274
Appendix F: U(1) Problem Solved by Gauge Group Topology
and since by locality the limit is reached for finite R, d/dλ may be interchanged
with lim R . Then, in order to obtain a relation involving the two-point function of a
local conserved current and the operator A, one needs that < dV
5
R (λ)/dλ| λ=0 A >
be well defined, so that
d
dλ
< β
λ
(A) > | λ=0 = lim
R
<
d
dλ
V
5
R (λ)| λ=0 A − A
d
dλ
V
5
R (λ)| λ=0 >=
= −2Im(−i) lim
R
<
d
dλ
V
5
R (λ)| λ=0 A >=< δ
5 A > .
(F.21)
Now, since j
5
μ is gauge invariant, for a localized gauge transformation α U n , n = 0,
satisfying the conditions of Eq. (F.19), one has
α U (V
5
R (λ)) = e
iλ 2n U V
5
R (λ).
(F.22)
On the other hand, by Proposition F.2 the (Gauss invariant) vacuum is invariant
under the full gauge group G and therefore for a gauge invariant symmetry breaking
operator A, as, e.g. ¯
ψ ψ, one has
< V
5
R (λ) A >=< α U n (V
5
R (λ) A) >= e
i2nλ
< V
5
R (λ) A > .
This proves that < V
5
R (λ) A > is a singular function of λ and its derivative with
respect to λ cannot be defined; hence Eq. (F.21) cannot be written and the Goldstone
theorem does not apply.
Clearly, the above mechanism relies on the locality of the gauge transformations
and on their non-trivial topology and it does not apply to the case of a local gauge
group with trivial topology, as in the abelian gauge theory. This shows that the presence of the anomaly is not enough, contrary to statements appeared in the literature.
More generally, the above Proposition proves that one cannot write the Ward
identities involving J
5
μ on a Gauss invariant vacuum state, equivalently on a gauge
invariant vacuum state (a so-called θ vacuum), and this solves the problems raised
by R. J. Crewther in his analysis of chiral Ward identities.
58
For the same reasons, the Goldstone dipole mechanism proposed by Kogut and
Susskind for evading the physical Goldstone bosons does not work in (the standard
formulation of) the temporal gauge.
It is worthwhile to remark that the above solution of the U (1) problem does not
make any reference to the instantons and the related semi-classical approximation.
Moreover, even if both V
C
R (λ) and V
5
R (λ) have vanishing matrix elements on
the physical states, thanks to the presence of fermions, V
5
R (λ) defines the chiral
symmetry β
λ of the observables and the effect of the non-trivial topology of G is
the impossibility of a local infinitesimal generation of β
λ in the physical states, in
particular in the expectations of the (Gauss invariant) vacuum.
58 R.J. Crewther, Chiral properties of quantum chromodynamics, in Field Theoretical methods in
particle Physics, W. Rühl ed., Reidel 1980, pp. 529–590.
Appendix F: U(1) Problem Solved by Gauge Group Topology
and since by locality the limit is reached for finite R, d/dλ may be interchanged
with lim R . Then, in order to obtain a relation involving the two-point function of a
local conserved current and the operator A, one needs that < dV
5
R (λ)/dλ| λ=0 A >
be well defined, so that
d
dλ
< β
λ
(A) > | λ=0 = lim
R
<
d
dλ
V
5
R (λ)| λ=0 A − A
d
dλ
V
5
R (λ)| λ=0 >=
= −2Im(−i) lim
R
<
d
dλ
V
5
R (λ)| λ=0 A >=< δ
5 A > .
(F.21)
Now, since j
5
μ is gauge invariant, for a localized gauge transformation α U n , n = 0,
satisfying the conditions of Eq. (F.19), one has
α U (V
5
R (λ)) = e
iλ 2n U V
5
R (λ).
(F.22)
On the other hand, by Proposition F.2 the (Gauss invariant) vacuum is invariant
under the full gauge group G and therefore for a gauge invariant symmetry breaking
operator A, as, e.g. ¯
ψ ψ, one has
< V
5
R (λ) A >=< α U n (V
5
R (λ) A) >= e
i2nλ
< V
5
R (λ) A > .
This proves that < V
5
R (λ) A > is a singular function of λ and its derivative with
respect to λ cannot be defined; hence Eq. (F.21) cannot be written and the Goldstone
theorem does not apply.
Clearly, the above mechanism relies on the locality of the gauge transformations
and on their non-trivial topology and it does not apply to the case of a local gauge
group with trivial topology, as in the abelian gauge theory. This shows that the presence of the anomaly is not enough, contrary to statements appeared in the literature.
More generally, the above Proposition proves that one cannot write the Ward
identities involving J
5
μ on a Gauss invariant vacuum state, equivalently on a gauge
invariant vacuum state (a so-called θ vacuum), and this solves the problems raised
by R. J. Crewther in his analysis of chiral Ward identities.
58
For the same reasons, the Goldstone dipole mechanism proposed by Kogut and
Susskind for evading the physical Goldstone bosons does not work in (the standard
formulation of) the temporal gauge.
It is worthwhile to remark that the above solution of the U (1) problem does not
make any reference to the instantons and the related semi-classical approximation.
Moreover, even if both V
C
R (λ) and V
5
R (λ) have vanishing matrix elements on
the physical states, thanks to the presence of fermions, V
5
R (λ) defines the chiral
symmetry β
λ of the observables and the effect of the non-trivial topology of G is
the impossibility of a local infinitesimal generation of β
λ in the physical states, in
particular in the expectations of the (Gauss invariant) vacuum.
58 R.J. Crewther, Chiral properties of quantum chromodynamics, in Field Theoretical methods in
particle Physics, W. Rühl ed., Reidel 1980, pp. 529–590.