170
H. Wittig
from the exponential fall-off of the correlation function at large Euclidean times.
The detailed functional form of the asymptotic behaviour depends on the choice of
boundary conditions. Thus, it is not always described by a cosh function, as in the
example of a pseudoscalar meson on a lattice with periodic boundary conditions
in time, c.f. Eq. (5.64). In the limit of infinite temporal lattice size T , the effect
of the boundary conditions is sufficiently weak, so that one may approximate the
functional form of the correlation function for a generic interpolating operator
φ had (x) by
C had (x 0 ; ;
p) =
x
e
i
p·· x
φ had (x)φ
†
had (0)
T →∞
=
α
w α (
p)e
− α (
p)x 0 .
(5.91)
Here, the quantity w α (
p) is referred to as the spectral weight of the state |α. A
large value for the spectral weight of the ground state, w 1 (
p), will lead to an early
domination of the correlation function by the ground state energy. The choice of
φ had in a given channel can be optimized such that
w 1 (
p) w i (
p),
i = 2, 3, . . . .
(5.92)
An optimal choice of interpolating operator is not only important to ensure a reliable
determination of the ground state energy: In order to determine the energies in
the excitation spectrum, the associated spectral weights must be maximized by
specifying appropriate operators.
Below we provide examples for interpolating operators in several mesonic and
baryonic channels:
K-meson : φ K = (sγ 5 ¯
u), (sγ 0 γ 5 ¯
u)
K ∗ -meson : φ K ∗ = (sγ j ¯
u),
j = 1, 2, 3
nucleon : φ N = ε abc {u a T Cγ 5 d b }u c
: φ = ε abc {u a T Cγ μ d b }u c
(5.93)
Here, parentheses indicate summation over spinor and colour indices, while curly
brackets denote that only spinor indices are summed over.
5.3.1 Light Hadron Spectrum
The determination of the spectrum of light hadrons was historically one of the first
attempts to compute hadronic properties on the lattice. Since the masses of the lowlying hadrons are known from experiment, such calculations serve as benchmarks
to test the intrinsic accuracy of the lattice approach.
H. Wittig
from the exponential fall-off of the correlation function at large Euclidean times.
The detailed functional form of the asymptotic behaviour depends on the choice of
boundary conditions. Thus, it is not always described by a cosh function, as in the
example of a pseudoscalar meson on a lattice with periodic boundary conditions
in time, c.f. Eq. (5.64). In the limit of infinite temporal lattice size T , the effect
of the boundary conditions is sufficiently weak, so that one may approximate the
functional form of the correlation function for a generic interpolating operator
φ had (x) by
C had (x 0 ; ;
p) =
x
e
i
p·· x
φ had (x)φ
†
had (0)
T →∞
=
α
w α (
p)e
− α (
p)x 0 .
(5.91)
Here, the quantity w α (
p) is referred to as the spectral weight of the state |α. A
large value for the spectral weight of the ground state, w 1 (
p), will lead to an early
domination of the correlation function by the ground state energy. The choice of
φ had in a given channel can be optimized such that
w 1 (
p) w i (
p),
i = 2, 3, . . . .
(5.92)
An optimal choice of interpolating operator is not only important to ensure a reliable
determination of the ground state energy: In order to determine the energies in
the excitation spectrum, the associated spectral weights must be maximized by
specifying appropriate operators.
Below we provide examples for interpolating operators in several mesonic and
baryonic channels:
K-meson : φ K = (sγ 5 ¯
u), (sγ 0 γ 5 ¯
u)
K ∗ -meson : φ K ∗ = (sγ j ¯
u),
j = 1, 2, 3
nucleon : φ N = ε abc {u a T Cγ 5 d b }u c
: φ = ε abc {u a T Cγ μ d b }u c
(5.93)
Here, parentheses indicate summation over spinor and colour indices, while curly
brackets denote that only spinor indices are summed over.
5.3.1 Light Hadron Spectrum
The determination of the spectrum of light hadrons was historically one of the first
attempts to compute hadronic properties on the lattice. Since the masses of the lowlying hadrons are known from experiment, such calculations serve as benchmarks
to test the intrinsic accuracy of the lattice approach.