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



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!





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...





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





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





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





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





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





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)!





Fermion species

Non-GW type:Wilson improvedStaggeredTwisted mass

Approximate GW type:Domain WallFixed PointChirally Improved

Exact GW type:Overlap





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)





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





Phase diagram for fermions (idealized)

β = 0

m

β = ∞ gg( = ∞) ( = 0)

m = 0

= ∞

constant physics

chiral limit

quenched limit

limitcontinuum

constant a




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 . . .





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 )

)





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





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





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





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!





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!





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)





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




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




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)




MesonsBaryons


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)




MesonsBaryons


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




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 (++–)?




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.)





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





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





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






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







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


Excited state(s) seen with present statistics only for larger valence quarkmasses...






Nucleon (negative parity)

For run C reasonable plateau signals only for higher valence quarkmasses (omitted in plot)






∆++ (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...






∆++ (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...






∆(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 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



163x32 run A

163x32 run B

163x32 run C

Delta (neg. parity)






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






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?)





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


lattice hadrons in zero- and two-flavor qcd

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