Lattice setup

Quark masses

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

NTFL workshop, YITP, Kyoto
Feb. 17, 2012
(PLB 705:477,2011, JHEP 1108:148,2011; PLB 701:265,2011)

Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

Kaon mixing

Lattice setup

Quark masses

Kaon mixing

LandscapeLandscape 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

200000.05

a[fm]

0.1

Christian Hoelbling (Wuppertal)

0.15

Lattice QCD with physical quark masses

Lattice setup

Quark masses

Kaon mixing

LandscapeLandscape 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)

65
L[fm]

43

2

0.1%
0.3%
1%

1100200300

400
Mπ[MeV]

500

Christian Hoelbling (Wuppertal)

600700Lattice QCD with physical quark masses

Lattice setup

Quark masses

Kaon mixing

LandscapeLandscape 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 '08

22 1/2(2MK-Mπ) [MeV]

800

600

400

200

0200

400
Mπ[MeV]

Christian Hoelbling (Wuppertal)

600

Lattice QCD with physical quark masses

Lattice setup

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

ren = 1 dmudZS 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

Lattice setup

Quark masses

Kaon mixing

Renormalization

Nonperturbative running0.9^ RI 2ZS (μ )

0.80.7β=3.8
β=3.7
β=3.61
β=3.5
β=3.310.6
0.5010Christian Hoelbling (Wuppertal)

20
22μ [GeV ]30

Lattice QCD with physical quark masses

40

Lattice setup

Quark masses

Kaon mixing

Finite volume

Tiny finite volume effects
0.19

MπL=3MπL=4

aMπ

0.18
0.17

FV effects tiny
Dedicated FVruns

ignored in final analysis

0.16

Perfectagreement withFV χPT (Colangelo et. al.

0.152005)

0.14

1624
L/a

Christian Hoelbling (Wuppertal)

32

Lattice QCD with physical quark masses

Lattice setup

Quark massesKaon mixing

mud and ms

Strange quark mass

00.01
Christian Hoelbling (Wuppertal)

0.02
αa[fm]

a=0.12 fm

0a=0.10 fm

a=0.06 fm

40

a=0.08 fm

80

RIms (4GeV) [MeV]

1200.03

0.04

Lattice QCD with physical quark masses

Lattice setup

Quark masses

Kaon mixing

mud and ms

Light quark masses

300.01
Christian Hoelbling (Wuppertal)

0.02
αa[fm]

a=0.12 fm

0a=0.10 fm

a=0.06 fm

1

a=0.08 fm

2RI

mud(4GeV) [MeV]

40.03

0.04

Lattice QCD with physical quark masses

Lattice setup

Quark masses

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

Kaon mixing

Lattice setup

Quark masses

Kaon mixing

Systematic errors

Final resultmsmudmumd

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

Lattice setup

Quark masses

Kaon mixing

Systematic errors

Comparison
mud456
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

6080

ms100

120140Nf=2+13

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=22

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
234
MeV

56

Christian Hoelbling (Wuppertal)

6080

100
MeV

120140Lattice QCD with physical quark masses

Lattice setup

Quark masses

Kaon mixing

DataWilson BK : Unphysical operator mixing
10080
contribution to BK[%]+ χSB induces mixing with 4unphysical operators+ Mixing terms chirally enhanced
3 Small even below physical mπ604020

3 Good chirality of our action

0Q10.002a≈0.077 fm

a≈0.077 fm

Q2Q3Q4a≈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Lattice setup

Quark masses

DataSignal
0.70.65

BKbare(o)0.60.550.50.4502468 10 12 14 16 18 20 22 24 26 28 30 32o/a

Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

Kaon mixing

Lattice setup

Quark masses

Kaon mixing

DataRunning

KK

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

1.06

_sa-scaling
2a -scaling

1.04
1.02
10.98
0.96
0.94

3456789101112µ2[GeV2]

Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

Lattice setup

Quark masses

Kaon mixing

Results

Physical point
0.6

a 0.093 fma 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
00.02

0.040.06

Christian Hoelbling (Wuppertal)

0.08
0.1
M 2[GeV2]

0.12

0.14

0.16

Lattice QCD with physical quark masses

Lattice setup

Quark masses

Kaon mixing

Results

Continuum extrapolation
0.56

BKRI(3.5 GeV)

0.55
0.54
0.53
0.52
0.51
0.5
0.49
00.0050.01

Christian Hoelbling (Wuppertal)

0.015
0.02
_sa[fm]

0.025

0.03

Lattice QCD with physical quark masses

0.035Lattice setup

Quark masses

Kaon mixing

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

Lattice setup

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)

^BK1

1.2

Lattice QCD with physical quark masses

Lattice setup

Quark masses

BACKUP

Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

Kaon mixing

Lattice setup

Quark masses

Kaon mixing

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

Lattice setup

Quark masses

Kaon mixing

Algorithm stability
2520

Gauge force
RHMC force
Pf0 force
Pf1 force

Pf2 force
Pf3 force
Pf4 force

1510502k4k

Christian Hoelbling (Wuppertal)

6k

8k

10k

Lattice QCD with physical quark masses

Lattice setup

Quark masses

Kaon mixing

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)

00.04

0.08

0.12

Lattice QCD with physical quark masses

Lattice setup

Quark masses

Kaon mixing

Topological sector sampling
0Topological charge β=3.8, mud=-0.02, ms=01000

2000

4000

5000

10 HYP30 HYP

4worst case

3000

42200-2-2-4
1-1-4

ΔEMD
01000

Christian Hoelbling (Wuppertal)

2000

3000

4000

5000

Lattice QCD with physical quark masses

Lattice setup

Quark masses

Kaon mixing

Autocorrelation time (finest lattice, small mass)
normalized autocorrelation for |qren| at β=3.8, mud=−0.02, ms=01ρ0.50τint = 27.3(7.4)−0.50int

2004-7)

100

τ

150

200

renτ

(MATLAB code from Wolff,

50with statistical errors for |q| at β=3.8, m =−0.02, m =0uds50τint403020
1000

Christian Hoelbling (Wuppertal)

50

100

150

fit window upper bound τmax

200

Lattice QCD with physical quark masses

Lattice setup

Quark masses

Kaon mixing

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

Lattice setup

Quark masses

Kaon mixing

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|/aChristian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

Lattice setup

Quark masses

Kaon mixingaM

Effective masses and correlated fits
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.10NK4812t/a
Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

Lattice setup

Quark masses

Individual mu and mdGoal:
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
mud3 Transform precise ms /mud into (md − mu )/mudWe use the conservative Q = 22.3(8) (Leutwyler, 2009)

Christian Hoelbling (Wuppertal)

Lattice QCD with physical quark masses

Kaon mixing