Kaon form factor and decay constant from lattice QCD Aida X. - - PowerPoint PPT Presentation

Kaon form factor and decay constant from lattice QCD

HC2NP workshop Puerto de la Cruz, Tenerife, 26-30 Sep 2016 Aida X. El-Khadra (University of Illinois)

Current and Future Status of the First-Row CKM Unitarity, 16-18 May 2019

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Fermilab Lattice and MILC collaboration

• 2

Fermilab Lattice Collaboration:


AXK, E. Freeland, E. Gámiz, S. Gottlieb, A. Kronfeld, J. Laiho, P . Mackenzie, E. Neil, J. Simone, R. Van de Water


• Z. Gelzer, W. Jay

MILC:


• A. Bazavov, C. Bernard, C. DeTar, S. Gottlieb, U. Heller, J. Osborn, R.

Sugar, D. Toussaint,


• J. Komijani, R. Li, Y. Liu, A. Vaquero


Computing done at: 
 Mira, Theta (Argonne under DOE INCITE, ALCC)
 Blue Waters (NCSA, UIUC under NSF PRAC and BW Illinois)
 BigRed 2+ (Indiana U), TACC, NCAR (XSEDE); NICS, ORNL (Teragrid); FNAL, BNL (USQCD), …

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Outline

Introduction and Motivation Lattice QCD introduction Leptonic and semileptonic K decay amplitudes Set-up and analysis (Kℓ3) Systematic error analysis Results in comparison Implications for …

✦ |Vus| ✦ first row unitarity

Summary and Outlook

• 3

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Γ(K0 → π−µ+νµ), Γ(K+ → µ+νµ), ...

Introduction and Motivation

⬆

∆md(s)

…

Lattice QCD

Experiment vs. SM theory: (experiment) = (known) x (CKM factors) x (had. matrix element)

⬆

parameterize the MEs in terms of form factors, decay constants, bag parameters, ...

• 4

example: Kℓ3 decay

dΓ(B → ⇡`⌫) dq2 , dΓ(B → K`+`−) dq2 , . . . dΓ(B → D`⌫) d! , dΓ(B → D⌧⌫) d! , . . .

K0

¯ u

d

π−

¯ s

W

µ+

νµ

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Γ(K0 → π−µ+νµ), Γ(K+ → µ+νµ), ...

Introduction and Motivation

⬆

∆md(s)

…

Lattice QCD

Experiment vs. SM theory: (experiment) = (known) x (CKM factors) x (had. matrix element)

⬆

parameterize the MEs in terms of form factors, decay constants, bag parameters, ...

• 4

example: Kℓ3 decay

dΓ(B → ⇡`⌫) dq2 , dΓ(B → K`+`−) dq2 , . . . dΓ(B → D`⌫) d! , dΓ(B → D⌧⌫) d! , . . .

Two main purposes: combine experimental measurements with LQCD results to determine CKM parameters. confront experimental measurements of rare processes or lepton flavor (universality) violating observables with SM theory using LQCD inputs.

K0

¯ u

d

π−

¯ s

W

µ+

νµ

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

K+ → µ+νµ

Introduction

• 5

example:

use experiment + LQCD input for determination of CKM element ratios such as : reduced statistical and systematic errors.

fK+/fπ+

Γ

• K+ → `+⌫`()
• = (known) × (1 + `

EM) × |Vus|2 × f 2 K+

Vus

¯ s

u

W

µ+

νµ

K+

Needed to relate pure QCD decay constant to experiment. Currently estimated phenomenologically. [Cirigliano et al, arXiv:

1107.6001, RMP 2012]

Davide Giusti talk

➠

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

K0 → ⇡−`+⌫`

• 6

example:

K0

¯ u

d

π−

¯ s

W

µ+

νµ

ΓK`3 = (known) × ✓ phase space ◆ × (1 + δK`

EM + δK⇡ SU(2)) × |Vus|2 × |fK0⇡− +

(0)|2

Needed to relate pure QCD form factor to experiment. Mode dependent.

Introduction

Needed to include charged kaon decay in the experimental average. Both are currently estimated phenomenologically. [Cirigliano et al, arXiv:1107.6001, RMP

2012].

Vus

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

adjustable parameters lattice spacing: finite volume, time: quark masses (mf):
 tune using hadron masses


extrapolations/interpolations

• 7

Lattice QCD Introduction

L a x

discrete Euclidean space-time (spacing a)
 derivatives ➙ difference operators, etc…
 finite spatial volume (L) finite time extent (T)

LQCD = X

f

¯ ψf(D / + mf)ψf + 1 4trFµνF µν

a ➙ 0 L ➙ ∞, T > L MH,lat = MH,exp mf ➙ mf,phys mud ms mc mb

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

adjustable parameters lattice spacing: finite volume, time: quark masses (mf):
 tune using hadron masses


extrapolations/interpolations

• 7

Lattice QCD Introduction

L a x

discrete Euclidean space-time (spacing a)
 derivatives ➙ difference operators, etc…
 finite spatial volume (L) finite time extent (T)

LQCD = X

f

¯ ψf(D / + mf)ψf + 1 4trFµνF µν

a ➙ 0 L ➙ ∞, T > L MH,lat = MH,exp mf ➙ mf,phys mud ms mc mb Integrals are evaluated numerically using monte carlo methods.

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

use monte carlo methods (importance sampling) to evaluate the integral. Note: Integrating over the fermion fields leaves det(D +m) in the integrand. The correlation functions, O, are then written in terms of (D+m)-1 and gluon fields.

/ /

• 1. generate gluon field configurations according to det(D+m) e-S
• 2. calculate quark propagators, (D+mq)-1, for each valence quark flavor and source point
• 3. tie together quark propagators into hadronic correlation functions (usually 2 or 3-pt

functions)

• 4. statistical analysis to extract hadron masses, energies, hadronic matrix elements, ….

from correlation functions

• 5. systematic error analysis

steps of a lattice QCD calculation:

/ /

• 8

L a x

Lattice QCD Introduction

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

...of lattice spacing, chiral, heavy quark, and finite volume effects is based on EFT (Effective Field Theory) descriptions of QCD ➙ ab initio

• finite a: Symanzik EFT
• light quark masses: Chiral Perturbation Theory
• heavy quarks: HQET
• finite L: finite volume EFT

• need large enough L and small enough a and simulations

with several a, L, …

• 9

systematic error analysis

L a x

Lattice QCD Introduction

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

• 10
• typical momentum scale of quarks gluons inside hadrons: ~ΛQCD
• make a small to separate the scales: ΛQCD ≪ 1/a
• Symanzik EFT: , n ≥ 2


provides functional form for extrapolation (depends on the details of the lattice action) can be used to build improved lattice actions can be used to anticipate the size of discretization effects

• to control and reliably estimate 


the error, repeat …

discretization effects — continuum extrapolation

hOilat = hOicont + O(aΛ)n

a (fm) L

L a x

Lattice QCD Introduction

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

One stable hadron (meson) in initial/final state: If L is large enough, FV error keep To quantify residual error: include FV effects in χPT compare results at several Ls (with other parameters fixed) The story changes completely with two or more hadrons in initial/final state

• r if there are two or more intermediate state hadrons.

➠ “simple quantities”: 


no more than one stable hadron in initial/final state
 If QED is included, FV effects also become more complicated…

• 11

finite volume effects

mπ L & 4

∼ e−mπ L

L a x

Lattice QCD Introduction

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

When mlight ≶ 1/2 (mu + md)phys 


𝜓PT guides the interpolation/extrapolation to the physical point.

combined chiral-continuum interpolation/extrapolation include (light quark) discretization effects 
 (for example, staggered 𝜓PT)

• 12

L a x

Lattice QCD Introduction

combined chiral-continuum interpolation/extrapolation

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

• 13

Growing number of collaborations have generated sets of ensembles that include sea quarks with physical light-quark masses and use improved lattice actions: 
 PACS-CS, BMW, MILC, RBC/UKQCD, ETM,…

L a x

Lattice QCD Introduction

combined chiral-continuum interpolation/extrapolation

MILC nf = 2+1+1

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Lattice QCD calculations of simple quantities (with at most one stable meson in initial/final state) that quantitatively account for all systematic effects (discretization, finite volume, renormalization,…) , in some cases with

• sub percent precision.
• total errors that are commensurate (or smaller) than corresponding

experimental uncertainties. Scope of LQCD calculations is increasing due to continual development

• f new methods:
• baryons
• nonleptonic decays (K → 𝜌𝜌, …)
• resonances, scattering, long-distance effects,
• QED effects
• inclusive decay rates
• …
• 14

The State of the Art

L a x

Lattice QCD Introduction

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Set-up

• 15
• MILC HISQ ensembles with Nf = 2+1+1 sea

✦ Size of disk: # of configurations ✦ For each lattice spacing, multiple light sea quark masses

MILC nf = 2+1+1

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

• 16

example:

Kℓ2

Vus

experimental average:

(M. Moulson @ CKM 2016, arXiv:1704.04104)

includes 


(V. Cirigliano & H. Neufeld, arXiv:1102.0563, PLB 2011) 


0.13% uncertainty

¯ s

u

W

µ+

νµ

K+

K+ → µ+νµ

|Vus| |Vud| fK+ fπ+ = 0.27599 (37)

δEM = 0.069 (17)

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Kℓ2 set-up

• 17
• Use the same action (HISQ) for valence and sea quarks

✦ use Ward identity to calculate decay constant from pseudo

scalar density matrix element (avoids current renormalization calculation)

K tsource tsource+t q s J

(mu + ms)h0|P|Ki = fKm2

K

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

K0 → ⇡−`+⌫`

• 18

example:

K0

¯ u

d

π−

¯ s

W

µ+

νµ

Vus

experimental averages:

(M. Moulson @ CKM 2016, arXiv:1704.04104)

All modes: K0 only: K± only:

|Vus|f K0π−

+

(0) = 0.21654 (41)

|Vus|f K0π−

+

(0) = 0.21633 (44)exp(24)δK0`

EM

|Vus|f K0π−

+

(0) = 0.21710 (58)exp(27)δK±`

EM (41)δK±⇡0 SU(2)

0.19% uncertainty

Kℓ3

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Kℓ3 set-up

• 19
• Use the same action (HISQ) for valence and sea quarks

✦ Kaon at rest, pion recoil momentum with twisted boundary

conditions so that q2 = 0.

✦ use Ward-Takashi identity to calculate scalar form factor

together with kinematic constraint

K tsource+T tsource tsource+t q (𝜄) q s 𝜌 S

f0(q2) = ms mu m2

K m2 π

hπ|S|Ki

f+(0) = f0(0)

• A. Bazavov et al [FNAL/MILC, arXiv:1809.02827]

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Kℓ3 analysis

• 20
• results for f+(0) at each lattice spacing and sea quark mass

• Correct the above form factors before the chiral-continuum fit for

✦ finite volume effects ✦ effects due to poorly sampled topology

!"#"$%$&'"() !"#"$%$*"() !"#"$%$+"() !"#"$%,'"() !"#"$%,-"() $%+* $%+*- $%+5 $%+5- $%+6 $%+6- $%++ $%++-

!)7""8!)9"

9:!

$%$- $%, $%,- $%' $%'-

• A. Bazavov et al [FNAL/MILC, arXiv:1809.02827]

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Kℓ3 finite volume corrections

• 21
• use ChPT to calculate the leading-order FV corrections for

each twisting angle and light-quark mass [Bernard et al, arXiv:

1702.03416, 2017 JHEP]

• the resulting corrections are ≤ 0.1% on all ensembles

∆V f+(0) ⌘ f V

+ (0) f 1 + (0)

= (ms md)∆V hπ|S|Ki (mV

K)2 (mV π )2

(ms md)hπ|S|KiV (∆V m2

K ∆V m2 π)

[(mV

K)2 (mV π )2] 2

,

• A. Bazavov et al [FNAL/MILC, arXiv:1809.02827]

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Kℓ3 chiral-continuum fit

• 22
• Fit error: statistical + chiral int. + disc. + fit parameters [O(p4) LECs, …]
• Interpolation line and band: isospin limit
• data points: corrected for FV effects
• yellow star: add NNLO isospin breaking corrections (Bijnens & Ghorbani, arXiv:0711.0148)

χ2/dof = 0.29 Q=0.98

a ≈ 0.042 fm a ≈ 0.06 fm a ≈ 0.09 fm a ≈ 0.12 fm a ≈ 0.15 fm Fit value Chiral int. in the cont. and isospin limit (stat. error) Chiral interpolation in the continuum and isospin limit

f+

K0π+ (q2=0)

0.96 0.97 0.98 0.99

aml /(ams )physical

0.05 0.1 0.15 0.2 0.25

f K0π−

+

(0) = 0.9696 (15)

• A. Bazavov et al [FNAL/MILC, arXiv:1809.02827]

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Kℓ3 chiral-continuum fit function

• 23
• for chiral interpolation and continuum extrapolation
• use ChPT for light-quark mass dependence, discretization

effects, finite volume, and isospin breaking effects.

• in isospin limit



 fi: chiral corrections of O(pi)

• f2: NLO PQSChPT


f4: NNLO continuum ChPT + + N3LO and N4LO analytic terms f Kπ

+ (0) = 1 + f2 + f4 + f6 + . . .

O(αsa2, α2

sa2, a4)

f Kπ

+ (0) = 1 + f PQSChPT 2

(a) + f cont.

4

+ g1,a + r4

1(m2 π − m2 K)2 h

˜ C4 + g2,a + hmπ i

• A. Bazavov et al [FNAL/MILC, arXiv:1809.02827]

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Kℓ3 systematic error analysis

• 24

f+

K

0π

• (0)

0.96 0.97 0.97 base NNLO N3LO fK vs fπ at two loops NNLO analytic N3LO analytic N4LO analytic no analytic a2 αs

2a2(mπ 2-mK 2)

αs

2a2(mπ 2-mK 2) + αs a2

no a≈ 0.15fm continuum, no a≈ 0.15fm continuum + analytic a2 no a≈ 0.042fm no physical mass data

• nly physical mass data

no FV ms

sea vs ms val

Q

0.2 0.5 0.8 1.0

• A. Bazavov et al [FNAL/MILC, arXiv:1809.02827]

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Kℓ3 systematic error analysis

• 24

f+

K

0π

• (0)

0.96 0.97 0.97 base NNLO N3LO fK vs fπ at two loops NNLO analytic N3LO analytic N4LO analytic no analytic a2 αs

2a2(mπ 2-mK 2)

αs

2a2(mπ 2-mK 2) + αs a2

no a≈ 0.15fm continuum, no a≈ 0.15fm continuum + analytic a2 no a≈ 0.042fm no physical mass data

• nly physical mass data

no FV ms

sea vs ms val

Q

0.2 0.5 0.8 1.0

chiral truncation

• A. Bazavov et al [FNAL/MILC, arXiv:1809.02827]

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

f+

K

0π

• (0)

0.96 0.97 0.97 base NNLO N3LO fK vs fπ at two loops NNLO analytic N3LO analytic N4LO analytic no analytic a2 αs

2a2(mπ 2-mK 2)

αs

2a2(mπ 2-mK 2) + αs a2

no a≈ 0.15fm continuum, no a≈ 0.15fm continuum + analytic a2 no a≈ 0.042fm no physical mass data

• nly physical mass data

no FV ms

sea vs ms val

Q

0.2 0.5 0.8 1.0

Kℓ3 systematic error analysis

• 25

discretization

• A. Bazavov et al [FNAL/MILC, arXiv:1809.02827]

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

f+

K

0π

• (0)

0.96 0.97 0.97 base NNLO N3LO fK vs fπ at two loops NNLO analytic N3LO analytic N4LO analytic no analytic a2 αs

2a2(mπ 2-mK 2)

αs

2a2(mπ 2-mK 2) + αs a2

no a≈ 0.15fm continuum, no a≈ 0.15fm continuum + analytic a2 no a≈ 0.042fm no physical mass data

• nly physical mass data

no FV ms

sea vs ms val

Q

0.2 0.5 0.8 1.0

Kℓ3 systematic error analysis

• 26

finite volume

• A. Bazavov et al [FNAL/MILC, arXiv:1809.02827]

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Kℓ3 systematic error analysis

• 27

cf 2014 PRL: f Kπ

+ (0) = 0.9704 (24)stat(22)sys = 0.9704 (32)

+

Source of uncertainty Error fK0π−

+

(0) (%) Statistical + discretization + chiral interpolation 0.154 Lr

7,8

0.079 Scale r1 0.080 mval

s

6= msea

s

0.013 Higher-order finite volume corrections 0.053 Higher-order isospin corrections 0.015 Isospin-breaking parameter R 0.002 Total Error 0.199

2014

0.24 0.08 0.03 0.2 0.016 0.33

f Kπ

+ (0) = 0.9696 (15)stat(12)sys = 0.9696 (19)

• A. Bazavov et al [FNAL/MILC, arXiv:1809.02827]

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Kℓ2 systematic error analysis

• 28

s mu/md ms/ml mc/ms fK+/fπ+ fK/fπ

1.20 0.17 0.12 0.13 0.10

+1.47 −0 +0.24 −0 +0 −0.47 +0 −0.14 +0 −0.12 +1.99 −1.45 +0.031 −0.024 +0.112 −0.115 +0.010 −0.007 +0.004 −0.003

0.040 0.061 0.001 0.012 0.012 0.081 0.059 0.002 0.021 0.016 0.010 0.004 0.051 0.023 0.024 0.283 0.000 0.000 0.001 0.000

K π

Error (%) Statistics Continuum extrapolation Electromagnetic corrections Topological-charge distribution Finite-volume corrections fπ,PDG ∆MK

fK+/fπ+ = 1.1950 (15)stat +4

−17

• sys (3)fπ,PDG(3)EM scheme

• A. Bazavov et al [FNAL/MILC, arXiv:1712.09262]

(lead by J. Komijani and Doug Toussaint)

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Kℓ3 form factor in comparison

• 29
• S. Aoki et al [FLAG review 2019, arXiv:1902.08191]

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Kℓ3 form factor in comparison

• 30

0.2%

• A. Bazavov et al [FNAL/MILC, arXiv:1809.02827]

(lead by E. Gámiz) small errors due to

✦ physical light quark

masses

✦ improved light-quark

actions

✦NPR or no

renormalization

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Kℓ3 form factor shape

• 31
• N. Carrasco et al [ETMC, arXiv:1602.04113, 2016 PRD]

0.02 0.04 0.06 0.08 0.1 0.12

q

2 (GeV 2)

0.22 0.23 0.24 0.25 0.26

Vusf

Vusf+(q

2)

Vusf0(q

2)

exp data fit

Comparison of lattice form factors to experimental data (from dispersive fit in Moulson [CKM 2014, arXiv:1411.5252]).

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Kaon decay constant results in comparison

• 32

0.16%

small errors due to

✦ physical light quark

masses

✦ improved light-quark

actions

✦NPR or no

renormalization

• S. Aoki et al [FLAG review 2019, arXiv:1902.08191]

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Kaon decay constant results in comparison

• 33
• S. Aoki et al [FLAG review 2019, arXiv:1902.08191]

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Implications for |Vus|

• 34

experimental averages:

(M. Moulson @ CKM 2016, arXiv:1704.04104)

Kℓ3, all modes: Kℓ3, K0 only: Kℓ2:

|Vus|f K0π−

+

(0) = 0.21654 (41)

|Vus|f K0π−

+

(0) = 0.21633 (44)exp(24)δK0`

EM

Theory error commensurate with experiment

|Vus| = 0.22333 (42)exp(44)f+(0)

|Vus|K0π = 0.22309 (44)exp(25)δK`

EM(44)f+(0)

|Vus| |Vud| fK+ fπ+ = 0.27599 (37)

|Vus|/|Vud| = 0.2310 (2)exp(4)fK/fπ(2)EM

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Implications for |Vus|

• 35

(1 − |Vud|2)1/2

• A. Bazavov et al [FNAL/MILC, arXiv:1809.02827]

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Implications for |Vus|

• 35

(1 − |Vud|2)1/2

Tensions with leptonic determinations:

• : 1.6𝜏
• : 2.2𝜏

Γexp

K`2 + fK±

Γexp

K`2 + fK±/fπ± + |Vud|

Tension with CKM unitarity: 2.6𝜏

• A. Bazavov et al [FNAL/MILC, arXiv:1809.02827]

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

First row CKM unitarity

• 36

Vub ∼ 0

∆u ≡ |Vud|2 + |Vus|2 + |Vub|2 − 1

Kℓ3 :
 Kℓ2, fK/f𝜌: Kℓ3,Kℓ2, f+(0), fK/f𝜌:

∆u = −0.00104 (27)Vus(41)Vud

∆u = −0.00151 (39)f+(0)(36)fK/fπ(36)exp(27)EM

Correlations between f+(0) and fK/f𝜌 not included |Vud| = 0.97420 (21) from nuclear 𝛾-decay

(Hardy & Towner @ CIPANP 2018, arXiv:1808.01146)

∆u = −0.00029 (46)

A. Bazavov et al [FNAL/MILC, arXiv:1809.02827]

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

First row CKM unitarity

• 36

Vub ∼ 0

∆u ≡ |Vud|2 + |Vus|2 + |Vub|2 − 1

Kℓ3 :
 Kℓ2, fK/f𝜌: Kℓ3,Kℓ2, f+(0), fK/f𝜌:

∆u = −0.00104 (27)Vus(41)Vud

∆u = −0.00151 (39)f+(0)(36)fK/fπ(36)exp(27)EM

2𝜏 tension

Correlations between f+(0) and fK/f𝜌 not included |Vud| = 0.97420 (21) from nuclear 𝛾-decay

(Hardy & Towner @ CIPANP 2018, arXiv:1808.01146)

∆u = −0.00029 (46)

A. Bazavov et al [FNAL/MILC, arXiv:1809.02827]

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

• 37

Implications for |Vus|

∆u = −0.00209 (27)Vus(29)Vud

Kℓ3 + |Vud|, RC from Seng et al (2018):

• A. Bazavov et al [FNAL/MILC, arXiv:1809.02827]

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

• 37

Implications for |Vus|

∆u = −0.00209 (27)Vus(29)Vud

Kℓ3 + |Vud|, RC from Seng et al (2018): Tension with CKM unitarity: 5.3𝜏

• A. Bazavov et al [FNAL/MILC, arXiv:1809.02827]

• 38

Amala Willenbrock

Conclusions and Outlook

• 39

Amala Willenbrock

Conclusions and Outlook

First LQCD calculation of with 0.20% total error — most precise result to date. Similarly, LQCD calculations of with 0.19% uncertainty. The determinations of from Kℓ3 are in tension with determinations from leptonic decays and with first row CKM unitarity and determined from 𝛾-decay at the 2-2.6𝜏 level. We plan to calculate the ratio of and with correlations. This will sharpen our calculation of and yield more precise unitarity tests from only kaon decay. Further improvements should include QED corrections based on lattice QCD+QED calculations. ➠ Davide Giusti talk


f Kπ

+ (0)

fK±/fπ±

|Vud|

|Vus|

f Kπ

+ (0)

fK+/fπ+

f Kπ

+ (0)

• 40

Farah Willenbrock

Thank you!

• 41

Appendix

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

• 42

≈ a(fm) ml/msea

s

mP

π L

Nconf × Nsrc amsea

s

amval

s

0.15 0.035 3.2 1000 × 4 0.0647 0.06905 0.12 0.2 4.5 1053 × 8 0.0509 0.0535 0.1 3.2 1020 × 8 0.0507 0.053 † 0.1 4.3 993 × 4 0.0507 0.053 0.1 5.4 1029 × 8 0.0507 0.053 * 0.035 3.9 945 × 8 0.0507 0.0531 0.09 0.2 4.5 773 × 4 0.037 0.038 0.1 4.7 853 × 4 0.0363 0.038 0.035 3.7 950 × 8 0.0363 0.0363 * 0.06 0.2 4.5 1000 × 8 0.024 0.024 * 0.035 3.7 692 × 6 0.022 0.022 † 0.042 0.2 4.3 432 × 12 0.0158 0.0158 †

†: New ensembles since our 2014 paper. *: Ensembles with increased statistics since 2014.

Simulation details

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

• 43
• Combined two- and three-point function fits

✦ 3+3 states, tmin ~ 0.6-0.7 fm with t ∈ [tmin, T - tmin ] ✦ For more details see 2014 paper (A. Bazavov et al, arXiv:1312.1224,

2014 PRL)

CP

2pt(~

pP; t) =

Nexp

X

m=0

(−1)m(t+1)(ZP

m)2

e−Em

P t + e−Em P (Lt−t)

CK→π

3pt

(~ pπ, ~ pK; t, T) =

N3pt

exp

X

m,n=0

(−1)m(t+1)(−1)n(T−t+1)Amn(q2)Zπ

mZK n

×

• e−Em

π t + e−Em π (Lt−t)

e−En

K(T−t) + e−En K(Lt−T+t)

Kℓ3 analysis

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

• 44
• Combined two- and three-point function fits

Kℓ3 analysis

5 10 15 20

t C3pt, normalized

Fitted correlator (fit range) Fitted correlator Correlator data T = 32 T = 27 T = 23

2 /dof = 0.75 Q = 0.95

χ

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

• 45
• fit stability

Kℓ3 analysis

tmin=8

f+(0)

0.964 0.966 0.968 0.970 0.972 0.974 0.976

Nexp

1 2 3 4 5 6 7

Nexp=3+3

f+(0)

0.964 0.966 0.968 0.970 0.972 0.974 0.976

tmin

3 4 5 6 7 8 9 10 11 12 13 14

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Chiral-continuum fit function

• 46

f Kπ

+

(0) = 1 + f PQSχPT

2

(a) + f cont

4

+ g1,a + r4

1(m2 π − m2 K)2 h

˜ C4 + g2,a + hmπ i * f PQSχPT

2

(a): one-loop (NLO) partially quenched SChPT Bernard, Bijnens, E.G., 1311.7511 * f cont

4

: Two-loop (NNLO) continuum ChPT Bijnens & Talavera, 0303103 * ˜ C4 ∝ ⇣ C12 + C34 − L2

5

⌘ (µ) * g1,a and g2,a account for higher order discretization effects:

g1,a = K1 s r2

1a2 ¯

∆ ✓

a r1

◆2 + K3 ✓

a r1

◆4 , g2,a = K2 s r2

1a2 ¯

∆ ✓

a r1

◆2 + K0

2 r2 1a2 ¯

∆ with r2

1a2 ¯

∆ used as a proxy of α2

sa2

* hmπ includes analytical terms that parametrize higher order chiral effects

hmπ = ˜ C6 m2

π + ˜

C8 m4

π

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Chiral-continuum fit function

• 47

Fit parameters: δ0

A,V , K1,2,3, K0 2, ˜

C4,6,8, L1,2,3,5,6, 2L6 − L4 Fixed parameters: fπ, taste splittings, r1/a, L7,8.

• 3. Take the continuum limit and interpolate to the QCD meson masses including

NNLO isospin corrections Gasser & Leutwyler, NPB250, 517 (1985), Bijnens & Ghorbani, 0711.0148 f K0π

+

(0) = 1 + f cont. isospin-break.

2

+ f cont. isospin-break.

4

+ (m2

π+ − m2 K0)

h ˜ C4 + hmπ i with hmπ = ˜

C6 m2

π + ˜

C8 m4

π

* Isospin breaking contributions: depend on R ≡

ms ˆ m mumd = 34.7(5)stat(+1.0 0.6)syst

with ms/ ˆ m and mu/md from FNAL/MILC 1712.09262 without correlations

• A. El-Khadra

FirstRow workshop 2019, 16-18 May 2019

Poorly sampled topology effects

• 48
• affects only the 0.042 fm ensemble at mℓ = 0.2ms
• following Bernard and Toussaint (arXiv:1707.05430, 2017 PRD) the

correction can be obtained by calculating in ChPT:

• The correction is < 1/2 stat error

a ≈ 0.042 fm a ≈ 0.06 fm a ≈ 0.09 fm a ≈ 0.12 fm a ≈ 0.15 fm

f+

K0π+ (q2=0)

0,96 0,965 0,97 0,975 0,98 0,985 0,99 0,995

aml /(ams )physical

0,05 0,1 0,15 0,2 0,25

f Kπ

+ (0)corrected = f Kπ + (0)sampled

1 2χTV (f Kπ

+ (0))00

✓ 1 hQ2isample χTV ◆

(f Kπ

+ (0))00 ≡ d2f

dθ2

• θ=0

f Kπ

+ (0)00 = 1

4 (ml ms)2 (ml + 2ms)2