Calculating BK , B7, B8 and Ward
identities using improved staggered
fermions
Weonjong Lee
Edinburgh 2005
Collaboration
• S. Sharpe (University of Washington, Seattle)
• W. Lee (Seoul National University)
• G. Fleming (Yale University, Jefferson Lab)
• T. Bhattacharya (Los Alamos National Lab)
R. Gupta (Los Alamos National Lab)
• G. Kilcup (Ohio State University)
• QCDSP: N. Christ, G. Liu, R. Mawhinney, L. Wu.
• SNU computing: T. Bae, K. Choi, J. Kim
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Re(A2), Re(A0), and 0/ (Exp)
• h(ππ)I |HW |K 0i = AI eiδI
• ω=
ReA2
ReA0
= 0.045
• ReA2 = 1.50 × 10−8GeV
• ReA0 = 33.3 × 10−8GeV
h
i
ImA0
ImA2
ω
−
• 0/ = − √2||
ReA0
ReA2
KTeV → Re(0/) = (20.7 ± 2.8) × 10−4
NA48 → Re(0/) = (16.7 ± 2.3) × 10−4
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Main goals of the staggered 0/
project.
• Check the results obtained using the quenched DomainWall fermions: → negative 0/ (CP-PACS, RBC).
• Check the NLO prediction in partially quenched chiral
perturbation.
• Calculate 0/ in the full QCD.
– Quenching → main systematic errors.
– Using the improved actions → HYP or Fat7.
• Comparison with experiment → new physics (?)
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Why staggered fermions?
• Advantage:
1.
2.
3.
4.
5.
Conserved U (1)V ⊗ U (1)A.
Quark mass is protected (No Residual Mass).
Computationally cheap (< 50 × 4).
Discretization error = O(a2).
Easy to improve (no extra cost).
• Disadvantage
1. Broken SU (4) Flavor (Taste) Symmetry.
2. Operator mixing matrix of 65536 × 65536.
3. NPR is impractical.
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Unimproved Staggered
Fermions
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The First Stage
Perturbative matching for unimproved staggered operators
• Penguin diagrams (Sharpe, Patel, 1993)
→ small (≈ 2%).
• Current-current diagrams (W. Lee, 2001)
→ large (≈ 100% for [P × P ][P × P ]).
• Hence, we must improve the staggered fermion action
and operators to reduce the perturbative correction.
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The Second Stage:
• Numerical study with unimproved
staggered
• Check the previous work
(Kilcup, Pekurovsky).
• Use fully one-loop matched operators.
• Nucl. Phys. B (Proc. Suppl.) 106, 311;
hep-lat/0111004.
• Large quenching uncertainty.
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Improved Staggered
Fermions
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The Third Stage:
Improving staggered fermions
• AsqTad: Fat7 + Lepage + Naik
• Fat7
• Fat7 + Lepage
• Hypercubic (HYP)
• Other possibilities: Fat7, APE
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Main goal of improvement:
Reduce perturbative corrections
• Key idea:
Suppress the taste changing interaction of high
momemtum gluons with quarks, using the Symanzik
improvement program and smearing.
• Numerical Studies find:
– Taste symmetry breaking substantially reduced.
(Hasenfratz, Knechtli)
– Largest reduction in HYP and APE.
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Bilinear Operator Renormalization
• W. Lee and S. Sharpe, PRD 66, (2002), 114501.
• Fat7 and HYP/Fat7 are the best in reducing the size of
one-loop corrections. (≤ 10% level)
• Comparable to those for Wilson and Domain Wall
Fermions.
• This leads us to choosing HYP/Fat7 for our numerical
study.
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The Fourth Stage
Study on HYP and Fat7
(W. Lee, PRD 66 (2002))
• Properties:
–
–
–
–
–
Theorem 1 SU(3) Projections.
Theorem 2 Triviality of renormalization.
Theorem 3 Multiple SU(3) Projection.
Theorem 4 Uniqueness
Theorem 5 Equivalence (HYP ↔ Fat7)
• |CHYP| |Cthin|, SU (3) Projection ↔ T.I. for doublers.
• Renormalization simplified by hAµAν i → hBµBν i
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Current-current Diagrams for
HYP/Fat7
• W. Lee and S. Sharpe,
PRD68, (2003) 054510;
hep-lat/0306016.
Finite correction to (O3)II
• HYP/Fat7 ≈ 10%.
Operators
UNIMPROVED
HYP/Fat7
[S × P ][S × P ]II
2 × (95.6 − 6CN )
2 × (19.1 − 6CH )
• Tadpole improvement:
CN = 13.159
CH = 1.4051
Operators
UNIMPROVED
HYP/Fat7
[P × P ][P × P ]II
2 × (111.3 − 2CN )
2 × (6.9 − 2CH )
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Penguin Diagrams for HYP/Fat7
• W. Lee, PRD 71 (2005), hep-lat/0409104.
• Theorem 1 (Equivalence):
At the one loop level, diagonal mixing coefficients are
identical between unimproved and improved staggered
operators of HYP(I), HYP(II), Fat7, Fat7+Lepage, and
Fat7.
• Note that AsqTad is NOT included in the list.
• Off-diagonal mixing vanishes for Fat7, HYP(II), and
Fat7.
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Numerical Study
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The Fifth Stage:
Numerical Study using HYP staggered (2003–present)
• Wilson Plaquette Action (quenched QCD)
• β = 6.0 → 1/a = 1.95 GeV
• 163 × 64 lattice → 218 confs.
• HYP Staggered Fermions (valence quarks)
• quark mass: (1/5)ms ∼ (4/5)ms
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Strange Quark Mass
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ms
• HYP Staggered Fermion, (one-loop matching, β = 6.0)
ms(M S, 2GeV ) = 101.2 ± 1.3 ± 4 M eV
• Unimproved Staggered Fermion (NPR, a → 0)
ms(M S, 2GeV ) = 106.0 ± 7.1 M eV
• Unimproved Staggered Fermion (β = 6.0)
ms(M S, 2GeV ) = 114.0 M eV
ms(M S, 2GeV ) = 84.0 M eV
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NPR
one-loop matching
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• Improvement in Renormalization Factor:
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Zm ≈ 2.5
(unimproved)
Zm ≈ 1.0
(HYP)
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BK
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BK
• HYP staggered fermions.
• U(1) Noise Source picks up only Goldstone Pion
(γ5 ⊗ ξ5).
• Set up the U(1) Noise Source at t = 0 and t = 26.
• We need a separation ∆t = 9 from the sources to
exclude the contamination from the excited states.
• The fitting range is 9 ≤ t ≤ 16.
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BK
• HYP staggered fermions.
• Chiral Perturbation with finite volume effect:
(
BK = b0 1 − 6.0
2
MK
(4πf )2
log
2
MK
(4πf )2
2 2
2
2
+ 2 −2g1(MK , 0, L) + MK g2(MK , 0, L)
f
)
2
4
+b1MK
+ b2 MK
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BK
• Fitting results in the limit of V = ∞.
BK (NDR, 2 GeV) = 0.578 ± 0.018 ± 0.042 ,
RGI
BK
= 0.806 ± 0.025 ± 0.058 ,
bRGI
= 0.314 ± 0.124 ± 0.176 .
0
• bRGI
is consistent with the 1/Nc results.
0
• BK (NDR, 2 GeV) is consistent with JLQCD results
at zero lattice spacing using unimproved staggered
femions. (OK if a2 removed but not OK if α2 removed)
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BK
• BK constrains the CKM matrix.
0
KL
1.5
0
KL
ε’/ε
π 0 νν
π 0 e+ e|Vub /Vcb|
η 1
B0 -B0
B
Xd νν
B
Xd ll
Bd
ll
A
0.5
α
γ
C
-1
(3/2)
• BK → B1
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π + νν
K+
ε
0
(3/2)
, B2
B
β
ρ
(3/2)
, B9
1
2
(3/2)
, B10
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hπ
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+
∆I
+
|Q(8,8)|K i
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Ward Identity of Q7 and Q8
• Q7, Q8 ∈ (8,8) of SU (3)L ⊗ SU (3)R
• Ward identity:
(1/2)
lim
hπ +|Q(8,8) |K +isub
mq →0
(3/2)
+
hπ |Q(8,8) |K +i
= 2
• Higher order corretions (quenched χPT):
(1/2)
hπ +|Q(8,8) |K +isub
(3/2)
hπ +|Q(8,8) |K +i
2
2
m
m
= 2 1+3
log
2
(4πf )
(4πf )2
+c1(m2) + c2(m2)2
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hπ
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+
∆I
+
|Q7 |K i
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hπ
+
∆I=3/2
+
|Q7
|K i
• Q7 ∈ (8,8) of SU (3)L ⊗ SU (3)R
• Chiral behavior (quenched χPT):
2
2
m
m
(3/2)
+
+
hπ |Q(8,8) |K i = c0 1 − 2
log
2
(4πf )
(4πf )2
+c1(m2) + c2(m2)2
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hπ
+
∆I=1/2
+
|Q7
|K isub
• Q7 ∈ (8,8) of SU (3)L ⊗ SU (3)R
• Chiral behavior (quenched χPT):
hπ
+
(1/2)
|Q(8,8) |K +isub
= c0 1 +
2
2
m
m
log
2
(4πf )
(4πf )2
+c1(m2) + c2(m2)2
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K
K
u,d,s
K
π
Eight
π
Eye
π
Eye Subtraction
X
K
u,d,s
Annihilation (I)
u,d,s
Annihilation (II)
X
K
K
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Subtraction
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Subtraction Term
• Regardless of the methods:
∂
8i r
0
lim
h0|Q(8,8)|K i = − c4B0
ms→md ∂ms
f
where B0 is defined as
m2K = B0(ms + md)
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• Size of Subtraction Term:
Eight
Eye
Sub
Sum
1/5ms
6.90(40)
15.12(86)
-8.52(47)
13.47(76)
2/5ms
0.718(34)
2.57(12)
-1.910(90)
1.372(63)
3/5ms
0.1532(62)
0.745(31)
-0.613(26)
0.286(12)
4/5ms
0.0448(16)
0.273(11)
-0.2350(93)
0.0825(30)
• Although the subtraction term is of higher order in χPT, it
is not small because the coefficient is divergent O(1/a2).
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hπ
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+
∆I
+
|Q8 |K i
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hπ
+
∆I=3/2
+
|Q8
|K i
• Q8 ∈ (8,8) of SU (3)L ⊗ SU (3)R
• Q8 is dominant in the ∆I = 3/2 contribution to 0/.
• Chiral behavior (quenched χPT):
2
2
m
m
(3/2)
+
+
hπ |Q(8,8) |K i = c0 1 − 2
log
2
(4πf )
(4πf )2
+c1(m2) + c2(m2)2
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hπ
+
∆I=1/2
+
|Q8
|K isub
• Q8 ∈ (8,8) of SU (3)L ⊗ SU (3)R
• Q8 is negligible in the ∆I = 1/2 contribution to 0/.
• Chiral behavior (quenched χPT):
hπ
+
(1/2)
|Q(8,8) |K +isub
= c0 1 +
2
2
m
m
log
2
(4πf )
(4πf )2
+c1(m2) + c2(m2)2
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Puzzle with Staggered χPT
• Staggered χPT predicts that the quenched chiral logs
are proportional to log(m2π (γ5 ⊗ 1)).
• We do not observe this behavior in our data.
• Are they absent for hπ +|Q(8,8)|K +i ???
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+
+
hπ |Q6|K isub
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hπ +|Q6|K +isub
• Q6 ∈ (8,1) of SU (3)L ⊗ SU (3)R
• Q6 dominate the ∆I = 1/2 contribution.
• Chiral behavior:
∆I=1/2
hπ +|Q6
|K +isub
m2 f 2
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= c0 + c1m2 + c2m2 log(m2)
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Lattice version of Q6
• Flavor symmetry of quenched QCD = SU (3|3)
• Flavor symmetry of full QCD = SU (3)
• Hence, one may choose a singlet in SU (3)R or a singlet
in SU (3|3)R in quenched QCD.
• The standard operator: singlet ∈ SU (3)R.
• Golterman-Pallante operator: singlet ∈ SU (3|3)R.
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∆I=1/2 +
+
hπ |Q6
|K isub
m2 f 2
Standard Operator
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Golterman-Pallante Op
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K
K
u,d,s
K
π
Eight
π
Eye
π
Eye Subtraction
X
K
u,d,s
Annihilation (I)
u,d,s
Annihilation (II)
X
K
K
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Subtraction
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0
/
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0/ (Theory)
• Standard model:
h
i
0/ = Im(Vts∗ Vtd) P (1/2) − P (3/2)
P (1/2) = r
10
X
yi(µ) < Qi >0 (µ)(1 − Ωη+η0 )
i=3
P (3/2)
10
X
r
yi(µ) < Qi >2 (µ)
=
ω i=3
GF ω
r =
2||ReA0
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0/ (Theory)
• NO contribution from hQ1iI and hQ2iI .
• P (1/2) is dominated by hQ6i.
• P (3/2) is dominated by hQ8i.
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(1/2)
P
(Standard op)
Linear fit in the chiral limit
15
5
P
(1/2)
10
0
−5
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1
2
3
4
5
6
7
Operator ID
8
9
10
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(1/2)
P
(Standard op)
chipt fit in the chiral limit
15
10
5
P
(1/2)
0
−5
−10
−15
−20
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1
2
3
4
5
6
7
Operator ID
8
9
10
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P
(1/2)
(Golterman−Pallante op)
Linear fit in the chiral limit
15
5
P
(1/2)
10
0
−5
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1
2
3
4
5
6
7
Operator ID
8
9
10
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(1/2)
P
(Golterman−Pallante op)
chipt fit in the chiral limit
40
30
P
(1/2)
20
10
0
−10
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1
2
3
4
5
6
7
Operator ID
8
9
10
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(3/2)
P
chipt fit in the chiral limit
18
16
14
12
8
P
(3/2)
10
6
4
2
0
−2
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2
3
4
5
6
7
Operator ID
8
9
10
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0/ (µ = 1/a, preliminary results)
• Standard Operator:
0/ = (−16.7 ± 17.6) × 10−4
(χPT fit)
→ consistent with RBC and CP-PACS.
• Golterman-Pallante Operator:
0/ = (+17.0 ± 14.5) × 10−4
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(χPT fit)
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Summary and Conclusion
• HYP/Fat7 resolves the long-standing problem of large
perturbative corrections.
• The perturbative calculation of matching factors for
HYP/Fat7 is extended to the current-current diagrams
and penguin diagrams.
• This allows full matching between lattice and continuum
matrix elements at the one-loop level.
• The size of one-loop corrections are under control (≈ 10
%).
• We performed a numerical study in quenched QCD
using HYP staggered fermions.
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• The main approximation is quenching and chiral
perturbation (leading order or NLO).
• The Golterman-Pallante method gives positive 0/.
• The standard method gives negative 0/.
• So uncertainty from the quenched approximation is so
large as to flip the sign of the estimate of 0/.
• We are extending the current calculation to include more
quark masses and to accumulate higher statistics (400
gauge configurations).
• In order to constrain the form of the fitting functions, we
need results of the staggered chiral perturbation.
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• We need to check the finite volume effect (1.6 fm → 2.4
fm → 3.2 fm).
• We need to extend it to weaker gauge couplings to
check the scaling violation, which is supposed to be
quite small.
• We need to calculate 0/ in partially quenched QCD
(mixed action: AsqTad sea + HYP valence).
• We need to extend numerical study to NLO in χPT.
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Calculating K → ππ on the lattice
• Calculate hπ +|Qi|K +i and h0|Qi|K 0i on the lattice
• Use χPT at leading order to obtain hπ +π −|Qi|K 0i
+
+
K →π
K0 → 0
=⇒ χP T =⇒ (K + → π +π −)
• Numerical study in quenched QCD.
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RG evolution for Nf = 3
• For Nf = 3, there is a removable singularity.
(0)
(0)
• 2β0 + γ8 − γ7 = 0 → singularity.
• Both the denomenator and numerator vanish.
• Final analytical form contains
αs(m2)
ln
αs(m1)
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,
αs(m2)
ln
αs(m1)
2
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Re(A2), Re(A0), and 0/ (Exp)
• h(ππ)I |HW |K 0i = AI eiδI
• ω=
ReA2
ReA0
= 0.045
• ReA2 = 1.50 × 10−8GeV
• ReA0 = 33.3 × 10−8GeV
h
i
ImA0
ImA2
ω
−
• 0/ = − √2||
ReA0
ReA2
KTeV → Re(0/) = (20.7 ± 2.8) × 10−4
NA48 → Re(0/) = (16.7 ± 2.3) × 10−4
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Standard model HW (Theory)
• HW =
G
√F VusVud
2
P i
zi(µ) + τ yi(µ) Qi(µ)
• Current-current: Q1, Q2 = (s̄u)L(ūd)L
P
• QCD Penguin: Q3, Q4, Q5, Q6 = (s̄d) q (q̄q)
P
• EW Penguin: Q7, Q8, Q9, Q10 = (s̄d) q eq (q̄q)
• Group representation:
Q1, Q2, Q9, Q10 ∈ (27L, 1R) + (8L, 1R)
Q3, Q4, Q5, Q6 ∈ (8L, 1R)
Q7, Q8 ∈ (8L, 8R)
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Re(A0) and Re(A2) (Theory)
• Standard model:
ReAI
10
i
h X
X
GF
= √ |VudVus|
zihQiiI + Re(τ )
yihQiiI
2
i=1,2
i=3
• hQiiI ≡ h(ππ)I |Qi|K 0i
• zi, yi ≈ 1 or α = 1/129
• Re(τ ) = −Re(λt/λu) = 0.002
• Hence, only hQ1iI and hQ2iI are important in ReAI .
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8
ReA2 X 10
Linear fit in the chiral limit
2
1.5
ReA2 X 10
8
1
0.5
0
−0.5
−1
1
2
3
4
5
6
7
Operator ID
8
9
10
ReA2 = 1.50 × 10−8GeV (Experiment), µ = 1/a
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8
ReA0 X 10
Linear fit in the chiral limit (std)
20
ReA0 X 10
8
15
10
5
0
−5
1
2
3
4
5
6
7
Operator ID
8
9
10
ReA0 = 33.3 × 10−8GeV (Experiment), µ = mc
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8
ReA0 X 10
Linear fit in the chiral limit (std)
20
ReA0 X 10
8
15
10
5
0
−5
1
2
3
4
5
6
7
Operator ID
8
9
10
ReA0 = 33.3 × 10−8GeV (Experiment), µ = 1/a
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8
ReA0 X 10
Lin−log fit in the chiral limit (std)
40
ReA0 X 10
8
30
20
10
0
−10
1
2
3
4
5
6
7
Operator ID
8
9
10
ReA0 = 33.3 × 10−8GeV (Experiment), µ = 1/a
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8
ReA0 X 10
Quadratic fit in the chiral limit (std)
30
25
ReA0 X 10
8
20
15
10
5
0
−5
1
2
3
4
5
6
7
Operator ID
8
9
10
ReA0 = 33.3 × 10−8GeV (Experiment), µ = 1/a
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