Physical predictions from lattice QCD Christian Hoelbling Bergische - - PowerPoint PPT Presentation
Introduction Hadron spectrum Quark masses etc. Summary Physical predictions from lattice QCD Christian Hoelbling Bergische Universitt Wuppertal Budapest: S. Katz Marseille: L. Lellouch, A. Portelli, A. Sastre Wuppertal: Sz. Borsanyi, S.
Introduction Hadron spectrum Quark masses etc. Summary
Budapest: S. Katz Marseille: L. Lellouch, A. Portelli, A. Sastre Wuppertal: Sz. Borsanyi, S. Durr, Z. Fodor, S. Krieg, T. Kurth,
Christian Hoelbling Bergische Universität Wuppertal FFP 14, Marseille July 17, 2014
1/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary How to stay clean in the brown muck
QCD fundamental objects: quarks and gluons QCD observed objects: protons, neutrons (π, K, . . . ) ! Huge discrepancy: not even the same particles observed as in the Lagrangean ➛ Perturbation theory has no chance Need to solve low energy QCD to:
Compute hadronic and nuclear properties “people who love QCD”
Masses, decay widths, scattering lengths, thermodynamic properties, . . .
Compute hadronic background “people who hate QCD”
Non-leptonic weak MEs, quark masses, g-2, . . .
2/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary How to stay clean in the brown muck
UV cutoff: space-time lattice Hypercubic, spacing a Momentum cutoff pµ < 2π/a IR cutoff on finite lattice
U (x+e )
μ
Ψ (x) a
μ
μ ν
☞ anti-commuting quark fields ψ(x) live on the sites ☞ gluon fields Uµ(x) = eig
x+ˆ
µ x
dzµ Aµ(z) ∈ SU(3) live on links
Essential: QCD perturbative on cutoff scale 1/a ≫ ΛQCD (asymptotic freedom) Perform Euclidean path integral stochastically
3/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary How to stay clean in the brown muck
Lattice QCD=QCD when Cutoff removed (continuum limit) Infinite volume limit taken At physical hadron masses (Especially π)
Numerically challenging to reach light quark masses
Statistical error from stochastic estimate of the path integral
4/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Lattice setup
0.05 0.1 0.15 a[fm] 200 400 600 Mπ[MeV]
ETMC '09 (2) ETMC '10 (2+1+1) MILC '10 MILC '12 QCDSF '10 (2) QCDSF-UKQCD '10 BMWc '10 BMWc'08 PACS-CS '09 RBC/UKQCD '10 JLQCD/TWQCD '09 HSC '08 BGR '10 CLS '10 (2)
5/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Lattice setup
100 200 300 400 500 600 700 Mπ[MeV] 1 2 3 4 5 6 L[fm]
ETMC '09 (2) ETMC '10 (2+1+1) MILC '10 MILC '12 QCDSF '10 (2) QCDSF-UKQCD '10 BMWc '10 BMWc'08 PACS-CS '09 RBC/UKQCD '10 JLQCD/TWQCD '09 HSC '08 BGR '10 (2) CLS '10(2)
0.1% 0.3% 1%
6/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Lattice setup
200 400 600 Mπ[MeV] 200 400 600 800 (2MK
2-Mπ 2) 1/2[MeV]
ETMC '10 (2+1+1) MILC '10 QCDSF-UKQCD '10 BMWc'10 PACS-CS '09 JLQCD/TWQCD '09 RBC-UKQCD '10 HSC '08
7/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Lattice setup
Compute target observable Extrapolate to physical point Renormalize if necessary
0.05 0.1 0.15 a[fm] 200 400 600 Mπ[MeV] 200 400 600 Mπ[MeV] 200 400 600 800 (2MK
2-Mπ 2) 1/2[MeV]
100 200 300 400 500 600 700 Mπ[MeV] 1 2 3 4 5 6 L[fm] 0.1% 0.3% 1%
8/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary We have done our homework
With operators that couple to the ground state (e.g.to the π+) Aµ(t) =
Ψdγµγ5Ψu ( x, t)
C(t) = A†
0(t)A0(0) t→∞
− → |π|A0|0|2 2Mπ e−Mπt ln C(t) C(t + 1)
t→∞
− → Meff
π
t
d u
A0 A0
9/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary We have done our homework
4 8 12 t/a 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 a M
10/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary We have done our homework
0.1 0.2 0.3 0.4 0.5
Mp
2 [GeV 2]
0.5 1 1.5 2 M [GeV] physical Mp
a~ ~0.085 fm a~ ~0.065 fm a~ ~0.125 fm
11/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary We have done our homework
The easy part: Virtual pion finite V effects Hadrons see mirror charges Exponential in lightest particle (pion) mass Leading effects MX (L)−MX
MX
= cM1/2
π
L−3/2eMπL
(Colangelo et. al., 2005)
12 16 20 24 28 32 36 L/a 0.21 0.215 0.22 0.225 aMp c1+ c2 e
L
L
Colangelo et. al. 2005
Pion
MpL=4 12 16 20 24 28 32 36 L/a 0.7 0.75 0.8 aMN c1+ c2 e
L
L
Colangelo et. al. 2005 MpL=4
Nucleon
More severe (if present): FV correction in resonances QED: 1/L terms from photons (Davoudi, Savage 2014; BMWc 2014)
12/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary We have done our homework
Method: Large number of analyses including“all reasonable” choices Construct (weighted) distribution of results
Median of this distribution ➛final result Central 68% ➛systematic error
Statistical error from bootstrap of the medians
900 920 940 960 980 MN [MeV] 0.05 0.1 0.15 0.2 0.25 median
Nucleon
1640 1660 1680 1700 1720 MΩ [MeV] 0.1 0.2 0.3 median
Omega
13/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary We have done our homework
p K r K* N L S X D S* X* O
experiment width input
Budapest-Marseille-Wuppertal collaboration
QCD
(BMWc, 2008) 14/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary We have done our homework
Extracting excited states is much tougher: ☞ Extraction of energy levels is harder: Die out at large t ➠ need to use small t correlators ☞ Once extracted, relation to V → ∞ is nontrivial: Disentangle resonances and scattering states at finite volume
Spectral density Finite volume energy levels L E
15/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Relevance of fine structure
200 400 600 800 1000 mass [MeV/c
2]
QCD QED
quark mass
Proton, neutron: 3 quarks Proton: uud Neutron: udd mu<md:Mp < Mn mu=md:Mp > Mn Proton decays Mp + Me− Mn No hydrogen
16/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Relevance of fine structure
935 936 937 938 939 940 mass [MeV/c
2]
QCD QED
quark mass
Proton, neutron: 3 quarks Proton: uud Neutron: udd mu<md:Mp < Mn mu=md:Mp > Mn Proton decays Mp + Me− Mn No hydrogen
16/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Relevance of fine structure
935 936 937 938 939 940 mass [MeV/c
2]
QCD QED
quark mass
Proton, neutron: 3 quarks Proton: uud Neutron: udd mu<md:Mp < Mn mu=md:Mp > Mn Proton decays Mp + Me− Mn No hydrogen
16/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Relevance of fine structure
935 936 937 938 939 940 mass [MeV/c
2]
QED
electron mass quark mass
Proton, neutron: 3 quarks Proton: uud Neutron: udd mu<md:Mp < Mn mu=md:Mp > Mn Proton decays Mp + Me− Mn No hydrogen
16/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Relevance of fine structure
1st generation: mu<md Why? 2nd generation: mc>ms 3rd generation: mt>mb
17/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Relevance of fine structure
,
K
3
p n
,
3
3
QCD: ∼ md−mu
ΛQCD
∼ 1% QED: ∼ α(Qu − Qd)2 ∼ 1%
On the lattice:
Include nondegenerate light quarks mu = md Include QED
18/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Finite volume
Effective theory only (UV completion unclear) π+, p, etc. no more gauge invariant QED (additive) mass renormalization Power law FV effects (soft photons) EM field of a point charge cannot be made periodic & continuous Remove p = 0 modes in fixed gauge (Hayakawa, Uno, 2008)
19/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Finite volume
Universal to O(1/L2) Divergent T dependence for p = 0 mode subtraction No T dependence for
subtraction
0.1858 0.186 0.1862 0.1864 0.1866 0.1868 0.187 0.1872 0.01 0.02 0.03 0.04 0.05 am a/L T/L=8 T/L=3 T/L=2 T=64 1-loop prediction A(p=0)=0 A(p=0)=0
δm = q2α κ 2mL
mL − 3π (mL)3
20/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Results
2 4 6 8 10
ΔM [MeV] ΔN ΔΣ ΔΞ ΔD ΔCG ΔΞcc experiment QCD+QED prediction
BMW 2014 HCH(BMWc 2014) 21/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary QCD and QED contributions
Problem: Disentangle QCD and QED contributions
Not unique, O(α2) ambiguities
Flavor singlet (e.g. π0) difficult (disconnected diagrams)
y x y x
Method: Use baryonic splitting Σ+-Σ− purely QCD
Only physical particles Exactly correct for pointlike particle Corrections below the statistical error
22/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary QCD and QED contributions
1 2
α/αphys
1 2
(md-mu)/(md-mu)phys physical point
1 MeV 2 MeV 3 MeV 4 MeV
I n v e r s e β d e c a y r e g i
(BMWc 2014) 23/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Quark masses
Goal: Compute light quark masses ab initio Alternative view: Translate fundamental parameters (Mπ, MK) into perturbatively useful quantities Method: Go to the physical point Read off input quark masses and renormalize Challenge: Minimize and control all systematics
2+1 dynamical fermion flavors Physical quark masses Continuum extrapolation Infinite volume Nonperturbative renormalization
24/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Quark masses
Quark masses logarithmically divergent (a → 0) ➛ renormalization Usually MS scheme: only perturbatively defined ☞ RI-MOM scheme matrix elements of off-shell quarks in fixed gauge
Renormalization condition: at p2 = µ2 tree level matrix element
10 20 30 40 μ
2[GeV 2]
0.5 0.6 0.7 0.8 0.9 Z ^
S RI(μ 2)
β=3.8 β=3.7 β=3.61 β=3.5 β=3.31
25/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Quark masses
a=0.06 fm a=0.08 fm a=0.10 fm a=0.12 fm 40 80 120 ms
RI(4GeV) [MeV]
0.01 0.02 0.03 0.04 αa[fm]
26/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Quark masses Collaboration p u b l i c a t i
s t a t u s c h i r a l e x t r a p
a t i
c
t i n u u m e x t r a p
a t i
fi n i t e v
u m e r e n
m a l i z a t i
r u n n i n g mud ms RBC/UKQCD 12 A
3.37(9)(7)(1)(2) 92.3(1.9)(0.9)(0.4)(0.8) PACS-CS 12 A b 3.12(24)(8) 83.60(0.58)(2.23) Laiho 11 C
3.31(7)(20)(17) 94.2(1.4)(3.2)(4.7) BMW 10A, 10B+ A c 3.469(47)(48) 95.5(1.1)(1.5) PACS-CS 10 A b 2.78(27) 86.7(2.3) MILC 10A C
3.19(4)(5)(16) – HPQCD 10∗ A
− 3.39(6) 92.2(1.3) RBC/UKQCD 10A A
3.59(13)(14)(8) 96.2(1.6)(0.2)(2.1) Blum 10† A
3.44(12)(22) 97.6(2.9)(5.5) PACS-CS 09 A b 2.97(28)(3) 92.75(58)(95) HPQCD 09A⊕ A
− 3.40(7) 92.4(1.5) MILC 09A C
3.25 (1)(7)(16)(0) 89.0(0.2)(1.6)(4.5)(0.1) MILC 09 A
3.2(0)(1)(2)(0) 88(0)(3)(4)(0) PACS-CS 08 A − 2.527(47) 72.72(78) RBC/UKQCD 08 A
3.72(16)(33)(18) 107.3(4.4)(9.7)(4.9) CP-PACS/ JLQCD 07 A − 3.55(19)(+56
−20)
90.1(4.3)(+16.7
−4.3 )
HPQCD 05 A
3.2(0)(2)(2)(0)‡ 87(0)(4)(4)(0)‡ MILC 04, HPQCD/ MILC/UKQCD 04 A
2.8(0)(1)(3)(0) 76(0)(3)(7)(0) (FLAG group, 2013) 27/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Quark masses Collaboration publication status chiral extrapolation continuum extrapolation finite volume renormalization running mu md mu/md PACS-CS 12 A a 2.57(26)(7) 3.68(29)(10) 0.698(51) Laiho 11 C ◦
1.90(8)(21)(10) 4.73(9)(27)(24) 0.401(13)(45) HPQCD 10‡ A ◦ − 2.01(14) 4.77(15) BMW 10A, 10B+ A b 2.15(03)(10) 4.79(07)(12) 0.448(06)(29) Blum 10† A ◦
2.24(10)(34) 4.65(15)(32) 0.4818(96)(860) MILC 09A C ◦
1.96(0)(6)(10)(12) 4.53(1)(8)(23)(12) 0.432(1)(9)(0)(39) MILC 09 A ◦
1.9(0)(1)(1)(1) 4.6(0)(2)(2)(1) 0.42(0)(1)(0)(4) MILC 04, HPQCD/ MILC/UKQCD 04 A ◦ ◦ ◦ − 1.7(0)(1)(2)(2) 3.9(0)(1)(4)(2) 0.43(0)(1)(0)(8) RM123 13 A ◦
2.40(15)(17) 4.80 (15)(17) 0.50(2)(3) RM123 11⊕ A ◦
2.43(11)(23) 4.78(11)(23) 0.51(2)(4) 1 1 r r u ¨ D
∗
A ◦
− 2.18(6)(11) 4.87(14)(16) RBC 07† A − 3.02(27)(19) 5.49(20)(34) 0.550(31) (FLAG group, 2013) 28/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Decay contants
With the axial vector current Aµ(t) =
Ψdγµγ5Ψu ( x, t)
A†
0(t)A0(0) t→∞
− → |π|A0|0|2 2Mπ e−Mπt = M2
πF bare π 2
2Mπ e−Mπt
t
d u
A0 A0
29/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Decay contants
90 100 110 120 130 140 10 20 30 40 50 60 70 80 90 Fπ [MeV] mRGI
ud
[MeV] β=3.5 β=3.61 β=3.7 β=3.8
χ 2 # dof = 1.52, #dof = 17, p-value = 0.08
Exp value (NOT INCLUDED)
30/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Decay contants
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 86 86.5 87 87.5 88 88.5 89 89.5 90 F [MeV ]
Fπ = 92.9(9)(2)
(BMWc 2014) 31/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Matrix elements
Matrix element of an effective weak operator (e.g. K †|O|K): JL
†(t+)O(0)JL 0(t−) t±→±∞
− → |K|JL
0|0|2
(2MK)2 K †|O|Ke−MK (t+−t−)
O
ΔS=2
P P t+ t-
where JL
0= [¯
sd]V−A = ¯ sγ0(1 − γ5)d O∆S=2= [¯ sd]V−A[¯ sd]V−A
W W u,c,t u,c,t d d s s s d s d
Norm from: JL
†(t)JL 0(0) t→∞
− → |K|JL
0|0|2
2MK e−MK t
32/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Matrix elements
☞ χSB induces mixing with 4 unphysical operators ☞ Mixing terms chirally enhanced ✓ Small even below physical mπ ✓ Good chirality of our action
20 40 60 80 100 Q1 Q2 Q3 Q4 Q5 contribution to BK[%]
0.001 0.002 5 10 15 20 25 30 35 40 45 Δsub
14
μ2[GeV2] a≈0.077 fm 5 10 15 20 25 30 35 40 45 μ2[GeV2] a≈0.077 fm 10 20 30 40 50 60 70 80 90 μ2[GeV2] a≈0.054 fm
33/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Matrix elements
0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 BK
RI(3.5 GeV)
M 2[GeV2] a 0.093 fm a 0.077 fm a 0.065 fm a 0.054 fm cont-limit
34/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Matrix elements
0.49 0.5 0.51 0.52 0.53 0.54 0.55 0.56 0.005 0.01 0.015 0.02 0.025 0.03 0.035 BK
RI(3.5 GeV)
sa[fm]
35/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Matrix elements
BK
RI(3.5 GeV)
0.51 0.52 0.53 0.54 0.55 0.51 0.52 0.53 0.54 0.55
36/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Matrix elements
BK UT prediction (CKM Fitter, 2012) BMW-c (2011, 2 HEX-CIW) SWME (2011, HYP-STAG/MILC) Laiho, Van de Water (2011, STAG/MILC) Aubin et al. (2010, DW/MILC) RBC-UKQCD (2010, DW) ETMC (2009, TM) 0.6 0.8 1 1.2 SWME (2014, HYP-STAG/MILC)
37/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary Progress since 2008
π K N Σ Ξ Ω
experiment input QCD+QED
Ξ Ξ
+
n p K
±
K
±
+ 40 + 40 + 40 +
QCD (2008)
(BMWc 2014) 38/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
39/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
40/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
Goal: Optimize physics results per CPU time Conceptually clean formulation Method: Dynamical 2 + 1 flavor, Wilson fermions at physical Mπ 3-5 lattice spacings 0.053 fm < a < 0.125 fm Tree level O(a2) improved gauge action (Lüscher, Weisz, 1985) Tree level O(a) improved fermion action (Sheikholeslami, Wohlert, 1985)
Why not go beyond tree level?
Keeping it simple (parameter fine tuning) No real improvement, UV mode suppression took care of this
This is a crucial advantage of our approach
UV filtering (APE coll. 1985; Hasenfratz, Knechtli, 2001; Capitani, Durr, C.H., 2006) ➛ Discretization effects of O(αsa, a2)
✓ We include both O(αsa) and O(a2) into systematic error
41/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
5 10 15 20 25 2k 4k 6k 8k 10k Gauge force RHMC force Pf0 force Pf1 force Pf2 force Pf3 force Pf4 force
42/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
Inverse iteration count (1000/Ncg)
β=3.31, Mπ≈135 MeV
0.04 0.08 0.12
β=3.5, Mπ≈130 MeV β=3.61, Mπ≈120 MeV
0.04 0.08 0.12
β=3.7, Mπ≈180 MeV β=3.8, Mπ≈220 MeV
43/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
worst case
Topological charge β=3.8, mud=-0.02, ms=0
2 4 1000 2000 3000 4000 5000 10 HYP 30 HYP
2 4
1 1000 2000 3000 4000 5000 ΔEMD
44/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
τint = 27.3(7.4)
(MATLAB code from Wolff, 2004-7)
50 100 150 200 −0.5 0.5 1
normalized autocorrelation for |qren| at β=3.8, mud=−0.02, ms=0 τ ρ
50 100 150 200 10 20 30 40 50
τ
int
with statistical errors for |qren| at β=3.8, m
ud
=−0.02, m
s
=0 τint fit window upper bound τmax
45/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
smearing
locality in position space: |D(x, y)| < const e−λ|x−y| with λ=O(a−1) for all couplings. Our case: D(x, y)=0 as soon as |x −y|>1 (despite smearing) locality of gauge field coupling: |δD(x, y)/δA(z)| < const e−λ|(x+y)/2−z| with λ=O(a−1) for all couplings. Our case: δD(x, x)/δA(z) < const e−λ|x−z| with λ≃2.2a−1 for 2≤|x −z|≤6
46/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
6-stout case:
1 2 3 4 5 6 7
|z|/a
10
10
10
10
10
10
10
a~ ~0.125 fm a~ ~0.085 fm a~ ~0.065 fm
47/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
Simultaneous fit to NLO SU(2) χPT (Gasser, Leutwyler, 1984) Consistent for Mπ 400 MeV
0.005 0.01
amud
PCAC
2.4 2.5 2.6 2.7 2.8
aMπ
2/mud PCAC
0.005 0.01
amud
PCAC
0.04 0.045 0.05 0.055
aFπ
➛ We use 2 safe chiral interpolation ranges: Mπ < 340, 380 MeV ➛ We use SU(2) χPT and Taylor interpolation forms
48/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
Nucleon ∆ Ω
(Bulava, et al., 2010)
✓ Qualitative understanding of experimental spectrum ✗ No extrapolation to physical point, continuum
49/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
SQCD+QED = Siso
QCD + 1
2(mu − md)
uu − ¯ dd) + ie
with jµ = ¯ qQγµq Method 1: operator insertion (RM123 ’12-’13)
O = Oiso
QCD−1
2(mu − md)O
uu − ¯ dd)iso
QCD+1
2e2O
jµ(x)Dµν(x − y)jν(y)iso
QCD+ .
✓ Use existing ensembles ✓ Going beyond O(mu − md) or O(α) difficult, but rarely necessary ✗ Complicated observables and renormalization ✗ Disconnected diagrams
50/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
Method 2: direct calculation ✓ Complete solution, observables “as usual” ✗ Only partially implemented yet
☞ mu = md valence only (MILC ’09, BMWc ’10-, Blum et al ’10, RBC/UKQCD ’12,. . .) ☞ QED valence only (Eichten ’97, Blum et al ’07-, BMWc ’10-, MILC ’10-) ☞ mu = md
(PACS-CS ’12) and QED (Blum et al ’12) reweighting
✓ Valence approximation reasonable
☞ Errors O(ααs), O((mu − md)2)
In this talk: QCD and QED isospin breaking, valence only Compact QED, Coulomb gauge ➛ linear
51/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
Problem: Parameterize QCD and QED splitting Method: Use ∆M2 = M2
uu − M2 dd to parameterize QCD splitting
Use αQED to parameterize QED splitting ∆MX = ∆M2CX + αQEDDX CX= c0
X + c1 X ˆ
M2
π + c2 X ˆ
M2
K + c3 Xf(a)
DX= d0
X + d1 X ˆ
M2
π + d2 X ˆ
M2
K + d3 Xa + d4 X
1 L WIP: use physical (hadronic) definition
52/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
− 5500 − 5000 − 4500 − 4000 − 3500 20 40 60 80 100
K [MeV 2] 1 L [MeV]
fit a = 0.11 fm a = 0.09 fm a = 0.07 fm a = 0.06 fm a = 0.05 fm
53/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
− 6000 − 5000 − 4000 − 3000 − 2000
αφ
0.003 0.006 0.009 0.012
K [MeV 2]
fit a = 0.11 fm a = 0.09 fm a = 0.07 fm a = 0.06 fm a = 0.05 fm
54/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
− 4200 − 4000 − 3800 − 3600 − 3400 − 3200 M φ2
π +
2002 2502 3002 3502 4002
K [MeV 2]
π + [MeV 2]
fit a = 0.11 fm a = 0.09 fm a = 0.07 fm a = 0.06 fm a = 0.05 fm
55/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
Goal: Compute light quark masses ab initio Relevance: Fundamental SM parameters Stability of matter depends on their values Not obtainable perturbatively Challenge: Minimize and control all systematics
2+1 dynamical fermion flavors Physical quark masses Continuum extrapolation Nonperturbative renormalization Infinite volume
56/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
Goal:
Full nonperturbative renormalization Optional accurate conversion to perturbative scheme
Method:
We use RI-MOM scheme (Martinelli et. al., 1993)
O(a) correction (Maillart, Niedermayer, 2008)
Compute mq at low scale µ ≪ 2π/a ∼ 11 − 24 GeV
µ = 2.1GeV µ = 1.3GeV
Do continuum non-perturbative running to high scale µ′ ≫ ΛQCD Further conversion in 4-loop PT
57/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
2 4 6 8 10 µ[GeV] 0.6 0.7 0.8 0.9 1
ZS(2-loop)/ZS(1-loop) ZS(3-loop)/ZS(2-loop) ZS(4-loop)/ZS(3-loop) ZS(4-loop/ana)/ZS(4-loop)
58/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
10 20 30 40 µ
2[GeV 2]
0.8 0.9 1 1.1 1.2 Z
RI S,nonpert(µ 2)/Z RI S,4-loop(µ 2)
µ=4GeV β=3.8
59/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
2 4 6 8 10 µ[GeV] 0.6 0.7 0.8 0.9 1
ZS(2-loop)/ZS(1-loop) ZS(3-loop)/ZS(2-loop) ZS(4-loop)/ZS(3-loop) ZS(4-loop/ana)/ZS(4-loop)
60/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
Lagrangian mass mbare mren =
1 ZS (mbare − mbare crit )
mPCAC from
∂0A0P P(t)P(0)
mren = ZA
ZP mPCAC
Better use d = mbare
s
− mbare
ud
dren =
1 ZS d
r = mPCAC
s
/mPCAC
ud
r ren = r and reconstruct mren
s
=
1 ZS rd r−1
mren
ud = 1 ZS d r−1
✓ No additive mass renormalization and ambiguity in mcrit ✓ Only ZS multiplicative renormalization (no pion poles) ☞ Works with O(a) improvement (we use this)
61/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
16 24 32 L/a 0.14 0.15 0.16 0.17 0.18 0.19 aMπ ignored in final analysis MπL=4 MπL=3
FV effects tiny Dedicated FV runs Perfect agreement with FV χPT (Colangelo
62/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
a=0.06 fm a=0.08 fm a=0.10 fm a=0.12 fm 1 2 3 4 mud
RI(4GeV) [MeV]
0.01 0.02 0.03 0.04 αa[fm]
63/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
Goal:
Compute mu and md separately
Method:
Need QED and isospin breaking effects in principle Alternative: use dispersive input -Q from η → πππ Q2 = 1
2
mud
2 md−mu
mud
✓ Transform precise ms/mud into (md − mu)/mud We use the conservative Q = 22.3(8) (Leutwyler, 2009)
64/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
Goal:
Reliably estimate total systematic error
Method:
288 full analyses (2000 bootstrap on each)
2 plateaux regions 2 continuum forms: O(αsa), O(a2) 3 chiral forms: 2 × SU(2), Taylor 2 chiral ranges: Mπ < 340, 380 MeV 3 renormalization matching procedures 2 NP continuum running forms 2 scale setting procedures
All analyses weighted by fit quality
Mean gives final result Stdev gives systematic error
Statistical error from 2000 bootstrap samples
65/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
RI @ 4 GeV RGI MS @ 2 GeV ms 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(04)(10) 2.86(05)(13) 2.15(03)(10) md 4.84(07)(12) 6.39(09)(15) 4.79(07)(12) Additional consistency checks: ✓ Use mPCAC only, no ratio-difference method ☞compatible, slightly larger error ✓ Unweighted final result and systematic error ☞negligible impact ✓ Additional Continuum, chiral and FV terms ☞all compatible with 0
66/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
2 2 3 3 4 4 5 5 6 6 MeV CP-PACS 01-03 JLQCD 02 SPQcdR 05 QCDSF/UKQCD 04-06 QCDSF/UKQCD 04-06 ETM 07 RBC 07 MILC 04 HPQCD 05 MILC 07 CP-PACS 07 RBC/UKQCD 08 PACS-CS 08 FLAG estimate Dominguez 09
mud
Narison 06 Maltman 01 This work MILC 09 MILC 09A HPQCD 09 PACS-CS 09 PDG 09 60 60 80 80 100 100 120 120 140 140 MeV CP-PACS 01-03 JLQCD 02 ALPHA 05 SPQcdR 05 QCDSF/UKQCD 04-06 QCDSF/UKQCD 04-06 ETM 07 RBC 07 MILC 04 HPQCD 05 MILC 07 CP-PACS 07 RBC/UKQCD 08 PACS-CS 08 FLAG estimate PDG 09
ms
Nf=2+1 Nf=2 Dominguez 09 Chetyrkin 06 Jamin 06 Narison 06 Vainshtein 78 MILC 09 MILC 09A HPQCD 09 PACS-CS 09 This work
67/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
Goal:
Check SM CP violation in neutral K system
Method:
Compute effective weak matrix element Relate kaon CP violation to CKM phase
Challenge:
Minimize and control all systematics
2+1 dynamical fermion flavors Physical quark masses Mixing of unphysical operators Continuum Infinite volume
68/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
BK
bare()
/a 0.45 0.5 0.55 0.6 0.65 0.7 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
69/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
0.94 0.96 0.98 1 1.02 1.04 1.06 3 4 5 6 7 8 9 10 11 12 RRI
BK(µ, 3.5 GeV)/RBK RI 2-loop(µ, 3.5 GeV)
µ2[GeV2] sa-scaling a2-scaling
70/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
90 100 110 120 130 140 10 20 30 40 50 60 70 80 90 Fπ [MeV] mRGI
ud
[MeV] β=3.5 β=3.61 β=3.7 β=3.8
χ 2 # dof = 1.52, #dof = 17, p-value = 0.08
Exp value (NOT INCLUDED)
71/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
90 100 110 120 130 140 10 20 30 40 50 60 70 80 90 Fπ [MeV] mRGI
ud
[MeV] β=3.5 β=3.61 β=3.7 β=3.8
χ 2 # dof = 1.16, #dof = 48, p-value = 0.21
Exp value (NOT INCLUDED)
72/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
0.00048 0.0005 0.00052 0.00054 0.00056 0.00058 100
2 200 2250 2 300 2
350
2
400
2
450
2
500
2
550
2
1/B RGI
π
[MeV− 1] M 2
π [MeV2]
β=3.5 β=3.61 β=3.7 β=3.8
χ 2 # dof = 1.54, #dof = 17, p-value = 0.07
73/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
0.00048 0.0005 0.00052 0.00054 0.00056 0.00058 100
2 200 2250 2 300 2
350
2
400
2
450
2
500
2
550
2
1/B RGI
π
[MeV− 1] M 2
π [MeV2]
β=3.5 β=3.61 β=3.7 β=3.8
χ 2 # dof = 1.19, #dof = 48, p-value = 0.18
74/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
84 86 88 90 92 94 F [MeV ] NL O x NL O ξ NNL O x NNL O ξ
1 2 200 250 300 350 400 450 500 550 600 ∆F [MeV ] M max
π
[MeV ]
75/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10 200 250 300 350 400 450 500 550 600 p-value M max
π
[MeV ] NL O x NL O ξ NNL O x NNL O ξ
75/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
p M min
π
x ξ
75/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD
Introduction Hadron spectrum Quark masses etc. Summary
F ξ ∆F M min
π
75/39 Christian Hoelbling (Wuppertal) Physical predictions from lattice QCD