Fermi's theory of weak interactions describes beta decay using a four-fermion interaction G_F (psi-bar psi)(psi-bar psi), with Fermi constant G_F ~ 10^{-5} GeV^{-2}. This is an effective field theory. What is the 'UV completion' that Fermi's theory approximates?
AQED with virtual photon exchange
BThe electroweak theory with W boson exchange — at energies much below M_W ~ 80 GeV, the W propagator 1/(q^2 - M_W^2) reduces to -1/M_W^2, collapsing the exchange to a point-like four-fermion vertex with G_F ~ g^2/M_W^2
CQCD with gluon exchange
DString theory
Fermi's theory is the low-energy limit of the electroweak theory. When the momentum transfer q is much less than M_W, the W boson propagator becomes approximately constant: 1/(q^2 - M_W^2) -> -1/M_W^2. The W exchange diagram then looks like a contact interaction between four fermions, with coupling G_F/sqrt(2) = g^2/(8M_W^2). Fermi's theory works beautifully for beta decay (q ~ MeV << M_W ~ 80 GeV) but fails at energies approaching M_W, where the momentum dependence of the W propagator matters and eventually the W boson appears as a real particle.
Question 2 Multiple Choice
In EFT, higher-dimension operators are suppressed by powers of 1/Lambda (where Lambda is the high-energy scale). A dimension-6 operator is suppressed by 1/Lambda^2 relative to a dimension-4 operator. Why does this make EFT a systematic expansion?
ABecause Lambda is always infinite
BBecause at energies E << Lambda, the ratio (E/Lambda)^n decreases rapidly with n — dimension-6 operators contribute corrections of order (E/Lambda)^2, dimension-8 operators of order (E/Lambda)^4, and so on, giving a controlled perturbative expansion in powers of E/Lambda
CBecause higher-dimension operators are always zero
DBecause the EFT Lagrangian contains only finitely many operators
This is the power of EFT: at low energies, you can achieve any desired precision by including operators up to a finite dimension. For E/Lambda = 0.1, dimension-6 operators give 1% corrections, dimension-8 give 0.01% corrections, etc. The unknown high-energy physics affects low-energy observables only through these suppressed operators, and the leading effects can be parameterized by a finite number of Wilson coefficients. The EFT breaks down when E approaches Lambda — at that scale, all operators contribute equally and the expansion is no longer useful.
Question 3 True / False
The Standard Model itself is now widely regarded as an effective field theory valid up to some unknown scale Lambda. This is not a failure but a feature.
TTrue
FFalse
Answer: True
Treating the Standard Model as an EFT (the 'Standard Model Effective Field Theory' or SMEFT) means adding all gauge-invariant higher-dimension operators to the Standard Model Lagrangian, suppressed by powers of a new physics scale Lambda. The leading new-physics effects come from dimension-6 operators. This framework allows systematic, model-independent searches for deviations from the Standard Model without committing to a specific beyond-the-Standard-Model theory. Current LHC data constrain many dimension-6 Wilson coefficients, pushing Lambda above several TeV for most operators.
Question 4 Short Answer
Explain why non-renormalizable theories are not 'sick' but are perfectly well-defined effective field theories, and what changes about their predictive power compared to renormalizable theories.
Think about your answer, then reveal below.
Model answer: A non-renormalizable theory contains operators of dimension > 4 whose couplings have negative mass dimension (e.g., G_F ~ 1/M_W^2 in Fermi theory). These couplings introduce new divergences at each loop order that require new counterterms, so the theory has infinitely many parameters in principle. However, at energies E << Lambda, operators of dimension 4+n contribute at order (E/Lambda)^n, so only finitely many operators contribute at any given precision. The theory is predictive to any specified accuracy — you just need more measured parameters for higher precision. A renormalizable theory is the special case where the leading-order Lagrangian alone (dimension <= 4) suffices for all-orders predictions. The EFT framework is strictly more general and includes renormalizable theories as a special case.
This perspective, developed by Wilson and others in the 1970s-80s, revolutionized how physicists think about quantum field theory. Rather than dividing theories into 'renormalizable (good)' and 'non-renormalizable (bad),' the modern view is that all theories are effective, and renormalizability is a feature of the leading-order Lagrangian, not a requirement for consistency.