It is well known that the electroweak phase transition (EWPhT) in extensions of the Standard Model with one real scalar singlet can be first-order for realistic values of the Higgs mass. We revisit this scenario with the most general renormalizable scalar potential systematically identifying all regions in parameter space that develop, due to tree-level dynamics, a potential barrier at the critical temperature that is strong enough to avoid sphaleron wash-out of the baryon asymmetry. Such strong EWPhTs allow for a simple mean-field approximation and an analytic treatment of the free-energy that leads to very good theoretical control and understanding of the different mechanisms that can make the transition strong. We identify a new realization of such mechanism, based on a flat direction developing at the critical temperature, which could operate in other models. Finally, we discuss in detail some special cases of the model performing a numerical calculation of the one-loop free-energy that improves over the mean-field approximation and confirms the analytical expectations. One-loop Numerical AnalysisSo far, we have identified choices for the T = 0 parameters that lead to strong electroweak phase transitions in the mean-field approximation. It is straightforward to refine these results starting from the same tree-level parameters but including in the scalar potential one-loop T = 0 corrections and the full one-loop thermal integrals (which correctly take into account Boltzmann decoupling effects) further improved by daisy resummation. Details of this standard procedure are given in Appendix A. To illustrate the impact of this refinement, we show in Fig. 9 the ratio v c /T c in the mean-field approximation (blue dashed line) compared with

If electroweak symmetry breaking arises via strong dynamics, electroweak precision tests and flavour physics experiments suggest that the minimal model should closely resemble the Standard Model at the LHC. I describe two directions going beyond the minimal model that result in radically different physics at the LHC. One direction extends the Higgs sector and the other involves composite leptoquark states. IntroductionStrong coupling provides a solution of the electroweak hierarchy problem that is natural in the literal sense of the word. That is to say, we already have an example in Nature where a hierarchy, namely the one between the proton mass and, say, the Planck scale, is generated by a stronglycoupled theory, QCD. As such, and given the problems suffered by the only weakly-coupled candidate that can stabilize the hierarchy, namely supersymmetry, strongly-coupled dynamics remains an attractive mechanism for electroweak symmetry breaking.But strongly-coupled dynamics has severe problems of its own, in the form of clashes with electroweak precision tests and flavour-changing neutral currents. The former can be solved, to some extent, by clever use of symmetries. To see how this may occur, let me first remark that we do not yet know what the 'symmetry' of electroweak symmetry is. Certainly, we do know that it contains the SU (3) × SU (2) L × U (1) Y of the Standard Model (SM) as a gauged subgroup, but it is quite possible that the true global symmetry of the strongly-coupled sector is somewhat larger. If we enlarge the SU (2) L × U (1) Y to SU (2) L × SU (2) R ≃ SO(4) (which is an accidental symmetry of the renormalizable SM Higgs potential), then we find that the W and Z bosons automatically obtain their measured mass ratio. 1 Furthermore, by adding the discrete parity that interchanges L ↔ R (or equivalently enlarging SO(4) to O(4)), we can suppress unwanted corrections to the coupling Zbb. 2 The remaining nuisance is the S-parameter, which, alas, no symmetry can forbid without simultaneously forbidding electroweak symmetry breaking. Nevertheless, one can still use symmetry to argue that S, although it cannot vanish, could be small.The argument goes as follows. If SU (2) L were a symmetry of the vacuum, then S would indeed vanish. 3 This implies that S must be proportional to some positive power of the electroweak vev, v; in fact, since S transforms as an SU (2) L triplet, whilst v transforms as a doublet, we have that S ∝ v 2 . Now, S is dimensionless, so we must have that S ≃ v 2 /f 2 , where the scale f is set by the strong dynamics; if we could arrange for v/f to be somewhat less than unity, by some dynamical accident, then we might end up with an acceptably-small value for S. One way to do this is to further enlarge the global symmetry of the strong sector to SO(5) and then to decree that strong dynamics breaks it to SO(4). 4 The theory then contains four

Run 2 LHC data show hints of a new resonance in the diphoton distribution at an invariant mass of 750 GeV. We analyse the data in terms of a new boson, extracting information on its properties and exploring theoretical interpretations. Scenarios covered include a narrow resonance and, as preliminary indications suggest, a wider resonance. If the width indications persist, the new particle is likely to belong to a strongly-interacting sector. We also show how compatibility between Run 1 and Run 2 data is improved by postulating the existence of an additional heavy particle, whose decays are possibly related to dark matter.

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