Finite dimension-six Warsaw coefficient contract

Use the fields, direct-product gauge group, hypercharge normalization, three-family content, exact $U(1{)}_B\times U(1{)}_L$ symmetries, natural units and coefficient scale of the preceding linearly realized model. A local operator is a finite polynomial in these fields and their covariant derivatives at one spacetime point, with all Lorentz, gauge and family indices contracted to a Lorentz and gauge scalar. Its canonical dimension assigns one to $H$ and each derivative, $3\mathrm{/}2$ to a fermion, and two to a field strength.

The Warsaw coefficient contract is the finite dimension-six source list of gauge-invariant representatives $Q_i^{(6)}$ in Tables 2 and 3 of Grzadkowski, Iskrzynski, Misiak and Rosiek (2010), under their stated field, hypercharge, Lorentz, color and $SU(2{)}_L$ conventions, with exact baryon- and lepton-number conservation. The list consists of the source's 15 bosonic, 19 single-fermion-current and 25 four-fermion structures before family multiplicities and Hermitian conjugates; it is not a count of independent three-family complex coefficients. Include all family-index components for three generations, with the source's stated permutation, Hermiticity and conjugation rules; impose no family-universality or diagonal-flavor restriction. No right-handed neutrino field or dimension-five Weinberg operator is included.

In particular, this contract includes the source representative $Q_{H\mathrm{\square }}=(H^{†}H)\mathrm{\square }(H^{†}H)$, with $\mathrm{\square }={\eta }^{\mu \nu }{\mathrm{\partial }}_{\mu }{\mathrm{\partial }}_{\nu }$ and the mostly-plus metric of the adopted model. It has canonical dimension six because each Higgs field has dimension one and $\mathrm{\square }$ has dimension two.

For $Q_i^{(6)}$ use the dimensionless Wilson coefficient convention ${\mathcal{L}}_{\mathrm{d}\mathrm{i}\mathrm{m}6}=\frac{1}{{\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}^2}{\sum }_i{C}_i{Q}_i^{(6)},{\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}>0,$ with the coefficient tensors chosen so the density is real after the required Hermitian conjugates are added. The retained operator contract has derivative-jet order at most two, maximum field degree six, canonical dimension at most six, and loop order zero. Compactly supported field variations make total divergences vanish in action-level uses.

The source's complete-basis assertion is adopted as a literature-defined coefficient list, not proved by this item. The separate finite invariant-operator quotient supplies only its declared complex-algebraic selection. In this contract that quotient is explicitly formal: no equality between its algebraic invariant space and the full physical compact-gauge invariant space, and no local proof of the EOM/IBP/Fierz classification, is claimed. The finite-cutoff quotient does not add a full Axiom of Choice assumption to the field-data suppliers' stated ${\mathrm{A}\mathrm{C}}_{\omega }$ base.

Linearly realized fields, gauge group, units and global symmetries

Finite-jet physical operator and coefficient contract

Finite-order invariant polynomial quotient

Grzadkowski et al., full dimension-six classification

Falkowski, full Lectures on SMEFT