David Tong Standard Model Solutions — Sheet 4: The Standard Model

These prompts paraphrase David Tong's The Standard Model, Example Sheet 4 (March 2026). Problem numbers, starred status, equations, and guidance follow the official sheet.

1 Sheet 4, Problem 1: Running Gauge Couplings

For $SU(N_c)$ with $N_f$ massless fundamental Dirac fermions and $N_s$ massless fundamental scalars, use \[\frac1{g^2(\mu)}=\frac1{g_0^2} -\frac{11N_c-2N_f-N_s/2}{3(4\pi)^2} \log\frac{\Lambda_{\rm UV}^2}{\mu^2}.\tag{1.1}\] \begin{enumerate}[label=\roman*)]
Prove that the Standard Model $SU(2)$ sector is asymptotically free.
Without a Higgs boson, estimate the strong-coupling scale $\Lambda_{\rm weak}$ from $\alpha_W(100\,\mathrm{GeV})\simeq1/30$.
If the Higgs did not condense, explain the effect on the weak force of a QCD condensate $\langle\bar q_{Li}q_{Rj}\rangle \simeq-\Lambda_{QCD}^3\delta_{ij}$.
Given $\alpha_s(100\,\mathrm{GeV})\simeq0.1$, estimate the scale $\mu\gg90\,\mathrm{GeV}$ where $\alpha_s=\alpha_W$.

For Standard Model $SU(2)$, we have $N_c = 2, N_s = 1$ and $N_f = 3\cdot (3+1)/2$, where the $3$ comes from three generations and the $4$ comes from $3$ colour of quarks and leptons. Therefore, we have \[\frac{1}{g^2(\mu_1)} - \frac{1}{g^2(\mu_2)} = -\frac{19}{6(4\pi)^2}\log\frac{\mu_2^2}{\mu_1^2}.\tag{1.2}\] For $\mu_1 > \mu_2$, we have $g^2(\mu_1) < g^2(\mu_2)$, and hence the theory is asymptotically free.
From the one-loop relation in Sheet 2, Problem 5, we have \[\Lambda_\text{weak} = (100\, {\rm GeV})\exp\left(-\frac{6\pi}{10} \cdot 30\right) = 2.76 \times 10^{-14} \, {\rm eV}.\tag{1.3}\]
Even if the $SU(2)$ symmetry is not broken by the Higgs, the condensate still breaks the $SU(2)$.
For $SU(3)$, $N_c = 3, N_f = 6, N_s = 0$. Then, using the one-loop relation in Sheet 2, Problem 5, we find \[\Lambda_\text{QCD} = (100\,\mathrm{GeV})\exp\left(-\frac{6\pi}{21}\cdot 10\right) = 1.26\times 10^{-2} \, {\rm GeV}.\tag{1.4}\] When $\alpha_W = \alpha_s$, we find \[\mu = \left(\frac{\Lambda_\text{weak}^{10}}{\Lambda_\text{QCD}^{21}}\right)^{-1/11} = 7.7 \times 10^{16}\, {\rm GeV}.\tag{1.5}\]

\item For $U(1)$ coupled to massless Weyl fermions of charges $Y_i$ and scalars of charges $\tilde Y_i$, use \[\frac1{g^2(\mu)}=\frac1{g_0^2} +\frac{2\sum_iY_i^2+\sum_i\tilde Y_i^2} {3(4\pi)^2} \log\frac{\Lambda_{\rm UV}^2}{\mu^2}.\tag{1.6}\] Decide whether hypercharge is asymptotically free. With $\alpha_Y(100\,\mathrm{GeV})\simeq1/98$, find the scales where $\alpha_s=\alpha_Y$ and $\alpha_W=\alpha_Y$.

Since this is always positive, the hypercharge is not asymptotically free. In particular, for one generation \[\sum_{\rm Weyl}Y_i^2 = 6\left(\frac16\right)^2 +3\left(\frac23\right)^2 +3\left(-\frac13\right)^2 +2\left(-\frac12\right)^2 +(-1)^2 = \frac{10}{3}.\tag{1.7}\] For the scalars, we only have the Higgs \[\sum_{\rm scalars}\tilde Y_i^2=2\left(\frac12\right)^2=\frac12.\tag{1.8}\] Therefore, we have \[2\sum_iY_i^2+\sum_i\tilde Y_i^2=20+\frac12=\frac{41}{2}\tag{1.9}\] Hence, through the one-loop relation in Sheet 2, Problem 5, we find \[\Lambda_Y=100\,\mathrm{GeV}\,\exp\left(\frac{1176\pi}{41}\right) = 1.36\times10^{41}\,\mathrm{GeV}.\tag{1.10}\] Following the same procedure as in the previous sections, we find \[\mu_{sY} = \left( \Lambda_{\rm QCD}^{42} \Lambda_Y^{41} \right)^{1/83} = 2.28 \times 10^{19} {\rm GeV},\tag{1.11}\] and \[\mu_{WY} = \left( \Lambda_{\rm weak}^{19} \Lambda_Y^{41} \right)^{1/60}=3.59\times10^{20}\,\mathrm{GeV}.\tag{1.12}\]

\end{enumerate}

2 Sheet 4, Problem 2: Charge of a New Multiplet

A hypothetical particle transforms as a $\mathbf3$ of $SU(2)$ and a $\mathbf6$ of $SU(3)$, with hypercharge $Y$. Find its electric charges after electroweak symmetry breaking.

3 Sheet 4, Problem 4: Counting Flavour Parameters

With $N$ generations, count the physical parameters in the quark Yukawa matrices and classify them as masses, mixing angles, and complex phases. Then assume neutrino masses arise from a dimension-five operator and perform the corresponding count for the charged-lepton Yukawa matrix together with that operator.

The two Yukawa matrices, $y^u, y^d$, have $2N^2$ complex parameters, and hence $2N^2$ real parameters and $2N^2$ phase angles. Since the kinetic term is invariant under $U(N)$ transformation, the physics remains under such transformations as long as the Yukawa matrices transform as \[y^u \to V^\dagger y^u U^u, \quad Y^d \to V^\dagger y^d U^d,\tag{3.1}\] where $Y,U^u,U^d \in U(N)$. $U(N)$ can be thought of as a decomposition of $O(N)$ and $U(1)^n$. $\dim U(n) - \dim O(n) = N^2 - \frac{1}{2}N(N-1) = \frac{1}{2}N(N+1) = n$. Therefore, there are \[2N^2 - \frac{3}{2}N(N-1) = \frac{1}{2}N^2 + \frac{3}{2}N\tag{3.2}\] real physical parameters, whereas $2N$ of them are physical mass parameters, and the rest are the mixing angles. Meanwhile, if all of $Y,U^u,U^d$ shifts by the same phase, the Yukawa potential remains unchanged, we must take away an $U(1)$ phase. Hence, there are \[2N^2 - \frac{3}{2}N(N+1) +1 = \frac{1}{2}N^2 - \frac{3}{2}N +1\tag{3.3}\] physical complex phases.

A similar analysis works in the electron-neutrino sector. The mass Lagrangian is given by \[-\bar L\,y^e e_R H + \frac{1}{2\Lambda}\kappa_{ij} (L_iH)(L_jH) +h.c.\tag{3.4}\] Here $y^e$ is an arbitrary complex $N\times N$ matrix, while $\kappa$ is a complex $N\times N$ symmetric matrix. Therefore, there are $N^2$ real parameters and $N^2$ complex phases in the electron Yukawa coupling and $\frac{1}{2}N(N+1)$ real parameters and $\frac{1}{2}N(N+1)$ complex phases in the dimension 5 operator. The Kinetic Lagrangian has a $U(N)_L \times U(N)_e$ symmetry, which the couplings transforms as \[y^e \to W^\dagger y^e U^e, \quad \kappa \to W^T \kappa W.\tag{3.5}\] Note that there is no overall $U(1)$ phase that preserves the dimension 5 coupling. Therefore, we have $N(N+3)/2$ real physical parameters and $N(N-1)/2$ physical phases. These are $N$ lepton masses, $N$ neutrino masses, $N(N-1)/2$ mixing angles and $N(N-1)/2$ complex phases.

4 Sheet 4, Problem 5*: A Two-Generation Cabibbo Angle

For two quark families, let \[\mathcal L_{\rm mass} =-m_{ij}\bar q^{i}q^j -\tilde m_{ij}\bar{\tilde q}^{i}\tilde q^{j}, \qquad m=\begin{pmatrix}0&a\\a^*&b\end{pmatrix}, \quad \tilde m=\begin{pmatrix}0&c\\c^*&d\end{pmatrix},\tag{4.1}\] with $a,c\in\mathbb C$ and $b,d\in\mathbb R$.

Rephase quarks to make $a,c$ real and positive.
Let rotations $R(\theta),R(\tilde\theta)$ diagonalise the two matrices into $(m_1,-m_2)$ and $(\tilde m_1,-\tilde m_2)$. Show \[\tan\theta=\sqrt{\frac{m_1}{m_2}}, \qquad \tan\tilde\theta=\sqrt{\frac{\tilde m_1}{\tilde m_2}}.\tag{4.2}\]
Derive $\theta_C=\theta-\tilde\theta$ from the weak charged-current mixing matrix.

$m$ can be written as \[m = \begin{pmatrix} 0 & |a|e^{i\alpha}\\ |a| e^{-i\alpha} & b \end{pmatrix}.\tag{4.3}\] The kinetic term is invariant under \[q^1 \to e^{-i\alpha}q^1,\tag{4.4}\] whereas $m$ transforms as \[m_{12} \to e^{i\alpha} m_{12}, \quad m_{21} \to e^{-i\alpha}m_{21}.\tag{4.5}\] Therefore, we can always make $a,c \in \mathbb{R}^+$.
We have \[m = R(\theta)^{-1}\operatorname{diag}(m_1, -m_2) R(\theta).\tag{4.6}\] Note that the top right entry is zero, we must have \[m_1 \cos^2\theta - m_2 \sin^2 \theta = 0 \implies \tan\theta = \sqrt{\frac{m_1}{m_2}},\tag{4.7}\] and similarly for $\tilde \theta$.
\[R(\theta_c) = R^{-1}(\tilde \theta)R(\theta) \implies \theta_c = \theta - \tilde \theta.\tag{4.8}\]

5 Sheet 4, Problem 6*: A Second Higgs Doublet

Alongside the Standard Model Higgs $H:(\mathbf2,\mathbf1)_{+1/2}$, introduce $\phi:(\mathbf2,\mathbf1)_{-1/2}$, with the entries indicating $SU(2)\times SU(3)$ representations and $U(1)_Y$ charge.

Find the electric charges of the two components of $\phi$.
Construct the most general renormalisable gauge-invariant potential of $H$ and $\phi$ and count its independent real parameters.
Suppose the vacuum is \[\langle H\rangle=\frac1{\sqrt2} \begin{pmatrix}0\\v\end{pmatrix}, \qquad \langle\phi\rangle=\frac1{\sqrt2} \begin{pmatrix}u+iw\\0\end{pmatrix}, \quad u,v,w\in\mathbb R.\tag{5.1}\] Does $\langle\phi\rangle$ break electromagnetism? Find the masses of $W_\mu^\pm=(W_\mu^1\mp iW_\mu^2)/\sqrt2$ and of the appropriate mixtures of $W_\mu^3,B_\mu$. Identify the charges of the scalar modes remaining after symmetry breaking.
List all permitted Yukawa couplings of $H$ and $\phi$ to the Standard Model fermions.

For $\mathbf{2}$ of $SU(2)$, $T^3$ has eigenvalues $\pm \frac{1}{2}$. Hence, the EM charges are $-1, 0$.
Since $[H] = [\phi] = 1$, we can at most have an order $4$ coupling. Meanwhile, we must have two scalar fields coming in pair as well. Finally, to respect $U(1)$ gauge symmetry, the only possible interactions we are left with are \[\begin{aligned} V={}&m_1^2H^\dagger H +m_2^2\phi^\dagger\phi -\left[m_{12}^2 H^T i\sigma^2\phi+\text{h.c.}\right]\\ &+\lambda_1(H^\dagger H)^2 +\lambda_2(\phi^\dagger\phi)^2 +\lambda_3(H^\dagger H)(\phi^\dagger\phi) +\lambda_4|H^T i\sigma^2\phi|^2\\ &+\left[ \lambda_5(H^T i\sigma^2\phi)^2 +\lambda_6(H^\dagger H)(H^T i\sigma^2\phi) +\lambda_7(\phi^\dagger\phi)(H^T i\sigma^2\phi) +\text{h.c.} \right], \end{aligned}\tag{5.2}\] Therefore, there are $6$ independent real parameters.
In the vacuum, we have \[Q\langle \phi\rangle = (T^3 +Y)\langle \phi\rangle = \begin{pmatrix} 0 & 0\\ 0 & -1 \end{pmatrix}\langle \phi\rangle = 0.\tag{5.3}\] Therefore, the $U(1)_\text{EM}$ symmetry survives.
Now, expanding the terms quadratic in the gauge fields around the vacuum, we find \[\mathcal L_{\rm mass} = \frac{v^2+u^2+w^2}{8} \left[ g^2\left((W_\mu^1)^2+(W_\mu^2)^2\right) +(gW_\mu^3-g'B_\mu)^2 \right].\tag{5.4}\] Therefore, \[m_W^2=\frac{g^2}{4}(v^2+u^2+w^2).\tag{5.5}\] Defining \[Z_\mu=\cos\theta_W W_\mu^3-\sin\theta_W B_\mu, \qquad A_\mu=\sin\theta_W W_\mu^3+\cos\theta_W B_\mu,\tag{5.6}\] where \[\cos\theta_W=\frac{g}{\sqrt{g^2+g'^2}},\tag{5.7}\] we obtain \[m_Z^2=\frac{g^2+g'^2}{4}(v^2+u^2+w^2), \qquad m_\gamma^2=0.\tag{5.8}\]
Since $\phi$ has hypercharge $Y_\phi=-1/2$, the allowed interactions are \[\mathcal L_{\phi f} = -\bar Q_L\, y_\phi^u\,\phi\,u_R -\bar Q_L\, y_\phi^d\,\widetilde\phi\,d_R -\bar L\, y_\phi^e\,\widetilde\phi\,e_R +\text{h.c.},\tag{5.9}\] where \[\widetilde\phi=i\sigma^2\phi^*.\tag{5.10}\]