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Physics 234C Lecture Notes

Jordan Smolinsky [email protected] Department of Physics & Astronomy, University of California, Irvine, ca 92697 Abstract These are lecture notes for Physics 234C: Advanced Elementary Physics as taught by Tim M.P. Tait during the spring quarter of 2015. This is a work in progress, I will try to update it as frequently as possible. Corrections or comments are always welcome at the above email address.

1 Goldstone

Goldstone’s Theorem states that there is a massless field for each spontaneously broken generator of a global . Rather than prove this, we will take it as an axiom but provide a supporting example. Take the Lagrangian given below:

N   X 1 1 λ 2 L = (∂ φ )(∂µφ ) + µ2φ φ − (φ φ ) (1) 2 µ i i 2 i i 4 i i i=1 This is a theory of N real scalar fields, each of which interacts with itself by a φ4 coupling. Note that the mass term for each of these fields is tachyonic, it comes with an additional − sign as compared to the usual real scalar field theory we know and love. This is not the most general such theory we could have: we can imagine instead a theory which includes mixing between the φi or has a nondegenerate mass spectrum. These can be accommodated by making the more general replacements

X 2 X µ φiφi → Mijφiφj i i,j (2) X 2 X λ (φiφi) → Λijklφiφjφkφl i i,j,k,l but this restriction on the potential terms of the Lagrangian endows the theory with a rich structure. The theory enjoys an O(N) symmetry, which we can see most clearly by repackaging the φi into a column vector

T Φ ≡ [φ1, ..., φN ] (3) The Lagrangian then takes the form

1 1 λ 2 L = ∂ ΦT  (∂µΦ) + µ2ΦT Φ − ΦT Φ (4) 2 µ 2 4 If we now transform the fields Φ → RΦ for some R ∈ O(N), we have

1 1 λ 2 L → ∂ (RΦ)T  (∂µ(RΦ)) + µ2(RΦ)T (RΦ) − (RΦ)T (RΦ) 2 µ 2 4 1 1 λ 2 = ∂ ΦT  RT R (∂µΦ) + µ2ΦT RT RΦ − ΦT RT RΦ (5) 2 µ 2 4 1 1 λ 2 = ∂ ΦT  (∂µΦ) + µ2ΦT Φ − ΦT Φ = L 2 µ 2 4 Notice also that the Lagrangian is invariant under the transformation Φ → −Φ, because each term is even in the fields. Such a transformation is isomorphic to the of the Z2. The symmetry group O(N) has N(N −1)/2 generators, so by Noether’s theorem there is an equal number of conserved currents.

µ µ T a Ja = ∂ Φ T Φ (6)

where T a indicates the ath generator of O(N). Because we want to perturb about the vacuum state, we will redefine our fields: φi(x) = πi(x) + hφii, where the hφii are the expectation values of the fields in the theory’s lowest energy state, which are not necessarily zero. To find out what these expectation values are, we can perform a Legendre transformation to find the Hamiltonian density of the theory, then minimize that with respect to the fields. The Hamiltonian density is

1 1 λ 2 H = ∇ΦT · ∇Φ − µ2ΦT Φ + ΦT Φ (7) 2 2 4 To minimize the energy of a field configuration we want ∇Φ to vanish, meaning that the vacuum state is uniform in spacetime. This tells us that we are indeed justified in making the above field redefinition because adding a constant to the field will not change the measure of integration in the path integral. The other thing we notice about this Hamiltonian density is that the potential contains a minimum for a nonzero value of the fields. This potential, containing a tachyonic mass term and a is known colloquially as a mexican hat potential. Examples for N = 1 and N = 2, shown in Figure (1), should make it clear how it earned this moniker.

a. b.

Figure 1: Mexican Hat Potentials for (a) N = 1 and (b) N = 2

In order to find the minima we set the first derivative of the potential with respect to the fields to 0.

∂V = −µ2φ + λ (φ φ ) φ = 0 ∂φ i j j i i (8) µ2 ⇒ φ φ = ≡ v2 j j λ Notice here that we really must have a positive-definite λ, because otherwise there would be no stable local minima in the potential! There an infinite number of valid vacuum states for N ≥ 2, which reflects the O(N) symmetry. This allows us to pick a particular vacuum state that will make the math easier without loss of generality. The most convenient one is

2 hΦiT = [0, ..., 0, v] (9)

where the first N − 1 hφiis are set to 0, and hφN i = v. We can plug this in to our field redefinition to find

T Φ(x) = [π1(x), ..., πN−1(x), v + πN (x)] (10)

It is conventional to denote the last field, πN , as σ. Now let us plug this into the Lagrangian.

1 µ 1 µ 1 2 2 2 L = (∂µπi)(∂ πi) + (∂µσ)(∂ σ) + µ πi + (v + σ) 2 2 2 (11) λ − π2π2 + (v + σ)4 + 2π π (v + σ)2 4 i j i i The first two terms are the usual kinetic terms for a real scalar field. If we expand the potential and collect terms, we obtain

1 λ 1 λ  1 λ  L ⊃ µ2v2 − v4 + σ µ22v − 4v3 + σ2 µ2 − 6v2 2 4 2 4 2 4 1 λ   λ   λ  +π2 µ2 − 4v2 + σ3 − 4v + π2σ − 4v (12) i 2 4 4 i 4 λ λ λ − π2π2 − σ4 − π2 2σ2 4 i j 4 4 i

Let’s simplify this a little further. If we plug v = pµ2/λ the term linear in σ vanishes, as does the term quadratic in πi. We are left with

1 1 λ L = (∂ π )(∂µπ ) + (∂ σ)(∂µσ) − µ2σ2 − π2π2 2 µ i i 2 µ 4 i j (13) λ λ 1 µ4 − σ4 − λvσ3 − λvπ2σ − π2σ2 − 4 i 2 i 2 λ There are several interesting things to notice about this Lagrangian

• The only mass term appearing in this Lagrangian has the right sign, we have eliminated the . • There is no term linear in σ. • We have generated a term cubic in σ, which breaks the Z2 symmetry. • We have generated a cosmological constant (the last term). • The theory remains invariant under O(N −1). We say that O(N) is spontaneously broken to O(N −1). • m2 = 0 πi These last three points are worth exploring in more detail. Usually when we find a term with no dependence on the fields in a Lagrangian, we ignore it, because it contributes a constant to the action. Recall, however, that this only holds if we are working with a static spacetime metric. If we include a dynamical spacetime, we have also the Einstein-Hilbert action, so that the complete action is (in Planck units)

3 Z √  1  S = d4x −g (R − 2Λ) + L (14) 16π M

µ where g is the determinant of the spacetime metric gµν , R ≡ Rµ is the Ricci scalar, and LM is the Lagrangian governing the fields living in the spacetime. I have temporarily indulged relativists by switching to their sign convention. The constant Λ is known as the cosmological constant, and we can absorb any constant terms appearing in the matter Lagrangian into Λ. Varying this action with respect to the metric gives us the Einstein Equations

1 R − Rg + Λg = 8πT (15) µν 2 µν µν µν where T is the stress-energy tensor. We can see from this that the constant term which appeared in our matter Lagrangian does in fact affect the dynamics of the spacetime. After that little detour into relativity, let’s return to the subject of this section: Goldstone’s theorem. The O(N) symmetry has broken to O(N − 1). Where before we had N(N − 1)/2 generators of a global symmetry, we now have (N − 1)(N − 2)/2, which leaves us with N − 1 broken generators. As we see, we are also left with N − 1 massless scalar fields πi. Goldstone’s theorem works! We may at this point want to take a step back from the more abstract mathematics of field redefinition and doing the algebra dance with Lagrangians to think about the physics of what we have accomplished. We want to think about as fluctuations in a field. Looking at the mexican hat potential we see that if we pick any point around the brim, creating an excitation moving radially costs us potential energy to move up and down the valley. This is an excitation in the massive σ field. On the other hand, creating excitations which move along the brim costs no potential energy, corresponding to the massless πi fields.

2 Spontaneous in Gauge Theories

Goldstone’s theorem can be evaded by theories which spontaneously break local symmetries rather than global symmetries. This is commonly known as the , and its most relevant and successful application is to the electroweak theory. As before we will work through some other simpler models before tackling this important building block of the .

Scalar QED Scalar QED is a U(1) where the gauge couples to a complex scalar field Φ, rather than the of regular QED. Its Lagrangian is

1 λ L = − F µν F + (D Φ)∗ (DµΦ) + µ2|Φ|2 + |Φ|4 (16) 4 µν µ 2

where Dµ is the gauge covariant derivative Dµ ≡ ∂µ + ieAµ. As its name implies, this theory has a U(1) gauge symmetry

Φ → eiα(x)Φ 1 (17) A → A − ∂ α(x) µ µ e µ but the potential terms for Φ contain a tachyonic mass term which will break the symmetry. The potential is minimized at the expectation value hΦi = v ≡ pµ2/λ, as we would expect. This spontaneously breaks the U(1) symmetry so we would expect a single massless scalar, which we can see by shifting our field variable 4 1 Φ = v + √ (σ(x) + iφ(x)) (18) 2 and plugging this in to the potential gives us

1  λv  λ V (Φ) → C + 2λv2 σ2 + √ σ3 + σφ2 + σ4 + φ4 + 2φ2σ2 2 2 8 (19)  λv  λ = µ2σ2 + √ σ3 + σφ2 + σ4 + φ4 + 2φ2σ2 2 8 As we can see there is no mass term for φ, so it is the , and σ now has a “good” mass term. We can also examine how this field redefinition affects the gauge-covariant kinetic term

1  1  DµΦ → √ (∂µσ + i∂µφ) + ieAµ v + √ (σ + iφ) 2 2 1 1 √ |∂ Φ|2 → (∂ σ)(∂µσ) + (∂ φ)(∂µφ) + e2v2A Aµ + 2evA Aµσ (20) µ 2 µ 2 µ µ µ 1 √ + e2A2σ2 + 2ev (∂ φ) Aµ + e (∂ φ) Aµ − e (∂ σ) Aµφ 2 µ µ µ This has given us a mass term for the , with mass 2e2v2. There is also a suite of interaction vertices between the gauge boson and the scalar fields, including two which seem problematic for calculations: mixing between the scalars and A. This can be fixed with a different clever field redefinition

σ(x)  Φ(x) = eiπ(x)/V √ + v (21) 2 This will still satisfy the vacuum criterion in that the potential is minimized for hσi = 0, with |Φ| = µ2/λ. With this new definition we associate the Goldstone boson with the field π. We can exploit our U(1) gauge freedom to “gauge away” π by the choice α(x) = −π/v. This choice is called “unitary gauge”, and because all physical quantities must be independent of our choice of gauge we know that the field π(x) must be unphysical. With these choices the Lagrangian becomes really simple.

1 1 1 1 λ L → − F F µν + M 2 A Aµ + (∂ σ)(∂µσ) − M 2σ2 + e2σ2A Aµ + 2e2vσA Aµ − σ4 + 2vλσ3 4 µν 2 A µ 2 µ 2 σ µ µ 2 (22)

Now there are no cross terms between π and A, because there is no longer any π in this Lagrangian! π is really just a gauge degree of freedom. We are left with normal kinetic terms for A and σ with non- pathological masses and totally normal-looking interactions. There is one new feature here worth noting: a vertex connecting a single σ with two A fields. This anticipates the h → WW decay in the electroweak theory, the observation of which tells our experimental chums at the LHC that what they have found is indeed (consistent with) the Standard Model Higgs. One more thing to note is that there is no in this theory. π is a “would-be Goldstone boson” (technical jargon). If the theory had only a global symmetry instead of a local gauge one π would be a Goldstone because we could not have eliminated π from the Lagrangian by a gauge choice. We didn’t have to choose this gauge, and it indeed may be convenient to do calculations in a different gauge, but of course all calculations should be equivalent to those done with this Lagrangian, and this Lagrangian just plain looks nicer anyway. Before moving to the electroweak theory we will consider a few more examples in less detail. 5 BCS In the BCS model of superconductivity lattice vibrations induce an attractive potential on such that they form bound states called “Cooper pairs” of two electrons. These cooper pairs have a negative mass-squared term leading to a nonzero . Inside a superconductor, couple to electrons and thus couple to these Cooper pairs, so the vev causes photons to acquire a mass.

Non-Abelian Gauge Theory Working our way closer to the full electroweak theory, consider a theory which is invariant under the in- finitesimal gauge transformation

a a Φi → (1 + iα T )ij Φj (23)

We can rewrite the Φ as a set of real fields and turn the gauge group generators into

a a Tij = itij (24)

where the ta are real, antisymmetric matrices. The covariant derivative is now

a a Dµ = ∂µ + gAµt (25)

When we write the φ fields in terms of deviations from the vev its gauge-covariant kinetic term will contain a mass term for the gauge fields

1 µ 1 2 a bµ a b  (DµΦ) (D Φ) ⊃ g AµA (t hΦi)i t hΦi i 2 2 (26) 1 = m2 Aa Abµ 2 ab µ

2 where we now have a mass matrix mab defined as

2 2 a b  mab ≡ g (t hΦi)i t hΦi i (27)

The diagonal entries of this matrix are positive semi-definite so we know that all of the eigenvalues are nonnegative. The gauge symmetry generators for which tahΦi = 0 correspond to the unbroken sector of the symmetry and have massless gauge bosons. A non-abelian model of particular interest is the Georgi-Glashow model, where the gauge symmetry is SU(2). Let us examine a Higgs real scalar triplet Φa of SU(2) in the adjoint representation. The gauge covariant derivative is

a a abc b c (DµΦ) = ∂µΦ + g AµΦ (28)

where abc are the structure constants of SU(2). The Lagrangian of the theory is

1 1 L = − F a  (F µνa) + (D Φ)a (DµΦ)a − V (Φ) (29) 4 µν 2 µ 6 Note that the Higgs triplet has a potential that we assume will induce a vev, and we can gauge the vev such that it takes the form

hΦai = δa3v (30)

As discussed above, the gauge-covariant kinetic term will induce a mass term for the gauge bosons of the form

1 1 g2v2  Ab   Adµ = g2v2 A1 Aµ1 + A2 Aµ2 (31) 2 ab3 µ ad3 2 µ µ There are now two massive gauge bosons out of the original three, leaving A3 massless. This single remaining gauge boson corresponds to the unbroken U(1) symmetry of the theory. We say that SU(2) is spontaneously broken to U(1). Georgi and Glashow identified the remaining gauge boson correctly with the and the massive bosons correspond to the W ± bosons of the correct electroweak theory, but unfortunately this model does not allow for flavor-changing neutral currents, which the electroweak theory contains in the form of the Z boson. The correct electroweak theory was found by Glashow independently from Weinberg and Salam and earned him a share of the 1979 Nobel Prize. It is perhaps lucky for the other prize winners that Georgi picked the wrong project on which to collaborate with Glashow, because the Nobel cannot be shared with more than three people.

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