RIKEN BNL Research Center 2018-06-18, Second Plenary Workshop of - - PowerPoint PPT Presentation

Interplay between Lattice and { Model and/or Dispersive Representation } for g-2 HLbL

Tom Blum, Norman Christ, Masashi Hayakawa, Taku Izubuchi, Luchang Jin, Chulwoo Jung, Chrisoph Lehner (RBC&UKQCD)

RIKEN BNL

Research Center 2018-06-18, “Second Plenary Workshop of the Muon g-2 Theory IniBaBve” Mainz, Germany

1

Introduction

2

[ HVP: Bernecker Meyer 2011]

Sweat spots of Lattice vs DR/Model

n Lattice, after take continuum/infinite volume limits with all

disconnected, short distance (high energy) : less noisy long distance (low energy) : very noisy

n DR / Model ( or experiments )

heavy particle / multiple hadron : less control light particle, pi0 pole or pion-loop : well controlled

• > Could cover sweat spots complementarily ?

n For HVP

, a good comparison/interplay is done in Eucliean coordinate space [ Christoph Lehner’s talk ]

3

First try

[ Luchang Jin’s talk ]

n LMD model in coordinate space n Fixed min {|x-y|,|x-z|,|y-z| } < R(min) n Plot as function of

max {|x-y|,|x-z|,|y-z| } = R(max)

n L = 9.6 fm, a=0.1fm, Nf=2+1 physical pion mass n Subtracted lepton part (to isolate the long-distant

part in this exercise)

n Connected only. Model is multiplied by

34/9 according to conn:disconn = 34:(-25) from charge factors

4

5

HLbL point source method

[L. Jin et al. 1510.07100]

• Anomalous magnetic moment, F2(q2) at q2 ! 0 limit

F cHLbL

2

(q2 = 0) m (s0,s)i 2 = P

x,y,z,xop

2V T ✏i,j,k (xop xref)j · i¯ us0(~ 0)FC

k (x, y, z, xop) us(~

0)

• Stochastic sampling of x and y point pairs. Sum over x and z.

FC

⌫ (x, y, z, xop)

= (ie)6G⇢,,(x, y, z)HC

⇢,,,⌫(x, y, z, xop),

xsrc xsnk α, ρ η, κ β, σ xop, ν z, κ x, ρ y, σ tsrc tsnk α, ρ η, κ β, σ z x y

6

Subtraction using current conservation

• From current conservation,

∂ρVρ(x) = 0, and mass gap, hxVρ(x)O(0)i ⇠ |x|n exp(mπ|x|) X

x

HC

ρ,σ,κ,ν(x, y, z, xop) =

X

x

hVρ(x)Vσ(y)Vκ(z)Vν(xop)i = 0 X

z

HC

ρ,σ,κ,ν(x, y, z, xop) = 0

at V ! 1 and a ! 0 limit (we use local currents).

• We could further change QED weight

G(2)

ρ,σ,κ(x, y, z)

= G(1)

ρ,σ,κ(x, y, z) G(1) ρ,σ,κ(y, y, z) G(1) ρ,σ,κ(x, y, y) + G(1) ρ,σ,κ(y, y, y)

without changing sum P

x,y,z Gρ,σ,κ(x, y, z)HC ρ,σ,κ,ν(x, y, z, xop).

• Subtraction changes discretization error and finite volume error.
• Similar subtraction is used for HVP case in TMR kernel, which makes FV error smaller.
• Also now G(2)

σ,κ,ρ(z, z, x) = G(2) σ,κ,ρ(y, z, z) = 0, so short distance O(a2) is suppressed.

• The 4 dimensional integral is calculated numerically with the CUBA library cubature

rules. (x, y, z) is represented by 5 parameters, compute on N 5 grid points and

• interpolates. (|x y| < 11 fm).

cHLbL

Integrand : Lattice vs LVD (preliminary)

−2 −1 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 integrand F2(0) (10−10) Rmax (fm) Rmin = 1.0 fm Pion TFF sub Pion TFF no-sub Lattice 48D sub Lattice 48D no-sub

7

Integrand (preliminary)

• 0.2

0.2 0.4 0.6 0.8 1 1.2 1.4 1 2 3 4 5 6 7 8 integrand F2(0) (1e-10) R(max) (fm) R(min) = 1.0 fm Pion TFF sub Lattice 48D sub

8

model integral is extrapolated to con2nuum/infinite volume limits extrapola2ons to be scru2nized

Patch-up example Preliminary

• 10

10 20 30 40 50 1 2 3 4 5 6 7 8 F2(0) (1e-10) R(max) (fm) 48D R(min) = 0.5 fm Pion TFF sub Lattice 48D sub Combine sub

9

Switching point ( long: model, short: lattice )

Preliminary

• 10

10 20 30 40 50 1 2 3 4 5 6 7 8 F2(0) (1e-10) R(max) (fm) 48D R(min) = 1.0 fm Pion TFF sub Lattice 48D sub Combine sub

10

Preliminary

• 10

10 20 30 40 50 1 2 3 4 5 6 7 8 F2(0) (1e-10) R(max) (fm) 48D R(min) = 2.0 fm Pion TFF sub Lattice 48D sub Combine sub

11

Preliminary

• 10

10 20 30 40 50 1 2 3 4 5 6 7 8 F2(0) (1e-10) R(max) (fm) 48D R(min) = 5.0 fm Pion TFF sub Lattice 48D sub Combine sub

12

Is this safe ?

n At given distance, there are other than pi0

contribution in DR and models [ truncation ]

n Probably not large for appropriate choice n To be safer, we could try to consider

subtracting pi0 contribution from Lattice GH = GH(Lat; all) - GH(Lat; pi0) + GH(DR; pi0)

n How to compute GH(Lat; pi0) is non-trivial

13

Similar problem in tau HVP

n In case of Vus analysis of tau -> up-strange

inclusive hadronic decay

n We subtract K-pole contribution from lattice by

fitting HVP in the on-shell long-distance, and evaluate the rest: C(t) = A exp(- mK t) + rest( t ) [ A, mK is from fit ] ( also tau-input for g-2 : [ Mattia Brunno’s talk ])

14

[ Hiroshi Ohki et al. arXiv:1803.07228 ]

15

Tau decay

τ −

ντ

¯ u s

W −

hadrons

¯ u

s

τ −

ντ

W −

• {

}

Im

V-A current (Hadronic) vacuum polarization function

• Π(Q2)
• Experiment side :τ → ν + had through V-A vertex. EW correction SEW

Rij = Γ(τ − → hadronsij ντ) Γ(τ − → e−¯ νeντ) = 12π|Vij|2SEW m2

τ

Z m2

τ

✓ 1 − s m2

τ

◆ ✓ 1 + 2 s m2

τ

◆ ImΠ(1)(s) + ImΠ(0)(s)

• |

{z }

≡ Im Π(s)

• Lattice side : The Spin=0 and 1, vacuum polarization, Vector(V) or Axial (A) current-

current two point Πµν

ij;V/A(q2) = i

Z d4xeiqx D 0|T Jµ

ij;V/A(x)J†µ ij;V/A(0)|0

E = (qµqν − q2gµν)Π(1)

ij;V/A(q2) + qµqνΠ(0) ij;V/A

16

1 2 3 4 s [GeV

2]

1e-05 0.0001 0.001 0.01 0.1 1

Belle K

• π

0, K 0π

• (Adematz)

Belle K

0π

• π

BaBar K

• π

+π

• ALEPH K
• 2π,Κ(3−5)π,Kη

pQCD, D=0 OPE (nf=3)

τ inclusive decay experiments

For K pole, we assume a delta function form

To compare with experiments, a conventional value of |Vus|=0.2253 is used

γKω(m2

K)

• btained from either experimental value of K→μ or τ→k decay width.

γK ∼ 2|Vus|2f 2

K

γK[τ → Kντ] =0.0012061(167)exp(13)IB [HFAG16] γK[Kµ2] =0.0012347( 29)exp(22)IB [PDG16]

˜ ρ(s) ≡ |Vus|2 ✓ 1 + 2 s m2

τ

◆ ImΠ1(s) + ImΠ0(s)

• s)˜

ρ(s)

8

[K. Maltman ]

Pi0 subtraction on Lattice [ N. Christ et al @ UConn ]

17

18

[ also Luchang Jin’s talk ]

Lattice implementation

n lattice pi0-gamma-gamma FF could be computed

separately, and if it’s accurately determined, we could replace for long-distance of the full HLbL

n Or compute pi0-pole contribution simultaneously with

the full HLbL on the same ensemble and subtract under the jack-knife

19

Discussion

n Interplay b/w Lattice and DR/model is a useful

“plan-B” for HVP . Could we apply to HLbL ?

n Lattice : disconnected, continuum/infinite V limit n Another interplay for HLbL possible ? n How about the box diagram in DR ? n Sum-rule for the full HLbL from Lattice to

constraint DR or model ? Int[ pole, cuts in DR ] = Int[ Euclidean Amp ]

n Use of GEVP in subtracting pi0 or other specific

contribution ? [ A. Meyer’s talk ]

20

21

The finite energy sum rule (FESR) w(s) is an arbitrary regular function such as polynomial in s.

• LHS : spectral function ρ(s) is related to the experimental τ inclusive decays
• RHS … Analytic calculation

with perturbative QCD (pQCD) and OPE

Finite Energy Sum Rule (FESR)

s0 ω(s)ρ(s)ds = − 1 2πi

• |s|=s0

ω(s)Π(s)ds,

Im(s) Re(s)

sth

τ experiment pQCD

s0

dRus;V/A ds =12π2|Vus|2SEW m2

τ

✓ 1 − s m2

τ

◆2 ✓ 1 + 2 s m2

τ

◆ ImΠ1(s) + ImΠ0(s)

• (s0 : finite energy)

[Shifman, Vainshtein, and Zakharov ’79]

˜ ρ(s) ≡ |Vus|2 ✓ 1 + 2 s m2

τ

◆ ImΠ1(s) + ImΠ0(s)

• 4

22

Our new method : Combining FESR and Lattice

• If we have a reliable estimate for Π(s) in Euclidean (space-like) points, s = −Q2

k < 0,

we could extend the FESR with weight function w(s) to have poles there, Z ∞

sth

w(s)ImΠ(s) = π

Np

X

k

Resk[w(s)Π(s)]s=−Q2

k

Π(s) = ✓ 1 + 2 s m2

τ

◆ ImΠ(1)(s) + ImΠ(0)(s) ∝ s (|s| → ∞)

• For Np ≥ 3, the |s| → ∞ circle integral vanishes.

Re(s) Im(s) pQCD OPE spectral data

XXX Lattice HVPs

(generalized dispersion relation )

If we have a reliable estimate for Π(s) in Euclidean (space-like) points, we could extend the FESR with weight function w(s) to have N poles there,

Our strategy

Im(s) Re(s)

sth

τ experiment

pQCD

Im(s) Re(s)

τ experiment & pQCD

s0

Lattice HVP

Collaborators / Machines

Part of related calculation are done by resources from USQCD (DOE), XSEDE, ANL BG/Q Mira (DOE, ALCC), Edinburgh BG/Q, BNL BG/Q, RIKEN BG/Q and Cluster (RICC, HOKUSAI) Support from RIKEN, JSPS, US DOE, and BNL 23

Tom Blum (Connecticut) Peter Boyle (Edinburgh) Norman Christ (Columbia) Vera Guelpers (Southampton) Masashi Hayakawa (Nagoya) James Harrison (Southampton) Taku Izubuchi (BNL/RBRC) Christoph Lehner (BNL) Kim Maltman (York) Chulwoo Jung (BNL) Andreas J¨ uttner (Southampton) Luchang Jin (BNL) Antonin Portelli (Edinburgh)

Peter Boyle (Edinburgh) Renwick James Hudspith (York) Taku Izubuchi (BNL/RBRC) Andreas Ju ̈Jner(Southampton) Christoph Lehner (BNL) Randy Lewis (Southampton) Kim Maltman (York) Hiroshi Ohki (RBRC/Nara Women) Antonin Portelli (Edinburgh) MaJhew Spraggs (Edinburgh) Taku Izubuchi (BNL/RBRC) Yoshinobu Kuramashi (Tsukuba/ AICS) Christoph Lehner (BNL) Eigo Shintani (RIKEN AICS) tau decay g-2 DWF HVP & HLbL HVP Clover

• n (8.5 fm)3

24

The RBC & UKQCD collaborations

BNL and RBRC Mattia Bruno T

• momi Ishikawa

T aku Izubuchi Luchang Jin Chulwoo Jung Christoph Lehner Meifeng Lin Hiroshi Ohki Shigemi Ohta (KEK) Amarjit Soni Sergey Syritsyn Columbia University Ziyuan Bai Norman Christ Duo Guo Christopher Kelly Bob Mawhinney David Murphy Masaaki T

• mii

University of Connecticut T

• m Blum

Dan Hoying Cheng Tu Edinburgh University Peter Boyle Guido Cossu Luigi Del Debbio Richard Kenway Julia Kettle Ava Khamseh Brian Pendleton Antonin Portelli T

• bias T

sang Oliver Witzel Azusa Yamaguchi University of Liverpool Nicolas Garron University of Southampton Jonathan Flynn Vera Guelpers James Harrison Andreas Juettner Andrew Lawson Edwin Lizarazo Chris Sachrajda York University (Toronto) Renwick Hudspith KEK Julien Frison Peking University Xu Feng Jiqun T u Bigeng Wang Tianle Wang