Nonlinear electroweak Goldstone and scalar-singlet EFT data

Use the same four-dimensional Minkowski spacetime, direct-product global gauge group, gauge connection, fermion representations, family content, and the preceding model's exact baryon- and lepton-number selection rules on formal local interactions. No anomaly-free quantum-current claim is made. In particular, the hypercharge-one-half Higgs doublet of that model is replaced in this realization by three angular Goldstone coordinates and a real scalar singlet; this is a distinct adopted model interface, not a deduction that the two models have identical field spaces.

Let $J=(-v,\mathrm{\infty })$, where $v>0$ is the mass-one reference radius from the preceding model. A scalar field is a smooth map $h:M\to J$ and is invariant under the electroweak gauge group. The Goldstone field is a smooth map $U:M\to SU(2)$, with $SU(2)$ represented by its defining $2\times 2$ unitary matrices of determinant one. For smooth gauge transformations $g_L:M\to SU(2{)}_L$ and $z:M\to U(1{)}_Y$, set $g_R(z)=\mathrm{diag}(z^3,z^{-3}),U(x)⟼g_L(x)U(x)g_R(z(x){)}^{-1},h(x)⟼h(x)\mathrm{.}$ Thus the left factor is the linearly represented $SU(2{)}_L$ gauge action, the right factor is the embedded hypercharge action, and $h$ is a gauge singlet. For explicitness, take ${\sigma }^1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }^2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }^3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),T_a=\frac{{\sigma }^a}{2}\mathrm{.}$ The exponential chart $U(\pi )=\exp \text{\hspace{0.17em}⁣}\left(\frac{2i{\pi }^a{T}_a}{v}\right),\pi \in {\mathbb{R}}^3,\mathrm{\parallel }\pi \mathrm{\parallel }<\pi v,$ is a smooth coordinate chart near the identity; $U$ itself is the globally valued group field and is not identified with one global exponential-coordinate chart.

Write $W_{\mu }=W_{\mu }^a{T}_a$ and use the local covariant derivative $D_{\mu }U={\mathrm{\partial }}_{\mu }U+ig{W}_{\mu }U-i{g}^{\mathrm{\prime }}{B}_{\mu }U{T}_3,$ with dimensionless gauge couplings $g,g^{\mathrm{\prime }}$ and the same right-principal connection convention as the shared gauge data. The adopted two-derivative scalar density is ${\mathcal{L}}_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{a}\mathrm{r}}^{(2)}=-\frac{1}{2}K(h){\mathrm{\partial }}_{\mu }h{\mathrm{\partial }}^{\mu }h-\frac{v^2}{4}{F}_C(h)\mathrm{Tr}\text{\hspace{0.17em}⁣}((D_{\mu }U{)}^{†}{D}^{\mu }U)-V(h),$ where $K$ and $F_C$ are supplied smooth real dimensionless functions on $J$, $K(h)>0$ and $F_C(h)>0$ on the field range used, $K(0)=F_C(0)=1$, and $V$ is a supplied real smooth potential of mass dimension four with $V^{\mathrm{\prime }}(0)=0$. Higher local interactions may use smooth coefficient functions of $h\mathrm{/}v$, the same gauge and matter fields, covariant derivatives, and curvature tensors, with B and L conserved. This statement adopts no finite derivative or chiral truncation; the separate power-counting definition specifies those cutoffs.

The assigned mass dimensions are $[h]=[{\pi }^a]=[v]=[{\mathrm{\Lambda }}_{\mathrm{H}\mathrm{E}\mathrm{F}\mathrm{T}}]=[{\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}]=1$, $[U]=[g]=[g^{\mathrm{\prime }}]=[K]=[F_C]=0$, and $[W_{\mu }]=[B_{\mu }]=1$; the potential $V$ has dimension four, ${\mathrm{\partial }}_{\mu }$ has dimension one, and any separately supplied regulator cutoff has dimension one. Thus the exponent defining $U(\pi )$ is dimensionless and every term in ${\mathcal{L}}_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{a}\mathrm{r}}^{(2)}$ has dimension four. The independent HEFT cutoff ${\mathrm{\Lambda }}_{\mathrm{H}\mathrm{E}\mathrm{F}\mathrm{T}}>0$, the radius $v$, the SMEFT coefficient scale ${\mathrm{\Lambda }}_{\mathrm{o}\mathrm{p}}$, and any finite-regulator cutoff are separate data; no equality or hierarchy among them is imposed here. Its low-energy use assumes external momenta small compared with ${\mathrm{\Lambda }}_{\mathrm{H}\mathrm{E}\mathrm{F}\mathrm{T}}$; this is an applicability condition and supplies no numerical remainder bound. It adopts no functional measure or regulator and implies no path integral, ultraviolet completion, infinite-volume limit, or empirical realization.

Shared gauge group, non-scalar fields, units and global symmetries

Buchalla, Catà and Krause, full electroweak chiral EFT paper

Alonso, full HEFT geometry lecture notes