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Standard Model

Field: Particle Physics

A theory concerning the electromagnetic, weak, and strong nuclear interactions.

Sequence of Expressions

The Standard Model Lagrangian LSM\mathcal{L}_{SM} is constructed using three generations of left-handed fermion doublets Li=(νi)L,(e)LL_i = \begin{pmatrix} \nu_i \end{pmatrix}_L, \begin{pmatrix} e \end{pmatrix}_L and right-handed singlets eiRe_{iR}, and three generations of quark doublets Qi=(u)L,(d)LQ_i = \begin{pmatrix} u \end{pmatrix}_L, \begin{pmatrix} d \end{pmatrix}_L and right-handed singlets uiR,diRu_{iR}, d_{iR}. The kinetic and interaction terms are given by the sum over generations i=1,2,3i=1, 2, 3: \nLfermion=∑i=13[LˉiiγμDμLi+QˉiiγμDμQi+eˉiRiγμDμeiR+uˉiRiγμDμuiR+dˉiRiγμDμdiR]+LYukawa\mathcal{L}_{fermion} = \sum_{i=1}^{3} \left[ \bar{L}_i i \gamma^{\mu} D_{\mu} L_i + \bar{Q}_i i \gamma^{\mu} D_{\mu} Q_i + \bar{e}_{iR} i \gamma^{\mu} D_{\mu} e_{iR} + \bar{u}_{iR} i \gamma^{\mu} D_{\mu} u_{iR} + \bar{d}_{iR} i \gamma^{\mu} D_{\mu} d_{iR} \right] + \mathcal{L}_{Yukawa}
Let L0\mathcal{L}_0 be the bare Lagrangian density containing bare parameters θ0\theta_0. Renormalization requires defining the renormalized Lagrangian LR\mathcal{L}_R such that L0=LR+LCT\mathcal{L}_0 = \mathcal{L}_R + \mathcal{L}_{CT}, where LCT\mathcal{L}_{CT} is the counterterm Lagrangian. The bare parameters are related to the physical (renormalized) parameters θR\theta_R and the counterterms δθ\delta\theta via θ0=θR+δθ\theta_0 = \theta_R + \delta\theta. The requirement that physical observables remain finite leads to the renormalization group equations (RGEs) for the running coupling constants αi\alpha_i: \nμdαidμ=βi(αi,… )\mu \frac{d\alpha_i}{d\mu} = \beta_i(\alpha_i, \dots) \nwhere μ\mu is the renormalization scale and βi\beta_i are the beta functions.
For a fundamental fermion ff with corresponding Yukawa coupling yfy_f, the mass term mfm_f arises from the interaction Lagrangian LYukawa=−yfψˉLΦψR+h.c.\mathcal{L}_{Yukawa} = -y_f \bar{\psi}_L \Phi \psi_R + h.c.. After spontaneous symmetry breaking, the Higgs field Φ\Phi acquires a vacuum expectation value ⟨Φ⟩=v/2\langle \Phi \rangle = v / \sqrt{2}. The resulting mass formula is:\nmf=yfv2m_f = \frac{y_f v}{\sqrt{2}} \nFor charged leptons, me=yev/2m_e = y_e v / \sqrt{2}. For quarks, mq=yqv/2m_q = y_q v / \sqrt{2}. The coupling yfy_f is determined by the ratio of the mass to the vacuum expectation value.
Let GFG_F be a proposed flavor symmetry group acting on the flavor indices i,j→i′,j′i, j \to i', j'. The Lagrangian density L\mathcal{L} must be invariant under this transformation: L→L′=L\mathcal{L} \to \mathcal{L}' = \mathcal{L}. This invariance implies that the mass matrices MijM_{ij} for quarks and leptons must transform covariantly under GFG_F. Specifically, if MM is the mass matrix, then GFG_F dictates that MM must be proportional to the identity matrix or satisfy specific relations derived from the symmetry generators TaT^a: \n∑aTaMTa=M\sum_{a} T^a M T^a = M \nThis constrains the structure of the Yukawa couplings YijY_{ij} such that Yij=Yij(GF)Y_{ij} = Y_{ij}(G_F).
Consider the weak interaction governed by the SU(2)LSU(2)_L symmetry. The Lagrangian density LWeak\mathcal{L}_{Weak} involves the coupling of the left-handed fermion doublet ψL=(νe,e)L\psi_L = (\nu_e, e)_L to the WμaW^a_{\mu} gauge bosons: LWeak=−12WμaWaμ+ψˉLγμτa2WμaψL+h.c.\mathcal{L}_{Weak} = - \frac{1}{2} W^a_{\mu} W^{a\mu} + \bar{\psi}_L \gamma^{\mu} \frac{\tau^a}{2} W^a_{\mu} \psi_L + \text{h.c.} After spontaneous symmetry breaking via the Higgs mechanism, the mass terms for the W±W^{\pm} and Z0Z^0 bosons are generated, leading to the effective interaction Lagrangian: Leff=g22ψˉγμτaψWμa+g′2ψˉγμψBμ−MW22Wμ+W−μ−MZ22ZμZμ\mathcal{L}_{eff} = \frac{g}{2 \sqrt{2}} \bar{\psi} \gamma^{\mu} \tau^a \psi W^a_{\mu} + \frac{g'}{2} \bar{\psi} \gamma^{\mu} \psi B_{\mu} - \frac{M_W^2}{2} W_{\mu}^+ W^{-\mu} - \frac{M_Z^2}{2} Z_{\mu} Z^{\mu} where τa\tau^a are Pauli matrices and g,g′g, g' are the coupling constants.
The strong interaction is described by Quantum Chromodynamics (QCD), based on the SU(3)SU(3) gauge group. The Lagrangian density LQCD\mathcal{L}_{QCD} is given by: LQCD=−14∑a=18GμaGaμ+ψˉ(iγμDμ−m)ψ\mathcal{L}_{QCD} = - \frac{1}{4} \sum_{a=1}^{8} G^a_{\mu} G^{a\mu} + \bar{\psi} (i \gamma^{\mu} D_{\mu} - m) \psi where ψ\psi are the quark fields, and Dμ=∂μ−igsTaGμaD_{\mu} = \partial_{\mu} - i g_s T^a G^a_{\mu} is the covariant derivative. The gluon field strength tensor GμνaG^a_{\mu\nu} is non-abelian: Gμνa=∂μGνa−∂νGμa+gsfabcGμbGνcG^a_{\mu\nu} = \partial_{\mu} G^a_{\nu} - \partial_{\nu} G^a_{\mu} + g_s f^{abc} G^b_{\mu} G^c_{\nu} The term fabcf^{abc} represents the structure constants of SU(3)SU(3), which dictates the self-interaction of gluons, leading to asymptotic freedom and color confinement.
Let G\mathcal{G} be a compact Lie group (e.g., U(1)×SU(2)×SU(3)U(1) \times SU(2) \times SU(3)) defining the gauge symmetry. The action SS is defined by the integral of the Lagrangian density L\mathcal{L} over spacetime: S=∫d4xL(ϕ,A,ψ;g)S = \int d^4x \mathcal{L}(\phi, A, \psi; g) where ϕ\phi are scalar fields, ψ\psi are matter fields, and AA are gauge fields. The Lagrangian must be invariant under local gauge transformations: ψ′=eigαaTaψ,A′=A−1g∂α\psi' = e^{i g \alpha^a T^a} \psi, \quad A' = A - \frac{1}{g} \partial \alpha The gauge field strength tensor FμνF_{\mu\nu} is constructed from the covariant derivative Dμ=∂μ−igTaAμaD_{\mu} = \partial_{\mu} - i g T^a A^a_{\mu} such that the kinetic term for the gauge bosons is Lgauge=−14Tr(FμνFμν)\mathcal{L}_{gauge} = - \frac{1}{4} \text{Tr}(F_{\mu\nu} F^{\mu\nu}), ensuring the theory is renormalizable and consistent with the underlying symmetry G\mathcal{G}.
Define the Lagrangian density LQED\mathcal{L}_{QED} for the interaction of the electron field ψ\psi (Dirac spinor) and the photon field AμA_{\mu} (gauge boson) under U(1)U(1) symmetry: LQED=ψˉ(iγμ∂μ−me)ψ−14FμνFμν\mathcal{L}_{QED} = \bar{\psi} (i \gamma^{\mu} \partial_{\mu} - m_e) \psi - \frac{1}{4} F_{\mu\nu} F^{\mu\nu} where Fμν=∂μAν−∂νAμF_{\mu\nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu} is the electromagnetic field strength tensor. The equation of motion for the gauge field is derived from the variation δS/δAμ=0\delta S / \delta A_{\mu} = 0, yielding the inhomogeneous Maxwell's equations: ∂μFμν=jν\partial^{\mu} F_{\mu\nu} = j_{\nu} where jν=eψˉγνψj_{\nu} = e \bar{\psi} \gamma_{\nu} \psi is the conserved electromagnetic current, ensuring charge conservation.
Define the QCD Lagrangian density LQCD\mathcal{L}_{QCD} for NfN_f flavors of quarks ψi\psi_i and the gluon field GμaG^a_{\mu}: LQCD=∑i=1Nfψˉi(iγμDμ−mi)ψi−14GμaGaμ\mathcal{L}_{QCD} = \sum_{i=1}^{N_f} \bar{\psi}_i (i \gamma^{\mu} D_{\mu} - m_i) \psi_i - \frac{1}{4} G^a_{\mu} G^{a\mu} where the covariant derivative DμD_{\mu} is defined as Dμ=∂μ−igsTaGμaD_{\mu} = \partial_{\mu} - i g_s T^a G^a_{\mu}. The gluon field strength tensor GμνaG^a_{\mu\nu} is given by: Gμνa=∂μGνa−∂νGμa+gsfabcGμbGνcG^a_{\mu\nu} = \partial_{\mu} G^a_{\nu} - \partial_{\nu} G^a_{\mu} + g_s f^{abc} G^b_{\mu} G^c_{\nu} The coupling constant gsg_s runs with the energy scale μ\mu according to the renormalization group equation, exhibiting asymptotic freedom: μdgsdμ=β(gs)=−gs316π2(11−23Nf)\mu \frac{d g_s}{d \mu} = \beta(g_s) = - \frac{g_s^3}{16\pi^2} (11 - \frac{2}{3} N_f).
Consider the Higgs doublet Φ\Phi with 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 minimum occurs at ⟨Φ†Φ⟩=v2/2\langle \Phi^{\dagger}\Phi \rangle = v^2/2, where v=−μ2/λv = \sqrt{-\mu^2 / \lambda}. Expanding Φ\Phi around its vacuum expectation value ⟨Φ⟩=12(0 v)\langle \Phi \rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 \ v \end{pmatrix}, the gauge boson kinetic terms Lkin=12∑A=W,Z(DμΦ)†(DμΦ)\mathcal{L}_{kin} = \frac{1}{2} \sum_{A=W,Z} (D_{\mu} \Phi)^{\dagger} (D^{\mu} \Phi) yield mass terms for the WW and ZZ bosons: \nLmass=12MW2Wμ ⁣Wμ+12MZ2ZμZμ\mathcal{L}_{mass} = \frac{1}{2} M_W^2 W_{\mu}\!W^{\mu} + \frac{1}{2} M_Z^2 Z_{\mu}Z^{\mu} \nwhere MW=12gvM_W = \frac{1}{2} g v and MZ=12g2+g′2vM_Z = \frac{1}{2} \sqrt{g^2 + g'^2} v.