Problem 1 (qiuzhen-qualify 2025-fall)
Section: General Relativity

Consider the following action for a system of a metric field $g_{\mu\nu}(x)$ and an electromagnetic field $A_\mu(x)$ in $3+1$ dimensions:

$$\int d^4x\,\sqrt{-g}\Big( R - \frac{1}{4e^2} F^{\mu\nu}F_{\mu\nu} \Big).$$

(a) From the above action, derive the equation of motion for $A_\mu(x)$.

(b) From the above action, derive the stress tensor $T_{\mu\nu}$ for the electromagnetic field and show that it is conserved.

(c) Derive the Einstein equation for the system by considering the variation $\delta g^{\mu\nu}$ of the inverse metric. (You may use $\delta\sqrt{-g} = -\tfrac{1}{2}\sqrt{-g}\,g_{\mu\nu}\,\delta g^{\mu\nu}$ and $\delta R_{\mu\nu} = \nabla_\sigma\delta\Gamma^{\sigma}_{\mu\nu} - \nabla_\nu\delta\Gamma^{\sigma}_{\mu\sigma}$.)

(d) Originally Einstein wrote an equation like $R_{\mu\nu} \propto T_{\mu\nu}$. Explain why this cannot be right. Also explain why the correct Einstein’s equation makes sense.

Problem 2 (qiuzhen-qualify 2025-fall)
Section: General Relativity

Consider the Schwarzschild metric:

$$ds^2 = -\Big(1 - \frac{2GM}{r}\Big) dt^2 + \Big(1 - \frac{2GM}{r}\Big)^{-1} dr^2 + r^2\,(d\theta^2 + \sin^2\!\theta\, d\phi^2). \quad (1)$$

(a) This is a black hole solution. Where is the horizon? How many Killing vectors does this solution have? Please write them down.

(b) Show that a massless particle moving with $\dot{\theta}=0$ can be reduced to a one-dimensional problem with an effective potential. Write down the effective potential in terms of the energy $E$ and angular momentum $L$ and $r$, and plot it.

(c) What is the radius of the circular orbit? Is it stable?

Problem 3 (qiuzhen-qualify 2025-fall)
Section: Quantum Field Theory

Consider the $\phi^4$ theory with Lagrangian:

$$\mathcal{L} = \tfrac{1}{2}\,\partial^{\mu}\phi\,\partial_{\mu}\phi - \tfrac{1}{2} m^2 \phi^2 - \frac{\lambda}{4!}\,\phi^4.$$

(Here we use the Minkowski metric with signature $(+, -, -, -)$.) Let us study the $2\to2$ scattering amplitude at the next-to-leading order $\mathcal{O}(\lambda^2)$.

(a) Draw all the one-particle irreducible (1PI) Feynman diagrams that contribute to the $2\to2$ scattering amplitude at order $\mathcal{O}(\lambda^2)$. For each diagram, indicate whether it is (i) divergent or finite, and (ii) dependent on the external momenta or a constant.

(b) Using the Feynman rules, write down the integral expression for the scattering amplitude $i\mathcal{M}(p_1 p_2 \to p_3 p_4)$ at order $\mathcal{O}(\lambda^2)$. Express your answer using the variables $s \equiv (p_1+p_2)^2$, $t \equiv (p_1-p_3)^2$ and $u \equiv (p_1-p_4)^2$, where $p_1, p_2$ are incoming external momenta and $p_3, p_4$ are outgoing external momenta.

(c) The divergent part of the amplitude is most clearly isolated using dimensional regularization ($d=4-\epsilon$). The divergent piece is a constant, independent of the external momenta. Calculate this constant at order $\mathcal{O}(\lambda^2)$.

(d) The total $\mathcal{O}(\lambda^2)$ amplitude must be made finite through renormalization. This requires introducing a counterterm $\delta\mathcal{L}$ from the renormalization of the coupling constant $\lambda$. Write down the relevant counterterm Lagrangian $\delta\mathcal{L}_\lambda$ needed to cancel the divergences found in the 1-loop.

Problem unknown (qiuzhen-qualify 2025-fall)

amplitude. Draw the corresponding Feynman diagram for this counterterm and write its contribution to $iM$.

Problem (e) (qiuzhen-qualify 2025-fall)

After renormalization, the physical, finite coupling constant $\lambda_{\text{phys}}$ is defined at a specific renormalization scale $\mu^2$. Briefly explain the purpose of this scale and how it appears in the renormalized amplitude.