hadrons in lattice qcd - institut für physik · daniel mohler (graz) hadrons in lattice qcd...

Hadrons in Lattice QCD

Daniel Mohler

Doktoratskolleg “Hadrons in Vacuum, Nuclei and Stars”Karl-Franzens-Universität Graz

Todtmoos,11.9. 2007

Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 1 / 24

Outline

1 IntroductionEuclidean correlatorsTowards a lattice simulation

2 Lattice actionsNaive lattice actionFermion doubling and Wilson termChiral lattice fermions

3 Mass spectra from lattice QCDVariational method and meson interpolatorsQuark smearingMeson interpolators with derivative quark souces(Some) preliminary results

Outline

Euclidean correlators

Euclidean correlator of two Hilbert-space operators O1 and O2.⟨

O2(t)O1(0)⟩

e−T HetHO2e−tHO1

< m|e−(T−t)HO2|n >< n|e−tHO1|m >

e−(T−t)Em < m|O2|n > e−tEn < n|O1|m >

with ZT =∑

e−TEn

T → ∞∑

e−t∆En < 0|O2|n >< n|O1|0 >

with ∆En = En − E0

Euclidean correlators

Euclidean correlator of two Hilbert-space operators O1 and O2.⟨

O2(t)O1(0)⟩

< m|e−(T−t)HO2|n >< n|e−tHO1|m >

e−(T−t)Em < m|O2|n > e−tEn < n|O1|m >

with ZT =∑

e−TEn

T → ∞∑

e−t∆En < 0|O2|n >< n|O1|0 >

with ∆En = En − E0

Euclidean correlators II

Can also be expressed as a Euclidean path integral

D[ψ, ψ,U]e−SE O2[ψ, ψ,U]O1[ψ, ψ,U],

D[ψ, ψ,U]e−SE .

No field operators appear on the right.

”Simple” integral over the classical Euclidean action

Can be evaluated with an (importance sampling) Markov chainMonte Carlo

D[ψ, ψ,U]e−SE .

Towards a lattice simulation I

ψquarks:

gluons: Uµ

Lattice Λ ={

~n = (n1,n2,n3,n4)|ni ∈ {0,1, . . . ,Li − 1}}

~x → a~n with lattice spacing a∫

d4x . . . → a4 ∑

n∈Λ . . .

∂µψ(~x) suitable covariant lattice derivative

ψquarks:

gluons: Uµ

Lattice Λ ={

~n = (n1,n2,n3,n4)|ni ∈ {0,1, . . . ,Li − 1}}

d4x . . . → a4 ∑

n∈Λ . . .

ψquarks:

gluons: Uµ

Lattice Λ ={

~n = (n1,n2,n3,n4)|ni ∈ {0,1, . . . ,Li − 1}}

d4x . . . → a4 ∑

n∈Λ . . .

ψquarks:

gluons: Uµ

Lattice Λ ={

~n = (n1,n2,n3,n4)|ni ∈ {0,1, . . . ,Li − 1}}

d4x . . . → a4 ∑

n∈Λ . . .

Towards a lattice simulation II

Gluon fields Uµ(n) are elements of the gauge group SU(3)(as opposed to elements of the Lie Algebra in the continuum)Uµ(n) is connected to the link between lattice site n and site n + µand referred to as ”link variable” or ”gauge link”Common notation:

U−µ(n) ≡ Uµ(n − µ)†

Gauge links: lattice version of the gauge transporter connecting nand n + µ

Uµ(n) = eıaAµ(n).

This leads to the form for the covariant lattice derivative:

Uµ(a~n)ψ(a~n + aµ) − Uµ(a~n − aµ)ψ(a~n − aµ))

The symmetric derivative has discretization errors in order a2.

U−µ(n) ≡ Uµ(n − µ)†

QCD continuum action

QCD action

SQCD[ψ, ψ,A] = SF + SG

d4xψ(f )(x)(

D/+ m(f ))

ψ(f )

2g2 Tr [Fµν(x)Fµν(x)]

Fµν(x) = ∂µAν(x) − ∂νAµ(x) + ı[Aµ(x),Aν(x)]

D/ = γµDµ with Dµ = ∂µ + ıAµ(x)

The Wilson gauge action

Required: gauge invariance of SG

Gauge invariant quantity: Trace over any closed loop L

L[U] = tr

(n,µ)∈L

Uµ(n)

Simplest possible such loop: plaquette

Uµν(n) = Uµ(n)Uν(n + µ)

· U−µ(n + µ+ ν)U−ν(n + ν)

Wilson gauge action: sum over all plaquettes

SG[U] =2g2

n∈Λ

ReTr(1− Uµν(n))

L[U] = tr

(n,µ)∈L

Uµ(n)

· U−µ(n + µ+ ν)U−ν(n + ν)

SG[U] =2g2

n∈Λ

ReTr(1− Uµν(n))

L[U] = tr

(n,µ)∈L

Uµ(n)

· U−µ(n + µ+ ν)U−ν(n + ν)

SG[U] =2g2

n∈Λ

ReTr(1− Uµν(n))

L[U] = tr

(n,µ)∈L

Uµ(n)

· U−µ(n + µ+ ν)U−ν(n + ν)

SG[U] =2g2

n∈Λ

ReTr(1− Uµν(n))

Naive fermion action

SF [ψ, ψ,U] = a4Nf∑

n∈Λ

ψf (n)

γµUµψ

f (n + µ) − U−µψf (n − µ)

+ mf ψf (n)ψ(n)

Naive lattice Dirac operator (one flavor)

D(n|m)αβab =

(γµ)αβ

Uµ(n)abδn+µ,m − U−µ(n)abδn−µ,m

+ mδαβδabδmn

Its inverse D−1(n,m) is the quark propagator

n∈Λ

ψf (n)

γµUµψ

f (n + µ) − U−µψf (n − µ)

+ mf ψf (n)ψ(n)

D(n|m)αβab =

(γµ)αβ

+ mδαβδabδmn

n∈Λ

ψf (n)

γµUµψ

f (n + µ) − U−µψf (n − µ)

+ mf ψf (n)ψ(n)

D(n|m)αβab =

(γµ)αβ

+ mδαβδabδmn

Fermion doubling

For the free theory (Uµ(n) = 1 ∀n ∈ Λ) with massless fermionsone obtains for the quark propagator in momentum space

D(p)−1∣

−ıa−1 ∑

µ γµ sin(pµa)

a−2∑

µ sin(pµa)2

On the lattice, this expression has a pole not only atp = (0,0,0,0) but also whenever sin(pµa) = 0

a,0,0,0), (0,

a,0,0), . . . , (

These 15 additional poles are called doublers

Fermion doubling

For the free theory (Uµ(n) = 1 ∀n ∈ Λ) with massless fermionsone obtains for the quark propagator in momentum space

D(p)−1∣

−ıa−1 ∑

µ γµ sin(pµa)

a−2∑

µ sin(pµa)2

On the lattice, this expression has a pole not only atp = (0,0,0,0) but also whenever sin(pµa) = 0

a,0,0,0), (0,

a,0,0), . . . , (

These 15 additional poles are called doublers

The Wilson fermion action

The doublers can be removed by adding a Wilson term to thelattice Dirac operator

D(f )(n|m)αβab = −

±4∑

µ=±1

(1− γµ)αβUµ(n)abδn+µ,m

m(f ) +4a

δαβδabδmn

with γ−µ = −γµ

Properties of the Wilson action

Gauge invariant; symmetric under translations and O(N) rotations(in continuum limit); symmetric under charge conjugation

Wilson-Term breaks chiral symmetry - even if all quark massesvanish (in the chiral limit)

Expansion of the gauge part gives:

SG =a4

n∈Λ

tr(Fµν(n)2) + O(a2)

Lattice error can be reduced with so called ”improved actions”→ talk by M. Limmer

SG =a4

n∈Λ

SG =a4

n∈Λ

Chiral fermions & Ginsparg-Wilson relation

Continuum chiral symmetry {D, γ5} = 0

Lattice implementations of above equation contain doublers(Nielsen-Ninomiya theorem)

Ginsparg-Wilson relation:

Dγ5 + γ5D = aDγ5D

→ Lattice version of chiral symmetry

Ginsparg-Wilson fermions: Fermions with full chiral symmetry.

Implementations of chiral fermions

Overlap fermions (Neuberger; hep-lat/9707022)

(1− sign[H])

H = γ5A with γ5-hermitian Kernel A (Wilson. . . )Only known exact solution of GWR

Domain wall fermions (Kaplan; hep-lat/9206013)exact solution for infinite extent of the fifth dimension - otherwiseapproximate

Chirally Improved (CI) fermions (Gattringer; hep-lat/0003005)construction of an approximate solution→ talk by M. Limmer

Fixed point fermions (P.Hasenfratz; hep-lat/9308004)

(1− sign[H])

A simple example

Correlator for an isotriplet meson with mass-degenerate quarks:⟨

O(n)O(m)⟩

F =⟨

d(n)Γu(n)u(m)Γd(m)⟩

= −Γα1β1Γα2β2

u(n)c1β1

u(m)c2α2

d(m)c2β2

d(n)c1α1

= −Γα1β1Γα2β2D−1u (n|m)β1α2

c1c2D−1

d (m|n)β2α1c2c1

= −tr[ΓD−1u (n|m)ΓD−1

d (m|n)]

Still both: quark propagators m → n and n → mDue to γ5 hermiticity: γ5D−1γ5 =

D−1)†

Different Dirac structures Γ for different mesons.For example Γ = γ5 for a pseudoscalar.

A simple example

O(n)O(m)⟩

F =⟨

u(n)c1β1

u(m)c2α2

d(m)c2β2

d(n)c1α1

c1c2D−1

d (m|n)β2α1c2c1

d (m|n)]

D−1)†

A simple example

O(n)O(m)⟩

F =⟨

u(n)c1β1

u(m)c2α2

d(m)c2β2

d(n)c1α1

c1c2D−1

d (m|n)β2α1c2c1

d (m|n)]

D−1)†

The variational method

Basis of interpolators Oi , i = 1, . . . ,N to compute the matrix ofcross correlations:

C(t)ij =⟨

Oi(t)Oj (0)⟩

The solutions to the generalized eigenvalue problem

C(t)~vi = λi(t)C(t0)~vi ,

behave as

λi(t) ∝ e(−tMi )(

1 + O(

e(−t∆Mi )))

∆Mi is the mass difference between the state i and the closestlying state.

Interpolators should be linearly independent, as orthogonal aspossible and possess a strong overlap with physical states

C(t)ij =⟨

Oi(t)Oj (0)⟩

behave as

1 + O(

e(−t∆Mi )))

C(t)ij =⟨

Oi(t)Oj (0)⟩

behave as

1 + O(

e(−t∆Mi )))

Quark smearing

So far: Point sources

S(α,a)0 (y ,0)ρ,c = δ(y ,0)δραδca

for each combination of Dirac and color indices.We would like to have: sources with a good overlap to physicalgroundstate and/or lowest excited states.Smeared sources (Jacobi smearing):

S = MS0

H(n,m) =3

(Uj(n, t)δ(n + j ,m) + Uj(n − j , t)†δ(n − j ,m))

Side remark: Not to be confused with ”link smearing”

Quark smearing

S = MS0

H(n,m) =3

Quark smearing

S = MS0

H(n,m) =3

Quark smearing

S = MS0

H(n,m) =3

Meson interpolators with derivative sources

Jacobi smearing can be used to construct Gaussian-type sourcesof different widths Sn , Sw

In addition: covariant derivatives which act upon a wide smearedsource to form our derivative quark sources Wdj

Pj(~x , ~y) = Uj(~x ,0)δ(~x + j , ~y) − Uj(~x − j,0)†δ(~x − j , ~y) ,

Wdj= PjSw .

Meson interpolators with derivative sources

Jacobi smearing can be used to construct Gaussian-type sourcesof different widths Sn , Sw

In addition: covariant derivatives which act upon a wide smearedsource to form our derivative quark sources Wdj

Pj(~x , ~y) = Uj(~x ,0)δ(~x + j , ~y) − Uj(~x − j,0)†δ(~x − j , ~y) ,

Wdj= PjSw .

Meson interpolators

JPC old interpolators new interpolators ♯

pseudoscalar 0−+ un/w γ5dn/w un/wγ4γ5dn/w udjγj γ4γ5dn/w udj

γ5ddjudj

γ4γ5ddj10

scalar 0++ un/w dn/w udjγj dn/w udj

γj γ4dn/w udjddj

vector 1−− un/wγj dn/w unγj γ4dn/w udjdn/w udj

γk ddj9

pseudovector 1++ un/wγj γ5dn/w udjγk γ5ddj

Table 1: Meson interpolators; n/w denote narrow and wide Jacobismearing and dj stands for a derivative source in the j-direction. Thelast column shows the total number of different interpolators.

Diagonal elements for pseudoscalar mesons (pions)

0 3 6 9 12 15t

nγ4γ

wγ4γ

amq= 0.04

Diagonal entries of the correlation matrix computed from 100quenched gauge configurations on a 163 × 32 lattice((2.4fm)3 × 4.8fm).Error bars from a jackknife procedure;Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 20 / 24

Effective masses

To display the results, one commonly plots the effective massesdetermined from ratios of eigenvalues:

aMi ,eff

λi(t)λi(t + 1)

0 3 6 9 12t

0th State1st State

pseudoscalar

amq= 0.12old

0 3 6 9 12t

0th State1st State2nd State

amq= 0.12new

Effective masses ground and excited state pions at amq = 0.12Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 21 / 24

A plot in physical units

0 0.4 0.8 1.2

Mπ2 [GeV

0th State1st State2nd StateExperimental valueExperimental valueExperimental value

Masses

Pion masses in physical units compared experimental excitations.Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 22 / 24

Disclaimer

I have not talked about...

Creation of the gauge fields (MC for quenched or HMC fordynamical data)idea of HMC → talk by M. Limmer

Calculation of the quark propagators

Continuum extrapolation; chiral extrapolation; finite size errors

Determination of the lattice spacing

Baryons or multiple particle states

The end...

So far:Derivative sources for pseudoscalar, scalar, vector and axial vectormesons using quenched data

To come:Derivatice sources on dynamical CI dataDerivative souces/different smearings for baryons ?Different interpolators/observables ?

Collaborators:Christof Gattringer, Christian Lang, Leonid Glozman, SasaPrelovsekMarkus Limmer, Julia Danzer, Regensburg group

The end...

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