6.10 Fermion Masses We have explained how the gauge bosons get their masses by the Higgs mechanism, but what about the quarks and leptons? Again, the answer is straightforward. Given a Higgs field transforming as a doublet of SU(2) with hypercharge one-half, we can write down the Yukawa couplings λuq Hcu λdq Hd λel He + h.c. (6.61) L⊃− L R − L R − L R where Hc iσ2H is an SU(2) doublet field with hypercharge minus one-half.59 These ⌘ ⇤ terms represent interactions, but when we plug in the Higgs VEV, lo and behold, we get fermion masses λuv λdv λev mu = ,md = ,me = . (6.62) p2 p2 p2 It just works.TM 6.11 Three Generations We have described what happens for the first generation of quarks and leptons. In fact there are three generations (we already know about the muon and the various flavours of quarks) and it turns out that the extension of the theory just described gives an elegant (and more to the point, correct) description of flavour physics (namely transitions between the generations). In particular, the Yukawa couplings in (6.61) can be complex, and this is what gives rise to CP violation, once we have three generations. We don’t have time to describe it here, but I encourage you to look it up. 6.12 The Standard Model and the Higgs boson We have almost finished our description of the Standard Model. To recap, we show in Table 1 the different fields and their representations under the SM gauge group SU(3) SU(2) ⇥ ⇥ U(1) (recall that SU(3) corresponds to QCD, or the strong nuclear force). We have worked out the properties of all of the particles, but one: the Higgs boson. What Higgs boson? Remember in the Abelian Higgs model that the Goldstone boson got eaten by the gauge field, but we were left with one massive scalar mode, corresponding to fluctuations in the radial direction in the complex plane of the field φ. For the Higgs field H in the Standard Model, we have four real scalar degrees of freedom (since H is a complex doublet); three of these get ‘eaten’ to form the longitudinal polarizations of the Wµ± and Zµ. One scalar remains: the Higgs boson. We can work out its properties by going to the unitary gauge, in which the three Goldstone bosons are manifestly eaten. In the SM, this amounts to choosing 1 0 H(x)= . (6.63) p2 v + h(x)! 59It is easy to see that Hc transforms with Y = 1 , since it involves the complex conjugate of H.Itisa − 2 doublet of SU(2) because the complex conjugate of SU(2) transforms as an anti-doublet of SU(2),which is equivalent to the doublet representation. The iσ2 is just the similarity transform that takes us from one rep to the other. Go and look in the group theory book if you’re worried about it. –57– Field SU(3)c SU(2)L U(1)Y g 810 W 130 B 110 T 1 qL =(uLdL) 32+ 6 2 uR 31+ 3 d 31 1 R − 3 l =(⌫ e )T 12 1 L L L − 2 eR 11 1 −1 H 12+ 2 Table 1.FieldsoftheStandardModelandtheirSU(3) SU(2) U(1) representations ⇥ ⇥ The Higgs boson, h(x), is a real scalar field. It is not charged under electromagnetism (it can’t be, since it is real). Its couplings to other fields can be worked out by replacing v with v + h in our previous expressions. Thus, from (6.62), we find a Yukawa coupling to fermion i given by mi h . (6.64) L⊃− v i i Similarly, from (6.59), we find couplings to gauge bosons given by 2 2 2 2 2h h + µ mZ 2h h µ m + W W − + + Z Z . (6.65) L⊃ W v v2 µ 2 v v2 µ ✓ ◆ ✓ ◆ Finally, the Higgs boson has self interactions, coming from the potential µ2 λ λ m2 m2 m2 + (v + h)2 (v + h)4 λv2h2 λvh3 h4 = h h2 h h3 h h4. L⊃ 2 − 4 ⊃− − − 4 − 2 − 2v − 8v2 (6.66) 2 2 Thus mh =2λv , such that we know the value of the coupling λ once we know the mass of the Higgs. The recent LHC measurement of m 125 GeV thus fixes λ 0.13. h ' ' With these couplings worked out, we can roughly work out the phenomenology of Higgs boson decays. The self interactions are not relevant here, because energy-momentum conservation obviously prevents the Higgs boson decaying to two or three Higgs bosons! For the same reason, if the Higgs is light, it will lie below the required mass threshold for decay to pairs of heavier particles, such as W +W or ZZ or top quarks (m 173 GeV, in − t ⇠ case you didn’t know). This consideration must be balanced against the fact that the Higgs boson couplings to particles all grow with the mass of the particle. Thus, for a lightish Higgs (above about 10 GeV), decays to bottom quark pairs will dominate (mb 4.1 GeV). + ' But by the time the Higgs has become very heavy (mh & 2mW ), decays to W W − and ZZ must dominate. Interestingly enough, the crossover does not occur near the mass threshold mh =2mW 160 GeV, but somewhat below, nearer mh 140 GeV. The reason is that ⇠ + ' QFT allows the Higgs boson to decay to a W W − or ZZ pair in which one of the gauge –58– FIG. 2: Select Standard Model Higgs boson partial widths, as afunctionofmass,MH .Individual partial widthsFigure are labeled, 8. while Higgs the boson total width partial (sum decay of all partial widths, widths, from some [9 minor]. ones not shown) is the black curve. Widths calculated with hdecay [22]. The astute reader will have noticed by now that Fig.E2 2 contain= p2 scurvesforHiggspartial+ m2 60 bosons is virtualwidths, to inmassless that thefinal mass-shellstates! (Have condition another look if you didn’t notice.)is We not kno satisfied.wthe The virtual gaugeHiggs boson couples then to particles decays proportional to real to (on-shell) their masses, quarks so this requires or leptons some explanation. by the usual gauge Recall that loop-induced transitions can occur at higher orders in perturbation theory. Such interaction.interactions The partial typically decays are important widths and to calculate branching only whe ratios,natree-levelinteractiondoesn’t as a function of mh, are shown in Figs. 8 andexist.9 They. Remarkably, are responsible at for the rare point decays ofm varioush = 125 mesonGeVs, for where instance, the and Higgs are in some was found, we cases sensitive to new physics which may appear in the loop. Here, we consider only SM see comparableparticles branching in the loop. ratios Which to ones a varietyare important? of final Recall states. also once This again has that the the disadvantage Higgs of making it veryboson di couplesfficult proportional to discover to particle the Higgs mass. Thus, in the the top first quark place, and EW since gauge the bosons number are of Higgs most important. For H gg,then,thatmeansonlythetopquark,whileforH γγ it is decays in aboth single the final top quark state and isW! suppressedloops (there is compared no ZZγ vertex). to the The fixedH gg backgroundexpression! (for of the things that ! look like theFeynman Higgs diagram decaying of Fig. that 3) is way, [24]: but are not. But it has the great advantage that it 2 3 2 makes it easy for usα tosGF makeMH a variety of experimental tests that the Higgs boson that we Γgg = τi 1+(1 τi)f(τi) (3) 16p2 π3 − i claim to have discovered really X does have the properties predicted in the SM. So far, the 4m2 1 2 LHC data seems to confirm that. f sin− 1/τ τ 1 with τi = 2 and f(τ)= 1 1+p1 τ 2 ≥ (4) M ln − iπ τ < 1 H ( 4 p1 p1 τ There is one thing that may be bothering you in the− Figures.− − − They suggest that the Higgs has a small coupling to both a pair of photons7 γγ and to a pair of gluons gg. How can this be, when the Higgs carries neither colour nor electric charge? The answer is that loop Feynman diagrams, like those in Fig. 10 generate such couplings. Though small, they are very important for Higgs boson phenomenology at the LHC. Indeed, the LHC is a proton- proton collider. Protons are mostly made of up and down quarks, but the coupling of the Higgs boson to these is very small (it doesn’t even appear in the Figures we just showed). But the proton also contains gluons, that bind the quarks together and these provide a way for us to produce the Higgs boson in pp collisions at the LHC. Similarly, the coupling to photons is small, but a pair of photons has a much lower background (from non-Higgs 60If you want to prove this for yourself, draw the Feynman diagram and show that the resulting amplitude is non-vanishing. –59– FIG. 4: Select Standard Model Higgs boson branching ratios asafunctionofmass,MH [22]. The Higgs prefers to decay to the most massive possible final state. The ratio of fermionic branching ratios are proportionalFigure to fermion 9.Higgsbosonbranchingratios,from[ masses squared, modulo color factors and radiative9]. corrections. γ 2. A brief word on statistics – the simple view Now that we understand the basics of Higgs decay, and production in electron-positron collisions, we should take a momenth to consider statistics.t The reason we must resort to statistics is that particle detectors are imperfect instruments.