Non-perturbative renormalization of
BK and quark mass in
SF scheme with quenched domain-wall QCD
Yousuke Nakamura, Yusuke Taniguchi
for CP-PACS Collaboration
External Review on Center for Computational Sciences(University of Tsukuba, October 30-31, 2007)

External Review on Center for Computational Sciences(@ University of Tsukuba 2007) – p.1/23

Plan of talkDefiniton of BK and its renormalization factor.
Strategy to derive RGI b
BK

SF scheme (definition of non-perturbative renormalization factor)
Alpha collaboration ’05

Simulation results
RGI B̂K
Scaling behavior of SSF at g2 (L) = 3.480Renormalization of quark mass
Summary

External Review on Center for Computational Sciences(@ University of Tsukuba 2007) – p.2/23

Definition of BK and its renormalization factorThe kaon B parameter: BK
hK¯0 | s̄γµ (1 − γ5 )d s̄γµ (1 − γ5 )d|K 0 i
BK ≡ 8¯0 | s̄γµ (1 − γ5 )d|0ih0| s̄γµ (1 − γ5 )d|K 0 ihK
3CP-PACS obtained BK with quenched domain-wall fermion 2001
⇒ We need non-perturbative renormalization for B K
Renormalization factor for BK
ZVV+AA (g0 , aµ) ZVA+AV (g0 , aµ)=ZBK (aµ) =2ZA (g0 )ZV2 (g0 )DWF has a good chiral symmetry even on latticeZV (g0 ) = ZA (g0 ) has been showen by CP-PACS 2003
ZVV+AA (g0 , aµ) = ZVA+AV (g0 , aµ)· ⇒this relation will be checked later

External Review on Center for Computational Sciences(@ University of Tsukuba 2007) – p.3/23

Our strategy
Lattice bare B(0)K with DWF⇒ Renormalization group invariant (RGI) B̂KPTNPZBK (g0 ) = ZVA+AV(∞, µmax ) ZVA+AV(µmax , µmin ) ZBNPK (g0 , aµmin )|{z}|{z}|{z}
IIIII

I

Three steps for RGI B̂K via SF scheme.

(SF)
I. B(0)⇒BK
K (aµmin ) at hadronic scale
suppress lattice artifact O(aµ) : aµ  1II.
III.

(SF)
Non-pert. RG running : B(SF)(µ)⇒BminK
K (µmax )Perturbative running to RGI: B(SF)K (µmax ) ⇒ B̂K

II. III. are reguralization independent.
Possible to evaluate with cheap Wilson fermion.
Already calculated by Alpha collab. for OVA+AV ’05⇒ OVA+AV has no mixing problem even for wilson fermion.
Our target: I. ZVA+AV (g0 , aµmin ) with DWF.

External Review on Center for Computational Sciences(@ University of Tsukuba 2007) – p.4/23

Set up for SF scheme Alpha ’05
SF formalism

0Oi jt=LP
6=a0 (R)Oi j00x)Γζ j(~y)~x~y ζ̄i (~parity odd0 (0)= Z ζ Oi j

PSfrag replacements

OV A+AV = (ψ̄1γµψ2)(ψ̄3γµγ5ψ4) + (ψ̄1γµγ5ψ2)(ψ̄3γµψ4)time(0)O(R)=ZO
V A+AV V A+AV
V A+AV

t=0space

Oi j = a 6Px)Γζ j(~y)~x~y ζ̄i(~(0)
O(R)
=ZO
ζijij

Correlation function (Alpha collaboration ’05)
F[Γ±A ,ΓB ,ΓC ] (x0 ) =1±0hO[Γ]O[Γ]O(x)O
21A45
B
VA+AV
53 [ΓC ]i
L3

Orbifolding construction for SF with DWF Y.T ’04,’06

External Review on Center for Computational Sciences(@ University of Tsukuba 2007) – p.5/23

In order to cancel divergence due to boundary operator
boundary-boundary correlation function
1
1 X 00f1 = − 6 hO12 [γ5 ]O21 [γ5 ]i, k1 = − 6hO12 [γk ]O21 [γk ]i2L2L kThree specific cases
Scheme 1F[γ±5 ,γ5 ,γ5 ] (x0 )h±1 (x0 ) =,3/2
f1

Scheme 3F[γ±5 ,γk ,γk ] (x0 )h±3 (x0 ) =,3/2
f1

Scheme 7F[γ±5 ,γk ,γk ] (x0 )h±7 (x0 ) =f11/2 k1

Renormalization condition±ZVA+AV;s (g0 , aµ)h±s (x0 ; g0 ) = hcont
s (x0 , g0 ) |g0 =0 ,s = 1, 3, 7

Renormalization factor±hs (x0 ; g0 )|g0 =0±ZVA+AV;s =h±s (x0 ; g0 )External Review on Center for Computational Sciences(@ University of Tsukuba 2007) – p.6/23

Simulation parameters
Same parameters as in previous CP-PACS calculation of B K
Phys.Rev. D64(2001)114506

Domain-wall fermion with M = 1.8 and N5 = 16
Iwasaki gauge action at β = 2.6, β = 2.9, β = 3.2(new,not in thepaper)corresponding to a−1 ∼ 2, a−1 ∼ 3, a−1 ∼ 4 GeV
Renormalization factor ZVA+AV (β, aµ)1/µmin = 2Lmax

Def. of Lmax : ḡ2SF (Lmax ) = 3.480

ZVA+AV at β = 2.6, β = 2.9, β = 3.2
Fine tuning of β to give aNT = aNL = 2Lmax = 1.498r0
Lmax /a68

1012141618

β2.4462.63392.78732.91753.03133

3.03133.2254# of conf

1000

1000

1000

1018

716556564

External Review on Center for Computational Sciences(@ University of Tsukuba 2007) – p.7/23

Simulation results
Numerical result for ZV and ZVA+AV;s

1.1

ZVZVA+AV;1(g0,aµmin)

L/a=16

L/a=16
1.3

1.05

1.2

1

1.1

0.95

1

0.9

0.9

0.85

0.8

0.8
02468t10

1214

16024

68t10

121416

We take the value at x0 = NT /2 (Alpha collaboration ’05)
ZBK (aµmin ) =ZVA+AV;s (x0 = 16/2, aµmin )= 1.2708(53)2ZV (x0 = 16/2, aµmin )External Review on Center for Computational Sciences(@ University of Tsukuba 2007) – p.8/23

Strategy to derive RGI b
BKLattice bare B(0)K with DWF⇒ Renormalization group invariant (RGI) B̂KPTNPZBK (g0 ) = ZVA+AV(∞, µmax ) ZVA+AV(µmax , µmin ) ZBNPK (g0 , aµmin )|{z}|{z}|{z}
III

III

External Review on Center for Computational Sciences(@ University of Tsukuba 2007) – p.9/23

Result for RGI ZBK and B̂K (quench)
RGI ZB

1.5

1.4

K

RGI BK

0.95

Scheme 1
Scheme 3
Scheme 7

scheme 1
scheme 3
scheme 7
tmQCD (Alpha)
DWF(RBC)

0.9
0.85

1.3
0.8
1.2

1.1
2.2

0.75

2.4

2.6

2.8

βFitting form

3

3.2

0.7

0.0

0.1

0.2

mVa

0.3

0.4

ẐBK (β) = a1 + b1 (β − 3) + c1 (β − 3)2

interpolated at β = 2.6, β = 2.9, β = 3.2
B̂K = 0.773 (7)(preliminary) : Scheme 1
B̂K = 0.760 (7)(preliminary) : Scheme 3
B̂K = 0.778 (7)(preliminary) : Scheme 7: RBC
B̂K = 0.786(31): AlphaB̂K = 0.789(46)Error is smaller
Scaling behaviour is better

Error of each data is smaller
The smallest lattice spacing
is finer

External Review on Center for Computational Sciences(@ University of Tsukuba 2007) – p.10/23

Conversion of RGI BK to that in MS scheme
Conversion from RGI to MS NDR at µ = 2GeV by NLO PT+ 2bγO
 2"Z g (µ)+ !#+ g (µ)  0MSγγ0 MS  exp
B̂K−BMS(NDR,µ)=dg
K 4π β(g) b0 g0Gauge couplingΛMS = µ(b0 g2MS )b− 122b01
exp −2b g20MS

 Z g
MS
 11b1 
 exp −+dg −β (0) b0 g3 b2 g0MS0

ΛMS = 0.586(48)/r0 Necco&Sommer ’02

External Review on Center for Computational Sciences(@ University of Tsukuba 2007) – p.11/23

Comparison of our result for BK (MS, 2GeV) with previous ones
BK(NDR,2GeV) vs. mVa

0.8

KS (non-invariant)
KS (invariant)
CP-PACS(perturbation)
CP-PACS(scheme 1)
CP-PACS(scheme 3)
CP-PACS(scheme 7)
Alpha (tmQCD)
RBC(DWF)

BK

0.7

0.6

0.5

0.00.2mVa

0.4+4 ) (preliminary)
BMS(NDR,2GeV)=0.557(5)(−10KBMS
K (NDR, 2GeV) = 0.567(4)BMS
K (NDR, 2GeV) = 0.563(21)BMS
K (NDR, 2GeV) = 0.573(34): CP-PACS(perturbative ren.): RBC: AlphaExternal Review on Center for Computational Sciences(@ University of Tsukuba 2007) – p.12/23

Further checks
Scaling behavior of Step Scaling Function
ΣVA+AV;s (u, Lmax /a) =ZVA+AV;s (g0 , a/2Lmax )ZVA+AV;s (g0 , a/Lmax )Chiral symmetry breaking effect in renormalization factor
ZVA+AV = ZVV+AA

External Review on Center for Computational Sciences(@ University of Tsukuba 2007) – p.13/23

Scaling behavior of SSF at g2 (L) = u = 3.480
ZVA+AV;s (g0 , a/2Lmax )σVA+AV;s (u) = lim ΣVA+AV;s (u, a/Lmax ) =a→0
ZVA+AV;s (g0 , a/Lmax ) ḡ2 =u
2++

Σ1 (u,a/L) vs. a/L with g (L)=3.480

1.6

5.5

CP-PACS (M=1.8)
Alpha (Clover action)
Alpha (Wilson action)
continuum (Alpha)

1.5
1.4

M=1.8
M=0.9

5
4.5

1.3

Σ1 (u,a/L)

4

1.2

3.5
3+

1.1+

Σ1 (u,a/L)

TREE LEVEL Σ1 (u,a/L)

1

2.5

0.9

2

0.8

1.5

0.7

10.6

0

0.1

a/L

0.2

0.5

00.1

a/L

0.2

Boundary O(a) effect is large at M = 1.8

External Review on Center for Computational Sciences(@ University of Tsukuba 2007) – p.14/23

Results at N5 = 8 and N5 = 32+

Σ1 (u,a/L) vs. Ns

1.6++

Σ1 (u,L/a=6)

1.2+

Σ1 (u,a/L)

1.4

Σ1 (u,L/a=4)

10.8

0.6

051015

Ns20

25

30

35

No N5 dependence

External Review on Center for Computational Sciences(@ University of Tsukuba 2007) – p.15/23

Our solution
SOLUTION 1: results at M = 1.4+

5.5

Σ1 (u,a/L) vs. a/L with g (L)=3.480

1.6

M=1.8
M=0.9

52+

TREE LEVEL Σ1 (u,a/L)

CP-PACS (M=1.4)
continuum (Alpha)

1.4

4.5

Σ1 (u,a/L)

3.5

1.2+3+

Σ1 (u,a/L)

42.5120.8

1.5
10.5

00.1

Mtadpole ∼ 1

a/L

0.2

0.6

00.1

a/L

0.2

Scaling violation is small
Continuum limit is consistent with Alpha’s

External Review on Center for Computational Sciences(@ University of Tsukuba 2007) – p.16/23

SOLUTION 2:

Tree-level improvement

ZVA+AV;s (g0 , aµ)h±s (x0 ; g0 ) = h s (x0 , g0 )± |g0 =0 CONT ⇒ h s (x0 , g0 )± |tree Lattice+1.6

M=1.8 (tree level imp.)
CP-PACS (M=1.4)
continuum (Alpha)

1.4

1.2+

Σ1 (u,a/L)

2Σ1 (u,a/L) vs. a/L with g (L)=3.480

10.8

0.6

00.1a/L

0.2

Scaling behavior is improved
Continuum limit is consistent with Alpha’s

External Review on Center for Computational Sciences(@ University of Tsukuba 2007) – p.17/23

Scaling behavior of SSF for Z BKZBK ;s (g0 , a/2Lmax )ΣBK (u, Lmax /a) =ZBK ;s (g0 , a/Lmax )ΣB (u,a/L)
K

1.3

ZB (M=1.8)
K

ZB (M=1.4)
K

1.25

Combined fit (linear)Continuum (combined fit)

1.2

1.15

1.1

1.05

100.1a/L

0.2Results at M = 1.4 and M = 1.8 are consistentO(a) error is patrly cancelled in Z BK

External Review on Center for Computational Sciences(@ University of Tsukuba 2007) – p.18/23

Chiral symmetry breaking effect
WT Identity
δq1 = −iγ5 q̃1 ,δζ1 = iγ5 ζ̃1 ,0δζ10 = iγ5 ζ˜1

hOVA+AV O[ζ]iS = hOVV+AA O0 [ζ]iS⇒ ZVV+AA = ZVA+AV

For the domain-wall caseS → S + Y :Y is the chiral symmetry breaking termhOVA+AV O[ζ]iS = hOVV+AA O0 [ζ]iS +Y, hOVV+AA O0 [ζ]iS

hOVA+AV O[ζ]iS ⇔ hOVV+AA O0 [ζ]iSWe investigate if ZVV+AA = ZVA+AV holds or not.

External Review on Center for Computational Sciences(@ University of Tsukuba 2007) – p.19/23

Result for ZVA+AV;1 and ZVV+AA;1 (N5 = 16)
ZVA+AV;1 VS ZVV+AA;11.6

L/a=16

ZVA+AVZVV+AA

1.4

1.2

10.8

02468t10121416

Good agreement between ZVA+AV;1 and ZVV+AA;1ZBK = ZVZA+AV: justified2V

External Review on Center for Computational Sciences(@ University of Tsukuba 2007) – p.20/23

Quark mass renormalization
RGI quark mass

M = m(µ) 2b0 g2 (µ)

− 2bd00exp −Z g(µ)
0dg

τ(g)
d0−β(g) b0 g!!

Zm (g0 ) = ZmPT (∞, µmax ) ZmNP (µmax , µmin ) ZmNP (g0 , aµmin ){z}| {z } |{z}|IIIIII

Lattice bare mass with DWF ⇒ RGI mass

II, III: given by Alpha collab. for PCAC mass
Zm (g0 , aµmin ) = 1/ZP (g0 , aµmin ) for DWF
RGI Zm(β)

RGI u,d mass

1.86
5.9m (MeV)

1.7
1.61.5
1.4
1.3

5.8
5.7
5.6
5.522.5β3

3.55.400.1

0.2

0.3

0.4
−1a (GeV )0.5

0.60.7External Review on Center for Computational Sciences(@ University of Tsukuba 2007) – p.21/23

MS scheme at µ = 2 GeV
Z MS

2 ! 2bd00
g(µ)

 τMS (g)d0MSMS
−m (µ) = M 2b0 g (µ)
exp dg 0βMS (g) b0 g
ΛMS = 0.586(48)/r0 as an input.
mud(µ=2 GeV) [MeV]

ms(µ=2 GeG) [MeV]140

5.0130120

4.0110DWF NPR
DWF PTR
VWI, qStd
AWI, qStd
VWI, qImp
AWI, qImp
KS, p=2.6

3.0

2.0

0.00.5−11.0

a [GeV ]1.5mu,d (µ = 2GeV) = 4.020(58) MeV,DWF, NPR
DWF, PTR
VWI, qStd
AWI, qStd
VWI, qImp
AWI, qImp
KS, p=2.6

100908070

0.00.5−11.0

a [GeV ]1.5m s (µ = 2GeV) = 105.4(15) MeV

External Review on Center for Computational Sciences(@ University of Tsukuba 2007) – p.22/23

SummaryEvaluation of NP renormalization factor for BK
Bare B(0)K on lattice with DWF ⇒ RGI B̂K
Investigation of chiral breaking effect
ZVA+AV = ZVV+AA in DWQCD
RGI B̂K = 0.773(7) (preliminary)+4 ) (preliminary)MS BK (NDR, 2GeV) = 0.557(5)( −10Consistent with previous results
Check of scaling behavior of SSF
Evaluation of NP renormalization factor for quark mass
mu,d (µ = 2GeV) = 4.020(58) MeV,m s (µ = 2GeV) = 105.4(15) MeV

External Review on Center for Computational Sciences(@ University of Tsukuba 2007) – p.23/23