nonperturbative renormalization in schrödinger functional ... filegoal of schrödinger functional...

Nonperturbative Renormalization

in Schrödinger Functional Scheme

Yusuke Taniguchifor

PACS-CS collaboration

14年2月20日木曜日

Position of renormalization worksComplement of the main project

Nf=2+1 QCD at physical quark mass (2008-2010)

• Strong coupling αs for Nf=3 QCD

• Quark mass renormalization factor for Nf=3 QCD

A challenge to larger volume L～9fm (2011-)

★ use of APE stout smeared link for Dirac operator

• Clover term coefficient cSW


Plan of the talk

1. Introduction

2. Strong coupling αs for Nf=3 QCD

3. Quark mass renormalization factor

4. Clover term coefficient

5. Conclusion

✔


Strong coupling αs for Nf=3 QCDGoal of Schrödinger functional scheme

Renormalize on the lattice and convert it to the MS scheme

or renormalization group invariant (RGI) quantity

An Example: ⇤QCD

can be evaluated precisely by perturbation theory

ifrenormalized coupling g(µ)

renormalization scale μare given precisely

for small g(µ)

⇤QCD = µ(b0g)� b1

2b20exp

✓� 1

2b0g

◆exp

�Z g

0dg

✓1

�(g)+

1

b0g3� b1

b20g

◆!


Strong coupling αs for Nf=3 QCDGoal of Schrödinger functional scheme

Renormalize on the lattice and convert it to the MS scheme

or renormalization group invariant (RGI) quantity

An Example: ⇤QCD

can be evaluated precisely by perturbation theory

ifrenormalized coupling g(µ)

renormalization scale μare given precisely

for small g(µ)

on latticeμ tends to be low energyg(µ) tends to be strong use of SF scheme


Strong coupling αs for Nf=3 QCDKey quantity: Step scaling function (SSF) (Luescher et al)

Discrete renormalization group flow: μ→２μg(µ) g(2µ)

1 10 100energy scale (GeV)

1

2

3

4

gaug

e co

uplin

g

•Follow the RG flow in discretized way

• Not a speciality of SF scheme

• Well suited for SF schemeUnique renormalization scale

Box size µ = 1/L

• Can reach at any high energy scale



Discrete renormalization group flow:

1 10 100energy scale (GeV)

1

2

3

4

gaug

e co

uplin

g

•Follow the RG flow in discretized way

• Not a speciality of SF scheme

• Well suited for SF schemeUnique renormalization scale

Box size µ = 1/L

L ! 2L

g(L) g(2L)

• Can reach at any high energy scale



Discrete renormalization group flow: L ! 2L

g(L) g(2L)

β

β



Discrete renormalization group flow: L ! 2L

g(L) g(2L)

larger β

SSF in the continuum limit14年2月20日木曜日

Strong coupling αs for Nf=3 QCDStep scaling function (SSF)

★ Iwasaki gauge action★ Nonperturbatively improved Wilson fermion

• three massless quarks

0 0.1 0.2a/L

1

2

3

4

5

6

Y(1

) (u,a

/L)

Const fit w/ L/a=6, 8Const fit w/ L/a=4, 6, 8Linear fit w/ L/a=4, 6, 8

SSF on lattice★Consistency between three

continuum extrapolations• Constant extrapolation

with two/three data• Linear extrapolation



★ Iwasaki gauge action★ Nonperturbatively improved Wilson fermion

• three massless quarks

0 0.1 0.2a/L

1

2

3

4

5

6

Y(1

) (u,a

/L)


0 1 2 3 4g2(L)

1

1.5g2 (2

L)/g

2 (L)

L/a=4L/a=6L/a=8Const fit w/ 6, 8PT 3 loopspolynomial fit

SSF on lattice SSF



0 0.1 0.2a/L

1

2

3

4

5

6

Y(1

) (u,a

/L)


0 1 2 3 4g2(L)

1

1.5g2 (2

L)/g

2 (L)

L/a=4L/a=6L/a=8Const fit w/ 6, 8PT 3 loopspolynomial fit

SSF on lattice SSF

★ Perturbative improvement of the SSF• One loop PT is not enough


Strong coupling αs for Nf=3 QCD↵S(MZ) ⇤MSand

Box size in physical unitg(L

max

)

Lmax

= aNmax

β=1.90 a=0.08995(40) fmtypically 4-6

perform step scaling n times

g(Lmax

/2n) and renormalization scale Lmax

/2n

are precisely determined

For n～10 perturbation theory is applicable


Strong coupling αs for Nf=3 QCD↵S(MZ) ⇤MSand

0 0.05 0.1 0.15a/L

0.11

0.115

0.12

0.125

CP−PACSConstant fit with three `Constant fit with `=2.05, 1.90Linear fit`=1.90 (PACS−CS)

0 0.05 0.1 0.15a/L

150

200

250

300

CP−PACSConstant fit with three `Constant fit with `=2.05, 1.90Linear fit`=1.90 (PACS−CS)

↵s(MZ) = 0.12047(81)(48)(�173)

⇤MS = 239(10)(6)(�22)MeV


Plan of the talk

1. Introduction




5. Conclusion

✔

✔


Quark mass renormalization factorGoal: RGI mass

M = m(µ)�2b0g

2(µ)�� d0

2b0exp

�Z g(µ)

0dg

✓⌧(g)

�(g)� d0

b0g

◆!

renormalized mass

renormalized coupling

if evaluated at large μRGI mass M can be given perturbatively

SSF works well m(L) m(2L)

Applying iteratively m(Lmax

) m(Lmax

/2n)


Quark mass renormalization factorQuark mass step scaling function (Luescher et al)

⌃P (u, a/L) =ZP (g0, 2L)

ZP (g0, L)

��g2(L)=u,m=0

0 0.1 0.2a/L

0.8

.85

0.9

.95

1 Const fit w/ L/a=6, 8Const fit w/ L/a=4, 6, 8Linear fit w/ L/a=4, 6, 8

SSF on lattice

★Consistency between three continuum extrapolations• Constant extrapolation

with two/three data• Linear extrapolation


Quark mass renormalization factorQuark mass step scaling function (Luescher et al)

⌃P (u, a/L) =ZP (g0, 2L)

ZP (g0, L)

��g2(L)=u,m=0

0 0.1 0.2a/L

0.8

.85

0.9

.95

1 Const fit w/ L/a=6, 8Const fit w/ L/a=4, 6, 8Linear fit w/ L/a=4, 6, 8

0 1 2 3 4g2(L)

0.7

0.8

0.9

1

Z P(2L

)/ZP(

L) L/a=4L/a=6L/a=8Const. fit w/ 6, 8PT 2 loopsNPT fit

SSF on lattice SSF


Quark mass renormalization factorRenormalization factor of axial vector current

M = m(bare)

PCAC

(�)ZA(�)

ZP (�, a/Lmax

)

��a!0

m(Lmax

/2n)

m(Lmax

)

��NP

M

m(Lmax

/2n)

��PT

1.5 2 2.5 3 3.5 4 4.5 5 5.5`

0.5

0.6

0.7

0.8

0.9

1

1.1

Z A

ZA(`)

DisconnectedConnectedPT

different definition of ZA

with different boundary operator

O(a) error

adopted


Quark mass renormalization factorNon-perturbative running mass for Nf=3 QCD

0 1 10 100 1000µ/RSF

0.2

0.4

0.6

0.8

1

mSF

(µ)/M

Non−perturbative running mass in SF scheme

NP running massPT 2/3−loops

mMSud (2 GeV) = 2.78(27)MeV

mMSs = 86.7(2.3)MeV


Plan of the talk

1. Introduction




5. Conclusion

✔

✔

✔


Evaluation of CSW by SF(Alpha)

T=L

L

Two different Dirichlet BC’s

Two PCAC masses

Two improved current correlators

O(a) indicator

�M = M (0) �M (T )

M (0)

M (T )

boundary + bulk O(a) effect

Fix cSW in order that O(a) effect vanish

�M = 0 ! cSW

M = 0 ! c

A(0)µ + cA@µP

(0)

A(T )µ + cA@µP

(T )

2

at tree levelO(10�4)14年2月20日木曜日

CSW dependence of O(a) indicator

1 1.1 1.2 1.3 1.4 1.5cSW

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0.004

0.005

0.006

Nsmear=1, β=1.95Nsmear=6, β=1.82

ΔM

0 2 4 6 8 10 12 14 16-0.03

-0.02

-0.01

0

0.01

0.02 MM’ΔM

cSW=1.0



1 1.1 1.2 1.3 1.4 1.5cSW

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0.004

0.005

0.006


ΔM

0 2 4 6 8 10 12 14 16-0.03

-0.02

-0.01

0

0.01

0.02 MM’ΔM

cSW=1.0

0 2 4 6 8 10 12 14 16-0.03

-0.02

-0.01

0

0.01

0.02 MM’ΔM

cSW=1.1



1 1.1 1.2 1.3 1.4 1.5cSW

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0.004

0.005

0.006


ΔM

0 2 4 6 8 10 12 14 16-0.03

-0.02

-0.01

0

0.01

0.02 MM’ΔM

cSW=1.0

0 2 4 6 8 10 12 14 16-0.03

-0.02

-0.01

0

0.01

0.02 MM’ΔM

cSW=1.1

0 2 4 6 8 10 12 14 16-0.03

-0.02

-0.01

0

0.01

0.02 MM’ΔM

cSW=1.2


Clover term coefficient

0 1 2 3 4 5 6 7Nsmear

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

c SW

0 1 2 3 4 5 6 7Nsmear

0.125

0.13

0.135

κ c

cSW c


Clover term coefficient

0 1 2 3 4 5 6 7Nsmear

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

c SW

0 1 2 3 4 5 6 7Nsmear

0.125

0.13

0.135

κ c

cSW c

0 1 2 3 4 5 6 7Nsmear

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

c A

cA


★Strong coupling αs for Nf=3 QCD

• SF scheme works well.

• Need a→0 limit, but is not available.

• Smearing may reduce O(a) error.

★Quark mass renormalization factor

• ZA has a large O(a) error.


★Clover term coefficient

•Csw～1.1

Conclusion


★Strong coupling αs for Nf=3 QCD

• SF scheme works well.

• Need a→0 limit, but is not available.


★Quark mass renormalization factor

• ZA has a large O(a) error.


★Clover term coefficient

•Csw～1.1

Conclusion

0 1 2 3 4 5 6 7Nsmear

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

c SW


nonperturbative renormalization in schrödinger functional ... filegoal of schrödinger functional...

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