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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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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 . . .
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
qqq qq
q
q q∆− ∆0 ∆+ ∆++
Ξ∗− Ξ∗0
Σ∗+Σ∗− Σ∗0
Ω−
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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)
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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*
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
|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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
our estimate for Nf = 2
Dominguez 09
mud
Nf=
2+1
Nf=
2
Narison 06Maltman 01
PDG 09
MILC 09MILC 09AHPQCD 09PACS-CS 09Blum 10
JLQCD/TWQCD 08A
our estimate for 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
QCDSF/UKQCD 06ETM 07RBC 07
MILC 04HPQCD 05MILC 07CP-PACS 07RBC/UKQCD 08PACS-CS 08
our estimate for Nf = 2
PDG 09
ms
Nf=
2+1
Nf=
2
Dominguez 09Chetyrkin 06Jamin 06Narison 06Vainshtein 78
MILC 09MILC 09AHPQCD 09PACS-CS 09Blum 10
our estimate for Nf = 2+1
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)
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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)
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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)
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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 . . .
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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)
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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]
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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)
... 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]
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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)
... 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]
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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)
... 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]
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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 !?
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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
Laurent Lellouch NCTS, Hsinchu, 26 Oct. 2010
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