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Flavor

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The Composite Nambu-Goldstone Higgs

Part of the book series: Lecture Notes in Physics ((LNP,volume 913))

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Abstract

As we saw in Chap. 2Goldstone Boson Higgs, a fundamental ingredient of the composite Higgs scenarios is the partial compositeness hypothesis, which provides a general framework to describe the Standard Model (SM) fermions and to generate their masses and couplings. In most of the previous discussions we focused our attention on the third-generations quarks, and in particular on the top. In fact, due to its large mass, the top is usually the elementary state with the largest mixing with the composite sector and is the one that almost completely determines the dynamics of Electro-Weak Symmetry Breaking (EWSB).

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Notes

  1. 1.

    In the UV, where the composite sector is close to the fixed point, the operators are characterized by their different scaling dimensions \(d_{u_{R}}^{\,j}\), which can be regarded as their eigenvalue under dilatation. The presence of this additional quantum number makes them distinguishable and does not allow to rotate them.

  2. 2.

    The SU(3) c color group is automatically respected, since all the operators are color triplets in order to mix with the elementary quarks.

  3. 3.

    For shortness we will not make an explicit distinction between the couplings to the Higgs field responsible to generate the mass matrices and the linear couplings of the Higgs fluctuations to the quarks. Obviously, due to the non-linear Higgs dynamics, the two things are in general different and only coincide at leading order in the v∕f expansion. In the following we will denote both couplings by “Yukawa’s” and leave the exact interpretation of the concept to the context.

  4. 4.

    Below we report the structure of the Yukawa matrices at leading-order in the \(\lambda /g_{{\ast}}\) expansion. Subleading effects from of order \(\lambda ^{2}\) modifications of the kinetic terms induced by the composite sector will be discussed in Sect. 4.1.2.

  5. 5.

    This is true in the absence of additional symmetries in the composite sector. For instance by imposing a P LR symmetry the number of invariants can be reduced to two and Higgs-mediated FCNC’s can be avoided.

  6. 6.

    The field redefinition can be numerically relevant for the top quark given its sizable degree of compositeness. It is instead typically negligible for all the other quarks.

  7. 7.

    The additional factor \(\lambda ^{2}\) comes from the quark mass factor that comes from the equations of motion.

  8. 8.

    In order to properly match the coefficients of the effective operators with the general estimates of partial compositeness, all the quantities must be evaluated at the m ∗ scale. To keep our discussion as simple as possible we avoid to explicitly include the running effects in our equations. We however include them in the numerical results (for this we assume \(m_{{\ast}}\sim 1\,\mathrm{TeV}\)). We refer the interested reader to the original literature [14–17].

  9. 9.

    See Sect. 2.2.2The Minimal Composite Higgs Model for a first description of the custodial group and section “The Custodial Symmetries” in Appendix in Chap. 7EW Precision Tests for a complete discussion.

  10. 10.

    See Eq. (7.24) for the exact definition of the bottom couplings.

  11. 11.

    In the cases we consider the \(\tilde{\mathcal{Q}}_{1}\) operator is generated with a smaller coefficient than \(\mathcal{Q}_{1}\), while the bounds on the two operators are comparable. The \(\mathcal{Q}_{2}\) and \(\mathcal{Q}_{3}\) operators are expected to have similar size, but the bounds on the former are always tighter. The same happens for \(\mathcal{Q}_{4}\) and \(\mathcal{Q}_{5}\) [27].

  12. 12.

    For instance, the experimental bounds on the \(\Delta F = 2\) transitions in the Kaon system require a suppression scale \(\Lambda \gtrsim 10^{5}\,\mathrm{TeV}\). The RS-GIM mechanism lowers this scale by four orders of magnitude (see Table 4.2).

  13. 13.

    The first proposals of flavor symmetric composite Higgs scenarios were developed in the extra-dimensional framework in [40–44].

  14. 14.

    This scenario can be easily realized also in the models with a single left-handed mixing with the choice \(\lambda _{q_{L}}^{{\prime}} =\lambda _{q_{L}} \propto\mathbb{1}\).

  15. 15.

    Mixed scenarios considering a reduced U(2) flavor symmetry only for one quark chirality and full U(3) symmetry for the others can also be constructed [46].

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Authors and Affiliations

  1. IFAE, Universitat Autònoma de Barcelona, Barcelona, Spain

    Giuliano Panico

  2. Dipartimento di Fisica e Astronomia, Università di Padova, Padova, Italy

    Andrea Wulzer

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  1. Giuliano Panico
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  2. Andrea Wulzer
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Panico, G., Wulzer, A. (2016). Flavor. In: The Composite Nambu-Goldstone Higgs. Lecture Notes in Physics, vol 913. Springer, Cham. https://doi.org/10.1007/978-3-319-22617-0_4

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