5 QCD on the Lattice
149
The hopping parameter κ is related to the bare mass m 0 via
κ =
1
2am 0 + 8r
,
(5.38)
while the dimensionless parameter r is usually set to one. Taken together with
the plaquette action of Eq. (5.25), the Wilson action for QCD is thus conveniently
parameterized in terms of the bare parameters (β, κ), with β = 6/g 2
0 and κ as above,
instead of the bare gauge coupling and quark mass (g 0 , m 0 ).
Another consequence of adding the Wilson term to the naïve lattice action is the
resulting additive renormalization of the quark mass. In other words, the point where
the quark mass vanishes is a priori unknown. The value that must be subtracted is
called the critical quark mass, which corresponds to the critical value of the hopping
parameter, κ c . The bare subtracted quark mass is then given by
m =
1
2a
1
κ
−
1
κ c
.
(5.39)
From Eq. (5.38) one easily infers that the critical value of κ in the free theory occurs
at
κ c =
1
8
,
r = 1,
(5.40)
while for non-zero g 0 the value of κ c must be determined, for instance, by adjusting
κ to the point where the pion mass vanishes.
We now turn to discussing one alternative to using Wilson’s solution to the
fermion doubling problem, namely the so-called “staggered” (or Kogut-Susskind)
fermions. One might think that the doubling problem arises since there are too
many fermion degrees of freedom in the discretized theory, if one associates
a four-component Dirac spinor with each individual lattice site. Pictorially, the
main idea of Kogut and Susskind was to “thin out” the degrees of freedom by
distributing single spinor components over different lattice sites. In their particular
formulation, the 16 corners of a four-dimensional hypercube serve to accommodate
the individual components of four Dirac spinors. Therefore, if these hypercubes
are regarded as the main building blocks for the fermionic discretization, rather
than the lattice sites themselves, this procedure will result in a partial lifting of
the degeneracy from 16 fermion species down to four. It is clear, though, that a
simple distribution of spinor components is not sufficient to define the action, since
the Dirac matrices mix different spinor components. Thus, the staggered fermion
action is only obtained after performing a diagonalization in spinor space, which
then decouples the individual components.
Rather than describing the details of this procedure, which can be found in most
textbooks, we simply state the result. Starting from the usual four-component spinor
and performing a spin-diagonalization, the lattice action for staggered fermions with
149
The hopping parameter κ is related to the bare mass m 0 via
κ =
1
2am 0 + 8r
,
(5.38)
while the dimensionless parameter r is usually set to one. Taken together with
the plaquette action of Eq. (5.25), the Wilson action for QCD is thus conveniently
parameterized in terms of the bare parameters (β, κ), with β = 6/g 2
0 and κ as above,
instead of the bare gauge coupling and quark mass (g 0 , m 0 ).
Another consequence of adding the Wilson term to the naïve lattice action is the
resulting additive renormalization of the quark mass. In other words, the point where
the quark mass vanishes is a priori unknown. The value that must be subtracted is
called the critical quark mass, which corresponds to the critical value of the hopping
parameter, κ c . The bare subtracted quark mass is then given by
m =
1
2a
1
κ
−
1
κ c
.
(5.39)
From Eq. (5.38) one easily infers that the critical value of κ in the free theory occurs
at
κ c =
1
8
,
r = 1,
(5.40)
while for non-zero g 0 the value of κ c must be determined, for instance, by adjusting
κ to the point where the pion mass vanishes.
We now turn to discussing one alternative to using Wilson’s solution to the
fermion doubling problem, namely the so-called “staggered” (or Kogut-Susskind)
fermions. One might think that the doubling problem arises since there are too
many fermion degrees of freedom in the discretized theory, if one associates
a four-component Dirac spinor with each individual lattice site. Pictorially, the
main idea of Kogut and Susskind was to “thin out” the degrees of freedom by
distributing single spinor components over different lattice sites. In their particular
formulation, the 16 corners of a four-dimensional hypercube serve to accommodate
the individual components of four Dirac spinors. Therefore, if these hypercubes
are regarded as the main building blocks for the fermionic discretization, rather
than the lattice sites themselves, this procedure will result in a partial lifting of
the degeneracy from 16 fermion species down to four. It is clear, though, that a
simple distribution of spinor components is not sufficient to define the action, since
the Dirac matrices mix different spinor components. Thus, the staggered fermion
action is only obtained after performing a diagonalization in spinor space, which
then decouples the individual components.
Rather than describing the details of this procedure, which can be found in most
textbooks, we simply state the result. Starting from the usual four-component spinor
and performing a spin-diagonalization, the lattice action for staggered fermions with