# Relationship between covariance of Wigner functions and transformation
noncontextuality

Lorenzo Catani  Schmid College of Science and Technology, Chapman University,
One University Drive, Orange, CA, 92866, USA<EMAIL_ADDRESS>
###### Abstract

We relate two notions of classicality for quantum transformations often
arising in popular subtheories of quantum mechanics: covariance of the Wigner
representation of the theory and the existence of a transformation
noncontextual ontological model of the theory. We show that covariance implies
transformation noncontextuality. The converse holds provided that the
underlying ontological model is the one given by the Wigner representation. In
addition, we investigate the relationships of covariance and transformation
noncontextuality with the existence of a positivity preserving
quasiprobability distribution for the transformations of the theory. We
conclude that covariance implies transformation noncontextuality, which
implies positivity preservation. Therefore the violation of the latter is a
stronger notion of nonclassicality than the violation of the former.

##  1 Introduction

Understanding what are the features of quantum theory that are inherently
nonclassical is crucial both from the foundational and the computational point
of view. Several no-go theorems have provided concrete contributions in this
sense [1, 2, 3, 4, 5, 6], as well as results concerning attempts of
reproducing quantum theory in a classical fashion [7, 8] and axiomatizations
of quantum theory in the framework of general probabilistic theories [9, 10,
11]. Among the notions of nonclassicality, in recent years contextuality [2,
12] has also been used to explain the origin of the quantum computational
speed-up for specific tasks and models of quantum computation [13, 14, 15, 16,
17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. Nevertheless, there is still a
neat discrepancy between what is considered to be truly nonclassical from a
foundational and a computational point of view. An example is provided by the
$n-$qubit stabilizer theory, the subtheory of quantum mechanics composed by
common eigenstates of Pauli operators, Clifford gates and Pauli measurements,
which has been proven to be efficiently simulatable by a classical computer
according to Gottesman-Knill theorem [28], despite it shows contextuality [29,
30]. It would be desirable to obtain a consistent picture connecting the
notions of nonclassicality in the two realms and distinguishing weaker and
stronger notions of nonclassicality.

In this work we move a step towards such understanding by studying two
properties of quantum transformations that are usually associated to classical
behaviors and therefore, when broken, to genuinely nonclassical features.
These are the covariance of the Wigner function under the transformations
allowed in the examined theory [31], and transformation noncontextuality –
i.e. the existence of a transformation noncontextual ontological model for the
theory [12]. The former is defined in the framework of Wigner functions [7] –
particular quasiprobability distributions that represent quantum theory in the
phase space, while the latter is defined in the framework of ontological
models (also known as hidden variable models) [32]. As we will report in the
next section, quasiprobability representations and ontological models of a
theory are strictly related notions [33].

Covariance indicates that transformations can be represented as symplectic
affine transformations in the phase space (a subset of the permutations in the
discrete dimensional case) [34, 31], while transformation noncontextuality
means that operationally equivalent experimental procedures for performing a
transformation must correspond to the same representation in the ontological
model [12]. The main reason for considering covariance a classical feature
derives from the Hamiltonian formulation of classical mechanics [35], where
the physical trajectories are represented – necessary and sufficient condition
– by symplectic transformations. Hence, covariantly represented quantum
transformations can be interpreted as classical trajectories in the phase
space.111For the readers familiar with Spekkens’ toy theory – a noncontextual
ontological model set in the phase space and developed to support the
epistemic view of quantum theory [8, 36, 37] , it is worth noticing that
covariance characterizes the Wigner representation of the quantum
transformations therein reproduced, as it coincides with the requirement of
preservation of the epistemic restriction. The existence of a transformation
noncontextual ontological model is a classical feature because, while holding
for classical mechanics, it is incompatible with the statistics of quantum
theory, as proven in [12].

Historically, most research on the role of nonclassical features in quantum
computation has focused on properties of preparations and measurements [38,
13, 14]. Instead, we here study properties of quantum transformations. They
have recently been shown to play a crucial role in information processing
tasks [20, 21], and in encoding nonclassical behaviors of subtheories of
quantum mechanics that were previously considered to behave classically, like
the single qubit stabilizer theory [39]. The latter provides one of the main
reasons why considering covariance and transformation noncontextuality as
connected: they are both broken by the presence of the Hadamard gate. Another
result that ignores transformations is due to Spekkens in [33]. He showed that
contextuality and negativity of quasiprobability representations are
equivalent notions of nonclassicality. Here we use this result – that
trivially extends to transformations – for proving theorems 1 and 3.

In this article we focus on subtheories of quantum mechanics defined by a set
of states, transformations and measurements, where we restrict the possibility
of nonclassicality to transformations only (_i.e._ we assume that states and
measurement elements are specified by a choice of Wigner function for which
they are non-negatively represented). The main theorem – theorem 1 – shows
that covariance – originally defined for unitaries, and here extended to more
general completely positive trace-preserving (CPTP) maps – implies
transformation noncontextuality. The converse holds only assuming that the
ontological model is the one corresponding to the Wigner representation.
Successively, we introduce the notion of positivity preservation. We say that
a subtheory is positivity preserving if there exists a quasiprobability
representation for which all the transformations of the theory map non-
negative states to non-negative states. Positivity preservation is a notion of
classicality since it holds in classical mechanics, where quasiprobabilities
are (non-negative) probabilities, and it does not hold in full quantum theory,
because of the unavoidable presence of negatively represented states and the
fact that there always exists a transformation between any two states. In
theorems 2 and 3 we relax the assumption that the subtheory is specified by
the choice of Wigner function and we prove that both covariance and
transformation noncontextuality imply positivity preservation, but not vice
versa. This shows that breaking positivity preservation is the strongest
notion of nonclassicality among the ones analyzed here.

The remainder of the article is structured as follows. In section 2 we define
quasiprobability distributions and Wigner functions – subsection 2.1,
covariance of Wigner functions – subsection 2.2, and transformation
noncontextuality – subsection 2.3. In section 3 we treat the example of the
single qubit stabilizer theory and then, in section 4, we prove the results
relating covariance, transformation noncontextuality and positivity
preservation. We conclude, in section 5, by discussing possible
generalizations of the results and future avenues.

##  2 Definitions

In this section we define Wigner functions and more general quasiprobability
distributions (with a special focus on the property of positivity
preservation), covariance of Wigner functions, and transformation
noncontextuality.

###  2.1 Wigner functions

Wigner functions are a way of reformulating quantum theory in the phase space
[7], which is the framework of classical Hamiltonian mechanics, and therefore
provide a tool to compare the two theories on the same ground. They are also
the most popular example of quasiprobability distributions [40, 34, 14]. The
latter are linear and invertible maps from operators on Hilbert space to real
distributions on a measurable space $\Lambda.$ Quasiprobability
representations of quantum mechanics associate i) a real-valued function
$\mu_{\rho}:\Lambda\rightarrow\mathbb{R}$ to any density operator $\rho$ such
that $\int d\lambda\mu_{\rho}(\lambda)=1$, ii) a real-valued function
$\xi_{\Pi_{k}}:\Lambda\rightarrow\mathbb{R}$ to any element $\Pi_{k}$ of the
POVM $\\{\Pi_{k}\\}$ such that $\sum_{k}\xi_{\Pi_{k}}(\lambda)=1$
$\;\forall\;\lambda\in\Lambda,$ and iii) a real-valued matrix
$\Gamma_{\varepsilon}:\Lambda\times\Lambda\rightarrow\mathbb{R}$ to any CPTP
map $\varepsilon,$ such that $\int
d\lambda^{\prime}\Gamma_{\varepsilon}(\lambda^{\prime},\lambda)=1.$ These
distributions provide the same statistics of quantum theory – given by the
Born rule – if

$\begin{split}p(k|\rho,\varepsilon,\\{\Pi_{k}\\})&=\textrm{Tr}[\varepsilon(\rho)\Pi_{k}]\\\
&=\int d\lambda
d\lambda^{\prime}\xi_{\Pi_{k}}(\lambda^{\prime})\Gamma_{\varepsilon}(\lambda^{\prime},\lambda)\mu_{\rho}(\lambda).\end{split}$
(1)

Quasiprobability distributions are named this way as they behave similarly to
probability distributions, with the crucial difference that they can take
negative values. The negativity is usually assumed to be a signature of
quantumness because it is unavoidable in order to reproduce the statistics of
quantum mechanics [41]. A quasiprobability distribution, for example
$\mu_{\rho}(\lambda),$ is non-negative if $\mu_{\rho}(\lambda)\geq
0\;\;\forall\lambda\in\Lambda.$ Non-negative quasiprobability representations
of quantum mechanics are an important tool to support the epistemic view of
quantum theory [32, 12], where quantum states are interpreted as states of
knowledge rather than states of reality, and are useful to perform classical
simulations of quantum computations [42]. We will focus on the following
property of quasiprobability distributions.

######  Definition 1.

Given a set $\mathcal{S}$ of quantum states $\rho$ that are _non-negatively_
represented by a quasiprobability distribution $\mu_{\rho}(\lambda),$ a
transformation $\varepsilon$ that maps $\rho$ to
$\rho^{\prime}=\varepsilon(\rho)$ is _positivity preserving_ if, for every
$\rho\in\mathcal{S},$ the quasiprobability distribution
$\mu_{\rho^{\prime}}(\lambda)$ is non-negative too.

A subtheory of quantum mechanics allows for a positivity preserving
quasiprobability representation if there exists a quasiprobability
distribution for which all the transformations of the subtheory are positivity
preserving.

We are interested in Wigner functions defined over the discrete phase space
$\Lambda=\mathbb{Z}^{2n}_{d},$ where the integers $d,n$ denote the
dimensionality of the system and the number of systems, respectively. We
follow the definition provided in [14] that includes the mostly used examples
of Wigner functions [34, 43].

######  Definition 2.

The Wigner function of a quantum state $\rho$ (and, analogously, of a POVM
element $\Pi_{k}$) is defined as

$W^{\gamma}_{\rho}(\lambda)=\textrm{Tr}[A^{\gamma}(\lambda)\rho],$ (2)

where the phase-point operator is

$A^{\gamma}(\lambda)=\frac{1}{N_{\Lambda}}\sum_{\lambda^{\prime}\in\Lambda}\chi([\lambda,\lambda^{\prime}])\hat{W}^{\gamma}(\lambda^{\prime}).$
(3)

The normalization $N_{\Lambda}$ is such that
$\textrm{Tr}(A^{\gamma}(\lambda))=1.$ The Weyl operators
$\hat{W}^{\gamma}(\lambda)$ are defined as
$\hat{W}^{\gamma}(\lambda)=w^{\gamma(\lambda)}Z(p)X(x),$ where the phase space
point is $\lambda=(x,p)\in\Lambda.$ The operators $X(x),Z(p)$ represent the
(generalized) Pauli operators,
$X(x)=\sum_{x^{\prime}\in\mathbb{Z}_{d}}\left|x^{\prime}-x\right\rangle\left\langle
x^{\prime}\right|$ and
$Z(p)=\sum_{x\in\mathbb{Z}_{d}}\chi(px)\left|x\right\rangle\left\langle
x\right|.$ The functions $w$ and $\chi$ are
$w^{\gamma(\lambda)}=i^{\gamma(\lambda)},\chi(a)=(-1)^{a}$ in the case of
qubits and
$w^{\gamma(\lambda)}=\chi(-2^{-1}\gamma(\lambda)),\chi(a)=e^{\frac{2\pi
i}{d}a}$ in the case of qudits of odd dimensions. By choosing the function
$\gamma:\Lambda\rightarrow\mathbb{Z}_{q},$ where $q$ is an integer, we can
specify which Wigner function to consider. The reason why choosing different
Wigner functions is that they are non-negative – thus providing a proof of
operational equivalence with classical theory – for different subtheories of
quantum mechanics. For example $\gamma=x\cdot p$ gives Gross’ Wigner function
[34, 44] for odd dimensional qudits, which non-negatively represents
stabilizer quantum mechanics in odd dimensions [36, 37], while $\gamma=x\cdot
p\;\mathrm{mod}4$ gives the Wigner function developed by Gibbons and Wootters
[43, 45] for qubits, which non-negatively represents the subtheory of quantum
mechanics composed by all the separable eigenstates of Pauli operators, Pauli
measurements and transformations between them [14, 17]. We will omit the
superscript $\gamma$ in the future in order to soften the notation.

Wigner functions satisfy the following properties.

  * •

The marginal of a Wigner function on the state $\rho$ behaves as a probability
distribution: $\sum_{p\in\mathbb{Z}_{d}}W_{\rho}(x,p)=|\langle
x|\rho|x\rangle|^{2}.$

  * •

The Wigner function of many systems in a product state is the tensor product
of the Wigner functions of each system, as a consequence of the factorability
of $A(\lambda),$ $A(x_{0},p_{0}\dots
x_{n-1},p_{n-1})=A(x_{0},p_{0})\otimes\dots\otimes A(x_{n-1},p_{n-1}).$

  * •

The phase-point operators form a complete basis of the Hermitian operators in
the Hilbert space with respect to the Hilbert-Schmidt inner product, thus
obeying Hermitianity, $A(\lambda)=A^{\dagger}(\lambda)$
$\forall\;\lambda\in\Lambda,$ and orthonormality,
$\textrm{Tr}[A(\lambda)A(\lambda^{\prime})]=\frac{1}{N_{\Lambda}}\delta_{\lambda,\lambda^{\prime}}.$
This implies that $\rho=\sum_{\lambda\in\Lambda}A(\lambda)W_{\rho}(\lambda)$
and
$\sum_{\lambda\in\Lambda}W_{\rho}(\lambda)W_{\sigma}(\lambda)=\textrm{tr}(\rho\sigma),$
where $\rho,\sigma$ are any two Hermitian operators.

  * •

$\sum_{\lambda\in\Lambda}A(\lambda)=\mathbb{I},$ thus implying that
$\textrm{Tr}[\rho]=1=\sum_{\lambda}W_{\rho}(\lambda).$

For consistency with equation (1), the Wigner function associated to a CPTP
map $\varepsilon$ is
$W_{\varepsilon}(\lambda^{\prime},\lambda)=\textrm{Tr}[\varepsilon(A(\lambda))A(\lambda)]$
and it is such that
$\sum_{\lambda\in\Lambda}W_{\varepsilon}(\lambda^{\prime},\lambda)=1.$

In proving the main theorem of this work – theorem 1 – we consider subtheories
of quantum mechanics defined by a set
$(\mathcal{S},\mathcal{T},\mathcal{M})_{\gamma}$ of quantum states,
transformations and measurements. The sets $\mathcal{S}$ and $\mathcal{M}$ are
composed by the states and measurements, respectively, that are non-negatively
represented by the Wigner function specified by $\gamma$ (this explains the
subscript), as in [14]. The set of transformations $\mathcal{T}$ is only
restricted by the fact that transformations must map between states (and
observables) in the subtheory. In this way the possibility of nonclassicality
is only confined to transformations. In the following we define and relate
properties that further restrict the set $\mathcal{T}.$

###  2.2 Covariance

Following [34], the Wigner function $W_{\rho}(\lambda)$ is covariant under the
unitary gate $U\in\mathcal{T}$ if

$W_{U\rho
U^{\dagger}}(\lambda)=W_{\rho}(S\lambda+a)\;\;\;\forall\;\rho\in\mathcal{S},\;\forall\;\lambda\in\Lambda,$
(4)

where $S$ is a symplectic transformation, i.e. $S^{T}JS=J,$ where
$J=\bigoplus_{j=1}^{n}\begin{bmatrix}0&1\\\ -1&0\end{bmatrix}_{j}$ is the
standard invertible matrix used in symplectic geometry, and $a$ is a
translation vector. We say that the unitary $U$ is covariantly represented, by
the Wigner function $W_{\rho}(\lambda)$, if equation (4) holds. Equation (4)
can also be written as,

$UA(\lambda)U^{\dagger}=A(S\lambda+a)\;\;\;\forall\;\lambda\in\Lambda.$ (5)

A _covariantly represented subtheory_
$(\mathcal{S},\mathcal{T},\mathcal{M})_{\gamma}$ is a theory where all
$T\in\mathcal{T}$ are covariantly represented. We now extend the above
definition of covariance to general quantum channels. We recall that a quantum
channel $\varepsilon$ – represented by a CPTP map – can in general be written
as

$\varepsilon(\rho)=\sum_{k}E_{k}\rho E_{k}^{\dagger},$ (6)

where the Kraus operators $E_{k}$ satisfy the completeness relation
$\sum_{k}E_{k}^{\dagger}E_{k}=\mathbb{I}.$ There are infinitely many
equivalent Kraus decompositions, and they are all mapped to each other via
unitary maps.

######  Definition 3.

A CPTP map $\varepsilon\in\mathcal{T}$ is _covariantly represented_ , by the
Wigner function $W_{\rho}(\lambda)$, if all its decompositions into unitary
Kraus operators, i.e. $E_{k}\in\mathcal{T}$ such that
$E_{k}^{\dagger}=E_{k}^{-1},$ have covariantly represented Kraus operators,

$E_{k}A(\lambda)E_{k}^{\dagger}=A(S_{k}\lambda+a_{k})\;\;\forall\lambda\in\Lambda,\;\;\forall
k.$ (7)

The motivation for definition 3 is that covariance is a concept conceived for
Wigner functions transformed under unitaries (see equation (4)). It would not
apply to irreversible transformations (it would no longer correspond to a
notion of classicality). We discuss another possible definition of covariance
under CPTP maps in section 5.

###  2.3 Transformation Noncontextuality

A natural way of justifying why quantum theory works is to provide an
ontological model that reproduces its statistics [12]. An ontological model
associates the physical state of the system at a given time – the ontic state
– to a point $\lambda$ in a measurable set $\Lambda,$ and the experimental
procedures – classified in preparations, transformations and measurements – to
probability distributions on the ontic space $\Lambda.$ A preparation
procedure $P$ of a quantum state $\rho$ is represented by a probability
distribution $\mu_{P}(\lambda)$ over the ontic space,
$\mu_{P}:\Lambda\rightarrow\mathbb{R}$ such that
$\int\mu_{P}(\lambda)d\lambda=1$ and $\mu_{P}(\lambda)\geq
0\;\;\forall\lambda\in\Lambda.$ A transformation procedure $T$ of a CPTP map
$\varepsilon$ is represented by a transition matrix
$\Gamma_{T}(\lambda^{\prime},\lambda)$ over the ontic space,
$\Gamma_{T}:\Lambda\times\Lambda\rightarrow\mathbb{R}$ such that
$\int\Gamma_{T}(\lambda^{\prime},\lambda)d\lambda^{\prime}=1$ and
$\Gamma_{T}(\lambda^{\prime},\lambda)\geq
0\;\;\forall\lambda,\lambda^{\prime}\in\Lambda.$ A measurement procedure $M$
with associated outcomes $k$ of a POVM $\\{\Pi_{k}\\}$ is represented by a set
of indicator functions $\\{\xi_{M,k}(\lambda)\\}$ over the ontic space,
$\xi_{M,k}:\Lambda\rightarrow\mathbb{R}$ such that
$\sum_{k}\xi_{M,k}(\lambda)=1$ and $\xi_{M,k}(\lambda)\geq
0\;\;\forall\lambda\in\Lambda,\;\forall k.$ The ontological model reproduces
the predictions of quantum theory according to the law of classical total
probability,

$\begin{split}p(k|P,T,M)&=\textrm{Tr}[\varepsilon(\rho)\Pi_{k}]\\\ &=\int
d\lambda
d\lambda^{\prime}\xi_{M,k}(\lambda^{\prime})\Gamma_{T}(\lambda^{\prime},\lambda)\mu_{P}(\lambda).\end{split}$
(8)

All the preparation procedures that prepare the state $\rho$ belong to the
equivalence class that we denote with $e_{\rho}(P).$ Analogous reasoning for
the equivalence classes $e_{\varepsilon}(T)$ associated to the CPTP map
$\varepsilon$ and $e_{\\{\Pi_{k}\\}}(M)$ associated to the POVM
$\\{\Pi_{k}\\}.$ The idea of the generalized notion of noncontextuality is
that operational equivalences – e.g. the different Kraus decompositions of a
CPTP map – are represented by identical probability distributions on the ontic
space.

######  Definition 4.

An ontological model of (a subtheory of) quantum mechanics is _transformation
noncontextual_ if

$\Gamma_{T}(\lambda^{\prime},\lambda)=\Gamma_{\varepsilon}(\lambda^{\prime},\lambda)\;\;\;\forall\;T\in
e_{\varepsilon}(T),\;\;\forall\;\varepsilon.$ (9)

The above definition can be analogously extended to preparation
noncontextuality and measurement noncontextuality [12],

$\begin{split}&\mu_{P}(\lambda)=\mu_{e_{\rho}(P)}(\lambda)\;\;\;\forall P\in
e_{\rho}(P),\;\;\forall\;\rho,\\\
&\xi_{M,k}(\lambda)=\xi_{e_{\Pi_{k}}(M)}(\lambda)\;\;\;\forall M\in
e_{\\{\Pi_{k}\\}}(M),\;\;\forall\;\\{\Pi_{k}\\}.\end{split}$ (10)

An ontological model is universally noncontextual if it is preparation,
transformation and measurement noncontextual. It can be shown that a
universally noncontextual ontological model of quantum mechanics is impossible
[12] and, in particular, that a transformation noncontextual ontological model
of quantum theory is impossible too.

From the definitions of ontological models above and the definitions of
quasiprobability representations provided previously, by substituting
equations (9) and (LABEL:PrepMeasNC) into equation (8), it is immediate to see
that the existence of a non-negative quasiprobability representation that
provides the statistics of quantum theory as in equation (1) coincides with
the existence of a noncontextual ontological model for it. This result was
proven in [33] and here stated also considering transformations.

##  3 The single qubit stabilizer theory

The stabilizer theory of one qubit is defined as the subtheory of one qubit
quantum mechanics that includes the eigenstates of $X,Y,Z$ Pauli operators,
the Clifford unitaries – generated by the Hadamard gate $H$ and the phase gate
$P$ – and $X,Y,Z$ Pauli observables. Its states and measurement elements are
non-negatively represented by Wootters-Gibbons’ Wigner functions [43], and its
transformations preserve positivity. More precisely, as shown in [45], there
are two possible Wigner functions for one qubit stabilizer states that are
non-negative and obey the desired properties stated previously. We denote the
two Wigner functions with $W_{+}$ and $W_{-}.$ The corresponding phase-point
operators are

$A_{+}(0,0)=\frac{1}{2}(\mathbb{I}+X+Y+Z),$
$A_{-}(0,0)=\frac{1}{2}(\mathbb{I}+X+Y-Z),$

and $A_{j}(0,1)=XA_{j}(0,0)X^{\dagger},$ $A_{j}(0,1)=YA_{j}(0,0)Y^{\dagger},$
$A_{j}(1,1)=ZA_{j}(0,0)Z^{\dagger}$ for $j\in\\{+,-\\},$ where we have denoted
with $X,Y,Z$ both the Pauli observables and the Pauli transformations.

The 8-state model [38] provides a natural preparation and measurement
noncontextual ontological model of the single qubit stabilizer theory. The
quantum states (and measurement elements) are represented as uniform
probability distributions over an ontic space of dimension $8$ and the
Clifford transformations are represented by permutations over the ontic space.
It can be seen as a model that corresponds to take into account both the
Wigner functions $W_{+}$ and $W_{-},$ as shown in figure 1.

Figure 1: Non-negative Wigner functions of one qubit stabilizer states. The
figure above shows the states that are non-negative according to the two
Wigner functions $W_{+}$ and $W_{-}$ for one qubit defined in [45]. The
intersection of the two non-negative sectors in the Bloch sphere forms the
stabilizer octahedron. The idea behind the 8-state model [38] is to consider a
model whose ontic space has twice the dimension of the standard phase space
and consider both the two Wigner functions. The Hadamard gate, as well as the
phase gate, maps between the two Wigner functions, thus causing the breaking
of covariance and the distinction between operationally equivalent
realizations of the completely depolarizing channel, i.e. the presence of
transformation contextuality.

This model may be viewed as another confirmation of the classical nature of
the theory. However, it is incompatible with covariance and transformation
noncontextuality. Covariance is broken by the Hadamard gate (or, similarly, by
the phase gate), that unavoidably maps a phase-point operator to a phase-point
operator belonging to a different basis set, e.g. $HA_{+}(0,0)H=A_{-}(0,1),$
as a consequence of its action on the Pauli operators, $HXH=Z,HZH=X$ and
$HYH=-Y.$ Moreover, the 8-state model and, more in general, any preparation
and measurement noncontextual model of the single qubit stabilizer theory,
shows transformation contextuality, as proven in [39]. This is due, again, to
the non-covariantly represented Hadamard gate (or, similarly, to the phase
gate): if we consider the operationally equivalent decompositions of the
completely depolarizing channel, $\varepsilon_{1}(\rho)=\sum_{R}R\rho
R=\mathbb{I}$ and
$\varepsilon_{2}(\rho)=\sum_{R}(HR)\rho(HR)^{\dagger}=\mathbb{I},$ where
$R\in\\{\mathbb{I},X,Y,Z\\}$ and $\rho$ is any qubit stabilizer state, they
are ontologically distinct in any preparation and measurement noncontextual
ontological model of the single qubit stabilizer theory. The ontological
distinctness can be pictured in the 8-state model, where $\varepsilon_{1}$ and
$\varepsilon_{2}$ are represented as different permutations in the ontic
space. The former only maps within a half of the ontic space, that can be seen
as the phase space where $W_{+}$ (or $W_{-}$) is defined, while the latter
maps between the two halves of the ontic space, so between the phase space of
$W_{+}$ and the phase space of $W_{-}$ – which is also a way of picturing the
breaking of covariance. This last consideration suggests the presence of a
tight connection between covariance and transformation noncontextuality.

##  4 Results

We now relate the notions of classicality for quantum transformations defined
so far.

######  Theorem 1.

Let us consider the subtheory of quantum mechanics
$(\mathcal{S},\mathcal{T},\mathcal{M})_{\gamma}.$ The subtheory is covariantly
represented if and only if the Wigner representation provides a transformation
noncontextual ontological model for it.

###### Proof.

Let us consider a transformation $\varepsilon\in\mathcal{T}$.

$\Rightarrow.$ Given covariance (equation (7)), the orthonormality of the
phase-point operators, and the linearity of the trace, then
$\textrm{Tr}[\varepsilon(A(\lambda))A(\lambda^{\prime})]=\textrm{Tr}[\sum_{k}A(S_{k}\lambda+a_{k})A(\lambda^{\prime})]=1/N_{\Lambda}\sum_{k}\delta_{S_{k}\lambda+a_{k},\lambda^{\prime}}\geq
0.$ A non-negative Wigner function for the transformation $\varepsilon$
implies a transformation noncontextual ontological model for it, as we have
shown at the end of subsection 2.3.

$\Leftarrow.$ Transformation noncontextuality for the ontological model given
by the Wigner representation of the theory coincides with the Wigner function
$\textrm{Tr}[\varepsilon(A(\lambda))A(\lambda^{\prime})]$ to be non-negative.
This implies also the Wigner representations of the unitary Kraus operators
$E_{k},$ $\textrm{Tr}[E_{k}A(\lambda)E_{k}^{\dagger}A(\lambda^{\prime})],$ to
be non-negative. If this was not the case, we could consider the multiple
representations of the $E_{k}$ to obtain multiple representations of
$\varepsilon$ too, which would contradict the original transformation
noncontextuality assumption. In order to show that $\varepsilon$ is
covariantly represented, let us define $B\equiv
E_{k}A(\lambda)E_{k}^{\dagger}$ and, by the fact that the phase-point
operators $A(\lambda)$ form a basis for the Hermitian operators in the Hilbert
space, let us write it as
$B=\sum_{\lambda^{\prime}}W_{E_{k}}(\lambda|\lambda^{\prime})A(\lambda^{\prime}).$
Successively, notice, from $\textrm{Tr}[A(\lambda)]=1,$ that
$\textrm{Tr}[B]=\sum_{\lambda^{\prime}}W_{E_{k}}(\lambda|\lambda^{\prime})=1.$
Moreover, from the orthonormality of the phase-point operators,
$\textrm{Tr}[B^{2}]=\sum_{\lambda^{\prime}}W^{2}_{E_{k}}(\lambda|\lambda^{\prime})=1.$
Therefore, given that $W_{E_{k}}(\lambda|\lambda^{\prime})\geq
0,\sum_{\lambda^{\prime}}W_{E_{k}}(\lambda|\lambda^{\prime})=1,$ and
$\sum_{\lambda^{\prime}}W^{2}_{E_{k}}(\lambda|\lambda^{\prime})=1,$ we
conclude that $W_{E_{k}}(\lambda|\lambda^{\prime})=0$
$\forall\lambda^{\prime}$ apart from one $\tilde{\lambda^{\prime}},$ for which
$W_{E_{k}}(\lambda|\tilde{\lambda^{\prime}})=1.$ This means that $B$ coincides
with one of the phase-point operators, i.e. $E_{k}$ is covariantly
represented. This is true for every $k,$ thus $\varepsilon$ is covariantly
represented. ∎

Notice, from the theorem above, that while covariance implies transformation
noncontextuality (as it is enough to have a transformation noncontextual model
to guarantee transformation noncontextuality), the converse implication may
not hold in general. In principle, it could be possible to have a
transformation noncontextual ontological model, even if the ontological model
corresponding to the Wigner representation is transformation contextual.

Let us now state the relationships of covariance and transformation
noncontextuality with the existence of a positivity preserving
quasiprobability distribution. We no longer refer to subtheories
$(\mathcal{S},\mathcal{T},\mathcal{M})_{\gamma}$ defined by a choice of Wigner
function for which states and measurement elements are non-negative, otherwise
positivity preservation would already be trivially satisfied. We refer to more
general subtheories that we denote with
$(\mathcal{S},\mathcal{T},\mathcal{M}),$ where the transformations in the set
$\mathcal{T}$ map between states in $\mathcal{S}$ and between observables in
$\mathcal{M}.$

######  Theorem 2.

Let us consider the subtheory of quantum mechanics
$(\mathcal{S},\mathcal{T},\mathcal{M}).$ If there exists a Wigner function
$W_{\rho}(\lambda)$ for which the subtheory is covariantly represented, then
the subtheory allows for a positivity preserving quasiprobability
representation. The converse implication does _not_ hold.

###### Proof of theorem 2.

Let us consider the states $\rho\in\mathcal{S}$ with non-negative Wigner
function $W_{\rho}(\lambda)$ and a covariantly represented transformation
$\varepsilon$ which maps $\rho$ to $\rho^{\prime}=\varepsilon(\rho)$ such that
$W_{\rho^{\prime}}(\lambda)=\sum_{k}W_{\rho}(S_{k}\lambda+a_{k}).$ Since all
$W_{\rho}(S_{k}\lambda+a_{k})$ are non-negative because $W_{\rho}(\lambda)\geq
0$ $\forall\lambda\in\Lambda$, then also $W_{\rho^{\prime}}(\lambda)$ is non-
negative. The converse implication does not hold, as proven by the
counterexample of the single qubit stabilizer theory, where the Hadamard gate
maps non-negative states to non-negative states (Wootters-Gibbons’ Wigner
function [43, 45]), but it is not covariantly represented. ∎

######  Theorem 3.

Let us consider the subtheory of quantum mechanics
$(\mathcal{S},\mathcal{T},\mathcal{M}).$ If the subtheory allows for a
transformation noncontextual ontological model, then it also allows for a
positivity preserving quasiprobability representation. The converse
implication does _not_ hold.

###### Proof of theorem 3.

From Spekkens’ result [33] extended to transformations, the existence of a
transformation noncontextual ontological model to represent any transformation
$\varepsilon\in\mathcal{T}$ implies that there exists a non-negative
quasiprobability distribution
$\Gamma_{\varepsilon}(\lambda,\lambda^{\prime})\geq
0\;\;\forall\lambda,\lambda^{\prime}\in\Lambda.$ Given a state
$\rho\in\mathcal{S}$ which is represented by the non-negative quasiprobability
distribution $\mu_{\rho}(\lambda),$ the state
$\rho^{\prime}=\varepsilon(\rho)$ is also non-negatively represented, as
$\mu_{\rho^{\prime}}(\lambda)=\sum_{\lambda^{\prime}}\mu_{\rho^{\prime}}(\lambda^{\prime})\Gamma_{\varepsilon}(\lambda,\lambda^{\prime}).$
This proves the first part of the theorem. The converse implication does not
hold. In the example of the single qubit stabilizer theory all the
transformations map non-negative states to non-negative states (Wootters-
Gibbons’ Wigner function [43, 45]), but the theory, with positivity preserving
transformations and non-negative states and measurement elements, does not
allow for a transformation noncontextual model. The core reason why
transformation contextuality, and, equivalently, the unavoidable presence of
some negativity in $\Gamma_{\varepsilon},$ is a weaker notion of
nonclassicality than the breaking of positivity preservation is that
$\Gamma_{\varepsilon},$ despite assuming some negative values, can still
preserve the positivity between the quasiprobability representations of the
states. ∎

The relations found are depicted in figure 2.

Figure 2: Relationship between three notions of classicality associated to
quantum transformations. The existence of a covariant Wigner function implies
the existence of a transformation noncontextual ontological model, which
implies the existence of a positivity preserving quasiprobability distribution
for a subtheory of quantum mechanics. The existence of a noncontextual
ontological model for the subtheory implies covariance if the model is the one
provided by the Wigner representation (starred arrow).

##  5 Discussion

The results obtained regard the relationships between a property of Wigner
functions – covariance, a property of ontological models – transformation
noncontextuality, and a property of quasiprobability representations –
positivity preservation. It would be interesting to extend the notion of
covariance to more general quasiprobability representations. However, we argue
that a relation between this extended covariance and transformation
noncontextuality is not expected to hold. Such covariance would hold in the
CSS rebit subtheory studied in [15], while transformation noncontextuality is
violated222The two decompositions, $\rho\rightarrow 1/2(\rho+Y\rho Y)$ and
$\rho\rightarrow 1/2(X\rho X+Z\rho Z),$ correspond to the same channel in the
CSS subtheory of [15], even if any ontological model of the theory represents
them as distinct (the proof mirrors the one contained in [39] for the single
qubit stabilizer theory). The author thanks Piers Lillystone for making him
aware of this fact.. Still, the latter is not expected to imply covariance,
considering that the orthonormality property of the phase point operators –
crucial for proving theorem 1 – does not hold for generic quasiprobability
representations.

One extra way to generalize covariance would be to consider the CPTP map
$\varepsilon$ as given by a unitary $U$ acting on a Hilbert space that
describes both the system and the environment,
$\varepsilon(\rho)=\sum_{k}E_{k}\rho
E_{k}^{\dagger}=\textrm{Tr}_{E}[U\rho_{SE}U^{\dagger}],$ where $\rho_{SE}$ is
the state of the system and environment, while $\rho$ is the state of the
system only. Covariance is then defined as
$UA_{SE}(\lambda)U^{\dagger}=A_{SE}(S_{SE}\lambda+a_{SE}),$ where $S_{SE}$ and
$a_{SE}$ are a symplectic matrix and a vector acting on the phase space
associated to the system and environment, for all the possible unitaries that
define $\varepsilon.$ We leave the study of this notion for future research.

Another open question is whether any quasiprobability representation on an
overcomplete frame [41] (like the one in [15] and unlike the Wigner
representation) always implies the corresponding model to be transformation
contextual. In an overcomplete frame representation the phase-point operators
are, by definition, more than the ones needed to form a basis. This is
expected to imply multiple distinct representations of each
transformation.333By the time this article first appeared this result has been
proven in [46] with minor extra assumptions. Showing that only complete frame
representations provide noncontextual models would be a first step to
ultimately try to prove that any noncontextual ontological model has to
correspond to a Wigner representation, thus also generalizing theorem $1$
(_i.e._ removing the red “star” in figure 2).

Our results show that breaking positivity preservation is a stronger notion of
nonclassicality than transformation contextuality and non-covariance. With
respect to positivity preservation, the single qubit stabilizer theory is
classical, even if it shows transformation contextuality and breaks
covariance. This fact motivates the study of the physical justifications for
considering positivity preservation a legitimate classical feature. It would
be also interesting to explore other subtheories in which positivity
preservation, unlike other notions of classicality, coincides with classical
computational simulability. With this work we aim to promote further research
on the reasons why the contextuality present in qubit stabilizer theory has,
computationally, a classical nature.444Meaningful results in this direction
have been found in [47], where it is shown that contextuality lower bounds the
number of classical bits of memory required to simulate subtheories of quantum
mechanics. In the case of multi-qubit stabilizer theory, it demands a
quadrating scaling in the number of qubits, which coincides with the scaling
of Gottesman-Knill algorithm [28]. If considering the noncontextual stabilizer
theory of odd dimensional qudits the scaling becomes linear.

Acknowledgements

The author thanks Huangjun Zhu for contributing in proving the second part of
theorem 1, and David Schmid and Matt Leifer for useful discussions. This
research was supported by the Fetzer Franklin Fund of the John E. Fetzer
Memorial Trust.

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$\displaystyle\Delta M_{1}$ $\displaystyle\equiv\left(\frac{\Delta
R_{h}}{2}+\frac{\Delta
v}{v}+\frac{(\sqrt{3}\pi-6)\lambda}{16\pi^{2}}+\frac{1}{16\pi^{2}}\left(\frac{\bar{g}_{1}^{2}}{4}+\frac{3\bar{g}_{2}^{2}}{4}+6\lambda\right)\log\left[\frac{\bar{m}_{h}^{2}}{\hat{\mu}^{2}}\right]\right)$
$\displaystyle+\frac{1}{16\pi^{2}}\left(\frac{\bar{g}_{1}^{2}}{4}\mathcal{I}[\bar{m}_{Z}]+(\frac{\bar{g}_{2}^{2}}{4}+\lambda)(\mathcal{I}[\bar{m}_{Z}]+2\mathcal{I}[\bar{m}_{W}])\right).$
(164)

This expression is formally dependent in individual terms on a gauge fixing
parameter which cancels in the common sum of terms present in $\Delta M_{1}$.
See Refs. Hartmann:2015oia ; Hartmann:2015oia  for details. We have set
$\xi=1$ in this expression for brevity of presentation. $\Delta v$ is defined
by the condition $T=0$ on Hartmann:2015oia  (with $\xi=1$)

$\displaystyle T$ $\displaystyle=$
$\displaystyle\,\bar{m}_{h}^{2}\,h\,\bar{v}_{T}\,\frac{1}{16\pi^{2}}\left[-16\pi^{2}\,\frac{\Delta
v}{\bar{v}_{T}}+3\,\lambda\left(1+\log\left[\frac{\hat{\mu}^{2}}{\bar{m}_{h}^{2}}\right]\right)+\frac{1}{4}\,\bar{g}_{2}^{2}\,\left(1+\log\left[\frac{\hat{\mu}^{2}}{\bar{m}_{W}^{2}}\right]\right)\right.$
$\displaystyle\hskip
65.44142pt\left.+\frac{1}{8}(\bar{g}_{1}^{2}+\bar{g}_{2}^{2})\,\left(1+\log\left[\frac{\hat{\mu}^{2}}{\bar{m}_{Z}^{2}}\right]\right)-2\sum_{\psi}y_{\psi}^{2}\,N_{c}\frac{\bar{m}_{\psi}^{2}}{\bar{m}_{h}^{2}}\left(1+\log\left[\frac{\hat{\mu}^{2}}{\bar{m}_{\psi}^{2}}\right]\right)\right.$
$\displaystyle\hskip
65.44142pt\left.+\frac{\bar{g}_{2}^{2}}{2}\frac{\bar{m}_{W}^{2}}{\bar{m}_{h}^{2}}\left(1+3\log\left[\frac{\hat{\mu}^{2}}{\bar{m}_{W}^{2}}\right]\right)+\frac{1}{4}(\bar{g}_{1}^{2}+\bar{g}_{2}^{2})\frac{\bar{m}_{Z}^{2}}{\bar{m}_{h}^{2}}\left(1+3\log\left[\frac{\hat{\mu}^{2}}{\bar{m}_{Z}^{2}}\right]\right)\right].$

The finite results for the Higgs wavefunction renormalization in the BFM are
Hartmann:2015oia

$\displaystyle 16\,\pi^{2}\,\Delta R_{h}$
$\displaystyle=2\,\lambda\,\Big{(}6-\sqrt{3}\pi-\mathcal{J}_{x}[\bar{m}_{Z}^{2}]-2\,\mathcal{J}_{x}[\bar{m}_{W}^{2}]\Big{)}+2\,\bar{g}_{2}^{2}\Bigg{(}\Big{(}\mathcal{J}_{x}[\bar{m}_{W}^{2}]-\frac{1}{2}\Big{)}\,\left(1-\frac{3\bar{m}_{W}^{2}}{\bar{m}_{h}^{2}}\right)-\mathcal{I}[\bar{m}_{W}^{2}]\Bigg{)}$
$\displaystyle+\left(\sum_{\psi}\,y_{\psi}^{2}\,N_{c}-\bar{g}_{1}^{2}-3\bar{g}_{2}^{2}\right)\log\left(\frac{\bar{m}_{h}^{2}}{\hat{\mu}^{2}}\right)+\left(\bar{g}_{1}^{2}+\bar{g}_{2}^{2}\right)\,\Bigg{(}\Big{(}\mathcal{J}_{x}[\bar{m}_{Z}^{2}]-\frac{1}{2}\Big{)}\left(1-\frac{3\bar{m}_{Z}^{2}}{\bar{m}_{h}^{2}}\right)-\mathcal{I}[\bar{m}_{Z}^{2}]\Bigg{)}$
$\displaystyle+\sum_{\psi}\,y_{\psi}^{2}\,N_{c}\Bigg{(}1+\left(1+\frac{2\bar{m}_{\psi}^{2}}{\bar{m}_{h}^{2}}\right)\,\mathcal{I}[\bar{m}_{\psi}^{2}]-\frac{2\bar{m}_{\psi}^{2}}{\bar{m}_{h}^{2}}\,\log\left(\frac{\bar{m}_{\psi}^{2}}{\bar{m}_{h}^{2}}\right)\Bigg{)}.$
(166)

We also use

$\displaystyle\Delta
R_{\hat{\mathcal{A}}}=\frac{\bar{g}_{1}^{2}\bar{g}_{2}^{2}}{(\bar{g}_{1}^{2}+\bar{g}_{2}^{2})}\,\left[-\frac{7}{16\pi^{2}}\log\left(\frac{\hat{\mu}^{2}}{\bar{m}_{W}^{2}}\right)+\sum_{\psi}\frac{N^{\psi}_{c}Q^{2}_{\psi}}{12\pi^{2}}\,\log\left(\frac{\hat{\mu}^{2}}{\bar{m}_{\psi}^{2}}\right)-\frac{1}{24\pi^{2}}\right].$
(167)

This result was successfully verified comparing to the explicit calculation
reported in in Ref. Dekens:2019ept . The Ward identities in the SMEFT in the
BFM Corbett:2019cwl  have been validated at one loop Corbett:2020ymv ;
Corbett:2020bqv . These identities also give

$\displaystyle\Delta Z_{e}$ $\displaystyle=-\frac{1}{2}\Delta
Z_{\hat{\mathcal{A}}},$ $\displaystyle\Delta R_{e}$
$\displaystyle=-\frac{1}{2}\Delta R_{\hat{\mathcal{A}}}.$ (168)

In the $\\{\hat{m}_{W},\hat{m}_{Z},\hat{G}_{F}\\}$ scheme one has
Corbett:2021cil

$\displaystyle\frac{\Delta g_{1}}{\hat{g}_{1}}$ $\displaystyle=\frac{\Delta
G_{F}}{2}+\frac{\Delta R_{m_{W}}\,\hat{m}_{W}^{2}-\Delta
R_{m_{Z}}\,\hat{m}_{Z}^{2}}{\hat{m}_{W}^{2}-\hat{m}_{Z}^{2}},$ (169)
$\displaystyle\frac{\Delta g_{2}}{\hat{g}_{2}}$ $\displaystyle=\frac{\Delta
G_{F}}{2}+\Delta R_{m_{W}},$ (170)

while the $\\{\hat{\alpha}_{ew},\hat{m}_{Z},\hat{G}_{F}\\}$ scheme defines
Corbett:2021cil

$\displaystyle\frac{\Delta g_{1}}{\hat{g}_{1}}$ $\displaystyle=\frac{\Delta
e}{\hat{e}}-\frac{\Delta c_{\theta}}{c_{\hat{\theta}}},$ (171)
$\displaystyle\frac{\Delta g_{2}}{\hat{g}_{2}}$ $\displaystyle=\frac{\Delta
e}{\hat{e}}-\frac{\Delta s_{\theta}}{s_{\hat{\theta}}},$ (172)

where

$\displaystyle\frac{\Delta
s_{\theta}}{s_{\hat{\theta}}}=\frac{1-s_{\hat{\theta}}^{2}}{2\,(1-2s_{\hat{\theta}}^{2})}\left[\frac{\Delta\alpha}{\alpha}-\Delta
G_{F}-2\,\Delta R_{M_{Z}}\right].$ (173)

The BFM expressions for $\Delta R_{m_{W},m_{Z}}$ are somewhat lengthy and
given in the Appendix Ref. Corbett:2021cil .

##  Appendix E $\Delta\delta\sigma(\mathcal{G}\mathcal{G}\rightarrow h)$ and
quadratic $\delta^{2}\sigma(\mathcal{G}\mathcal{G}\rightarrow h)$ results

Explicitly, in the $m_{t}\rightarrow\infty$ limit, the leading results for the
interference with the SM one loop amplitude are Georgi:1977gs ;
Gounaris:1998ni ; Manohar:2006gz

$\displaystyle\Delta^{2}\hat{\sigma}^{SM,m_{t}\rightarrow\infty}_{LO}(\mathcal{G}\mathcal{G}\rightarrow
h)$ $\displaystyle\equiv$ $\displaystyle\frac{\pi}{4}\lim_{\epsilon\rightarrow
0}\left|\Delta
C^{SM,m_{t}\rightarrow\infty}_{h\mathcal{G}\mathcal{G}}\right|^{2},$ (174)
$\displaystyle=$
$\displaystyle\frac{(\alpha_{s}^{(r)})^{2}}{576\,\pi\,\bar{v}_{T}^{2}},$

while

$\displaystyle\frac{\Delta\delta\hat{\sigma}(\mathcal{G}\mathcal{G}\rightarrow
h)[\tilde{C}^{(6)}_{HG}]}{\Delta^{2}\hat{\sigma}^{SM,m_{t}\rightarrow\infty}_{LO}(\mathcal{G}\mathcal{G}\rightarrow
h)}$ $\displaystyle=$
$\displaystyle\frac{24\,\pi}{\alpha_{s}^{(r)}}\,\tilde{C}_{HG},$ (175)

and a contribution at $\delta^{2}$ order (in the $m_{t}\rightarrow\infty$
limit) is

$\displaystyle\frac{\delta^{2}\hat{\sigma}(\mathcal{G}\mathcal{G}\rightarrow
h)[(\tilde{C}^{(6)}_{HG})^{2}]}{\Delta^{2}\hat{\sigma}^{SM}_{LO}(\mathcal{G}\mathcal{G}\rightarrow
h)}$ $\displaystyle=$ $\displaystyle
9\left(\frac{4\,\pi}{\alpha_{s}^{(r)}}\right)^{2}(\tilde{C}^{(6)}_{HG})^{2}.$
(176)

The results for these ratios reported in in Ref. Corbett:2021cil  were further
scaled by a correction factor of $(1+11\,\alpha_{s}^{(r)}/2\pi)^{-1}\simeq
1.21$, using $\alpha_{s}\simeq 0.118$ due to the inclusion of the partial NLO
result easily retaining by the two loop matching correction to the SM result.
Using Ref. Anastasiou:2020qzk  we can improve this rough approximation (while
still in the $m_{t}\rightarrow\infty$ limit) using

$\displaystyle\frac{\Delta^{3}\sigma^{SM}}{\Delta^{2}\,\hat{\sigma}^{SM}_{LO,\epsilon\rightarrow
0}}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}^{(r)}}{4\pi}\left[4\pi^{2}+22\right]\,\delta(1-z)+\frac{6\,\alpha_{s}^{(r)}\,f^{2}_{1}(z)}{\pi}\log\left(\frac{s}{\hat{\mu}^{2}}\right)\left(\frac{1}{1-z}\right)_{+}-11\frac{\alpha_{s}^{(r)}}{2\pi}\,(1-z)^{3}$
(177) $\displaystyle+$
$\displaystyle\frac{3\,\alpha_{s}^{(r)}}{\pi}\,z\,p_{\mathcal{G}\mathcal{G}}(z)\,\log\left(\frac{\hat{\mu}^{2}}{\mu_{F}^{2}}\right)+\frac{12\,\alpha_{s}^{(r)}}{\pi}\,f^{2}_{1}(z)\,\left(\frac{\log(1-z)}{1-z}\right)_{+}.$

Here we used the AP counterterm that accounts for $L_{\hat{m}_{t}}$ dependence

$\displaystyle\Delta^{2}\delta\sigma_{DR\,c.t}^{AP}$ $\displaystyle\equiv$
$\displaystyle\Delta^{2}\hat{\sigma}^{SM}_{LO,\epsilon\rightarrow
0}(\mathcal{G}\,\mathcal{G}\rightarrow
h)\frac{3\alpha^{(r)}_{s}}{2\,\pi}\left[\left(\frac{\mu^{2}}{\mu_{F}^{2}}\right)^{\epsilon}\right]$
(178) $\displaystyle\times$
$\displaystyle(4\pi)^{\epsilon}\frac{\Gamma(1+\epsilon)\Gamma(1-\epsilon)^{2}}{\Gamma(1-2\epsilon)}\left[\frac{1}{\epsilon}+1-2L_{\hat{m}_{t}}\right]\,z\,p_{\mathcal{G}\mathcal{G}}(z).$

###  E.1 $\Delta\delta\sigma(\mathcal{G}\mathcal{G}\rightarrow h)$

The contributions to
$\langle\mathcal{G}\mathcal{G}|h\rangle^{1}_{\mathcal{O}(v^{2}/\Lambda^{2})}$
that need to be added to Eqn. (3.1.5) follow from the following perturbations.
We express these various terms as Corbett:2021cil

$\displaystyle\langle\mathcal{G}\mathcal{G}|h\rangle^{1}_{\mathcal{O}(v_{T}^{2}/\Lambda^{2})}=-4[\Delta
G_{F}+\Delta
M_{1}]\,\frac{\tilde{C}_{HG}^{(6)}}{\hat{v}_{T}}\,K_{ab}-4\left(\frac{\tilde{C}_{i}\,\Delta
f_{i}}{16\,\pi^{2}\,\hat{v}^{2}_{T}}\right)\,\hat{v}_{T}\,K_{ab}.$ (179)

where $\tilde{C}_{i}\,\Delta f_{i}$ contains all corrections – from operator
mixing and $\mathcal{O}(\bar{v}^{2}_{T}/\Lambda^{2})$ corrections to the SM –
that are not proportional to $\tilde{C}^{(6)}_{HG}$. The $\Delta f_{i}$ are
Hartmann:2015aia ; Deutschmann:2017qum ; Grazzini:2016paz  (using
$\tau_{p}=4m_{p}^{2}/\bar{m}_{h}^{2}$)

$\displaystyle\Delta f_{H\Box}$ $\displaystyle=$
$\displaystyle-\sum_{f}\,\alpha_{s}\pi\,A_{1/2}(\tau_{f}),$ (180)
$\displaystyle\Delta f_{HD}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\,\sum_{f}\,\alpha_{s}\pi\,A_{1/2}(\tau_{f}),$ (181)
$\displaystyle\Delta f_{\delta G_{F}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\,\sum_{f}\,\alpha_{s}\pi\,A_{1/2}(\tau_{f}),$
(182) $\displaystyle\left[Y_{\psi^{\prime}_{ff}}\right]\,\Delta
f_{\begin{subarray}{c}\psi^{\prime}H\\\ ff\end{subarray}}$ $\displaystyle=$
$\displaystyle\sum_{f}\,\alpha_{s}\pi\,A_{1/2}(\tau_{f}).$ (183)

In practice, contributions from light fermions to the $\Delta f_{i}$ are
suppressed since $A_{1/2}(\tau_{f}\ll
1)\sim\tau\sim\left[Y_{\psi^{\prime}_{ff}}\right]^{2}$, so we will only
include effects from the top and bottom quarks. The dipole operators enter at
one loop Hartmann:2015aia ; Deutschmann:2017qum ; Grazzini:2016paz , the only
term which enter at $\mathcal{O}(v^{2}_{T}/16\pi^{2}\Lambda^{2})$ (again
retaining only the $Y_{t,b}$ terms) are the $\mathcal{L}^{(6)}$ operators
$\tilde{C}_{\begin{subarray}{c}uG\\\ tt\end{subarray}}$ and
$\tilde{C}_{\begin{subarray}{c}dG\\\ bb\end{subarray}}$:

$\displaystyle\langle\mathcal{G}\mathcal{G}|h\rangle^{1}_{\mathcal{O}(v^{2}/\Lambda^{2})}\supset-\frac{\sqrt{\kappa}\,\bar{g}_{3}}{8\pi^{2}\,\bar{v}_{T}}\,\left(\Delta
f_{uG}\,\tilde{C}_{\begin{subarray}{c}uG\\\ tt\end{subarray}}[Y]_{tt}+\Delta
f_{dG}\,\tilde{C}_{\begin{subarray}{c}dG\\\
bb\end{subarray}}[Y]_{bb}+h.c.\right)K_{ab},$ (184)

where

$\displaystyle\Delta f_{uG}$ $\displaystyle=$
$\displaystyle\left[-1+2\,\log\left(\frac{\hat{\mu}^{2}}{\hat{m}_{h}^{2}}\right)+\,\log\left(\frac{4}{\tau_{t}}\right)\right]-2\,\mathcal{I}_{y}[m_{t}^{2}]-\mathcal{I}[m_{t}^{2}],$
$\displaystyle\Delta f_{dG}$ $\displaystyle=$
$\displaystyle\left[-1+2\,\log\left(\frac{\hat{\mu}^{2}}{\hat{m}_{h}^{2}}\right)+2\,\log\left(\frac{4}{\tau_{b}}\right)\right]-\tau_{b}\,f(\tau_{b})-4\,i\,\sqrt{1-\tau_{b}}\,f^{1/2}(\tau_{b}).$
(185)

This set of $\Delta^{2}\delta$ corrections in the $m_{t}\rightarrow\infty$
limit are Corbett:2021cil

$\displaystyle\frac{\Delta^{2}\delta\sigma(\mathcal{G}\,\mathcal{G}\rightarrow
h)}{\Delta^{2}\,\hat{\sigma}^{SM}_{LO,\epsilon\rightarrow
0}(\mathcal{G}\,\mathcal{G}\rightarrow
h)}=\frac{24\pi}{\alpha_{s}^{(r)}}\,\left[[\Delta G_{F}+\Delta
M_{1}]\,\tilde{C}_{HG}^{(6)}+{\rm{Re}}\left(\frac{\tilde{C}_{i}\,\Delta
f_{i}}{16\,\pi^{2}}\right)\right]\delta(1-z).$ (186)

###  E.2 $\delta^{2}\sigma(\mathcal{G}\mathcal{G}\rightarrow h)$ geoSMEFT
terms

The $\mathcal{O}(v^{4}_{T}/\Lambda^{4})$ terms are denoted as $\delta^{2}$
terms. There are two sets of terms of this order. One that follows from the
“self-square” or “quadratic” term of the tree level $\tilde{C}_{HG}$
dependence, and a further set of terms that are obtained consistently
expanding to $\delta^{2}$ order. In this subsection we report the later set of
terms.

For the $\mathcal{G}\,\mathcal{G}\rightarrow h$ amplitude these corrections
are Corbett:2021cil

$\displaystyle\langle\mathcal{G}\mathcal{G}|\phi_{4}\rangle^{0}_{\mathcal{O}(v^{4}/\Lambda^{4})}$
$\displaystyle=\langle\sqrt{h}^{44}\rangle_{\mathcal{O}(v^{2}/\Lambda^{2})}\langle\mathcal{G}\mathcal{G}|\phi_{4}\rangle^{0}_{\mathcal{O}(v^{2}/\Lambda^{2})}$
(187)
$\displaystyle\quad\quad\quad\quad\quad\quad+2\,\frac{v_{T}[\langle\mathcal{G}\mathcal{G}|\phi_{4}\rangle^{0}_{\mathcal{O}(v^{2}/\Lambda^{2})}]^{2}}{\langle\mathcal{G}\,\mathcal{G}|\phi_{4}\rangle^{0}}+\,(\langle\mathcal{G}\mathcal{G}|\phi_{4}\rangle^{0}_{\mathcal{O}(v^{2}/\Lambda^{2})})\Big{|}_{\tilde{C}^{(6)}_{HG}\rightarrow\tilde{C}^{(8)}_{HG}}$

where
$\langle\sqrt{h}^{44}\rangle_{\mathcal{O}(v^{2}/\Lambda^{2})}=\tilde{C}^{(6)}_{H\Box}-\frac{1}{4}\tilde{C}^{(6)}_{HD}$.
A term from the redefinition of $\bar{v}_{T}$ in its relation to input
observables is formally present but suppressed as it cancels when the SM
amplitude is interfered with, which is $\propto 1/\bar{v}_{T}$. These
dimension eight interference corrections in the $m_{t}\rightarrow\infty$ limit
are Corbett:2021cil

$\displaystyle\frac{\Delta\delta^{2}\sigma(\mathcal{G}\,\mathcal{G}\rightarrow
h)}{\Delta^{2}\,\hat{\sigma}^{SM}_{LO,\epsilon\rightarrow
0}(\mathcal{G}\,\mathcal{G}\rightarrow
h)}=\frac{24\pi}{\alpha_{s}^{(r)}}\,\left[\left(\langle\sqrt{h}^{44}\rangle_{\mathcal{O}(v^{2}/\Lambda^{2})}+2\tilde{C}^{(6)}_{HG}\right)\tilde{C}^{(6)}_{HG}+\tilde{C}^{(8)}_{HG}\right]\delta(1-z).$
(188)

Note that taking the “quadratic” dependence on $\tilde{C}_{HG}^{2}$ (the
square of the $\delta$ correction due to this operator) does not generate all
terms dependent on $\tilde{C}_{HG}^{2}$ in the observable. See Ref.
Hays:2020scx  for more discussion.

##  Appendix F $\Delta\delta\Gamma(h\rightarrow\mathcal{G}\mathcal{G})$ and
quadratic $\delta^{2}\Gamma(h\rightarrow\mathcal{G}\mathcal{G})$ results

The results unchanged from Ref. Corbett:2021cil  are as follows. The leading
order result in the $m_{t}\rightarrow\infty$ limit is

$\displaystyle\Delta^{2}\Gamma(h\rightarrow\mathcal{G}\,\mathcal{G})_{SM}\equiv\frac{2}{\pi}\,\hat{m}_{h}^{3}\lim_{\epsilon\rightarrow\infty}|\Delta
C^{SM,m_{t}\rightarrow\infty}_{h\mathcal{G}\mathcal{G}}|^{2},$ (189)

leading to

$\displaystyle\frac{\Delta\delta\Gamma(h\rightarrow\mathcal{G}\,\mathcal{G})}{\Delta^{2}\Gamma(h\rightarrow\mathcal{G}\,\mathcal{G})_{SM}}\equiv\frac{24\pi}{\alpha_{s}^{(r)}}\tilde{C}^{(6)}_{HG},$
(190)

and

$\displaystyle\frac{\delta^{2}\Gamma(h\rightarrow\mathcal{G}\,\mathcal{G})}{\Delta^{2}\Gamma(h\rightarrow\mathcal{G}\,\mathcal{G})_{SM}}\equiv
9\left(\frac{4\pi}{\alpha_{s}^{(r)}}\right)^{2}\,(\tilde{C}^{(6)}_{HG})^{2}.$
(191)

###  F.1 $\Delta\,\delta\Gamma(h\rightarrow\mathcal{G}\,\mathcal{G})$

As previously reported in Ref. Corbett:2021cil , the EW correction is
identical to the case of $\sigma(\mathcal{G}\mathcal{G}\rightarrow h)$,

$\displaystyle\frac{\Delta\delta\Gamma(h\rightarrow\mathcal{G}\,\mathcal{G})^{m_{t}\rightarrow\infty}_{EW}}{\Delta^{2}\Gamma(h\rightarrow\mathcal{G}\,\mathcal{G})_{SM}}=\frac{24\pi}{\alpha_{s}}\times\left(\left[\Delta
G_{F}+\Delta M_{1}+\Delta
R_{\mathcal{G}}\right]\,\tilde{C}^{(6)}_{HG}+\sum_{i}\,\frac{{\rm
Re}\,\tilde{C}_{i}^{(6)}\Delta f_{i}^{(6)}}{16\pi^{2}}\right).$ (192)

In this expression we also include the BFM wavefunction renormalization finite
factor of the final state gluons

$\displaystyle\Delta
R_{\mathcal{G}}=\frac{1}{24\pi^{2}}\sum_{f}\log\left(\frac{m_{f}^{2}}{\hat{\mu}^{2}}\right),$
(193)

as the $\tilde{C}^{(6)}_{HG}$ operator was not redefined to rescale it by
$g_{3}^{2}$. Note that in the BFM this has the result of the $\Delta
R_{\mathcal{G}}$ contribution not canceling against a corresponding finite
term for $g_{3}^{2}$, but contributing.

###  F.2 $\delta^{2}\Gamma(h\rightarrow\mathcal{G}\,\mathcal{G})$ geoSMEFT
terms

The $\Delta\delta^{2}$ terms for this decay that follow from the geoSMEFT and
added to the naive Quadratic terms are

$\displaystyle\frac{\Delta\delta^{2}\Gamma(h\rightarrow\mathcal{G}\,\mathcal{G})}{\Delta^{2}\Gamma(h\rightarrow\mathcal{G}\,\mathcal{G})_{SM}}=\frac{24\pi}{\alpha_{s}^{(r)}}\,\left[\left(\langle\sqrt{h}^{44}\rangle_{\mathcal{O}(v^{2}/\Lambda^{2})}+2\tilde{C}^{(6)}_{HG}\right)\tilde{C}^{(6)}_{HG}+\tilde{C}^{(8)}_{HG}\right].$
(194)

##  Appendix G $\Delta\delta\Gamma(h\rightarrow\mathcal{A}\mathcal{A})$ and
quadratic $\delta^{2}\Gamma(h\rightarrow\mathcal{A}\mathcal{A})$ results

We define Corbett:2021cil

$\displaystyle\langle
h|\mathcal{A}\mathcal{A}\rangle_{SM}^{1}=\frac{-\hat{g}_{2}\,\hat{e}^{2}}{64\,\pi^{2}\,\hat{m}_{W}}\Bigg{(}A_{1}(\tau_{W})+\sum_{i}\,N_{c}^{i}\,Q^{2}_{i}\,A_{1/2}(\tau_{\psi^{i}})\Bigg{)}\,\langle
h\mathcal{A}^{\mu\nu}\mathcal{A}_{\mu\nu}\rangle^{0},$ (195)

with $\psi^{i}$ a mass eigenstate fermion and the loop functions are reported
in Appendix B. For notational convenience we define a short hand notation

$\displaystyle
A_{\mathcal{A}\mathcal{A}}\equiv\Bigg{(}A_{1}(\tau_{W})+\sum_{i}\,N_{c}^{i}\,Q^{2}_{i}\,A_{1/2}(\tau_{\psi^{i}})\Bigg{)}.$
(196)

For later convenience we define

$\displaystyle
A^{\mathcal{L}^{(6)}}_{\mathcal{A}\mathcal{A}}\equiv\left[\frac{\hat{g}_{2}^{2}\,\tilde{C}_{HB}^{(6)}+\hat{g}_{1}^{2}\,\tilde{C}_{HW}^{(6)}-\hat{g}_{1}\,\hat{g}_{2}\,\tilde{C}_{HWB}^{(6)}}{(\hat{g}^{\rm
SM}_{Z})^{2}}\right].$ (197)

Directly one has

$\displaystyle\langle
h|\mathcal{A}\mathcal{A}\rangle^{0}_{\mathcal{O}(\bar{v}^{2}/\Lambda^{2})}$
$\displaystyle=\frac{A^{\mathcal{L}^{(6)}}_{\mathcal{A}\mathcal{A}}}{\hat{v}_{T}}\,\langle
h\mathcal{A}^{\mu\nu}\mathcal{A}_{\mu\nu}\rangle^{0},$ (198)

leading to the $\Delta\delta$ contribution

$\displaystyle\frac{\Delta\delta\Gamma(h\rightarrow\mathcal{A}\mathcal{A})}{\Delta^{2}\Gamma_{SM}(h\rightarrow\mathcal{A}\mathcal{A})}=-64\pi^{2}\,\frac{{\rm
Re}A_{\mathcal{A}\mathcal{A}}}{|A_{\mathcal{A}\mathcal{A}}|^{2}}\left[\frac{\tilde{C}_{HB}^{(6)}}{\hat{g}_{1}^{2}}+\frac{\tilde{C}_{HW}^{(6)}}{\hat{g}_{2}^{2}}-\frac{\tilde{C}_{HWB}^{(6)}}{\hat{g}_{1}\,\hat{g}_{2}}\right],$
(199)

and a $\delta^{2}$ contribution

$\displaystyle\frac{\delta^{2}\Gamma(h\rightarrow\mathcal{A}\mathcal{A})}{\Delta^{2}\Gamma_{SM}(h\rightarrow\mathcal{A}\mathcal{A})}=\frac{1024\pi^{4}}{|A_{\mathcal{A}\mathcal{A}}|^{2}}\left[\frac{\tilde{C}_{HB}^{(6)}}{\hat{g}_{1}^{2}}+\frac{\tilde{C}_{HW}^{(6)}}{\hat{g}_{2}^{2}}-\frac{\tilde{C}_{HWB}^{(6)}}{\hat{g}_{1}\,\hat{g}_{2}}\right]^{2}.$
(200)

###  G.1 $\Delta^{2}\delta\Gamma(h\rightarrow\mathcal{A}\mathcal{A})$

$\displaystyle\langle
h|\mathcal{A}\mathcal{A}\rangle^{1}_{\mathcal{O}(v_{T}^{2}/\Lambda^{2})}$
$\displaystyle=$ $\displaystyle\langle
h|[C^{(6)}_{i}]|\mathcal{A}\mathcal{A}\rangle^{1}\frac{\langle
h\mathcal{A}^{\mu\nu}\mathcal{A}_{\mu\nu}\rangle_{0}}{\hat{v}_{T}}+\left(\frac{A^{\mathcal{L}^{(6)}}_{\mathcal{A}\mathcal{A}}}{\bar{v}_{T}^{2}}\,\Delta
M_{1}+\frac{\tilde{C}_{i}\,\Delta
f_{i}}{16\,\pi^{2}\,\hat{v}_{T}^{2}}\right)\,\hat{v}_{T}\,\langle
h\mathcal{A}^{\mu\nu}\mathcal{A}_{\mu\nu}\rangle^{0}.$

Here we have redefined notation slightly from Ref. Corbett:2021cil  and
explicitly

$\displaystyle\langle h|[C^{(6)}_{i}]|\mathcal{A}\mathcal{A}\rangle^{1}$
$\displaystyle=$
$\displaystyle\frac{\hat{e}^{2}}{32\,\pi^{2}}\,A_{\mathcal{A}\mathcal{A}}\left[\frac{\delta
G_{F}}{\sqrt{2}}-\frac{\delta\alpha}{\alpha}\right]+\frac{2.1\,\hat{e}^{2}}{16\,\pi^{2}}\,\frac{\delta
M_{W}^{2}}{\hat{M}_{W}^{2}}$ $\displaystyle+$ $\displaystyle\left(\Delta
R_{A}+\Delta G_{F}+\frac{2\hat{g}_{1}(\Delta g_{2}\hat{g}_{1}-\Delta
g_{1}\hat{g}_{2})}{\hat{g}_{2}\,(\hat{g}^{\rm
SM}_{Z})^{2}}\right)\frac{\hat{g}_{2}^{2}\,\tilde{C}^{(6)}_{HB}}{(\hat{g}^{\rm
SM}_{Z})^{2}}$ $\displaystyle+$ $\displaystyle\left(\Delta R_{A}+\Delta
G_{F}+\frac{2\hat{g}_{2}(\Delta g_{1}\hat{g}_{2}-\Delta
g_{2}\hat{g}_{1})}{\hat{g}_{1}\,(\hat{g}^{\rm
SM}_{Z})^{2}}\right)\frac{\hat{g}_{1}^{2}\,\tilde{C}^{(6)}_{HW}}{(\hat{g}^{\rm
SM}_{Z})^{2}}$ $\displaystyle-$ $\displaystyle\left(\Delta R_{A}+\Delta
G_{F}+\frac{(\hat{g}_{1}^{2}-\hat{g}_{2}^{2})(\Delta g_{2}\hat{g}_{1}-\Delta
g_{1}\hat{g}_{2})}{\hat{g}_{1}\hat{g}_{2}(\hat{g}^{\rm
SM}_{Z})^{2}}\right)\frac{\hat{g}_{1}\,\hat{g}_{2}\,\tilde{C}^{(6)}_{HWB}}{(\hat{g}^{\rm
SM}_{Z})^{2}}.$

The remaining $\Delta f_{i}$’s are in Refs. Hartmann:2015oia ;
Hartmann:2015aia  in terms of the one loop functions are

$\displaystyle\frac{\hat{g}_{1}\,\hat{g}_{2}}{\hat{e}^{2}}\Delta f_{HWB}$
$\displaystyle=$
$\displaystyle\left(-3\,\hat{g}_{2}^{2}+4\,\lambda\right)\,\log\left(\frac{\bar{m}_{h}^{2}}{\hat{\mu}^{2}}\right)+(4\lambda-\hat{g}_{2}^{2})\,\mathcal{I}[\bar{m}_{W}^{2}]-4\,\hat{g}_{2}^{2}\,\mathcal{I}_{y}[\bar{m}_{W}^{2}]-2\,\hat{g}_{2}^{2}\left[1+\log\left(\frac{\tau_{W}}{4}\right)\right]$
(203) $\displaystyle+$
$\displaystyle\hat{e}^{2}\,(2+3\,\tau_{W})+6\,\hat{e}^{2}(2-\tau_{W})\,\,\mathcal{I}_{y}[\bar{m}_{W}^{2}],$
$\displaystyle\frac{\hat{g}_{2}^{2}}{\hat{e}^{2}}\Delta f_{HW}$
$\displaystyle=-\hat{g}_{2}^{2}\Bigg{[}3\,\tau_{W}+\left(16-\frac{16}{\tau_{W}}-6\,\tau_{W}\right)\,\mathcal{I}_{y}[\bar{m}_{W}^{2}]\Bigg{]},$
(204) $\displaystyle\frac{\hat{g}_{2}^{3}}{\hat{e}^{2}}\Delta f_{W}$
$\displaystyle=-9\,\hat{g}_{2}^{4}\,\log\left(\frac{\bar{m}_{h}^{2}}{\hat{\mu}^{2}}\right)-9\,\hat{g}_{2}^{4}\,\mathcal{I}[\bar{m}_{W}^{2}]-6\,\hat{g}_{2}^{4}\,\mathcal{I}_{y}[\bar{m}_{W}^{2}]+6\,\hat{g}_{2}^{4}\,\mathcal{I}_{xx}[\bar{m}_{W}^{2}]\,\left(1-1/\tau_{W}\right)-12\,\hat{g}_{2}^{4},$
(205) $\displaystyle\frac{\hat{g}_{1}}{\hat{e}^{2}}\Delta
f_{\begin{subarray}{c}eB\\\ ss\end{subarray}}$
$\displaystyle=2\,Q_{\ell}\,[Y_{\ell}]_{ss}\left[-1+2\,\log\left(\frac{\hat{\mu}^{2}}{\bar{m}_{h}^{2}}\right)+\,\log\left(\frac{4}{\tau_{s}}\right)\right]-2\,Q_{\ell}\,[Y_{\ell}]_{ss}\,\Big{[}2\,\mathcal{I}_{y}[m_{s}^{2}]+\mathcal{I}[m_{s}^{2}]\Big{]},$
(206) $\displaystyle\frac{\hat{g}_{1}}{\hat{e}^{2}}\Delta
f_{\begin{subarray}{c}uB\\\ ss\end{subarray}}$
$\displaystyle=2\,N_{c}\,Q_{u}\,[Y_{u}]_{ss}\left[-1+2\,\log\left(\frac{\hat{\mu}^{2}}{\bar{m}_{h}^{2}}\right)+\,\log\left(\frac{4}{\tau_{s}}\right)\right]-2\,Q_{u}\,[Y_{u}]_{ss}\,\Big{[}2\,\mathcal{I}_{y}[m_{s}^{2}]+\mathcal{I}[m_{s}^{2}]\Big{]},$
(207) $\displaystyle\frac{\hat{g}_{1}}{\hat{e}^{2}}\Delta
f_{\begin{subarray}{c}dB\\\ ss\end{subarray}}$
$\displaystyle=2\,N_{c}\,Q_{d}\,[Y_{d}]_{ss}\left[-1+2\,\log\left(\frac{\hat{\mu}^{2}}{\bar{m}_{h}^{2}}\right)+\,\log\left(\frac{4}{\tau_{s}}\right)\right]-2\,Q_{d}\,[Y_{d}]_{ss}\,\Big{[}2\,\mathcal{I}_{y}[m_{s}^{2}]+\mathcal{I}[m_{s}^{2}]\Big{]}.$
(208)

Note $\hat{g}_{2}\Delta f_{\begin{subarray}{c}eW\\\
ss\end{subarray}}\rightarrow-\hat{g}_{1}\,\Delta f_{\begin{subarray}{c}eB\\\
ss\end{subarray}}$. In the case of up quarks $\hat{g}_{2}\Delta
f_{\begin{subarray}{c}uW\\\ ss\end{subarray}}\rightarrow\hat{g}_{1}\Delta
f_{\begin{subarray}{c}uB\\\ ss\end{subarray}}$, while in the case of down
quarks $\hat{g}_{2}\Delta f_{\begin{subarray}{c}dW\\\
ss\end{subarray}}\rightarrow-\hat{g}_{1}\Delta f_{\begin{subarray}{c}dB\\\
ss\end{subarray}}$. Here, $s=\\{1,2,3\\}$ sums over the flavors. The Wilson
coefficients are summed with their Hermitian conjugates, and the normalization
is such that $\Delta f_{\begin{subarray}{c}eB\\\ ss\end{subarray}}$ multiplies
${\rm Re}\,C_{\begin{subarray}{c}eB\\\ ss\end{subarray}}$. The remaining
contributions proportional to the SM loop functions are

$\displaystyle[Y_{e}]_{ss}\,\Delta f_{\begin{subarray}{c}eH\\\
ss\end{subarray}}$
$\displaystyle=\hat{e}^{2}\,\frac{Q_{\ell}^{2}}{2}A_{1/2}(\tau_{s}),$
$\displaystyle[Y_{u}]_{ss}\,\Delta f_{\begin{subarray}{c}uH\\\
ss\end{subarray}}$
$\displaystyle=N_{c}\,\hat{e}^{2}\,\frac{Q_{u}^{2}}{2}A_{1/2}(\tau_{s}),$
$\displaystyle[Y_{d}]_{ss}\,\Delta f_{\begin{subarray}{c}dH\\\
ss\end{subarray}}$
$\displaystyle=N_{c}\,\hat{e}^{2}\,\frac{Q_{d}^{2}}{2}A_{1/2}(\tau_{s}),$
$\displaystyle\Delta f_{H\Box}$
$\displaystyle=-\hat{e}^{2}\frac{Q_{\ell}^{2}}{2}A_{1/2}(\tau_{p})-N_{c}\,\hat{e}^{2}\,\frac{Q_{u}^{2}}{2}A_{1/2}(\tau_{r})-N_{c}\,\hat{e}^{2}\,\frac{Q_{d}^{2}}{2}A_{1/2}(\tau_{s})-\frac{1}{2}\,\hat{e}^{2}\,A_{1}(\tau_{W}),$
(209)

and $\Delta f_{HD}=-\Delta f_{H\Box}/4$. Here $p,r,s$ run over $1,2,3$ as
flavor indices. Several of these results have been cross checked against Ref.
Ghezzi:2015vva .

These input parameter scheme dependent corrections perturb
$\Gamma(h\rightarrow\mathcal{A}\mathcal{A})$ as

$\displaystyle\frac{\Delta^{2}\delta\Gamma(h\rightarrow\mathcal{A}\mathcal{A})}{\Delta^{2}\Gamma_{SM}(h\rightarrow\mathcal{A}\mathcal{A})}\simeq\frac{-16\pi^{2}}{\hat{\alpha}_{ew}\,|A_{\mathcal{A}\mathcal{A}}|^{2}}{\rm
Re}A_{\mathcal{A}\mathcal{A}}\left[\langle
h|[C^{(6)}_{i}]|\mathcal{A}\mathcal{A}\rangle^{1}+A^{\mathcal{L}^{(6)}}_{\mathcal{A}\mathcal{A}}\,\Delta
M_{1}+\frac{\tilde{C}_{i}\,\Delta f_{i}}{16\,\pi^{2}}\right].$ (210)

###  G.2 $\delta^{2}\Gamma(h\rightarrow\mathcal{A}\,\mathcal{A})$ geoSMEFT
terms

The ${\cal{O}}(v^{4}/\Lambda^{4})$ terms in the full three-point function are
Hays:2020scx

$\displaystyle\langle
h|\mathcal{A}\mathcal{A}\rangle_{{\cal{O}}(v^{4}/\Lambda^{4})}^{0}$
$\displaystyle=\langle\sqrt{h}^{44}\rangle_{{\cal{O}}(v^{2}/\Lambda^{2})}\,\langle
h|\mathcal{A}\mathcal{A}\rangle^{0}_{\mathcal{O}(\bar{v}^{2}/\Lambda^{2})}+2\,\frac{\bar{v}_{T}\,[\langle
h|\mathcal{A}\mathcal{A}\rangle_{\mathcal{O}(\bar{v}^{2}/\Lambda^{2})}^{0}]^{2}}{\langle
h\mathcal{A}^{\mu\nu}\mathcal{A}_{\mu\nu}\rangle^{0}}$ (211)
$\displaystyle+2\,\langle
h|\mathcal{A}\mathcal{A}\rangle_{\mathcal{O}(\bar{v}^{2}/\Lambda^{2})}^{0}|_{C_{i}^{(6)}\rightarrow
C_{i}^{(8)}}.$

Leading to the $\Delta\delta^{2}$ interference term

$\displaystyle\frac{\Delta\delta^{2}\Gamma(h\rightarrow\mathcal{A}\mathcal{A})}{\Delta^{2}\Gamma_{SM}(h\rightarrow\mathcal{A}\mathcal{A})}\simeq\frac{-16\pi^{2}}{\hat{\alpha}_{ew}\,|A_{\mathcal{A}\mathcal{A}}|^{2}}{\rm
Re}A_{\mathcal{A}\mathcal{A}}\left[\langle\sqrt{h}^{44}\rangle_{{\cal{O}}(v^{2}/\Lambda^{2})}\,A^{\mathcal{L}^{(6)}}_{\mathcal{A}\mathcal{A}}+2(A^{\mathcal{L}^{(6)}}_{\mathcal{A}\mathcal{A}})^{2}+2A^{\mathcal{L}^{(6)}}_{\mathcal{A}\mathcal{A}}|_{C_{i}^{(6)}\rightarrow
C_{i}^{(8)}}\right].$

Here we have used the short-hand notation

$\displaystyle\langle\sqrt{h}^{44}\rangle_{{\cal{O}}(v^{2}/\Lambda^{2})}$
$\displaystyle=\tilde{C}_{H\Box}^{(6)}-\frac{1}{4}\tilde{C}_{HD}^{(6)},$
$\displaystyle\quad C_{HB}^{(6)}$
$\displaystyle\rightarrow\frac{1}{2}C_{HB}^{(8)},$ (213) $\displaystyle
C_{HW}^{(6)}$
$\displaystyle\rightarrow\frac{1}{2}\left(C_{HW}^{(8)}+C_{HW,2}^{(8)}\right),$
$\displaystyle\quad C_{HWB}^{(6)}$
$\displaystyle\rightarrow\frac{1}{2}C_{HWB}^{(8)}.$ (214)

##  Appendix H Past literature results

We have included a significant set of numerical and analytic detail in this
work to aid reproducibility. However, a complete reproduction of the numerical
results requires the following additional literature expressions:

  * •

The two loop QCD corrections for $C_{1}^{H}(\tau_{p})$ is lengthy and directly
given in Ref. Harlander:2005rq . Specifically Eqn. 2.8 in this work.

  * •

The explicit expression for $\delta G_{F}^{(8)}$ can be derived from Appendix
C, Eqn. C.12 in Ref. Hays:2020scx .

  * •

The explicit expression for $C_{H,kin}^{(8)}$ can be derived from Eqn. 3.10 in
Ref. Helset:2020yio .

  * •

The one loop corrections to the $W,Z$ masses in the BFM reported in Appendix
A, Eqns. A.1, A.2 in Ref. Corbett:2021cil .

## References

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  * (2) A. V. Manohar and M. B. Wise, _Modifications to the properties of the Higgs boson_ , _Phys. Lett._ B636 (2006) 107 [hep-ph/0601212].
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