The 5^th of July marks the fifth anniversary since the discovery of a Higgs boson at the LHC. Summarizing notes for a recent set of lectures at the Asia-Europe-Pacific School of High-Energy Physics, John Ellis reviews events leading to the discovery of the Higgs boson and what we know about it now. This article explains the main motivations for expecting new physics beyond the Standard Model and discusses the possibilities for future Higgs factories including plans for future linear and circular colliders.

The mechanism for giving masses to gauge bosons was introduced into particle physics by François Englert, Rober Brout and Peter Higgs in 1964, and the latter pointed out that there should also be a massive scalar boson. In 1975 Mary Gaillard, Dimitri Nanopoulos and I made the first attempt at a systematic survey of the possible phenomenological profile of this 'Higgs boson¹. At that time, the Standard Model was not established, the Englert-Brout-Higgs idea not generally accepted, there was general scepticism about scalar particles and, even if one bought all that, nobody had any idea how heavy a Higgs boson might be. For all these reasons, we were rather cautious in the final paragraph of our paper, writing "we do not want to encourage big experimental searches for the Higgs boson, but we do feel that people doing experiments vulnerable to the Higgs boson should know how it may turn up."

Subsequently, searches for the Higgs boson were placed on the experimental agendas of the LEP² and LHC accelerators at CERN. For example, a review of the possibilities for new particle searches presented at the first workshop on prospective LHC physics in 1984³ discussed various ways of producing the Standard Model Higgs boson at the LHC. There were also studies of Higgs production at the ill-fated SSC and an influential review by Sally Dawson, Jack Gunion, Howie Haber, Gordy Kane. However, in the 1980s there was still no indication what the Higgs mass might be. The first clues about m_H emerged from the high-precision measurements at LEP and the SLC in the 1990s. These and other experiments found excellent overall agreement with the predictions of the Standard Model, if the Higgs mass was well below 1 TeV.

 In 2011, just before the Higgs boson was discovered, the precision electroweak data suggested a range m_H = 100 ± 30 GeV. In parallel, unsuccessful searches at LEP had implied that mH ≥ 114 GeV, and searches at the Fermilab Tevatron collider had excluded a range around 160 to 170 GeV.

The discovery of the Higgs boson in 2012 was a tremendous success for the Standard Model at the quantum level. It was based primarily on the observation of excesses of events in the γγ and 4-lepton channels, interpreted as being due to H → ZZ^∗ , together with a broad excess of 2-lepton + missing transverse energy events, interpreted as being due to H → WW^∗ . Measurements of the γγ and 4−lepton final states have enabled the mass of the Higgs boson to be determined with high precision. The final combined results from ATLAS and CMS LHC Run 1 data yield m_H = 125.09 ± 0.21 (statistical) ± 0.11 (systematic) GeV, a measurement at the level of 2 per mille. An accurate measurement of m_H is a sine qua non for precision tests of the Standard Model, since it enters in the Higgs production cross section and decay branching ratios. Moreover, it is crucial for the discussion below of the stability of electroweak vacuum. The measurement is fully in line with previous indications from precision electroweak data and previous searches at LEP and the Fermilab Tevatron collider.

The stakes in the Higgs search were very high. How do gauge bosons get their masses, by hand or by an elegant theoretical mechanism? Assuming the latter, does it involve an elementary scalar field, a novelty that raises perhaps more questions than it answers? The Higgs is very likely a portal towards many issues in physics beyond the Standard Model. It would have been associated with a phase transition in the Universe when it was about 10⁻¹² seconds old, which might have been when the baryon asymmetry of the Universe was generated. The Higgs or a related scalar field might have caused the Universe to expand near-exponentially in a bout of cosmological inflation when it was about 10⁻³⁵ seconds old. And a Higgs field should contribute a factor ∼ 10⁶⁰ too much to the dark energy measured in the Universe today. The stakes in the search for the Higgs boson were undoubtedly high!

Shortly after the discovery of the Higgs boson, Peter Higgs was quoted in the Times of London as saying: “A discovery widely acclaimed as the most important scientific advance in a generation has been overhyped" [58]. I would very humbly and respectfully beg to disagree. Without the Higgs boson (or something to do its job), there would be no atoms because electrons would escape from nuclei at the speed of light, the weak interactions responsible for radioactivity would not be weak, and the universe would be totally unliveable. It was a big deal.

Despite the continuing absence of any direct evidence for the new physics beyond the Standard Model at the LHC, one should not become disheartened. History abounds with examples of people who thought they knew it all, but did not. In 1894, just before the discoveries of radioactivity and the electron, Albert Michelson declared that “The more important fundamental laws and facts of physical science have all been discovered". More recently, prior to the string revolution, Stephen Hawking asked “Is the End in Sight for Theoretical Physics?". However, my favourite example of a lack of ability to think outside the box is the Spanish Royal Commission that rejected a proposal by Christopher Columbus to sail west before 1492: “So many centuries after the Creation, it is unlikely that anyone could find hitherto unknown lands of any value" . Many of us have seen referees’ reports with a similar flavour.

Today one could quote 007 reasons for anticipating physics beyond the Standard Model. 001) Within the Standard Model, the electroweak vacuum is unstable against decay to high Higgs field values. 002) The Standard Model has no candidate for the astrophysical dark matter. 003) The Cabibbo-Kobayashi-Maskawa (CKM) Model does not explain the origin of the matter in the universe. 004) The Standard Model does not have a satisfactory mechanism for generating neutrino masses. 005) The Standard Model does not explain or stabilize the hierarchy of mass scales in physics. 006) The Standard Model does not have a satisfactory mechanism for cosmological inflation. 007) We need a quantum theory of gravity. Several of these issues will be addressed by the LHC experiments during Run 2: there will be more accurate measurements of the Standard Model parameters, there will be searches for dark matter particles, there will be searches for CP violation and other flavour physics beyond the CKM model, as well as for new particles that could help stabilize the electroweak scale.

“Beyond any reasonable doubt", the LHC has discovered a (possibly the) Higgs boson. Whilst being a tremendous success for theoretical physics, it also represents a tremendous challenge. Even in the minimal elementary Higgs model, its mass and field value are problematic. How come the Higgs mass is so small compared to the scale of gravity? Is the Higgs field value unstable?

The LHC may yet discover new physics beyond the Standard Model during Run 2. If it does, the global priority for high-energy physics will surely be to study it. If it does not discover new physics at the TeV scale, it will be natural to study the Higgs boson in detail in future accelerator experiments. Either way, future circular colliders may offer the best experimental prospects, being able to probe the 10 TeV scale indirectly via high-precision low-energy experiments, and directly via the production of new heavy particles.

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