chris kelly university of edinburgh rbc & ukqcd collaborations · bk chiral fit forms we use...
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![Page 1: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/1.jpg)
BK for 2+1 flavour domain wall fermions from243 and 323 × 64 lattices
Chris Kelly
University of EdinburghRBC & UKQCD Collaborations
Mon 14th July
![Page 2: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/2.jpg)
Outline
The neutral kaon mixing amplitude BK
![Page 3: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/3.jpg)
Outline
The neutral kaon mixing amplitude BK
Ensemble details
![Page 4: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/4.jpg)
Outline
The neutral kaon mixing amplitude BK
Ensemble details
Measurement of BK
![Page 5: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/5.jpg)
Outline
The neutral kaon mixing amplitude BK
Ensemble details
Measurement of BK
The chiral extrapolation of BK
![Page 6: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/6.jpg)
Outline
The neutral kaon mixing amplitude BK
Ensemble details
Measurement of BK
The chiral extrapolation of BK
The non-perturbative renormalisation of BK
![Page 7: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/7.jpg)
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
![Page 8: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/8.jpg)
The neutral kaon mixing amplitude BK
![Page 9: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/9.jpg)
Kaon mixing
◮ Indirect CP violation in neutral kaon sector
![Page 10: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/10.jpg)
Kaon mixing
◮ Indirect CP violation in neutral kaon sector
◮ Neutral kaon mixing amplitude:
![Page 11: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/11.jpg)
Kaon mixing
◮ Indirect CP violation in neutral kaon sector
◮ Neutral kaon mixing amplitude:
![Page 12: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/12.jpg)
Kaon mixing
◮ Indirect CP violation in neutral kaon sector
◮ Neutral kaon mixing amplitude:
![Page 13: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/13.jpg)
Kaon mixing
◮ Indirect CP violation in neutral kaon sector
◮ Neutral kaon mixing amplitude:
![Page 14: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/14.jpg)
BK
◮ BK parameterises the matrix element of the four-quarkK 0 → K̄ 0 operator
![Page 15: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/15.jpg)
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
![Page 16: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/16.jpg)
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)
![Page 17: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/17.jpg)
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.
![Page 18: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/18.jpg)
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.
![Page 19: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/19.jpg)
Ensemble details
![Page 20: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/20.jpg)
Details of ensembles
243 × 64 323 × 64
![Page 21: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/21.jpg)
Details of ensembles
243 × 64
◮ 2+1f domain wall fermionensemble with Ls = 16
323 × 64
◮ 2+1f domain wall fermionensemble with Ls = 16
![Page 22: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/22.jpg)
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
![Page 23: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/23.jpg)
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
![Page 24: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/24.jpg)
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
![Page 25: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/25.jpg)
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
![Page 26: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/26.jpg)
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
![Page 27: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/27.jpg)
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
![Page 28: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/28.jpg)
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
![Page 29: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/29.jpg)
Measurement of BK
![Page 30: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/30.jpg)
Method comparison
243 × 64 323 × 64
![Page 31: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/31.jpg)
Method comparison
243 × 64
◮ 2 gauge-fixed wall sources att = 5, 59 for propagators
323 × 64
![Page 32: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/32.jpg)
Method comparison
243 × 64
◮ 2 gauge-fixed wall sources att = 5, 59 for propagators
◮ Use p + a boundaryconditions → removesunwanted round-the-worldcontributions.
323 × 64
![Page 33: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/33.jpg)
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
![Page 34: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/34.jpg)
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
![Page 35: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/35.jpg)
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.
![Page 36: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/36.jpg)
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.
![Page 37: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/37.jpg)
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
![Page 38: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/38.jpg)
The chiral extrapolation of BK
![Page 39: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/39.jpg)
BK chiral fit forms
◮ We use NLO SU(2) × SU(2) partially-quenched chiralperturbation theory (PQChPT) for maximum use ofensembles.
![Page 40: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/40.jpg)
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
![Page 41: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/41.jpg)
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
![Page 42: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/42.jpg)
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).
![Page 43: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/43.jpg)
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
) ]
![Page 44: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/44.jpg)
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
![Page 45: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/45.jpg)
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
![Page 46: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/46.jpg)
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
![Page 47: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/47.jpg)
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
![Page 48: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/48.jpg)
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
![Page 49: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/49.jpg)
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
![Page 50: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/50.jpg)
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)
![Page 51: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/51.jpg)
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
![Page 52: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/52.jpg)
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
![Page 53: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/53.jpg)
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
![Page 54: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/54.jpg)
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
![Page 55: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/55.jpg)
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
![Page 56: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/56.jpg)
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
![Page 57: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/57.jpg)
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
![Page 58: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/58.jpg)
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
![Page 59: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/59.jpg)
243 × 64 BK chiral limit results – Allton et al
[arXiv:0804.0473]
◮ 243 × 64 B latK = 0.565(10).
![Page 60: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/60.jpg)
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
![Page 61: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/61.jpg)
The non-perturbative renormalisation of BK
![Page 62: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/62.jpg)
Why NPR?
◮ Lattice perturbative (DWF) calculations exist but:
![Page 63: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/63.jpg)
Why NPR?
◮ Lattice perturbative (DWF) calculations exist but:◮ exist only at low order
![Page 64: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/64.jpg)
Why NPR?
◮ Lattice perturbative (DWF) calculations exist but:◮ exist only at low order◮ are poorly convergent
![Page 65: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/65.jpg)
Why NPR?
◮ Lattice perturbative (DWF) calculations exist but:◮ exist only at low order◮ are poorly convergent◮ involve prescription dependent ambiguities such as MF
improvement
![Page 66: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/66.jpg)
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
![Page 67: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/67.jpg)
Bilinear vertices
◮ ZV = ZA due to good chiral symmetry
![Page 68: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/68.jpg)
Bilinear vertices
◮ ZV = ZA due to good chiral symmetry
◮ At high momenta, ΛA = ΛV =Zq
ZA=
Zq
ZVshould hold.
![Page 69: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/69.jpg)
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.
![Page 70: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/70.jpg)
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.
![Page 71: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/71.jpg)
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
![Page 72: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/72.jpg)
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
![Page 73: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/73.jpg)
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.
![Page 74: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/74.jpg)
Chiral symmetry breaking and exceptional momenta
◮ Generic bilinear vertex graph
![Page 75: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/75.jpg)
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.
![Page 76: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/76.jpg)
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.
![Page 77: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/77.jpg)
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:
![Page 78: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/78.jpg)
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
![Page 79: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/79.jpg)
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
![Page 80: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/80.jpg)
Chiral symmetry breaking and exceptional momenta
◮ However in case p2 − p1 = 0 then highmomenta do not enter internal subgraphs
![Page 81: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/81.jpg)
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
![Page 82: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/82.jpg)
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
![Page 83: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/83.jpg)
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
![Page 84: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/84.jpg)
BK NPR with RI/MOM and exceptional momenta
◮ Calculate four-quark vertex matrix element in Landau gauge.
![Page 85: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/85.jpg)
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
![Page 86: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/86.jpg)
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
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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
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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
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Method comparison
243 × 32 323 × 64
![Page 90: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/90.jpg)
Method comparison
163 × 32
◮ Use point sources, 4 quarkvertex formed at sourcelocation.
323 × 64
![Page 91: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/91.jpg)
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
![Page 92: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/92.jpg)
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
![Page 93: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/93.jpg)
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)
![Page 94: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/94.jpg)
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.
![Page 95: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/95.jpg)
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.
![Page 96: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/96.jpg)
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
![Page 97: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/97.jpg)
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
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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
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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
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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
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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
![Page 102: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/102.jpg)
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
![Page 103: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/103.jpg)
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
![Page 104: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/104.jpg)
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
![Page 105: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/105.jpg)
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
![Page 106: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/106.jpg)
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
![Page 107: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/107.jpg)
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
![Page 108: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/108.jpg)
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
![Page 109: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/109.jpg)
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
![Page 110: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/110.jpg)
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
![Page 111: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/111.jpg)
243 × 64 result – Aoki et al [arXiv:0712.1061]
◮ Reapply RI/MOM perturbative running to ZSIBK and scale to
conventional µ = 2 GeV.
![Page 112: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/112.jpg)
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
![Page 113: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/113.jpg)
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).
![Page 114: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/114.jpg)
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:
![Page 115: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/115.jpg)
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
![Page 116: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/116.jpg)
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
![Page 117: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/117.jpg)
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
![Page 118: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/118.jpg)
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.
![Page 119: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/119.jpg)
Conclusions and Outlook
![Page 120: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/120.jpg)
243 × 64 final value and 323 outlook
◮ Combining chirally extrapolated BK with aforementioned ZBK
result
![Page 121: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/121.jpg)
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]
![Page 122: Chris Kelly University of Edinburgh RBC & UKQCD Collaborations · BK chiral fit forms We use NLO SU(2) ×SU(2) partially-quenched chiral perturbation theory (PQChPT) for maximum](https://reader035.vdocument.in/reader035/viewer/2022081522/5f144f19f39fc56de07a6cef/html5/thumbnails/122.jpg)
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!