Domain wall fermion calculation of the kaon B-parameter BK
Tom Blum
University of Connecticut
RIKEN BNL Research Center
[RBC Collaboration]
“From Actions to Experiment”
The 2nd International Lattice Field Theory Network Workshop
Edinburgh
1
Introduction
• BK is the low energy QCD matrix element relevant to indirect CP-violation in
neutral kaon mixing. If CP unbroken (by the Weak interaction) then
K1 =
K2 =
1
√ K 0 − K 0 ; CP +
2
1
0
0
√ K + K ; CP −
2
• But, CP is broken by the Weak interactions; Actually observe
KS
=
KL
=
K1 + K2
p
1 + ||2
K2 + K1
p
1 + ||2
2
Effective theory of Weak decays
Weak decays handled well in perturbation theory, but not on the lattice ; vise versa
for low energy QCD.
• Integrate out heavy particles; use (2-loop) perturbation theory to run the
effective Hamiltonian down to low a scale accessible to the lattice ( 2 GeV ).
|K | = C A2λ6 η −η1S(xc ) + η2 S(xt )(A2λ4(1 − ρ) + η3 S(xc , xt ) B̂K
• η and ρ are elements of the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix,
the Standard Model paradigm for CP violation.
• η = 0: CP not broken. Experiment finds K 6= 0, so η 6= 0 in the Standard
Model.
• BK is the long-distance, low energy matrix element in QCD, the part of the
mixing that we compute on the lattice. (B̂K is the scale invariant B-parameter).
BK
hK 0 |OV V +AA|K 0 i
=
8 2 2
m f
3 K K
3
Operator Mixing and BK
• If chiral and flavour symmetry hold at finite lattice spacing, then their is only
one operator that contributes to BK :
OV V +AA = sγµ (1 − γ5)d sγµ (1 − γ5 )d
• If chiral symmetry is broken (Wilson fermions) four other operators may mix
(the other possible gamma structures).
X
0
0
0
0
hK |OV V +AA|K iren = Z11 hK |OV V +AA |K ilatt +
z1ihK 0 |OMIX,i |K 0ilatt
i≥2
• If flavour symmetry is broken (Staggered fermions) many other operators may
mix
– actually under better control than the Wilson situation.
4
BK ...
• The lattice matrix element mixes with wrong chirality operators:
X
0
0
0
0
hK |OV V +AA |K ilatt ∝ hK |OV V +AA |K iren +
ci hK 0 |Oi|K 0 iren
i≥2
where the ci are small mixing coefficients.
• First order chiral perturbation theory predicts that
and, unfortunately, that
hK 0|OV V +AA |K 0i ∝ Mk2
hK 0 |OTHE REST |K 0i ∝ 1
so... as mf → 0, wrong chirality operators dominate .
• Work at relatively large values of the kaon mass, but still important to understand the expected size of ci coefficients.
5
Quenched Results for all B-parameters
30
VV+AA
VV-AA
SS-PP
SS+PP
TT
20
• up to ≈ 50 times larger than the
operator we are interested in.
10
B-parameters
Not the most sensible quantity ever,
but can calculate “B-parameters” for
these other operators.
0
-10
-20
ms/2
-30
0.01
0.02
0.03
mf
0.04
0.05
DWF allows us to simply ignore this mixing problem.
6
Domain wall fermion explicit chiral symmetry breaking
q(L)
1
q(R)
2
...
Ls/2
...
Ls
• Each trip through the
bulk flips a left-handed
quark to a right-handed
quark but comes with
a supression factor of
O(mres ) .
mf
• Mixing of chiral modes due to finite Ls enters exactly like a mass, mres
• Since our operator has chiral structure “left-left”, two flips are needed for wrong
chirality operators to mix. Hence, the intrinsic level of chiral symmetry breaking
induced mixing is O(m2res) (∼ 10−6 in the current case).
• Not quite true: exact zero modes of the 4d Wilson Dirac operator behave
differently, but still suppressed by more than one factor of mres . [Golterman and
Shamir (2004), but their large size estimate has been retracted]
7
BK on the lattice
• To calculate
BK
=
hK 0|OV V +AA |K 0i
8 2 2
m f
3 K K
=
hK 0 |OV V +AA|K 0 i
8
hK 0|Aµ|0ih0|Aµ|K 0i
3
• Construct a three-point correlation function of interpolating operators for the
K 0 , K 0 , and the effective weak operator OV V +AA , with the K 0 at some small
time, the K 0 at large time, and OV V +AA in between.
• Divide by the two-point correlation functions for the K 0 and K 0 .
• When all three operators are separated by large time, a plateau develops corresponding to the desired ground states of the neutral kaons.
8
Simulation Summary
• New quenched [J. Noaki, et al., in preparation] and Nf = 2 [hep-lat/4110006]
simulations using DBW2 gauge action and domain wall fermions
conf/traj
202
106
94/5361
94/6195
94/5605
β
1.04
1.22
0.80
0.80
0.80
a−1
1.982(30)
2.914(54)
1.691(53)
volume
163 × 32
243 × 32
163 × 32
163 × 32
163 × 32
msea
∞
∞
0.02
0.03
0.04
Ls
16
10
12
12
12
mres
1.86(12) × 10−5
9.722(27) × 10−5
1.372(49) × 10−3
• Quenched: scaling study
• Nf = 2: sea quark effects, non-degenerate quark mass effects
• Non-perturbative renormalization of operators for both studies
9
Update algorithm for DWB2 β = 1.22
• Tunneling between topological sectors is suppressed as a → 0
• DBW2 suppresses small topological “dislocations” → improved chiral symmetry
• Tunneling at a−1 ≈ 3 GeV slow
• Use hybrid Wilson+DBW2 algorithm to improve distribution
DBW2
Wilson
5k sweeps
10
Wilson initial
10k sweeps
5
0
−5
−10
10
DBW2 5k sweeps
5
0
−5
−10
10
DBW2 10k sweeps
5
0
−5
−10
0
10
20
30
40
#sweep (x10k)
50 0
5
10
15
10
Quenched results
Renormalization constant, ZBK (exptrapolated to mf = −mres).
Extrapolate to (aplatt)2 = 0 to remove lattice spacing error.
β = 1.04
β = 1.22
1.5
1.5
1.4
1.4
1.3
1.3
ZB (platt)
ZB (platt)
1.2
K
−1
K
1.1
1.0
1.0
0.9
0.9
0.8
0.0
0.5
1.0
platt
2
1.5
2.0
w (platt)ZB (platt)
K
w (platt)ZB (platt)
1.1
K
−1
1.2
2.5
0.8
0.0
0.5
1.0
p
2
latt
1.5
2.0
2.5
11
Matrix element of OV V +AA
• Chiral symmetry: should vanish at mf = −mres
(enforced in fit, but consistent when left as free parameter)
• Curvature well described by continuum chiral log with known coefficient
β = 1.04
β = 1.22
0.004
0.0008
DBW2 β=1.04
DBW2 β=1.22
0.0006
<PS|Q|PS> (lattice)
<PS|Q|PS> (lattice)
0.003
0.002
0.001
0.0004
0.0002
0.0000
0.000
0
0.1
0.2
0.3
2
0.4
2
mPS [GeV ]
0.5
0.6
0.7
0
0.1
0.2
0.3
2
0.4
2
0.5
0.6
0.7
mPS [GeV ]
12
Bare BK plateaus
β = 1.04
β = 1.22
0.8
0.8
mfa=0.03
0.7
0.6
0.6
0.5
0.8
0.5
0.8
mfa=0.02
0.7
0.6
0.5
0.7
0.5
0.7
mfa=0.01
mfa=0.008
0.6
0.5
0.4
mfa=0.016
0.7
0.6
0.6
mfa=0.024
0.7
0.5
0
10
t
20
30
0.4
0
10
20
t
30
40
13
Chiral fits, extracting BK
• BK also described well by known chiral log coefficient
• BK extracted from physical point m2P S = m2K ; m2P S corresponds to meson made
of degenerate quarks
β = 1.04
β = 1.22
0.8
0.8
mPS=mK
constrainted chiral log.
free chiral log.
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0
0.1
0.2
0.3
0.4
2
2
mPS [GeV ]
mPS=mK
constrainted chral log.
free chiral log.
0.5
0.6
0.7
0.3
0
0.1
0.2
0.3
0.4
2
2
mPS [GeV ]
0.5
0.6
0.7
14
Quenched continuum limit of BK
• Apparent scaling violation for DBW2 (∼ 2 σ), though a constant fit has acceptable χ2.
• Continuum limit consistent with CP-PACs Iwasaki/DWF calculation using perturbative renormalization
0.8
0.64
DBW2 β=1.22
DBW2 β=1.04
Wilson β=6.0
0.60
BK(MS, µ=2GeV)
BPS (MS, µ=2GeV)
0.7
DBW2 β=1.22
DBW2 β=1.04
Wilson β=6.0
CP−PACS (Iwasaki)
0.62
0.6
0.5
0.58
0.56
0.54
0.52
0.4
0.50
0.3
0
0.1
0.2
0.3
2
PS
m
0.4
2
[GeV ]
0.5
0.6
0.7
0.48
0
0.1
2
−2
0.2
0.3
a [GeV ]
15
Nf = 2 results
• Average values from range 14 ≤ t ≤ 17
msea = 0.02
msea = 0.04
0.8
0.8
mval=0.05
mval=0.04
mval=0.03
mval=0.02
mval=0.01
0.75
0.75
0.7
0.65
BK
BK
0.7
0.65
0.6
0.6
0.55
0.55
0.5
0.5
0.45
0
4
8
12
16
timeslice
mval=0.05
mval=0.04
mval=0.03
mval=0.02
mval=0.01
20
24
28
32
0.45
0
4
8
12
16
20
24
28
32
timeslice
16
Bare BK
• Degenerate and non-degenerate valence quark results
(plotted as mval = (m1 + m2)/2)
Bare BP ; amsea = 0.02
degenerate
non-degenerate
0.65
0.6
0.55
0.5
0.45
0.4
0
0.01
0.02
0.03
amvalence
0.04
0.05
0.06
17
Chiral fits, extracting BK
Two options for interpolating/extrapolating to the physical point:
1. Degenerate: The NLO chiral perturbation theory formula for degenerate valence quark masses is
!!
2
M
1
2
2
2
BP S = b0 1 −
6
M
log
+
b
M
+
b
M
1
2
SS
(4πf )2
Λ2χ
M 2 = 2B0 (mval + mres ) → m2K
2
MSS
= 2B0 (msea + mres) → m2π
i.e. three (unknown) parameters; up to 15 data-points.
2. Non-degenerate: The
quark mass is
NLO chiral perturbation for
non-degenerate valence
complicated.....
Four (unknown) parameters; up to 45 data-points.
18
Degenerate fit
• msea → m̄ extrapolation, known chiral logarithm used
• msea dependence not well resolved between msea = 0.03, 0.04.
• msea = 0.02 is clearly lower
• lighest/heaviest valence points aren’t fit well
0.65
Bp
lat
0.6
0.55
msea=0.02
msea=0.03
msea=0.04
0.5
msea=m
0.45
0.4
0
0.01
0.02
0.03
mval
0.04
0.05
0.06
19
Nf = 2 renormalization constant
2 -1
Elements of (Z/Zq ) (chiral limit)
1.5
1.4
1.3
1.2
1.1
1
0.9
vv+aa : vv+aa
vv+aa : vv-aa
vv+aa : tt
vv+aa : ss+pp
vv+aa : ss-pp
vv-aa : vv+aa
tt : vv+aa
ss+pp : vv+aa
ss-pp : vv+aa
0.8
0.7
0.6
0.5
0.4
0.3
• Mixing with wrong chirality
operators very small
• Combining ZBK with perturbative (continuum) matching
calculation gives
ZBK = 0.93(2), µ = 2 GeV,
M S scheme.
0.2
0.1
0
-0.1
0
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8
(ap)^2
2
2.2 2.4
20
Results for BK
Fitting for valence and dynamical masses such that
0.02 ≤ amsea , amval ≤ 0.04
as for low valence masses the plateau quality is bad, and we wish to stay in the
(relatively) low mass region to fit to NLO chiral perturbation theory
Fit
Degenerate
Non-degenerate
Bare Number
0.547(15)
0.533(14)
M S, 2GeV
0.509(18)
0.495(18)
The difference between the degenerate and non-degenerate fits is within the quoted
statistical error, but due to these errors being correlated it is actually statistically
well resolved as a 2.8 ± 0.03% effect.
21
Estimates of systematic uncertainties for DWF BK calculation
• Quenched finite volume: +(2-3)% (CP-PACS)
• Quenched lattice spacing error: ∼ −2, +5% at a−1 ≈ 2 GeV (CP-PACS, RBC)
• Quenched renormalization: ∼ −4% at a−1 ≈ 2 GeV (RBC)
• Quenched finite Ls: negligible (CP-PACS, RBC)
• Nf = 2 Non-degenerate quarks: ∼ −3% (RBC)
• Nf = 2: Quenching error ∼ −3% (RBC)
staggered, inv. (JLQCD)
staggered, non-inv. (JLQCD)
DWF, large vol (CP-PACS)
DWF, NPR, Wilson gauge (RBC)
DWF, NPR, DBW2 (RBC)
DWF, NPR, DBW2, Nf = 2 (RBC)
BK(NDR, 2GeV)
0.8
0.7
0.6
0.5
0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
mρa
22
Outlook
• Proposed Nf = 2 + 1 DWF calculations (RBC, UKQCD, ...) should have a
large impact on this plot, CKM mixing matrix
• 95% CL’s (∼ 2σ)
• Central value is KS, quenched (still reasonable?). DWF give larger value of η̄.
0.7
0.5
η
0.4
0.3
excluded area has CL < 0.05
0.6
∆md
0
-0.4
CKM
fitter
ICHEP 2004
εK
α
α
εK
0.2
0.1
∆ms & ∆md
|Vub/Vcb|
-0.2
γ
0
β
0.2
0.4
0.6
0.8
1
ρ
23