Theoretical strategies for ε'/εKaon09June 10, 2009

Norman H. Christ
RBC and UKQCD Collaborations

Outline•

Introduction•

Challenges for lattice QCD– Operator renormalization– Quadratic divergences–π – π final states[2008 RBC/UKQCD results using ChPT]– Disconnected graphs•

Outlook–Δ I = 3/2–

Δ I = 1/2

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Introduction

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Low Energy Effective Theory•

Represent weak interactions by local
four-quark Lagrangian

GF
H.S D1/ D p Vud Vus2( 10 XiD1•

Vqq′ – CKM matrix elements•zi and yi – Wilson Coefficients•

Qi – four-quark operators)

V td Vuszi ./  yi ./V ts Vud

Qi

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Four quark operators• Current-current operators
Q1  .Ns˛ d˛ /V A.uN ˇ uˇ /V A
Q2  .Ns˛ dˇ /V A.uN ˇ u˛ /V A• QCD Penguins
Q3  .Ns˛ d˛ /V A

X.qNˇ qˇ /V AqDu;d;s

Q4  .Ns˛ dˇ /V AX.qNˇ q˛ /V AqDu;d;sQ5  .Ns˛ d˛ /V AX.qNˇ qˇ /V CAqDu;d;s

Q6  .Ns˛ dˇ /V AX• Electro-Weak PenguinsX
3Q7  .Ns˛ d˛ /V A
eq .qNˇ qˇ /V CA
2qDu;d;sX
3Q8  .Ns˛ dˇ /V A
eq .qNˇ q˛ /V CA
2qDu;d;sX
3Q9  .Ns˛ d˛ /V A
eq .qNˇ qˇ /V A
2qDu;d;sX
3Q10  .Ns˛ dˇ /V A
eq .qNˇ q˛ /V A
2qDu;d;s.qNˇ q˛ /V CAqDu;d;s
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Status• The ΔI = ½ rule and ε ′/ε are long-standing
problems in particle physics.
• Accurate experimental result allows test ofstandard model CP violation.
re(ε ′/ε) = 16.8 (1.4) x 10-4
• Natural target for lattice QCD.
• Even 10-20% errors would be of great value.

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Challenges• Match lattice and continuum operators• Eye diagrams contain quadratic divergences}

Ultraviolet• Difficult π – π final state– SU(3) x SU(3) ChPT fails– Physical decay: p ~ 205 MeV– Euclidean, large time limit: p ~ 0 MeV•}Infrared

ΔI = 1/2 amplitudes require disconnectedgraphs

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ComputationalChallenges

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OperatorNormalization

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Operator Renormalization•

RI/MOM scheme, gauge-fixed off-shell Green’sfunctions.

•Earlier quenched and recent 2+1 flavor calculation
demonstrate errors ~few % errors are feasible.

•Sub-percent statistical errors possible from 5-10
configurations (Dirk Broemmel, Chris Kelley, Jan Wennekers)•Non-exceptional kinematics gives sub-percent infraredeffects at μ = 1.7 GeV.

•Largest uncertainty comes from μ = 2 GeV QCD
perturbation theory. Remove by step-scaling– Compare RI/MOM Green’s functions or Schrodinger
functional amplitudes on a sequence of ensembles withsmall physical volumes, L ~ 1/2N
– Match with continuum perturbation theory atμ = 1.7⋅ 2N GeV Æ error ~1/N
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Operator Renormalization (con’t)• Seven ∆S = 1 operators divide into threegroups which mix:
– O(27,1)– O7 and O8– O2, O3, O5, O6• Accurately handled by RI/MOM (ChrisDawson, Shu Li)• Mixing with lower dimension operators is asmall effect and easily treated.
• Effects of a single gluonic operator not yet
included.

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Operator Renormalization (con’t)
1
2
3
5
6
7
8

1
1.218(33)0.0(0.0)
0.0(0.0)
0.0(0.0)
0.0(0.0)-0.00030(89)0.0002(14)20.0(0.0)
1.062(84)0.13(20)-0.08(24)-0.64(72)0.006(20)0.052(55)30.0(0.0)
0.076(77)1.30(27)-0.03(21)-0.31(92)0.024(25)0.138(76)5
0.0(0.0)0.001(33)-0.180(89)1.00(12)-0.67(37)-0.0012(75)0.007(29)6
0.0(0.0)
0.016(29)0.120(99)0.269(93)1.97(38)-0.0074(92)-0.010(21)70.0033(53)0.0026(80)0.044(22)-0.016(23)0.130(93)1.084(26)0.060(22)8-0.0063(33)0.0026(68)-0.037(26)-0.034(24)-0.14(10)0.294(29)1.711(97)[Shu Li]
Inverse of renormalization matrix in 7 operator basis
for unitary mass 0.01 and μ = 1.92 GeV. Done with75 configurations, 163 x 32 1/a=1.73 GeV
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Operator Renormalization (con’t)(ΛV - ΛA)/((ΛV + ΛA)/2)
00.0050(ΛA - ΛV) / [(ΛA + ΛV) / 2]

(ΛA - ΛV) / [(ΛA + ΛV) / 2]

-0.02

-0.04

-0.06
ml = 0.01
ml = 0.02
ml = 0.03-0.08-0.005-0.01-0.015

ml = 0.01
ml = 0.02
ml = 0.03-0.02

chiral limit

chiral limit-0.025-0.1

00.5

11.5(ap)

22.5

00.5211.5(ap)Exceptional momenta

22.52

Nonexceptional momenta
Y. Aoki, et al

Phys.Rev.D78:054510,2008,
arXiv:0712.1061 [hep-lat]
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Quadratic Divergences

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Quadratic Divergences• Penguin diagrams
have quadraticallydivergent part ~ 1/a2
• Easily determinedand subtracted with
sub percent errors.

0.4
unsubtracted
divergent part
0.3

RBC:

subtracted

Q6

0.2

CP-PACS:
0.1• Easily controlled at
the percent level!

000.010.02
mx = mz

0.030.04

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Two pion final state

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SU(3) x SU(3) Chiral
Perturbation Theory• Use “soft-pion” methods to related K Æ ππ toKÆ π and K Æ vac.
• Earlier 2001 quenched calculations suggested
this was promising (but gave ε'/ε = ??).
• However, quenched ChPT highly unphysical(Golterman and Pallante).
• Quenched calculation now repeated in 2+1
flavor QCD again using chiral (domain wall)
fermions.

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RBC and UKQCD Collaboration
First description presented by Bob Mawhinney at
Kaon07
Tom Blum (Connecticut)

Shu Li (Columbia)

Norman Christ (Columbia)

Robert Mawhinney (Columbia)

Chris Dawson (Virginia)

Enno Scholz (BNL)
Amarjit Soni (BNL)

2+1 Flavor partially quenched chiralperturbation theory
Christopher Aubin (W&M)
Jack Laiho (St Louis)
Shu Li (Columbia)
Meifeng Lin (Columbia)
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Determination of α27
Fit to points with (mval + mres)avg ≤ 0.013
PQChPT describes this data
Large, ~100% correction!?
Same large ChPT corrections as RBC/UQKCD,arXiv:0804.0473• Fit does not work without mK mπ fK fπ division.

•••
•

1.6
ml=0.005 mx=0.001
ml=0.005 mx=0.01

1.4

ml=0.01 mx=0.001
ml=0.01 mx=0.005
1.2

ml=0.01 mx=0.01
SU(3) extrapolation

1+

<π |O

(27,1)(3/2)+| K > / mK mπ fK fπ

ml=0.005 mx=0.005

0.8

0.6

00.01

0.02mz + mres

0.03

0.04

0.05

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Relative size of LO and NLO terms• LO and NLO log terms are the same size.
• Consistent results if we divide by mKmπ ( fKfπ )2• Double the difference between two fits toestimate systematic error.
2
ml=0.005 mx=0.005
log

1.5

analytical

10.5

+

<π |O

(27,1)(3/2)+| K > / mK mπ fK fπleading order

00

0.01

0.02

mz + mres

0.03

0.04

0.05

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Determination of α6• NLO fit not possible, insufficient data to
determine 8 LEC’s.
• LO fit works well for large mass range.
• Omitted NLO logs are important!
ml=0.005 mx=0.001
ml=0.005 mx=0.005-0.02

ml=0.005 mx=0.01
ml=0.01 mx=0.005-0.025ml=0.01 mx=0.01+(1/2)+< π | Q6,latt | K >sub

ml=0.01 mx=0.001-0.03-0.03500.01

0.02
mz + mres

0.03

0.04

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Effect of NLO logs on α6••••

Chose mmax = 0.005.
Use linear fit for mmax ≤ m
Use chiral log for m ≤ mmax
Match value, slope and curvature at m = mmax
-0.01
mz = 0.001
mz = 0.005
mz = 0.01
mz = 0.02
mz = 0.03-0.02mz = 0.04+

< π | Q6,latt | K >sub(1/2)

linear422

linear + mX ln( mX / Λ )+Slope doubled
by including
NLO chiral log

-0.03

-0.04

00.02

0.04

mxz mX

0.06

0.08

0.1

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Results for LEC’s.1=2/Qi
˛i;ren1
6:6.15/.66/  1052
9:9.21/.99/  1053
0:8.31/.21/  1054 1:62.44/.162/  1045 1:52.29/.152/  1046
4:1.7/.41/  1047 1:11.17/.18/  1058 4:92.72/.75/  1059
9:8.20/.98/  10510
6:8.15/.68/  105.3=2/

˛i;ren2:48.24/.39/  106
2:47.24/.39/  1060.0
0.0
0.0
0.0
5:53.85/.91/  106
2:46.37/.37/  105
3:72.37/.59/  106
3:69.37/.59/  106

• Q1– Q6, Q9, Q10 in (GeV)4 Q7, Q8 in (GeV)6• Heroic 7-operator NPR performed.
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SU(3) x SU(3) ChPTCritique• Difficult to extrapolate to chiral limit and extractneeded LEC’s (240 MeV ≤ mπ ≤ 430 MeV)
• Highly unrealistic to then use those LEC’s toreconstruct physical 495 MeV kaon.
• Soft-pion methods are too unreliable to be used.
• While not a positive result, this reflects a major
RBC/UQKCD effort since Kaon07 and is an
important conclusion.

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Calculate π - π final state directly• Lellouch-Luscher method:– Correct normalization for mixing of different l coming fromcubic box.
– Correctly include π - π interactions and Euclidean space
Watson theorem.
– Defeat Maiani-Testa theorem by tuning finite volume so that
1st or 2nd excited state has physical relative momentum.

• Further refinements:– G-parity boundary conditions – force π - π to carry physical205 MeV momentum. (Changhoan Kim)
– Non-zero cm mass momentum adjusted to make π - πrelative momentum physical. (Takeshi Yamazaki)
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Disconnected
diagrams

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Disconnected graphs• Exponential e – Eππ t fall off produced bystochastic average rather than explicit
quark propagators• Many-source, high statistics methods
needed• Reliable signals must be extracted fromsmall time separations:– Multi-state fits
– Luscher-Wolff• A serious challenge for LQCD
• The π –π system is likely the easiest!
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Disconnected graphs
Two pion correlation functions
1e+18DC
RV0V

1e+17

1e+16Corr(t)• Current testing:– 2+1 flavors– 163 x 32
– mπ = 430 MeV

I = 0 π – π scattering

1e+151e+141e+1302468t101214•[Qi Liu]
π - π correlator•

32 wall sources, one for each t.

•

146 configurations•

12 hrs/config. at 1/8 Tflops.
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16

Outlook

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Direct calculation of K Æ π π amajor RBC/UKQCD project• Collaborators:
RBC
Tom Blum
Norman Christ
Taku Izubuchi
Changhoan Kim
Matthew Lightman
Qi LiuBob Mawhinney
Amarjit Soni

UQKCD
Dirk Broemmel
Jonathan Flynn
Elaine Goode
Chris Sachrajda• Ready to start USQCD 80 M core-hour BG/PArgonne Incite allocation:– 4.5 fm box, 1/a = 1.4 GeV, AuxDet action– mπ = 240 and 180 MeV
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Outlook Δ I = 3/2(Matthew Lightman)

Two Pion Effective Mass

24 cubed, Quenched DBW2, 79 configurations, ms=0.08 ml=0.0055

•

•

0.5

0.4
Effective Mass•

1/a=1.3 GeV,mπ = 228 MeV tests underway.
Anti-periodic d quark.
– p = 0, 170, 240, 295 MeV.
– pphys = 205 MeV– Only needed for valence d’s
Use AuxDet large volume lattices– mres = 0.0018/a ~ 3 MeV– 1/a = 1.4 GeV– L = 4.5 fm– mπ = 180 and 240 MeV
Computing re A2 and im A2– ~15% accuracy– Practical 2-year goal

0.30.20.100102030405060tπ–πK
K->PiPi Effective Mass
24 cubed, Quenched DBW2, 79 configurations, ms=0.08 ml=0.0055

0.4

0.2

Effective Mass•

0.6

Quenched 243 x 64,

0-0.2-0.40A2= 2.17(12) 10-8 GeV, p = 010

2030

405060tKaon09, June 10, 2009(31)Outlook: Δ I = ½(Qi Liu)
1e+18– mπ = 427 MeV

1e+17Corr(t)• 2+1 flavor, 163 x 32experiments underway:

isospin0– 1st π – π scattering

1e+161e+15– 2nd K Æ π π

1e+140– Eigenmode projection + CG(Ran Zhou)

46810

121416t
isospin0 correlation function
1e+18

isospin0V
difference

1e+17

Corr(t)connected graphs

disconnected graph

2

1e+16

1e+15

1e+1402

4

68t

10

1214Kaon09, June 10, 200916(32)

Outlook: Δ I = ½(Qi Liu)
1.4
1.2

– Improved short-time resolution– More efficient inversions

Without disconnectgraphs

10.8
0.6

– Non-zero cm momentum– G-parity boundary conditions

0.4
0.2
0024

68t

10

121416f(x)isospin0

1.4
1.2

With disconnect
graphs10.8m• Complete K Æ π π code
written and first 83 x 12calculations underway.
• 163 x 32 Æ 323 x 64 requires:f(x)isospin0Vm• Disconnected graphs introduce
large errors into π – πscattering for t ≥ 5• Non-zero momentum:

π – π effective mass

0.6
0.4
0.2
0024

6810

121416t

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Conclusion• Calculation of re A2 and im A2 to ~15% a
realistic 1 - 2 year goal
• re A0 and im A0 more difficult– Theoretical issues are resolved.
– Disconnected diagrams easiest in this π – π case.
– Next generation of computer hardware likely neededfor definitive results: Next generation IBM BG/?
machine should be sufficient!

• Expect 20% result for Δ I = ½ rule and ε'/ε in~3 years!

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