An effective field theory describes phenomena below a chosen energy scale by retaining the relevant light fields and organizing local interactions by increasing suppression at that scale.
An effective operator is a local interaction built from the retained fields in an effective field theory. Its coefficient records effects of heavier degrees of freedom, and allowed operators respect the relevant symmetries. In four dimensions an operator of mass dimension has the displayed coefficient for dimensionless and a heavy mass scale . For example a dimension-six proton-decay operator has a coefficient proportional to .
A low-energy effective action retains the light degrees of freedom and describes the effects of heavier ones through local interactions and a derivative expansion. It can be obtained by matching an effective field theory or by a low-energy compactification. A Wilsonian effective action implements integration of high-energy modes with a scale cutoff. The quantum effective action that generates one-particle-irreducible vertices is a different construction; the two need not coincide.
An operator basis in effective field theory is an independent set of allowed local field operators at a chosen order, modulo relevant algebraic identities, integration by parts and permitted field redefinitions. Redundant operators can be removed without changing on-shell amplitudes, provided their coefficient shifts are accounted for. The basis depends on the light fields and symmetries.
Matching in effective field theory fixes Wilson coefficients by requiring a low-energy theory and its ultraviolet completion to give the same low-energy amplitudes through a specified order. Both calculations must treat shared infrared physics consistently. Contributions from retained light particles are reproduced inside the effective field theory; their common nonanalytic parts are not replaced by arbitrary local heavy coefficients.
A Wilson coefficient weights a local operator in an effective field theory. For an operator of mass dimension in four spacetime dimensions, a dimensionless coefficient is conventionally accompanied by . The coefficient encodes short-distance dynamics and can be a matrix in flavour space.
An ultraviolet completion specifies a theory valid at higher energies whose low-energy limit gives an effective field theory. For large-field cosmic inflation, it may also need to control the potential over a large field range, which is distinct from the energy scale of a process.
In an effective field theory, higher-power terms in an inflaton scalar potential can be important when is large, even if is small. An assumed simple monomial inflation potential over such a field range needs a symmetry or ultraviolet completion controlling these operators and radiative corrections. A large field value alone does not establish Planckian energy density or invalidate every possible completion.
A local field operator is built from fields and finitely many of their derivatives at one spacetime point. Its integral can deform an action, with the operator's scaling dimension determining the engineering dimension of the associated coupling.
Heavy-field decoupling says that a field of mass affects processes with external momenta through renormalizations of existing low-energy couplings and local higher-dimensional operators suppressed by powers of .
An ultraviolet cutoff removes modes above a chosen energy or momentum scale. An effective field theory encodes the influence of those omitted short-distance modes in its couplings.
An infrared divergence is a divergence caused by contributions from arbitrarily low energies or long wavelengths.
An observable unchanged by adding a particle with vanishing energy or replacing one massless particle by exactly collinear particles with the same total momentum. This property permits cancellation of real and virtual infrared divergences and collinear divergences in its perturbative cross-section. Fully inclusive electron-positron hadronic rates and suitably defined jet observables are examples.
A singularity of a massless scattering or radiation integral when momenta of two particles become parallel. In Quantum chromodynamics, unresolved quark-gluon splittings cause such divergences. A mass regulator or dimensional regularization exposes them; they cancel between suitable inclusive real and virtual contributions or are absorbed into factorized distributions where appropriate.
A quadratic Lagrangian contains terms of second order in field fluctuations and determines their linearized propagators and dispersion relations.
A Higgs mode is a fluctuation in the magnitude of a complex or multicomponent order parameter, as distinct from a phase or orientational fluctuation.
Integrating out a field means performing its path integral while retaining the other fields as backgrounds. The result is generally a nonlocal quantum effective action; for a field much heavier than all external momenta, a derivative expansion turns it into local effective interactions.
A heavy real scalar with quadratic mass and interaction can be removed at tree level using . Substituting into both its source and quadratic terms gives the displayed interaction. Its derivative expansion starts with . For a light quartic written as , the matched quartic is . The expansion is valid for momentum invariants small compared with .
A derivative expansion expresses a nonlocal effective interaction as a series of local operators with successively more derivatives. For a field of mass , each additional pair of derivatives is typically suppressed by .
A four-fermion interaction is a local operator containing four fermion fields. In four spacetime dimensions it has mass dimension six, so its coefficient has mass dimension minus two; writing it as the square of a current multiplied by a coefficient inside the square gives that coefficient mass dimension minus one.
Approximate a hadronic matrix element of a product of two currents by a product of separate current matrix elements. For a decay to two mesons, one factor can create a pion from the vacuum while the other gives a pseudoscalar-to-pseudoscalar form factor. Nonfactorizable Quantum chromodynamics interactions are omitted; tree-level weak vertices do not alone justify this approximation.
The Fermi interaction is the low-energy four-fermion approximation to the weak charged current. Integrating out a W boson gives its coefficient at tree level.
For a massive charged vector with interaction , neglect derivatives at momenta small compared with its mass. Completing the square in leaves . With this yields the Fermi constant relation . Corrections contain extra derivatives and are suppressed by momentum squared over the heavy mass squared.
For a left-handed incoming neutrino, unpolarized electrons, and electron vector/axial couplings , the four-fermion interaction gives . Integrating massless two-body Lorentz-invariant phase space yields . The electron spin average is included, while no sterile neutrino helicity is averaged in.
Crossing a massless neutrino to an antineutrino reverses the antisymmetric part of its spin tensor. The and weights exchange, giving for the same electron couplings. This is a low-energy contact statement rather than a replacement for a full weak-boson propagator.
Electron-neutrino scattering has a charged-current contribution absent for muon neutrinos. Fierz rearrangement combined with the external fermionic sign adds a left-electron current: , . The right-electron coupling stays unchanged; the left-electron coupling increases by two.
A Fierz rearrangement rewrites a product of fermion bilinears with one spinor pairing as a sum of products with another pairing. For left-handed vector currents, it lets a charged-current four-fermion operator be compared directly with a neutral-current operator.
The Fermi constant is the coefficient conventionally used for the low-energy charged-current weak interaction. Tree-level exchange of a boson gives .
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Effective field theory (EFT) is a framework in theoretical physics used to describe physical systems at specific energy scales while accounting for the effects of higher energy processes in a systematic way. The main idea behind EFT is that, at a given energy scale, we can ignore the details of physics that occurs at much higher energy scales, focusing instead on the degrees of freedom and interactions relevant to the low-energy behavior of the system.
www.youtube.com/watch?v=WB8r7CU7clk&list=PLUl4u3cNGP60TvpbO5toEWC8y8w51dtvm by Iain Stewart. Basically starts by explaining how quantum field theory is so generic that it is hard to get any numerical results out of it :-)