Lattice QCD with physically light up and down quarks · Blum 10 FLAG 10v1 N f = 2+1 mMS s (2 GeV) =...

Lattice QCD with physically light up and downquarks

Laurent Lellouch

CPT Marseille

Budapest-Marseille-Wuppertal collaboration (BMWc)

lavinet

Laurent Lellouch STONGnet 2011, 4-7 october 2011

Motivation

Verify that QCD is theory of strong interaction at low energies↔ verify the validity of the computational framework

→ light hadron spectrum (BMWc, Science 322 (2008))

→ hadron interactions→ . . .

Fix fundamental parameters and help search for new physics

→ mu, md , ms, . . . (BMWc, PLB 701 (2011); JHEP 1108 (2011))

→ 〈N|mq qq|N〉, q = u,d , s for dark matter (BMWc, arXiv:1109.4265 [hep-lat])

→ FK/Fπ ↔ GqGµ

[|Vud |2 + |Vus|2 + |Vub|2

]= 1 ? (BMWc, PRD 81 (2010))

→ BK ↔ consistency of CPV in K and B decays ? (BMWc, arXiv:1106.3230

[hep-lat])

→ . . .

Make predictions in nuclear physics?Full description of low energy particle physics→ include QED


Introduction

Tool

→ ab initio QCD calculations on the lattice

Challenge

Minimize and control all systematics

+ compute hugely expensive fermion determinant+ fight fast increasing cost of simulations as:

mud mphud ⇒ reach physical mass point in controlled way

a 0⇒ controlled continuum extrapolationL→∞⇒ controlled infinite volume extrapolation

+ nonperturbative renormalization⇒ eliminate all perturbative uncertainties

⇒ true nonperturbative QCD predictions


The calculation that I’ve been dreaming of doing

Nf = 2 + 1 simulations to include u, d and s sea quark effectsSimulations all the way down to Mπ <∼ 135 MeV to allow smallinterpolation to physical mass pointLarge L >∼ 5 fm to have sub-percent finite V errorsAt least three a <∼ 0.1 fm for controlled continuum limitReliable determination of the scale w/ a well measured physicalobservableUnitary, local gauge and fermion actionsFull nonperturbative renormalization and nonperturbativecontinuum running if necessaryComplete analysis of systematic uncertainties


Dream comes true in 2010Dürr et al (BMWc) PLB 701 (2011); JHEP 1108 (2011)

47 large scale Nf = 2 + 1 Wilson fermion simulations

Mπ >∼ 120 MeV 5a’s ≈ 0.054÷ 0.116 fm L→ 6 fm

00 2002

3002

4002

5002

6002

Mπ

2 [MeV

2]

0

0.0502

0.1002

0.1502

a2 [

fm2]

BMWc ’08BMWc ’10

1002

2002

3002

4002

5002

6002

Mπ

2 [MeV

2]

2

3

4

5

6

7

L[f

m]

0.1%

BMWc ’08BMWc ’10

0.3%

1%


Dream comes true in 2010

Dürr et al (BMWc) PLB 701 (2011); JHEP 1108 (2011)

47 large scale Nf = 2 + 1 Wilson fermion simulations

Mπ >∼ 120 MeV 5a’s ≈ 0.054÷ 0.116 fm L→ 6 fm

00 2002

3002

4002

Mπ

2 [MeV

2]

0

0.0502

0.1002

0.1502

a2 [

fm2]

MILC ’10JLQCD/TW. ’09

QCDSF/UK. ’10

PACS-CS ’09RBC/UK. ’10HSC ’08ETMC ’10BMWc ’08BMWc ’10

1002

2002

3002

4002

Mπ

2 [MeV

2]

2

3

4

5

6

7

L[f

m]

0.1%

MILC ’10JLQCD/TW. ’09

QCDSF/UK. ’10

PACS-CS ’09RBC/UK. ’10HSC ’08ETMC ’10BMWc ’08BMWc ’10

0.3%

1%

Still only ones there! (only Nf ≥ 2 + 1 simulations are shown)


Action and algorithm details

Dürr et al (BMWc), PRD 79 (2009)

Nf = 2 + 1 QCD: degenerate u & d w/ mass mud and s quark w/ mass ms ∼ mphyss

1) Conceptually clean discretization which balances improvement and CPU cost:

tree-level O(a2)-improved gauge action (Lüscher et al ’85)

tree-level O(a)-improved Wilson fermion (Sheikholeslami et al ’85) w/ 2 HEX smearing(Morningstar et al ’04, Hasenfratz et al ’01, Capitani et al ’06)

⇒ O(αsa, a2) instead of O(a)

2) Highly optimized algorithms (see also Urbach et al ’06):

HMC for u and d and RHMC for s

mass preconditioning (Hasenbusch ’01)

multiple timescale integration of MD (Sexton et al ’92)

higher-order (Omelyan) integrator for MD (Takaishi et al ’06)

mixed precision acceleration of inverters via iterative refinement

3) Highly optimized codes for Blue Gene


Algorithm stability

Spacetime max. of MD forces for β = 3.31at physical mud

0

5

10

15

20

25

2k 4k 6k 8k 10k

Gauge forceRHMC forcePf0 forcePf1 force

Pf2 forcePf3 forcePf4 force

No killer spikes

−→ good acceptance >∼ 90%

Ensembles w/ smallest mq per β; lightestpseudofermion

Inverse iteration count (1000/Ncg)

β=3.31, Mπ≈135 MeV

0 0.04 0.08 0.12

β=3.5, Mπ≈130 MeV

β=3.61, Mπ≈120 MeV

0 0.04 0.08 0.12

β=3.7, Mπ≈180 MeV

β=3.8, Mπ≈220 MeV

103/NCG distribution is approx.Gaussian

NCG remains clearly bounded fromabove


Topological charge sampling on finest lattice

Long autocorrelations in top. charge as a→ 0 have been observed (Schaefer et al (2010))

5000 trajectoryautocorrelation-check run

Absolute worst case:a ' 0.054 fm and Mπ ' 260 MeVon 483 × 64 lattice

Qnaive =a4

(4π)2

∑x

Tr[F HYPµν F HYP

µν ](x)

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

-4

-2

0

2

4

0 1000 2000 3000 4000 5000

10 HYP30 HYP

-4

-2

0

2

4

-1

1

0 1000 2000 3000 4000 5000

∆EMD

Q fluctuates and evolves: τint ∼ 30

Q falls into integer centered bins

Q distribution is reasonablysymmetric

⇒ no obvious ergodicity problem


Scaling study

Nf = 3 w/ 2 HEX action, 4 lattice spacings (a ' 0.06÷ 0.15fm), MπL > 4 fixed and

Mπ/Mρ =

√2(Mph

K )2 − (Mphπ )2/Mph

φ ∼ 0.67

i.e. mq ∼ m phs

1450

1500

1550

1600

1650

1700

1750

1800

1850

0 0.01 0.02 0.03 0.04 0.05 0.06

M[M

eV

]

αa[fm]

0.1

5 fm

0.1

0 fm

0.0

6 fm

M∆

MN

MN and M∆ are linear in αsaout to a ∼ 0.15 fm

⇒ very good scaling:discretization errors <∼ 2%out to a∼0.15 fm

Results perfectly consistentw/ analogous 6 stout analysisin BMWc PRD 79 (2009)


Finite volume studies

In large volumes FVE ∼ e−MπL

MπL >∼ 4 expected to give L→∞ masses within our statistical errors

16 24 32L/a

0.14

0.15

0.16

0.17

0.18

0.19

aMπ ignored in final analysis

MπL=4MπL=3

2HEX, a ≈ 0.116 fm, Mπ ≈ 0.25, 0.30 GeV,

MπL = 2.4→ 5.6

12 16 20 24 28 32 36L/a

0.7

0.75

0.8

aM

N

c1+ c

2e

-Mp L

-3/2 fit

L

Colangelo et. al. 2005

MpL=4 volume dependence

6STOUT, a ≈ 0.125 fm, Mπ ≈ 0.33 GeV,

MπL = 3.5→ 7

Well described by (and Colangelo et al, 2005)

MX (L)−MX

MX= C

(Mπ

πFπ

)2 1(MπL)3/2 e−MπL

Very small, for the volumes that we consider


Light quark masses: motivation

Determine mu, md , ms ab initio

Fundamental parameters of naturePrecise values→ stability of matter, N-N scattering lengths,presence or absence of strong CP violation, etc.Information about BSM: theory of fermion masses must reproducethese valuesNonperturbative (NP) computation is requiredWould be needle in a haystack problem if not for χSB

⇒ interesting first “measurement” w/ physical point LQCD


Light quark masses circa Aug. 2010FLAG→ analysis of unquenched lattice determinations of light quark masses(arXiv:1011.4408v1)

2

2

3

3

4

4

5

5

6

6

MeV

CP-PACS 01-03JLQCD 02

SPQcdR 05QCDSF/UKQCD 04

QCDSF/UKQCD 06ETM 07RBC 07

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

FLAG 10v1 Nf = 2

Dominguez 09

mud

Nf=

2+

1N

f=2

Narison 06Maltman 01

PDG 09

MILC 09MILC 09AHPQCD 09PACS-CS 09Blum 10

JLQCD/TWQCD 08A

FLAG 10v1 Nf = 2+1

mMSud (2 GeV) =

3.4(4) MeV [12%] FLAG2.5÷ 5.0 MeV [30%] PDG

60

60

80

80

100

100

120

120

140

140

MeV

CP-PACS 01-03JLQCD 02

ALPHA 05SPQcdR 05

QCDSF/UKQCD 04


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

FLAG 10v1 Nf = 2

PDG 09

ms

Nf=

2+

1N

f=2

Dominguez 09Chetyrkin 06Jamin 06Narison 06Vainshtein 78


FLAG 10v1 Nf = 2+1

mMSs (2 GeV) =

95.(10) MeV [11%] FLAG70÷ 130 MeV [30%] PDG

Even extensive study by MILC still has:

MRMSπ ≥ 260 MeV ⇒ mMILC,eff

ud ≥ 3.7×mphysud

perturbative renormalization (albeit 2 loops)


Quark mass definitions

Standard

Lagrangian mass mbare

mren = 1ZS

(mbare −mcrit)

mPCAC from 〈∂0A0P〉〈P(t)P(0)〉

mren = ZAZP

mPCAC

Better use . . .

d = mbares −mbare

ud

d ren = 1ZS

dr = mPCAC

s /mPCACud

r ren = r

. . . and reconstruct

mrens = 1

ZS

rdr−1 mren

ud = 1ZS

dr−1

3 No additive mass renormalization3 Only ZS multiplicative renormalization w/ no pion poles+ Use O(a)-improved version


Renormalization strategy

Goal

Convert bare lattice masses to finite renormalized ones . . .. . . fully nonperturbatively . . .. . . with optional accurate conversion to other schemes

Method

Use RI/MOM scheme nonperturbative renormalization (NPR)(Martinelli et al ’95)

. . . with S(p)→ S(p) = S(p)− TrD[S(p)]/4 (Becirevic et al ’00)

Compute ZS(aµ,g0) for µ 2π/a ∼ 11÷ 24 GeV

3 µ = 1.3 GeV 3 µ = 2.1 GeVContinuum nonperturbative running to high scale µ′ ΛQCD

Further conversions in 4-loop PT21 additional Nf = 3 RI/MOM simulations at same 5 β’s


RI/MOM nonperturbative renormalization

(Martinelli et al ’95)Have

[q1Γq2](µ) = Z RIΓ (aµ,g0)[q1Γq2](a)

w/ Z RIΓ defined by ratio of quark Green’s fns in Landau gauge

=ZRIΓ (ap,g0)

ZRIq (ap,g0)

Γ

p p

p p( )

2

Γ

p p

p p( )

2

Cancel Z RIq (aµ,g0) by normalizing with LHS of conserved vector

current


Choice of target RI/MOM scale

2 4 6 8 10µ[GeV]

0.6

0.7

0.8

0.9

1

ZS(2-loop)/Z

S(1-loop)

ZS(3-loop)/Z

S(2-loop)

ZS(4-loop)/Z

S(3-loop)

ZS(4-loop/ana)/Z

S(4-loop)

⇒ σPT <∼ 1% for µ >∼ 4 GeV


Nonperturbative running to 4 GeV

Determine nonperturbative running in continuum limit from (see also

Constantinou et al ’10, Arthur et al ’10)

RRIS (µ,4 GeV) = lim

a→0

Z RIS (4 GeV,a)

Z RIS (µ,a)

0 10 20 30 40

µ2[GeV

2]

0.5

0.6

0.7

0.8

0.9

Z^SRI (

µ2)

β=3.8β=3.7β=3.61β=3.5β=3.31

Rescaled Z RIS (aµ, β) for β < 3.8 to∼ match Z RI

S (aµ, β = 3.8)

3 β up to µ = 4 GeV

Running very similar atall 5 β

⇒ flat and controlled a→ 0extrapolation


Running beyond 4 GeV

For µ > 4 GeV, 4-loop PT and NP running agrees on finest lattice

10 20 30 40

µ2[GeV

2]

0.8

0.9

1

1.1

1.2

ZRI

S,nonpert(

µ2)/ZRI

S,4-loop(µ

2)

µ=4GeV

β=3.8

⇒ get RGI masses w/ negligible PT error⇒ masses in other schemes w/ only errors proper to that scheme


Combined mass interp. and continuum extrap. (1)

Project onto mud axis: chiral interpolation to Mphπ

Illustration of chiral behavior• Fixed a'0.09 fm and Mπ∼130÷ 410 MeV

0.005 0.01

amud

PCAC

2.4

2.5

2.6

2.7

2.8

aM

π2/m

ud

PC

AC

131 MeV

258 MeV368 MeV

409 MeV

• Fit to NLO SU(2) χPT (Gasser et al ’84)

0.005 0.01

amudPCAC

0.04

0.045

0.05

0.055

aFπ

3 Consistent w/ NLO χPT for Mπ <∼ 410 MeV

⇒ 2 safe interpolation ranges: Mπ < 340, 380 MeV

⇒ SU(2) NLO χPT & Taylor interpolations to physical point


Combined mass interp. and continuum extrap. (2)

Project onto a axis: continuum extrapolation

Leading order is O(αsa)

Allow also domination of sub-leading O(a2)a=

0.0

6 f

m

a=

0.0

8 f

m

a=

0.1

0 f

m

a=

0.1

2 f

m

0

1

2

3

4

mud

RI (4

GeV

) [M

eV

]

0 0.01 0.02 0.03 0.04α

sa[fm]

a=

0.0

6 f

m

a=

0.0

8 f

m

a=

0.1

0 f

m

a=

0.1

2 f

m

0

40

80

120

msR

I (4G

eV

) [M

eV

]0 0.01 0.02 0.03 0.04

αsa[fm]

(continuum extrapolation examples – errors on points are statistical)

⇒ fully controlled continuum limitLaurent Lellouch STONGnet 2011, 4-7 october 2011

Individual mu and md

Calculation performed in isospin limit:

mu = md NO QED⇒ leave ab initio realm

Use dispersive Q from η → πππ

Q2 ≡m2

s −m2ud

m2d −m2

u

* Precise mud and ms/mud ⇒

mu/d = mud

1∓ 1

4Q2

[(ms

mud

)2

− 1

]

Use conservative Q = 22.3(8) (Leutwyler ’09)


Systematic error treatment

288 full analyses on 2000 boostrap samples

2 correlator time fit ranges3 NPR procedures2 continuum extrap. forms for NP running3 chiral interp. forms: 2× SU(2) χPT, Taylor2 chiral interp. ranges: Mπ < 340, 380 MeV2 chiral interp. ranges for scale setting channel MΩ:Mπ < 340, 480 MeV2 continuum forms

Analyses weighted by fit quality⇒ systematic error distribution

Mean→ final resultStd. dev. → systematic error

Statistical error from distribution of means over 2000 samples


Results

RI 4 GeV RGI MS 2 GeVms 96.4(1.1)(1.5) 127.3(1.5)(1.9) 95.5(1.1)(1.5)mud 3.503(48)(49) 4.624(63)(64) 3.469(47)(48)mu 2.17(4)(3)(10) 2.86(5)(4)(13) 2.15(4)(3)(9)md 4.84(7)(7)(10) 6.39(9)(9)(13) 4.79(7)(7)(9)

ms

mud= 27.53(20)(8)

mu

md= 0.449(6)(2)(29)

Additional consistency checks

3 Additional continuum, chiraland FV terms+ all compatible with 0

3 Unweighted final result andsystematic error+ negligible impact

3 Use mPCAC only+ compatible, slightly largererror

3 Full quenched check ofprocedure + cf. referencecomputation (Garden et al ’00)


Comparison

2

2

3

3

4

4

5

5

6

6

MeV

CP-PACS 01JLQCD 02

SPQcdR 05QCDSF/UKQCD 04


MILC 04, HPQCD/MILC/UKQCD 04HPQCD 05MILC 07CP-PACS/JLQCD 07RBC/UKQCD 08PACS-CS 08

FLAG 10v1 Nf = 2

Dominguez 09

mud

Nf=

2+

1N

f=2

Narison 06Maltman 01

PDG 10


JLQCD/TWQCD 08A

FLAG 10v1 Nf = 2+1

HPQCD 10RBC/UKQCD 10A

ETM 10B

MILC 10APACS-CS 10BMWc 10A

70

70

80

80

90

90

100

100

110

110

120

120

MeV

CP-PACS 01JLQCD 02

ALPHA 05SPQcdR 05

QCDSF/UKQCD 04


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

FLAG 10v1 Nf = 2

PDG 10

ms

Nf=

2+

1N

f=2

Dominguez 09Chetyrkin 06Jamin 06Narison 06Vainshtein 78


FLAG 10v1 Nf = 2+1

HPQCD 10RBC/UKQCD 10A

ETM 10B

PACS-CS 10BMWc 10A

mud and ms are now known to 2%. . . mu to 5% and md to 3% w/ help of phenomenology


Quark flavor mixing constraints in the SM and beyond

Test SM paradigm of quark flavor mixing and CP violation and look for new physics

Unitary CKM matrix

V

u

b W

ub

d s b

→ V =

u

c

t

1− λ

2 λ Aλ3(ρ− iη)

−λ 1− λ2 Aλ2

Aλ3(1− ρ− iη) −Aλ2 1

+O(λ4)

Test CKM unitarity/quark-lepton universality and constrain NP using, e.g.

1st row unitarity:G2

q

G2µ

|Vud |2[1 + |Vus/Vud |2 + |Vub/Vud |2

]= 1 + O

(M2

W

Λ2NP

)

Unitarity triangle:G2

q

G2µ

(Vcd V ∗cb)

[1 +

Vud V ∗ub

Vcd V ∗cb+

Vud V ∗ub

Vcd V ∗cb

]= O

(M2

W

Λ2NP

)


CPV and the K 0-K 0 system

Two neutral kaon flavor eigenstates: K 0(ds) & K 0(sd)

In experiment, have predominantly:

K 0S → ππ ⇒ K 0

S ∼ K1, the CP even combination

K 0L → πππ ⇒ K 0

L ∼ K2, the CP odd combination

However, CP violation⇒ K 0L → ππ

Experimentally:(PDG ’11)

∆MK ≡ MKL −MKS = 3.483(6)× 10−12 MeV [0.2%]

∆ΓK ≡ ΓKS − ΓKL = 7.339(4)× 10−12 MeV [0.05%]

|ε| = 2.228(11) · 10−3 [0.5%]

Re(ε′/ε) = 1.65(26) · 10−3 [16%]


K 0-K 0 mixing in the SM

c, t W W

sc, ts dd d

s

c, t

W d

sdsWs d c, t

M12 −i2

Γ12 =〈K 0|H∆S=2

eff (0)|K 0〉2MK

− i2MK

∫d4x〈K 0|H∆S=1

eff (x)H∆S=1eff (0)|K 0〉︸︷︷︸

long-distance contributions to M12 & Γ12

+O(G3F )

To LO in the OPE

2MK M∗12LO= 〈K 0|H∆S=2

eff |K 0〉 = CSM1 (µ)〈K 0|O1(µ)|K 0〉

O1 = (sd)V−A(sd)V−A 〈K 0|O1(µ)|K 0〉 =163

M2K F 2

K BK (µ)


Indirect CPV in K → ππ

Parametrized by

Re ε = cosφε sinφε[

ImM12

2 ReM12− ImΓ12

2 ReΓ12

]w/ φε = tan−1 (2∆MK/∆ΓK ) = 43.51(5)o, ∆MK ' 2 Re M12, ∆ΓK ' −2 Re Γ12

−→ ε = κεeiφε

√2

ImM12

∆MK

w/ κε = 0.94(2) (Buras et al. (2010))

To NLO in αs

|ε| = κεCε Imλt Reλc [η1Scc

−η3Sct ]− Reλtη2Stt BK

∝ A2λ6η [cst + cst

×A2λ4(1− ρ)]

BK

w/ λq = Vqd V ∗qs

γ

α

α

dm∆

Kε

Kε

sm∆ & dm∆

SLub

V

ν τubV

βsin 2

(excl. at CL > 0.95)

< 0βsol. w/ cos 2

α

βγ

ρ

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

η

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

exclu

ded a

rea h

as C

L >

0.9

5

ICHEP 10

CKMf i t t e r


BK with Wilson fermions

(Dürr et al [BMWc], arXiv:1106:3230 [hep-lat])

χSB of Wilson fermions→ O1(a) mixes w/ ops of different chirality:

〈K 0|O1(µ)|K 0〉 = Z11(g0, aµ)Q1(a)

w/

Q1(a) = Q1(a) +5∑

i=2

∆1i (g0)Qi (a), Qi (a) ≡ 〈K 0|Oi (a)|K 0〉

and complete parity conserving basis:

O1 = γµ ⊗ γµ + γµγ5 ⊗ γµγ5

O2 = γµ ⊗ γµ − γµγ5 ⊗ γµγ5

O3 = I ⊗ I + γ5 ⊗ γ5

O4 = I ⊗ I − γ5 ⊗ γ5

O5 = σµν ⊗ σµν

w/ Γ⊗ Γ = [sΓd ][sΓd ]

Perform calculation on BMWc 2010 dataset


Multiplicative renormalization of BK

DefineZBK (g0, aµ) ≡ Z11(g0, aµ)/Z 2

A (g0)

w/ ZA, axial current renormalization

Use similar RI/MOM methods as for ZS(g0, aµ)

Chiral behavior

0.9

0.95

1

1.05

1.1

3 4 5 6 7 8 9 10 11 12

Z1

1(µ

2)/

ZA

2

β=

3.5

µ2[GeV

2]

ambare=-0.006ambare=-0.01

ambare=-0.012ambare=-0.02

ambare=-0.035chiral limit

0.9

0.95

1

1.05

1.1

3 4 5 6 7 8 9 10 11 12

Z1

1(µ

2)/

ZA

2

β=

3.8

µ2[GeV

2]

ambare=0.0ambare=-0.004ambare=-0.008ambare=-0.012ambare=-0.014

chiral limit

Flat chiral extrapolation, more so at larger µ


Nonperturbative continuum running of BK

Continuum, nonperturbative running to µ = 3.5 GeV is given by

RRIBK

(µ0,3.5 GeV) = limg0→0

Z RIBK

(g0,aµ)/Z RIBK

(g0,aµ0)

Divided by 2-loop PT running

⇒ good agreement in range1.75÷ 3.5 GeV


Mixing coefficients ∆1i

Determined in standard way (Donini et al. (1999)), w/ RI/MOM Goldstone poles removedthrough (Giusti et al. (2000))

∆sub1i ≡

m1∆1i (a,m1)−m2∆1i (a,m2)

m1 −m2

Subtract remaining 1/p4 and (ap)2 w/ fits

→ mixing coeffs are well determined and small thanks to smearing→ still O(10)× mixing for DWF


Extraction of bare matrix elements

Bi (g0; t) =(L/a)3∑

~x〈W (T/2)Oi (g0; t , ~x)W +(0)〉83

∑~x,~y 〈W (T/2)Aµ(t , ~x)〉〈Aµ(t , ~y)W +(0))〉

Q1(

)

/a

0.45

0.5

0.55

0.6

0.65

0.7

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

(β = 3.7, Mπ ∼ 245 MeV)


Operator contributions to BK

(β = 3.61, Mπ ∼ 120 MeV)

→ contributions from Q2,...,5 small and often consistent with zero

→ nevertheless, dominate systematic error


Combined chiral interp. and continuum extrap. (1)

Project onto chiral axes:interpolation to Mπ = 134.8(3) MeV and MK = 494.2(5) MeV

→ nearly flat mud -dependence near mphysud

→ much steeper ms-dependence near mphyss


Combined chiral interp. and continuum extrap. (2)Project onto a axis: continuum extrapolation

→ mild extrapolation for β ≥ 3.5

→ β ≥ 3.31 was found to be outside scaling regime


Systematic and statistical error2 time-fit ranges for π and K masses

2 time-fit ranges for Q1,...,5

O(αsa) or O(a2) for running

3 intermediate renormalization scales

2 fit fns and 4 ranges in p2 for ∆1i

5 fit fns for mass interpolation

2 pion mass cuts (Mπ < 340, 380 MeV)

BKRI

(3.5 GeV)

0.51 0.52 0.53 0.54 0.55

BKRI

(3.5 GeV)

0.51 0.52 0.53 0.54 0.55

→ 5760 analyses, each of which is a reasonablechoice, weighted by fit quality

→ median and central 68% give central value andsystematic uncertainty

2000 bootstraps of the median give statisticalerror

Many cross checks


Results

Procedure gives BRIK (3.5 GeV) fully nonperturbatively

Can convert to other schemes w/ perturbation theory (PT)→ perturbative uncertainty

conv. RGI MS-NDR 2 GeV4-loop β, 1-loop γ 1.427 1.0474-loop β, 2-loop γ 1.457 1.062

ratio 1.021 1.01376

Take blanket 1% for 3 loop uncertainty

RI 3.5 GeV RGI MS-NDR 2 GeVBK 0.5308(56)(23) 0.7727(81)(34)(77) 0.5644(59)(25)(56)

Total error 1-1.5%, statistical (and PT) dominated


BK comparison

Dominant in CKMfitter global fit results is |Vcb|4 ∼ (Aλ2)4

→ our result is an encouragement to reduce uncertainties in otherparts of the calculation of ε


Conclusion

After > 40 years we are finally able to perform fully controlled LQCDcomputations all the way down to Mπ = 135 MeV

Presented fully controlled results for light quark masses and BK w/ σtot < 2%

Also results on BMWc 08 ensembles for light hadron masses, FK/Fπ and sigmaterms, and preliminary results on BMWc 10 ensembles for E+M corrections,ρ-width, . . .

Lattice QCD is undergoing a major shift in paradigm

it is now possible to control and reliably quantify all systematicerrors with “data” (for 1 or 2 hadron states)

⇒ we are getting QCD NOT LQCD predictionsrequires numerous simulations with Mπ < 200 MeV and preferably 135 MeV, more than 3 a < 0.1 fm and lattice sizes L→ 4÷ 6 fmrequires trying all reasonable analyses of “data” and combiningresults in sensible way to obtain a reliable systematic error

Expect many more very interesting nonperturbative QCD predictions in comingyears


Autocorrelation of topological charge

Autocorrelation of Q2ren (Dürr et al, JHEP 0704) on same sample

ρ(t) ≡ Γ(t)Γ(0)

Γ(t) ' 1T − t

T−t∑i=1

(Q2ren(i)−〈Q2

ren〉)

×(Q2ren(i + t)− 〈Q2

ren〉)

τint '12

+

t0∑t=1

ρ(t)

τint = 29(8)

(Wolff ’04)

0 50 100 150 200−0.5

0

0.5

1

normalized autocorrelation for qren

2 at β=3.8, m

ud=−0.02, m

s=0

τ

ρ

0 50 100 150 2000

20

40

60

τint

with statistical errors for qren

2 at β=3.8, m

ud=−0.02, m

s=0

τ int

fit window upper bound τmax


Scaling and overall method check: quenched

Check scaling of short-distance quantities and of important ingredients⇒ repeat ALPHA’s quenched milestone determination ofr0(ms + mud )MS(2 GeV)

Perform quenched calculation w/ Wilson glue and 2 HEX fermions

5 β w/ a ∼ 0.06÷ 0.15 fm

At least 4 mq per β w/ MπL > 4 and fixed L ' 1.84fm

Calculate

m(µ) =(1− amW/2)mW

ZS(µ)

w/ mW = mbare −mcrit

Determine ZS(µ) using RI/MOM NPR (Martinelli et al ’95) and run nonperturbatively incontinuum to µ = 3.5 GeV (see below)

Interpolate in r0MPS to r0MphysK

mRI(3.5 GeV) −→ mMS(2 GeV) perturbatively


Quenched test: determination of r0(ms + mud)

Continuum extrapolation of r0(ms + mud )MS(2 GeV)

0.24

0.26

0.28

0.3

0.32

0.34

0 0.02 0.04 0.06 0.08 0.1

(ms+

mu

d)r

0

αa/r0

αa-extrapolation

With full systematic analysis

r0(ms + mud )MS(2 GeV) = 0.261(4)(3)

Perfect agreement w/ ALPHA r0(ms + mud )MS(2 GeV) = 0.261(9)


What is 2 HEX smearing?

2 HEX smearing:

Elementary smearing algorithm is stout (EXponential) smearing (Morningstar et al ’04)

Embedded into 2 steps of HYPercubic smearing (Hasenfratz et al ’01)

a) b)

Couples q(x) to Aµ(x + (3.5a)e) (e.e = 1) w/ weight ∼ 3× 10−5

→ effective range:√〈r 2〉 = 1.1a

Ultralocal and effectively extends barely more than nearest neighbor

Only differs from regular improved Wilson fermions by O(αsa)

More local than previously used 6 stout smearing (Dürr et al, Science 322 (2008))


Does our smearing compromise locality of Dirac op.?

Two different forms of locality: our Dirac operator is ultralocal in both senses

1∑

xy ψ(x)D(x , y)ψ(y) and D(x , y) ≡ 0 for |x − y | > a→ no problem

2 D(x , y) depends on Uµ(x + z) for |z| > a→ potential problem

0 1 2 3 4 5 6 7

|z|/a

10-6

10-5

10-4

10-3

10-2

10-1

100

||¶D

(x,y

)/¶Um(x

+z) ||

a~~0.125 fma~~0.085 fma~~0.065 fm However,

||∂D(x , y)/∂Uµ(x + z)|| ≡ 0 for |z| ≥ 7.1a

fall off ∼ e−2.2|z|/a

2.2 a−1 physical masses of interest

⇒ not a problem here


VWI and AWI masses: ratio-difference method

With Nf = 2 + 1, O(a)-improved Wilson fermions, can construct the followingrenormalized, O(a)-improved quantities (using Bhattacharya et al ’06)

(ms−mud )VWI = (mbares −mbare

ud )1

ZS

[1− bS

2a(mW

ud + mWs )− bS a(2mW

ud + mWs )

]+O(a2)

w/ mW = mbare −mcrit and

mAWIs

mAWIud

=mPCAC

s

mPCACud

[1 + (bA − bP) a(mbare

s −mbareud )]

w/

mPCAC ≡ 12

∑~x〈∂µ [Aµ(x) + acA∂µP(x)] P(0)〉∑

~x〈P(x)P(0)〉

and bA,P,S = 1 + O(αs), bA,P,S = O(α2s), cA = O(αs)


Ratio-difference method (cont’d)

Define

d ≡ ambares − ambare

ud , r ≡ mPCACs

mPCACud

and subtracted bare masses

amsubud ≡

dr − 1

, amsubs ≡

rdr − 1

Then, with our tree-level O(a)-improvement, renormalized masses can be written

mud =msub

ud

ZS

[1− a

2(msub

ud + msubs )]

+ O(αsa)

ms =msub

s

ZS

[1− a

2(msub

ud + msubs )]

+ O(αsa)

Benefits:

Only ZS (non-singlet) is required and difficult RI/MOM ZP is circumvented

No need to determine mcrit


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Lattice QCD with physically light up and down quarks · Blum 10 FLAG 10v1 N f = 2+1 mMS s (2 GeV) =...

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