What is the instantaneous acceleration (2nd derivative of time) of the field? Sol. The Euler-Lagrange equations quickly yield:

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1 PHYSICS 75: The Standard Model Midter Exa Solution Key. [3 points] Short Answer (6 points each (a In words, explain how to deterine the nuber of ediator particles are generated by a particular local gauge syetry. (E.g., there are 3 weak force ediators. Why that nuber? Each syetry has a nuber of degrees of freedo, each corresponding to a generator. For SU(2 there are 3 (2 2, and siilarly, there are 8 for SU(3. Each corresponds to a unique ediator for the corresponding force. (b Knowing that the energy density of an electroagnetic field is 2 E2 + 2 B2, and looking at the Lagrangian, what are the units of the 4-potential, A µ in ters of [Energy] n? Don t just give an answer. Please argue fro the relationship between E-field and potential. The units of E are clearly: [Energy] 2. The E- and B- fields are generated via: E µ A ν so [A µ ] [Energy] 2 [T ] [Energy] (c Consider a particle oving with the 4-oentu: p µ = 8 GeV What is the ass of the particle? Recall: 2 = p p = ( GeV 2 so = 6GeV (d In the previous part, what happens to the quantity, p p in a frae that is boosted by v =.8 in the x-direction? N othing. That s kind of the point. (e In class, we encountered the Higgs potential for a scalar field: L = 2 µφ µ φ µ 2 φ 2 + λφ 4 What is the instantaneous acceleration (2nd derivative of tie of the field? The Euler-Lagrange equations quickly yield: µ µ φ = 2µ 2 φ + 4λφ 3 φ = 2 φ 2µ 2 φ + 4λφ 3

2 2. [3 points] Consider a universe consisting of two real valued scalar fields, φ and with the total Lagrangian: L = L + L 2 + L int where L and L 2 are the free-field Lagrangians of the two fields: L = 2 µφ µ φ 2 2 φ 2 and L 2 = 2 µ µ There is also an interaction Lagrangian: L int = λφ 2 2 (a Copute the Euler-Lagrange equations and the general for of the solution for the free φ field. We ve done this any ties, so there s no need to go through the steps again. The Euler-Lagrange equations are: which have the generalized solution: ( µ µ + 2 φ = φ = p a p exp( ip x + a p exp(ip x such that: p p = 2 (b Copute the coponent of the stress-energy tensor for the φ and free fields. Again, we have done these any ties. Starting with: T µν = ( µ φ ν φ g µν L we get : and siilarly, φ = 2 φ φ φ φ 2 = (c Copute the coponent of the stress-energy tensor for the interaction Lagrangian. Clearly, in this case, \ ( µ φ = and siilarly for. int = L int = λφ 2 2 (d What (if anything can you say about the electrical charges of the φ and particles? They are real-valued and thus electrically neutral.

3 (e The interaction Lagrangian can be represented by a fundaental Feynan Vertex. Please draw it, labeling the φ and particles. λ φ φ There really shouldn t be any arrows on any of this, since there s no distinction between a φ and φ. (f E.C. For the final part, consider a coposite vector coposed of: ( φ ψ = I d like to consider the possibility that there is a transforation atrix of the for: ( a b M = c a Such that the transforation: leaves: ψ T ( ψ Mψ ψ = 2 φ reains unchanged. Give at least one non-trivial atrix (e.g. the identity atrix doesn t count for which the ass ters reain invariant under the transfor. You can ultiply it out, but the upshot is that the generalized for of the transforation atrix can be written as: ( cos θ 2 M = sin θ 2 sin θ cos θ which has deterinant ( Special!, and the generator: ( i 2 X = i 2 Soething like this will turn out to be iportant when thinking about neutrino oscillation. 3. [2 points] The SO(3 syetry (rotation in 3-d has the generators: i X = i ; X 2 = ; X 3 = (a Copute the coutator of [X, X 2 ]. Yes, I just want you to do a bit of atrix ultiplying: X X 2 = i i = i i

4 and X 2 X = i [X, X 2 ] = i = = ix 3 (b As you ll recall, the eleents of Lie group can be coputed fro the generators via: M = exp( iθ i X i For a rotation in the 3 direction only, and recalling: e x x n = n! expand the atrix M to first order in θ 3. (c Now consider a vector in this 3-d space: n= V i = Applying a sall rotation, θ 3, copute the st order estiate of the transfored vector. The transfor is: θ 3 M = I iθ 3 X 3 = θ 3 V i (d In a short sentence (or an equation if you like, what about the vector (or other 3-d vectors is going to be conserved in an SO(3 transforation? It s just a rotation, so: V V will be a conserved quantity on all SO(3 rotations. 4. [2 points] Consider the Lagrangian of a free, charged, assive vector field such as ight describe the W ± particles: θ 3 L = 2 F µνf µν + M 2 A µ A µ It s close to the description of the W fields, but not exact. We ll see later in the course what the corrections are. As with scalar fields (which were treated siilarly, we can treat the unstarred and starred fields independently for the purpose of partial derivatives. Recalling that: if α = µ and β = ν, and - if α = ν and β = µ, F µν ( α A β =

5 (a Copute ( α A β We re going to use this a lot, but we ve already done it (nearly on the hoework: αβ ( α A = F β (b Fro there, copute the Euler-Lagrange equations for the free field A-particle. For a couple of points of extra credit, siplify this expression in Lorentz Gauge. We quickly get: Writing this out: Lorentz gauge yields: and thus we get: which I think you ll agree, is uch tidier. (c Consider the global U( transforation: and siilarly: Copute δa µ for a sall change in θ. One step: and thus: β F αβ = M 2 A α β α A β + β β A α = M 2 A α β A β = A µ = M 2 A µ A µ A µ exp( iθ A µ A µ exp(iθ A µ ( iθa µ δa µ = ia µ (d Use the previous result to copute the conserved current in the field associated with a global U( transforation. This, naturally, will look siilar to what we saw with scalar fields: J µ = ( µ A ν δaν + ( µ A ν δaν = F µ ν ( ia ν F µ ν(ia ν = i [F µν A ν F µν A ν]