the rst two fermion generations in twisted mass lattice qcd€¦ · the rst two fermion generations...
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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
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• 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
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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
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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
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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
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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