JLQCD's dynamical overlap fermion project Norikazu Yamada (KEK/GUAS) for JLQCD Collaboration Seminar@Toyama Univ. 2008.04.18
JLQCD Collaboration KEK: S. Hashimoto, H. Ikeda, T. Kaneko, H. Matsufuru, J. Noaki, E. Shintani, N.Y. Tsukuba: S. Aoki, K. Kanaya, N. Ishizuka, K. Takeda, A. Ukawa, T. Yoshie NBI: H. Fukaya (KEK) YITP: H. Ohki, T. Onogi Hiroshima: K.-I. Ishikawa, M. Okawa (Taiwan: T.W. Chiu, T.H. Hsieh, K. Ogawa)
Machines at KEK (since 2006) Hitachi SR11000/K1 (2.15 TFlops) IBM BlueGene/L (57.3 TFlops) × ~50 upgrade compared to the previous system!
Goals of the project • Understanding S χ SB of QCD from a microscopic point of view. - How is the QCD vacuum formed? Lattice formulation with the exact chiral symmetry is clearly suitable. • Precise determinations of hadron matrix elements like B K , B B , form factors and etc. - The exact chiral symmetry plays a important role in two ways. ‣ The symmetry often prohibits unwanted operator mixings, which do not exist in the continuum. ‣ We can simulate very light quarks. The uncertainty due to the chiral extrapolation is reduced.
Strategy • Use the overlap fermion formalism which has the exact chiral symmetry on the lattice. - Run 1: “Super light” quarks in a small box ( ε -regime) - Run II: Ordinary lattice calculations with relatively light quarks (normal regime) • Computational cost is very demanding... (Supuer-computer + improvements of algorithms) makes it possible! Keyword : Chiral Symmetry on the lattice
Publications - Published • Lattice gauge action suppressing near-zero modes of H W • Two-flavor lattice QCD in the epsilon-regime and chiral Random Matrix Theory • Two-flavor lattice QCD simulation in the epsilon-regime with exact chiral symmetry • Lattice study of meson correlators in the epsilon-regime of two-flavor QCD • B K with two flavors of dynamical overlap fermions - Submitted • Topological susceptibility in two-flavor lattice QCD with exact chiral symmetry • Two-flavor QCD simulation with exact chiral symmetry - Coming soon • Meson spectrum of two-flavor QCD and the light quark masses • S-parameter and pseudo-Nambu-Goldstone boson mass • Sea quark content of the nucleon sigma term • pion form factors
Fields on a lattice Discretize the space-time and define “fields” on a lattice x µ = an µ = a × ( n x , n y , n z , n t ) a − 3 / 2 ¯ a − 3 / 2 ψ ( x ) , ψ ( x ) , µ ( x ) t b ∈ SU ( N c ) U µ ( x ) = e igaA b a x µ ) U † ν ) U † � � P µ, ν ( x ) = Tr U µ ( x ) U ν ( x + ˆ µ ( x + ˆ ν ( x ) ν Gauge transformation: ψ ( x ) → V ( x ) ψ ( x ) , ¯ ψ ( x ) → ¯ ψ ( x ) V † ( x ) , μ U µ ( x ) → V ( x ) U µ ( x ) V † ( x + ˆ µ ) Gauge action (Plaquette gauge action) d 4 x 1 S g = β � � � 4 F b µ ν F b µ ν + O ( a 2 ) (where β = 6 /g 2 ) [3 − Re P µ, ν ( x )] − → 6 x µ � = ν �
⇒ Fermionic action • Naive fermion action: � ¯ S naive = 1 � µ ) − ¯ d 4 x ¯ � ψ ( x ) γ µ U † � ψ ( x ) γ µ U µ ( x ) ψ ( x + ˆ µ ( x − ˆ µ ) ψ ( x − ˆ µ ) ψ ( x ) D / ψ ( x ) ? − → 2 a • Doubler problem: 16 poles in the Brillouin zone ⇔ 16 particles x,µ propagator: 1 aS ( p ) = − i γ µ sin( ap µ ) µ sin 2 ( ap µ ) � • Usually the Wilson term is added to avoid the doublers � ¯ µ ) − 2 ψ ( x ) + U † � � S wt = − r ψ ( x ) U µ ( x ) ψ ( x + ˆ µ ( x − ˆ µ ) ψ ( x − ˆ µ ) 1 − i γ µ sin( ap µ ) aS ( p ) = x,µ � 2 � µ sin 2 ( ap µ ) + � � r � − ar µ (cos( ap µ ) − 1) � � ¯ d 4 x ψ ( x ) D 2 ψ ( x ) + O ( a 3 ) − → 2 This gives you a gauge-invariant regularization of Quantum Field Theory. Wilson fermion action / − ar � � 2 D 2 + O ( a 2 ) � d 4 x ¯ S Wilson = S naive + S wt − ψ ( x ) D ψ ( x ) →
Chiral Symmetry Define the chiral transformation by ψ ( x ) e i θ b t a P L/R , ψ ( x ) → e − i θ b t a P R/L ψ ( x ) , P L/R = (1 ∓ γ 5 ) / 2 ψ ( x ) → ¯ ¯ In the continuum, QCD is invariant for massless quarks. ⇒ Chiral Symmetry SU(N f ) L × SU(N f ) R Since we want to study S χ SB using Lattice QCD, it is clearly better that it has the symmetry. If the symmetry is violated explicitly from the beginning, the study will encounter many difficulties.
Lack of Chiral Symmetry • Wilson fermion: / + m q − ar � � 2 D 2 + O ( a 2 ) � d 4 x ¯ S Wilson = ψ ( x ) D ψ ( x ) The axial part of chiral transformation, ψ ( x ) e i θ b t a γ 5 , ψ ( x ) → e i θ b t a γ 5 ψ ( x ) , ψ ( x ) → ¯ ¯ is not the symmetry because of the Wilson term even in the massless llimit. Chiral symmetry is explicitly violated for Wilson fermion! This difficulty is rather general, and known as “No-go theorem”. [Nielsen,Ninomiya(1981,1981)] Long standing problem in Lattice QCD for ~25 years.
Overlap fermion [Neuberger (1998)] Overlap-Dirac operator: ( m q is lattice quark mass. ) m 0 + m q m 0 − m q � � � � D ov = + γ 5 sgn [ H W ( − m 0 )] � � � 2 2 − 2 H W ( − m 0 ) = γ 5 ( D W − m 0 ) X sgn[ X ] = √ X † X where 0 < m 0 < 2 ( m 0 =1.6 ). • The action is invariant under the lattice variant of chiral rotation: δ ¯ ψ ( x ) = ¯ ψ ( x ) i γ 5 θ b T b , δψ ( x ) = i θ b T b γ 5 (1 − aD ov ) ψ ( x ) , which is equivalent to satisfying Ginsberg-Wilson relation, D ov γ 5 + γ 5 D ov = aD ov γ 5 D ov
Overlap fermion [Neuberger (1998)] With G-W relation D ov γ 5 + γ 5 D ov = aD ov γ 5 D ov Suppose that D ov u k = λ k u k , one can prove that - eigenvalue necessarily appears in a pair ( λ k , λ k * ), and then eigenvectors for those are given by ( u k , γ 5 u k ) (except for zero- modes) - eigenvalues are distributed on a circle in complex plane. ‣ In lattice calculations, we need to calculate the propagator 1 /( D ov + m q ) . ‣ With this constraint, the propagator can be calculated for very light quark mass. ‣ For Wilson-Dirac op., no such a constraint ⇒ algorithm is breakdown for light quarks.
Lattice simulation Lattice simulation is essentially doing Path Integral numerically. e.g.) pion two-point correlation function ⇒ f π & m π A 4 ( x ) = u ( x ) γ 4 γ 5 d ( x ) , ¯ � [ DU ]det[ D ov [ U ]] N f e − S g [ U ] � � � A 4 ( x ) A † A 4 ( x ) A † Z − 1 4 (0) � = 4 (0) x � � x Tr[ D − 1 ov ( x, 0) γ 4 γ 5 D − 1 � � = ov (0 , x ) γ 4 γ 5 ] U f 2 π m π e − m π t + (excited states) − → 2 We can thus calculate various hadron masses, decay constants, transition matrix elements through lattice calculation.
An example Simulating lighter quark masses becomes possible. Quark mass dependence of the decay const
Physics Results All results are obtained in 2-flavor QCD (degenerate u & d). [Phys.Rev.D74:094505,2006,arXiv:0803.3197] 2+1-flavor (real QCD) simulation is on going. ‣ Run I ( ε -regime) - Chiral condensate Σ and f π in the chiral limit [PRL98(2007)172001,PRD76(2007)054503,arXiv:0711.4965 (appear in PRD)] ‣ Run II (normal regime) - Topological susceptibility [arXiv:0710.1130] - B K [arXiv:0801.4186 (appear in PRD)] - Sea quark content of the nucleon sigma term - S-parameter and pseudo-NG boson mass - Meson spectrum and the light quark masses - pion form factors
Physics Results All results are obtained in 2-flavor QCD (degenerate u & d). [Phys.Rev.D74:094505,2006,arXiv:0803.3197] 2+1-flavor (real QCD) simulation is on going. ‣ Run I ( ε -regime) - Chiral condensate Σ and f π in the chiral limit [PRL98(2007)172001,PRD76(2007)054503,arXiv:0711.4965 (appear in PRD)] ‣ Run II (normal regime) - Topological susceptibility [arXiv:0710.1130] - B K [arXiv:0801.4186 (appear in PRD)] - Sea quark content of the nucleon sigma term - S-parameter and pseudo-NG boson mass - Meson spectrum and the light quark masses - pion form factors
B K
B K ǫ K = A ( K L → ( ππ ) I =0 ) | ǫ K | exp = 2 . 28(2) × 10 − 3 A ( K S → ( ππ ) I =0 ) , η A 2 ˆ 1 . 11(5) A 2 (1 − ¯ � � ǫ K = ¯ B K × ρ ) + 0 . 31(5) , V DD B K = C ( µ ) � K 0 |O ∆ S =2 ( µ ) | K 0 � ˆ 8 3 f 2 K m 2 X ! X ! K • Lattice QCD ⇒ B K • Constraint on ρ and η
Current status of Unitarity Introduction 1.5 1.5 excluded at CL > 0.95 excluded area has CL > 0.95 # 3 1 1 m & m $ $ sin2 # s d • Check consistency among 1 constraints from | ε | and other 0.5 0.5 $ m experiments using ( ρ , η )-plane. d # # % 2 2 • Inconsistency ⇒ the effects of NP . K # # " " 3 1 0 0 approximately. The dominant • Currently, V use � B K = 0 . 79 ± 0 . 04 ± 0 . 09 ub Sizable error is one of the dominant 4 ) [ i.e. , proportional to ( ( # -0.5 -0.5 2 uncertainties in light green band. • Improving the error is important. % -1 -1 K CKM sol. w/ cos2 < 0 # # 1 f i t t e r (excl. at CL > 0.95) 3 Summer 2007 -1.5 -1.5 -1 -1 -0.5 -0.5 0 0 0.5 0.5 1 1 1.5 1.5 2 2 ! !
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