Electroweak Theory, Spontaneous Symmetry Breaking, and the Higgs Mechanism PHYS 461

Electroweak Theory, Spontaneous Symmetry Breaking, and the Higgs Mechanism PHYS 461 The generation of mass for the weak gauge bosons in the Standard Model comes from a process known as spontaneous symmetry breaking, also known more popularly as the Higgs mechanism. Just like the example in class, a constant shift in the value of a scalar boson field interacting with the gauge particles causes the Lagrangian symmetry to be destroyed, with the net result being the appearance of new mass terms. In the following few pages, we'll see exactly how this happens. 1 Nuclear Decays and Fermi's Four Fermion In- teraction The earliest evidence of a fundamental force beyond electromagnetism and gravitation came in the early 1900s, with the study of nuclear decay processes. It was known that certain radioactive elements could undergo \beta decay", a process whereby the nucleus emits (or absorbs) a beta particle. Without going too much into the specifics, suffice it to say that a beta particle was nothing more than a lepton { the electron, or its antiparticle the positron. The two possible modes included negative beta decay, in which a neutron decays into a proton and an electron, n −! p+ + e− + Energy and positive beta decay, where the proton decays into a neutron and a positron: p+ −! n + e+ + Energy Note that electric charge is conserved in each case { but spin is not! Protons, neutrons, electrons, and positrons are all fermions, so the spin before the decay must 1 be ± 2 , but clearly the total spin after can either by 0 or 1. This suggests that the process as written is incomplete. In addition to this non-conservation of spin, it was observed that momentum and energy were also not conserved. Something seemed to be carrying away an ever-so- tiny component of both quantities, but whatever it was could not be detected with the traditional tools. Since this mystery particle also had to be neutral, Enrico Fermi proposed that it should be called the neutrino, Italian for \small neutral one"1. 1All good particle physicists must come up with catchy names for things before any other property is known. Sure enough, the predicted neutrino was eventually discovered, and was found to have spin-1/2 { hence also a fermion, restoring spin conservation in the reactions: + + + − p ! n + e + νe + Energy n ! p + e +ν ¯e + Energy As it turns out, the neutrino accompanying the first reaction is an electron anti- neutrino, while that in the second is a plain old electron neutrino. The reason has to do with Feynman diagrams, which we'll discuss a bit later. For the time being, accept it as gospel! Fermi's prediction of the neutrino was backed in part by the requirement that spin be conserved, but also by his attempt to write down a Lagrangian describing the weak nuclear decay interaction, which looked something like this: Lweak ∼ GF (¯pneν¯ e +npe ¯ ν¯e) The first term represents an incoming proton that decays into an outgoing neutron, outgoing positron, and outgoing electron neutrino, while the latter is an incoming neutron decaying into an outgoing proton, outgoing electron, and outgoing anti-neutrino. Each interaction has strength GF , which is called Fermi's Constant (go figure). Nu- merically, it's value is −5 −5 −2 10 GF = 1:17 × 10 GeV ≈ 2 mp i.e. it is a dimensionful constant that is about 10−5 inverse-square proton masses. 1.1 Weak Decays of Leptons and Quarks Since protons and neutrons are actually composite particles made up of up (u) and down (d) quarks, the weak decay described above should really be one of quark decays. Recall that the proton is p = uud, and the neutron is n = udd, and the charge of 2 1 each quark (qu = + 3 ; qd = − 3 ) ensure the charge of the corresponding particles. So, the decays described above are actually + + + p ! n + e + νe =) u ! d + e + νe + − − n ! p + e +ν ¯e =) d ! u + e +ν ¯e It turns out that there are other decay processes that can be described by Fermi's theory, and these involve the other generations of leptons. In addition to the family (e; νe), there are the muon and mu neutrino, (µ, νµ), and the tau and tau neutrino (τ; ντ ). Each generation is successively heavier than the last (but the neutrinos are effectively massless): me = 0:5 MeV ; mµ = 100 MeV ; mτ = 1:8 GeV So, a muon decay process looks like µ ! e +ν ¯e + νµ where the final product includes a mu neutrino. Similar reactions can happen with the other particles. Since there are three generations of quarks, (u; d); (s; c); (b; t), we could have similar decays between these as well (e.g. c-quarks decaying into muons, etc...). In the weak vernacular, these interactions are called charged currents, because an electric charge is transferred during the boson exchange. 1.2 Toward a Gauge Theory of Weak Interactions The electrostatic interaction of two charged particles (say, two electrons, or two muons, or an electron and an anti-muon, etc....) could technically be modelled as a similar four-fermion vertex similar to that discussed above. As we know, however, the correct theory describing this interaction is quantum electrodynamics, a gauge theory in which the boson mediating the interaction of two fermions is massless (i.e. the photon). In reality, the fermions do not interact at a point, but instead exchange a photon that couples to each with strength e (for a total interaction strength e2). Could the weak decays in Fermi's theory also be described by a gauge theory? If so, the gauge boson could not be the photon, for several reasons. First, there is a transfer of electric charge in the decay, so it can't be neutral. Second, unlike electromagnetism the decay happens over a very short length scale { say less than the size of the proton but greater than the electron { and thus the gauge boson must have a non-zero mass to account for this. Remembering our length-energy conversions, −15 `proton ∼ 10 m corresponds to a mass M ∼ 0:1 GeV. We can assume a lower −18 bound of è ∼ 10 m (a typical value for electron confinement scales). So if the weak interactions take place on a scale è < `W < `p, this corresponds to a particle ± of mass 0:1 GeV < MW < 100 GeV. As it turns out, the W bosons have a mass of about 80 GeV, and the Z0 has a mass of 91 GeV. Not a bad guess!2 1.3 Neutral Currents In addition to the decays described above, there is another form of weak interaction. Neutrinos are very weakly interacting particles, in the sense that they aren't really affected by the presence of matter. They aren't impervious to it, however. They do interact, albeit barely. Thus, it's possible to have neutrino scattering off charged leptons like the electron (or the muon, but these are even't weaker interactions), e.g. e− + ν ! e− + ν Since no charge is transferred in the boson, these are called neutral current inter- ractions. This is the basis for neutrino detection experiments, because it's the only way to observe neutrinos! 2 Electroweak Theory We know that electromagnetism { or rather, quantum electrodynamics { is a particle theory whose gauge symmetry is U(1). As we've seen, this demands that the theory contain one gauge boson, which we know is the photon Aµ. The weak force has an + − 0 SU(2) symmetry, and thus contains three gauge bosons: Wµ ;Wµ , and Zµ. But these four bosons are all different! In the quest to understand grand unified theories, we would like to believe that at some higher energy level, they become indistinguish- able, and the electromagnetic force merges with the weak force. This unification was realized in the late 1960s by Abdus Salaam, Sheldon Glashow, and Stephen Weinberg. Known as the GSW electroweak theory, it is a gauge theory that contains four identical gauge bosons. In a lower-energy limit, it breaks up into the two forces mentioned above. The gauge symmetry for the theory is SU(2) × U(1), which is just a fancy way of saying that the covariant derivative term includes four boson fields. The trio was awarded the Nobel Prize in 1979 for \... their contributions to the theory of the unified weak and electromagnetic interaction between elementary particles, including, inter alia, the prediction of the weak neutral current." Their theory explained several existing experimental mysteries, and of course predicted the existence of the W and Z bosons. The latter were experimentally observed at CERN in 1983. 2 Note that this same reasoning explains why the photon is massless: since `photon ! 1, the associated mass scale must be Mphoton ! 0. 2.1 Weak Hypercharge and Isospin The charges associated with these symmetries are somewhat different than electromagnetism. In the latter, we understand the charge to be the electric charge Q that the photon feels. In electroweak theory, however, the charges are ones which are felt by all bosons. An early attempt at understanding the weak and electromagnetic forces led to the introduction of a set of characteristics common to all particles, called weak isospin I3 and weak hypercharge YW . These conspire to give the well-known electric charge as follows: Y Q = I + W (1) 3 2 Each particle that can feel the (electro)weak force has such charges. And what particles are these? They include all leptons and quarks. The leptons are the electron, muon, and tau, as well as their neutrinos.

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