news from the lattice - national tsing hua universityhep/ncts_jc2010fall/lellouch... · 2010. 11....

News from the lattice

Laurent Lellouch

CPT Marseille

withDürr, Fodor, Frison, Hoelbling, Katz, Krieg, Kurth, Lellouch, Lippert,

Portelli, Ramos, Szabo, Vulvert(Budapest-Marseille-Wuppertal Collaboration)

lavinet

Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010

Some of the questions that we would like to answerCan one show that the mass of ordinary matter comes from QCD?

What are the masses of the light u, d (and s) quarks which are the building blocks ofordinary matter?

Is our understanding of quark flavor mixing and the fundamental asymmetry betweenmatter and antimatter, which it leads to, correct?

Does dark matter couple strongly enough to ordinary matter to make it visible withcurrent detectors?

Can one show that QCD and QED explain why the proton is lighter than the neutron?(If Mp > MN , there would be no atoms . . .)

Heisenberg’s uncertainty principle allows the creation of particle-antiparticle pairswhich can induce the decay of other particles (e.g. vac.→ uu ⇒ ρ→ ππ)Does QCD describe ρ→ ππ correctly?

Can a confining gauge theory such as QCD explain electroweak symmetry breakingand the masses of elementary particles? (Technicolor at LHC?)→ ask C.-J. David Lin!

All require quantitative understanding of nonperturbative strong interaction effects⇒ only known ab initio approach is Lattice QCD


What is Lattice QCD (LQCD)?

Lattice gauge theory −→ mathematically sound definition of NP QCD:

UV (and IR) cutoffs and a well defined pathintegral in Euclidean spacetime:

〈O〉 =

∫DUDψDψ e−SG−

∫ψD[M]ψ O[U, ψ, ψ]

=

∫DU e−SG det(D[M]) O[U]Wick

DUe−SG det(D[M]) ≥ 0 and finite # of dof’s→ evaluate numerically using stochasticmethods

rrrrrrrrrr

rrrrrrrrrr

rrrrrrrrrr

rrrrrrrrrr

rrrrrrrrrr

rrrrrrrrrr

rrrrrrrrrr

rrrrrrrrrr

-6?

6

?

T

-L

?6a

Uµ(x) = eiagAµ(x) ψ(x)

NOT A MODEL: LQCD is QCD when a→ 0, V →∞ and stats→∞

In practice, limitations . . .


Limitations: statistical and systematic errors

Limited computer resources→ a, L and mq are compromises andstatistics finite

Quenching: in past, det(D[M])→ cstBeing removed w/ difficult 2+1 calculations

Statistical: 1/√

Nconf

Eliminate w/ Nconf →∞Discretization: aΛQCD, amq , a|~p|, with a−1 ∼ 2− 4 GeV

Eliminate w/ a→ 0: need at least three a’s

Chiral extrapolation: m phud barely reachable⇒ mq[> m ph

ud ]→ m phud

Difficult calculations w/ Mπ <∼ 350 MeV + χPT or Taylor expansionsBetter: simulate directly at m ph

ud

Finite volume: simple quantities ∼(

MππFπ

)2e−MπL

(MπL)3/2 → MπL >∼ 4 usually safeResonant states more complicatedEliminate with L→∞ (+ χPT)

Renormalization: in all field theories must renormalize;Can be done in PT, best done nonperturbatively


Why is LQCD so numerically difficult?

# of d.o.f. ∼ O(109) and large overhead for det(D[M]) (∼ 109 × 109 matrix)

cost of simulations increases rapidly when mu,d → mphysu,d & a→ 0

0 0.25 0.5 0.75M

π/M

ρ

0

2

4

6

8

10

Tfl

ops

x y

ear/

500 c

onfi

gs

Wilson

Staggered (R-algorithm)

L = 2.5 fm, T = 8.6 fm, a = 0.09 fm

Physical point

Wilson fermions (≤ 2004)

cost ∼ NconfV 5/4m−3u,d a−7

(Ukawa ’02)

Serious cost wall

⇒ can physical mu,d ever be reached?

Observe very long-lived autocorrelations oftopological charge vs MC time (Schaefer et al

’09, quenched)

⇒ a→ 0 may be even harder thananticipated

-20

-15

-10

-5

0

5

10

15

20

0 100 200 300 400 500

Q

τ

a=0.04fm


How we overcome these problemsDürr et al (BMW), PRD79 2009

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

1) Discretization which balances improvement in gauge/fermionic sector and CPUcost:

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

tree-level O(a)-improved Wilson fermion (Sheikholeslami et al ’85) w/ 6-stout or 2-HEXsmearing (Morningstar et al ’04, Hasenfratz et al ’01, Capitani et al ’06)

⇒ approach to continuum is improved (O(αsa, a2) instead of O(a))

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

Hybrid Monte Carlo (HMC) for u and d and Rational HMC (RHMC) for s

mass preconditioning (Hasenbusch ’01)

multiple timescale integration of molecular dynamics (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


Where does the mass of ordinary matter come from?

→ Higgs? SEWSB? . . . ?⇒ mass of fundamental particles/⇒ mass of ordinary matter

> 99% of mass of visible universe is in the form of p & n

Mass of object usually sum of mass of constituents: not true for light hadrons

Light hadron masses are generated by QCD through energy imparted to q & g via:

m = E/c2


Ab initio calculation of the light hadron spectrumDürr et al, Science 322 (2008) 1224

Aim: determine light hadron spectrum, at few percent level, directly in QCD incalculation w/ all sources of systematic errors under control

⇒ i. Inclusion of Nf = 2 + 1 sea quark effects w/ an exact algorithm and w/ aunitary, local action whose universality class is known to be QCD (→ see above)

⇒ ii. 3 parameters of Nf =2 + 1 QCD (mud , ms & ΛQCD) by observables w/ smallundisputed errors, transparent connection to experiment and no hiddenassumptions

⇒ iii. Complete spectrum for the light mesons and octet and decuplet baryons

⇒ iv. Large volumes to guarantee negligible finite-size effects (→ check)

⇒ v. Controlled interpolations to m phs (straightforward) and extrapolations to m ph

ud(difficult)Of course, simulating directly around m ph

ud would be better!

⇒ vi. Controlled extrapolations to the continuum limit→ at least 3 a’s in the scaling regime

⇒ vii. Complete analysis of systematic uncertainties


Simulation parameters: BMW ’08

β, a [fm] amud Mπ [GeV] ams L3 × T # traj.3.3 -0.0960 0.65 -0.057 163 × 32 10000

-0.1100 0.51 -0.057 163 × 32 1450∼ 0.125 -0.1200 0.39 -0.057 163 × 64 4500

-0.1233 0.33 -0.057 163 × 64 | 243 × 64 | 323 × 64 5000 | 2000 | 1300-0.1265 0.27 -0.057 243 × 64 700

3.57 -0.03175 0.51 0.0 243 × 64 1650-0.03175 0.51 -0.01 243 × 64 1650

∼ 0.085 -0.03803 0.42 0.0 243 × 64 1350-0.03803 0.41 -0.01 243 × 64 1550-0.044 0.31 0.0 323 × 64 1000-0.044 0.31 -0.07 323 × 64 1000-0.0483 0.20 0.0 483 × 64 500-0.0483 0.19 -0.07 483 × 64 1000

3.7 -0.007 0.65 0.0 323 × 96 1100-0.013 0.56 0.0 323 × 96 1450

∼ 0.065 -0.02 0.43 0.0 323 × 96 2050-0.022 0.39 0.0 323 × 96 1350-0.025 0.31 0.0 403 × 96 1450

# of trajectories given is after thermalization

autocorrelation times (plaquette, nCG) less than ≈ 10 trajectories

2 runs with 10000 and 4500 trajectories −→ no long-range correlations found


ad ii, iii: QCD parameters and light hadron massesNf =2+1 QCD in isospin limit has 3 parameters which have to be fixed w/ expt:

ΛQCD: fixed w/ Ω or Ξ mass

don’t decay through the strong interactionhave good signalhave a weak dependence on mud

→ 2 separate analyse and compare

(mud ,ms): fixed using Mπ and MK

Determine masses of remaining non-singlet light hadrons in

TTT

TTT

r

r

r

rr rr rr

K∗0 K∗−

K∗− K∗0

ρ+ρ−ρ0 ω

φ

TTT

TTT

r

r

r

rr rrr

n p

Ξ− Ξ0

Σ+Σ−Σ0

Λ

TTTTTT

qq

qqq qq

q

q q∆− ∆0 ∆+ ∆++

Ξ∗− Ξ∗0

Σ∗+Σ∗− Σ∗0

Ω−


ad iii: fits to 2-point functions in different channelse.g. in pseudoscalar channel, Mπ from correlated fit

CPP(t) ≡ 1(L/a)3

∑~x

〈[dγ5u](x)[uγ5d ](0)〉 0tT−→ 〈0|dγ5u|π+(~0)〉〈π+(~0)|uγ5d |0〉2Mπ

e−Mπ t

Effective mass aM(t + a/2) = log[C(t)/C(t + a)]

4 8 12

t/a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

a M

K

N

Effective masses for simulation ata ≈ 0.085 fm and Mπ ≈ 0.19 GeV

Gaussian sources and sinks withr ∼ 0.32 fm (BMW ’08)


ad iv: (I) Virtual pion loops around the world

In large volumes FVE ∼ e−MπL

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

For a ≈ 0.125 fm and Mπ ≈ 0.33 GeV, perform FV study MπL = 3.5→ 7

12 16 20 24 28 32 36L/a

0.21

0.215

0.22

0.225

aMp

c1+ c

2e

-Mp L

-3/2 fit

L

Colangelo et. al. 2005

volume dependenceMpL=4

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

Well described by (Dürr and Colangelo et al, 2005)

MX (L)−MX

MX= C

(Mπ

πFπ

)2 1(MπL)3/2 e−MπL

Though very small, we fit them out


ad iv: (II) Finite volume effects for resonances

Important since 5/12 of hadrons studied are resonances

Systematic treatment of resonant states in finite volume (Lüscher, ’85-’91)

e.g., the ρ↔ ππ system in the COM frame

Non-interacting:E2π = 2(M2

π + k2)1/2, ~k = 2π~n/L,~n ∈ Z 3

Interacting case: k solution of

nπ−δ11(k) = φ(q), n ∈ Z , q = kL/2π

2 3 4 5 6 7 8

MπL

2

2.5

3

3.5

4

E2

π/M

π

Mρ/M

π = 3

Know L and lattice gives E2π and Mπ

⇒ infinite volume mass of resonance and coupling to decay products

in this calculation, low sensitivity to width (compatible w/ expt w/in large errors)

small but dominant FV correction for resonances


ad v: extrapolation to m phud and interpolation to m ph

s

Consider two approaches to the physical QCD limit for a hadron mass MX :

1 Mass-independent scale setting2 Simulation-by-simulation normalization

For both normalization procedures, use parametrization

MX = M(0)X + αK M2

K + απM2π + h.o.t.

linear term in M2K is sufficient for interpolation to m ph

s

curvature in M2π is visible in extrapolation to m ph

ud in some channels

→ two options for h.o.t.:

ChPT: expansion about M2π = 0 and h.o.t. ∝ M3

π (Langacker et al ’74)

Flavor: expansion about center of M2π interval considered and h.o.t. ∝ M4

π

Further estimate of contributions of neglected h.o.t. by restricting fit interval:Mπ ≤ 650→ 550→ 450 MeV

→ use 2× 2× 3 combinations of options for error estimate


ad vi: including the continuum extrapolationCutoff effects formally O(αsa) and O(a2)

Small and cannot distinguish a and a2

Include through

MphX → Mph

X [1 + γX a] or MphX [1 + γX a2]

→ difference used for systematic error estimation

not sensitive to ams or amud

0.1 0.2 0.3 0.4 0.5

Mπ

2 [GeV

2]

0

0.5

1

1.5

2

M [G

eV

]

physical Mπ

Ω

N

a~~0.085 fm

a~~0.065 fm

a~~0.125 fm

K*


ad vii: systematic and statistical error estimate

Uncertainties associated with:

Continuum extrapolation→ O(a) vs O(a2)

Extrapolation to physical mass point

→ ChPT vs Taylor expansion→ 3 Mπ ranges ≤ 650 MeV, 550 MeV, 450 MeV

Normalization→ MX vs RX

⇒ contributions to physical mass point extrapolation (and continuumextrapolation) uncertainties

Excited state contamination→ 18 time fit ranges for 2pt fns

Volume extrapolation→ include or not leading exponential correction

⇒ 432 procedures which are applied to 2000 bootstrap samples, for each of Ξ and Ωscale setting


ad vii: systematic and statistical error estimate

→ distribution for MX : weigh each of the 432 results for MX in original bootstrapsample by fit quality

900 920 940 960 980M

N [MeV]

0.05

0.1

0.15

0.2

0.25

median

Nucleon

1640 1660 1680 1700 1720M

Ω [MeV]

0.1

0.2

0.3

median

Omega

Median→ central value

Central 68% CI→ systematic error

Central 68% CI of bootstrap distribution of medians→ statistical error


Post-dictions for the light hadron spectrum

0

500

1000

1500

2000M

[Me

V]

p

K

rK* N

LSXD

S*X*O

experiment

width

input

Budapest-Marseille-Wuppertal collaboration

QCD


|Vus/Vud | from K , π → µν(γ)

In experiment see

_u

d

_

Wµ

ν

π∝ Vud 〈0|uγµγ5d |π−〉 ∝ Vud Fπ

Have (Marciano ’04, Flavianet ’08)

Γ(K → µν(γ))

Γ(π → µν(γ))−→ |Vus|

|Vud |FK

Fπ= 0.2760(6) [0.22%]

⇒ need high precision nonperturbative calculation of FK/Fπ

Use our ’08 data sets (Mπ → 190 MeV, a ≈ 0.065÷ 0.125 fm, L→ 4 fm) to computeFK/Fπ

Perform 1512 independent full analyses of our data⇒ systematic error distribution for FK/Fπ (as above)

Get statistical error from bootstrap analysis on 2000 samples


FK/Fπ in QCD

1002

2002

3002

4002

M2

π[MeV2]

1

1.05

1.1

1.15

1.2

1.25

FK

/Fπ

β=3.3β=3.57β=3.7Taylor extrapolation

Dürr et al, PRD81 ’10

0

0.02

0.04

0.06

0.08

0.1

0.12

1.17 1.175 1.18 1.185 1.19 1.195 1.2 1.205 1.21

Distribution of valuesTaylorSU(2)SU(3)

FK

Fπ= 1.192(7)stat(6)syst [0.8%]

Main source of sytematic error: extrapolation mud → mphysud ; then a→ 0


FK/Fπ summary and CKM unitarity

1.14

1.14

1.16

1.16

1.18

1.18

1.20

1.20

1.22

1.22

1.24

1.24

1.26

1.26

FK/F

π

ETM 09

ETM 07

QCDSF/UKQCD 07

NPLQCD 06

RBC/UKQCD 07

HPQCD/UKQCD 07

PACS-CS 08, 08B

ALVdW 08

MILC 09

our estimate for Nf = 2+1

x

Nf=

2+

1N

f=2

MILC 09A

JLQCD/TWQCD 09A

BMW 10

MILC 04

our estimate for Nf = 2

x

(Flavianet Lattice Averaging Group (FLAG) ’10)

FLAG ’10 estimates (direct)

FK

Fπ= 1.193(6) Nf = 2 + 1

FK

Fπ= 1.210(6)(17) Nf = 2

(x) FLAG determinations assuming the SM: expt for ΓK`2,

ΓK`3and |Vub| (neglible contribution), plus SM

universality and CKM unitarity(|Vud |2 + |Vus|2 + |Vub|2 = 1)

⇒ FK /Fπ , |Vud | and |Vus| in terms of lattice f+(0)

FindG2

q

G2µ

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

]= 1.0001(9) [0.09%]

If assume true result within 2σ then, naively, ΛNP ≥ 1.9 TeV


Light quark masses at the physical mass point

Masses of light u, d and s quarks→ fundamental parameters of the StandardModel (SM)

Stability of atoms, nuclear reactions which power stars, presence or absence ofstrong CP violation, etc. depend critically on precise values

Values carry information about flavor structure of BSM physics

Quarks are confined w/in hadrons⇒ a nonperturbative computation is required

Deviation of mud ≡ (mu + md )/2 from zero brings only very small corrections tomost hadronic observables⇒ its determination is a needle in haystack problem

Fortunately, QCD spontaneously breaks chiral symmetry⇒ masses of resulting Nambu-Goldstone mesons very sensitive the light

quark masses⇒ presence of chiral logs near physical point⇒ must get very close to mph

ud to control mass dependence precisely

→ quark masses are an interesting first “measurement” to make w/ physical pointLQCD simulations


Current knowledge of the light quark massesFLAG has performed a detailed analysis of unquenched lattice determinations of thelight quark masses (“our estimate” = FLAG estimate)

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

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


Dominguez 09

mud

Nf=

2+1

Nf=

2

Narison 06Maltman 01

PDG 09

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

JLQCD/TWQCD 08A


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

QCDSF/UKQCD 06ETM 07RBC 07

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


PDG 09

ms

Nf=

2+1

Nf=

2

Dominguez 09Chetyrkin 06Jamin 06Narison 06Vainshtein 78

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


mMSs (2 GeV) =

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

Even extensive study by MILC does not control all systematics:

MRMSπ ≥ 260 MeV ⇒ mMILC,eff

ud ≥ 3.7×mphysud

perturbative renormalization (albeit 2 loops)


My dream calculation

Requirements for ab initio calculations (i-vii) given earlier

+ 2 ingredients which guarantee added precision:

Nf = 2 + 1 calculations all the way down to Mπ ∼ 130 MeV toallow small interpolation to physical mass point(Mπ = 134.8(3)MeV)Full nonperturbative renormalization and nonperturbativecontinuum extrapolated running for determiningrenormalization group invariant (RGI) quark masses


Where do we stand?

All simulations w/ Nf ≥ 2 + 1 and Mπ ≤ 400 MeV . . . (points for our currently running,next-to-finest simulations at β = 3.7 are estimates)

00 2002

3002

4002

Mπ

2 [MeV

2]

0

0.0502

0.1002

a2 [

fm2]

MILC ’10QCDSF ’10

PACS-CS ’09RBC/UK. ’10HSC ’08ETMC ’10BMW ’08This work

(from C. Hoelbling, Lattice 2010)


Where do we stand?

. . . and w/ unitary, local gauge and fermion actions. . .

00 2002

3002

4002

Mπ

2 [MeV

2]

0

0.0502

0.1002

a2 [

fm2]

QCDSF ’10

PACS-CS ’09RBC/UK. ’10HSC ’08ETMC ’10BMW ’08This work


Where do we stand?

. . . and w/ sea u and d quarks clearly in the chiral regime, i.e. Mminπ ≤ 250 MeV. . .

00 2002

3002

4002

Mπ

2 [MeV

2]

0

0.0502

0.1002

a2 [

fm2]

PACS-CS ’09BMW ’08This work


Where do we stand?

. . . and w/ sea u and d quarks at or below physical mass point . . .

00 2002

3002

4002

Mπ

2 [MeV

2]

0

0.0502

0.1002

a2 [

fm2]

PACS-CS ’09This work


Where do we stand?

. . . and w/ volumes such that FV errors ≤ 0.5% – PACS-CS has LMπ = 1.97 . . .

00 2002

3002

4002

Mπ

2 [MeV

2]

0

0.0502

0.1002

a2 [

fm2]

This work


Where do we stand?

. . . and w/ at least three a ≤ 0.1 fm – PACS-CS has only 1 a ∼ 0.09 fm

00 2002

3002

4002

Mπ

2 [MeV

2]

0

0.0502

0.1002

a2 [

fm2]

This work


Does our smearing enhance discretization errors?⇒ 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

BMW ’10

1500

1550

1600

1650

1700

1750

1800

1850

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

M[M

eV

]

a2[fm

2]

0.2

0 f

m

0.1

8 f

m

0.1

5 f

m

0.1

0 f

m

0.0

6 f

m

M∆

MN

BMW ’08

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

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


Does our smearing enhance discretization errors?

Perhaps 2 HEX works for spectral quantities but not for short distance dominatedquantities⇒ repeat ALPHA’s 2000 quenched milestone determination of r0(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 = 4 GeV (see below)

Interpolate in r0MPS to r0MphysK

mRI(4 GeV) −→ mMS(2 GeV) perturbatively


Quenched check: determination of r0(ms + mud)

Perform continuum extrapolation of r0(ms + mud )MS(2 GeV) (preliminary)

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0 0.02 0.04 0.06 0.08 0.1

(mu

d+

ms)r

0

αa/r0

Garden et al., [Nucl.Phys.B 2000]

αa-extrapolation

With full systematic analysis

r0(ms + mud )MS(2 GeV) = 0.262(4)(3)

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


Nf = 2 + 1 simulation parameters

38 + 9, Nf = 2 + 1 phenomenological runs:

5 a ' 0.054÷ 0.116 fm

Mminπ ' 135, 130, 120, 180, 220 MeV

L up to 6 fm and such that δFV ≤ 0.5% on Mπ for all runs10 + 3 different values of ms around mphys

s

Determine lattice spacing using MΩ

17 + 4, Nf = 3 RI/MOM runs at same β as phenomenological runs:

At least 4 mq ∈ [mphyss /3,mphys

s ] per β for chiral extrapolationL ≥ 1.7 fm in all runs


Do we see chiral logs?Simultaneous fit of M2

π and Fπ vs mud to NLO SU(2) χPT expressions (Gasser et al, ’84)

M2π = M2

[1− 1

2x log

(Λ2

3

M2

)]Fπ = F

[1 + x log

(Λ2

4

M2

)]w/ M2 = 2Bmud and x = M2/(4πF )2

Fixed a ' 0.09 fm and Mπ ' 130→ 400 MeV (preliminary)

0.005 0.01

amudPCAC

2.4

2.5

2.6

2.7

2.8

aMπ2 /m

udPC

AC

0.005 0.01

amudPCAC

0.04

0.045

0.05

0.055

aFπ

Consistent w/ NLO χPT . . .


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


Improved RI/MOM for ZS

Determine Z RIS (µ, a) nonperturbatively in RI/MOM scheme, from truncated, forward

quark two-point functions in Landau gauge (Martinelli et al ’95), computed on specificallygenerated Nf = 3 gauge configurations

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

⇒ tree-level O(a) improvement

⇒ significant improvement in S/N

⇒ recover usual massless RI/MOM scheme for mRGI → 0

For controlled errors, require:

(a) µ 2π/a for a→ 0 extrapolation

(b) µ ΛQCD if masses are to be used in perturbative context

i.e. the window problem, which we solve as follows


Ad (a): RI/MOM at sufficiently low scale

Controlled continuum extrapolation of renormalized mass

⇒ renormalize at µ where RI/MOM O(αsa) errors are small for all β

For coarsest (β = 3.31) lattice, 2π/a ' 11 GeV

Restrict study of Z RIS (µ, a) to µ <∼ π/2a ' 2.7 GeV (β = 3.31)

Pick µren ∼ 2 GeV as common renormalization point for all β

Can take a→ 0⇒ continuum mRI(µren) determined fully nonperturbatively . . .

. . . but not very useful for phenomenology since RI/MOM perturbative error stillsignificant at such µren


Ad (b): nonperturbative continuum running to 4 GeV

To make result useful, run nonperturbatively in continuum limit up to perturbative scale

For µ : µren → 4 GeV, always have at least 3 a w/ µ <∼ π/2a

⇒ can determine nonperturbative running in continuum limit

RRI(µren, 4 GeV) = lima→0

Z RIS (4 GeV, a)

Z RIS (µren, 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 (µ, aβ ) for β < 3.8 to∼ match Z RI

S (µ, aβ=3.8)

Preliminary

Running is very similar at all 4 β⇒ flat a→ 0 extrapolation


Ad (b): running above 4 GeV

For µ > 4 GeV, 4-loop perturbative running agrees w/ nonperturbative running on our finerlattices

Preliminary

10 20 30 40

µ2[GeV

2]

0.8

0.9

1

1.1

1.2ZRI

S,nonpert(

µ2)/ZRI

S,4-loop(µ

2)

µ=4GeV

β=3.8


Continuum extrapolation of renormalized massesRenormalized quark masses interpolated in M2

π & M2K to physical point using:

SU(2) χPT

or low-order polynomial anszätze

w/ cuts on pion mass Mπ < 340, 380 MeV

Example of continuum extrapolations (only statistical errors on data)

Preliminary

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 0.04αa[fm]

Preliminary

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 0.04αa[fm]




SU(2) χPT




... and syst. error due to chiral interp.

Preliminary

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 0.04αa[fm]

σχ w/ M

π >= 120 MeV

Preliminary

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 0.04αa[fm]




SU(2) χPT




... and syst. error due to chiral extrap. if Mπ ≥ Mvalπ |min

MILC ' 180 MeV

Preliminary

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 0.04αa[fm]

σχ w/ M

π >= 120 MeV

σχ w/ M

π >= 180 MeV

Preliminary

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 0.04αa[fm]




SU(2) χPT




... and syst. error due to chiral extrap. if Mπ ≥ MRMSπ |min

MILC ' 260 MeV (very small range)

Preliminary

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 0.04αa[fm]

σχ w/ M

π >= 120 MeV

σχ w/ M

π >= 180 MeV

σχ w/ M

π >= 260 MeV

Preliminary

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 0.04αa[fm]


Conclusions

Ab initio calculation of light hadrons masses→ excellent agreement w/ experiment⇒ vindication of (lattice) QCD in nonperturbative domain

FK/Fπ in same approach→ very competitive determination of |Vus|→ stringent tests of the SM and constraints on NP

Nf = 2 + 1 simulations have been performed all the way down to mphysud and below w/

ms ' mphyss :

5 a ' 0.054÷ 0.116 fmMminπ ' 135, 130, 120, 180, 220 MeV

L up to 6 fm and such that δFV ≤ 0.5% on Mπ for all runs

→ eliminates large systematic error associated w/ reaching mphysud

Described an RI/MOM procedure which includes continuum limit, nonperturbativerunning

→ eliminates large systematic error associated w/ the “window” problem

Currently finalizing analysis of light quark masses


Conclusions

Systematic error will be estimated following an extended frequentist approach (Dürr et

al, Science ’08)

→ expect total uncertainty on mud and ms to be of order 2%

⇒ will significantly improve knowledge of mud and ms whose errors are, at present,12% [FLAG]÷ 30% [PDG]

HPQCD published results on mud w/ similar uncertainties, but these are obtained byfixing Nf = 2 + 1 QCD parameters w/:

r1 for the scale – their r0 is 3.5 σ away from PACS-CS ’09

mc for the scale (?) and for renormalization – their mc has an error 13 timessmaller than PDG!Mss for ms – but ms is needed to determine Mss!?MILC ms/mud for mud – not their ms/mud !?


Conclusions

In addtion, these results are obtained from simulations w/ MRMSπ ≥ 260 MeV

Imposing the cut Mπ ≥ 260 MeV on our results

⇒ δχmud ∼ 0.1% −→ δχmud ∼ 15%

Imposing the cut Mπ ≥ 180 MeV (lightest MILC valence pion) on our results

⇒ δχmud ∼ 0.1% −→ δχmud ∼ 2%

⇒ smaller error requires assumptions on mass dependence of results which gobeyond (partially quenched) NLO SU(2) χPT

Fully controlled LQCD calculations can now be envisaged w/out any assumptionson light quark mass dependence of results

The dream of simulating QCD w/ no ifs nor buts is finally becoming a reality


Our “particle accelerators”

IBM Blue Gene/P (Babel), GENCI-IDRISParis

139 Tflop/s peak

IBM Blue Gene/P (JUGENE), FZ Jülich1. Pflop/s peak

BULL cluster (1024 Nehalem 8 corenodes), GENCI-CCRT Bruyère-le-Châtel

100 Tflop/s peak

And computer clusters at Uni. Wuppertal and CPT Marseille


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