Flavored-mass terms for naive and staggered fermions Tatsuhiro - PowerPoint PPT Presentation

Flavored-mass terms for naive and staggered fermions Tatsuhiro MISUMI YITP/BNL M. Creutz, T. Kimura, T. Misumi, JHEP 1012 :041 (2010) M. Creutz, T. Kimura, T. Misumi, PRD 83 :094506 (2011) T. Kimura, S. Komatsu, T. Misumi, T. Noumi, S. Torii, S. Aoki, JHEP 1201 :048 (2012) T. Misumi, Ph.D Thesis , Kyoto University (2012) 02/09/2012 NTFL workshop@Yukawa Institute, Kyoto

Introduction ◆ Wilson fermion ( ψ n +ˆ µ − 2 ψ n + ψ n − ˆ µ ) S W = − ar � a 4 ¯ � ~ d 4 x ¯ S = S nf + S W with ψ ( x ) D 2 ψ n µ ψ ( x ) a 2 a 2 n,µ D naive D wilson Im ! Im ! as m = 0, 16 1 4 6 4 1 Re ! ! Re ! § 15 species are decoupled → doubler-less § additive mass renormalization → Fine-tune for chiral limit Overlap & Domain-wall fermion H W ( m ) D W ( m ) D ov 15 D ov = 1 + γ 5 = 1 + 1 � � H 2 W ( m ) D † W ( m ) D W ( m ) Ginsparg-Wilson : { γ 5 , D ov } = aD ov γ 5 D ov ,

◆ Staggered fermion ¯ ψ n = γ n 1 1 γ n 2 2 γ n 3 3 γ n 4 χ n γ n 4 4 γ n 3 3 γ n 2 2 γ n 1 Spin diagonalization : ψ n = ¯ 4 χ n , 1 S nf = 4 S st = 4[1 µ ) + m � � P η µ ( n )¯ χ n ( χ n +ˆ χ n χ n ] ¯ ν <µ n ν µ − χ n − ˆ η µ ( n ) = ( − 1) 2 2 n,µ n One naive fermion → 4 Staggered fermions Properties ・ 4-flavor Dirac fermions " # !" " #$%$& ・ Flavored chiral symmetry &' ' � n = ( − 1) n 1 + n 2 + n 3 + n 4 ! !"$ ! !"#"$ ~ as Γ 55 = γ 5 ⊗ γ 5 . " #$% " # spin flavor ! ! ! !"# § chiral symmetry + one-component → suitable for calculations § 4 species → more than 3.....

Naive Wilson Overlap Chiral broken GW symmetry Wilson term Overlap form. #=16 #=1 #=1 Fine tuning Numerical expense 4 copies Staggered #=4 4 tastes

Naive Wilson Overlap Chiral broken GW symmetry Wilson term Overlap form. #=16 #=1 #=1 Fine tuning Numerical expense Flavored-mass 4 copies Generalization term Staggered #=4 4 tastes

Flavored mass terms ~ Generalized Wilson terms ~ � � � D nf e.g.) 2-split flavored mass (8,8) 4 2 � � M P = C µ , 1.5 1 µ =1 sym. 0.5 Naive Im [ λ ] 0 -0.5 -1 -1.5 -2 -1 -0.5 0 0.5 1 Re [ λ ] Creutz, Kimura, TM, JHEP1012,041 [1011.0761] 4 copies D st × → (2,2) 4 2 2 � � M A = ζ 5 C µ 1.5 Staggered µ =1 sym. 1 1 0.5 0 Golterman, Smit (1984) 0 -0.5 -1 -1 Adams, PRL104, 141602 [0912.2850] -1.5 -2 -2 -1 0 1 -0.5 0.5 de Forcrand, Kurkela, Panero, [1102.1000] -1 -0.5 0 0.5 1

Generalized Wilson&overlap Naive Wilson Overlap Chiral broken GW symmetry Wilson term Overlap form. #=16 #=1 #=1 Fine tuning Numerical expense Flavored-mass 4 copies Generalization term Staggered ??? #=4 4 tastes

Generalized Wilson&overlap Naive Wilson Overlap Chiral broken GW symmetry Wilson term Overlap form. #=16 #=1 #=1 Fine tuning Numerical expense Flavored-mass 4 copies Generalization term St.Wilson St.Overlap Staggered #=1 #=4 #=1 4 tastes Faster Wilson & Overlap

1. Flavored-mass terms ~ general terms to lift degenerate species ~  Naïve fermion M. Creutz, T. Kimura, TM, JHEP 1012:041 (2010) ・ 16 species ( i ) γ µ Γ ( i ) = γ ( i ) as Γ − 1 label position χ charge type Γ µ 1 (0 , 0 , 0 , 0) + S 1 2 ( π , 0 , 0 , 0) A i γ 1 γ 5 − 3 (0 , π , 0 , 0) A i γ 2 γ 5 − 4 ( π , π , 0 , 0) + T i γ 1 γ 2 5 (0 , 0 , π , 0) A i γ 3 γ 5 ・ 16-flavor multiplet − 6 ( π , 0 , π , 0) + T i γ 1 γ 3   7 (0 , π , π , 0) + T i γ 2 γ 3 ψ (1) ( p − p (1) )   8 ( π , π , π , 0) V γ 4 − ψ (2) ( p − p (2) )     Ψ ( p ) = .   9 (0 , 0 , 0 , π ) A i γ 4 γ 5 . − .     10 ( π , 0 , 0 , π ) + T i γ 1 γ 4   ψ (16) ( p − p (16) ) 11 (0 , π , 0 , π ) + T i γ 2 γ 4 12 ( π , π , 0 , π ) V γ 3 − Flavor mass matrix 13 (0 , 0 , π , π ) + T i γ 3 γ 4 14 ( π , 0 , π , π ) V γ 2 − ¯ Ψ ( 1 ⊗ X ) Ψ 15 (0 , π , π , π ) V γ 1 − Mass matrix 16 ( π , π , π , π ) + P γ 5

◆ Point-split fields M. Creutz (2010), for minimally doubled fermions. 1 ψ (1) ( p − p (1) ) = 2 4 (1 + cos p 1 )(1 + cos p 2 )(1 + cos p 3 )(1 + cos p 4 ) Γ (1) ψ ( p ) , 1 ψ (2) ( p − p (2) ) = 2 4 (1 − cos p 1 )(1 + cos p 2 )(1 + cos p 3 )(1 + cos p 4 ) Γ (2) ψ ( p ) , 1 ψ (3) ( p − p (3) ) = 2 4 (1 + cos p 1 )(1 − cos p 2 )(1 + cos p 3 )(1 + cos p 4 ) Γ (3) ψ ( p ) , . . . 1 ψ (16) ( p − p (16) ) = 2 4 (1 − cos p 1 )(1 − cos p 2 )(1 − cos p 3 )(1 − cos p 4 ) Γ (16) ψ ( p ) , → Independent fields in low energy limit   ψ (1) ( p − p (1) )   ψ (2) ( p − p (2) )   ¯ Ψ ( 1 ⊗ X ) Ψ   Ψ ( p ) = .   . .   Mass matrix     ψ (16) ( p − p (16) ) 16-flavor multiplet

・ Conditions on flavored-mass terms (1) gamma-5 hermiticity : D † = γ 5 D γ 5 det( D ) ≥ 0 essential for euclidian vector-like theory   ψ (1) ( p − p (1) )   ψ (2) ( p − p (2) )   ※ to γ 5 ⊗ ( τ 3 ⊗ τ 3 ⊗ τ 3 ⊗ τ 3 ) for   Ψ ( p ) = .   . spin flavor .       ψ (16) ( p − p (16) ) (2) O( a ) irrelevant term � ~ d 4 x ¯ ψ ( x ) D 2 µ ψ ( x ) a dim-5 operator vanishes in a → 0 ・ Physical modes in the continuum limit ・ Rotational symmetry

◆ Flavored-mass terms ¯ cos p 1 ¯ V : Ψ ( 1 ⊗ ( τ 3 ⊗ 1 ⊗ 1 ⊗ 1 )) Ψ = � ψψ M V = C µ , µ ¯ cos p 1 cos p 2 ¯ T : Ψ ( 1 ⊗ ( τ 3 ⊗ τ 3 ⊗ 1 ⊗ 1 )) Ψ = ψψ � � M T = C µ C ν , � 4 perm. sym. � ¯ � ¯ A : Ψ ( 1 ⊗ ( 1 ⊗ τ 3 ⊗ τ 3 ⊗ τ 3 )) Ψ = cos p µ ψψ � � � M A = C ν , µ =2 perm. sym. ν � 4 � 4 ¯ � ¯ P : Ψ ( 1 ⊗ ( τ 3 ⊗ τ 3 ⊗ τ 3 ⊗ τ 3 )) Ψ = cos p µ ψψ � � M P = C µ , µ =1 µ =1 sym. � ・ O( a ) irrelevant terms ¯ d 4 x ¯ � ψ ( x ) D 2 µ ψ ( x ) + O ( a 2 ) ψ n ( M P − 1) ψ n → − a n ・ low-energy species-splitting terms Im ! ・ M V ( M A ) → Wilson term ! Re !

: : Dirac spectra with flavored mass terms (8,8) (4,8,4) D nf − M ( i ) D nf − M P T (1,15) M V +M A : (1,14,1) M P +M T : (4,12) M P +M V : (5,1,10) M T +M V : (10,5,1) M A +M V +M T1+ M T2 : (3,12,1) D nf − ( M V + M T + M A + M P ) → Multi-flavor Wilson & Overlap (although we need care about renormalization)

◆ Pseudo-scalar type (8,8) � M P = C 1 C 2 C 3 C 4 sym. consistent ¯ P : Ψ ( 1 ⊗ ( τ 3 ⊗ τ 3 ⊗ τ 3 ⊗ τ 3 )) Ψ 8 (+) and 8 (-) masses D nf − M P ・ Index theorem from spectral flow cf.) For staggered, Adams (2009) H = γ 5 ( D nf − rM P ) 36 × 36 lattice, randomness δ =0.25, Q=1 λ ( r ) Index( D nf ) = - Spectral flow( H ) doubled Index( D nf ) = 2 d ( − 1) d/ 2 Q Index( D gw ) = -4 λ ( r )

Adams-type flavored mass D. Adams (2009) ・ spin diagonalization ¯ χ x γ x 4 4 γ x 3 3 γ x 2 2 γ x 1 1 γ x 1 +1 γ x 2 +1 γ x 3 +1 γ x 4 +1 4 = ¯ ψ x ψ x +ˆ χ x +ˆ 1+ˆ 2+ˆ 3+ˆ 1+ˆ 2+ˆ 3+ˆ 1 2 3 4 4 = ( − 1) x 2 + x 4 ¯ χ x γ 5 χ x +ˆ ( γ 5 diagonalized) 1+ˆ 2+ˆ 3+ˆ 4 → ± ¯ χ x �η 1 η 2 η 3 η 4 χ x +ˆ 1+ˆ 2+ˆ 3+ˆ 4 4 Adams fermions derived up to sign ¯ ± ¯ χ x ( �η 1 η 2 η 3 η 4 C 1 C 2 C 3 C 4 ) χ x . ψ x C 1 C 2 C 3 C 4 ψ x → (2,2) (8,8) × → 2 1 M P st ( M A ) 0 -1 -2 -1 0 1 -0.5 0.5 de Forcrand, Kurkela, Panero, [1102.1000] S nf ( M P ) S st ( M A ) →

◆ Tensor type (4,8,4) M T = M (1) T + M (2) T + M (3) T , = 1 2( C 1 C 2 + C 2 C 1 ) + 1 M (1) 2( C 3 C 4 + C 4 C 3 ) , T = 1 2( C 1 C 3 + C 3 C 1 ) + 1 M (2) 2( C 2 C 4 + C 4 C 2 ) , T = 1 2( C 1 C 4 + C 4 C 1 ) + 1 M (3) 2( C 2 C 3 + C 3 C 2 ) . T Double rotation symmetric units : x → R ( µ ν ) R ( ρσ ) x D nf − M ( i ) T ・ Index theorem from spectral flow 36 × 36 lattice, randomness δ =0.25, Q=1 λ ( r ) H = γ 5 ( D nf − rM ( i ) T ) Index( D ) = - Spectral flow( H ) Index( D ) = -2 λ ( r ) Index( D ) = 2 d − 1 ( − 1) d/ 2 Q

Hoelbling-type flavored mass Hoelbling PLB696, 422(2011) [1009.5362]. de Forcrand (2010) ・ spin diagonalization 4 = ( − 1) x 2 ¯ 2 + ( − 1) x 4 ¯ ¯ 2 + ¯ ψ x ψ x +ˆ ψ x ψ x +ˆ χ x γ 1 γ 2 χ x +ˆ χ x γ 3 γ 4 χ x +ˆ 1+ˆ 3+ˆ 1+ˆ 3+ˆ 4 → ± ¯ 2 ± ¯ χ x i � 12 η 1 η 2 χ x +ˆ χ x i � 34 η 3 η 4 χ x +ˆ 1+ˆ 3+ˆ 4 ※ two terms simultaneously diagonalizable : [ σ 12 , σ 34 ] = 0 4 Hoelbling fermions (3 units) up to sign ¯ ψ x [( C 1 C 2 + C 2 C 1 ) + ( C 3 C 4 + C 4 C 3 )] ψ x ± ¯ χ x [ i � 12 η 1 η 2 ( C 1 C 2 + C 2 C 1 ) ± i � 34 η 3 η 4 ( C 3 C 4 + C 4 C 3 )] χ x → (4,8,4) (1,2,1) 2 for the d 1 − M ( i ) M ( i ) 0 T H -1 -2 1 4 0 2 3 4 Hoelbling, PLB696, 422(2011) [1009.5362]. S nf ( M ( i ) S st ( M ( i ) T ) H ) →