Outline
Introduction
Mass extrapolation & chiral lagrangianVus
fK /fπKl3
Kaon bag parameters
BK
BSM
Other topics
Current simulations

Lattice Kaon physics – p.1/46

Introduction
Wholeheartedly support FLAG’s aims
perhaps some disagreement in detail, but it is a very important effort!
Unfortunately can not match this level of detailWill rather try to emphasize the most interesting activity in the field
Also recomend excellent Lattice 2008 review by Lellouch arXiv:0902.4545

Lattice Kaon physics – p.2/46

Introduction
Lattice QCD simulations not (currently) at physical masses in a large volume.
Interpolations are performed in ms
Extrapolations are performed in mud from larger masses mπ ∈ [200 − 500]M eV down.

valence
Some calculations follow unitary line msea
π = mπmπsea
Lattice QCD

finite volume
danger zone
QCD

Some calculations explore orthogonal direction and include unitary data as a subset: “partially quenched”
quenched 6= unitary ⊂ partially quenched

Partially quenched chiral perturbation theory Finite volume corrections sometimes appliedm π valence

affects smallest masses. Most popular model:
Colangelo, Durr, Haefeli (hep-lat/0503014).

Lattice Kaon physics – p.3/46

Introduction
List simulations covering broad physics programme relevant to this conference
Collaboration

Actionnf

Chirality

Cost

JLQCD

Overlap

2 (2+1)

Exact

$ 50

RBC-UKQCD

DWF

2+1

Good

$ 16

ETMC

Twisted mass

2

Poor

$1

PACS-CS

Clover

2+1

Poor

$1

BMW

smeared clover

2+1

Poor

$1

MILC

Staggered

2+1

Poor

$ 0.25

Also: QCDSF, CLS collaborations.
Overlap and DWF have good lattice chiral symmetry but are expensive→ weak matrix elements have continuum renormalisation structure→ widest range of chiral weak operators→ “V - A ” does exactly what it says
PACS-CS use perturbative renormalisation for ZA
BMW have not yet calculated ZA but plan to do so non-perturbatively.

Lattice Kaon physics – p.4/46

Introduction
List simulations covering broad physics programme relevant to this conference
Collaboration

Action

L(fm)

mmin
(MeV)
π

a−1 GeVJLQCD

Overlap

1.7

310

1.8

RBC-UKQCD

DWF

2.0,2.7330

1.7,2.3

ETMC

Twisted mass

2.0

300

1.9,2.2,2.8

PACS-CS

Clover

3.1

156

2.2

BMW

smeared clover

4.0

190

1.6,2.3,3.03

MILC

Staggered

2.4,2.9,3.4

320

1.1,1.3,1.6,2.2,3.3,4.4

Also: QCDSF, CLS collaborations.
PACS-CS and BMW are getting very close the physical point
Exciting times!

Lattice Kaon physics – p.4/46

Introduction
List simulations covering broad physics programme relevant to this conference
Collaboration

ActionChPT

Unitary

Finite vol. corr.

JLQCD

Overlap

SU(2)nfU

CDH

RBC-UKQCD

DWF

SU(2)nf , SU(3)nf

PQ

-

ETMC

Twisted massSU(2)nfU

CDH

PACS-CS

CloverSU(2)nf , SU(3)nf

U

CDH

BMW

smeared cloverpolynomial

-

-

MILC

Staggered

rs-SU(3)nf

PQ-

Also: QCDSF, CLS collaborations.
Colangelo, Durr, Haefeli used to “correct” volume effects

Lattice Kaon physics – p.4/46

Outline
Introduction
Mass extrapolation & chiral lagrangianVus
fK /fπKl3
Kaon bag parameters
BK
BSM
Other topics

χq

=

MP2

=

FP2

=

F ππ (q 2 )

=

(1)

ΠV −A

=

2Bmq
χ2q

»
–
χq
χq +
log 2 − l̄3
32π 2 F 2
mπ
»
–
χq
χq
log 2 − l̄4F−
16π 2 F 2
mπ
i
1 h
2
2
r 2
1 + 2 −2l6 q + 4H̃(χq , q , µ )Fm2log( µ2π + 13 − H(x))
fπ2− 2 − 8Lr10 (µ) −q
24π 2

Lattice Kaon physics – p.5/46

2+1 flavors : SU (3) or SU (2) ?
Debate: best mass extrapolation method for 2+1f?
SU(3)nf – Treats the Kaon as a dynamical chiral pseudoscalar
Least convergent terms expand in (Mη /4πf )2
SU(2)nf RBC & UKQCD hep-ph/0702042,arXiv:0804.0473
Treats the Kaon as a heavy spectator
Least convergent terms expand in (Mπ /MK )2
Analytic expansion around all physical quark masses BMW
Discussed Lellouch, Kaon review Lattice 2008 arXiv:0902.4545
Argument compelling near physical point if volume big enough
Chiral effective theory guides extrapolations to physical masses (and in some cases volumes).
Convergence radius of chiral effective theory critical component of physical predictions.
Knowledge of LEC’s yields good predictive tool with different scope than direct lattice simulation.
Debate:Inclusion of partial NNLO analytic terms?
NNLO with non-lattice input for LEC’s?
Finite volume correction of data?

Lattice Kaon physics – p.6/46

2+1 flavors : SU (3) or SU (2) ?
Both RBC-UKQCD and PACS-CS have recently found (arXiv:0804.0473 , arXiv:0811.2630):
Kaons cannot be well described by NLO χPT
3
PTBoth χPT
3 and χ2 can fit pionic lattice data 160 ≤ mπ ≤ 400 MeV

Collaboration

F0 (MeV)

F/F0

MILC

106(8)

RBC/UKQCD

93.5(7.3)´`
1.15(5) +13−3

PACS-CS

118.5(9)1.078(58)1.229(59)Pseudoscalar decay constant F0 in 3 flavor chiral limit may be quite low
Simulated pions (≃ 310 MeV) receive worryingly large NLO correction to this low F0Some spread. PACS-CS use a LPT one-loop ZA - however cancels in F/F0
symmetrised MILC sys err in quad w. stat

Lattice Kaon physics – p.7/46

2+1f DWF RBC-UKQCD fπfPS [MeV]
160

140
fπ
120
f
mll = 331 MeV
mll = 419 MeV
SU(2) fit
SU(3) fit22
mPS [MeV ]

100
f0
80

0 1402

2

250

2

330

2

420

Partially quenched & simultaneous fit with pion mass
DWF axial current is conserved ⇒ renormalisation controlled
Single lattice spacing

Lattice Kaon physics – p.8/46

2+1f DWF RBC-UKQCD fπ
1.8
1.6
1.4
1.2

fll/f0, mh=0.04
total
LO
NLOl
NLOlh

1
0.8
0.6
0.4
0.2
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
χl/GeV2

Partially quenched & simultaneous fit with pion mass
DWF axial current is conserved ⇒ renormalisation controlled
Single lattice spacing

Lattice Kaon physics – p.8/46

2+1f Clover PACS-CS fπ
0.10

0.10

fπ

0.09
0.08

0.08
κs=0.13640
κs=0.13660
SU(2)SU(2)-FSE
SU(2)@phSU(2)-FSE@ph

0.07
0.06
0.05

fπ

0.09

0

0.01

0.02

κs=0.13640
κs=0.13660
SU(3)
SU(3)-FSE
SU(3)@phSU(3)-FSE@ph

0.07
0.06

0.03

0.05

AWI

m ud

0

0.01

0.02

0.03

AWI

m ud

Unitary & simultaneous fit with pion mass
Not far from direct simulation at the physical mass (but small vol)Finite volume CDH adjust at lightest masses

Lattice Kaon physics – p.9/46

2+1f Clover PACS-CS fπ
0.100

0.100

0.095

0.095

fK

0.090

0.090

0.085

0.085
κs=0.13640
κs=0.13660
SU(2)SU(2)-FSE
SU(2)@phSU(2)-FSE@ph

0.080
0.075
0.070

fK

0

0.01

0.02
AWI

m ud

κs=0.13640
κs=0.13660
SU(3)
SU(3)-FSE
SU(3)@phSU(3)-FSE@ph

0.080
0.075
0.03

0.070

0

0.01

0.02

0.03

AWI

m ud

Kaon is poorly described by SU(3)Lattice Kaon physics – p.9/46

2+1f Clover PACS-CS fπ
1.0

NLO/LO

0.8

0.6

fK (su(3)-fit)
fπ (su(3)-fit)

0.4

0.2

fπ (su(2)-fit)

fK (su(2)-fit)

0.0

-0.2

0

0.005

0.01

0.015
AWI

0.02

0.025

0.03

mud

PACS-CS see a smaller F/F0 ratio than MILC and RBC-UKQCD
SU(3) corrections to pionic data correspondingly smaller
ZA is only 1-loop LPT
Physical values likely too high due to higher orders
Does not affect decay constant ratios & convergence discussion

Lattice Kaon physics – p.9/46

SU (3) vs SU (2)Conclusions:

Broad agreement that SU(2) is better for KaonsSome disagreement about just how low F0 is
Convergence of SU (3) χPT questioned even for pionic lattice data
As SU(3) reduces to SU(2) in large strange mass limit

SU(3) form fits pionic data well but with large NLO terms
F0 & other LEC’s from lattice data describe “real world” pions well
These may not be true LEC’s massless 3-flavour theory
Lookout for:JLQCD’s 2+1f overlap results will be very interesting
RBC-UKQCD’s second lattice spacing later this year

Lattice Kaon physics – p.10/46

Chiral effective lagrangian
Quote SU(2) LEC’s as more meaningful – perturbative conversion of SU(3) results
From S. Necco arXiv:0901.4257 (excellent recent review)
F in convention where fπ ≃ 92MeV
F0 (MeV)Σ1/3 (MeV)

Collaboration

Nf

F (MeV)ETM

2

86.03(5)267(2)(9)(4)JLQCD/TWQCD

2

”
“
+4.2
79.0(2.5)(0.7) −0.0

MILC

2+1

”
“
235.7(5.0)(2.0) +12.7“ ” −0.0+2(5)278(1) −3

RBC/UKQCD

2+1

81.2(2.9)(5.7)

PACS-CS

2+1

90.3(3.6)

Giusti, Lüscher

2

ETM

2

86.6(4)(7)

Colangelo, Dürr

phen.

86.2(5)

1/3

Σ0

(MeV)

”
“
+5
(4)
242(9) −17

255(8)(8)(13)
83.8(6.4)309(7)

290(15)276(3)(4)(5)
264(2)(5)+ omitted large body of ǫ-regime work. See Necco.

Lattice Kaon physics – p.11/46

Chiral effective lagrangian
Quote SU(2) LEC’s as more meaningful – perturbative conversion of SU(3) results
From S. Necco arXiv:0901.4257 (excellent recent review)
F in convention where fπ ≃ 92MeV
CollaborationNfl̄3 (SU(2))

CERN-TOV2

3.0(5)(1)ETM

2

3.42(8)(10)(27)

4.59(4)(2)(13)ETM

2JLQCD/TWQCD2

3.2(4)(2)
”
”“
“
+32
+0
3.44(57) −68−04.4(1)(1)
”
”“
“
+32
4.14(26) +49
−0
−0

MILC

2+1

RBC/UKQCD

2+1

3.13(33)(24)

PACS-CS

2+1

Gasser, Leutwyler

phen.

Colangelo et alphen.

l̄4 (SU(2))

l̄3 (SU(3))

l̄4 (SU(3))

”
“
+1.0
1.1(6) −1.5

“ ”
+4
4.4(4) −1

4.43(14)(77)

2.87(28)

4.10(5)3.14(23)

4.04(19)

3.47(11)

4.21(11)

2.9(2.4)

4.3(9)
4.4(2)+ omitted large body of ǫ-regime work. See Necco.

Lattice Kaon physics – p.11/46

Pion form factors & l6hπ + (p′ )|Vµ |π + (p)i = F ππ (q 2 )(pµ + p′µ )ETMC arXiv:0812.4042
RBC-UKQCD arXiv:0804.3971

ETM (N = 2)f

JLQCD (N = 2, overlap)

set A
set B
set C
pole fit
QCDSF/UKQCD

1

f

0.95

JLQCD (N = 2, Clover)f

0.9

ylabelQCDSF/UKQCD (N = 2)
f

0.85
0.8

UKQCD/RBC (N = 2 + 1)
f0.75

LHP (N = 2 + 1)

0.7

f

0.65

0.2

0.3

0.4

0.5
2 phys.

<r >

0.6

0.7

0.8

0

0.05

0.1

0.15

0.2

0.25

0.3

xlabel2(fm )Simulation required at low mass and q 2
Twisted BC’s (ETMC’07,RBC-UKQCD’08) get low q 2 region
ETMC concluded NNLO fits needed in mass extrap

Lattice Kaon physics – p.12/46

Pion form factors & l6hπ + (p′ )|Vµ |π + (p)i = F ππ (q 2 )(pµ + p′µ )

2
i
hrπ

=
=

d ππ 2f (q )|q2 =0
dq 2
«
„
12l6r
1
m2π
− 2 −
log 2 + 1f
8π 2 f 2µ

6

Collab

Action

2i
hrπ
V

2i
hrπ
S

ETMC

2f TM

0.456(38)JLQCD

2f Overlap

0.409(23)(37)

JLQCD

2f Clover

0.396(10)QCDSF

2f Clover

0.441(19)

RBC-UKQCD

2+1f DWF

0.418(31)LHPC

2+1f MILC/DWF

0.310(46)0.617(79)(66)

Lattice Kaon physics – p.12/46

Pion form factors & l6hπ + (p′ )|Vµ |π + (p)i = F ππ (q 2 )(pµ + p′µ )Scalar form factor:
JLQCD arXiv:0905.2465 have determined the scalar form factor.
Useful complement to fπ to determine l̄4
ETMC used experiment & χPT as input to better constrain NNLO fits.
Both use NNLO χPT analyses with non-lattice input

Lattice Kaon physics – p.12/46

r
L10 and chiral fermions
Determine Lr10 from difference in vector and axial vacuum polarisation
JLQCD arXiv:0806.4222, Nf = 2
ΠVµν − ΠAµν

=

(1)

=

ΠV −A

´ (1)
` 2
(0)
q δµν − qµ qν ΠV −A − qµ qν ΠV −A

m2
1
π
2
log(
2 + 3 − H(x))
fπµ− 2 − 8Lr10 (µ) −q
24π 2Only accessible with lattice chiral fermions (Overlap, DWF)JLQCD, Nf = 2−3
Lr10 mρ = −5.2(2)+5−3 × 10

Pion mass squared e-m splitting (in chiral limit):2m2π± = 993(12)(+0−135 )(149)MeVPDG (physical masses): 1261MeV2

Lattice Kaon physics – p.13/46

r
L10 and chiral fermions
Determine Lr10 from difference in vector and axial vacuum polarisation
JLQCD arXiv:0806.4222, Nf = 2
ΠVµν − ΠAµν
(1)

ΠV −A

=
=

´ (1)
` 2
(0)
q δµν − qµ qν ΠV −A − qµ qν ΠV −A

m2
1
π
2
log(
2 + 3 − H(x))
fπµ− 2 − 8Lr10 (µ) −q
24π 2Approach copied by RBC-UKQCD: preliminary Nf = 2 + 1 (Wennekers)

Lr10 = −4.7(3) × 10−3
Error is statistical only

Lattice Kaon physics – p.13/46

Outline
Introduction
Mass extrapolation & chiral lagrangianVus
fK /fπKl3
Kaon bag parameters
BK
BSM
Other topics
Vus/Vud

Lattice Kaon physics – p.14/46

Vus
Two best lattice constraints for Vus involve SU(3) breaking effects
quantity

SU(3) breaking

fKfπ
f0Kπ (q 2 = 0)

20%
4%.

While less mature Kl3 looks very promising and is rapidly becoming better studied.
Debate: Systematic error estimation:
Exactly unity in the ms = mu limit for a 6= 0 and V 6= ∞ and even configuration by configuration
Lattice errors should be suppressed and are likely a fraction of the deviation from unity
Associating a precise error is difficult

Lattice Kaon physics – p.15/46

BMW
1.25
a≈0.125 fm
a≈0.085 fm
a≈0.065 fm
a=0

1.2

FK/Fπ

1.15
1.1

1.05
0.1

0.2

2
2
Mπ[GeV ]0.3

No detailed exposition in preprints yet, but exciting
2+1f clover & a−1 ∈ {1.6, 2.3, 3.0} GeV
No renormalisation yet
Is axial current improved non-peturbatively ?
190 MeV pion masses ... wow!

Lattice Kaon physics – p.16/46

ETMC
ETMC have updated (arXiv:0904.0954) for Nf = 2 twisted mass:fK /fπ = 1.210(18)Analysis uses measured pseudoscalar meson masses in chiral formulae rather than quark masses.
Rearranges the χPT beyond LO
Easier to fit multiple lattice spacings simultaneously without computing change in Zm
ETMC also follow the RBC-UKQCD approach and find SU(2)-ChPT better describes the kaon sector.

Lattice Kaon physics – p.17/46

fK /fπ
Lellouch Lattice 2008 world average:
ETMC ’07
Nf = 2ETMC ’08
NPLQCD ’06 (*)
HPQCD/UKQCD ’07 (*)

Nf = 2+1(MILC)

MILC ’07 (*)
ALV ’08
RBC/UKQCD ’08 (*)

Nf = 2+1

PACS-CS ’08
BMW ’08

FK/Fπ = 1.194(3)(10)
1.15

1.2

1.25

1.3

1.35

FK/Fπ

Here RBC-UKQCD estimated discretisation error as 4% of 1.2
Likely pessimistic

Lattice Kaon physics – p.18/46

fK /fπUpdate of Lellouch Lattice 2008 table:
Ref.

Nf actiona[ fm]LMπminMπmin [ MeV]

typ/valtyp/val

PDG’08FK /Fπ1.193(6)ETM’082 tmQCD

0.07, 0.09, 0.10[Fπ ]3.6/3.6

260/260

1.196(13)(7)(8)ETM’092 tmQCD

0.07, 0.09, 0.10[Fπ ]3.6/3.6

260/260

1.210(6)(15)(9)0.13[r0 ]5.1/3.5

420/290

1.218(2)+11−24

5.3/4.2

300/240

1.197(3)+6
−13

DWF/
NPLQCD’06 2+1 KSMILC

0.06, 0.09, 0.012,

MILC’04-07

2+1 KSAsqTad
MILC

HPQCD/’07
UKQCD

2+1 KSHISQ
MILC

0.09, 0.12, 0.15[Υ]4.8/4.1

360/3101.189(2)(7)

RBC/’08UKQCD

2+1 DWF

0.11[Ω]

4.1/3.4

290/240

1.205(18)(62)

ALV’082+1 DWF/KSMILC

0.09, 0.12[Υ/Fπ ]5.3/4.2

300/2401.191(16)(17)

PACS-CS’08

2+1 NP-SW

0.09[Ω]

2.3/2.3

160/160

1.189(20)4/4

190/190

1.19(1)(1)

BMW’082+1 SW

0.15[Fπ ]0.065, 0.085,
0.125[Ξ]Lattice Kaon physics – p.19/46

Outline
Introduction
Mass extrapolation & chiral lagrangianVus
fK /fπKl3
Kaon bag parameters
BK
BSM
Other topics
Vus/Vud

Lattice Kaon physics – p.20/46

Kl3
hπ(p′ )|Vµ |K(p)i = f+ (q 2 )(pµ + p′µ ) + f− (q 2 )(pµ − p′µ )Several “double ratios” employed (hep-ph/0403217)
e.g.
hK(~0)|V0 |π(~0)ihK(~0)|V0 |π(~0)i(mK + mπ )22)|2|f0 (qmax=
4mK mπhK(~0)|V0 |K(~0)ihπ(~0)|V0 |π(~0)i
Lattice methods undergone two related improvements
Twisted boundary conditions enable simulation directly at q 2 = 0
Previously constrained to discrete Fourier momenta
Model dependent interpolation to q 2 = 0 can be eliminatedL3 volume average can be taken improving statistical precision
t0

t

t0

t1

θ1

Σ eiq.yyΣz eip.y

Σ
x

t1

t

θ2

Σ
y

Σz

Lattice Kaon physics – p.21/46

Kl3
Kπ (again from Lellouch)
Recent results for f+
Not aware of any updates in literatureNfaction

JLQCD

2

RBC

Mπmin [ MeV]

f+ (0)

a[ fm]

L[ fm]

NP SW

0.09

1.8

550/5500.967(6)

2

DWF

0.12

2.5

490/4900.968(9)(6)

ETM

2

tmQCD

0.11

2.7

260/260

0.957(5)

FNAL/MILC

2+1

KS+Wil

RBC/UKQCD

2+1

DWF

Ref.

typ/val0.962(6)(9)0.11

1.8, 2.8

290/2400.9644(33)(34)(14)Best constraint is from RBC-UKQCD

Lattice Kaon physics – p.22/46

RBC-UKQCD Kl3
Phys.Rev.Lett.100:141601,2008.

Combined q 2 and chiral extrapolation w. hybrid pole dominance/Ademollo-Gatto constrained model.
f0 (q

2

1 + f2 + (m2K − m2π )2 (A0 + A1 (m2K + m2π ))22, mπ , mK ) =1 − q 2 /(M0 + M1 (m2K + m2π ))2

f+ (0) = 0.9644(33)stat (34)extrapolation (14)disc
model dependence in q 2 interpolation & chiral extrapolation & strange mass adjust
no continuum limit; budget 4% of 1 − f+ (0)Is difference from f2 poor convergence of χPT ???

Lattice Kaon physics – p.23/46

RBC-UKQCD Kl3
1

0.99

Hang on... SU(3) breaking is only 4%
0.98

Replace fπ with F0 = 93.5 in f2

0.97

Looks much more convergent &commensurate with degree of breaking

0.96

RBC+UKQCD 243 data
RBC+UKQCD
PRL phys. point)
f2(fphysπf2(fχ)

0.95

Favours positive ∆f

π

0.94

0

0.05

0.1

0.15

0.2

0.25
(0)
fKπ
0

0.3

0.35

0.4

0.45

0.5

Lattice Kaon physics – p.24/46

Independent determination of F0?
FLAG have SM determination of f+ (0) from: (Colangelo & Sachrajda, private communication)
CKM unitarity
Vud from Towner & Hardy arXiv:0812.1202
fK /fπ Vus /Vudf+ (0)Vus

f+ (0) = 0.9620(20)(37)
assume convergence: f2 is dominant contribution to 1 − f+ (0) &F0 = 101.9(2.6)(4.8)MeV
interesting given spread of lattice determinations
Including positive NNLO ∆f predictions would lead to even slightly lower F0
may explain preference of lattice f+ (0) for Leutwyler-Roos→ f2 should have been lower in the first place

Lattice Kaon physics – p.25/46

RBC-UKQCD preliminaryThanks to Zanotti, Juettner, De Lima
Recalculation for our lightest point (ml = 0.005) using twisted BC approach(Z2 wall noise sources; color-spin trace performed stochastically)→ should better constrain chiral limit & address model dep. & stat errorsnimLines are prediction from global fit in PRL (blackdata)
not a fit to blue and red

i
l
e
r
P
y

r
a

Global fit does correctly describe strange massdependence
Remove model dependence by direct q 2 = 0

n
i
m

r
a

Improved statistical error with volume average (compare blue/red & black)323 data complete and under analysis. Will address the sub-dominant discretisation error.

i
l
e
r
P

Lattice Kaon physics – p.26/46

Outline
Introduction
Mass extrapolation & chiral lagrangianVus
fK /fπKl3
Kaon bag parameters
BK
BSM

0.6

Other topics

0.5

η

0.4

excluded area has CL > 0.95

0.7

γ∆md & ∆ms

∆md

CKM

fitterSummer 08

εK

sin 2βsol. w/ cos 2β < 0(excl. at CL > 0.95)

α

εK

0.3

α
0.2

Vub

0.1
0.0
−0.4

β

γ

α
−0.2

0.0

0.2

0.4

0.6

0.8

1.0

ρ

Lattice Kaon physics – p.27/46

BK
hK 0 |OVV+AA |K̄ 0 i
BK = 8hK 0 |A0 ihA0 |K 0 i
3Chiral symmetry gives multiplicatively renormalised calculation→ non-symmetric actions deal with unphysical taste/chirality mixings which inflate errors→ mixed actions sometimes used (e.g. ALV, ETMC)
t

t0

t1

t

OR

peri−anti (t<0)Σ
xperi+anti (t>0)

t=0

Σ
x

Σ
y

Σz

Σ

x,y,z,w

Lattice Kaon physics – p.28/46

Recent BK results
Lellouch 2008 table
Mπmin [ MeV]

Ref.

Nf

actiona[ fm]

L[ fm]JLQCD’08

2

Overlap

0.12

1.9

290/2900.734(5)(55)

ETM’082

OS/tmQCD

0.07,0.09

2.1,2.7

300/300

0.78(3)

HPQCD/UKQCD’06

2+1

KSHYP
MILC

0.125

2.5

460/360

0.85(2)(18)

RBC/UKQCD’07-08

2+1

DWF

0.11

1.8,2.8

290/2400.720(13)(37)

Bae et al ’08

2+1

KSHYP
MILC

≥ 0.064

300/240

δ B̂K → 3%

typ/valB̂K

Lellouch average was weighted combination of RBC/UKQCD and HPQCD
B̂K = 0.723(11)(35)

Lattice Kaon physics – p.29/46

Recent BK results
Update Lellouch 2008 table to include recent ALV arXiv:0905.3947 DWF valence on MILC configurations.

NfactionJLQCD’08

2

Overlap

ETM’082

HPQCD/UKQCD’06

Mπmin [ MeV]

B̂Ka[ fm]

L[ fm]

0.12

1.9

290/2900.734(5)(55)

OS/tmQCD

0.07,0.09

2.1,2.7

300/300

0.78(3)

2+1

KSHYP
MILC

0.125

2.5

460/360

0.85(2)(18)

RBC/UKQCD’07-08

2+1

DWF

0.11

1.8,2.8

290/2400.720(13)(37)

Bae et al ’08

2+1

KSHYP
MILC

≥ 0.064

300/240

δ B̂K → 3%

Aubin et al ’09

2+1

DWF/KS

0.12,0.09

2.5-3.6

300/230

0.724(8)(28)

Ref.

typ/val

Lattice Kaon physics – p.29/46

Recent BK results
Update precise 0f DWF continuum limit from CP-PACS arXiv:0803.2569
NfactionJLQCD’08

2

Overlap

ETM’082

HPQCD/UKQCD’06

Mπmin [ MeV]

B̂Ka[ fm]

L[ fm]

0.12

1.9

290/2900.734(5)(55)

OS/tmQCD

0.07,0.09

2.1,2.7

300/300

0.78(3)

2+1

KSHYP
MILC

0.125

2.5

460/360

0.85(2)(18)

RBC/UKQCD’07-08

2+1

DWF

0.11

1.8,2.8

290/2400.720(13)(37)

Bae et al ’08

2+1

KSHYP
MILC

≥ 0.064

300/240

δ B̂K → 3%

Aubin et al ’09

2+1

DWF/KS

0.12,0.09

2.5-3.6

300/230

0.724(8)(28)

JLQCD ’08

0f

DWF

0.05-0.10

2.3-2.6∞/340

0.782(5)(7) )Ref.

typ/valBeautiful result with non-perturbative SF renormalisation and running
three lattice spacing extrapolation continuum limit.

Lattice Kaon physics – p.29/46

Averaging BK
Lellouch estimated BK as the weighted average of HPQCD and RBC-UKQCD 2+1f results,
while noting discrepancy between staggered nf = 0 and nf = 2 DWF surprisingly large.
Already clarified by CP-PACS 0f DWF continuum limit
In quenched approximation JLQCD got a similar high staggered result
Has simply not agreed with simpler formulations in continuum limit.
one loop LPT staggered operator mixing is fragile
Nn LO lattice taste mixing easily underestimated & amplified through mixing matrix cancellation
0.75
DBW2 + DWF 0f; RImom, PT running
CP-PACS Iwasaki + DWF 0f; 1-loop LPT
CP-PACS 0f continuum; SF, NP running
JLQCD 0f stag continuum; 1-loop LPT

0.7

Tension recent lattice unquenched BK induces
between SM and |ε| is created by
the theoretically better actions.

BK

0.65
0.6
0.55
0.5
0

0.01

0.005
2

0.015

2

a (fm)

Lattice Kaon physics – p.30/46

Lessons ?
FLAG should include a Universality “star”

The most important “star” for dynamical simulation may be passing a robust universality test.
A universality test could be to obtain a true, agreed, quenched continuum result with comparable
simulation parameters.
Simple sociology ⇒ lower quality 2+1f preferred to careful quenched continuum limit.
Particularly important for b-quarks to verify the various effective methods.
nf=2?

Several (JLQCD, ETMC) high quality Nf = 0 and Nf = 2 calculations should have created astrong bias against the high BK result.
sys-errors are subjective estimates

Lattice Kaon physics – p.31/46

Average BK
We now have two well renormalised 2+1f BK calculations in remarkable agreement
RBC-UKQCD ’07, ’08

Aubin, Laiho, Van de Water ’09

B̂K = 0.720(13)(37)

B̂K = 0.724(8)(28)

Discretisation

4%

NP renormalisation

3.3%

NP renormalisation

2%

Chiral/cont extrapolation

1.9%

Chiral extrapolation2%

statistical

1.2%

Statistical

1.8%

Scale & masses

0.8%

Sea strange mass adjust

1%

Finite volume

0.6%

Finite volume1%

total (quad)

4.1%

total

5.4%

Pros:
SU(2)
Controlled sea content
Cons:Single lattice spacing
discretisation estimated from quenched scaling

Pros:
Two lattice spacings
Cons:SU(3) removal of taste unitarity violations∆I = (460MeV)2 ⇒ ≤ 670 MeV pi-taste-ons &even heavier taste-η terms
Only NLO Taste mixing despite partial NNLO fit.

Lattice Kaon physics – p.32/46

Average BK
We now have two well renormalised 2+1f BK calculations in remarkable agreement
Averaging difficult due to subjective nature of error estimates
extrapolation

RBC-UKQCD SU(2) approach more convergent but estimate larger chiral extrap. error
than ALV combined chiral/continuum rs-χPT
3 extrap. error
volumeVolumes similar for most data points; ALV use CDH model to adjust data and estimate remnant
error, RBC directly fit data and estimate FV.
renormalisation

Same renormalisation method but again different errors;
ALV estimate larger error mainly from using difference between NPR & LPT.

Lattice Kaon physics – p.32/46

Average BK
We now have two well renormalised 2+1f BK calculations in remarkable agreement
Averaging difficult due to subjective nature of error estimates
extrapolation

RBC-UKQCD SU(2) approach more convergent but estimate larger chiral extrap. error
than ALV combined chiral/continuum rs-χPT
3 extrap. error
volumeVolumes similar for most data points; ALV use CDH model to adjust data and estimate remnant
error, RBC directly fit data and estimate FV.
renormalisation

Same renormalisation method but again different errors;
ALV estimate larger error mainly from using difference between NPR & LPT.
ProposeB̂K = 0.722(40)and emphasize that most components of the errors are subjective

Lattice Kaon physics – p.32/46

What’s next for BK
RBC-UKQCD second lattice spacing (thanks to Scholz, Kelly)
0.6

Bxy

0.59

PRELIMINARY

0.58

bare, unrenormalised BK extrapolation for
a−1 = 2.4 GeV DWF

0.57

my=mh=0.03
unitary, ml=mx
ml=0.004
ml=0.006
ml=0.008

0.56

0.55
0

0.002

0.004
mx

0.006

0.008

Preliminary, and will change with strange
mass reweighting
0.01

Soon (if not already) error dominated by renormalisation
Statistical & extrapolation uncertainties will be sub-leading

Lattice Kaon physics – p.33/46

Non-perturbative renormalisation
RBC-UKQCD arXiv:0712.1061

q Γµ (0)qk

RI-mom scheme traditionally uses p = p′kChiral symmetry ⇒ ΛA = ΛVkSpontaneous χSB at low p2
2% systematic introduced ;obscures good chiral properties of DWF

ΛA − ΛV with p = p′
0

k

p−k

p

p

ΛA − ΛV with non exceptional
momenta p2 = (p′ )2 = (p − p′ )20.005

0(ΛA - ΛV) / [(ΛA + ΛV) / 2]

(ΛA - ΛV) / [(ΛA + ΛV) / 2]

-0.02

-0.04

-0.06
ml = 0.01
ml = 0.02-0.005-0.01-0.015

ml = 0.01
ml = 0.02

ml = 0.03-0.08

ml = 0.03-0.02

chiral limit

chiral limit

-0.1

0

0.5

1

1.5
(ap)

2

2

2.5

-0.025
0

0.5

1

1.5
(ap)

2

2

2.5

Lattice Kaon physics – p.34/46

NPR with non-exceptional momenta0

Non-exceptional BK chirality mixing consistent with zero
New: 1-loop conversion known for Zm , OVV+AA
Sturm, Sachrajda arXiv:0901.2599 & preliminary

-0.02

-1

FMVV+AA,VV-AA-0.01

mdyn = 0.01

-0.03

mdyn = 0.02

Higher order perturbative conversion to M S is needed

mdyn = 0.03

-0.04

chiral limit
-0.05

0

0.5

1

1.5
(ap)

2

2.5

2

Jan Wennekers, RBC-UKQCD, preliminary

Five 1-loop mom schemes to estimate systematics
1.10

Improved renormalisation may shift central value toaround 0.54 from 0.52

1.05

1.00

Expect good scaling from RBC-UKQCD data
0.95

Two loop MOM scheme matching for
non-exceptional kinematics required to reduce
dominant error in the next few years.

0.90

0.85

0

1

2

3

4

Who will do this?

Lattice Kaon physics – p.35/46

Outline
Introduction
Mass extrapolation & chiral lagrangianVus
fK /fπKl3
Kaon bag parameters
BK
BSM
Other topics

Matrix elements of BSM ∆S = 2 effective operators
hK0 |Qi |K0 i
Q1=s¯a γµ PL da s̄b γµ PL db

Q2=s¯a γµ PL da s̄b γµ PR db

Q3=

s¯a PL da s̄b PR db

Q4=s¯a PL da s̄b PL db

Q5=s¯a PL da s̄b PL db

Lattice Kaon physics – p.36/46

BSM Kaon Bag parameters
Matrix elements of Q2 , . . . , Q5 allow ǫK to constrain certain BSM ∆S = 2 processes
Mixing matrix becomes 5 × 5 in absence of chiral symmetry.
→ Overlap and DWF actions required for control.
Current best study remains quenched overlap hep-lat/0605016 Boston/Marseille/Wuppertal
SPQCD arXiv:0401033
β=6.2

0.5

∆34

w/o PGB Subt.
Single PGB Subt.
Double PGB Subt.

0.4

Standard RI-mom renormalisation has
1
infra-red pole.
mp2

0.3

residue in p2 diverges as m → 0
Difficult to subtract precisely; affects all existing calculations

0.2

0.1

0

0.5

1

2 2

1.5

2

2.5

µa

Lattice Kaon physics – p.37/46

BSM bag parameters in 2+1f DWF
RBC-UKQCD (Wennekers) arXiv:0810.1841

First calculation of SUSY bag parameters with dynamical chiral fermionsNon-exceptional momentum eliminates pion pole
Includes all lattice components of NPR
Perturbative conversion to MS in progress (1-loop, Sachrajda & Sturm)

Lattice Kaon physics – p.38/46

BSM bag parameters in 2+1f DWF
RBC-UKQCD (Wennekers) arXiv:0810.1841

Bare matrix elements of Q2 (left) and Q5 (right) as function of mass
When 1-loop calculation is complete expect:matrix elements to 2% – comparable accuracy to RBC-UKQCD BK
Dramatic systematic improvement in renormalisation

Lattice Kaon physics – p.39/46

Outline
Introduction
Mass extrapolation & chiral lagrangianVus
fK /fπKl3
Kaon bag parameters
BK
BSM
Other topics

Lattice Kaon physics – p.40/46

Other topics
Unfortunately there was insufficient time to cover :K → ππNorman Christ’s talk this conferenceElectromagnetic effects
Taku Izubuchi’s talk this conferencem̄s RI-mom approach is best so far used in 2+1f simulations
Suffers poor perturbative convergence despite 3-loops
Yasumichi Aoki’s forthcoming talk at Lattice 2009
Kaon distribution amplitudes
RBC-UKQCD & QCDSF

Lattice Kaon physics – p.41/46

Thanks
I wish to thank:
Kaon 2009 organising committee
KEK & Tsukuba University for their support and hospitality

Lattice Kaon physics – p.42/46

BACKUP

Lattice Kaon physics – p.43/46

Non-perturbative renormalisation
Matrix elements of non-conserved operators renormalise between any lattice simulation and MS.
Recent progress on Zm and ZV V +AA using the gauge-fixed off-shell Rome-Southampton RImom
scheme.
Bilinear amputated vertex functionΛ(p, p′ )in some scheme (Lattice or MS) is related to the RI mom scheme defined at this momentum point by
RI
Λscheme (p, p′ ) = Λtree (p, p′ )ZschemeMulti-loop calculations long available for p = p′ for bilinears
OVV+AA known only to 1-loop.
Conversion between Lattice and MS is then
MS
RI
RI
ZLat
= ZLat/ZMS= ΛMS (p, p′ )/ΛLat (p, p′ )For simultaneous discretisation and perturbative accuracy we requireΛ2QCD ≪ p2 ≪“ π ”2aLattice Kaon physics – p.44/46

Strange mass
arXiv:0901.2599

From vertex function ΛS (p1 , p2 ); p21 = p22 = −µ2 and q 2 = −ωµ2 obtain
Cm =

MS
Zm

=1−
RI−ω

Zm

αs
(1)
CF cm (ω)
4π

ω = 1 is the symmetric momentum point (smom)
6

ξ=1
ξ=0

4

c(1),RI/SMOMm

ξ=-1
2 ξ=-6

ξ=-2

0

ξ=-3

-2

ξ=-4

-4
-6
0

ξ=-5

ξ=1

ξ=-6
0.5

1

1.5

2

ω

2.5

3

3.5

4

` ´
RI = 1 − αs C 4 + O α2
Cm
s
4π F
` 2´
αs
RIsmomCm= 1 − 4π CF 0.4841391 + O αs
Is this reduction preserved at higher orders?
5% perturbative error from RImom approach to ms
could be greatly reduced
Step-scaling alternative has not been performed with
2+1f to date

Lattice Kaon physics – p.45/46

ETMC NNLOETMC NNLO fits poorly constrained where there is no data due to many parameters
0.20/ m (GeV)

2

0.18

NNLO fit2

with <r >

5.0

exp.
S

0.16

4.6

ETMC

NNLO fit
0.02

0.04

0.06

0.12
0.00

0.08

0.02

0.04

NNLO fit

0.06

4.2
0.00

0.08

0.02

0.04

^

^

0.02

0.04

0.06

0.006

0.5

(d)

(c)

0.005

0.005

0.4

ETMC

0.002

0.004
0.3

0.2

physical point
0.1

ETMC

0.1

0.001

0.02

4
exp.

2

with <r >

S

exp.

2

0.003

with <r >

S

0.002

ETMC

0.001

NNLO fit
0.04^

NNLO fit

physical point

NNLO fit
0.0
0.00

c (fm )

2

0.003

2

0.2

<r > (fm )NNLO fit4c (fm )

2

2

<r > (fm )

ETMC
0.004
0.3

0.08

^

m (GeV)

(d)

(c)
0.4

0.08

m (GeV)

0.006

0.5

0.06

0.12
0.00

^

m (GeV)m (GeV)S

physical point

0.14

ETMC

4.2
0.00

exp.

2

with <r >M!

physical point

0.14

4.6

ETMC

f

^

(GeV)
!

f

^

/ m (GeV)

0.16

(b)

physical point

5.4

M

2

0.18

NNLO fit

5.0

0.20
(a)

!

ETMC

5.4

5.8
(b)

physical point

!

(a)

(GeV)

5.8

m (GeV)

0.06

0.08

0.000
0.00

0.02

0.04
^

m (GeV)

0.06

0.08

0.0
0.00

0.02

0.04
^

m (GeV)

0.06

0.08

0.000
0.00

0.02

0.04

0.06

0.08

^

m (GeV)

Both ETMC include FULL NNLO but need additional constraint on parameters (from scalar pion form
factor).
Does this mean anything for partial NNLO analyses?

Lattice Kaon physics – p.46/46