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Beyond Standard Model

Field: Particle Physics

Physics that explains the shortcomings of the Standard Model.

Sequence of Expressions

Definition

Higgs Mechanism

Consider a complex scalar doublet Φ\Phi with a potential V(Φ)=μ2∣Φ∣2+λ∣Φ∣4V(\Phi) = \mu^2 |\Phi|^2 + \lambda |\Phi|^4. Spontaneous symmetry breaking occurs when μ2<0\mu^2 < 0, leading to a vacuum expectation value (VEV) v=−μ2/λv = \sqrt{-\mu^2 / \lambda}. The Lagrangian density for the Higgs field is: LHiggs=(DμΦ)†(DμΦ)−V(Φ) \mathcal{L}_{Higgs} = (D_{\mu} \Phi)^{\dagger} (D^{\mu} \Phi) - V(\Phi) The resulting mass squared for the physical Higgs boson hh is mh2=2λv2=−2μ2m_h^2 = 2 \lambda v^2 = -2 \mu^2.
The running of a coupling constant αi\alpha_i with the energy scale μ\mu is governed by the Renormalization Group Equation (RGE): μdαidμ=βi(αi) \mu \frac{d\alpha_i}{d\mu} = \beta_i(\alpha_i) For a general coupling α\alpha, the beta function is defined as β(α)=μdαdμ\beta(\alpha) = \mu \frac{d\alpha}{d\mu}. In perturbation theory, this is expanded as: β(α)=∑n=0∞βnαn+1 \beta(\alpha) = \sum_{n=0}^{\infty} \beta_n \alpha^{n+1} where β0\beta_0 is the one-loop coefficient.
Define the effective potential V(ϕ)V(\phi) for a scalar field ϕ\phi around its vacuum expectation value ⟨ϕ⟩=v\langle \phi \rangle = v. The mass squared m2m^2 of the physical scalar excitation is determined by the second derivative of the potential evaluated at the minimum: m2=d2V(ϕ)dϕ2∣ϕ=v m^2 = \left. \frac{d^2 V(\phi)}{d\phi^2} \right|_{\phi=v} For a general potential V(ϕ)V(\phi), the mass term in the Lagrangian is 12m2ϕ2\frac{1}{2} m^2 \phi^2.
Let the spacetime manifold be M=M4×Kn\mathcal{M} = \mathbb{M}^4 \times \mathcal{K}^n, where M4\mathbb{M}^4 is the observed 4D spacetime and Kn\mathcal{K}^n is the compact internal manifold. The higher-dimensional action is given by the Einstein-Hilbert action: SD=116πGD∫d4+nx−G(RD+Lmatter)S_{D} = \frac{1}{16\pi G_{D}} \int d^{4+n}x \sqrt{-G} \left( R_{D} + \mathcal{L}_{matter} \right) where GDG_{D} is the metric tensor in D=4+nD=4+n dimensions. Assuming a compactification radius RR for Kn\mathcal{K}^n, the effective 4D metric gμNg_{\mu N} and the resulting effective 4D action S4S_{4} are obtained via dimensional reduction, leading to the effective gravitational coupling G4=GD/(VK)G_{4} = G_{D}/(V_{\mathcal{K}}), where VK=Vol(Kn)V_{\mathcal{K}} = \text{Vol}(\mathcal{K}^n) is the volume of the internal space.
Define the Lagrangian density L\mathcal{L} for a generic Weakly Interacting Massive Particle (WIMP) candidate χ\chi interacting with the Standard Model fields Φ\Phi: L=LSM+Lχ+Lint\mathcal{L} = \mathcal{L}_{SM} + \mathcal{L}_{\chi} + \mathcal{L}_{int} where Lχ\mathcal{L}_{\chi} describes the free propagation of χ\chi (e.g., Lχ=12Tr(∂μχ∂μχ)−12mχ2Tr(χ2)\mathcal{L}_{\chi} = \frac{1}{2} \text{Tr}(\partial_{\mu} \chi \partial^{\mu} \chi) - \frac{1}{2} m_{\chi}^2 \text{Tr}(\chi^2)). The interaction term Lint\mathcal{L}_{int} must be suppressed relative to the SM interactions, typically involving couplings gχg_{\chi} to SM gauge bosons or fermions: Lint=gχχψˉψ+12gVχ2 Tr(FμνFμν)+…\mathcal{L}_{int} = g_{\chi} \chi \bar{\psi} \psi + \frac{1}{2} g_{V} \chi^2 \ \text{Tr}(F_{\mu\nu} F^{\mu\nu}) + \dots The relic density Ωχh2\Omega_{\chi} h^2 is determined by solving the Boltzmann equation for the particle number density nχ(t)n_{\chi}(t) in the early universe.
Consider the Lagrangian density L\mathcal{L} extended to include three right-handed Majorana neutrinos NR,iN_{R, i} with masses MiM_i: L=LSM+NˉR(i∂ˉγμ∂μ−12Mi)NR+Yukawa terms\mathcal{L} = \mathcal{L}_{SM} + \bar{N}_R \left( \text{i} \bar{\partial} \gamma^{\mu} \partial_\mu - \frac{1}{2} M_i \right) N_R + \text{Yukawa terms} The neutrino mass generation is governed by the effective mass matrix MνM_{\nu} derived from the seesaw mechanism. In the basis (νL,NRT)(\nu_L, N_R^T), the full mass matrix M\mathcal{M} is: M=(0MDMDTMR)\mathcal{M} = \begin{pmatrix} 0 & M_D \\ M_D^T & M_R \end{pmatrix} where MDM_D is the Dirac mass matrix (coupling νL\nu_L to NRN_R) and MRM_R is the Majorana mass matrix for NRN_R. The light neutrino mass matrix is then approximated by Mν≈−MDMR−1MDTM_{\nu} \approx -M_D M_R^{-1} M_D^T. The resulting mass eigenvalues mim_i satisfy ∑mi=0\sum m_i = 0 (if neutrinos are Majorana).
Define the supercharge operator QQ and the super-Poincaré algebra by the anti-commutation relation: {QA,QˉB}=2×(γμPμ)δAB \{ Q_A, \bar{Q}_B \} = 2 \times (\gamma^{\mu} P_{\mu}) \delta_{AB} . The action for a supermultiplet Φ\Phi is given by the superfield integral: S=∫d4x(12(∂μΦ)(∂μΦ)+V(Φ)) S = \int d^4x \left( \frac{1}{2} \left( \partial_{\mu} \Phi \right) \left( \partial^{\mu} \Phi \right) + V(\Phi) \right) where the transformation relates bosonic components ϕ\phi and fermionic components ψ\psi via the supercovariant derivative DαD_{\alpha}.
Consider a gauge group GGUTG_{GUT} that unifies the Standard Model groups GSM=SU(3)C×SU(2)L×U(1)YG_{SM} = SU(3)_C \times SU(2)_L \times U(1)_Y. The running of the coupling constants αi\alpha_i is governed by the beta function βi\beta_i: dαidln⁡(μ)=βi(αi)=12×πbiαi2 \frac{d\alpha_i}{d\ln(\mu)} = \beta_i(\alpha_i) = \frac{1}{2\times \pi} b_i \alpha_i^2 where bib_i are the coefficients. GUT unification requires the couplings to meet at a single point ΛGUT\Lambda_{GUT}: α1(ΛGUT)=α2(ΛGUT)=α3(ΛGUT)=αGUT \alpha_1(\Lambda_{GUT}) = \alpha_2(\Lambda_{GUT}) = \alpha_3(\Lambda_{GUT}) = \alpha_{GUT}
Consider the differential cross-section dσdPS\frac{d\sigma}{d\text{PS}} for a scattering process A+B→X+Y\text{A} + \text{B} \to \text{X} + \text{Y} at the LHC center-of-mass energy s\sqrt{s}. The cross-section is calculated using perturbative Quantum Field Theory (QFT) and is generally expressed as: dσdPS=12s^12kPS∑spins∣M∣2⋅Flux(s^,PS)\frac{d\sigma}{d\text{PS}} = \frac{1}{2\sqrt{\hat{s}}} \frac{1}{2^k \text{PS}} \sum_{\text{spins}} |\mathcal{M}|^2 \cdot \text{Flux}(\hat{s}, \text{PS}) where M\mathcal{M} is the scattering amplitude derived from the Lagrangian, s^\hat{s} is the partonic center-of-mass energy squared, and PS\text{PS} denotes the phase space. Searches for BSM physics involve analyzing deviations from the Standard Model prediction, often parameterized by effective dimension-6 operators O6\mathcal{O}_{6} in the Lagrangian: LBSM=LSM+∑iciΛ2O6,i\mathcal{L}_{BSM} = \mathcal{L}_{SM} + \sum_i \frac{c_i}{\Lambda^2} \mathcal{O}_{6, i}.
Define the Higgs doublet Φ\Phi and the underlying global symmetry G/HG/H, where GG is the global symmetry and HH is the custodial symmetry. In the Composite Higgs Model, the Higgs boson hh is identified as a pseudo-Nambu-Goldstone boson (pNGB) arising from the spontaneous breaking of G→HG \to H. The effective Lagrangian is parameterized by the vacuum expectation value vv and the scale ff of the symmetry breaking. The Higgs potential V(h)V(h) is typically derived from the underlying strong dynamics, leading to the effective potential: V(h)=12mh2h2+14λf2h4+…V(h) = \frac{1}{2} m_{h}^2 h^2 + \frac{1}{4} \frac{\lambda}{f^2} h^4 + \dots The coupling of the Higgs to a generic operator O\mathcal{O} is then determined by the derivative expansion, yielding a coupling strength C=12v2f2CSM\mathcal{C} = \frac{1}{2} \frac{v^2}{f^2} \mathcal{C}_{SM}, where CSM\mathcal{C}_{SM} is the SM coupling, and the ratio ξ=v2/f2\xi = v^2/f^2 quantifies the deviation from the SM prediction.