For instance, among many possible examples ranging from phenomenology to formal theory, positivity has been applied to quantum gravitational modifications of general relativity and their implications for the Weak Gravity Conjecture [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21].

We note that it is never possible for any combination of SMEFT-like and HEFT-like UV physics to produce Wilson coefficients that, upon projection to the SMEFT, land in the forbidden dark blue region in Fig. 1. This is clear since every UV completion that transforms linearly under the SM gauge group (producing the SMEFT) can be written as a special case of a HEFT completion.

For a collection of $n$ scalars ${\mathrm{\Phi }}_I$, there are $n(n+1)(n^2+n+2)/8$ EFT operators of the form $c_{IJKL}{\partial }_{\mu }{\mathrm{\Phi }}_I{\partial }^{\mu }{\mathrm{\Phi }}_J{\partial }_{\nu }{\mathrm{\Phi }}_K{\partial }^{\nu }{\mathrm{\Phi }}_L$. The $CP$-violating operator ${\epsilon }^{\mu \nu \rho \sigma }{\partial }_{\mu }{\mathrm{\Phi }}_I{\partial }_{\nu }{\mathrm{\Phi }}_J{\partial }_{\rho }{\mathrm{\Phi }}_K{\partial }_{\sigma }{\mathrm{\Phi }}_K$ can be written as a total derivative upon integration by parts.

Let us briefly expand on this point. We write the Higgs doublet in terms of four real scalar fields as $$H=\frac{1}{\sqrt{2}}\left(\begin{array}{c}{\varphi }_2+i{\varphi }_1\\ {\varphi }_4-i{\varphi }_3\end{array}\right)$$and associate the vev with $⟨{\varphi }_I⟩=v{\delta }_{I4}$. In the notation of Ref. [45], we can then decompose $H$ into polar coordinates determined by a radius, $v+h$, and unit direction $\mathbf{n}$ on the 3-sphere, constructed such that ${\varphi }_I=(v+h)n_I$. Being a unit vector, we have $\mathbf{n}·\mathbf{n}=1$, and it is parametrized by the three angular coordinates ${\pi }_i/v$, in detail $$\mathbf{n}=\left(\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right)\exp [{\pi }_i{\tau }_i/v],$$where again ${\tau }_i$ are the generators of the rotations broken when the Higgs vev reduces custodial symmetry from $\mathrm{O}(4)\to \mathrm{O}(3)$. Using this language, the operators in the HEFT can be explicitly written in terms of $\mathbf{n}$ rather than ${\pi }_i$. As a single example, we have ${\mathcal{O}}_1^{h\pi }=v^2(\partial h{)}^2({\partial }_{\mu }\mathbf{n}·{\partial }^{\mu }\mathbf{n})$, up to terms of higher multiplicity in fields.

More explicitly, the Wilson coefficients of SMEFT form a self-contained vector subspace within the space of HEFT coefficients. While a violation of the SMEFT bounds, and the observation of nonzero Wilson coefficients in the HEFT-allowed region of Fig. 1, would arguably correspond to a smoking gun for the HEFT, the converse is not true; it is possible for the correct EFT to be the HEFT, but for its Wilson coefficients to be such that they do not allow the theory to be distinguished from the SMEFT via positivity bounds, e.g., if they land in the SMEFT-allowed region of Fig. 1.

Throughout, we write all inequalities as strict for brevity. However, if we are working at fixed order in the coupling expansion in a weakly coupled theory, all of these results reduce to weak inequalities [1, 4, 22, 67].

See Refs. [22, 30] for discussion of tree-level UV extensions of the SMEFT Higgs operators.

That is, all of these interactions can be simply written as couplings of heavy states to the unit vector $\mathbf{n}$. The only subtlety in writing the $\mathrm{O}(4)$ analogue of our UV extensions is that for the $B$ vector one first needs to dualize to an $\mathrm{O}(4)$ valued two-form, i.e., $B_{\mu }^{ab}{n}^a{\partial }^{\mu }{n}^b$.

See the discussion in Sec. 2.3.1 of Ref. [32].