dynamical simulations with twisted mass fermions including the … · 2006-08-03 · ∗...

Dynamical simulations with twisted mass fermions

including the strange quark

Enno E. Scholz

in collaboration with:

European Twisted Mass (ETM) Collaboration

DESY

Hamburg Universitat MunsterINFN Rome, Milano NIC Zeuthen

ECT∗

Trento

Univ. of Liverpool

DESY

Zeuthen

Lattice Hadron Physics — LHP 2006

Jefferson Laboratory, August 2006

• twisted-mass Wilson fermions for Nf = 2 viable alternative

• “realistic” QCD-simulations should include the strange quark

• no “single twisted fermion”

• different approaches possible with twisted mass

∗ use untwisted (standard Wilson) strange-quark

∗ use another doublet (Nf = 2 + 2) (works only for quenched studies)

• use mass-splitting a’la Frezzotti-Rossi for 2nd doublet

Nf = 2 + 1 + 1

degenerate up, down; non-degenerate strange, charm (“charm as bonus”)

E. E. Scholz — Dynamical twisted mass fermions including the strange quark 1

Mass-split doublets

fermionic action with twisted mass term and mass-splitting term [Frezzotti, Rossi, 2004]

χ-basis: S =P

xy χx Qχx,y χy

Qχx,y = δx,y

h

µκ + iγ5τ1aµσ + τ3aµδi

− 1

2

4X

µ=±1

δx,y+µγµUyµ

| {z }≡Nx,y

− r

2

4X

µ=±1

δx,y+µUyµ

| {z }≡Rx,y

ψ-basis: ψx = 1√2(1 + iγ5τ1)χx, ψx = χx

1√2(1 + iγ5τ1)

Qψx,y =

1

2(1 − iγ5τ1)Q

χx,y(1 − iγ5τ1) = aµσ + τ3aµδ + N − iγ5τ1(µκ + R)

physical basis ψphys: ψphysx = exp

hi2

“

ω − π2

”

γ5τ1

i

ψx, ψphysx = ψx exp

hi2

“

ω − π2

”

γ5τ1

i

twist angle ω: (tuning to full twist. . . )

“ tanω =aµσ

µκ − µκcrit

”


Set-up for dynamical Nf = 2 + 1 + 1-simulations

• light doublet: µδ = 0 representing degenerate up- and down-quarks (Nf = 2)

∗ parameters: µκ,l = 1/(2κl), aµl = aµσ,l more convenient: τ1 → τ3

∗ tune κl to κl,crit

• heavy doublet: µδ 6= 0 representing non-degenerate strange- and charm-quarks

∗ parameters: µκ,h = 1/(2κh), aµσ, aµδ∗ tune κh to κh,crit∗ based on (parity) × (µ → −µ)-symmetry: PCAC-defined critical quark mass only influenced by

O(a)-effects due to TM-terms [Farchioni et al., 2005; Chiarappa et al., 2006]

⇒ κl,crit ≈ κh,crit

• gauge action: gauge coupling β = 6/g2 (improved gauge actions tree-level Symanzik, Iwasaki,

DBW2 preferable to pure Wilson)

• 5 parameters:

β, κ (= κl = κh), aµl, aµσ, aµδ

ultimate goal: κ = κcrit & µ’s such that renormalized masses close to physical masses


tuning to κcrit

• in light-sector measure untwisted PCAC quark mass

amPCACχl ≡

〈∂∗µA

+l,xµP

−l,y〉

2〈P+l,xP

−l,y〉

• analogously in heavy sector . . . but

• using symmetry of the action:

∗ parity × (µ → −µ)

∗“

χh → exp(iπ2τ1)χh, χh → χh exp(−iπ2τ1)”

×“

µδ → −µδ”

mPCACχh = mPCAC

χl + O(a)


extracting the K’s, D’s, . . .

• pion-mass and -decay constant like Nf = 2

• including the heavy-doublet: Kaon-D-meson sector

CK+ = χsΓCχu CD0 = χcΓCχu

CK0 = χsΓCχd CD− = χcΓCχd

C = S, P, V,K ΓC = 1, γ5, γµ, γµγ5

• expect better signal from scalar correlators

V = (ZPPK+, ZPPD0, ZSSK+, ZSSD0)T

V = (−ZPPK−,−ZPPD0,−ZSSK−,−ZSSD0)

• correlator-matrix C = 〈V ⊗ V〉• fully renormalised (physical) matrix C from

V = MV ¯V = VM−1


M(ωl, ωh) =

0

BBBBBBBB@

cosωl2 cos

ωh2 − sin

ωl2 sin

ωh2 i sin

ωl2 cos

ωh2 i sin

ωh2 cos

ωl2

− sinωl2 sin

ωh2 cos

ωl2 cos

ωh2 i sin

ωh2 cos

ωl2 i sin

ωl2 cos

ωh2

i sinωl2 cos

ωh2 i sin

ωh2 sin

ωl2 cos

ωl2 cos

ωh2 − sin

ωh2 sin

ωl2

i sinωh2 cos

ωl2 i sin

ωl2 cos

ωh2 − sin

ωl2 sin

ωh2 cos

ωh2 cos

ωl2

1

CCCCCCCCA

restore parity- and flavour-symmetry: C = MCM−1 diagonal

⇒ ωh

extract masses from C


Mass-splitting in K-D-sector

using combined parity-isospin symmetry of the heavy-light action

Qψl = aµl +N − iγ5τ3(µκl + R) Qψ

h = aµσ + τ3aµδ +N − iγ5τ1(µκh + R)

light : heavy :

Parity ⊗ τ1 Parity ⊗ τ3

u(x) → γ0d(Px) c → γ0c(Px)

d(x) → γ0u(Px) s → −γ0s(Px)

no mass splittings in Kaon- or D-meson-doublets

recent (quenched) study by Abdel-Rehim, Lewis et al.:

same isospin-direction in both doublets leads to observed splitting


Polynomial Hybrid Monte Carlo (PHMC)-algorithm

• HMC with mass preconditioning efficient algorithm for unsplit-TM. . .

• . . . but unfortunately not applicable to µδ 6= 0-case.

det(Q) cannot be written as single flavor det(Q′2)

• use polynomial approximation P1(Q2) ≃ (Q2)−

12 of order n1 [Frezzotti, Jansen, 1997-99]

• combine with stochastic correction step (noisy correction) [Montvay, EES, 2005]

P1(x)P2(x) ≃ x−12, n2 > n1

(further correction steps and/or “polynomial mass preconditioning” may be useful at smaller masses. . . )

• adjusting step-size(s) in HMC-step plus polynomial orders n1, n2 allows for good tuning properties

• improvement: mixed (P)HMC: heavy doublet–PHMC, light doublet–HMC[Chiarappa et al., 2005]

• other possible algorithms for heavy doublet

∗ Rational-HMC (did not try yet for TM)

∗ Two-Step Multi-Boson or Multi-Step Multi-Boson (less efficient than PHMC)

[Montvay, 1995; Montvay, EES, 2005]

∗ in principle every algorithm capable of odd Nf . . .


Dynamical Simulations

Chiarappa,. . . , EES, . . . , Urbach, hep-lat/0606011

• tree-level Symanzik gauge action, two lattice spacings (fixed physical volume: aL ≃ 2.4fm)

∗ a ≃ 0.20fm (β = 3.25, L3 × T = 123 × 24)

aµl = 0.01, aµσ = 0.315, aµδ = 0.285, κ ∈ [0.1740, 0.1755] (7 values, 10 runs)

∗ a ≃ 0.15fm (β = 3.35, L3 × T = 163 × 32)

aµl = 0.0075, aµσ = 0.2363, aµδ = 0.2138, κ ∈ [0.1690, 0.1710] (9 values)

• varied κ(= κl = κh) to find κcrit, explore phase-structure

• lattice-spacing, light-doublet similar to previous studies (DBW2 and pure Wilson gauge action)

• performed at

ZAM Julich

(IBM-p690) atNIC Julich

and PC-Cluster atDESY

Hamburg


a ≈ 0.20fm, 123× 24

0.49

0.5

0.51

0.52

0.53

0.174 0.1745 0.175 0.1755 0.176

aver

age

plaq

uette

κ = κl = κh

123x24 latticetlSym gauge actionβ = 3.25aµl = 0.01aµσ = 0.315aµδ = 0.285

0

0.2

0.4

0.6

0.8

1

1.2

-0.1 -0.05 0 0.05 0.1

(am

π,K)2

amχlPCAC

123x24 latticetlSym gauge actionβ = 3.25

aµl = 0.01aµσ = 0.315aµδ = 0.285

(amπ)2

(amK)2

• strong metastability

• minimal pion mass ≈ 670MeV

• minimal mK ≃ 920MeV


. . . just a reminder

β

µ

κ = (2µκ)-1

β1


a ≈ 0.15fm, 163× 32

0.5

0.51

0.52

0.53

0.54

0.169 0.1695 0.17 0.1705 0.171

aver

age

plaq

uette

κ = κl = κh

163x32 latticetlSym gauge actionβ = 3.35aµl = 0.0075aµσ = 0.2363aµδ = 0.2138

0.52

0.525

0.53

0.535

0.54

0 500 1000 1500 2000 2500

1e-04

1e-03

1e-02

1e-01

aver

age

plaq

uette

min

imal

eig

enva

lue

trajectory

163x32 latticetlSym gauge actionβ = 3.35κl=κh = 0.1706aµl = 0.0075aµσ = 0.2363aµδ = 0.2138

⟨ ❒ ⟩ = 0.53700(7)τint = 12.6

average plaquetteminimal eigenvalue (κl, aµl)

• sharp rise in 〈�〉? weaker first order phase transition

or

? cross-over

∗ not distinguishable in finite volume

• not an algorithmic imperfection

∗ transition low → high plaquette phase

∗ “crossing near origin”

∗ lowest EV fluctuating in high plaquette phase


0

0.2

0.4

0.6

0.8

1

1.2

-0.05 0 0.05 0.1 0.15

(am

π,K)2

amχlPCAC

163x32 latticetlSym gauge actionβ = 3.35

aµl = 0.0075aµσ = 0.2363aµδ = 0.2138

(amπ)2

(amK)2

• minimal pion mass ≈ 450MeV

• keep in mind: varying κ at fixed µl

⇒ mπ > 0

• minimal mK ≃ 850MeV

• tuning mK possible by changing µσ, µδ


. . . using χPT

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-20 -15 -10 -5 0 5 10 15 20

(mπ/

mK)2

amχlPCAC/µl

tlSym gauge action, β, aµl, aµσ, aµδ 3.25, 0.01, 0.315, 0.285

3.35, 0.0075, 0.2363, 0.21383.35, 0.0075, 0.15, 0.15 m2

π

m2K

=2mud

mud +ms

mud =q

(ZAmPCACχl )2 + µ2

l

ms =q

(ZAmPCACχh )2 + µ2

σ − ZP

ZSµδ

fitted ZP/ZS ≃ 0.45

take ZA as input

mPCACχl ≈ mPCAC

χl

minimum close to physical value


Summary

• Nf = 2 + 1 + 1-mechanism understood

• dynamical simulations with PHMC-algorithm

• stronger phase-structure compared Nf = 2

• minimal pion mass 450MeV

• going to a ≃ 0.10fm on 243 × 48 should allow for pion masses around 300MeV


dynamical simulations with twisted mass fermions including the … · 2006-08-03 · ∗...

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