LLMs Can Tackle High Energy Theoretical Physics, yet Gaps Remain: Part I, Quantum Field Theory
Paper • 2610.33134 • Published
Datasets:
num int64 | question string | category string | reference_answer string | reference_solution string | specific_requirement string | reference_answer_alt list | numeric_tolerance float64 |
|---|---|---|---|---|---|---|---|
1 | Consider gamma matrices in the chiral representation using the Pauli matrices
$$
\gamma^\mu\equiv\begin{pmatrix}0&\sigma^\mu\\\overline{\sigma}^\mu&0\end{pmatrix},\quad \text{with}\quad\sigma^\mu\equiv(1,\sigma^i),\quad\overline{\sigma}^\mu\equiv(1,-\sigma^i).
$$
What is the value of $\mathrm{tr}(\overline{\sigma}^{\mu... | Classical | $2(\eta^{\mu\nu}\eta^{\rho\sigma}-\eta^{\mu\rho}\eta^{\nu\sigma}+\eta^{\mu\sigma}\eta^{\nu\rho}+i\epsilon^{\mu\nu\rho\sigma})$ | One can check that
$$
\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma=\begin{pmatrix}\sigma^\mu\overline{\sigma}^\nu\sigma^\rho\overline{\sigma}^\sigma&0\\0&\overline{\sigma}^\mu\sigma^\nu\overline{\sigma}^\rho\sigma^\sigma\end{pmatrix}.
$$
Taking the trace of this equation provides a first identity:
$$
\mathrm{tr}(\sigma... | When judging the answer of this question, the Levi-Civita term may also be written with lower indices as $-2i\epsilon_{\mu\nu\rho\sigma}$, i.e. $2(\eta^{\mu\nu}\eta^{\rho\sigma}-\eta^{\mu\rho}\eta^{\nu\sigma}+\eta^{\mu\sigma}\eta^{\nu\rho}-i\epsilon_{\mu\nu\rho\sigma})$ is also correct. This is equivalent to $+2i\epsil... | null | null |
2 | Consider 4-dimensional Yang-Mills theory $\mathcal{L}_{\mathrm{YM}}=-\frac{1}{2}\operatorname{tr}(F^{2})=-\frac{1}{4}F^{a\mu\nu}F_{\mu\nu}^{a}$ in the temporal gauge $A_0=0$. Based on that, calculate the value of
$$
\partial_0 F_a^{0i}+\left[D_j,F^{ji}\right]_a .
$$
Here $0$ means the time direction and Latin indices m... | Classical | $0$ | By using the temporal gauge $A^0=0$, we circumvent the problem that $A_{a}^{0}$ has a vanishing conjugate momentum, because $A^{0}$ is not a dynamical variable in this gauge. Regarding the other components of the gauge field, the conjugate momentum $\Pi^{i}$ of $A^{i}$ is given by
$$
\Pi_{a}^{i}\equiv\frac{\partial\mat... | null | null | null |
3 | Consider a non-Abelian gauge field $A_\mu^a$ in $d$-dimensional spacetime, with action
$$
S = \int \mathrm{d}^d x\, \left( -\frac{1}{4} F_{\mu\nu}^a F^{a\,\mu\nu} + A_\mu^a j^{a\,\mu} \right),
$$
where
$$
F_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + g f^{abc} A_\mu^b A_\nu^c
$$
and $f^{abc}$ are the str... | Classical | $\frac{4-d}{2}$ | The action is dimensionless and $[\mathrm{d}^dx]=-d$, so the Lagrangian density has mass dimension $d$. The kinetic term $(\partial_\mu A_\nu^a)^2\subset F^a_{\mu\nu}F^{a\,\mu\nu}$ then fixes the dimension of the gauge field:
$$
2\,(1+[A])=d \implies [A]=\frac{d-2}{2}.
$$
Now consider the non-Abelian term in $F_{\mu\nu... | null | null | null |
4 | Consider the Lagrangian in 4 spacetime dimensions:
$$
\mathcal{L} = -i\bar{\psi}\gamma^\mu\partial_\mu \psi+\lambda\left((\bar\psi\psi)^2-(\bar\psi \gamma_5\psi)^2\right)
$$
where $\psi$ is a spinor and $\lambda$ is a constant. Here $\bar{\psi} = \psi^\dagger \gamma_0$. Requiring that the Lagrangian is Lorentz invarian... | Classical | yes | Yes. If $\psi$ transforms as a Dirac spinor, $\psi\to\Lambda_{1/2}\psi$ with $\Lambda_{1/2}=\exp(-\tfrac{i}{2}\omega_{\mu\nu}S^{\mu\nu})$, $S^{\mu\nu}=\tfrac{i}{4}[\gamma^\mu,\gamma^\nu]$, then $\bar\psi\to\bar\psi\Lambda_{1/2}^{-1}$. The kinetic term is Lorentz invariant, and $\bar\psi\psi$ is a scalar. Since $\gamma_... | null | null | null |
5 | Consider the Lagrangian in 4 spacetime dimensions:
$$
\mathcal{L} = -i\bar{\psi}\gamma_\mu\partial^\mu \psi+\lambda\left((\bar\psi\psi)^2-(\bar\psi \gamma_5\psi)^2\right)
$$
where $\psi$ is a spinor and $\lambda$ is a constant. Here $\bar{\psi} = \psi^\dagger \gamma_0$. Consider the $\psi\psi\to\psi\psi$ scattering pro... | Classical | $8\lambda^2(t^2+u^2)$ | Consider the interaction Hamiltonian $H_{\text{int}} = -\lambda[(\bar{\psi}\psi)(\bar{\psi}\psi)-(\bar{\psi}\gamma_5\psi)(\bar{\psi}\gamma_5\psi)]$. The following contractions will contribute to the process:
$$
\langle e(p_3)e(p_4)|:\bar{\psi}\psi\bar{\psi}\psi:|e(p_1)e(p_2)\rangle-\langle e(p_3)e(p_4)|:\bar{\psi}\gamm... | When judging the answer of this question, equivalent forms obtained using $s+t+u=0$ are also correct. | null | null |
6 | Consider the action of a free massive complex scalar field in four-dimensional spacetime with the Lorentzian metric
$$
S=\int \mathrm{d}^{4} x\left[\partial_{\mu} \phi^{*} \partial^{\mu} \phi-m^{2} \phi^{*} \phi\right].
$$
The mode decomposition of the field operator reads
$$
\hat{\phi}(x)=\int \frac{\mathrm{d}^{3} \ve... | Quantization | $$
\hat{\Pi}(x)=-\frac{i}{2} \int \frac{\mathrm{d}^3 \vec{p}}{(2\pi)^3} \left[\hat{b}(\vec{p}) e^{-ip \cdot x}-\hat{a}^{\dagger}(\vec{p}) e^{i p \cdot x} \right]
$$ | The mode decomposition of the field operator reads
$$
\hat{\phi}(x)=\int \frac{\mathrm{d}^3 \vec p}{(2\pi)^3 2\omega_{\vec p}} \left[\hat{a} (\vec p) e^{-i p \cdot x} + \hat{b}^{\dagger} (\vec p) e^{i p \cdot x} \right], \quad \text{where} \quad p \cdot x= \omega_{\vec p}\, t - \vec p \cdot \vec x
$$
Then it immediatel... | null | null | null |
7 | Consider the action of a free massive complex scalar field in four-dimensional spacetime with the Lorentzian metric
$$
S=\int \mathrm{d}^{4} x\left[\partial_{\mu} \phi^{*} \partial^{\mu} \phi-m^{2} \phi^{*} \phi\right].
$$
The mode decomposition of the field operator reads
$$
\hat{\phi}(x)=\int \frac{\mathrm{d}^{3} \ve... | Quantization | $(2\pi)^3 2 \omega_{\vec p_1} \delta^{(3)} (\vec p_1 - \vec p_2)$ | We write $\hat\phi=(\hat\phi_1+i\hat\phi_2)/\sqrt2$ in terms of two independent real scalar fields, as in the previous problem, so that the ladder operators are related by
$$
\hat{a}(\vec p)=\frac{\hat{a}_1 (\vec p)+i \hat{a}_2 (\vec p)}{\sqrt{2}}, \qquad \hat{b}(\vec p)=\frac{\hat{a}_1 (\vec p)-i \hat{a}_{2}(\vec p)}{... | null | null | null |
8 | Consider the action
$$
S= \int d^4x \, \left( -\frac{1}{4} F_{\mu\nu} F^{\mu \nu}-\frac{1}{2}m^2A_\mu A^\mu\right)
$$
where $F_{\mu \nu} = \partial_\mu A_\nu-\partial_\nu A_\mu$ and $m$ is the mass of the vector field. The metric is $\eta_{\mu \nu}=\mathrm{diag}(-1,+1,+1,+1)$. The momentum-space Feynman propagator $\ti... | Quantization | $-\frac{i}{p^2+m^2} \left(\eta^{\nu \rho} +\frac{p^\nu p^\rho}{m^2} \right)$ | Considering $F_{\mu\nu} F^{\mu \nu} = 2(\partial_\mu A_\nu \partial^\mu A^\nu-\partial_\mu A_\nu \partial^\nu A^\mu)$, the action can be rewritten (after integrating by parts) as
$$
S=\int d^4x \,\left[-\frac{1}{2}(\partial_\mu A_\nu \partial^\mu A^\nu-\partial_\mu A_\nu \partial^\nu A^\mu)-\frac{1}{2}m^2A^\mu \eta_{\m... | null | null | null |
9 | Consider the path integral of a gauge invariant action:
$$
\int \mathcal{D}A\, e^{i\int \mathrm{d}^4x\, \mathcal{L}(A(x),x)}
$$
in which $A$ is the gauge field. Let $G(A(x),\lambda(x))-C(x)$ be a gauge fixing equation and $\lambda$ be the gauge parameter. We perform the Faddeev-Popov trick and insert
$$
1=\Delta_{\text... | Quantization | $\mathcal{L}_{gauge-fixed} = \mathcal{L}(A(x),x)-\frac{\xi}{2}G(A(x))^2$ | Here $G(A(x),\lambda(x))\equiv G(A^\lambda(x))$ denotes the gauge-fixing function evaluated on the gauge-transformed field $A^\lambda$. Inserting the identity into the path integral gives
$$
\int \mathrm{d}\lambda\int \mathcal{D}A\, \Delta_{\text{FP}}[A]\, \delta\big(G(A^\lambda(x))-C(x)\big)\, e^{i\int \mathrm{d}^4x\,... | null | null | null |
10 | Consider the action
$$
S = \int d^4x \,(i \bar{\psi} \gamma^\mu \partial_\mu \psi -m\bar{\psi}\psi)
$$
where $\bar{\psi}=\psi^\dagger\gamma_0$ and $m$ is the real mass of the fermion. Consider the transformation $\psi \to \psi'=e^{i\alpha} \psi$ with $\alpha$ a constant. Promote $\alpha \to \alpha(x)$; the Noether curr... | QED | $-\bar{\psi} \gamma^\mu \psi$ | $\psi \to \psi'=e^{i\alpha} \psi$ means $\bar{\psi} \to \bar{\psi}'=\bar{\psi}e^{-i\alpha}$, so for constant $\alpha$ the action is invariant. Now promote $\alpha\to\alpha(x)$. The mass term $\bar\psi\psi$ is still invariant, and in the kinetic term the only change comes from the derivative acting on the phase:
$$
i\ba... | null | null | null |
11 | From the action
$$
S = \int d^4x \,(i \bar{\psi} \gamma^\mu \partial_\mu \psi -m\bar{\psi}\psi)
$$
where $\bar{\psi}=\psi^\dagger\gamma_0$ and $m$ is the real mass of the fermion, consider the transformation
$$
\psi \to \psi'=e^{i\alpha\gamma_5}\psi
$$
with $\alpha$ a constant and $\gamma_5=i\gamma_0\gamma_1\gamma_2\ga... | QED | 0 | From $\psi \to \psi'=e^{i\alpha\gamma_5} \psi$ we have
$$
\psi^\dagger\to \psi^{\dagger\prime} = (e^{i\alpha \gamma_5}\psi)^\dagger=\psi^\dagger e^{-i\alpha\gamma_5}
$$
since $\gamma_5=\gamma_5^\dagger$. Using $\gamma_5^2=1$ and $\{\gamma_5,\gamma^0\}=0$,
$$
\begin{aligned}
\bar{\psi}' &= \psi^{\dagger\prime} \gamma^0=... | null | null | null |
12 | Consider the following theory describing the cubic interaction of a real field $\phi$ with a complex scalar $\chi$, with action
$$
S = \int d^4x \,\left[\frac{1}{2}(\partial_\mu \phi)^2-\frac{1}{2}m_\phi^2 \phi^2+|\partial_\mu \chi|^2-m^2_\chi |\chi|^2-g\phi |\chi|^2\right]
$$
where $g$ is a real parameter with mass di... | QED | $\mathcal{A}=-\frac{ig}{\sqrt{8V^3\omega_{\vec{r}}\omega_{\vec{p}}\omega_{\vec{q}}}}(2\pi)^4\delta^{(4)}(p+q-r)$ | The evolution operator at leading order reads
$$
U(t,t_0)=1-i\int_{t_0}^{t} dt' \, H_{\text{int}}+\cdots
$$
with $H_{\text{int}}=g\int d^3x\,\phi|\chi|^2$. The transition amplitude from the initial state $|i\rangle$ to the final state $|f\rangle$ at first order is therefore $-i\langle f|\int dt \, H_{\text{int}}|i\rang... | null | null | null |
13 | Consider the QED Lagrangian
$$
\mathcal{L} = -\frac{1}{4}F_{\mu \nu}F^{\mu \nu} +\bar{\psi}(i\slashed{\partial}-m)\psi -e\bar{\psi}\slashed{A}\psi
$$
where $\psi$ is a Dirac fermion with mass $m$ and $A$ is a massless photon, with
$$
\psi(x)=\int_p \sum_i \left(u_i(\vec{p}) a_i(\vec{p}) e^{-ipx}+v_i(\vec{p}) b_i^\dagge... | QED | $(2\pi)^4\delta^{(4)}(k-p-q)\, e\, \bar{v}_{s_2}(\vec{q}) \gamma^\mu \epsilon_{\mu,r}\, u_{s_1}(\vec{p})$ | We have
$$
\langle f|\int dt\, H_{\text{int}}|i\rangle = e\int d^4x \, \langle 0| c_r(\vec{k})\, \bar\psi(x) \gamma_\mu A^\mu(x) \psi(x)\, a^\dagger_{s_1}(\vec{p})b^\dagger_{s_2}(\vec{q})|0\rangle .
$$
Here $\int_p$ denotes the Lorentz-invariant measure, so that the ladder operators obey $\{a_i(\vec p),a_j^\dagger(\vec... | When judging the answer of this question, an additional overall factor $\frac{1}{\sqrt{8E_pE_q\omega_k}}$ is also correct, because the question does not specify the measure of $\int_p$. The question asks for the matrix element, so answering $0$ or stating that it equals $0$ is wrong. | null | null |
14 | Compute the differential cross section $\frac{d\sigma}{d\cos\theta}$ for Bhabha scattering at tree level, $e^+ e^- \to e^+ e^-$. Here we ignore the electron mass. Express the result with the fine-structure constant $\alpha$ and the Mandelstam variables $\{s,t,u\}$. | QED | $\frac{\pi \alpha^2}{s}\left[\frac{t^2}{s^2}+\frac{s^2}{t^2}+u^2\left(\frac{1}{s}+\frac{1}{t}\right)^2\right]$ | At tree level the process consists of two diagrams: the $s$-channel annihilation diagram and the $t$-channel exchange diagram (entering with a relative minus sign). The minus sign before the $t$-channel diagram comes from the exchange of two fermion field operators when contracting with in and out states. In the high e... | null | null | null |
15 | Consider the QED Lagrangian
$$
\mathcal{L} = i\bar{\psi}\gamma^\mu D_\mu \psi-m\bar\psi\psi-\frac{1}{4}F_{\mu\nu}F^{\mu \nu}
$$
where
$$
D_\mu= \partial_\mu-ieA_\mu, \quad F_{\mu \nu}=\partial_\mu A_\nu-\partial_\nu A_\mu.
$$
How many degrees of freedom are propagated by $A_\mu$? | QED | 2 | We start initially with a field with 4 components. However, we have the Gauss law and the gauge redundancy (1 constraint + 1 invariance), giving in total 2 propagating degrees of freedom. | null | null | null |
16 | Consider the Proca Lagrangian
$$
\mathcal{L} = i\bar{\psi}\gamma^\mu D_\mu \psi-m\bar\psi\psi+\frac{1}{2}m^2_A A_\mu A^\mu-\frac{1}{4}F_{\mu\nu}F^{\mu \nu}
$$
where
$$
D_\mu= \partial_\mu-ieA_\mu, \quad F_{\mu \nu}=\partial_\mu A_\nu-\partial_\nu A_\mu.
$$
How many degrees of freedom are propagated by $A_\mu$? | QED | 3 | The equations of motion for the free massive vector read
$$
\partial^\mu F_{\mu \nu}+m^2_A A_\nu=0
$$
from which we obtain the constraint
$$
\partial^\mu A_\mu=0
$$
The equations of motion for the massive vector in the presence of a source term read
$$
\partial^\mu F_{\mu \nu}+m^2_A A_\nu=j_\nu
$$
from which we obtain ... | null | null | null |
17 | Consider the pseudoscalar Yukawa Lagrangian:
$$
\mathcal{L} = \frac{1}{2}(\partial_\mu \phi)^2-\frac{1}{2}m^2 \phi^2+\bar{\psi} (i\slashed{\partial}-M)\psi-ig\bar{\psi} \gamma^5 \psi \phi
$$
where $\phi$ is a real scalar and $\psi$ is a Dirac fermion. Which extra term can we add to make the theory renormalizable?
A. N... | Renormalization | C | Here we consider the superficial degree of divergence $D$. In 4d we have
$$
D = 4L -2I_\phi -I_\psi
$$
where $I_\phi$ is the number of internal lines for the scalar and $I_\psi$ the number of internal lines for the fermion. In a Feynman diagram we have
$$
L = I_\phi+I_\psi-V+1
$$
where $V$ is the number of vertices. Fo... | null | null | null |
18 | The massless Yukawa Lagrangian is written as
$$
\mathcal{L} = \frac{1}{2}(\partial_\mu \phi)^2-\frac{\lambda}{4!} \phi^4+\sum_{j=1}^N\bar{\psi}_j (i\slashed{\partial})\psi_j-\sum_{j=1}^N ig\bar{\psi}_j \gamma^5 \psi_j \phi
$$
where $\phi$ is a real scalar and $\psi_j$ with $j=1,\cdots,N$ are $N$ Dirac fermions coupled ... | Renormalization | $\beta_g(\lambda,g)=\frac{(2N+3)g^3}{(4\pi)^2}$ | The Yukawa vertex factor is $i\times(-ig\gamma^5)=g\gamma^5$, and all $N$ fermions couple identically to $\phi$. The divergent 1-loop diagrams that enter $\beta_g$ are the following (in $d=4-\epsilon$). First, the fermion-loop contribution to the scalar self-energy, for a single flavour $j$:
$$
[\text{fermion loop on s... | null | null | null |
19 | Consider massless two-dimensional QED, the so-called Schwinger model. The one-loop 1PI photon two-point function is denoted $-i \Pi_{\mu \nu}(p)$; compute it. | Renormalization | $-\frac{ie^2}{\pi p^2}(p_\mu p_\nu -p^2g_{\mu \nu})$ | With the QED vertex $-ie\gamma^\mu$ and the massless fermion propagator $\frac{i\slashed{q}}{q^2}$, the one-loop 1PI two-point function (including the $(-1)$ for the fermion loop) is
$$
-i \Pi_{\mu \nu}(p) = -(-ie)^2 \int \frac{d^D q}{(2\pi)^D} \frac{\text{tr}[\gamma_\mu\, i\slashed{q}\,\gamma_\nu\, i(\slashed{q}-\slas... | null | null | null |
20 | Consider massless two-dimensional QED, the so-called Schwinger model. Find the mass of the photon from the full photon propagator. Hint: consider the vacuum polarization at one loop, $-i \Pi_{\mu \nu}(p)$. | Renormalization | $\frac{e}{\sqrt{\pi}}$ | Write the one-loop 1PI two-point function of the previous problem as
$$
-i\Pi_{\mu\nu}(p)= i\,(p^2g_{\mu\nu}-p_\mu p_\nu)\,\Pi(p^2),\qquad \Pi(p^2)=\frac{e^2}{\pi p^2},
$$
and introduce the transverse projector $P^T_{\mu\nu}=g_{\mu\nu}-\frac{p_\mu p_\nu}{p^2}$, which satisfies $P^T_{\mu}{}^{\rho}P^T_{\rho\nu}=P^T_{\mu\... | null | null | null |
21 | Consider the Lagrangian
$$
\mathcal{L} = -\frac{1}{2} \mathrm{Tr}[G_{\mu \nu} G^{\mu \nu}]+i\bar{\psi}_j(\slashed{D}\psi)_j-m\bar{\psi}_j \psi_j.
$$
Here the fields $\psi$ form a vector in the fundamental $N$-dimensional representation of $SU(N)$ and the invariant form is normalized such that $\mathrm{Tr}\,T_aT_b=\frac... | Non-Abelian | $g \bar{\psi}_j T_a \gamma^\nu\psi_j$ | From $[D_\mu,D_\nu]\psi=igG_{\mu\nu}\psi$ we have $G_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu+ig[A_\mu,A_\nu]$. The fermion kinetic term contains the interaction $i\bar\psi_j\gamma^\mu(igA^a_\mu T_a\psi)_j=-gA^a_\mu\,\bar{\psi}_jT_a\gamma^\mu \psi_j$. The variation with respect to the gauge field is therefore
$$
... | null | null | null |
22 | Consider the pure Yang-Mills Lagrangian
$$
\mathcal{L} = -\frac{1}{4} G_{\mu \nu}^a G^{\mu \nu}_a
$$
where, due to our normalization of the gauge fields, we use the metric $\delta_{ab}$ to raise and lower the indices in the adjoint representation. It is in particular invariant under global gauge transformations. Find t... | Non-Abelian | $G_a^{\mu \nu}A_\nu^c C_{cb}^a$ | For a symmetry of the form
$$
\mathcal{L}(\phi_j+\epsilon\delta\phi_j,\partial(\phi_j+\epsilon\delta\phi_j)) = \mathcal{L}(\phi_j,\partial \phi_j)+\epsilon\partial_\mu K^\mu (\phi_j,\partial \phi_j)+ \mathcal{O}(\epsilon^2)
$$
there is a conserved current
$$
J^\mu = \frac{\partial \mathcal{L}}{\partial(\partial_\mu \ph... | When judging the answer of this question, both overall signs are correct. | [
"-G^{\\mu\\nu}_a \\, C^a_{bc} \\, A_\\nu^c"
] | null |
23 | Consider the Lagrangian for the electroweak interactions in the Standard Model:
$$
\begin{aligned}
\mathcal{L} &= -\frac{1}{4}W^a_{\mu \nu}W^{\mu \nu\,a}-\frac{1}{4}B_{\mu \nu}B^{\mu \nu} +i\sum_j\bar{\psi}_j \slashed{\partial}\psi_j +D_\mu H^\dagger D^\mu H+\mu^2H^\dagger H-\lambda(H^\dagger H)^2 \\
&\quad -\bar{E}_L ... | Electroweak | $\sqrt{2\lambda}\, v$ | $$
\begin{aligned}
V&= -\mu^2 H^\dagger H+\lambda(H^\dagger H)^2=-\frac{\mu^2}{2}(v+h)^2+\frac{\lambda}{4}(v+h)^4 \\
&=\left(-\mu^2+\frac{\lambda v^2}{2}\right) \frac{v^2}{2}+(-\mu^2+\lambda v^2)vh+(-\mu^2+3\lambda v^2)\frac{h^2}{2}+\lambda vh^3+\frac{\lambda}{4}h^4
\end{aligned}
$$
Considering $\frac{\partial V}{\part... | null | null | null |
24 | Consider the Lagrangian for the electroweak interactions in the Standard Model:
$$
\begin{aligned}
\mathcal{L} &= -\frac{1}{4}W^a_{\mu \nu}W^{\mu \nu\,a}-\frac{1}{4}B_{\mu \nu}B^{\mu \nu} +i\sum_j\bar{\psi}_j \slashed{\partial}\psi_j +D_\mu H^\dagger D^\mu H+\mu^2H^\dagger H-\lambda(H^\dagger H)^2 \\
&\quad -\bar{E}_L ... | Electroweak | $\frac{\sqrt{g^2+g'^2}\,v}{2}$ | With $\tau^a=\sigma^a/2$ and hypercharge $Y=1$ for the Higgs doublet (so that $\frac{Y}{2}B_\mu$ acts as $\frac12 B_\mu\mathbb{1}$), the gauge-covariant derivative of $H$ is
$$
\begin{aligned}
D_\mu H &=
\partial_\mu H
- \frac{ig}{2}
\left[
W_\mu^1
\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}
+ W_\mu^2
\begin{pmatrix} ... | null | null | null |
25 | Compute the partial decay widths of the $W^+$ boson into pairs of fermions (including leptons and quarks) with numerical values. Assume that the top quark mass $m_t$ is larger than $m_W$, and ignore the other quark masses. The decay widths to quarks are enhanced by QCD corrections. Use $\sin^2\theta_w = 0.23$, and take... | Electroweak | 2.06 GeV | In the Lagrangian, $i$ runs over all three generations of leptons, $j$ over the first two generations of quarks (the decay $W^+\to t\bar b$ is kinematically forbidden since $m_t>m_W$), and $c$ over the 3 colours. All final-state fermions are treated as massless. The amplitude of the decay into a pair of fermions is
$$
... | When judging the answer of this question, a relative error of 2% is allowed. | null | 0.02 |
26 | Compute the partial decay widths of the $Z^0$ boson into pairs of fermions (including hadrons, charged leptons, and the invisible modes $\nu \bar{\nu}$) with numerical values, treating the quarks in the same way as in the $W^+$ case. Take the relevant Lagrangian to be
$$
\mathcal{L} = \frac{g}{\cos\theta_w}Z_\mu \sum_i... | Electroweak | 2.47 GeV | The corresponding amplitude is
$$
i\mathcal{M} = \frac{ig}{\cos \theta_w} \epsilon_\mu(k)\bar{u}(p_1)\gamma^\mu\big(I^3P_L-\sin^2\theta_wQ\big) v(p_2)
=\frac{ig}{\cos \theta_w} \epsilon_\mu(k)\bar{u}(p_1)\gamma^\mu\big(g_LP_L+g_RP_R\big) v(p_2),
$$
with $P_{L,R}=\frac{1\mp\gamma^5}{2}$ and
$$
g_L=I^3-\sin^2\theta_wQ,\q... | When judging the answer of this question, a relative error of 2% is allowed. | null | 0.02 |
27 | Consider the Abelian Higgs Lagrangian
$$
\mathcal{L}[A_\mu,H] = -\frac{1}{4}F_{\mu\nu}F^{\mu \nu}+(D_\mu H)^\dagger D^\mu H-\frac{\lambda^2}{2}\left(H^\dagger H-\frac{v^2}{2}\right)^2
$$
where
$$
D_\mu= \partial_\mu-igA_\mu, \quad F_{\mu \nu}=\partial_\mu A_\nu-\partial_\nu A_\mu, \quad H=\rho(x)e^{i\theta(x)/v}.
$$
Id... | Electroweak | $m_A= gv$ | The Lagrangian has the gauge redundancy
$$
A_\mu \to A_\mu+\frac{1}{g} \partial_\mu \alpha, \quad H \to e^{i\alpha}H \quad(\text{i.e. } \theta\to\theta+v\alpha),
$$
under which $D_\mu H\to e^{i\alpha}D_\mu H$. The potential is minimized at $H^\dagger H=\rho^2=\frac{v^2}{2}$, so we expand around this vacuum with $\rho=\... | null | null | null |
28 | Consider the Abelian Higgs Lagrangian
$$
\mathcal{L}[A_\mu,H] = -\frac{1}{4}F_{\mu\nu}F^{\mu \nu}+(D_\mu H)^\dagger D^\mu H-\frac{\lambda^2}{2}(H^\dagger H-v^2)^2
$$
where
$$
D_\mu= \partial_\mu-igA_\mu, \quad F_{\mu \nu}=\partial_\mu A_\nu-\partial_\nu A_\mu, \quad H=\rho(x)e^{i\theta(x)/v}.
$$
Assume that the Higgs i... | Electroweak | $\xi^2= \sqrt{2}g\frac{m_A}{m_h}$ | For the potential $\frac{\lambda^2}{2}(H^\dagger H-v^2)^2$ the vacuum is at $\rho=v$. We expand with a canonically normalized Higgs field, $\rho=v+\frac{h}{\sqrt2}$, and define $\tilde A_\mu\equiv A_\mu-\frac{1}{gv}\partial_\mu\theta$ as in the previous problem. The same steps as there give
$$
\begin{aligned}
\mathcal{... | When judging the answer of this question, note that here $m_A=\sqrt{2}gv$ and $m_h=\sqrt{2}\lambda v$; equivalent forms obtained using these relations are also correct. | null | null |
29 | The Higgs Yukawa coupling to fermions is given by
$$
\mathcal{L}_Y =\frac{g}{2}\frac{m_f}{M_W}h\bar{f}f .
$$
Find the decay rate $\Gamma$ for the Higgs into a fermion-antifermion pair at tree level. Take the mass of $h$ to be $m_h$. Account for the situation $f=\text{quark}$; the number of colors can be written as $N_c... | Electroweak | $\Gamma=m_h \frac{N_c g^2m_f^2}{32\pi M_W^2}\left(1-\frac{4m_f^2}{m_h^2}\right)^{\frac{3}{2}}$ | To keep the expression short we introduce $y=\frac{g}{2}\frac{m_f}{M_W}$; the tree-level amplitude reads
$$
\mathcal{M} = y\bar{u}_{s_1}(\vec{p}_1)v_{s_2}(\vec{p}_2)
$$
with $\{\vec{p}_i,s_i\}$ the momenta and spins of the outgoing fermion and anti-fermion. Then
$$
\begin{aligned}
\sum_{\text{spin}} |\mathcal{M}|^2 &=y... | null | null | null |
30 | Assume that the mass matrices for the up- and down-type quarks have the following forms (in the basis of weak interaction eigenstates)
$$
M^{(u)}=\begin{pmatrix}
m_u & 0 & 0 \\
0 & m_c & 0 \\
0 & 0 & m_t
\end{pmatrix},\quad\text{and} \quad M^{(d)}=m\begin{pmatrix}
1+a^2 & ab & 0 \\
ab & 1+b^2 & 0 \\
0 & 0 & 1
\end{pmat... | Electroweak | $\frac{4a^2b^2}{(a^2+b^2)^2}$ | First we consider the CKM matrix $V_{CKM} = (L^u)^\dagger L^d$, where $L^u$ and $L^d$ are the transformation matrices of $u_L$ and $d_L$ respectively. Since $M^{(u)}$ is already diagonal in the weak eigenbasis, the CKM matrix is just $L^d$ from diagonalizing $M^{(d)}$. Since $M^{(d)}$ is Hermitian, it can be diagonaliz... | null | null | null |
31 | Consider an $O(N)$ $\phi^4$ scalar model in the large $N$ limit. Suppose all two-point functions of said theory are proportional to the propagator and can hence be renormalized into the mass. How many connected Feynman diagrams would contribute to the four-point function $\langle\phi^i\phi^i\phi^j\phi^j\rangle$ at all ... | SUSY | 3 | The $O(N)$ $\phi^4$ scalar model has Lagrangian
$$
\mathcal{L} = \frac{1}{2} (\partial_\mu \phi^i)^2 + \frac{m^2}{2} \phi^i \phi^i + \frac{\lambda}{2N} (\phi^i \phi^i)^2
$$
in which the index $i$ is summed over.
At tree level, only 1 cross diagram contributes to the 4-point function.
At 1-loop level, only 1 bubble di... | null | null | null |
32 | Suppose the four-point correlator $\langle O_\Delta(x_1)O_\Delta(x_2)O_\Delta(x_3)O_\Delta(x_4)\rangle$ of a conformal field theory can be written in spectral representation as
$$
\int_0^\infty \mathrm{d}\nu\, B^{\text{free}}(\nu)\, g^{\Delta,\Delta,\Delta,\Delta}_{\nu}(x_1,x_2,x_3,x_4)+\int_0^\infty \mathrm{d}\nu\, B^... | SUSY | $\frac{B^{\text{int}}_{-2}(\nu_n)}{\text{Res}B^{free}(\nu_n)}$ (it is OK to differ by a factor of any combination of $2\pi$, $-1$ and $i$) | First we consider only the free theory and compare the spectral representation to the conformal block expansion:
$$
\begin{aligned}
&\langle O_\Delta(x_1)O_\Delta(x_2)O_\Delta(x_3)O_\Delta(x_4)\rangle^{\text{free}} \\
=&\int_0^\infty \mathrm{d}\nu\, B^{\text{free}}(\nu) g^{\Delta,\Delta,\Delta,\Delta}_{\nu}(x_1,x_2,x_3... | When judging the answer of this question, the numerator must be the coefficient of the second-order pole; only differences by factors of $2\pi$, $-1$ and $i$ are allowed. Read the expression literally. | null | null |
33 | Consider two anticommuting CFT fields $b(z)$ and $c(z)$,
$$
b(z)c(w) = -c(w)b(z), \quad c(z)c(w) = -c(w)c(z), \quad b(z)b(w) = -b(w)b(z)
$$
with the following OPE between them
$$
\langle c(z)b(w) \rangle = \frac{1}{z-w}+\cdots, \qquad \langle b(z)b(w)\rangle=\cdots, \qquad \langle c(z)c(w) \rangle=\cdots
$$
Here the do... | SUSY | $-3(2\lambda -1)^2+1$ | The central charge is read off from the fourth-order pole, $T(z)T(w)\sim\frac{c/2}{(z-w)^4}+\dots$, which comes from the double contractions. Write $T^{bc}=-\lambda\,\mathcal B+(\lambda-1)\,\mathcal C$ with $\mathcal B\equiv:b\partial c:$ and $\mathcal C\equiv:c\partial b:$. Then
$$
T^{bc}(z) T^{bc}(w) = \lambda^2\,\ma... | null | null | null |
34 | Consider two anticommuting CFT fields $b(z)$ and $c(z)$,
$$
b(z)c(w) = -c(w)b(z), \quad c(z)c(w) = -c(w)c(z), \quad b(z)b(w) = -b(w)b(z)
$$
with the following OPE between them
$$
\langle c(z)b(w) \rangle = \frac{1}{z-w}+\cdots, \qquad \langle b(z)b(w)\rangle=\cdots, \qquad \langle c(z)c(w) \rangle=\cdots
$$
Here the do... | SUSY | $\{c_n,b_m\} = \delta_{m+n,0}$ | $$
\begin{aligned}
\{c_n,b_m\} &= \left\{\oint_{C_0} \frac{dz}{2\pi i} z^{n-2}c(z),\oint_{C_0} \frac{dw}{2\pi i} w^{m+1}b(w)\right\} \\
&=\oint_{C_0} \frac{dw}{2\pi i} w^{m+1}\oint_{C_0} \frac{dz}{2\pi i} z^{n-2} \{c(z),b(w)\} \\
&= \oint_{C_0} \frac{dw}{2\pi i} w^{m+1}\left( \oint_{|z|>|w|}\frac{dz}{2\pi i} z^{n-2} c(... | null | null | null |
35 | Consider a supermanifold $M$ with a sheaf that is locally isomorphic to $C^\infty(\mathbb{R}) \otimes \Lambda^\cdot (\mathbb{R}^2)$, with commutative coordinate $x$ and Grassmannian coordinates $\theta_1$ and $\theta_2$. Let $f(x,\theta_1,\theta_2)$ be a function on $M$. Make use of the Dirac delta function and write d... | SUSY | $f(x,\theta_1,\theta_2) \text{d}x\delta(d\theta_1)\delta(d\theta_2)$ | On a supermanifold the parity of the differentials is opposite to that of the coordinates. $\mathrm{d}x$ is odd, so $(\mathrm{d}x)^2=0$, while $\mathrm{d}\theta_1$ and $\mathrm{d}\theta_2$ are even (commuting) variables. A differential form, being polynomial in the even variables $\mathrm{d}\theta_i$, therefore cannot ... | null | null | null |
36 | Consider a primary field $\phi(z,\bar{z})$ of conformal weight $(h,\bar{h})$. By considering infinitesimal conformal transformations $f(z)=z+\epsilon(z)$ and $\bar{f}(\bar{z})=\bar{z}+\bar{\epsilon}(\bar{z})$ with $\epsilon(z), \bar{\epsilon}(\bar{z}) \ll 1$, compute the corresponding infinitesimal transformation $\del... | CFT | $$
\delta_{\epsilon,\bar{\epsilon}}\phi(z,\bar{z})
=\epsilon(z)\,\partial_z\phi(z,\bar{z})
+h\,\partial_z\epsilon(z)\,\phi(z,\bar{z})+\bar{\epsilon}(\bar{z})\,\partial_{\bar{z}}\phi(z,\bar{z})
+\bar{h}\,\partial_{\bar{z}}\bar{\epsilon}(\bar{z})\,\phi(z,\bar{z}).
$$ | A primary field $\phi(z, \bar{z})$ with conformal weight $(h,\bar{h})$ is defined by its transformation under a conformal map $z\mapsto f(z), \bar{z}\mapsto \bar{f}(\bar{z})$ as
$$
\phi(z,\bar{z}) \mapsto \phi'(z,\bar{z})=\left(\frac{\partial f}{\partial z}\right)^{h} \left(\frac{\partial \bar{f}}{\partial \bar{z}}\rig... | When judging the answer of this question, an answer written as $(\epsilon\partial+h\partial\epsilon+\cdots)\phi$ is wrong, because it cannot be determined which function each derivative acts on. | null | null |
37 | Consider a conformal field theory on flat spacetime in the absence of conformal anomalies. Let $T_{\mu\nu}$ be its energy-momentum tensor. Determine its trace $T^\mu{}_{\mu}$. | CFT | $T^\mu{}_{\mu}=0$ | For every continuous symmetry in a field theory, there exists a conserved current $j^{\mu}$ satisfying $\partial_{\mu}j^{\mu}=0$. In particular, for a theory invariant under conformal transformations $x^{\mu} \mapsto x^{\mu}+\epsilon^{\mu}(x)$, the conserved current can be written as $j^{\mu}=T^{\mu\nu}\epsilon_{\nu}$,... | null | null | null |
38 | Consider the Riemann sphere $\mathbb{CP}^{1}:=\mathbb{C}\cup\{\infty\}$. Möbius transformations act on it by
$$
z\mapsto \frac{az+b}{cz+d},
\qquad ad-bc\neq 0.
$$
What is the global conformal group of the Riemann sphere? Select the correct option and answer with the corresponding letter only.
A. $GL_2(\mathbb{C})$
B. ... | CFT | $\text{C}$ | Every Möbius transformation of the Riemann sphere has the form
$$
f_g(z)=\frac{az+b}{cz+d},
\qquad
g=
\begin{pmatrix}
a&b\\
c&d
\end{pmatrix}\in GL_2(\mathbb C).
$$
Composition of Möbius transformations corresponds to matrix multiplication $f_{g_1}\circ f_{g_2}=f_{g_1g_2}$. Thus we obtain a group homomorphism
$$
GL_2(\... | null | null | null |
39 | Let $|h\rangle$ be a highest-weight state of the Virasoro algebra, satisfying
$$
L_0|h\rangle=h|h\rangle,
\qquad
L_n|h\rangle=0 \quad (n>0),
$$
with commutation relations
$$
[L_m,L_n]=(m-n)L_{m+n} +\frac{c}{12}m(m^2-1)\delta_{m+n,0}.
$$
Suppose that at level $2$ there is a nonzero singular (null) state of the form $\le... | CFT | $h=\frac{1}{16}(5-c\pm \sqrt{(c-1)(c-25)})$ | Let $|\chi\rangle=\left(L_{-1}^2+aL_{-2}\right)|h\rangle$ be the null state; it must be annihilated by the positive Virasoro modes. In particular, we require $L_1|\chi\rangle=0$, $L_2|\chi\rangle=0$. First, acting with $L_1$, we use
$$
[L_1,L_{-2}]=3L_{-1},
\qquad
[L_1,L_{-1}]=2L_0.
$$
Since $L_1|h\rangle=0$, we find
$... | null | null | null |
40 | Let $|h\rangle$ be a normalized highest-weight state of the Virasoro algebra satisfying
$$
\langle h|h\rangle=1,\qquad
L_0|h\rangle=h|h\rangle,\qquad
L_n|h\rangle=0\quad(n>0).
$$
The Virasoro generators satisfy
$$
[L_m,L_n]
=
(m-n)L_{m+n}
+\frac{c}{12}m(m^2-1)\delta_{m+n,0},
\qquad m,n\in\mathbb Z.
$$
Assume that $L_n^... | CFT | $\det M_{2}(h,c)=32h\left(h^2-\frac{5h}{8}+\frac{hc}{8}+\frac{c}{16}\right)$ | At level $2$, using the Virasoro algebra and the highest-weight conditions, we obtain
$$
\langle h|L_2L_{-2}|h\rangle
=
4h+\frac{c}{2}, \qquad \langle h|L_2L_{-1}^2|h\rangle
=
\langle h|L_1^2L_{-2}|h\rangle = 6h
$$
and
$$
\langle h|L_1^2L_{-1}^2|h\rangle
=
4h+8h^2.
$$
Therefore
$$
M_2(h,c)
=
\begin{pmatrix}
4h+\frac{c}... | null | null | null |
41 | Let $Q$ be a nilpotent symmetry, $Q^{2}=0$, and suppose the action of a field theory is $Q$-exact, $S_t=t\{Q,V\}$, where $t>0$ is a real parameter. Assume the path-integral measure is $Q$-invariant. Compute $\frac{dZ(t)}{dt}$ for the partition function $Z(t)=\int \mathcal{D}X\, e^{-S_t}$. | Quantization | $\frac{dZ(t)}{dt}=0$ | For the partition function $Z(t)$, differentiating with respect to $t$ gives
$$
\frac{dZ(t)}{dt}
=
-\int \mathcal{D}X\, \{Q,V\}\, e^{-t\{Q,V\}}=\langle QV+VQ \rangle =0 .
$$ | null | null | null |
42 | Consider the path integral for a free particle of mass $m$ on the interval $0\le t\le T$, with fixed endpoints $x(0)=x(T)=0$. The propagator can be written as a Gaussian path integral
$$
K(0,T;0,0)
=
\int_{x(0)=0}^{x(T)=0}\mathcal{D}x(t)\,
\exp\left[
\frac{i}{2}\int_0^T dt\,dt'\,
x(t')A(t',t)x(t)
\right].
$$
Starting f... | Quantization | $\det A = \prod_{n=1}^{\infty}\frac{m n^2\pi^2}{T^2}$ | For fixed endpoints $x(0)=x(T)=0$, integration by parts gives
$$
S[x]
=
\frac{m}{2}\int_0^T dt\,\dot x^2
=
-\frac{m}{2}\int_0^T dt\,x(t)\frac{d^2}{dt^2}x(t).
$$
Thus the quadratic operator appearing in the Gaussian path integral is $A=-m\frac{d^2}{dt^2}$. Equivalently, in kernel notation,
$$
A(t',t)=-m\,\delta(t'-t)\fr... | null | null | null |
43 | Consider a theory described by the Lagrangian
$$
\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{m^{2}}{2}A_{\mu}A^{\mu}.
$$
For a plane-wave solution of the form $A^{\mu}=\epsilon^{\mu}e^{\pm i p\cdot x}$, where $\epsilon^{\mu}$ is the polarization vector, use the equation of motion to determine the constraint rela... | Classical | $p_\mu\epsilon^\mu=0$ | The equation of motion following from the Lagrangian is
$$
\partial_\mu\partial^\mu A^\nu -\partial^\nu(\partial_\mu A^\mu) +m^2 A^\nu=0
$$
or equivalently,
$$
\left[\eta^{\mu\nu}(\Box+m^2)-\partial^\mu\partial^\nu\right]A_\nu=0.
$$
Applying $\partial_{\mu}$ to the equation of motion and contracting the free Lorentz in... | null | null | null |
44 | Consider a complex scalar field $\phi$ with a global $U(1)$ phase symmetry generated by
$$
U_\alpha=e^{-i\alpha Q},
\qquad
Q=i\int d^3x\,(\pi_{\phi^*}\phi^*-\pi_{\phi}\phi).
$$
Here, $\pi_\phi$ and $\pi_{\phi^*}$ are the canonical momenta conjugate to $\phi$ and $\phi^*$, respectively. Assume that the vacuum is invaria... | Quantization | $\langle 0|\phi(x)\phi(x')|0\rangle=0$ | The charge $Q$ generates the global $U(1)$ symmetry. Using the canonical commutation relations, we find $[Q,\phi(x)]=-\phi(x)$. The Baker-Campbell-Hausdorff formula then gives
$$
U_\alpha^\dagger\phi(x)U_\alpha
=
e^{-i\alpha}\phi(x).
$$
Since the vacuum is invariant,
$$
U_\alpha|0\rangle=|0\rangle,
\qquad
\langle0|U_\a... | null | null | null |
45 | Consider the one-loop renormalization group equation for the Higgs quartic coupling
$$
Q^2\frac{d\lambda}{dQ^2}
=
\frac{3}{4\pi^2}\lambda^2 .
$$
Assume the boundary condition
$$
\lambda(v^2)=\frac{M_H^2}{2v^2}.
$$
Determine the Landau pole scale $\Lambda_c$, defined by the value of $Q$ at which $\lambda(Q^2)$ diverges. | Renormalization | $\Lambda_{c}=v\exp\left(\frac{4\pi^2v^2}{3M_H^2}\right)$ | The renormalization group equation is
$$
Q^2\frac{d\lambda}{dQ^2}
=
\frac{3}{4\pi^2}\lambda^2 .
$$
Separating variables and integrating from $Q=v$ to $Q$, we get
$$
\int_{\lambda(v^2)}^{\lambda(Q^2)}
\frac{d\lambda}{\lambda^2}
=
\frac{3}{4\pi^2}
\int_{v^2}^{Q^2}\frac{dQ^2}{Q^2}.
$$
Thus
$$
-\frac{1}{\lambda(Q^2)}
+
\fr... | null | null | null |
46 | In four-dimensional $\mathcal{N}=1$ superspace, let $V$ be a real vector superfield in Wess-Zumino gauge and define the gauge field-strength superfield by $W_\alpha:=-\frac14 \overline D^2 D_\alpha V$, $\overline D^2:=\overline D^{\dot\beta}\overline D_{\dot\beta}$, where
$$
V(x,\theta,\bar\theta)
=
(\theta\sigma^\mu\b... | SUSY | $\overline D_{\dot\alpha}W_\alpha=0$ | Using $\overline D^2=\overline D_{\dot\beta}\overline D^{\dot\beta}$, we have
$$
\overline D_{\dot\alpha}W_\alpha
=
-\frac14
\overline D_{\dot\alpha}
\overline D_{\dot\beta}
\overline D^{\dot\beta}
D_\alpha V .
$$
The operators $\overline D_{\dot\alpha}$ anticommute:
$$
\{\overline D_{\dot\alpha},\overline D_{\dot\beta... | null | null | null |
47 | Consider an $SU(2)$ Yang-Mills theory whose Lie algebra $\mathfrak{su}(2)$ has structure constants $f^{a}{}_{bc}=\epsilon^{a}{}_{bc}$. Compute the quadratic Casimir $C_2(\text{adj})$ in the adjoint representation, defined by $f^{acd}f^{b}{}_{cd}=C_2(\text{adj})\delta^{ab}$. | Non-Abelian | $C_2(\text{adj})=2$ | For the Lie algebra $\mathfrak{su}(2)$, the structure constants are $f^a{}_{bc}=\epsilon^a{}_{bc}$. Therefore,
$$
f^{acd}f^b{}_{cd}
=
\epsilon^{acd}\epsilon^b{}_{cd}
=
2\delta^{ab}.
$$
Comparing this with the definition $f^{acd}f^b{}_{cd}=C_2(\mathrm{adj})\delta^{ab}$, we obtain $C_2(\mathrm{adj})=2$. | null | null | null |
48 | Consider a non-Abelian Yang-Mills theory in Minkowski spacetime with metric signature $(-,+,+,+)$. Impose the axial gauge $A_3^a=0$ by adding the gauge-fixing term $\mathcal{L}_{\mathrm{gf}}=-\frac{1}{2\xi}(A_3^a)^2$ to the Yang-Mills Lagrangian $\mathcal{L}=-\frac14(F_{\mu\nu}^a)^2+\cdots$. Derive the tree-level momen... | Quantization | $D_{\mu\nu}^{ab}(k)=-\frac{i}{k^2-i\epsilon} \left[ \eta_{\mu\nu} + \frac{k_\mu k_\nu}{k_3^2} - \frac{k_\mu \delta_\nu^3+k_\nu \delta_\mu^3}{k_3} \right]\delta^{ab}$ | To implement the axial gauge, we use a slightly generalized gauge condition
$$
A_3^a(x)=\omega^a(x),
$$
where $\omega^a(x)$ is an arbitrary auxiliary function. This condition is imposed in the path integral by inserting
$$
\delta\!\left(A_3^a(x)-\omega^a(x)\right).
$$
We then average over $\omega^a$ with a Gaussian wei... | null | null | null |
49 | Consider a non-Abelian Yang-Mills theory with gauge field $A_{\mu}^{a}$ and field strength
$$
F_{\mu\nu}^a
=
\partial_\mu A_\nu^a-\partial_\nu A_\mu^a
+g f^{abc}A_\mu^b A_\nu^c .
$$
Choose the convention $\epsilon^{0123}=1$. The $\theta$-term is given by $\mathcal{L}_\theta=\epsilon^{\mu\nu\lambda\rho}F_{\mu\nu}^a F_{\... | Non-Abelian | $2\epsilon^{\mu\nu\lambda\rho}\left(A_\nu^aF_{\lambda\rho}^a-\frac{g}{3}f^{abc}A_\nu^aA_\lambda^bA_\rho^c\right)$ | We want to write $\epsilon^{\mu\nu\lambda\rho}F_{\mu\nu}^aF_{\lambda\rho}^a$ as a total derivative. A natural first guess is
$$
K_0^\mu
=
2\epsilon^{\mu\nu\lambda\rho}A_\nu^aF_{\lambda\rho}^a .
$$
Taking the divergence gives
$$
\partial_\mu K_0^\mu
=
2\epsilon^{\mu\nu\lambda\rho}
(\partial_\mu A_\nu^a)F_{\lambda\rho}^a... | null | null | null |
50 | Consider a theory in $d$ spacetime dimensions with a Dirac fermion $\psi$ coupled to a massive vector field $A_\mu$ through the interaction term $-e A_\mu \bar{\psi}\gamma^\mu \psi$. At large momentum, the massive vector propagator scales as $\mathcal{O}(1)$. Derive the superficial degree of divergence $D$ for a connec... | Renormalization | $D = d+\frac{d-2}{2}V-\frac{d-1}{2}E_e-\frac{d}{2}E_A$ | Let $E$ denote external lines and $I$ internal lines; we have $D=dL-I_e$ with $d$ the number of spacetime dimensions and $L$ the number of loops. For a connected diagram,
$$
L=I_e+I_A-V+1 .
$$
We use $2V=2I_e+E_e$ and $V=2I_A+E_A$ to obtain
$$
I_e=V-\frac{E_e}{2},
\qquad
I_A=\frac{V-E_A}{2}.
$$
Substituting into $L$, w... | null | null | null |
51 | Consider a connected vacuum diagram in a large-$N$ matrix gauge theory. Let $E$ be the number of propagators, $V$ the number of vertices, and $F$ the number of faces. Suppose that the diagram contributes a factor
$$
A \sim (g^2)^E (g^2)^{-V} N^F .
$$
Introduce the 't Hooft coupling
$$
\lambda=g^2N,
$$
and suppose that ... | Non-Abelian | $\log Z = \sum_{h=0}^{\infty}N^{2-2h}f_h(\lambda)$ | The vacuum diagram has
$$
A \sim (g^2)^E (g^2)^{-V} N^F=\lambda^{L-1}N^{\chi}=\lambda^{L-1}N^{2-2h}
$$
where $L$ is the number of loops and $\chi$ is the Euler characteristic. Since $\log Z$ is the sum over connected vacuum diagrams, we sum over all genera $h$ and all loop numbers $L$:
$$
\log Z =
\sum_{h=0}^{\infty}\s... | null | null | null |
52 | Consider a four-dimensional $\mathcal{N}=1$ Wess-Zumino model with one chiral superfield $\Phi$ and superpotential
$$
W(\Phi)=\frac{m}{2}\Phi^2+\frac{\lambda}{3}\Phi^3 ,
$$
where $m\neq0$ and $\lambda\neq0$. Determine the values of the scalar field $\phi$ corresponding to all classical supersymmetric vacua of the theor... | SUSY | $\phi=0, \,\, \phi=-\frac{m}{\lambda}$ | For a theory of chiral superfields, the scalar potential is determined by the superpotential:
$$
V(\phi)=\left|W'(\phi)\right|^2 .
$$
A supersymmetric vacuum must have zero energy, so it must satisfy $V(\phi)=0$. For the given superpotential
$$
W(\phi)=\frac{m}{2}\phi^2+\frac{\lambda}{3}\phi^3,
$$
we find
$$
W'(\phi)=m... | null | null | null |
53 | Consider a finite-dimensional supersymmetric multiplet of states with fixed four-momentum $p_\mu$ and positive energy $p_0>0$. Let the supercharges satisfy
$$
\{Q_\alpha,\overline Q_{\dot\beta}\}
=
2(\sigma^\mu)_{\alpha\dot\beta}P_\mu .
$$
Let $(-1)^F$ be the fermion number operator, acting as
$$
(-1)^F|B\rangle=|B\ran... | SUSY | $n_B=n_F$ | Since the supercharges map bosonic states to fermionic states and vice versa, they anticommute with the fermion number operator $(-1)^F$:
$$
\{(-1)^F,Q_\alpha\}=0,
\qquad
\{(-1)^F,\overline Q_{\dot\beta}\}=0.
$$
Now consider the trace over a supersymmetric multiplet $\operatorname{Tr}\left[(-1)^F\{Q_\alpha,\overline Q_... | null | null | null |
54 | Let $S_1$ and $S_2$ be superfields. A superfield transforms under supersymmetry as $\delta S=i[\epsilon Q+\bar\epsilon \bar Q,S]$. Is the product $S_1S_2$ also a superfield? Justify your answer by computing $\delta(S_1S_2)$. Please give the final answer as either "yes" or "no". | SUSY | yes | Let the supersymmetry variation be $\delta S=i[\epsilon Q+\bar\epsilon \bar Q,S]$. Then $\delta(S_1S_2)=i[\epsilon Q+\bar\epsilon\bar Q,S_1S_2]$. Using the Leibniz rule for the supersymmetry generator, we get
$$
\delta(S_1S_2)
=
(\delta S_1)S_2+S_1(\delta S_2).
$$
Equivalently,
$$
\delta(S_1S_2)
=
i(\epsilon Q+\bar\eps... | null | null | null |
55 | Consider a renormalizable four-dimensional $\mathcal{N}=1$ theory with a single chiral superfield
$$
\Phi=\phi+\sqrt{2}\theta\psi+\theta\theta F .
$$
The supersymmetric Lagrangian can be written in terms of a Kähler potential $K(\Phi,\Phi^\dagger)$ and a superpotential $W(\Phi)$ as
$$
\mathcal{L}
=
\left. K(\Phi,\Phi^\... | SUSY | $W(\Phi) = c_{0}+c_{1}\Phi+c_{2}\Phi^2+c_{3}\Phi^3$ | For a renormalizable theory in four dimensions, the Lagrangian must have mass dimension $4$. The lowest component of the chiral superfield is the scalar field. Therefore, from $\Phi=\phi+\sqrt2\,\theta\psi+\theta\theta F$, we assign $[\Phi]=[\phi]=1$. In four dimensions, the fermion has the standard mass dimension $[\p... | null | null | null |
56 | For a supersymmetric gauge theory, define the one-loop coefficient $b$ by
$$
b=\frac{11}{6}T(\mathrm{adj})
-\frac{1}{3}\sum_f T(\mathbf R_f)
-\frac{1}{6}\sum_b T(\mathbf R_b),
$$
where the sums run over the Weyl fermions and the complex bosons. In an $\mathcal{N}=1$ supersymmetric gauge theory, the vector multiplet con... | SUSY | $b=\frac{3}{2}T(\mathrm{adj})-\frac{1}{2}\sum_c T(\mathbf R_c)$ | For an $\mathcal{N}=1$ SUSY gauge theory, the vector multiplet contains a gauge boson $A_{\mu}$ and a gaugino $\lambda_{\alpha}$, both transforming in the adjoint representation. Therefore its contribution is
$$
b_{\text{vect}}=\frac{11}{6}T(\mathrm{adj})
-\frac{1}{3} T(\mathrm{adj})=\frac{3}{2}T(\mathrm{adj})
$$
An $\... | null | null | null |
57 | For $\mathcal{N}=1$ SQCD with gauge group $SU(N)$ and $N_f$ flavors $(\mathcal{Q}_i,\widetilde{\mathcal{Q}}^{\,i})$, assume that the IR theory is superconformal. Define the anomalous dimension by $\Delta(\mathcal{Q})=1+\gamma(\mathcal{Q})$. Determine $\gamma(\mathcal{Q})$. | SUSY | $\gamma(\mathcal{Q})=\frac{N_{f}-3N}{2N_{f}}$ | We use the convention that the superspace coordinate $\theta^{\alpha}$ carries $R$-charge $+1$. We have the superfield $W_{\alpha}$ with the expansion
$$
W_{\alpha}=\lambda_{\alpha}+F_{\alpha\beta}\theta^{\beta}+\cdots,
$$
With this convention, we are forced to assign $R$-charge $1$ to the gaugino $\lambda_{\alpha}$ an... | null | null | null |
58 | Consider an $\mathcal{N}=1$ supersymmetric gauge theory. Let the microscopic holomorphic coupling be
$$
\tau_0=\frac{4\pi i}{g^2}+\frac{\theta_{\mathrm{YM}}}{2\pi},
$$
and define the dynamical scale by
$$
\Lambda=\mu e^{2\pi i\tau_0/b},
\qquad b\in\mathbb{Z}_{>0}.
$$
Suppose that the effective holomorphic coupling has ... | SUSY | $\tau_{\mathrm{eff}}(\Lambda,\mu)=\frac{b}{2\pi i }\text{log}\left(\frac{\Lambda}{\mu}\right)+\sum_{k\geq 0}a_{k}\left(\frac{\Lambda}{\mu}\right)^{bk}$ | The periodicity of the theta angle implies $\tau_{0} \rightarrow \tau_{0} +1$. Under this transformation the dynamical scale transforms as $\Lambda \rightarrow e^{2\pi i/b}\Lambda$, and the logarithmic term changes as
$$
\frac{b}{2\pi i }\log\left(\frac{\Lambda}{\mu}\right) \mapsto \frac{b}{2\pi i }\log\left(\frac{e^{2... | null | null | null |
59 | Consider $\mathcal{N}=1$ SQCD with gauge group $SU(N)$ and $N_{f}$ flavors. The NSVZ beta function is
$$
\beta(g)=\frac{-g^{3}}{16\pi^{2}}\frac{3N-N_{F}(1-\gamma(g^{2}))}{1-\frac{Ng^{2}}{8\pi^{2}}},
$$
with
$$
\gamma(g^{2})=-\frac{g^{2}}{8\pi^{2}}\frac{N^{2}-1}{N}+\mathcal{O}(g^{4}).
$$
Assume
$$
\frac{N_{F}}{N}=3-\eps... | SUSY | $g_*^2=\frac{8\pi^{2}}{3}\frac{N}{N^{2}-1}\epsilon$ | Using $N_f=N(3-\epsilon)$, we first expand the numerator of the NSVZ beta function:
$$
\begin{aligned}
3N-N_f\bigl(1-\gamma(g^2)\bigr)
&=
3N-N_f
-\frac{N_f(N^2-1)}{8\pi^2N}g^2
+\mathcal O(g^4)\\
&=
N\epsilon
-\frac{(3-\epsilon)(N^2-1)}{8\pi^2}g^2
+\mathcal O(g^4).
\end{aligned}
$$
The denominator gives
$$
\frac{1}{1-\f... | null | null | null |
60 | In pure $SU(2)$ Seiberg-Witten theory, suppose $a\sim \sqrt{2u}$, $a_{D} \sim \frac{2i}{\pi}a\log a$. Analytically continue once along the path $u\mapsto e^{2\pi i}u$. Compute the monodromy matrix $M_\infty$ defined by
$$
\begin{pmatrix}a_D\\a\end{pmatrix}
\longmapsto
M_\infty
\begin{pmatrix}a_D\\a\end{pmatrix}.
$$ | SUSY | $M_{\infty}=\begin{pmatrix} -1 & 2\\ 0 & -1 \end{pmatrix}$ | Under a large loop around infinity $u\mapsto e^{2\pi i}u$, since $a\sim \sqrt{2u}$, then $a\mapsto -a$. Now use the asymptotic formula $a_{D} \sim \frac{2i}{\pi}a\log a$; we have
$$
a_{D}\mapsto \frac{2i}{\pi}(-a)(\log a+i\pi)=-\frac{2i}{\pi}a\log a+2a=-a_{D}+2a
$$
Thus the vector transforms as
$$
\begin{pmatrix}
a_D\\... | null | null | null |
61 | In pure $SU(2)$ Seiberg-Witten theory, the Coulomb branch has a $\mathbb{Z}_{2}$ symmetry under $u\mapsto -u$. The moduli space has three singularities located at
$$
u=\infty, \qquad u=u_{+}, \qquad u=u_{-}=-u_{+}
$$
At $u=u_{+}$, a monopole with electromagnetic charge $(n_{m},n_{e})=(1,0)$ becomes massless, while at $... | SUSY | $(M_{u_{+}},M_{u_{-}})=\left(\begin{pmatrix} 1 & 0\\ -2 & 1 \end{pmatrix},\,\begin{pmatrix} -1 & 2\\ -2 & 3 \end{pmatrix} \right)$ | If $u=\infty$ were the only singularity, then the monodromy around infinity could be continuously deformed to a trivial contour in the finite $u$-plane. This would imply that the periods $(a_{D},a)^{T}$ have no nontrivial monodromy, so $a$ would be a good global coordinate on the whole moduli space. However, this is no... | null | null | null |
62 | In pure $SU(2)$ Seiberg-Witten theory, the BPS central charge is $Z=n_{m}a_{D}+n_{e}a$. Suppose a dyon with charge $(n_{m},n_{e})$ becomes massless at a singularity. Using the period basis $(a_D,a)^T$ and a counterclockwise monodromy loop around the singularity, determine the associated monodromy matrix $M_{(n_m,n_e)}$... | SUSY | $M_{(n_{m},n_{e})}=\begin{pmatrix} 1+2n_{m}n_{e} & 2n_{e}^{2}\\ -2n_{m}^{2} & 1-2n_{m}n_{e} \end{pmatrix}$ | We write the period vector as $\Pi=(a_{D},a)^{T}$. Then the central charge $Z=(n_{m},n_{e})\Pi$. Suppose that a BPS state with charge $(n_{m},n_{e})$ becomes massless at a singularity. Geometrically, this means that the corresponding cycle is a vanishing cycle. By the Picard-Lefschetz formula, when one goes around such... | null | null | null |
63 | Consider an $SU(2)$ gauge theory with gauge-field Lagrangian
$$
\mathcal{L}
=
-\frac{1}{4}F_{a}^{\mu\nu}F^{a}_{\mu\nu}
-\frac{\theta e^{2}}{32\pi^{2}}
F_{a}^{\mu\nu}\widetilde{F}^{a}_{\mu\nu}
+\cdots .
$$
Using the gauge transformation
$$
\delta A_\mu^a=\frac{1}{ev}D_\mu\phi^a
$$
and the conventions
$$
F_a^{0i}=-(E_a)_... | Non-Abelian | $\mathcal{Q}=\frac{Q}{e}+\frac{\theta e g}{8\pi^2}$ | The Noether charge can be written as $\mathcal{Q}=\int d^3x\,\frac{\partial\mathcal{L}}{\partial(\partial_0A_\mu^a)}\delta A_\mu^a$. The gauge-field Lagrangian is
$$
\mathcal{L}
=
-\frac{1}{4}F_{a}^{\mu\nu}F^{a}_{\mu\nu}
-\frac{\theta e^{2}}{32\pi^{2}}
F_{a}^{\mu\nu}\widetilde{F}^{a}_{\mu\nu}
+\cdots .
$$
Then, we have... | null | null | null |
64 | Consider a four-dimensional gauge theory in Minkowski spacetime with theta angle $\theta$ and coupling $\tau=\frac{\theta}{2\pi}+\frac{4\pi i}{e^2}$. Define
$$
F^2:=F_{\mu\nu}F^{\mu\nu},
\qquad
F{*}F:=F_{\mu\nu}({*}F)^{\mu\nu},
$$
and use the convention $({*}F)^2=-F^2$. The gauge kinetic terms are written in the compac... | Classical | $\mathcal{L}=-\frac{1}{4e^{2}}F^{2}-\frac{\theta}{32\pi^{2}}F*F$ | We can compute
$$
(F+i*F)^{2}=F^{2}+2iF*F-(*F)^{2}=2F^{2}+2iF*F
$$
where we have used the identity $(*F)^{2}=-F^{2}$ in four-dimensional Minkowski spacetime. Taking the imaginary part, we have
$$
\operatorname{Im}(\tau(F+i*F)^{2})= 2\operatorname{Im}(\tau F^{2})+2\operatorname{Re}(\tau F*F).
$$
Since $\tau=\frac{\theta... | null | null | null |
65 | In a four-dimensional $\mathcal{N}=1$ SCFT, the central charges are
$$
a=\frac{3}{32}\left(3\operatorname{Tr}R^3-\operatorname{Tr}R\right),
\qquad
c=\frac{1}{32}\left(9\operatorname{Tr}R^3-5\operatorname{Tr}R\right),
$$
where $\operatorname{Tr}R$ and $\operatorname{Tr}R^3$ denote sums over all left-handed Weyl fermions... | SUSY | $(a,c)=\left(\frac{1}{48},\frac{1}{24}\right)$ | If the theory has a Lagrangian description, then $\operatorname{Tr}$ denotes a sum over all left-handed Weyl fermions. For the free chiral multiplet $\Phi$, the scalar has scaling dimension $\Delta(\phi)=1$. Since a chiral primary satisfies $\Delta(\phi)=\frac{3}{2}R(\phi)$, we obtain $R(\phi)=\frac{2}{3}$. Since the f... | null | null | null |
66 | Consider a primary operator $\mathcal{O}^{a}$ in a $d$-dimensional Euclidean CFT. It has scaling dimension $\Delta$ and transforms in an irreducible representation $R$ of $SO(d)$. In radial quantization
$$
P^{\dagger}_{\mu}=K_{\mu}, \qquad [K_{\mu},P_{\nu}]=2\delta_{\mu\nu}D-2M_{\mu\nu}
$$
The primary state satisfies
$... | CFT | $\Delta \geq s_{R}$ | We compute the nonnegative norm squared of an arbitrary linear combination $|\Psi\rangle=\xi^\nu{}_aP_\nu|\mathcal O^a\rangle$. Then
$$
\begin{aligned}
0
&\leq \langle\Psi|\Psi\rangle\\
&=
(\xi^\mu{}_b)^*\xi^\nu{}_a
\bigl(P_\mu|\mathcal O^b\rangle\bigr)^\dagger
P_\nu|\mathcal O^a\rangle\\
&=
(\xi^\mu{}_b)^*\xi^\nu{}_a
... | null | null | null |
67 | Let $\mathcal O_{\mu_1\cdots\mu_\ell}$ be a spin-$\ell$ symmetric traceless primary operator in a unitary $d$-dimensional CFT with $d\geq3$, and let its scaling dimension be $\Delta$. Determine the unitarity bound on $\Delta$ for $\ell>0$. | CFT | $\Delta \geq \ell+d-2$ | To compute the eigenvalues of $(S_{\mu\nu})^b{}_a$, we can rewrite it in terms of the generators of $SO(d)$. Let $L^{\alpha\beta}$ be the generators acting on the vector representation $V_1$ of $SO(d)$, with
$$
(L^{\alpha\beta})_{\mu\nu}
=
\delta^\alpha_\mu \delta^\beta_\nu
-
\delta^\alpha_\nu \delta^\beta_\mu, \qquad ... | null | null | null |
68 | Let $\phi$ be a real scalar primary operator of scaling dimension $\Delta_\phi$ in a $d$-dimensional CFT, normalized by
$$
\langle \phi(x_1)\phi(x_2)\rangle
=
\frac{1}{x_{12}^{2\Delta_\phi}},
\qquad
x_{ij}:=|x_i-x_j|.
$$
The $\phi\times\phi$ OPE is $\phi(x_1)\phi(x_2)=\sum_{\mathcal O}f_{\phi\phi\mathcal O}\,C_a(x_{12}... | CFT | $\langle \phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4)\rangle=\frac{g(u,v)}{x_{12}^{2\Delta_\phi}x_{34}^{2\Delta_\phi}}$ | Applying the OPE in the $(12)$ and $(34)$ channels, we obtain
$$
\begin{aligned}
\langle \phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4)\rangle &= \sum_{\mathcal O,\mathcal O'} f_{\phi\phi\mathcal O} f_{\phi\phi\mathcal O'} C_a(x_{12},\partial_2) C_b(x_{34},\partial_4) \langle \mathcal O^a(x_2)\mathcal O'^b(x_4)\rangle .
\end{ali... | null | null | null |
69 | Let $\phi$ and $\mathcal{O}$ be scalar primary operators in a $d$-dimensional CFT with scaling dimensions $\Delta_\phi$ and $\Delta$, respectively. Assume that the two-point function is normalized as
$$
\langle \mathcal O(x)\mathcal O(y)\rangle
=
\frac{1}{|x-y|^{2\Delta}}.
$$
Write the $\phi\times\phi$ OPE as
$$
\phi(x... | CFT | $(\alpha,\beta,\gamma)=\left(\frac12,\frac{\Delta+2}{8(\Delta+1)},-\frac{\Delta}{16\left(\Delta-\frac{d-2}{2}\right)(\Delta+1)}\right)$ | By conformal invariance, the three-point function of scalar operators is fixed to be
$$
\langle \phi(x)\phi(0)\mathcal O(z)\rangle = \frac{f_{\phi\phi\mathcal O}} {|x|^{2\Delta_\phi-\Delta}|z|^\Delta |z-x|^\Delta} = \frac{f_{\phi\phi\mathcal O} |x|^{\Delta-2\Delta_\phi}}{|z|^\Delta |z-x|^\Delta}.
$$
On the other hand, ... | null | null | null |
70 | Let $\mathcal{O}_1(x)$ and $\mathcal{O}_2(x)$ be scalar primary operators in a $d$-dimensional Euclidean conformal field theory, with scaling dimensions $\Delta_1$ and $\Delta_2$, respectively. Using conformal invariance, determine the two-point function $\left\langle \mathcal{O}_1(x_1)\mathcal{O}_2(x_2)\right\rangle$ ... | CFT | $$
\left\langle \mathcal{O}_1(x_1)\mathcal{O}_2(x_2)\right\rangle
=
\begin{cases}
\dfrac{C}{|x_1-x_2|^{2\Delta_1}},
& \Delta_1=\Delta_2,\\[6pt]
0,
& \Delta_1\neq\Delta_2.
\end{cases}
$$ | We know that translation, rotation, and scale invariance fix the two-point function of scalars up to a constant
$$
\langle \mathcal{O}_{1}(x_{1}) \mathcal{O}_{2}(x_{2}) \rangle = \frac{C}{|x_{1}-x_{2}|^{\Delta_{1}+\Delta_{2}}}.
$$
For a conformal transformation,
$$
(x_{1}-x_{2})^{2}\Omega(x_{1}')\Omega(x_{2}')=(x_{1}'-... | null | null | null |
71 | Consider a two-dimensional CFT with central charge $(c,\bar{c})=(c,c)$ on the complex plane with stress tensors $T(z)$ and $\overline{T}(\bar{z})$. Define the Virasoro modes by
$$
T(z)=\sum_{n\in\mathbb Z}L_n z^{-n-2},
\qquad
\overline T(\bar z)
=
\sum_{n\in\mathbb Z}\overline L_n\bar z^{-n-2}.
$$
Define the torus $T^2... | CFT | $Z(\tau,\bar{\tau})=\operatorname{Tr}_{\mathcal H}\left(q^{L_0-\frac{c}{24}}\overline q^{\,\overline L_0-\frac{c}{24}}\right)$ | Let $w$ be the coordinate on the covering space of the torus. The first identification
$$
w\sim w+\omega_1,
\qquad
\omega_1=L,
$$
defines a cylinder with spatial circumference $L$. We map this cylinder to the complex plane by
$$
z
=
e^{2\pi i w/\omega_1}
=
e^{2\pi i w/L}.
$$
The stress tensor transforms as
$$
T_{\mathr... | null | null | null |
72 | Consider the free boson CFT with left- and right-moving central charges $(c,\bar c)=(1,1)$. It has a continuum of primary operators
$$
V_\alpha(z,\bar z):=e^{i\alpha\phi(z,\bar z)},
\qquad
\alpha\in\mathbb R,
$$
with corresponding states $|V_\alpha\rangle$ satisfying
$$
L_0|V_\alpha\rangle=\frac{\alpha^2}{2}|V_\alpha\r... | CFT | $Z_{\mathrm{Boson}}(\tau,\bar{\tau})=\frac{1}{\sqrt{2\,\operatorname{Im}\tau}\,|\eta(\tau)|^2}$ | Since we know the value of the central charge and the conformal weights of the primary states, we can compute the partition function explicitly. Let $k=(k_1,\ldots,k_r)$ denote a partition with $1\leq k_1\leq\cdots\leq k_r$, and define $|k|:=\sum_{i=1}^{r}k_i$. We use the analogous notation for $\bar{k}$. Then,
$$
\beg... | null | null | null |
73 | Consider the free boson CFT with torus partition function
$$
Z_{\mathrm{Boson}}(\tau,\bar{\tau})=\frac{1}
{\sqrt{2\,\operatorname{Im}\tau}\,|\eta(\tau)|^2}, \qquad \eta(\tau)=q^{1/24}
\prod_{n=1}^{\infty}(1-q^n),
$$
where $\tau=\tau_1+i\tau_2$, $\tau_2>0$, $q=e^{2\pi i\tau}$. Determine the relation between $Z_{\mathrm{... | CFT | $Z_{\mathrm{Boson}}\!\left(-\frac{1}{\tau},-\frac{1}{\bar{\tau}}\right)=Z_{\mathrm{Boson}}(\tau,\bar{\tau})$ | Under the modular transformation $S:\tau\mapsto -\frac{1}{\tau},\bar{\tau}\mapsto -\frac{1}{\bar{\tau}}$, the Dedekind $\eta$ function satisfies
$$
\left|\eta\!\left(-\frac{1}{\tau}\right)\right|^2
=
|\tau|\,|\eta(\tau)|^2,
$$
and
$$
\operatorname{Im}\!\left(-\frac{1}{\tau}\right)
=
\operatorname{Im}\!\left(-\frac{\ove... | null | null | null |
74 | For $n$ left-moving free complex fermions $\psi_i(z)$ with action
$$
S=\int d^2z\,\psi_i\bar{\partial}\psi^i,
$$
where $\psi^i=(\psi_i)^*$, work in the Neveu-Schwarz sector and adopt the oscillator normalization
$$
\{\psi_{i,r},\psi_s^j\}
=
\delta_i^{\,j}\delta_{r+s,0},\qquad \{\psi_{i,r},\psi_{j,s}\}=0,
\qquad
\{\psi^... | CFT | $\langle \psi_i(z)\psi^j(w)\rangle=\frac{\delta_i^{\,j}}{z-w}$ | The mode expansions are
$$
\psi_i(z)=\sum_{r\in\mathbb Z+\frac12}\psi_{i,r}\,z^{-r-\frac12},
\qquad
\psi^i(z)=\sum_{r\in\mathbb Z+\frac12}\psi^i_r\,z^{-r-\frac12}.
$$
The fermionic modes satisfy the anti-commutation relations
$$
\{\psi_{i,r},\psi^j_s\}
=
\delta_i^{\,j}\delta_{r+s,0}, \qquad \{\psi_{i,r},\psi_{j,s}\}=0,... | null | null | null |
75 | Let $\psi_i(z)$, $i=1,\ldots,n$, be $n$ left-moving free complex fermions, and let $\psi^i(z)$ be their complex conjugates. Let $\psi_{i,r}$ and $\psi_r^i$, with $r\in\mathbb Z+\frac12$, denote their oscillator modes in the Neveu-Schwarz sector, normalized by
$$
\{\psi_{i,r},\psi_s^j\}
=
\delta_i^{\,j}\delta_{r+s,0},
\... | CFT | $\mathfrak u(n)$ | Using the creation-annihilation normal ordering, the current can be expanded as
$$
J_i^{\,j}(z)
=
:\psi_i(z)\psi^j(z):
=
\sum_{r,s\in\mathbb Z+\frac12}
:\psi_{i,r}\psi^j_s:\,z^{-r-s-1}
=
\sum_{m\in\mathbb Z}J_{i,m}^{\,j}z^{-m-1}.
$$
Hence $J_{i,m}^{\,j}=\sum_{r\in\mathbb Z+\frac12}:\psi_{i,m-r}\psi^j_r:$. To compute th... | null | null | null |
LLMs Can Tackle High Energy Theoretical Physics, yet Gaps Remain:
Part I, Quantum Field Theory
QFT-Bench is a benchmark designed to evaluate the reasoning capabilities of large language models (LLMs) in quantum field theory (QFT), a fundamental theoretical framework underlying modern particle physics.
The benchmark consists of 75 problems covering eight QFT core areas:
- Classical Field Theory (Classical)
- Quantization of Scalar, Spinor, and Vector Fields (Quantization)
- Quantum Electrodynamics (QED)
- Regularization and Renormalization (Renormalization)
- Non-Abelian Gauge Theory and QCD (Non-Abelian)
- Electroweak Theory and the Higgs Mechanism (Electroweak)
- Supersymmetric Field Theories (SUSY)
- Conformal Field Theories (CFT)
Dataset Structure
The dataset is provided in test.json. Each problem contains the following fields:
| Field | Description |
|---|---|
num |
Problem number |
question |
Problem statement |
category |
QFT topic category |
reference_answer |
Reference final answer |
reference_solution |
Reference derivation and solution |
numeric_tolerance, reference_answer_alt, specific_requirement |
Additional evaluation instructions when applicable |
Mathematical expressions are represented using LaTeX.
Load the dataset
from datasets import load_dataset
dataset = load_dataset("LLMsHEP/QFT-Bench", split="test")
print(f"Number of problems: {len(dataset)}")
print(dataset[0]["question"])
Citation
If you find it useful, feel free to cite our paper (that's an encouragement to us):
@article{llmsqft,
title = {LLMs Can Tackle High Energy Theoretical Physics, yet Gaps Remain: Part I, Quantum Field Theory},
author = {Liu, Tong and Li, Bohan and Cheng, Gongrui and Yu, Zixiong and He, Yulan and Tresp, Volker},
journal = {arXiv preprint arXiv:2610.33134},
url = {https://qft-bench.github.io/},
year = {2026}
}
License
This dataset is released under the Creative Commons Attribution 4.0 International (CC BY 4.0) license.
- Downloads last month
- 118