# On the structure of anomalous composite Higgs models

## Abstract

We describe the anomaly structure of a composite Higgs model in which the *SO*(5) / *SO*(4) coset structure of the minimal model is extended by an additional, non-linearly realised \(U(1)_{\eta }\). In addition, we show that the effective Lagrangian admits a term that, like the Wess–Zumino–Witten term in the chiral Lagrangian for QCD, is not invariant under the non-linearly realised symmetries, but rather changes by a total derivative. This term is unlike the Wess–Zumino–Witten term in that it does not arise from anomalies. If present, it may give rise to the rare decay \(\eta \rightarrow h W^+ W^-Z\). The phenomenology of the singlet in this model differs from that in a model based on *SO*(6) / *SO*(5), in that couplings to both gluons and photons, arising via anomalies, are present. We show that while some tuning is needed to accommodate flavour and electroweak precision constraints, the model is no worse than the minimal model in this regard.

## Keywords

Higgs Boson Yukawa Coupling Goldstone Boson Partial Compositeness Coset Space## 1 Introduction

In recent years, theorists have devoted much attention to models in which the electroweak hierarchy problem is solved by postulating that the Higgs boson arises as a composite pseudo-Goldstone boson of some new, TeV-scale strong dynamics [1, 2, 3].

If this is really what happens in Nature, then it is interesting to ask how we might go about figuring out what the underlying UV dynamics is, given our current rather poor theoretical understanding of strongly coupled dynamics.

One way in which may we may do so is via triangle anomalies, which are not renormalised and so, if present in the UV, must be reproduced in the IR, either by massless fermions or by terms involving the pseudo-Goldstone bosons. Such anomalies are not only not renormalised, but they are also topological in nature. This means that by measuring them in the IR, we may gain concrete information as regards the UV dynamics. The classic example, of course, is in QCD, where the measurement of the decay rate \(\pi ^0 \rightarrow \gamma \gamma \) (which arises via the electromagnetic anomaly [4, 5]) enables us to infer that \(N_c =3\).

In order to make such spectacular inferences, one must be lucky enough to have a low-energy Lagrangian that admits a non-trivial anomaly structure. The minimal, and by far the most popular, composite Higgs model, based on *SO*(5) / *SO*(4) [6] does not feature anomalies. However, the ‘next-to-minimal’ model based on *SO*(6) / *SO*(5) [7], which is just as good from the phenomenological point of view, does. Compared to the minimal model, it features only an additional electroweak singlet scalar, which couples to electroweak gauge bosons via a single \(SO(6)^3\) triangle anomaly.

Here we wish to describe yet another model, based on \(SO(5)\times U(1)/SO(4)\). It is just as minimal as the *SO*(6) / *SO*(5) model, in the sense that it features only an additional electroweak singlet scalar. But it turns out to have a much richer anomaly structure, with several novel features.

A first novel feature is that there are now three distinct triangle anomalies, which give rise, at leading order, to couplings of the singlet to both gluons and electroweak bosons.

A second novel feature is that the higher-order structure of the anomalous effective action is not unique. Indeed, we exhibit two solutions to the Wess–Zumino consistency conditions. As far as we are aware, this phenomenon has not been observed before in the literature on sigma models.

A third novel feature is that the effective Lagrangian admits a term that is not invariant, but rather changes by a total derivative, under the non-linearly realised symmetries. Such a term is much like the Wess–Zumino–Witten (WZW) term in the chiral Lagrangian of QCD, which allows processes violating a putative internal symmetry under which Goldstone bosons change sign, such as \(K+\overline{K} \rightarrow 3\pi \) [8, 9]. But there is one noteworthy distinction between the WZW-like term presented here and WZW term in QCD. In the latter, the presence of the anomaly implies the presence of the WZW term, in the sense that the low-energy effective action reproducing the anomaly reduces to the WZW term when the gauge fields vanish. In the model presented here, this is not so. This phenomenon is also, we believe, unknown in the sigma-model literature. The WZW-like term is also of phenomenological interest, in that it may lead to a rare decay of the singlet via \(\eta \rightarrow h W^+ W^- Z\).

The outline is as follows. In the next section, we present the pattern of symmetry breaking and sketch the concomitant anomalies. We then present a full discussion of the anomaly structure and the WZW-like term in Sect. 3. In Sect. 4, we describe the couplings to fermions and the implications for flavour physics. In Sect. 5, we discuss the form of the scalar potential that is induced by the couplings to gauge fields and fermions. We conclude in Sect. 6. Two more technical discussions are relegated to appendices.

## 2 The model

We wish to consider composite Higgs models based on a homogeneous space *G* / *H* that feature triangle anomalies.^{1}

The minimal model [6], based on *SO*(5) / *SO*(4) (or *SO*(5) / *O*(4) with custodial protection of \(Z\rightarrow b\overline{b}\) [11]), features no triangle anomalies. The ‘next-to-minimal’ model based on *SO*(6) / *SO*(5) [7] does, however, feature triangle anomalies. Indeed the Goldstone bosons transform as the 5-d irrep of *SO*(5), which, on restriction to the *SO*(4) subgroup, yields both a 4-d irrep (viz. the Higgs field) and a singlet. Moreover, since *SO*(6) is locally isomorphic to *SU*(4), we have the possibility of an \(SU(4)^3\) triangle anomaly.^{2} This anomaly leads to an interaction, at leading order, of the form \(\frac{1}{16 \pi ^2} \frac{\eta }{f} (g^2_2 \, W_{\mu \nu }\tilde{W}^{\mu \nu }- g_1^2 \, B_{\mu \nu }\tilde{B}^{\mu \nu }) \), with neither a coupling to gluons nor to photons [12].

The absence of a coupling to photons in this model is something of a group-theoretical accident, in that there are couplings to *ZZ*, \(\gamma Z\), and *WW*. But the absence of a coupling to gluons looks, at first sight, to be a generic problem in a composite Higgs model, given that the rôle of the new strong dynamics is to break the electroweak symmetry, independently of the \(SU(3)_C\) dynamics. In fact, this is not so, since a consequence of partial compositeness is that the new strong sector must be charged under \(SU(3)_C\) [13]. So it seems quite plausible that the elementary fermions of the UV theory could generate an anomaly involving \(SU(3)_C\).

*G*. But such a strategy will lead to additional coloured Goldstone bosons, with potentially dangerous phenomenological implications.

^{3}A safer, and simpler, strategy is to modify the minimal model by adding a non-linearly realised \(U(1)_{\eta }\) factor, such that the symmetry breaking pattern in the strong sector becomes

*U*(1) needed in composite Higgs models to give the correct hypercharge assignments to SM fermions. This model features an additional SM singlet compared to the minimal composite Higgs model. We remark that, unlike the

*SO*(6) /

*SO*(5) model, this coset space allows for two distinct decay constants,

*f*and \(f_{\eta }\), associated with the Higgs boson and the \(\eta \), respectively. We assume henceforth that these are generated by the same strong dynamics, and hence are of the same order of magnitude.

Let us now consider the possible triangle anomalies in this model. As we shall see in Sect. 3, triangle anomalies in *G* are admissible only if they vanish on restriction to *H*. Thus, our model admits three possible sources of triangle anomalies, namely \(SU(3)_C^2 \, U(1)_{\eta }\) and \(SO(5)^2 \, U(1)_{\eta }\) anomalies, and anomalies involving \(U(1)_{\eta }\) and \(U(1)_X\).

## 3 Anomalies and WZW-like terms

We now discuss the anomaly structure of the model in more detail, together with the phenomenological consequences. Let us begin with a general discussion. A theory with internal global symmetry group *G* may be anomalous, in the sense^{4} that there is no way to regularise the theory such that the divergences of 3-point functions of conserved currents are all vanishing. Such anomalies are not renormalised and must be reproduced at all energies, with consequences for low-energy physics.

One consequence is a consistency condition on the possible pattern of symmetry breaking at low energy: if a subgroup \(H \subset G\) is linearly realised at low energy, then *H* must be anomaly free. The reason [16] is that we could perturb the theory in an arbitrarily small way by gauging the whole of *G*, but choosing the gauge coupling to be arbitrarily small. If there were anomalies in *H*, the gauge bosons in *H* could get masses via a loop diagram formed out of two anomalous vertices, implying that *H* could not be linearly realised.

*G*/

*H*can be found (see also [18]). As in [17], in this Section we employ the langauge of differential forms and omit normalisation factors, giving the result only for the special case of a symmetric space, which is sufficient for our needs. The formula is most conveniently written in the fully gauged case; the result for gauging a subset \(F \subset G\) can be obtained by setting the corresponding gauge fields to zero in the formula.

^{5}Let \(\mathfrak {g}\) and \(\mathfrak {h}\) be the Lie algebras of

*G*and

*H*. Since

*G*/

*H*is reductive and symmetric, there exists a split, \(\mathfrak {g} = \mathfrak {h} + \mathfrak {k}\), such that \([\mathfrak {h},\mathfrak {k}] \subseteq \mathfrak {k}\) and \([\mathfrak {k},\mathfrak {k}] \subseteq \mathfrak {h}\), together with an ‘internal parity’ automorphism of \(\mathfrak {g}\) given by \(\mathfrak {h} \rightarrow \mathfrak {h}\) and \(\mathfrak {k} \rightarrow -\mathfrak {k}\). Letting

*A*be a \(\mathfrak {g}\)-valued 1-form representing the gauge fields and letting the coset representative be \(e^\xi \), with \(\xi \in \mathfrak {k}\), we have

*h*and

*k*denote projections onto the corresponding subspaces, such that \(F_h = dA_h + A_h^2 +A_k^2, F_k = dA_k + A_hA_k +A_k A_h\).

The action (3.1) is unique in the sense that it is the only action which vanishes when the Goldstone bosons vanish and whose anomaly is given by \(\delta _\alpha \Gamma = \sum _\pm {c_\pm \mathrm {tr} \alpha G^\pm [A]}\) [18]. But it is not unique in the sense that the anomaly can take many forms, corresponding to the addition of local counterterms to the effective action. (For a counterexample, it suffices to choose \(H = 0\), for which *any* form *G*[*A*] for the anomaly is reproduced by the effective action \(\Gamma = \int _0^1 \mathrm{d}t \int _x \mathrm {tr} \xi G [A_t]\).) The action (3.1) is the one obtained by starting from the canonical form of the anomaly (which is symmetric with respect to *G*) and subtracting a counterterm that enforces the vanishing of the anomaly on *H* [17]. Hadronic data suggest that this is the option chosen by the strong interactions, but we are unaware of an argument that it is the only consistent option.

*raison d’être*is to reproduce anomalies that arise due to gauging, (3.1) may not vanish in the limit that gauge fields vanish. In that limit, we obtain

*G*transformation, but rather changes by a total derivative. We will call such non-invariant Lagrangian terms ‘WZW terms’, in honour of their prototype in the chiral Lagrangian. It was shown in [20] that, for compact

*G*in \(d=4\), and for field configurations in the trivial fourth homotopy class,

^{6}such terms are in 1-1 correspondence with the generators of the fifth de Rham cohomology group of

*G*/

*H*.

We caution the reader that not all such terms can arise from effective actions reproducing triangle anomalies. By way of a counterexample, consider the homogeneous space \(SU(2)\times SU(2)/U(1)\), where the *U*(1) is included in one of the *SU*(2)s. This space is equivalent as a smooth manifold to \(S^3 \times S^2\) and a straightforward generalisation of the arguments presented below shows that \(H^5_{dR} (S^3 \times S^2) = \mathbb {R}\). Thus, there is a WZW term in this case, but since *SU*(2) has no triangle anomalies, it cannot arise from reproducing them.^{7}

### 3.1 Composite Higgs model anomalies

*U*(1)s, there are no higher-order corrections to the effective action. There are, however, higher-order corrections for the \(SO(5)^2 U(1)\) anomaly, the detailed calculation of which we relegate to Appendix 1. The next-to-leading order corrections arise at dimension 7, up to which order the effective action is given, in the operator basis of [21], by

*B*are the field strength 2-forms and

*f*is the non-linear scale.

These corrections to the leading-order action appear to constitute a definite prediction of the model, once \(c_5\) has been determined from measurements at leading order. Unfortunately, the issue of non-uniqueness discussed above now rears its ugly head. Indeed, it is easy to check that the leading-order action (2.2) alone also provides a solution of the Wess–Zumino consistency conditions that vanishes on *SO*(4) and so is, *ceteris paribus*, just as good a candidate for the anomalous action. It corresponds to a regularisation of the \(SO(5)^2U(1)\) anomaly such that it is appears entirely in the *U*(1) symmetry, whereas our action corresponds to an anomaly that is symmetric with respect to the broken generators in *U*(1) and *SO*(5). Whether there exist yet more consistent effective actions is an open question.

The two anomalous effective actions that we have found differ structurally only in their higher-order terms. But this does not mean that the non-uniqueness is phenomenologically inconsequential. Indeed, different choices of regulator for the anomaly lead to different values of the coefficient of the leading-order term. In particular, the coefficient that corresponds to an anomaly that is symmetrised amongst all three currents is \(\frac{1}{3}\) that of the coefficient that corresponds to the anomaly that is contained wholly in a single current. So the resolution of the non-uniqueness issue will be crucial, if we want to make inferences about the UV structure of the theory (in particular its fermionic representation content), using experimental data.

Even if this non-uniqueness can be resolved, one should also bear in mind that the couplings of the Goldstone bosons to SM fermions will also generate loop contributions to the couplings in the anomalous effective action.

### 3.2 The WZW term

There is a possible WZW term in the model, as we can see by computing \(H^5_{dR} (SO(5) \times U(1)/SO(4))\). Recalling that \(SO(n+1)/SO(n)\) and \(S^n\) are equivalent as smooth manifolds, we thus have \(H^5_{dR} (SO(5) \times U(1)/SO(4)) = H^5_{dR} (S^4 \times S^1) = H^4_{dR} (S^4) \otimes H^1_{dR} (S^1) \simeq \mathbb {R} \otimes \mathbb {R} \simeq \mathbb {R}\) (where we used the Künneth formula and the fact that \(H^i_{dR} (S^n)\) vanishes unless \(i=0\) or \(i=n\), in which case it is isomorphic to \(\mathbb {R}\)). Thus, the theory admits a WZW term.

*G*-invariant 5-form, whose existence is guaranteed by the non-vanishing fifth de Rham cohomology group.

^{8}For \(G/H \simeq S^4 \times S^1\) it is just the product of the usual volume forms on the hyperspheres. At leading order in the fields, we can integrate over the 5-ball to get

^{9}In \(SU(2)\times U(1)\) language, the LO WZW term is \( \eta dH^\dagger \sigma ^i dH dH^\dagger \sigma ^i dH\).

As expected, the leading-order term is invariant under the linearly realised subgroup *SO*(4) and changes by a total derivative under a shift of the Goldstone bosons, corresponding to an infinitesimal \(SO(5)\times U(1)\) transformation.

As we see in Appendix 1, the WZW term does not arise from (3.1), which vanishes when the gauge fields vanish. Thus, unlike in QCD, the WZW term and the anomaly are independent, at least for this choice of regularisation of the anomaly.

The WZW term is in fact the leading-order term coupling all 5 Goldstone bosons to each other. This can be seen by forming Lagrangian invariants of the sigma model in the usual way out of the objects \(\mathrm{d}\eta \) and \(e^\xi d e^{-\xi }\), which transform as adjoints under *H*. By Lorentz invariance, all terms involve an even number of derivatives. Terms with no derivatives are forbidden by the non-linearly realised symmetry, while terms with two derivatives are forbidden, because such a term must take the form \(\partial _\mu \eta \, \mathrm {tr} \, e^\xi \partial _\mu e^{-\xi } =0\). A possible term with 4 derivatives takes the form \(\partial _\mu \eta \, \mathrm {tr} \, (e^\xi \partial _\nu e^{-\xi }) (e^\xi \partial _\sigma e^{-\xi }) (e^\xi \partial _\rho e^{-\xi })\). Since *SO*(5) is free of triangle anomalies, the trace term must be antisymmetric in its three entries and so a non-vanishing Lorentz invariant can be obtained only by contracting with \(\epsilon ^{\mu \nu \sigma \rho }\), such that we can revert to the language of differential forms. We have \(e^\xi d e^{-\xi } = \mathrm{d}\xi + \frac{1}{2} [\xi , \mathrm{d}\xi ] + \cdots \), such that the leading-order term involving all Goldstone bosons takes the form \(\frac{3}{2}\mathrm{d}\eta \mathrm {tr} \mathrm{d}\xi \mathrm{d}\xi [\xi ,\mathrm{d}\xi ]\). We need this to be non-vanishing when each \(\xi \) corresponds to a distinct Goldstone boson and one easily check using the basis in (B.11) that this is not so.

To explore the physics of the WZW term, we first gauge the SM subgroup. Since this is a subgroup of *H*, under which the WZW term transforms linearly, we may follow the usual prescription of promoting derivatives to covariant derivatives, obtaining \(\eta DH^\dagger \sigma ^i DH DH^\dagger \sigma ^i DH\).

Being of high dimension, the WZW leads to small contributions to low-energy physics. They may, nevertheless, be observable at a future high-precision collider, if sufficiently exotic. As an example, by the Goldstone boson equivalence theorem and by the antisymmetry in the fields, the WZW term leads, after electroweak-symmetry breaking, to a coupling involving \(\eta , h, W^+,W^-,\) and *Z* and hence a possible decay mode \(\eta \rightarrow hW^+W^-Z\).

We remark that, whilst the WZW term is the leading-order term coupling all five Goldstone bosons to one another, this does not necessarily imply that it gives the dominant contribution to this decay mode. Indeed, once we switch on the gauging and other symmetry-breaking couplings, we may well get contributions to this decay at lower orders, albeit paying the price of small, symmetry breaking couplings instead.

### 3.3 Discrete symmetries and \(Z\rightarrow b \overline{b}\)

As we have already remarked, the fact that \(SO(5) \times U(1)/SO(4)\) is a symmetric space means that the Lie algebra possesses the ‘internal parity’ automorphism \(\mathfrak {h} \rightarrow \mathfrak {h}, \mathfrak {k} \rightarrow -\mathfrak {k}\). The terms in the effective action giving rise to production and decay of the \(\eta \) are odd under this automorphism, so it could only be a symmetry of the dynamics if it were accompanied by a spatial inversion. In any case, the internal parity is broken in the vacuum by the Higgs VEV.

A more desirable symmetry to have, perhaps, is one that protects the decay rate for \(Z\rightarrow b \overline{b}\) [11]. In the minimal model based on \(G=SO(5)\), this is achieved by enlarging the linearly realised subgroup from *SO*(4) to *O*(4).^{10} The same enlargement could, of course, be carried out in the model described here, but it has the consequence that the WZW term is forced to vanish. Indeed, the usual action of *SO*(5) on \(\mathbb {R}^5\) gives rise to transitive actions on both \(S^4\) (included in \(\mathbb {R}^5\) as the set of points equidistant from the origin) and \(\mathbb {R}P^4\) (given as the set of lines through the origin in \(\mathbb {R}^5\) and which we may also think of as the sphere with antipodal points identified). The stability subgroup in the former case is isomorphic to *SO*(4), while in the latter case it is *O*(4). Thus *SO*(5) / *SO*(4) is homeomorphic to \(S^4\), while *SO*(5) / *O*(4) is homeomorphic to \(\mathbb {R}P^4\). Now, \(H^4_{dR} (\mathbb {R}P^4)\) vanishes,^{11} as do its other de-Rham cohomology groups (excepting of course \(H^0_{dR}\)), and so the Künneth formula tells us that with *O*(4) included in this way, \(H^5_{dR} (SO(5) \times SO(2)/O(4)) = 0\), such that there can be no WZW term.

The WZW term may, however, be resurrected by changing the inclusion of the custodial *O*(4) in *G*. To understand this, it is useful to see more explicitly why the leading-order WZW term is forbidden in the standard implementation. To this end, choose co-ordinates \((\varvec{h} , 1 )\) on the unit 4-sphere included in \(\mathbb {R}^5\) in the neighbourhood of the stability point \((\varvec{0}, 1)\). The stability group of the sphere is then \(\{{O^+ \quad 0 \atopwithdelims ()0\quad +1}\}\), where \(O^+\) is any 4x4 orthogonal matrix of determinant \(+1\), and hence is isomorphic to *SO*(4). But if we identify antipodal points, \((-\varvec{h}, -1 )\sim (\varvec{h}, 1)\), then the stability subgroup is enhanced to \(\{{O^\pm \quad 0 \atopwithdelims ()0\quad \pm 1}\}\), where \(O^-\) is any 4x4 orthogonal matrix of determinant \(\pm 1\), and hence is indeed isomorphic to *O*(4), as we claimed earlier. Now, under the action of an element of *O*(4) that is disconnected from the identity, \(( \varvec{h} , 1) \rightarrow (O^-\varvec{h}, -1 )\sim (-O^-\varvec{h} , +1 )\). Thus the putative leading-order WZW term, which is proportional to \(\epsilon _{ijkl} h^i h^j h^k h^l\) is sent to \((-1)^4 \mathrm {det} O^- = -1\) times itself, and is not invariant under such transformations. But the leading-order WZW term should be invariant under *O*(4) and so must vanish.

*O*(4) custodial group to be included in

*G*in such a way that the action of elements in

*O*(4) disconnected from the identity also sends \(\eta \rightarrow -\eta \). To achieve this, set \(G= SO(5) \times O(2)\) and let

*H*be the subgroup

*H*is still isomorphic to

*O*(4), but now the action of elements in

*O*(4) disconnected from the identity sends \(\eta \rightarrow -\eta \). We conjecture therefore that \(H^5_{dR} \ne 0\) in this case, such that there is a WZW term.

## 4 Couplings to fermions and flavour violation

*i*is the family index) couple linearly to fermionic operators \(\overline{\mathcal {O}}^f_i\) of the strong sector:

We remark that we have tacitly assumed, for simplicity, that every elementary field \(f^{a}_i\) couples to a *single* operator of the strong sector. In that case, it is easy to derive the coupling of the goldstone boson \(\eta \) to the fermions \(f^i\). Indeed, it is enough to replace \(f^i \rightarrow f^i \exp \left( i \frac{\sqrt{2}}{f_{\eta }} \eta \, Z_{f_i} \right) \) in the EFT of the usual composite Higgs model based on \(SO(5) \times U(1)_X\), where \(Z_{f_i}\) is the \(U(1)_\eta \) charge. As we shall see in Sect. 5, there is a price to be paid for this assumption, namely that one then requires an additional source of explicit \(U(1)_\eta \) breaking in the model in order to generate a potential for the singlet. We expect, however, that relaxing this assumption will lead to comparable bounds.

*H*/

*f*and \(\eta /f_{\eta }\), we get (in complete generality) that

*Z*-charges of the fields; in particular we have defined \(\hat{Z}_{f} \equiv \text {diag}(Z_{f^1}, Z_{f^2},Z_{f^3})\).

With these expressions in hand, let us consider to what extent the suppression provided by the partial compositeness ansatz is sufficient to protect the model from dangerous flavour- and CP-violating contributions to physical observables.

*Z*and \(f_{\eta }\) are unknown and depend on the details of the strongly coupled sector. However, the most natural expectation is that \(f_{\eta } \sim f\) and \(Z \sim 1\), because the composite Higgs and the composite \(\eta \) are generated from the same strong dynamics. If this is the case, an extra source of flavour protection is required. An easy fix to this problem is to assume that the \(\eta \) PNGB couples to flavour in a universal way. More specifically, we can impose \(Z_{f^i} = Z_f\) for \(i=\{ 1,2,3 \}\). In this case the \(\eta \) and the Higgs boson couplings to fermions are aligned in each sector, such that

^{12}It is, moreover, rather predictive. Indeed the \(\eta \), like the Higgs, couples predominantly to the third generation. This could have important implications for the production and decay mechanisms of the singlet, as we now discuss.

We conclude this section noticing that the simple flavour structure that we have just described, while guaranteeing immunity from flavour problems, does not allow one to generate a scalar potential (and hence a mass) for the singlet from fermionic couplings. As we discuss in the next Section, to do so requires that at least one of the elementary fermions in the partial compositeness scenario mixes with multiple strong-sector operators. Even if one tries to do so in a way that is as safe as possible (for example by allowing the right-handed up quarks to couple to strong-sector operators with just two values of the \(U(1)_\eta \) charge), one ends up re-introducing flavour violation in the right-handed up sector at a level comparable to that obtained with anarchic charge assignments in Fig. 2, which is itself comparable to that obtained in the minimal composite Higgs model. Thus, if one wishes to generate the scalar potential from fermionic couplings, it would seem that either a mild tuning or some kind of flavour-alignment mechanism (such as those advanced in [30]) is required.

## 5 The scalar potential

Since the \(\eta \) singlet is protected by a shift symmetry, its mass and non-derivative interactions must be proportional to \(U(1)_\eta \)-breaking couplings. The elementary fermion couplings to the strong sector are the main source of such global symmetry violations, and the \(\eta \) singlet then obtains a potential via the same Coleman–Weinberg mechanism that radiatively generates the pseudo-Goldstone Higgs potential at one loop. This must originate from fermion couplings, since no potential is generated by gauge couplings in the absence of anomalies, because \(U(1)_\eta \) commutes with the rest of *G*.^{13} The particular form of the symmetry breaking from Yukawa couplings is, in general, model-dependent.

To illustrate the mechanism in a minimal phenomenological model, we take an elementary top-right coupling to two strong-sector operators with different \(U(1)_\eta \) charges such that the symmetry is explicitly broken by a collective mechanism.^{14} The doubling of the top-right operator is necessary to break the \(U(1)_\eta \) symmetry, since with only one operator, we can restore it by assigning a suitable \(U(1)_\eta \) charge to the elementary fermion.

*SO*(5), under which the corresponding composite operators \(\mathcal {O}_q, \mathcal {O}_{u_1}, \mathcal {O}_{u_2}\) and \(\mathcal {O}_{d}\) transform. Summing implicitly over three flavours, the relevant Lagrangian terms may be written as

*SO*(5) decompose as \(\mathbf{4} = (\mathbf{2}, \mathbf{1}) + (\mathbf{1}, \mathbf{2})\) under \(SU(2)_L \times SU(2)_R\). By completing the representation with spurious fermions they can be represented by fields transforming under this symmetry as

*Z*represents the \(U(1)_\eta \) charges and the subscript is the \(U(1)_X\) charge assigned by requiring \(Y = T^3_R + X\). We have set to zero the non-dynamical spurions that complete the \(SU(2)_L\) (\(SU(2)_R\)) representation in the upper (lower) two components of the multiplet, though they are formally required to restore the global

*SO*(5) symmetry.

*H*the usual complex Higgs doublet and

The potential has extrema occurring at \(s_\eta ^{12} = 0 \implies c_\eta ^{12} = \pm 1\) ^{15} and \(c_h = - \frac{1}{2} \frac{\alpha \pm \alpha _{12}}{\beta \pm \beta _{12}}\). As is usual in composite Higgs models, we find that with *O*(1) values for the coefficients, \(v \sim f\) is expected and so a slight tuning is needed to obtain the required suppression of the weak scale for compatibility with electroweak precision tests.

*f*, we will also obtain a corresponding suppression of the Higgs mass-squared, exactly as one finds in [6]. The mass of \(\eta \), however, is unsuppressed, so we obtain a hierarchy of scales, of parametric size

*v*/

*f*(assuming \(f_\eta \sim f\)) between the \(\eta \) mass and either the electroweak scale or the mass of the Higgs boson.

An identical conclusion is reached if we instead embed the elementary fermions in the fundamental, 5-d representation of *SO*(5), as we describe in Appendix 1. (In this case, as we discuss in Appendix 3, we can also protect the \(Zb_L\overline{b}_L\) coupling by a custodial symmetry.) The hierarchy of scales is, in fact, generic and follows from the fact that the scalar potential is an even function of *h*. Indeed, with \(V(h,\eta ) = f(h^2, \eta )\), we obtain \({\partial V}{\partial h^2} = 4v^2 {\partial f}/{\partial h^2}|_v\), at an electroweak-symmetry-breaking minimum. One can also see that any mixing between the mass eigenstates of the singlet and the Higgs will also be *v* / *f* suppressed.

Although this hierarchy of scales is generic, it may be affected by the well-known difficulty (see e.g. [31] for a comprehensive discussion) of accommodating a Higgs mass as low as 125 GeV in composite Higgs models, given the size of contributions to the Higgs potential from top quark loops. If the required additional suppression is an accidental tuning, then we expect no corresponding suppression in the \(\eta \) mass. But if it is achieved by the presence of light top partners that cut off all contributions to the scalar potential, then one should find a corresponding suppression of the \(\eta \) mass.

## 6 Conclusion

Composite Higgs models remain viable possibilities for solving the electroweak hierarchy problem. Here we introduced the most minimal extension of the coset structure allowing a non-trivial anomaly structure and discussed the details of the low-energy action reproducing the anomalies. We showed that there can be higher-order corrections, beyond dimension 5, to the action reproducing the \(SO(5)^2 U(1)\) anomaly, but also pointed out that the effective action is not unique. We also showed that the structure of the coset space admits a possible Wess–Zumino–Witten term, by which we mean a term in the effective Lagrangian which is not invariant under the non-linearly realised symmetries, but rather shifts by a total derivative. Unlike in QCD, this term is not contained in the anomalous effective action that we consider. If present, the term leads to an exotic phenomenological signature in the form of the singlet decay \(\eta \rightarrow h W^+ W^- Z\).

The discussion of the anomaly structure in this specific model highlights three questions that it would be interesting to resolve in models based on a general coset space, *G* / *H*. Firstly: is there a way to resolve the non-uniqueness issue of the low-energy anomalous effective action? Secondly: do Wess–Zumino–Witten terms that are not required to reproduce triangle anomalies have some other purpose? Thirdly, is there an elegant way to write the Wess–Zumino–Witten term for coset spaces whose fourth homotopy group is non-vanishing?

The anomaly-induced production and decays of the singlet may induce flavour violation of its couplings to fermions and we have shown how they can be kept under control without fine-tuning if the \(\eta \) couples in a flavour-universal way through the mechanism of partial compositeness. For natural \(\mathcal {O}(1)\) charge assignments, this pattern of coupling predicts a large decay width through the \(t\overline{t}\) final state.

We also showed how the potential for the PNGB Higgs and singlet can be generated by elementary fermion couplings to the strong sector that break the global symmetry, though this requires a slight departure from the flavour-universal pattern of couplings, because of the need for a collective breaking mechanism to give mass to the singlet. We find that the singlet mass is naturally unsuppressed relative to the Higgs mass and electroweak scale, thus requiring no additional tuning beyond the usual ones needed for a small electroweak scale and light Higgs mass in composite models. Since the form of the potential contains no mixing between the Higgs and the singlet there are no further bounds from the Higgs sector.

Should the Higgs arise as a pseudo-Nambu–Goldstone boson, it will be imperative to determine the new strong sector responsible for it. Given our current limited understanding of strongly coupled theories, the anomaly structure, if present, may be crucial in gaining some insight as to the nature of the underlying UV dynamics. We hope that the model described here, or some variant thereof, may be useful in this regard.

## Footnotes

- 1.
- 2.
Since \(H_{dR}^5 (SO(6)/SO(5)) = H_{dR}^5 (S^5) = \mathbb {R}\), there is also a possible WZW term. As explained in the next sections, \(H_{dR}^5\) denotes the fifth de Rham cohomology group.

- 3.
- 4.
We consider only triangle anomalies here.

- 5.
We caution the reader that the symmetry group of the resulting theory is not

*G*, even at the classical level, but rather it is the normaliser of*F*in*G*[19]. - 6.
As usual, we identify spacetime, with fields thereon tending to a constant value at infinity, with \(S^4\).

- 7.
Moreover, since \(\pi _4 (S^3 \times S^2) = \mathbb {Z}/2\), one cannot use Witten’s trick to write the WZW term as an integral over a 5-disk in this case.

- 8.
Unfortunately, this trick does not work for a general field configuration, because the fourth homotopy is \(\pi _4 (S^4 \times S^1) \simeq \pi ^4 (S^4) \oplus \pi ^4(S^1) \simeq \mathbb {Z} \oplus \{e\} \simeq \mathbb {Z} \ne 0\).

- 9.
This term was also singled out in [22], but for different reasons.

- 10.
In fact, if we wish to include matter fields in the theory in spinor representations, then we should consider not

*SO*(5) but rather its universal cover*Sp*(2). As described in [23], the relevant homogeneous spaces without and with custodial protection of \(Z\rightarrow b \overline{b}\) are \(Sp(2)/(Sp(1)\times Sp(1))\) and \(Sp(2)/(Sp(1)\times Sp(1) \rtimes \mathbb {Z}_2)\), where the homomorphism in the semi-direct product maps the non-trivial element in \(\mathbb {Z}_2\) to the outer automorphism of \(Sp(1)\times Sp(1)\) that interchanges the two*Sp*(1)s. The homogeneous spaces are homeomorphic to*SO*(5) /*SO*(4) and*SO*(5) /*O*(4), respectively, and the discussion given here can be carried over straightforwardly. - 11.
The reason for this is that the volume form on \(S^4\), which is given by the pull-back to \(S^4\) via the inclusion map \(i:S^4 \rightarrow \mathbb {R}^5\) of the form \(\sum _{i=1}^4 (-1)^i x^i \mathrm{d}x^1 \dots \mathrm{d}x^4\) (where in the ellipsis we omit \(\mathrm{d}x^i\)), is not identical at antipodal points; this is consistent with the non-orientability of \(\mathbb {R}P^4\).

- 12.
There remain, however, sub-dominant flavour-violating contributions from possible derivative operators, analogous to those described in [28].

- 13.
In the presence of anomalies and without other sources of \(U(1)_\eta \)-breaking, the \(\eta \) plays the role of an electroweak axion. The resulting contributions to its mass are thus completely negligible compared to those considered here.

- 14.
This is in contrast to various composite Higgs models where the top right is a fully composite state.

- 15.
We remark that a vacuum with \(c_\eta ^{12} = - 1\) does not imply spontaneous violation of

*CP*, because*CP*sends \(\eta \rightarrow -\eta \) and because physics is periodic in the argument of the cosine. - 16.
The term multiplied by \(\frac{1}{2}\) simplifies to \([[F^2,\xi ],\xi ]\), but we prefer to write it in a form that leaves the Lie algebra structure manifest.

- 17.
We have removed erroneous factors of \(\pm i\), which appear in [7].

## Notes

### Acknowledgements

BG acknowledges the support of the Science and Technology Facilities Council (Grant ST/L000385/1) and King’s College, Cambridge and thanks O. Randal-Williams for discussions. TY is supported by a Research Fellowship from Gonville and Caius College, Cambridge.

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