Introduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Lattice QCD with physical quark masses
Christian Hoelbling
Budapest-Marseille-Wuppertal collaboration
Bergische Universität Wuppertal

NTFL workshop, YITP Kyoto
Feb. 22, 2012
(PLB 705:477,2011, JHEP 1108:148,2011; PLB 701:265,2011;
PRD 81:054507,2010; Science 322:1224,2008)

Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Purpose of lattice QCD
QCD fundamental objects: quarks and gluons
QCD observed objects: protons, neutrons (π, K, . . . )! Huge discrepancy: not even the same particles observed
as in the Lagrangean
Û Perturbation theory has no chance
Need to solve low energy QCD to:Compute hadronic and nuclear properties“people who love QCD”
Masses, decay widths, scattering lengths, thermodynamic
properties, . . .

Compute hadronic background“people who hate QCD”
Non-leptonic weak MEs, quark masses, g-2, . . .
Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Strategy outline
Goal:
Make ab initio QCD predictions
Step 1: validate (lattice) QCD
Compute light hadron spectrum
Step 2: Make QCD predictions
Pseudoscalar decay constant ratio (CKM)
Light quark masses
Neutral kaon mixing
Challenge:
Minimize and control all systematics
2+1 dynamical fermion flavors
Physical quark masses
Continuum extrapolation
Nonperturbative renormalization (where applicable)
Infinite volume
Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Summary

Lattice setup

Landscape Mπ vs. a
ETMC '09 (2)ETMC '10 (2+1+1)
MILC '10
QCDSF '10 (2)QCDSF-UKQCD '10
BMWc '10
PACS-CS '09
RBC/UKQCD '10
JLQCD/TWQCD '09
HSC '08
BGR '10
CLS '10 (2)Mπ[MeV]

600

400

200

0
0

0.05

a[fm]

0.1

Christian Hoelbling (Wuppertal)

0.15

Lattice QCD with physical quark masses

Introduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Summary

Lattice setup

Landscape L vs. Mπ
ETMC '09 (2)ETMC '10 (2+1+1)
MILC '10
QCDSF '10 (2)QCDSF-UKQCD '10
BMWc '10
PACS-CS '09
RBC/UKQCD '10
JLQCD/TWQCD '09
HSC '08
BGR '10 (2)CLS '10(2)

6
5
L[fm]

4
3

2

0.1%
0.3%
1%

1
100

200

300

400
Mπ[MeV]

500

Christian Hoelbling (Wuppertal)

600

700

Lattice QCD with physical quark masses

Introduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Summary

Lattice setup

Landscape MK vs. Mπ
ETMC '10 (2+1+1)
MILC '10
QCDSF-UKQCD '10
BMWc'10
PACS-CS '09
JLQCD/TWQCD '09
RBC-UKQCD '10
HSC '082

2 1/2(2MK-Mπ) [MeV]

800

600

400

200

0

200

400
Mπ[MeV]

Christian Hoelbling (Wuppertal)

600

Lattice QCD with physical quark masses

Introduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Summary

Technical details

Action details
Goal:
Optimize physics results per CPU time
Conceptually clean formulation
Method:Dynamical 2 + 1 flavor, Wilson fermions at physical Mπ
3-5 lattice spacings 0.053 fm < a < 0.125 fm
Tree level O(a2 ) improved gauge action (Lüscher, Weisz, 1985)
Tree level O(a) improved fermion action (Sheikholeslami, Wohlert, 1985)
Why not go beyond tree level?
Keeping it simple (parameter fine tuning)
No real improvement, UV mode suppression took care of thisThis is a crucial advantage of our approach

UV filtering (APE coll. 1985; Hasenfratz, Knechtli, 2001; Capitani, Durr, C.H., 2006)
Û Discretization effects of O(αs a, a2 )3 We include both O(αs a) and O(a2 ) into systematic error
Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

Introduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Technical details

Algorithm stability
25
20

Gauge force
RHMC force
Pf0 force
Pf1 force

Pf2 force
Pf3 force
Pf4 force

15
10
5
0

2k

4k

Christian Hoelbling (Wuppertal)

6k

8k

10k

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Technical details

No exceptional configs
Inverse iteration count (1000/Ncg)0β=3.31, Mπ≈135 MeV

β=3.61, Mπ≈120 MeV

β=3.5, Mπ≈130 MeV

β=3.7, Mπ≈180 MeV
β=3.8, Mπ≈220 MeV

0.04

0.08

0.12

Christian Hoelbling (Wuppertal)

0

0.04

0.08

0.12

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Summary

Technical details

Topological sector sampling0

Topological charge β=3.8, mud=-0.02, ms=0

1000

2000

4000

5000

10 HYP
30 HYP

4

worst case

3000

4

2

2

0

0

-2

-2

-4
1
-1

-4

ΔEMD
0

1000

Christian Hoelbling (Wuppertal)

2000

3000

4000

5000

Lattice QCD with physical quark masses

Introduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Summary

Technical details

Autocorrelation time (finest lattice, small mass)
normalized autocorrelation for |qren| at β=3.8, mud=−0.02, ms=0
1

ρ

0.5

0

τint = 27.3(7.4)

−0.5

0

int

2004-7)

100

τ

150

200

ren

τ

(MATLAB code from Wolff,

50with statistical errors for |q| at β=3.8, m =−0.02, m =0
ud

s

50

τint

40
30
20
10
0

0

Christian Hoelbling (Wuppertal)

50

100

150

fit window upper bound τmax

200

Lattice QCD with physical quark masses

Introduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Summary

Technical details

Locality properties
smearing

locality in position space:|D(x, y )| < const e−λ|x−y | with λ = O(a−1 ) for all couplings.
Our case: D(x, y ) = 0 as soon as |x −y | > 1(despite smearing)
locality of gauge field coupling:|δD(x, y )/δA(z)| < const e−λ|(x+y )/2−z| with λ = O(a−1 ) for
all couplings.
Our case: δD(x, x)/δA(z) < const e−λ|x−z| with λ ' 2.2a−1
for 2 ≤ |x −z| ≤ 6
Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

Introduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Technical details

Gauge field coupling locality
6-stout case:010

a~0.125 fm
a~0.085 fm
a~0.065 fm-1||¶D(x,y)/¶Um(x+z)||

10

-2

10

-3

10

-4

10

-5

10

-6

10

0

1

2

3

4

5

6

7

|z|/a

Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Technical detailsaM

Effective masses and correlated fits
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

N
K

4

8

12

t/a
Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Summary

Technical details

Chiral interpolation
Simultaneous fit to NLO SU(2) χPT (Gasser, Leutwyler, 1984)
Consistent for Mπ . 400 MeV
2.8

0.055

aFπ

0.05

2.6

2

aMπ/mud

PCAC

2.7

0.045

2.5

2.4

0.005

PCAC

0.01

amud

0.04

0.005

PCAC

amud

Û We use 2 safe chiral interpolation ranges:
Mπ < 340, 380 MeV
Û We use SU(2) χPT and Taylor interpolation forms
Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

0.01

Introduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Motivation

Light hadron spectrum
Goal:
Firmly establish (or invalidate?) QCD as the theory ofstrong interaction in the low energy region

Method:Post-diction of light hadron spectrum
Octet baryons
Decuplet baryons
Vector mesons

Challenge:
Minimize and control all systematics
2+1 dynamical fermion flavors
Physical quark masses
Continuum
Infinite volume (treatment of resonant states)
Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Motivation

Scale setting
Goal:
Unambiguous, precise scale setting
Method:We set the scale via a baryon mass
Desirable properties:
experimentally well known
small lattice error (Octet better than Decuplet)
independent of light quark mass → large strange content

Best candidates:
Ξ: largest strange content of the octetΩ: member of the decuplet, but no light quarks

Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Summary

Motivation

Quark mass dependence
Goal:
Extra-/Interpolate MX (baryon/vector meson mass) tophysical point (Mπ , MK )Method:Fundamental parameters: g, mud , ms
Experimentally inaccessible (confinement!)
Must be set via 3 experimentally accessible quantities

Use MΞ or MΩ and Mπ , MK to set parameters
Variables to parametrize Mπ2 and MK2 dependence of MX :Use bare masses aMy , y ∈ {X , π, K } and a (bootstrapped)My
Use dimensionless ratios ry := MΞ/Ω
(cancellations)

We use both procedures Ûsystematic error
Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

Introduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Summary

Motivation

Quark mass dependence (ctd.)
Method (ctd.):(0)

Parametrization: MX = Mx + αMπ2 + βMK2 + higher orders
Leading order sufficinet for MK2 dependence
We include higher order term in Mπ2
Next order χPT (around Mπ2 = 0): ∝ Mπ3
Taylor expansion (around Mπ2 6= 0): ∝ Mπ4

Both procedures fine Ûsystematic error
No sensitivity to any order beyond these

Vector mesons: higher orders not significant
Baryons: higher orders significant
Restrict fit range to further estimate systematics:
full range, Mπ < 550/450MeV

We use all 3 ranges Ûsystematic error
Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

Introduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Motivation

M [GeV]

Chiral fit
2

O

1.5

N

1
a~0.125 fm
a~0.085 fm~0.065 fm
a~0.5
physical Mp
0

0.1

0.2
0.3
2
2
Mp [GeV ]Christian Hoelbling (Wuppertal)

0.4

0.5

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Motivation

Chiral fit using ratios

O

1.4
1.2

N

M/MX

1
0.8
0.6

a~0.125 fm
a~0.085 fm
a~~0.065 fm

0.4
0.2
0

physical Mp

0.05

Christian Hoelbling (Wuppertal)

0.1
2(Mp/MX)

0.15

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

ContinuumContinuum extrapolation

Goal:
Eliminate discretization effects
Method:Formally in our action: O(αs a) and O(a2 )Discretization effects are tiny
Not possible to distinguish between O(a) and O(a2 )Ûinclude both in systematic error

Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Infinite volume

Finite volume effects from virtual pions
Goal:
Eliminate virtual pion finite V effects
Hadrons see mirror charges
Exponential in lightest particle (pion) mass

Method:Best practice: use large V
Rule of thumb: Mπ L & 4
1/2
X
Leading effects MX (L)−M= cMπ L−3/2 eMπ L
MX
0.8

Pion

aMp

-Mp L

-M L

L
fit
c1+ c2 eColangelo et. al. 2005

0.22

Nucleon

MpL=4

-3/2

fit
c1+ c2 e p L
Colangelo et. al. 2005

-3/2

aMN

MpL=4

0.225

(Colangelo et. al., 2005)

0.75

0.215

0.7
0.21
12

16

20

24
L/a

28

32

36

Christian Hoelbling (Wuppertal)

12

16

20

24
L/a

28

32

36

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Summary

Infinite volume

Finite volume effects in resonances
Goal:
Eliminate spectrum distortions from resonances mixing
with scattering states
Method:Stay in region where resonance is ground state
Otherwise no sensitivity to resonance mass in ground state
Finite volume energy levels

Spectral density

Treatment as scattering
problem(Lüscher, 1985-1991)

E

L
Christian Hoelbling (Wuppertal)

Parameters: mass andcoupling (width)
Alternative approaches
suggested
Lattice QCD with physical quark masses

Introduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Summary

Systematics

Systematic uncertainties
Goal:
Accurately estimate total systematic error
Method:We account for all the above mentioned effects
When there are a number of sensible ways to proceed, wetake them: Complete analysis for each of
18 fit range combinations
ratio/nonratio fits (rX resp. MX )O(a) and O(a2 ) discretization terms
NLO χPT Mπ3 and Taylor Mπ4 chiral fit
3 χ fit ranges for baryons: Mπ < 650/550/450 MeV

resulting in 432 (144) predictions for each baryon (vector
meson) mass with each 2000 bootstrap samples for eachΞ and Ω scale setting
Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

Introduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Summary

Systematics

Systematic uncertainties II
Method (ctd.):
Weigh each of the 432 (144) central values by fit quality Q
Median of this distribution Ûfinal result
Central 68% Ûsystematic error

Statistical error from bootstrap of the medians
medianmedian0.25

Nucleon

0.3

Omega

0.2

0.2

0.15
0.1

0.1
0.05
900

920

940
MN [MeV]960

980

Christian Hoelbling (Wuppertal)

1640

1660 1680
MO [MeV]

1700

1720

Lattice QCD with physical quark masses

Introduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Result

The light hadron spectrum
2000

Budapest-Marseille-Wuppertal collaboration

M[MeV]

1500
L

1000

r

500

0

K*

K
p

Christian Hoelbling (Wuppertal)

S

X

D

S*

X*

O

N
experiment
width
input
QCD

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Introduction

Pseudoscalar decay constant ratio

Goal:
Check first row unitarity of CKM matrix

Method:Compute FK /Fπ
Perturbative relation to |Vus |2 /|Vud |2 with 0.4% accuracy

Challenge:
Minimize and control all systematics
2+1 dynamical fermion flavors
Physical quark masses
Continuum
Infinite volume

Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Introduction

Observable
With the axial vector current
Aµ (t) =XΨ̄d γµ γ5 Ψu (~x , t)~xone obtains
22
Mπ2 Fπbare −Mπ tt→∞ |hπ|A0 |0i|†
−Mπ t
hA0 (t)A0 (0)i −→
e
=
e
2Mπ
2Mπ
u

t

0

A0
d

Christian Hoelbling (Wuppertal)

A0

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Results

Chiral extrapolation
1.25
β=3.3
β=3.57β=3.7Taylor extrapolation

1.2

FK/Fπ

1.15
1.1

1.05
1

100

2

200

2

2

300
2
2
Mπ[MeV ]Christian Hoelbling (Wuppertal)

400

2

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Results

Errors0.12
Distribution of values
Confidence interval
Taylor
SU(2)
SU(3)

0.1

0.08

0.06

0.04

0.02

0
1.17

1.175

1.18

1.185

Christian Hoelbling (Wuppertal)

1.19

1.195

1.2

1.205

1.21

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Results

Comparison
Prediction from CKM universality (|Vus |/|Vud|)N f = 2+1+1

ETM '10
NPLQCD '06
HPQCD/UKQCD '07

N f = 2+1

MILC '10(MILC)

ALV '08
RBC/UKQCD '10
PACS-CS '09

N f = 2+1

PACS-CS '10
THIS WORKQCDSF-UKQCD '10
JLQCD/TWQCD '09

1.15

1.2

Christian Hoelbling (Wuppertal)

1.25

1.3

1.35

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Strategy outline
Goal:
Compute light quark masses ab initio
Relevance:
Fundamental SM parameters
Stability of matter depends on their values
Not obtainable perturbatively
Challenge:
Minimize and control all systematics
2+1 dynamical fermion flavors
Physical quark masses
Continuum extrapolation
Nonperturbative renormalization
Infinite volume
Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Summary

Renormalization

Renormalization strategy

Goal:
Full nonperturbative renormalization
Optional accurate conversion to perturbative scheme

Method:We use RI-MOM scheme (Martinelli et. al., 1993)
O(a) correction (Maillart, Niedermayer, 2008)

Compute mq at low scale µ  2π/a ∼ 11 − 24 GeVµ = 2.1GeVµ = 1.3GeV

Do continuum non-perturbative running to high scale µ0  ΛQCD
Further conversion in 4-loop PT

Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

Introduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Renormalization

Desired scale in RI-MOM scheme
1

0.9

0.8

0.7

0.6

ZS(2-loop)/ZS(1-loop)
ZS(3-loop)/ZS(2-loop)
ZS(4-loop)/ZS(3-loop)
ZS(4-loop/ana)/ZS(4-loop)

24µ[GeV]

Christian Hoelbling (Wuppertal)

6

8

10

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Summary

Renormalization

Nonperturbative running
0.9^ RI 2ZS (μ )

0.8
0.7

β=3.8
β=3.7
β=3.61
β=3.5
β=3.31

0.6
0.5
0

10

Christian Hoelbling (Wuppertal)

20
2
2
μ [GeV ]30

Lattice QCD with physical quark masses

40

Introduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Summary

Renormalization

Reaching the perturbative regime
β=3.8µ=4GeV2

ZS,nonpert(µ )/ZS,4-loop(µ )

1.2

2

1

RI

RI

1.1

0.9
0.8
10

Christian Hoelbling (Wuppertal)

20
2
2µ [GeV ]30

Lattice QCD with physical quark masses

40

Introduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Renormalization

Optional conversion to MS
1

0.9

0.8

0.7

0.6

ZS(2-loop)/ZS(1-loop)
ZS(3-loop)/ZS(2-loop)
ZS(4-loop)/ZS(3-loop)
ZS(4-loop/ana)/ZS(4-loop)

24µ[GeV]

Christian Hoelbling (Wuppertal)

6

8

10

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Ratio-difference method

Quark mass definitions
Lagrangian mass mbareh∂0 A0 Pi
mPCAC from hP(t)P(0)ibare )
mren = Z1S (mbare − mcrit
mren = ZZPA mPCAC
Better usebared = msbare − mudd ren = Z1S d

PCAC
r = msPCAC /mudr ren = r
and reconstructdmsren = Z1S rr−1

ren = 1 d
mud
ZS r −1

3 No additive mass renormalization and ambiguity in mcrit
3 Only ZS multiplicative renormalization (no pion poles)+ Works with O(a) improvement (we use this)Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Summary

Finite volume

Tiny finite volume effects
0.19

MπL=3

MπL=4

aMπ

0.18
0.17

FV effects tiny
Dedicated FV
runs

ignored in final analysis

0.16

Perfect
agreement withFV χPT (Colangelo et. al.

0.15

2005)

0.14

16

24
L/a

Christian Hoelbling (Wuppertal)

32

Lattice QCD with physical quark masses

Introduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

mud and ms

Strange quark mass

00.01
Christian Hoelbling (Wuppertal)

0.02
αa[fm]

a=0.12 fm

0

a=0.10 fm

a=0.06 fm

40

a=0.08 fm

80

RI

ms (4GeV) [MeV]

120

0.03

0.04

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

mud and ms

Light quark masses

3

0

0.01
Christian Hoelbling (Wuppertal)

0.02
αa[fm]

a=0.12 fm

0

a=0.10 fm

a=0.06 fm

1

a=0.08 fm

2

RI

mud(4GeV) [MeV]

4

0.03

0.04

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

mu and md

Individual mu and md

Goal:
Compute mu and md separately

Method:Need QED and isospin breaking effects in principle
Alternative: use dispersive input -Q from η → πππ 2
md −muQ 2 = 12 mmuds
mud
3 Transform precise ms /mud into (md − mu )/mud
We use the conservative Q = 22.3(8) (Leutwyler, 2009)

Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Systematic errors

Systematic error treatment
Goal:
Reliably estimate total systematic error

Method:288 full analyses (2000 bootstrap on each)
2 plateaux regions
2 continuum forms: O(αs a), O(a2 )3 chiral forms: 2 × SU(2), Taylor
2 chiral ranges: Mπ < 340, 380 MeV
3 renormalization matching procedures
2 NP continuum running forms
2 scale setting procedures

All analyses weighted by fit quality
Mean gives final result
Stdev gives systematic error

Statistical error from 2000 bootstrap samples
Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Systematic errors

Final result
ms
mud
mu
md

RI @ 4 GeV
96.4(1.1)(1.5)
3.503(48)(49)
2.17(04)(10)
4.84(07)(12)RGI
127.3(1.5)(1.9)
4.624(63)(64)
2.86(05)(13)
6.39(09)(15)

MS @ 2 GeV
95.5(1.1)(1.5)
3.469(47)(48)
2.15(03)(10)
4.79(07)(12)

Additional consistency checks:
3 Use mPCAC only, no ratio-difference method+compatible, slightly larger error
3 Unweighted final result and systematic error+negligible impact
3 Additional Continuum, chiral and FV terms+all compatible with 0
Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Summary

Systematic errors

Comparison
mud

4

5

6
PACS-CS 09
HPQCD 09
MILC 09A
MILC 09
PACS-CS 08
RBC/UKQCD 08
CP-PACS 07
MILC 07
HPQCD 05
MILC 04
RBC 07
ETM 07QCDSF/UKQCD 04-06
QCDSF/UKQCD 04-06
SPQcdR 05
JLQCD 02
CP-PACS 01-03

60

80

ms

100

120

140

Nf=2+1

3

PACS-CS 09
HPQCD 09
MILC 09A
MILC 09
PACS-CS 08
RBC/UKQCD 08
CP-PACS 07
MILC 07
HPQCD 05
MILC 04

Nf=2

2

RBC 07
ETM 07QCDSF/UKQCD 04-06
QCDSF/UKQCD 04-06
SPQcdR 05
ALPHA 05JLQCD 02
CP-PACS 01-03
FLAG estimate

FLAG estimate
Dominguez 09
Narison 06
Maltman 01

Dominguez 09
Chetyrkin 06
Jamin 06
Narison 06
Vainshtein 78

PDG 09PDG 09This work

This work
2

3

4
MeV

5

6

Christian Hoelbling (Wuppertal)

60

80

100
MeV

120

140

Lattice QCD with physical quark masses

Introduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Introduction

Standard model neutral K mixing
Goal:
Check SM CP violation in neutral K system

Method:Compute effective weak matrix element
Relate kaon CP violation to CKM phase

Challenge:
Minimize and control all systematics
2+1 dynamical fermion flavors
Physical quark masses
Mixing of unphysical operators
Continuum
Infinite volume

Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Summary

Introduction

Observable
Matrix element of an effective weak operator (e.g. hK † |O|K i):
L2
t± →±∞ |hK |J0 |0i|hK † |O|K ie−MK (t+ −t− )(2MK )2†

hJ0L (t+ )O(0)J0L (t− )i −→
L

γ0

t+

t-

0
O

ΔS=2

L

γ0

where
d

J0L = [s̄d]V −A = s̄γ0 (1 − γ5 )dO∆S=2 = [s̄d]V −A [s̄d]V −A
Norm from:
†

u,c,t

L
2
t→∞ |hK |J0 |0i|

Christian Hoelbling (Wuppertal)

2MK

d

s

s

d

W

W

s

hJ0L (t)J0L (0)i −→

s

u,c,t

d

e−MK tLattice QCD with physical quark masses

Introduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Summary

Data

Unphysical operator mixing
100

80contribution to BK[%]+ χSB induces mixing with 4
unphysical operators+ Mixing terms chirally enhanced
3 Small even below physical mπ

60

40

20

3 Good chirality of our action

0
Q1

0.002
a≈0.077 fm

a≈0.077 fm

Q2

Q3

Q4

a≈0.054 fm

0.001

Δsub140-0.001-0.002-0.0035

10 15 20 25 30 35 40 45
μ2[GeV2]5

10 15 20 25 30 35 40 45

Christian Hoelbling (Wuppertal)

μ2[GeV2]

10 20 30 40 50 60 70 80 90
μ2[GeV2]

Lattice QCD with physical quark masses

Q5

Introduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Data

Signal0.7

0.65

BKbare(o)

0.6

0.55

0.5

0.45
0

2

4

6

8 10 12 14 16 18 20 22 24 26 28 30 32
o/aChristian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Data

RunningK

K

RRIB (µ, 3.5 GeV)/RB RI 2-loop(µ, 3.5 GeV)

1.06

_sa-scaling
2
a -scaling

1.04
1.02
1
0.98
0.96
0.94

3

4

5

6

7

8

9

10

11

12

µ2[GeV2]

Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Results

Physical point
0.6

a 0.093 fm
a 0.077 fm
a 0.065 fm
a 0.054 fm
cont-limit

0.58

BKRI(3.5 GeV)

0.56
0.54
0.52
0.5
0.48
0.46
0

0.02

0.04

0.06

Christian Hoelbling (Wuppertal)

0.08
0.1
M 2[GeV2]

0.12

0.14

0.16

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Summary

Results

Continuum extrapolation
0.56

BKRI(3.5 GeV)

0.55
0.54
0.53
0.52
0.51
0.5
0.49
0

0.005

0.01

Christian Hoelbling (Wuppertal)

0.015
0.02
_sa[fm]

0.025

0.03

Lattice QCD with physical quark masses

0.035

Introduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Summary

Results

Errorsstatistical

0.51

0.52

0.53

0.54

systematic

0.55

0.51

0.52

0.53

0.54

BKRI (3.5 GeV)

Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

0.55

Introduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Results

Comparison
CKM Fitter (2010)
ETMC (2009, TM)
RBC-UKQCD (2010, DW)
Aubin et al. (2010, DW/MILC)SWME (2010, HYP-STAG/MILC)
BMWc (2011, 2HEX-CIW)

0.6

0.8

Christian Hoelbling (Wuppertal)

^

BK

1

1.2

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Budapest-Marseille-Wuppertal collaboration

L

1000

r

500

S*

Summary

NPLQCD '06

ALV '08
PACS-CS '09

N f = 2+1

PACS-CS '10
THIS WORK

width
input

6

6

MILC '10(MILC)

experiment

QCDSF-UKQCD '10
JLQCD/TWQCD '09

60

80

ms

1.2

1.3

1.25

1.35

CKM Fitter (2010)
100

120

ETMC (2009, TM)

140
PACS-CS 09
HPQCD 09
MILC 09A
MILC 09
PACS-CS 08
RBC/UKQCD 08
CP-PACS 07
MILC 07
HPQCD 05
MILC 04
RBC 07
ETM 07QCDSF/UKQCD 04-06
QCDSF/UKQCD 04-06
SPQcdR 05
ALPHA 05JLQCD 02
CP-PACS 01-03

RBC-UKQCD (2010, DW)
Aubin et al. (2010, DW/MILC)SWME (2010, HYP-STAG/MILC)FLAG estimate

Dominguez 09
Narison 06
Maltman 01

Dominguez 09
Chetyrkin 06
Jamin 06
Narison 06
Vainshtein 78

PDG 09PDG 09BMWc (2011, 2HEX-CIW)

This work

This work
5

HPQCD/UKQCD '07

N f = 2+1

1.15

5

ETM '10

RBC/UKQCD '10

FLAG estimate

4
MeV

N f = 2+1+1

QCD

RBC 07
ETM 07QCDSF/UKQCD 04-06
QCDSF/UKQCD 04-06
SPQcdR 05
JLQCD 02
CP-PACS 01-03

3

Kaon mixingNp

PACS-CS 09
HPQCD 09
MILC 09A
MILC 09
PACS-CS 08
RBC/UKQCD 08
CP-PACS 07
MILC 07
HPQCD 05
MILC 042

O

Nf=2+1

4

D

K

0
mud

K*

S

X

X*

Nf=2

M[MeV]

1500

3

Quark masses

Prediction from CKM universality (|Vus |/|Vud|)

2000

2

Decay constants

60

80

100
MeV

120

140

Christian Hoelbling (Wuppertal)

0.6

0.8

^

BK

1

Lattice QCD with physical quark masses

1.2

Introduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

THE END

Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Chiral cuts
30
29

ms/mud

28
27
26
25

120

200
240
minimal mπ [MeV]

Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Chiral cuts
4
3.9
3.8
3.7
mud [MeV]

3.6
3.5
3.4
3.3
3.2
3.1
3

120

200
240
minimal mπ [MeV]

Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Chiral cuts
100
99
98
ms [MeV]

97
96
95
94
93
92
91
90

120

200
240
minimal mπ [MeV]

Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

SummaryIntroduction

Light hadron spectrum

Decay constants

Quark masses

Kaon mixing

Error budgetmudms
ms /mudcen. val.
3.503
96.43
27.531

σstat
0.048
1.13
0.196

σsyst
0.049
1.47
0.083

Relative error contributions
plateau
0.330
0.207
0.513

scale set
0.034
0.005
0.200

fit form
0.030
0.031
0.023

Christian Hoelbling (Wuppertal)

mass cut
0.157
0.085
0.320

renorm.
0.080
0.085
—

cont.
0.926
0.970
0.771

Lattice QCD with physical quark masses

Summary