How far do we understand the 125 GeV Higgs boson (and its consequences)? A 2017 perspective
Run 1 accumulated striking evidence that the Higgs field is the cause of the screening of the weak interaction at long distances. Indeed, the observation and measurement of the H → ZZ → 4 channel indicate that the Higgs field develops a vacuum expectation value (vev) that is not invariant under the SU(2)L × U(1)Y gauge symmetry of the SM. Furthermore, this vev seems to be the common source of the Z-boson mass and the coupling between the Higgs boson and the Z boson. However, this evidence only addresses the question of how the symmetry of the weak interaction is broken. It does not address the question of why the symmetry is broken or why the Higgs field acquires an expectation value. The situation is simply summarized in the following tautology :
Why is electroweak symmetry broken?Because the Higgs potential is unstable at the origin.
Why is the Higgs potential unstable at the origin?Because otherwise EW symmetry would not be broken.
The discovery of a Higgs boson allowed first glimpses into a new sector of the microscopic world. Now comes the time of the detailed exploration of this new Higgs sector. And some key questions about the Higgs boson emerge:
1. Is it the SM Higgs?
2. Is it an elementary or a composite particle?
3. Is it unique and solitary? Or are there additional states populating the Higgs sector?
4. Is it eternal or only temporarily living in a metastable vacuum?
5. Is its mass natural following the criteria of Dirac, Wilson or ’t Hooft?
6. Is it the first superparticle ever observed?
7. Is it really responsible for the masses of all the elementary particles?
8. Is it mainly produced by top quarks or by new heavy vector-like particles?
9. Is it a portal to a hidden world forming the dark matter component of the Universe?
10. Is it at the origin of the matter-antimatter asymmetry?
11. Has it driven the primordial inflationary expansion of the Universe?
The answers to these questions will have profound implications on our understanding of the fundamental laws of physics...
... to global point of view
The recent discovery of the Higgs boson [1, 2] and the ongoing measurements of its properties  are in good agreement with the hypothesis that this particle is a remnant of the Brout-Englert-Higgs mechanism, i.e. the spontaneous breaking of SU(2)L×U(1)Y → U(1)QED.
While the precise determination of the Higgs and gauge boson masses, as well as the interactions of the Higgs boson with elementary particles, including itself, will continue to improve our understanding of the scalar potential’s local structure in the vicinity of the vacuum, its global structure, which can possibly explain the nature of electroweak symmetry breaking, is very difficult to probe experimentally.
For example, the nature of the Higgs, whether elementary or composite, is still an open question. Even if the Higgs is assumed to be elementary, the shape of its potential remains unknown. It could be of mexican-hat shape as in the Standard Model (SM), or it could be deformed by strong quantum corrections due to virtual effects of additional fields. Were the Higgs boson to be a composite pseudo-Nambu-Goldstone boson of a strongly-coupled sector, one would expect a periodic potential involving trigonometric functions. In all cases, the Higgs mass is fixed by the curvature of the potential at its minimum, and so in the vicinity of the latter the shape of the potential will be similar in all possible models. Nevertheless, deviations are allowed away from the minimum. For example, one could have a barrier at zero temperature between the vacuum and the origin of field-space. Moreover, in composite Higgs models the relation between the Higgs field’s vacuum expectation value (VEV) and the gauge boson masses differs from its SM counterpart, and thus the location of the minimum in field-space may vary.
Discriminating between the different possibilities is of fundamental importance for our understanding of nature and, hence, the embedding of the effective Standard Model in an underlying UV theory. This motivates to consider possible observables which could be sensitive to the Higgs potential beyond its minimum. A possible candidate is the energy scale of baryon-number-violating processes. If baryon number is only violated by the anomaly under the weak interactions, then it follows that processes that violate baryon-number are associated with transitions between vacua classified by their weak topological charge. The minimum energy barrier between these vacua thus sets the expected scale of baryon-violating processes, which is an observable that could potentially be probed by experiments, either at colliders [4–9] or cosmic ray and neutrino detectors [10–15]. Getting accurate predictions for the rates of baryon-number-violating interactions is a difficult problem, due to a possible breakdown of the semiclassical expansion used to compute vacuum transitions. After extensive discussion in the literature (see for example [16–24]) the latest estimates point towards rates that could be probed by future experiments [25, 26]; for recent analyses of measurement prospects at colliders, cosmic ray and neutrino detectors, see for example [27–29].
(Submitted on 16 Nov 2016)
A very personal (thus naive) spectral perspective
...to go beyond the above tautology
Because spacetime geometry has a fine structure or more crudely a "discrete" dimension at the zeptometer scale that the discovery of the Higgs boson makes it possible to uncover provided one understands it through the spectral noncommutative point of view.
Why is the Higgs potential unstable at the origin?
This is a consequence of the spectral action principle applied to the proper almost commutative 4D manifold. The latter is a small (but topologically highly nontrivial) extension of our ordinary continuous and commutative geometric model of spacetime while the former is a stronger hypothesis than the usual diffeomorphism invariance of the action of general relativity.
...and propose tentative answers or rather educated guess
1. Is it the SM Higgs? Yes
2. Is it an elementary or a composite particle? It is elementary in the spectral model of particle physics compatible with current experiments and observations.
3. Is it unique and solitary? Or are there additional states populating the Higgs sector? There should be more scalars responsible for several gauge symmetry breaking but they may be at partial or grand unification energy scales inacessible to terrestrial particle accelerators. The 125 GeV Higgs boson could be very weakly mixed with (but strongly coupled to) a big brother ("big broson") responsible for the type I seesaw mechanism that gives very low masses to left-handed neutrinos.
4. Is it eternal or only temporarily living in a metastable vacuum? The coupling with the above very massive scalar "big broson" should stabilise the vacuum.
5. Is its mass natural following the criteria of Dirac, Wilson or ’t Hooft? This question could be settled once the proper fine structure of spacetime is established and the quantum dynamics of scalars in this new arena is better understood.
6. Is it the first superparticle ever observed? There might be no need for that hypothesis to paraphrase a famous Laplace quote.
7. Is it really responsible for the masses of all the elementary particles? There could exist a dilaton scalar ruling all the masses so to speak and responsible for a spontaneous symmetry breaking of Weyl invariance.
8. Is it mainly produced by top quarks or by new heavy vector-like particles? Noncommutative geometry and the spectral action principle provide a conceptual explanation for the standard model algebra and the number of fundamental fermions by generation so there should be no need for heavy vector-like particles to explain the production rate of Higgs boson at the LHC.
9. Is it a portal to a hidden world forming the dark matter component of the Universe? In a metaphorical way the answer could be yes. Indeed, understanding how to accommodate the measured Higgs mass in the spectral noncommutative framework has helped to uncover new mathematical structures to build 4D spin manifolds from a higher degree Heisenberg commutation relation. This provides new perspectives on the dark component as not composed of unknown particles or fields but mimicking some kind of new quanta of geometry.
10. Is it at the origin of the matter-antimatter asymmetry? In an indirect way one cold say yes. The phenomenological consequences of the spectral standard model have not been extensively probed but they might be close to the predictions of a minimal nonsupersymmetric SO(10) model which values of the parameters obtained from the low energy observables yield a baryon assymetry in agreement with observations.
11. Has it driven the primordial inflationary expansion of the Universe? This question has not been investigated serioulsy in the most advanced spectral modelisation of spacetime and matter at my knowledge but I think some people responsible for an interesting extension of the standard model not that far from the spectral one work on this...
Fortune favors the prepared mind.
La chance ne sourit qu'aux esprits bien préparés.