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Matching, Decoupling, and Threshold Evolution

Matching determines how short-distance physics enters a low-energy theory. The central operation is a controlled comparison: calculate the same low-energy object in a full theory and in its EFT, with the light states, infrared prescription, field normalization, basis, and renormalization scheme aligned. Their difference fixes Wilson coefficients; renormalization-group evolution then transports those coefficients between thresholds and the scale of observation.

Enter this chapter

Section titled “Enter this chapter”

Begin with Integrating Out Heavy Fields to distinguish removing a heavy field from making a local low-energy expansion.

Topics

Section titled “Topics”
  1. Integrating Out Heavy Fields — Derives the light-field action after the heavy field is integrated out.
  2. Matching with Amplitudes, Green Functions, and Background Fields — Chooses matching data with compatible gauge, field, and basis conventions.
  3. Tree-Level Matching and Classical Elimination — Uses classical elimination to reproduce virtual heavy exchange.
  4. Matching Conditions Beyond Tree Level — Determines the hard remainder of full-theory and EFT loop calculations.
  5. Infrared Cancellation, Regulators, and Matching Consistency — Checks cancellation of shared low-energy singularities.
  6. Decoupling Theorems and Threshold Corrections — States the assumptions under which heavy particles decouple.
  7. Running and Matching across Multiple Thresholds — Orders running and matching across several particle thresholds.
  8. Nondecoupling Effects and Matching Validation — Distinguishes genuine heavy-mass effects from inconsistent comparisons.

How the ideas fit together

Section titled “How the ideas fit together”

Tree-level elimination and loop matching implement the same requirement: the full theory and EFT must reproduce the same low-energy quantity. Their shared infrared behavior cancels in the difference, leaving short-distance coefficients. Running subsequently transports those coefficients within a fixed theory; it cannot replace matching when the particle content changes. Decoupling is conditional on the heavy-mass limit: couplings that grow with that mass can invalidate a simple inverse-mass suppression argument. Georgi 1993, §§ 2–4.

Where to go next

Section titled “Where to go next”

Continue to Operator Bases and Field Redefinitions to distinguish independent interactions from an unreduced operator list.

References

Section titled “References”
  • Georgi, Howard. “Effective Field Theory.” Annual Review of Nuclear and Particle Science 43 (1993): 209–252. DOI