150
H. Wittig
bare mass m 0 coupled to the gauge field is derived as
S
stagg
F
[U, ¯
χ , χ] = a
4
x∈ E
4
α=1
m 0 ¯
χ α (x)χ α (x)
+
1
2a
3
μ=0
η μ (x)
¯
χ α (x)U μ (x)χ α (x + a ˆ
μ) − ¯
χ α (x + a ˆ
μ)U μ (x)
−1 χ α (x)
, (5.41)
where χ α denotes a one-component Grassmann variables. The spin-diagonalization
has thus replaced the Dirac matrices γ μ by real, position-dependent phase factors
η μ (x), which are given by
η 0 (x) = 1,
η j (x) = (−1)
n 0 +...+n j−1 , n j = x j /a.
(5.42)
At the level of the classical action, the spinor components are completely decoupled,
and the action is decomposed into four identical pieces. In order to occupy all 16
corners of a four-dimensional hypercube with one-component Grassmann variables,
one needs four Dirac spinors, each of which contributes a term like Eq. (5.41) to the
overall action. This produces the fourfold degeneracy of staggered fermions, with
the remnant doubler states being referred to as “tastes”, in order to distinguish them
from physical flavours. The formulation using the one-component fields within a
hypercube can be re-expressed in terms of the spin-taste basis [18], from which one
can infer directly that the taste symmetry is broken. However, one axial generator of
the taste symmetry remains unbroken. The fermion mass in the staggered approach
is therefore protected against any additive renormalization through the associated
global axial U(1) symmetry, unlike the case of the Wilson action. While the various
tastes decouple in the continuum limit, non-vanishing interactions between the tastes
at O(a 2 ) in the lattice spacing are induced, leading to large lattice artefacts. The
Symanzik improvement programme can be employed to reduce these taste-changing
interactions [19], and the resulting “improved staggered fermions” (the so-called
“Asqtad”-action being one particular example [20]) have been widely used in a
series of simulations.
For a long time lattice physicists have struggled to find a fermionic discretization
which would both solve the doubling problem and be compatible with chiral
symmetry. In fact, physicists grew increasingly doubtful that this could be achieved,
following the proof of a “No-Go theorem” by Nielsen and Ninomiya [21], which
stated that the conditions (a)–(d) mentioned above could not be satisfied simultaneously. Since one does not want to give up locality and property (b), this
would imply that either (c) or (d) must be violated. Indeed, the Wilson and
staggered discretizations seem to confirm this expectation: while the Wilson fermion
action removes all doublers, it breaks chiral symmetry, leading to an additive
renormalization of the quark mass, as well as several other consequences. By
contrast, the staggered formulation preserves a U(1) subgroup of chiral symmetry at
the price of only partially removing the spurious degrees of freedom.
H. Wittig
bare mass m 0 coupled to the gauge field is derived as
S
stagg
F
[U, ¯
χ , χ] = a
4
x∈ E
4
α=1
m 0 ¯
χ α (x)χ α (x)
+
1
2a
3
μ=0
η μ (x)
¯
χ α (x)U μ (x)χ α (x + a ˆ
μ) − ¯
χ α (x + a ˆ
μ)U μ (x)
−1 χ α (x)
, (5.41)
where χ α denotes a one-component Grassmann variables. The spin-diagonalization
has thus replaced the Dirac matrices γ μ by real, position-dependent phase factors
η μ (x), which are given by
η 0 (x) = 1,
η j (x) = (−1)
n 0 +...+n j−1 , n j = x j /a.
(5.42)
At the level of the classical action, the spinor components are completely decoupled,
and the action is decomposed into four identical pieces. In order to occupy all 16
corners of a four-dimensional hypercube with one-component Grassmann variables,
one needs four Dirac spinors, each of which contributes a term like Eq. (5.41) to the
overall action. This produces the fourfold degeneracy of staggered fermions, with
the remnant doubler states being referred to as “tastes”, in order to distinguish them
from physical flavours. The formulation using the one-component fields within a
hypercube can be re-expressed in terms of the spin-taste basis [18], from which one
can infer directly that the taste symmetry is broken. However, one axial generator of
the taste symmetry remains unbroken. The fermion mass in the staggered approach
is therefore protected against any additive renormalization through the associated
global axial U(1) symmetry, unlike the case of the Wilson action. While the various
tastes decouple in the continuum limit, non-vanishing interactions between the tastes
at O(a 2 ) in the lattice spacing are induced, leading to large lattice artefacts. The
Symanzik improvement programme can be employed to reduce these taste-changing
interactions [19], and the resulting “improved staggered fermions” (the so-called
“Asqtad”-action being one particular example [20]) have been widely used in a
series of simulations.
For a long time lattice physicists have struggled to find a fermionic discretization
which would both solve the doubling problem and be compatible with chiral
symmetry. In fact, physicists grew increasingly doubtful that this could be achieved,
following the proof of a “No-Go theorem” by Nielsen and Ninomiya [21], which
stated that the conditions (a)–(d) mentioned above could not be satisfied simultaneously. Since one does not want to give up locality and property (b), this
would imply that either (c) or (d) must be violated. Indeed, the Wilson and
staggered discretizations seem to confirm this expectation: while the Wilson fermion
action removes all doublers, it breaks chiral symmetry, leading to an additive
renormalization of the quark mass, as well as several other consequences. By
contrast, the staggered formulation preserves a U(1) subgroup of chiral symmetry at
the price of only partially removing the spurious degrees of freedom.