, G_V and F_A

Fix values by Weinberg sum rules and VMD in γ^∗ππ With a₁-loop (is different plot!!)

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ChPT for HVP and a

Introduction

HLbL Disconnected/

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a

₁

-loop: cases with good L

₉

and L

₁₀

0.1

1

10 0.1

1

10 0

5e-11 1e-10 1.5e-10 2e-10

-a_µ^LLQ

π loop

bare a₁ no a₁-loop,VMD

P₁ = P₂

Q -a_µ^LLQ

Add a1 with F_A² = +F_π²

Add the full VMD as done earlier for the bare pion loop

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ChPT for HVP and a

Introduction

HLbL Disconnected/

Integration results

0 5e-11 1e-10 1.5e-10 2e-10 2.5e-10 3e-10 3.5e-10 4e-10

0.1 1 10

-a_µ^Λ

Λ a₁ F_A²= -2F²

a₁ F_A² = -1 a₁-loop HLS HLS a=1 VMD a₁ VMD a₁ Weinberg

P1, P2, Q ≤ Λ

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Integration results with a

₁

Problem: get high energy behaviour good enough

But all models with reasonable L₉ and L₁₀ fall way inside the error quoted earlier (−1.9 ± 1.3) 10⁻¹⁰

Tentative conclusion: Use hadrons only below about 1 GeV:a^π−loopµ = (−2.0 ± 0.5) 10⁻¹⁰

Note thatEngel and Ramsey-Musolf, arXiv:1309.2225is a bit more pessimistic quoting numbers from(−1.1 to −7.1) 10⁻¹⁰

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ChPT for HVP and a

Introduction HLbL

HVP Disconnected/

Connected Finite Volume Conclusions

Two-point: Why

Muon: aµ= (g − 2)/2 and a^LO,HVP_µ = Z ∞

0

dQ²f Q²Π Qˆ ²

0.00 0.05 0.10 0.15 0.20

0.000 0.002 0.004 0.006 0.008 0.010 0.012

plot: f Q²Π Qˆ ² with Q² = −q² in GeV² Figure and data: Aubin, Blum, Chau, Golterman, Peris, Tu,

Phys. Rev. D93 (2016) 054508 [arXiv:1512.07555]

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ChPT for HVP and a

Introduction HLbL

HVP Disconnected/

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Two-point: Connected versus disconnected

Connected Disconnected

yellow=lots of quarks/gluons

Π^µν_ab(q) ≡ i Z

d⁴xe^iq·xT (j_a^µ(x )j_a^ν†(0)) j_π^µ+ = ¯d γ^µu

j_u^µ= ¯uγ^µu, j_d^µ= ¯d γ^µd , j_s^µ= ¯sγ^µs j_e^µ= 2

3¯uγ^µu −1

3d γ¯ ^µd−1 3¯sγ^µs Study in ChPT at one-loop:

Della Morte, J¨uttner, JHEP 1011 (2010) 154 [arXiv:1009.3783]

Extend to two-loops and some more:

JB, J. Relefors, JHEP 1611 (2016) 086 [arXiv:1609.01573]

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ChPT for HVP and a

Introduction HLbL

HVP Disconnected/

Two-point: Connected versus disconnected

Include also singlet part of the vector current There are new terms in the Lagrangian p⁴ only one more: hLµνi hL^µνi + hR_µνi hR^µνi

=⇒ The pure singlet vector current does not couple to mesons until p⁶

=⇒ Loop diagrams involving the pure singlet vector current only appear at p⁸ (implies relations)

=⇒ Loop diagrams with singlet vector and WZW only at p¹⁰

p⁶ (no full classification, just some examples) hD_ρL_µνi hD^ρL^µνi + hD_ρR_µνi hD^ρR^µνi, hL_µνiL^µνχ^†U + hR_µνiR^µνχU^†,. . .

Results at two-loop order, unquenched isospin limit

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Two-point: Connected versus disconnected

Π^µν_π+π⁺: only connected Π^µν_ud: only disconnected Π^µν_uu = Π^µν_π+π⁺ + Π^µν_ud Π^µν_ee = 5

9Π^µν_π+π⁺+1 9Π^µν_ud

Infinite volume (and the ab considered here):

Π^µν_ab = q^µq^ν− q²g^µν Π⁽¹⁾_ab

Large N_c + VMD estimate: Π⁽¹⁾_π+π⁺ = 4F_π² M_V² − q²

Plots on next pages are for Π⁽¹⁾_ab0(q²) = Π⁽¹⁾_ab(q²) − Π⁽¹⁾_ab(0) At p⁴ the extra LEC cancels, at p⁶ there are new LEC contributions, but no new ones in the loop parts

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ChPT for HVP and a

Introduction HLbL

HVP Disconnected/

Two-point: Connected versus disconnected

-0.006 -0.005 -0.004 -0.003 -0.002 -0.001 0

-0.1 -0.08 -0.06 -0.04 -0.02 0

^ Ππ+

q² [GeV²] VMD p⁴+p⁶ p⁴ p⁶ R p⁶ L

• Connected

• p⁶ is large

• Due to the L^r_i loops

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Two-point: Connected versus disconnected

0 0.0005 0.001 0.0015 0.002 0.0025 0.003

-0.1 -0.08 -0.06 -0.04 -0.02 0

^ ΠUD

q² [GeV²] p⁴+p⁶

p⁴ p⁶ R p⁶ L

• Disconnected

• p⁶ is large

• Due to the L^r_i loops

• about

−¹₂ connected

• −₁₀¹ is from Π⁽¹⁾_ee =

5

9Π⁽¹⁾_π+π⁺+ ¹₉Π⁽¹⁾_ud

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ChPT for HVP and a

Introduction HLbL

HVP Disconnected/

Two-point: Connected versus disconnected

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

-0.1 -0.08 -0.06 -0.04 -0.02 0

^ ΠUD/^ Ππ+

q² [GeV²] p⁴+p⁶+VMD

p⁴+p⁶ p⁴ p⁶ R p⁶ L

• p⁴ and p⁶ pion part

exactly −¹₂

• not true for unsubtracted at p⁴(LEC)

• not true for pure LEC at

p⁶

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Two-point: Including strange

-0.0015 -0.001 -0.0005 0 0.0005

-0.1 -0.08 -0.06 -0.04 -0.02 0

^ ΠS

q² [GeV²] p⁴+p⁶

p⁴ p⁶ R p⁶ L VMDφ

• π

connected u,d

• ud

disconnected u,d

• ss

strange current

• us mixed strange-u,d

• strange part is very small:

q²= 0 subtraction (only kaon loops) p⁴ and p⁶ cancel largely

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ChPT for HVP and a

Introduction HLbL

HVP Disconnected/

Two-point: with strange, electromagnetic current

-0.0025 -0.002 -0.0015 -0.001 -0.0005 0 0.0005

-0.1 -0.08 -0.06 -0.04 -0.02 0 q²

5/9 π 1/9 ud 1/9 ss -2/9 us sum

• π

connected u,d

• ud

disconnected u,d

• ss

strange current

• us

mixed s–u,d

• new p⁶ LEC cancels

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Comparison with other results

Time moments, Taylor expansion Π(q)ˆ ² = Π1q²− Π₂q⁴+ · · ·

Lattice data from Miura presentation Lattice2106 and HPQCD

Phenomenological HLS fit (Benayoun et al)

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ChPT for HVP and a

Introduction HLbL

HVP Disconnected/

Comparing with other results

Reference Π_A Π₁ (GeV⁻²) Π₂ (GeV⁻⁴)

Π^VMD Πˆ_π⁺ 0.0967 −0.163

p⁴ Πˆ_π⁺ 0.0240 −0.091

p⁶ R Πˆ_π⁺ 0.0031 −0.014

p⁶ L Πˆ_π⁺ 0.0286 −0.067

sum Πˆ_π⁺ 0.152 −0.336

Miura Lattice2016 Πˆ_π⁺ 0.1657(16)(18) −0.297(10)(05) HPQCD Πˆ_π⁺ 0.1460(22) −0.2228(65)

p⁴ Πˆ_UD −0.0116 0.045

p⁶ R Πˆ_UD −0.0015 0.007

p⁶ L Πˆ_UD −0.0146 0.032

sum ΠˆUD −0.0278 0.085

Miura Lattice2016 Πˆ_UD −0.015(2)(1) 0.046(10)(04) Connectedand Disconnected:

size difference (-few%) understood from ChPT and VMD