5 QCD on the Lattice
211
where we have included the vacuum angle θ and assumed that M ≡ m1. If the
quark mass m is tuned so that
mΣ F
2
0 p
2
min ∼ F
2
0 /L
2 ,
(5.198)
the statistical weight of fields with ∂ μ U = 0 will be strongly suppressed in the path
integral. In other words, the mass term will dominate over the kinetic term, except
for fields U with ∂ μ U = 0. Since 2mΣ/F 2
0 = m 2
PS , the conditions in Eq. (5.196),
which define the kinematical situation of the -regime, are equivalent to
mΣV 1.
(5.199)
The zero-momentum part can be represented by a constant SU(3) matrix U 0 such
that
U(x) = U 0 e
2iξ(x)/F 0 ,
U 0 ∈ SU(3),
(5.200)
where the field ξ incorporates the fluctuations about the zero momentum mode.
According to Leutwyler and Smilga [108], the path integral of the theory in
topological sector ν can be written in the form
Z
(0)
ν =
D[U 0 ] (det U 0 )
ν exp (mΣV ReTr U 0 ) .
(5.201)
After this somewhat lengthy preparatory discussion, the connection between QCD
in the -regime and chiral RMT can finally be established. An important result
derived by Shuryak and Verbaarschot [109] states that the path integral Z
(0)
ν can be
mapped exactly onto the partition function Z ν of RMT. One therefore expects that
the low-lying eigenvalues of QCD in the -regime are distributed in the same way
as those in RMT. By computing the former in a lattice simulation and performing
a comparison to the analytically known distributions in RMT, one may verify the
Banks–Casher scenario of spontaneous chiral symmetry breaking.
The Neuberger-Dirac operator D N of Eq. (5.47) is ideally suited for this task.
Since it satisfies the Ginsparg-Wilson relation, chiral symmetry is preserved at the
level of the discretized theory. Furthermore, D N can be shown to satisfy an exact
index theorem, so that it sustains |ν| exact zero modes on gauge configurations
with topological charge ν. This allows for an unambiguous identification of
topological sectors to which the path integral Z
(0)
ν is restricted [110]. Therefore, the
investigation of spontaneous chiral symmetry breaking is a prime example where it
is absolutely vital that the lattice-regularized theory obeys the same symmetries that
are present in the continuum.
Before we proceed we must elucidate the relation of the spectra of the random
matrix ˆ
D and the Neuberger-Dirac operator. While the eigenvalues of ˆ
D are purely
imaginary, the operator D N is unitary, and hence its eigenvalues lie on a circle with
radius 1/a in the complex plane, centered around the point 1/a on the real axis.
211
where we have included the vacuum angle θ and assumed that M ≡ m1. If the
quark mass m is tuned so that
mΣ F
2
0 p
2
min ∼ F
2
0 /L
2 ,
(5.198)
the statistical weight of fields with ∂ μ U = 0 will be strongly suppressed in the path
integral. In other words, the mass term will dominate over the kinetic term, except
for fields U with ∂ μ U = 0. Since 2mΣ/F 2
0 = m 2
PS , the conditions in Eq. (5.196),
which define the kinematical situation of the -regime, are equivalent to
mΣV 1.
(5.199)
The zero-momentum part can be represented by a constant SU(3) matrix U 0 such
that
U(x) = U 0 e
2iξ(x)/F 0 ,
U 0 ∈ SU(3),
(5.200)
where the field ξ incorporates the fluctuations about the zero momentum mode.
According to Leutwyler and Smilga [108], the path integral of the theory in
topological sector ν can be written in the form
Z
(0)
ν =
D[U 0 ] (det U 0 )
ν exp (mΣV ReTr U 0 ) .
(5.201)
After this somewhat lengthy preparatory discussion, the connection between QCD
in the -regime and chiral RMT can finally be established. An important result
derived by Shuryak and Verbaarschot [109] states that the path integral Z
(0)
ν can be
mapped exactly onto the partition function Z ν of RMT. One therefore expects that
the low-lying eigenvalues of QCD in the -regime are distributed in the same way
as those in RMT. By computing the former in a lattice simulation and performing
a comparison to the analytically known distributions in RMT, one may verify the
Banks–Casher scenario of spontaneous chiral symmetry breaking.
The Neuberger-Dirac operator D N of Eq. (5.47) is ideally suited for this task.
Since it satisfies the Ginsparg-Wilson relation, chiral symmetry is preserved at the
level of the discretized theory. Furthermore, D N can be shown to satisfy an exact
index theorem, so that it sustains |ν| exact zero modes on gauge configurations
with topological charge ν. This allows for an unambiguous identification of
topological sectors to which the path integral Z
(0)
ν is restricted [110]. Therefore, the
investigation of spontaneous chiral symmetry breaking is a prime example where it
is absolutely vital that the lattice-regularized theory obeys the same symmetries that
are present in the continuum.
Before we proceed we must elucidate the relation of the spectra of the random
matrix ˆ
D and the Neuberger-Dirac operator. While the eigenvalues of ˆ
D are purely
imaginary, the operator D N is unitary, and hence its eigenvalues lie on a circle with
radius 1/a in the complex plane, centered around the point 1/a on the real axis.