chris kelly university of edinburgh rbc & ukqcd collaborations · bk chiral ﬁt forms we use...

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


Ensemble details

Outline


Ensemble details

Measurement of BK

Outline


Ensemble details

Measurement of BK

The chiral extrapolation of BK

Outline


Ensemble details

Measurement of BK


The non-perturbative renormalisation of BK

Outline


Ensemble details

Measurement of BK



Conclusions and Outlook

Kaon mixing

◮ Indirect CP violation in neutral kaon sector

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 ≡〈K 0|OVV+AA|K̄

0〉83 f 2

KM2K

BK



0〉83 f 2

KM2K

OVV+AA = (s̄γµd)(s̄γµd) + (s̄γ5γµd)(s̄γ5γµd)

BK



0〉83 f 2

KM2K


◮ BK related to measure of indirect CP violation ǫK = KL→(ππ)KS→(ππ)

you →relation contains unknown direct CP violatingparameters.

BK



0〉83 f 2

KM2K


◮ 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


243 × 64

◮ 2+1f domain wall fermionensemble with Ls = 16

323 × 64



243 × 64


◮ Iwasaki gauge actionβ = 2.13

323 × 64



243 × 64



323 × 64




243 × 64



◮ a−1 = 1.729(28) GeV →(2.74 fm)3 lattice volume

323 × 64




243 × 64




323 × 64



◮ a−1 = 2.42(4) 0.47r0 (fm) GeV →

(2.61 fm)3 lattice volume


243 × 64




◮ Strange sea quark mass 0.04lattice units

323 × 64



◮ a−1 = 2.42(4) 0.47r0 (fm) GeV →



243 × 64





323 × 64



◮ a−1 = 2.42(4) 0.47r0 (fm) GeV →



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


◮ Use p + a boundaryconditions → removesunwanted round-the-worldcontributions.

323 × 64

Method comparison

243 × 64



◮ Costs 4 inversions perconfiguration.

323 × 64

Method comparison

243 × 64




323 × 64

◮ 1 gauge-fixed wall source att = 0

Method comparison

243 × 64




323 × 64


◮ Use p + a and p − a

boundary conditions → givesforwards and backwardspropagating quarks.

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

BK chiral fit forms

◮ We use NLO SU(2) × SU(2) partially-quenched chiralperturbation theory (PQChPT) for maximum use ofensembles.

BK chiral fit forms


◮ Kaon sector is coupled to SU(2) soft pion loops at lowestorder in non-relativistic expansion

BK chiral fit forms


◮ Kaon sector is coupled to SU(2) soft pion loops at lowestorder in non-relativistic expansiontext→ direct connection to HMχPT

BK chiral fit forms


◮ 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

) ]


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


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


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

) ]



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

) ]



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

) ]


◮ Use simultaneous pure SU(2) × SU(2) PQChPT fit (nocoupling to Kaon sector) to FPS and MPS to determine B andf (E. Scholz)


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

) ]


◮ 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


◮ 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



◮ 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



◮ 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

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


◮ At high momenta, ΛA = ΛV =Zq

ZA=

Zq

ZVshould hold.

Bilinear vertices



ZA=

Zq

ZVshould hold.

◮ Therefore can use ΛA or ΛV as a measure ofZq

ZA.

Bilinear vertices



ZA=

Zq

ZVshould hold.


ZA.

◮ ΛA and ΛV expected to differ at low momenta due to QCDspontaneous chiral symmetry breaking.

Bilinear vertices



ZA=

Zq

ZVshould hold.


ZA.


◮ However, even at high momenta we find ΛA 6= ΛV at 2% level

Bilinear vertices



ZA=

Zq

ZVshould hold.


ZA.


◮ However, even at high momenta we find ΛA 6= ΛV at 2% leveltext→ difference caused by kinematic choice : Exceptionalmomentum configuration

Bilinear vertices



ZA=

Zq

ZVshould hold.


ZA.


◮ 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



◮ p2 → ∞ behaviour governed by subgraphwith least negative degree of divergencethrough which we can route all hard

external momenta.




external momenta.

◮ For p1 6= p2 this is the entire graph.




external momenta.


◮ Can connect to low-energy subgraphswhich are affected by spont. chiralsymmetry breaking, but:




external momenta.


◮ Can connect to low-energy subgraphswhich are affected by spont. chiralsymmetry breaking, but:text→ Low energy subgraph notcontained within circled subgraph




external momenta.


◮ 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


◮ 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




◮ 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.



◮ 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





ZVV+AA

Z 2q

ΛOVV+AA= Otree

VV+AA

◮ DefineZ

RI/MOM

BK ≡ ZVV+AA

Z2A

=(

Z2q

Z2A

)

ZVV+AA

Z2q





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


◮ Average over 4 sourcelocations on 75configurations on ourml = 0.03, 0.02 and 0.01ensembles.

323 × 64

Method comparison

163 × 32



◮ Momentum applied byapplying phase differencebetween propagator sourceand sink. Solution can begiven arbitrary momentum.

323 × 64

Method comparison

163 × 32




323 × 64

◮ Use lattice volume sources,vertex formed at propagatorsink. (D. Broemmel)

Method comparison

163 × 32




323 × 64


◮ Average over all sinklocations, lattice volumefactor gain over pointapproach.

Method comparison

163 × 32




323 × 64


◮ 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




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


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


◮ 323 lever-arm forextrapolation smallcompared to extrapolationdistance


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


◮ 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



◮ 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




◮ 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




◮ 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.




◮ Apply conversion factor ZRI/MOM

BK → ZMSBK





BK → ZMSBK

◮ ZMSBK (2 GeV) = 0.9276 ± 0.0052(stat) ± 0.0220(sys).





BK → ZMSBK


◮ Sys errors:





BK → ZMSBK


◮ Sys errors:◮ O(αs) ⇒ 0.0177 corrections due to truncation of perturbative

analysis





BK → ZMSBK



analysis◮ ⇒ 0.0007 unphysical strange mass correction





BK → ZMSBK



analysis◮ ⇒ 0.0007 unphysical strange mass correction◮ ⇒ 0.0131 correction for use of exceptional momenta





BK → ZMSBK



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



result text→ BMSK (2GeV) = 0.524(10)stat(13)ren(25)sys

resulttext→mmmp[arXiv:0804.0473]



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!

chris kelly university of edinburgh rbc & ukqcd collaborations · bk chiral ﬁt forms we use...

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