264
Appendix F: U(1) Problem Solved by Gauge Group Topology
(see the excellent discussion in S. Weinberg [1996], Chap. 19). This is the (axial)
U (1) problem.
In the following, we shall critically review the standard solution, point out its
weak points and offer an alternative non-perturbative solution.
F.1 The Axial Anomaly
At the classical level, the invariance of the Lagrangian under U (1) A transformations
implies the conservation of the gauge invariant axial current j
5
μ = i ¯
ψ(x)γ
5
γ μ ψ(x),
but at the quantum level the product of fields at the same point x is ill defined and
needs a point splitting regularization.
A well established result is that a gauge invariant point splitting regularization
leads to a gauge invariant axial current j
5
μ which does not satisfy the continuity
equation:
∂
μ j
5
μ (x) =
g
2
8π 2 ε
μνρσ Tr [F μν F ρσ ],
(F.3)
where g is the gauge coupling constant, F μν ≡
a F
a
μν T
a , T
a denoting the Hermitian
representation matrices T
a of the Lie algebra of the global colour gauge group G,
normalized so that Tr [T
a T
b
] = δ ab .
45
The occurrence of the anomaly has been regarded as the solution of the U (1)
problem, with the argument that the non-conservation of the gauge invariant chiral
current means that the chiral symmetry of the classical Lagrangian does not survive
quantization; then, there is no symmetry to be spontaneously broken and the Goldstone theorem does not apply. Such an argument would apply equally well to the
abelian case where, for chiral symmetry breaking, Goldstone bosons are expected
compatibly with the anomaly of the axial current.
As a matter of fact, the anomaly only excludes the existence of a time-independent
chiral symmetry of the observables generated by j
5
μ , but it does not preclude its existence if such a condition is removed. To this purpose, one introduces the conserved
(gauge dependent) current
J
5
μ = j
5
μ − (16π
2
)
−1
ε μνρσ Tr [F
νρ A
σ
− (2/3)A
ν A
ρ A
σ
] ≡ j
5
μ + K
5
μ ,
(F.4)
where A μ =
a A
a
μ T
a and we put g = 1, for simplicity.
Now, by the current conservation and the locality of the quark fields the limit
i lim R [ J
5
0 ( f R , α), ψ(x, x 0 ) ], where lim R ≡ lim R→∞ , exists, is reached for finite
R and is independent of α ( ˜
α(0) = 1); therefore, by the argument in Chap. 27,
Remark 3, (see also F. Strocchi [13, 16], Chap. 7, Sect. 2), such a limit is governed
45 The proof of the anomaly is rather simple and instructive in the case of an abelian gauge theory;
see, e.g. F. Strocchi [13, 16], Chap. 4, Sect. 6.2. A general proof based on the functional integral,
which covers also the case of non-abelian gauge theory, has been given by K. Fujikawa, Phys. Rev.
D 21, 2848 (1980). See also S. Weinberg [1996], Chap. 22.
Appendix F: U(1) Problem Solved by Gauge Group Topology
(see the excellent discussion in S. Weinberg [1996], Chap. 19). This is the (axial)
U (1) problem.
In the following, we shall critically review the standard solution, point out its
weak points and offer an alternative non-perturbative solution.
F.1 The Axial Anomaly
At the classical level, the invariance of the Lagrangian under U (1) A transformations
implies the conservation of the gauge invariant axial current j
5
μ = i ¯
ψ(x)γ
5
γ μ ψ(x),
but at the quantum level the product of fields at the same point x is ill defined and
needs a point splitting regularization.
A well established result is that a gauge invariant point splitting regularization
leads to a gauge invariant axial current j
5
μ which does not satisfy the continuity
equation:
∂
μ j
5
μ (x) =
g
2
8π 2 ε
μνρσ Tr [F μν F ρσ ],
(F.3)
where g is the gauge coupling constant, F μν ≡
a F
a
μν T
a , T
a denoting the Hermitian
representation matrices T
a of the Lie algebra of the global colour gauge group G,
normalized so that Tr [T
a T
b
] = δ ab .
45
The occurrence of the anomaly has been regarded as the solution of the U (1)
problem, with the argument that the non-conservation of the gauge invariant chiral
current means that the chiral symmetry of the classical Lagrangian does not survive
quantization; then, there is no symmetry to be spontaneously broken and the Goldstone theorem does not apply. Such an argument would apply equally well to the
abelian case where, for chiral symmetry breaking, Goldstone bosons are expected
compatibly with the anomaly of the axial current.
As a matter of fact, the anomaly only excludes the existence of a time-independent
chiral symmetry of the observables generated by j
5
μ , but it does not preclude its existence if such a condition is removed. To this purpose, one introduces the conserved
(gauge dependent) current
J
5
μ = j
5
μ − (16π
2
)
−1
ε μνρσ Tr [F
νρ A
σ
− (2/3)A
ν A
ρ A
σ
] ≡ j
5
μ + K
5
μ ,
(F.4)
where A μ =
a A
a
μ T
a and we put g = 1, for simplicity.
Now, by the current conservation and the locality of the quark fields the limit
i lim R [ J
5
0 ( f R , α), ψ(x, x 0 ) ], where lim R ≡ lim R→∞ , exists, is reached for finite
R and is independent of α ( ˜
α(0) = 1); therefore, by the argument in Chap. 27,
Remark 3, (see also F. Strocchi [13, 16], Chap. 7, Sect. 2), such a limit is governed
45 The proof of the anomaly is rather simple and instructive in the case of an abelian gauge theory;
see, e.g. F. Strocchi [13, 16], Chap. 4, Sect. 6.2. A general proof based on the functional integral,
which covers also the case of non-abelian gauge theory, has been given by K. Fujikawa, Phys. Rev.
D 21, 2848 (1980). See also S. Weinberg [1996], Chap. 22.