Chiral behavior of pion properties from lattice QCD Chiral behavior of pion properties from lattice

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Transcript of Chiral behavior of pion properties from lattice QCD Chiral behavior of pion properties from lattice

  • 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)V 3 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)V 3 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 = lim mq→0

    Fπ and B = lim mq→0

    〈0| q̄q |0〉 F 2π

    Next to Leading Order (NLO) O(p4) Seven new terms in the Lagrangian O(p4): l̄i , 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 Fπ Colangelo,Gasser,Leutwyler Nucl.Phys.B603(01)

    x-expansion ( x = M

    2

    (4πF )2

    ) , M2 = 2Bmud

    M2π = M 2

    ( 1− x

    2 log

    ( Λ23 M2

    ) + x2

    ( 17

    8 log(

    Λ2M M2

    ) + kM

    ) +O(x3)

    ) Fπ = F

    ( 1 + x log

    ( Λ24 M2

    ) + x2

    ( −5

    4 log(

    Λ2F M2

    ) + kF

    ) +O(x3)

    ) log

    ( Λ2M M2

    ) =

    1

    51

    ( 4

    ( 7 log

    ( Λ21 M2

    ) + 8 log

    ( Λ22 M2

    )) − 9 log

    ( Λ23 M2

    ) + 49

    ) log

    ( Λ2F M2

    ) =

    1

    30

    ( 2

    ( 7 log

    ( Λ21 M2

    ) + 8 log

    ( Λ22 M2

    )) + 6

    ( log

    ( Λ23 M2

    ) − log

    ( Λ24 M2

    )) + 23

    )

    l̄i ≡ log (

    Λ2i M2π

    )

    Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 5 / 32

  • Quark-mass dependence of M2π and Fπ Colangelo,Gasser,Leutwyler Nucl.Phys.B603(01)

    ξ-expansion ( ξ =

    M2π (4πFπ)2

    ) M2 = M2π

    ( 1 +

    ξ

    2 log

    ( Λ23 M2π

    ) + ξ2

    ( log(

    Ω2M M2π

    ) + κM

    ) +O(ξ3)

    )

    F = Fπ

    ( 1− ξ log

    ( Λ24 M2

    ) + ξ2

    ( log(

    Ω2F M2π

    ) + κF

    ) +O(ξ3)

    )

    log

    ( Ω2M M2

    ) =

    1

    15

    ( − (

    4 log

    ( 7

    Λ21 M2

    ) + 8 log

    ( Λ22 M2

    )) − 33 log

    ( Λ23 M2

    ) − 12 log

    ( Λ24 M2

    ) + 52

    )

    log

    ( Ω2F M2

    ) =

    1

    3

    ( − (

    7 log

    ( Λ21 M2

    ) + 8 log

    ( Λ22 M2

    )) + 18 log

    ( Λ24 M2

    ) −

    29

    2

    )

    l̄i ≡ log (

    Λ2i M2π

    )

    Alfonso Sastre (CPT, BMWc) Pion properties from lattice QCD Mainz, August 2, 2013 6 / 32

  • BMWc 2HEX. Action

    Gauge Action Lüscher, 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

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

    00 200 2

    300 2

    400 2

    500 2

    600 2

    M π

    2 [MeV

    2 ]

    0

    0.050 2

    0.100 2

    0.150 2

    a 2 [

    fm 2 ]

    BMWc ’08 BMWc ’10

    100 2

    200 2

    300 2

    400 2

    500 2

    600 2

    M π

    2 [MeV

    2 ]

    2

    3

    4

    5

    6

    7

    L [f

    m ]

    0.1%

    BMWc ’08 BMWc ’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 = amPCACs amPCACud

    , d = ambares − ambareud

    r imp = r , d imp = d

    ( 1− d

    2

    r + 1

    r − 1

    ) amschemeud =

    1

    Z schemeS

    d imp

    r imp − 1 , amschemes =

    1

    Z schemeS

    r impd imp

    r imp − 1

    amPCAC1 + am PCAC 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

    〈 (d̄γ5u)(~x , t)(ūγ5d)(0)

    〉 = CPP cosh

    ( (aMπ)

    ( T

    2 − t ))

    ∑ ~x

    〈 (d̄γ0γ5u)(~x , t)(ūγ5d)(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

  • Strange quark effects and scale setting

    M2SS ≡ 2M2k −M2π∑ ~x

    〈(s̄γ5u)(~x , t)(ūγ5s)(0)〉 = C ′PP cosh (

    (aMK )

    ( T

    2 − t ))

    Scale setting Mπ MΩ , MKMΩ → 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 ≡ M 2 π

    2mud and Fπ in terms of mud

    ξ-expansion: B−1π ≡ 2mud M2π

    and Fπ in terms of M 2 π

    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

    F π [M

    eV ]

    mRGIud [MeV]

    β=3.5 β=3.61 β=3.7 β=3.8

    χ2

    #dof = 1.52, #dof = 17, p-value = 0.08 Exp value (NOT INCLUDED)

    1700

    1750

    1800

    1850

    1900

    1950

    2000

    2050

    2100

    0 20 40 60 80 100

    B R G I

    π [M

    eV ]

    mRGIud [MeV]

    β=3.5 β=3.61 β=3.7 β=3.8

    χ2

    #dof = 1.52, #dof = 17, p-value = 0.08

    90

    100

    110

    120

    130

    140

    0 20 40 60 80 100

    F π [M

    eV ]

    mRGIud [MeV]

    β=3.5 β=3.61 β=3.7 β=3.8

    χ2

    #dof = 1.16, #dof = 48, p-val