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)
• T. Kimura, S. Komatsu, T. Misumi, T. Noumi, S. Torii, S. Aoki, JHEP 1201:048 (2012)
• T. Misumi, Ph.D Thesis, Kyoto University (2012)
• M. Creutz, T. Kimura, T. Misumi, PRD 83:094506 (2011)

02/09/2012 NTFL workshop@Yukawa Institute, Kyoto

◆Wilson fermion

§ additive mass renormalization → Fine-tune for chiral limit

a

• d4x ¯

ψ(x)D2

µψ(x)

~

Dov = 1 + γ5 HW(m)

• H2

W(m)

= 1 + DW(m)

• D†

W(m)DW(m)

Overlap & Domain-wall fermion

{γ5, Dov} = aDovγ5Dov, S = Snf + SW with SW = −ar 2

• n,µ

a4 ¯ ψn (ψn+ˆ

µ − 2ψn + ψn−ˆ µ)

a2

Ginsparg-Wilson : § 15 species are decoupled → doubler-less

Im ! Re !

Dnaive Dwilson

!

Im ! Re !

1 4 4 6 1 16

as m = 0,

1 15 Dov

Introduction

◆Staggered fermion

§ 4 species → more than 3..... One naive fermion → 4 Staggered fermions

ηµ(n) = (−1)

P

ν<µ nν

Spin diagonalization :

!! !!"# !!"$ !!"#"$ "# "#!" "#$% "#$%$&

' &'

・4-flavor Dirac fermions

spin flavor

as Γ55 = γ5 ⊗ γ5.

~

§ chiral symmetry + one-component → suitable for calculations

ψn = γn1

1 γn2 2 γn3 3 γn4 4 χn,

¯ ψn = ¯ χnγn4

4 γn3 3 γn2 2 γn1 1

n = (−1)n1+n2+n3+n4

Snf = 4Sst = 4[1 2

• n,µ

ηµ(n)¯ χn (χn+ˆ

µ − χn−ˆ µ) + m

2

• n

¯ χnχn]

・Flavored chiral symmetry

Properties

Naive

#=16

Staggered

#=4

Wilson

#=1

Overlap

#=1

Overlap form.

4 tastes Fine tuning Numerical expense

Wilson term Chiral broken GW symmetry

4 copies

Naive

#=16

Staggered

#=4

Wilson

#=1

Overlap

#=1

Overlap form.

4 copies Fine tuning Numerical expense

Wilson term Chiral broken GW symmetry Flavored-mass term

Generalization 4 tastes

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2

• 1
• 0.5

0.5 1

Dnf

Flavored mass terms

× →

• 1
• 0.5

• 2
• 1

Naive Staggered

4 copies

Creutz, Kimura, TM, JHEP1012,041 [1011.0761] de Forcrand, Kurkela, Panero, [1102.1000] Adams, PRL104, 141602 [0912.2850] Golterman, Smit (1984)

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2

• 1
• 0.5

0.5 1

Dst

e.g.) 2-split flavored mass (2,2) (8,8)

~ Generalized Wilson terms ~

MP =

• sym.

4

• µ=1

Cµ,

Re [λ] Im [λ]

MA = ζ5

• sym.

4

• µ=1

Cµ

Naive

#=16

Staggered

#=4

Wilson

#=1

Overlap

#=1

Overlap form.

4 copies Fine tuning Numerical expense

Wilson term Chiral broken GW symmetry

???

Flavored-mass term

Generalized Wilson&overlap 4 tastes Generalization

Naive

#=16

Staggered

#=4

Wilson

#=1

Overlap

#=1

Overlap form.

4 copies

St.Wilson

#=1

St.Overlap

#=1

Fine tuning Numerical expense

Wilson term Chiral broken GW symmetry

Faster Wilson & Overlap

Flavored-mass term

Generalized Wilson&overlap 4 tastes Generalization

• 1. Flavored-mass terms

 Naïve fermion

• M. Creutz, T. Kimura, TM, JHEP1012:041 (2010)

label position χ charge Γ type 1 (0, 0, 0, 0) + 1 S 2 (π, 0, 0, 0) − iγ1γ5 A 3 (0, π, 0, 0) − iγ2γ5 A 4 (π, π, 0, 0) + iγ1γ2 T 5 (0, 0, π, 0) − iγ3γ5 A 6 (π, 0, π, 0) + iγ1γ3 T 7 (0, π, π, 0) + iγ2γ3 T 8 (π, π, π, 0) − γ4 V 9 (0, 0, 0, π) − iγ4γ5 A 10 (π, 0, 0, π) + iγ1γ4 T 11 (0, π, 0, π) + iγ2γ4 T 12 (π, π, 0, π) − γ3 V 13 (0, 0, π, π) + iγ3γ4 T 14 (π, 0, π, π) − γ2 V 15 (0, π, π, π) − γ1 V 16 (π, π, π, π) + γ5 P

~ general terms to lift degenerate species ~

as Γ−1

(i) γµΓ(i) = γ(i) µ

・16 species ・16-flavor multiplet Flavor mass matrix

Ψ(p) =         ψ(1)(p − p(1)) ψ(2)(p − p(2)) . . . ψ(16)(p − p(16))        

¯ Ψ(1 ⊗ X)Ψ

Mass matrix

ψ(1)(p − p(1)) = 1 24(1 + cos p1)(1 + cos p2)(1 + cos p3)(1 + cos p4)Γ(1)ψ(p), ψ(2)(p − p(2)) = 1 24(1 − cos p1)(1 + cos p2)(1 + cos p3)(1 + cos p4)Γ(2)ψ(p), ψ(3)(p − p(3)) = 1 24(1 + cos p1)(1 − cos p2)(1 + cos p3)(1 + cos p4)Γ(3)ψ(p), . . . ψ(16)(p − p(16)) = 1 24(1 − cos p1)(1 − cos p2)(1 − cos p3)(1 − cos p4)Γ(16)ψ(p),

→ Independent fields in low energy limit

Ψ(p) =         ψ(1)(p − p(1)) ψ(2)(p − p(2)) . . . ψ(16)(p − p(16))        

16-flavor multiplet

¯ Ψ(1 ⊗ X)Ψ

Mass matrix

◆Point-split fields

• M. Creutz (2010), for minimally doubled fermions.

・Conditions on flavored-mass terms

(1) gamma-5 hermiticity : D† = γ5Dγ5

to γ5 ⊗ (τ3 ⊗ τ3 ⊗ τ3 ⊗ τ3)

spin flavor

(2) O(a) irrelevant term

a

• d4x ¯

ψ(x)D2

µψ(x)

det(D) ≥ 0

~

essential for euclidian vector-like theory dim-5 operator vanishes in a→0

※ ・Physical modes in the continuum limit

Ψ(p) =         ψ(1)(p − p(1)) ψ(2)(p − p(2)) . . . ψ(16)(p − p(16))        

for

・Rotational symmetry

◆ Flavored-mass terms

・ MV (MA) → Wilson term

!

Im ! Re !

• n

¯ ψn(MP − 1)ψn → −a

• d4x ¯

ψ(x)D2

µψ(x) + O(a2)

V : ¯ Ψ (1 ⊗ (τ3 ⊗ 1 ⊗ 1 ⊗ 1)) Ψ = cos p1 ¯ ψψ T : ¯ Ψ (1 ⊗ (τ3 ⊗ τ3 ⊗ 1 ⊗ 1)) Ψ = cos p1 cos p2 ¯ ψψ A : ¯ Ψ (1 ⊗ (1 ⊗ τ3 ⊗ τ3 ⊗ τ3)) Ψ = 4

• µ=2

cos pµ

• ¯

ψψ P : ¯ Ψ (1 ⊗ (τ3 ⊗ τ3 ⊗ τ3 ⊗ τ3)) Ψ = 4

• µ=1

cos pµ

• ¯

ψψ

MV =

• µ

Cµ, MT =

• perm.
• sym.

CµCν, MA =

• perm.
• sym.
• ν

Cν, MP =

• sym.

4

• µ=1

Cµ,

・ O(a) irrelevant terms ・ low-energy species-splitting terms

Dirac spectra with flavored mass terms

(8,8) (4,8,4) → Multi-flavor Wilson & Overlap (although we need care about renormalization)

Dnf − MP

MP+MT : (4,12) MP+MV : (5,1,10) MT+MV : (10,5,1) (1,15)

: :

MV+MA : (1,14,1) MA+MV +MT1+MT2 : (3,12,1)

Dnf − (MV + MT + MA + MP )

Dnf − M (i)

T

◆ Pseudo-scalar type

MP =

• sym.

C1C2C3C4

(8,8)

Dnf − MP

・Index theorem from spectral flow

Index(Dnf) = - Spectral flow(H)

36×36 lattice, randomness δ=0.25, Q=1

doubled

Index(Dgw) = -4

H = γ5(Dnf − rMP)

Index(Dnf) = 2d(−1)d/2Q

cf.) For staggered, Adams (2009)

λ(r)

λ(r)

8 (+) and 8 (-) masses

P : ¯ Ψ (1 ⊗ (τ3 ⊗ τ3 ⊗ τ3 ⊗ τ3)) Ψ

consistent

× →

• 1
• 0.5

• 2
• 1

(2,2) (8,8) MP

¯ ψxC1C2C3C4ψx → ±¯ χx(η1η2η3η4C1C2C3C4)χx.

Adams-type flavored mass

・spin diagonalization 4 Adams fermions derived up to sign

Snf(MP) → Sst(MA)

st(MA)

• D. Adams (2009)

¯ ψxψx+ˆ

1+ˆ 2+ˆ 3+ˆ 4 = ¯

χxγx4

4 γx3 3 γx2 2 γx1 1 γx1+1 1

γx2+1

2

γx3+1

3

γx4+1

4

χx+ˆ

1+ˆ 2+ˆ 3+ˆ 4

= (−1)x2+x4 ¯ χxγ5χx+ˆ

1+ˆ 2+ˆ 3+ˆ 4

→ ±¯ χxη1η2η3η4χx+ˆ

1+ˆ 2+ˆ 3+ˆ 4

(γ5 diagonalized)

de Forcrand, Kurkela, Panero, [1102.1000]

◆Tensor type

(4,8,4)

MT = M (1)

T + M (2) T + M (3) T ,

M (1)

T

= 1 2(C1C2 + C2C1) + 1 2(C3C4 + C4C3), M (2)

T

= 1 2(C1C3 + C3C1) + 1 2(C2C4 + C4C2), M (3)

T

= 1 2(C1C4 + C4C1) + 1 2(C2C3 + C3C2).

36×36 lattice, randomness δ=0.25, Q=1

Index(D) = -2

・Index theorem from spectral flow

Index(D) = - Spectral flow(H)

Dnf − M (i)

T

Double rotation symmetric units : x → R(µν)R(ρσ)x

H = γ5(Dnf − rM (i)

T )

Index(D) = 2d−1(−1)d/2Q λ(r)

λ(r)

(1,2,1)

4 1 2 3 4

• 2
• 1

1 2

(4,8,4)

¯ ψx[(C1C2 + C2C1) + (C3C4 + C4C3)]ψx → ±¯ χx[i12η1η2(C1C2 + C2C1) ± i34η3η4(C3C4 + C4C3)]χx

for the d − M (i)

T

Snf(M (i)

T )

→ Sst(M (i)

H )

Hoelbling-type flavored mass

・spin diagonalization 4 Hoelbling fermions (3 units) up to sign M (i)

H

¯ ψxψx+ˆ

1+ˆ 2 + ¯

ψxψx+ˆ

3+ˆ 4 = (−1)x2 ¯

χxγ1γ2χx+ˆ

1+ˆ 2 + (−1)x4 ¯

χxγ3γ4χx+ˆ

3+ˆ 4

→ ±¯ χxi12η1η2χx+ˆ

1+ˆ 2 ± ¯

χxi34η3η4χx+ˆ

3+ˆ 4

[σ12, σ34] = 0

※ two terms simultaneously diagonalizable :

Hoelbling PLB696, 422(2011) [1009.5362]. de Forcrand (2010) Hoelbling, PLB696, 422(2011) [1009.5362].

MT = M (1)

T + M (2) T + M (3) T

※ Direct decomposition is impossible, unlike Adams’ case.

• 2
• 1.5
• 1
• 0.5

• 2
• 1

MH = M (1)

H + M (2) H + M (3) H ,

M (1)

H =

i 2 √ 3[12η1η2(C1C2 + C2C1) + 34η3η4(C3C4 + C4C3)], M (2)

H =

i 2 √ 3[13η1η3(C1C3 + C3C1) + 42η4η2(C4C2 + C2C4)], M (3)

H =

i 2 √ 3[14η1η4(C1C4 + C4C1) + 23η2η3(C2C3 + C3C2)].

Three units of Hoelbling flavored mass

MT → MH

M (i)

T

→ M (i)

H

MH = M (1)

H + M (2) H + M (3) H

[σµν, σνρ] = 0

×

(6,8,2) (1,2,1)

Hoelbling, PLB696, 422(2011) [1009.5362].

• 2. Symmetries of St.Wilson

broken to 2-link shift for SA broken to 4-link shift for SH ・Axis reversal broken to shifted axis reversal remain in SA broken to subgroup in SH ・Conjugation

Sρ : χx → ζρ(x)χx+ˆ

ρ,

¯ χx → ζρ(x)¯ χx+ˆ

ρ,

Uµ,x → Uµ,x+ˆ

ρ,

Iρ : χx → (−1)xρχIx, ¯ χx → (−1)xρ ¯ χIx, Uµ,x → Uµ,Ix,

Rρσ : χx → SR(R−1x)χR−1x, ¯ χx → SR(R−1x)¯ χR−1x, Uµ,x → Uµ,Rx,

・Shift symmetry ・Rotation remain in SA broken in SH

C : χx → x ¯ χT

x , ¯

χx → −x ¯ χT

x , Uµ,x → U ∗ µ,x

Sµ : φ(p) → exp(ipµ)Ξµ φ(p).

Iρ : φ(p) → ΓρΓ5ΞρΞ5 φ(Ip)

: spinor-space gamma : taste-space gamma

and Ξµ.

matrices Γµ

× erties {Γµ, Γν} = 2δµν, {Ξµ, Ξν} = 2δµν and {Γµ, Ξν} = 0.

Rρσ : φ(p) → exp(π 4 ΓρΓσ) exp(π 4 ΞρΞσ) φ(R−1p)

as φ(p)A ≡ χ(p + πA) (−π/2 ≤ pµ < π/2)

§ Separating spinor & taste in momentum space

Shift Axis inv. Rotation Conjugation C : φ(p) → ¯

φ(−p)T

・Parity → 4th-shift × spatial axis ・Charge conjugation → triple-rotation × conjugation for Hoelbling-type ・Shifted square rotation → µν-rot × νµ-rot × µ-shift × ν-shift SνSµRνµRµν ∼ exp(ipµ + ipν)ΓµΓν φ(˜ p)

◆ Discrete symmetries in Staggered Wilson

IsS4 ∼ exp(ip4)Γ4 φ(−p, p4)

Re-interpretation of Golterman-Smit (1984)

These symmetries hold for . Indicates restoration of essential symmetries in the continuum limit.

MA, MH, M (i)

H

• 3. Central cusps

◆ Wilson fermion without on-site terms Extra U(1)V symmetry emerge !

ψx → eiθ(−1)x1+x2+x3+x4, ¯ ψx → ¯ ψxeiθ(−1)x1+x2+x3+x4

S = 1 2

• x,µ

¯ ψx[γµ(ψx+µ − ψx−µ) − (ψx+µ + ψx−µ)]

・prohibits additive mass renormalization ! ・will be spontaneously broken due to pion condensation !

condensate, ¯ ψγ5ψ

Aoki phase

§ Strong-coupling meson potential

cosh(mSPA ) = 1 + 2M 2

W(16 + M 2 W)

16 − 15M 2

W

p = (π, π, π, π + imSPA ),

Massless NG boson

It is expected to describe 6-flavor Twisted-mass QCD.

¯ ψψ ↔ ¯ ψγ5ψ different bases

MW ≡ m + 4r = 0

Kimura, Komatsu, Misumi, Noumi, Torii, Aoki, JHEP 1201:048 (2012) Creutz, Kimura, Misumi, PRD 83:094506 (2011),

◆ For other naive flavored mass terms MA : U(1)V restored MT : None MP : None ◆ For staggered flavored mass terms MA : None MH : Naive conjugation

Restoration of U(1)V is peculiar to odd-link flavored mass.

Odd-link flavored mass for staggered fermions possible ? ・gamma5-hermiticity breaks down. ・chiral symmetry remains.............

• 2
• 1.5
• 1
• 0.5

• 0.25
• 0.2
• 0.15
• 0.1
• 0.05

Isospin-type possible? Automatically overlap !? Isospin-type may work.

T.Kimura and TM (2011) in communication with de Forcrand (2011)

M1L =

• µ

ξµCµ ∼

• µ

(1 ⊗ γµ) + O(a)

C : χx → ¯ χT

x ,

¯ χx → χT

x ,

Uµ,x → U ∗

µ,x

• 4. Summary
• 1. Flavored-mass terms give us new types of Wilson and
• verlap fermions.
• 2. Staggered-Wilson can be derived from generalized

Wilson fermions through spin-diagonalization.

• 3. Central cusps are expected to describe twisted-mass QCD

without any parameter tuning.

case [13]: we start with a smooth U(1) gauge field with topological charge Q, Ux,x+e1 = eiωx2, Ux,x+e2 =    1 (x2 = 1, 2, · · · , L − 1) eiωLx1 (x2 = L) , (30) where L is the lattice size and ω is the curvature given by ω = 2πQ. Then, to emulate a typical gauge configuration of a practical simulation, we introduce disorder effects to link variables by random phase factors, Ux,y → eirx,yUx,y, where rx,y is a random number uniformly distributed in [−δπ, δπ]. The parameter δ determines the magnitude of disorder.

◆gauge configuration

2 − (cos ap1 + cos ap2) = a2p2

1 + a2p2 2

2 + O(a3)

1 − cos ap1 cos ap2 = a2p2

1 + a2p2 2

2 + O(a3)

cf.)

MA, MH