fundamental qcd parameters from the lattice
TRANSCRIPT
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Fundamental QCD parameters from the lattice
Jochen Heitger
Westfalische Wilhelms-Universitat MunsterWilhelm-Klemm-Str. 9, D-48149 Munster
Sino-German Workshop on Frontiers in QCDDeutsches Elektronen-Synchrotron DESY, Hamburg
20. September 2006
Jochen Heitger Fundamental QCD parameters from the lattice
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Does QCD describe all strong interaction phenomena ?
Complementary properties: Confinement & Asymptotic freedom
0.5
0
− 0.5
0 0.2 0.4 0.6 0.8
Lattice Gauge Theory
Model Calculation
Distance in fm
Pote
ntial in
GeV
Confinement Potential
between Quarks
0.1
0.2
0.3
0.4
0.5
!s
1 10 100
Measured
QCD
Energy in GeV
Uncovering the connectionbetween the low- and high-energy regimes of QCD:Demands non-perturbativetreatment of the theory (NP)
Jochen Heitger Fundamental QCD parameters from the lattice
![Page 3: Fundamental QCD parameters from the lattice](https://reader030.vdocuments.site/reader030/viewer/2022012605/619a7ac553c4755e7f02e961/html5/thumbnails/3.jpg)
Does QCD describe all strong interaction phenomena ?
Complementary properties: Confinement & Asymptotic freedom
0.5
0
− 0.5
0 0.2 0.4 0.6 0.8
Lattice Gauge Theory
Model Calculation
Distance in fm
Pote
ntial in
GeV
Confinement Potential
between Quarks
0.1
0.2
0.3
0.4
0.5
!s
1 10 100
Measured
QCD
Energy in GeV
Uncovering the connectionbetween the low- and high-energy regimes of QCD:Demands non-perturbativetreatment of the theory (NP)
Scope of lattice QCD computations:Determination of fundamental parameters
→ ΛQCD, quark massesStringent tests of QCD
→ e.g. Fπ/ΛQCD, (light) hadron spectrumPredict experimentally inaccessible physical quantities
→ e.g. hadron masses, decay constants, mixing amplitudesJochen Heitger Fundamental QCD parameters from the lattice
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Lattice QCD‘Ab initio’ approach to determine phenomenologically relevant key parameters
LQCD [g0, mf] = −1
2g20
Tr FµνFµν +∑
f=u,d,s,...
ψf γµ (∂µ + g0Aµ) + mf ψf
Fπ
mπ
mK
mD
mB
︸ ︷︷ ︸
Experiment
LQCD [g0,mf]=⇒
ΛQCD12(Mu + Md)
Ms
Mc
Mb
︸ ︷︷ ︸
QCDparameters (RGIs)
+
FD
FB
BB
ξ
· · ·
︸ ︷︷ ︸
Predictions
LQCD [g0,mf]=⇒ means formulation of QCD on a Euclidean lattice with:
Gauge invariance
Locality
Unitarity
Applicable for all scalesTechnical issues/obstacles
I Continuum limit & RenormalizationI Computing resources of > 1TFlop/s
Jochen Heitger Fundamental QCD parameters from the lattice
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Outline
1 Lattice QCD basics
2 Determining fundamental QCD parametersThe running of the coupling in two-flavour QCDQuenched versus unquenched: The strange quark’s mass
3 New perspectives for B-physicsNon-perturbative Heavy Quark Effective TheoryApplication: The b-quark’s mass
4 Summary & Outlook
Jochen Heitger Fundamental QCD parameters from the lattice
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Lattice QCD basics
Lattice QCD basics
Jochen Heitger Fundamental QCD parameters from the lattice
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Lattice QCD basics
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-¾
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L
T
ψ(x)Uµ(x) = e iag0Aµ(x)
a
Lattice cutoff a−1 ∼ ΛUV
Finite volume L3 × T
Lattice action
S[U, ψ, ψ] = SG[U] + SF[U, ψ, ψ]
SG = 1g2
0
∑
p
Tr 1 − U(p)
SF = a4∑
x
ψ(x)D[U]ψ(x)
EVs: represented as path integrals
Z =
∫
D[U]D[ψ,ψ] e−S[U,ψ,ψ] =
∫
D[U]∏
f det(D/ + mf
)e−SG[U]
〈O〉 =1
Z
∫∏
x,µdUµ(x)O∏
f det(D/ + mf
)e−SG[U] = thermal average
Stochastic evaluation with Monte Carlo methods
→ Observables 〈O〉 = 1N
∑Nn=1 On ± ∆O from numerical simulations
Jochen Heitger Fundamental QCD parameters from the lattice
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Lattice QCD basics
Towards realistic QCD simulations — Systematics
(1) Effects of dynamical fermionsValence quark approximation (quenching):det
(D/ + mf
)= 1
→ neglection of quark loops
Costs: ‘full’ QCDquenched QCD & 100 − 1000
(A) Quenched QCD: quark loops neglected
(B) Full QCD
Jochen Heitger Fundamental QCD parameters from the lattice
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Lattice QCD basics
Towards realistic QCD simulations — Systematics
(1) Effects of dynamical fermionsValence quark approximation (quenching):det
(D/ + mf
)= 1
→ neglection of quark loops
Costs: ‘full’ QCDquenched QCD & 100 − 1000
(A) Quenched QCD: quark loops neglected
(B) Full QCD
(2) Lattice artefacts cause discretization errors
〈O〉lattice = 〈O〉continuum+ O (ap)
removable by extrapolating to the continuum limit: a → 0
accelerated convergence for p ′ > p → O(ap) improvement
Jochen Heitger Fundamental QCD parameters from the lattice
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Lattice QCD basics
Towards realistic QCD simulations — Systematics
(1) Effects of dynamical fermionsValence quark approximation (quenching):det
(D/ + mf
)= 1
→ neglection of quark loops
Costs: ‘full’ QCDquenched QCD & 100 − 1000
(A) Quenched QCD: quark loops neglected
(B) Full QCD
(2) Lattice artefacts cause discretization errors
〈O〉lattice = 〈O〉continuum+ O (ap)
removable by extrapolating to the continuum limit: a → 0
accelerated convergence for p ′ > p → O(ap) improvement
(3) Quark mass restrictions
a ¿ m−1q ¿ L
→ Extrapolations to mu,d and mb , guided by ChPT and HQET
Jochen Heitger Fundamental QCD parameters from the lattice
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Determining fundamental QCD parameters
Determining fundamental QCD parameters
Jochen Heitger Fundamental QCD parameters from the lattice
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Determining fundamental QCD parameters
Scale dependence of QCD parameters
Renormalization group (RG) equations, g ≡ g(µ)
µ∂g
∂µ= β(g)
g→0∼ −g3
b0 + b1g
2 + b2g4 + . . .
µ∂m
∂µ= τ(g)m τ(g)
g→0∼ −g2
d0 + d1g
2 + . . .
Jochen Heitger Fundamental QCD parameters from the lattice
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Determining fundamental QCD parameters
Scale dependence of QCD parameters
Renormalization group (RG) equations, g ≡ g(µ)
µ∂g
∂µ= β(g)
g→0∼ −g3
b0 + b1g
2 + b2g4 + . . .
µ∂m
∂µ= τ(g)m τ(g)
g→0∼ −g2
d0 + d1g
2 + . . .
Solution leads to exact equations in a mass-independent scheme
Λ ≡ µ(b0g
2)−b1/(2b2
0) e−1/(2b0g2) exp
−
∫ g
0
dg[ 1
β(g)+
1
b0g3−
b1
b20g
]
M ≡ m(µ)(2b0g
2)−d0/(2b0)
exp
−
∫ g
0
dg[ τ(g)
β(g)−
d0
b0g
] RG invariantquark mass
Jochen Heitger Fundamental QCD parameters from the lattice
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Determining fundamental QCD parameters
Scale dependence of QCD parameters
Renormalization group (RG) equations, g ≡ g(µ)
µ∂g
∂µ= β(g)
g→0∼ −g3
b0 + b1g
2 + b2g4 + . . .
µ∂m
∂µ= τ(g)m τ(g)
g→0∼ −g2
d0 + d1g
2 + . . .
Solution leads to exact equations in a mass-independent scheme
Λ ≡ µ(b0g
2)−b1/(2b2
0) e−1/(2b0g2) exp
−
∫ g
0
dg[ 1
β(g)+
1
b0g3−
b1
b20g
]
M ≡ m(µ)(2b0g
2)−d0/(2b0)
exp
−
∫ g
0
dg[ τ(g)
β(g)−
d0
b0g
] RG invariantquark mass
Simple relations between different renormalization schemes:
Λ ′/Λ = exp c/(4πb0) , α ′ = α + cα2 + O(α3) and M ′ = M
⇒ Choose suitable physical scheme/coupling to compute Λ and M
Jochen Heitger Fundamental QCD parameters from the lattice
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Determining fundamental QCD parameters The Nf = 2 running coupling
The running of the coupling in two-flavour QCDLPHAA
Collaboration , NPB713(2005)378Physical couplingsI are defined for all energies µ and fix a renormalization schemeI should be independent of the regularization procedureI in a lattice computation: Problem of a hierarchy of disparate scales
L−1 ¿ 0.2 GeV ¿ µ ' 10 GeV ¿ a−1
Jochen Heitger Fundamental QCD parameters from the lattice
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Determining fundamental QCD parameters The Nf = 2 running coupling
The running of the coupling in two-flavour QCDLPHAA
Collaboration , NPB713(2005)378Physical couplingsI are defined for all energies µ and fix a renormalization schemeI should be independent of the regularization procedureI in a lattice computation: Problem of a hierarchy of disparate scales
L−1 ¿ 0.2 GeV ¿ µ ' 10 GeV ¿ a−1
Solution [ Luscher, Weisz & Wolff, 1991; LPHAACollaboration , 1993-2005 ]
Identify µ = 1/L =⇒ One is left with L/a À 1 =⇒ Finite-size scaling
Jochen Heitger Fundamental QCD parameters from the lattice
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Determining fundamental QCD parameters The Nf = 2 running coupling
The running of the coupling in two-flavour QCDLPHAA
Collaboration , NPB713(2005)378Physical couplingsI are defined for all energies µ and fix a renormalization schemeI should be independent of the regularization procedureI in a lattice computation: Problem of a hierarchy of disparate scales
L−1 ¿ 0.2 GeV ¿ µ ' 10 GeV ¿ a−1
Solution [ Luscher, Weisz & Wolff, 1991; LPHAACollaboration , 1993-2005 ]
Identify µ = 1/L =⇒ One is left with L/a À 1 =⇒ Finite-size scaling
Realization: QCD Schrodinger functional (SF) as an intermediate scheme
C
time
space
T
0
C’ T × L3 QCD with Dirichlet boundary conditions in time
Take finite-size effect as physical observable:
1/ g2SF(L) ≡ δ(free energy)/δ(boundary conditions)
NP handle on β(g) via the step scaling function:
σ(u) ≡ g2(2L)∣∣g2(L)=u ,m (L)=0
Jochen Heitger Fundamental QCD parameters from the lattice
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Determining fundamental QCD parameters The Nf = 2 running coupling
Recursive strategy to obtain the step scaling functionFix g2(L0 ≡ Lmax), compute g2(L) for latticesalternating in L = 2−kL0 resp. a = 2−ka0
. . .
12
12
a = a
a
0
L = 0a = 0L
,L = L0
a → 0 limit of Σ(u,a/L) = σ(u) + O(a/L)
Comparison to PT and NP fit of σ(u) toσ(u) = u + s0u
2 + s1u3 + s2u
4 + . . .
Jochen Heitger Fundamental QCD parameters from the lattice
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Determining fundamental QCD parameters The Nf = 2 running coupling
Nf = 2 β–function in the SF scheme
Comparison to PT and Nf = 2 :
Non-perturbative deviations from3–loop β for α > 0.25
Perturbative series by itself doesn’tshow signs of failure at e.g. α ≈ 0.4
Jochen Heitger Fundamental QCD parameters from the lattice
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Determining fundamental QCD parameters The Nf = 2 running coupling
Nf = 2 β–function in the SF scheme
Comparison to PT and Nf = 2 :
Non-perturbative deviations from3–loop β for α > 0.25
Perturbative series by itself doesn’tshow signs of failure at e.g. α ≈ 0.4
Nf = 0 (SF scheme) & Experiment + PT ( MS scheme, [ Bethke, 2004 ] )
Jochen Heitger Fundamental QCD parameters from the lattice
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Determining fundamental QCD parameters The Nf = 2 running coupling
Nf = 2 β–function in the SF scheme
Comparison to PT and Nf = 2 :
Non-perturbative deviations from3–loop β for α > 0.25
Perturbative series by itself doesn’tshow signs of failure at e.g. α ≈ 0.4
Non-perturbative running of αSF and the Λ–parameter for Nf = 2
Jochen Heitger Fundamental QCD parameters from the lattice
![Page 22: Fundamental QCD parameters from the lattice](https://reader030.vdocuments.site/reader030/viewer/2022012605/619a7ac553c4755e7f02e961/html5/thumbnails/22.jpg)
Determining fundamental QCD parameters The Nf = 2 running coupling
Nf = 2 β–function in the SF scheme
Comparison to PT and Nf = 2 :
Non-perturbative deviations from3–loop β for α > 0.25
Perturbative series by itself doesn’tshow signs of failure at e.g. α ≈ 0.4
Non-perturbative running of αSF and the Λ–parameter for Nf = 2
Start from g2(Lmax) ≡ 5.5 and evolvek steps with NP step scaling function:g2(Lmax/2k) = σ(g2(Lmax/2k+1))
I g2 6 uk small: continue with pert. β
I Result: − ln(ΛLmax) = 1.09(7)
Jochen Heitger Fundamental QCD parameters from the lattice
![Page 23: Fundamental QCD parameters from the lattice](https://reader030.vdocuments.site/reader030/viewer/2022012605/619a7ac553c4755e7f02e961/html5/thumbnails/23.jpg)
Determining fundamental QCD parameters The Nf = 2 running coupling
Nf = 2 β–function in the SF scheme
Comparison to PT and Nf = 2 :
Non-perturbative deviations from3–loop β for α > 0.25
Perturbative series by itself doesn’tshow signs of failure at e.g. α ≈ 0.4
Non-perturbative running of αSF and the Λ–parameter for Nf = 2
Start from g2(Lmax) ≡ 5.5 and evolvek steps with NP step scaling function:g2(Lmax/2k) = σ(g2(Lmax/2k+1))
I g2 6 uk small: continue with pert. β
I Result: − ln(ΛLmax) = 1.09(7)
To set the scale for Λ in MeV, useI ideally: large volume computation of FK
I here: r0 = 0.5 fm , FQQ(r0)r20 = 1.65
Jochen Heitger Fundamental QCD parameters from the lattice
![Page 24: Fundamental QCD parameters from the lattice](https://reader030.vdocuments.site/reader030/viewer/2022012605/619a7ac553c4755e7f02e961/html5/thumbnails/24.jpg)
Determining fundamental QCD parameters The Nf = 2 running coupling
Nf = 2 β–function in the SF scheme
Comparison to PT and Nf = 2 :
Non-perturbative deviations from3–loop β for α > 0.25
Perturbative series by itself doesn’tshow signs of failure at e.g. α ≈ 0.4
Non-perturbative running of αSF and the Λ–parameter for Nf = 2
Start from g2(Lmax) ≡ 5.5 and evolvek steps with NP step scaling function:g2(Lmax/2k) = σ(g2(Lmax/2k+1))
I g2 6 uk small: continue with pert. β
I Result: − ln(ΛLmax) = 1.09(7)
To set the scale for Λ in MeV, useI ideally: large volume computation of FK
I here: r0 = 0.5 fm , FQQ(r0)r20 = 1.65
ΛMSr0 = 0.62(8) ⇒ ΛMS = 245(32)MeVJochen Heitger Fundamental QCD parameters from the lattice
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Determining fundamental QCD parameters The Nf = 2 running coupling
Discussion
I Running close to PT below αSF ≈ 0.2, non-perturbative above 0.25
I ΛMSr0 : Nf dependence & Comparison to phenomenology
Nf 0 2 4 5
[ LPHAACollaboration ] 0.60(5) 0.62(4)(4)
[ Bethke, Loops&Legs04 ]‘experiment’ 0.74(10) 0.54(8)
[ Blumlein et al., Loops&Legs04 ]DIS NNLO 0.57(8)
Irregular Nf dependence ? But errors relatively large . . .
Jochen Heitger Fundamental QCD parameters from the lattice
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Determining fundamental QCD parameters The Nf = 2 running coupling
Discussion
I Running close to PT below αSF ≈ 0.2, non-perturbative above 0.25
I ΛMSr0 : Nf dependence & Comparison to phenomenology
Nf 0 2 4 5
[ LPHAACollaboration ] 0.60(5) 0.62(4)(4)
[ Bethke, Loops&Legs04 ]‘experiment’ 0.74(10) 0.54(8)
[ Blumlein et al., Loops&Legs04 ]DIS NNLO 0.57(8)
Irregular Nf dependence ? But errors relatively large . . .
I Improvement of lattice results: Via replacing Lmax/r0 by FK × Lmax
with small quark masses and small lattice spacingsAlso: Nf = 3, 4 (Nf = 3 on agenda of JLQCD and CP-PACS)Lattice will yield best controlled and most precise results for Λ
(minimal assumptions !)Jochen Heitger Fundamental QCD parameters from the lattice
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Determining fundamental QCD parameters The strange quark’s mass
A precision calculation of ms
Garden, H., Sommer & Wittig, NPB571(2000)237StrategyCombination of lattice QCD with chiral perturbation theory→ relate LQCD with experimentally known (spectroscopic) quantity, here:
Jochen Heitger Fundamental QCD parameters from the lattice
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Determining fundamental QCD parameters The strange quark’s mass
A precision calculation of ms
Garden, H., Sommer & Wittig, NPB571(2000)237StrategyCombination of lattice QCD with chiral perturbation theory→ relate LQCD with experimentally known (spectroscopic) quantity, here:
m2PS(Mref)r
20 = (mKr0)
2∣∣exp
ChPT:
2Mref ' Ms + M` , Ms
M`= 24.4(1.5)
where M` = 12(Mu + Md) and M ≡ lim
µ→∞
[ 2b0g
2(µ) ]−d0/(2b0) m(µ)
: RGI quark mass
Jochen Heitger Fundamental QCD parameters from the lattice
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Determining fundamental QCD parameters The strange quark’s mass
A precision calculation of ms
Garden, H., Sommer & Wittig, NPB571(2000)237StrategyCombination of lattice QCD with chiral perturbation theory→ relate LQCD with experimentally known (spectroscopic) quantity, here:
m2PS(Mref)r
20 = (mKr0)
2∣∣exp
ChPT:
2Mref ' Ms + M` , Ms
M`= 24.4(1.5)
where M` = 12(Mu + Md) and M ≡ lim
µ→∞
[ 2b0g
2(µ) ]−d0/(2b0) m(µ)
: RGI quark mass
Determination of Mref through numerical simulations
(1)PCAC relation :
FKm2K = (mu + ms) 〈 0 | uγ5s | K 〉
(uγ5s)MS = ZP(g0, aµ) (uγ5s) lattice
(2) ZP(g0, aµ) is poorly convergent in PT→ non-perturbative values for ZP & M/m(µ) needed [ LPHAA
Collaboration 1999 ]
(3) Mref fixed by K-meson matrix elements to be taken from the lattice
Ms + M`
FK
=M
m(µ)
1
ZP(g0, aµ)× m2
PS
〈 0 | `γ5s | PS 〉
∣∣∣∣mPS=mK
+ O(a2)
Jochen Heitger Fundamental QCD parameters from the lattice
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Determining fundamental QCD parameters The strange quark’s mass
Hadron physics from the Schrodinger FunctionalH., NPB557(1999)309
Guagnelli, H., Sommer & Wittig, NPB560(1999)465Volume: L & 1.5 fm , T = 2L
SF = QCD partition function with Dirichlet BCs in time(finite-V renormalization scheme, but has also some practical benefits for large-V physics)
Jochen Heitger Fundamental QCD parameters from the lattice
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Determining fundamental QCD parameters The strange quark’s mass
Hadron physics from the Schrodinger FunctionalH., NPB557(1999)309
Guagnelli, H., Sommer & Wittig, NPB560(1999)465Volume: L & 1.5 fm , T = 2L
SF = QCD partition function with Dirichlet BCs in time(finite-V renormalization scheme, but has also some practical benefits for large-V physics)
Fermionic correlation functions
fX(x0) ∝⟨ ∑
y,z
X(x) ζ(y)γ5ζ(z)⟩
X =
Aµ = ψiγµγ5ψj : axial vector currentP = ψiγ5ψj : pseudoscalar density
f1 ∝⟨ ∑
u,v,y,z
ζ ′
i(u)γ5ζ′
j(v) ζj(y)γ5ζi(z)⟩
ζ, ζ : boundary (anti-)quark fields
construct ratios with f1 to eliminate Z ζ,ζ
Jochen Heitger Fundamental QCD parameters from the lattice
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Determining fundamental QCD parameters The strange quark’s mass
Hadron physics from the Schrodinger FunctionalH., NPB557(1999)309
Guagnelli, H., Sommer & Wittig, NPB560(1999)465Volume: L & 1.5 fm , T = 2L
SF = QCD partition function with Dirichlet BCs in time(finite-V renormalization scheme, but has also some practical benefits for large-V physics)
Fermionic correlation functions
fX(x0) ∝⟨ ∑
y,z
X(x) ζ(y)γ5ζ(z)⟩
X =
Aµ = ψiγµγ5ψj : axial vector currentP = ψiγ5ψj : pseudoscalar density
f1 ∝⟨ ∑
u,v,y,z
ζ ′
i(u)γ5ζ′
j(v) ζj(y)γ5ζi(z)⟩
ζ, ζ : boundary (anti-)quark fields
construct ratios with f1 to eliminate Z ζ,ζ
⇒ for 0 ¿ x0 ¿ T :
fA√f1
≈ 〈0|A|PS〉 e−(x0−T/2)mPSfA
fP
≈ 〈0|A|PS〉〈0|P|PS〉
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Determining fundamental QCD parameters The strange quark’s mass
Result in quenched QCD [ Garden, H., Sommer & Wittig, NPB571(2000)237 ]
mMSs (2 GeV) = 97 ± 4 MeV
NP renormalization & Continuum limitI scale set by FK = 160(2)MeV
I 3 all errors except quenching (∼ 15%?)
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Determining fundamental QCD parameters The strange quark’s mass
Result in quenched QCD [ Garden, H., Sommer & Wittig, NPB571(2000)237 ]
mMSs (2 GeV) = 97 ± 4 MeV
NP renormalization & Continuum limitI scale set by FK = 160(2)MeV
I 3 all errors except quenching (∼ 15%?)
Computation in similar spirit:I Nf = 0 mcharm, input: r0 = 0.5 fm, mDs
I mMSc (mc) = 1.30(3)GeV [Rolf&Sint, ’02]
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Determining fundamental QCD parameters The strange quark’s mass
Result in quenched QCD [ Garden, H., Sommer & Wittig, NPB571(2000)237 ]
mMSs (2 GeV) = 97 ± 4 MeV
NP renormalization & Continuum limitI scale set by FK = 160(2)MeV
I 3 all errors except quenching (∼ 15%?)
Computation in similar spirit:I Nf = 0 mcharm, input: r0 = 0.5 fm, mDs
I mMSc (mc) = 1.30(3)GeV [Rolf&Sint, ’02]
Crucial for precise predictions: Inclusion of dynamical (sea) quark effects
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Determining fundamental QCD parameters The strange quark’s mass
Result in quenched QCD [ Garden, H., Sommer & Wittig, NPB571(2000)237 ]
mMSs (2 GeV) = 97 ± 4 MeV
NP renormalization & Continuum limitI scale set by FK = 160(2)MeV
I 3 all errors except quenching (∼ 15%?)
Computation in similar spirit:I Nf = 0 mcharm, input: r0 = 0.5 fm, mDs
I mMSc (mc) = 1.30(3)GeV [Rolf&Sint, ’02]
Crucial for precise predictions: Inclusion of dynamical (sea) quark effects
Non-perturbative running of m for Nf = 2 [ LPHAACollaboration , NPB729(2005)117 ]
mi(µ) = Zm(g0, aµ) × mbarei (g0)
whereby Zm is flavour independent
Scale evolution & RGI mass M via thesame recursive strategy as for α(µ):I exp. input: r0 = 0.5 fm, mK = 495MeV
I mMSs (2GeV) = 97 ± 22MeV
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Determining fundamental QCD parameters The strange quark’s mass
Unquenched mstrange (Nf = 2, 2 + 1) [ Status November 2005 ]
0 0.005 0.01 0.015 0.02 0.025 0.03
0
20
40
60
80
100
120
mM
Ss
(µ=
2G
eV)
[MeV
]
a2 [fm2]
ALPHA, Nf =2, W/CW, nonpert. renorm. (SF)
ALPHA, Nf =2, W/CW, nonpert. renorm. (SF), Zcon
A
CP-PACS, Nf =2, I/CW, pert. renorm.
JLQCD, Nf =2, W/CW, pert. renorm.
QCDSF/UKQCD, Nf =2, W/CW, nonpert. renorm. (RI-MOM)
SPQCDR, Nf =2, W/W, nonpert. renorm. (RI-MOM)
CP-PACS/JLQCD, Nf =3, I/CW, pert. renorm., stat. err. only
HPQCD/MILC/UKQCD, Nf =3, LW/KS, pert. renorm. 1-loop
HPQCD/MILC/UKQCD, Nf =3, LW/KS, pert. renorm. 2-loop
← Nf = 0
I Unquenched estimates have not yet stabilizedLarge cutoff effects at a ≈ 0.1 fm
Better control over continuum limit requiredI Impact of perturbative versus non-perturbative renormalization ?
Results with pert. renormalization tend to be smaller than NP onesI Presently: Hard to claim a definitive Nf dependence of mstrange
Note: QCD sum rules in agreement with range of lattice results
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New perspectives for B-physics
New perspectives for B-physics
Jochen Heitger Fundamental QCD parameters from the lattice
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New perspectives for B-physics
B-physics & Lattice QCDI Precision CKM-physics involves hadronic matrix elements:
Determination of CKM matrix parameters, tests of its unitarityCP violation
I b-quark mass, spectrum and lifetimes of beauty-hadrons, . . .
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New perspectives for B-physics
B-physics & Lattice QCDI Precision CKM-physics involves hadronic matrix elements:
Determination of CKM matrix parameters, tests of its unitarityCP violation
I b-quark mass, spectrum and lifetimes of beauty-hadrons, . . .
Illustration: Unitarity triangle analysis (‘CKM fit’) [ Okamoto @ Lattice 2005 ]
〈M | O∆M=2 |M 〉 =4
3m2
MF2MBM
〈 0 | bγµγ5q |Bq 〉 = ipµFBq, q = d, s
∆md ∝ F2Bd
BBd|VtdV
∗
tb|2
∆ms
∆md
∝F2
BsBBs
F2Bd
BBd
|Vts|2
|Vtd|2= ξ2 |Vts|
2
|Vtd|2
UT with lattice input from BK, FBd,BBd
,ξ: Today versus expected (5-3)% accuracy
−0.4 −0.2 0 0.2 0.4 0.6 0.8 1
ρ−0.6
−0.4
−0.2
0.0
0.2
0.4
η β
~|Vtd |
~|V
ub|
|εK|∆M
B
Lattice’05
−0.4 −0.2 0 0.2 0.4 0.6 0.8 1
ρ−0.6
−0.4
−0.2
0.0
0.2
0.4
η β
~|Vtd |
~|V
ub|
|εK|∆M
B
Lattice 201X
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New perspectives for B-physics Non-perturbative HQET
Lattice HQET — Why ?
Consider a large lattice as possible in the quenched approximation
λπ = 1/mπ ≈ L
λB ∼ 1/mb < a
I Light quarks: too lightWidely spread objectsFinite-volume errors due tolight pions
I b-quark: too heavyExtremely localized objectB-mesons need very fineresolutions, otherwise:
I discretization errorsI ‘fall through the lattice’
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New perspectives for B-physics Non-perturbative HQET
Lattice HQET — Why ?
Consider a large lattice as possible in the quenched approximation
λπ = 1/mπ ≈ L
λB ∼ 1/mb < a
I Light quarks: too lightWidely spread objectsFinite-volume errors due tolight pions
I b-quark: too heavyExtremely localized objectB-mesons need very fineresolutions, otherwise:
I discretization errorsI ‘fall through the lattice’
⇒ Propagating b on the lattice beyond today’s computing resources
⇒ Recourse to an effective theory for the b-quark:
Heavy Quark Effective Theory[ Eichten, 1988; Eichten & Hill, 1990 ]
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New perspectives for B-physics Non-perturbative HQET
HQET — An asymptotic expansion of QCD
Physical picture
q, gQ
m−1Q
a
L
Λ−1QCD
PµQ = mQvµ + kµ
v2 = 1residual momentum:
k ∼ O(ΛQCD) ¿ mQv
Momentum scales in heavy-light (Qq) mesons
Q almost at rest at bound state’s center,surrounded by the light DOFs
Motion of the heavy quark is suppressedby ΛQCD/mQ
Formally, in case of the B-system:LHQET = 1/mb–expansion of continuum QCD
ψb γµDµ + mbψb → Lstat + O(1/mb)
Lstat(x) = ψh(x) D0 + mb ψh(x)
P+ψh = ψh, P+ = 12(1+γ0) ⇒ 2 d.o.f.
Systematic & accurate for mh/ΛQCD À 1
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New perspectives for B-physics Non-perturbative HQET
Non-perturbative formulation of HQETH. & Sommer, JHEP0402(2004)022Beyond the static approximation
SHQET = a4∑
x
Lstat(x) +
n∑
ν=1
L(ν)(x)
, L(ν)(x) =∑
i
ω(ν)
i L(ν)
i (x)
Lstat = ψh [∇∗0 + δm ] ψh → Eichten-Hill action
L(1)1 = ψh
(− 1
2σ · B
)ψh ≡ Ospin →
chromomagnetic interactionwith the gluon field
L(1)2 = ψh
(− 1
2D2
)ψh ≡ Okin →
kinetic energy from heavyquark’s residual motion
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New perspectives for B-physics Non-perturbative HQET
Non-perturbative formulation of HQETH. & Sommer, JHEP0402(2004)022Beyond the static approximation
SHQET = a4∑
x
Lstat(x) +
n∑
ν=1
L(ν)(x)
, L(ν)(x) =∑
i
ω(ν)
i L(ν)
i (x)
Lstat = ψh [∇∗0 + δm ] ψh → Eichten-Hill action
L(1)1 = ψh
(− 1
2σ · B
)ψh ≡ Ospin →
chromomagnetic interactionwith the gluon field
L(1)2 = ψh
(− 1
2D2
)ψh ≡ Okin →
kinetic energy from heavyquark’s residual motion
Effective theory regularized on a space-time latticeδm, ω = ω(g0, m) to be determined such that HQET matches QCDAt classical level: ωspin = ωkin = 1/m + O(g2
0) , δm = 0 + O(g20)
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New perspectives for B-physics Non-perturbative HQET
Non-perturbative formulation of HQETH. & Sommer, JHEP0402(2004)022Beyond the static approximation
SHQET = a4∑
x
Lstat(x) +
n∑
ν=1
L(ν)(x)
, L(ν)(x) =∑
i
ω(ν)
i L(ν)
i (x)
Lstat = ψh [∇∗0 + δm ] ψh → Eichten-Hill action
L(1)1 = ψh
(− 1
2σ · B
)ψh ≡ Ospin →
chromomagnetic interactionwith the gluon field
L(1)2 = ψh
(− 1
2D2
)ψh ≡ Okin →
kinetic energy from heavyquark’s residual motion
Effective theory regularized on a space-time latticeδm, ω = ω(g0, m) to be determined such that HQET matches QCDAt classical level: ωspin = ωkin = 1/m + O(g2
0) , δm = 0 + O(g20)
Analogously: Composite fields in the effective theory, e.g.
AHQET0 (x) = ZHQET
A︸ ︷︷ ︸1+O(g2
0)
ψl(x)γ0γ5ψh(x)︸ ︷︷ ︸
Astat0 (x)
+ cHQETA︸ ︷︷ ︸
∝ 1/m
ψl(x)γjγ5
←−Djψh(x) + . . .
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New perspectives for B-physics Non-perturbative HQET
Expectation valuesPath integral representation at the quantum level
〈O〉 =1
Z
∫
D[ϕ] O[ϕ] e−(Srel+SHQET) Z =
∫
D[ϕ] e−(Srel+SHQET)
Now the integrand is expanded in a power series in 1/m
exp −SHQET =
exp−a4
∑xLstat(x)
1 − a4∑
xL(1)(x) + 12
[a4
∑xL(1)(x)
]2− a4
∑xL(2)(x) + . . .
⇒ 〈O〉 =1
Z
∫
D[ϕ] e−Srel−a4∑
x Lstat(x) O
1 − a4∑
xL(1)(x) + . . .
Important (but not automatic) implications of this definition of HQET
1/m – terms appear only as insertions of local operators⇒ Power counting: Renormalizability at any given order in 1/m
⇔ Existence of the continuum limit with universality
Effective theory = Continuum asymptotic expansion in 1/m
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New perspectives for B-physics Non-perturbative HQET
RenormalizationWe assume massless quarks except the b and no 1/m – term in O
(fA)R (Mb/Λ, x0Λ) =
ZHQETA e−mbarex0
⟨Astat
0 (x)O[1 + ωkin
∑yOkin(y) + ωspin
∑yOspin(y)
]⟩
stat
+ cHQETA
⟨δAstat
0 O⟩stat
=
ZHQETA e−mbarex0
fstatA + ωkinf
kinA + ωspinf
spinA + cHQET
A fstatδA
0
LxLxL
x
x0
0
= T
=⇒ Effective theory receives renormalizations in
mbare = mb + δm ZHQETA cHQET
A ωkin ωspin
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New perspectives for B-physics Non-perturbative HQET
RenormalizationWe assume massless quarks except the b and no 1/m – term in O
(fA)R (Mb/Λ, x0Λ) =
ZHQETA e−mbarex0
⟨Astat
0 (x)O[1 + ωkin
∑yOkin(y) + ωspin
∑yOspin(y)
]⟩
stat
+ cHQETA
⟨δAstat
0 O⟩stat
=
ZHQETA e−mbarex0
fstatA + ωkinf
kinA + ωspinf
spinA + cHQET
A fstatδA
0
LxLxL
x
x0
0
= T
=⇒ Effective theory receives renormalizations in
mbare = mb + δm ZHQETA cHQET
A ωkin ωspin
The Problem: Power-law divergences due to operator mixing
ORd=5 = ZO
Od=5 +
∑k ck Ok
d=4
, ck = a−1c
(0)
k + c(1)
k g20 + . . .
⇒ for a → 0 divergent remainder, if ck are only perturbatively calculated
Example at the static level:Linearly divergent counterterm δm ∝ 1/a originates from mixing of ψhD0ψh with ψhψh
⇒ Non-pert. renormalization of HQET needed to have a continuum limit
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New perspectives for B-physics Non-perturbative HQET
Solution: Non-perturbative matching of HQET and QCDH. & Sommer, JHEP0402(2004)022Effective theory approximates QCD,
if the coefficients ck are chosen correctly such that
ΦHQET(M) = ΦQCD(M) + O(1/[ r0M ]n+1
)
M = RG-invariant (heavy) quark mass(free of renormalization scheme dependence)
Strategy to guarantee this equivalence of HQET and QCD
Fix the coefficients of regularized HQET by imposing matching conditions¨
§
¥
¦ΦHQET
k (M) = ΦQCDk (M) k = 1, . . . ,NHQET (?)
(?) determines the set ck for any lattice spacing (bare coupling)
ΦQCDk = Renormalized quantities, computable for a → 0 in QCD
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New perspectives for B-physics Non-perturbative HQET
Solution: Non-perturbative matching of HQET and QCDH. & Sommer, JHEP0402(2004)022Effective theory approximates QCD,
if the coefficients ck are chosen correctly such that
ΦHQET(M) = ΦQCD(M) + O(1/[ r0M ]n+1
)
M = RG-invariant (heavy) quark mass(free of renormalization scheme dependence)
Strategy to guarantee this equivalence of HQET and QCD
Fix the coefficients of regularized HQET by imposing matching conditions¨
§
¥
¦ΦHQET
k (M) = ΦQCDk (M) k = 1, . . . ,NHQET (?)
(?) determines the set ck for any lattice spacing (bare coupling)
ΦQCDk = Renormalized quantities, computable for a → 0 in QCD
Question
How to treat the heavy quark as particle with finite mass on the lattice ?
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New perspectives for B-physics Non-perturbative HQET
Idea: Employing finite volumes
Goal: Non-perturbative matching of HQET & QCD
QCD
1/mb À a
Matching conditions
ΦQCDk = ΦHQET
k
for observables Φk
(e.g. matrix elements)
HQET
1/mb ¿ L
I HQET parameters fixed by relating them to QCD observables in small V
I Legitimate: Underlying Lagrangian does not know about the finite V !
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New perspectives for B-physics Non-perturbative HQET
Idea: Employing finite volumes
Goal: Non-perturbative matching of HQET & QCDObjection: Requires to simulate the b-quark as a relativistic particle
QCD
1/mb À a
Matching conditions
ΦQCDk = ΦHQET
k
for observables Φk
(e.g. matrix elements)
HQET
1/mb ¿ L
I HQET parameters fixed by relating them to QCD observables in small V
I Legitimate: Underlying Lagrangian does not know about the finite V !
Jochen Heitger Fundamental QCD parameters from the lattice
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New perspectives for B-physics Non-perturbative HQET
Idea: Employing finite volumes
Goal: Non-perturbative matching of HQET & QCDObjection: Requires to simulate the b-quark as a relativistic particle⇒ Trick: Start with QCD in small volume V = L4, L ≡ L0 ' 0.2 fm
QCD
1/mb À a
Matching conditions
ΦQCDk = ΦHQET
k
for observables Φk
(e.g. matrix elements)
HQET
1/mb ¿ L
I HQET parameters fixed by relating them to QCD observables in small V
I Legitimate: Underlying Lagrangian does not know about the finite V !
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New perspectives for B-physics Non-perturbative HQET
Connecting small and large volumes
(?) in finite V : ΦHQETk (L, M) = ΦQCD
k (L, M) k = 1, . . . ,NHQET
Typical choice: L = L0 ' 0.2 − 0.4 fm, very small lattice spacingsLarge volumes required to extract physical observables (e.g. mB, FBs
)→ How can we bridge the gap to practicable lattice spacings ?
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New perspectives for B-physics Non-perturbative HQET
Connecting small and large volumes
(?) in finite V : ΦHQETk (L, M) = ΦQCD
k (L, M) k = 1, . . . ,NHQET
Typical choice: L = L0 ' 0.2 − 0.4 fm, very small lattice spacingsLarge volumes required to extract physical observables (e.g. mB, FBs
)→ How can we bridge the gap to practicable lattice spacings ?
Recursive finite-size scaling
p p p pp p p pp p p pp p p p
-
ΦHQET(L0)
p p p pp p p pp p p p
p p p p
-
ΦHQET(8L0)
p p p p
p p p p
p p p p
p p p p
-
p p p p p p pp p p p p p pp p p p p p pp p p p p p pp p p p p p pp p p p p p p
p p p p p p p
p p p p p p pp p p p p p pp p p p p p pp p p p p p pp p p p p p pp p p p p p p
p p p p p p p
p p p p p p p
p p p p p p p
p p p p p p p
p p p p p p p
p p p p p p p
p p p p p p p
p p p p p p p
[ Luscher, Weisz & Wolff, 1991; LPHAACollaboration , 1993-2005 ]
Non-perturbative, continuum limit exists
Use of the QCD Schrodinger Functional
Change L → 2L via step scaling functions
ΦHQETk (2L) = σk
(ΦHQET
j (L), j = 1, . . . ,N)
→ Large V , where the B fits comfortably
⇒ First fully non-perturbative formulation of HQET, including matchingJochen Heitger Fundamental QCD parameters from the lattice
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New perspectives for B-physics The b-quark’s mass
Application: Computation of Mb in lowest-order HQETH. & Sommer, JHEP0402(2004)022
H. & Wennekers, JHEP0402(2004)064Introduce L– dependent energiesfrom correlators in the B-channel (and with SF boundary conditions):
Γ(L, M)
Γstat(L)
=
B-meson mass in a finite volume of extent L4
energy of a state with B-meson quantum numbers in L4
C(x0,M) :
x0 = 0
ζl2
ζl1
x0 = L
A0 → Γ(L,M) ≡ −d
dx0
ln [C(x0,M)]∣∣∣x0=
L
2
Cstat(x0) :
x0 = 0
ζh
ζl
x0 = L
Astat0 → Γstat(L) ≡ −
d
dx0
ln [Cstat(x0)]∣∣∣x0=
L
2
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New perspectives for B-physics The b-quark’s mass
⇒ Matching condition by equating in small volume with linear extent L0 :
Γstat(L0) + mbare = Γ(L0, Mb) [ Recall: mbare = m + 1a
ln(1 + a δm) ]
As C(x0)x0→∞
∼ e−mBx0 and Cstat(x0)x0→∞
∼ e−Estatx0 in the large – L
limit, we have to connect this condition (by finite-size scaling) to
Estat + mbare = mB
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New perspectives for B-physics The b-quark’s mass
⇒ Matching condition by equating in small volume with linear extent L0 :
Γstat(L0) + mbare = Γ(L0, Mb) [ Recall: mbare = m + 1a
ln(1 + a δm) ]
As C(x0)x0→∞
∼ e−mBx0 and Cstat(x0)x0→∞
∼ e−Estatx0 in the large – L
limit, we have to connect this condition (by finite-size scaling) to
Estat + mbare = mB
Sketch of the method
Experiment Lattice with amb ¿ 1
mB = 5.4 GeV Γ(L0, M)
Γstat(L2) Γstat(L1) Γstat(L0)
ui = g2(Li)σm(u1) σm(u0)? ?¾ ¾
A step scaling function σm(u) ≡ 2L [ Γstat(2L) − Γstat(L) ] bridgesbetween small & larger volumes by evolving L0 → L2 = 22L0 ' 1 fm
For L ' 2 fm @ same resolution:Physical situation to accommodate (& calculate) B-meson properties
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New perspectives for B-physics The b-quark’s mass
⇒ Equation to solve for the b-quark mass:
L0Γ(L0,M)︸ ︷︷ ︸
a→0≡ ΩQCD in QCD (L40)
= L0m(exp)
B − L0
[ Γstat(L2) − Γstat(L0) ] − [Estat − Γstat(L2) ]
︸ ︷︷ ︸a→0≡ ΩHQET in HQET (contains σm)
Divergent static quark’s self-energy δm cancels in differences !Γ(L0, M) carries entire (relativistic) heavy quark mass dependence
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New perspectives for B-physics The b-quark’s mass
⇒ Equation to solve for the b-quark mass:
L0Γ(L0,M)︸ ︷︷ ︸
a→0≡ ΩQCD in QCD (L40)
= L0m(exp)
B − L0
[ Γstat(L2) − Γstat(L0) ] − [Estat − Γstat(L2) ]
︸ ︷︷ ︸a→0≡ ΩHQET in HQET (contains σm)
Divergent static quark’s self-energy δm cancels in differences !Γ(L0, M) carries entire (relativistic) heavy quark mass dependence
QCD ( z ≡ L0M À 1 , L0 ' 0.2 fm ) Quenched static result
O(a) improved QCD & HQET
NP renormalization & Cont. limit
mMSb
(mMS
b
)= 4.12(8)GeV + O( 1
Mb)
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New perspectives for B-physics The b-quark’s mass
Inclusion of 1/m–terms in the computation of Mb
Della Morte, Garron, Sommer & Papinutto, PoS LAT2005(2005)223&224LPHAA
Collaboration , in preparationmB at next-to-leading order of HQET
mB = Estat + mbare + ωkinEkin + ωspinEspin
Ekin, Espin associated with ψh(−12D2)ψh and ψh(−
12σ · B)ψh in L(1)
→ Three observables Φ1, Φ2, Φ3 required in the matching stepConsidering the spin-averaged B-meson instead, ωspin cancels:
m(av)
B = 14mB + 3
4m∗
B = Estat + mbare + ωkinEkin
→ Only two obervables Φ1, Φ2 necessary
The previous strategy now extends to
Experiment Lattice with amb ¿ 1
mB = 5.4 GeV Φ1(L1, M), Φ2(L1, M)
ΦHQET1 (L2), Φ
HQET2 (L2) ΦHQET
1 (L1), ΦHQET2 (L1)
L1 ' 0.4fm,L2 = 2L1
u1 = g2(L1)
σm(u1)
σkin1 (u1), σ
kin2 (u1)
? ?¾
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New perspectives for B-physics The b-quark’s mass
Matching formula = Static part + 1/m – correction :
L2m(av)
B = L2mstatB
= L2
[Estat − Γ stat
1 (L2)]+ σm(u1) + 2Φ2(L1,M)
+ L2m(1)
B
= σkin2 (u1)Φ1(L1,M) + L2
[Ekin − Γ kin
1 (L2)]ωkin
ImplementationAs in the static case, but technical details more involved (& omitted here)
I SF boundary conditions, i.e. T × L3, Dirichlet at x0 = 0, T , fermionfields periodic in space modulo a phase: ψ(x + Lk) = e iθψ(x)
I Avoid extra term ( 1m
fstatδA ) in boundary-to-A0 correlator by boundary-
to-boundary ones with pseudoscalar or vector quantum numbers
(f1)R (T) = Zboundary e−mbareT
fstat1 + ωkinf
kin1 + ωspinf
spin1
0
LxLxL
x
x0
0
= T
=
I Φ1 : Suitable ratios of f1(θ), f1(θ′) such that Z– factors drop out
Φ2 : Spin-averaged energy Γ1 = −∂T ln f(av)1 = mbare + Γ stat
1 + ωkinΓkin1
plus corresponding step scaling functions σkin1 (u), σkin
2 (u)
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New perspectives for B-physics The b-quark’s mass
Examples for continuum limit extrapolations (quenched)
Note: 1/m – terms approach a = 0 only with a rate ∝ a
0 0.0005 0.001 0.0015 0.002 0.0025
(a/L) 2
8
8.5
9
9.5
10
Φ2
0 0.05 0.1 0.15 0.2a/L
0
2
4
6
8
10
Σ2
kin
HYP 2HYP 10.1 % uncertainty on M
b
Left:Dimensionless spin-averaged finite-volume energy from f1 in QCD
Right:Cancellation of 1/a2 power-divergence in σkin
2 = lima/L→0 Σkin2
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New perspectives for B-physics The b-quark’s mass
10 11 12 13 14M L
1
12
13
14
15
16
17
18
19
L2 m
B
exp - L
2 (E
stat-Γ
1
stat(L
2)) - σ
m
2 Φ2(L
1, M)
Mb
stat
Left: Leading order (= static)
I mlight = 0 & L1 ' 0.4 fm
I Input: r0m(exp)
B , r0 = 0.5 fm
I Error dominated by that on ZM inLM = ZM Z (1 + bmamq) × Lmq
( NP known incl. O(a) [ LPHAACollaboration ] )
Quenched b-quark’s mass to order Λ2/mb in HQET
Result in the MS scheme (reported @ Lattice Conference 2005):
mb (mb) = mstatb + m
(1)
b mstatb = 4.35(6)GeV , m(1)
b = −0.05(3)GeV
= 4.30(7)
Though quenched approximation, well within range quoted by PDG
Difficult piece: Large-volume HQET matrix element [Ekin − Γkin1 (L2)]
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Summary & Outlook
Summary & Outlook
I Predictive power of lattice QCD demands high precision:→ Overcoming the error from the quenched approximation is under way→ Several & fine enough resolutions required to quantify cutoff effects
I New quality of the computations employing lattice HQET:X Non-perturbative formulation including the matchingX Continuum limit at large quark masses (small-volume setup !)X Renormalization of 1/m – terms can be performed non-perturbatively
[ Della Morte et al., PoS LAT2005 (2005) 223&224, LPHAACollaboration 3 H. ]
X Physics results still quenched, but extension of the methods to QCDwith dynamical quarks (Nf > 0) straightforward and already started
[ first steps towards FBs for Nf = 2: Della Morte, Fritzsch & H. ]
I Next steps, resp. related work in progress:→ Decay constant FB
→ Spin-splitting mB∗ − mB
→ NP matching for Nf = 2
Jochen Heitger Fundamental QCD parameters from the lattice