^{*}

^{†}

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^{3}.

We propose a unified model of scalar particles that addresses the flavor hierarchies, solves the strong

Although the Standard Model (SM) of particle physics provides an excellent description of nature around the weak scale, it has several shortcomings that lead us to the conclusion that it is rather an effective low-energy parametrization of a more fundamental theory of nature.

Amongst the most pressing issues are the missing candidate to generate the dark matter (DM) populating our Universe, and the failure to explain large hierarchies present in the fermion masses and mixings. Beyond that, the apparent conservation of

The flavor hierarchies in the SM can be addressed via the Froggatt-Nielsen (FN) mechanism

As shown in a previous work

See also Refs.

In this paper, we take a further step to unify the scalar degrees of freedom in the theory by combining the axiflavon field,

From a different perspective, the proposed model can be seen as adding a flavor story to the recently proposed elementary-Goldstone-Higgs scenario

Before working out the setup and its predictions in detail, we summarize the main model-building steps. We formulate the theory at the scale

This article is organized as follows. In Sec.

In the following, we present the explicit model setup. The symmetry-breaking pattern that leads to the unified realization of the Higgs doublet and the axion as pNGBs reads

The FN messengers are denoted by

The SM fermions,

The Lagrangian of the system includes renormalizable operators made out of

Before presenting the computation of the Higgs potential, let us discuss an example to show how the FN mechanism is explicitly realized in our setup.

Consider two chiral fermions,

By integrating out

Note that for the apparently more minimal chain between two light fermions that differ only by

In this section, we compute the Higgs potential generated by the interaction with the top quark and the FN messengers which directly couple to it. A charge assignment that is compatible with the top mass must satisfy

Let us first compute the top mass according to Eq.

We use the same symbol for the classical background as for the Higgs field earlier in Eq.

We compute the Higgs potential by matching the SM effective potential renormalized at the scale

In the axiflavon–Higgs picture, the Higgs potential arises at the loop level, and it is given in terms of the field-dependent masses of the physical eigenstates, namely the SM particles and the FN messengers. Considering only the top sector in Eq.

To compute the rhs of Eq.

By direct computation, the terms

In Fig.

Matching of the Higgs quartic coupling at the scale

It is useful to confront this region with constraints following from the flavor-violating couplings of the axiflavon. In fact, limits from searches for the decay

Note that the axion couplings to fermions differ by approximately a factor of 2 with respect to the axiflavon case of Ref.

We now discuss the impact of including right-handed (RH) neutrinos. Let us consider one family first. The left-handed doublet

Matching of the Higgs quartic coupling at the scale

Such values of

We presented a model framework where the flavor puzzle, the strong

We showed that successfully reproducing the SM-like Higgs potential at low energies fixes the axion decay constant to

We also demonstrated that by including RH neutrinos, the symmetry-breaking scale could be lowered bringing the axion decay constant down to the TeV range.

Due to the large domain-wall number

Still, it is tempting to realize inflation in our setup by identifying the flavon, or a combination of the scalar components of the

We are grateful to Giorgio Arcadi, Pablo Quilez, Kai Schmitz, Stefan Vogl and Kei Yagyu for useful discussions.

Let us discuss the implication of relaxing the degeneracy among the FN messengers. We parametrize the departure from the degenerate case by introducing

Similarly to the top eigenstate, the contribution of the gauge bosons will appear on both sides of Eq.

For concreteness, let us consider the up-type quarks. The Yukawa interactions before diagonalization read