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 → ∞, 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. The quarks are up, down, charm, strange, bottom, and top. Table 1 shows the weak isospin and hypercharges associated with these particles. Although a somewhat antiquated description of what we now know as gauge couplings, this was a good attempt at finding a unifying structure for several different types of interactions.

Particle Weak isospin (I3) Weak hypercharge (Y ) Electric charge (Q) − − − 1 e , µ , τ − 2 −1 −1 1 νe, νµ, ντ 2 −1 0 1 1 2 u, c, t quarks + 2 + 3 + 3 1 1 1 d, s, b quarks − 2 + 3 − 3 Table 1: Weak isospin and hypercharge values for leptons (top two rows) and quarks (bottom two rows). The particle’s electric charge Q is calculated as in Equation 1. In weak interactions and decays, these three quantities must be conserved.

3 The Electroweak Lagrangian and Covariant Deriva- tive

Gauge symmetries are introduced into particle theories by changing the regular derivative in the Lagrangian to a covariant one,

X a a ∂µ −→ Dµ = ∂µ + ig T Aµ a a a where the T are the group generators and the Aµ are the gauge fields. For U(1), the generator is just 1, and there is one field (the photon). For SU(2), the generators are the Pauli spin matrices σ1, σ2, σ3 and the gauge fields the weak bosons. But this is only after symmetry breaking! Before, the fields are something more generic. 1 2 3 So, let’s introduce the U(1) field Bµ and the three SU(2) fields Wµ ,Wµ ,Wµ . These are massless gauge fields. The covariant derivative for this theory is then

3 gB gW X D = ∂ + i B + i σaW a µ µ 2 µ 2 µ a=1

The couplings gB and gW are related to the charges, but are not the speciﬁc charges of the broken theories. In fact, in the GSW electroweak theory, they are related to each other by the weak mixing angle (or Weinberg mixing angle) θW , deﬁned as

gB tan θW = (2) gW which is a measure of how the two ﬁelds mix together. Ultimately, we should expect this mixing to play a role in how the weak force relates to electromagnetism (in fact, it’s related to the masses of the gauge bosons!).

4 The Higgs Mechanism

At this point, we can introduce the Higgs mechanism required to break the symmetry of the Lagrangian. The particle we choose is called a doublet, which is a pair of complex scalar particles much like a vector: φA φH = (3) φB

We have some freedom to choose the values of φA and φB (for reasons that are a bit more complicated to explain here), so we set

0 φ = 0 , φ = v =⇒ φ = (4) A B H v

This now looks more like a scalar, since there’s only one non-zero component that can transform under complex rotations. The value v is called the vacuum expectation value of the Higgs ﬁeld, which is basically the value the ﬁeld takes on in its lowest energy state. It is real, since it corresponds to a measurable energy, namely v = 246 GeV. Note this is not the Higgs mass, which is about MH = 125 GeV. The fact that the vev has a non-zero value in the vacuum is what will cause the symmetry breaking of the theory, and ultimately give the particles mass.

So, introduce the Higgs particle to the theory through the Lagrangian:

µ † 2 † LHiggs = (D φH ) (DµφH ) + mH φH φH + other terms We actually only care about the covariant derivative term, since that’s where the symmetry breaking comes from. Expanding it out, we ﬁnd g g D φ = ∂ φ + i B B φ + i W σ1W 1 + σ W 2 + σ3W 3 φ µ H µ H 2 µ H 2 µ 2 µ µ H Since the Pauli spin matrices are 2 × 2, and the Higgs doublet is a 1 × 2 vector, the above equation is simply a matrix equation! We can expand this out piece by piece as follows: g i W σ1W 1 + σ W 2 + σ3W 3 2 µ 2 µ µ g 0 1 0 −i 1 0 = i W W 1 + W 2 + W 3 2 µ 1 0 µ i 0 µ 0 −1 gW 3 gW 1 2 i 2 Wµ i 2 (Wµ − iWµ ) = gW 1 2 gW 3 i 2 (Wµ + iWµ ) −i 2 Wµ

The term involving Bµ can also be written as a matrix by multiplying it by the identity:

g g gB B B i 2 Bµ 0 i Bµ = i Bµ I = gB 2 2 0 i 2 Bµ Since we only care about the interaction terms of the Higgs with the gauge ﬁelds, we won’t worry about the kinetic term for the Higgs. Writing this interaction as a complete matrix, we ﬁnd

3 1 2 i gBBµ + gW Wµ gW (Wµ − iWµ ) Interaction term = 1 2 3 (5) 2 gW (Wµ + iWµ ) gBBµ − gW Wµ

1 Now let’s study what each of these terms means. Since the combination of Wµ and 2 Wµ in the oﬀ-diagonal blocks looks like some variety of complex conjugate, we can associate that with a charge particle and its antiparticle! So, we deﬁne

− 1 2 + 1 2 Wµ = Wµ + iWµ ,Wµ = Wµ − iWµ

± 3 3 These are our W bosons! We can postulate that the other states Bµ and Wµ must somehow combine to become the photon and Z boson. This is in fact true, and we deﬁne

3 Aµ = Bµ cos θW + Wµ sin θW 3 Zµ = −Bµ sin θW + Wµ cos θW We see that the photon and Z are literally mixtures of the two via a rotation (in Hilbert space!) by an angle equal to the Weinberg angle.

3 + 1 2 − 1 2 You might wonder why W = Wµ − iWµ and Wµ = Wµ + iWµ , and not the other way around. Good question! Anyway, back to the Lagrangian. The mass terms that arise come from the fact that the bosonic interaction terms are of the form

µ † 2 (D φH ) (DµφH ) = kinetic terms + interaction terms × |φH |

Again, we concentrate only on the interaction terms, which will end up being φH coupled with Equation 5, times their complex conjugate transposes,

µ 3,µ +,µ † 3 + 1 † gBB + gW W gW W gBBµ + gW Wµ gW Wµ φH −,µ µ 3,µ − 3 φH 4 gW W gBB − gW W gW Wµ gBBµ − gW Wµ This seems yucky, but it becomes incredibly simpliﬁed when we remember that the Higgs doublet has only one term (see Equation 3). So, the ﬁrst part of the above equation becomes

µ 3,µ +,µ gBB + gW W gW W (0 , v) −,µ µ 3,µ gW W gBB − gW W − 3 = v gW Wµ , gBBµ − gW Wµ The second part is just the complex-conjugate transpose of this (changing W − to W +),

+ +gW Wµ (v) v 3 (6) gBBµ − gW Wµ Their product then gives then full interaction term,

2 + v − 3 gW Wµ (v) Int. term = gW Wµ , gBBµ + gW Wµ 3 4 gBBµ − gW Wµ v2g2 v2 = W W +,µW − + (g Bµ − g W 3,µ)(g B − g W 3) (7) 4 µ 4 B W B µ W µ This looks less icky, but we can seem some things emerging. First, we’re getting boson cross-terms that should be recognizable as mass terms. In fact, the ﬁrst term is the mass term for the W boson! That is, v2g2 vg M 2 = W =⇒ M = W W 4 W 2

The mass of the W boson is determined by the SU(2) coupling constant gW and the Higgs vacuum-expectation value – the Higgs has given the W mass!

In a similar fashion, we can find the mass of the photon and Z boson by expanding out the second term in the interaction expression above. First, we write it in terms 3 of the fields Bµ and Wµ , 1 g2 BµB + g2 W 3,µW 3 − 2g g BµW 3 4 B µ W µ B W µ These aren’t the bosons we’re looking for, however, since they both have mass! Instead, we refer to our mixture in Equation 6, and re-write it in terms of these fields:

Bµ = Aµ cos θW − Zµ sin θW 3 Wµ = Aµ sin θW + Zµ cos θW Plugging these back in to the interaction term, we get (term by term, and ignoring cross-terms):

v2g2 Term 1 = B BµB 4 µ v2g2 = B (Aµ cos θ − Zµ sin θ )(A cos θ − Z sin θ ) 4 W W µ W µ W v2g2 v2g2 = B cos2 θ AµA + B sin2 θ ZµZ + cross terms (8) 4 W µ 4 W µ Aha! We’re starting to see mass terms for the photon and Z! Next, we do the second term: v2g2 Term 2 = W W 3,µW 3 4 µ v2g2 = W (Aµ sin θ + Zµ cos θ )(A sin θ + Z cos θ ) 4 W W µ W µ W v2g2 v2g2 = W sin2 θ AµA + W cos2 θ ZµZ + cross terms (9) 4 W µ 4 W µ

µ 3 Lastly, the cross-term between B and Wµ gives

v2g g Term 3 = − B W BµW 3 2 µ v2g g = − B W (Aµ cos θ − Zµ sin θ )(A sin θ + Z cos θ ) 2 W W µ W µ W v2g g g g = − B W sin θ cos θ AµA + B W sin θ cos θ ZµZ + cross terms(10) 2 W W µ 2 W W µ µ Perfect! Now we collect like terms – speciﬁcally, we want the terms involving A Aµ µ (photon mass) and Z Zµ (Z boson mass). Starting with the photon, the terms from Equations 8, 9, and 10 are

v2g2 v2g2 v2g g B cos2 θ AµA + B sin2 θ AµA − B W sin θ cos θ AµA 4 W µ 4 W µ 2 W W µ v2g2 v2g2 v2g g = B cos2 θ + W sin2 θ − B W sin θ cos θ AµA 4 W 4 W 2 W W µ v2 = (g cos θ − g sin θ )2AµA 4 B W W W µ This looks like a mass term – for the photon??? But the photon is massless! So, let’s think about the deﬁnition of the mixing angle θW again (Equation 2). Since the ratio gB of the couplings is tan θW = . But we learned (or should have!) long ago that gW sin θW tan θW = . So, combining these, we ﬁnd cos θW

sin θW gB = =⇒ gW sin θW = gB cos θB cos θW gW Inserting those in the expression above gives us v2 (g cos θ − g cos θ )2AµA = 0 4 B W B W µ THE PHOTON IS MASSLESS!!

µ What about the Z particle? We can show by collecting the terms in Z Zµ that the mass must be v2 (g sin θ + g cos θ )2ZµZ = M 2 ZµZ 4 B W W W µ Z µ To evaluate this term, we can think of gB and gW as sides of a right-angle triangle p 2 2 whose hypothenuse is gB + gW (because that gives the right tan θW ratio). So, we can also deﬁne g g sin θ = B , cos θ = W W p 2 2 W p 2 2 gB + gW gB + gW so the term above becomes g2 + g2 q g sin θ + g cos θ = B W = g2 + g2 B W W W p 2 2 B W gB + gW which means the mass of the Z boson is v q M = g2 + g2 Z 2 B W 5 Measured Masses and Values of the Couplings

Experimentally, we can measure the mass of the W and Z bosons. They are (as of the most recent report)

MW = 80.4 GeV ,MZ = 91.2 GeV

This means that we can place rigid constraints on the values of the couplings gB and gW , and moreover on the value of the mixing angle θW . Note that the ratio of the two masses is M g v g W = W 2 = W = cos θ v p 2 2 p 2 2 W MZ 2 gB + gW gW + gB The ratio of the boson masses is essentially the cosine of the weak mixing angle! Plugging in the masses, we ﬁnd 80 GeV cos θ ≈ = 0.88 W 91 GeV Typically in particle physics, we refer to the square of the sine instead, so we get

2 2 2 MW sin θW = 1 − cos θW = 1 − ≈ 0.22 MZ

4 2 The Review of Particle Properties lists the most recently-measured value as sin θW = 0.2312.

4Citation: J. Beringer et al. (Particle Data Group), Phys. Rev. D86, 010001 (2012).