The first Two Fermion Generations inTwisted Mass Lattice QCD

Karl Jansen ��������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

for the European Twisted Mass Collaboration

• Maximally twisted mass fermions, Nf = 2 results

• Introducing Nf = 2 + 1 + 1 flavours

• Using Nf = 2 + 1 + 1 flavours of quarks

– light meson physics– Osterwalder-Seiler valence quarks– non-perturbative renormalization

Why the first two generations?

• want to include as many quarks as possible: charm still realistic

• charm quark mass

• needed for charmed mesons, e.g. η, η′, ηc and baryonsyesterdays talk by M. Shepherd

• decay constants fD, fDs

• heavy quark effects in operator matrix elements

• running of αs with Nf = 4

• very natural to include strange/charm doublet for twisted mass

1

• Cyprus (Nicosia)C. Alexandrou, M. Constantinou

• France (Orsay, Grenoble)R. Baron, B. Bloissier, Ph. Boucaud, M. Brinet, J. Carbonell,P. Guichon, P.A. Harraud, O. Pene

• Italy (Rome I,II,III, Trento)P. Dimopoulos, R. Frezzotti, V. Lubicz, G. Martinelli, G.C. Rossi, L. Scorzato,S. Simula, C. Tarantino

• Netherlands (Groningen)E. Pallante, S. Reker

• Poland (Poznan)K. Cichy, A. Kujawa

• Spain (Huelva, Madrid, Valencia)V. Gimenez, D. Palao, J. Rodriguez-Quintero, A. Shindler

• Switzerland (Bern)A. Deuzemann, U. Wenger

• United Kingdom (Glasgow, Liverpool)G. McNeile, C. Michael

• Germany (Berlin/Zeuthen, Bonn, Hamburg, Munster)V. Drach, F. Farchioni, J. Gonzalez Lopez, G. Herdoiza, K. Jansen, I. Montvay,G. Munster, M. Petschlies, T. Sudmann, C. Urbach, M. Wagner

2

Wilson (Frezzotti, Rossi) twisted mass QCD (Frezzotti, Grassi, Sint, Weisz)

Fermion action of twisted mass fermions

Sl =∑lx χx

[mq + 1

2γµ[∇µ +∇∗µ

]− ar12∇∗µ∇µ + iµtmτ3γ5

]χlx

Sh =∑x χ

hx

[mq + 1

2γµ[∇µ +∇∗µ

]− ar12∇∗µ∇µiγ5τ1µσ + τ3µδ

]χhx

• quark mass parameter mq , twisted mass parameter µtm

• strange and charm quark masses

ms,c = Z−1P (µσ ± ZP/ZSµδ)

simulation: ZP/ZS ≈ 0.65

• note, mq the same in Sl and Sh

3

Pros and Cons of a generic fermion action

Pro maximal twisted mass:

• O(a)-improvement for all physical quantities automatically

• helps to simplify mixing patterns in non-perturbative renormalization

• explicit infrared regularization through µtm

Con twisted mass:

• isospin violation at any a 6= 0

• observe large O(a2) effectin neutral pion mass(a similar large O(a2) effectexpected for Wilson fermions inanother quantity)

mπ ≈ 430MeVstout

mπ ≈ 330MeVmπ ≈ 260MeV

(−∆) Nf = 2

(+∆) Nf = 2 + 1 + 1

(a/r0)2

∆=

r2 0×( (m

± PS)2

−(m

0 PS)2)

0.050.040.030.020.010.00

0.6

0.4

0.2

0.0

-0.2

-0.4

-0.6

Controlling the effect theoretically:Frezzotti, Rossi (2007), Dimopoulos et.al (2010), Colangelo, Wenger, Wu (2010), Bar (2010)

what counts in the end: Universality

4

Tuning to maximal twist

Maximal twist: tune mq such that

mPCAC =∑

x〈∂0Aa0(x)Pa(0)〉2∑

x〈Pa(x)Pa(0)〉= 0

• tuning of mq at each µtm used

• demand mPCAC . 0.1µtm

• demand ∆(mPCAC) . 0.1µtm

5

Selected results for Nf = 2

Simulation results versus PDG Low energy constants

6

I=2 Pion scattering length(X. Feng, D. Renner, K.J.)

energy determined fromR(t) = 〈(π+π+)†(t+ ts)(π

+π+)(ts)〉/〈(π+)†(t+ ts)π+(ts)〉2

→ ∆E = c/L3 · aI=2ππ (1 +O(1/L)

E865 (BNL) mπaI=0ππ = 0.203 (33) and mπa

I=2ππ = −0.055 (23) .

NA48/2 (CERN) mπaI=0ππ = 0.221 (5) and mπa

I=2ππ = −0.0429 (47).

our work mπaI=2ππ = −0.04385 (28)(38)

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The ρ-meson resonance: dynamical quarks at work(X. Feng, D. Renner, K.J.)

• usage of three Lorentz frames

0.3 0.35 0.4 0.45 0.5 0.55 0.6aE

CM

0

0.5

1

sin2 (δ

)

CMFMF1MF2sin

2(δ)=1=>aM

R

0 0.05 0.1 0.15 0.2mπ

2(GeV

2)

0

0.05

0.1

0.15

0.2

0.25

Γ R(G

eV)

a=0.086fma=0.067fmPDG data

mπ+ = 330 MeV, a = 0.079 fm, L/a = 32 fitting z = (Mρ + i12Γρ)2

mρ = 1033(31) MeV, Γρ = 123(43) MeV

8

Simulation setup for Nf = 2 + 1 + 1Configurations available through ILDG

β a[fm] L3T/a4 mπ[MeV] status

1.9 ≈ 0.085 24348 300 – 500 ready1.95 ≈ 0.075 32364 300 –500 ready2.0 ≈ 0.065 32364 300 ready2.1 ≈ 0.055 48396 300 – 500 running/ready

643128 230 thermalizing643128 200 planned963192 160 planned

• trajectory length always one

• 1000 trajectores for thermalization

• ≥ 5000 trajectores for measurements

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Nf = 2 + 1 + 1 light quark sector: scaling

Nf = 2 r0mPS = 0.614Nf = 2 r0mPS = 1.100

Nf = 2 + 1 + 1 r0mPS = 0.728

r0fPS

(a/r0)2

0.060.040.020

0.42

0.38

0.34

0.30

0.26

Nf = 2 + 1 + 1 r0mPS = 0.614Nf = 2 + 1 + 1 r0mPS = 0.728

r0mN

(a/r0)2

0.040.030.020.010

3.1

2.9

2.7

2.5

2.3

2.1

1.9

pseudoscalar decay constant fPS nucleon mass

10

Nf = 2 + 1 + 1 light quark sector: χPT fit

• central values + stat. error : fπ = 130.4(2) MeV ; scale

• estimate systematic effects : lattice artifacts, FSE

Nf = 2 Nf = 2 + 1 + 1¯3 3.70(27) 3.50(31)

¯4 4.67(10) 4.66(33)fπ/f0 1.076(3) 1.076(9)B0 [MeV] 2437(120) 2638(200)

〈r2〉NLOs [fm2] 0.710(28) 0.715(77)

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Nf = 2 + 1 + 1 light quark sector: adding strange quark

fπ (input)

fK

fK(a = 0.060 fm)

fK(a = 0.079 fm)

fπ(a = 0.060 fm)

fπ(a = 0.079 fm)

Fit

m2π [GeV2]

fPS[M

eV]

0.250.20.150.10.050

190

180

170

160

150

140

130

120

110

• fit β = 1.95 and β = 2.10 simultaneously

• from setting m2PS(µ`, µs, µs) = 2m2

K −m2π

• mπ = 135 MeV, fπ = 130.7 MeV,mK = 497.7 MeV

prelinimary fit results:

• fK/fπ = 1.224(13), fK = 160(2) MeV, ¯4 = 4.78(2)

• errors statistical only

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Nf = 2 + 1 + 1 heavy quark sector

Wilson twisted mass Dirac operator for (c, s) pair:

Dh =

(γµ∇µ + µσ + µδ iγ5

(a2∇∗µ∇µ −mq

)

iγ5(a2∇∗µ∇µ −mq

)γµ∇µ + µσ − µδ

)

• mixing of c and s flavour and of parity

• Kaon is the ground state : good precision

• D meson appears as an excited state

• three independent methods:

– generalised eigenvalue problem– multi-exponential fits– imposing parity and flavour restoration

at finite a

• they provide consistent results for mD

• overcome mixing of flavour ; mixed action

cs

cO(a)O(a)

c

j = {strange, Γ = 1}j = {strange, Γ = γ5}j = {charm, Γ = 1}j = {charm, Γ = γ5}

t/a

eigenvectors

|v(c,γ

5)

j|2

20151050

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

symmetry restaurationmulti-exponential

GEP

method

am

D

321

0.95

0.9

0.85

0.8

0.75

13

Nf = 2 + 1 + 1 approaching the charm quark

• introduce Wilson twisted mass doublets in the valence sector

Dtm(µval) = D +mcrit + i µvalγ5τ3

(Osterwalder, Seiler (1990), Pena et al. (2004); Frezzotti, Rossi (2004))

• mcrit from unitary set-up

• 4− 6 values for µval in the strange µs and the charm µc regioninversions with multi-mass solver

• matching to unitary set-up using mK and mD

⇒ obtain simulated µs and µc

14

Unitary versus Osterwalder-Seiler: fK

• the unitary fK can be computed from: fK = (m` +ms)〈0|PK|K〉m2K

with ms = µσ − (ZP/ZS)µδ

• similar formula for fD

• PK is the physical Kaon projecting operator

• the mixed action fK computed from: fPS =(µ(1)val + µ

(2)val

)|〈0|P |PS〉|

mPS sinhmPS,

fPS(OS) mixedfPS(OS,Mtm) unit.

(af0)2

0.0050.0040.0030.0020.0010.000

1.70

1.65

1.60

1.55

1.50

1.45

1.40

1.35

1.30

Test for Nf = 2 situation for Nf = 2 + 1 + 1

15

Projection operator

• unitary kaon decay constant

fK = µ`+µσ−(ZP/ZS)µδ2m2

K· 〈0|(PK − PD) + i(ZS/ZP )(SK + SD)|K〉

• Kaon is lowest state, so flavour mixings should play no role

• mixing of scalar and pseudoscalar

16

Preliminary analysis of fD and fDs in MA set-up

• SU(2) heavy meson χPT fit to our datafor fDs

√mDs and fDs

√mDs/(fD

√mD)

(ETMC, Blossier et al. (2009))

• including terms proportional to a2m2Ds

and 1/mDs

• results very encouragingfDs = 250(3) MeV, fD = 204(3) MeV, fDs/fD = 1.230(6)

• very preliminary but very first results from Nf = 2 + 1 + 1 !

17

Nucleon structure for Nf = 2 + 1 + 1(C. Alexandrou, M. Constantinou, S. Dinter, V. Drach, D. Renner, K.J.)

First calculation for 〈x〉 comparison to Nf = 2

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.05 0.1 0.15 0.2 0.25

<x>

u−

d

m2π[GeV2]

N f = 2, a = 0.079 fmN f = 2, a = 0.063 fmN f = 2, a = 0.051 fm

N f = 2 + 1 + 1, a = 0.078 fmABKM09

same effect as for Nf = 2: need to explore smaller quark mass region

simulations are underway

18

Non-perturbative renormalization for Nf = 2 + 1 + 1

renormalisation factors computed from dedicated Nf = 4 flavoursimulations of Wilson fermions

• RI-MOM scheme at non zero values of both the standardand twisted mass parameters

MR = 1ZP

√(ZAmPCAC)2 + µ2

q → 0

• O(a) improvement via average of simulations with +mPCAC and −mPCAC

0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012(aMsea)

2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0Z_AZ_SZ_V(WI)Z_P

study at β = 1.95

a = 0.08 fm, L = 1.9 fm

• linear mass dependence

• allows for chiral extrapolation

19

Topology

20

Summary

• successful simulations with Nf = 2 flavours

– using maximally twisted mass fermions– automatic O(a)- improvement

• First simulations with Nf = 2 + 1 + 1 flavours

– using maximally twisted mass fermions in the sea– use Osterwalder-Seiler fermions in the heavy valence sector– already precise results for fK/fπ, fD, fDs– non-perturbative renormalization under way

• our conclusion: adding stange and charm as dynamical degrees of freedomperfectly feasible

21