real scalar action, Klein–Gordon action, canonical scalar action
Scope
A canonically normalized real scalar field in flat d-dimensional Lorentzian spacetime.
Assumptions
Boundary conditions justify the integrations by parts used by the owning derivation.; The interaction potential and its stability or EFT interpretation are specified locally.
J is a bosonic source, phi_c is the source-dependent classical field, W generates connected correlators, and Gamma generates one-particle-irreducible vertices.
Dimensional check
The source term integral and action are dimensionless in natural units.
generating functional, effective action Legendre transform, W and Gamma
Scope
Lorentzian bosonic source functional with the declared plus source coupling.
Assumptions
The functional Legendre transform exists in the bounded sense used by the owning treatment.; Fermionic, Euclidean, background-field, and closed-time-path variants declare their signs locally.
J is the plus-sign Lorentzian source; JD_FJ includes both spacetime integrals. D_F^(n) is the raw time-ordered n-point function. D_ij=D_F(x_i-x_j), and the numerals label distinct spacetime arguments.
Dimensional check
In four dimensions [phi]=1, [J]=3, [D_F(x)]=2 and the integrated exponent is dimensionless.
The connected fourth moment vanishes in the centered Gaussian vacuum. Interactions, non-Gaussian states and renormalized coincident composite products require their own treatment.
Centered free real scalar vacuum with the Feynman boundary prescription.
Assumptions
Use a finite regulator before interpreting the functional expression.; D_F is the raw time-ordered two-point function; G_F=iD_F is the delta-normalized inverse of the Klein–Gordon operator.
H is the internal heavy mediator, phi the light field, [g]=1, and R=M_full-M_EFT^[1]. The superscript [1] retains one power of q/M^2 in each heavy propagator.
Dimensional check
The contact, exchange amplitudes and remainder are dimensionless. The leading derivative operator has coefficient g^2/M^4 of dimension -2.
Application
Scalar capstone, stage 7: complete the potential square and use 1/(1-z)-(1+z)=z^2/(1-z).
Limitation
The bound is absolute. A relative error with respect to the full amplitude needs a nonzero lower bound on that amplitude; cancellations with the contact can invalidate an exchange-relative estimate. This is tree matching, without loop threshold corrections.
Tree-level light-scalar scattering below a heavy scalar mass M.
Assumptions
The full potential is U=m^2 phi^2/2+M^2 H^2/2+g H phi^2/2+lambda_full phi^4/24, with m^2>0 and lambda_full>3g^2/M^2.; Use the same contact in full and effective amplitudes, M much larger than m, g^2/M^2 much smaller than 1 and lambda_full much smaller than 1.; For each q in {s,t,u}, require |q|<=rho M^2 with 0<rho<1.; This massive worked extension retains one power of each invariant; the canonical massless fixed-angle example instead retains the quadratic terms.
s is the squared center-of-mass energy, v_s=sqrt(1-4m^2/s), m is the physical scalar mass, F is the invariant flux and dPhi_2 is the labeled phase-space measure.
Dimensional check
The cross section has mass dimension -2; the flux has dimension 2 and the four-dimensional two-body phase space is dimensionless.
Application
Scalar capstone, stage 4: derive the radial delta-function Jacobian and compare the two angular-counting conventions.
Limitation
The total-rate formula uses the angle-independent contact amplitude. At exact threshold the incident flux vanishes; the stated limit is approached from s>4m^2. A permutation-ordered hemisphere instead uses no final factorial.
Elastic two-to-two scattering of equal-mass identical scalars, s>4m^2.
Assumptions
Integrate over the full labeled center-of-mass sphere and include exactly one final 1/2!.; The two incoming beams are specified separately; there is no incoming factorial.; At tree level the contact amplitude is M=-lambda.
lambda_0=mu^(2epsilon)[lambda+delta lambda+...] is the bare quartic coupling; gamma_E is the Euler constant and mu_0 is the boundary-data scale.
Dimensional check
lambda and its four-dimensional beta function are dimensionless; the bare coupling in d=4-2epsilon has dimension 2epsilon.
Application
Scalar capstone, stage 6: the three finite bubble logarithms cancel the leading scale variation of the tree amplitude.
Limitation
The solution integrates the one-loop beta function. Its formal Landau singularity signals failure of that approximation and does not by itself prove scalar triviality.
One-component scalar phi^4 coupling with interaction -lambda phi^4/4! near four dimensions.
Assumptions
Use the MSbar coupling coordinate and d=4-2epsilon.; Hold bare data fixed and retain the canonical -2epsilon lambda term until after differentiation.; Use the running solution only while the coupling remains perturbative.
The factorial in the interaction cancels the attachments of four labeled external fields.
Expression
Lint=−4!λϕ4,V4=−iλ,Mtree=−λ.
Variables
lambda is the real quartic coupling. V_4 is the vertex including its i; the scattering contribution is iM, with the momentum-conserving delta function factored out.
Dimensional check
The coupling, vertex and four-dimensional two-to-two amplitude are dimensionless.
Application
Scalar capstone, stage 3: count the 4! attachments before applying LSZ.
Limitation
This factorial cancellation does not remove a graph automorphism factor or an identical final-state phase-space factor.
Tree-level quartic interaction of a canonically normalized real scalar in four dimensions.
Assumptions
Use the normalized vacuum perturbation expansion with the inherited +iS weight.; The four external insertions are labeled; state normalization is relativistic.
Continuous functions or distributions on d-dimensional spacetime for which the transforms are defined.
Assumptions
Distributional transforms use an appropriate test-function or tempered-distribution setting.; Specialized transforms state their own measure and phases.
Q is Euclidean external momentum, m is the positive mass and mu is the subtraction scale. B is the subtracted scalar integral.
Dimensional check
B is dimensionless in the four-dimensional limit; the logarithm contains a ratio of squared mass scales.
Application
Scalar calculation, stages 5–6: continue the integral with its boundary prescription and assemble the three-channel amplitude, local counterterm and two-particle cut.
Limitation
Analytic continuation to a timelike amplitude requires the Feynman boundary prescription, vertex and symmetry factors, and all required channels. The Euclidean expression alone is not a complete scattering amplitude.
The scalar two-propagator integral at Euclidean momentum Q, with Q^2≥0 and m^2>0.
Assumptions
Use d=4-2epsilon and modified minimal subtraction.; The integral excludes interaction vertices, graph symmetry factors and external-state normalization.
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