Peccei-Quinn-like symmetries for nonabelian axions

Axions were first introduced in connection with chiral symmetry but are now being looked for mainly as dark matter. In this paper we introduce a nonabelian analogue of axions which can also be potential candidates for dark matter. Their nonabelian symmetries, which are generalizations of the Peccei-Quinn symmetry, are interesting in their own right. Detailed analysis, using fermion measure and zeta function approaches shows that these symmetries are not anomalous.


Introduction
The chiral symmetry which holds in classical Dirac theory with massless fermions interacting with gauge fields is broken by what is called an anomaly [1]: the transformation is a symmetry of the kinetic termψ[i / ∂]ψ and also of the interaction termψ[i / A]ψ with the gauge field A µ , but the axial currentψγ µ γ 5 ψ, which is classically conserved, is found to violate this conservation when the fermion triangle diagram is regularized and evaluated. The divergence of the axial current is finite and proportional to trFF [2].
In general, a classical symmetry may or may not survive quantization. The simple phase symmetry, whereby ψ is multiplied by a phase factor, does survive quantization for all masses m. To see whether a symmetry survives quantization, the action has to be regularized. If the regularized action still has the symmetry, the symmetry obviously has no anomaly. If the regularized action does not possess the symmetry, one tends to think that the symmetry has an anomaly, but there are different ways of regularizing fermion field theories and one must check whether a different regularization can preserve the symmetry. Familiarity with the chiral anomaly may make one suspect all classical symmetries involving chiral rotations in any manner to be anomalous. But whether a symmetry is anomalous or not has to be checked individually for each symmetry.
Below we review the example of the Peccei-Quinn symmetry which occurs in the presence of the hypothetical field called the axion and was introduced by these authors. It has been shown to survive quantization [3]. We point out that the symmetry can even be made local. In section II, a new nonabelian analogue of axions is introduced: the new Peccei-Quinn symmetry too has no anomaly, as shown by a measure analysis and a zeta function approach.

Peccei-Quinn symmetry and the axion
Chiral symmetry is explicitly broken by the mass term mψψ and also by quantum effects, i.e. the anomaly. However, an artificial chiral symmetry for massive fermions works by letting a new field ϕ absorb the chiral transformation. The mass term is replaced by [4]ψ which is invariant if the field ϕ transforms under This transformation leaves the action invariant provided the new field ϕ is massless. This is the Peccei-Quinn symmetry. The particle [5] corresponding to the new field ϕ introduced by them is called the axion, but it has not been seen in any experiment [6]. It is being studied extensively because it is expected to contribute to the elusive dark matter. For a discussion of strong CP symmetry in the absence of axions, one may look at [7]. Careful regularization has been shown to respect the Peccei-Quinn symmetry, which accordingly is not anomalous but survives quantization [3].
In spite of this subtlety about the Peccei-Quinn symmetry, the axion can still be used to remove any FF term in the action dynamically by coupling the axion directly to FF so that its vacuum expectation value cancels the coefficient of the FF term.
Observe that the axial symmetry can be made local by introducing an extra gauge field B µ for this purpose: where under a local chiral transformation. Here F is a constant of mass dimension such that the axion kinetic term is 1 2 F 2 ∂ µ ϕ∂ µ ϕ and there are additional kinetic terms of the gauge fields. It is to be noted that this provides a formulation of a chiral gauge theory similar to but different from the Wess-Zumino formulation suggested in [8]. The similarity is that in both cases there is an extra degree of freedom. The difference is that gauge invariance occurs at both the classical and quantum levels in the present formulation, while in the old formulation gauge invariance occurs only at the quantum level [9].

Nonabelian chiral symmetry and nonabelian axions
The usual chiral symmetry is under a transformation of the fermion in spinor space. If the fermion is an SU (N ) multiplet, there exist nonabelian chiral symmetries. The kinetic piecē is invariant under the chiral transformations where U L , U R are spacetime independent SU (N ) matrices acting on the two chiral projections of ψ. The gauge interactions will also be invariant under these provided the matrix A µ commutes with U L , U R . For instance, the SU (N ) could be a flavour group and the colour SU (3) or the U (1) could be gauged. The usual mass term m(ψ L ψ R +ψ R ψ L ) is not invariant under (7) unless U L = U R , in which case of course the transformation is not a chiral transformation. An analogue of the Peccei-Quinn mass term can be introduced: m(ψ L W ψ R + ψ R W † ψ L ). Here W is a hypothetical SU (N ) matrix field analogous to the axion. Considering that the original axion has not been detected, we must be cautious about such an object. However, just as the usual axion is expected to be a kind of dark matter, this nonabelian object too could be relevant as dark matter. Note that it is visualized as a new degree of freedom and not as mesonic matter. The mathematical construction may in any case be useful for calculations because of the symmetry. This term is invariant under (7) if W transforms as As an SU (N ) matrix it involves N 2 − 1 parameters which become fields. The kinetic term for this matrix field has to be of the form T r[∂ µ W ∂ µ W † ], familiar from chiral models. This is invariant under (8). Thus the full action is invariant under the generalized Peccei-Quinn symmetry The question now is whether this nonabelian symmetry survives quantization. Anomalies arise when regularizations break some symmetries of classical actions. In the functional integral approach, it is said that the action has a symmetry which is broken by the measure [10].

Fermion measure approach
To formulate the fermion measure, it is customary to expand the fermion field in eigenfunctions of some operator. To maintain gauge invariance, the covariant Dirac operator is considered. The eigenvalue equation is where the subscript labels the eigenvalue and the eigenfunction. Under a gauge transformation, so that The field is expanded as ψ = n a n f n ,ψ = nā n f † n .
Each a,ā is gauge invariant because ψ and f transform the same way under gauge transformations andψ and f † also transform like each other. The gauge invariant measure n da n dā n is used for the fermion integration. It is well known that the measure is not chirally invariant: chiral transformations alter a,ā and the change of the measure is a Jacobian which can be evaluated after some regularization and yields the chiral anomaly. One needs measures for other fields too, but these do not break symmetries. Given this situation, it would appear that the Peccei-Quinn transformation would also alter the measure. The above measure would certainly be altered, but remembering that the requirement of gauge invariance led to the use of a fermion measure involving the eigenfunctions of the Dirac operator which contains the gauge field, we can involve the axion field now. First for the abelian axion, we consider the new expansion Although the fermion field changes under the Peccei-Quinn transformation, the exponential factor too changes and cancels it, because of (3), leaving b,b invariant. Hence the measure n db n db n is invariant under the transformation. In other words, although the naïve fermion measure is altered by the Peccei-Quinn transformation, there does exist a fermion measure which is left invariant. This is very similar to what happens with regularizations. It is also similar to the alteration occurring in the fermion measure in the presence of a twisted fermion mass term [11]. It may be added that the measure for ϕ is translation invariant. For the SU (N ) version of axions, the construction of the measure is a bit complicated. First, note that eigenvalues and eigenfunctions of i / D come in pairs: i / Df n = λ n f n , i / Dγ 5 f n = −λ n γ 5 f n .
Hence it is possible to consider expansions in f nL , f nR , which are chiral combinations of the f n , γ 5 f n , though they are not eigenfunctions of i / D. We expand Of course, the range of n is implicitly altered here. The measure n da L n da R n dā L n dā R n is not invariant under an SU (N ) chiral transformation (7) because a,ā have to change unless U L = U R , in which case the common vector transformation may be absorbed in f . However, a new measure can be constructed using the generalized axion field. Consider the expansions As W is invertible, it may also be transferred to the right if desired. This construction is not unique, but serves the purpose. Note the asymmetric use of W here. Because of this asymmetry, the left hand sides of both equations in the first line acquire U R under (9) and the left hand sides in the second line acquire U † R , so that the common vector factor U R may be absorbed in f : leaving the b,b invariant. This means that there exists a measure n db L n db R n db L n db R n invariant under the SU (N ) version of the Peccei-Quinn transformation, exactly as before. As regards the measure for W , it can be chosen to be SU (N ) invariant. So the measure respects the symmetry and the new nonabelian Peccei-Quinn symmetry is not anomalous.

Zeta function approach
Instead of considering a regularized action, one may also look at the fermion determinant which then has to be regularized. The most convenient way to do this in this context is the zeta function regularization [12]. The determinant is that of the Dirac operator, which in the simple case of a singlet axion is The zeta function regularization works for a hermitian, positive definite operator, which has to be obtained by constructing the Laplacian, Here the gamma matrices have been taken to be antihermitian in euclidean spacetime. The determinant of the Dirac operator is defined as the square root of the determinant of ∆. The anomaly has been checked in this framework [13]. Now one can write This can be written after some formal manipulations as Apart from the initial and final exponential factors, this depends on the axion field ϕ only through its derivative. When the determinant is calculated, those exponential factors cancel. Hence, its determinant will also involve only derivatives of this field and will be invariant under translations thereof. But after the fermion is integrated out, the Peccei-Quinn transformation is just a constant translation of the axion field, so the determinant is invariant under Peccei-Quinn transformations and it has no anomaly when the product of its eigenvalues is regularized through the zeta function.
The case of the nonabelian Peccei-Quinn symmetry is more complicated. Here, the Dirac operator is So one needs the Laplacian This can be recast as where V, Y are some SU (N ) matrices to be constrained later. This becomes which can all be satisfied by requiring V, Y to obey the single relation When the determinant is calculated, the initial and final factors cancel out. The remaining expression involves V, Y only in the combinations V ∂V † , Y † ∂Y, ∂V V † , ∂Y † Y . These are invariant under the Peccei-Quinn-like transformations of W whose U R actions are taken as left actions on Y and U L actions as right actions on V because of the constraint on V W Y . Thus the determinant is invariant under Peccei-Quinn-like transformations. This persists upon zeta function regularization of the product of its eigenvalues. Hence it is seen once again that these symmetries are not anomalous.
As there is no anomaly in the Peccei-Quinn-like symmetries, gauge fields can again be used to extend these global chiral symmetries to local ones. For example, for the left handed chiral symmetry, one needs an SU (N ) gauge field matrix B µ :ψ Here the transformation of the new gauge field is given by Gauge field kinetic terms have to be added. The right handed chiral symmetry too can be gauged if desired in a similar way. This reformulation of a chiral gauge theory in a gauge invariant way is again reminiscent of [8].

Conclusion
Symmetries which appear to be anomalous because they are not consistent with obvious regularizations may turn out to be consistent if regularized with care and therefore may be non-anomalous. The first known case of such a faux anomaly is the Peccei-Quinn symmetry which arises in QCD in the presence of axions, which are now being looked for as dark matter but have not so far been found. Our new SU (N ) version of this symmetry, which holds if SU (N ) analogues of axions are introduced, provides the second example: a fermion measure invariant under this symmetry has been explicitly constructed and the determinant in the zeta function approach is also invariant under nonabelian transformations of W . We hope these mathematical observations will be of as much interest as ordinary axions in the context of dark matter. While axions have been expected to be useful in the context of the strong CP issue, there is no such possibility with the nonabelian analogues discussed here because they cannot be coupled in a natural way to the gluon FF term. However, as they couple to quarks, they may also transform into other gauge bosons in the same way as the ordinary axions get feebly converted to photons. They could be detected for example as decaying Z bosons just as ordinary axions are sought to be caught in the form of photons. Apart from this, nonabelian axions can be of use in chiral gauge theories where they are unphysical and get swallowed up while providing mass to gauge bosons [9].