5 QCD on the Lattice
157
of the quark determinant is still large compared with the quenched approximation.
More details can be found in Sect. 5.2.6 below.
Correlation functions, i.e. the expectation values of polynomials in the quark and
gluon fields, are the most important quantities, since they determine implicitly the
particle spectrum of the theory. As was discussed already in Sect. 5.2.1, the link
between correlation functions and the particle spectrum is provided by the transfer
matrix T. For lattice QCD with Wilson fermions, the existence of a positive transfer
matrix was rigorously established [32].
As a concrete example we shall discuss the two-point correlation function of a
charged kaon. A polynomial of quark fields with the quantum numbers of the kaon
is given by
φ K (x) = ( ¯
uγ 5 s) (x),
(5.61)
where the parentheses indicate summation over spinor and colour components of the
fields. Mostly one is interested in correlation functions in which all spatial points
have been summed over and which therefore only depend on the Euclidean time
separation. We define
C K (x 0 ; ;
p) =
x
e
i
p·· x
φ K (x)φ
†
K (0)
.
(5.62)
The inclusion of the phase factor in conjunction with the summation over
x amounts
to a projection onto spatial momentum
p. On a finite lattice with periodic boundary
conditions C K (x 0 ; ;
p) must be symmetric under x 0 ↔ T −x 0 . Therefore, the spectral
decomposition of C K (x 0 ; ;
p) reads
C K (x 0 ; ;
p) =
α
0
φ K (0)
α
2
2 α (
p)
e
− α (
p)x 0 + e
− α (
p)(T −x 0 )
,
(5.63)
where the sum runs over all states in the kaon channel with fixed momentum
p, and
α (
p) is the mass gap (see Sect. 5.2.1). 4 For large Euclidean times x 0 the ground
state dominates. If we further set
p = 0, then the asymptotic form of the two-point
function reads
lim
x 0 →∞
C K (x 0 ; ;
p) =
0
φ K (0)
K
2
m K
e
−m K T /2 cosh (m K (T /2 − x 0 )) ,
(5.64)
where m K = 0 (
p)|
p=0 is the mass of the kaon, and the sum of the two exponentials
has been re-expressed using the cosh function. Owing to the ordering 0 (
p) <
1 (
p) < . . ., the higher excited states are exponentially suppressed. The functional
4 In the commonly normalization of hadron states one includes a factor 2 α (
p) in the denominator.
157
of the quark determinant is still large compared with the quenched approximation.
More details can be found in Sect. 5.2.6 below.
Correlation functions, i.e. the expectation values of polynomials in the quark and
gluon fields, are the most important quantities, since they determine implicitly the
particle spectrum of the theory. As was discussed already in Sect. 5.2.1, the link
between correlation functions and the particle spectrum is provided by the transfer
matrix T. For lattice QCD with Wilson fermions, the existence of a positive transfer
matrix was rigorously established [32].
As a concrete example we shall discuss the two-point correlation function of a
charged kaon. A polynomial of quark fields with the quantum numbers of the kaon
is given by
φ K (x) = ( ¯
uγ 5 s) (x),
(5.61)
where the parentheses indicate summation over spinor and colour components of the
fields. Mostly one is interested in correlation functions in which all spatial points
have been summed over and which therefore only depend on the Euclidean time
separation. We define
C K (x 0 ; ;
p) =
x
e
i
p·· x
φ K (x)φ
†
K (0)
.
(5.62)
The inclusion of the phase factor in conjunction with the summation over
x amounts
to a projection onto spatial momentum
p. On a finite lattice with periodic boundary
conditions C K (x 0 ; ;
p) must be symmetric under x 0 ↔ T −x 0 . Therefore, the spectral
decomposition of C K (x 0 ; ;
p) reads
C K (x 0 ; ;
p) =
α
0
φ K (0)
α
2
2 α (
p)
e
− α (
p)x 0 + e
− α (
p)(T −x 0 )
,
(5.63)
where the sum runs over all states in the kaon channel with fixed momentum
p, and
α (
p) is the mass gap (see Sect. 5.2.1). 4 For large Euclidean times x 0 the ground
state dominates. If we further set
p = 0, then the asymptotic form of the two-point
function reads
lim
x 0 →∞
C K (x 0 ; ;
p) =
0
φ K (0)
K
2
m K
e
−m K T /2 cosh (m K (T /2 − x 0 )) ,
(5.64)
where m K = 0 (
p)|
p=0 is the mass of the kaon, and the sum of the two exponentials
has been re-expressed using the cosh function. Owing to the ordering 0 (
p) <
1 (
p) < . . ., the higher excited states are exponentially suppressed. The functional
4 In the commonly normalization of hadron states one includes a factor 2 α (
p) in the denominator.