Download - Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Lattice Hadrons in Zero- and Two-Flavor QCD
Christian B. Lang
Inst. f. Physik, FB Theoretische Physik
Universität Graz
Sept. 2008
Collaborators in these projects:C. Gattringer, L. Glozman, M. Limmer, T. Maurer,
D. Mohler, S. Prelovsek, A. Schäfer
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Nonperturbative challengesFermionsChiral symmetryChirally Improved quarks
QCD problems
Problems that cannot be attacked with perturbation theory:
Chiral symmetry breakingExplicit: Non-zero quark massesSpontaneous: The pion is a Goldstone boson
Confinement and the low energy properties ofhadrons
Hadron massesLow energy parameters (decay constants, currentquark masses, LEC of Chiral Perturbation Theory)Form factors, matrix elements, structure functions
We need non-perturbative methods!
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Nonperturbative challengesFermionsChiral symmetryChirally Improved quarks
Lattice QCD
Kenneth Wilson suggested 1974 to regularize QCDby introducing a 4-d (Euclidean) space-time lattice.
Gauge field variables Uµ(x) ∈ SU(3)(3x3 complex, unitary matrices oneach link)
Quark field variables ψ(x), ψ(x)
(ψ(f )α,c(x) are color 3-vectors, Dirac
4-spinors, nf vectors and Grassmannvariables, on each lattice site)
... and Mike Creutz initiated computer simulationswith this formulation of QCD in 1979...
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Nonperturbative challengesFermionsChiral symmetryChirally Improved quarks
The three limits of LQCD
Continuum limit: a(g,m) → 0 (g → 0)Lattice artifacts should become small
→ Improvement programme
Thermodynamic limit: L → ∞ (L · a = const.)Hadron physics in a box of a few fm→ Finite volume effects can be utilized
Chiral limit: m → m0 (Mπ → Mπ,exp)Physical u, d quark masses are small→ We want to understand chiral symmetry breaking
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Nonperturbative challengesFermionsChiral symmetryChirally Improved quarks
Monte-Carlo simulation of fermions
“Full QCD”:
C(t) ∝∫
D[U] D[ψ, ψ] e−SG[U]−ψ D[U]ψ N(t)N(0)
=∫
D[U] e−SG[U] (det Du det Dd . . .)
×[
D−1u D−1
d . . .+ . . .]
Set det D ≡ 1 (no dynamical fermionvacuum, i.e. no sea quarks)
Gauge field vacuum is fully dynamical(Monte Carlo)
Consider only the valence quarks
Hadron correlation functions are builtfrom the quark propagators
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Nonperturbative challengesFermionsChiral symmetryChirally Improved quarks
Monte-Carlo simulation of fermions
Quenched approximation:
C(t) ∝∫
D[U] D[ψ, ψ] e−SG[U]−ψ D[U]ψ N(t)N(0)
=∫
D[U] e−SG[U] (det Du det Dd . . .)
×[
D−1u D−1
d . . .+ . . .]
Set det D ≡ 1 (no dynamical fermionvacuum, i.e. no sea quarks)
Gauge field vacuum is fully dynamical(Monte Carlo)
Consider only the valence quarks
Hadron correlation functions are builtfrom the quark propagators
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Nonperturbative challengesFermionsChiral symmetryChirally Improved quarks
Dynamical fermions
Bosons: det[A]−1 = π−N∫
D[φ]D[φI ] e−φ†Aφ
Fermions: det[A] =
∫
D[ψ]D[ψ] e−ψAψ
Replace fermions by pseudofermions:∫
D[ψ]D[ψ] e−ψuDψu−ψd Dψd = π−N∫
D[φR]D[φI ] e−φ†(D D†)−1φ .
Doubling is necessary in order to ensure positivity:
det[D] det[D] = det[D] det[D†] = det[D D†] ≥ 0
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Nonperturbative challengesFermionsChiral symmetryChirally Improved quarks
Hybrid Monte Carlo (HMC) algorithm
Simulation with Hybrid Monte Carlo (HMC) algorithm:
〈O〉Q =
∫
D[U] exp(−S[U]) O[U]∫
D[U] exp(−S[U])
=
∫
D[U]D[P] exp(− 12P2 − S[U]) O[U]
∫
D[U]D[P] exp(− 12P2 − S[U])
= 〈O〉P,U .
For the dynamical fermion simulation S[U] and D[U] include thepseudofermion terms.
Microcanonical ensemble → canonical ensemble
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Nonperturbative challengesFermionsChiral symmetryChirally Improved quarks
Chiral symmetry
Chiral symmetry is a problem for LQCD!
The formulation should allow explicite chiral symmetry, such that itcan be broken spontaneously!
No-go theorem (Nielsen, Ninomiya, 1982):Lattice theories do not allow simultaneously chiral invariance,locality, and correct continuum behavior of quark propagators.
Finally excavated (Hasenfratz):
D γ5 + γ5 D =12
a D γ5 D
Ginsparg-Wilson condition (1982!) for chiral lattice fermions.
Consequences for the spectrum of D: zero modes, Banks-Casher!
“Lattice chiral symmetry” transformation (Lüscher).
The GWC is violated for simple Dirac operators (simple fermionactions)!
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Nonperturbative challengesFermionsChiral symmetryChirally Improved quarks
Chiral symmetry
Chiral symmetry is a problem for LQCD!
The formulation should allow explicite chiral symmetry, such that itcan be broken spontaneously!
No-go theorem (Nielsen, Ninomiya, 1982):Lattice theories do not allow simultaneously chiral invariance,locality, and correct continuum behavior of quark propagators.
Finally excavated (Hasenfratz):
D γ5 + γ5 D =12
a D γ5 D
Ginsparg-Wilson condition (1982!) for chiral lattice fermions.
Consequences for the spectrum of D: zero modes, Banks-Casher!
“Lattice chiral symmetry” transformation (Lüscher).
The GWC is violated for simple Dirac operators (simple fermionactions)!
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Nonperturbative challengesFermionsChiral symmetryChirally Improved quarks
Chiral symmetry
Chiral symmetry is a problem for LQCD!
The formulation should allow explicite chiral symmetry, such that itcan be broken spontaneously!
No-go theorem (Nielsen, Ninomiya, 1982):Lattice theories do not allow simultaneously chiral invariance,locality, and correct continuum behavior of quark propagators.
Finally excavated (Hasenfratz):
D γ5 + γ5 D =12
a D γ5 D
Ginsparg-Wilson condition (1982!) for chiral lattice fermions.
Consequences for the spectrum of D: zero modes, Banks-Casher!
“Lattice chiral symmetry” transformation (Lüscher).
The GWC is violated for simple Dirac operators (simple fermionactions)!
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Nonperturbative challengesFermionsChiral symmetryChirally Improved quarks
Fermion species
Non-GW type:Wilson improvedStaggeredTwisted mass
Approximate GW type:Domain WallFixed PointChirally Improved
Exact GW type:Overlap
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Nonperturbative challengesFermionsChiral symmetryChirally Improved quarks
Chirally Improved Dirac operator
General ansatz for fermion action:
Dmn =
16∑
α=1
Γα∑
p∈Pα
m,n
cαp∏
l∈p
Ul δn,m+p
1s s4s2 s3+ + + ....
Wilson
v1 v3v2
a1γµγν t1+ ....+
+
++
−−
−−
γµγνγρ γ5
p1
+ + + ....+ +
+
+
−
−
− −µ
....
γ
+ + ....
(Gattringer, PRD63(2001)114501)
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Nonperturbative challengesFermionsChiral symmetryChirally Improved quarks
Chirally Improved Dirac operator
Insert the ansatz in theGinsparg-Wilson-equation, truncatethe length of the contributions (to,e.g.,4) and compare the coefficients!
Leads to a set of (e.g. 50) algebraicequations, which can be solved (normminimization).
Quenched experience:
DCI with hypercubic smearing
Small O(a2) corrections for mass spectra
Renormalization ZA/ZV ≈ 1.04
Eigenvalues close to circle
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Nonperturbative challengesFermionsChiral symmetryChirally Improved quarks
Chirally Improved Dirac operator
Insert the ansatz in theGinsparg-Wilson-equation, truncatethe length of the contributions (to,e.g.,4) and compare the coefficients!
Leads to a set of (e.g. 50) algebraicequations, which can be solved (normminimization).
Quenched experience:
DCI with hypercubic smearing
Small O(a2) corrections for mass spectra
Renormalization ZA/ZV ≈ 1.04
Eigenvalues close to circle
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Nonperturbative challengesFermionsChiral symmetryChirally Improved quarks
Phase diagram for fermions (idealized)
β = 0
m
β = ∞ gg( = ∞) ( = 0)
m = 0
= ∞
constant physics
chiral limit
quenched limit
limitcontinuum
constant a
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Correlation functionImproving the field operatorsVariational analysis
How to determine hadron masses?
Masses are measured through real space propagators of the form
CX (~p, t) = 〈X(~p, t) X (~p, 0)〉 with X(~p, t) =∑
~x
X(~x , t) e−i~x ·~p .
Construct operators with correct Lorentz-, Dirac-, flavor-, and colorsymmetries ("interpolating field operators")
Projected the real space operators to a fixed 3-momentum ~p, oftento ~p = 0.Usually inspired by the heavy quark limit
(Almost) any operator with correct quantum number should be okContinue from heavy to light masses
Example: Pion π+:d(x)γ5u(x) or d(x)γ5U(x , y)u(y) or . . .
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Correlation functionImproving the field operatorsVariational analysis
Effective mass
0 5 10 15 20 25 30n
t
10-6
10-4
10-2
100
C(nt)
m = 0.02m = 0.05m = 0.10m = 0.20
−→0 5 10 15 20 25 30
nt
0.0
0.2
0.4
0.6
0.8
1.0
1.2
meff
m = 0.02m = 0.05m = 0.10m = 0.20
First idea: “effective mass”
meff
(
nt +12
)
= lnC(nt )
C(nt + 1)
Then fit to exponential (or cosh-) behavior in the plateau range.
C(nt )
C(nt + 1)=
cosh(
meff(nt −NT2 )
)
cosh(
meff(nt + 1 − NT2 )
)
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Correlation functionImproving the field operatorsVariational analysis
Interpolating field operators
Examples for mesons:us Γ ds
where us, ds denote smeared and/or derivative smeared quarks
Examples for hadrons, e.g. Nucleon:
N(i) = ǫabc Γ(i)2 ua
(
uTb Γ
(i)2 dc − dT
b Γ(i)2 uc
)
with the choices
Γ(i)1 Γ
(i)2
i = 1 1 Cγ5
i = 2 γ5 Ci = 3 i Cγ4γ5
where us, ds denote smeared quarks
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Correlation functionImproving the field operatorsVariational analysis
Quark sources
The correlation function is built from quark propagators.
Pointlike, e.g., d x Γ ux
Extended , e.g., dx Γ Uµ(x)ux+µ
Combining different such terms according to the cubic group, buildinglattice group representations (LHPC collab.)
Smeared quark sources, e.g., (S d)x Γ (S ux )
Fewer quark propagatorsCombination allows nodes in the interpolating operators (BGR collab.)
x,y,z
t
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Correlation functionImproving the field operatorsVariational analysis
Interpolating fields: Built from smeared quark sources
Jacobi smeared quark sources, e.g., us ≡ (S u)x
S = M S0 with M =∑N
n=0 κn Hn
H(~n, ~m ) =∑3
j=1
[
Uj(~n, 0) δ(~n + j , ~m )
+ Uj(~n− j , 0)† δ(~n − j , ~m )]
.
Fewer quark propagators
Combination allows nodes in the interpolatingoperatorsDerivative quark sources Wdi :
Di(~x , ~y) = Ui(~x , 0)δ(~x + i, ~y) − Ui(~x − i , 0)†δ(~x − i , ~y) ,
Wdi = Di Sw .
0 5 10r/a
0.0
0.2
0.4
0.6
0.8
P(r) σ = 0.27 fm
Narrow source
0 5 10r/a
σ = 0.41 fm
Wide source
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Correlation functionImproving the field operatorsVariational analysis
The problem with excited states
Exited states are unstable (no problem in quenched simulations)
Energy spectrum volume dependent (cf. Lüscher) → allows todetermine scattering lengths and phase shifts below the inelasticitythreshold.
Many states contribute to a quantum channel!
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Correlation functionImproving the field operatorsVariational analysis
Correlation function
Insertion of complete set of states gives a sum of exponentials
〈X(t)X (0)〉 =∑
n 〈X(t)|n〉e−mn t 〈n|X (0)〉
∼ a1 e−m1 t + a2 e−m2 t + a3 e−m3 t + . . .
Leading term with smallest mass: ground state (dominant at large t).
Higher mass states observed at smaller t
Fit to several exponentials is usually unstable!
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Correlation functionImproving the field operatorsVariational analysis
Disentangle the states
Bayesian analysis (stepwise reduction ofexponential with biased estimators)Minimize
F = χ2 + λφ ,
where φ is a stabilizing function of the fitparameters (prior).
Reconstruction of spectral density withmaximum entropy method
Variational analysis
→ Mathur et al. (05), Leeet al (03), Juge et al.(06), Zanotti et al. (03),Melnichouk et al.(03)
→ Sasaki et al. (05),
→ Burch et al. (03-06),Basak et al. (05, 06)
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Correlation functionImproving the field operatorsVariational analysis
Hadron masses
Variational analysis (Michael, Lüscher/Wolff)
Use several interpolators Oi
Compute all cross-correlations
C(t)ij = 〈Oi(t)Oj(0)〉 =∑
n
〈0|Oi |n〉〈∣
∣O†j
∣
∣0〉e−t Mn .
Solve the generalized eigenvalue problem:
C(t)~vi = λi(t) C(t0)~vi
Thenλi(t) ∝ e−t Mi
(
1 + O(
e−t ∆Mi))
,
where ∆Mi is the mass difference between the state i and theclosest lying state.
The eigenvectors are “fingerprints” of the state
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
MesonsBaryons
Some results for quenched ensembles: Mesons
0
2
4
6
r 0 M
π ρ φK K* a1 b1 a0
Burch et al.,PR D 73 (2006) 094505
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
MesonsBaryons
Some results for quenched ensembles
Mesons with derivative sources: Pion channel
0 3 6 9 12 15t
-1
0
1
Eig
enve
ctor
com
pone
nts
16910
Ground state
0 3 6 9 12 15t
-1
0
1
16910
First excited state
0 3 6 9 12 15t
-1
0
1
16910
Second excited state
Eigenvector componentsGattringer et al., ArXiv:0802.2020 [hep-lat], PRD(2008)
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
MesonsBaryons
Some results for quenched ensembles
Mesons with derivative sources: Pion channel
0.0 0.4 0.8 1.2
Mπ2 [GeV
2]
1.0
1.2
1.4
1.6
1.8
2.0
2.2
MP [G
eV]
systematic error1,4,5,61,4,6,9,12π(1300)
0.0 0.4 0.8 1.2
Mπ2 [GeV
2]
1.6
1.8
2.0
2.2
2.4
MP [G
eV]
systematic error1,4,6,9,12π(1800)
1st and 2nd excitation of π
Gattringer et al., ArXiv:0802.2020 [hep-lat], PRD(2008)
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
MesonsBaryons
Some results for quenched ensembles
Chiral extrapolations (quenched): Baryons
1.0
1.5
2.0
2.5
MB [G
eV]
N Σ Ξ Λ ∆++ Ω− −
positive parity
1.0
1.5
2.0
2.5
MB [G
eV]
N Σ Ξ Λ ∆++ Ω− −
negative parity
Burch et al., PR D74 (2006) 014504
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
MesonsBaryons
Some results for quenched ensembles: The Roper puzzle
0.00 0.40 0.80 1.20
(mπ)2 [GeV]
2
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
MN [G
eV]
λ(1), a = 0.148fm
λ(2), a = 0.148fm
λ(3), a = 0.148fm
λ(1), a = 0.119fm
λ(2), a = 0.119fm
λ(3), a = 0.119fm
positive parity
N(938)
N(1440)
N(1710)
0.00 0.40 0.80 1.20
(mπ)2 [GeV]
2
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
MN [G
eV]
λ(1), a = 0.148fm
λ(2), a = 0.148fm
λ(1), a = 0.119fm
λ(2), a = 0.119fm
λ(3), a = 0.119fm
negative parity
N(1535)
N(1650)
N(2090)
Roper?
Burch et al., PRD 74 (2006) 014504
Level crossing (+-+-) to (++–)?
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Simulation parametersBaryonsMesons
Study with 2 dynamical CI fermions
Lüscher-Weisz gauge actionChirally improved fermions, nf = 2 light quarksStout smearingHybrid Monte Carlo simulation
Earlier results (2005/2006): 123 × 24 (L/Majumdar/Ortner)Presently: Three ensembles for 163 × 32
ensemble βLW m0 HMC time Nc a[fm] mπ
A 4.70 -0.05 500 100 0.151(2) fm 526(7) MeVB 4.65 -0.06 1000 200 0.150(1) fm 469(4) MeVC 4.58 -0.077 1000 200 0.144(1) fm 318(5) MeV
(Every 5th config. to be analyzed; preliminary: only subset analyzed andpresented here; scale set with r0,exp = 0.48 fm.)
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Simulation parametersBaryonsMesons
Autocorrelation
Plaquette per configuration (tint ≈ 3)
0 200 4000.448
0.450
0.452
0.454
plaq
uette
+ sp/dp+ HB
0 200 400 600
0.446
0.448
0.450pl
aque
tte
0 200 400 600 800 1000config number
0.442
0.444
0.446
plaq
uette
+ sp/dp
Pion mass a mπ per configuration (plotted every 5th configuration)
0 20 40 60 80configuration number
0
0.1
0.2
0.3
0.4
0.5
a*m
π
0 40 80 120configuration number
0
0.1
0.2
0.3
0.4
0.5
a*m
π
0 40 80 120 160 200configuration number
0
0.1
0.2
0.3
0.4
0.5
a*m
π
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Simulation parametersBaryonsMesons
Parameters: Lattice spacing and AWI mass
Lattice spacing: From staticpotential where r2
0 F (r0) = 1.65(Sommer parameter r0 = 0.48fm).
Axial Ward identity: mAWI
defined from the ratio
〈 ∂t A4(~p = ~0, t) P(0) 〉
〈P(~p = ~0, t) P(0) 〉≡ 2 mAWI ,
(
= 2 m(ren) ZP/ZA)
0 5 10 15 20 25 30t
0
0.01
0.02
0.03
0.04
0.05
a*m
AW
I
Run A
0 5 10 15 20 25 30t
0
0.01
0.02
0.03
0.04
0.05
a*m
AW
I
Run B
0 5 10 15 20 25 30t
0
0.01
0.02
0.03
0.04
0.05
a*m
AW
I
Run C
Plateau behavior of the ratios
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Simulation parametersBaryonsMesons
Results for dynamical quarks (preliminary)
GMOR-plot
0 50 100 150 200m
AWI [MeV]
0
0.5
1
1.5
2m
π2 [GeV
2 ]
run Arun Brun C
0 2 4 6 80
0.01
0.02
0.03
0.04
Full symbols: dynamical; open symbols: mvalence > msea
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Simulation parametersBaryonsMesons
Results for dynamical quarks (preliminary)
Nucleon (positive parity)
0 0.4 0.8 1.2mπ
2 [GeV
2]
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2m
N [G
eV]
quenched, 16x32quenched, 20x32run Arun Brun Cexp. value
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Simulation parametersBaryonsMesons
Results for dynamical quarks (preliminary)
Nucleon (positive parity)
0 0.4 0.8 1.2
Mπ2 [GeV
2]
0
0.5
1
1.5
2
2.5
3
mas
s [G
eV]
Ground state (C)1st excitation (C)ExperimentExperiment
Nucleon (positive parity)
Excited state(s) seen with present statistics only for larger valence quarkmasses...
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Simulation parametersBaryonsMesons
Results for dynamical quarks (preliminary)
Nucleon (negative parity)
For run C reasonable plateau signals only for higher valence quarkmasses (omitted in plot)
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Simulation parametersBaryonsMesons
Results for dynamical quarks (preliminary)
∆++ (positive parity)
0 0.4 0.8 1.2mπ
2 [GeV
2]
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4m
∆ [GeV
]
quenched, 16x32quenched, 20x32run Arun Brun Cexp. value
..too high at small quark masses...
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Simulation parametersBaryonsMesons
Results for dynamical quarks (preliminary)
∆++ (positive parity)
0 0.1 0.2 0.3(mπ/mN
)2
1.0
1.1
1.2
1.3
1.4m
∆/mN
run Arun Brun Cexp. value
High masses in N and ∆++ cancel...
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Simulation parametersBaryonsMesons
Results for dynamical quarks (preliminary)
∆(1700) (negative parity)
0 0.4 0.8 1.2
Mπ2 [GeV
2]
1
1.5
2
2.5
3
mas
s [G
eV]
Exp.
203x32 quenched - a=0.119
163x32 quenched - a=0.147
163x32 dyn. A
163x32 dyn. B
163x32 dyn. C
Delta (neg. parity)
0 0.1 0.2 0.3 0.4
Mπ2/M
N
2
1
1.2
1.4
1.6
1.8
2
2.2
2.4
M /
MN
Experiment
203x32 quenched - a=0.119
163x32 quenched - a=0.147
163x32 run A
163x32 run B
163x32 run C
Delta (neg. parity)
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Simulation parametersBaryonsMesons
Results for dynamical quarks (preliminary)
ρ(770) meson
0 0.4 0.8 1.2mπ
2 [GeV
2]
0.6
0.8
1.0
1.2
1.4m
V [G
eV]
quenched, 16x32run Arun Brun Cexp. value
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Simulation parametersBaryonsMesons
Results for dynamical quarks (preliminary)
Scalar meson a0(980)
Quenched: Extrapolates to a0(1300)Dynamical: Unclear signal; operator dependent 10% variation.
0 0.4 0.8 1.2
Mπ2 [GeV
2]
0
0.5
1
1.5
2
mas
s [G
eV]
Experimentdyn. Adyn. Bdyn C
Mass of a0 (interpolators 9, 10)
Do we see a level crossing here (πη-channel?)
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD
Lattice formulationHow to determine hadron masses?
Quenched resultsDynamical sea quarks
Simulation parametersBaryonsMesons
Summary
Good ground state masses for (in lattice units) small lattices
Weak signals for excited states (for dynamical quarks; improvementwith better observables?)
Possibly indications for level crossing due to decay channel(s) (e.g.in a0 sector)
Resonance regime and scattering: a long way to go...
Some References:
Excited hadrons on the lattice: Mesons; T. Burch et al., Phys. Rev. D 73 (2006) 094505;
Excited hadrons on the lattice: Baryons; T. Burch et al. Phys. Rev. D 74 (2006) 014504
The hadron spectrum from lattice QCD; C. B. Lang, Prog. Part. Nucl. Phys. 61 (2008) 35
Derivative sources in lattice spectroscopy of excited mesons; Ch. Gattringer et al., Phys. Rev. D78 (2008) 034501
Spectroscopy with dynamical Chirally Improved quarks; D. Mohler et al., Lattice 2008
Christian B. Lang Lattice Hadrons in Zero- and Two-Flavor QCD