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Topic 22 of 22

Effective Field Theory

Effective Field Theory is the systematic description of physics at a given scale without requiring knowledge of shorter-distance details. Heavy particles with mass M ≫ E are "integrated out," generating an effective Lagrangian organized as an expansion in E/M with Wilson coefficients encoding the high-energy physics. This shows renormalizability is not fundamental — it is an emergent property of any low-energy description. EFT is the language of modern precision tests and new-physics searches.

Key Concepts

  • Wilson's picture: integrate out fields with mass M → ∞ and get effective couplings at scale E ≪ M
  • ℒ_eff = Σ_n (C_n/M^{n−4}) O_n — expansion in operators of increasing dimension d=n
  • Renormalizable = d ≤ 4 operators dominate; d > 4 suppressed by powers of E/M
  • Matching: set C_n by requiring UV and EFT amplitudes agree at E ∼ M (tree or loop level)
  • Fermi theory: ℒ = (GF/√2) J_μ J^μ is EFT below MW; GF/√2 = g²/8M²_W
  • SMEFT: all BSM physics parametrized as d≥5 operators built from SM fields suppressed by new-physics scale Λ

Key Equations

EFT Lagrangian
Leff=LSM+1Λ∑iCi(5)Oi(5)+1Λ2∑jCj(6)Oj(6)+⋯\mathcal{L}_{\rm eff}=\mathcal{L}_{\rm SM}+\frac{1}{\Lambda}\sum_i C_i^{(5)}\mathcal{O}_i^{(5)}+\frac{1}{\Lambda^2}\sum_j C_j^{(6)}\mathcal{O}_j^{(6)}+\cdots
Fermi constant from W
GF2=g28MW2=12v2,GF≈1.166×10−5 GeV−2\frac{G_F}{\sqrt{2}}=\frac{g^2}{8M_W^2}=\frac{1}{2v^2},\quad G_F\approx1.166\times10^{-5}\text{ GeV}^{-2}
Weinberg operator (dim 5)
O5=(LˉH~)(H~TLc)Λ⇒mν=y2v22Λ\mathcal{O}_5=\frac{(\bar L\tilde H)(\tilde H^T L^c)}{\Lambda}\quad\Rightarrow\quad m_\nu=\frac{y^2v^2}{2\Lambda}
Power counting
ANPASM∼(EΛ)d−4\frac{\mathcal{A}_{\rm NP}}{\mathcal{A}_{\rm SM}}\sim\left(\frac{E}{\Lambda}\right)^{d-4}
Worked Example

Example Problem

Problem

Given GF/√2 = g²/(8M²W) with g = 0.653 and MW = 80.4 GeV, calculate GF in units of 10⁻⁵ GeV⁻².

Solution

GF = √2 × g²/(8M²W) = 1.414 × (0.653)²/(8 × 80.4²) = 1.414 × 0.4264/(8 × 6464) = 0.6029/51712 = 1.166×10⁻⁵ GeV⁻² ✓.

Key Takeaways

  • EFT gives the correct low-energy description: heavy particles decouple and their effects appear only through Wilson coefficients of higher-dimension operators
  • Renormalizability is automatic: d ≤ 4 operators dominate at E ≪ M; non-renormalizable interactions are power-law suppressed
  • Matching fixes Wilson coefficients at E ∼ M; RGE then runs them to low energies where measurements constrain them
  • SMEFT systematically parametrizes all new physics via higher-dimension operators — the universal framework for interpreting precision measurements and LHC data
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