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Physics 234C Lecture Notes

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Physics 234C Lecture Notes 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 Particle 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 Bosons Goldstone's Theorem states that there is a massless scalar field for each spontaneously broken generator of a global symmetry. 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 2 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 action of the group 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 = rΦT · rΦ − µ2ΦT Φ + ΦT Φ (7) 2 2 4 To minimize the energy of a field configuration we want rΦ 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 quartic interaction 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 tachyons. • 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 p 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 Field 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 particles 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 Symmetry Breaking 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 Higgs mechanism, 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 Standard Model. Scalar QED Scalar QED is a U(1) gauge theory where the gauge boson couples to a complex scalar field Φ, rather than the fermions of regular QED. Its Lagrangian is 1 λ L = − F µν F + (D Φ)∗ (DµΦ) + µ2jΦj2 + jΦj4 (16) 4 µν µ 2 where Dµ is the gauge covariant derivative Dµ ≡ @µ + ieAµ.
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