Questions: Effective Field Theory in Particle Physics

Question 1

The SMEFT Lagrangian is L = L_SM + sum_i (C_i / Lambda^2) * O_i^{(6)} + sum_j (C_j / Lambda^4) * O_j^{(8)} + ..., where O_i^{(d)} are operators of dimension d built from SM fields. At dimension 6, there are 2499 independent operators (for one generation of fermions, 59 operators; for three generations, 2499). Why are dimension-6 operators the leading BSM effects?

• A. Because there are no dimension-5 operators
• B. Because there is exactly one dimension-5 operator (the Weinberg operator, which generates Majorana neutrino masses), and after accounting for it, the leading new effects come from dimension-6 operators suppressed by 1/Lambda^2 — dimension-7 and higher operators are further suppressed by additional powers of 1/Lambda and are typically negligible if Lambda >> v (the Higgs vev)
• C. Because dimension-6 operators are renormalizable
• D. Because only dimension-6 operators conserve gauge symmetry

Answer: (B)

The dimension-5 Weinberg operator L = (C_5/Lambda) * (LH)(LH) (where L is the lepton doublet and H is the Higgs) generates neutrino masses m_nu ~ C_5 * v^2/Lambda after electroweak symmetry breaking. For Lambda ~ 10^{14} GeV and C_5 ~ 1, this gives m_nu ~ 0.05 eV, consistent with oscillation data. Aside from this unique operator, the leading BSM effects are from dimension-6 operators, which modify Higgs couplings, gauge boson self-interactions, fermion couplings, and produce new 4-fermion interactions. Each operator has a Wilson coefficient C_i that encodes the strength and sign of the new physics contribution.

Question 2

A specific BSM model (e.g., a heavy Z' boson) can be 'matched' onto the SMEFT by integrating out the heavy particle and expressing the resulting effects as Wilson coefficients of SMEFT operators. Why is this matching useful?

Model answer: Matching separates the model-specific UV physics (the Z' mass, couplings, and quantum numbers) from the model-independent low-energy effects (shifts in SM observables). Once a BSM model is matched onto the SMEFT, its predictions for all low-energy observables are encoded in a finite set of Wilson coefficients. Conversely, experimental measurements can constrain the Wilson coefficients model-independently, and these constraints can then be translated to any specific BSM model. This factorization means that each experimental measurement needs to be interpreted only once (in terms of Wilson coefficients), and each BSM model needs to be matched only once, rather than confronting every model with every measurement individually.

The SMEFT framework has become the standard for LHC Higgs coupling measurements, electroweak precision tests, and top quark measurements. Global fits to SMEFT Wilson coefficients combine hundreds of measurements and constrain the new physics scale Lambda to be above ~1-10 TeV for O(1) couplings, depending on the operator.

Question 3

The SMEFT is valid when the new physics scale Lambda is well above the energies being probed (E << Lambda). At the LHC, some processes probe energies of several TeV. Under what conditions does the SMEFT description break down?

• A. When the number of operators becomes too large to fit
• B. When E approaches Lambda, the expansion in E/Lambda converges poorly or breaks down entirely — dimension-8 operators become as important as dimension-6, the truncation is no longer valid, and one must use the full UV-complete model; additionally, if the new particles can be directly produced (E > M_new), they appear as resonances rather than contact interactions, and the SMEFT description misses this qualitatively different signature
• C. When the Wilson coefficients become negative
• D. When more than one operator contributes to the same observable

Answer: (B)

The SMEFT is a perturbative expansion in E/Lambda. If Lambda = 2 TeV and the LHC probes E = 1 TeV, then E/Lambda = 0.5 and higher-order terms may not be negligible. In practice, SMEFT analyses must check the validity of the truncation (e.g., by including dimension-8 operators and verifying that their contributions are small). For processes at the highest LHC energies (tails of distributions), the SMEFT may be unreliable, and dedicated searches for resonances are complementary.