TTopics in Higgs Physics
John Ellis * Theoretical Particle Physics & Cosmology Group, Department of Physics, King’s College London,Strand, London WC2R 2LS, United Kingdom
Abstract
These lecture notes review the theoretical background to the Higgs boson, pro-vide an introduction to its phenomenology, and describe the experimental teststhat lead us to think that “beyond any reasonable doubt, it is a Higgs boson".Motivations for expecting new physics beyond the Standard Model are re-called, and the Standard Model effective field theory is advocated as a toolto help search for it. The phenomenology of N = 1 and N = 2 supersym-metric Higgs bosons is reviewed, and the prospects for possible future Higgsfactories are previewed.KCL-PH-TH/2017-09, CERN-TH/2017-039 Keywords
Standard Model; Higgs boson; LHC; supersymmetry; future colliders. Background to the Higgs Discovery
The fundamental equations of physics have a high degree of symmetry - think the rotation and translationsymmetry of Newton’s equations, the gauge invariance of Maxwell’s equations for electrodynamics, theboost symmetry of special relativity, or the the general coordinate invariance of general relativity - andthese theories are generally considered to be very beautiful. However, the solutions to these equationsoften conceal these symmetries, and they appear asymmetric - people are not spherical, for example.Sometimes these asymmetric solutions may even appear more beautiful than symmetric solutions - theimage of the Mona Lisa, for example, would not be so interesting if it were symmetric. Regardless ofthese aesthetic considerations, the rich variety of physical phenomena clearly requires the potential tobreak symmetries.However, breaking a symmetry must be done with care - for example, the gauge invariance ofelectrodynamics guarantees the renormalisability of quantum electrodynamics (QED) and hence its cal-culability. The trick in formulating theories with ‘broken’ symmetry is often to hide the symmetry sothat it is not manifest, while maintaining it at a fundamental level and thereby preserving its attractivefeatures such as renormalizability. This can be done by postulating a lowest-energy (‘vacuum’) state ofthe symmetric equations that does not possess the full symmetry of the underlying equations - an ideaknown as spontaneously-broken or ‘hidden’ symmetry.This idea originated in condensed-matter physics - an early example being the superfluidity thatplays an essential rôle in the LHC magnet system. In this case, the spontaneously-broken symmetry isglobal, i.e., the symmetry transformations are independent of the spatial location within the superfluid.This type of hidden symmetry was introduced into particle physics by Yoichiro Nambu [1], who used it tounderstand the (relatively) low mass and the low-energy dynamics of the pion. According to his theory, * Also Theoretical Physics Department, CERN, CH 1211 Geneva 23, Switzerland a r X i v : . [ h e p - ph ] F e b osonsGauge bosons Higgs boson γ , W + , W − , Z , g ... φ FermionsQuarks Leptons / − / (cid:18) ud (cid:19) , (cid:18) cs (cid:19) , (cid:18) tb (cid:19) − (cid:18) ν e e − (cid:19) , (cid:18) ν µ µ − (cid:19) , (cid:18) ν τ τ − (cid:19) Table 1:
Particle content of the Standard Model. Each quark comes in 3 colours, and the electric charges of thefermions are listed in the Table. the underlying symmetry (in this case chiral symmetry) is not manifest, but is reflected in the couplingsof the pion, which would have no mass if the up and down quarks were strictly massless. Jeffrey Gold-stone subsequently published a simple elementary field-theoretical model of this phenomenon [2] andhe, Abdus Salam and Steven Weinberg [3] subsequently proved rigorously that in a relativistic theoryevery spontaneously-broken global symmetry would be reincarnated in a massless particle with specificcouplings, called a Nambu-Goldstone boson.All well and good, but there are interesting cases where the spontaneously-broken symmetry is lo-cal, as in a gauge theory - the prime example being the superconductivity that also plays an essential rôlein the LHC magnet system. The theory of spontaneous gauge symmetry breaking in this non-relativisticsituation was first developed by Philip Anderson [4] and Nambu [5]. According to their theory, inside asuperconductor the (externally massless) photon acquires a medium-dependent mass by ‘eating’ Cooperpairs of electrons in the lowest-energy (‘vacuum’) state inside the medium. Anderson also conjecturedthat a similar phenomenon might be possible in a relativistic theory, but did not develop this idea. In-deed, Walter Gilbert [6] argued that this would not be possible, because spontaneous symmetry breakingseemed to require the presence of a vector breaking Lorentz symmetry explicitly.However, in 1964 this argument was circumvented in papers by François Englert and RobertBrout [7], and by Peter Higgs [8, 9]. The Englert-Brout paper was received by the journal where itwas published on June 26th, 1964, and a first paper by Higgs was received on July 27th, 1964. Unawareof the paper by Englert and Brout, he pointed out a potential loophole in the Gilbert argument, and in asecond paper he constructed an explicit example. Curiously, whereas the first paper was accepted quicklyby the journal Physics Letters, that journal refused the second paper. It was subsequently accepted byPhysical Review Letters after an anonymous referee (generally held to be Nambu) suggested to Higgsthat he emphasise more the physical implications of his theory. Later in 1964, a more detailed descrip-tion of this idea appeared in a paper by Gerald Guralnik, Carl Hagen and Tom Kibble [10]. Among allthese 1964 papers, the only one to point out explicitly on the appearance in the theory of a massive scalarboson was Higgs, in his second paper [9], which is why it is generally referred to as the Higgs boson.These authors considered the spontaneous breaking of an Abelian U(1) gauge theory. The analo-gous phenomenon in a non-Abelian theory was first studied by Sashas Migdal and Polyakov [11], whowere unaware of the earlier papers. Publication of their paper was delayed because Soviet academicianscould not believe that two young students, as they were then, could come up with such a ground-breakingtheory. Partial breaking of a non-Abelian gauge symmetry was subsequently rediscovered by Kibble [12],and this is the form of spontaneous symmetry breaking that is central to the Standard Model.
Table 1 summarizes the particle content of the Standard Model (SM) [13, 14]. The weak and electro-magnetic interactions are described by a Lagrangian that is symmetric under gauge transformations in aSU(2) L × U(1) Y group, where the subscript L recalls that the weak SU(2) group acts only on left-handed2ermions, and Y is the hypercharge. We can write the SU(2) L × U(1) Y part of the SM Lagrangian as L = − F aµν F aµν + iψ/Dψ + h.c. + ψ i y ij ψ j φ + h.c. + | D µ φ | − V ( φ ) , (1)which is short enough to write on a T-shirt or a pullover!The first line in (1) contains the kinetic terms for the gauge bosons of the electroweak theory, wherethe index a runs over the single U(1) Y gauge field, A µ , and the three gauge fields W , , µ associated withSU(2) L . The U(1) field-strength tensor is the familiar f µν = ∂ ν A µ − ∂ µ A ν , (2)and the SU(2) L field-strength tensor is F aµν = ∂ ν W aµ − ∂ µ W aν + ig(cid:15) abc W bµ W cν for a = 1 , , . (3)where g in (3) is the gauge coupling of SU(2) L and the (cid:15) abc are its structure constants. The last term in(3) arises from the non-Abelian nature of the SU(2) group. At first sight, all the gauge fields are massless,in conflict with the massive nature of the weak bosons W ± and Z . As we see later, they acquire massesthrough the Englert-Brout-Higgs mechanism, whose physical manifestation is the Higgs boson.The second line in (1) contains the interactions between the spin- / matter fields ψ and the gaugefields via the covariant derivatives D µ = ∂ µ + ig (cid:48) A µ Y + ig τ · W µ , (4)where g (cid:48) is the U(1) coupling constant, Y is the generator of the U(1) hypercharge, and τ ≡ ( τ , τ , τ ) is the set of SU(2) Pauli matrices that represent the SU(2) algebra.The third line in (1) describes the interactions between the matter fields and the Higgs field, φ ,via the Yukawa couplings y ij , which give fermions their masses when the Higgs field acquires a vacuumexpectation value (vev) (cid:104) φ (cid:105) (cid:54) = 0 . In the SM the Higgs field φ is a complex doublet of SU(2) with non-zeroU(1) hypercharge Y , so this vev breaks electroweak symmetry.The fourth and final line in (1) describes dynamics of the Higgs sector. The first term is the kineticterm for the Higgs field, which also includes a covariant derivative D µ (4), and the second term in thefinal line of (1) is the Higgs potential V ( φ ) : V ( φ ) = − µ | φ | + λ | φ | . (5)The negative sign of the first, quadratic term in (5) destabilizes the symmetric case (cid:104) φ (cid:105) = 0 , and thesecond, quartic term in (5) ensures that there is a stable minimum of the potential with (cid:104) φ (cid:105) ≡ v √ µ √ λ (cid:54) = 0 , (6) if λ > . The requirements that the coefficient of the quadratic term is negative and that of the quarticterm is positive are both problematic in the SM, as we shall see later.Many different experiments have confirmed with high precision theoretical predictions derivedfrom the first two lines in the SM Lagrangian (1). However, until 2012 there was no experimentalevidence for the last two lines, and there was considerable theoretical doubt whether it could be correct.3owever, during Run 1 of the LHC the ATLAS [15] and CMS [16] Collaborations discovered a particlewith properties resembling those of the Higgs boson in the SM, as discussed later in this Lecture, albeitwith much less accuracy than, e.g., the precision electroweak tests based on properties of the W ± and Z boson. The major tasks for future experiments at the LHC and elsewhere will include probing whetherthe Higgs and other sectors of the SM Lagrangian in (1) hold up under more detailed scrutiny, whetherthere are additional interactions between SM particles, and whether there is any evidence for new physicsbeyond the SM, as discussed in the second Lecture. As a warm-up exercise, we consider the simplest Abelian model for spontaneous gauge symmetry break-ing, with just a U(1) gauge field A µ and a single complex field φ described by the Lagrangian L ( A, φ ) = − f µν f µν + ( D µ φ † ) ( D µ φ ) − V ( φ ) , (7)where f µν is given by (2), D µ ≡ ∂ µ − ieA µ and V ( φ ) has the ‘Mexican hat’ form (5) illustrated in Fig. 1.The U(1) gauge invariance implies that the theory is invariant under the local transformations φ → φ (cid:48) = e iα ( x ) φ = e iα ( x ) e iθ ( x ) η ( x ) ,A µ → A (cid:48) µ = A µ ( x ) + 1 e ∂ µ α ( x ) , (8)where η ( x ) and θ ( x ) are the magnitude and phase of φ ( x ) , respectively. Fig. 1:
The ‘Mexican hat’ potential (5). The lowest-energy state may be described by a random point around thebase of the hat. In the case of a global symmetry, motion around the bottom of the hat corresponds to a masslessspin-zero Nambu-Goldstone boson [1, 2]. In the case of a local (gauge) symmetry, this boson combines with amassless spin-one gauge boson to yield a massive spin-one particle with three polarization states. The Higgsboson [9] is a massive spin-zero particle corresponding to quantum fluctuations in the radial direction, up the sideof the hat.
We can exploit this gauge invariance to choose α ( x ) = − θ ( x ) , in which case φ (cid:48) ( x ) = η ( x ) andthe Lagrangian (7) takes the form L ( A (cid:48) µ , η ) = − f (cid:48) µν f (cid:48) µν + | ( ∂ µ − ieA (cid:48) µ ) η | − V ( η ) , (9)4he minimum of the potential V ( η ) occurs at the following value of η : η = v √ ≡ µ √ λ . (10)We can then rewrite (9) writing η = ( v + H ) / √ and simplifying the notation: A (cid:48) µ → A µ , to obtain L ( A µ , H ) = − f µν f µν + | ( ∂ µ − ieA µ )( v + H √ | − V ( v + H √ . (11)Expanding (11) to quadratic order in A µ and H , we find L ( A µ , H ) (cid:51) − f µν f µν + e v A µ A µ + 12 [( ∂ µ H ) − m H H ] + . . . , (12)where the gauge boson has acquired a mass m A = ev/ and m H = √ µ = √ λv . The mass of thevector boson results from the spontaneous symmetry gauge breaking mechanism, with the phase degreeof freedom θ of the complex scalar field φ , which is a Nambu-Goldstone boson, being ‘eaten’ by thepreviously massless gauge boson - which has two polarization degrees of freedom - to become the thirdpolarization state needed for a massive gauge boson.The simultaneous appearance of a massive scalar boson H is an inescapable feature of this mech-anism, since it is related to the positivity of the curvature in the scalar potential around the minimum,which is needed to fix the vev v . This insight was made explicit in the second 1964 paper by Higgs [9],but was not mentioned in any of the other 1964 papers. The principle of spontaneous gauge symmetry breaking can easily be extended to the case of a non-Abelian group, in particular the SU(2) L × U(1) Y of the Standard Model. In this case, using the expres-sions (2) and (3) for the gauge field strengths and the expression (4) for the covariant derivative of whatis now an isospin doublet of Higgs fields φ , we can expand the first (kinetic) term in the bottom line ofequation (1), | D µ φ | , to obtain L (cid:51) − g v W + µ W µ − − g (cid:48) v B µ B µ + gg (cid:48) B µ W µ − g v W µ W µ + . . . . (13)where v is the vev of the scalar field, which is determined in the same way as the previous Abelianexample.The first term in (13) yields masses for the charged vector bosons W ± : m W = g v . (14)The other three terms yield a mass matrix for the two neutral gauge fields ( B, W ) , which can be diago-nalized to yield m Z = (cid:113) g + g (cid:48) v Z ≡ gW − g (cid:48) B (cid:113) g + g (cid:48) ,m A = 0 : A ≡ g (cid:48) W + g (cid:48) B (cid:113) g + g (cid:48) . (15)5he first of these mass eigenstates is the massive Z studied in detail in experiments at the LEP accelerator,in particular, and the second, massless eigenstate is identified with the photon. It is useful to introducethe weak mixing angle θ W : tan θ W ≡ g (cid:48) g → m Z = m W cos θ W , Z = cos θ W W − sin θ W B, A = sin θ W W + cos θ W A . (16)Measuring θ W in different ways with high precision has provided important consistency tests of theStandard Model, and provided a clue about the mass of the Higgs boson before its discovery, as wediscuss later.As in the previous Abelian case, there is again a massive scalar (Higgs) boson whose mass isrelated to the curvature of the potential V ( φ ) in the radial direction, and has the value M H = √ µ . (17)The couplings of the Higgs boson to other Standard Model particles are predicted exactly. Expanding thefirst (kinetic) term in the bottom line of equation (1), | D µ φ | , beyond quadratic order, we find trilinearcouplings of the Higgs boson to the massive gauge bosons: g HW W = 2 m W v , g HZZ = = 2 m Z v , (18)and there are also important trilinear and quartic Higgs couplings. As was first pointed out be Wein-berg [14], the third line in (1) links the Higgs-fermion couplings to their masses: m f = y f v ↔ y f = m f v . (19)The couplings (18, 19) lead to characteristic predictions for the partial decay rates of the Standard ModelHiggs boson: Γ( H → f ¯ f ) = N c G F m H π √ m f , (20)where the number of colours N c = 3 for quarks, 1 for leptons, and (for a sufficiently heavy Higgsboson) [18] Γ( H → W + W − ) = G F m H π √ F ( m W m H ) , (21)where F ( m W /m H ) is a phase-space factor, and there is a corresponding formula for Γ( H → ZZ ) witha prefactor / . Experimentally, m H < m W , so the Higgs boson cannot decay into pairs of on-shellgauge bosons, but the decays H → W W ∗ , ZZ ∗ are quite distinctive, and have been measured.Measurements of the couplings of the boson discovered in 2012 and checking their consistencywith the predictions (18, 19) have led to the general acceptance that it is indeed a Higgs boson, as wealso discuss later. Future higher-precision measurements will see whether it is consistent with being thesingle Higgs boson of the Standard Model, or whether the couplings exhibit deviations characteristic ofsome scenario for new physics beyond the Standard Model. In 1975 Mary Gaillard, Dimitri Nanopoulos and I made the first attempt at a systematic survey of thepossible phenomenological profile of the Higgs boson [17,18]. At that time, the Standard Model was notestablished, idea of spontaneous gauge symmetry breaking was far from being generally accepted, therewas general scepticism about scalar particles and, even if one bought all that, nobody had any idea howheavy a Higgs boson might be. For all these reasons, we were rather cautious in the final paragraph of6ur paper, writing "we do not want to encourage big experimental searches for the Higgs boson, but wedo 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 theLEP [19, 20] and LHC accelerators at CERN. For example, a review of the possibilities for new par-ticle searches presented at the first workshop on prospective LHC physics in 1984 [21] discussed variousways of producing the Standard Model Higgs boson at the LHC. There were also studies of Higgs pro-duction at the ill-fated SSC [22], and the state of play was described extensively in [23]. However, in the1980s 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 SLCthat started in 1989. These and other experiments found excellent overall agreement with the predictionsof the Standard Model. However, this consistency depended on the existences if the top quark (whichwas discovered several years later) and the Higgs boson. The accuracy of the LEP et al. measurementspointed towards (what seemed at that time) a relatively heavy top quark [24] and a Higgs boson weighing (cid:46) GeV [26].These indications came about through quantum (loop) corrections to electroweak observables,such as the W ± and Z masses: m W sin θ W = m Z sin θ W cos θ W = πα √ G F (1 + ∆ r ) , (22)where ∆ r is the leading one-loop radiative correction, which exhibits the following dependences on thetop and Higgs masses: ∆ r (cid:51) G F π √ m t + . . . , G F π m W (cid:32)
113 ln m H m W + . . . (cid:33) , (23)where we have exhibited the leading dependences on m t and m H for large masses. These are relicsof the divergences that would appear if either the top quark or the Higgs boson were absent from theStandard Model, which would render it non-renormalizable.In the early 1990s, even before the top quark was discovered, the precision electroweak data wereproviding indications that the Higgs mass was probably well below the unitarity limit of TeV [25],which were strengthened when the top quark mass was measured [26]. Back in 2011, just before theHiggs boson was discovered, the precision electroweak data suggested a range m H = 100 ± GeV.In parallel, unsuccessful searches at LEP had implied that m H ≥ GeV [27], and searches at theFermilab Tevatron collider had excluded a range around (160 , GeV [28]. Combining all the infor-mation available in 2011, the Gfitter Group obtained the χ likelihood function shown in Fig. 2 [29],corresponding to the estimate m H = 125 ±
10 GeV . (24)The success of this prediction was a tremendous success for the Standard Model at the quantum level. Fig. 3 displays various leading-order diagrams contributing to Higgs production at a proton-proton col-lider: those for gg → H , vector boson fusion and associated V + H production are shown in the upperrow, and diagrams for associated ¯ tt + H and some of those for single t + H production are shown inthe lower row. The left panel of Fig. 4 displays the most important Higgs production cross sections atthe LHC at 13 TeV in the centre of mass, as functions of the Higgs mass [30]. The dominant crosssection for m H (cid:46) TeV is that for gluon fusion: gg → H via intermediate quark loops [31], the mostimportant in the Standard Model being the top quark. The next most important processes at low masses m H (cid:46) GeV are the associated-production mechanisms q ¯ q → W + H, Z + H [32], whereas thevector-boson fusion processes W + W − , ZZ → H [33] are more important for m H (cid:38) GeV. Next in7 ig. 2:
The ∆ χ as a function of m H for a complete fit to the data available in mid-2011 [29], including precisionelectroweak data and the negative results of searches at LEP [27] and the Fermilab Tevatron [29] (grey bands).The solid (dashed) lines represent results that include (omit) theoretical uncertainties. the hierarchy of cross sections are the associated-production processes gg, q ¯ q → b ¯ bH (which is difficultto distinguish) and t ¯ tH [34] (which is more distinctive). Lowest in the hierarchy for m H (cid:46) GeVis the cross section for producing H in association with a single t or ¯ t [35]. The right panel of Fig. 4displays is a zoom of the cross sections in a limited range of Higgs mass around 125 GeV [30]. The goodnews is that for m H ∼ GeV most of these cross sections are potentially measurable at the LHC, andseveral of them have already been observed, as discussed later. gg → H Vector boson fusion Associated V + H production ¯ tt + H production s -channel diagrams for t + H production Fig. 3:
Leading-order diagrams for Higgs production. Upper row: gg → H , vector boson fusion and associated V + H production. Lower row: ¯ tt + H production and s -channel diagrams for single t + H production. As can be seen in Fig. 4, the dominant gg → H cross section has a relatively large uncertainty.This is because it is a strong-interaction process, which has relatively large perturbative corrections, andis induced at the loop level, implying that the calculation of these corrections is more arduous than for8 ig. 4: Calculations of the dominant production cross sections for a Standard Model Higgs boson, for a wide rangeof masses (left panel) and for with a m H ∈ [120 , GeV (right panel) [30]. The uncertainties are representedby the widths of the coloured bands for each production mechanism displayed.
Scale Truncation PDFs (TH) Electroweak m t,b,c /m t +0 . − . ± . ± . ± . ± . ± . Table 2:
Breakdown of the theoretical uncertainties (in pb) in (25) that are associated with different approxima-tions in the calculation of the gg → H cross section [30]. a tree-level process. Nevertheless, a complete calculation of the gg → H cross section at the next-to-next-to-leading order (NNLO) is available, as is a heroic N LO calculation in the limit of a heavy topquark [36]. The result of these efforts is the following estimate of the gg → H cross section [30]: σ = 48 . +2 . − . (theory) ± .
56 (PDF , α s ) pb , (25)corresponding to a total uncertainty of < %. It is worth noting that the NLO correction more thandoubled the cross section, that the the NNLO correction was about 20% of the final estimate (25). How-ever, the N LO correction was only ∼ %, indicating that the perturbative QCD corrections are undercontrol. Table 2 compiles the principal theoretical uncertainties in the calculation (25) of the gg → H cross section. The first is associated with the choice of scale in the perturbative QCD calculation, and thesecond is an estimate of the uncertainty due to the truncation of the perturbative expansion. The third isan estimate of the theoretical uncertainty in the use of the parton distribution functions (PDFs) and α s ,and the fourth is an estimate of the uncertainty in higher-order mixed electroweak and QCD perturbativecorrections. The fifth is the parametric uncertainty in the values of m t,b,c to be used, and the sixth andlast is an estimate of the uncertainty in the heavy-top approximation in the N LO calculation. We seethat many uncertainties are comparable at the level of ±O (0 . pb, indicating that a struggle on manyfronts will be needed to reduce substantially the theoretical error in (25).Concerning the PDF uncertainties in (25), Fig. 5 shows that there is now good consistency betweenthe gg collision luminosities estimated by different PDF fitting groups with the recommendation of thePDF4LHC Working Group [37]. The uncertainty from this source is currently estimated at ∼ %, whichis comparable to the parametric uncertainty associated with α s . These are currently the largest individualsources of uncertainty in the gg → H cross section.There are smaller uncertainties in the cross sections for vector boson fusion (shown in the upperpanels of Fig. 6) and H production in association with a W ± or Z boson (lower left panel of Fig. 6), both9 ig. 5: Comparison between the parton-parton production luminosities calculated using different PDF sets [30],compared to the PDF4LHC recommendation [37]. of which have been calculated at NNLO including electroweak corrections at NLO [30]. In both cases,there is good convergence of the perturbation expansion, and there are also quite small uncertainties inthe relevant quark parton PDFs. On the other hand, the uncertainties in the cross section for associated t ¯ t + H production (shown in the lower right panel of Fig. 6) are significantly greater. This is a stronginteraction cross section that has been calculated only at NLO, so there are considerable uncertaintiesassociated with the perturbation expansion. Also, there are greater uncertainties associated with thechoices of quark, antiquark and gluon PDFs. The situation is similar for single t or ¯ t + H associatedproduction. Since Higgs couplings to other particles are proportional to their masses, as seen in (18) and (19), it isexpected to decay predominantly into the heaviest particles that are kinematically accessible [17]. Thisis apparent in the left panel of Fig. 7, which gives an overview of Higgs decay branching ratios fora wide range of Higgs masses. In the specific case of interest when m H (cid:39) GeV, as seen in theright panel of Fig. 7 [30], we see that H → b ¯ b decays are expected to dominate, with H → c ¯ c decayssuppressed by ( m c /m b ) , and H → τ + τ − decays suppressed by a missing colour factor as well. Since m H < m W,Z , only the off-shell decays H → W W ∗ , ZZ ∗ followed by W ∗ , Z ∗ → f ¯ f decays arepossible. Nevertheless, because of the large vector-boson mass factors in (18), these three-body decayshave branching ratios comparable to the leading H → f ¯ f two-body decays, as seen in the right panel ofFig. 7.Also comparable is the rate for H → gg decay, which is a loop-induced process: see Fig. 8. It isdifficult to observe H → gg decay directly, but is related to the rate for the dominant gg → H fusionproduction process. The electroweak loop-induced decays H → γγ and Zγ occur with somewhat lowerbranching ratios. However, the H → γγ mode is very clean, and was one of the discovery modes at10 ig. 6: The cross sections for vector boson fusion production of the Higgs boson with the LHC at 7 TeV (upper leftpanel), and at 14 TeV (upper right panel), for associated
W H production at 7 and 14 TeV (lower left panel), andfor ¯ ttH production at 7 and 14 TeV (lower right panel), all as functions of m H [38]. the LHC. The amplitude for H → γγ decay is generated by loops of massive charged particles [17] asshown in Fig. 8. The most important contributors in the Standard Model are the top quark and the W ± boson. Their contributions may be written as follows: Γ( H → γγ ) = G F α m H π √ (cid:12)(cid:12)(cid:12) Σ f N c Q f A / ( r f ) + A ( r W ) (cid:12)(cid:12)(cid:12) , (26)where A / and A are known functions of r f ≡ m f /m H and r f ≡ m W /m H that have opposite signs,so that the top and W ± contributions interfere destructively. In the Standard Model, the ggH amplitudereceives contributions only from the top and (less important) lighter quarks.The last decay mode shown in the right panel of Fig. 7 is H → µ + µ − , which is suppressed relativeto H → τ + τ − by a factor ( m µ /m τ ) . Nevertheless, it also has quite a clean experimental signature, andis expected to be the first decay of H into second-generation fermions to be probed.There have been strong theoretical efforts to calculate perturbative corrections to H decays [30],leading to the relatively small uncertainties shown in Fig. 7. The largest relative uncertainties are in H → gg decay, because it is a loop-induced decay into strongly-interacting particles, and H → Zγ decay, where high-order calculations are complicated by the masses of the initial-state H and the final-state Z .Echoing what was said earlier about Higgs production mechanisms, another piece of good newsis that for m H (cid:39) GeV many Higgs decay modes ares measurable at the LHC. These happy chancesprovide many opportunities to measure distinctive properties of the Higgs boson. Also, it is an amusing11 ig. 7:
Calculations of the dominant decay branching ratios for a Standard Model Higgs boson over a largerange of masses (left panel) and with mass m H ∈ [120 , GeV (right panel) [30], where the uncertainties arerepresented by the widths of the coloured bands for each decay mode displayed.
Fig. 8:
Loop diagrams for H couplings to massless gauge bosons. In the Standard Model, the dominant fermiondiagram (left) for H → gg and γγ involves the top quark. The W ± diagrams (middle and right) contribute onlyto H → γγ . There are similar diagrams for H → Zγ . irony that the largest H production mechanism, gg → H , and one of the cleanest H decay channels, H → γγ , are both loop-induced processes. Thus, LHC data already give us access to quantum aspectsof Higgs physics, including the possible existence of new heavy particles beyond the Standard Model.If the Higgs boson had weighed 750 GeV (just saying) [39], gathering a lot of information aboutit would have been more difficult. In that case, observing other production modes besides gg → H andvector-boson fusion at the LHC would have been difficult, and observing any other decays besides H → W + W − , ZZ and t ¯ t would probably have been impossible in the absence of any other new particles [40].Obviously, we regret the passing of the late lamented X (750) particle, which would have required newphysics to explain its production and decay, but we should thank our lucky stars for the openness of the H (125) ! The stakes in the Higgs search were very high. How is gauge symmetry broken: spontaneously (ele-gantly) or explicitly (ugly and uncalculably)? Assuming that it is broken spontaneously, is it broken byan elementary scalar field, which would be a novelty that raises perhaps more questions than it answers,many of which are related to the hierarchy of mass scales in physics? The Higgs is very likely a por-tal towards many issues in physics beyond the Standard Model. It would have been associated with aphase transition in the Universe when it was about − seconds old, which maight have been when thebaryon asymmetry of the Universe was generated. The Higgs or a related scalar field might have causedthe Universe to expand (near-)exponentially in a bout of cosmological inflation when it was about − ∼ too much to the dark energy measuredin the Universe today. The stakes in the search for the Higgs boson were undoubtedly high! The discovery of the Higgs boson in 2012 [15, 16] was primarily based on the observation of excessesof events in the γγ and (cid:96) + (cid:96) − channels (where (cid:96) = µ or e ), interpreted as being due to H → ZZ ∗ ,together with a broad excess of (cid:96) + (cid:96) − + missing transverse energy events, interpreted as being due to H → W W ∗ .Measurements of the γγ and (cid:96) + (cid:96) − final states have enabled the mass of the Higgs boson to bedetermined with high precision. The final combined results from ATLAS and CMS LHC Run 1 datayield [41] m H = 125 . ± .
21 (statistical) ± .
11 (systematic) GeV , (27)a measurement at the level of 2 per mille that is dominated by the statistical error and hence can be furtherreduced. 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, as seen in the rightpanels of Figs. 4 and 7. Moreover, it is crucial for the discussion below of the stability of electroweakvacuum. As already mentioned, the measurement (27) is fully in line with previous indications fromprecision electroweak data and previous searches at LEP [27] and the Fermilab Tevatron collider [28],see Fig. 2. Fig. 9 shows a direct comparison between the ATLAS and CMS measurements of m H andthe χ function from a global analysis of the precision electroweak data, omitting the LEP and Tevatronconstraints [42]. We see that the measured value of m H agrees with the indication from the electroweakdata at the ∆ χ ∼ . level. This may seem like a disaster for the quest for physics beyond the Standard Model, but not so fast!
Fig. 9:
The ∆ χ as a function of m H for a fit to the precision electroweak data compared with the measurementsof m H by the ATLAS and CMS Collaborations [42]. Let us consider the second, λφ , term in the Higgs potential (5) It is essential for the Mexican hat formof the potential seen in Fig. 1 and the existence of the non-zero vev (6) that λ > . However, like anyother coupling in a quantum field theory, λ is subject to renormalization. In the Standard Model, there13re two important sources of this renormalization at the one-loop level. One is that due to λ itself, whichtends to increase λ as the renomalization scale Q increases: λ ( Q ) (cid:39) λ ( v )1 − π λ ( v ) ln (cid:16) Q /v (cid:17) . (28)Left to itself, this self-renormalization would cause λ to blow up at some high renormalization scale Q .However, there is also importantant one-loop renormalization of λ due to loops of top quarks: λ ( Q ) (cid:39) λ ( v ) − m t π v ln (cid:16) Q /v (cid:17) , (29)which tends to decrease λ as the renomalization scale Q increases, driving it towards negative values. If λ indeed turns negative, there soon appears a field value with lower energy than our electroweak vacuum,which becomes unstable or at least metastable.The left panel of Fig. 10 illustrates how the negative renormalization by the top quark drives λ < in the Standard Model [43, 44], though this is subject to uncertainties in m t , in particular. As seen in theright panel of Fig. 10, the current world averages of m t and m H suggest that these parameters indeedlie within the region where the Standard Model electroweak vacuum is metastable. In my view, this is apotential disaster (pun intended) that would require new physics to avert it.
Fig. 10:
Left panel: The negative renormalization of the Higgs self-coupling by the top quark within the StandardModel leads to an instability in the Higgs potential for field values ∼ GeV [43]. Right panel: Experimentalmeasurements of m t and m H suggest that the electroweak vacuum of the Standard Model would be metastable,modulo uncertainties in m t , in particular [44]. As seen in Fig. 10, the location and indeed existence of the instability scale Λ I are particularlysensitive to m t , and also to α s as well as to m H . One calculation including higher-order effects yieldsthe following dependences on these parameters [44]: log (cid:18) Λ I GeV (cid:19) = 9 . . (cid:16) m H GeV − . (cid:17) − . (cid:16) m t GeV − . (cid:17) + 0 . (cid:18) α s ( m Z ) − . . (cid:19) . (30)Inserting the world average value (27) for m H , m t = 173 . ± . GeV and α s ( m Z ) = 0 . ± . ,we estimate log Λ I ) = 9 . ± . , (31)14ndicating that (in the immortal words of the Apollo 13 astronauts) we have a problem [45].Several words of caution are in order. The first is that the experimental measurement of m H (27) isalready so accurate that it is the smallest source of uncertainty in (30). Concerning the larger uncertaintydue to m t , subsequent to the compilation of the world average value [46], several new measurementshave been published, including by D0: m t = 174 . ± . GeV [47], by ATLAS: m t = 172 . ± . GeV [48] and by CMS: m t = 172 . ± . GeV [49]. These scatter above and below the officialworld average, and a new compilation awaits better understanding of the discrepancies between them.However, all these measurements lie in the unstable region of m t . A second comment concerns theinterpretation of the value of m t that the experiments measure. This is defined within a specific MonteCarlo event simulation code, and one issue concerns the relation between this and the pole mass. Presentunderstanding is that the difference between these definitions ∼ Λ QCD < GeV, so this correction seemsmanageably small at the present time, though it will require more detailed analysis as the experimentalprecision improves. Another issue concerns the relation between the pole mass and the
M S mass that isused in the loop calculations of vacuum stability. This relation has been calculated to O ( α s ) [50]: m t | pole − m t | MS m t | MS ≡ δmm t | MS ≡ ∆ = 0 . α s ( m t ) + 0 . α s ( m t )+ 2 . α s ( m t ) + (8 . ± . α s ( m t ) + . . . , (32)leading to the numerical result m t | pole = m t | MS + 7 .
557 + 1 . .
501 + 0 .
195 + . . . , (33)where the numbers are for the O ( α ns ) corrections calculated for n = 1 , , , assuming m t | MS =163 . GeV. These corrections decrease systematically in magnitude, so the QCD perturbation seriesseems to be well-behaved. The sum of the uncalculated higher-order terms has been estimated to be ∼ GeV, with uncertainty of ∼ GeV [51]. These effects are also below within the current experi-mental precision.Another comment concerns the length of the lifetime of our (in principle) unstable electroweakvacuum, which may be much longer than the age of the Universe to date. This is certainly a necessaryconsistency condition for the existence of physicists capable of recognizing the problem, but also leadssome of them to disregard it as unimportant. I disagree with this attitude for two reasons. One is becausethe present vacuum energy is very small and positive. Arguments have been proposed how this mightcome about if our vacuum is (one of) the lowest-energy state(s) in an extensive landscape, or one mightimagine that some approximate symmetry could yield a small positive value. Personally, I would findthe small value of the present vacuum energy much more difficult to understand if it is only a temporarystate, and if our universe will eventually decay into an anti-De Sitter state with negative vacuum energyof much larger magnitude. The other reason for taking vacuum stability seriously as a requirement isthat, if it were not, fluctuations in the Higgs field in the hot and dense early universe would have takenmost of it into the anti-De Sitter “Big Crunch" phase, and the conventional expansion of the universewould never have occurred [52, 53]. On the other hand, one could argue anthr*p*c*lly that if even aninfinitesimal part of the universe escaped the “Big Crunch", that would have been enough for sentientphysicists to come into being.My own take on the instability problem is that we should take it seriously, and that it motivatessome form of new physics to stabilize the electroweak vacuum . Clearly, any such physics shouldappear at some energy scale below Λ I ∼ GeV, but might lie far beyond the reach of conceivableaccelerators. However, in my view it is the best hint for some new physics beyond the Standard Modelprovided by Run 1 of the LHC. Assuming that we have not misunderstood the experimental measurements of m t . φ , whose loop would have a positive sign, and which could in general have couplings of the forms [54]: L (cid:51) M | φ | + M v | H | | φ | , (34)where M and M are two mass parameters. Indeed, if one chooses M (cid:46) GeV, the effective Higgspotential can be stabilized. However, avoiding a blow-up in λ as well as a negative value typicallyrequires some fine-tuning of M , i.e., the coupling of the new scalar φ to the Higgs field H , at the permille level. The simplest way to stabilize the coupling is to postulate new fermions to counteract the φ − H coupling in (34). But now we have introduced scalar partners of the top and fermionic partners of theHiggs that make the theory reminiscent of supersymmetry [54]. So, why not postulate supersymmetry,as we discuss in the second Lecture? The ATLAS and CMS Collaborations have published a joint analysis of their measurements of Higgsproduction and decay in various channels [55], as shown in Fig. 11. Several Higgs decay modes havebeen established with high significance, including γγ , W W ∗ , ZZ ∗ and τ + τ − , and there are importantconstraints on other Higgs decay models. The gg → H and vector boson fusion production mechanismshave been established, and there are interesting constraints on H production in association with W ± , Z and t ¯ t pairs. Fig. 11:
The products of cross sections σ and branching ratios B measured by the ATLAS and CMS Collaborationsin various channels, normalized to the Standard Model predictions [55]. However, many H couplings remain to be established. Most prominently, the H → b ¯ b decaythat is expected to dominate has been seen only at the 2.6- σ level at the LHC and the 2.8- σ level at theTevatron [56]. Moreover, although there is indirect evidence for a t ¯ tH coupling from measurements ofthe induced ggH and γγH couplings, there is no direct evidence in the absence of measurements of t ¯ tH (or single t/ ¯ tH ) associated production. Also, there is as yet no evidence for H → µ + µ − decay, thoughupper limits on it already provide interesting information, as we see in the second Lecture.16he first analyses of Higgs data from Run 2 of the LHC have also been shown by ATLAS andCMS [57]. The distinctive H → γγ and ZZ ∗ decays have been seen again with 10- σ significance, theHiggs production cross section at 13 TeV is in line with theoretical calculations, and the searches are onfor H → µ + µ − and associated t ¯ tH and single t/ ¯ tH production. Shortly after the discovery of the Higgs boson, Peter Higgs was quoted in the Times of London assaying: “A discovery widely acclaimed as the most important scientific advance in a generation has beenoverhyped" [58]. I would very humbly and respectfully beg to disagree. Without the Higgs boson (orsomething to do its job), there would be no atoms because electrons would escape from nuclei at thespeed of light, the weak interactions responsible for radioactivity would not be weak, and the universewould be totally unliveable.
It was a big deal. The Particle Physics Higgsaw Puzzle
The first piece of the particle physics jigsaw puzzle to be discovered was the electron in 1997, and ittook 115 years until a candidate for the final missing piece of the Standard Model, the Higgs boson,was discovered in 2012 [15, 16]. In the first Lecture, I was jumping the gun, blithely assuming that theparticle discovered in 2012 is indeed the Higgs boson. In this Lecture we first review the experimentaland theoretical justifications for this assumption. In the language of jigsaw puzzles, is it the right shape,and does it have the right size, in the language of particle physics, does it have the right spin, parity andcouplings? We then discuss what physics may lie behind and beyond it, and review how to probe theseideas with possible future accelerators.
Since the H (125) particle decays into pairs of photons, Lorentz invariance assures us that it cannot havespin 1, but it might a priori have spin 0, 2 or higher and, in each case, it might have either positive ornegative parity. Many tests of the H (125) spin and parity have been proposed theoretically and carriedout by the LHC experiments. Examples include the polar angle distribution in H → γγ decays andfinal-state angular correlations in H → W W ∗ → (cid:96) + (cid:96) − + missing transverse energy decays and in H → ZZ ∗ → (cid:96) + 2 (cid:96) − decays [59]. Also, the kinematics of H (125) production mechanisms such asproduction in association with a W or Z boson would differ for different spin-parity assignments [60],which have been probed by the Fermilab Tevatron experiments [61].One example of a spin-parity analysis of the H (125) in the X → ZZ → (cid:96) + (cid:96) final stateis shown in Fig. 12 [62]. This and all the other published analyses are in excellent agreement withthe J P = 0 + spin-parity assignment predicted for the Higgs boson, and all the alternative spin-parityassignments studied have been strongly excluded. These include the pseudoscalar − possibility andvarious spin-2 possibilities. The H (125) passed these first important experimental tests with flyingcolours. It is a fundamental property of the Higgs boson that, since the field vev gives masses to the other elemen-tary particles, the H couplings to them should be proportional to their masses, see (18) and (19). Oneway to test this is to analyze the H (125) production and decay data assuming couplings to other particlesthat are proportional to some nonlinear power of their masses [63]: y f = (cid:16) m f M (cid:17) (cid:15) , g V = (cid:32) m (cid:15) ) V M (cid:15) (cid:33) , (35)17 ig. 12: Comparison between the Standard Model Higgs boson hypothesis in the X → ZZ → (cid:96) + (cid:96) − final statewith various spin-two J P hypotheses [62]. where the unknown power (cid:15) and mass scale M are to be fitted to the data, the Standard Model expec-tations being (cid:15) = 0 and M = v = 246 GeV. The result of such an analysis performed jointly by theATLAS and CMS Collaborations is shown in Fig. 13 [55], including the best fit and the 68 and 95% CLbands. The joint analysis yields (cid:15) = 0 . +0 . − . , M = 233 ±
13 GeV , (36)which are highly compatible with the Standard Model predictions. In this way, the H (125) passedanother crucial experimental test. We can also see explicitly in Fig. 13 that the decay rates for H → µ + µ − and τ + τ − must be very different, a first strong violation of lepton universality, as expected in themass-dependent couplings of the Higgs boson. Flavour-changing couplings of the Higgs boson are expected to be very suppressed in the StandardModel, though they might be present in extensions with multiple Higgs multiplets. Upper limits onflavour-changing interactions at low energies and dipole moments can be used to constrain the the pos-sible flavour-changing interactions of the H (125) [64]. Examples of relevant tree and loop diagramsinvolving H are shown in Fig. 14. Upper limits on flavour-changing quark interactions exclude theobservability of quark-flavour-violating H (125) decays, but lepton-flavour-violating decays could berelatively large. We found that the branching ratio for either H → τ µ or τ e (but not both) could be O (10) %, comparable to the Standard Model prediction for BR ( H → τ + τ − ) , whereas the branchingratio for H → µe must be < × − . Analyses of LHC Run 1 data yielded results [65, 66] compatiblewith these upper limits: CMS : BR( H → τ µ ) = 0 . +0 . − . % , BR( H → τ e ) < . , BR( H → eµ ) < . , ATLAS : BR( H → τ µ ) = 0 . ± . . (37)That said, the CMS result for BR ( H → τ µ ) , in particular, whetted theoretical appetites for additionaldata from Run 2 of the LHC. A first preliminary result from CMS does not indicate any deviation fromthe Standard Model [67], but this is definitely une affaire à suivre !18 ig. 13: A fit to a parametrization of the form (35) to Higgs coupling measurements by the ATLAS and CMSCollaborations in various channels. The dotted line connects the Standard Model predictions, the best fit is shownas a red line, and the 68 and 95% CL ranges are shown as green and yellow bands [55].
Fig. 14:
Left panel: Tree-level H -exchange diagram that may contribute to a generic flavour-changing ampli-tude. Right panel: One-loop H -exchange diagram that may contribute to anomalous magnetic and electric dipolemoments of charged leptons ( i = j ), or to radiative lepton-flavour-violating decays ( i (cid:54) = j ) [64]. As mentioned in the first Lecture, two of the most important Higgs couplings are induced by loop dia-grams, namely the ggH vertex responsible for the dominant H production mechanism, which is mainlygenerated by the top quark in the Standard Model, and the Hγγ vertex responsible for one of the mostdistinctive H (125) decays, which is mainly generated by loops of top quarks and W ± bosons in theStandard Model, as shown in Fig. 8. Via these vertices, the Run 1 LHC data have already providedimportant consistency checks on the Standard Model predictions for the H couplings at the quantumlevel. Fig. 15 displays the combined ATLAS and CMS constraints on the magnitudes of the Hγγ and ggH couplings relative to their Standard Model values [55]. We see good consistency at the 10 to 20%level, which also provides significant restrictions on possible extensions of the Standard Model such as19 fourth generation, supersymmetric particles and heavy vector-like quarks.
Fig. 15:
A fit by the ATLAS and CMS Collaborations to the magnitudes of the
Hγγ and ggH couplings, normalizedby factors ( κ γ , κ g ) relative to their Standard Model values [55]. Broadly speaking, there are two schools of theoretical thought about this question.On the one hand, many theorists are attracted by the idea of an elementary Higgs scalar field, asin the original formulation, but are concerned by the problems connected with loop corrections to theHiggs mass. Quantum corrections to the mass parameter µ in the effective potential (5) due to, e.g., thetop quark or the Higgs self-coupling, exhibit quadratic divergences. If one cuts the loop integrals off atsome momentum scale Λ , one is left with large residual contributions if Λ is identified with some highnew physics scale such as that of grand unification or the Planck mass. In the case of a loop of fermions f such as the top quark, shown in Fig. 16(a), one finds ∆ m H = − y f π [2Λ + 6 m f ln(Λ /m f ) + ... ] , (38)where y f is the Yukawa coupling and the . . . represent non-divergent mass-dependent terms, and in thecase of a loop of scalars S , shown in Fig. 16(b), one finds similar divergent contributions: ∆ m H = λ S π [Λ − m S ln(Λ /m S ) + ... ] . (39)If the Standard Model were to remain valid up to the Planck scale, M P (cid:39) GeV, so that
Λ = M P ,each of the quadratic “corrections" would be (cid:39) times larger than the physical mass-squared of theHiggs, namely (10 ) GeV .The relatively small physical value of the Higgs mass is not protected by any symmetry of theStandard Model, and keeping it small seems to require some unnatural fine-tuning unless there is somesuitable new physics at the TeV scale. The favoured example of such new physics in an elementary20 a)H f (b)H S Fig. 16:
One-loop quantum corrections to the mass-squared of the Higgs boson due to (a) the loop of a genericfermion f , (b) a generic scalar S . Higgs scenario is supersymmetry, which exploits the opposite signs of loop corrections due to fermionsand bosons in (38, 39). If these occur in pairs with related couplings: λ S = 2 λ f (40)as in supersymmetric models [68], and if the differences between the masses of supersymmetric partnersare O (1) TeV so that the subleading terms in (38, 39) are not large, the quadratic term µ in the Higgspotential (5) is kept naturally small, and hence also the Higgs mass and the electroweak scale. One ofthe miracles of supersymmetry is that the symmetry between fermions and bosons cancels not only theone-loop quadratic divergences (38, 39), but also all quadratic divergences at higher order in perturbationtheory, as well as many logarithmic divergences [69].On the other hand, many other theorists believe that the Higgs is composite, a bound state offermions like Cooper pairs in BCS superconductivity and pions in QCD. Such a theory has a naturalcut-off at the scale of the strongly-interacting composite dynamics, analogous to Λ QCD . The StandardModel does not contain any candidate for this new strong dynamics, and attention focused initially onsome scaled-up version of QCD [70]. However, simple models of this type were incompatible withthe precision electroweak data mentioned in the first Lecture, and predicted a heavy strongly-interactingscalar particle unlike the Higgs boson discovered in 2012. Accordingly, attention has shifted to analternative idea that the Higgs boson is analogous to the pion of QCD, namely that it is a pseudo-Nambu-Goldstone boson of some larger chiral symmetry that is broken down to the Standard Model, much likethe pion in QCD [71]. In such a model, the lightness of the Higgs boson is enforced by this approximatechiral symmetry. Generic features of such theories include a coloured top partner fermion that cancelsthe one-loop Higgs mass corrections due to the top quark, some new scalars and/or gauge bosons withrelatively low masses (cid:46) TeV, and a strongly-interacting ultraviolet completion at a mass scale that is O (10) TeV.A convenient way to parametrize the phenomenology of such a theory is to assume that the Higgssector has an underlying SU(2) × SU(2) structure that is broken down to a custodial SU(2) symmetryso as to retain the successful tree-level relation ρ ≡ m W /m Z cos θ W (cid:39) . The Goldstone bosons π a : a = 1 , , of this symmetry-breaking pattern that are ‘eaten’ by the W ± and Z to become theirlongitudinal polarization states are then parametrized by a traceless × matrix Σ = exp( iσ a π a /v ) ,with the following couplings to the Higgs boson H : L = v D µ Σ D µ Σ (cid:32) a Hv + b H v + . . . (cid:33) − m i ¯ ψ iL Σ (cid:18) c Hv + . . . (cid:19) + h . c . + 12 ∂ µ H∂ µ H + 12 H + d (cid:32) m H v (cid:33) H + d (cid:32) m H v (cid:33) H + ... , (41)where the coefficients a, b, c, . . . are normalized so that they are all unity in the Standard Model. The taskof experiments is then to measure these coefficients and see whether they differ from these predictions, as21hey may in composite Higgs models. For example, in two minimal composite Higgs models MCHM , one has MCHM : a = (cid:112) − ζ, c = (cid:112) − ζ , MCHM : a = (cid:112) − ζ, c = 1 − ζ √ − ζ , (42)where ζ is a model parameter that is not specified a priori .Fig. 17 shows two experimental analyses of the data, using a different notation: a → κ V , c → κ F .The left panel compares the constraints from Higgs data alone (yellow and orange ellipses) with theresult of a global analysis including also precision electroweak data (blue ellipses) [72]. We see thatthese are largely complementary, with the Higgs data constraining κ F = c and the electroweak dataconstraining κ V = a . The Standard Model prediction κ F = c = 1 , κ V = a = 1 is close to the best-fitpoint and well within the global 68% CL contour. The right panel compares ATLAS and CMS Higgsmeasurements with the predictions (42) of the MCHM , models [73]. We see that the data require ζ (cid:46) . , necessitating some tuning of these models so that their predictions resemble those of theStandard Model. Fig. 17:
Left panel: A fit by the Gfitter Group to the LHC H coupling measurements (orange and yellow ellipses)and in combination with precision electroweak data (blue ellipses) [72]. Right panel: Comparison of ATLAS andCMS constraints on H couplings with predictions of the MCHM , models [73]. Fig. 18 shows how the different Higgs coupling measurements by ATLAS and CMS combine togive their overall constraints on ( κ V , κ F ) [55]. Most of the measurements are relatively insensitive tothe sign of κ F , the exception being that of the Hγγ coupling. Its sensitivity is due to the interferencebetween the top and W ± loop contributions to the coupling, which interfere destructively for the StandardModel (positive) sign of κ F and constructively for the non-standard (negative) sign. Largely as a resultof this asymmetry, the combined fit decisively favours the Standard Model sign of κ F . Measuring thecross section for single t/ ¯ tH production could also provide a direct determination of this sign, and couldprobe the possible existence of a CP-violating t ¯ tH coupling [74].The general conclusion of these analyses using the parametrization (41) is that there is no indica-tion of any deviation from the Standard Model predictions for the H couplings of the form that mighthave arisen in a composite Higgs scenario. Introducing a single global modification factor µ for the H couplings to Standard Model particles, the combined ATLAS and CMS data imply that [55] µ = 1 . +0 . − . = 1 . ± .
07 (stat . ) ± .
04 (expt) ± .
03 (thbgrd) +0 . − . (thsig) , (43)where the last three uncertainties are systematics. Thus, overall the strength of the Higgs couplings agreeswith the Standard Model at the ∼ % level, though individual couplings have larger uncertainties.22 ig. 18: A fit by the ATLAS and CMS Collaborations to the magnitudes of the
HV V and H ¯ f f couplings, normal-ized by factors ( κ V , κ F ) relative to their Standard Model values [55]. Moreover, there is no evidence for any decays of H to unknown particles, in particular invisible decays.The H (125) particle looks very much the way it was predicted in the Standard Model.For this reason, the Physics Class of the Swedish Academy stated in its citation for the 2013 NobelPrize [75] that Today we believe that “Beyond any reasonable doubt, it is a Higgs boson." . At this point a popular approach is to assume that the H (125) particle is exactly Standard Model-like, anduse it in a model-independent search for new physics that could manifest itself via higher-dimensionaleffective interactions between Standard Model fields, in particular those of dimension 6 [76]: L eff = (cid:88) n c n Λ O n , (44)where Λ is some characteristic scale of new physics and the c n are unknown dimensionless coefficients.These can be constrained by a combination of data on Higgs properties, precision electroweak data,triple-gauge couplings (TGCs), etc.. The beauty of this approach is that it provides an integrated frame-work for analyzing all these categories of data in a unified and consistent way.This attractive approach is, however, unwieldy when applied in full generality, because of thelarge number of possible dimension-6 operators, even if one assumes the SU(2) × U(1) symmetry ofthe Standard Model. For this reason, one often makes simplifying assumptions, e.g., about the flavourstructure of the operators. Furthermore, if one restricts attention to precision electroweak observables,Higgs and TGC measurements, global fits to these data become manageable. Table 3 lists the CP-evendimension-6 operators [77] relevant for these measurements. In each case, we also indicate the categoriesof observables that provide the greatest sensitivities to the operator coefficients. This quotation was taken from the preprint version of [63]. They apparently did not notice that this phrase was removedfrom the published version of [63] at the insistence of the anonymous referee, who considered that “Beyond any reasonabledoubt" is not a scientific statement. WPTs Higgs Physics TGCs O W = ig (cid:16) H † σ a ↔ D µ H (cid:17) D ν W aµν O B = ig (cid:48) (cid:16) H † ↔ D µ H (cid:17) ∂ ν B µν O W = g (cid:15) abc W a νµ W bνρ W c ρµ O T = (cid:16) H † ↔ D µ H (cid:17) O HW = ig ( D µ H ) † σ a ( D ν H ) W aµν O (3) lLL = ( ¯ L L σ a γ µ L L ) ( ¯ L L σ a γ µ L L ) O HB = ig (cid:48) ( D µ H ) † ( D ν H ) B µν O eR = ( iH † ↔ D µ H )(¯ e R γ µ e R ) O g = g s | H | G Aµν G Aµν O uR = ( iH † ↔ D µ H )(¯ u R γ µ u R ) O γ = g (cid:48) | H | B µν B µν O dR = ( iH † ↔ D µ H )( ¯ d R γ µ d R ) O H = ( ∂ µ | H | ) O (3) qL = ( iH † σ a ↔ D µ H )( ¯ Q L σ a γ µ Q L ) O f = y f | H | ¯ F L H ( c ) f R + h.c. O qL = ( iH † ↔ D µ H )( ¯ Q L γ µ Q L ) O = λ | H | Table 3:
The relevant CP-even dimension-6 operators in the basis [77] that we use. For each operator, we list thecategories of observables that provide the greatest sensitivities to the operator.
The left panel of Fig. 19 shows results from a fit to precision data on leptonic electroweak observ-ables [78]. The lower horizontal axis shows the possible numerical values of these coefficients, and theupper horizontal axis shows the corresponding new physics scales. Here and in subsequent plots, thegreen bars are for fits to individual operator coefficients assuming the other operators are absent, and thebars of other colours are for global fits marginalizing of all the operators that could contribute. In general,these coloured bars extend further than the green bars. The right panel of Fig. 19 extends this analysisto include hadronic electroweak observables [78]. The vertical dashed lines in the two panels are for thesame new physics scale, and serve to emphasize the point that the constraints on hadronic observablesare, in general, weaker than those on leptonic observables and, moreover, some exhibit deviations fromthe Standard Model predictions that remain to be understood. However, in both cases the new-physicsconstraints are in the multi-TeV range.
Fig. 19:
The 95% CL ranges from an analysis [78] of precision leptonic electroweak observables (left panel) andincluding also hadronic electroweak observables (right panel). The upper (green) bars denote fits to individualoperator coefficients, and the lower (red) bars are for marginalized multi-operator fits. The upper axis should beread with factor m W /v ∼ / for the combination ¯ c W + ¯ c B . The left panel of Fig. 20 shows results from global fits to data on Higgs production strengths andkinematics (blue bars), to data on TGCs (red bars), and their combination (black bars) [78]. The greenbars again show the results of fits to individual operator coefficients, which are generally smaller thanthe other bars. We note that the constraints on the new-physics scale from Higgs and TGC data shown24n the left panel of Fig. 20 are, in general, weaker than from the precision electroweak data: they aretypically only a fraction of a TeV. The right panel Fig. 20 emphasizes the complementarity of Higgs andTGC measurements, as reflected in anomalous TGC couplings. The orange and yellow ellipses showthe constraints from direct TGC measurements, whereas the green ellipses show the indirect constraintsfrom Higgs measurements, and the blue ellipses show the results of a global fit [79].
Fig. 20:
Left panel: The 95% CL ranges for individual operator fits (green bars), and the marginalised 95% rangesfor multi-operator fits. The blue bars combine the LHC signal-strength data with the kinematic distributions forassociated H + V production measured by the ATLAS and D0 Collaborations, the red bars includeh the LHC TGCdata, and the black bars show results from a global combination with both the associated production and TGCdata [78]. Note that the coefficients ¯ c γ,g are shown magnified by factors of 100, so for these coefficients the upperaxis should be read with a factor of 10. Right panel: The 68 and 95% CL ranges in the plane of anomalous TGCs ( δg ,z , δκ γ ) including LEP TGC constraints, LHC Higgs data and their combination [79]. The Standard Model effective field theory is the preferred framework for analyzing future LHCdata. Fig. 21 shows the results of global fits to LHC Higgs data using only production rates (left panel)and including production kinematics (right panel) [30]. In each case, the blue bars are obtained from ananalysis of present data, the green bars illustrate the prospective sensitivities with 300/fb of data, and thered bars those with 3000/fb of data. The prospective sensitivities are impressive, particularly when thekinematical information is included.
Despite the continuing absence of any direct evidence for the new physics beyond the Standard Model atthe LHC, one should not become disheartened. History abounds with examples of people who thoughtthey knew it all, but did not. In 1894, just before the discoveries of radioactivity and the electron, AlbertMichelson declared that “The more important fundamental laws and facts of physical science have allbeen discovered" [80]. More recently, prior to the string revolution, Stephen Hawking asked “Is the Endin Sight for Theoretical Physics?" [81]. However, my favourite example of a lack of ability to thinkoutside the box is the Spanish Royal Commission that rejected a proposal by Christopher Columbus tosail west before 1492: “So many centuries after the Creation, it is unlikely that anyone could find hithertounknown lands of any value" [82]. Many of us have seen referees’ reports with a similar flavour.25 ig. 21:
Results of present and prospective global fits to LHC Higgs data using only production rates (left panel)and including production kinematics (right panel). The blue bars are obtained from an analysis of present data,the green bars illustrate the prospective sensitivities with 300/fb of data, and the red bars those with 3000/fb ofdata [30].
The title of this Subsection is a paraphrase of the title of a James Bond movie [83] and, in deference tohim, one may cite 007 reasons for anticipating physics beyond the Standard Model. 001) As discussedin Lecture 1, within the Standard Model the electroweak vacuum is unstable against decay to high H field values. 002) The Standard Model has no candidate for the astrophysical dark matter. 003) TheCabibbo-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) TheStandard Model does not have a satisfactory mechanism for cosmological inflation. 007) We need aquantum theory of gravity.Several of these issues will be addressed by LHC measurements during Run 2, e.g., the top quarkmass will be determined more accurately, there will be searches for dark matter particles, there will besearches for CP violation and other flavour physics beyond the CKM model, as well as new particles thatcould help stabilize the electroweak scale. Personally, I am a fan of supersymmetry as a framework thatcould solve or at least mitigate many of the problems on James Bond’s list, so I focus now on that theory. Supersymmetry is an extension of the Standard Model that has long been favoured by many theorists [84].Some are disappointed that it has not yet appeared at the LHC, but then neither has any other proposedextension of the Standard Model such as compositeness or extra dimensions. Rather, I would argue thatRun 1 of the LHC has provided three new additional reasons to favour supersymmetry.One is the apparent instability of the electroweak vacuum within the Standard Model, which canbe stabilized by a theory resembling supersymmetry, as discussed in Subsection 1.10. Specifically, ina supersymmetric theory the negative running of the Higgs quartic self-coupling λ due to the top quarkloop is exactly cancelled by stop squark loops. Moreover, the negative sign of the quadratic term in theHiggs potential (5) and hence the appearance of the electroweak vacuum can be understood dynamically26s a different effect of renormalization by the heavy top quark via the logarithmic terms in (38 39).A second Run 1 motivation for supersymmetry is the mass of the Higgs boson. Minimal super-symmetric models predicted that it should weigh (cid:46) GeV [85], as discussed in more detail in the nextSubsection, in agreement with the measurement (27). This is because supersymmetry actually predictsthe magnitude of the Higgs quartic self-coupling: λ ∼ g + g (cid:48) , where g and g (cid:48) are the SU(2) and U(1)couplings of the Standard Model.The third Run-1 motivation for supersymmetry is that simple supersymmetric models also pre-dicted that the Higgs couplings should be with a few % of the Standard Model values, in perfect consis-tency with the measurements to date [86]. In fact, in a supersymmetric model it may be very difficult tomeasure any deviations from the Standard Model predictions for the Higgs couplings [87], as discussedlater. These new reasons for liking supersymmetry are in addition to the many traditional reasons tolike it, such as its ability to stabilize the mass hierarchy via the cancellation of the quadratic divergencesin loop corrections to the Higgs mass (38 39) [68], the fact that it naturally predicts a cold dark matterparticle [88], the fact that it improves the accuracy of the grand unification prediction for sin θ W [89],and its essential rôle in the superstring framework for a theory of quantum gravity. Even the minimal supersymmetric extension of the Standard Model (MSSM) requires two complexHiggs doublets, in order to cancel out anomalous triangle diagrams with higgsinos that would other-wise destroy the renomalizability of the theory, and in order to give masses to all the quarks. These twocomplex Higgs doublets contain 8 degrees of freedom, of which 3 are combined with the massless W ± and Z fields to give them masses, as discussed in Subsection 1.4. There remain 5 degrees of freedomthat manifest themselves as massive Higgs bosons, two neutral scalars h, H , one neutral pseudoscalar A and two charged bosons H ± .The tree-level Higgs mass-squared matrix has the following form in the MSSM: M ,N =1 tree = (cid:18) m Z cos β + m A sin β − ( m A + m Z ) cos β sin β − ( m A + m Z ) cos β sin β m Z sin β + m A cos β (cid:19) . (45)Diagonalizing (45), we find that the masses of the two scalars at the classical (tree) level can be writtenas m h,H = 12 (cid:18) m A + m Z ∓ (cid:113) ( m A + m Z ) − m Z m A cos β (cid:19) , (46)where tan β is the ratio of the vevs of the 2 Higgs doublets. At face value, the formula (46) impliesthat the lighter neutral scalar Higgs boson h should have a mass < m Z . However, there is an importantone-loop correction to m h due to the stop squarks ˜ t , : ∆ m h = 3 m t π v ln (cid:18) m ˜ t m ˜ t m t (cid:19) + . . . , (47)which can increase m h by (cid:46) GeV, as seen in Fig. 22. As also seen there, if the other Higgs bosons
H, A and H ± are heavy, they are expected to all be quite degenerate in mass. A curiosity of Fig. 22 is thepossibility that the heavier neutral scalar H might weigh (cid:46) GeV, with the h even lighter [90]. It maybe difficult to reconcile this possibility with the LHC measurements of the couplings of the H (125) , butthe possibility of a lighter Higgs boson should not be discounted completely, and further experimentalsearches in the low-mass range are welcome! 27 ig. 22: A representative calculation of MSSM Higgs masses in the m max h scenario with tan β = 5 , using the FeynHiggs 2.2 code [91]. Note a region at larger m A where the lighter neutral scalar Higgs boson may weigh ∼ GeV [85], and a smaller region at small m A where the heavier neutral scalar Higgs boson may weigh ∼ GeV [90].
The Standard Model contains chiral fermions, i.e., the left- and right-handed fermion states live in in-equivalent representations of the SU(2) × U(1) gauge group. As such, they can be accommodated onlywithin supermultiplets of simple N = 1 supersymmetry. Theories with N ≥ supersymmetries wouldrequire left- and right-handed fermions to transform identically under SU(2) × U(1), so phenomenolog-ical supersymmetric models such as the MSSM are usually restricted to N = 1 . However, left-rightsymmetric (vector-like) fermions appear in many extensions of the Standard Model, such as models withextra dimensions, string compactifications and some grand unified theories. These extensions of theStandard Model could accommodate N = 2 supersymmetry.In the MSSM, although the quarks and leptons are chiral, the Higgs representations forma vector-like pair, and so could in principle also accommodate N = 2 supersymmetry [92]. One could considerthe possibility that the Higgs sector is just the tip of an N = 2 supersymmetric iceberg, which wouldalso include an N = 2 gauge sector and possibly vector-like fermion supermultiplets, as occurs in somestring compactifications and grand unified theories. So, what would an N = 2 Higgs sector look like?
In this case, differently from (47), the tree-level mass-squared matrix is: M ,N =2 tree = (cid:18) m Z cos β + m A sin β − ( m A − m Z ) cos β sin β − ( m A − m Z ) cos β sin β m Z sin β + m A cos β (cid:19) . (48)The crucial difference: is the replacement: m A + m Z → m A − m Z in the off-diagonal terms between the N = 1 and N = 2 cases (45) and (48). The corresponding tree-level Higgs masses after diagonalizationare shown in Fig. 23. In the N = 1 case (left panel) we see level repulsion for m A ∼ GeV. However,28n the N = 2 case (right panel) we see linear level crossing with m N =2 h = m Z , m N =2 H = m A (49)at the tree level. Moreover, at the tree level the N = 2 Higgs sector is ‘aligned’, so that the lighter neutralHiggs boson has exactly the same couplings as in the Standard Model [92]. m H m h tan β = β = MSSM @ Tree - Level m A [ GeV ] m Φ [ G e V ] m H m h SUSY N = @ Tree - Level m A [ GeV ] m Φ [ G e V ] Fig. 23:
A comparison between the tree-level values of Higgs masses in the MSSM (left panel) and in the N = 2 supersymmetric model (right panel) [92]. The dominant one-loop corrections ε to (45) and (48) are those due to top quarks and stop squarks,which appear in their [22] entries. Let us assume that they are such as to give the measured Higgs bosonmass m h = 125 GeV. In the N = 1 case the required loop correction is [93] ε N =1 = ∆ M ,N =122 = m h ( m A + m Z − m h ) − m A m Z cos βm Z cos β + m A sin β − m h , (50)whereas in the N = 2 case it is [92] ε N =2 = ∆ M ,N =222 = 2( m A − m h )( m h − m Z )cos 2 β (cid:16) m Z − m A (cid:17) + m A − m h + m Z . (51)Fig. 24 compares (left panel) the required value of m H for tan β = 1 in the two cases, (middle panel)the value of m H − m A as a function of m A for tan β = 3 , and (right panel) the value of m H − m A as a function of tan β for m A = 300 GeV [92]. We see that m H can be lighter in the N = 2 casethan when N = 1 , and that m H − m A is also smaller, in general. Another effect of doubling up onsupersymmetry in the Higgs sector is seen in Fig. 25, where we compare the supersymmetry-breakingmass scale M SUSY that is needed in the N = 1 and N = 2 cases for different values of the squark massmiixing parameter X t [92]. We see a consistent pattern that smaller values of M SUSY are required when N = 2 than when N = 1 for any fixed values of m A and tan β .On the other hand, the sensitivities of the LHC searches are also different, as seen in the ( m A , tan β ) plane in Fig. 26 [92]. The upper part of this plane (shaded grey) is excluded by direct LHC searches for H, A → τ + τ − , which have similar sensitivities in the N = 1 and N = 2 cases. The red (green) curves29 SSM SUSY N = tan β =
200 300 400 500 600 700 800200300400500600700800 m A [ GeV ] m H [ G e V ] MSSM SUSY N = tan β =
200 300 400 500 600 700 80005101520 m A [ GeV ] m H - m A [ G e V ] MSSM SUSY N = m A =
300 GeV tan β [ GeV ] m H - m A [ G e V ] Fig. 24:
Left panel: The value of m H required to obtain m h = 125 GeV via one-loop radiative corrections for tan β = 1 in the MSSM and N = 2 supersymmetry [92]. Middle panel: The value of m H − m A as a function of m A for tan β = 3 . Right panel: The value of m H − m A as a function of tan β for m A = 300 GeV.
Fig. 25:
Contours of the supersymmetry-breaking mass scale M SUSY that are required as functions of m A and tan β to yield m h = 125 GeV in the MSSM scenario (dotted lines) and the N = 2 scenario (full lines) [92]. Theleft panel is for X t = 0 , and the right panel is for the maximal-mixing scenario with X t = √ M SUSY . show the ranges of m A that are excluded indirectly by the LHC. We see that m A (cid:38) GeV is allowedin the N = 2 case, whereas m A (cid:38) GeV is required when N = 1 . The bottom line is that bothsupersymmetry and supersymmetric Higgs bosons may be closer in an N = 2 supersymmetric modelthan has been suggested by the N = 1 MSSM.
Now that a (the?) Higgs boson has been discovered, there is naturally a lot of interest in studying itin detail. The LHC has considerable potential in this respect, with a target of eventually accumulating3000/fb of data with the HL-LHC that has now been approved by the CERN Council [94]. Ideas forfuture Higgs factories should take this into account, and should be able to demonstrate how much betterthey can measure the Higgs boson, as well as look for other possible new physics. Two proposals for30
50 200 250 300 350 400 450 5002468101214 m A ( GeV ) t an β Direct H / A →ττ S U SY N = M SS M Fig. 26:
The direct exclusion from searches for heavy scalars in the
H/A → τ τ final state (grey shading), and theindirect bounds from measurements of Higgs couplings to fermions and massive bosons at Run 1 of the LHC in theMSSM (green) and the N = 2 model (red) [92]. linear e + e − colliders are on the market: the ILC that aims initially at a centre-of-mass energy of 500 GeVwith a planned upgrade to 1 TeV [95], and CLIC that aims at centre-of-mass energies between 350 GeVand 3 TeV [96], as seen in Fig. 27. Designs for circular e + e − colliders, CEPC in China [97] and FCC-ee near CERN [98], are now also being discussed. As also seen in Fig. 27, these are more limited incentre-of-mass energy, but have the potential for higher luminosities for probing the Higgs boson via the e + e − → ZH process, and for precision electroweak studies at the Z peak and the W + W − threshold.The capabilities of the ILC [95] and FCC-ee [99] for Higgs coupling measurements are shown inthe left and right panels of Fig. 28, respectively. The capabilities of the LHC, HL-LHC, ILC and FCC-ee to probe the Hγγ, HZZ, HW W and
Hgg couplings are shown in Fig. 29, and compared with thedeviations form the Standard Model that are expected in different supersymmetric models whose param-eters were chosen to be consistent with LHC data [87]. As mentioned previously, these typically predictvery small deviations from the Standard Model that will be very difficult to distinguish experimentally.Moreover, the current theoretical uncertainties in these predictions, indicated by the green bars, are largecompared with the prospective experimental accuracies at the FCC-ee, in particular. More theoreticalwork will be needed!As discussed earlier, a favoured approach is to use future Higgs, precision electroweak and TGCmeasurements to constrain the coefficients of possible dimension-6 operators constructed out of StandardModel fields. Fig. 30 compares the LHC constraints (left panel) and the prospective FCC-ee sensitivities(right panel) in global fits to the scale Λ in dimension-6 operator coefficients [100]. Fig. 31 comparesthe sensitivities of the ILC and FCC-ee in fits combining Higgs and precision electroweak data (leftpanel), and combining Higgs and TGC data (right panel) [100]. In the left panel, the shadings compareresults obtained with and without estimates of the theoretical uncertainties in the precision electroweakobservables: we see again the importance of minimizing these. In the right panel, the effects of including31 ig. 27: The design luminosities at various centre-of-mass energies of projects for future high-energy e + e − col-liders [95–98]. Fig. 28:
Left panel: Prospective measurements of Higgs couplings at the ILC [95]. Right panel: Prospectivemeasurements of Higgs couplings at FCC-ee [99].
TGC measurements at the ILC are also indicated by shading.By virtue of its higher centre-of-mass energy, the CLIC proposal for an e + e − collider has particu-lar advantages in looking for the effects of dimension-6 operators, since the effects of their interferenceswith Standard Model amplitudes typically increase ∝ E [101]. The following are some examples of thesensitivities to dimension-6 operator coefficients at 350 GeV and 3 TeV for associated HZ production: ∆ σ ( HZ ) σ ( HZ ) (cid:12)(cid:12)(cid:12)(cid:12)
350 GeV = 16¯ c HW + 4 . c HB + 35¯ c W + 11¯ c B − ¯ c H + 5 . c γ , ∆ σ ( HZ ) σ ( HZ ) (cid:12)(cid:12)(cid:12)(cid:12) = 2130¯ c HW + 637¯ c HB + 2150¯ c W + 193¯ c B − ¯ c H + 7 . c γ , (52)32 ( BR − BR SM ) /BR SM (%) h → ggh → WWh → ZZh → γγ Best Fit Predictions cmssm highcmssm lownuhm1 highnuhm1 lowLHCHL-LHCILCTLEPSM unc. Higgs WG
Fig. 29:
Comparison between prospective measurements of Higgs branching ratios at future colliders, low- andhigh-mass CMSSM and NUHM1 predictions (red and purple symbols) and the current uncertainties within theStandard Model (turquoise bars) [87].
Fig. 30:
Left panel: Constraints on dimension-6 operator coefficients from measurements at the LHC. Right panel:Prospective corresponding constraints from measurements at FCC-ee [100]. and for e + e − → W + W − production : ∆ σ ( W + W − ) σ ( W + W − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
350 GeV = 0 . c HW + 0 . c HB + 4 . c W − . c W , ∆ σ ( W + W − ) σ ( W + W − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 13¯ c HW + 7 . c HB + 17¯ c W − . c W . (53)The sensitivities to most of the operator coefficients indeed increase substantially with the centre-of-massenergy, for both the associated H + Z and W + W − cross-sections, confirming the expected competitive33 ig. 31: Summary of the reaches for the dimension-6 operator coefficients, in individual fits (green) and whenmarginalised in a global fit including all operators (red), from projected ILC250 (lighter shades) and FCC-ee(darker shades) precision measurements [100]. The left plot shows the operators that are most strongly constrainedby electroweak precision observables and Higgs physics, and the different shades of dark green and dark redillustrate the effects of theoretical uncertainties at FCC-ee. The right plot is constrained primarily by Higgsphysics and TGCs, and the different shades of light green demonstrate the improved sensitivity when TGCs areincluded in the ILC250 analysis. advantage of the high energies attainable with CLIC. Fig. 32 shows the increase in sensitivity of CLICoperating at 3 TeV compared with 350 GeV or 1.4 TeV for a number of dimension-6 operator coeffi-cients [101]. The green bars are for fits including individual operators, and the red bars are for global fitswith the coefficients marginalized.
Fig. 32:
The estimated sensitivities of CLIC measurements at centre-of-mass energies of 350 GeV, 1.4 TeV and3.0 TeV to the scales of various (combinations of) dimension-6 operator coefficients [101]. The results of individual(marginalised) fits are shown as green (red) bars. The lighter (darker) green bars in the left panel include (omit)the prospective HZ Higgsstrahlung constraint.
Studies of the sensitivities to Higgs properties of a 100-TeV pp collider such as FCC-hh are still atan early stage. However, as seen in Fig. 33, the Higgs production cross sections will be much larger thanat the LHC, and there will be extensive opportunities for kinematical measurements as well as overallproduction rates [102]. Moreover, such a machine might provide the first opportunity to measure directlythe triple-Higgs coupling with respectable accuracy, since it contributes to the HH cross section that34ncreases by almost two orders of magnitude compared to the LHC, as seen in Fig. 33 (grey line). Fig. 33:
The most important cross sections for Higgs productions in pp collisions, as functions of the centre-of-mass energy up to 100 TeV [102]. “Beyond any reasonable doubt", the LHC has discovered a (possibly the) Higgs boson [75]. Whilstbeing a tremendous success for theoretical physics, it also represents a tremendous challenge. Even inthe minimal elementary Higgs model, both the terms in the Higgs potential present problems. Does thequartic term turn negative at high scales, implying vacuum instability? How come the quadratic termis so small compared to plausible fundamental mass scales in physics such as the Planck mass? TheLHC may yet discover new physics beyond the Standard Model during Run 2. If it does, the globalpriority for high-energy physics will surely be to study it. If it does not discover new physics at the TeVscale, it will be natural to focus future accelerator experiments on the Higgs boson. Either way, in mypersonal opinion future circular colliders may offer the best experimental prospects, being able to probethe 10 TeV scale indirectly via high-precision low-energy experiments, and directly via the productionof new heavy particles.
Acknowledgements
The author thanks the UK STFC for support via the research grant ST/J002798/1.
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