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Page 1: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 2: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 3: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 4: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 5: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 6: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 7: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 8: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 9: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

BMWc 2HEX. Landscape BMWc Phys.Lett.B701(11),JHEP 1108(11)

00 2002

3002

4002

5002

6002

2 [MeV

2]

0

0.0502

0.1002

0.1502

a2 [

fm2]

BMWc ’08BMWc ’10

1002

2002

3002

4002

5002

6002

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

Page 10: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 11: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 12: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

))

(aFπ) =CAP√

2(aMπ)CPP

e(aMπ)T

4

Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 11 / 32

Page 13: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 14: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 15: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 16: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 17: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 18: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 19: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 20: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

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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

Page 22: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

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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

Page 24: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 25: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 26: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

thank you!

Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 22 / 32

Page 27: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 28: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 29: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 30: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 31: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

Log?

Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 27 / 32

Page 32: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 33: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 34: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 35: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 36: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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

Page 37: Chiral behavior of pion properties from lattice QCD · Chiral behavior of pion properties from lattice QCD Alfonso Sastre CPT, Marseille Budapest-Marseille-Wuppertal collaboration

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