The Standard Model Lagrangian — Full Structure
The complete Lagrangian of particle physics: gauge kinetic, fermion kinetic, Higgs, and Yukawa sectors built on the gauge group SU(3)×SU(2)×U(1)
1. The SM Gauge Group
The Standard Model of particle physics is a Yang–Mills gauge theory with gauge group:
$$G_{\mathrm{SM}} = \mathrm{SU}(3)_C \times \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$$
Each factor contributes its own gauge bosons: $\mathrm{SU}(3)_C$ provides 8 gluon fields $G^a_\mu$ ($a = 1, \ldots, 8$),$\mathrm{SU}(2)_L$ provides 3 weak bosons $W^i_\mu$ ($i = 1, 2, 3$), and $\mathrm{U}(1)_Y$ provides the hypercharge boson $B_\mu$. The total dimension of the gauge group is $8 + 3 + 1 = 12$.
Compare this with general relativity, where the relevant “gauge group” is the diffeomorphism group Diff(M), an infinite-dimensional group. The SM has a finite-dimensional internal symmetry group, while GR has an infinite-dimensional spacetime symmetry group.
2. Gauge Kinetic Terms
The gauge kinetic sector contains one term for each factor of the gauge group. The field strength tensors are defined by the structure constants of each Lie algebra:
$$\mathcal{L}_{\mathrm{gauge}} = -\frac{1}{4}G^a_{\mu\nu}G^{a\mu\nu} - \frac{1}{4}W^i_{\mu\nu}W^{i\mu\nu} - \frac{1}{4}B_{\mu\nu}B^{\mu\nu}$$
The individual field strengths are:
$$G^a_{\mu\nu} = \partial_\mu G^a_\nu - \partial_\nu G^a_\mu + g_s f^{abc} G^b_\mu G^c_\nu$$
$$W^i_{\mu\nu} = \partial_\mu W^i_\nu - \partial_\nu W^i_\mu + g\,\epsilon^{ijk} W^j_\mu W^k_\nu$$
$$B_{\mu\nu} = \partial_\mu B_\nu - \partial_\nu B_\mu$$
The non-abelian field strengths ($G^a_{\mu\nu}$ and $W^i_{\mu\nu}$) contain self-interaction terms due to the non-commutativity of their Lie algebras, exactly as the Riemann tensor contains terms quadratic in the Christoffel symbols.
3. Fermion Kinetic Terms and the Covariant Derivative
Each fermion field couples to the gauge bosons through the covariant derivative. The fermion kinetic Lagrangian is:
$$\mathcal{L}_{\mathrm{fermion}} = i\bar{\psi}\gamma^\mu D_\mu \psi$$
The full SM covariant derivative, acting on a left-handed quark doublet for example, is:
$$D_\mu = \partial_\mu - ig_s G^a_\mu T^a - ig\,W^i_\mu \frac{\tau^i}{2} - ig'\,B_\mu\,Y$$
Here $T^a$ are the SU(3) generators (Gell-Mann matrices divided by 2),$\tau^i/2$ are the SU(2) generators (Pauli matrices divided by 2), and $Y$ is the weak hypercharge. The three coupling constants $g_s$, $g$, and $g'$ parametrize the strength of each interaction.
Compare with the GR covariant derivative on a spinor:
$$\nabla_\mu \psi = \partial_\mu \psi + \frac{1}{4}\omega_\mu^{\ ab}\gamma_a\gamma_b\,\psi$$
The spin connection $\omega_\mu^{\ ab}$ plays the same role as the gauge fields, with the Lorentz group SO(1,3) replacing the internal gauge group.
4. The Higgs Sector
The Higgs field is a complex SU(2) doublet with hypercharge $Y = 1/2$:
$$\Phi = \begin{pmatrix} \phi^+ \\ \phi^0 \end{pmatrix}, \qquad \mathcal{L}_{\mathrm{Higgs}} = (D_\mu\Phi)^\dagger(D^\mu\Phi) - V(\Phi)$$
The Higgs potential is the Mexican hat potential:
$$V(\Phi) = -\mu^2 \Phi^\dagger\Phi + \lambda(\Phi^\dagger\Phi)^2$$
For $\mu^2 > 0$, the minimum occurs at $|\Phi| = v/\sqrt{2}$ where the vacuum expectation value is $v = \mu/\sqrt{\lambda} \approx 246$ GeV. The vacuum manifold is $S^3 \cong \mathrm{SU}(2)$, and choosing a vacuum breaks the symmetry:
$$\mathrm{SU}(2)_L \times \mathrm{U}(1)_Y \longrightarrow \mathrm{U}(1)_{\mathrm{em}}$$
Three of the four Higgs degrees of freedom become the longitudinal modes of $W^\pm$and $Z^0$, while the fourth becomes the physical Higgs boson $H$with mass $m_H = \sqrt{2\lambda}\,v \approx 125$ GeV.
5. Yukawa Couplings and Fermion Masses
Fermion masses arise from Yukawa couplings to the Higgs field. For quarks:
$$\mathcal{L}_{\mathrm{Yukawa}} = -y^u_{ij}\,\bar{Q}_L^i\,\tilde{\Phi}\,u_R^j - y^d_{ij}\,\bar{Q}_L^i\,\Phi\,d_R^j - y^e_{ij}\,\bar{L}_L^i\,\Phi\,e_R^j + \mathrm{h.c.}$$
where $\tilde{\Phi} = i\tau^2\Phi^*$ is the charge conjugate Higgs doublet,$Q_L$ is the left-handed quark doublet, and $L_L$ is the left-handed lepton doublet. After symmetry breaking, each fermion acquires mass:
$$m_f = \frac{y_f\,v}{\sqrt{2}}$$
The Yukawa matrices $y^u_{ij}$, $y^d_{ij}$, and $y^e_{ij}$are $3 \times 3$ complex matrices (one for each generation). Diagonalizing them produces the CKM quark mixing matrix and the fermion mass eigenstates.
6. Counting Degrees of Freedom
The full Standard Model Lagrangian brings together all sectors:
$$\mathcal{L}_{\mathrm{SM}} = \mathcal{L}_{\mathrm{gauge}} + \mathcal{L}_{\mathrm{fermion}} + \mathcal{L}_{\mathrm{Higgs}} + \mathcal{L}_{\mathrm{Yukawa}}$$
The bosonic degrees of freedom: 8 gluons $\times$ 2 polarizations = 16,$W^\pm$ with 3 polarizations each = 6, $Z^0$ with 3 = 3, photon with 2 = 2, Higgs with 1 = 1. Total bosonic: 28.
The fermionic degrees of freedom: 3 generations $\times$ (2 quarks $\times$ 3 colors$\times$ 2 chiralities $\times$ 2 spin + 2 leptons$\times$ 2 chiralities $\times$ 2 spin) = 3 $\times$ 30 = 90. Total fermionic: 90.
By contrast, the Einstein–Hilbert action for pure gravity contains only the metric$g_{\mu\nu}$ with its 2 physical (transverse-traceless) degrees of freedom:
$$S_{\mathrm{EH}} = \frac{1}{16\pi G}\int R\,\sqrt{-g}\,d^4x \quad \longleftrightarrow \quad S_{\mathrm{SM}} = \int\mathcal{L}_{\mathrm{SM}}\,\sqrt{-g}\,d^4x$$
The structural parallel is clear: both are built from curvatures of connections, but the SM curvature lives in an internal fibre bundle while the GR curvature lives in the tangent bundle of spacetime itself.
Simulation: Standard Model Particle Spectrum
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