ETMC results for Kaon Physics:
dynamical twisted mass lattice QCD

Federico Mescia
Universitat de Barcelona
on behalf of the European Twisted Mass Collaboration (ETMC)

KAON 2009, June 7th-12th, Tsukuba, Japan• Simplest Kaon Observables on the lattice:1) Tree-level mediated decays -> CKM unitarity, SM tests

Hadronic uncertainties froml | s u | K   f ,0 (q2 )f+(0), +0, c+0 ?

2) Helicity suppressed decays -> sensitivity to the Higgs sector/r.h. quarks
Hadronic uncertainties from0| s  5u | K   p f K
fK, fK/ f ?

3) FCNC processes -> SUSY, Little Higgs ….

Hadronic uncertainties from K | s L ds L d | K   8/3 f K mK  BK


2

BK ?
Warning: sub-leading effects, no longer negligible in K
A.Buras & D.Guadagnoli ’08,’09• Lattice simulations with dynamical quarks are now well established
Already available important results:
Kl2 HPQCD et al, 2005
PRL100:062002,2008

fK/f
 m  MeV,a->0:Staggered fermions?

Kl3

RBC/UKQCD, 2007
PRL100:141601,2008

f+(0) m  MeV, a=0.1fm: DW fermions!

RBC/UKQCD, 2006
PRD78:114509,2008.
Aubin et al. 0905.3947

BK m  MeV, DWF fermions!

K




But, setup not yet optimal: Staggered f. unsatisfactory: DW f. at potentially large
residual mass mres/msea ~ 0.6

 m range too heavy: mm  continuum limit & finite volume effects
 Renormalisation in mass-independent scheme NF=3

Better Unquenched setup is being generated:
RBC: DW f. at smaller lattice spacing mres/msea~0.02, m=300 MEV

Not cheap

Thanks to recent theoretical developments, very light regime 260  m 400 MeV havebeen reached by Wilson-like and Overlap fermionscheaperBMW, ETMC, PACS-CS & JLQCD/TWQCDPlots by G. Herdoiza

RecentRecentoutstanding
outstandingachievements
achievements: :• Wilson: CERN-TOV.’05,• Twisted-mass: ETMC‘07,•“Clover”: BMW‘08,• Clover: PACS-CS’08,• Overlap: JLQCD/TW’08,m~260 MeVm~260 MeVm~190 MEVm~156 MEVm~290 MEV(NF=2  NF=2+1+1)(NF=2+1)(NF=2+1)(NF=2 & 2+1)

• Domain Wall: RBC-’06,• Staggered: MILC-2002,m~330 MEVm~280 MeV(NF=2+1)(NF=2+1)

sea
QUENCHED QCD
mq 
NF=0

NF=2+1
mu=md < ms

(NF=2)

mcsea  !!
??

Chiral Regime of QCD

A problem with (old) lattice simulations:
Incompatibility with ChPT?

JLQCD’01m~720 MeV

m~ MeVNo more problems with (new) lattice simulations:
Chiral Regime of QCD
Good compatibility with ChPTETMC arXiV:0803.0224(amamq

Getting lighter:
Good compatibility with ChPTamqm~500 MeVm~260 MeV

amq

MareNostrum

BlueGene

BlueGene

QCDOC

APENEXT

Similar plots by other collaborations (using different approaches)(amamq
Hints of chiral Logs

PACS-CS’08, NF=2+1, Clover:a=0.09 fm, MMeV, L M ~2.3!!

JLQCD/TWQCD’08, NF=2+1, Overlap:
a=0.11 fm, MMeV, L M ~2.6!!

QCDSF’08, NF=2, Clover:a=0.07 fm, MMeV, LM ~4

RBC/UKQCD’06, NF=2+1, DWF
a=0.114 fm, MLM ~3-4

Twisted Mass Fermions (tmLQCD): Properties
Frezzotti, Grassi, Sint & Weisz 1999

STWsea   q f    mq f  ia 5 rf Wterm  q fx

 Discretized QCD Action + a “generalized” Wilson term,Wterm: Wilson 1974
 up and down flavors as doublets in the r-Wilson parameter space: ru=1, rd=-1
 Successful studies for both NF=2 and NF=2+1+1

 protection against unphysical small eigenvalues:f low computation coststable simulations for light pions

 only (a2) lattice discretization errors

Frezzotti & Rossi 2003

 simple renormalization pattern => fK, BK

Frezzotti & Rossi 2005

 explicit breaking of parity and isospin:

however, this breaking is an (a2) effect in physical quantities

Vus determinations and the CKM Unitarity
 Axial Weak Universality Vus / Vud f K / f  0.2760(6) Vus / Vud  0.2281(5)(35)
Vus  0.2222(34)th:1.5%

exp:0.21%

fK/ffitsyst

ETMC arXiV:0904.0954

 Vector Weak Universality Vus f  (0)  0.21668(45)  VusKl 3  0.2267(5)(22)
exp:0.21%2

Vud  Vus  1.0004(15)2f+(0)fitsyst

th:0.9%!!

ETMC in preparation

Determination
Determination ofof ffKK/f/f
 ETMC determination of fK/f

ETMC arXiV:0904.0954

 NF=2  the strange quark is quenched 3 values of the lattice spacing: a≈0.10fm, a≈0.085fm, a≈0.065fm
 different volumes

FSEstudies (several) light sea quark masses

M    l , l 

(unitarity pions l =sea)

0.2  msphys  l   sea  0.5  msphys

M K   l ,  s 0.9  msphys   s  1.5  msphysContinuum
LimitSystematics Overview: finite L and a

FSE under control
L≈2.9 fm

L≈2.16 fm

a≈0.10fm
a≈0.085fm
a≈0.065fm
a≈0.000 fm

Discretisationerrors

ETMC arXiV:0904.0954

ETMC arXiV:0904.0954

 Combined fit of chiral dependence + a2 and L corrections► Chiral-fit: SU(2) ChPt for l + Linear dep. of LECS in “strange quarks”

ETMC arXiV:0904.0954

A fit based on NLO SU(3)-ChPT finds good compatibilityfK/f SU(3) SU(2)However SU(3)-ChPT looks less robust that SU(2)Stat/chiral

(a2)

FSE

ETMC arXiV:0904.0954

Warning from ETMC: Continuum Limit important!
2007→2009fK/f:1.227(9)(24) →1.210(18)Determination
Determination ofof ff++(0)(0)
 ETMC determination of f+(q2) at q2=0

ETMC in preparation

 NF=2  the strange quark is quenched one lattice spacing: a≈0.09fm, other values in progress:
 to estimate systematics we exploit a≈0.07fm for M≈ 300 & 470 MeV
 LM  to keep finite size effects negligible M≈ 260, 300, 375, 435, 470, 575 MeV
L=2.9fm
V=323x64

L=2.2fm
V=243x48(unitarity pions l =sea) several q2 in order to extract f+,0(q2) at q2=0
 K | su |    f  (q 2 )  ( p
22M K2  M 2  
2 M K  M q  f 0 (q )qK  p ) 2

q2q



f  (0)  f 0 (0)221)Extracting(q))1)Extractingff++(0)(0)fromfromff0+(q0+(f0+(q2) by appropriate double-ratios, FNAL’02, SPQCD’04)

f+(0)=f0(0)Systematic uncertainty in the determination of f+(0)Chiral Extrapolation for f+(0)SU(3)-CHPt

SU(2) CHPT

f+(0)=1 + (MKM

Expansion for MM

f+(0) = 1 + f2 + ff2=-0.023(NO UNCERTAINTY!)NO LECsf on the lattice?
=>f+(0)F+ and C+ on the lattice?
=> f+(0)ChiralChiral extrapolation
extrapolation I:
I: SU(3)-ChPT
SU(3)-ChPT

physstrange  strange

f  0.0233(61)(32)
Stat. errorf SU (3) (0)  0.9599(61)(32)chiral & q2 fit uncertainties

ChiralChiral extrapolation
extrapolation II:
II: SU(2)-ChPT
SU(2)-ChPTphysstrange  strangef SU (2) (0)  0.9563(53)(13)
Stat. error

chiral & q2 fit uncertainties

(Preliminary)f SU (3) (0)  0.9599(61)(32)f

SU (2)
(0)  0.9563(53)(13)f  (0)  0.9581(57)(35)

Estimate of remnant systematics: L, a & ms-quenched(awaiting for a complete systematics study) Finite size effects: add (in quadrature) 0.0019 to sys..
by comparing M≈ 300 between L=24 & L=32

 Finite lattice spacing: add +0.0037 to c. value & sys.
by comparing M≈ 300 & 470 MeV between a=0.09 & 0.07 fm

 Quenching ms (NF=2 vs NF=2+1): -0.0058 to c. value, 0.0029 to sys.
f2NF=2+1-f2NF=2 known in SU(3)-CHPT

f  (0)  0.9560(57) stat (62) syst  0.9560  0.0084

f  (0)f  (0)  0.9560(57) stat (62) syst  0.9560  0.0084(Preliminary)BK from NF=2 tmQCD: (Preliminary)

BKRGIETMC: NF = 2:  TmQCD
 continuum limit=> a-1=>1.97, 2.19,2.81 Gev,
L=2.2/2.9 fm, m260 MeV
 sea q. masses dep.
 Quenching the strange
RBC: NF=2+1:  DWF
 no continuum limit
 a-1=1.73 Gev,
L=2.74 fm, m>=330MeV
 2 sea q. masses!!!
Aubin et al. 0905.3947
 Staggered/DWF continuum limit
JLQCD: NF = 2: Overlap  no continuum limit=> a-1=1.67 Gev,
L=1.9 fm, m>=290

BKRGI  0.739(25)(?)(Very Preliminary)28 K | s L ds L d | K    f K mK  BK3Determination
Determination ofof BBKK ETMC determination of BK Notoriously, Wilson fermions are problematic for BK ;chiral symmetry mixing with dim-6 operators
Wilson FermionsnoFermions
Way Out: combining Wilson fermions with different r-parameters
Frezzotti & Rossi hep-lat/0407002

s L ds L dMultiplicative renormalization as in the continuum(a2) Cut-off effects

► Chiral-fit Setup: SU(2)Lx SU(2)R fit (l) + static strange quarks (h)Unitarity case
2B0BKbare (h , v  sea )  Bbare (h ) 1   Pv (h )  Psea (h )  sea logseasea 32 2 f02
a≈0.10fm
a≈0.085fm
a≈0.065fm
a≈0.000 fm

Small discretisation
errorsl

BKRGI  0.739(25)(Very Preliminary)► Renormalization Constants in RI-MOM scheme: PRELIMINARY

RGI However, (a2g2) looks sizable(a2Substraction of (a2g2) by explicit calculation
H. Papadopoulos et al. ‘09RGIBBKRGIK NF = 2:  continuum limit

 sea q. masses dep.
NP-renormalisation
 Quenching the strange

BKRGI  0.739(25)(Very Preliminary)Summary of Kaon Physics from ETMC

f K (0) / f (0)f  (0)BKRGIRecent developments will allow soon high precision Kaon physics
a~0.05 fm, L ³ 4.5 fm and 200  m 300 MeV is not far away

Recent developments will soon allow precision Kaon phys.
a~0.05 fm, L ³ 4.5 fm and 200  m 300 MeV is not far away

Determination
Determination ofof ffKK/f/f

 ETMC determination of fK/f

ETMC arXiV:0904.0954

 NF=2  the strange quark is quenched; 3 values of the lattice spacing: a≈0.10fm, a≈0.085fm, a≈0.065fm

Continuum
Limit

Moreover for Twisted Mass fermions (at maximal twist)
Advantages:• Pseudoscalar meson masses and decay constants are automatically (a)-improved• Axial currents do not require renormalization constant:Chiral Regime of QCD - summary

Colangelo, Gasser & Leutwyler ’01• First hints of chiral Logs by different approaches and groups:
 great success from Lattice approaches based on first principle;

Wilson-like, Overlap and Domain-Wall fermions information qualitatively deeper than using Staggered fermions;no clear clue of chiral logs because of fits with ~40 parameters

 a slight lower value of l3-bar with staggered fermions (MILC)!!!

Twisted Mass QCD Fermions: isospin breaking

- a, L & m Systematics at 1%. - How far are we?

Recent developments will soon allow precision Kaon phys.
a~0.05 fm, L ³ 4.5 fm and 200  m 300 MeV is not far away

Plots by G. Herdoiza

fK/f
f+(0)-1 , BK

Qlat = Qphys´(1+a ) (1+L )(1+m)sys. at 1% ? How far are we?

 discretization errors:a £ 0.05 fm (4 GeV), finite size effects: L*m³L³ 4.5 fm
 chiral regime: 200  m 300 MeV(a³ 0.07 fm)(L £ 3 fm)