Axions are blind to anomalies
Abstract
The axion couplings to SM gauge bosons are derived in various models, and shown to always arise entirely from non-anomalous fermion loops. They are thus independent of the anomaly structure of the model. This fact is without consequence for vector gauge interactions like QCD and QED, but has a major impact for chiral gauge theories. For example, in the DFSZ axion model, the couplings of axions to electroweak gauge bosons do not follow the pattern expected from chiral anomalies, as we prove by an explicit calculation. The reason for this mismatch is traced back to triangle Feynman diagrams sensitive to the anomalous breaking of the vector Ward identity, and is ultimately related to the conservation of baryon and lepton numbers. Though our analyses are entirely done for true axion models, this observation could have important consequences for axion-like particle searches.
1 Introduction
Though this picture is correct, the nature of the couplings in Eq. (1) is often wrongly ascribed to the anomaly of the \(U(1)_{PQ}\) fermionic current. This idea comes from Noether’s theorem: the Goldstone boson \(a^{0}\) is coupled to its symmetry current, \(vp^{\mu }=\langle 0|J_{PQ}^{\mu }|a^{0}(p)\rangle \), and the current is anomalous, \(\partial _{\mu }J_{PQ}^{\mu }\sim g_{s}^{2}\mathcal {N}_{C}G_{\mu \nu }^{a}\tilde{G}^{a,\mu \nu }+e^{2}\mathcal {N}_{em}F_{\mu \nu }\tilde{F}^{\mu \nu }\), hence it seems Eq. (1) is immediately recovered. It has been known for a long time that this derivation of the axion couplings is not correct but sufficient for practical purpose [10, 11]. It permits to identify the couplings of axions to gluons and photons as induced by heavy fermions. So, by a common abuse of language, the \(a\rightarrow \gamma \gamma \) and \(a\rightarrow gg\) processes are said to be induced by the anomaly in the \(U(1)_{PQ}\) current.
The paper is organized as follow. To set the stage, we start in the next section by presenting in details an axion toy model [1, 2]. Though simple, this model illustrates many important physical features of more realistic axion models. In particular, it will be clear that the axion couplings to gauge bosons are not anomalous, but that this has no quantitative consequence for photons or gluons. Then, before turning our attention to full-fledged axion models, we derive a number of important results for anomalies in Sect. 3. Specifically, to be able to treat chiral gauge theories, the Ward identities applied to AVV or AAA triangle graphs have to be properly calculated, where \(V_{\mu }=\bar{\psi }\gamma _{\mu }\psi \) and \(A_{\mu }=\bar{\psi }\gamma _{\mu }\gamma _{5}\psi \) denote the vector and axial currents. Indeed, it is necessary to go beyond simple regularization procedures to be able to locate the anomaly of the AVV triangle in one of the vector Ward identities or to break explicitly the Bose symmetry of the AAA triangle. Equipped with these results, we turn to the Peccei-Quinn axion model in Sect. 4, derive the correct axion couplings to gauge bosons, and identify precisely where the naive procedure leading to Eq. (2) fails. This analysis is then trivially extended to invisible axion models. Finally, our results are summarized in Sect. 5, along with their implications for axion-like particle searches.
2 An axionic toy model
When the global \(U(1)_{PQ}\) symmetry is broken spontaneously, a Goldstone boson is left behind and will be identified with the axion [3, 4]. So, we choose the potential as \(V(\phi ^{\dagger }\phi )=\mu ^{2}\phi ^{\dagger }\phi +\lambda (\phi ^{\dagger }\phi )^{2}\) with \(\mu ^{2}<0\). Let us analyze the resulting theory using two different representations for the scalar field.
2.1 Linear representation
2.2 Polar representation
- In the polar representation, the appearance of the anomalous local term \(a^{0}F_{\mu \nu }\tilde{F}^{\mu \nu }\) is spurious. When computing specific amplitudes, it only serves to cancel out the anomalous term arising from the derivative interaction \(\partial _{\mu }a^{0}\bar{\psi }\gamma ^{\mu }\gamma _{5}\psi \). At the end, this is nothing but an application of the well-known axial current Ward identity:where \(P=\bar{\psi }\gamma _{5}\psi \), \(A^{\mu }=\bar{\psi }\gamma ^{\mu }\gamma _{5}\psi \). The right-hand side corresponds to the amplitude in the linear representation, and the left-hand side to the two contributions arising in the polar representation. It is thus clear that it would be wrong to understand the \(a^{0}\rightarrow \gamma \gamma \) coupling as induced by the anomaly. The anomalous interaction of Eq. (14) only arises because of the reparametrization in Eq. (12), and necessarily comes together with the appropriate derivative interactions.$$\begin{aligned} \partial _{\mu }A^{\mu }-\frac{e^{2}}{8\pi ^{2}}F_{\mu \nu }\tilde{F}^{\mu \nu }=2imP, \end{aligned}$$(20)
- In the \(m\rightarrow 0\) limit, the \(a^{0}\rightarrow \gamma \gamma \) amplitude vanishes exactly,In the linear representation, this trivially follows from the vanishing of the coupling of \(a^{0}\) to fermions, see Eq. (6). In the polar representation, it requires an exact cancellation of the local anomalous contribution with the non-local triangle amplitudes. Again, this can be understood in terms of the axial current Ward identity. In this limit, the spurious nature of the contact interaction is manifest.$$\begin{aligned} \mathcal {M}(a^{0}\rightarrow \gamma \gamma )_{\mathrm {Linear}}=\mathcal {M}(a^{0}\overset{}{\rightarrow }\gamma \gamma )_{_{\mathrm {Polar}}}\overset{m\rightarrow 0}{=}0. \end{aligned}$$(21)
- The reason why \(a^{0}\rightarrow \gamma \gamma \) is often misinterpreted as induced by the anomaly can be understood looking at the \(m\rightarrow \infty \) limit. Indeed, the contribution of the derivative term vanishes,as can be trivially seen in Eq. (17). Since then all that remains in the non-linear theory is the local term from \(\delta \mathcal {L}_{\mathrm {Jac}}\), it necessarily corresponds to the contribution surviving in the \(m\rightarrow \infty \) limit in the linear theory, Eq. (10). Yet, this is only a parametric correspondence, and certainly not a physical identification. The anomaly still cancels in \(\mathcal {M}(a^{0}\rightarrow \gamma \gamma )_{\mathrm {Der}}+\mathcal {M} (a^{0}\rightarrow \gamma \gamma )_{\mathrm {Jac}}\), and the only surviving contribution is actually the first term of Eq. (17). Ultimately, the local anomalous term \(\delta \mathcal {L}_{\mathrm {Jac}}\) in Eq. (14) is no more than a convenient book-keeping device tracking all the fields that have been integrated out [10, 11].$$\begin{aligned} \mathcal {M}(a^{0}\rightarrow \gamma \gamma )_{\mathrm {Der}}\overset{m\rightarrow \infty }{=}0, \end{aligned}$$(22)
Furry’s theorem together with the vector current Ward identity ensures the absence of any dependence on the free parameter \(\alpha \) in the non-linear theory, in agreement with the absence of this parameter in the corresponding linear representation.
As an aside, it should be clear that though the anomalous coupling in Eq. (14) is not affected by radiative corrections [15], neither is the anomalous part of the triangle graph in Eq. (17). So, the two exactly cancel at all orders, but the \(a^{0}\rightarrow \gamma \gamma \) amplitude does get corrected by higher order effects since it is actually not induced by the anomaly. Further, note that the theory in Eq. (3) is obviously renormalizable, so radiative corrections can be calculated perturbatively using standard techniques. In this respect, if the photons are also coupled to some other fermions \(\chi \), the two-loop process \(a^{0}\rightarrow \gamma \gamma \rightarrow \bar{\chi }\chi \) is even UV finite in the linear representation [16]. This fact would clearly be difficult to guess using the polar representation, where the local \(a^{0}\rightarrow \gamma \gamma \) vertex from \(\delta \mathcal {L}_{\mathrm {Jac}}\) leads to a UV divergent diagram.
- As said earlier, the present toy model can be generalized to more complicated, and more realistic KSVZ-like axion models [13, 14]. Consider for instance the SM, to which a gauge-singlet scalar \(\phi \) and a set of vector-like fermions \(\psi \) are added: with the same scalar potential as in Eq. (3). A priori, the covariant derivative can include all three SM interactions,where \(T^{a}\) and \(T^{i}\) are the \(SU(3)_{C}\) and \(SU(2)_{L}\) generators in the representation carried by \(\psi \), and Y its hypercharge. Then, if \(yv \gg v_{EW}\) and \(v\gg v_{EW}\), with \(v_{EW}\) the electroweak vacuum expectation value, the effect of the heavy fermion is to induce (see Eq. (10))$$\begin{aligned} D^{\mu }=\partial ^{\mu }-ig_{s}G_{a}^{\mu }T^{a}-igW_{i}^{\mu }T^{i}-ig^{\prime }\frac{Y}{2}B^{\mu }\;, \end{aligned}$$(24)with the quadratic invariants \(C_{C}^{\psi }\delta ^{ab}=\mathrm{Tr}[T^{a}T^{b}]\), \(C_{L}^{\psi }\delta ^{ij}=\mathrm{Tr}[T^{i}T^{j}]\), and \(C_{Y}^{\psi }=Y^{2}/4\), and \(d_{C,L}^{\psi }\) the corresponding dimensions of the \(SU(3)_{C}\) and \(SU(2)_{L}\) representations. Yet, several points must be clear: (1) These couplings are not anomalous, but result from the appropriately generalized Eq. (9) in the \(y\rightarrow \infty \) limit. (2) The integrated fermion must have vector couplings to gauge fields, otherwise the free parameter \(\alpha \) corresponding to fermion number does not cancel in Eq. (14). We will see later on how to deal with chiral theories. (3) Even if vector-like, no mass term is allowed for \(\psi \) because it would explicitly break the \(U(1)_{PQ}\) symmetry. (4) The couplings in \(\mathcal {L}_{\mathrm {Linear}}^{\mathrm {eff,KSVZ}}\) are not protected from radiative corrections since they are not anomalous. (5) Finally, the nature of the divergence arising when using this effective Lagrangian to compute e.g. the couplings of \(a^{0}\) to SM fermions is clear. In the UV complete theory, Eq. (23), the \(a^{0}\rightarrow \gamma \gamma \), WW, and gg vertices are never local. Instead, the pseudoscalar triangle of Eq. (9) acts as a form factor and is sufficient to regulate the UV behavior of the vector boson loops, making them finite.$$\begin{aligned} \mathcal {L}_{\mathrm {Linear}}^{\mathrm {eff,KSVZ}}= & {} -\frac{1}{16\pi ^{2}v} a^{0}(g_{s}^{2}d_{L}^{\psi }C_{C}^{\psi }G_{\mu \nu }^{a}\tilde{G}^{a,\mu \nu }\nonumber \\&+g^{2}d_{C}^{\psi }C_{L}^{\psi }W_{\mu \nu }^{i}\tilde{W}^{i,\mu \nu }\nonumber \\&+g^{\prime 2}d_{L}^{\psi }d_{C}^{\psi }C_{Y}^{\psi }B_{\mu \nu }\tilde{B}^{\mu \nu }), \end{aligned}$$(25)
3 On the consistent use of anomalies
4 The Peccei-Quinn axion and its couplings
To further explore this simple model, we follow the same strategy as for the toy model of Sect. 2, and perform the analysis using either a linear or a polar representation for the scalar fields \(\Phi _{1,2}\).
4.1 Linear representation
Often, one interprets the non-decoupling of these amplitudes when \(m_{u,d,e}\rightarrow \infty \) as an indirect manifestation of the underlying anomalies, but it is important to understand this statement correctly. The pseudoscalar triangles do decouple in that limit, \(\mathcal {T}_{PVV}^{\alpha \beta }(m_{f})\rightarrow 0\) and \(\mathcal {T}_{PAA}^{\alpha \beta }(m_{f})\rightarrow 0\) when \(m_{f}\rightarrow \infty \), but this is compensated by the mass-dependent couplings of the \(A^{0}\) to fermions. The fact that this compensation works, leaving a constant remainder in the \(m_{f}\rightarrow \infty \), is guaranteed by the anomalous axial Ward identity of Eq. (20). Yet, there is no anomalous contribution of any kind to the \(A^{0}\rightarrow V_{1}V_{2}\) amplitudes, and those are driven entirely by non-local and well-behaved fermion loops. The situation is quite similar for the loop-induced Higgs decay \(h\rightarrow \gamma \gamma \) in the SM, whose non-decoupling behavior in the \(m_{u,d,e}\rightarrow \infty \) limit is interpreted as a manifestation of the trace anomaly [22]. Yet, there is of course no direct contribution of that anomaly to \(h\rightarrow \gamma \gamma \), which can be safely computed perturbatively.
4.2 Polar representation
4.3 Matching the polar and linear representations
The physics must not depend on the representation chosen for the scalar fields or on the parametrization of the fermion fields. So, the one-loop \(A^{0}\rightarrow V_{1}V_{2}\) amplitudes of Eq. (43) must match the \(a^{0}\rightarrow V_{1}V_{2}\) amplitudes computed using the local terms \(\delta \mathcal {L}_{{\text {Jac}}}\) together with the triangle graphs arising from the derivative interactions in \(\delta \mathcal {L}_{{\text {Der}}}\) (these two contributions are of the same order in the coupling constants). At this stage, it is immediately clear that the triangle graphs will play a crucial role. Indeed, taken alone, the contributions from the anomalous local terms of \(\delta \mathcal {L}_{{\text {Jac}}}\) match those found in the linear case in the \(m_{u,d,e}\rightarrow \infty \) limit, Eq. (47), only for the \(\gamma \gamma \) and gg final states. For the others, there is no dependence on \(\alpha \) and \(\beta \) in Eq. (47), but there is one through \(\mathcal {N}_{L}\) in \(\delta \mathcal {L}_{{\text {Jac}}}\). Even worse, no choice of these parameters could make the two results compatible because the relative strengths of the \(\gamma \gamma \), \(\gamma Z\), ZZ, and \(W^{+}W^{-}\) decays are irremediably different. As we will now detail, it is only once the anomalous contributions from \(\delta \mathcal {L}_{{\text {Jac}}}\) precisely cancel out with those hidden in the triangle graphs that the \(a^{0}\rightarrow V_{1}V_{2}\) amplitudes match the \(A^{0}\rightarrow V_{1}V_{2}\) amplitudes.
Throughout this section, the momentum flow is defined as \(a^{0}(q_{1}+q_{2})\rightarrow V_{1}(q_{1},\alpha )V_{2}(q_{2},\beta )\), with \(V_{1,2}=g,\gamma ,Z,W\), and neither the final gauge bosons nor the initial axion are necessarily on-shell. The final amplitudes thus depend on \(q_{1}^{2}\), \(q_{2}^{2}\), and \((q_{1}+q_{2})^{2}\), and are to be compared to those obtained in the linear case in Eq. (43).
4.3.1 The \(a^{0}\rightarrow \gamma \gamma \) and \(a^{0}\rightarrow gg\) decays
As for the toy model, the polar representation is thus interesting only to make the shift symmetry manifest, and because the contact \(a^{0}\gamma \gamma \) and \(a^{0}gg\) interactions read off \(\delta \mathcal {L}_{{\text {Jac}}}\) are reliable book-keeping of the effects of heavy fermions. Specifically, \(\mathcal {M}(a^{0}\rightarrow \gamma \gamma ,gg)_{{\text {Der}}}^{AVV}\overset{m\rightarrow \infty }{=}0\) implies that \(\mathcal {M}(A^{0}\overset{}{\rightarrow }\gamma \gamma ,gg)_{\mathrm {Linear}}\overset{m\rightarrow \infty }{=}\mathcal {M}(a^{0}\overset{}{\rightarrow }\gamma \gamma ,gg)_{{\text {Jac}}}\). Finally, remark that the cancellation of the local anomalous terms ensures \(\mathcal {M}(a^{0}\overset{}{\rightarrow }\gamma \gamma ,gg)_{_{\mathrm {Polar}}}=0\) in the \(m_{u,d,e}\rightarrow 0\) limit. So, though interpreting the axion coupling to photons or gluons as induced by the anomaly is incorrect, this misidentification does not lead to serious consequences for those final states. For heavy fermions, the coupling to gluons is tuned by \(\mathcal {N}_{C}\), and that to photons by \(\mathcal {N}_{em}\), and their ratio, when restricted to quarks, give back the usual \(\mathcal {N}_{em}^{q}/\mathcal {N}_{C}=8/3\). However, as we will see in the next subsection, interpreting the axion coupling involving at least one electroweak gauge boson as induced by the anomaly is not only wrong in principle but also leads to incorrect couplings.
4.3.2 The \(a^{0}\rightarrow \gamma Z\) decay
Actually, the reason why the VAV triangle plays a role can be understood directly from the fermion reparametrization, given the charges in Eq. (52). When \(\alpha \) and/or \(\beta \) are different from zero, the fermionic current associated to \(U(1)_{PQ}\) has a component aligned with baryon and/or lepton number, respectively. The fermionic reparametrization thus generates the anomalous \(\mathcal {B}+\mathcal {L}\) interactions: those correspond to the \(N_{C}\alpha +\beta \) terms of \(\mathcal {N}_{L}\) and \(\mathcal {N}_{Y}\) in Eq. (56). But as for the toy model, these interactions are spurious and must cancel with the anomalies present in the triangle graphs induced by the derivative interactions \(\delta \mathcal {L}_{{\text {Der}}}\). This must necessarily come from a breakdown of the vector Ward identity for the VAV triangle since \(\mathcal {B}\) and \(\mathcal {L}\) are purely vectorial symmetries [24].
4.3.3 The \(a^{0}\rightarrow ZZ\) and \(a^{0}\rightarrow W^{+}W^{-}\) decays
Turning first to the \(a^{0}\rightarrow ZZ\) amplitude, the derivative interactions induce new types of diagrams: the AAA triangles and graphs with neutrinos circulating in the loop, see Fig. 5.
4.3.4 Impact of heavy neutrinos
As a final note, it is interesting to remark that once certain that \(\alpha \) and \(\beta \) disappear from the physical amplitudes, one can chose to fix them as one wishes. In particular, as said before, they can be set to eliminate the electroweak \(\theta \) term. Alternatively, one can set \(\alpha =\beta =0\), thereby getting rid of the \(a^{0}W_{\mu \nu }^{i}\tilde{W}^{i,\mu \nu }\) coupling altogether. As we have seen, this does not forbids \(a^0\rightarrow W^+W^-\) since the amplitude is independent of \(\alpha \) and \(\beta \). Finally, this freedom can also be used to allow for the presence of a lepton number violating Majorana mass term, as well as an effective dimension-five \((\bar{\ell }_{L}^{C}H_{1})(\ell _{L}H_{1})/\Lambda \) operator. We will not explore such settings here, see Ref. [25] for studies involving axions together with neutrino Majorana mass terms.
4.4 Application to the DFSZ axion model
The Peccei-Quinn model is ruled out experimentally because the coupling of the axion to fermions, tuned by the electroweak vacuum expectation value, is too large. One way to extend the model and render the axion invisible was proposed not long after the original idea and is called the DFSZ axion model [26, 27]. Our goal here is to derive the electroweak couplings for that model.
4.5 Anomaly cancellation in generic axion models
Coefficients of the effective axion to gauge boson couplings of Eq. (95) in the \(m_{u,d,e} \rightarrow \infty \) limit. For the linear representation, those are found directly from the THDM amplitudes of Ref. [21]. For the polar representation, the contributions of the local anomalous terms and that of the triangle amplitudes built on the axial (A) and vector (V) derivative interactions have to be added together. The fact that only the three independent coefficients \(\mathcal {N}^{gg}\), \(\mathcal {N}^{\gamma \gamma }\), and \(\mathcal {N}_L\) occur for the local anomalous interactions comes from their \(SU(2)_L \otimes U(1)_Y\) invariance. The coefficient \(\mathcal {N}_L = -1/2 (3 \alpha + \beta )\), with \(\alpha \) and \(\beta \) being the free parameters tuning the \(U(1)_{\mathcal {B}}\) and \(U(1)_{\mathcal {L}}\) components of \(U(1)_{PQ}\). The explicit presence of \(\beta \) in the \(a^0 \rightarrow ZZ, WW\) triangles is due to the peculiar nature of the neutrinos, which are kept massless. They do not contribute in the linear representation but have to explicitly appear in the anomalous interactions and derivative terms since those are \(SU(2)_L \otimes U(1)_Y\) invariant
Linear | Anomalous interactions | Polar | ||
---|---|---|---|---|
\(a^0\bar{\psi }\gamma _{5}\psi \) | \(\partial _{\mu }a^0\bar{\psi }\gamma ^{\mu }\gamma _{5}\psi \) | \(\partial _{\mu }a^0\bar{\psi }\gamma ^{\mu }\psi \) | ||
AVV | AAA | VAV | ||
\(\mathcal {N}^{gg}=\frac{1}{2}\left( x+\frac{1}{x}\right) \) | \(\mathcal {N}^{gg}\) | 0 | − | − |
\(\mathcal {N}^{\gamma \gamma }=\frac{4}{3}\left( x+\frac{1}{x}\right) \) | \(\mathcal {N}^{\gamma \gamma }\) | 0 | − | − |
\(\mathcal {N}_{1}^{\gamma Z}=\frac{1}{2}\left( x+\frac{1}{x}\right) \) | \(\mathcal {N}_{L}\) | 0 | − | \(\mathcal {N}_{1}^{\gamma Z}-\mathcal {N}_{L}\) |
\(\mathcal {N}_{2}^{\gamma Z}=\mathcal {N}^{\gamma \gamma }\) | \(\mathcal {N}^{\gamma \gamma }\) | 0 | − | 0 |
\(\mathcal {N}_{1}^{ZZ}=\frac{1}{4}x+\frac{1}{3x}\) | \(\mathcal {N}_{L}\) | \(\frac{\beta }{16}\ \) | \(-\frac{1}{2}\mathcal {N}_{1}^{ZZ}+\frac{\beta }{16}\) | \(\frac{3}{2}\mathcal {N}_{1}^{ZZ}-\mathcal {N}_{L}-\frac{\beta }{8}\) |
\(\mathcal {N}_{2}^{ZZ}=\mathcal {N}_{1}^{\gamma Z}\) | \(\mathcal {N}_{L}\) | 0 | 0 | \(\mathcal {N}_{2}^{ZZ}-\mathcal {N}_{L}\) |
\(\mathcal {N}_{3}^{ZZ}=\mathcal {N}^{\gamma \gamma }\) | \(\mathcal {N}^{\gamma \gamma }\) | 0 | 0 | 0 |
\(\mathcal {N}^{WW}=\frac{x}{4}+\frac{3}{8x}\) | \(\mathcal {N}_{L}\) | \(\frac{3}{2}\mathcal {N}^{WW}-\frac{3}{2}\mathcal {N}_{1}^{\gamma Z}+\frac{\beta }{16}\) | \(-\frac{1}{2}\mathcal {N}^{WW}+\frac{\beta }{16}\) | \(\frac{3}{2}\mathcal {N}_{1}^{\gamma Z}-\mathcal {N}_{L}-\frac{\beta }{8}\) |
5 Conclusion and perspective
- In the linear representation, the axion couplings to gauge bosons are not induced by the anomaly, but by non-anomalous pseudoscalar triangle graphs with fermions circulating in the loop. Though this has to our knowledge not been exploited before, these amplitudes can be identified with those of the pseudoscalar Higgs to gauge bosons calculated in the THDM, which have been known for a long time [21]. In the \(m_{u,d,e}\rightarrow \infty \) limit, they match onto the effective interactionswith the coefficients shown in the first column of Table 1. In the opposite limit \(m_{u,d,e}\rightarrow 0\), all these amplitudes vanish since the axion couplings to fermions are proportional to the fermion masses.$$\begin{aligned} \mathcal {L}^{\mathrm {eff}}&=\frac{a^{0}}{16\pi ^{2}v}\left( g_{s}^{2} \mathcal {N}^{gg}G_{\mu \nu }^{a}\tilde{G}^{a,\mu \nu }\right. \nonumber \\&\quad \left. +e^{2}\mathcal {N} ^{\gamma \gamma }F_{\mu \nu }\tilde{F}^{\mu \nu }+\frac{2e^{2}}{c_{W}s_{W}} \left( \mathcal {N}_{1}^{\gamma Z}-s_{W}^{2}\mathcal {N}_{2}^{\gamma Z}\right) Z_{\mu \nu }\tilde{F}^{\mu \nu }\right. \nonumber \\&\quad \left. +\frac{e^{2}}{ c_{W}^{2}s_{W}^{2}}\left( \mathcal {N}_{1}^{ZZ}-2s_{W}^{2}\mathcal {N} _{2}^{ZZ}+s_{W}^{4}\mathcal {N}_{3}^{ZZ}\right) Z_{\mu \nu }\tilde{Z}^{\mu \nu }\right. \nonumber \\&\quad \left. +2\mathcal {N}^{WW}g^{2}W_{\mu \nu }^{+}\tilde{W}^{-,\mu \nu }\right) , \end{aligned}$$(95)
In the context of axion models, it is customary to adopt a polar representation for the scalar fields. The pseudoscalar axion couplings to fermions are replaced by contact anomalous interactions to the gauge bosons and axion derivative interactions to the fermions. All these interactions are entirely fixed in terms of the assigned PQ charges of the SM fermions. As shown in Table 1, this parametrization is particularly convenient for \(a^{0}\rightarrow \gamma \gamma \) and \(a^{0}\rightarrow gg\) because the derivative interactions do not contribute in the \(m_{u,d,e}\rightarrow \infty \) limit [10, 11], and thus the strengths of the \(a^{0}\rightarrow \gamma \gamma \) and \(a^{0}\rightarrow gg\) processes can be read off the anomalous couplings \(a^{0}F_{\mu \nu }\tilde{F}^{\mu \nu }\) and \(a^{0}G_{\mu \nu }^{a}\tilde{G}^{a,\mu \nu }\). By contrast, and contrary to what is conjectured in Ref. [10], the central result of this paper is that for chiral gauge theories in which chiral fermions have non-trivial PQ charges, the strengths of the \(a^{0}\rightarrow \gamma Z\), ZZ, and \(W^{+}W^{-}\) processes do not match the anomalous couplings, even in the \(m_{u,d,e}\rightarrow \infty \) limit, as evident comparing the first and second columns of Table 1. They thus cannot be encoded into the \(SU(2)_{L}\otimes U(1)_{Y}\) invariant effective interactions \(a^{0}B_{\mu \nu }\tilde{B}^{\mu \nu }\ \)and \(a^{0}W_{\mu \nu }^{i}\tilde{W}^{i,\mu \nu }\).
The main reason for this mismatch is due to the presence of triangle graphs arising from the derivative interactions \(\partial _{\mu }a^{0}\bar{\psi }\gamma ^{\mu }\gamma _{5}\psi \) and \(\partial _{\mu }a^{0}\bar{\psi }\gamma ^{\mu }\psi \), which do not vanish in the \(m_{u,d,e}\rightarrow \infty \) limit for chiral gauge theories. Importantly, even the vector couplings play a role since the anomalous breaking of the axionic vector current conservation enters through the VAV triangle graphs (see Figs. 4, 5). One important result of this paper is the proof that once all these triangle contributions (last three columns of Table 1) are summed with the local anomalous amplitudes (second column of Table 1), the THDM results are recovered. Without surprise, physical observables do not depend on the chosen parametrization, and this further confirms that the THDM calculation is correct.
Our analysis also proves that the axion to gauge boson processes are not induced by the anomaly. This is obvious in the linear representation since the fermion loops driving these processes are not anomalous. In the polar representation, what happens is that the local anomalous interactions precisely cancel with the anomalies present in triangle graphs induced by the vector and axial derivative interactions, leaving the non-anomalous pseudoscalar triangles of the THDM as the only surviving mechanism driving all the axion to gauge bosons processes. Ultimately, this precise cancellation is ensured by the anomalous vector and axial Ward identities. It is required because all the amplitudes must vanish in the \(m_{u,d,e,\nu }\rightarrow 0\) limit, but the anomalous contributions are independent of the fermion mass.^{6}
In all axion models where chiral fermions have PQ charges but conserve baryon and lepton numbers, both the vector couplings \(\partial _{\mu }a^{0}\bar{\psi }\gamma ^{\mu }\psi \) and the \(a^{0}W_{\mu \nu }^{i}\tilde{W}^{i,\mu \nu }\), \(a^{0}B_{\mu \nu }\tilde{B}^{\mu \nu }\) anomalous contact interactions depend on some free parameters [10]. In Table 1, these free parameters enter into \(\mathcal {N}_{L}\), and are related to the freedom to choose the PQ charge of left-handed fermions (there are also explicit occurrence of the free parameter \(\beta \), but this is an artifact of keeping the neutrino massless). It is only once the failure of the naive vector Ward identity in the VAV triangle graphs is properly accounted for that these free parameters cancel out, as they should [24]. To understand how this cancellation is connected with the conservation of baryon and lepton numbers \(\mathcal {B}\) and \(\mathcal {L}\), notice first that the freedom to choose the PQ charges of the fermions is actually due to the invariance under \(U(1)_{\mathcal {B}}\) and \(U(1)_{\mathcal {L}}\). In general, the \(U(1)_{PQ}\) symmetry has two arbitrary components aligned with \(U(1)_{\mathcal {B}}\) and \(U(1)_{\mathcal {L}}\). But then, these components translate into derivative couplings of the axion to the \(\mathcal {B}\) and \(\mathcal {L}\) fermionic vectorial currents together with a local \(a^{0}W_{\mu \nu }^{i}\tilde{W}^{i,\mu \nu }\) coupling from the \(\mathcal {B}+\mathcal {L}\) anomaly. Clearly, all these couplings are spurious when the axion is a Goldstone boson living in a \(\mathcal {B}\) and \(\mathcal {L}\)-invariant vacuum. Since \(\mathcal {B}\) and \(\mathcal {L}\) are not spontaneously broken, \(\mathcal {N}_{L}\) must systematically drop out of physical observables, and Table 1 shows that this indeed occurs.
In the linear representation, the calculation is done in the electroweak broken phase throughout. In the \(m_{u,d,e}\rightarrow \infty \) limit, the amplitudes do not match onto \(SU(2)_{L}\otimes U(1)_{Y}\) invariant operators, as shown in Table 1. Though this is somewhat expected, what is not obvious is the very peculiar way in which the \(SU(2)_{L}\otimes U(1)_{Y}\) symmetry breaking seeps in. Indeed, in the non-linear representation, the anomalous interactions and the derivative interactions are manifestly \(SU(2)_{L}\otimes U(1)_{Y}\) invariant once the SM fermions are assigned PQ charges in an \(SU(2)_{L}\otimes U(1)_{Y}\) invariant way. Further, evidently, the anomalies present in the triangle graphs built on the derivative interactions do not break \(SU(2)_{L}\otimes U(1)_{Y}\) since that would put the SM gauge invariance itself in jeopardy. So, in that representation, the fact that the chiral \(SU(2)_{L}\otimes U(1)_{Y}\) symmetry is spontaneously broken enters only and entirely through the fermion masses. The triangle graph explicitly break the \(SU(2)_{L}\otimes U(1)_{Y}\) symmetry because the \(m_{u,d,e}\rightarrow \infty \) limit is incompatible with that symmetry. In the opposite limit \(m_{u,d,e}\rightarrow 0\), the \(SU(2)_{L}\otimes U(1)_{Y}\) symmetry is recovered but in a very trivial way: the triangle graphs precisely cancel the local anomalous term and all the amplitudes simply vanish.
We derived generalized forms for all the triangle anomalies, including the terms proportional to the fermion masses, see Eqs. (30) and (31). This requires going beyond usual regularization procedures. To our knowledge, and except those based on dispersion relations [19], only that described by Weinberg in Ref. [17] offers sufficient freedom to choose which current in the AVV and AAA triangles is to carry the anomaly. This is crucial to deal with chiral gauge theories, in which some axial currents have to remain anomaly-free to preserve gauge invariance, as well as for treating the \(\mathcal {B}\) and \(\mathcal {L}\) vectorial currents [24].
Footnotes
- 1.
We use the notations and conventions of Ref. [5]
- 2.
This non-decoupling is to be understood in the formal sense, since strictly speaking the \(m \rightarrow \infty \) limit is not compatible with our perturbative treatment of the Yukawa couplings.
- 3.
Note that the polar representation still allows the pseudoscalar of having a small mass which could be treated as a perturbation as this is usually done in axion phenomenology.
- 4.
These results apply to axion like particles but also to heavier pseudoscalars.
- 5.
- 6.
This interpretation is compatible with the matching observed in Table 1 in the \(m_{u,d,e}\rightarrow \infty \) limit for \(a^{0}\rightarrow \gamma \gamma ,gg\). Indeed, in the polar representation, these amplitudes receive two contributions, \(\mathcal {M}_{ano}\) from the local anomalous term, and \(\mathcal {M}_{der}\) from the triangle amplitudes built from \(\partial _{\mu }a^0\bar{\psi }\gamma ^{\mu }\gamma _{5}\psi \). But in this case, the axial Ward identity translates as \(\mathcal {M}_{der} = \mathcal {M}_{lin} - \mathcal {M}_{ano}\), with \(\mathcal {M}_{lin}\) the THDM amplitude. So, even if parametrically, \(\mathcal {M}_{lin} = \mathcal {M}_{ano}\) and \(\mathcal {M}_{der}=0\) when \(m_{u,d,e}\rightarrow \infty \), \(a^{0}\rightarrow \gamma \gamma ,gg\) are not induced by the anomaly since it cancels out in \(\mathcal {M}_{ano}+\mathcal {M}_{der}\).
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