Axion field induces exact symmetry

While no regularization is consistent with the anomalous chiral symmetry which occurs for massless fermions, the artificial axion-induced symmetry for massive fermions is shown here to be consistent with a standard regularization, even in curved spacetime, so that it can be said to have no anomaly in gauge or gravitational fields. Implications for theta terms are pointed out.


Introduction
Chiral symmetry, which is exact for massless fermions at the classical level and approximate for light quarks, has been very useful in particle physics. In analogy with this chiral symmetry, an artificial chiral-like symmetry [1] was introduced some time back for the strong interactions with massive quarks. It implies the occurrence of a light pseudoscalar particle, the axion [2], which has however not been detected in spite of elaborate searches [3]. This symmetry has been suspected to be fraught with an anomaly because of the involvement of a chiral transformation. However, this is not obvious and has to be examined by careful regularization of the theory. This is important because of implications for the symmetry of the theory.
While chiral symmetry of the action, corresponding to the transformation is broken by a non-vanishing quark mass term mψψ, with mass m, the artificial chiral symmetry for massive fermions works by letting a new field ϕ absorb the chiral transformation. The mass term is replaced bȳ which is classically invariant under the transformation which is also a symmetry of the other terms of the action. The original interaction introduced by Peccei and Quinn [1] was of the form where Φ is a complex scalar field with a symmetry breaking potential. The artificial chiral symmetry transformation here is Φ may be taken to be of the form ρe iϕ . The amplitude ρ of the scalar field acquires a vacuum expectation value because of symmetry breaking, which provides a massive boson. The phase ϕ is the zero mode of the potential and provides a Goldstone boson. This is the axion, which should acquire a mass because of the quark masses, but does not appear to exist. The kinetic term of Φ yields 1 2 ρ 2 0 ∂ µ ϕ∂ µ ϕ as the kinetic term for ϕ, with ρ 0 the vacuum expectation value of ρ. The mass m comes from ρ 0 and the coupling constant of Φ.
The anomaly, which occurs for the usual chiral symmetry and makes the divergence of the axial currentψγ µ γ 5 ψ nonvanishing, appears when the singularities in the theory are handled by regularization. Even the measure picture of anomalies requires a regularization for actual calculations. We shall therefore use an explicit regularization for studying chiral transformations of fermions in the presence of the axion coupling. This will tell us whether the continuous symmetry (3) introduced classically for massive quarks is anomalous or not.
The new symmetry is expected to be broken spontaneously by the vacuum because of the translation of the spinless field by a c-number. The field must have a definite value in a fixed vacuum: let it be denoted by ϕ 0 . Then is the shifted field with vanishing vacuum expectation value. This is a pseudoscalar field, so thatψϕ ′ γ 5 ψ and (ϕ ′ tr F µνF µν ) are scalars. On the other hand, (ϕ 0 tr F µνF µν ) is a pseudoscalar, like the θ term (θ tr F µνF µν ). If a (ϕ 0 tr F µνF µν ) term is generated when the fermion is integrated out, it can modify the θ term. The possibility of this occurrence is related to the question of an anomaly in the symmetry (3).
If the symmetry (3) is anomalous, no regularization of the fermions will be consistent with it. On the other hand, if any regularization is found to be consistent with it, the symmetry has no anomaly in this quantization and such symmetric regularizations are the ones to be preferred. Pauli-Villars regularization will be studied in flat and in curved spacetime.

Pauli-Villars regularization
We recall first the simple form of the Pauli-Villars regularization. For a fermion with mass m, the Lagrangian densitȳ involves regulator spinor fields χ,χ which however are assigned Bose statistics. The regulator mass M is ultimately taken to infinity when the regulator fields decouple. The regulator fields couple to the gauge fields in the standard way. The regulator mass term breaks the chiral symmetry and yields the chiral anomaly in the M → ∞ limit. This is the original form of the Pauli-Villars regularization. A more general form with several species of regulator fields is also available [4]:ψ Here c j are integers whose signs are related to the statistics assigned to χ jk . They have to satisfy some conditions to ensure regularization of the divergences [4]: Chiral transformations work on χ as well as ψ. Hence the axion coupling has to be introduced for both ψ, χ, like the gauge coupling. One has This regularization (10) is invariant under the combined transformation which is the Pauli-Villars extension of (3). This means the symmetry survives when the regularization is removed by taking the limit M → ∞ and hence is not anomalous. Unlike the regularized axial current, which has a pseudoscalar divergence arising from the masses, the Noether current for the extended symmetry, namelyψ is conserved. The non-vanishing divergence of the axial current is cancelled by that of the ϕ piece by virtue of the axion equation of motion.
It is interesting to note that the ϕ phases in (10) can be removed by a joint chiral transformation of both the physical fermions and the regulator spinors: The derivative operators in the action produce derivatives of ϕ, which appear after the transformation. Now the transformation has a trivial Jacobian because the contribution of the fermion field is cancelled by that of the bosonic regulator field as in the latter case the determinant arising from functional integration comes in the denominator. The argument for the generalized Pauli-Villars regularization, with (10) replaced bȳ involves a Jacobian with the factor 1 + j c j = 0 in the exponent [5]. Consequently the effective action does not contain any ϕ tr F µνF µν in this regularization. The conserved Noether current here is Note that the argument may be easily extended to curved spacetime, where the Dirac operator comes with a tetrad e µ l and a spin connection A mn µ in addition to the gauge field, but it continues to anticommute with γ 5 . The regularized action is again invariant under the Pauli-Villars extension of (3). This means that there is not even an RR anomaly [6] in this symmetry. Similar results have also been found by others [7].

Conclusion
We have examined the chiral symmetry of the fermion action including ϕ. This classical symmetry is preserved by the explicit regularization (10). When an acceptable regularization preserves a symmetry, one can conclude that the quantum theory defined by the limit of that regularization will also satisfy it. Hence, after the fermion is integrated out, the effective action must be invariant under translations of ϕ 0 , i.e., independent of this variable. This is indeed true of this regularization, as seen above, and there is no ϕ tr F µνF µν term. This makes the modification of a θ term by a contribution from ϕ 0 impossible. The field ϕ occurs only in the form of derivatives. The extension of the argument to curved spacetime means that an existing RR term [8] too cannot be modified.
There is another way of seeing this. The exact classical chiral symmetry could be expected to play a rôle like the chiral symmetry that holds for massless fermions. With massless fermions, the θ term can be modified by a chiral transformation which does not alter the fermion action but produces an FF term because of the chiral anomaly. However, the chiral symmetry which occurs in the presence of mass and ϕ is not anomalous, as we have seen. So θ cannot be modified in that way.
It should be pointed out here that any effect of θ terms can be removed by setting θ = 0 [5] or by making it explicitly dynamical, which forces the topological charge to vanish. A dynamical θ-like term is essentially a Peccei-Quinn (ϕ 0 tr F µνF µν ) term without an axion particle. However, as only fields are dynamical in field theory, the field ϕ was used in [1] and a direct (ϕ tr F µνF µν ) term can remove θ.
One may wonder how much freedom one has in choosing regularizations. Different regularizations have been used in the past and it is known that all do not lead to the same result. A key point is that symmetries of the action are generally sought to be preserved by the regularization. Thus one is always looking for Lorentz invariant and gauge invariant regularizations for actions having such symmetries. Regularizations which maintain all symmetries are technically natural. We have seen that the Pauli-Villars regularization conveniently respects the symmetry (3) of the fermion action. Non-symmetric regularizations are possible, but are not preferable in any way. Regularizations that break an existing classical symmetry like (3) may to that extent be termed technically unnatural. The artificial chiral symmetry here is not anomalous as a regularization consistent with it has been demonstrated to exist. In these circumstances, working with regularizations inconsistent with it would be a needless and avoidable violation.
We end with the hope that the exact chiral symmetry in the presence of axions will be useful also in other calculations and even in curved spacetime.