Positivity restrictions on the mixing of
dimension-eight SMEFT operators

Mikael Chala${}^{\,\bm{a}}$ and Xu Li${}^{\,\bm{b,c}}$
${}^{a}\,\,$Departamento de Física Teórica y del Cosmos, Universidad de
Granada, Campus de Fuentenueva, E–18071 Granada, Spain
${}^{b}\,\,$ Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China
${}^{c}\,\,$ School of Physical Sciences, University of Chinese Academy of
Sciences, Beijing 100049, China

###### Abstract

We discuss the structure of the mixing among dimension-eight operators in the
SMEFT relying on the positivity of two-to-two forward scattering amplitudes.
We uncover tens of new non-trivial zeros as well as hundreds of terms with
definite sign in (a particular basis of) the corresponding anomalous dimension
matrix. We highlight that our results are not immediately apparent from the
Feynman diagrammatic perspective, nor from on-shell amplitude methods. As a
byproduct of this work, we provide positivity bounds not previously derived in
the literature, as well as explicit values of certain elements of the
anomalous dimension matrix that serve for cross-check of our results.

##  1 Introduction

The SMEFT is the extension of the Standard Model (SM) with operators of
dimension higher than four, suppressed by powers of the cutoff $\Lambda$; see
Refs. [1, 2] for recent reviews. During the last decades, it has increasily
become one of the most popular descriptions of physics beyond the SM, with
different theoretical aspects of this framework being better and better
understood. Among the most important ones, we could highlight the following
three.

(i) The number of independent physical operators up to dimension 15 has been
computed using different techniques, including Hilbert series methods [3, 4],
standard group theory techniques [5, 6] or on-shell amplitudes [7, 8, 9]. The
explicit form of these operators have been also computed in several instances,
and different software tools (partially) automatising this construction are
already available [5, 6, 10, 11]. Similar considerations have been also
investigated for operators independent off-shell [12, 13, 9, 14].

(ii) The physical parameter space of the SMEFT, particularly at dimension
eight, has been considerably constrained in light of positivity bounds. These
are restrictions on the sign of Wilson coefficients ensuing from very robust
principles, including the analiticity, unitarity and the large-energy
behaviour of the S-matrix [15]. Many of these constraints are strongly
competitive with experimental limits [16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29, 30].

(iii) The full renormalisation group equations (RGEs) of the SMEFT are known
up to dimension six [31, 32, 33, 34, 15], including finite terms (or matching
corrections) induced by evanescent operators [35]; as well as to dimension
seven [36, 37, 38, 14]. And the matching of different models onto the SMEFT to
this order has been also investigated [39, 12, 40, 41, 42, 43, 44, 45, 46,
47].

Despite the significant progress made also towards renormalising the SMEFT to
dimension eight [48, 7, 49, 50, 51, 52, 53], this endevour is still far from
complete, due mostly to the enourmous amount of operators [54, 55]. However,
there is a reasonable understanding of the structure of mixings based on
generalised unitarity and on-shell amplitude techniques, which highlight
certain non-renormalisation results otherwise obscure within the realm of
Feynman diagrams [56, 57]. More recently, it has been proposed [58] that the
sign of some mixing terms, as well as certain zeros that are not even apparent
from the on-shell amplitude perspective, can be unveiled upon studying the
positivity of two-to-two scattering amplitudes in the forward limit at very
large distances. The basic idea is that, as indicated before, certain Wilson
coefficients $c_{i}$ are constrained to be $c_{i}\geq 0$. For this to hold
irrespective of the running induced by any other Wilson coefficient $c_{j}$ at
a scale $\mu$, $c_{i}(\mu)\sim\gamma_{ij}c_{j}\log{\mu/\Lambda}$, where
$\gamma$ is the so-called anomalous dimension, the condition $\gamma_{ij}\leq
0$ must be satisfied if $c_{j}$ is also bounded by positivity, $c_{j}\geq 0$;
otherwise $\gamma_{ij}=0$.

In Ref. [58], a precise formulation of this idea was applied to the
electroweak and leptonic sectors of the SMEFT. In this paper, we extend the
methods of that reference to the full SMEFT including quarks and colour. We
organise the article as follows. In section 2, we describe the structure of
the SMEFT and provide an example of vanishing mixing that is neither apparent
from the Feynman-diagrammatic approach nor from the point of view of on-shell
amplitude methods. In section 3, we discuss briefly the constraints on
anomalous dimensions implied by the positivity of the S-matrix, making special
emphasis on the derivation of complete positivity bounds. We apply these
methods to uncover the structure of the anomalous dimension matrix (ADM) of
the SMEFT in section 4. Finally, we conclude in section 5. Appendices A, B and
C complement the previous discussions.

##  2 Theoretical background

Our convention for the SM Lagrangian is the following:

$\displaystyle\mathcal{L}_{\text{SM}}$
$\displaystyle=-\frac{1}{4}G_{\mu\nu}^{A}G^{A\,\mu\nu}-\frac{1}{4}W^{I}_{\mu\nu}W^{I\,\mu\nu}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}$
(1)
$\displaystyle+\overline{q}\mathrm{i}\not{D}q+\overline{l}\mathrm{i}\not{D}l+\overline{u}\mathrm{i}\not{D}u+\overline{d}\mathrm{i}\not{D}d+\overline{e}\mathrm{i}\not{D}e$
$\displaystyle+(D_{\mu}\phi)^{\dagger}(D^{\mu}\phi)+\mu_{\phi}^{2}|\phi|^{2}-\lambda_{\phi}|\phi|^{4}-(\overline{q}\tilde{\phi}Y_{u}u+\overline{q}\phi
Y_{d}d+\overline{l}\phi Y_{e}e+\text{h.c.})\,.$

As usual, $G$, $W$ and $B$ denote the $SU(3)_{c}$, $SU(2)_{L}$ and $U(1)_{Y}$
gauge fields with gauge couplings $g_{1}$, $g_{2}$ and $g_{3}$, respectively;
$q$ and $l$ stand for the left-handed quarks and leptons, respectively, and
$u,d$ and $e$ for the right-handed counterparts; $\phi$ represents the Higgs
doublet and $Y_{u}$, $Y_{d}$ and $Y_{e}$ are the Yukawa couplings. We work in
the limit of one family of fermions.

The SMEFT Lagrangian to dimension eight, ignoring lepton-number violation,
reads:

$\mathcal{L}_{\text{SMEFT}}=\mathcal{L}_{\text{SM}}+\frac{1}{\Lambda^{2}}\sum_{i}c_{i}^{(6)}\mathcal{O}_{i}^{(6)}+\frac{1}{\Lambda^{4}}\sum_{j}c_{j}^{(8}\mathcal{O}_{j}^{(8)}\,,$
(2)

where $\Lambda$ represents the cutoff of the effective-field theory (EFT). The
first and second sums above run over bases of dimension-six [59] and
dimension-eight [54, 55] operators, respectively. We are only interested on
the dimension-eight ones, for which we use the self-explanatory notation of
Ref. [54]. The mixing of these operators under renormalisation group running
is governed by the corresponding RGEs:

$\dot{c}_{i}^{(8)}\equiv
16\pi^{2}\mu\frac{dc_{i}^{(8)}}{d\mu}=\gamma_{ij}c_{j}^{(8)}+\gamma_{ijk}^{\prime}c_{j}^{(6)}c_{k}^{(6)}\,.$
(3)

The anomalous dimension $\gamma$ characterises the mixing between dimension-
eight operators due to SM terms, while $\gamma^{\prime}$ describes the
renormalisation of dimension-eight operators by pairs of dimension-six ones.
Our goal in this paper is constraining the shape of $\gamma$ from the
positivity of the S-matrix [58]. Previous restrictions on $\gamma$, and in
particular the occurrence of certain non-obvious zeros, can be found in Refs.
[60, 56, 61, 62, 63, 64]. See also Ref. [50] for complementary results based
on a geommetric description of the SMEFT.

As a simple motivation for this work, let us consider the example of the
mixing of $e^{2}\phi^{2}D^{3}$ into $e^{2}B^{2}D$. There are two operators in
the first class, and only one in the second; see Ref. [54]. For convenience,
we write them here explicitly:

$\displaystyle\mathcal{O}_{e^{2}B^{2}D}$
$\displaystyle=\mathrm{i}(\overline{e}\gamma^{\mu}D^{\nu}e)B_{\mu\rho}B_{\nu}^{\,\rho}+\text{h.c.}\,$
(4) $\displaystyle\mathcal{O}_{e^{2}\phi^{2}D^{3}}^{(1)}$
$\displaystyle=\mathrm{i}(\overline{e}\gamma^{\mu}D^{\nu}e)(D_{(\mu}D_{\nu)}\phi^{\dagger}\phi)+\text{h.c.}\,,$
(5) $\displaystyle\mathcal{O}_{e^{2}\phi^{2}D^{3}}^{(2)}$
$\displaystyle=\mathrm{i}(\overline{e}\gamma^{\mu}D^{\nu}e)(\phi^{\dagger}D_{(\mu}D_{\nu)}\phi)+\text{h.c.}\,.$
(6)

Figure 1: Feynman diagrams relevant for the one-loop renormalisation of
$e^{2}B^{2}D$ by $e^{2}\phi^{2}D^{3}$. Diagrams with massless bubbles, which
vanish in dimensional-regularisation, are not shown.

It is obvious that there exists some linear combination of
$\mathcal{O}_{e^{2}\phi^{2}D^{3}}^{(1)}$ and
$\mathcal{O}_{e^{2}\phi^{2}D^{3}}^{(2)}$ that does not mix into
$\mathcal{O}_{e^{2}B^{2}D}$. Indeed, if any of the two anomalous dimensions,
say $\gamma_{c_{e^{2}B^{2}D},\,c_{e^{2}\phi^{2}D^{3}}^{(2)}}$, is non-zero,
then we can simply define:

$\widetilde{O}_{e^{2}\phi^{2}D^{3}}^{(1)}=\mathcal{O}_{e^{2}\phi^{2}D^{3}}^{(1)}-\frac{\gamma_{c_{e^{2}B^{2}D},\,c_{e^{2}\phi^{2}D^{3}}^{(1)}}}{\gamma_{c_{e^{2}B^{2}D},\,c_{e^{2}\phi^{2}D^{3}}^{(2)}}}\mathcal{O}_{e^{2}\phi^{2}D^{3}}^{(2)}\,,$
(7)

which by construction does not renormalise $\mathcal{O}_{e^{2}B^{2}D}$.

The exact form of $\widetilde{\mathcal{O}}_{e^{2}\phi^{2}D^{3}}^{(1)}$ can be
obtained by explicit computation of $\dot{c}_{e^{2}B^{2}D}$. From the Feynman-
diagrammatic perspective, we must compute the diagrams in Fig. 1. Using
dimensional regularisation with space-time dimension $d=4-2\epsilon$, and with
the help of FeynArts [65] and FormCalc [66], we find that most of them vanish;
the rest, despite the many different terms involving a variety of products of
momenta, polarizations and gamma matrices, shrink to give the compact result:

$\dot{c}_{e^{2}B^{2}D}=-\frac{1}{3}g_{1}^{2}(c_{e^{2}\phi^{2}D^{3}}^{(1)}+c_{e^{2}\phi^{2}D^{3}}^{(2)})\,.$
(8)

From here, it is clear that
$\widetilde{\mathcal{O}}_{e^{2}\phi^{2}D^{3}}^{(1)}$ is the linear combination
of $\mathcal{O}_{e^{2}\phi^{2}D^{3}}^{(1)}$ and
$\mathcal{O}_{e^{2}\phi^{2}D^{3}}^{(2)}$ with different signs. In other words,
let us make, for example, the basis transformation defined by:

$\begin{pmatrix}c_{e^{2}\phi^{2}D^{3}}^{(1)}\\\
c_{e^{2}\phi^{2}D^{3}}^{(2)}\end{pmatrix}=P_{e^{2}\phi^{2}D^{3}}\begin{pmatrix}\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(1)}\\\
\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(2)}\end{pmatrix}\,,\quad
P_{e^{2}\phi^{2}D^{3}}=\begin{pmatrix}1&0\\\ -1&1\end{pmatrix}\,.$ (9)

Then, we have that:

$\dot{c}_{e^{2}B^{2}D}=-\frac{1}{3}g_{1}^{2}\begin{pmatrix}1&1\end{pmatrix}P_{e^{2}\phi^{2}D^{3}}\begin{pmatrix}\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(1)}\\\
\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(2)}\end{pmatrix}=(\fcolorbox{black}{LightCyan}{$\,0\,$}\,\,-\frac{1}{3}g_{1}^{2})\begin{pmatrix}\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(1)}\\\
\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(2)}\end{pmatrix}\,.$ (10)

What is not obvious at all is how one can anticipate a matrix like
$P_{e^{2}\phi^{2}D^{3}}$ without computing explicitly the anomalous dimensions
involved.

This result is not inmediate from the on-shell amplitude approach either. To
see why, let us express the aforementioned operators in the amplitude basis:

$\displaystyle\widetilde{\mathcal{A}}_{e^{2}\phi^{2}D^{3}}^{(1)}(1_{\overline{e}},2_{e},3_{\phi^{\dagger}},4_{\phi})$
$\displaystyle=\langle 12\rangle\langle 13\rangle[12][23]\,,$ (11)
$\displaystyle\widetilde{\mathcal{A}}_{e^{2}\phi^{2}D^{3}}^{(2)}(1_{\overline{e}},2_{e},3_{\phi^{\dagger}},4_{\phi})$
$\displaystyle=\langle 13\rangle\langle 23\rangle[23]^{2}\,,$ (12)
$\displaystyle\mathcal{A}_{e^{2}B^{2}D}(1_{\overline{e}},2_{e},3_{B_{-}},4_{B_{+}})$
$\displaystyle=\langle 13\rangle^{2}[14][24]\,.$ (13)

We can compute the mixing of the first two amplitudes into the second
performing the usual unitarity double cuts [67, 68, 69, 7]; see Fig. 2. The SM
amplitude in the cut reads:

$\mathcal{A}_{\text{SM}}(1_{\phi},2_{\phi^{\dagger}},3_{B_{-}},4_{B_{+}})=-\frac{1}{2}g_{1}^{2}\frac{\langle
13\rangle\langle 23\rangle}{\langle 14\rangle\langle 24\rangle}\,.$ (14)

Figure 2: Double cut for the mixing of $e^{2}\phi^{2}D^{3}$ into
$e^{2}B^{2}D$. The cross indicates the EFT (minimal) amplitude, while the
bubble stands for the SM (non-necessarily minimal) amplitude.

Thus, we have:

$\gamma_{c_{e^{2}B^{2}D},\,\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(1)}}\mathcal{A}_{e^{2}B^{2}D}(1_{\overline{e}},2_{e},3_{B_{-}},4_{B_{+}})=-\frac{4}{\pi}\mathcal{I}_{1}\,,$
(15)

with

$\displaystyle\mathcal{I}_{1}$ $\displaystyle=\int
d\text{LIPS}\,\widetilde{\mathcal{A}}_{e^{2}\phi^{2}D^{3}}^{(1)}(1_{\overline{e}},2_{e},3_{\phi^{\dagger}},4_{\phi})\times\mathcal{A}_{\text{SM}}(-3^{\prime}_{\phi},-4^{\prime}_{\phi^{\dagger}},3_{B_{-}},4_{B_{+}})$
$\displaystyle=-\frac{1}{2}g_{1}^{2}\int d\text{LIPS}\,\langle
12\rangle\langle 13^{\prime}\rangle[12][23^{\prime}]\,\frac{\langle
3^{\prime}3\rangle\langle 4^{\prime}3\rangle}{\langle
3^{\prime}4\rangle\langle 4^{\prime}4\rangle}$
$\displaystyle=-\frac{1}{4}g_{1}^{2}\int_{0}^{\pi/2}\int_{0}^{2\pi}d\varphi\,d\theta\,s_{\theta}\,c_{\theta}\,e^{\mathrm{i}\varphi}\,\langle
12\rangle[12]\left(e^{\mathrm{i}\varphi}c_{\theta}[23]-s_{\theta}[24]\right)\left(e^{\mathrm{i}\varphi}c_{\theta}[23]-s_{\theta}[24]\right)$
$\displaystyle=0\,,$ (16)

which implies

$\gamma_{c_{e^{2}B^{2}D},\,\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(1)}}=0\,.$ (17)

Note that, contrary to those zeros highlighted by non-renormalisation theorems
[56, 57], in this case the SM amplitude does not vanish. The zero arises
because all terms in the product of tree-level on-shell amplitudes are
proportional to $e^{\mathrm{i}n\varphi}$ with $n=1,2,...$, for which
$\int_{0}^{2\pi}d\varphi e^{\mathrm{i}n\varphi}$ vanishes.

The negative value of
$\gamma_{c_{e^{2}B^{2}D},\,\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(2)}}$ seems also
accidental from this perspective. In fact:

$\displaystyle\gamma_{c_{e^{2}B^{2}D},\,\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(2)}}\mathcal{A}_{e^{2}B^{2}D}(1_{\overline{e}},2_{e},3_{B_{-}},4_{B_{+}})=-\frac{4}{\pi}\mathcal{I}_{2}\,$
(18)

with

$\displaystyle\mathcal{I}_{2}$ $\displaystyle=\int
d\text{LIPS}\,\widetilde{\mathcal{A}}_{e^{2}\phi^{2}D^{3}}^{(2)}(1_{\overline{e}},2_{e},3_{\phi^{\dagger}},4_{\phi})\times\mathcal{A}_{\text{SM}}(3^{\prime}_{\phi},4^{\prime}_{\phi^{\dagger}},3_{B_{-}},4_{B_{+}})$
$\displaystyle=-\frac{1}{2}g_{1}^{2}\int d\text{LIPS}\,\langle
13^{\prime}\rangle\langle 23^{\prime}\rangle[23^{\prime}]^{2}\,\frac{\langle
3^{\prime}3\rangle\langle 4^{\prime}3\rangle}{\langle
3^{\prime}4\rangle\langle 4^{\prime}4\rangle}$ $\displaystyle=\frac{1}{2}\pi
g_{1}^{2}\int_{0}^{\pi/2}\,d\theta\,s_{\theta}\,c_{\theta}^{4}s_{\theta}^{4}\langle
13\rangle\langle 23\rangle[24]^{2}=-\frac{1}{12}\pi g_{1}^{2}\langle
13\rangle\langle 23\rangle[24]^{2}\,,$ (19)

where in this case we have ommited the details about the integration over
$\varphi$.

The final amplitude does not look exactly like
$\mathcal{A}_{e^{2}B^{2}D}(1_{\overline{e}},2_{e},3_{B_{-}},4_{B_{+}})$,
however:

$\displaystyle\langle 13\rangle\langle 23\rangle[24]^{2}=\langle
13\rangle\langle 32\rangle[24][42]=-\langle 13\rangle\langle
31\rangle[14][42]=-\mathcal{A}_{e^{2}B^{2}D}(1_{\overline{e}},2_{e},3_{B_{-}},4_{B_{+}})\,,$
(20)

where in the first equality we have used the antisymmetry of angles and
squares, while in the second we have used momentum conservation:

$|2\rangle\langle 2|=-|1\rangle\langle 1|-|3\rangle\langle 3|-|4\rangle\langle
4|\,.$ (21)

From here, we obtain that:

$\displaystyle\gamma_{c_{e^{2}B^{2}D},\,\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(2)}}=-\frac{g_{1}^{2}}{3}\,.$
(22)

In the next section, we show that both the zero in Eq. (17) and the negative
value of $\gamma$ in Eq. (22) are apparent when looking at these loops from
the perspective of the positivity of two-to-two forward scattering amplitudes.

##  3 Constraints from positivity

Let us consider the tree-level amplitude $\mathcal{A}$ for the elastic process
$e\phi\to e\phi$ within some well-defined (local and unitary) quantum-field
theory (QFT). In the forward limit, this amplitude is an analytic function of
the Mandesltan invariant $s$ only, with at most single poles corresponding to
particle thresholds. This permits relating the IR and UV behaviour of
$\mathcal{A}$ through dispersion relations [15] which, together with robust
bounds on the growing of the amplitude at $s\to\infty$ [70] and the optical
theorem, implies that $\mathcal{A}^{\prime\prime}|_{s=0}\geq 0$. This latter
quantity can be computed within the EFT, implying:

$-\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(2)}\geq 0\,,$ (23)

while $\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(1)}$ remains completely unconstrained.
(In the amplitude basis this result is very intuitive because the forward
limit amounts to the relations $2\to 1$ and $3\to 4$, which implies that
$\langle 12\rangle\langle 13\rangle[12][23]$ vanishes.)

Within this same QFT, we could compute the one-loop amplitude for $eB\to eB$,
and apply an analogous reasoning [71, 24, 72, 58], obtaining that:

$c_{e^{2}B^{2}D}(\mu)\sim
c_{e^{2}B^{2}D}(\Lambda)+\dot{c}_{e^{2}B^{2}D}\log{\frac{\mu}{\Lambda}}\leq
0\Rightarrow\dot{c}_{e^{2}B^{2}D}\geq 0\,.$ (24)

In the last step we have taken into account that $e^{2}B^{2}D$ operators do
not arise at tree level in weakly-coupled UV completions of the SMEFT [57].
Now, dimension-six tree-level interactions that might be also present in the
IR do not renormalise $\mathcal{O}_{e^{2}B^{2}D}$, so we simply have:

$\dot{c}_{e^{2}B^{2}D}=\alpha_{1}\gamma_{c_{e^{2}B^{2}D},\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(1)}}+\alpha_{2}\gamma_{c_{e^{2}B^{2}D},\tilde{c}_{e^{2}B^{2}D^{3}}^{(2)}}\,,$
(25)

with $\alpha_{1}$ and $\alpha_{2}$ some real numbers.

In order for this expression to be positive in any physical UV completion of
the SMEFT, or equivalently for all values of
$\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(1)}$ and
$\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(1)}$ compatible with the bound in Eq. (23),
the only possibility is that $\alpha_{1}=0$, $\alpha_{2}=-\alpha$, with
$\alpha\geq 0$:

$\dot{c}_{e^{2}B^{2}D}=(\fcolorbox{black}{LightCyan}{$\,0\,$}-\alpha)\begin{pmatrix}\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(1)}\\\
\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(2)}\end{pmatrix}\,.$ (26)

This is essentially Eqs. (10), (17) and (22), though the explicit values of
the anomalous dimensions have not been obtained. (And, of course, there has
been no need for computing any loop!)

This same reasoning can not be applied to constraining the mixing of, for
example, $e^{2}\phi^{2}D^{3}$ into $\phi^{4}D^{4}$. The reason is that, in
this case, one expects also the presence of dimension-six $e^{2}\phi^{2}D$
terms which, in pairs, renormalise also $\phi^{4}D^{4}$; see Fig. 3. Hence,
the positivity of $\phi^{4}D^{4}$ can not be directly attributed to the
positivity of $\gamma$ but rather to $\gamma^{\prime}$ [24, 58]. In general,
this problem arises whenever all fields in the renormalised operator are
already present in the renormalising interaction.

Figure 3: Example diagrams for the mixing of $e^{2}\phi^{2}D^{n}$ operators
into $\phi^{4}D^{4}$. Vertices like
$\,{\color[rgb]{0.8,0.8,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0.8,0.8}i}\,$
represent dimension-six terms ($n=1$), while crosses $\boldmath{\times}$ are
dimension-eight ones ($n=3$).

One further comment is in order. The crucial step from Eq. (25) to Eq. (26)
relies on the conviction that $\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(1)}$ is not
constrained at all by positivity (it can have either sign). This can be only
guaranteed if the set of independent positivity restrictions is fully
exhausted. For this matter, studying the elastic positivity bounds with non-
superposed asymptotic states does not suffice. We must instead supplement the
analysis with the scattering of states with certain degree of superposition,
that we quantify as $x$; see Appendix A. 111Nonetheless, we have certain
evidence, based on some explicit computations as well as on generalised
unitarity methods, rather than on an analysis of the positivity of amplitudes
in all possible UV completions of the SMEFT, that restricting to bounds valid
only when $x\to 0$ is still a valid approach. This would allow us to establish
more stringent constraints on the ADM of the SMEFT than those that we present
in Appendix B. We have no indisputable proof of this observation, though. In
the case of $e\phi\to e\phi$, the bounds obtained either way are the same. In
other classes of operators, however, the differences are apparent.

On the other hand, for constraining the running functions themselves, the
bounds to be considered are only those that survive in the limit $x\rightarrow
0$. (Again, in the case at work, namely $eB\to eB$, the bounds are
equivalent.) To explain this issue, let us imagine that we rather focus on the
mixing $e^{2}\phi^{2}D^{3}\to e^{2}u^{2}D^{2}$. In this case, if we take into
account all independent bounds on $e^{2}u^{2}D^{3}$, we can conclude that
$\dot{c}_{e^{2}u^{2}D^{2}}^{(1)}-3\dot{c}_{e^{2}u^{2}D^{2}}^{(2)}\leq 0$ and
$-\dot{c}_{e^{2}u^{2}D^{2}}^{(1)}-\dot{c}_{e^{2}u^{2}D^{2}}^{(2)}\leq 0$,
meaning that $c_{e^{2}u^{2}D^{2}}^{(1)}$ can never be the only renormalised
coupling. However, an explicit computation yields
$\dot{c}_{e^{2}u^{2}D^{2}}^{(1)}=\frac{2}{9}g_{1}^{2}(c_{e^{2}\phi^{2}D^{2}}^{(2)}-c_{e^{2}\phi^{2}D^{2}}^{(1)})$
and $\dot{c}_{e^{2}u^{2}D^{2}}^{(2)}=0$ in the limit of vanishing Yukawas. The
reason is that, in the derivation of the most general positivity bounds for
$e^{2}u^{2}D^{2}$ operators (see Eq. (66)), we find processes of the sort
$ee\to ee$, which at one loop receive contributions from pairs of dimension-
six terms $e^{2}\phi^{2}D$. The way out for ignoring the contamination from
dimension-six terms is considering only those bounds on anomalous dimensions
that arise in the limit $x=0$.

##  4 Structure of the anomalous dimension matrix

Figure 4: Schematic structure of the SMEFT ADM. The different blocks are given
in Tabs. 1, 2, 3 and 4

In light of the previous discussion, we aim to constrain anomalous dimensions
$\gamma_{ij}$ of four-field interactions, where $i$ represents a Wilson
coefficient that is bounded in the limit $x=0$, and $j$ is any other
coefficient of a tree-level operator.

To this aim, we choose a basis in which all $x$-independent positivity bounds
are decoupled. This amounts to picking one (in general there might be many)
rotation matrix $R$ such that $PR=\mathbb{I}_{P}$, where $P$ is the matrix of
$x$-independent positivity bounds, namely that satisfying $P\vec{c}\geq 0$;
and $\mathbb{I}_{P}$ is the matrix consisting in as many rows of the identity
matrix as number of positivity bounds.

For example, we can have:

$\displaystyle P_{u^{2}\phi^{2}D^{3}}$
$\displaystyle=\begin{pmatrix}-1&-1\end{pmatrix}\,,\quad
R_{u^{2}\phi^{2}D^{3}}=\begin{pmatrix}2&3\\\ -3&-3\end{pmatrix}\,,$ (27)
$\displaystyle P_{l^{2}u^{2}D^{2}}$ $\displaystyle=\begin{pmatrix}1&-3\\\
-1&-1\end{pmatrix}\,,\quad R_{l^{2}u^{2}D^{2}}=\begin{pmatrix}2&3\\\
-1&0\end{pmatrix}\,.$ (28)

In this new basis, in which we denote the Wilson coefficients with a tilde,
$\tilde{c}$, the $x$-independent bound for $u^{2}\phi^{2}D^{2}$ reads simply
$\tilde{c}_{u^{2}\phi^{2}D^{3}}^{(1)}\geq 0$. The bound for $l^{2}u^{2}D^{2}$
valid in the limit $x\to 0$ is $\tilde{c}_{l^{2}u^{2}D^{2}}^{(1)}\geq 0$. From
here, it follows that

$\displaystyle 0\geq\dot{c}_{l^{2}u^{2}D^{2}}^{(1)}$
$\displaystyle=-\alpha\tilde{c}_{u^{2}\phi^{2}D^{3}}^{(1)}+\cdots$ (29)

with $\alpha\geq 0$ and where the ellipses represent operators of other
classes which are completely independent. Hence,
$\gamma_{\tilde{c}_{l^{2}u^{2}D^{2}}\,,\,\tilde{c}_{u^{2}\phi^{2}D^{3}}^{(1)}}\leq
0$ while
$\gamma_{\tilde{c}_{l^{2}u^{2}D^{2}}\,,\,\tilde{c}_{u^{2}\phi^{2}D^{3}}^{(2)}}=0$.

The process of obtaining a full colection of rotation matrices can be easily
automatised. We provide one in the ancillary file matrices.txt. Some of them
are completely determined by the requirement $PR=\mathbb{I}_{P}$; for the rest
we have simply fixed one of the many possibilities that fulfill
$\text{Det}(R)=1$. (In particular, the matrix $P_{e^{2}\phi^{2}D^{3}}$ is
different from that shown in Eq. (9).)

The reader will notice that in certain classes of operators, for instance in
$G^{2}\phi^{2}D^{2}$, the matrices are smaller then the number of operators.
This is simply because we ignore all those Wilson coefficients beyond the last
one intervening in the positivity bounds in Appendix A, such as for example
$c_{G^{2}\phi^{2}D^{2}}^{(3)}$.

In the basis defined by these rotations, the different blocks of the SMEFT ADM
(see Fig. 4) are described by Tabs. 1, 3, 2 and 4 in Appendix B. It should be
noticed that operators not constrained by positivity are not shown in the left
rows, while operators that only appear at one-loop in UV completions of the
SMEFT are absent from the top columns. Besides, we provide cross-checks of
some of these entries in Appendix C.

##  5 Discussion and outlook

Even if two different classes of dimension-8 SMEFT operators mix under
renormalisation group running at one-loop, there are always certain
interactions in one class that do not mix into certain other interactions in
the second class, or that they do it in such a way that the running has always
definite sign, irrespective of the SM couplings. In the appropriate basis of
operators, these effects manifest as zeros or definite-sign entries in the
ADM.

We have shown that such a basis, or equivalently the corresponding rotation
matrix $\mathcal{R}$ that must be applied to the commonly used basis of Ref.
[54], can not be easily found without computing explicitly the one-loop RGEs,
either using Feynman diagrams or on-shell amplitude methods. However,
following the results of Ref. [58], we have derived $\mathcal{R}$ basing
solely on positivity bounds obtained from tree-level computations in the
forward limit. These bounds, which extend those previously computed in the
literature [19, 18, 21, 73, 74, 23, 75, 76], particularly in the
$G^{2}\phi^{2}D^{2}$ and $X^{2}\psi^{2}D$ sectors, are of major importance for
constraining the SMEFT parameter space. Hence, our work adds also to this
aspect of the SMEFT phenomenology.

In the new basis defined by $\cal{R}$, we highlight 84 non-trivial zeros and
much more negative entries in the ADM. We emphasise that, following our same
methods, different bases involving different number of zeros and negative
entries can be obtained; in this paper we have simply singled out one for the
sake of example. It would be actually interesting to know which is such basis
in which the number of zeros is maximised. The main point is that we can do so
without performing any involved one-loop calculation. It would be also
important to cross-check these results by explicit computations, some of which
we provide in Appendix C, and eventually complete the renormalisation of the
dimension-8 SMEFT.

## Acknowledgments

We thank Jose Santiago for useful discussions. MC is supported by the Ramón y
Cajal grant RYC2019-027155-I funded by the MCIN /AEI/10.13039/501100011033 and
by “El FSE invierte en tu futuro”, and acknowledges support from the grant
CNS2022-136024 funded by the European Union NextGenerationEU/PRTR, from grants
PID2021-128396NB-I00 and PID2022-139466NB-C22 funded by “FEDER Una manera de
hacer Europa” as well as from the Junta de Andalucía grants FQM 101 and
P21-00199. XL is supported in part by the National Natural Science Foundation
of China under grant No. 11835013.

##  Appendix A Positivity bounds

The most stringent positivity bounds can be derived from the elastic
scattering of two superposition states $|u\rangle=\sum_{i}u_{i}|i\rangle$ and
$|v\rangle=\sum_{j}v_{j}|j\rangle$:

$\mathcal{A}(|u\rangle|v\rangle\to|u\rangle|v\rangle)=\sum_{ijkl}u_{i}v_{j}u_{k}^{*}u_{l}^{*}\mathcal{A}(|i\rangle|j\rangle\to|i\rangle|j\rangle)\,.$
(30)

Assuming $u_{i}$ and $v_{j}$ to be real, and adopting the self-conjugate
particle basis, this amplitude in the forward limit fulfills [77, 78, 79]

$\displaystyle\mathcal{A}^{uvuv}\equiv\frac{d^{2}}{ds^{2}}\mathcal{A}(|u\rangle|v\rangle\to|u\rangle|v\rangle)=\sum_{ijkl}u_{i}v_{j}u_{k}v_{l}\mathcal{A}^{ijkl}\geq
0\,.$ (31)

In practice, we parameterise the superposition via a quantity $x$. For
example, for constraining operators of the type $B^{2}\phi^{2}D^{2}$ we
consider $|u\rangle=|\phi\rangle+x|B\rangle$ as well as
$|v\rangle=|\phi\rangle+|B\rangle$.

Below, for each class of operators, we quote the set of independent positivity
bounds valid in general (namely for any superposition), as well as those valid
only for $x=0$, which we show in gray. (When both coincide, we show only one.)

###  A.1 $\phi^{4}D^{4}$

$\displaystyle c_{\phi^{4}D^{4}}^{\left(2\right)}\geq 0\,,\quad
c_{\phi^{4}D^{4}}^{\left(2\right)}+c_{\phi^{4}D^{4}}^{\left(2\right)}\geq
0\,,\quad
c_{\phi^{4}D^{4}}^{\left(1\right)}+c_{\phi^{4}D^{4}}^{\left(2\right)}+c_{\phi^{4}D^{4}}^{\left(3\right)}\geq
0\,.$ (32)

###  A.2 $G^{2}\phi^{2}D^{2}$

$\displaystyle-
c_{G^{2}\phi^{2}D^{2}}^{\left(1\right)}-2c_{G^{2}\phi^{2}D^{2}}^{\left(2\right)}\geq
0\,,\ \ c_{G^{2}\phi^{2}D^{2}}^{\left(2\right)}\geq 0\,,$ (33)
$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}-c_{G^{2}\phi^{2}D^{2}}^{\left(1\right)}\geq
0}\,.$ (34)

###  A.3 $W^{2}\phi^{2}D^{2}$

$\displaystyle-
c_{W^{2}\phi^{2}D^{2}}^{\left(1\right)}-2c_{W^{2}\phi^{2}D^{2}}^{\left(2\right)}\geq
0\,,\quad c_{W^{2}\phi^{2}D^{2}}^{\left(2\right)}\geq 0\,,$ $\displaystyle-
c_{W^{2}\phi^{2}D^{2}}^{\left(1\right)}+c_{W^{2}\phi^{2}D^{2}}^{\left(4\right)}\geq
0\,,\quad-
c_{W^{2}\phi^{2}D^{2}}^{\left(1\right)}-c_{W^{2}\phi^{2}D^{2}}^{\left(4\right)}\geq
0\,,$ (35)
$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}-c_{W^{2}\phi^{2}D^{2}}^{\left(1\right)}\geq
0\;.}$ (36)

###  A.4 $B^{2}\phi^{2}D^{2}$

$\displaystyle-
c_{B^{2}\phi^{2}D^{2}}^{\left(1\right)}-2c_{B^{2}\phi^{2}D^{2}}^{\left(2\right)}\geq
0,\ \ c_{B^{2}\phi^{2}D^{2}}^{\left(2\right)}\geq 0\,,$ (37)
$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}-c_{B^{2}\phi^{2}D^{2}}^{\left(1\right)}\geq
0\;.}$ (38)

###  A.5 $l^{2}\phi^{2}D^{3}$

$\displaystyle-
c_{l^{2}\phi^{2}D^{3}}^{\left(1\right)}-c_{l^{2}\phi^{2}D^{3}}^{\left(2\right)}-c_{l^{2}\phi^{2}D^{3}}^{\left(3\right)}-c_{l^{2}\phi^{2}D^{3}}^{\left(4\right)}\geq
0\,,$ $\displaystyle-
c_{l^{2}\phi^{2}D^{3}}^{\left(1\right)}-c_{l^{2}\phi^{2}D^{3}}^{\left(2\right)}+c_{l^{2}\phi^{2}D^{3}}^{\left(3\right)}+c_{l^{2}\phi^{2}D^{3}}^{\left(4\right)}\geq
0\;.$ (39)

###  A.6 $e^{2}\phi^{2}D^{3}$

$\displaystyle-
c_{{e^{2}\phi}^{2}D^{3}}^{\left(1\right)}-c_{{e^{2}\phi}^{2}D^{3}}^{\left(2\right)}\geq
0\,.$ (40)

###  A.7 $q^{2}\phi^{2}D^{3}$

$\displaystyle-
c_{q^{2}\phi^{2}D^{3}}^{\left(1\right)}-c_{q^{2}\phi^{2}D^{3}}^{\left(2\right)}+c_{q^{2}\phi^{2}D^{3}}^{\left(3\right)}+c_{q^{2}\phi^{2}D^{3}}^{\left(4\right)}\geq
0,$ $\displaystyle-
c_{q^{2}\phi^{2}D^{3}}^{\left(1\right)}-c_{q^{2}\phi^{2}D^{3}}^{\left(2\right)}-c_{q^{2}\phi^{2}D^{3}}^{\left(3\right)}-c_{q^{2}\phi^{2}D^{3}}^{\left(4\right)}\geq
0\;.$ (41)

###  A.8 $u^{2}\phi^{2}D^{3}$

$\displaystyle-
c_{u^{2}\phi^{2}D^{3}}^{\left(1\right)}-c_{u^{2}\phi^{2}D^{3}}^{\left(2\right)}\geq
0\,.$ (42)

###  A.9 $d^{2}\phi^{2}D^{3}$

$\displaystyle-
c_{d^{2}\phi^{2}D^{3}}^{\left(1\right)}-c_{d^{2}\phi^{2}D^{3}}^{\left(2\right)}\geq
0\,.$ (43)

###  A.10 $q^{2}G^{2}D$

$\displaystyle-3c_{q^{2}G^{2}D}^{\left(1\right)}-c_{q^{2}G^{2}D}^{(3)}\geq
0\,,\quad-3c_{q^{2}G^{2}D}^{\left(1\right)}+c_{q^{2}G^{2}D}^{(3)}\geq 0\;.$
(44)

###  A.11 $q^{2}W^{2}D$

$\displaystyle-c_{q^{2}W^{2}D}^{\left(1\right)}\ \geq 0\;.$ (45)

###  A.12 $q^{2}B^{2}D$

$\displaystyle-c_{q^{2}B^{2}D}^{\left(1\right)}\ \geq 0\,.$ (46)

###  A.13 $u^{2}G^{2}D$

$\displaystyle-3c_{u^{2}G^{2}D}^{(1)}-c_{u^{2}G^{2}D}^{(3)}\geq
0,\quad-3c_{u^{2}G^{2}D}^{\left(1\right)}+c_{u^{2}G^{2}D}^{(3)}\geq 0\;.$ (47)

###  A.14 $u^{2}W^{2}D$

$\displaystyle-c_{u^{2}W^{2}D}\geq 0\,.$ (48)

###  A.15 $u^{2}B^{2}D$

$\displaystyle-c_{u^{2}B^{2}D}\geq 0\,.$ (49)

###  A.16 $d^{2}G^{2}D$

$\displaystyle-3c_{d^{2}G^{2}D}^{(1)}-c_{d^{2}G^{2}D}^{(3)}\geq
0\,,\quad-3c_{d^{2}G^{2}D}^{\left(1\right)}+c_{d^{2}G^{2}D}^{(3)}\geq 0\,.$
(50)

###  A.17 $d^{2}W^{2}D$

$\displaystyle-c_{d^{2}W^{2}D}\geq 0\,.$ (51)

###  A.18 $d^{2}B^{2}D$

$\displaystyle-c_{d^{2}B^{2}D}\geq 0\,.$ (52)

###  A.19 $l^{2}G^{2}D$

$\displaystyle-c_{l^{2}G^{2}D}\geq 0\,.$ (53)

###  A.20 $l^{2}W^{2}D$

$\displaystyle-c_{l^{2}W^{2}D}^{\left(1\right)}\geq 0\,.$ (54)

###  A.21 $l^{2}B^{2}D$

$\displaystyle-c_{l^{2}B^{2}D}\geq 0\,.$ (55)

###  A.22 $e^{2}G^{2}D$

$\displaystyle-c_{e^{2}G^{2}D}\geq 0\,.$ (56)

###  A.23 $e^{2}W^{2}D$

$\displaystyle-c_{e^{2}W^{2}D}\geq 0\,.$ (57)

###  A.24 $e^{2}B^{2}D$

$\displaystyle-c_{e^{2}B^{2}D}\geq 0\,.$ (58)

###  A.25 $l^{4}D^{2}$

$\displaystyle-c_{l^{4}D^{2}}^{\left(2\right)}\geq 0,\ \
-c_{l^{4}D^{2}}^{\left(1\right)}-3c_{l^{4}D^{2}}^{\left(2\right)}\geq 0\,.$
(59)

###  A.26 $q^{4}D^{2}$

$\displaystyle-
c_{q^{4}D^{2}}^{\left(1\right)}-3c_{q^{4}D^{2}}^{\left(2\right)}-c_{q^{4}D^{2}}^{\left(3\right)}-3c_{q^{4}D^{2}}^{\left(4\right)}$
$\displaystyle\geq$ $\displaystyle 0\,,$ $\displaystyle-
c_{q^{4}D^{2}}^{\left(2\right)}-c_{q^{4}D^{2}}^{\left(3\right)}$
$\displaystyle\geq$ $\displaystyle 0\,,$ $\displaystyle-
c_{q^{4}D^{2}}^{\left(2\right)}-c_{q^{4}D^{2}}^{\left(4\right)}$
$\displaystyle\geq$ $\displaystyle 0,$ $\displaystyle-
c_{q^{4}D^{2}}^{\left(2\right)}+c_{q^{4}D^{2}}^{\left(4\right)}$
$\displaystyle\geq$ $\displaystyle 0\,.$ (60)

###  A.27 $l^{2}q^{2}D^{2}$

$\displaystyle
c_{l^{2}q^{2}D^{2}}^{\left(1\right)}-c_{l^{2}q^{2}D^{2}}^{\left(2\right)}+c_{l^{2}q^{2}D^{2}}^{\left(3\right)}-c_{l^{2}q^{2}D^{2}}^{\left(4\right)}$
$\displaystyle\geq$ $\displaystyle 0\,,$ $\displaystyle
c_{l^{2}q^{2}D^{2}}^{\left(1\right)}-c_{l^{2}q^{2}D^{2}}^{\left(2\right)}-c_{l^{2}q^{2}D^{2}}^{\left(3\right)}+c_{l^{2}q^{2}D^{2}}^{\left(4\right)}$
$\displaystyle\geq$ $\displaystyle 0\,,$ $\displaystyle-
c_{l^{2}q^{2}D^{2}}^{\left(1\right)}-3c_{l^{2}q^{2}D^{2}}^{\left(2\right)}-c_{l^{2}q^{2}D^{2}}^{\left(3\right)}-3c_{l^{2}q^{2}D^{2}}^{\left(4\right)}$
$\displaystyle\geq$ $\displaystyle 0\,,$ $\displaystyle-
c_{l^{2}q^{2}D^{2}}^{\left(1\right)}-3c_{l^{2}q^{2}D^{2}}^{\left(2\right)}+c_{l^{2}q^{2}D^{2}}^{\left(3\right)}+3c_{l^{2}q^{2}D^{2}}^{\left(4\right)}$
$\displaystyle\geq$ $\displaystyle 0\,,$ $\displaystyle-
c_{l^{2}q^{2}D^{2}}^{\left(2\right)}+c_{l^{2}q^{2}D^{2}}^{\left(3\right)}+2c_{l^{2}q^{2}D^{2}}^{\left(4\right)}$
$\displaystyle\geq$ $\displaystyle 0\,,$ $\displaystyle-
c_{l^{2}q^{2}D^{2}}^{\left(2\right)}-c_{l^{2}q^{2}D^{2}}^{\left(3\right)}$
$\displaystyle\geq$ $\displaystyle 0\,,$ (61)
$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}-c_{l^{2}q^{2}D^{2}}^{\left(2\right)}\
-c_{l^{2}q^{2}D^{2}}^{\left(4\right)}\geq 0\,,\quad\
-c_{l^{2}q^{2}D^{2}}^{\left(2\right)}\ +c_{l^{2}q^{2}D^{2}}^{\left(4\right)}}$
$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}0\,.}$
(62)

###  A.28 $e^{4}D^{2}$

$\displaystyle-c_{e^{4}D^{2}}\geq 0\,.$ (63)

###  A.29 $u^{4}D^{2}$

$\displaystyle{-c_{u^{4}D^{2}}^{\left(1\right)}-3c_{u^{4}D^{2}}^{\left(2\right)}\geq
0,\quad-c_{u^{4}D^{2}}^{\left(2\right)}\geq 0\,.}$ (64)

###  A.30 $d^{4}D^{2}$

$\displaystyle{-c_{d^{4}D^{2}}^{\left(1\right)}-3c_{d^{4}D^{2}}^{\left(2\right)}\geq
0\,,\quad-c_{d^{4}D^{2}}^{\left(2\right)}\geq 0\,.}$ (65)

###  A.31 $e^{2}u^{2}D^{2}$

$\displaystyle{-c_{e^{2}u^{2}D^{2}}^{\left(1\right)}-3c_{e^{2}u^{2}D^{2}}^{\left(2\right)}\geq
0\,,\quad
c_{e^{2}u^{2}D^{2}}^{\left(1\right)}-c_{e^{2}u^{2}D^{2}}^{\left(2\right)}\geq
0\,,}$ (66)
$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}-c_{e^{2}u^{2}D^{2}}^{\left(2\right)}\geq
0\;.}$ (67)

###  A.32 $e^{2}d^{2}D^{2}$

$\displaystyle{-c_{e^{2}d^{2}D^{2}}^{\left(1\right)}-3c_{e^{2}d^{2}D^{2}}^{\left(2\right)}\geq
0\,,\quad
c_{e^{2}d^{2}D^{2}}^{\left(1\right)}-c_{e^{2}d^{2}D^{2}}^{\left(2\right)}\geq
0\,,}$ (68)
$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}-c_{e^{2}d^{2}D^{2}}^{\left(2\right)}\geq
0\;.}$ (69)

###  A.33 $u^{2}d^{2}D^{2}$

$\displaystyle-3c_{u^{2}d^{2}D^{2}}^{\left(1\right)}-9c_{u^{2}d^{2}D^{2}}^{\left(2\right)}-c_{u^{2}d^{2}D^{2}}^{\left(3\right)}-3c_{u^{2}d^{2}D^{2}}^{\left(4\right)}$
$\displaystyle\geq$ $\displaystyle 0\,,$ $\displaystyle
3c_{u^{2}d^{2}D^{2}}^{\left(1\right)}-3c_{u^{2}d^{2}D^{2}}^{\left(2\right)}+c_{u^{2}d^{2}D^{2}}^{\left(3\right)}-c_{u^{2}d^{2}D^{2}}^{\left(4\right)}$
$\displaystyle\geq$ $\displaystyle 0\,,$
$\displaystyle-12c_{u^{2}d^{2}D^{2}}^{\left(2\right)}+3c_{u^{2}d^{2}D^{2}}^{\left(3\right)}+5c_{u^{2}d^{2}D^{2}}^{\left(4\right)}$
$\displaystyle\geq$ $\displaystyle 0\,,$
$\displaystyle-12c_{u^{2}d^{2}D^{2}}^{\left(2\right)}-3c_{u^{2}d^{2}D^{2}}^{\left(3\right)}-c_{u^{2}d^{2}D^{2}}^{\left(4\right)}$
$\displaystyle\geq$ $\displaystyle 0\;.$ (70)
$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}-3c_{u^{2}d^{2}D^{2}}^{\left(2\right)}\
-c_{u^{2}d^{2}D^{2}}^{\left(4\right)}\geq
0\,,\quad-6c_{u^{2}d^{2}D^{2}}^{\left(2\right)}+c_{u^{2}d^{2}D^{2}}^{\left(4\right)}}$
$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}0\;.}$
(71)

###  A.34 $l^{2}e^{2}D^{2}$

$\displaystyle-
c_{l^{2}e^{2}D^{2}}^{\left(1\right)}-c_{l^{2}e^{2}D^{2}}^{\left(2\right)}\geq
0\,,\ \ \
c_{l^{2}e^{2}D^{2}}^{\left(1\right)}-3c_{l^{2}e^{2}D^{2}}^{\left(2\right)}\geq
0\,,$ (72)
$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}-c_{l^{2}e^{2}D^{2}}^{\left(2\right)}\geq
0\;.}$ (73)

###  A.35 $l^{2}u^{2}D^{2}$

$\displaystyle
c_{l^{2}u^{2}D^{2}}^{\left(1\right)}-3c_{l^{2}u^{2}D^{2}}^{\left(2\right)}\geq
0\,,\ \
-c_{l^{2}u^{2}D^{2}}^{\left(1\right)}-c_{l^{2}u^{2}D^{2}}^{\left(2\right)}\geq
0\,,$ (74)
$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}-c_{l^{2}u^{2}D^{2}}^{\left(2\right)}\geq
0\,,}$ (75)

###  A.36 $l^{2}d^{2}D^{2}$

$\displaystyle
c_{l^{2}d^{2}D^{2}}^{\left(1\right)}-3c_{l^{2}d^{2}D^{2}}^{\left(2\right)}\geq
0\,,\ \
-c_{l^{2}d^{2}D^{2}}^{\left(1\right)}-c_{l^{2}d^{2}D^{2}}^{\left(2\right)}\geq
0\,,$ (76)
$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}-c_{l^{2}d^{2}D^{2}}^{\left(2\right)}\geq
0\;.}$ (77)

###  A.37 $q^{2}e^{2}D^{2}$

$\displaystyle
c_{q^{2}e^{2}D^{2}}^{\left(1\right)}-3c_{q^{2}e^{2}D^{2}}^{\left(2\right)}\geq
0\,,\ \
-c_{q^{2}e^{2}D^{2}}^{\left(1\right)}-c_{q^{2}e^{2}D^{2}}^{\left(2\right)}\geq
0\,,$ (78)
$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}-c_{q^{2}e^{2}D^{2}}^{\left(2\right)}\geq
0\;.}$ (79)

###  A.38 $q^{2}u^{2}D^{2}$

$\displaystyle
3c_{q^{2}u^{2}D^{2}}^{\left(1\right)}-9c_{q^{2}u^{2}D^{2}}^{\left(2\right)}+c_{q^{2}u^{2}D^{2}}^{\left(3\right)}-3c_{q^{2}u^{2}D^{2}}^{\left(4\right)}$
$\displaystyle\geq$ $\displaystyle 0\,,$
$\displaystyle-3c_{q^{2}u^{2}D^{2}}^{\left(1\right)}-3c_{q^{2}u^{2}D^{2}}^{\left(2\right)}-c_{q^{2}u^{2}D^{2}}^{\left(3\right)}-c_{q^{2}u^{2}D^{2}}^{\left(4\right)}$
$\displaystyle\geq$ $\displaystyle 0\,,$
$\displaystyle-6c_{q^{2}u^{2}D^{2}}^{\left(2\right)}+c_{q^{2}u^{2}D^{2}}^{\left(4\right)}$
$\displaystyle\geq$ $\displaystyle 0\,,$ (80)
$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}-3c_{q^{2}u^{2}D^{2}}^{\left(2\right)}\
-c_{q^{2}u^{2}D^{2}}^{\left(4\right)}\geq
0\,,\quad-6c_{q^{2}u^{2}D^{2}}^{\left(2\right)}+c_{q^{2}u^{2}D^{2}}^{\left(4\right)}}$
$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}0\;.}$
(81)

###  A.39 $q^{2}d^{2}D^{2}$

$\displaystyle
3c_{q^{2}d^{2}D^{2}}^{\left(1\right)}-9c_{q^{2}d^{2}D^{2}}^{\left(2\right)}+c_{q^{2}d^{2}D^{2}}^{\left(3\right)}-3c_{q^{2}d^{2}D^{2}}^{\left(4\right)}$
$\displaystyle\geq$ $\displaystyle 0\,,$
$\displaystyle-3c_{q^{2}d^{2}D^{2}}^{\left(1\right)}-3c_{q^{2}d^{2}D^{2}}^{\left(2\right)}-c_{q^{2}d^{2}D^{2}}^{\left(3\right)}-c_{q^{2}d^{2}D^{2}}^{\left(4\right)}$
$\displaystyle\geq$ $\displaystyle 0\,,$
$\displaystyle-6c_{q^{2}d^{2}D^{2}}^{\left(2\right)}+c_{q^{2}d^{2}D^{2}}^{\left(4\right)}$
$\displaystyle\geq$ $\displaystyle 0\,,$ (82)
$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}-3c_{q^{2}d^{2}D^{2}}^{\left(2\right)}-c_{q^{2}d^{2}D^{2}}^{\left(4\right)}\geq
0\,,\quad-6c_{q^{2}d^{2}D^{2}}^{\left(2\right)}+c_{q^{2}d^{2}D^{2}}^{\left(4\right)}}$
$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>$\displaystyle{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}0\;.}$
(83)

##  Appendix B Blocks of the anomalous dimension matrix

In the following tables, we represent with a blue minus sign ($-$) ADM entries
that are neccessarily negative in the basis provided in the ancillary file
matrices.txt. Trivial zeros, arising for example in cases in which the
renormalising and the renormalised operators have no fields in common, are
specified with a simple zero ($0$). On the contrary, non-trivial zeros are
represented with a blue-boxed zero ($\,0\,$).

ADM elements that we can not constrain from our positivity arguments are shown
with a gray cross ($\times$).

| $\tilde{c}_{\phi^{4}D^{4}}^{(1)}$ | $\tilde{c}_{\phi^{4}D^{4}}^{(2)}$ | $\tilde{c}_{\phi^{4}D^{4}}^{(3)}$ | $\tilde{c}_{l^{2}\phi^{2}D^{3}}^{(1)}$ | $\tilde{c}_{l^{2}\phi^{2}D^{3}}^{(2)}$ | $\tilde{c}_{l^{2}\phi^{2}D^{3}}^{(3)}$ | $\tilde{c}_{l^{2}\phi^{2}D^{3}}^{(4)}$ | $\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(1)}$ | $\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(2)}$ | $\tilde{c}_{q^{2}\phi^{2}D^{3}}^{(1)}$ | $\tilde{c}_{q^{2}\phi^{2}D^{3}}^{(2)}$ | $\tilde{c}_{q^{2}\phi^{2}D^{3}}^{(3)}$ | $\tilde{c}_{q^{2}\phi^{2}D^{3}}^{(4)}$ | $\tilde{c}_{u^{2}\phi^{2}D^{3}}^{(1)}$ | $\tilde{c}_{u^{2}\phi^{2}D^{3}}^{(2)}$ | $\tilde{c}_{d^{2}\phi^{2}D^{3}}^{(1)}$ | $\tilde{c}_{d^{2}\phi^{2}D^{3}}^{(2)}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\tilde{c}_{\phi^{4}D^{4}}^{(1)}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{\phi^{4}D^{4}}^{(2)}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{\phi^{4}D^{4}}^{(3)}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{G^{2}\phi^{2}D^{2}}^{(1)}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | $\bm{-}$ | $\,0\,$ | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | $\bm{-}$ | $\,0\,$ | $\bm{-}$ | $\,0\,$
$\tilde{c}_{W^{2}\phi^{2}D^{2}}^{(1)}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | $\bm{-}$ | $\,0\,$ | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | $\bm{-}$ | $\,0\,$ | $\bm{-}$ | $\,0\,$
$\tilde{c}_{B^{2}\phi^{2}D^{2}}^{(1)}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | $\bm{-}$ | $\,0\,$ | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | $\bm{-}$ | $\,0\,$ | $\bm{-}$ | $\,0\,$
$\tilde{c}_{l^{2}\phi^{2}D^{3}}^{(1)}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\bm{-}$ | $\,0\,$ | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | $\bm{-}$ | $\,0\,$ | $\bm{-}$ | $\,0\,$
$\tilde{c}_{l^{2}\phi^{2}D^{3}}^{(2)}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\bm{-}$ | $\,0\,$ | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | $\bm{-}$ | $\,0\,$ | $\bm{-}$ | $\,0\,$
$\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(1)}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | $\times$ | $\times$ | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | $\bm{-}$ | $\,0\,$ | $\bm{-}$ | $\,0\,$
$\tilde{c}_{q^{2}\phi^{2}D^{3}}^{(1)}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | $\bm{-}$ | $\,0\,$ | $\times$ | $\times$ | $\times$ | $\times$ | $\bm{-}$ | $\,0\,$ | $\bm{-}$ | $\,0\,$
$\tilde{c}_{q^{2}\phi^{2}D^{3}}^{(2)}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | $\bm{-}$ | $\,0\,$ | $\times$ | $\times$ | $\times$ | $\times$ | $\bm{-}$ | $\,0\,$ | $\bm{-}$ | $\,0\,$
$\tilde{c}_{u^{2}\phi^{2}D^{3}}^{(1)}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | $\bm{-}$ | $\,0\,$ | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | $\times$ | $\times$ | $\bm{-}$ | $\,0\,$
$\tilde{c}_{d^{2}\phi^{2}D^{3}}^{(1)}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | $\bm{-}$ | $\,0\,$ | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | $\bm{-}$ | $\,0\,$ | $\times$ | $\times$
$\tilde{c}_{q^{2}G^{2}D}^{(1)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | 0 | 0 | 0 | 0
$\tilde{c}_{q^{2}G^{2}D}^{(2)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | 0 | 0 | 0 | 0
$\tilde{c}_{q^{2}W^{2}D}^{(1)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | 0 | 0 | 0 | 0
$\tilde{c}_{q^{2}B^{2}D}^{(1)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | 0 | 0 | 0 | 0
$\tilde{c}_{u^{2}G^{2}D}^{(1)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\,0\,$ | 0 | 0
$\tilde{c}_{u^{2}G^{2}D}^{(2)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\,0\,$ | 0 | 0
$\tilde{c}_{u^{2}W^{2}D}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\,0\,$ | 0 | 0
$\tilde{c}_{u^{2}B^{2}D}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\,0\,$ | 0 | 0
$\tilde{c}_{d^{2}G^{2}D}^{(1)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\,0\,$
$\tilde{c}_{d^{2}G^{2}D}^{(2)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\,0\,$
$\tilde{c}_{d^{2}W^{2}D}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\,0\,$
$\tilde{c}_{d^{2}B^{2}D}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\,0\,$
$\tilde{c}_{l^{2}G^{2}D}$ | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\tilde{c}_{l^{2}W^{2}D}$ | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\tilde{c}_{l^{2}B^{2}D}$ | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\tilde{c}_{e^{2}G^{2}D}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\,0\,$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\tilde{c}_{e^{2}W^{2}D}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\,0\,$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\tilde{c}_{e^{2}B^{2}D}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\,0\,$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0

Table 1: Mixing between operators containing at most two fermions.

| $\tilde{c}_{l^{2}\phi^{2}D^{3}}^{(1)}$ | $\tilde{c}_{l^{2}\phi^{2}D^{3}}^{(2)}$ | $\tilde{c}_{l^{2}\phi^{2}D^{3}}^{(3)}$ | $\tilde{c}_{l^{2}\phi^{2}D^{3}}^{(4)}$ | $\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(1)}$ | $\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(2)}$ | $\tilde{c}_{q^{2}\phi^{2}D^{3}}^{(1)}$ | $\tilde{c}_{q^{2}\phi^{2}D^{3}}^{(2)}$ | $\tilde{c}_{q^{2}\phi^{2}D^{3}}^{(3)}$ | $\tilde{c}_{q^{2}\phi^{2}D^{3}}^{(4)}$ | $\tilde{c}_{u^{2}\phi^{2}D^{3}}^{(1)}$ | $\tilde{c}_{u^{2}\phi^{2}D^{3}}^{(2)}$ | $\tilde{c}_{d^{2}\phi^{2}D^{3}}^{(1)}$ | $\tilde{c}_{d^{2}\phi^{2}D^{3}}^{(2)}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\tilde{c}_{l^{4}D^{2}}^{(1)}$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\tilde{c}_{l^{4}D^{2}}^{(2)}$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\tilde{c}_{q^{4}D^{2}}^{(1)}$ | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0
$\tilde{c}_{q^{4}D^{2}}^{(2)}$ | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0
$\tilde{c}_{q^{4}D^{2}}^{(3)}$ | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0
$\tilde{c}_{q^{4}D^{2}}^{(4)}$ | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0
$\tilde{c}_{l^{2}q^{2}D^{2}}^{(1)}$ | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | 0 | 0 | 0 | 0
$\tilde{c}_{l^{2}q^{2}D^{2}}^{(2)}$ | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | 0 | 0 | 0 | 0
$\tilde{c}_{e^{4}D^{2}}$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\tilde{c}_{u^{4}D^{2}}^{(1)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | 0 | 0
$\tilde{c}_{u^{4}D^{2}}^{(2)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | 0 | 0
$\tilde{c}_{d^{4}D^{2}}^{(1)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$
$\tilde{c}_{d^{4}D^{2}}^{(2)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$
$\tilde{c}_{e^{2}u^{2}D^{2}}^{(1)}$ | 0 | 0 | 0 | 0 | $\bm{-}$ | $\,0\,$ | 0 | 0 | 0 | 0 | $\bm{-}$ | $\,0\,$ | 0 | 0
$\tilde{c}_{e^{2}d^{2}D^{2}}^{(1)}$ | 0 | 0 | 0 | 0 | $\bm{-}$ | $\,0\,$ | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\,0\,$
$\tilde{c}_{u^{2}d^{2}D^{2}}^{(1)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\,0\,$ | $\bm{-}$ | $\,0\,$
$\tilde{c}_{u^{2}d^{2}D^{2}}^{(2)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\,0\,$ | $\bm{-}$ | $\,0\,$
$\tilde{c}_{l^{2}e^{2}D^{2}}^{(1)}$ | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | $\bm{-}$ | $\,0\,$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\tilde{c}_{l^{2}u^{2}D^{2}}^{(1)}$ | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\,0\,$ | 0 | 0
$\tilde{c}_{l^{2}d^{2}D^{2}}^{(1)}$ | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\,0\,$
$\tilde{c}_{q^{2}e^{2}D^{2}}^{(1)}$ | 0 | 0 | 0 | 0 | $\bm{-}$ | $\,0\,$ | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | 0 | 0 | 0 | 0
$\tilde{c}_{q^{2}u^{2}D^{2}}^{(1)}$ | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | $\bm{-}$ | $\,0\,$ | 0 | 0
$\tilde{c}_{q^{2}u^{2}D^{2}}^{(2)}$ | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | $\bm{-}$ | $\,0\,$ | 0 | 0
$\tilde{c}_{q^{2}d^{2}D^{2}}^{(1)}$ | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | 0 | 0 | $\bm{-}$ | $\,0\,$
$\tilde{c}_{q^{2}d^{2}D^{2}}^{(2)}$ | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\,0\,$ | $\,0\,$ | 0 | 0 | $\bm{-}$ | $\,0\,$

Table 2: Mixing of two fermion operators into four-fermion ones.

| $\tilde{c}_{l^{4}D^{2}}^{(1)}$ | $\tilde{c}_{l^{4}D^{2}}^{(2)}$ | $\tilde{c}_{q^{4}D^{2}}^{(1)}$ | $\tilde{c}_{q^{4}D^{2}}^{(2)}$ | $\tilde{c}_{q^{4}D^{2}}^{(3)}$ | $\tilde{c}_{q^{4}D^{2}}^{(4)}$ | $\tilde{c}_{l^{2}q^{2}D^{2}}^{(1)}$ | $\tilde{c}_{l^{2}q^{2}D^{2}}^{(2)}$ | $\tilde{c}_{l^{2}q^{2}D^{2}}^{(3)}$ | $\tilde{c}_{l^{2}q^{2}D^{2}}^{(4)}$ | $\tilde{c}_{e^{4}D^{2}}$ | $\tilde{c}_{u^{4}D^{2}}^{(1)}$ | $\tilde{c}_{u^{4}D^{2}}^{(2)}$ | $\tilde{c}_{d^{4}D^{2}}^{(1)}$ | $\tilde{c}_{d^{4}D^{2}}^{(2)}$ | $\tilde{c}_{e^{2}u^{2}D^{2}}^{(1)}$ | $\tilde{c}_{e^{2}u^{2}D^{2}}^{(2)}$ | $\tilde{c}_{e^{2}d^{2}D^{2}}^{(1)}$ | $\tilde{c}_{e^{2}d^{2}D^{2}}^{(2)}$ | $\tilde{c}_{u^{2}d^{2}D^{2}}^{(1)}$ | $\tilde{c}_{u^{2}d^{2}D^{2}}^{(2)}$ | $\tilde{c}_{u^{2}d^{2}D^{2}}^{(3)}$ | $\tilde{c}_{u^{2}d^{2}D^{2}}^{(4)}$ | $\tilde{c}_{l^{2}e^{2}D^{2}}^{(1)}$ | $\tilde{c}_{l^{2}e^{2}D^{2}}^{(2)}$ | $\tilde{c}_{l^{2}u^{2}D^{2}}^{(1)}$ | $\tilde{c}_{l^{2}u^{2}D^{2}}^{(2)}$ | $\tilde{c}_{l^{2}d^{2}D^{2}}^{(1)}$ | $\tilde{c}_{l^{2}d^{2}D^{2}}^{(2)}$ | $\tilde{c}_{q^{2}e^{2}D^{2}}^{(1)}$ | $\tilde{c}_{q^{2}e^{2}D^{2}}^{(2)}$ | $\tilde{c}_{q^{2}u^{2}D^{2}}^{(1)}$ | $\tilde{c}_{q^{2}u^{2}D^{2}}^{(2)}$ | $\tilde{c}_{q^{2}u^{2}D^{2}}^{(3)}$ | $\tilde{c}_{q^{2}u^{2}D^{2}}^{(4)}$ | $\tilde{c}_{q^{2}d^{2}D^{2}}^{(1)}$ | $\tilde{c}_{q^{2}d^{2}D^{2}}^{(2)}$ | $\tilde{c}_{q^{2}d^{2}D^{2}}^{(3)}$ | $\tilde{c}_{q^{2}d^{2}D^{2}}^{(4)}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\tilde{c}_{l^{2}\phi^{2}D^{3}}^{(1)}$ | $\bm{-}$ | $\bm{-}$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\tilde{c}_{l^{2}\phi^{2}D^{3}}^{(2)}$ | $\bm{-}$ | $\bm{-}$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(1)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | 0 | 0 | 0 | 0 | $\bm{-}$ | $\times$ | $\bm{-}$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\tilde{c}_{q^{2}\phi^{2}D^{3}}^{(1)}$ | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{q^{2}\phi^{2}D^{3}}^{(2)}$ | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{u^{2}\phi^{2}D^{3}}^{(1)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | 0 | 0 | $\bm{-}$ | $\times$ | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0
$\tilde{c}_{d^{2}\phi^{2}D^{3}}^{(1)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | 0 | 0 | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{q^{2}G^{2}D}^{(1)}$ | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{q^{2}G^{2}D}^{(2)}$ | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{q^{2}W^{2}D}^{(1)}$ | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{q^{2}B^{2}D}^{(1)}$ | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{u^{2}G^{2}D}^{(1)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | 0 | 0 | $\bm{-}$ | $\times$ | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0
$\tilde{c}_{u^{2}G^{2}D}^{(2)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | 0 | 0 | $\bm{-}$ | $\times$ | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0
$\tilde{c}_{u^{2}W^{2}D}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | 0 | 0 | $\bm{-}$ | $\times$ | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0
$\tilde{c}_{u^{2}B^{2}D}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | 0 | 0 | $\bm{-}$ | $\times$ | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0
$\tilde{c}_{d^{2}G^{2}D}^{(1)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | 0 | 0 | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{d^{2}G^{2}D}^{(2)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | 0 | 0 | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{d^{2}W^{2}D}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | 0 | 0 | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{d^{2}B^{2}D}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | 0 | 0 | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{l^{2}G^{2}D}$ | $\bm{-}$ | $\bm{-}$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\tilde{c}_{l^{2}W^{2}D}$ | $\bm{-}$ | $\bm{-}$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\tilde{c}_{l^{2}B^{2}D}$ | $\bm{-}$ | $\bm{-}$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\tilde{c}_{e^{2}G^{2}D}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | 0 | 0 | 0 | 0 | $\bm{-}$ | $\times$ | $\bm{-}$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\tilde{c}_{e^{2}W^{2}D}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | 0 | 0 | 0 | 0 | $\bm{-}$ | $\times$ | $\bm{-}$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\tilde{c}_{e^{2}B^{2}D}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | 0 | 0 | 0 | 0 | $\bm{-}$ | $\times$ | $\bm{-}$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0

Table 3: Mixing of four-fermion operators into two-fermion ones.

| $\tilde{c}_{l^{4}D^{2}}^{(1)}$ | $\tilde{c}_{l^{4}D^{2}}^{(2)}$ | $\tilde{c}_{q^{4}D^{2}}^{(1)}$ | $\tilde{c}_{q^{4}D^{2}}^{(2)}$ | $\tilde{c}_{q^{4}D^{2}}^{(3)}$ | $\tilde{c}_{q^{4}D^{2}}^{(4)}$ | $\tilde{c}_{l^{2}q^{2}D^{2}}^{(1)}$ | $\tilde{c}_{l^{2}q^{2}D^{2}}^{(2)}$ | $\tilde{c}_{l^{2}q^{2}D^{2}}^{(3)}$ | $\tilde{c}_{l^{2}q^{2}D^{2}}^{(4)}$ | $\tilde{c}_{e^{4}D^{2}}$ | $\tilde{c}_{u^{4}D^{2}}^{(1)}$ | $\tilde{c}_{u^{4}D^{2}}^{(2)}$ | $\tilde{c}_{d^{4}D^{2}}^{(1)}$ | $\tilde{c}_{d^{4}D^{2}}^{(2)}$ | $\tilde{c}_{e^{2}u^{2}D^{2}}^{(1)}$ | $\tilde{c}_{e^{2}u^{2}D^{2}}^{(2)}$ | $\tilde{c}_{e^{2}d^{2}D^{2}}^{(1)}$ | $\tilde{c}_{e^{2}d^{2}D^{2}}^{(2)}$ | $\tilde{c}_{u^{2}d^{2}D^{2}}^{(1)}$ | $\tilde{c}_{u^{2}d^{2}D^{2}}^{(2)}$ | $\tilde{c}_{u^{2}d^{2}D^{2}}^{(3)}$ | $\tilde{c}_{u^{2}d^{2}D^{2}}^{(4)}$ | $\tilde{c}_{l^{2}e^{2}D^{2}}^{(1)}$ | $\tilde{c}_{l^{2}e^{2}D^{2}}^{(2)}$ | $\tilde{c}_{l^{2}u^{2}D^{2}}^{(1)}$ | $\tilde{c}_{l^{2}u^{2}D^{2}}^{(2)}$ | $\tilde{c}_{l^{2}d^{2}D^{2}}^{(1)}$ | $\tilde{c}_{l^{2}d^{2}D^{2}}^{(2)}$ | $\tilde{c}_{q^{2}e^{2}D^{2}}^{(1)}$ | $\tilde{c}_{q^{2}e^{2}D^{2}}^{(2)}$ | $\tilde{c}_{q^{2}u^{2}D^{2}}^{(1)}$ | $\tilde{c}_{q^{2}u^{2}D^{2}}^{(2)}$ | $\tilde{c}_{q^{2}u^{2}D^{2}}^{(3)}$ | $\tilde{c}_{q^{2}u^{2}D^{2}}^{(4)}$ | $\tilde{c}_{q^{2}d^{2}D^{2}}^{(1)}$ | $\tilde{c}_{q^{2}d^{2}D^{2}}^{(2)}$ | $\tilde{c}_{q^{2}d^{2}D^{2}}^{(3)}$ | $\tilde{c}_{q^{2}d^{2}D^{2}}^{(4)}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\tilde{c}_{l^{4}D^{2}}^{(1)}$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\tilde{c}_{l^{4}D^{2}}^{(2)}$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\tilde{c}_{q^{4}D^{2}}^{(1)}$ | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{q^{4}D^{2}}^{(2)}$ | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{q^{4}D^{2}}^{(3)}$ | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{q^{4}D^{2}}^{(4)}$ | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{l^{2}q^{2}D^{2}}^{(1)}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{l^{2}q^{2}D^{2}}^{(2)}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{e^{4}D^{2}}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\tilde{c}_{u^{4}D^{2}}^{(1)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | 0 | 0 | $\times$ | $\times$ | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0
$\tilde{c}_{u^{4}D^{2}}^{(2)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | 0 | 0 | $\times$ | $\times$ | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0
$\tilde{c}_{d^{4}D^{2}}^{(1)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{d^{4}D^{2}}^{(2)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{e^{2}u^{2}D^{2}}^{(1)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | 0 | 0 | $\times$ | $\times$ | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0
$\tilde{c}_{e^{2}d^{2}D^{2}}^{(1)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{u^{2}d^{2}D^{2}}^{(1)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\times$ | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{u^{2}d^{2}D^{2}}^{(2)}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\times$ | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{l^{2}e^{2}D^{2}}^{(1)}$ | $\bm{-}$ | $\bm{-}$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\bm{-}$ | 0 | 0 | 0 | 0 | $\bm{-}$ | $\times$ | $\bm{-}$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\tilde{c}_{l^{2}u^{2}D^{2}}^{(1)}$ | $\bm{-}$ | $\bm{-}$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | $\bm{-}$ | $\bm{-}$ | 0 | 0 | $\bm{-}$ | $\times$ | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0
$\tilde{c}_{l^{2}d^{2}D^{2}}^{(1)}$ | $\bm{-}$ | $\bm{-}$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | 0 | 0 | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{q^{2}e^{2}D^{2}}^{(1)}$ | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\bm{-}$ | 0 | 0 | 0 | 0 | $\bm{-}$ | $\times$ | $\bm{-}$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{q^{2}u^{2}D^{2}}^{(1)}$ | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | $\bm{-}$ | $\bm{-}$ | 0 | 0 | $\bm{-}$ | $\times$ | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | $\times$ | $\times$ | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{q^{2}u^{2}D^{2}}^{(2)}$ | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | $\bm{-}$ | $\bm{-}$ | 0 | 0 | $\bm{-}$ | $\times$ | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | $\times$ | $\times$ | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{q^{2}d^{2}D^{2}}^{(1)}$ | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | 0 | 0 | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$\tilde{c}_{q^{2}d^{2}D^{2}}^{(2)}$ | 0 | 0 | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | $\bm{-}$ | $\bm{-}$ | 0 | 0 | $\bm{-}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$

Table 4: Mixing between four-fermion operators.

##  Appendix C Some cross-checks

To further cross-check the validity of our approach towards restricting the
ADM of the dimension-eight SMEFT, we have computed explicitly a number of
elements of this matrix based on the approach of Ref. [80]. Some exmaples are:

$\displaystyle
4\gamma_{\tilde{c}_{q^{2}\phi^{2}D^{3}}^{(1)}\,,\,\tilde{c}_{q^{4}D^{2}}^{(1)}}$
$\displaystyle=\gamma_{\tilde{c}_{q^{2}\phi^{2}D^{3}}^{(1)}\,,\,\tilde{c}_{q^{4}D^{2}}^{(3)}}=2\gamma_{\tilde{c}_{q^{2}\phi^{2}D^{3}}^{(2)}\,,\,\tilde{c}_{q^{4}D^{2}}^{(2)}}=\gamma_{\tilde{c}_{q^{2}\phi^{2}D^{3}}^{(2)}\,,\,\tilde{c}_{q^{4}D^{2}}^{(4)}}=-16|Y_{u}|^{2}\,,$
(84) $\displaystyle
2\gamma_{\tilde{c}_{q^{2}\phi^{2}D^{3}}^{(1)}\,,\,\tilde{c}_{q^{4}D^{2}}^{(2)}}$
$\displaystyle=\gamma_{\tilde{c}_{q^{2}\phi^{2}D^{3}}^{(1)}\,,\,\tilde{c}_{q^{4}D^{2}}^{(4)}}=4\gamma_{\tilde{c}_{q^{2}\phi^{2}D^{3}}^{(2)}\,,\,\tilde{c}_{q^{4}D^{2}}^{(1)}}=\gamma_{\tilde{c}_{q^{2}\phi^{2}D^{3}}^{(2)}\,,\,\tilde{c}_{q^{4}D^{2}}^{(3)}}=-16|Y_{d}|^{2}\,,$
(85) $\displaystyle
4\gamma_{\tilde{c}_{q^{2}B^{2}D^{1}}^{(1)}\,,\,\tilde{c}_{q^{4}D^{2}}^{(1)}}$
$\displaystyle=2\gamma_{\tilde{c}_{q^{2}B^{2}D^{1}}^{(1)}\,,\,\tilde{c}_{q^{4}D^{2}}^{(2)}}=\gamma_{\tilde{c}_{q^{2}B^{2}D^{1}}^{(1)}\,,\,\tilde{c}_{q^{4}D^{2}}^{(3)}}=\gamma_{\tilde{c}_{q^{2}B^{2}D^{1}}^{(1)}\,,\,\tilde{c}_{q^{4}D^{2}}^{(4)}}=-\frac{8}{27}g_{1}^{2}\,,$
(86)
$\displaystyle\gamma_{\tilde{c}_{l^{2}\phi^{2}D^{3}}^{(1)}\,,\,\tilde{c}_{l^{4}D^{2}}^{(1)}}$
$\displaystyle=2\gamma_{\tilde{c}_{l^{2}\phi^{2}D^{3}}^{(2)}\,,\,\tilde{c}_{l^{4}D^{2}}^{(2)}}=-4|Y_{l}|^{2}\,,$
(87)
$\displaystyle\gamma_{\tilde{c}_{l^{2}B^{2}D^{1}}^{(1)}\,,\,\tilde{c}_{l^{4}D^{2}}^{(1)}}$
$\displaystyle=2\gamma_{\tilde{c}_{l^{2}B^{2}D^{1}}^{(1)}\,,\,\tilde{c}_{l^{4}D^{2}}^{(2)}}=-\frac{4}{3}g_{1}^{2}\,,$
(88) $\displaystyle
4\gamma_{\tilde{c}_{u^{2}\phi^{2}D^{3}}^{(1)}\,,\,\tilde{c}_{u^{4}D^{2}}^{(1)}}$
$\displaystyle=\gamma_{\tilde{c}_{u^{2}\phi^{2}D^{3}}^{(1)}\,,\,\tilde{c}_{u^{4}D^{2}}^{(2)}}=-16|Y_{u}|^{2}\,,$
(89) $\displaystyle
4\gamma_{\tilde{c}_{u^{2}B^{2}D}\,,\,\tilde{c}_{u^{4}D^{2}}^{(1)}}$
$\displaystyle=\gamma_{\tilde{c}_{u^{2}B^{2}D}\,,\,\tilde{c}_{u^{4}D^{2}}^{(2)}}=-\frac{128}{27}g_{1}^{2}\,,$
(90)
$\displaystyle\gamma_{\tilde{c}_{l^{2}B^{2}D}^{(1)}\,,\,\tilde{c}_{l^{2}\phi^{2}D^{3}}^{(1)}}$
$\displaystyle=\gamma_{\tilde{c}_{l^{2}B^{2}D}^{(1)}\,,\,\tilde{c}_{l^{2}\phi^{2}D^{3}}^{(2)}}=-\frac{1}{24}g_{1}^{2}\,,$
(91)
$\displaystyle\gamma_{\tilde{c}_{l^{2}B^{2}D}^{(1)}\,,\,\tilde{c}_{l^{2}\phi^{2}D^{3}}^{(3)}}$
$\displaystyle=\gamma_{\tilde{c}_{l^{2}B^{2}D}^{(1)}\,,\,\tilde{c}_{l^{2}\phi^{2}D^{3}}^{(4)}}=\fcolorbox{black}{LightCyan}{\,0\,}\,,$
(92)
$\displaystyle\gamma_{\tilde{c}_{e^{2}B^{2}D}\,,\,\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(1)}}$
$\displaystyle=-\frac{1}{3}g_{1}^{2}\,,$ (93)
$\displaystyle\gamma_{\tilde{c}_{e^{2}B^{2}D}\,,\,\tilde{c}_{e^{2}\phi^{2}D^{3}}^{(2)}}$
$\displaystyle=\fcolorbox{black}{LightCyan}{\,0\,}\,.$ (94)

They all agree with the results in Tabs. 1–4.

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