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Electroweak Theory

Field: Quantum Field Theory

The unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction.

Sequence of Expressions

Consider a divergent quantity O\mathcal{O} calculated via loop diagrams, yielding O=Ofinite+Odivergent\mathcal{O} = \mathcal{O}_{finite} + \mathcal{O}_{divergent}. Renormalization requires defining the bare parameters (λ0,m0,Zi\lambda_0, m_0, Z_i) such that the physical, measurable quantities (λ,m,Zi\lambda, m, Z_i) are finite. This is achieved by introducing counterterms LCT\mathcal{L}_{CT} into the Lagrangian: \nLbare=Lrenormalized+LCT\mathcal{L}_{bare} = \mathcal{L}_{renormalized} + \mathcal{L}_{CT} \nWhere LCT\mathcal{L}_{CT} cancels the infinities arising from the loop integrals, e.g., LCT=∑iδZiLi\mathcal{L}_{CT} = \sum_i \delta Z_i \mathcal{L}_i, ensuring that the renormalized Green's functions are finite.
The massive neutral and charged gauge bosons, W±W^{\pm} and ZZ, are linear combinations of the underlying SU(2)LSU(2)_L and U(1)YU(1)_Y fields (Wμ3W^3_{\mu} and BμB_{\mu}). Their masses are derived from the vacuum expectation value (VEV) of the Higgs field, ⟨ϕ⟩=v/2\langle \phi \rangle = v / \sqrt{2}. The mass matrices yield the physical masses: \nM_W = \frac{1}{2} g v \quad \text{and} \nM_Z = \frac{1}{2} \sqrt{g^2 + g'^2} v$$ \nThese masses confirm that the $W$ and $Z$ bosons are massive force carriers, unlike the massless photon $\gamma$.
The Higgs mechanism is described by the potential V(ϕ)=μ2ϕ†ϕ+λ(ϕ†ϕ)2V(\phi) = \mu^2 \phi^{\dagger} \phi + \lambda (\phi^{\dagger} \phi)^2. If μ2<0\mu^2 < 0 and λ>0\lambda > 0, the potential is minimized at a non-zero vacuum expectation value (VEV), ⟨ϕ⟩=v/2\langle \phi \rangle = v / \sqrt{2}. This VEV breaks the symmetry, generating mass terms for the gauge bosons WW and ZZ through the kinetic term Lkin=(Dμϕ)∗(Dμϕ)\mathcal{L}_{kin} = (D_{\mu} \phi)^* (D^{\mu} \phi), and for fermions via Yukawa couplings LYukawa=−yfψˉL ϕ~ τa σa2 → ψˉL mfv ψˉR\mathcal{L}_{Yukawa} = -y_f \bar{\psi}_L \ \tilde{\phi} \ \tau^a \ \frac{\sigma^a}{2} \ \rightarrow \ \bar{\psi}_L \ \frac{m_f}{v} \ \bar{\psi}_R.
The full electroweak Lagrangian density LEW\mathcal{L}_{EW} is constructed from the gauge fields WμaW_{\mu}^a and BμB_{\mu}, the Higgs doublet ϕ\phi, and the matter fields ψ\psi: \nLEW=−14WμaWμa−14BμBμ+(Dμϕ)∗(Dμϕ)−V(ϕ)+ψˉiγμDμψ\mathcal{L}_{EW} = -\frac{1}{4} W_{\mu}^a W^{\mu a} - \frac{1}{4} B_{\mu} B^{\mu} + (D_{\mu} \phi)^* (D^{\mu} \phi) - V(\phi) + \bar{\psi} i \gamma^{\mu} D_{\mu} \psi \nWhere Dμ=∂μ−igτ⃗2⋅W⃗μ−ig′Y2BμD_{\mu} = \partial_{\mu} - i g \frac{\vec{\tau}}{2} \cdot \vec{W}_{\mu} - i g' \frac{Y}{2} B_{\mu}. This expression is invariant under the local gauge transformations of U(1)Y×SU(2)LU(1)_Y \times SU(2)_L.
The effective low-energy Fermi interaction, describing weak processes at energy scales E≪MW,MZE \ll M_W, M_Z, is represented by the four-fermion Lagrangian density LFermi\mathcal{L}_{Fermi}: \nLFermi=−GF2[JCCμJCCμ+JNCμJNCμ]\mathcal{L}_{Fermi} = -\frac{G_F}{\sqrt{2}} \left[ J^{\mu}_{CC} J^{\mu}_{CC} + J^{\mu}_{NC} J^{\mu}_{NC} \right] \nWhere GFG_F is the Fermi constant, and the charged and neutral current four-fermion currents are defined as: \nJCCμ=νˉeγμ1−γ52e+μˉγμ1−γ52μ†+…J^{\mu}_{CC} = \bar{\nu}_e \gamma^{\mu} \frac{1 - \gamma_5}{2} e + \bar{\mu} \gamma^{\mu} \frac{1 - \gamma_5}{2} \mu^{\dagger} + \dots \nThis approximation arises from the WW and ZZ boson propagators being approximated by g2M2−q2≈g2M2\frac{g^2}{M^2 - q^2} \approx \frac{g^2}{M^2}.
The Weinberg angle θW\theta_W defines the mixing between the neutral gauge bosons BμB_{\mu} (hypercharge) and Wμ3W^3_{\mu} (weak isospin) to form the physical photon AμA_{\mu} and the ZZ boson ZμZ_{\mu}. The transformation is given by the rotation matrix: \n(Aμ Zμ)=(cos⁡θWsin⁡θW)(Bμ Wμ3)\begin{pmatrix} A_{\mu} \ Z_{\mu} \end{pmatrix} = \begin{pmatrix} \cos \theta_W & \sin \theta_W \end{pmatrix} \begin{pmatrix} B_{\mu} \ W^3_{\mu} \end{pmatrix} \nThis angle relates the coupling constants and masses: cos⁡θW=MZMZMWMW11−MW2/MZ2\cos \theta_W = \frac{M_Z}{M_Z} \frac{M_W}{M_W} \frac{1}{\sqrt{1 - M_W^2/M_Z^2}} (or simply cos⁡θW=MW2/MZ2\cos \theta_W = \sqrt{M_W^2/M_Z^2} in the limit of small mixing) and determines the electric charge e=gsin⁡θWe = g \sin \theta_W.
The electric charge QQ is quantized and is related to the generators of the gauge group U(1)YU(1)_Y and SU(2)LSU(2)_L by the Gell-Mann–Nishijima formula: \nQ=T3+Y2Q = T_3 + \frac{Y}{2} \nWhere T3T_3 is the third component of the weak isospin (eigenvalue of SU(2)LSU(2)_L), and YY is the weak hypercharge (eigenvalue of U(1)YU(1)_Y). The quantization condition dictates that QQ must be an integer multiple of the fundamental unit ee, i.e., Q=neQ = n e, where n∈Zn \in \mathbb{Z}.
The Standard Model Lagrangian density LSM\mathcal{L}_{SM} is given by the sum of kinetic terms for the gauge fields and the matter fields, respecting the gauge group G=U(1)Y×SU(2)LG = U(1)_Y \times SU(2)_L: \nLSM=LGauge+LHiggs+LFermion\mathcal{L}_{SM} = \mathcal{L}_{Gauge} + \mathcal{L}_{Higgs} + \mathcal{L}_{Fermion} \nWhere LGauge=−14W⃗μ⋅W⃗μ−14BμBμ\mathcal{L}_{Gauge} = -\frac{1}{4} \vec{W}_{\mu} \cdot \vec{W}^{\mu} - \frac{1}{4} B_{\mu} B^{\mu}, LHiggs=(Dμϕ)∗(Dμϕ)−V(ϕ)\mathcal{L}_{Higgs} = (D_{\mu} \phi)^* (D^{\mu} \phi) - V(\phi), and LFermion=ψˉ(iγμDμ−M)ψ\mathcal{L}_{Fermion} = \bar{\psi} (i \gamma^{\mu} D_{\mu} - M) \psi. The covariant derivative DμD_{\mu} incorporates the gauge couplings gg and g′g'.
The electroweak theory is based on the local gauge symmetry group G=U(1)Y×SU(2)LG = U(1)_Y \times SU(2)_L. Under a gauge transformation parameterized by α(x)=αY(x)1+τ⃗⋅α(x)⃗\alpha(x) = \alpha_Y(x) \mathbf{1} + \vec{\tau} \cdot \vec{\alpha(x)}, the matter fields ψ\psi transform as: \nψ′=eig′αY(x)1+igτ⃗⋅α(x)⃗/2ψ\psi' = e^{i g' \alpha_Y(x) \mathbf{1} + i g \vec{\tau} \cdot \vec{\alpha(x)} / 2} \psi \nThis transformation dictates the covariant derivative Dμ=∂μ−ig′Y2Bμ−igτ⃗2⋅W⃗μD_{\mu} = \partial_{\mu} - i g' \frac{Y}{2} B_{\mu} - i g \frac{\vec{\tau}}{2} \cdot \vec{W}_{\mu}, ensuring the Lagrangian remains invariant.
Consider a scalar field ϕ\phi governed by a potential V(ϕ)V(\phi) with a Mexican hat shape, V(ϕ)=μ2ϕ†ϕ+λ(ϕ†ϕ)2V(\phi) = \mu^2 \phi^{\dagger} \phi + \lambda (\phi^{\dagger} \phi)^2, where μ2<0\mu^2 < 0. The system minimizes its energy by acquiring a non-zero vacuum expectation value (VEV): \n⟨ϕ⟩=v/2, where v=−μ2/λ\langle \phi \rangle = v / \sqrt{2}, \text{ where } v = \sqrt{-\mu^2 / \lambda} \nThis process breaks the symmetry G→HG \to H (e.g., SU(2)L×U(1)Y→U(1)EMSU(2)_L \times U(1)_Y \to U(1)_{EM}), resulting in mass generation for the gauge bosons and fermions.