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Expand the string embedding map along target geodesics as $X={\exp }_{\bar{X}}Y$. The quadratic Euclidean fluctuation action has a covariant derivative term and a curvature term $-R_{acbd}{Y}^a{Y}^b\partial {\bar{X}}^c\partial {\bar{X}}^d$. The logarithmically divergent coincident contraction is proportional to ${\alpha }^{\prime }{g}^{ab}$; its curvature contraction is the Ricci tensor. A compensating metric counterterm gives ${\beta }_{ab}^g={\alpha }^{\prime }{R}_{ab}+O({\alpha }^{\prime 2})$. This is why the target gravitational equation arises from quantum worldsheet Weyl anomaly cancellation, rather than from the classical worldsheet metric variation. Infrared regulation and removal of equation-of-motion terms by field redefinitions specify the usual one-loop scheme.

A CP-violating phase is a complex phase in couplings or mixing matrices that cannot be removed by field redefinitions and changes sign under CP conjugation.

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.

The equivalence theorem states that an invertible local field redefinition leaves on-shell S-matrix elements unchanged, although it changes off-shell correlation functions and individual interaction vertices.

A field redefinition is an invertible change of variables among fields. Local field redefinitions can rearrange higher-derivative operators without changing on-shell scattering amplitudes.

An off shell field configuration is not required to satisfy the Euler-Lagrange field equation. An external four-momentum is off shell if it is not constrained to its physical mass shell. On shell identities can therefore fail for general untransformed correlation functions, even when the associated S-matrix is unchanged by a field redefinition.

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.

Vary the action, taking the field variation to vanish at the boundary. Integration by parts givesThus the Euler-Lagrange field equation isFor the field redefinition in part f, define the functional derivative $E(\varphi )=\delta S_0/\delta \varphi =-(\square \varphi +m^2\varphi +\lambda {\varphi }^3/6)$. The sign does not affect the equation of motion, but it fixes the sign of an added redundant operator.

Use the sign convention $E(\varphi )=\delta S_0/\delta \varphi$ from part b, and letIn the path integral, make the local field redefinition ${\varphi }_{\mathrm{old}}=\chi -\eta f(\chi )$. The functional Taylor expansion of the action givesThe two first-order terms cancel. Neglecting the functional Jacobian as permitted,Thus $fE$ is a redundant operator at first order, by the equivalence theorem for field redefinitions. The same argument works for a local derivative-dependent $f$, which is needed to remove $(\square \varphi {)}^2$. Insertions and sources must be transformed too: arbitrary untransformed off shell correlation functions need not be equal.

Sign qualification. If instead $E_{+}=\square \varphi +m^2\varphi +\lambda {\varphi }^3/6$ is used, then $E_{+}=-\delta S_0/\delta \varphi$. An added $+\eta f{E}_{+}$ is removed by ${\varphi }_{\mathrm{old}}=\chi +\eta f(\chi )$. The minus shift requested in the source therefore fixes which sign is meant by its otherwise unspecified $E$.

Let $a=\lambda /6$. Within terms already suppressed by ${\Lambda }^{-2}$, the leading equation of motion gives $\square \varphi =-m^2\varphi -a{\varphi }^3$. Equality here means equivalence under a perturbative field redefinition, together with integration by parts, rather than a pointwise identity inside arbitrary off shell correlation functions. HenceFor the second relation, factor the difference as $(\square \varphi -m^2\varphi -a{\varphi }^3)(\square \varphi +m^2\varphi +a{\varphi }^3)$, which is explicitly a redundant operator. Ifthe reduced correction isThe first two coefficients shift the already present mass and quartic coupling constant. After these shifts,This is the equation-of-motion reduction of scalar operators through $O({\Lambda }^{-2})$; terms of $O({\Lambda }^{-4})$ are outside that truncation. The lower-dimensional shifts should not be silently dropped when $m\ne 0$.

Assume the standard canonical kinetic action $\int d^4\theta {\Phi }^{†}\Phi$. A constant field redefinition $\Phi =\Psi +s$ is invertible and preserves chirality. Its kinetic action differs by $s{\Psi }^{†}+s^{\ast }\Psi +|s{|}^2$, whose full superspace integrals vanish up to a boundary term: a chiral or antichiral field has no independent full-superspace component. Equivalently, spacetime derivatives of a constant are zero. There is no nontrivial Jacobian for this translation of the field. Thus physical predictions are unchanged when all parameters and the coordinate of the vacuum are transformed together.

The transformed superpotential iswhere $A=W(s)$. Removing the linear term means choosing a critical point $W^{\prime }(s)=0$, not merely choosing $s=-\kappa /m$ while ignoring the cubic term. For $g\ne 0$,The resulting superpotential is $A+M{\Psi }^2/2+g{\Psi }^3/6$. Its constant term has no effect in global supersymmetry, since its chiral superspace integral is zero. It would matter in supergravity, which is not the model here.

A real shift is physically allowed, but it does not always suffice to remove the linear term. With real couplings it requires $\Delta \ge 0$ for $g\ne 0$. For example $m=0$, $\kappa =1$, $g=2$ gives $W^{\prime }(s)=1+s^2$, which has no real root. A complex shift $s=\pm i$ is still a valid redefinition of the complex chiral scalar and removes the term. Thus the printed instruction needs either the real-root hypothesis or permission for a complex shift.

If $g=0$ and $m\ne 0$, choose $s=-\kappa /m$. If $g=m=0$, no shift removes a nonzero $\kappa$; if $\kappa =0$ it is already absent. These exceptional branches will also distinguish the supersymmetry-breaking possibilities below.

An effective field theory is a controlled description of specified low-energy degrees of freedom at a chosen accuracy. It does not require knowing all physics at arbitrarily short distances. If new particles or strong dynamics enter at a scale $M$, processes with characteristic energy and momentum transfers $E\ll M$ can be described using the light fields and interactions consistent with their symmetries. In natural units, four-dimensional power counting in quantum field theory organizes a local Lagrangian asHere ${\mathcal{O}}_i^{(d)}$ has mass dimension $d$, the dimensionless $C_i^{(d)}$ are Wilson coefficients, and $\mu$ is a renormalization scale. In a relativistic vacuum with canonical fields, a typical insertion of a dimension-$d$ interaction contributes an additional power $(E/M{)}^{d-4}$, multiplied by its couplings and any light mass ratios. Additional expansions, such as a loop expansion, must also be specified. Symmetry or on-shell identities can postpone a particular observable's first correction.

The theory is useful when its retained states and expansion parameters adequately describe the experiment. It ceases to be a reliable truncated local description near an omitted particle's production threshold, near a heavy propagator pole, or where the retained dynamics become too strongly coupled for an assumed perturbative expansion. A light particle cannot be removed merely because one wants fewer variables: its propagation can produce nonanalytic momentum dependence that must be represented by retained light fields. Heavy-field decoupling can shift renormalizable masses and couplings as well as generate suppressed interactions; those low-energy parameters must be measured or matched, not assumed unchanged. The ultraviolet cutoff is an organizational scale, while a calculation may use a regulator other than a hard momentum cutoff.

Construction starts by identifying the light particles, the hierarchy of scales, and the exact or approximate symmetries relevant to the problem. Form all allowed local field operators through the desired order in power counting in quantum field theory. Choose an operator basis in effective field theory: remove equivalent terms by integration by parts, algebraic identities and allowed field redefinitions. Operators proportional to lower-order equations of motion can be redundant operators for on-shell amplitudes, provided coefficients are consistently transformed. This reduces bookkeeping without imposing additional physics assumptions.

If an ultraviolet completion is known, determine the Wilson coefficients by matching in effective field theory: calculate low-energy amplitudes or appropriate correlation functions in both descriptions using the same infrared conventions, and adjust coefficients so they agree to the chosen order. Equivalently, integrating out a field performs its path integral while retaining the light fields as backgrounds. Heavy propagators have an analytic expansion below their singularities, producing a derivative expansion; heavy loops also generate local terms and logarithms in their coefficients. Without a specified ultraviolet completion, the coefficients are parameters to be constrained by data.

An effective field theory remains predictive even when it contains nonrenormalizable interactions. At each fixed order in energy and loops there are finitely many required coefficients and counterterms. Renormalization absorbs divergences into that order's allowed operators. The renormalization group evolves the Wilson coefficients between matching and measurement scales, compensating scale dependence in matrix elements and, when appropriate, resumming large logarithms. The truncation error in effective field theory is estimated from the first omitted orders, under a stated coupling-size assumption; it is separate from parameter uncertainty and cannot be inferred merely by writing down infinitely many terms.

A concrete example is heavy scalar exchange in effective field theory. Take a light real scalar field $\phi$ and a heavy real scalar field $S$, withThe coupling $a$ has mass dimension one. The light-field symmetry is $\phi \mapsto -\phi$. For $m^2\ge 0$ and $\lambda >3a^2/M^2$, the displayed scalar potential is bounded below: completing the square in $S$ leaves a positive light quartic. Thus the example can be treated as a stable theory around $S=\phi =0$, with weak enough couplings for the tree approximation.

At tree level the heavy equation of motion is $(M^2+\square )S=-a{\phi }^2/2$. Substitute its solution back into the action, including both its quadratic and source terms, to obtainThis is a derivative expansion valid for small momentum transfers, not an exact local replacement near the pole. The first term gives ${\lambda }_{\mathrm{eff}}=\lambda -3a^2/M^2$ in the $-{\lambda }_{\mathrm{eff}}{\phi }^4/4!$ convention. After integration by parts, the next term is $+a^2({\partial }_{\mu }{\phi }^2)({\partial }^{\mu }{\phi }^2)/(8M^4)$; it is a dimension-six local operator. Writing $a=g_{\ast }M$ puts its coefficient in the usual $g_{\ast }^2/M^2$ form. These coefficients are a tree-level matching in effective field theory result.

One can directly check the matching through the on-shell scattering amplitude for $\phi \phi \to \phi \phi$. With $s,t,u$ the Mandelstam variables, the full theory has three heavy-exchange channels:For $|s|,|t|,|u|\ll M^2$,The effective field theory reproduces these terms in order: a shifted quartic, then local derivative interactions. Since $s+t+u=4m^2$ on shell, the first derivative correction is a light-mass-dependent constant; it vanishes for $m=0$. This illustrates why an operator basis in effective field theory can trade some derivative operators for mass-dependent or higher-field interactions using field redefinitions. For massless external particles, the first nonconstant correction in this four-point tree amplitude starts at the following order.

The example exhibits the central logic: keep the light field, encode virtual heavy exchange in matched local coefficients, and control the error by expanding in momentum divided by the heavy scale. No heavy particle is actually produced in the domain of the approximation. Near $s=M^2$, the full propagator is resonant and the truncated effective field theory fails; retaining $S$ or adopting a different description is then necessary. Light loops are computed within the effective field theory, while higher-order matching supplies the corresponding heavy corrections.

A momentum-shell renormalization group step has three parts. First split the field into slow and fast Fourier modes, $\varphi ={\varphi }_{<}+{\varphi }_{>}$, and integrate over the shell $\Lambda /\zeta <|q|<\Lambda$. Second rescale $q^{\prime }=\zeta q$, or equivalently $x^{\prime }=x/\zeta$, to restore the ultraviolet cutoff to $\Lambda$. Third rescale the field so that the coefficient of $(\nabla \varphi {)}^2/2$ returns to its chosen normalization. The resulting free energy has the same operator expansion but new couplings; iteration traces a renormalization-group flow in coupling space.

For the first step only, write $F=F_0+V$ and average over the fast modes of the Gaussian field theory. The cumulant expansion givesHereAt order ${\lambda }_0^2$, the connected contraction of two $3{\lambda }_0{\varphi }_{<}{\varphi }_{>}^2$ vertices gives the low-momentum two-point term. Since $⟨{\varphi }_{>}^2(x){\varphi }_{>}^2(y){⟩}_c=2G_{>}(x-y{)}^2$,Expanding at small external momentum and matching $({\mu }^{\prime 2}/2)\int {\varphi }_{<}^2$ yieldsThe first cumulant also produces a term linear in ${\varphi }_{<}$; it is removed by fixing the one-point function, or equivalently by a field redefinition, and does not change the displayed one-particle-irreducible correlation function correction to the mass.

The leading vertex correction is order ${\lambda }_0^3$. Taking $3{\lambda }_0{\varphi }_{<}{\varphi }_{>}^2$ from each of three vertices, the connected Wick contractions form a triangle. There are eight contractions, so the third cumulant contributes $27\times 8/3!=36$ times the triangle integral. At zero external momentum,For nonzero external momenta the three propagators carry the corresponding shifted loop momenta, with every internal line restricted to the fast shell.

An Einstein-Hilbert action contains two derivatives, so its cubic graviton vertex produces only the $k\times k$ part of the product of the two $T$ tensors. The string amplitude also contains cross terms of order ${\alpha }^{\prime }{k}^4$ and a term of order ${\alpha }^{\prime 2}{k}^6$. These cannot arise from pure Einstein gravity. They require higher-curvature corrections, schematicallywith index contractions and coefficients fixed by string amplitudes up to field redefinitions. The full bosonic-string effective action also contains the dilaton and Kalb-Ramond field. Thus Einstein gravity is only its leading low-energy approximation.

A redundant operator is an interaction that can be removed by an invertible local field redefinition or by integration by parts together with the equations of motion. It can change off-shell formulas while leaving the on-shell S-matrix unchanged.

The string-frame metric is the target metric appearing directly in the kinetic term of the string nonlinear sigma model. With conventional Euclidean dilaton $\phi$, its leading spacetime gravitational action contains $\sqrt{-g^{(s)}}{e}^{-2\phi }(R+4|\nabla \phi {|}^2)$. The dilaton-dependent Ricci coefficient distinguishes it from the Einstein-frame metric. Frame changes are local field redefinitions where the conformal factor is finite and nonzero; the Lorentzian dilaton coupling sign convention fixes which named field is $\phi$.