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Recent Progress in Staggered Chiral Perturbation TheoryStaggered Fermion Formulation Staggered...

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Recent Progress in Staggered Chiral Perturbation Theory Weonjong Lee Lattice Gauge Theory Research Center Department of Physics and Astronomy Seoul National University Chiral Dynamics 2012, Jlab, 08/07/2012 Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 1 / 45
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  • Recent Progress in Staggered Chiral PerturbationTheory

    Weonjong Lee

    Lattice Gauge Theory Research CenterDepartment of Physics and Astronomy

    Seoul National University

    Chiral Dynamics 2012, Jlab, 08/07/2012

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 1 / 45

  • Outline

    1 Project: 1998 – Present

    2 Staggered Fermion FormulationStaggered Fermion Formulation

    3 SChPTStaggered Chiral Perturbation Theory

    4 AppilcationPion MassPion Decay ConstantsBKπ − π Scattering Phase Shift

    5 Summary and Conclusion

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 2 / 45

  • Project: 1998 – Present

    SWME Collaboration1998 — Present

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 3 / 45

  • Project: 1998 – Present

    SWME Collaboration

    Seoul National University (SNU):Prof. Weonjong LeeDr. Jon Bailey and Dr. Nigel Cundy (RA Prof.)11 graduate students.

    Brookhaven National Laboratory (BNL):Dr. Chulwoo JungDr. Hyung-Jin Kim (Postdoc)

    University of Washington, Seattle (UW):Prof. Stephen R. Sharpe.

    KISTI: Dr. Taegil Bae (Postdoc).

    University of Arizona, Tucson: Dr. Jongjeong Kim (Postdoc).

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 4 / 45

  • Project: 1998 – Present

    Lattice Gauge Theory Research Center (SNU)

    Center Leader: Prof. Weonjong Lee. (***)

    Research Assistant Prof.: Dr. Jon Bailey (***)

    Research Assitant Prof.: Dr. Nigel Cundy

    11 graduate students (***)

    Secretary: Ms. Sora Park.

    more details on http://lgt.snu.ac.kr/.

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 5 / 45

  • Project: 1998 – Present

    Group Photo

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 6 / 45

  • Staggered Fermion Formulation Staggered Fermion Formulation

    Staggered Fermion Formulation

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 7 / 45

  • Staggered Fermion Formulation Staggered Fermion Formulation

    How to put quarks on the lattice ?

    Wilson Fermions:

    1 Clover Action: (**)2 Twisted mass fermions: (**)3 Domain Wall Fermions: (***)4 Overlap Fermions:

    Staggered Fermions;

    1 Asqtad action: (*)2 HYP staggered fermions: (***)3 Fat7 staggered fermions:4 HISQ action: (***)

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 8 / 45

  • Staggered Fermion Formulation Staggered Fermion Formulation

    Cons and Pros for Staggered Fermions

    Advantages (Pros):

    1 Preserve part of exact chiral symmetries.2 Numerically cheapest on the lattice.3 No residual quark mass (no additive renormalization).4 Easy to improve with almost no extra cost.5 Staggered Chiral Perturbation Theory.

    Possess 4 degenerate tastes (pure lattice artifacts).

    Disadvantages (Cons):

    1 Born with taste symmetry breaking by construction.2 Theoretically more challenging to interprete the data.

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 9 / 45

  • SChPT Staggered Chiral Perturbation Theory

    Staggered Chiral Perturbation Theory

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 10 / 45

  • SChPT Staggered Chiral Perturbation Theory

    What is Staggered ChPT ?

    ChPT designed to analyze the data produced using staggeredfermions.

    Dual expansion in powers of p2 ≈ mq and a2.

    It incorporates all the taste symmetry breaking effects into the LECsorder by order in a perturbative series.

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 11 / 45

  • SChPT Staggered Chiral Perturbation Theory

    Birth of Staggered ChPT

    At the leading order of p2 ≈ mq ≈ m2π ≈ a2, we can prove that thepion spectrum respects SO(4) taste symmetry out of the full SU(4)taste symmetry.

    Lee and Sharpe proved it for single flavor case (1999).

    Aubin and Bernard proved it for multiple flavor case (2003).

    Power counting rules are established through the numerical study onthe lattice.

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 12 / 45

  • SChPT Staggered Chiral Perturbation Theory

    Splittings of Pion Multiplet Spectrum

    (1) Coarse lattice (a = 0.12fm) (2) Fine lattice (a = 0.09fm)

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 13 / 45

  • SChPT Staggered Chiral Perturbation Theory

    Scaling of the Splittings

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 14 / 45

  • SChPT Staggered Chiral Perturbation Theory

    Sea quark mass dependence of splittings

    0

    2

    4

    6

    8

    10

    12

    0 0.01 0.02 0.03 0.04

    ∆(B

    )[×

    10−2GeV

    2]

    amℓ

    A

    T

    V

    S

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 15 / 45

  • SChPT Staggered Chiral Perturbation Theory

    Staggered Chiral Perturbation Theory (SChPT)

    1 We need to incorporate this effect of pion multiplet splittings into thedata analysis.

    2 Staggered fermion formulation introduces mixing with extra operatorsin addition to the physical mixing. We can also incorporate this effectinto the data analysis using SChPT.

    3 The systematic tool is the SChPT.

    4 Using the SChPT, we obtain the fitting functional form exactly orderby order.

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 16 / 45

  • SChPT Staggered Chiral Perturbation Theory

    Staggered Chiral Perturbation Theory (SChPT)

    1 We need to incorporate this effect of pion multiplet splittings into thedata analysis.

    2 Staggered fermion formulation introduces mixing with extra operatorsin addition to the physical mixing. We can also incorporate this effectinto the data analysis using SChPT.

    3 The systematic tool is the SChPT.

    4 Using the SChPT, we obtain the fitting functional form exactly orderby order.

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 16 / 45

  • SChPT Staggered Chiral Perturbation Theory

    Staggered Chiral Perturbation Theory (SChPT)

    1 We need to incorporate this effect of pion multiplet splittings into thedata analysis.

    2 Staggered fermion formulation introduces mixing with extra operatorsin addition to the physical mixing. We can also incorporate this effectinto the data analysis using SChPT.

    3 The systematic tool is the SChPT.

    4 Using the SChPT, we obtain the fitting functional form exactly orderby order.

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 16 / 45

  • SChPT Staggered Chiral Perturbation Theory

    Staggered Chiral Perturbation Theory (SChPT)

    1 We need to incorporate this effect of pion multiplet splittings into thedata analysis.

    2 Staggered fermion formulation introduces mixing with extra operatorsin addition to the physical mixing. We can also incorporate this effectinto the data analysis using SChPT.

    3 The systematic tool is the SChPT.

    4 Using the SChPT, we obtain the fitting functional form exactly orderby order.

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 16 / 45

  • SChPT Staggered Chiral Perturbation Theory

    Staggered chiral perturbation theory

    Power counting

    O(a2Λ2QCD) ≈ O(p2/Λ2χ) ≈ O(m2π/Λ2χ) ≈ O(mq/ΛQCD)

    Lee & Sharpe Lagrangian for multiple flavors[Aubin and Bernard, 2003]

    LLO =f 2

    8Tr(∂µΣ∂µΣ

    †)− 14µf 2Tr(MΣ + MΣ†)

    +2m20

    3(UI + DI + SI )

    2 + a2V

    - M = diag(mu,md ,ms)⊗ ξI- V : taste symmetry breaking potential [Lee and Sharpe, 1999]

    SO(4)× SU(4)T a 6=0−−→ SW4,diag⊂

    p�ΛχSO(4)× SO(4)T

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 17 / 45

  • Appilcation

    Application of SChPT

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 18 / 45

  • Appilcation Pion Mass

    Pion Mass (Quark Mass)

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 19 / 45

  • Appilcation Pion Mass

    Pion Flow Diagrams

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 20 / 45

  • Appilcation Pion Mass

    Quark Flow Diagrams (1)

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 21 / 45

  • Appilcation Pion Mass

    Quark Flow Diagrams (2)

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 22 / 45

  • Appilcation Pion Mass

    Results at NLO

    Pion self energy:

    M2πF = m2πF

    + Σ(m2πF ) + NNLO

    Σ(p2) =1

    (4πf )2[σconn(p

    2) + σdisc(p2)] + σanal(p

    2)

    Connected Part:

    σconn = a2∑B

    (δconnBF `(π

    +B ) +

    ∆connBF48

    [`(UB) + 2`(π+B ) + `(DB)]

    )Disconnected Part:

    σdisc =1

    12

    [2(−12X5 + a4(∆connVF + . . .)δ′V

    (RπXη(XV )˜̀(XV ) + . . .

    )+ . . .)

    ]Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 23 / 45

  • Appilcation Pion Mass

    Progress History (Pion Mass)

    Goldstone pion sector at NLO : Aubin & Bernard (2003)

    Non-Goldstone pion sectors at NLO : Bailey & Kim & Lee (2012)

    Extension to mixed actions : underway by Yoon & Bailey & Lee(YBL) (2012)※ Example of a mixed action :• valence quarks = HYP staggered fermions• sea quarks = asqtad staggered fermions

    Results have been used for the numerical study by MILC.

    We plan to apply the mixed action results to the data analysis.

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 24 / 45

  • Appilcation Pion Decay Constants

    Pion Decay Constants

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 25 / 45

  • Appilcation Pion Decay Constants

    Pion Flow Diagrams

    (13) Wavefunction Cont. (14) Current Cont.

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 26 / 45

  • Appilcation Pion Decay Constants

    Quark Flow Diagrams (Current Contribution)

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 27 / 45

  • Appilcation Pion Decay Constants

    Results at NLO

    Example:- Pion decay constant for fully dynamical case (xy = ud)

    - SU(2) chiral perturbation theory (mu,md � ms)- 2+1 flavors (mu = md = m` 6= ms)

    fπF

    =f

    {1 +

    1

    32π2f 2

    [− 1

    4

    ∑B

    gB`(πB)

    + (4−ΘVF ){`(πV )− `(ηV )

    }+ (V → A)

    ]

    + L416µ

    f 2(2m` + ms) + L5

    16µ

    f 2m` + a

    2FF

    }

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 28 / 45

  • Appilcation Pion Decay Constants

    Progress Report

    Goldstone Pion Sector: Aubin & Bernard (2003)

    Non-Goldstone Pion Sectors: Yoon & Bailey & Lee (2012)

    Extension to the mixed action: underway by YBL (2012)

    Results have been used for the numerical study by MILC.

    We plan to apply the mixed action results to our data analysis.

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 29 / 45

  • Appilcation BK

    BK (Indicrect CP Violation)

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 30 / 45

  • Appilcation BK

    BK definition in standard model

    BK =〈K̄0|[s̄γµ(1− γ5)d ][s̄γµ(1− γ5)d ]|K0〉

    83〈K̄0|s̄γµγ5d |0〉〈0|s̄γµγ5d |K0〉

    B̂K = C (µ)BK (µ),

    C (µ) = αs(µ)− γ0

    2b0 [1 + αs(µ)J3]

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 31 / 45

  • Appilcation BK

    Pion Flow Diagrams for BK

    (20) OK (21) OK (22) OK

    (23) X (24) X

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 32 / 45

  • Appilcation BK

    Quark Flow Diagrams for BK

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 33 / 45

  • Appilcation BK

    SU(2) Results at NLO

    BK : (mu = md = m` � ms)

    BK = d1Q1 + d2XPΛ2χ

    + d3LPΛ2χ

    + NNLO

    Q1 = 1 +1

    32π2f 2

    [(LI − XI)˜̀(XI) + `(XI)− 2

    ∑B

    τB`(XB)

    ]

    XP = [mxxπ (ξ5)]

    2

    LP = [m``π (ξ5)]

    2

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 34 / 45

  • Appilcation BK

    Progress History

    BK at NLO : Sharpe & Van de Water (2006)

    Extension to the mixed action : Sharpe (2008)

    BSM operators at NLO : Bailey & Kim & Lee & Sharpe (2012)

    Application to the numerical study : SWME (2010 ∼ present)

    The SWME result of BK is posted to FLAG officially (2012).

    B̂K = 0.727± 0.004(stat)± 0.038(sys)εK = (1.56± 0.22)× 10−3 (Exclusive Vcb)

    = (1.88± 0.22)× 10−3 (Inclusive Vcb)

    We must reduce the errors of BK and Vcb simultaneously.

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 35 / 45

  • Appilcation π − π Scattering Phase Shift

    π − π Scattering

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 36 / 45

  • Appilcation π − π Scattering Phase Shift

    π − π Scattering and S–matrix

    Five channels of two pion states in staggered fermion formulation:

    π(P)− π(P), π(A)− π(A), π(T )− π(T ),π(V )− π(V ), π(S)− π(S),

    The trouble is that their energy eigenvalues are non-degenerate.

    Recently, Hansen & Sharpe make it possible to study multi-channelscattering problem by modifying the Luscher formula.

    Now, it is possible to study the N = 5 multi-channel π − π scatteringproblem on the lattice using staggered fermions.

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 37 / 45

  • Appilcation π − π Scattering Phase Shift

    Quark Flow Diagrams for π − π Scattering

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 38 / 45

  • Appilcation π − π Scattering Phase Shift

    Unitarity Violation by Rooting Technique

    If the SU(4) taste symmetry is exactly conserved, then rooting cannotmake a trouble of unitarity violation.

    However, if the SU(4) taste symmetry is broken, then rooting makes aunitarity violation.

    The staggered fermion formulation has taste symmetry breaking byconstruction.

    Hence, the rooting triggers the unitarity violation for staggeredfermions.

    As a consequence, there are two kinds of unitarity violation on thelattice using staggered fermions: one from partially quenched QCDand the other from the rooting.

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 39 / 45

  • Appilcation π − π Scattering Phase Shift

    Rooting Technique

    Fermion Determinant of Staggered Fermions:∫[dψ][dψ̄] exp[

    ∫ψ̄(D + m1)ψ] = det(D + m1)

    Here, the Dirac operator (D + m1) contains 4 copies of degeneratetastes.

    In order to reduce the number of tastes to one, we use the rootingtechnique in the numerical study.

    det(D + m1) −→ [det(D + m1)]14

    However, if the SU(4) taste symmetry is broken, then the rootingcauses a unitarity violation since sea quarks and valence quarks havedifferent Dirac operators.

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 40 / 45

  • Appilcation π − π Scattering Phase Shift

    How to get around the trouble: SChPT

    SChPT can, in principle, trace the rooting part and the unitarityviolation terms.

    Hence, we fit the numerical data to the functional form suggested bySChPT.

    Then, we can remove the unitarity violating terms by hand.

    Then, the remaining part will be unitary, which corresponds to theS-matrix defined by Hansen & Sharpe.

    The SChPT calculation is underway by Yoon & Bailey & Lee.

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 41 / 45

  • Summary and Conclusion

    Summary

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 42 / 45

  • Summary and Conclusion

    Summary of Current Status in SChPT

    physics Goldstone Non-Goldstone mixed numerical

    m2π © © 4 ©fπ © © 4 ©BK © × © ©

    BSM op © × © 4π − π 4 × 4

    K → ππ × × ×Vcb © × ©

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 43 / 45

  • Summary and Conclusion

    Sincere apologiesfor omitting some topics

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 44 / 45

  • Summary and Conclusion

    Thank you very much !!!

    Weonjong Lee (SNU) Lattice QCD Chiral Dynamics 2012 45 / 45

    Project: 1998 – PresentStaggered Fermion FormulationStaggered Fermion Formulation

    SChPTStaggered Chiral Perturbation Theory

    AppilcationPion MassPion Decay ConstantsBK - Scattering Phase Shift

    Summary and Conclusion


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