Abstract
In the previous chapters we focused on the broad qualitative features of the composite Higgs scenarios that follow directly from the Nambu-Goldstone boson nature of the Higgs and from partial fermion compositeness. As we discussed at length, the Goldstone structure determines many important aspects of the elementary and composite dynamics and, when supplemented by a power-counting rule, can be exploited to obtain a semi-quantitative understanding of the new-physics effects. The full generality of this approach is at the same time the source of its advantages and of its main limitations. Most of the results we derived are indeed valid only as order of magnitude estimates and important numerical corrections could be present in explicit models. Moreover, so far we mainly focused on the dynamics of the Standard Model (SM) fields, but we did not consider in details the properties of the composite resonances that unavoidably arise from the composite sector and constitute one of the most distinctive features of the composite Higgs scenarios.
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Notes
- 1.
- 2.
It is important to stress that the deconstructed models are deeply different from a naive truncation of the KK tower (for an effective model based on this approach see [16]). A naive truncation, indeed, implies a breaking of the symmetries that protect the Higgs dynamics in the holographic models, thus not allowing to implement calculability in the effective model.
- 3.
We momentarily neglect the presence of the extra U(1) X charge in the definition of the hypercharge and we set \(Y = T_{R}^{3}\). The U(1) X subgroup does not play any role until the matter fermion fields are introduced.
- 4.
The derivation is not particularly enlightening and thus it will not be reported here. The reader is referred to [13].
- 5.
Thinking backwards, this could have been a way to establish the bound g ∗ ≤ 4π in the operator estimate.
- 6.
The classification of invariant operators constructed with the gauge spurions was already carried on in Sect. 3.3.1 with a different and more general technique based on dressed spurions. Two operators were found for each of the two \(\mathcal{G}\) and \(\mathcal{G}^{{\prime}}\) spurions, however only the ones that are even under the P LR symmetry are generated by radiative corrections and are reported in the equations that follow. This is because the 2-derivative non-linear σ-model Lagrangian is accidentally P LR -invariant, thus it can not generate odd operators.
- 7.
Notice that, due to gauge invariance, the elementary/composite mixing arises from terms of the form \(f^{2}(g_{0}W_{\mu }^{\alpha } - g_{\rho }\rho _{\mu }^{\alpha })^{2}\), see for instance Eq. (5.16).
- 8.
We could equivalently assume that the ψ field transforms under the vector combination of \(\mbox{ SO}(5)_{R}^{1}\) and \(\mbox{ SO}(5)_{L}^{2}\), in analogy with the ρ μ gauge fields. The choice made in the main text, however, helps in clarifying the pattern of symmetry breaking induced by the fermions and makes simpler the introduction of spurions.
- 9.
For simplicity here we associate all the σ models to a single f decay constant and a single m ∗.
- 10.
The expansion is not valid in the limit \(p \rightarrow 0\), in which the argument of the logarithm diverges. However in this limit the factor p 3 in front of the logarithm compensate for the divergence and the approximate integrand vanishes for \(p \rightarrow 0\). The error introduced by this approximation is thus small. As discussed in Sect. 3.3.2, the presence of the divergence is related to the IR contribution to the Coleman–Weinberg potential coming from the top quark. A fully consistent computation of the potential can be obtained by first isolating the top contribution and then expanding the remaining terms which are regular for \(p \rightarrow 0\).
- 11.
Additional subleading corrections to the Higgs mass can also come from loops of vector partners. In particular the two-loop contribution due to gluon partners can be non-negligible in some regions of the parameter space [24].
- 12.
- 13.
If, as in the three-site case, the Higgs potential is completely finite at one loop, an analogous condition holds for the logarithmic term, i.e. \(\sum _{i}m_{i}^{4}(H) =\sum _{i}m_{i}^{4}(H = 0) =\mathrm{ const.}\)
- 14.
The reader is referred to [25] for an implementation of the same setup in the framework of 5-d holographic models.
- 15.
The mass scale of the resonances m ψ has been identified in the scan with the quadratic average of the fermion mass parameters present in the composite sector Lagrangian.
- 16.
For this purpose it is sufficient to replace y t with y L in Eq. (5.91).
- 17.
The integral in the expression for β has a spurious IR divergence arising from the expansion of the potential. It can be cured by inserting a small IR cut-off on the integration domain. Given that we are interested only in the UV behavior we will ignore this subtlety.
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Panico, G., Wulzer, A. (2016). Phenomenological Models. In: The Composite Nambu-Goldstone Higgs. Lecture Notes in Physics, vol 913. Springer, Cham. https://doi.org/10.1007/978-3-319-22617-0_5
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