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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
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 2 / 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
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 2 / 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
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 2 / 24
Euclidean correlators
Euclidean correlator of two Hilbert-space operators O1 and O2.⟨
O2(t)O1(0)⟩
T=
1ZT
tr(
e−T HetHO2e−tHO1
)
=1
ZT
∑
m,n
< m|e−(T−t)HO2|n >< n|e−tHO1|m >
=1
ZT
∑
m,n
e−(T−t)Em < m|O2|n > e−tEn < n|O1|m >
with ZT =∑
n
e−TEn
T → ∞∑
n
e−t∆En < 0|O2|n >< n|O1|0 >
with ∆En = En − E0
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 3 / 24
Euclidean correlators
Euclidean correlator of two Hilbert-space operators O1 and O2.⟨
O2(t)O1(0)⟩
T=
1ZT
tr(
e−T HetHO2e−tHO1
)
=1
ZT
∑
m,n
< m|e−(T−t)HO2|n >< n|e−tHO1|m >
=1
ZT
∑
m,n
e−(T−t)Em < m|O2|n > e−tEn < n|O1|m >
with ZT =∑
n
e−TEn
T → ∞∑
n
e−t∆En < 0|O2|n >< n|O1|0 >
with ∆En = En − E0
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 3 / 24
Euclidean correlators II
Can also be expressed as a Euclidean path integral
1ZT
tr(
e−T HetHO2e−tHO1
)
=1
ZT
∫
D[ψ, ψ,U]e−SE O2[ψ, ψ,U]O1[ψ, ψ,U],
Z =
∫
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
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 4 / 24
Euclidean correlators II
Can also be expressed as a Euclidean path integral
1ZT
tr(
e−T HetHO2e−tHO1
)
=1
ZT
∫
D[ψ, ψ,U]e−SE O2[ψ, ψ,U]O1[ψ, ψ,U],
Z =
∫
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
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 4 / 24
Euclidean correlators II
Can also be expressed as a Euclidean path integral
1ZT
tr(
e−T HetHO2e−tHO1
)
=1
ZT
∫
D[ψ, ψ,U]e−SE O2[ψ, ψ,U]O1[ψ, ψ,U],
Z =
∫
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
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 4 / 24
Euclidean correlators II
Can also be expressed as a Euclidean path integral
1ZT
tr(
e−T HetHO2e−tHO1
)
=1
ZT
∫
D[ψ, ψ,U]e−SE O2[ψ, ψ,U]O1[ψ, ψ,U],
Z =
∫
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
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 4 / 24
Towards a lattice simulation I
ψquarks:
gluons: Uµ
}
a
Λ
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
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 5 / 24
Towards a lattice simulation I
ψquarks:
gluons: Uµ
}
a
Λ
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
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 5 / 24
Towards a lattice simulation I
ψquarks:
gluons: Uµ
}
a
Λ
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
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 5 / 24
Towards a lattice simulation I
ψquarks:
gluons: Uµ
}
a
Λ
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
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 5 / 24
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:
12a
(
Uµ(a~n)ψ(a~n + aµ) − Uµ(a~n − aµ)ψ(a~n − aµ))
The symmetric derivative has discretization errors in order a2.
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 6 / 24
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:
12a
(
Uµ(a~n)ψ(a~n + aµ) − Uµ(a~n − aµ)ψ(a~n − aµ))
The symmetric derivative has discretization errors in order a2.
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 6 / 24
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:
12a
(
Uµ(a~n)ψ(a~n + aµ) − Uµ(a~n − aµ)ψ(a~n − aµ))
The symmetric derivative has discretization errors in order a2.
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 6 / 24
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:
12a
(
Uµ(a~n)ψ(a~n + aµ) − Uµ(a~n − aµ)ψ(a~n − aµ))
The symmetric derivative has discretization errors in order a2.
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 6 / 24
QCD continuum action
QCD action
SQCD[ψ, ψ,A] = SF + SG
=
Nf∑
f=1
∫
d4xψ(f )(x)(
D/+ m(f ))
ψ(f )
+1
2g2 Tr [Fµν(x)Fµν(x)]
Fµν(x) = ∂µAν(x) − ∂νAµ(x) + ı[Aµ(x),Aν(x)]
D/ = γµDµ with Dµ = ∂µ + ıAµ(x)
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 7 / 24
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
∑
µ<ν
ReTr(1− Uµν(n))
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 8 / 24
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
∑
µ<ν
ReTr(1− Uµν(n))
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 8 / 24
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
∑
µ<ν
ReTr(1− Uµν(n))
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 8 / 24
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
∑
µ<ν
ReTr(1− Uµν(n))
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 8 / 24
Naive fermion action
Naive fermion action
SF [ψ, ψ,U] = a4Nf∑
f=1
∑
n∈Λ
(
ψf (n)
4∑
µ=1
γµUµψ
f (n + µ) − U−µψf (n − µ)
2a
+ mf ψf (n)ψ(n)
)
Naive lattice Dirac operator (one flavor)
D(n|m)αβab =
4∑
µ=1
(γµ)αβ
Uµ(n)abδn+µ,m − U−µ(n)abδn−µ,m
2a
+ mδαβδabδmn
Its inverse D−1(n,m) is the quark propagator
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 9 / 24
Naive fermion action
Naive fermion action
SF [ψ, ψ,U] = a4Nf∑
f=1
∑
n∈Λ
(
ψf (n)
4∑
µ=1
γµUµψ
f (n + µ) − U−µψf (n − µ)
2a
+ mf ψf (n)ψ(n)
)
Naive lattice Dirac operator (one flavor)
D(n|m)αβab =
4∑
µ=1
(γµ)αβ
Uµ(n)abδn+µ,m − U−µ(n)abδn−µ,m
2a
+ mδαβδabδmn
Its inverse D−1(n,m) is the quark propagator
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 9 / 24
Naive fermion action
Naive fermion action
SF [ψ, ψ,U] = a4Nf∑
f=1
∑
n∈Λ
(
ψf (n)
4∑
µ=1
γµUµψ
f (n + µ) − U−µψf (n − µ)
2a
+ mf ψf (n)ψ(n)
)
Naive lattice Dirac operator (one flavor)
D(n|m)αβab =
4∑
µ=1
(γµ)αβ
Uµ(n)abδn+µ,m − U−µ(n)abδn−µ,m
2a
+ mδαβδabδmn
Its inverse D−1(n,m) is the quark propagator
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 9 / 24
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∣
∣
∣
∣
mq=0=
−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
{
(π
a,0,0,0), (0,
π
a,0,0), . . . , (
π
a,π
a,π
a,π
a)}
These 15 additional poles are called doublers
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 10 / 24
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∣
∣
∣
∣
mq=0=
−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
{
(π
a,0,0,0), (0,
π
a,0,0), . . . , (
π
a,π
a,π
a,π
a)}
These 15 additional poles are called doublers
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 10 / 24
The Wilson fermion action
The doublers can be removed by adding a Wilson term to thelattice Dirac operator
D(f )(n|m)αβab = −
12a
±4∑
µ=±1
(1− γµ)αβUµ(n)abδn+µ,m
+
(
m(f ) +4a
)
δαβδabδmn
with γ−µ = −γµ
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 11 / 24
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
2g2
∑
n∈Λ
∑
µ<ν
tr(Fµν(n)2) + O(a2)
Lattice error can be reduced with so called ”improved actions”→ talk by M. Limmer
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 12 / 24
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
2g2
∑
n∈Λ
∑
µ<ν
tr(Fµν(n)2) + O(a2)
Lattice error can be reduced with so called ”improved actions”→ talk by M. Limmer
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 12 / 24
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
2g2
∑
n∈Λ
∑
µ<ν
tr(Fµν(n)2) + O(a2)
Lattice error can be reduced with so called ”improved actions”→ talk by M. Limmer
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 12 / 24
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.
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 13 / 24
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.
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 13 / 24
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.
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 13 / 24
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.
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 13 / 24
Implementations of chiral fermions
Overlap fermions (Neuberger; hep-lat/9707022)
D =1a
(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)
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 14 / 24
Implementations of chiral fermions
Overlap fermions (Neuberger; hep-lat/9707022)
D =1a
(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)
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 14 / 24
Implementations of chiral fermions
Overlap fermions (Neuberger; hep-lat/9707022)
D =1a
(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)
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 14 / 24
Implementations of chiral fermions
Overlap fermions (Neuberger; hep-lat/9707022)
D =1a
(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)
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 14 / 24
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)⟩
F
= −Γα1β1Γα2β2
⟨
u(n)c1β1
u(m)c2α2
⟩
u
⟨
d(m)c2β2
d(n)c1α1
⟩
d
= −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.
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 15 / 24
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)⟩
F
= −Γα1β1Γα2β2
⟨
u(n)c1β1
u(m)c2α2
⟩
u
⟨
d(m)c2β2
d(n)c1α1
⟩
d
= −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.
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 15 / 24
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)⟩
F
= −Γα1β1Γα2β2
⟨
u(n)c1β1
u(m)c2α2
⟩
u
⟨
d(m)c2β2
d(n)c1α1
⟩
d
= −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.
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 15 / 24
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
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 16 / 24
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
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 16 / 24
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
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 16 / 24
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
=N
∑
n=0
κnHn
H(n,m) =3
∑
j=1
(Uj(n, t)δ(n + j ,m) + Uj(n − j , t)†δ(n − j ,m))
Side remark: Not to be confused with ”link smearing”
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 17 / 24
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
=N
∑
n=0
κnHn
H(n,m) =3
∑
j=1
(Uj(n, t)δ(n + j ,m) + Uj(n − j , t)†δ(n − j ,m))
Side remark: Not to be confused with ”link smearing”
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 17 / 24
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
=N
∑
n=0
κnHn
H(n,m) =3
∑
j=1
(Uj(n, t)δ(n + j ,m) + Uj(n − j , t)†δ(n − j ,m))
Side remark: Not to be confused with ”link smearing”
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 17 / 24
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
=N
∑
n=0
κnHn
H(n,m) =3
∑
j=1
(Uj(n, t)δ(n + j ,m) + Uj(n − j , t)†δ(n − j ,m))
Side remark: Not to be confused with ”link smearing”
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 17 / 24
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 .
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 18 / 24
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 .
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 18 / 24
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
8
vector 1−− un/wγj dn/w unγj γ4dn/w udjdn/w udj
γk ddj9
pseudovector 1++ un/wγj γ5dn/w udjγk γ5ddj
4
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.
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 19 / 24
Diagonal elements for pseudoscalar mesons (pions)
0 3 6 9 12 15t
0.001
0.01
0.1
1
mag
nitu
de o
f th
e co
rrel
ator
nγ5n
nγ5w
wγ5w
nγ4γ
5n
nγ4γ
5w
wγ4γ
5w
diγ
iγ
4γ
5n
diγ
iγ
4γ
5w
diγ
4d
id
iγ
4γ
5d
i
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
(
t +12
)
= ln(
λi(t)λi(t + 1)
)
0 3 6 9 12t
0
0.5
1
1.5
2
2.5
Eff
ectiv
e m
asse
s -
am
0th State1st State
pseudoscalar
amq= 0.12old
0 3 6 9 12t
0
0.5
1
1.5
2
2.5
Eff
ectiv
e m
asse
s -
am
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
2]
0
0.5
1
1.5
2
2.5m
ass
[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
. . .
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 23 / 24
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
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 24 / 24
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
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 24 / 24
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
Daniel Mohler (Graz) Hadrons in Lattice QCD Todtmoos, 11.9. 2007 24 / 24