News from the lattice
Laurent Lellouch
CPT Marseille
with
Dürr, Fodor, Frison, Hoelbling, Katz, Krieg, Kurth, Lellouch, Lippert,
Portelli, Ramos, Szabo, Vulvert
(Budapest-Marseille-Wuppertal Collaboration)
lavi
net
Laurent Lellouch
NCTS, Hsinchu, 26 Oct. 2010
Some of the questions that we would like to answer
Can 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 of
ordinary matter?
Is our understanding of quark flavor mixing and the fundamental asymmetry between
matter and antimatter, which it leads to, correct?
Does dark matter couple strongly enough to ordinary matter to make it visible with
current 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 pairs
which can induce the decay of other particles (e.g. vac. → ūu ⇒ ρ → ππ)
Does QCD describe ρ → ππ correctly?
Can a confining gauge theory such as QCD explain electroweak symmetry breaking
and 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:
Uµ (x) = eiagAµ (x)
UV (and IR) cutoffs and a well defined path
6r r r r r r
integral in Euclidean spacetime:
r r r r r r
Z
r r r r r r
R
−SG − ψ̄D[M]ψ
r r r r r r
hOi =
DUDψ̄Dψ e
O[U, ψ, ψ̄]
T r r r rr r
Z
?
r r6 r
r r r =
DU e−SG det(D[M]) O[U]Wick
DUe−SG det(D[M]) ≥ 0 and finite # of dof’s
→ evaluate numerically using stochastic
methods
r r
r r
r r
?r r
r
r
r
r
r
r
r
r
NOT A MODEL: LQCD is QCD when a → 0, V → ∞ and stats → ∞
In practice, limitations . . .
Laurent Lellouch
NCTS, Hsinchu, 26 Oct. 2010
r
r
r
r
L
r
r
r
r
r
r
r
r
r
r
r
r
r
r
ψ(x)
r
a
6
r ?
r
r
r
r
r
r
r
r
-
Limitations: statistical and systematic errors
Limited computer resources → a, L and mq are compromises and
statistics finite
Quenching: in past, det(D[M]) → cst
Being 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
ph
ph
ph
Chiral extrapolation: mud
barely reachable ⇒ mq [> mud
] → mud
Difficult calculations w/ Mπ <
∼ 350 MeV + χPT or Taylor expansions
ph
Better: simulate directly at mud
2 −M L
π
Mπ
e
→ Mπ L >
Finite volume: simple quantities ∼ πF
∼ 4 usually safe
(Mπ L)3/2
π
Resonant states more complicated
Eliminate 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)
phys
cost of simulations increases rapidly when mu,d → mu,d
&a→0
L = 2.5 fm, T = 8.6 fm, a = 0.09 fm
10
Tflops x year/500 configs
Physical point
8
Wilson fermions (≤ 2004)
Wilson
−3 −7
a
cost ∼ Nconf V 5/4 mu,d
6
4
Serious cost wall
Staggered
(R-algorithm)
⇒ can physical mu,d ever be reached?
2
0
0
(Ukawa ’02)
0.25
0.5
0.75
Mπ/Mρ
a=0.04fm
20
15
Observe very long-lived autocorrelations of
topological charge vs MC time (Schaefer et al
5
Q
’09, quenched)
10
⇒ a → 0 may be even harder than
anticipated
0
-5
-10
-15
-20
0
100
200
300
τ
Laurent Lellouch
NCTS, Hsinchu, 26 Oct. 2010
400
500
How we overcome these problems
Dürr et al (BMW), PRD79 2009
Nf = 2 + 1 QCD: degenerate u & d w/ mass mud and s quark w/ mass ms ∼ msphys
1) Discretization which balances improvement in gauge/fermionic sector 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/ 6-stout or 2-HEX
smearing (Morningstar et al ’04, Hasenfratz et al ’01, Capitani et al ’06)
⇒ approach to continuum is improved (O(αs a, 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/c 2
Laurent Lellouch
NCTS, Hsinchu, 26 Oct. 2010
Ab initio calculation of the light hadron spectrum
Dürr et al, Science 322 (2008) 1224
Aim: determine light hadron spectrum, at few percent level, directly in QCD in
calculation w/ all sources of systematic errors under control
⇒ i. Inclusion of Nf = 2 + 1 sea quark effects w/ an exact algorithm and w/ a
unitary, 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/ small
undisputed errors, transparent connection to experiment and no hidden
assumptions
⇒ iii. Complete spectrum for the light mesons and octet and decuplet baryons
⇒ iv. Large volumes to guarantee negligible finite-size effects (→ check)
ph
⇒ v. Controlled interpolations to msph (straightforward) and extrapolations to mud
(difficult)
ph
Of course, simulating directly around mud
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]
3.3
∼ 0.125
3.57
∼ 0.085
3.7
∼ 0.065
amud
-0.0960
-0.1100
-0.1200
-0.1233
-0.1265
-0.03175
-0.03175
-0.03803
-0.03803
-0.044
-0.044
-0.0483
-0.0483
-0.007
-0.013
-0.02
-0.022
-0.025
Mπ [GeV]
0.65
0.51
0.39
0.33
0.27
0.51
0.51
0.42
0.41
0.31
0.31
0.20
0.19
0.65
0.56
0.43
0.39
0.31
ams
-0.057
-0.057
-0.057
-0.057
-0.057
0.0
-0.01
0.0
-0.01
0.0
-0.07
0.0
-0.07
0.0
0.0
0.0
0.0
0.0
L3 × T
163 × 32
163 × 32
163 × 64
163 × 64 | 243 × 64 | 323 × 64
243 × 64
243 × 64
243 × 64
243 × 64
243 × 64
323 × 64
323 × 64
483 × 64
483 × 64
323 × 96
323 × 96
323 × 96
323 × 96
403 × 96
# traj.
10000
1450
4500
5000 | 2000 | 1300
700
1650
1650
1350
1550
1000
1000
500
1000
1100
1450
2050
1350
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 masses
Nf =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 interaction
have good signal
have 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
K ∗0
r
− ρ r
T
T
ρ0
rK
T
rrr
ω
φ
Tr
r
K
K̄
∗−
∗−
T +
Tr ρ
∗0
r
rp
T
n
−
Σ r
T
T
Tr
Ξ
Laurent Lellouch
−
Σ0
rr
Λ
r
q
T
q
∆−
T +
Tr Σ
TqΣ
T
Ξ0
NCTS, Hsinchu, 26 Oct. 2010
q∆+
∆0
∗−
qΣ
∗0
q∆++
qΣ∗+
∗− q ∗0
q
TΞ Ξ
TT
qΩ−
ad iii: fits to 2-point functions in different channels
e.g. in pseudoscalar channel, Mπ from correlated fit
+ ~
+ ~
1 X
0tT h0|d̄γ5 u|π (0)ihπ (0)|ūγ5 d|0i −Mπ t
CPP (t) ≡
h[d̄γ5 u](x)[ūγ5 d](0)i −→
e
(L/a)3
2Mπ
~
x
aM
Effective mass aM(t + a/2) = log[C(t)/C(t + a)]
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Effective masses for simulation at
a ≈ 0.085 fm and Mπ ≈ 0.19 GeV
N
K
4
8
Gaussian sources and sinks with
r ∼ 0.32 fm (BMW ’08)
12
t/a
Laurent Lellouch
NCTS, Hsinchu, 26 Oct. 2010
ad iv: (I) Virtual pion loops around the world
In large volumes FVE ∼ e−Mπ L
Mπ L >
∼ 4 expected to give L → ∞ masses within our statistical errors
For a ≈ 0.125 fm and Mπ ≈ 0.33 GeV, perform FV study Mπ L = 3.5 → 7
0.8
volume dependence
MpL=4
0.225
volume dependence
MpL=4
-M L -3/2
0.22
aMN
aMp
-M L -3/2
c1+ c2 e p L
fit
Colangelo et. al. 2005
c1+ c2 e p L
fit
Colangelo et. al. 2005
0.75
0.215
0.7
0.21
12
16
20
24
L/a
28
32
36
12
16
20
24
L/a
28
Well described by (Dürr and Colangelo et al, 2005)
2
MX (L) − MX
Mπ
1
−Mπ L
=C
e
MX
πFπ
(Mπ L)3/2
Though very small, we fit them out
Laurent Lellouch
NCTS, Hsinchu, 26 Oct. 2010
32
36
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)
4
e.g., the ρ ↔ ππ system in the COM frame
Non-interacting:
E2π = 2(Mπ2 + k 2 )1/2 , ~k = 2π~n/L,
~n ∈ Z 3
E2π/Mπ
3.5
3
Interacting case: k solution of
2.5
nπ−δ11 (k ) = φ(q), n ∈ Z , q = kL/2π
2
Mρ/Mπ = 3
2
3
4
5
6
7
MπL
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
8
ph
ad v: extrapolation to mud
and interpolation to msph
Consider two approaches to the physical QCD limit for a hadron mass MX :
1
Mass-independent scale setting
2
Simulation-by-simulation normalization
For both normalization procedures, use parametrization
(0)
MX = MX + αK MK2 + απ Mπ2 + h.o.t.
linear term in MK2 is sufficient for interpolation to msph
ph
curvature in Mπ2 is visible in extrapolation to mud
in some channels
→ two options for h.o.t.:
ChPT: expansion about Mπ2 = 0 and h.o.t. ∝ Mπ3 (Langacker et al ’74)
Flavor: expansion about center of Mπ2 interval considered and h.o.t. ∝ Mπ4
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 extrapolation
Cutoff effects formally O(αs a) and O(a2 )
Small and cannot distinguish a and a2
Include through
MXph → MXph [1 + γX a]
or MXph [1 + γX a2 ]
→ difference used for systematic error estimation
not sensitive to ams or amud
2
Ω
N
M [GeV]
1.5
K
1
a~0.125 fm
a~0.085 fm
a~
~0.065 fm
0.5
physical Mπ
0
0.1
*
0.2
0.3
2
2
Mπ [GeV ]
Laurent Lellouch
0.4
0.5
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 continuum
extrapolation) 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 Ω
scale 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 bootstrap
sample by fit quality
median
median
0.25
Nucleon
0.3
Omega
0.2
0.2
0.15
0.1
0.1
0.05
900
920
940
MN [MeV]
960
980
1640
1660 1680
MΩ [MeV]
1700
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
1720
Post-dictions for the light hadron spectrum
2000
Budapest-Marseille-Wuppertal collaboration
O
X*
1500
M[MeV]
X
S
S*
D
L
1000
r
500
K*
N
experiment
K
width
input
p
QCD
0
Laurent Lellouch
NCTS, Hsinchu, 26 Oct. 2010
|Vus /Vud | from K , π → µν̄(γ)
In experiment see
_
u
1111
0000
0000
_1111
0000
1111
0000
π 1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000 d
1111
µ
W
∝
Vud h0|ūγµ γ5 d|π − i ∝ Vud Fπ
ν
Have (Marciano ’04, Flavianet ’08)
Γ(K → µν̄(γ))
|Vus | FK
−→
= 0.2760(6) [0.22%]
Γ(π → µν̄(γ))
|Vud | Fπ
⇒ 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 compute
FK /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
Dürr et al, PRD81 ’10
1.25
β=3.3
β=3.57
β=3.7
Taylor extrapolation
FK/Fπ
1.2
0.12
Distribution of values
Taylor
SU(2)
SU(3)
0.1
0.08
1.15
0.06
1.1
0.04
0.02
1.05
0
1.17
1
100
2
200
2
2
300
2
2
Mπ[MeV ]
400
1.175
1.18
1.185
1.19
1.195
1.2
1.205
2
FK
= 1.192(7)stat (6)syst [0.8%]
Fπ
phys
Main source of sytematic error: extrapolation mud → mud
; then a → 0
Laurent Lellouch
NCTS, Hsinchu, 26 Oct. 2010
1.21
FK /Fπ summary and CKM unitarity
FK/Fπ
1.14 1.16 1.18 1.20 1.22 1.24 1.26
Nf=2
Nf=2+1
BMW 10
JLQCD/TWQCD 09A
MILC 09A
MILC 09
ALVdW 08
PACS-CS 08, 08B
HPQCD/UKQCD 07
RBC/UKQCD 07
NPLQCD 06
MILC 04
x
our estimate for Nf = 2+1
ETM 09
ETM 07
QCDSF/UKQCD 07
our estimate for Nf = 2
x
FLAG ’10 estimates (direct)
FK
Fπ
FK
Fπ
=
1.193(6)
=
1.210(6)(17)
⇒ FK /Fπ , |Vud | and |Vus | in terms of lattice f+ (0)
(Flavianet Lattice Averaging Group (FLAG) ’10)
h
i
Gq2
2
2
2
|Vud | 1 + |Vus /Vud | + |Vub /Vud | = 1.0001(9) [0.09%]
Gµ2
If assume true result within 2σ then, naively, ΛNP ≥ 1.9 TeV
Laurent Lellouch
Nf = 2
(x) FLAG determinations assuming the SM: expt for ΓK ` ,
2
ΓK ` and |Vub | (neglible contribution), plus SM
3
universality and CKM unitarity
(|Vud |2 + |Vus |2 + |Vub |2 = 1)
1.14 1.16 1.18 1.20 1.22 1.24 1.26
Find
Nf = 2 + 1
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 Standard
Model (SM)
Stability of atoms, nuclear reactions which power stars, presence or absence of
strong 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 to
most 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
ph
⇒ must get very close to mud
to control mass dependence precisely
→ quark masses are an interesting first “measurement” to make w/ physical point
LQCD simulations
Laurent Lellouch
NCTS, Hsinchu, 26 Oct. 2010
Current knowledge of the light quark masses
FLAG has performed a detailed analysis of unquenched lattice determinations of the
light quark masses (“our estimate” = FLAG estimate)
mud
4
5
60
6
Nf=2+1
Blum 10
PACS-CS 09
HPQCD 09
MILC 09A
MILC 09
PACS-CS 08
RBC/UKQCD 08
CP-PACS 07
MILC 07
HPQCD 05
MILC 04
our estimate for Nf = 2+1
Nf=2
JLQCD/TWQCD 08A
RBC 07
ETM 07
QCDSF/UKQCD 06
SPQcdR 05
QCDSF/UKQCD 04
JLQCD 02
CP-PACS 01-03
our estimate for Nf = 2
2
3
MS
mud (2 GeV)
4
MeV
=
5
80
ms
100
120
Blum 10
PACS-CS 09
HPQCD 09
MILC 09A
MILC 09
PACS-CS 08
RBC/UKQCD 08
CP-PACS 07
MILC 07
HPQCD 05
MILC 04
our estimate for Nf = 2+1
RBC 07
ETM 07
QCDSF/UKQCD 06
SPQcdR 05
ALPHA 05
QCDSF/UKQCD 04
JLQCD 02
CP-PACS 01-03
our estimate for Nf = 2
Dominguez 09
Narison 06
Maltman 01
Dominguez 09
Chetyrkin 06
Jamin 06
Narison 06
Vainshtein 78
PDG 09
PDG 09
6
60
3.4(4) MeV [12%] FLAG
2.5 ÷ 5.0 MeV [30%] PDG
80
MS
ms (2 GeV)
100
MeV
=
120
⇒
phys
MILC,eff
mud
≥ 3.7 × mud
perturbative renormalization (albeit 2 loops)
Laurent Lellouch
140
95.(10) MeV [11%] FLAG
70 ÷ 130 MeV [30%] PDG
Even extensive study by MILC does not control all systematics:
MπRMS ≥ 260 MeV
140
Nf=2+1
3
Nf=2
2
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 to
allow small interpolation to physical mass point
(Mπ = 134.8(3) MeV)
Full nonperturbative renormalization and nonperturbative
continuum extrapolated running for determining
renormalization 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)
MILC ’10
QCDSF ’10
PACS-CS ’09
RBC/UK. ’10
HSC ’08
ETMC ’10
BMW ’08
This work
2
2
a [fm ]
2
0.100
2
0.050
0
0
2
200
2
300
2
2
Mπ [MeV ]
(from C. Hoelbling, Lattice 2010)
Laurent Lellouch
NCTS, Hsinchu, 26 Oct. 2010
2
400
Where do we stand?
. . . and w/ unitary, local gauge and fermion actions. . .
QCDSF ’10
PACS-CS ’09
RBC/UK. ’10
HSC ’08
ETMC ’10
BMW ’08
This work
2
2
a [fm ]
2
0.100
2
0.050
0
0
2
200
2
300
2
2
Mπ [MeV ]
Laurent Lellouch
NCTS, Hsinchu, 26 Oct. 2010
2
400
Where do we stand?
. . . and w/ sea u and d quarks clearly in the chiral regime, i.e. Mπmin ≤ 250 MeV. . .
PACS-CS ’09
BMW ’08
This work
2
2
2
a [fm ]
0.100
2
0.050
0
0
2
200
2
300
2
2
Mπ [MeV ]
Laurent Lellouch
NCTS, Hsinchu, 26 Oct. 2010
2
400
Where do we stand?
. . . and w/ sea u and d quarks at or below physical mass point . . .
PACS-CS ’09
This work
2
2
2
a [fm ]
0.100
2
0.050
0
0
2
200
Laurent Lellouch
2
300
2
2
Mπ [MeV ]
NCTS, Hsinchu, 26 Oct. 2010
2
400
Where do we stand?
. . . and w/ volumes such that FV errors ≤ 0.5% – PACS-CS has LMπ = 1.97 . . .
This work
2
2
2
a [fm ]
0.100
2
0.050
0
0
2
200
Laurent Lellouch
2
300
2
2
Mπ [MeV ]
NCTS, Hsinchu, 26 Oct. 2010
2
400
Where do we stand?
. . . and w/ at least three a ≤ 0.1 fm – PACS-CS has only 1 a ∼ 0.09 fm
This work
2
2
2
a [fm ]
0.100
2
0.050
0
0
2
200
Laurent Lellouch
2
300
2
2
Mπ [MeV ]
NCTS, Hsinchu, 26 Oct. 2010
2
400
Does our smearing enhance discretization errors?
⇒ scaling study: Nf = 3 w/ 2 HEX action, 4 lattice spacings (a ' 0.06 ÷ 0.15fm),
Mπ L > 4 fixed and
q
Mπ /Mρ = 2(MKph )2 − (Mπph )2 /Mφph ∼ 0.67
1750
1750
1700
1700
1650
M∆
MN
1600
1650
0.20 fm
0.18 fm
0.15 fm
0.15 fm
1800
M[MeV]
M[MeV]
1800
0.10 fm
0.06 fm
1850
0.10 fm
1850
0.06 fm
i.e. mq ∼ msph
M∆
MN
1600
1550
1550
1500
1500
1450
0
0.01
0.02
0.03
αa[fm]
0.04
0.05
0
0.06
0.005
0.01
0.015
0.02
2
a [fm ]
BMW ’10
BMW ’08
MN and M∆ are linear in αs a out to a ∼ 0.15 fm
⇒ very good scaling: discret. errors <
∼ 2% out to a ∼ 0.15 fm
Laurent Lellouch
2
NCTS, Hsinchu, 26 Oct. 2010
0.025
0.03
0.035
0.04
Does our smearing enhance discretization errors?
Perhaps 2 HEX works for spectral quantities but not for short distance dominated
quantities
⇒ 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/ Mπ L > 4 and fixed L ' 1.84fm
Calculate
(1 − amW /2)mW
m(µ) =
ZS (µ)
w/ mW = mbare − mcrit
Determine ZS (µ) using RI/MOM NPR (Martinelli et al ’95) and run nonperturbatively in
continuum to µ = 4 GeV (see below)
Interpolate in r0 MPS to r0 MKphys
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)
0.34
0.32
(mud+ms)r0
0.3
0.28
0.26
0.24
Garden et al., [Nucl.Phys.B 2000]
αa-extrapolation
0.22
0
0.02
0.04
0.06
0.08
0.1
αa/r0
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
Mπmin ' 135, 130, 120, 180, 220 MeV
L up to 6 fm and such that δFV ≤ 0.5% on Mπ for all runs
phys
10 + 3 different values of ms around ms
Determine lattice spacing using MΩ
17 + 4, Nf = 3 RI/MOM runs at same β as phenomenological runs:
phys
At least 4 mq ∈ [ms
phys
/3, ms
] per β for chiral extrapolation
L ≥ 1.7 fm in all runs
Laurent Lellouch
NCTS, Hsinchu, 26 Oct. 2010
Do we see chiral logs?
Simultaneous fit of Mπ2 and Fπ vs mud to NLO SU(2) χPT expressions (Gasser et al, ’84)
2 2 1
Λ
Λ4
3
Mπ2 = M 2 1 − x log
F
=
F
1
+
x
log
π
2
M2
M2
w/ M 2 = 2Bmud and x = M 2 /(4πF )2
Fixed a ' 0.09 fm and Mπ ' 130 → 400 MeV (preliminary)
2.8
0.055
aFπ
0.05
2.6
2
aMπ/mud
PCAC
2.7
0.045
2.5
2.4
0.04
0.005
0.01
0.005
Consistent w/ NLO χPT . . .
Laurent Lellouch
0.01
PCAC
amud
PCAC
amud
NCTS, Hsinchu, 26 Oct. 2010
VWI and AWI masses: ratio-difference method
With Nf = 2 + 1, O(a)-improved Wilson fermions, can construct the following
renormalized, O(a)-improved quantities (using Bhattacharya et al ’06)
(ms − mud )VWI
bS
bare 1
W
W
= (msbare − mud
)
1−
a(mud
+ msW ) − b̄S a(2mud
+ msW ) + O(a2 )
ZS
2
w/ mW = mbare − mcrit and
i
msPCAC h
msAWI
bare
bare
= PCAC 1 + (bA − bP ) a(ms − mud )
AWI
mud
mud
w/
mPCAC
P ¯
1 ~x h∂µ [Aµ (x) + acA ∂µ P(x)] P(0)i
P
≡
2
~
x hP(x)P(0)i
and bA,P,S = 1 + O(αs ), b̄A,P,S = O(αs2 ), cA = O(αs )
Laurent Lellouch
NCTS, Hsinchu, 26 Oct. 2010
Ratio-difference method (cont’d)
Define
d≡
amsbare
−
bare
amud
,
msPCAC
r ≡ PCAC
mud
and subtracted bare masses
sub
amud
≡
d
,
r −1
amssub ≡
rd
r −1
Then, with our tree-level O(a)-improvement, renormalized masses can be written
mud
i
sub h
a sub
mud
sub
1 − (mud + ms ) + O(αs a)
=
ZS
2
i
mssub h
a sub
sub
ms =
1 − (mud + ms ) + O(αs a)
ZS
2
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 ZSRI (µ, a) nonperturbatively in RI/MOM scheme, from truncated, forward
quark two-point functions in Landau gauge (Martinelli et al ’95), computed on specifically
generated Nf = 3 gauge configurations
Use S(p) → S̄(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(αs a) errors are small for all β
For coarsest (β = 3.31) lattice, 2π/a ' 11 GeV
Restrict study of ZSRI (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 still
significant 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
ZSRI (4 GeV, a)
R (µren , 4 GeV) = lim
a→0 Z RI (µren , a)
S
RI
0.9
^ RI 2
ZS (µ )
0.8
Preliminary
0.7
Running is very similar at all 4 β
β=3.8
β=3.7
β=3.61
β=3.5
β=3.31
0.6
⇒ flat a → 0 extrapolation
0.5
0
10
20
2
µ [GeV ]
30
40
2
Rescaled ZSRI (µ, aβ ) for β < 3.8 to ∼ match ZSRI (µ, aβ=3.8 )
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 finer
lattices
Preliminary
β=3.8
µ=4GeV
2
ZS,nonpert(µ )/ZS,4-loop(µ )
1.2
RI
2
RI
1.1
1
0.9
0.8
10
20
2
µ [GeV ]
30
2
Laurent Lellouch
NCTS, Hsinchu, 26 Oct. 2010
40
Continuum extrapolation of renormalized masses
Renormalized quark masses interpolated in Mπ2 & MK2 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
Preliminary
120
0
0
0
0.01
0.02
αa[fm]
0.03
0.04
Laurent Lellouch
0
0.01
NCTS, Hsinchu, 26 Oct. 2010
0.02
αa[fm]
0.03
a=0.12 fm
a=0.10 fm
a=0.06 fm
40
a=0.08 fm
80
RI
a=0.12 fm
a=0.10 fm
a=0.06 fm
1
a=0.08 fm
2
ms (4GeV) [MeV]
3
RI
mud(4GeV) [MeV]
4
0.04
Continuum extrapolation of renormalized masses
Renormalized quark masses interpolated in Mπ2 & MK2 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
Preliminary
σχ w/ Mπ >= 120 MeV
120
0
0
0.01
0.02
αa[fm]
0.03
0.04
Laurent Lellouch
0
0
0.01
NCTS, Hsinchu, 26 Oct. 2010
0.02
αa[fm]
0.03
a=0.12 fm
a=0.10 fm
a=0.06 fm
40
a=0.08 fm
80
RI
a=0.12 fm
a=0.10 fm
a=0.06 fm
1
a=0.08 fm
2
ms (4GeV) [MeV]
3
RI
mud(4GeV) [MeV]
4
0.04
Continuum extrapolation of renormalized masses
Renormalized quark masses interpolated in Mπ2 & MK2 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 ≥ Mπval |min
MILC ' 180 MeV
Preliminary
Preliminary
σχ w/ Mπ >= 120 MeV
0
0.01
0.02
αa[fm]
0.03
0.04
Laurent Lellouch
0
0
0.01
NCTS, Hsinchu, 26 Oct. 2010
0.02
αa[fm]
0.03
a=0.12 fm
a=0.10 fm
a=0.06 fm
40
a=0.08 fm
80
RI
a=0.12 fm
a=0.10 fm
a=0.06 fm
1
a=0.08 fm
2
ms (4GeV) [MeV]
3
RI
mud(4GeV) [MeV]
4
0
120
σχ w/ Mπ >= 180 MeV
0.04
Continuum extrapolation of renormalized masses
Renormalized quark masses interpolated in Mπ2 & MK2 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 ≥ MπRMS |min
MILC ' 260 MeV (very small range)
Preliminary
Preliminary
σχ w/ Mπ >= 120 MeV
0
0.01
0.02
αa[fm]
0.03
0.04
Laurent Lellouch
0
0
0.01
NCTS, Hsinchu, 26 Oct. 2010
0.02
αa[fm]
0.03
a=0.12 fm
a=0.10 fm
40
a=0.08 fm
a=0.06 fm
a=0.12 fm
a=0.10 fm
a=0.06 fm
1
a=0.08 fm
2
80
RI
3
ms (4GeV) [MeV]
σχ w/ Mπ >= 260 MeV
RI
mud(4GeV) [MeV]
4
0
120
σχ w/ Mπ >= 180 MeV
0.04
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
phys
Nf = 2 + 1 simulations have been performed all the way down to mud
and below w/
ms ' msphys :
5 a ' 0.054 ÷ 0.116 fm
Mπmin ' 135, 130, 120, 180, 220 MeV
L up to 6 fm and such that δFV ≤ 0.5% on Mπ for all runs
phys
→ eliminates large systematic error associated w/ reaching mud
Described an RI/MOM procedure which includes continuum limit, nonperturbative
running
→ 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 by
fixing 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 times
smaller than PDG!
Ms̄s for ms – but ms is needed to determine Ms̄s !?
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/ MπRMS ≥ 260 MeV
Imposing the cut Mπ ≥ 260 MeV on our results
⇒ δχ mud ∼ 0.1% −→ δχ mud ∼ 15%
Imposing the cut Mπ ≥ 180 MeV (lightest MILC valence pion) on our results
⇒ δχ mud ∼ 0.1% −→ δχ mud ∼ 2%
⇒ smaller error requires assumptions on mass dependence of results which go
beyond (partially quenched) NLO SU(2) χPT
Fully controlled LQCD calculations can now be envisaged w/out any assumptions
on 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-IDRIS
Paris
139 Tflop/s peak
IBM Blue Gene/P (JUGENE), FZ Jülich
1. Pflop/s peak
BULL cluster (1024 Nehalem 8 core
nodes), 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