Fundamental QCD parameters from the lattice

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

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)


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

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


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


Lattice QCD basics

Lattice QCD basics


Lattice QCD basics

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-¾

6

?

- 6¾?

6?

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


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


Lattice QCD basics



(D/ + mf

)= 1




(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


Lattice QCD basics



(D/ + mf

)= 1




(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


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 + . . .





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





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


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





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





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



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 + . . .



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 ] )







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











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


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 . . .



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

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





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)



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






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〉



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









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



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


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, . . .



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


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’



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 ]



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



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




SHQET = a4∑

x

Lstat(x) +

n∑

ν=1

L(ν)(x)

, L(ν)(x) =∑

i

ω(ν)

i L(ν)

i (x)


L(1)1 = ψh

(− 1

2σ · B

)ψh ≡ Ospin →


L(1)2 = ψh

(− 1

2D2

)ψh ≡ Okin →


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)




SHQET = a4∑

x

Lstat(x) +

n∑

ν=1

L(ν)(x)

, L(ν)(x) =∑

i

ω(ν)

i L(ν)

i (x)


L(1)1 = ψh

(− 1

2σ · B

)ψh ≡ Ospin →


L(1)2 = ψh

(− 1

2D2

)ψh ≡ Okin →


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) + . . .



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



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



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



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



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 ?



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 !




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


HQET

1/mb ¿ L






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


HQET

1/mb ¿ L





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 ?



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

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



⇒ 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



⇒ 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



⇒ 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



⇒ 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)



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



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)



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



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)]


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


Fundamental QCD parameters from the lattice

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