5 QCD on the Lattice
175
i.e. the long-distance properties. It should be stressed that the links in the temporal
direction do not undergo the fuzzing procedure: Fuzzed temporal links will alter the
transfer matrix and the spectral information it contains.
In order to obtain detailed information on the glueball spectrum one also seeks to
determine the masses of the excited states in a given channel. This requires another
level of refinement, since one normally hopes that excited state contributions die out
quickly, while they now become the very focus of interest. A widely used method to
gain information on the higher excitations is to construct a whole set of interpolating
operators {O 1 , . . . , O r } in a given channel, say, A
++
1 . This is achieved either by
considering different shapes of Wilson loops that share the same transformation
properties, or by applying several different smearing levels to one particular Wilson
loop. Thus, each individual member of the set {O 1 , . . . , O r } is a perfectly valid
operator in a given channel, but the projection properties, i.e. the associated spectral
weights w
(i)
α for a particular state α in the spectral sum will in general be different
for each member i = 1, . . . , r. One then computes the matrix
C ij (x 0 ) :=
x
O i (x)O
†
j (0)
,
i,j = 1, . . . , r,
(5.96)
whose elements consist of the correlations of all combinations of operators in the
set. The diagonalization of the matrix correlator then yields the appropriate linear
combination of operators which correspond to the states α = 1, 2, . . . in the spectral
decomposition. Diagonalization is achieved by solving the generalized eigenvalue
problem
C ij (x 0 )φ j = λ i (x 0 , x
0 )C ik (x
0 )φ k , x
0 < x 0 ,
(5.97)
where φ denotes a vector, x
0 is fixed, and C(x 0 ), C(x
0 ) denote the matrix correlators
taken at Euclidean times x 0 and x
0 , respectively. As shown in [56], the set of
eigenvalues λ(x 0 , x
0 ) converges rapidly towards
λ α (x 0 , x
0 ) = e
−(x 0 −x
0 )) α ,
α = 1, . . . , r,
(5.98)
where α is the mass (energy) of the state α in the spectral sum.
After all these technicalities, we now report on the status of glueball calculations.
Recent results obtained in the quenched approximation were published in [53, 57–
60]. In Fig. 5.8 we show the results from Ref. [57]. The three lowest-lying states are
the scalar (0 ++ ), tensor (2 ++ ) and the 0 −+ glueballs, whose masses are determined
as
m 0 ++ = 1710(50)(80) MeV, m 2 ++ = 2390(30)(120) MeV,
m 0 −+ = 2560(35)(120) MeV.
(5.99)
175
i.e. the long-distance properties. It should be stressed that the links in the temporal
direction do not undergo the fuzzing procedure: Fuzzed temporal links will alter the
transfer matrix and the spectral information it contains.
In order to obtain detailed information on the glueball spectrum one also seeks to
determine the masses of the excited states in a given channel. This requires another
level of refinement, since one normally hopes that excited state contributions die out
quickly, while they now become the very focus of interest. A widely used method to
gain information on the higher excitations is to construct a whole set of interpolating
operators {O 1 , . . . , O r } in a given channel, say, A
++
1 . This is achieved either by
considering different shapes of Wilson loops that share the same transformation
properties, or by applying several different smearing levels to one particular Wilson
loop. Thus, each individual member of the set {O 1 , . . . , O r } is a perfectly valid
operator in a given channel, but the projection properties, i.e. the associated spectral
weights w
(i)
α for a particular state α in the spectral sum will in general be different
for each member i = 1, . . . , r. One then computes the matrix
C ij (x 0 ) :=
x
O i (x)O
†
j (0)
,
i,j = 1, . . . , r,
(5.96)
whose elements consist of the correlations of all combinations of operators in the
set. The diagonalization of the matrix correlator then yields the appropriate linear
combination of operators which correspond to the states α = 1, 2, . . . in the spectral
decomposition. Diagonalization is achieved by solving the generalized eigenvalue
problem
C ij (x 0 )φ j = λ i (x 0 , x
0 )C ik (x
0 )φ k , x
0 < x 0 ,
(5.97)
where φ denotes a vector, x
0 is fixed, and C(x 0 ), C(x
0 ) denote the matrix correlators
taken at Euclidean times x 0 and x
0 , respectively. As shown in [56], the set of
eigenvalues λ(x 0 , x
0 ) converges rapidly towards
λ α (x 0 , x
0 ) = e
−(x 0 −x
0 )) α ,
α = 1, . . . , r,
(5.98)
where α is the mass (energy) of the state α in the spectral sum.
After all these technicalities, we now report on the status of glueball calculations.
Recent results obtained in the quenched approximation were published in [53, 57–
60]. In Fig. 5.8 we show the results from Ref. [57]. The three lowest-lying states are
the scalar (0 ++ ), tensor (2 ++ ) and the 0 −+ glueballs, whose masses are determined
as
m 0 ++ = 1710(50)(80) MeV, m 2 ++ = 2390(30)(120) MeV,
m 0 −+ = 2560(35)(120) MeV.
(5.99)