Linearly realized Standard Model EFT field and scale data

Adopt four-dimensional affine Minkowski spacetime $M={\mathbb{R}}^{1,3}$ with coordinates $x^0=ct$ and metric $\eta =\mathrm{diag}(-1,1,1,1)$. Use natural units $c=\mathrm{\hslash }=1$ for engineering dimensions, so ${\mathrm{\partial }}_{\mu }$ has mass dimension one, a scalar field has dimension one, a spinor field has dimension $3\mathrm{/}2$, and a gauge field strength has dimension two.

The gauge group is the direct product $G_{\mathrm{S}\mathrm{M}}=SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$, with this global form and not a quotient by a subgroup of its center. Write an element of $U(1{)}_Y$ as $z\in U(1)$ and normalize its integer charge as $y=6Y$, so its representation acts by $z^y$. The three-family matter fields and scalar are smooth fields in the representations $q_L:(3,2{)}_1,u_R:(3,1{)}_4,d_R:(3,1{)}_{-2},{\mathrm{\ell }}_L:(1,2{)}_{-3},e_R:(1,1{)}_{-6},H:(1,2{)}_3\mathrm{.}$ There are no right-handed neutrino fields. Fix the finite Grassmann coefficient algebra $\mathcal{G}={\mathrm{\Lambda }}_{\mathbb{C}}({\zeta }_1,\dots ,{\zeta }_N)$ for some finite $N\ge 1$, graded by exterior parity, with the conjugate-linear involution ${\zeta }_a^{†}={\zeta }_a$ and $(uv{)}^{†}=v^{†}{u}^{†}$. On the trivial spin structure of $M$, use the spinor space $S={\mathbb{C}}^4$ with matrices ${\gamma }^{\mu }\in {\mathrm{End}}_{\mathbb{C}}(S)$ satisfying ${\gamma }^{\mu }{\gamma }^{\nu }+{\gamma }^{\nu }{\gamma }^{\mu }=2{\eta }^{\mu \nu }I,({\gamma }^0{)}^{†}=-{\gamma }^0,({\gamma }^j{)}^{†}={\gamma }^j(j=1,2,3)\mathrm{.}$ Set ${\gamma }_5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$, $P_L=(I-{\gamma }_5)\mathrm{/}2$, and $P_R=(I+{\gamma }_5)\mathrm{/}2$. Thus the left- and right-handed Weyl spaces are $S_L=P_{L}S$ and $S_R=P_{R}S$; the infinitesimal Lorentz spin action on $S$ is generated by ${\mathsf{S}}^{\mu \nu }=\frac{1}{4}[{\gamma }^{\mu },{\gamma }^{\nu }]$. For a spinor field define its Dirac adjoint by $\bar{\psi }={\psi }^{†}i{\gamma }^0$, with conjugation also acting on its internal and Grassmann components. The fermion components take values in ${\mathcal{G}}_{\bar{1}}$, while $H$ is a Grassmann-even complex doublet. The global charges are $B(q_L)=B(u_R)=B(d_R)=1\mathrm{/}3$, $L({\mathrm{\ell }}_L)=L(e_R)=1$, and zero for $H$ and the gauge fields; the global circles are parametrized by $b,\mathrm{\ell }\in U(1)$ and act on a field of charges $(B,L)$ by $b^{3B}{\mathrm{\ell }}^L$. Thus their characters have integer exponents: quarks have baryon-circle weight one, leptons have lepton-circle weight one, and conjugate fields have opposite weights. Impose these as exact selection rules on the retained formal local interactions; no anomaly-free quantum-current claim is made. There is no flavor symmetry beyond the displayed gauge action: Yukawa and Wilson coefficients may carry unrestricted three-family indices subject to the stated reality and Hermitian-conjugation conditions.

Here $\mathbf{3}={\mathbb{C}}^3$ and $\mathbf{2}={\mathbb{C}}^2$ are the defining representations of $SU(3{)}_c$ and $SU(2{)}_L$, respectively, and $\mathbf{1}$ is the trivial representation. The family space ${\mathbb{C}}_{\text{\hspace{0.17em}⁣}f}^3$ has trivial gauge action. In the order displayed above, the internal family-and-gauge carriers are ${\mathbb{C}}_{\text{\hspace{0.17em}⁣}f}^3\otimes {\mathbb{C}}^3\otimes {\mathbb{C}}^2$, ${\mathbb{C}}_{\text{\hspace{0.17em}⁣}f}^3\otimes {\mathbb{C}}^3$, ${\mathbb{C}}_{\text{\hspace{0.17em}⁣}f}^3\otimes {\mathbb{C}}^3$, ${\mathbb{C}}_{\text{\hspace{0.17em}⁣}f}^3\otimes {\mathbb{C}}^2$, ${\mathbb{C}}_{\text{\hspace{0.17em}⁣}f}^3$, and ${\mathbb{C}}^2$. On these carriers the element $(g_3,g_2,z)\in G_{\mathrm{S}\mathrm{M}}$ acts by ${\rho }_{q_L}=z(I_f\otimes g_3\otimes g_2),{\rho }_{u_R}=z^4(I_f\otimes g_3),{\rho }_{d_R}=z^{-2}(I_f\otimes g_3),{\rho }_{{\mathrm{\ell }}_L}=z^{-3}(I_f\otimes g_2),{\rho }_{e_R}=z^{-6}{I}_f,{\rho }_H=z^3{g}_2,$ where $I_f$ is the identity on ${\mathbb{C}}_{\text{\hspace{0.17em}⁣}f}^3$ and omitted group factors act trivially. The listed carriers are the internal factors; the matter fields, including their spacetime spin type, are the smooth maps $\begin{array}{rlrl}{\displaystyle q_L}& {\displaystyle :M\to {\mathcal{G}}_{\bar{1}}\otimes S_L\otimes {\mathbb{C}}_{\text{\hspace{0.17em}⁣}f}^3\otimes {\mathbb{C}}^3\otimes {\mathbb{C}}^2,}& {\displaystyle u_R,d_R}& {\displaystyle :M\to {\mathcal{G}}_{\bar{1}}\otimes S_R\otimes {\mathbb{C}}_{\text{\hspace{0.17em}⁣}f}^3\otimes {\mathbb{C}}^3,}\\ {\displaystyle {\mathrm{\ell }}_L}& {\displaystyle :M\to {\mathcal{G}}_{\bar{1}}\otimes S_L\otimes {\mathbb{C}}_{\text{\hspace{0.17em}⁣}f}^3\otimes {\mathbb{C}}^2,}& {\displaystyle e_R}& {\displaystyle :M\to {\mathcal{G}}_{\bar{1}}\otimes S_R\otimes {\mathbb{C}}_{\text{\hspace{0.17em}⁣}f}^3,}& {\displaystyle H}& {\displaystyle :M\to {\mathbb{C}}^2\mathrm{.}}\end{array}$ Each Grassmann coefficient function is smooth on $M$; the Lorentz spin action is on $S_{L\mathrm{/}R}$ and the displayed gauge action is on the internal factors.

Choose a reference radius $v>0$ of mass dimension one; it denotes a vacuum radius only under the optional full-potential condition below. Define ${\sigma }^1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }^2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }^3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),T^I=\frac{{\sigma }^I}{2},$ and $\tilde{H}=i{\sigma }^2{H}^{\ast }$.

Choose the trivial smooth right principal bundle $P=M\times G_{\mathrm{S}\mathrm{M}}$ and only this bundle sector. A connection is a smooth right-principal connection with the convention $\omega (Z_P)=Z$ and $R_g^{\ast }\omega ={\mathrm{Ad}}_{g^{-1}}\omega$. In a local section write its form as $A=A_{\mu }d{x}^{\mu }$. Choose Hermitian generators $t^A$ for the defining $SU(3{)}_c$ representation with $\mathrm{tr}(t^A{t}^B)={\delta }^{AB}\mathrm{/}2$ and $[t^A,t^B]=i{f}^{ABC}{t}^C$; the $T^I$ satisfy $[T^I,T^J]=i{ϵ}^{IJK}{T}^K$, with ${ϵ}^{123}=1$. Let $G^A=G_{\mu }^{A}d{x}^{\mu }$, $W^I=W_{\mu }^{I}d{x}^{\mu }$, and $B=B_{\mu }d{x}^{\mu }$ be smooth real gauge-potential one-forms with components of mass dimension one, and take $g_s,g,g^{\mathrm{\prime }}>0$ dimensionless. In the product Lie algebra, the local connection components are $(i{g}_s{G}_{\mu }^A{t}^A,ig{W}_{\mu }^I{T}^I,i{g}^{\mathrm{\prime }}{B}_{\mu }\mathrm{/}6)$, where the final entry uses the generator $i$ of $\mathfrak{u}(1)$ for the coordinate $z=e^{i\vartheta }$. Thus in a charge-$y$ representation ${\rho }_{\ast }(A_{\mu })=i{g}_s{G}_{\mu }^A{t}^A+ig{W}_{\mu }^I{T}^I+i{g}^{\mathrm{\prime }}{B}_{\mu }Y,Y=y\mathrm{/}6,$ with absent factors acting as zero. A field in representation $\rho$ has $D_{\mu }={\mathrm{\partial }}_{\mu }+{\rho }_{\ast }(A_{\mu })$; this acts trivially on its Lorentz spin factor in the global flat frame. In particular $D_{\mu }H={\mathrm{\partial }}_{\mu }H+ig{W}_{\mu }^I{T}^{I}H+\frac{i}{2}{g}^{\mathrm{\prime }}{B}_{\mu }H$. Define the curvature components by $G_{\mu \nu }^A={\mathrm{\partial }}_{\mu }{G}_{\nu }^A-{\mathrm{\partial }}_{\nu }{G}_{\mu }^A-g_s{f}^{ABC}{G}_{\mu }^B{G}_{\nu }^C,W_{\mu \nu }^I={\mathrm{\partial }}_{\mu }{W}_{\nu }^I-{\mathrm{\partial }}_{\nu }{W}_{\mu }^I-g{ϵ}^{IJK}{W}_{\mu }^J{W}_{\nu }^K,B_{\mu \nu }={\mathrm{\partial }}_{\mu }{B}_{\nu }-{\mathrm{\partial }}_{\nu }{B}_{\mu };$ they are the coefficients of $F_A=dA+\frac{1}{2}[A\wedge A]$ under the supplied graded-bracket convention. The representation differential and gauge couplings are dimensionless, so $A_{\mu }$ has mass dimension one. The ordinary component hypercharges are $Y=y\mathrm{/}6$; in particular $Y(H)=1\mathrm{/}2$. The formal local interaction set imposes the exact $U(1{)}_B$ and $U(1{)}_L$ selection rules stated above; no anomaly-free quantum-current claim is made. Yukawa matrices are otherwise arbitrary.

To instantiate the finite-jet algebraic operator data, let $T={\mathbb{C}}^4$ with the complex-bilinear extension ${\eta }_{\mathbb{C}}$ of $\eta$. Its Clifford algebra is the unital complex algebra generated by $T$ with $uv+vu=2{\eta }_{\mathbb{C}}(u,v)1$. Let $\mathrm{Spin}({\eta }_{\mathbb{C}})$ be the complex spin group, realized as even products of vectors $u\in T$ satisfying ${\eta }_{\mathbb{C}}(u,u)=1$ inside this algebra. It acts on $T$ by $v\mapsto sv{s}^{-1}$ and on $S$ through the Clifford representation $e_{\mu }\mapsto {\gamma }^{\mu }$, preserving $S_L$ and $S_R$. Use the complex reductive affine algebraic group $G_{\mathrm{a}\mathrm{l}\mathrm{g}}=\mathrm{Spin}({\eta }_{\mathbb{C}})\times S{L}_3(\mathbb{C})\times S{L}_2(\mathbb{C})\times {\mathbb{C}}_Y^{\times }\times {\mathbb{C}}_B^{\times }\times {\mathbb{C}}_L^{\times }\mathrm{.}$ Here $\mathrm{Spin}({\eta }_{\mathbb{C}})\cong S{L}_2(\mathbb{C})\times S{L}_2(\mathbb{C})$. The physical connected Lorentz spin group is the subgroup generated by the real-parameter exponentials of the bivectors $\frac{1}{4}[e_{\mu },e_{\nu }]$; its inclusion, together with the defining matrix inclusions of the gauge factors and the circle inclusions, gives $\iota :{\mathrm{Spin}}^{+}(1,3)\times G_{\mathrm{S}\mathrm{M}}\times U(1{)}_B\times U(1{)}_L⟶G_{\mathrm{a}\mathrm{l}\mathrm{g}},(s,g_3,g_2,z,b,\mathrm{\ell })⟼(s,g_3,g_2,z,b,\mathrm{\ell })\mathrm{.}$ On each matter carrier, extend the displayed internal matrix formula by replacing $g_3,g_2,z$ by the corresponding complex factors and multiplying by $b^{3B}{\mathrm{\ell }}^L$; retain its indicated Clifford spin factor. Family factors and all supplied coefficient parameters have trivial action; there is no additional flavor group. Treat $H^{†}$ as an independent algebraic dual doublet, and $\bar{\psi }$ as an independent dual Dirac-spinor/internal generator restricted by $\bar{\psi }=\bar{\psi }{P}_R$ for left-handed $\psi$ and $\bar{\psi }=\bar{\psi }{P}_L$ for right-handed $\psi$. Evaluation on physical fields imposes the stated conjugation. Curvatures have the adjoint internal action and the ${\mathrm{\Lambda }}^2{T}^{\ast }$ spacetime action, with trivial abelian adjoint action and zero $B,L$ weights.

At each point take the finite graded generator space $J_2$ consisting of the components of $D_{{\mu }_1}\cdots D_{{\mu }_a}\varphi$ for $0\le a\le 2$, where $\varphi$ runs over these matter, conjugate-matter and curvature generators. Derivative indices carry $T^{\ast }$; the action is the tensor product of that index action and the indicated representation of $\varphi$, with dual actions on dual factors. Derivative words are initially independent formal generators, and evaluation sends them to the ordered covariant derivatives of smooth fields. Undifferentiated gauge potentials are not homogeneous jet generators: they enter through the connection in $D$ and the curvatures. Restriction along $\iota$ is the physical pointwise action on these covariant tensors; the connection transformation makes the same rule valid for position-dependent gauge transformations. The generators are even for Higgs and curvatures and odd for fermions, and their weights are $[\varphi ]+a$. These finite tensor, dual and adjoint representations are rational and preserve weights and parity. Form $P_2=\mathrm{Sym}(J_{2,\bar{0}}^{\ast })\otimes \mathrm{\Lambda }(J_{2,\bar{1}}^{\ast })$, with $(g\cdot f)(j)=f(g^{-1}j)$, and retain degree at most six and weight at most six. Any invariant quotient used here is explicitly a formal algebraic selection only. No equality with all physical invariant functions, completeness of covariant-jet relations, or Warsaw classification follows from this construction; the dimension-six representatives remain the separately adopted source list.

The adopted object is a formal local Lagrangian density made from finitely many fields, covariant derivatives, and field strengths at one spacetime point, contracted to Lorentz scalars and invariant under the stated gauge and global actions. The renormalizable part has the gauge kinetic terms $-\frac{1}{4}{G}_{\mu \nu }^A{G}^{A\mu \nu }-\frac{1}{4}{W}_{\mu \nu }^I{W}^{I\mu \nu }-\frac{1}{4}{B}_{\mu \nu }{B}^{\mu \nu }$, fermion kinetic terms $-{\sum }_{\psi }\bar{\psi }{\gamma }^{\mu }{D}_{\mu }\psi$, Higgs kinetic term $-(D_{\mu }H{)}^{†}{D}^{\mu }H$, and potential contribution $-V_4(H)$, where the real gauge-invariant potential is $V_4(H)=V_0-{\mu }_H^2{H}^{†}H+\lambda (H^{†}H{)}^2,V_0,{\mu }_H^2,\lambda \in \mathbb{R},$ and Yukawa terms $-({\bar{q}}_L{Y}_{d}H{d}_R+{\bar{q}}_L{Y}_u\tilde{H}{u}_R+{\bar{\mathrm{\ell }}}_L{Y}_{e}H{e}_R+\mathrm{h}\mathrm{.}\mathrm{c}\mathrm{.}),$ where each $Y_f$ is a complex $3\times 3$ family matrix and h.c. is the Hermitian conjugate. With the stated Clifford and adjoint conventions the kinetic prefactor is $-1$: writing $B_S=i{\gamma }^0$, one has $(B_S{\gamma }^{\mu }{)}^{†}=-B_S{\gamma }^{\mu }$, so $\int \bar{\psi }{\gamma }^{\mu }{D}_{\mu }\psi$ is Hermitian after integration by parts under the stated boundary convention. Equivalently the density may be replaced by $-\frac{1}{2}[\bar{\psi }{\gamma }^{\mu }{D}_{\mu }\psi -(D_{\mu }\bar{\psi }){\gamma }^{\mu }\psi ]$, where $D_{\mu }\bar{\psi }={\mathrm{\partial }}_{\mu }\bar{\psi }-\bar{\psi }{\rho }_{\ast }(A_{\mu })$. For translating source four-component spinor factors from the mostly-minus metric, use ${\eta }_{\mathrm{s}\mathrm{r}\mathrm{c}}=-\eta$, ${\mathrm{\Gamma }}^{\mu }=-i{\gamma }^{\mu }$ and ${\bar{\psi }}_{\mathrm{s}\mathrm{r}\mathrm{c}}={\psi }^{†}{\mathrm{\Gamma }}^0=-\bar{\psi }$, with ${\mathrm{\Gamma }}_5=i{\mathrm{\Gamma }}^0{\mathrm{\Gamma }}^1{\mathrm{\Gamma }}^2{\mathrm{\Gamma }}^3={\gamma }_5$; source spinor factors are read with this explicit change of conventions. Scalar and gauge contractions use the explicit local conventions, including the separately specified Warsaw scalar representatives; numerical source coefficients require the corresponding convention conversion before identification. In particular $i{\bar{\psi }}_{\mathrm{s}\mathrm{r}\mathrm{c}}{\mathrm{\Gamma }}^{\mu }{D}_{\mu }\psi =-\bar{\psi }{\gamma }^{\mu }{D}_{\mu }\psi$. Set the gauge theta terms to zero in this selected model. Add finite sets $I_d$ of local gauge-invariant, $B$- and $L$-neutral polynomial operators ${\mathcal{O}}_{d,i}$ at each retained dimension $d\le 6$. Retain derivative-jet order at most two, maximum polynomial field degree six, and loop order zero. Use $C_{d,i}{\mathcal{O}}_{d,i}\mathrm{/}{\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}^{d-4}$, where ${\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}>0$ has mass dimension one and each $C_{d,i}$ is dimensionless. The dimension-five set is empty under the adopted exact lepton-number symmetry. The dimension-six set is fixed by the separate Warsaw coefficient contract. Wilson coefficients are supplied parameters, not calculated or fitted here.

In these natural units $Y_f$ has mass dimension zero, $V_0$ has dimension four, ${\mu }_H^2$ has dimension two, and $\lambda$ has dimension zero; the Lagrangian density has mass dimension four. To define the retained scalar potential, set $s=H^{†}H\ge 0$ and evaluate the full retained Lagrangian on constant $H$, with fermions, gauge fields and all derivatives set to zero. Let ${\mathcal{L}}_{\mathrm{a}\mathrm{d}\mathrm{d}}^{(0)}(s)$ be the sum of all retained operator contributions surviving this evaluation, including their Wilson coefficients, powers of ${\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}$, and the Hermitian-conjugate terms required for a real density, each counted as in the adopted Lagrangian. Define ${{\mathcal{L}}_{\mathrm{r}\mathrm{e}\mathrm{t}}\mid }_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}H,\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{s}=0}=-V_{\mathrm{r}\mathrm{e}\mathrm{t}}(s),V_{\mathrm{r}\mathrm{e}\mathrm{t}}(s)=V_4(s)-{\mathcal{L}}_{\mathrm{a}\mathrm{d}\mathrm{d}}^{(0)}(s)\mathrm{.}$ In the dimension-six Warsaw scalar-potential sector, $Q_H=(H^{†}H{)}^3$ has real coefficient $C_H$ and Lagrangian contribution $+C_H{s}^3\mathrm{/}{\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}^2$, hence contributes $-C_H{s}^3\mathrm{/}{\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}^2$ to $V_{\mathrm{r}\mathrm{e}\mathrm{t}}$. The derivative operators $Q_{H\mathrm{\square }}$ and $Q_{HD}$ vanish in this constant-field evaluation. In particular, when the only addition to $V_4$ is this Warsaw potential term, $V_{\mathrm{r}\mathrm{e}\mathrm{t}}(s)=V_0-{\mu }_H^{2}s+\lambda s^2-\frac{C_H}{{\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}^2}{s}^3\mathrm{.}$

The optional interpretation of $v$ as a vacuum radius assumes that $s_v=v^2\mathrm{/}2>0$ is a local minimum of this full $V_{\mathrm{r}\mathrm{e}\mathrm{t}}$ within the field range where the truncated EFT is being used. Thus $V_{\mathrm{r}\mathrm{e}\mathrm{t}}^{\mathrm{\prime }}(s_v)=0$ is necessary. If a nondegenerate radial minimum is selected, impose also $V_{\mathrm{r}\mathrm{e}\mathrm{t}}^{\mathrm{\prime }\mathrm{\prime }}(s_v)>0$; the radial potential Hessian for $H=(0,r\mathrm{/}\sqrt{2}{)}^T$ at $r=v$ is then $v^2{V}_{\mathrm{r}\mathrm{e}\mathrm{t}}^{\mathrm{\prime }\mathrm{\prime }}(s_v)>0$, while gauge-orbit directions need not have positive Hessian. Degenerate local minima remain permitted under the local-minimum condition itself. For the displayed cubic potential the stationarity and nondegeneracy conditions read $-{\mu }_H^2+\lambda v^2-\frac{3C_H{v}^4}{4{\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}^2}=0,2\lambda -\frac{3C_H{v}^2}{{\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}^2}>0.$ No minimum or stationarity condition is imposed on $V_4$ separately: retained higher-dimension terms may shift or supply the local minimum. Without the full-potential assumption $v$ is only a reference value. No global stability, large-field behavior, or control of omitted higher-dimension terms is inferred from a minimum of the truncated potential.

This is an operator-level model on smooth fields and compactly supported variations; total divergences are therefore zero in the associated action comparison. Its low-energy use is restricted to external momenta small compared with ${\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}$, with the finite dimension-six truncation understood formally and without a numerical remainder bound. No regulator or functional measure is adopted, and no finite graph cutoff is identified with ${\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}$. The postulate asserts no path integral, continuum limit, ultraviolet completion, or empirical realization.

Countable-choice assumption inherited by the gauge construction

Finite-jet physical EFT operator and scale convention

Right-principal gauge group, global form, bundle and representation data

Associated-bundle covariant derivative

Curvature and local field-strength convention

Grzadkowski et al., full dimension-six operator article

Falkowski, full Lectures on SMEFT