CSIS thesis Alois Grimbach

IntroductionThe static-light current on the lattice

The static-light current in the Lattice SFDetermination of cstat

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Minimisation of the self energySummary

Master thesis in CSiSImprovement of the static-light axial current on the lattice

Alois Grimbach

Institut fuer Theoretische PhysikBergische Universität Wuppertal

Author Short Paper Title



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Outline

1 Introduction

2 The static-light current on the latticeActions and CurrentsO(a) improvementHYP smearing

3 The static-light current in the Lattice SFThe Schrödinger FunctionalPertubation Theory in the SFHYP smearing in the SF

4 Determination of cstatA

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5 Minimisation of the self energy

6 Summary




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Introduction

Lattice QCD

Strong interaction is described by SU(3) colour group

Pertubation theory successful at small distances (High energyregime)

1974, Wilson: Lattice QCD (Low energy regime):hadronic spectra and matrix elements between hadronic statescan be investigated

Principle:Euclidean (Wick-rotated) hypercubic lattice with lattice spacing aallows application of statistical methods

- Investigation of static-light axial current is helpful in b-physics- Improvement accelerates approach to continuum limit




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Introduction

Lattice QCD









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Introduction

Lattice QCD









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Introduction

Lattice QCD









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Introduction

Lattice QCD









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Actions and CurrentsO(a) improvementHYP smearing

ActionsGauge Action

Gauge Action

Action consists of gauge action and fermionic actionS = SG[U] + SF [U,Ψ,Ψ]

Gauge links Uµ(x)- connect x with x + aµ- are members of SU(3) group

Gauge Action is described by sum over plaquettesSG[U] = 1

g20

∑

ptr 1− U(p)

Formulation is gauge invariant and yields Yang-Mills theory in thecontinuum limit




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ActionsGauge Action

Gauge Action




g20

∑

ptr 1− U(p)





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ActionsGauge Action

Gauge Action




g20

∑

ptr 1− U(p)





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ActionsFermion Action

Two theories for light and static quarks

Light quarks

Fermionic Action for light quarksSl [ψl , ψl ] = a4 ∑

xΨl(x)(D + m0)Ψl(x)

D is Wilson-Dirac operatorD = 1

2

γµ(∇∗µ +∇µ)− a∇∗

µ∇µ

Wilson term- removes fermion doublers- vanishes in the continuum limit a→ 0- breaks chiral symmetry for massless fermions




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ActionsFermion Action

Two theories for light and static quarks

Light quarks

Fermionic Action for light quarksSl [ψl , ψl ] = a4 ∑

xΨl(x)(D + m0)Ψl(x)

D is Wilson-Dirac operatorD = 1

2

γµ(∇∗µ +∇µ)− a∇∗

µ∇µ

Wilson term- removes fermion doublers- vanishes in the continuum limit a→ 0- breaks chiral symmetry for massless fermions




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ActionsFermionic Action

Heavy quarks

Heavy quarks are described by HQET- static approximation at m0 →∞- higher contributions organised as powers of inverse quark mass

Static quarks

Static quarks- have only temporal dynamics- are described by decoupled pair of fermion fieldsSh[ψh, ψh] = a4 ∑

x ψh(x)∇∗0ψh(x)

Sh[ψ—h, ψh] = −a4 ∑

x ψh(x)∇0ψ—h(x)




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ActionsFermionic Action

Heavy quarks

Heavy quarks are described by HQET- static approximation at m0 →∞- higher contributions organised as powers of inverse quark mass

Static quarks

Static quarks- have only temporal dynamics- are described by decoupled pair of fermion fieldsSh[ψh, ψh] = a4 ∑

x ψh(x)∇∗0ψh(x)

Sh[ψ—h, ψh] = −a4 ∑

x ψh(x)∇0ψ—h(x)




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Static-light Axial Current

Axial Current

Isovector Axial Current for SU(2) isospinAα

µ(x) = Ψ(x)γµγ512τ

αΨ(x)

Static-light Axial Current- is defined by Astat

0 = Ψl(x)γ0γ5Ψh(x)- is induced by a static quark and a light anti-quark




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Static-light Axial Current

Axial Current

Isovector Axial Current for SU(2) isospinAα

µ(x) = Ψ(x)γµγ512τ

αΨ(x)

Static-light Axial Current- is defined by Astat

0 = Ψl(x)γ0γ5Ψh(x)- is induced by a static quark and a light anti-quark




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O(a) improvementSymanzik improvement scheme

Discretisation error proportional to lattice spacing a- can be improved to O(a2)

Symanzik improvement scheme

- consider momentum cutoff as scale of new physics- describe lattice action by continuum effective theory

Seff =∫

d4x[

L0(x) +∞∑

k=1akLk (x)

]

- lowest order describes continuum field theory- cancel term proportional to a by counterterms




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Seff =∫

d4x[

L0(x) +∞∑

k=1akLk (x)

]





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Seff =∫

d4x[

L0(x) +∞∑

k=1akLk (x)

]





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O(a) improvementAction and Current

Counterterms can be found by- considering dimensions and symmetries- taking into account EOM

Action

- counterterm is Sheikoleslami Wohlert clover term- proportional constant cSW

Static-light axial current

- countertermδAstat

0 = Ψlγjγ512 (←−∇ j +

←−∇∗j )Ψh

- proportional constant cstatA may be expanded in PT by

cstatA =

∞∑

k=0cstat

A

(k)g2k0




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Action




0 = Ψlγjγ512 (←−∇ j +

←−∇∗j )Ψh


cstatA =

∞∑

k=0cstat

A

(k)g2k0




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Action




0 = Ψlγjγ512 (←−∇ j +

←−∇∗j )Ψh


cstatA =

∞∑

k=0cstat

A

(k)g2k0




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Action




0 = Ψlγjγ512 (←−∇ j +

←−∇∗j )Ψh


cstatA =

∞∑

k=0cstat

A

(k)g2k0




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Smearing techniques - APE

consider gauge links

APE smearing

APE smearing- decorate the gauge link with staples- parameter α weigthing the staples




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Smearing techniques - HYP

HYP smearing

- 3 levels of recursive APE smearing- use only links that stay within the hypercubes attached to theoriginal link- project onto SU(3) after each step- parameters α1, α2, α3 weigthing the smearing steps




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HYP smearing - Properties

- preserves locality- improves signal-to-noise ratio- origin: reduction of static self-energy

cstatA for HYP smeared action

- estimated values for cstatA known from hybrid methods

- error ∝ (O)(g40), but unknown

sought quantities

sought(1):- one-loop expansion of cstat

A with HYP smearing

sought(2):- mimimum of self-energy w.r.t. smearing parameters




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sought quantities


A with HYP smearing





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sought quantities


A with HYP smearing





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sought quantities


A with HYP smearing





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The Schrödinger FunctionalPertubation Theory in the SFHYP smearing in the SF

Tool: The Schrödinger Functional (SF)

The SF - sketchP−ψ(x)|x0=T = ρ′(x)

P+ψ(x)|x0=0 = ρ(x)

x0 = T

Uk(x)|x0=T = W ′

k(x)

Uk(x)|x0=0 = Wk(x)

x0 = 0

Uµ(x)|x0>T = 1

ψ(x)|x0>T = 0

Uµ(x)|x0<0 = 1

ψ(x)|x0<0 = 0Author Short Paper Title



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O(a) improvement in the SF

The Schrödinger Functional

- Dirichlet boundary conditions for fermionic fields at x0 = 0 andx0 = T- PBC in spatial directions described by a phase shift Θk


- contains an additionally boundary term for the light action:Wilson Dirac operator in the SF δD = δDV + δDb

- static quark action does not contain boundary term due to EOM- static axial current does not contain a boundary term- free theory is already O(a) improved→ cstat

A

(0)= 0




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The Schrödinger Functional

- Dirichlet boundary conditions for fermionic fields at x0 = 0 andx0 = T- PBC in spatial directions described by a phase shift Θk


- contains an additionally boundary term for the light action:Wilson Dirac operator in the SF δD = δDV + δDb

- static quark action does not contain boundary term due to EOM- static axial current does not contain a boundary term- free theory is already O(a) improved→ cstat

A

(0)= 0




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Correlation functions in the SF

expectation value of operator O< O >= 1

Z

∫

fieldsOe−S

- integrate over fermionic and gluonic fields

fermionic fields

- compute fermionic fields analytically- correlation functions can be reduced tobasic correlation functions for light and static quarksby Wick contraction

gluonic fields

- gluonic fields can be evaluated in pertubation theory




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Pertubation Theory in the SF-1

Pertubation theory - approach

- describe link variable by gauge vector field qµ(x)Uµ(x) = exp(g0aqµ(x))- expand in terms of coupling constant g0

correlation functions

define correlation functionsf stat

A (x0) = −a6 ∑

y,z

12

⟨

Astat0 (x)ζh(y)γ5ζl(z)

⟩

and

f stat1 = − 1

2a12

L6

∑

u,v,y,z< ζ ′l (u)γ5ζ

′h(v)ζh(y)γ5ζ(z) >

- expand them in pertubation theory




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Pertubation Theory in the SF-1

Pertubation theory - approach

- describe link variable by gauge vector field qµ(x)Uµ(x) = exp(g0aqµ(x))- expand in terms of coupling constant g0

correlation functions

define correlation functionsf stat

A (x0) = −a6 ∑

y,z

12

⟨

Astat0 (x)ζh(y)γ5ζl(z)

⟩

and

f stat1 = − 1

2a12

L6

∑

u,v,y,z< ζ ′l (u)γ5ζ

′h(v)ζh(y)γ5ζ(z) >

- expand them in pertubation theory




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Feynman Diagrams for f statA at one-loop order

setting-sun tadpoles gluon exchange




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Feynman Diagrams for f stat1 at one-loop order

setting-sun tadpoles gluon exchange




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HYP links in the SF - 1

sought:

relation between HYP link and original thin link in time-momentumspace

known

- result on the full torus in momentum space:B(3)

µ (p) =∑

ν fµν(p)qν(p) + O(g0)

solution:

- anti FT in time- is feasible du to Dirichlet BC




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HYP links in the SF - 2

result

B(3)0 (x0; p) =

∑6i=0 h0;i(p)qµ(i)(x0 + as(i); p)

with

i µH(i) sH(i) h0;i(p)

0 0 0 1− α16

∑3k=1 a2p2

kΩ0k (p)

1,2,3 i 0 + iα16 apiΩ0i(p)

4,5,6 i − 3 1 − iα16 apµ(i)Ω0µ(i)(p)

- result was checked by direct spatial FT- computation of spatial HYP links is more involved→ in publication




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Determination of cstatA

(1)

take the ratio X (g0,La ,

TL ,Θ) =

f statA√f stat1

cstatA

(1) can be extracted

eliminates divergent part δm of the self-energy - wave functionrenormalistion constants at the boundaries cancel

continuum extrapolation

cstatA

(1) may be extracted from the computed correlation functionsas

cstatA

(1)=

lim aL →0

L22a (∂+∂∗)Xlat

(1)( La )|ct =1−lim a

L →0 ct(1)LX (1)

b ( La )

lim aL →0 LXδA

(0)( La )




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(1)

take the ratio X (g0,La ,

TL ,Θ) =

f statA√f stat1

cstatA

(1) can be extracted

eliminates divergent part δm of the self-energy - wave functionrenormalistion constants at the boundaries cancel

continuum extrapolation

cstatA

(1) may be extracted from the computed correlation functionsas

cstatA

(1)=

lim aL →0

L22a (∂+∂∗)Xlat

(1)( La )|ct =1−lim a

L →0 ct(1)LX (1)

b ( La )

lim aL →0 LXδA

(0)( La )




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(1)

HYP1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

a/L

c Astat

(1)

cAstat(1) for the HYP1 action

Theta=0.5

Theta=1.0

cstatA

(1)HYP1 = 0.0025(3)

HYP2

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.04

0.05

0.06

0.07

0.08

0.09

0.1

a/L

c Astat

(1)

cAstat(1) for the HYP2 action

Theta=0.5

Theta=1.0

cstatA

(1)HYP2 = 0.0516(3))




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Results for the self-energy

self-energy

- The self energy can be determined by summing up the 1-loopFeynman diagrams- comparison with known results provides a check of the diagrams

results

Action e(1)

EH 0.168502(1)

HYP1 0.048631(1)

HYP2 0.035559(1)

- results differ less than 0.3% from the linear divergent contribution tothe static propagator at 1-loop order inM. Della Morte, A. Shindler and R. Sommer, [arXiv:hep-lat/0506008]




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smearing parameters

- The self-energy has a functional dependence upon the smearingparameters, i.e.

e(1) =2

∑

k1,k2,k3=0e(1)

k1k2k3αk1

1 αk22 α

k33

- coefficients can be determined out of the of the Feynman diagrams- coefficients have a triangular structure, only for0 ≤ k3 ≤ k2 ≤ k1 ≤ 2 non-zero- Results are align with the one-loop expansion of the staticself-energy won from the static potential by R.Hoffmann




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Function of e(1) w.r.t the smearing parameters

3D plot of e(1) at α1 = 1

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

0.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075

0.08

α2

α3

e(1)

Minimum at

~α∗ = (α∗1, , α

∗2, α

∗3)

= (1.0000, 0.9011, 0.5196)

with

e(1)(~α∗) = 0.03520(1)

1− loop result for HYP2

e(1)(~αHYP2) = 0.03544(1)




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Summary

Theoretical topic of PT in the SF was reviewed

Feynman rules were extended to HYP smearing

cstatA

(1)HYPx at 1-loop order was determined

The self-energy was minimised w.r.t. the HYP parameters




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Summary



cstatA






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Summary



cstatA






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Summary



cstatA




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CSIS thesis Alois Grimbach

Education