chris kelly university of edinburgh rbc & ukqcd collaborations · bk chiral fit forms we use...
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BK for 2+1 flavour domain wall fermions from243 and 323 × 64 lattices
Chris Kelly
University of EdinburghRBC & UKQCD Collaborations
Mon 14th July
Outline
The neutral kaon mixing amplitude BK
Outline
The neutral kaon mixing amplitude BK
Ensemble details
Outline
The neutral kaon mixing amplitude BK
Ensemble details
Measurement of BK
Outline
The neutral kaon mixing amplitude BK
Ensemble details
Measurement of BK
The chiral extrapolation of BK
Outline
The neutral kaon mixing amplitude BK
Ensemble details
Measurement of BK
The chiral extrapolation of BK
The non-perturbative renormalisation of BK
Outline
The neutral kaon mixing amplitude BK
Ensemble details
Measurement of BK
The chiral extrapolation of BK
The non-perturbative renormalisation of BK
Conclusions and Outlook
The neutral kaon mixing amplitude BK
Kaon mixing
◮ Indirect CP violation in neutral kaon sector
Kaon mixing
◮ Indirect CP violation in neutral kaon sector
◮ Neutral kaon mixing amplitude:
Kaon mixing
◮ Indirect CP violation in neutral kaon sector
◮ Neutral kaon mixing amplitude:
Kaon mixing
◮ Indirect CP violation in neutral kaon sector
◮ Neutral kaon mixing amplitude:
Kaon mixing
◮ Indirect CP violation in neutral kaon sector
◮ Neutral kaon mixing amplitude:
BK
◮ BK parameterises the matrix element of the four-quarkK 0 → K̄ 0 operator
BK
◮ BK parameterises the matrix element of the four-quarkK 0 → K̄ 0 operator
BK ≡〈K 0|OVV+AA|K̄
0〉83 f 2
KM2K
BK
◮ BK parameterises the matrix element of the four-quarkK 0 → K̄ 0 operator
BK ≡〈K 0|OVV+AA|K̄
0〉83 f 2
KM2K
OVV+AA = (s̄γµd)(s̄γµd) + (s̄γ5γµd)(s̄γ5γµd)
BK
◮ BK parameterises the matrix element of the four-quarkK 0 → K̄ 0 operator
BK ≡〈K 0|OVV+AA|K̄
0〉83 f 2
KM2K
OVV+AA = (s̄γµd)(s̄γµd) + (s̄γ5γµd)(s̄γ5γµd)
◮ BK related to measure of indirect CP violation ǫK = KL→(ππ)KS→(ππ)
you →relation contains unknown direct CP violatingparameters.
BK
◮ BK parameterises the matrix element of the four-quarkK 0 → K̄ 0 operator
BK ≡〈K 0|OVV+AA|K̄
0〉83 f 2
KM2K
OVV+AA = (s̄γµd)(s̄γµd) + (s̄γ5γµd)(s̄γ5γµd)
◮ BK related to measure of indirect CP violation ǫK = KL→(ππ)KS→(ππ)
you →relation contains unknown direct CP violatingparameters.
◮ ǫK known experimentally to high precision ⇒ BK constrainsunknown direct CP violating parameters.
Ensemble details
Details of ensembles
243 × 64 323 × 64
Details of ensembles
243 × 64
◮ 2+1f domain wall fermionensemble with Ls = 16
323 × 64
◮ 2+1f domain wall fermionensemble with Ls = 16
Details of ensembles
243 × 64
◮ 2+1f domain wall fermionensemble with Ls = 16
◮ Iwasaki gauge actionβ = 2.13
323 × 64
◮ 2+1f domain wall fermionensemble with Ls = 16
Details of ensembles
243 × 64
◮ 2+1f domain wall fermionensemble with Ls = 16
◮ Iwasaki gauge actionβ = 2.13
323 × 64
◮ 2+1f domain wall fermionensemble with Ls = 16
◮ Iwasaki gauge actionβ = 2.25
Details of ensembles
243 × 64
◮ 2+1f domain wall fermionensemble with Ls = 16
◮ Iwasaki gauge actionβ = 2.13
◮ a−1 = 1.729(28) GeV →(2.74 fm)3 lattice volume
323 × 64
◮ 2+1f domain wall fermionensemble with Ls = 16
◮ Iwasaki gauge actionβ = 2.25
Details of ensembles
243 × 64
◮ 2+1f domain wall fermionensemble with Ls = 16
◮ Iwasaki gauge actionβ = 2.13
◮ a−1 = 1.729(28) GeV →(2.74 fm)3 lattice volume
323 × 64
◮ 2+1f domain wall fermionensemble with Ls = 16
◮ Iwasaki gauge actionβ = 2.25
◮ a−1 = 2.42(4) 0.47r0 (fm) GeV →
(2.61 fm)3 lattice volume
Details of ensembles
243 × 64
◮ 2+1f domain wall fermionensemble with Ls = 16
◮ Iwasaki gauge actionβ = 2.13
◮ a−1 = 1.729(28) GeV →(2.74 fm)3 lattice volume
◮ Strange sea quark mass 0.04lattice units
323 × 64
◮ 2+1f domain wall fermionensemble with Ls = 16
◮ Iwasaki gauge actionβ = 2.25
◮ a−1 = 2.42(4) 0.47r0 (fm) GeV →
(2.61 fm)3 lattice volume
Details of ensembles
243 × 64
◮ 2+1f domain wall fermionensemble with Ls = 16
◮ Iwasaki gauge actionβ = 2.13
◮ a−1 = 1.729(28) GeV →(2.74 fm)3 lattice volume
◮ Strange sea quark mass 0.04lattice units
323 × 64
◮ 2+1f domain wall fermionensemble with Ls = 16
◮ Iwasaki gauge actionβ = 2.25
◮ a−1 = 2.42(4) 0.47r0 (fm) GeV →
(2.61 fm)3 lattice volume
◮ Strange sea quark mass 0.03lattice units
Up/down sea quark masses
243 × 64
latt. units mπ (MeV)
0.03 6260.02 5580.01 3450.005 331
323 × 64
latt. units mπ (MeV)
0.008 ∼ 4200.006 ∼ 3600.004 ∼ 300
Highly preliminary data asdatasets only partially complete
Measurement of BK
Method comparison
243 × 64 323 × 64
Method comparison
243 × 64
◮ 2 gauge-fixed wall sources att = 5, 59 for propagators
323 × 64
Method comparison
243 × 64
◮ 2 gauge-fixed wall sources att = 5, 59 for propagators
◮ Use p + a boundaryconditions → removesunwanted round-the-worldcontributions.
323 × 64
Method comparison
243 × 64
◮ 2 gauge-fixed wall sources att = 5, 59 for propagators
◮ Use p + a boundaryconditions → removesunwanted round-the-worldcontributions.
◮ Costs 4 inversions perconfiguration.
323 × 64
Method comparison
243 × 64
◮ 2 gauge-fixed wall sources att = 5, 59 for propagators
◮ Use p + a boundaryconditions → removesunwanted round-the-worldcontributions.
◮ Costs 4 inversions perconfiguration.
323 × 64
◮ 1 gauge-fixed wall source att = 0
Method comparison
243 × 64
◮ 2 gauge-fixed wall sources att = 5, 59 for propagators
◮ Use p + a boundaryconditions → removesunwanted round-the-worldcontributions.
◮ Costs 4 inversions perconfiguration.
323 × 64
◮ 1 gauge-fixed wall source att = 0
◮ Use p + a and p − a
boundary conditions → givesforwards and backwardspropagating quarks.
Method comparison
243 × 64
◮ 2 gauge-fixed wall sources att = 5, 59 for propagators
◮ Use p + a boundaryconditions → removesunwanted round-the-worldcontributions.
◮ Costs 4 inversions perconfiguration.
323 × 64
◮ 1 gauge-fixed wall source att = 0
◮ Use p + a and p − a
boundary conditions → givesforwards and backwardspropagating quarks.
◮ Costs 2 inversions perconfiguration.
BK example plateaux
243 × 64 ml = 0.005
12 16 20 24 28 32 36 40 44 48 52t
0.5
0.55
0.6
0.65
BK
mx = 0.04 m
y =0.04
mx = 0.04 m
y =0.001
Preliminary 323 × 64 ml = 0.004
12 16 20 24 28 32 36 40 44 48 52t
0.5
0.55
0.6
0.65
BK
mx = 0.03 m
y = 0.03
mx = 0.03 m
y = 0.002
The chiral extrapolation of BK
BK chiral fit forms
◮ We use NLO SU(2) × SU(2) partially-quenched chiralperturbation theory (PQChPT) for maximum use ofensembles.
BK chiral fit forms
◮ We use NLO SU(2) × SU(2) partially-quenched chiralperturbation theory (PQChPT) for maximum use ofensembles.
◮ Kaon sector is coupled to SU(2) soft pion loops at lowestorder in non-relativistic expansion
BK chiral fit forms
◮ We use NLO SU(2) × SU(2) partially-quenched chiralperturbation theory (PQChPT) for maximum use ofensembles.
◮ Kaon sector is coupled to SU(2) soft pion loops at lowestorder in non-relativistic expansiontext→ direct connection to HMχPT
BK chiral fit forms
◮ We use NLO SU(2) × SU(2) partially-quenched chiralperturbation theory (PQChPT) for maximum use ofensembles.
◮ Kaon sector is coupled to SU(2) soft pion loops at lowestorder in non-relativistic expansiontext→ direct connection to HMχPT
◮ 243 analysis [Allton et al arXiv:0804.0473] indicatedSU(3) × SU(3) PQChPT has large higher order correctionsand doesn’t fit data well up to physical strange quark mass(R. Mawhinney).
SU(2) × SU(2) PQChPT fit form for BK
BK = B0K
[
1 + 2B(md+mres)c0
f 2 +2B(my +mres)c1
f 2
−2B(my+mres)
32π2f 2 log(
2B(my +mres)Λ2
χ
) ]
SU(2) × SU(2) PQChPT fit form for BK
BK = B0K
[
1 + 2B(md+mres)c0
f 2 +2B(my +mres)c1
f 2
−2B(my+mres)
32π2f 2 log(
2B(my +mres)Λ2
χ
) ]
◮ 5 free parameters: B0K
SU(2) × SU(2) PQChPT fit form for BK
BK = B0K
[
1 + 2B(md+mres)c0
f 2 +2B(my +mres)c1
f 2
−2B(my+mres)
32π2f 2 log(
2B(my +mres)Λ2
χ
) ]
◮ 5 free parameters: B0K , B , f , c0, c1
SU(2) × SU(2) PQChPT fit form for BK
BK = B0K
[
1 + 2B(md+mres)c0
f 2 +2B(my+mres)c1
f 2
−2B(my+mres)
32π2f 2 log(
2B(my+mres)Λ2
χ
) ]
◮ 5 free parameters: B0K , B , f , c0, c1
SU(2) × SU(2) PQChPT fit form for BK
BK = B0K
[
1 + 2B(md+mres)c0
f 2 +2B(my +mres)c1
f 2
−2B(my+mres)
32π2f 2 log(
2B(my +mres)Λ2
χ
) ]
◮ 5 free parameters: B0K , B , f , c0, c1
SU(2) × SU(2) PQChPT fit form for BK
BK = B0K
[
1 + 2B(md+mres)c0
f 2 +2B(my +mres)c1
f 2
−2B(my+mres)
32π2f 2 log(
2B(my +mres)Λ2
χ
) ]
◮ 5 free parameters: B0K , B , f , c0, c1
SU(2) × SU(2) PQChPT fit form for BK
BK = B0K
[
1 + 2B(md+mres)c0
f 2 +2B(my +mres)c1
f 2
−2B(my+mres)
32π2f 2 log(
2B(my +mres)Λ2
χ
) ]
◮ 5 free parameters: B0K , B , f , c0, c1
SU(2) × SU(2) PQChPT fit form for BK
BK = B0K
[
1 + 2B(md+mres)c0
f 2 +2B(my +mres)c1
f 2
−2B(my+mres)
32π2f 2 log(
2B(my +mres)Λ2
χ
) ]
◮ 5 free parameters: B0K , B , f , c0, c1
◮ Use simultaneous pure SU(2) × SU(2) PQChPT fit (nocoupling to Kaon sector) to FPS and MPS to determine B andf (E. Scholz)
SU(2) × SU(2) PQChPT fit form for BK
BK = B0K
[
1 + 2B(md+mres)c0
f 2 +2B(my +mres)c1
f 2
−2B(my+mres)
32π2f 2 log(
2B(my +mres)Λ2
χ
) ]
◮ 5 free parameters: B0K , B , f , c0, c1
◮ Use simultaneous pure SU(2) × SU(2) PQChPT fit (nocoupling to Kaon sector) to FPS and MPS to determine B andf (E. Scholz)
→ perform frozen 3-parameter fit to BK
Simultaneous PQChPT fits to FPS and MPS : fPS
243 × 64
0.08
0.09
0.1
0.11
0 0.01 0.02 0.03 0.04my
ml = 0.005, ms = 0.04fit: mavg ≤ 0.01
fxy
mx=0.001mx=0.005mx=0.01 mx=0.02 mx=0.03 mx=0.04
323 × 64
0 0.005 0.01 0.015 0.02 0.025 0.03m
y+m
res
0.055
0.06
0.065
0.07
0.075
0.08
mx = 0.03
mx = 0.025
mx = 0.008
mx = 0.006
mx = 0.004
mx = 0.002
fxy
ml = 0.004, m
s = 0.03
fit: mavg
<= 0.008
texttext
Simultaneous PQChPT fits to FPS and MPS : fPS
◮ For fixed ml chiral fit formsnon-analytic as mx/y → 0
323 × 64
0 0.005 0.01 0.015 0.02 0.025 0.03m
y+m
res
0.055
0.06
0.065
0.07
0.075
0.08
mx = 0.03
mx = 0.025
mx = 0.008
mx = 0.006
mx = 0.004
mx = 0.002
fxy
ml = 0.004, m
s = 0.03
fit: mavg
<= 0.008
texttext
Simultaneous PQChPT fits to FPS and MPS : fPS
◮ For fixed ml chiral fit formsnon-analytic as mx/y → 0
◮ Perform full PQChPT fit toall data points thenextrapolate to chiral limitalong unitary curvemx = my = ml → 0 toobtain physical fPS.
323 × 64
0 0.005 0.01 0.015 0.02 0.025 0.03m
y+m
res
0.055
0.06
0.065
0.07
0.075
0.08
mx = 0.03
mx = 0.025
mx = 0.008
mx = 0.006
mx = 0.004
mx = 0.002
fxy
ml = 0.004, m
s = 0.03
fit: mavg
<= 0.008
texttext
Simultaneous PQChPT fits to FPS and MPS : fPS
◮ For fixed ml chiral fit formsnon-analytic as mx/y → 0
◮ Perform full PQChPT fit toall data points thenextrapolate to chiral limitalong unitary curvemx = my = ml → 0 toobtain physical fPS.
◮ Unitary curve is finite valuedat chiral limit.
323 × 64
0 0.005 0.01 0.015 0.02 0.025 0.03m
y+m
res
0.055
0.06
0.065
0.07
0.075
0.08
mx = 0.03
mx = 0.025
mx = 0.008
mx = 0.006
mx = 0.004
mx = 0.002
fxy
ml = 0.004, m
s = 0.03
fit: mavg
<= 0.008
texttext
Simultaneous PQChPT fits to FPS and MPS : MPS
243 × 64
4.2
4.4
4.6
4.8
5
0 0.01 0.02 0.03 0.04my
ml = 0.005, ms = 0.04fit: mavg ≤ 0.01
mxy2 / (mavg+mres)
mx=0.001mx=0.005mx=0.01 mx=0.02 mx=0.03 mx=0.04
323 × 64
0 0.005 0.01 0.015 0.02 0.025 0.03m
y+m
res
3.3
3.4
3.5
3.6
3.7
3.8
mx = 0.03
mx = 0.025
mx = 0.008
mx = 0.006
mx = 0.004
mx = 0.002
m2
xy/(m
avg+m
res)
ml = 0.004, m
s = 0.03
fit: mavg
<= 0.008
texttext
PQChPT fits to BK
243 × 64
0.54
0.56
0.58
0.6
0.62
0.64
0.66
0.68
0 0.01 0.02 0.03 0.04mx
my = 0.04fit: mx ≤ 0.01
Bxy
ml=0.005 ml=0.01 mx=ml mx=ml=mud
323 × 64
0 0.002 0.004 0.006 0.008 0.01m
x+m
res
0.54
0.56
0.58
0.6
0.62
ml = 0.008
ml = 0.006
ml = 0.004
mx = m
l
Bxy
my = 0.03
fit: mx <= 0.008
◮ Unitary curve is fixed ms , mx = ml → mphysl
PQChPT fits to BK
243 × 64
0.54
0.56
0.58
0.6
0.62
0.64
0.66
0.68
0 0.01 0.02 0.03 0.04mx
my = 0.04fit: mx ≤ 0.01
Bxy
ml=0.005 ml=0.01 mx=ml mx=ml=mud
323 × 64
0 0.002 0.004 0.006 0.008 0.01m
x+m
res
0.54
0.56
0.58
0.6
0.62
ml = 0.008
ml = 0.006
ml = 0.004
mx = m
l
Bxy
my = 0.03
fit: mx <= 0.008
Stat err?
◮ Unitary curve is fixed ms , mx = ml → mphysl
243 × 64 BK chiral limit results – Allton et al
[arXiv:0804.0473]
◮ 243 × 64 B latK = 0.565(10).
243 × 64 BK chiral limit results – Allton et al
[arXiv:0804.0473]
◮ 243 × 64 B latK = 0.565(10).
◮ 323 × 64 extrapolation not yet available, dataset only partiallycompletetext– Stat uncertainties in data sets, unknown physical quarkmasses
The non-perturbative renormalisation of BK
Why NPR?
◮ Lattice perturbative (DWF) calculations exist but:
Why NPR?
◮ Lattice perturbative (DWF) calculations exist but:◮ exist only at low order
Why NPR?
◮ Lattice perturbative (DWF) calculations exist but:◮ exist only at low order◮ are poorly convergent
Why NPR?
◮ Lattice perturbative (DWF) calculations exist but:◮ exist only at low order◮ are poorly convergent◮ involve prescription dependent ambiguities such as MF
improvement
Why NPR?
◮ Lattice perturbative (DWF) calculations exist but:◮ exist only at low order◮ are poorly convergent◮ involve prescription dependent ambiguities such as MF
improvement
◮ Use Rome-Southampton RI/MOM scheme
Bilinear vertices
◮ ZV = ZA due to good chiral symmetry
Bilinear vertices
◮ ZV = ZA due to good chiral symmetry
◮ At high momenta, ΛA = ΛV =Zq
ZA=
Zq
ZVshould hold.
Bilinear vertices
◮ ZV = ZA due to good chiral symmetry
◮ At high momenta, ΛA = ΛV =Zq
ZA=
Zq
ZVshould hold.
◮ Therefore can use ΛA or ΛV as a measure ofZq
ZA.
Bilinear vertices
◮ ZV = ZA due to good chiral symmetry
◮ At high momenta, ΛA = ΛV =Zq
ZA=
Zq
ZVshould hold.
◮ Therefore can use ΛA or ΛV as a measure ofZq
ZA.
◮ ΛA and ΛV expected to differ at low momenta due to QCDspontaneous chiral symmetry breaking.
Bilinear vertices
◮ ZV = ZA due to good chiral symmetry
◮ At high momenta, ΛA = ΛV =Zq
ZA=
Zq
ZVshould hold.
◮ Therefore can use ΛA or ΛV as a measure ofZq
ZA.
◮ ΛA and ΛV expected to differ at low momenta due to QCDspontaneous chiral symmetry breaking.
◮ However, even at high momenta we find ΛA 6= ΛV at 2% level
Bilinear vertices
◮ ZV = ZA due to good chiral symmetry
◮ At high momenta, ΛA = ΛV =Zq
ZA=
Zq
ZVshould hold.
◮ Therefore can use ΛA or ΛV as a measure ofZq
ZA.
◮ ΛA and ΛV expected to differ at low momenta due to QCDspontaneous chiral symmetry breaking.
◮ However, even at high momenta we find ΛA 6= ΛV at 2% leveltext→ difference caused by kinematic choice : Exceptionalmomentum configuration
Bilinear vertices
◮ ZV = ZA due to good chiral symmetry
◮ At high momenta, ΛA = ΛV =Zq
ZA=
Zq
ZVshould hold.
◮ Therefore can use ΛA or ΛV as a measure ofZq
ZA.
◮ ΛA and ΛV expected to differ at low momenta due to QCDspontaneous chiral symmetry breaking.
◮ However, even at high momenta we find ΛA 6= ΛV at 2% leveltext→ difference caused by kinematic choice : Exceptionalmomentum configurationtext→ Gives weak 1/p2 suppression of low energy chiralsymmetry breaking.
Chiral symmetry breaking and exceptional momenta
◮ Generic bilinear vertex graph
Chiral symmetry breaking and exceptional momenta
◮ Generic bilinear vertex graph
◮ p2 → ∞ behaviour governed by subgraphwith least negative degree of divergencethrough which we can route all hard
external momenta.
Chiral symmetry breaking and exceptional momenta
◮ Generic bilinear vertex graph
◮ p2 → ∞ behaviour governed by subgraphwith least negative degree of divergencethrough which we can route all hard
external momenta.
◮ For p1 6= p2 this is the entire graph.
Chiral symmetry breaking and exceptional momenta
◮ Generic bilinear vertex graph
◮ p2 → ∞ behaviour governed by subgraphwith least negative degree of divergencethrough which we can route all hard
external momenta.
◮ For p1 6= p2 this is the entire graph.
◮ Can connect to low-energy subgraphswhich are affected by spont. chiralsymmetry breaking, but:
Chiral symmetry breaking and exceptional momenta
◮ Generic bilinear vertex graph
◮ p2 → ∞ behaviour governed by subgraphwith least negative degree of divergencethrough which we can route all hard
external momenta.
◮ For p1 6= p2 this is the entire graph.
◮ Can connect to low-energy subgraphswhich are affected by spont. chiralsymmetry breaking, but:text→ Low energy subgraph notcontained within circled subgraph
Chiral symmetry breaking and exceptional momenta
◮ Generic bilinear vertex graph
◮ p2 → ∞ behaviour governed by subgraphwith least negative degree of divergencethrough which we can route all hard
external momenta.
◮ For p1 6= p2 this is the entire graph.
◮ Can connect to low-energy subgraphswhich are affected by spont. chiralsymmetry breaking, but:text→ Low energy subgraph notcontained within circled subgraphtext→ Adding extra external legs tocircled subgraph increases suppression ofthe graph
Chiral symmetry breaking and exceptional momenta
◮ However in case p2 − p1 = 0 then highmomenta do not enter internal subgraphs
Chiral symmetry breaking and exceptional momenta
◮ However in case p2 − p1 = 0 then highmomenta do not enter internal subgraphs
◮ Graph free to couple to low-energy chiralsymmetry breaking subgraphs with nofurther suppression
Chiral symmetry breaking and exceptional momenta
◮ However in case p2 − p1 = 0 then highmomenta do not enter internal subgraphs
◮ Graph free to couple to low-energy chiralsymmetry breaking subgraphs with nofurther suppression
◮ This is an exceptional momentumconfiguration
Chiral symmetry breaking and exceptional momenta
◮ However in case p2 − p1 = 0 then highmomenta do not enter internal subgraphs
◮ Graph free to couple to low-energy chiralsymmetry breaking subgraphs with nofurther suppression
◮ This is an exceptional momentumconfiguration
◮ Chiral symmetry breaking inducesdifference between ΛA and ΛV → use12(ΛA + ΛV ) ≈
Zq
ZA
BK NPR with RI/MOM and exceptional momenta
◮ Calculate four-quark vertex matrix element in Landau gauge.
BK NPR with RI/MOM and exceptional momenta
◮ Calculate four-quark vertex matrix element in Landau gauge.
◮ Amputate vertex with ensemble averaged unrenormalisedpropagator, giving ΛOVV+AA
BK NPR with RI/MOM and exceptional momenta
◮ Calculate four-quark vertex matrix element in Landau gauge.
◮ Amputate vertex with ensemble averaged unrenormalisedpropagator, giving ΛOVV+AA
◮ Renormalisation condition: Fix to tree level value at µ2 = p2
ZVV+AA
Z 2q
ΛOVV+AA= Otree
VV+AA
BK NPR with RI/MOM and exceptional momenta
◮ Calculate four-quark vertex matrix element in Landau gauge.
◮ Amputate vertex with ensemble averaged unrenormalisedpropagator, giving ΛOVV+AA
◮ Renormalisation condition: Fix to tree level value at µ2 = p2
ZVV+AA
Z 2q
ΛOVV+AA= Otree
VV+AA
◮ DefineZ
RI/MOM
BK ≡ ZVV+AA
Z2A
=(
Z2q
Z2A
)
ZVV+AA
Z2q
BK NPR with RI/MOM and exceptional momenta
◮ Calculate four-quark vertex matrix element in Landau gauge.
◮ Amputate vertex with ensemble averaged unrenormalisedpropagator, giving ΛOVV+AA
◮ Renormalisation condition: Fix to tree level value at µ2 = p2
ZVV+AA
Z 2q
ΛOVV+AA= Otree
VV+AA
◮ DefineZ
RI/MOM
BK ≡ ZVV+AA
Z2A
=(
Z2q
Z2A
)
ZVV+AA
Z2q
◮ Use 12(ΛA + ΛV ) ≈
Zq
ZA
Method comparison
243 × 32 323 × 64
Method comparison
163 × 32
◮ Use point sources, 4 quarkvertex formed at sourcelocation.
323 × 64
Method comparison
163 × 32
◮ Use point sources, 4 quarkvertex formed at sourcelocation.
◮ Average over 4 sourcelocations on 75configurations on ourml = 0.03, 0.02 and 0.01ensembles.
323 × 64
Method comparison
163 × 32
◮ Use point sources, 4 quarkvertex formed at sourcelocation.
◮ Average over 4 sourcelocations on 75configurations on ourml = 0.03, 0.02 and 0.01ensembles.
◮ Momentum applied byapplying phase differencebetween propagator sourceand sink. Solution can begiven arbitrary momentum.
323 × 64
Method comparison
163 × 32
◮ Use point sources, 4 quarkvertex formed at sourcelocation.
◮ Average over 4 sourcelocations on 75configurations on ourml = 0.03, 0.02 and 0.01ensembles.
◮ Momentum applied byapplying phase differencebetween propagator sourceand sink. Solution can begiven arbitrary momentum.
323 × 64
◮ Use lattice volume sources,vertex formed at propagatorsink. (D. Broemmel)
Method comparison
163 × 32
◮ Use point sources, 4 quarkvertex formed at sourcelocation.
◮ Average over 4 sourcelocations on 75configurations on ourml = 0.03, 0.02 and 0.01ensembles.
◮ Momentum applied byapplying phase differencebetween propagator sourceand sink. Solution can begiven arbitrary momentum.
323 × 64
◮ Use lattice volume sources,vertex formed at propagatorsink. (D. Broemmel)
◮ Average over all sinklocations, lattice volumefactor gain over pointapproach.
Method comparison
163 × 32
◮ Use point sources, 4 quarkvertex formed at sourcelocation.
◮ Average over 4 sourcelocations on 75configurations on ourml = 0.03, 0.02 and 0.01ensembles.
◮ Momentum applied byapplying phase differencebetween propagator sourceand sink. Solution can begiven arbitrary momentum.
323 × 64
◮ Use lattice volume sources,vertex formed at propagatorsink. (D. Broemmel)
◮ Average over all sinklocations, lattice volumefactor gain over pointapproach.
◮ Volume source has fixedmomentum as phase mustbe applied to source latticesites before inversion.
Method comparison
163 × 32
◮ Use point sources, 4 quarkvertex formed at sourcelocation.
◮ Average over 4 sourcelocations on 75configurations on ourml = 0.03, 0.02 and 0.01ensembles.
◮ Momentum applied byapplying phase differencebetween propagator sourceand sink. Solution can begiven arbitrary momentum.
323 × 64
◮ Currently calculated 5independent momenta (10total) on 10 configurationson our ml = 0.006 and0.004 ensembles
ZRI/MOM
BK (µ)
163 × 32
0 0.5 1 1.5 2 2.5(aµ)2
0.8
0.9
1
1.1
ZB
K
ml = 0.01
ml = 0.02
ml = 0.03
chiral limit
texttext
323 × 64
texttext
ZRI/MOM
BK (µ)
163 × 32
0 0.5 1 1.5 2 2.5(aµ)2
0.8
0.9
1
1.1
ZB
K
ml = 0.01
ml = 0.02
ml = 0.03
chiral limit
texttext
323 × 64
0 0.5 1 1.5 2 2.5
(aµ)2
0.8
0.9
1
1.1
ZB
K
ml = 0.006
ml = 0.004
chiral limitPoint m
l = 0.006
texttext
ZRI/MOM
BK (µ)
163 × 32
0 0.5 1 1.5 2 2.5(aµ)2
0.8
0.9
1
1.1
ZB
K
ml = 0.01
ml = 0.02
ml = 0.03
chiral limit
texttext
323 × 64
0 0.5 1 1.5 2 2.5
(aµ)2
0.91
0.92
0.93
0.94
0.95
ZB
K
ml = 0.006
ml = 0.004
chiral limit
texttext
Chiral extrapolation – 323 × 64
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008m
l
0.91
0.92
0.93
0.94
ZB
K
0.6939570.9252751.310811.619231.92766
texttexttext
◮ For each ZBK (µ), perform alinear chiral extrapolation tom = −mres
Chiral extrapolation – 323 × 64
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008m
l
0.91
0.92
0.93
0.94
ZB
K
0.6939570.9252751.310811.619231.92766
texttexttext
◮ For each ZBK (µ), perform alinear chiral extrapolation tom = −mres
◮ 323 lever-arm forextrapolation smallcompared to extrapolationdistance
Chiral extrapolation – 323 × 64
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008m
l
0.91
0.92
0.93
0.94
ZB
K
0.6939570.9252751.310811.619231.92766
texttexttext
◮ For each ZBK (µ), perform alinear chiral extrapolation tom = −mres
◮ 323 lever-arm forextrapolation smallcompared to extrapolationdistancetext→ Future: Addml = 0.008 dataset
Exceptional momenta systematic error
◮ 323 stat errors smallcompared to systematicerror from exceptionalmomenta.
323 × 64
0 0.5 1 1.5 2 2.5
(aµ)2
0.91
0.92
0.93
0.94
0.95
0.96
ZB
K
ZBK
using (ΛA
+ΛV
)/2
ZBK
using ΛA
ZBK
using ΛV
Exceptional momenta systematic error
◮ 323 stat errors smallcompared to systematicerror from exceptionalmomenta.
◮ On 243 we attributed a1.5% sys error to this alone.
323 × 64
0 0.5 1 1.5 2 2.5
(aµ)2
0.91
0.92
0.93
0.94
0.95
0.96
ZB
K
ZBK
using (ΛA
+ΛV
)/2
ZBK
using ΛA
ZBK
using ΛV
Exceptional momenta systematic error
◮ 323 stat errors smallcompared to systematicerror from exceptionalmomenta.
◮ On 243 we attributed a1.5% sys error to this alone.
◮ Difference greatly reducedby using non-exceptionalmomentum configurationp1 6= p2
323 × 64
0 0.5 1 1.5 2 2.5
(aµ)2
0.91
0.92
0.93
0.94
0.95
0.96
ZB
K
ZBK
using (ΛA
+ΛV
)/2
ZBK
using ΛA
ZBK
using ΛV
Exceptional momenta systematic error
◮ 323 stat errors smallcompared to systematicerror from exceptionalmomenta.
◮ On 243 we attributed a1.5% sys error to this alone.
◮ Difference greatly reducedby using non-exceptionalmomentum configurationp1 6= p2
◮ Unfortunately noperturbative calculationavailable for non-exceptional(Y. Aoki)
323 × 64
0 0.5 1 1.5 2 2.5
(aµ)2
0.91
0.92
0.93
0.94
0.95
0.96
ZB
K
ZBK
using (ΛA
+ΛV
)/2
ZBK
using ΛA
ZBK
using ΛV
Removal of lattice artefacts
0 0.5 1 1.5 2 2.5
(aµ)2
0.8
1
1.2
1.4
1.6
ZB
K
chiral limitSI
(aµ)2 fit
texttexttext
◮ Divide out perturbativerunning: Quantity is scaleinvariant up to latticeartefacts
Removal of lattice artefacts
0 0.5 1 1.5 2 2.5
(aµ)2
0.8
1
1.2
1.4
1.6
ZB
K
chiral limitSI
(aµ)2 fit
texttexttext
◮ Divide out perturbativerunning: Quantity is scaleinvariant up to latticeartefacts
◮ Expect quadraticdependence of latticeartefacts on lattice spacing
→ fit to form ZSIBK + B(aµ)2
Extrapolation of Z SIBK
163 × 32 323 × 64
0 0.5 1 1.5 2 2.5
(aµ)2
0.8
1
1.2
1.4
1.6
ZB
K
chiral limitSI
(aµ)2 fit
Extrapolation of Z SIBK
163 × 32
0 0.5 1 1.5 2 2.5
(aµ)2
0.8
1
1.2
1.4
1.6
ZB
K
RI/MOM
SI
(aµ)2 fit
323 × 64
0 0.5 1 1.5 2 2.5
(aµ)2
0.8
1
1.2
1.4
1.6
ZB
K
chiral limitSI
(aµ)2 fit
243 × 64 result – Aoki et al [arXiv:0712.1061]
◮ Reapply RI/MOM perturbative running to ZSIBK and scale to
conventional µ = 2 GeV.
243 × 64 result – Aoki et al [arXiv:0712.1061]
◮ Reapply RI/MOM perturbative running to ZSIBK and scale to
conventional µ = 2 GeV.
◮ Apply conversion factor ZRI/MOM
BK → ZMSBK
243 × 64 result – Aoki et al [arXiv:0712.1061]
◮ Reapply RI/MOM perturbative running to ZSIBK and scale to
conventional µ = 2 GeV.
◮ Apply conversion factor ZRI/MOM
BK → ZMSBK
◮ ZMSBK (2 GeV) = 0.9276 ± 0.0052(stat) ± 0.0220(sys).
243 × 64 result – Aoki et al [arXiv:0712.1061]
◮ Reapply RI/MOM perturbative running to ZSIBK and scale to
conventional µ = 2 GeV.
◮ Apply conversion factor ZRI/MOM
BK → ZMSBK
◮ ZMSBK (2 GeV) = 0.9276 ± 0.0052(stat) ± 0.0220(sys).
◮ Sys errors:
243 × 64 result – Aoki et al [arXiv:0712.1061]
◮ Reapply RI/MOM perturbative running to ZSIBK and scale to
conventional µ = 2 GeV.
◮ Apply conversion factor ZRI/MOM
BK → ZMSBK
◮ ZMSBK (2 GeV) = 0.9276 ± 0.0052(stat) ± 0.0220(sys).
◮ Sys errors:◮ O(αs) ⇒ 0.0177 corrections due to truncation of perturbative
analysis
243 × 64 result – Aoki et al [arXiv:0712.1061]
◮ Reapply RI/MOM perturbative running to ZSIBK and scale to
conventional µ = 2 GeV.
◮ Apply conversion factor ZRI/MOM
BK → ZMSBK
◮ ZMSBK (2 GeV) = 0.9276 ± 0.0052(stat) ± 0.0220(sys).
◮ Sys errors:◮ O(αs) ⇒ 0.0177 corrections due to truncation of perturbative
analysis◮ ⇒ 0.0007 unphysical strange mass correction
243 × 64 result – Aoki et al [arXiv:0712.1061]
◮ Reapply RI/MOM perturbative running to ZSIBK and scale to
conventional µ = 2 GeV.
◮ Apply conversion factor ZRI/MOM
BK → ZMSBK
◮ ZMSBK (2 GeV) = 0.9276 ± 0.0052(stat) ± 0.0220(sys).
◮ Sys errors:◮ O(αs) ⇒ 0.0177 corrections due to truncation of perturbative
analysis◮ ⇒ 0.0007 unphysical strange mass correction◮ ⇒ 0.0131 correction for use of exceptional momenta
243 × 64 result – Aoki et al [arXiv:0712.1061]
◮ Reapply RI/MOM perturbative running to ZSIBK and scale to
conventional µ = 2 GeV.
◮ Apply conversion factor ZRI/MOM
BK → ZMSBK
◮ ZMSBK (2 GeV) = 0.9276 ± 0.0052(stat) ± 0.0220(sys).
◮ Sys errors:◮ O(αs) ⇒ 0.0177 corrections due to truncation of perturbative
analysis◮ ⇒ 0.0007 unphysical strange mass correction◮ ⇒ 0.0131 correction for use of exceptional momenta
◮ Current 323 ZMSBK stat error ∼ 0.0013.
Conclusions and Outlook
243 × 64 final value and 323 outlook
◮ Combining chirally extrapolated BK with aforementioned ZBK
result
243 × 64 final value and 323 outlook
◮ Combining chirally extrapolated BK with aforementioned ZBK
result text→ BMSK (2GeV) = 0.524(10)stat(13)ren(25)sys
resulttext→mmmp[arXiv:0804.0473]
243 × 64 final value and 323 outlook
◮ Combining chirally extrapolated BK with aforementioned ZBK
result text→ BMSK (2GeV) = 0.524(10)stat(13)ren(25)sys
resulttext→mmmp[arXiv:0804.0473]
◮ Improved techniques for 323 in use; results expected soon:Watch this space!