Effective field theory (EFT)

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.

Soft-collinear effective theory is an effective field theory of quantum chromodynamics for processes containing energetic particles moving close to lightlike directions together with lower-momentum radiation. It retains the modes needed for that hierarchy and organizes interactions by power counting. Fleming's overview describes energetic collinear particles interacting with soft quanta and applies the framework to meson-decay observables.^[ref]

The geometry is clearer in light-cone coordinates. Choose the Minkowski metric with signature $(+,-,-,-)$ and defineA particle moving nearly along the positive $z$ direction has one large component $p^{-}$, a smaller transverse momentum and a still smaller $p^{+}$. A common light-cone power counting assignment isIndividual terms in $p^2$ then scale as $Q^2{\lambda }^2$; they can cancel for a particle on its mass shell. Virtuality and energy are therefore different measures of scale.

For an original exact example in natural units, take $p^{-}=100\mathrm{GeV}$, $p^{+}=1\mathrm{GeV}$ and $|{\mathbf{p}}_{\perp }|=10\mathrm{GeV}$. Then $E=50.5\mathrm{GeV}$, $p_z=49.5\mathrm{GeV}$ andThe momentum is energetic and exactly lightlike in this example. Removing all energetic fields from a low-virtuality theory would discard this mode. The numerical components illustrate the stated convention and are not data from a collider.

The original2001 construction by Christian W. Bauer, Sean Fleming, Dan Pirjol and Iain W. Stewart identifies separated scales and determines effective-theory currents and coefficient evolution.^[ref] Comparing calculations with the underlying theory fixes Wilson coefficients; renormalization group evolution then organizes scale dependence. The resulting effective Lagrangian is an expansion with a specified domain of validity, rather than a replacement for QCD in every kinematic regime. Precise assignments for lower-energy modes depend on which hierarchy the application requires.

Light-cone power counting organizes terms in a multiscale field theory by assigning powers of a small parameter to light-cone momentum components. In soft-collinear effective theory, energetic nearly lightlike momenta can have one large longitudinal component, smaller transverse components and a still smaller opposite longitudinal component. Fleming's review explains why these collinear degrees of freedom must remain in a theory that also describes lower-momentum radiation.^[ref]

Fix natural units and metric signature $(+,-,-,-)$, and define $p^{+}=E-p_z$, $p^{-}=E+p_z$. A common assignment isThe symbol $\sim$ specifies parametric size, not an exact equality. Sinceeach term is generically of order $Q^2{\lambda }^2$. Their difference can be smaller, including zero on the mass shell. The assignment therefore concerns momentum regions and generic virtuality, not a claim that every momentum in the region has the same invariant mass.

For an original example, take $Q=200\mathrm{GeV}$ and $\lambda =0.05$. The assigned component scales are200,10and0.5GeV. Values $p^{-}=200$, $|{\mathbf{p}}_{\perp }|=10$ and $p^{+}=0.5$ give $p^2=0$; changing the last value to1 gives $p^2=100\mathrm{Ge}{\mathrm{V}}^2$. Both small components have the same parametric order, differing by a factor of2, while the exact cancellation changes.

Power assignments also organize integration measures. Two transverse components each scale as $Q\lambda$, soA generic scalar factor $1/p^2$ in this region scales as $Q^{-2}{\lambda }^{-2}$. Together these contribute $Q^2{\lambda }^2$ before any numerator, couplings or further denominators are included. This illustrative count is not by itself a complete scattering amplitude.

Precise assignments depend on the scale hierarchy under study and the chosen light-cone directions. An isotropic mode with all components of order $Q{\lambda }^2$ has a different generic virtuality scale, $Q^2{\lambda }^4$. Explicit counting keeps such contributions distinct and states what it means to retain a given order in an effective Lagrangian.

An effective Lagrangian encodes the interactions of retained degrees of freedom after others have been removed, often by a path integral. For external energies below a massive field's excitation threshold, its effects admit a local expansion in external momenta divided by its mass. Matching scattering amplitudes fixes the coefficients of the resulting local operators; truncation has a specified order in that small ratio.

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 ${\Delta }_i$ has the displayed coefficient for dimensionless $C_i$ and a heavy mass scale $M$. For example a dimension-six proton-decay operator has a coefficient proportional to $M^{-2}$.

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.

In four dimensions, a real scalar field and a partial derivative each have mass dimension one. Seven scalar monomials of dimension six can be listed before integration by parts: ${\varphi }^6$, ${\varphi }^2(\partial \varphi {)}^2$, ${\varphi }^3\square \varphi$, $\varphi {\square }^2\varphi$, ${\partial }_{\mu }\varphi {\partial }^{\mu }\square \varphi$, $(\square \varphi {)}^2$, and $({\partial }_{\mu }{\partial }_{\nu }\varphi {)}^2$. They reduce to the three displayed representatives modulo total derivatives. Leading equations of motion reduce them further to a single new ${\varphi }^6$ interaction, together with shifts of the existing mass and quartic coupling constant.

At first order in the suppression of an effective operator, a factor of the leading Euler-Lagrange field equation defines a redundant operator removable by a local field redefinition. For ${\mathcal{L}}_0=(\partial \varphi {)}^2/2-m^2{\varphi }^2/2-\lambda {\varphi }^4/24$, one obtains ${\varphi }^3\square \varphi \doteq -m^2{\varphi }^4-\lambda {\varphi }^6/6$ and $(\square \varphi {)}^2\doteq m^4{\varphi }^2+m^2\lambda {\varphi }^4/3+{\lambda }^2{\varphi }^6/36$. Lower-dimensional parameter shifts must accompany this reduction when $m\ne 0$; it is not an identity for untransformed off shell correlation functions.

Two Lagrangian densities differing by a total derivative yield the same action under suitable boundary conditions. For example ${\partial }_{\mu }({\varphi }^3{\partial }^{\mu }\varphi )=3{\varphi }^2(\partial \varphi {)}^2+{\varphi }^3\square \varphi$. Likewise $\varphi {\square }^2\varphi \simeq -\partial \varphi \cdot \partial \square \varphi \simeq (\square \varphi {)}^2\simeq ({\partial }_{\mu }{\partial }_{\nu }\varphi {)}^2$. These are algebraic relations modulo boundary terms, not uses of the equation of motion.

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 $d$ in four spacetime dimensions, a dimensionless coefficient is conventionally accompanied by ${\Lambda }^{4-d}$. 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 $\varphi /M_{\mathrm{Pl}}$ is large, even if $V/M_{\mathrm{Pl}}^4$ 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 $M$ affects processes with external momenta $p\ll M$ through renormalizations of existing low-energy couplings and local higher-dimensional operators suppressed by powers of $p/M$.

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 arises from contributions at arbitrarily low energy or long wavelength. A massless mode can produce such a problem because many soft excitations are available at very small energy, or because a long-distance fluctuation integral fails to converge.

Its interpretation depends on the observable. In scattering, an exclusive amplitude with no additional arbitrarily soft radiation need not be separately measurable; combining appropriately inclusive real-emission and virtual contributions can remove the relevant divergence. In equilibrium field theory, an infrared-divergent order parameter fluctuation can instead diagnose failure of an assumed ordered-state approximation. A finite size or detector resolution may regulate the calculation, but the dependence on that regulator must be interpreted rather than silently discarded.

An infrared regulator suppresses the small-momentum region responsible for an infrared divergence. Examples are a small positive mass or a finite volume. It must be distinguished from an ultraviolet cutoff: setting external momenta and masses to zero to extract an ultraviolet divergence can introduce a spurious infrared divergence. Keep a positive auxiliary mass until the divergent local coefficients have been separated, then take the desired limit in infrared-safe quantities.

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.

For $-\gamma \ddot{\psi }=-{\psi }^{\prime \prime }+\beta (|\psi {|}^2-1)\psi$ with positive $\beta ,\gamma$, expand about $\psi =1$ as $1+\rho /2+i\theta$. The real and imaginary equations become $\gamma \ddot{\rho }={\rho }^{\prime \prime }-2\beta \rho$ and $\gamma \ddot{\theta }={\theta }^{\prime \prime }$. Plane waves give a gapped radial amplitude branch and a gapless tangential phase branch. This second-order model has two independent real mode coordinates, unlike density-phase conjugacy in first-order dynamics.

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 $S$ with quadratic mass $M$ and interaction $-aS{\phi }^2/2$ can be removed at tree level using $(M^2+\square )S=-a{\phi }^2/2$. Substituting into both its source and quadratic terms gives the displayed interaction. Its derivative expansion starts with $a^2{\phi }^4/(8M^2)-a^2{\phi }^2\square {\phi }^2/(8M^4)$. For a light quartic written as $-\lambda {\phi }^4/4!$, the matched quartic is $\lambda -3a^2/M^2$. The expansion is valid for momentum invariants small compared with $M^2$.

A derivative expansion expresses a nonlocal effective interaction as a series of local operators with successively more derivatives. For a field of mass $M$, each additional pair of derivatives is typically suppressed by $M^{-2}$.

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.

For ${\mathcal{L}}_{\mathrm{int}}=g^2(\bar{\psi }\psi {)}^2/2$ with one species of Dirac field, the tree-level Feynman diagrams for $f(p_1)\bar{f}(p_2)\to f(p_3)\bar{f}(p_4)$ giveThe fermionic sign is the antisymmetry of the two external contractions. The initial spin average is one quarter of the spin sum. The two diagonal contributions are $g^4(t-4m^2{)}^2$ and $g^4(s-4m^2{)}^2$; the interference Dirac trace is $R=(s-4m^2{)}^2+(t-4m^2{)}^2-(u-4m^2{)}^2$, giving the displayed result after subtracting $g^{4}R/2$. Here the Mandelstam variables satisfy $s+t+u=4m^2$. At threshold this gives $16g^4{m}^4$; for zero mass it gives $g^4(s^2+t^2+u^2)/2$. The formula describes the leading scattering amplitude of an effective interaction and is not a statement of perturbative renormalizability.

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 $G_F/\sqrt{2}=g^2/(8M_W^2)$ at tree level.

For a tree-level left-handed three-body decay with one massive final fermion and two massless ones, the integrated width relative to the all-massless result includes $f(r)=1-8r+8r^3-r^4-12r^2\ln r$, where $r$ is the squared final-to-parent mass ratio. It satisfies $f(0)=1$ and $f(1)=0$. It is not a universal factor for channels with several massive daughters.

For a massive charged vector with interaction $-\kappa (W^{+}J+W^{-}{J}^{†})$, neglect derivatives at momenta small compared with its mass. Completing the square in $m_W^2{W}^{+}{W}^{-}$ leaves $-{\kappa }^2{J}^{†}J/m_W^2$. With $\kappa =g/(2\sqrt{2})$ this yields the Fermi constant relation $G_F/\sqrt{2}=g^2/(8m_W^2)$. 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 $c_V,c_A$, the four-fermion interaction gives $\overline{|\mathcal{M}{|}^2}=4G_F^2[(c_V+c_A{)}^2{s}^2+(c_V-c_A{)}^2{u}^2]$. Integrating massless two-body Lorentz-invariant phase space yields $\sigma =G_F^{2}s(c_V^2+c_V{c}_A+c_A^2)/(3\pi )$. 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 $s^2$ and $u^2$ weights exchange, giving ${\sigma }_{\bar{\nu }}=G_F^{2}s[(c_V-c_A{)}^2+(c_V+c_A{)}^2/3]/(4\pi )$ 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: $c_V\to c_V+1$, $c_A\to c_A+1$. 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 $W$ boson gives $G_F/\sqrt{2}=g^2/(8M_W^2)$.