Post on 13-May-2020
Chiral behavior of pion properties from lattice QCD
Alfonso Sastre
CPT, Marseille
Budapest-Marseille-Wuppertal collaboration
Mainz, August 2, 2013
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 1 / 32
Motivation
SU(3)c + two massless quarks (u,d)
SU(2)R × SU(2)L × U(1)V −−−→SχSB
SU(2)V × U(1)V3 Goldstone Bosons (π±, π0)
Real World: Quarks masses are small but different from zero
Pseudo Goldstone Bosons (PGB) with Mπ > 0
Chiral Perturbation Theory (χPT )
Effective theory of PGBs.
X Perturbative expansion in term s of PGB momenta and masses.
× Infinite new parameters.
X Only a finite number at each order.
X All these parameters are determined by QCD.
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 2 / 32
Motivation
SU(3)c + two massless quarks (u,d)
SU(2)R × SU(2)L × U(1)V −−−→SχSB
SU(2)V × U(1)V3 Goldstone Bosons (π±, π0)
Real World: Quarks masses are small but different from zero
Pseudo Goldstone Bosons (PGB) with Mπ > 0
Chiral Perturbation Theory (χPT )
Effective theory of PGBs.
X Perturbative expansion in term s of PGB momenta and masses.
× Infinite new parameters.
X Only a finite number at each order.
X All these parameters are determined by QCD.
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 2 / 32
SU(2) Chiral Perturbation Theory (χPT)Gasser,Leutwyler Nucl.Phys.B(85), Ann.Phys.158(84)
Leading Order (LO) O(p2)
F = limmq→0
Fπ and B = limmq→0
〈0| qq |0〉F 2π
Next to Leading Order (NLO) O(p4)
Seven new terms in the Lagrangian O(p4): li , i = 1, .., 7
Next to Next to Leading Order (NNLO) O(p6)
Many other terms ....
Are we able to compute these numbers from QCD?
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 3 / 32
Different approaches to LEC determination.
Nf = 2
ETM, MILC, JLQCD/TWQCD, CERN, HHS.
Nf = 2 + 1
MILC, JLQCD/TWQCD, RBC/UKQCD, PACS-CS, BMWc.
Necessary to fix the ms dependence.
Nf = 2 + 1 + 1
ETM
Necessary to fix the ms ,mc dependence.
Results are reviewed by FLAG Eur.Phys.J. C71 (2011) 1695
Apologies if I forget someone
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 4 / 32
Quark-mass dependence of M2π and FπColangelo,Gasser,Leutwyler Nucl.Phys.B603(01)
x-expansion(x = M2
(4πF )2
), M2 = 2Bmud
M2π = M2
(1− x
2log
(Λ2
3
M2
)+ x2
(17
8log(
Λ2M
M2) + kM
)+O(x3)
)Fπ = F
(1 + x log
(Λ2
4
M2
)+ x2
(−5
4log(
Λ2F
M2) + kF
)+O(x3)
)log
(Λ2M
M2
)=
1
51
(4
(7 log
(Λ2
1
M2
)+ 8 log
(Λ2
2
M2
))− 9 log
(Λ2
3
M2
)+ 49
)log
(Λ2F
M2
)=
1
30
(2
(7 log
(Λ2
1
M2
)+ 8 log
(Λ2
2
M2
))+ 6
(log
(Λ2
3
M2
)− log
(Λ2
4
M2
))+ 23
)
li ≡ log
(Λ2i
M2π
)
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 5 / 32
Quark-mass dependence of M2π and FπColangelo,Gasser,Leutwyler Nucl.Phys.B603(01)
ξ-expansion(ξ =
M2π
(4πFπ)2
)M2 = M2
π
(1 +
ξ
2log
(Λ2
3
M2π
)+ ξ2
(log(
Ω2M
M2π
) + κM
)+O(ξ3)
)
F = Fπ
(1− ξ log
(Λ2
4
M2
)+ ξ2
(log(
Ω2F
M2π
) + κF
)+O(ξ3)
)
log
(Ω2
M
M2
)=
1
15
(−(
4 log
(7
Λ21
M2
)+ 8 log
(Λ2
2
M2
))− 33 log
(Λ2
3
M2
)− 12 log
(Λ2
4
M2
)+ 52
)
log
(Ω2
F
M2
)=
1
3
(−(
7 log
(Λ2
1
M2
)+ 8 log
(Λ2
2
M2
))+ 18 log
(Λ2
4
M2
)−
29
2
)
li ≡ log
(Λ2i
M2π
)
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 6 / 32
BMWc 2HEX. Action
Gauge Action Luscher, Weisz Phys.Lett.B158(85)
SG =1
β
c0
3
∑plaq
ReTr (1− Uplaq) +c1
3
∑rect
ReTr(1− Urect)
Fermionic action Sheikholeslami, Wohlert Nucl.Phys.B259(85)
SF = SWilson −cSW
2
∑x
∑µ<ν
(ψσµνFµν [V ]ψ
)(x)
where σµν = i2 [γµ, γν ] and ci are set to their tree level values.
V refers to links which have undergone two HEX smearing.
Hasenfratz, Knechtli Phys.Rev.D64(01)Morningstar, Peardon Phys.Rev.D69(04)
Capitani, Durr, Holbling JHEP11(06)
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 7 / 32
BMWc 2HEX. Landscape BMWc Phys.Lett.B701(11),JHEP 1108(11)
00 2002
3002
4002
5002
6002
Mπ
2 [MeV
2]
0
0.0502
0.1002
0.1502
a2 [
fm2]
BMWc ’08BMWc ’10
1002
2002
3002
4002
5002
6002
Mπ
2 [MeV
2]
2
3
4
5
6
7
L[f
m]
0.1%
BMWc ’08BMWc ’10
0.3%
1%
⇒ 47 points at five different lattice spacing: a−1 ∼ (1.697, 2.131, 2.561, 3.026, 3.662) GeV.
⇒ Pion masses around the physical point: MMINπ ∼ (136, 131, 120, 182, 219) MeV.
⇒ Spacial volumes as large as ∼ (6 fm)3 and temporal extents up ∼ 8 fm.
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 8 / 32
What do we need from the lattice to compute LEC?
Inputs for χPT theory
(amud), (aMπ), (aFπ)
Strange mass corrections
(aMSS)
Scale Setting
lattice spacing: a
Renormalization
ZS and ZA
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 9 / 32
Quark masses
Improved ratio-different method BMWc Phys.Lett.B701(11),JHEP 1108(11)
r =amPCAC
s
amPCACud
, d = ambares − ambare
ud
r imp = r , d imp = d
(1− d
2
r + 1
r − 1
)amscheme
ud =1
Z schemeS
d imp
r imp − 1, amscheme
s =1
Z schemeS
r impd imp
r imp − 1
amPCAC1 + amPCAC
2 =
∑~x 〈∂µ[Aµ(x)]O(0)〉∑
~x 〈P(x)O(0)〉
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 10 / 32
Mπ and Fπ constant from the Lattice
Mπ and Fπ are extracted from the combined fit of correlators∑~x
⟨(dγ5u)(~x , t)(uγ5d)(0)
⟩= CPP cosh
((aMπ)
(T
2− t
))∑~x
⟨(dγ0γ5u)(~x , t)(uγ5d)(0)
⟩= CAP sinh
((aMπ)
(T
2− t
))
Fπ
(aFπ) =CAP√
2(aMπ)CPP
e(aMπ)T
4
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 11 / 32
Strange quark effects and scale setting
M2SS ≡ 2M2
k −M2π∑
~x
〈(sγ5u)(~x , t)(uγ5s)(0)〉 = C ′PP cosh
((aMK )
(T
2− t
))
Scale settingMπMΩ, MK
MΩ→ Experimental value
Renormalization ZS and ZA
RI/MOM scheme Martinelli, Pittori, Sachrajda, Testa, Vladikas Nucl.Phys.B445(95)
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 12 / 32
Fitting procedure
Chiral Fits
LECs are computed by a fully correlated combined fit:
x-expansion: Bπ ≡ M2π
2mudand Fπ in terms of mud
ξ-expansion: B−1π ≡ 2mud
M2π
and Fπ in terms of M2π
Statistical errors:
2000 bootstrap samples are used in each fit
Systematics errors: Final results: 2× 2× 3× 6× 2× 2 = 288
2 different time fitting ranges for correlators.
2 Mπ cuts for interpolating scale fixing quantity MΩ (380/480 MeV).
3 ZA determinations and 6 ZS determinations.
2 Mπ cuts for chiral fits (250/300 MeV).
2 chiral expansions: x, ξ
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 13 / 32
Typical Fit x-expansion
90
100
110
120
130
140
0 20 40 60 80 100
Fπ[M
eV]
mRGIud [MeV]
β=3.5β=3.61β=3.7β=3.8
χ2
#dof = 1.52, #dof = 17, p-value = 0.08Exp value (NOT INCLUDED)
1700
1750
1800
1850
1900
1950
2000
2050
2100
0 20 40 60 80 100
BRGI
π[M
eV]
mRGIud [MeV]
β=3.5β=3.61β=3.7β=3.8
χ2
#dof = 1.52, #dof = 17, p-value = 0.08
90
100
110
120
130
140
0 20 40 60 80 100
Fπ[M
eV]
mRGIud [MeV]
β=3.5β=3.61β=3.7β=3.8
χ2
#dof = 1.16, #dof = 48, p-value = 0.21Exp value (NOT INCLUDED)
1700
1750
1800
1850
1900
1950
2000
2050
2100
0 20 40 60 80 100
BRGI
π[M
eV]
mRGIud [MeV]
β=3.5β=3.61β=3.7β=3.8
χ2
#dof = 1.16, #dof = 48, p-value = 0.21
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 14 / 32
Typical Fit x-expansion
90
100
110
120
130
140
0 20 40 60 80 100
Fπ[M
eV]
mRGIud [MeV]
β=3.5β=3.61β=3.7β=3.8
χ2
#dof = 1.52, #dof = 17, p-value = 0.08Exp value (NOT INCLUDED)
1700
1750
1800
1850
1900
1950
2000
2050
2100
0 20 40 60 80 100
BRGI
π[M
eV]
mRGIud [MeV]
β=3.5β=3.61β=3.7β=3.8
χ2
#dof = 1.52, #dof = 17, p-value = 0.08
90
100
110
120
130
140
0 20 40 60 80 100
Fπ[M
eV]
mRGIud [MeV]
β=3.5β=3.61β=3.7β=3.8
χ2
#dof = 1.16, #dof = 48, p-value = 0.21Exp value (NOT INCLUDED)
1700
1750
1800
1850
1900
1950
2000
2050
2100
0 20 40 60 80 100
BRGI
π[M
eV]
mRGIud [MeV]
β=3.5β=3.61β=3.7β=3.8
χ2
#dof = 1.16, #dof = 48, p-value = 0.21
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 14 / 32
Typical Fit ξ-expansion
90
100
110
120
130
140
0 50000 100000 150000 200000 250000 300000 350000
Fπ[M
eV]
M2π [MeV 2]
β=3.5β=3.61β=3.7β=3.8
χ2
#dof = 1.54, #dof = 17, p-value = 0.07Exp value (NOT INCLUDED)
0.00048
0.0005
0.00052
0.00054
0.00056
0.00058
0 50000 100000 150000 200000 250000 300000 350000
(BRGI
π)−
1[M
eV−1]
M2π [MeV 2]
β=3.5β=3.61β=3.7β=3.8
χ2
#dof = 1.54, #dof = 17, p-value = 0.07
90
100
110
120
130
140
0 50000 100000 150000 200000 250000 300000 350000
Fπ[M
eV]
M2π [MeV 2]
β=3.5β=3.61β=3.7β=3.8
χ2
#dof = 1.19, #dof = 48, p-value = 0.18Exp value (NOT INCLUDED)
0.00048
0.0005
0.00052
0.00054
0.00056
0.00058
0 50000 100000 150000 200000 250000 300000 350000
(BRGI
π)−
1[M
eV−1]
M2π [MeV 2]
β=3.5β=3.61β=3.7β=3.8
χ2
#dof = 1.19, #dof = 48, p-value = 0.18
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 15 / 32
Typical Fit ξ-expansion
90
100
110
120
130
140
0 50000 100000 150000 200000 250000 300000 350000
Fπ[M
eV]
M2π [MeV 2]
β=3.5β=3.61β=3.7β=3.8
χ2
#dof = 1.54, #dof = 17, p-value = 0.07Exp value (NOT INCLUDED)
0.00048
0.0005
0.00052
0.00054
0.00056
0.00058
0 50000 100000 150000 200000 250000 300000 350000
(BRGI
π)−
1[M
eV−1]
M2π [MeV 2]
β=3.5β=3.61β=3.7β=3.8
χ2
#dof = 1.54, #dof = 17, p-value = 0.07
90
100
110
120
130
140
0 50000 100000 150000 200000 250000 300000 350000
Fπ[M
eV]
M2π [MeV 2]
β=3.5β=3.61β=3.7β=3.8
χ2
#dof = 1.19, #dof = 48, p-value = 0.18Exp value (NOT INCLUDED)
0.00048
0.0005
0.00052
0.00054
0.00056
0.00058
0 50000 100000 150000 200000 250000 300000 350000
(BRGI
π)−
1[M
eV−1]
M2π [MeV 2]
β=3.5β=3.61β=3.7β=3.8
χ2
#dof = 1.19, #dof = 48, p-value = 0.18
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 15 / 32
Fit quality in terms of Mmaxπ
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
10
200 250 300 350 400 450 500 550 600
p-value
Mmaxπ (MeV)
NLO x-expansion
NLO ξ-expansion
NNLO x-expansion
NNLO ξ-expansion
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 16 / 32
LO LEC in terms of Mmaxπ
84
86
88
90
92
94
F[M
eV]
NLO x
NLO ξ
NNLO x
NNLO ξ
-4
-2
0
2
4
200 250 300 350 400 450 500 550 600
∆F
[MeV
]
Mmaxπ [MeV]
1800
1850
1900
1950
2000
2050
2100
BRGI[M
eV]
NLO x
NLO ξ
NNLO x
NNLO ξ
-50
0
50
100
150
200 250 300 350 400 450 500 550 600
∆BRGI[M
eV]
Mmaxπ [MeV]
2
2.5
3
3.5
4
4.5
5
5.5
6
l 4
NLO x
NLO ξ
NNLO x
NNLO ξ
-3-2-10123
200 250 300 350 400 450 500 550 600
∆l 4
Mmaxπ [MeV]
0
1
2
3
4
5
6
7
8
l 3
NLO x
NLO ξ
NNLO x
NNLO ξ
-4
-2
0
2
4
200 250 300 350 400 450 500 550 600
∆l 3
Mmaxπ [MeV]
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 17 / 32
LO LEC in terms of Mmaxπ
84
86
88
90
92
94
F[M
eV]
NLO x
NLO ξ
NNLO x
NNLO ξ
-4
-2
0
2
4
200 250 300 350 400 450 500 550 600
∆F
[MeV
]
Mmaxπ [MeV]
1800
1850
1900
1950
2000
2050
2100
BRGI[M
eV]
NLO x
NLO ξ
NNLO x
NNLO ξ
-50
0
50
100
150
200 250 300 350 400 450 500 550 600
∆BRGI[M
eV]
Mmaxπ [MeV]
2
2.5
3
3.5
4
4.5
5
5.5
6
l 4
NLO x
NLO ξ
NNLO x
NNLO ξ
-3-2-10123
200 250 300 350 400 450 500 550 600
∆l 4
Mmaxπ [MeV]
0
1
2
3
4
5
6
7
8
l 3
NLO x
NLO ξ
NNLO x
NNLO ξ
-4
-2
0
2
4
200 250 300 350 400 450 500 550 600
∆l 3
Mmaxπ [MeV]
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 17 / 32
Fπ and other interesting quantities in terms of Mπ
88
90
92
94
96
98
Fπ[M
eV]
NLO x
NLO ξ
NNLO x
NNLO ξ
-4
-2
0
2
4
200 250 300 350 400 450 500 550 600
∆Fπ[M
eV]
Mmaxπ [MeV]
1.03
1.04
1.05
1.06
1.07
1.08
Fπ/F
[MeV
]
NLO x
NLO ξ
NNLO x
NNLO ξ
-0.03-0.02-0.01
00.010.020.03
200 250 300 350 400 450 500 550 600
∆Fπ/F
Mmaxπ [MeV]
235
240
245
250
255
260
ΣRGI[M
eV]
NLO x
NLO ξ
NNLO x
NNLO ξ
-8-6-4-202468
200 250 300 350 400 450 500 550 600
∆ΣRGI[M
eV]
Mmaxπ [MeV]
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 18 / 32
LECs from x,ξ expansions
ξ-expansion x-expansion
LO
BRGI [GeV] 1.93± 0.05± 0.02 1.93± 0.05± 0.01F [MeV] 87.9± 1.4± 0.3 88.1± 1.3± 0.3
[ΣRGI ]1/3 [MeV] 246± 4± 1 247± 3± 1
NLO
l3 2.6± 0.6± 0.3 2.3± 0.4± 0.2l4 4.0± 0.4± 0.1 3.7± 0.3± 0.1
Other quantities
Fπ [MeV] 92.9± 0.9± 0.2 92.9± 0.9± 0.2Fπ/F 1.057± 0.007± 0.001 1.054± 0.006± 0.001
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 19 / 32
Combined results
Combined FLAG + PDG
LO
BRGI [GeV] 1.93± 0.05± 0.01F [MeV] 88.0± 1.3± 0.3 86.4± 0.7
[ΣRGI ]1/3 [MeV] 246± 4± 1 246± 14
NLO¯3 2.5± 0.5± 0.3 3.2± 0.7
¯4 3.8± 0.4± 0.2 4.4± 0.6
Other quantities
Fπ [MeV] 92.9± 0.9± 0.2 92.2± 0.02± 0.14Fπ/F 1.057± 0.007± 0.003 1.067± 0.009
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 20 / 32
Conclusion and outlook
SU(2) χPT has been compared to lattice QCD simulations all theway down to the physical pion mass point [see also Borsanyi et al,
arXiv:1205.0788 for an analysis with staggered fermion simulations]
Results for both x and ξ expansions look promising:
→ Within our O(1%) statisttical errors, NLO SU(2) χPT describes theFπ and M2
π well up to Mπ ≈ 300 MeV→ LO and NLO LECs generally consistent with expectations, but some
errors are substantially reduced
The era of precision on Lattice QCD has started
Ab-initio computations of QCD at low energy are possible now
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 21 / 32
thank you!
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 22 / 32
l12 and NNLO LEC
-15
-10
-5
0
5
10
15
350 400 450 500 550 600
Mmaxπ [MeV]
kMkF
-20
-10
0
10
20
30
40
50
60
70
350 400 450 500 550 600
Mmaxπ [MeV]
κMκF
-1
0
1
2
3
4
5
6
7
8
350 400 450 500 550 600
l 12
Mmaxπ [MeV]
NLO x
NLO ξ
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 23 / 32
Fit quality in terms of Mmaxπ
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
10
100 150 200 250 300 350
p-value
Mminπ (MeV)
NLO x-expansion
NLO ξ-expansion
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 24 / 32
LO LEC in terms of Mmaxπ
76
78
80
82
84
86
88
90
92
F(M
eV)
NLO x
NLO ξ
-20246810
100 150 200 250 300 350
∆F
(MeV
)
Mminπ (MeV)
1800
1850
1900
1950
2000
2050
2100
BRGI(M
eV)
NLO x
NLO ξ
-150-100-50050100150
100 150 200 250 300 350
∆BRGI(M
eV)
Mminπ (MeV)
3
3.5
4
4.5
5
5.5
6
l 4
NLO x
NLO ξ
-3-2-10123
100 150 200 250 300 350
∆l 4
Mminπ (MeV)
0
1
2
3
4
5
l 3
NLO x
NLO ξ
-4
-2
0
2
4
100 150 200 250 300 350
∆l 3
Mminπ (MeV)
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 25 / 32
Fπ and other interesting quantities in terms of Mπ
84
86
88
90
92
94
96
Fπ(M
eV)
NLO x
NLO ξ
-20246810
100 150 200 250 300 350
∆Fπ(M
eV)
Mminπ (MeV)
1.04
1.05
1.06
1.07
1.08
1.09
1.1
Fπ/F
(MeV
)
NLO x
NLO ξ
-0.06-0.05-0.04-0.03-0.02-0.01
00.01
100 150 200 250 300 350
∆Fπ/F
Mminπ (MeV)
225
230
235
240
245
250
255
260
ΣRGI(M
eV)
NLO x
NLO ξ
0
5
10
15
20
100 150 200 250 300 350
∆ΣRGI(M
eV)
Mminπ (MeV)
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 26 / 32
Log?
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 27 / 32
Fitting χPT formulae with 2HEX data
x-expansion
χ2 =
Npoints∑i
XTi S−1
i Xi +∑β
(apβ− aβ )2
σ2aβ
+(Z
pS,β− ZS,β )2
σ2ZS,β
+(Z
pA,β− ZA,β )2
σ2ZA,β
X =
amud − apZ
pSm
pud
(1− γ1(ap)2)
(aMSS )2 − (ap)2(MpSS
)2
(aMπ )2
2(amud )− (apZ
pS
)(1 + γp1 (ap)2 + γ
p2 ((M
pSS
)2 − M2Φ,SS ))Bπ(m
pud, Bp , Fp , Λ
p3 |l12, kM , kF )
(aFπ)− ap
Zpa
(1 + γp3 ((M
pSS
)2 − M2Φ,SS ))Fπ(m
pud, Bp , Fp , Λ
p4 |l12, kM , kF )
Sab =
NBOOTS∑m=1
(yma − y0
a
) (ymb − y0
b
), y t =
(amud , (aMSS )2
,(aMπ)2
2(amud ), aFπ
)
A term, γ1αa + γ2a2 + γ3((M
pSS
)2 − M2Φ,SS ), was tried for each component of X.
l12 = 715
log
(Λ2
1M2π
)+ 8
15log
(Λ2
2M2π
), kM and kF are included only for NNLO fits
Fits are repeated for each bootstrap sample.
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 28 / 32
Fitting χPT formulae with 2HEX data
ξ-expansion
χ2 =
Npoints∑i
XTi S−1
i Xi +∑β
(apβ− aβ )2
σ2aβ
+(Z
pS,β− ZS,β )2
σZ2S,β
+(Z
pA,β− ZA,β )2
σ2ZA,β
X =
(aMπ)2 − (ap)2(Mp
π)2
(aMSS )2 − (ap)2(MpSS
)2
2(amud )
(aMπ )2 −1
(apZpS
)(1 + γ
p1 (ap)2 + γ
p2 ((M
pSS
)2 − M2Φ,SS ))B−1
π ((Mpπ)2, Bp , Fp , Λ
p3 |l12, κM , κF )
(aFπ)− ap
Zpa
(1 + γp3 ((M
pSS
)2 − M2Φ,SS ))Fπ((Mp
π)2, Bp , Fp , Λp4 |l12, κM , κF )
Sab =
NBOOTS∑m=1
(yma − y0
a
) (ymb − y0
b
), y t =
((aMπ)2
, (aMSS )2,
2(amud )
(aMπ)2, aFπ
)
A term, γ1αa + γ2a2 + γ3((M
pSS
)2 − M2Φ,SS ), was tried for each component of X.
l12 = 715
log
(Λ2
2M2π
)+ 8
15log
(Λ2
1M2π
), κM and κF are included only for NNLO fits
Fits are repeated for each bootstrap sample.
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 29 / 32
LO LEC in terms of Mπ (prior l1 and l2)
84
86
88
90
92
94
F(M
eV)
NLO xNLO ξ
NNLO xNNLO ξ
-4
-2
0
2
4
200 250 300 350 400 450 500 550 600
∆F
(MeV
)
Mmaxπ (MeV)
1800
1850
1900
1950
2000
2050
2100
BRGI(M
eV)
NLO xNLO ξ
NNLO xNNLO ξ
-50
0
50
100
150
200 250 300 350 400 450 500 550 600
∆BRGI(M
eV)
Mmaxπ (MeV)
0
1
2
3
4
5
6
7
8
l 3
NLO xNLO ξ
NNLO xNNLO ξ
-4
-2
0
2
4
200 250 300 350 400 450 500 550 600
∆l 3
Mmaxπ (MeV)
2
2.5
3
3.5
4
4.5
5
5.5
6
l 4
NLO xNLO ξ
NNLO xNNLO ξ
-3-2-10123
200 250 300 350 400 450 500 550 600
∆l 4
Mmaxπ (MeV)
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 30 / 32
LO LEC in terms of Mπ (prior l1 and l2)
84
86
88
90
92
94
F(M
eV)
NLO xNLO ξ
NNLO xNNLO ξ
-4
-2
0
2
4
200 250 300 350 400 450 500 550 600
∆F
(MeV
)
Mmaxπ (MeV)
1800
1850
1900
1950
2000
2050
2100
BRGI(M
eV)
NLO xNLO ξ
NNLO xNNLO ξ
-50
0
50
100
150
200 250 300 350 400 450 500 550 600
∆BRGI(M
eV)
Mmaxπ (MeV)
0
1
2
3
4
5
6
7
8
l 3
NLO xNLO ξ
NNLO xNNLO ξ
-4
-2
0
2
4
200 250 300 350 400 450 500 550 600
∆l 3
Mmaxπ (MeV)
2
2.5
3
3.5
4
4.5
5
5.5
6
l 4
NLO xNLO ξ
NNLO xNNLO ξ
-3-2-10123
200 250 300 350 400 450 500 550 600
∆l 4
Mmaxπ (MeV)
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 30 / 32
Fπ and other interesting quantities in terms of Mπ (prior l1and l2)
88
90
92
94
96
98
Fπ(M
eV)
NLO xNLO ξ
NNLO xNNLO ξ
-4
-2
0
2
4
200 250 300 350 400 450 500 550 600
∆Fπ(M
eV)
Mmaxπ (MeV)
1.03
1.04
1.05
1.06
1.07
1.08
Fπ/F
(MeV
)
NLO xNLO ξ
NNLO xNNLO ξ
-0.03-0.02-0.01
00.010.020.03
200 250 300 350 400 450 500 550 600
∆Fπ/F
Mmaxπ (MeV)
235
240
245
250
255
260
ΣRGI(M
eV)
NLO xNLO ξ
NNLO xNNLO ξ
-8-6-4-202468
200 250 300 350 400 450 500 550 600
∆ΣRGI(M
eV)
Mmaxπ (MeV)
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 31 / 32
l12 and NNLO LEC
-15
-10
-5
0
5
10
15
350 400 450 500 550 600
Mmaxπ [MeV]
kMkF
-40
-20
0
20
40
60
80
100
350 400 450 500 550 600
Mmaxπ [MeV]
κMκF
1
1.5
2
2.5
3
350 400 450 500 550 600
l 12
Mmaxπ [MeV]
NLO x
NLO ξ
Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 32 / 32