Appendix F: U(1) Problem Solved by Gauge Group Topology
267
It is a four divergence, see eqs. (F.3)), i.e. a boundary term, and even if it does not
change the equations of motion and should have no effect for (regular) configurations
rapidly vanishing when r → ∞, it has been argued that the (non-local) instanton
configurations, decreasing at infinity as O(1/r
2
), yield a non-zero contribution to the
expectations of such a term, leading to a θ dependence of the Euclidean correlation
functions. The argued θ dependence of the Euclidean correlation functions of the
observable fields implies that they define a vacuum sector labelled by the θ angle (θ
vacua).
For the solution of the U (1) problem, the relevant consequence of the topological
term is that, as shown by Fujikawa, the resulting functional measure (defined by a
gauge invariant regularization) is not invariant under the chiral transformations of
eqs. (F.2)), which induce the following change in the functional measure: θ → θ + α.
This means that each θ vacuum (and the corresponding θ sector) is not invariant under
chiral transformations and therefore chiral symmetry is lost in each θ sector.
Such results led G. ’t Hooft to conclude that “chiral U (1) symmetry is explicitly
broken by instantons”, so that there is no time-independent chiral symmetry and
Goldstone theorem does not apply.
Another argument for the absence of a massless Goldstone boson associated with
U (1) A breaking accepts the existence of a time-independent chiral symmetry, but,
on the basis of a mechanism displayed by the Schwinger model, claims that also in
QCD the Goldstone modes appearing in the spectrum of the θ vacuum expectation
< J
5
0 (x) F > θ (F a local field giving the symmetry breaking order parameter) are
unphysical dipole ghosts.
50
The problematic aspect of the standard solution of the U (1) problem is its strong
dependence on the semi-classical instanton approximation, since already for free
fields it has been proved that the set of Euclidean configurations with finite Euclidean
action has zero functional measure. In contrast with the quantum mechanical case,
such a vanishing measure holds more generally for the set of continuous configurations
51 and if the relevant configurations are not continuous, it is impossible to
classify their topological structure and derive the corresponding vacuum structure.
The somewhat generic definition of a θ vacuum on the basis of the instanton semiclassical approximation does not shed much light on its concrete realization and on
its structural properties; in particular, the problem arises about the representation
of the local field algebra given by a θ vacuum (a crucial issue for discussing the
two-point function < J
5
0 (x) F > θ ).
In view of the important physical consequences of the θ vacuum structure, as
emphasized by Jackiw, a derivation which does not rely on the instanton approximation is desirable, if not needed. Jackiw’s farsighted proposal is to exploit the topology
50 J. Kogut and L. Susskind, Phys. Rev. D 10, 3468 (1974); S. Weinberg, Phys. Rev. 11, 3583 (1975);
S. Coleman [1985], Chap. 7, Sect. 5.
51 See, e.g. the brief account in F. Strocchi [13, 16], Chap. 5, Sect. 8.
267
It is a four divergence, see eqs. (F.3)), i.e. a boundary term, and even if it does not
change the equations of motion and should have no effect for (regular) configurations
rapidly vanishing when r → ∞, it has been argued that the (non-local) instanton
configurations, decreasing at infinity as O(1/r
2
), yield a non-zero contribution to the
expectations of such a term, leading to a θ dependence of the Euclidean correlation
functions. The argued θ dependence of the Euclidean correlation functions of the
observable fields implies that they define a vacuum sector labelled by the θ angle (θ
vacua).
For the solution of the U (1) problem, the relevant consequence of the topological
term is that, as shown by Fujikawa, the resulting functional measure (defined by a
gauge invariant regularization) is not invariant under the chiral transformations of
eqs. (F.2)), which induce the following change in the functional measure: θ → θ + α.
This means that each θ vacuum (and the corresponding θ sector) is not invariant under
chiral transformations and therefore chiral symmetry is lost in each θ sector.
Such results led G. ’t Hooft to conclude that “chiral U (1) symmetry is explicitly
broken by instantons”, so that there is no time-independent chiral symmetry and
Goldstone theorem does not apply.
Another argument for the absence of a massless Goldstone boson associated with
U (1) A breaking accepts the existence of a time-independent chiral symmetry, but,
on the basis of a mechanism displayed by the Schwinger model, claims that also in
QCD the Goldstone modes appearing in the spectrum of the θ vacuum expectation
< J
5
0 (x) F > θ (F a local field giving the symmetry breaking order parameter) are
unphysical dipole ghosts.
50
The problematic aspect of the standard solution of the U (1) problem is its strong
dependence on the semi-classical instanton approximation, since already for free
fields it has been proved that the set of Euclidean configurations with finite Euclidean
action has zero functional measure. In contrast with the quantum mechanical case,
such a vanishing measure holds more generally for the set of continuous configurations
51 and if the relevant configurations are not continuous, it is impossible to
classify their topological structure and derive the corresponding vacuum structure.
The somewhat generic definition of a θ vacuum on the basis of the instanton semiclassical approximation does not shed much light on its concrete realization and on
its structural properties; in particular, the problem arises about the representation
of the local field algebra given by a θ vacuum (a crucial issue for discussing the
two-point function < J
5
0 (x) F > θ ).
In view of the important physical consequences of the θ vacuum structure, as
emphasized by Jackiw, a derivation which does not rely on the instanton approximation is desirable, if not needed. Jackiw’s farsighted proposal is to exploit the topology
50 J. Kogut and L. Susskind, Phys. Rev. D 10, 3468 (1974); S. Weinberg, Phys. Rev. 11, 3583 (1975);
S. Coleman [1985], Chap. 7, Sect. 5.
51 See, e.g. the brief account in F. Strocchi [13, 16], Chap. 5, Sect. 8.