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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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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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%
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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)
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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)
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
QCDSF/UKQCD 06ETM 07RBC 07
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
MILC 09MILC 09AHPQCD 09PACS-CS 09Blum 10
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)
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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)
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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)
Laurent Lellouch STONGnet 2011, 4-7 october 2011
Comparison
2
2
3
3
4
4
5
5
6
6
MeV
CP-PACS 01JLQCD 02
SPQcdR 05QCDSF/UKQCD 04
QCDSF/UKQCD 06ETM 07RBC 07
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
MILC 09MILC 09AHPQCD 09PACS-CS 09Blum 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
QCDSF/UKQCD 06ETM 07RBC 07
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
MILC 09MILC 09AHPQCD 09PACS-CS 09Blum 10
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
)
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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%]
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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)
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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)
Laurent Lellouch STONGnet 2011, 4-7 october 2011
Operator contributions to BK
(β = 3.61, Mπ ∼ 120 MeV)
→ contributions from Q2,...,5 small and often consistent with zero
→ nevertheless, dominate systematic error
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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)
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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))
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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
Laurent Lellouch STONGnet 2011, 4-7 october 2011
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 STONGnet 2011, 4-7 october 2011
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 STONGnet 2011, 4-7 october 2011