aainstitutetext: School of Physics and Astronomy, University of Minnesota,
Minneapolis, Minnesota 55455, USAbbinstitutetext: Deutsches Elektronen-
Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germanyccinstitutetext:
Institut für Physik, Humboldt-Universität zu Berlin, 12489 Berlin,
Germanyddinstitutetext: Theoretical Physics Department, CERN, 1211 Geneva 23,
Switzerland

# Small instanton-induced flavor invariants and
the axion potential

Ravneet Bedi<EMAIL_ADDRESS>a Tony Gherghetta<EMAIL_ADDRESS>b,c,d
Christophe Grojean<EMAIL_ADDRESS>b Guilherme Guedes
<EMAIL_ADDRESS>b,c Jonathan Kley<EMAIL_ADDRESS>b Pham Ngoc
Hoa Vuong<EMAIL_ADDRESS>
###### Abstract

Small instantons which increase the axion mass due to an appropriate
modification of QCD at a UV scale $\Lambda_{\rm SI}$, can also enhance the
effect of CP-violating operators to shift the axion potential minimum by an
amount, $\theta_{\rm ind}$, proportional to the flavorful couplings in the
SMEFT. Since physical observables must be flavor basis independent, we
construct a basis of determinant-like flavor invariants that arise from
instanton calculations containing the effects of dimension-six CP-odd
operators at the scale $\Lambda_{\cancel{\rm CP}}$. This new basis provides a
more reliable estimate of the shift $\theta_{\rm ind}$, that is severely
constrained by neutron electric dipole moment experiments. In particular, for
the case of four-quark, semi-leptonic and gluon dipole operators, these
invariants are then used to provide improved limits on the ratio of scales
$\Lambda_{\rm SI}/\Lambda_{\cancel{\rm CP}}$ for different flavor scenarios.
The CP-odd flavor invariants also provide a classification of the leading
effects from Wilson coefficients, and as an example, we show that a semi-
leptonic four-fermion operator is subdominant compared to the four-quark
operators. More generally, the flavor invariants, together with an instanton
NDA, can be used to more accurately estimate small instanton effects in the
axion potential that arise from any SMEFT operator.

CERN-TH-2024-022

DESY-24-019

HU-EP-24/05

UMN–TH–4313/24

##  1 Introduction

The Standard Model (SM) has provided a remarkably successful description of
the elementary particles and non-gravitational interactions. However, one
unresolved issue is CP violation which remains to be completely understood. In
particular, there are two sources of CP violation in the Standard Model at the
renormalizable level: a weak CP phase from the CKM matrix and the strong CP
phase $\bar{\theta}=\theta_{\rm QCD}-{\rm arg}\,{\rm det}[Y_{\rm u}Y_{\rm
d}]$, where $\theta_{\rm QCD}$ is the QCD vacuum angle and $Y_{\rm u,d}$ are
the up, down Yukawa coupling matrices. Limits from not observing a neutron
electric dipole moment (EDM) imply an upper bound $\bar{\theta}\lesssim
10^{-10}$ Abel _et al._ (2020). This unexpectedly small value for
$\bar{\theta}$ compares with the order-one weak CP phase and leads to the
well-known strong CP problem. The strong CP problem occurs at the
renormalizable level in the SM Lagrangian and remains rather stable under
renormalization group flow where radiative corrections to $\bar{\theta}$,
induced by the weak CP phase, first appear at seven loops Ellis and Gaillard
(1979); Khriplovich and Vainshtein (1994) (finite contributions appear at four
loops although still highly suppressed Khriplovich (1986)). However, the
problem can be exacerbated by phases arising from higher-dimensional terms in
the UV theory, which can cause order-one shifts in $\bar{\theta}$ thereby
potentially invalidating solutions to the strong CP problem Bigi and Uraltsev
(1991a).

Such a solution to the strong CP problem is the Peccei–Quinn (PQ) mechanism
Peccei and Quinn (1977) where a global $U(1)$ PQ symmetry is spontaneously
broken giving rise to the axion Weinberg (1978); Wilczek (1978). The PQ
symmetry is anomalous because it is explicitly broken by non-perturbative QCD
effects, generating the axion potential with a minimum that exactly cancels
$\bar{\theta}$. However, an underlying assumption of the PQ mechanism is that
the explicit breaking of the global PQ symmetry must be dominated by non-
perturbative QCD effects. Generically, shift-symmetry violating terms (which
may be CP-violating) arising from interactions of the axion with gravity spoil
the axion solution by misaligning the axion potential minimum. These
contributions must therefore be sufficiently suppressed, leading to the so-
called axion quality problem, which can be addressed by one of several
mechanisms in the literature (see for example Refs. Witten (1984); Randall
(1992); Choi (2004); Lillard and Tait (2018); Gavela _et al._ (2019); Hook _et
al._ (2020); Contino _et al._ (2022); Cox _et al._ (2023)).

However, even under the assumption that shift-violating operators in the axion
EFT are sufficiently suppressed, another aspect of the axion quality problem
occurs within the Standard Model Effective Field Theory (SMEFT) where higher-
dimensional CP-violating terms induced at a scale $\Lambda_{\cancel{\rm CP}}$
can shift $\bar{\theta}$ and misalign the axion potential Dine and Seiberg
(1986); Dine (2022); Bedi _et al._ (2022). These new sources of CP violation
depend on the specific UV completion. If QCD is not modified in the UV
completion then the effects on the axion potential and the neutron EDM
constrain $\Lambda_{\cancel{\rm CP}}$ and these effects decouple as
$\Lambda_{\cancel{\rm CP}}$ $\rightarrow\infty$. Alternatively, there has been
renewed interest in the old idea to modify QCD at a UV scale, $\Lambda_{\rm
SI}$ that can increase the axion mass while still solving the strong CP
problem Dimopoulos (1979); Holdom and Peskin (1982); Holdom (1985); Dine and
Seiberg (1986); Flynn and Randall (1987); Rubakov (1997); Fukuda _et al._
(2015); Gherghetta _et al._ (2016); Agrawal and Howe (2018); Csáki _et al._
(2020); Gherghetta and Nguyen (2020). In this case, new CP-violating sources
can also be enhanced by small (UV) instantons whose effects are no longer
suppressed due to the assumed larger QCD coupling at the scale $\Lambda_{\rm
SI}$. The leading contributions to $\bar{\theta}$ then scale as
$\Lambda^{2}_{\rm SI}/\Lambda_{\cancel{\rm CP}}^{2}$ Bedi _et al._ (2022)
which do not necessarily decouple (i.e. when $\Lambda_{\rm
SI},\Lambda_{\cancel{\rm CP}}\to\infty$ with a finite ratio $\Lambda_{\rm
SI}/\Lambda_{\cancel{\rm CP}}$) and give rise to important constraints on CP
violation in certain UV scenarios. For a sufficiently small (although large)
QCD gauge coupling these effects can be computed by performing a one-instanton
calculation which provides the dominant contribution to the action. However,
when the QCD gauge coupling becomes non-perturbative, the dilute instanton gas
approximation breaks down and non-perturbative methods must be used.

The effects of CP-violation arising from higher-dimension operators in an
instanton background, including a four-quark SMEFT operator were computed in
Ref. Kitano and Yin (2021); Demirtas _et al._ (2023); Bedi _et al._ (2022) and
estimated using an instanton naive dimensional analysis (NDA) in Ref. Csáki
_et al._ (2023). However, different CP-violating UV scenarios can give rise to
many other operators de Blas _et al._ (2018); Guedes _et al._ (2023) and
therefore previous estimates of the contributions to $\bar{\theta}$ should be
generalized for the complete list of SMEFT operators. These new contributions
to $\bar{\theta}$ must be independent of the flavor basis, and hence can be
written in terms of flavor-invariant quantities constructed from the Wilson
coefficients. Flavor-invariant quantities allow for an estimation of the
physical consequences of the Wilson coefficients – especially when used
together with other NDA techniques Csáki _et al._ (2023) – prior to any
explicit computation. Indeed, knowledge of CP-violating invariants has
previously been used for physical estimates Ellis and Gaillard (1979);
Khriplovich and Vainshtein (1994); Farrar and Shaposhnikov (1994); Smith and
Touati (2017); Denton and Parke (2019); Yu and Zhou (2022). This is
particularly relevant in instanton calculations where the computation can
become quite cumbersome. Besides providing an order parameter to estimate CP-
violating physical effects, CP-odd invariants also provide selection rules on
the contribution from a particular Wilson coefficient, such as the number of
Yukawa coupling insertions or loop factor suppressions. For example, we will
show that semi-leptonic operators generate a $\overline{\theta}$ but are
suppressed by one extra loop-order and an extra lepton Yukawa factor compared
with four-quark operators. Furthermore, knowing the number of up, down and
lepton Yukawa couplings needed to construct the invariants, we will be able to
classify the leading contributions for arbitrary Wilson coefficients. The use
of flavor invariants also helps to explain the size of the instanton effects
in different flavor scenarios. Moreover, most computations in the literature
have focused on the limit of completely diagonal Yukawa matrices but, with the
help of flavor invariants, the extension to general flavorful matrices becomes
simpler.

Due to the CP-nature of $\bar{\theta}$, the induced $\bar{\theta}$ in the
presence of different SMEFT operators can be parameterized via the CP-
violating invariants introduced in Refs. Bonnefoy _et al._ (2022, 2023a),
where a basis of CP-odd flavor invariants was proposed at leading order in the
corresponding Wilson coefficient. The invariants introduced were all
constructed as the imaginary part of a trace of flavorful matrices. However,
as will be made clear throughout this work, we find that the instanton
calculations can be more directly captured by a different basis, built out of
determinant-like structures, which were first pointed out in Appendix F of
Ref. Bonnefoy _et al._ (2022). Indeed, we find that the instanton computations
of the enhanced $\bar{\theta}$ directly result in quantities proportional to
invariants in our new basis, instead of a complicated combination of CP-even
and -odd invariants if the results were projected onto the trace-like basis.
While any CP-odd quantity can clearly be projected into both bases, the
invariants built in this paper allow for an immediate estimation of the CP-
violating effects in the SMEFT, in the presence of instanton backgrounds.

While the new flavor invariants introduced in this paper improve the estimate
of $\bar{\theta}$ induced in the SMEFT, we will also perform the detailed
computation of $\bar{\theta}$ in the presence of small instantons, using the
one-instanton (or dilute instanton gas) approximation, to explicitly show how
the new invariants appear and more naturally describe physical processes that
arise from instanton computations. In particular, we generalize previous
results on the insertion of the CP-violating four-quark operator ${\cal
O}^{(1)}_{\rm quqd}$ and calculate the effects of the semi-leptonic, ${\cal
O}^{(1)}_{\rm lequ}$ and gluon dipole, ${\cal O}_{\rm dG}$ operators. These
results can then be used to translate the stringent upper bound on
$\bar{\theta}$ into limits on the scale of new physics generating these
operators. For the leading order operators, such as the four-quark and gluon
dipole operators, we find that $\Lambda_{\cancel{\rm CP}}\gtrsim
10^{6}\,\Lambda_{\rm SI}$, assuming a minimally flavor violating (MFV)
scenario for the SMEFT couplings. Under these same flavor assumptions, the
loop suppressed contributions, arising from ${\cal O}^{(1)}_{\rm lequ}$, lead
to the weaker constraint $\Lambda_{\cancel{\rm CP}}\gtrsim
10^{4}\,\Lambda_{\rm SI}$. The bounds become more stringent if there is no
flavor structure in the Wilson coefficients, such as the anarchic flavor
scenario. In this case, assuming all the Wilson coefficients are order one, we
obtain $\Lambda_{\cancel{\rm CP}}\gtrsim 10^{11}\,\Lambda_{\rm SI}$ for the
four-quark operator ${\cal O}^{(1)}_{\rm quqd}$, $\Lambda_{\cancel{\rm
CP}}\gtrsim 10^{8}\,\Lambda_{\rm SI}$ for the gluon dipole operator ${\cal
O}_{\rm dG}$ and $\Lambda_{\cancel{\rm CP}}\gtrsim 10^{7}\,\Lambda_{\rm SI}$
for the semi-leptonic operator ${\cal O}^{(1)}_{\rm lequ}$. As a non-trivial
check, we also verify the expected independence of the renormalization scheme
as well as the cancellation of divergences in the renormalized effective
theory that arise from loop integrals. To arrive at this cancellation, one has
to include the appropriate counterterms which lead to the renormalization
group equations (RGEs) of the SMEFT.

The paper is organized as follows. In Section 2, we introduce the determinant-
like invariants, showing that they form a basis of CP-violating invariants,
and then build such a basis for certain SMEFT operators. The instanton
calculations are presented in Section 3 where we show explicitly that the
determinant-like invariants appear directly from the topological
susceptibility computation; furthermore, we also explore why the determinant
structures appear in this sort of calculation and the information it contains
prior to any actual computation. In Section 4, we compute the enhanced
$\bar{\theta}$ and obtain new bounds on the ratio between the small-instanton
scale $\Lambda_{\rm SI}$ and the CP-violating scale $\Lambda_{\cancel{\rm
CP}}$ . We conclude and suggest future directions for this work in Section 5.
The Appendices review background information and contain further details of
the calculation. In Appendix A, we introduce our conventions and the relevant
SMEFT operators. A complete basis of determinant-like invariants is given in
Appendix B, while in Appendix C we briefly review the basics of instanton
calculations that are relevant for this paper. Finally, in Appendix D we
perform the integrals over collective coordinates used throughout the main
text; in particular, we verify the cancellation of divergences with the
appropriate counterterms, which is a new result obtained from an instanton
calculation.

##  2 Flavor invariants featuring $\theta_{\text{QCD}}$

Perturbative CP violation in the SM induced by the CKM phase is an intricate
collective effect that can only be properly described by a combination of
Lagrangian parameters. The most effective way to capture this effect in the SM
is by using the Jarlskog invariant $J_{4}$, defined as Jarlskog (1985a, b);
Bernabeu _et al._ (1986)

$J_{4}=\text{Im}\left(\text{Tr}\left[X_{\rm u},X_{\rm d}\right]^{3}\right)\,,$
(2.1)

where $X_{\rm u,d}^{\vphantom{\dagger}}\equiv Y^{\vphantom{\dagger}}_{\rm
u,d}Y_{\rm u,d}^{\dagger}$. The quantity $J_{4}$ captures, in a flavor basis-
invariant way (c.f. Table 1), the single physical phase that appears in the
renormalizable part of the SM Lagrangian and hence is the order parameter of
CP violation in the SM. Here, we have used the fact that $U(3)^{5}\equiv
U(3)_{\rm Q}\times U(3)_{\rm u}\times U(3)_{\rm d}\times U(3)_{\rm L}\times
U(3)_{\rm e}$ is the largest possible flavor group allowed by the SM fermion
kinetic terms and is only broken by the SM Yukawa couplings and global
anomalies. The Lagrangian can be formally made invariant under this symmetry
by promoting the Yukawa couplings to spurions transforming under $U(3)^{5}$ as
given in the Table 1.

| $U(3)_{\rm Q}$ | $U(3)_{\rm u}$ | $U(3)_{\rm d}$ | $U(3)_{\rm L}$ | $U(3)_{\rm e}$
---|---|---|---|---|---
$e^{i\theta_{\text{QCD}}}$ | $\mathbf{1}_{+6}$ | $\mathbf{1}_{-3}$ | $\mathbf{1}_{-3}$ | $\mathbf{1}_{0}$ | $\mathbf{1}_{0}$
$Y_{\rm u}$ | $\mathbf{3}_{+1}$ | $\mathbf{\bar{3}}_{-1}$ | $\mathbf{1}_{0}$ | $\mathbf{1}_{0}$ | $\mathbf{1}_{0}$
$Y_{\rm d}$ | $\mathbf{3}_{+1}$ | $\mathbf{1}_{0}$ | $\mathbf{\bar{3}}_{-1}$ | $\mathbf{1}_{0}$ | $\mathbf{1}_{0}$
$Y_{\rm e}$ | $\mathbf{1}_{0}$ | $\mathbf{1}_{0}$ | $\mathbf{1}_{0}$ | $\mathbf{3}_{+1}$ | $\mathbf{\bar{3}}_{-1}$
Table 1: Flavor transformation properties of $\theta_{\rm QCD}$ and the Yukawa
coupling matrices $Y_{\rm u,d,e}$. The subscripts of the $SU(3)$
representations denote the charge under the $U(1)$ part of the flavor
symmetry.

Following the same logic, Refs. Bonnefoy _et al._ (2022, 2023a) constructed
the flavor invariants which capture new CP-violating phases present at lowest
order in the SMEFT parametrization of UV physics. In Appendix A, we briefly
introduce the SMEFT and the relevant operators which will be considered
throughout our work. The construction of these invariants specifically adopted
the philosophy of the Jarlskog invariant in Eq. (2.1) in the sense that they
are built out of traces of flavorful matrices. Similarly, we can construct an
invariant for the QCD theta angle, $\theta_{\rm QCD}$. Using the charges
introduced in Table 1, the flavor invariant given by111 For instance, the
correction to the axion mass in the SM is proportional to
$K_{\theta}=\text{Re}[e^{-i\theta_{\text{QCD}}}\det(Y_{\rm u}Y_{\rm
d})]=\cos\bar{\theta}$, leading to the well-known cosine potential Marsh
(2016); Di Luzio _et al._ (2020), whereas the linear term of the axion
potential generated via non-perturbative QCD effects in the SM is proportional
to $J_{\theta}=-\sin\bar{\theta}$.

$J_{\theta}=\text{Im}[e^{-i\theta_{\text{QCD}}}\det(Y_{\rm u}Y_{\rm d})]\,,$
(2.2)

captures the _non_ -perturbative source of CP violation in the SM Lagrangian,
specifically $\bar{\theta}=\theta_{\rm QCD}-{\rm arg}\,{\rm det}[Y_{\rm
u}Y_{\rm d}]$. Indeed, contributions to the $\theta$ potential in the presence
of an instanton background gives the following dependence ’t Hooft (1976);
Flynn and Randall (1987):

$V\left(\theta_{\text{QCD}},Y_{\rm u},Y_{\rm d}\right)\propto
e^{-i\theta_{\text{QCD}}}\prod_{i=1}^{3}\hat{y}_{{\rm u},i}\,\hat{y}_{{\rm
d},i}\,,$ (2.3)

where $\hat{y}_{{\rm u},i},\,\hat{y}_{{\rm d},i}$ are the Yukawa matrix
eigenvalues and $i$ labels the quark flavors. Later, the eigenvalues will
sometimes be referred to by their particle name, i.e. for instance
$\hat{y}_{{\rm u},3}=y_{\rm t}$. This result does not appear to be flavor
invariant. However, the result can be reproduced from $J_{\theta}$ or
$K_{\theta}$ (depending on the CP parity of the contribution to $V$ that is
calculated) by expanding the invariants in the limit of diagonal SM Yukawa
matrices. This suggests that the flavor invariants can appear directly in the
instanton calculations provided general flavorful couplings are used in the
computation.

In this paper, our goal is to consider the contribution of SMEFT operators to
the topological susceptibility in the presence of (small) instantons. We begin
by considering the effect of the four-fermion operator $\mathcal{O}_{\rm
quqd}^{(1)}=\bar{Q}u\bar{Q}d$. The effect of inserting this effective operator
in an instanton background to the topological susceptibility, $\chi_{\rm
quqd}$, or equivalently the potential, was calculated in Ref. Bedi _et al._
(2022) which found, in the limit of diagonal SM Yukawa couplings,222The CKM
matrix is assumed to be unity here, which is possible in the SM below the
$W$-boson mass, since all effects of the CKM matrix can be put into effective
operators. a contribution proportional to

$V\left(\theta_{\text{QCD}},Y_{\rm u},Y_{\rm d},C_{{\rm
quqd}}^{(1)}\right)\propto
e^{-i\theta_{\text{QCD}}}\frac{\prod_{k=1}^{3}\hat{y}_{{\rm u},k}\hat{y}_{{\rm
d},k}}{\hat{y}_{{\rm u},i}\hat{y}_{{\rm d},j}}c_{ij}\,,$ (2.4)

where $c_{ij}$ captures the contribution from the two possible flavor
structures $C_{{\rm quqd},iijj}^{(1)}$ or $C_{{\rm quqd},ijji}^{(1)}$ and $k$
labels the six entries of the diagonal Yukawa matrices. The proportionality
factor in Eq. (2.4) depends on the details of the instanton calculation and
these factors will be derived in Section 3.

It is not immediately apparent that Eq. (2.4) is related in any way to the
trace-like invariants introduced in Ref. Bonnefoy _et al._ (2022). As
mentioned in the Introduction, determinant-like invariants are much better
suited to describe instanton calculations. Respecting the charge assignments
introduced in Table 1, we can build the following simplest leading order (in
the EFT power counting) invariant

$\mathcal{I}(C_{\rm
quqd}^{(1,8)})=\text{Im}\left[e^{-i\theta_{\text{QCD}}}\epsilon^{ABC}\epsilon^{abc}\epsilon^{DEF}\epsilon^{def}Y_{{\rm
u},Aa}^{\phantom{(}}Y_{{\rm u},Bb}^{\phantom{(}}C_{{\rm
quqd},CcDd}^{(1,8)}Y_{{\rm d},Ee}^{\phantom{(}}Y_{{\rm
d},Ff}^{\phantom{(}}\right]\,,$ (2.5)

which contains the Wilson coefficient $C_{{\rm quqd},ijkl}^{(1,8)}$ and we sum
over repeated indices. Note that this exact invariant had already been
proposed in Appendix F of Ref. Bonnefoy _et al._ (2022). In the limit of
diagonal Yukawa couplings and vanishing $\theta_{\text{QCD}}$ (assumed in Ref.
Bedi _et al._ (2022)), Eq. (2.5) can be expanded as

$\mathcal{I}(C_{\rm quqd}^{(1,8)})=4\left(\prod_{k=1}^{3}\hat{y}_{{\rm
u},k}\hat{y}_{{\rm d},k}\right)\sum_{i,j=1}^{3}\frac{\mathrm{Im}[C_{\rm
quqd}^{(1,8)}]_{iijj}}{\hat{y}_{{\rm u},i}\hat{y}_{{\rm d},j}}\,,$ (2.6)

which matches the result obtained in Eq. (2.4) and explains the flavor
structure appearing in the instanton contribution. Similarly, the $C_{{\rm
quqd},ijji}^{(1,8)}$ flavor structure arises if we consider a second invariant
where the indices $C$ and $D$ are interchanged.

At this point several questions arise regarding these determinant-like
invariants: why do they seem to more naturally appear in instanton
calculations? How can their knowledge help in these computations? Can we
construct similar determinant-like invariants for all effective operators and
how do they appear in instanton calculations? Furthermore, one should also
connect with the previously constructed trace-like basis of invariants: can a
complete basis be built out of the determinant-like invariants and how do they
relate with the previously constructed basis? While the former questions will
be addressed in detail in Section 3, the remainder of Section 2 will answer
the latter questions regarding the construction of a determinant-like basis
suitable for instanton calculations.

###  2.1 A basis of determinant-like flavor invariants

In this section, we will discuss how one can, in principle, construct a
complete basis of flavor invariants for all SMEFT operators that are suitable
for instanton calculations featuring $\theta_{\text{QCD}}$. By complete, we
mean that the basis captures all CP-violating effects from UV phases in the
Wilson coefficients of SMEFT operators. Hence, we do not include opportunistic
effects where the interference of the real part of a Wilson coefficient with
the CKM phase induces CP violation, as previously considered in Ref. Bonnefoy
_et al._ (2023a). We also want to emphasize that we only work at leading order
in the EFT, i.e., all flavor invariants will be linear in the Wilson
coefficients.

To test our set for completeness we will use the transfer matrix method
introduced in Ref. Bonnefoy _et al._ (2022). The linearity of the invariants
in the Wilson coefficients implies that there is a linear map between the
flavor invariants and the entries of the Wilson coefficients $C^{(6)}$

${\cal I}_{a}^{\vphantom{(}}\left(C^{(6)}\right)={\cal
T}_{ai}^{\vphantom{(}}\,\vec{C}_{i}^{(6)}\,,$ (2.7)

where we have defined the transfer matrix ${\cal T}$ and the vector of Wilson
coefficients
$\vec{C}_{i}^{(6)}=\left(\left(\text{Re}\,C^{(6)}\right)_{1},\left(\text{Re}\,C^{(6)}\right)_{2},\dots,\left(\text{Im}\,C^{(6)}\right)_{1},\left(\text{Im}\,C^{(6)}\right)_{2},\dots\right)$
which is a list of the real and imaginary parts of the entries of the Wilson
coefficients. The index $a$ ranges from 1 to the total number of independent
flavor invariants built out of $C^{(6)}$ and the index $i$ ranges from 1 to
the total number of parameters in the theory which cannot be removed by field
redefinitions and that can appear in an interference amplitude with the SM.
The transfer matrix has a block-diagonal form ${\cal T}=\left({\cal T}^{R}\
{\cal T}^{I}\right)$ where the block ${\cal T}^{R}$ will be ignored because it
captures the interference of the real part of the Wilson coefficients with the
phases of the SM. To check if the set captures all sufficient and necessary
conditions of CP violation at the leading order in the EFT, we simply check if
the block ${\cal T}^{I}$ of the transfer matrix has full rank, i.e., if the
rank equals the number of phases which can interfere with the SM (see Ref.
Bonnefoy _et al._ (2022), where the maximal ranks are given for all operators
in the Warsaw basis). Note also that, in this paper, we do not consider Yukawa
matrices with any special values, e.g. degenerate masses, zero masses or
texture zeros in the CKM matrix, that enlarge the flavor symmetry of the SM
left unbroken by the Yukawa couplings.

#####  $\mathbf{{\cal O}_{\rm uH}}$ operator:

As a first example for building a complete basis with determinant-like
invariants, we consider the higher-dimensional Yukawa interactions of the up-
type quarks, ${\cal O}_{\rm uH}=|H|^{2}\bar{Q}\widetilde{H}u$. This requires
constructing an object that simultaneously removes the $U(1)$ transformations
of $e^{-i\theta_{\text{QCD}}}$ (which appears in instanton calculations) and
is invariant under the remaining non-Abelian part of the flavor symmetry,
while at the same time being linear in the Wilson coefficients. Following the
previous discussion, the simplest flavor invariant object that fulfills all
these requirements is

$\text{Im}\left[e^{-i\theta_{\text{QCD}}}\epsilon^{IJK}\epsilon^{ijk}Y_{{\rm
u},Ii}Y_{{\rm u},Jj}C_{{\rm uH},Kk}\det Y_{\rm d}\right]\,,$ (2.8)

where the rephasings of the Yukawa couplings and the Wilson coefficients
precisely cancel those of $e^{-i\theta_{\text{QCD}}}$ and the determinant-like
structure of the Levi-Civita symbols allows the construction of
$SU(3)$-invariant structures. Starting from the form in Eq. (2.8), we can now
systematically construct flavor invariants that can capture all phases in the
Wilson coefficient $C_{\rm uH}$ by using the matrices $X_{\rm
u,d}^{\vphantom{\dagger}}=Y^{\vphantom{\dagger}}_{\rm u,d}Y_{\rm
u,d}^{\dagger}$, transforming in the adjoint of $SU(3)_{\rm Q}$, to project
out different entries of the Wilson coefficients.

With the help of the transfer matrix method, one can check that a set of
flavor invariants which captures all the sources of CP violation for the
operator ${\cal O}_{\rm uH}$, for $J_{4}=J_{\theta}=0$, is

$\begin{split}&{\cal I}_{0000}(C_{\rm uH}),\,{\cal I}_{1000}(C_{\rm
uH}),\,{\cal I}_{0100}(C_{\rm uH}),\,{\cal I}_{1100}(C_{\rm uH}),\,{\cal
I}_{0110}(C_{\rm uH}),\\\ &\qquad\quad{\cal I}_{2200}(C_{\rm uH}),\,{\cal
I}_{0220}(C_{\rm uH}),\,{\cal I}_{1220}(C_{\rm uH}),\,{\cal I}_{0122}(C_{\rm
uH})\,,\end{split}$ (2.9)

where we have defined

${\cal I}_{abcd}(C_{\rm
uH})\equiv\text{Im}\left[e^{-i\theta_{\text{QCD}}}\epsilon^{IJK}\epsilon^{ijk}Y_{{\rm
u},Ii}^{\phantom{d}}Y_{{\rm u},Jj}^{\phantom{d}}\left(X_{\rm u}^{a}X_{\rm
d}^{b}X_{\rm u}^{c}X_{\rm d}^{d}C_{\rm uH}\right)_{Kk}\det Y_{\rm
d}\right]\,.$ (2.10)

#####  $\mathbf{{\cal O}_{\rm quqd}^{(1,8)}}$ operators:

After discussing this simple complete example for an operator that only
contains 9 CPV phases, we next return to the four-fermion operator ${\cal
O}_{\rm quqd}^{(1)}$, and its $SU(3)$ adjoint form ${\cal O}_{\rm
quqd}^{(8)}$, that appeared in the previous section. A complete invariant
basis can also be built for this operator by defining the following two
structures

$\begin{split}{\cal
A}_{a_{2},b_{2},c_{2},d_{2}}^{a_{1},b_{1},c_{1},d_{1}}(C_{\rm
quqd}^{(1,8)})&=\text{Im}\left[e^{-i\theta_{\text{QCD}}}\epsilon^{ABC}\epsilon^{abc}\epsilon^{DEF}\epsilon^{def}Y^{\phantom{\dagger}}_{{\rm
u},Aa}Y^{\phantom{\dagger}}_{{\rm u},Bb}\left(X_{\rm u}^{a_{1}}X_{\rm
d}^{b_{1}}X_{\rm u}^{c_{1}}X_{\rm
d}^{d_{1}}\right)_{C}^{\hphantom{C}C^{\prime}}\right.\\\
&\qquad\quad\left.\times\,C_{{\rm
quqd},C^{\prime}cD^{\prime}d}^{(1,8)}\left(X_{\rm u}^{a_{2}}X_{\rm
d}^{b_{2}}X_{\rm u}^{c_{2}}X_{\rm
d}^{d_{2}}\right)_{D}^{\hphantom{D}D^{\prime}}Y^{\phantom{\dagger}}_{{\rm
d},Ee}Y^{\phantom{\dagger}}_{{\rm d},Ff}\right]\,,\\\ {\cal
B}_{a_{2},b_{2},c_{2},d_{2}}^{a_{1},b_{1},c_{1},d_{1}}(C_{\rm
quqd}^{(1,8)})&=\text{Im}\left[e^{-i\theta_{\text{QCD}}}\epsilon^{ABC}\epsilon^{abc}\epsilon^{DEF}\epsilon^{def}Y^{\phantom{\dagger}}_{{\rm
u},Aa}Y^{\phantom{\dagger}}_{{\rm u},Bb}\left(X_{\rm u}^{a_{1}}X_{\rm
d}^{b_{1}}X_{\rm u}^{c_{1}}X_{\rm
d}^{d_{1}}\right)_{D}^{\hphantom{D}C^{\prime}}\right.\\\
&\qquad\quad\left.\times\,C_{{\rm
quqd},C^{\prime}cD^{\prime}d}^{(1,8)}\left(X_{\rm u}^{a_{2}}X_{\rm
d}^{b_{2}}X_{\rm u}^{c_{2}}X_{\rm
d}^{d_{2}}\right)_{C}^{\hphantom{C}D^{\prime}}Y^{\phantom{\dagger}}_{{\rm
d},Ee}Y^{\phantom{\dagger}}_{{\rm d},Ff}\right]\,.\end{split}$ (2.11)

Here, the index assignment ${\cal A}_{0000}^{0000}(C_{\rm quqd}^{(1,8)})$
corresponds to the invariant in Eq. (2.5) and ${\cal B}_{0000}^{0000}(C_{\rm
quqd}^{(1,8)})$ corresponds to the second invariant mentioned in the last
section, where the indices C and D are interchanged. The operator ${\cal
O}_{\rm quqd}^{(1,8)}$ has 81 phases that can interfere with the dimension-4
terms of the SM. We list a full set of 81 invariants that capture all these
phases in non-perturbative calculations in Appendix B.1.

#####  $\mathbf{{\cal O}_{\rm lequ}^{(1,3)}}$ operators:

One can also build determinant-like invariants for (semi-)leptonic operators.
For instance, the invariants capturing the 27 CP-odd phases of the Wilson
coefficients $C_{\rm lequ}^{(1,3)}$ of the semi-leptonic operator ${\cal
O}_{\rm lequ}^{(1)}=\left(\bar{L}e\right)\left(\bar{Q}u\right)$ and its
$SU(2)$ adjoint form ${\cal O}_{\rm lequ}^{(3)}$ are

${\cal I}_{abcd}^{f}(C_{\rm
lequ}^{(1,3)})\equiv\text{Im}\left[e^{-i\theta_{\text{QCD}}}\epsilon^{IJK}\epsilon^{ijk}Y^{\phantom{\dagger}}_{{\rm
u},Ii}Y^{\phantom{\dagger}}_{{\rm u},Jj}\left(X_{\rm u}^{a}X_{\rm d}^{b}X_{\rm
u}^{c}X_{\rm d}^{d}\right)_{K}^{\hphantom{K}L}\left(Y_{\rm e}^{\dagger}X_{\rm
e}^{f}\right)^{mN}C_{{\rm lequ},NmLk}^{(1,3)}\det Y^{\phantom{\dagger}}_{\rm
d}\right].$ (2.12)

Here again the index assignments for the insertion of $X_{\rm u,d,e}$ are the
same as those in the trace invariants of Ref. Bonnefoy _et al._ (2022) for the
same operators, which are also mentioned in Section B.1.

#####  $\mathbf{{\cal O}_{\rm Hq}^{(1,3)}}$ operators:

As a last example, one can also build invariants for SMEFT operators that are
not charged under the $U(1)$ rephasings of the flavor group. For instance, for
the phases in the Wilson coefficients $C_{\rm Hq}^{(1,3)}$ of the operator
${\cal O}_{\rm
Hq}^{(1)}=\left(H^{\dagger}i\overleftrightarrow{D}_{\mu}H\right)\left(\bar{Q}\gamma^{\mu}Q\right)$
and its $SU(2)$ adjoint form ${\cal O}_{\rm Hq}^{(3)}$, one can write down the
following invariants

${\cal I}_{1100}(C_{\rm Hq}^{(1,3)}),\,{\cal I}_{2200}(C_{\rm
Hq}^{(1,3)}),\,{\cal I}_{1122}(C_{\rm Hq}^{(1,3)})\,,$ (2.13)

where

${\cal I}_{abcd}(C_{\rm
Hq}^{(1,3)})\equiv\text{Im}\left[e^{-i\theta_{\text{QCD}}}\epsilon^{IJK}\epsilon^{ijk}Y_{{\rm
u},Ii}^{\phantom{d}}Y_{{\rm u},Jj}^{\phantom{d}}\left(X_{\rm u}^{a}X_{\rm
d}^{b}X_{\rm u}^{c}X_{\rm d}^{d}\,C_{\rm
Hq}^{(1,3)}\,Y^{\phantom{\dagger}}_{\rm u}\right)_{Kk}\det
Y^{\phantom{\dagger}}_{\rm d}\right].$ (2.14)

Following this procedure, a complete set of flavor invariants capturing all
CP-violating effects at leading order in the SMEFT expansion can be built for
all operators in the Warsaw basis Grzadkowski _et al._ (2010). We present a
few more complete examples in Appendix B.1.

Let us once again emphasize that these new invariants are redundant with
respect to the trace-like invariants introduced in Ref. Bonnefoy _et al._
(2022), as those were already a complete basis of invariants which fully
characterize the CP-violating phases of the theory. Therefore, the
determinant-like invariants _must_ be redundant in regards to the trace-like
invariants. For example, the invariants in Eq. (2.13) can be rewritten as

$\displaystyle{\cal I}_{abcd}(C_{\rm Hq}^{(1,3)})=$
$\displaystyle\text{Im}\left[\left(e^{-i\theta_{\text{QCD}}}\det\left(Y_{\rm
u}Y_{\rm d}\right)\right)\epsilon^{IJK}\epsilon_{IJL}\left(X_{\rm u}^{a}X_{\rm
d}^{b}X_{\rm u}^{c}X_{\rm d}^{d}C_{\rm
Hq}^{(1,3)}\right)_{K}^{\hphantom{K}L}\right]$ $\displaystyle=$ $\displaystyle
2\,\left(J_{\theta}\,R_{abcd}(C_{\rm Hq}^{(1,3)})+K_{\theta}\,L_{abcd}(C_{\rm
Hq}^{(1,3)})\right)~{},$ (2.15)

where $R_{abcd}(C)=\text{Re}\,\left[\text{Tr}\left(X_{\rm u}^{a}X_{\rm
d}^{b}X_{\rm u}^{c}X_{\rm d}^{d}C\right)\right]$ and
$L_{abcd}(C)=\text{Im}\,\left[\text{Tr}\left(X_{\rm u}^{a}X_{\rm d}^{b}X_{\rm
u}^{c}X_{\rm d}^{d}C\right)\right]$, as defined in Ref. Bonnefoy _et al._
(2022). There are also similar relations for other operators such as

${\cal I}_{abcd}(C_{\rm uH})=2\,\left(J_{\theta}\,R_{(a-1)bcd}(C_{\rm
uH}Y_{\rm u}^{\dagger})+K_{\theta}\,L_{(a-1)bcd}(C_{\rm uH}Y_{\rm
u}^{\dagger})\right)~{},$ (2.16) ${\cal I}_{abcd}^{f}(C_{\rm
lequ}^{(1,3)})=2\left(J_{\theta}\,\text{Im}\,A^{f}_{(a-1)bcd}(C_{\rm
lequ}^{(1,3)})+K_{\theta}\,\text{Re}\,A^{f}_{(a-1)bcd}(C_{\rm
lequ}^{(1,3)})\right)~{},$ (2.17)

where $A^{f}_{abcd}(C_{\rm lequ}^{(1,3)})=X_{{\rm e},{ji}}^{f}\left(X_{\rm
u}^{a}X_{\rm d}^{b}X_{\rm u}^{c}X_{\rm d}^{d}\right)_{lk}Y_{{\rm
e},{{mj}}}^{\dagger}Y^{\vphantom{\dagger}}_{{\rm u},{nl}}C_{{\rm
lequ},imkn}^{(1,3)}.$ This procedure allows us to map all determinant-like
invariants directly to the trace invariants of Ref. Bonnefoy _et al._ (2022)
for all operators up to the invariants of the form ${\cal I}_{0bcd}(C_{\rm
uH})$, where inverse Yukawa couplings appear in the trace invariants (c.f.
${\cal I}_{0000}(C_{\rm uH})$ in Eq. (2.16)). We will show in Appendix B.2 how
these latter invariants can also be mapped to the old basis, or alternatively
a different basis from the one given in Ref. Bonnefoy _et al._ (2022) would be
required, including invariants with inverse Yukawa couplings. It also becomes
apparent that ${\cal I}_{abcd}$ captures both the CP-violation due to the
phases introduced by SMEFT operators and that due to the interference between
these SMEFT operators and the SM strong CP phase.

In the next section we will explore why (and when) the determinant-like
invariants are better suited to describe CP violation.

##  3 The interplay of Topological susceptibilities and Flavor invariants

As mentioned earlier, the determinant-like invariants capture the interference
between the strong CP phase $\bar{\theta}$ and the Wilson coefficients of
various SMEFT operators. In particular, we focus on CP-violating SMEFT
operators

$\displaystyle{\cal
L}\supset\dfrac{C_{{}_{\mathcal{O}}}^{ij\cdots}}{\Lambda_{\cancel{\rm
CP}}^{D-4}}\mathcal{O}^{D,\,ij\cdots}~{},$ (3.1)

where $D$ is the mass dimension of the EFT operator $\cal O$ and $i,j,\dots$
are the flavor indices. Since both the parameters $\bar{\theta}$ and
$C_{{}_{\mathcal{O}}}$ are odd under $CP$, the vacuum energy is given by Bigi
and Uraltsev (1991b)

$\displaystyle
V(\bar{\theta},C_{{}_{\mathcal{O}}})=\frac{1}{2}\left(u_{\theta}\,\bar{\theta}\,^{2}+2u_{\theta{\cal
O}}\,\bar{\theta}\,C_{{}_{\cal O}}+u_{{\cal O}}\,C_{{}_{\cal
O}}^{2}\right)~{},$ (3.2)

where $u_{\theta},u_{\theta\,{\cal O}},u_{{\cal O}}$ can be computed in terms
of SM and SMEFT operators. We can arrive at Eq. (3.2) by computing the terms
quadratic in the $\theta$ term and the SMEFT operator $\cal O$ in the
effective action, which implies

$\displaystyle
u_{\theta}\sim\left\langle\left(G\tilde{G}\right)^{2}\right\rangle,\qquad
u_{\theta{\cal O}}\sim\frac{1}{\Lambda_{\cancel{\rm CP}}^{D-4}}\left\langle
G\tilde{G}\,{\cal O}\right\rangle,\qquad u_{{\cal
O}}\sim\frac{1}{\Lambda_{\cancel{\rm CP}}^{2D-8}}\left\langle{\cal
O}^{2}\right\rangle~{}.$ (3.3)

As such, the $u_{{\cal O}}$ term can be neglected compared to the other two
terms. Note that Eq. (3.2) introduces a linear term in $\bar{\theta}$,
implying that the minimum of the energy is shifted. In the PQ mechanism, i.e.,
when $\bar{\theta}$ is promoted to $a/f_{a}$, a dynamical degree of freedom,
the effective value of $\bar{\theta}\equiv\langle a/f_{a}\rangle$ is
determined by Eq. (3.2). To be concrete, the potential can be re-written in
terms of the axion field $a$

$\displaystyle
V(a)=\chi_{{}_{\mathcal{O}}}(0)\dfrac{a}{f_{a}}+\dfrac{1}{2}\chi(0)\left(\dfrac{a}{f_{a}}\right)^{2}~{},$
(3.4)

where we have introduced $\chi(0)$ and $\chi_{{}_{\mathcal{O}}}(0)$ to replace
$u_{\theta}$ and $u_{\theta\,{\cal O}}$, respectively, which can be defined as
Witten (1979); Shifman _et al._ (1980); Pospelov (1998); Pospelov and Ritz
(2005)

$\displaystyle\chi(0)$ $\displaystyle=-i\lim_{k\rightarrow 0}\int
d^{4}x\,e^{ikx}\left\langle
0\left|T\left\\{\dfrac{g^{2}}{32\pi^{2}}G\tilde{G}(x)\,,\,\dfrac{g^{2}}{32\pi^{2}}G\tilde{G}(0)\right\\}\right|0\right\rangle\,,$
(3.5)

known as the QCD topological susceptibility and

$\displaystyle\chi_{{}_{\mathcal{O}}}(0)$ $\displaystyle=-i\lim_{k\rightarrow
0}\int d^{4}x\,e^{ikx}\left\langle
0\left|T\left\\{\dfrac{g^{2}}{32\pi^{2}}G^{a}_{\,\mu\nu}\tilde{G}_{a}^{\,\mu\nu}(x)\,,\,\dfrac{C_{{}_{\mathcal{O}}}^{ij\cdots}}{\Lambda_{\cancel{\rm
CP}}^{D-4}}\mathcal{O}^{D,\,ij\cdots}(0)\right\\}\right|0\right\rangle\,.$
(3.6)

The shift in the axion potential of Eq. (3.4) is then given by

$\displaystyle\theta_{\text{ind}}\equiv-\dfrac{\chi_{{}_{\mathcal{O}}}(0)}{\chi(0)}\,.$
(3.7)

Experimental bounds on the neutron EDM lead to the constraint
$\theta_{\text{ind}}\lesssim 10^{-10}$, which can then be used to obtain
limits on any UV parameters contained in $\chi_{{}_{\mathcal{O}}}(0)$.

Usually, models of axions or axion-like particles (ALPs) are constructed with
a $U(1)$ Peccei–Quinn symmetry in mind, which dictates the ALP couplings to
the SM particles – either directly or in an EFT after integrating out the
heavy modes from the theory. If one allows for some explicit breaking of the
$U(1)$ symmetry,333Even if this explicit breaking is not introduced by hand it
will be generated by quantum gravity effects Georgi _et al._ (1981); Holman
_et al._ (1992); Kamionkowski and March-Russell (1992); Barr and Seckel
(1992); Ghigna _et al._ (1992). responsible for the Nambu–Goldstone boson
nature of the ALP, an axion potential can be generated in ordinary
perturbation theory. The interactions of the ALP with the SM particles,
including those breaking the shift symmetry, can be captured in an EFT in a
relatively model-independent way. In this case, the axion potential can be
determined by calculating the Coleman–Weinberg potential in the ALP EFT
including operators that break the shift symmetry of the ALP explicitly. The
tadpole term of the resulting potential should be proportional to the
invariants presented in Ref. Bonnefoy _et al._ (2023b) that capture all
sources of shift symmetry breaking in the effective theory (see also Ref.
Bonnefoy (2023)).

In this work, we focus on the dimension-six SMEFT operators in the Warsaw
basis Grzadkowski _et al._ (2010) – see also Appendix A for definitions and
conventions. We will show in Section 3.2 how the determinant-like CP-odd
invariants introduced in Section 2 arise in the vacuum-to-vacuum amplitude in
Eq. (3.6) in the presence of the one-(anti)instanton background. To illustrate
how the invariants for different types of SMEFT operators appear in the
instanton calculations, we will focus on the operators $\mathcal{O}_{\rm
quqd}^{(1)},\,\mathcal{O}^{(1)}_{\rm lequ},\,\mathcal{O}_{\rm dG}$. The full
instanton computation to estimate the contribution of the SMEFT operators to
$\chi_{{}_{\mathcal{O}}}(0)$ requires the substitution of the zero mode
profile of fermions and gauge fields, which are reviewed in Appendix C.1, as
well as the evaluation of loop and collective coordinates integrals. These
final steps will be carried out in detail in Appendix D, where the calculated
expressions will then be used in Section 4 to obtain phenomenological bounds
on the CP-odd invariants.

###  3.1 Topological Susceptibilities

In quantum field theory, much of the essential information (e.g. the S-matrix
elements, power spectrum) can be accessed by evaluating correlation functions
of operators. In this section, we schematically evaluate the two-point
correlation functions which are relevant for our calculations in the presence
of a one-(anti)-instanton background by using the path integral formalism
together with the result derived by ’t Hooft ’t Hooft (1976) (see Eq. (C.13)
in Appendix C.).

As an example to illustrate how to evaluate the correlator defined in Eq.
(3.6), we consider a generic dimension-six operator $\mathcal{O}[\varphi_{\rm
I},\varphi]$, where $\varphi_{\rm I}$ are fields with instanton solutions
(e.g. gluon and quark fields), and $\varphi$ denotes the other fields
unrelated to instanton dynamics (e.g. Higgs or lepton fields). The
susceptibility associated with $\cal O$ is given by

$\displaystyle\chi_{{}_{\mathcal{O}}}(0)$ $\displaystyle=-i\lim_{k\rightarrow
0}\int d^{4}x\,e^{ikx}\left\langle
0\left|T\left\\{\dfrac{g^{2}}{32\pi^{2}}G\tilde{G}(x)\,,\,\dfrac{C_{{}_{\mathcal{O}}}}{\Lambda_{\cancel{\rm
CP}}^{2}}\mathcal{O}[\varphi_{\rm
I},\varphi](0)\right\\}\right|0\right\rangle\,,$
$\displaystyle=e^{-i\theta_{\text{QCD}}}\int
d^{4}x_{0}\int\dfrac{d\rho}{\rho^{5}}d_{N}(\rho)\int\prod_{f=1}^{N_{f}}\big{(}\rho\,d\xi^{(0)}_{f}d\bar{\xi}^{(0)}_{f}\big{)}$
$\displaystyle\times\int\mathcal{D}\varphi\,e^{-S_{0}[\varphi]-S_{\rm
int}[\varphi_{\rm I},\varphi]}\int
d^{4}x\dfrac{g^{2}}{32\pi^{2}}G\tilde{G}(x)\times\dfrac{C_{{}_{\mathcal{O}}}}{\Lambda_{\cancel{\rm
CP}}^{2}}\mathcal{O}[\varphi_{\rm
I},\varphi](0)\bigg{|}_{\text{1-(a.-)inst.}}\,,$ (3.8)

where $S_{0}[\varphi]$, $S_{\rm int}[\varphi_{\rm I},\varphi]$ describe the
action containing the $\varphi$ kinetic terms and the interactions between
$\varphi_{\rm I}$ and $\varphi$, respectively. The key points to evaluate Eq.
(3.8) depend on the different treatments of the fields $\varphi_{\rm
I},~{}\varphi$ and their corresponding path integrals. The essential steps are
summarized as follows:

  * •

Fields with instanton solutions $\varphi_{\rm I}$. The field $\varphi_{\rm I}$
is expanded in eigenmodes, where only the zero modes of $\varphi_{\rm I}$ are
replaced by the instanton solutions. In particular, the fermion fields are
expanded as

$\displaystyle\psi_{f}(x)=\sum_{k}\xi_{f}^{(k)}\psi^{(k)}\,;\quad\bar{\psi}_{f}(x)=\sum_{k}\bar{\xi}_{f}^{(k)}\bar{\psi}^{(k)}\,,$
(3.9)

where $\xi_{f}^{(k)}$, $\bar{\xi}_{f}^{(k)}$ are Grassmann variables, $f$ is a
fermion flavor index and the explicit form of $\psi^{(0)}$ is given in Eq.
(C.10). Importantly, the non-zero modes of $\varphi_{\rm I}$ are integrated
out and the path integral over the zero modes is interpreted as an integration
over collective coordinates (see Appendix C for further details). Thus, we can
directly replace the path integral of $\varphi_{\rm I}$ using ’t Hooft’s
result ’t Hooft (1976)

$\displaystyle\int\mathcal{D}\varphi_{\rm I}\,e^{-S_{E}[\varphi_{\rm
I}]}\rightarrow e^{-i\theta_{\text{QCD}}}\int
d^{4}x_{0}\int\dfrac{d\rho}{\rho^{5}}d_{N}(\rho)\int\prod_{f=1}^{N_{f}}\big{(}\rho\,d\xi_{f}^{(0)}d\bar{\xi}_{f}^{(0)}\big{)}\,,$
(3.10)

where $d\xi^{(0)},\,d\bar{\xi}^{(0)}$ are Grassmann integration measures
associated with the fermion zero modes and $d_{N}(\rho)$ is the instanton
density in the $SU(N)$ theory (see Eq. (C.14)) with $\rho$ denoting the
instanton size.

  * •

Fields without instanton solutions $\varphi$. The remaining fields, $\varphi$,
are integrated over without performing the eigenmode expansion. This procedure
can be diagrammatically seen as closing the external legs $\varphi_{\rm I}$
(e.g. quark fields) which are coupled to the instanton vertex by using the
fields $\varphi$ (e.g. Higgs fields), e.g. see Fig. 1(a). The crucial step is
to expand the interaction terms of $e^{-S_{\rm int}[\varphi_{\rm
I},\varphi]}$, obtaining the contributions of the $\varphi_{\rm I}$ zero
modes.

  * •

The remaining steps require substituting the zero mode profiles of
$\varphi_{\rm I}$ (given by Eqs. (C.4), (C.10)) and evaluating the remaining
loop integrals induced by $\varphi$ and collective coordinate integrals. Most
of these calculations are carried out in Appendix D. Finally, the integral
over instanton size, $\rho$, is performed in Section 4, where some UV
scenarios responsible for the instanton dynamics are specified.

Another relevant correlation function is the QCD topological susceptibility
defined in Eq. (3.5). This two-point correlation function has been computed in
the literature, assuming $\chi(0)$ only receives contributions from the
Standard Model, and within the perturbative regime and one-instanton
approximation is given by Callan _et al._ (1978); Flynn and Randall (1987)

$\displaystyle\chi(0)=-3!\left(2K_{\theta}\right)\,i\int\dfrac{d\rho}{\rho^{5}}d_{N}(\rho)\dfrac{1}{(6\pi^{2})^{3}}\,,$
(3.11)

where $K_{\theta}=\text{Re}\left[e^{-i\theta_{\text{QCD}}}\det\left(Y_{\rm
u}Y_{\rm d}\right)\right]$ as introduced in the footnote 1.

###  3.2 Relevance of determinant-like flavor invariants

The appearance of determinant-like structures in instanton computations, which
are more directly parameterized by the invariants introduced in Section 2.1,
is related to the technicalities introduced in the previous section which are
explored in further detail here.

The main point relates to the treatment of the fermionic contributions in the
instanton background. The fermion fields are expanded in their eigenmodes,
such that one can then isolate the zero modes as in Eq. (C.9). The Grassmann
integration measures $d\xi^{(0)}_{f}$ will effectively project out the zero
modes of the fermions, since $\int d\xi^{(0)}_{f}\xi^{(0)}_{f}=1$. In the full
path integral calculation, the following Grassmann integral relations are
useful

$\begin{split}\int d^{3}\xi_{1}d^{3}\xi_{2}~{}e^{\xi_{1}A\xi_{2}}&=\det
A\,,\\\ \int
d^{3}\xi_{1}d^{3}\xi_{2}~{}e^{\xi_{1}A\xi_{2}}\xi_{1}B\xi_{2}&=\frac{1}{2}\epsilon^{i_{1}i_{2}i_{3}}\epsilon^{j_{1}j_{2}j_{3}}A_{i_{1}j_{1}}A_{i_{2}j_{2}}B_{i_{3}j_{3}}\,,\\\
\int
d^{3}\xi_{1}d^{3}\xi_{2}d^{3}\xi_{3}d^{3}\xi_{4}~{}e^{\xi_{1}A\xi_{2}+\xi_{3}B\xi_{4}}\xi_{1}C\xi_{2}&=\frac{1}{2}\epsilon^{i_{1}i_{2}i_{3}}\epsilon^{j_{1}j_{2}j_{3}}A_{i_{1}j_{1}}A_{i_{2}j_{2}}C_{i_{3}j_{3}}\det
B\,,\\\ \int
d^{3}\xi_{1}d^{3}\xi_{2}d^{3}\xi_{3}d^{3}\xi_{4}~{}e^{\xi_{1}A\xi_{2}+\xi_{3}B\xi_{4}}\xi_{1}C\xi_{2}\,\xi_{3}D\xi_{4}&=\frac{1}{4}\epsilon^{i_{1}i_{2}i_{3}}\epsilon^{j_{1}j_{2}j_{3}}A_{i_{1}j_{1}}A_{i_{2}j_{2}}C_{i_{3}j_{3}}\\\
&~{}~{}~{}\times\epsilon^{k_{1}k_{2}k_{3}}\epsilon^{l_{1}l_{2}l_{3}}B_{k_{1}l_{1}}B_{k_{2}l_{2}}D_{k_{3}l_{3}}\,,\end{split}$
(3.12)

where $\xi_{1,\dots,4}$ are three-dimensional Grassmann variables and
$A,B,C,D$ are $3\times 3$ matrices. These identities are at the origin of the
appearance of flavorful objects contracted with Levi-Civita symbols in the
calculation, which we describe as determinant-like.

As such, in computations where the fermion zero modes in an effective operator
are integrated over, the determinant-like invariants introduced in Section 2.1
are better suited at describing CP-violation. This happens not only because
the final result is more easily connected to them, but also because they
capture the full dependency of the final result in terms of the flavorful
couplings of the theory. In other words, the result obtained will be
proportional to a determinant-like invariant times instanton-related
quantities; all the dependence on the rest of the theory (in this case Yukawa
couplings) is captured by the invariant. This would not be the case if we were
considering the trace-like basis of invariants. While the results could be
projected into this basis, this would occur as complicated combinations of the
invariants and with coefficients which include other SM flavor invariants as
we explicitly show in Appendix B.1.

$H$$H$I$\mathcal{O}_{\rm quqd}^{(1)}$ (a) Instanton diagram with an insertion
of the effective operator $\mathcal{O}_{\rm quqd}^{(1)}$.

$H$$H$$H$I$\mathcal{O}_{\rm quqd}^{(1)}$ (b) Instanton diagram with an
insertion of the effective operator $\mathcal{O}_{\rm quqd}^{(1)}$ and
additional Yukawa coupling insertions, corresponding to invariants of higher
order in the Yukawa couplings.

$H$$H$I$\mathcal{O}_{\rm Hq}^{(1)}$ (c) Instanton diagram with an insertion of
a non-chirality-flipping effective operator $\mathcal{O}_{\rm Hq}^{(1)}$.

$\mathcal{O}_{\rm lequ}^{(1)}$$H$$H$$H$I (d) Instanton diagram with an
insertion of an effective operator, giving rise to mixed trace and
determinant-like invariants. One example is the insertion of the effective
operator $\mathcal{O}_{\rm lequ}^{(1)}$.

Figure 1: Examples of instanton diagrams corresponding to invariants discussed
in the text. Here, the gray blob depicts the instanton background that the
fermions (solid lines) are coupled to. The fermion lines are closed via Yukawa
interactions with the Higgs (dashed lines).

Furthermore, a direct relation between diagrammatic contributions and
invariants seems to be clear. Consider the example of the calculation
performed in Ref. Bedi _et al._ (2022), where the topological susceptibility
from an insertion of the effective operator $\mathcal{O}_{\rm quqd}^{(1)}$ was
studied. Diagrammatically this process can be understood as that of Fig. 1(a);
one can observe in the diagram that since all fermion legs of the effective
operator are directly connected to the instanton background, this corresponds
to their zero modes being projected out in the path integral calculations.
Therefore, as previously shown, the resulting contribution will follow a
determinant-like structure on all indices of the Wilson coefficient. Indeed,
as we will prove explicitly in the next section, this diagram gives a
contribution proportional to the introduced invariants ${\cal
A}_{0000}^{0000}(C_{\rm quqd}^{(1)})$ and ${\cal B}_{0000}^{0000}(C_{\rm
quqd}^{(1)})$.

Another interesting contribution which illustrates the previous points is the
contribution from the insertion of the operator $\mathcal{O}_{\rm
lequ}^{(1)}$. The corresponding diagram is that of Fig. 1(d) and only the
quarks emerging from the effective operator have zero modes, as the leptons
are assumed not to be charged under the symmetry group responsible for the
instanton dynamics. Indeed, looking at the constructed invariants
$I_{abcd}^{f}(C_{\rm lequ}^{(1)})$, we see exactly that only the quark indices
are contracted with the anti-symmetric $\epsilon$-structure (determinant-like)
whereas the lepton indices are contracted in a trace-like manner over a matrix
product with a lepton Yukawa coupling.

A final illustrative example is that of rephasing invariant operators such as
$\mathcal{O}_{\rm Hq}^{(1)}$. In this case, even at the lowest order in Yukawa
couplings, one cannot build an invariant where both quark indices are directly
contracted with an $\epsilon$-structure; at most one index is contracted, as
shown in the invariant ${\cal I}_{abcd}(C_{\rm Hq}^{(1)})$. This means that
diagrammatically only one fermionic propagator is directly connected to the
instanton background, that is, only one zero mode is projected out from the
effective operator. This case is illustrated in Fig. 1(c).

We have so far argued that the result of instanton calculations are
proportional to the determinant-like invariants (and no extra flavor
structures). Next, we will show how these patterns arise explicitly.

###  3.3 Four-quark operator

Diagrammatically speaking, having effective operators in the theory allows for
a different way to contract the open fermion legs coupled to the instanton
vertex apart from utilizing mass terms or Yukawa couplings Csáki _et al._
(2020). Let us start by considering the operator $\mathcal{O}_{\rm
quqd}^{(1)}$, which can give rise to the instanton diagram in Fig. 1(a). Since
the SM $SU(2)$ gauge group is unrelated to the instanton dynamics, the $SU(2)$
quark structure can be treated in the same way as a flavor index. As such,
zero modes in the instanton background will not depend on the $SU(2)$ index.
The topological susceptibility, Eq. (3.6), induced by the four-fermion
operator can be calculated as444Note, that all computations are done in
Euclidean space by Wick-rotating the time coordinate everywhere in the
calculations.

$\begin{split}&\chi_{\rm quqd}^{(1)}(0)^{1-\text{inst.}}=-i\lim_{k\rightarrow
0}\int d^{4}x\,e^{ikx}\left\langle
0\left|T\left\\{\frac{1}{32\pi^{2}}G\widetilde{G}(x),\frac{C^{(1)}_{\rm
quqd}}{\Lambda_{\cancel{\rm CP}}^{2}}\mathcal{O}^{(1)}_{\rm
quqd}(0)\right\\}\right|0\right\rangle\,,\\\ &=e^{-i\theta_{\text{QCD}}}\int
d^{4}x_{0}\int\frac{d\rho}{\rho^{5}}d_{N}(\rho)\int\mathcal{D}H\mathcal{D}H^{\dagger}\,e^{-S_{0}[H,H^{\dagger}]}\int\prod_{f=1}^{3}\left(\rho^{2}\,d\xi_{u_{f}}^{(0)}d\xi_{d_{f}}^{(0)}d^{2}\bar{\xi}_{Q_{f}}^{(0)}\right)\\\
&\times e^{\int d^{4}x(\bar{Q}Y_{\rm u}\widetilde{H}u+\bar{Q}Y_{\rm
d}Hd+\text{h.c.})(x)}\frac{1}{32\pi^{2}}\int
d^{4}x\,G\widetilde{G}(x)\left(\frac{C^{(1)}_{\rm quqd}}{\Lambda_{\cancel{\rm
CP}}^{2}}\bar{Q}u\bar{Q}d(0)+\text{h.c.}\right)\,,\end{split}$ (3.13)

where $d^{2}{\bar{\xi}}_{Q_{f}^{\vphantom{1}}}^{(0)}\equiv
d{\bar{\xi}}_{Q_{f}^{1}}^{(0)}\,d{\bar{\xi}}_{Q_{f}^{2}}^{(0)}$ for the two
components of the $SU(2)$ quark doublet. The fermions are expanded in their
eigenmodes (c.f. Eq. (C.9)) and only those terms containing fields with zero
modes in the instanton background have been kept. As can be seen in Eq. (C.10)
this will be $u,d,Q^{\dagger}$ in the instanton background and the conjugates
will contribute to the anti-instanton scenario.

The next step is to expand the exponential of the interacting action of the
fermions and Higgs, such that precisely enough fermion fields appear in the
Grassmann integral to obtain a non-vanishing result. We will also make the
$SU(2)$ indices of all $SU(2)$ doublets explicit in the following calculations
by giving all $SU(2)$ doublets upper case indices. We find

$\displaystyle\chi_{\rm
quqd}^{(1)}(0)^{1-\text{inst.}}=e^{-i\theta_{\text{QCD}}}\int
d^{4}x_{0}\int\frac{d\rho}{\rho^{5}}d_{N}(\rho)\int\mathcal{D}H\mathcal{D}H^{\dagger}\,e^{-S_{0}[H,H^{\dagger}]}\prod_{f=1}^{3}\left(\rho^{2}\,d\xi_{u_{f}}^{(0)}d\xi_{d_{f}}^{(0)}d^{2}\bar{\xi}_{Q_{f}}^{(0)}\right)$
$\displaystyle\times\int
d^{4}x_{1}d^{4}x_{2}d^{4}x_{3}d^{4}x_{4}\,\frac{1}{4!}\left[\sum_{\begin{subarray}{c}\text{perm.
over}\\\ \text{fermion
fields}\end{subarray}}\bar{\xi}_{Q_{i_{1}}^{I}}^{(0)}(\bar{\psi}^{(0)}Y_{{\rm
u},i_{1}j_{1}}\widetilde{H}^{I}P_{R}\psi^{(0)})(x_{1})\xi_{u_{j_{1}}}^{(0)}\right.$
$\displaystyle\left.\times\bar{\xi}_{Q_{i_{2}}^{J}}^{(0)}(\bar{\psi}^{(0)}Y_{{\rm
u},i_{2}j_{2}}\widetilde{H}^{J}P_{R}\psi^{(0)})(x_{2})\xi_{u_{j_{2}}}^{(0)}\quad\bar{\xi}_{Q_{k_{1}}^{K}}^{(0)}(\bar{\psi}^{(0)}Y_{{\rm
d},k_{1}l_{1}}H^{K}P_{R}\psi^{(0)})(x_{3})\xi_{d_{l_{1}}}^{(0)}\right.$ (3.14)
$\displaystyle\left.\times\bar{\xi}_{Q_{k_{2}}^{L}}^{(0)}(\bar{\psi}^{(0)}Y_{{\rm
d},k_{2}l_{2}}H^{L}P_{R}\psi^{(0)})(x_{4})\xi_{d_{l_{2}}}^{(0)}\,\int
d^{4}x\frac{G\widetilde{G}(x)}{32\pi^{2}}\left(\frac{C_{{\rm
quqd},mnop}^{(1)}}{\Lambda_{\cancel{\rm
CP}}^{2}}\bar{\xi}_{Q_{m}^{M}}^{(0)}(\bar{\psi}^{(0)}P_{R}\psi^{(0)})\xi_{u_{n}}^{(0)}\,\epsilon_{MN}\right.\right.$
$\displaystyle\left.\left.\times\bar{\xi}_{Q_{o}^{N}}^{(0)}(\bar{\psi}^{(0)}P_{R}\psi^{(0)})\xi_{d_{p}}^{(0)}\right)(0)\right]\,,$

where the indices $m,M$ of $\xi_{Q_{m}^{M}}^{(0)}$ denote the $SU(2)$ and
flavor indices, respectively, of the zero mode Grassmann vector $\xi_{Q}$,
which in this case is six-dimensional. After integrating over all Grassmann
variables of the zero modes and considering all the permutations over the
fermion fields, we find that the flavor invariants constructed in Section 2
appear explicitly

$\begin{split}&\chi_{\rm
quqd}^{(1)}(0)^{1-\text{inst.}}=\frac{1}{4\Lambda_{\cancel{\rm
CP}}^{2}}\left[e^{-i\theta_{\text{QCD}}}\epsilon^{i_{1}i_{2}m}\epsilon^{j_{1}j_{2}n}Y_{{\rm
u},i_{1}j_{1}}Y_{{\rm u},i_{2}j_{2}}C_{{\rm
quqd},mnop}^{(1)}\epsilon^{k_{1}k_{2}o}\epsilon^{l_{1}l_{2}p}Y_{{\rm
d},k_{1}l_{1}}Y_{{\rm d},k_{2}l_{2}}\right.\\\
&\left.+e^{-i\theta_{\text{QCD}}}\epsilon^{i_{1}i_{2}m}\epsilon^{j_{1}j_{2}n}Y_{{\rm
u},i_{1}j_{1}}Y_{{\rm u},i_{2}j_{2}}C_{{\rm
quqd},onmp}^{(1)}\epsilon^{k_{1}k_{2}o}\epsilon^{l_{1}l_{2}p}Y_{{\rm
d},k_{1}l_{1}}Y_{{\rm d},k_{2}l_{2}}\right]\,\int
d^{4}x_{0}\int\frac{d\rho}{\rho^{5}}d_{N}(\rho)\rho^{6}\\\
&\times\underbrace{\int\mathcal{D}H\mathcal{D}H^{\dagger}\,e^{-S_{0}[H,H^{\dagger}]}\left[\int
d^{4}x_{1}d^{4}x_{2}(\bar{\psi}^{(0)}H_{I}^{\dagger}\epsilon^{IJ}P_{R}\psi^{(0)})(x_{1})\,(\bar{\psi}^{(0)}\epsilon_{JK}H^{K}P_{R}\psi^{(0)})(x_{2})\right]^{2}}_{=\,2!\,\left[\int
d^{4}x_{1}d^{4}x_{2}(\bar{\psi}^{(0)}P_{R}\psi^{(0)})(x_{1})\Delta_{H}(x_{1}-x_{2})\epsilon_{IJ}\epsilon^{JI}(\bar{\psi}^{(0)}P_{R}\psi^{(0)})(x_{2})\right]^{2}\equiv\,2!\,\mathcal{I}^{2}}\\\
&\times\left(\epsilon_{MN}\epsilon^{MN}\bar{\psi}^{(0)}P_{R}\psi^{(0)}\,\bar{\psi}^{(0)}P_{R}\psi^{(0)}\right)(0)\int
d^{4}x\frac{G\widetilde{G}(x)}{32\pi^{2}}\,.\end{split}$ (3.15)

The factor of $1/4$ at the beginning of Eq. (3.15) appears because the
integral over the fermion zero modes is expressed in terms of the Levi-Civita
symbols (see also Eq. (3.12)). The last step is to integrate over the Higgs
field in the Euclidean path integral; using the definition of the Higgs
propagator in position space

$\int{\cal D}H{\cal
D}H^{\dagger}e^{-S_{0}[H,H^{\dagger}]}H_{I}^{\vphantom{\dagger}}(x_{1})H_{J}^{\dagger}(x_{2})=\Delta_{H}(x_{1}-x_{2})\,\delta_{IJ}\,,$
(3.16)

we are left with the integral

$\mathcal{I}=2\int
d^{4}x_{1}d^{4}x_{2}(\bar{\psi}^{(0)}P_{R}\psi^{(0)})(x_{1})\Delta_{H}(x_{1}-x_{2})(\bar{\psi}^{(0)}P_{R}\psi^{(0)})(x_{2})\,,$
(3.17)

multiplying the invariant that we set out to find. After some simplifications,
we finally arrive at

$\begin{split}&\chi_{\rm quqd}^{(1)}(0)^{1-\text{inst.}}=\frac{A_{\rm
quqd}^{(1)}+B_{\rm quqd}^{(1)}}{\Lambda_{\cancel{\rm CP}}^{2}}\int
d^{4}x_{0}\int\frac{d\rho}{\rho^{5}}d_{N}(\rho)\rho^{6}\mathcal{I}^{2}\left(\bar{\psi}^{(0)}P_{R}\psi^{(0)}\bar{\psi}^{(0)}P_{R}\psi^{(0)}\right)(0)\,,\end{split}$
(3.18)

where we have defined

$\begin{split}A_{\rm
quqd}^{(1)}&=e^{-i\theta_{\text{QCD}}}\epsilon^{i_{1}i_{2}m}\epsilon^{j_{1}j_{2}n}Y_{{\rm
u},i_{1}j_{1}}Y_{{\rm u},i_{2}j_{2}}C_{{\rm
quqd},mnop}^{(1)}\epsilon^{k_{1}k_{2}o}\epsilon^{l_{1}l_{2}p}Y_{{\rm
d},k_{1}l_{1}}Y_{{\rm d},k_{2}l_{2}}\,,\\\ B_{\rm
quqd}^{(1)}&=e^{-i\theta_{\text{QCD}}}\epsilon^{i_{1}i_{2}m}\epsilon^{j_{1}j_{2}n}Y_{{\rm
u},i_{1}j_{1}}Y_{{\rm u},i_{2}j_{2}}C_{{\rm
quqd},onmp}^{(1)}\epsilon^{k_{1}k_{2}o}\epsilon^{l_{1}l_{2}p}Y_{{\rm
d},k_{1}l_{1}}Y_{{\rm d},k_{2}l_{2}}\,.\end{split}$ (3.19)

The same calculation can be performed with the anti-instanton solution. In
this case, the non-vanishing contributions will arise from the Hermitian
conjugate terms in the calculation. Furthermore, the winding number $\int
d^{4}x\,G\widetilde{G}(x)$ will flip its sign, which also induces a sign flip
in the exponential of $\theta_{\text{QCD}}$. Therefore, the full one-instanton
and anti-instanton contribution to the topological susceptibility induced by a
four-fermion operator reads

$\begin{split}&\chi_{\rm quqd}^{(1)}(0)=\chi_{\rm
quqd}^{(1)}(0)\big{|}_{1-\text{inst.}}+\chi_{\rm
quqd}^{(1)}(0)\big{|}_{1-\text{a.-inst.}}\\\ &=\frac{1}{\Lambda_{\cancel{\rm
CP}}^{2}}\left(A_{\rm quqd}^{(1)}+B_{\rm quqd}^{(1)}\right)\int
d^{4}x_{0}\int\frac{d\rho}{\rho^{5}}d_{N}(\rho)\rho^{6}\mathcal{I}^{2}\left(\bar{\psi}^{(0)}P_{R}\psi^{(0)}\,\bar{\psi}^{(0)}P_{R}\psi^{(0)}\right)(0)\bigg{|}_{\text{1-inst.}}\\\
&-\frac{1}{\Lambda_{\cancel{\rm CP}}^{2}}\left(A_{\rm quqd}^{(1)}+B_{\rm
quqd}^{(1)}\right)^{*}\int
d^{4}x_{0}\int\frac{d\rho}{\rho^{5}}d_{N}(\rho)\rho^{6}\mathcal{I}^{2}\left(\bar{\psi}^{(0)}P_{L}\psi^{(0)}\,\bar{\psi}^{(0)}P_{L}\psi^{(0)}\right)(0)\bigg{|}_{\text{1-a.-inst.}}\,.\end{split}$
(3.20)

Substituting the explicit form of the fermion zero modes from Eq. (C.10) gives

$\bar{\psi}_{i}^{(0)}P_{R}\psi^{(0)}_{i}\bar{\psi}_{j}^{(0)}P_{R}\psi^{(0)}_{j}\bigg{|}_{\text{1-inst.}}=\dfrac{4\rho^{4}}{\pi^{4}}\dfrac{1}{(x_{0}^{2}+\rho^{2})^{6}}=\bar{\psi}_{i}^{(0)}P_{L}\psi^{(0)}_{i}\bar{\psi}_{j}^{(0)}P_{L}\psi^{(0)}_{j}\bigg{|}_{\text{1-a.-inst.}}\,,$
(3.21)

which in turn leads to the result

$\displaystyle\chi_{\rm quqd}^{(1)}(0)$
$\displaystyle=\frac{2i}{\Lambda_{\cancel{\rm CP}}^{2}}\,\text{Im}\left(A_{\rm
quqd}^{(1)}+B_{\rm quqd}^{(1)}\right)\int
d^{4}x_{0}\int\frac{d\rho}{\rho^{5}}d_{N}(\rho)\rho^{6}\mathcal{I}^{2}\left[\dfrac{4\rho^{4}}{\pi^{4}}\dfrac{1}{(x_{0}^{2}+\rho^{2})^{6}}\right]\,.$
(3.22)

As expected, the final result depends explicitly on the determinant-like
invariants introduced in Eq. (2.11), since using Eq. (3.19) we find

$\text{Im}(A_{\rm quqd}^{(1)})={\cal A}_{0000}^{0000}\left(C_{\rm
quqd}^{(1)}\right)\,,\quad\text{Im}(B_{\rm quqd}^{(1)})={\cal
B}_{0000}^{0000}\left(C_{\rm quqd}^{(1)}\right)\,.$ (3.23)

It is instructive to compare our results with the NDA estimates introduced in
Ref. Csáki _et al._ (2023), which states that the loop factor suppression,
$(4\pi)^{-\alpha}$, can be predicted by

$\alpha=z-2v+2p\,,$ (3.24)

where $z$ is the number of fermion zero modes, $v$ the number of vertices and
$p$ the number of propagators in the instanton calculation. From Fig. 1(a), we
verify that for an insertion of ${\cal O}^{(1)}_{\rm quqd}$ we have
$\alpha=12-10+4=6$. After appropriately substituting the fermion zero modes,
our result in Eq. (D.6) gives a suppression of $1/(45\pi^{6})$. Clearly, the
power of $\pi$ matches the NDA prediction, but the numerical factor is smaller
than $4^{6}$ obtained from NDA. This difference arises from the fact that we
have assumed an unbroken $SU(2)$ group and there are also combinatoric
factors. Taking these effects into account and summing both the instanton and
anti-instanton configurations, the estimation from NDA would predict a
suppression factor $1/(256\pi^{6})$ in Eq. (D.6), which is within one order of
magnitude compared to the full calculation.

###  3.4 Semileptonic four-fermion operator

In Section 2.1, we showed that invariants featuring $\theta_{\text{QCD}}$ can
also be constructed for the semileptonic operator ${\cal O}_{\rm lequ}^{(1)}$,
so it is important to verify whether they arise in instanton calculations. As
the leptons are not charged under the gauge group generating the instantons,
they are not coupled to the instanton vertex directly and should, hence, be
treated perturbatively like the Higgs field in the last section. The
invariants of a semi-leptonic operator can therefore only enter by treating
the leptons perturbatively on top of the instanton background giving the
special functional form to the quark zero modes (c.f. Eq. (C.10)), as we will
now show. The topological susceptibility will be calculated with an insertion
of the operator ${\cal O}_{\rm lequ}^{(1)}$, where the leptons will be kept in
the path integral. As before, we will split off the zero modes and integrate
over the non-zero modes of the quark fields. This gives

$\begin{split}&\chi_{\rm lequ}^{(1)}(0)^{1-\text{inst.}}=-i\lim_{k\rightarrow
0}\int d^{4}xe^{ikx}\left\langle
0\left|T\left\\{\frac{1}{32\pi^{2}}G\widetilde{G}(x),\frac{C^{(1)}_{\rm
lequ}}{\Lambda_{\cancel{\rm CP}}^{2}}\mathcal{O}^{(1)}_{\rm
lequ}(0)\right\\}\right|0\right\rangle\,,\\\ &=e^{-i\theta_{\text{QCD}}}\int
d^{4}x_{0}\int\frac{d\rho}{\rho^{5}}d_{N}(\rho)\int\mathcal{D}H\mathcal{D}H^{\dagger}\mathcal{D}L\mathcal{D}\bar{L}\mathcal{D}e\mathcal{D}\bar{e}\,e^{-S_{0}[H,H^{\dagger}]}\,e^{-S_{0}[L,\bar{L}]}\,e^{-S_{0}[e,\bar{e}]}\\\
&\times\int\prod_{f=1}^{3}\left(\rho^{2}\,d\xi_{u_{f}}^{(0)}d\xi_{d_{f}}^{(0)}d^{2}\bar{\xi}_{Q_{f}}^{(0)}\right)e^{\int
d^{4}x(\bar{Q}Y_{\rm u}\widetilde{H}u+\bar{Q}Y_{\rm e}Hd+\bar{L}Y_{\rm
e}He+\text{h.c.})(x)}\\\ &\times\frac{1}{32\pi^{2}}\int
d^{4}x\,G\widetilde{G}(x)\left(\frac{C^{(1)}_{\rm lequ}}{\Lambda_{\cancel{\rm
CP}}^{2}}\bar{L}e\bar{Q}u(0)+\text{h.c.}\right)\,.\end{split}$ (3.25)

As previously, we will now expand the exponential of the action containing the
fermion and Higgs field. We expand the exponential over the quark Yukawa
couplings in the zero modes as before and neglect the quark non-zero modes.
Then, as is usually done in perturbation theory, we expand the exponential of
the lepton Yukawa interaction order by order in the small Yukawa coupling,
since expanding the exponential to first order will be sufficient to obtain a
a non-vanishing result.

$\begin{split}&\chi_{\rm
lequ}^{(1)}(0)^{1-\text{inst.}}=e^{-i\theta_{\text{QCD}}}\int
d^{4}x_{0}\int\frac{d\rho}{\rho^{5}}d_{N}(\rho)\int\mathcal{D}H\mathcal{D}H^{\dagger}\mathcal{D}L\mathcal{D}\bar{L}\mathcal{D}e\mathcal{D}\bar{e}\,e^{-S_{0}[H,H^{\dagger}]}\\\
&\times\,e^{-S_{0}[L,\bar{L}]}\,e^{-S_{0}[e,\bar{e}]}\prod_{f=1}^{3}\left(\rho^{2}\,d\xi_{u_{f}}^{(0)}d\xi_{d_{f}}^{(0)}d^{2}\bar{\xi}_{Q_{f}}^{(0)}\right)\int
d^{4}x_{1}d^{4}x_{2}d^{4}x_{3}d^{4}x_{4}d^{4}x_{5}\\\
&\times\frac{1}{6!}\left[\sum_{\begin{subarray}{c}\text{perm. over}\\\
\text{fermion
fields}\end{subarray}}\bar{\xi}_{Q_{i_{1}}^{I}}^{(0)}(\bar{\psi}^{(0)}Y_{{\rm
u},i_{1}j_{1}}\widetilde{H}^{I}P_{R}\psi^{(0)})(x_{1})\xi_{u_{j_{1}}}^{(0)}\quad\bar{\xi}_{Q_{i_{2}}^{J}}^{(0)}(\bar{\psi}^{(0)}Y_{{\rm
u},i_{2}j_{2}}\widetilde{H}^{J}P_{R}\psi^{(0)})(x_{2})\xi_{u_{j_{2}}}^{(0)}\right.\\\
&\left.\times~{}\bar{\xi}_{Q_{k_{1}}^{K}}^{(0)}(\bar{\psi}^{(0)}Y_{{\rm
d},k_{1}l_{1}}H^{K}P_{R}\psi^{(0)})(x_{3})\xi_{d_{l_{1}}}^{(0)}\quad\bar{\xi}_{Q_{k_{2}}^{L}}^{(0)}(\bar{\psi}^{(0)}Y_{{\rm
d},k_{2}l_{2}}H^{L}P_{R}\psi^{(0)})(x_{4})\xi_{d_{l_{2}}}^{(0)}\right.\\\
&\left.\times~{}\bar{\xi}_{Q_{k_{3}}^{M}}^{(0)}(\bar{\psi}^{(0)}Y_{{\rm
d},k_{3}l_{3}}H^{M}P_{R}\psi^{(0)})(x_{5})\xi_{d_{l_{3}}}^{(0)}\,\int
d^{4}x\frac{G\widetilde{G}(x)}{32\pi^{2}}\left(\frac{C_{{\rm
lequ},mnop}^{(1)}}{\Lambda_{\cancel{\rm
CP}}^{2}}\left(\bar{L}_{m}^{N}e_{n}\right)\,\epsilon_{NO}\right.\right.\\\
&\left.\left.\times~{}\bar{\xi}_{Q_{o}^{O}}^{(0)}(\bar{\psi}^{(0)}P_{R}\psi^{(0)})\xi_{u_{p}}^{(0)}\right)(0)\right]\int
d^{4}x_{6}\,\left(\bar{e}_{q}Y_{{\rm
e},qr}^{\dagger}H^{\dagger,P}L_{r}^{P}\right)(x_{6})\,.\end{split}$ (3.26)

Note that in comparison to the computation of ${\cal O}_{\rm quqd}^{(1)}$, an
extra down Yukawa coupling is needed to complete the zero mode expansion
because ${\cal O}_{\rm lequ}^{(1)}$ has no down-quark bilinear. In the next
step, we will perform the integration in the path integral over the Higgs and
lepton fields, as well as the zero mode integrals. Using

$\int\mathcal{D}\psi\mathcal{D}\bar{\psi}\,e^{-S_{0}[\psi,\bar{\psi}]}\,\psi_{I}(x_{1})\bar{\psi}_{J}(x_{2})\equiv\Delta_{F}(x_{1}-x_{2})\,\delta_{IJ}\,,$
(3.27)

the lepton fields are contracted to form a loop.555The indices $I,J$ represent
all internal indices, like the flavor and $SU(2)$ gauge group indices. The
resulting expression reads

$\begin{split}&\chi_{\rm
lequ}^{(1)}(0)^{1-\text{inst.}}=\frac{1}{2\Lambda_{\cancel{\rm
CP}}^{2}}\underbrace{e^{-i\theta_{\text{QCD}}}\epsilon^{i_{1}i_{2}m}\epsilon^{j_{1}j_{2}n}Y_{{\rm
u},i_{1}j_{1}}^{\phantom{(}}Y_{{\rm u},i_{2}j_{2}}^{\phantom{(}}C_{{\rm
lequ},opmn}^{(1)}Y_{\vphantom{l}{\rm e},po}^{\dagger}\det Y_{\rm
d}}_{\equiv\,I_{\rm lequ}^{(1)}}\\\ &\times 3!\,\int
d^{4}x_{0}\int\dfrac{d\rho}{\rho^{5}}d_{N}(\rho)\rho^{6}\,\mathcal{I}^{2}\,\int
d^{4}x_{5}d^{4}x_{6}\left(\bar{\psi}^{(0)}P_{R}\psi^{(0)}\right)(x_{5})\Delta_{H}(x_{5}-x_{6})\\\
&\times\tr\Big(P_{R}\Delta_{F}(x_{6}-0)P_{L}\Delta_{F}(0-x_{6})\Big{missing})\left(\bar{\psi}^{(0)}P_{R}\psi^{(0)}\right)(0)\,\epsilon_{OP}\epsilon^{PO}\int
d^{4}x\frac{G\widetilde{G}(x)}{32\pi^{2}}\,.\end{split}$ (3.28)

As before, we add the anti-instanton contribution to obtain the full result,
which leads to the complex conjugate invariants appearing with the opposite
sign in the final result. Thus, the final result is proportional to the
invariant

$\begin{split}\text{Im}\left(I_{\rm
lequ}^{(1)}\right)&=\text{Im}\left[e^{-i\theta_{\text{QCD}}}\epsilon^{i_{1}i_{2}m}\epsilon^{j_{1}j_{2}n}Y_{\vphantom{l}{\rm
u},i_{1}j_{1}}^{\phantom{(}}Y_{\vphantom{l}{\rm
u},i_{2}j_{2}}^{\phantom{(}}C_{{\rm lequ},opmn}^{(1)}Y_{\vphantom{l}{\rm
e},po}^{\dagger}\det Y_{\rm d}\right]={\cal I}_{0000}^{0}\left(C_{\rm
lequ}^{(1)}\right)\,,\end{split}$ (3.29)

that we have defined in Section 2.1 multiplied by a complicated integral. In
Appendix D.2, the integrals in Eq. (3.28) are evaluated; in particular the
integral over the leptonic loop is divergent. We have explicitly verified that
when considering the appropriate renormalized effective field theory, the
counterterms cancel this divergence, as expected. Furthermore, the NDA
estimation of Ref. Csáki _et al._ (2023) also works in this case: following
Eq. (3.24), we expect a suppression of $(4\pi)^{-8}$ which matches the $\pi$
suppression of the result obtained in Eq. (D.19), where the numerical factor
$\simeq 1/(450\pi^{8})$. For this factor, if we take into account the
combinatoric factors, the unbroken $SU(2)$ and the sum over instanton and
anti-instanton configurations, the NDA estimation of $4^{-8}$ becomes
approximately half of the full result.

This analysis can be repeated for all other operators in the SMEFT following
the same procedure. We present calculations for the insertion of the gluon
dipole operator ${\cal O}_{\rm dG}$ in Appendix D.3, that will also be
considered in a phenomenological study in Section 4. For some SMEFT operators,
their leading contribution might not arise from projecting the zero modes out
of all the fermion legs. Indeed, considering non-zero modes of the quarks in
the effective operators is also needed to obtain the invariants with more
powers of Yukawa couplings introduced in Section 2.1. We next discuss the
calculations in these cases.

###  3.5 Higher-order invariants and selection rules

In our calculations we have only considered contributions at leading order
with the least possible power of the Yukawa couplings. However, one could
compute higher-loop diagrams with more powers of Yukawa couplings, which would
be captured by higher-order invariants, such as the diagram depicted in Fig.
1(b). The explicit calculation of higher-loop diagrams works differently than
what was performed in Sections 3.3 and 3.4, because some interactions mix the
zero and non-zero modes of the fermions charged under the instanton group.
Explicitly performing this calculation is beyond the scope of this paper but
we will comment on how these calculations work in principle. Instead of just
including the quark zero modes in the calculations, one would have to include
the non-zero mode interactions in the action as well. These should be treated
perturbatively as was done for the leptons in Section 3.4. As a consequence,
one can no longer simply integrate over the free part of the action containing
just the non-zero modes to remove them from the path integral as done in Eq.
(3.10), since their interactions with the zero modes necessarily appear in the
action as well. Hence, we would have to reevaluate ’t Hooft’s result for the
non-zero mode integration (i.e., remove the factor of $e^{0.292N_{f}}$ from
the instanton density in Eq. (C.15)) and treat the non-zero modes of the
colored fields as perturbations around the instanton background.

In summary, including extra powers of Yukawas from the interacting part of the
action forces us to consider terms mixing zero- and non-zero modes of the
quarks. The contraction of the flavor indices of the non-zero mode fermions in
the Yukawa interactions is done by Kronecker deltas as a result of flavorful
propagators introduced in the perturbative calculation (c.f. the calculation
with the semi-leptonic operator in Section 3.4). This results in the higher-
order invariants introduced in Section 2.1, where the extra Yukawas are
contracted in a matrix product. The indices corresponding to non-zero modes
remain contracted in a determinant-like manner, i.e. via the $\epsilon$
symbol.

In addition, flavor invariants offer an explanation as to why operators
invariant under the flavor-diagonal $U(1)$ quark rephasings (shown in Table 1)
cannot enter through zero mode contributions in instanton calculations. One
example for such an operator is ${\cal O}_{\rm Hq}^{(1)}$, which is invariant
under $Q\to e^{i\alpha_{Q}}Q$, where all flavors are rephased with the same
parameter. Because the operator is rephasing invariant, other flavorful
objects besides the Wilson coefficient are needed to counteract the rephasing
of $e^{-i\theta_{\text{QCD}}}$ that necessarily appears in instanton
calculations. Due to the linearity of the flavor invariants in the Wilson
coefficient, this object can only be constructed by SM Yukawa couplings. There
are two options to construct a flavor invariant given these constraints. The
object counteracting the rephasing of $e^{-i\theta_{\text{QCD}}}$ can either
be $\det Y_{\rm u}Y_{\rm d}$ with the Wilson coefficient appearing in a trace
invariant or a determinant-like invariant where, even at lowest order, the
Wilson coefficient multiplies one of the Yukawa couplings (c.f. Eq.
(2.13)).666These two types of invariants are equivalent as we show explicitly
in Eq. (2.15) for the operator ${\cal O}_{\rm Hq}^{(1)}$ and all arguments
presented here work for both forms. As we have discussed previously, both
traces and matrix products can only appear through propagators in perturbative
calculations of the non-zero modes of quarks around the instanton background.
Hence, the flavor invariants imply a selection rule on all operators that are
invariant under rephasings to only contribute in instanton calculations when
the non-zero modes of its fermions are considered.

As a more general statement, flavor invariants can be used to understand how
the contribution from any SMEFT operator will contract with the flavor
structure of the theory without explicitly doing the path integral
computations, as we have anticipated in Section 3.2. Furthermore, knowing how
the non-vanishing contributions scale and connect with an instanton diagram,
allows to correctly account for the loop factors coming from the zero modes
and from the rest of the perturbative calculation; invariants can therefore
allow for a more refined NDA estimate of the instanton effects in the spirit
of what was done in Ref. Csáki _et al._ (2023).

##  4 Constraints on dimension-six CP-violating operators

The results derived in the previous sections can now be used to place bounds
on the scale $\Lambda_{\cancel{\rm CP}}$ associated with dimension-six CP
violating operators. We will assume that QCD is modified at a scale
$\Lambda_{\rm SI}$ where the one-instanton approximation remains valid and
small instantons induce a shift in $\bar{\theta}$ proportional to
$\Lambda^{2}_{\rm SI}/\Lambda_{\cancel{\rm CP}}^{2}$. Using the determinant-
like invariants arising from the one-instanton calculation, we will then
obtain limits on the ratio $\Lambda_{\rm SI}/\Lambda_{\cancel{\rm CP}}$, under
the assumption that this induced $\bar{\theta}$ in the small instanton
background saturates the experimental bound from the neutron EDM,
$\bar{\theta}\lesssim 10^{-10}$ Abel _et al._ (2020).

The effects from small (UV) instantons are enhanced, provided the gauge
coupling becomes larger in the UV. In this limit, the one instanton
calculation remains consistent and the invariants derived in Section 2 can be
directly used. As such, we can reliably compute the effects of various
dimension-six operators and compare them to the topological susceptibility
$\chi(0)$ shown in Eq. (3.11). There are two UV models that modify QCD UV
dynamics to enhance the topological susceptibility for which the instanton
computation is relevant and these will be briefly reviewed below:

##### Product group models

A natural way to explicitly compute the instanton integrals discussed in
Section 3 is to Higgs an enlarged gauge group containing QCD color. This
modifies the instanton measure $d_{N}(\rho)$ by an exponential factor

$\displaystyle d_{N}(\rho)\to
d_{N}(\rho)\,e^{-2\pi^{2}\rho^{2}\sum|\langle\sigma\rangle|^{2}}~{},$ (4.1)

where the sum extends over all the scalars $\sigma$ that Higgs the gauge
group. This provides a cutoff $\sim 1/|\langle\sigma\rangle|$ for the
instanton integrals. In particular, we consider the product group model
introduced in Refs. Agrawal and Howe (2018); Csáki _et al._ (2020), where the
gauge group $SU(3)_{1}\times SU(3)_{2}\times\dots\times SU(3)_{k}$ is Higgsed
to the diagonal $SU(3)_{c}$ via bifundamental scalars $\sigma$. The gauge
couplings of the individual gauge groups at the UV scale can be chosen to be
larger than the QCD coupling while still in the perturbative regime so that
small instantons are enhanced and instanton computations are applicable.

##### 5D Instantons

Another model where small instantons provide a significant contribution to the
topological susceptibility was studied in Ref. Gherghetta _et al._ (2020).
This involves uplifting the BPST instanton presented in Eq. (C.4) to a compact
extra dimension of size $R$, which modifies the running of the effective gauge
coupling in Eq. (C.16) above the compactification scale $1/R$. The effective
action then becomes

$S_{\rm
eff}\simeq\dfrac{8\pi^{2}}{g^{2}(1/R)}-\frac{R}{\rho}+b_{0}\ln\frac{R}{\rho}~{},$
(4.2)

where $b_{0}$ is the $\beta$-function coefficient of the zero modes and the
linear term $R/\rho$ is due to additional contributions in Eq. (C.11) from the
Kaluza-Klein modes. As such, the instanton measure becomes modified by an
amount

$\displaystyle d_{N}(\rho)\to d_{N}(\rho)\,e^{R/\rho}~{}.$ (4.3)

The dilute instanton gas approximation can then be used to compute the
topological susceptibility in this model by imposing the 5D perturbativity
condition

$\displaystyle\Lambda_{\rm SI}R\lesssim\frac{24\pi^{2}}{g^{2}}~{},$ (4.4)

where $\Lambda_{\rm SI}$ is identified with the cutoff scale of the 5D gauge
theory.

###  4.1 Bounds from induced $\bar{\theta}$

As discussed in Section 3, new sources of CP violation in the SMEFT can induce
a shift in $\bar{\theta}$, which leads to observable effects such as the
neutron EDM. In principle, all the invariants discussed in Section 2.1 and
Appendix B.1 will give contributions to $\bar{\theta}$. However, due to the
different flavor structures, there are only a few invariants that contribute
to leading order.

In the following, we consider three different flavor scenarios to study their
impact on the bounds obtained from the induced $\bar{\theta}$-angle. We will
briefly introduce them here.

  1. 1.

The simplest is the anarchic flavor scenario, in which all Wilson coefficients
have an ${\cal O}(1)$ value. Compared to the SM, this will in particular lead
to large flavor-changing interactions.

  2. 2.

A slightly more restrictive flavor assumption is the MFV Buras _et al._
(2001); D’Ambrosio _et al._ (2002); Isidori and Straub (2012) scenario. As we
have noted earlier, in the SM the only breaking of the $U(3)^{5}$ flavor
symmetry of the fermion kinetic term is due to the SM Yukawa couplings. Taking
the Yukawa couplings to be spurions under this symmetry (c.f. Table 1), makes
the Lagrangian formally invariant under this approximate symmetry. In MFV we
assume that the non-renormalizable operators of the SMEFT follow the same
symmetry scheme. Thus, all SMEFT Wilson coefficients are polynomials in the
Yukawa couplings dictated by the spurious transformations of the Wilson
coefficients under the flavor group.

  3. 3.

Lastly, we consider a Froggatt–Nielsen (FN) scenario Froggatt and Nielsen
(1979) that offers an explanation for the size of the SM lepton and quark
masses as well as the parameters in the CKM matrix. In this scenario, the SM
fields are extended by a complex scalar field $\phi$ which is a singlet under
the SM gauge group. The new scalar field has charge $-1$ under a global $U(1)$
symmetry. Constructing a Lagrangian invariant under the SM gauge group and the
newly postulated $U(1)$ yields

${\cal L}=-\left(\frac{\phi^{\star}}{\Lambda_{\rm
FN}}\right)^{q_{Q_{i}}+q_{u_{j}}}C_{ij}^{u}\bar{Q}_{i}^{\vphantom{u}}\tilde{H}u_{j}^{\vphantom{u}}-\left(\frac{\phi^{\star}}{\Lambda_{\rm
FN}}\right)^{q_{Q_{i}}+q_{d_{j}}}C_{ij}^{d}\bar{Q}_{i}^{\vphantom{d}}Hd_{j}^{\vphantom{d}}-\left(\frac{\phi^{\star}}{\Lambda_{\rm
FN}}\right)^{q_{L_{i}}+q_{e_{j}}}C_{ij}^{e}\bar{L}_{i}^{\vphantom{e}}He_{j}^{\vphantom{e}}\,,$
(4.5)

where the FN charges of the left-handed fermions $Q$, $u^{\dagger}$,
$d^{\dagger}$, $L$, $e^{\dagger}$ are denoted as $q_{Q}$, $q_{u}$, $q_{d}$,
$q_{L}$, $q_{e}$, respectively, $q_{H}=0$, $\Lambda_{\rm FN}$ is the effective
scale where the Froggatt–Nielsen scenario is UV completed and the coefficients
$C_{ij}^{u,d,e}$ are ${\cal O}(1)$ complex numbers. Eventually, the global
$U(1)$ symmetry is broken by the VEV of the complex scalar, which yields
hierarchical Yukawa couplings as powers of
$\lambda=\frac{\langle\phi\rangle}{\Lambda_{\rm FN}}\sim 0.2$ dictated by the
FN charges. One set of charge assignments that can reproduce the SM Yukawa
couplings to large accuracy is

$q_{Q}=\\{3,2,0\\},\quad q_{u}=\\{5,2,0\\},\quad q_{d}=\\{4,3,3\\}~{},$ (4.6)

for the quarks and

$q_{L}=\\{9,5,3\\},\quad q_{e}=\\{0,0,0\\}~{},$ (4.7)

for the leptons. This construction can be extended to the effective operators
of the SMEFT Bordone _et al._ (2020), resulting in hierarchical entries for
the Wilson coefficients.

We begin by identifying the leading order invariants amongst those given in
Section 2. This can be easily achieved by studying the FN scaling of the
invariant with the least number of Yukawa matrices for each operator. Consider
the topological susceptibility of QCD, $\chi(0)\propto K_{\theta}$, e.g.,
which scales as $\propto\lambda^{27}.$ This compares with the SMEFT invariants
in Section 2.1 which scale as

$\displaystyle{\cal I}_{0000}(C_{\rm uH}),\,{\cal A}_{0000}^{0000}(C_{\rm
quqd}^{(1,8)}),\,{\cal B}_{0000}^{0000}(C_{\rm
quqd}^{(1,8)})\propto\lambda^{27}~{},$ (4.8) $\displaystyle{\cal
I}_{1100}(C_{\rm Hq}^{(1,3)}),\,{\cal I}_{0000}^{0}(C_{\rm
lequ}^{(1,3)})\propto\lambda^{33}~{}.$ (4.9)

This scaling helps to determine which invariants are important and hence
phenomenologically the most interesting. For instance, Eqs. (4.8) and (4.9)
indicate that the operators ${\cal O}_{\rm uH}$ and ${\cal O}_{\rm
quqd}^{(1,8)}$ lead to larger effects compared to ${\cal O}_{\rm Hq}^{(1,3)}$
or ${\cal O}_{\rm lequ}^{(1,3)}$, i.e. if the Wilson coefficients are assumed
to be of the same order (up to the appropriate power of the FN parameter
$\lambda$), the contribution of the operator ${\cal O}_{\rm quqd}^{(1,8)}$ (or
${\cal O}_{\rm uH}$) to $\theta_{\rm ind}$ dominates over that of ${\cal
O}_{\rm lequ}^{(1,3)}$ (or ${\cal O}_{\rm Hq}^{(1,3)}$). This can also be
understood from Fig. 1(a) and Fig. 1(d) – the latter figure contains
additional loops and Yukawa couplings, compared to the former figure and the
leading order contribution from $\chi(0)$.

Below, we study in more detail how these invariants contribute to the shift in
the axion potential minimum, $\theta_{\rm ind}$, for two leading-order
operators – ${\cal O}_{\rm quqd}^{(1)}$ as well as the dipole operator ${\cal
O}_{\rm dG}$, and the sub-leading semi-leptonic operator ${\cal O}_{\rm
lequ}^{(1)}$.

For this analysis, the MFV (at leading order) and FN flavor scenarios result
in the same scaling for the Wilson coefficients, which occurs because we are
only considering the contribution from chirality-flipping operators to one
observable. For instance, the scaling of ${\cal O}^{(1)}_{\rm quqd}$ is

$C^{(1)}_{\mathrm{quqd},ijkl}\,\overset{\rm
MFV}{\sim}c_{1}Y_{\mathrm{u},ij}Y_{\mathrm{d},kl}+{\cal O}(Y^{3}_{\rm
u,d})\,,\qquad C^{(1)}_{\mathrm{quqd},ijkl}\,\overset{\rm
FN}{\sim}c_{1,ijkl}\lambda^{q_{Q_{i}}+q_{u_{j}}+q_{Q_{k}}+q_{d_{l}}}\,,$
(4.10)

where $c_{1}$ are ${\cal O}(1)$ coefficients. Since by the FN construction,
Eq. (4.5), $Y_{\rm u}\sim\lambda^{q_{Q}+q_{u}}$ and $Y_{\rm
d}\sim\lambda^{q_{Q}+q_{d}}$, we explicitly see the same scaling in both
scenarios. Therefore, we will only present constraints on
$\Lambda_{\cancel{\rm CP}}$ (for a given $\Lambda_{\rm SI}$) under anarchic
and MFV scenarios for the considered operators in the following.

####  4.1.1 Four-quark operators

For the four-quark operator ${\cal O}_{\rm quqd}^{(1)}$, the topological
susceptibility is computed in Appendix D.1 and the result is given in Eq.
(D.6). Performing the integral over $\rho$ in the product group model
$SU(3)^{k}\to SU(3)_{c}$ and assuming $|\langle\sigma\rangle|=\Lambda_{\rm
SI}$, we obtain

$\displaystyle\theta_{\rm
ind}=\dfrac{16\pi^{2}}{5(b_{0}-6)K_{\theta}}\left({\cal
A}_{0000}^{0000}\left(C_{\rm quqd}^{(1)}\right)+{\cal
B}_{0000}^{0000}\left(C_{\rm quqd}^{(1)}\right)\right)\dfrac{\Lambda_{\rm
SI}^{2}}{\Lambda_{\cancel{\rm CP}}^{2}}~{},$ (4.11)

where $b_{0}=13/2$ for $SU(3)_{1}$ and $b_{0}=21/2$ for $SU(3)_{k}$ . For
$SU(3)_{2},\,\dots\,SU(3)_{k-1}$ , $b_{0}=10$ and we get an additional factor
of $2$ on the RHS of Eq (4.11). In the case of 5D instantons, we obtain

$\displaystyle\theta_{\rm ind}=\dfrac{2}{5K_{\theta}}\left({\cal
A}_{0000}^{0000}\left(C_{\rm quqd}^{(1)}\right)+{\cal
B}_{0000}^{0000}\left(C_{\rm quqd}^{(1)}\right)\right)\dfrac{\Lambda_{\rm
SI}^{2}}{\Lambda_{\cancel{\rm CP}}^{2}}~{}.$ (4.12)

Note that as a consequence of the Higgsed theory, the product group model has
a smooth cutoff on the instanton size $\rho$, which leads to a mild dependence
on the $\beta$ function coefficient, $b_{0}$ in Eq. (4.11).

In contrast, Eq. (4.3) implies that the integral over $\rho$ is dominated by
instantons of size $\rho\sim 1/\Lambda_{\rm SI}$. Therefore, all the
susceptibilities for the 5D model only depend on $\Lambda_{\rm SI}$, up to an
overall factor. This factor cancels when taking the ratio of susceptibilities,
implying that $\theta_{\rm ind}$ is independent of the $\beta$ function
coefficient, $b_{0}$.

Figure 2: Limits on the UV scale $\Lambda_{\cancel{\rm CP}}$ of the four-quark
operator ${\cal O}_{\rm quqd}^{(1)}$ in different flavor scenarios as a
function of the small instanton scale $\Lambda_{\rm SI}$. The shaded regions
are excluded by the non observation of the neutron EDM.

The constraints arising from Eqs. (4.11) and (4.12) are shown in Fig. 2, where
for the product group model, we use $b_{0}=13/2$ since it gives the most
stringent constraints. Note that the same value of $b_{0}$ will be used in
constraining the semi-leptonic and gluon dipole operators. In the anarchic
flavor scenario, we find that $\Lambda_{\cancel{\rm CP}}\gtrsim
10^{10}(10^{11})\,\Lambda_{\rm SI}$ for the 5D (product group) model. On the
other hand, the MFV scenario provides a much weaker constraint –
$\Lambda_{\cancel{\rm CP}}\gtrsim 5\times 10^{5}(10^{6})\,\Lambda_{\rm SI}$
for the 5D (product group) model, which differs exactly by a factor of
$\sim\sqrt{y{\rm{}_{u}}\,y_{\rm d}}$, as indicated by Eq (2.6). This matches
the MFV scenario considered in Ref. Bedi _et al._ (2022), up to an overall
factor due to the Higgs doublet structure. In addition, the invariants help us
to easily incorporate off-diagonal Yukawa couplings which changes the previous
estimate of $\theta_{\rm ind}$ in Ref. Bedi _et al._ (2022) by $\sim 6\%$.

####  4.1.2 Semi-leptonic operator

For the four-fermion operator ${\cal O}_{\rm lequ}^{(1)}$, the susceptibility
is given by Eq. (D.19). In the case of the product group model, performing the
integral over $\rho$ gives

$\displaystyle\theta_{\rm ind}$ $\displaystyle=\dfrac{{\cal
I}^{0}_{0000}(C_{\rm
lequ}^{(1)})}{(b_{0}-6)K_{\theta}}\left[\dfrac{11}{25}+\frac{6}{5}\left(\log\left(\frac{\Lambda_{\cancel{\rm
CP}}}{\Lambda_{\rm SI}}\right)+\gamma_{\rm E}-\log
4\pi\right)+\frac{3}{5}\Psi\left(\frac{b_{0}}{2}-3\right)\right]\frac{\Lambda_{\rm
SI}^{2}}{\Lambda_{\cancel{\rm CP}}^{2}}\,,$ (4.13)

where $\Psi(z)$ is the digamma function, while in the 5D instanton model we
obtain

$\displaystyle\theta_{\rm ind}$ $\displaystyle=\dfrac{{\cal
I}^{0}_{0000}(C_{\rm
lequ}^{(1)})}{8\pi^{2}K_{\theta}}\,\left[\frac{11}{25}+\frac{6}{5}\left(\log\left(\frac{\Lambda_{\cancel{\rm
CP}}}{\Lambda_{\rm SI}}\right)+\gamma_{\rm E}-\log
2\right)\right]\frac{\Lambda_{\rm SI}^{2}}{\Lambda_{\cancel{\rm CP}}^{2}}\,.$
(4.14)

Figure 3: Limits on the UV scale $\Lambda_{\cancel{\rm CP}}$ of the semi-
leptonic operator ${\cal O}_{\rm lequ}^{(1)}$ in different flavor scenarios as
a function of the small instanton scale $\Lambda_{\rm SI}$. The shaded regions
are excluded by the non observation of the neutron EDM

We present the constraints arising from Eqs. (4.13) and (4.14) in Figure 3. As
expected, the constraints in this case are much weaker than the four quark
operators. For the anarchic and MFV cases, we obtain $\Lambda_{\cancel{\rm
CP}}\gtrsim 10^{6}(10^{7})\,\Lambda_{\rm SI}$ and $\Lambda_{\cancel{\rm
CP}}\gtrsim 5\times 10^{3}(10^{4})\,\Lambda_{\rm SI}$, respectively, for the
5D (product group) model. From Eq. (2.12), we see that the largest term in
${\cal I}^{0}_{0000}(C_{\rm lequ}^{(1)})$ is approximately $\sim
K_{\theta}\,y_{\tau}/y_{\rm u}$ for the anarchic case ($C_{\rm
lequ}^{(1)}\sim{\cal O}(1)$), and $\sim K_{\theta}\,y_{\tau}^{2}$ for the MFV
scenario ($C_{{\rm lequ},NmLk}^{(1)}\sim Y_{{\rm e},Nm}Y_{{\rm u},Lk}$). This
results in a difference by a factor of $\sim\sqrt{y_{{\rm u}}\,y_{\tau}}$ in
the two flavor scenarios. In comparison to the result for the four-quark
operator, the difference can again be understood in terms of a loop factor and
different Yukawa couplings entering the invariants ${\cal
A}_{0000}^{0000}(C_{\rm quqd}^{(1)})$, ${\cal B}_{0000}^{0000}(C_{\rm
quqd}^{(1)})$ and ${\cal I}^{0}_{0000}(C_{\rm lequ}^{(1)})$ – for MFV, there
is a relative factor of
$\sim\sqrt{y_{\tau}^{2}/16\pi^{2}}\equiv\sqrt{\lambda^{6}/16\pi^{2}}$ whereas
for the anarchic case the factor is $\sim\sqrt{y_{\tau}y_{\rm d}/16\pi^{2}}$.

####  4.1.3 Gluon dipole operator

Next, we consider the gluon dipole operator ${\cal O}_{\rm
dG}=(\bar{Q}\sigma^{\mu\nu}T^{A}d)H\,G_{\mu\nu}^{A}$. This operator
contributes to the topological susceptibility at the same order as ${\cal
O}_{\rm quqd}^{(1,8)}$, and has the same functional form in terms of instanton
parameters. The flavor structure of this operator is similar to ${\cal O}_{\rm
uH}$ presented in Eq. (2.10), and the leading order invariant is given by

${\cal I}_{0000}(C_{\rm
dG})\equiv\text{Im}\left[e^{-i\theta_{\text{QCD}}}\epsilon^{IJK}\epsilon^{ijk}Y_{{\rm
d},Ii}Y_{{\rm d},Jj}C_{{\rm dG},\,Kk}\det Y_{\rm u}\right]\,,$ (4.15)

The computation of the susceptibility for this operator is given in App. D.3
and the result is presented in Eq. (D.26). In this case, the product group
model gives the result

$\displaystyle\theta_{\rm ind}=\dfrac{{\cal I}_{0000}\left(C_{\rm
dG}\right)}{K_{\theta}}\dfrac{144\pi^{2}}{5(b_{0}-6)}\dfrac{\Lambda_{\rm
SI}^{2}}{\Lambda_{\cancel{\rm CP}}^{2}}~{},$ (4.16)

where $b_{0}$ and $|\langle\sigma\rangle|$ are similarly defined as in Eq.
(4.11). In the case of the 5D instanton model we obtain

$\displaystyle\theta_{\rm ind}=\dfrac{{\cal I}_{0000}\left(C_{\rm
dG}\right)}{K_{\theta}}\dfrac{18}{5}\dfrac{\Lambda_{\rm
SI}^{2}}{\Lambda_{\cancel{\rm CP}}^{2}}~{}.$ (4.17)

Figure 4: Limits on the UV scale $\Lambda_{\cancel{\rm CP}}$ of the gluon
dipole operator ${\cal O}_{\rm dG}$ in different flavor scenarios as a
function of the small instanton scale $\Lambda_{\rm SI}$. The shaded regions
are excluded by the non observation of the neutron EDM

The constraints coming from $\theta_{\rm ind}$ in Eqs. (4.16) and (4.17) are
presented in Figure 4. In the anarchic scenario, the constraint is
approximately, $\Lambda_{\cancel{\rm CP}}\gtrsim 5\times
10^{7}(10^{8})\,\Lambda_{\rm SI}$ for the 5D (product group) model. It is
worth noting that the constraints arising from both the leading order
operators ${\cal O}_{\rm quqd}^{(1)}$ and ${\cal O}_{\rm dG}$ are similar in
the MFV scenario, while those from ${\cal O}_{\rm lequ}^{(1)}$ are the weakest
among the three, as expected. The anarchic and MFV flavor scenarios are
defined by $C_{\rm dG}\sim 1$ and $C_{\rm dG}\sim Y_{\rm d}$, respectively,
resulting in bounds differing by a factor of $\sim\sqrt{y_{\rm d}}$ (see Eq.
(4.15)). This is much less pronounced compared to ${\cal O}_{\rm quqd}^{(1)}$
and ${\cal O}_{\rm lequ}^{(1)}$ which have multiple Yukawas.

Finally, note that for simplicity we have assumed the small-instanton induced
$\overline{\theta}$ provides the entire contribution to the neutron EDM.
However, in principle, there can be direct contributions to the neutron EDM
from the SMEFT operators which should also be taken into account. These
contributions from parameters other than $\overline{\theta}$ have been
considered in Refs. Pospelov and Ritz (2005); Dekens and de Vries (2013);
Cirigliano _et al._ (2016); Alioli _et al._ (2017); de Vries _et al._ (2019);
Kley _et al._ (2022).

##  5 Conclusion

The enhanced effect of small (UV) instantons due to new high-energy dynamics
that modifies QCD at a UV scale $\Lambda_{\rm SI}$ can be used to increase the
QCD axion mass while still solving the strong CP problem Holdom and Peskin
(1982); Holdom (1985); Dine and Seiberg (1986); Flynn and Randall (1987);
Agrawal and Howe (2018); Csáki _et al._ (2020); Gherghetta _et al._ (2020).
However, in the presence of higher-dimension CP-odd operators at a scale
$\Lambda_{\cancel{\rm CP}}$, an extra contribution to $\bar{\theta}$ can also
be induced from these small instantons which can misalign the axion potential
and give rise to a large neutron EDM, leading to stringent constraints on the
ratio $\Lambda_{\rm SI}/\Lambda_{\cancel{\rm CP}}$ Bedi _et al._ (2022). In
this paper, we have further explored these contributions for generic UV CP-
violating scenarios parameterized by SMEFT operators. In order to estimate the
induced shift $\theta_{\rm ind}$, the calculations have been performed in the
one-(anti-)-instanton (or dilute instanton gas) approximation, valid when the
instanton dynamics can be treated as perturbative. This assumes that the QCD
coupling near the scale $\Lambda_{\rm SI}$ is large or semi-perturbative (in
order to amplify the effect of small instantons). The calculation is not
applicable for non-perturbatively large QCD couplings, where non-perturbative
methods must be used.

Given that the instanton calculations involve complicated integrals and are
usually considered as estimations Csáki _et al._ (2023), a more accurate
estimation can be obtained by including the effect of flavorful couplings in
the theory. However, since physical observable are flavor basis independent,
the flavorful couplings should be arranged into rephasing flavor invariants.
In particular, the topological susceptibility from small instantons can be
described in terms of SMEFT CP-odd invariants introduced in Ref. Bonnefoy _et
al._ (2022). However, the basis of trace invariants presented in Ref. Bonnefoy
_et al._ (2022) is not well-suited to characterize the results from the
instanton calculation of topological susceptibilities. The results, when
projected into the basis of trace invariants, yield complicated linear
combinations of the invariants with coefficients that may contain inverse
powers of Yukawa couplings, therefore making the task of estimating physical
effects with these invariants impractical. Instead, in this work we have
proposed a new basis of CP-odd SMEFT invariants, built from determinant-like
structures which are much better suited to describe instanton computations. We
have explicitly shown and argued that the instanton calculations give results
that are directly proportional to elements of our new basis with no extra
powers of flavorful couplings. The flavor invariants derived in this paper
therefore allow a more refined estimation of the effects of including CP-
violating new physics in small-instanton calculations, complementing the
instanton NDA estimates in Ref. Csáki _et al._ (2023).

Furthermore, the new invariants directly imply selection rules on which kind
of operators can appear at the leading order in the instanton calculation
since they determine the number of Yukawa couplings and loop factors. We have
also shown that rephasing invariant operators cannot contribute only via
fermion zero modes which usually give the dominant contribution. For example,
we show in the case of the semi-leptonic operator contribution how invariants
encapsulate the expected lepton Yukawa dependence and extra loop suppression.
Performing the computations in a flavor-invariant fashion also allows us to
easily test different flavor assumptions for the SMEFT Wilson coefficients.
For instance, we find that for the four-quark operator ${\cal O}_{\rm
quqd}^{(1)}$, the leading order invariant in the MFV scenario contains an
extra product of the up and down Yukawa couplings (or just the down Yukawa
coupling for the gluon dipole operator ${\cal O}_{\rm dG}$) compared to the
anarchic scenario. Using the experimental bound on $\overline{\theta}$, we
obtain constraints on the scale of the higher-dimension CP-violating operators
by assuming that the contribution to $\overline{\theta}$ is entirely due to
the calculated effect of the small instantons. We also show that for the
leading order operators ${\cal O}_{\rm quqd}^{(1)}$ and ${\cal O}_{\rm dG}$
the invariants are approximately $\sim\sin\bar{\theta}$ and result in similar
bounds $\Lambda_{\cancel{\rm CP}}\gtrsim 10^{6}\,\Lambda_{\rm SI}$ for the MFV
scenario. However, for the anarchic scenario, the limits are operator
dependent and become much more stringent– for ${\cal O}_{\rm quqd}^{(1)}$, we
obtain $\Lambda_{\cancel{\rm CP}}\gtrsim 10^{11}\,\Lambda_{\rm SI}$ while for
${\cal O}_{\rm dG}$, the bound becomes $\Lambda_{\cancel{\rm CP}}\gtrsim
10^{8}\,\Lambda_{\rm SI}$.

The cancellation of divergences appearing in the instanton loop integrals can
be used as a non-trivial cross-check of our calculations. The divergences in
the correlation functions are canceled by including the counterterms of the
SMEFT in a Green’s basis. This cancellation has been explicitly shown for the
semi-leptonic operator ${\cal O}_{\rm lequ}^{(1)}$ to obtain a divergence-free
result that is then used for a phenomenological study.

Our work can be extended in several directions. The most immediate one is to
perform similar calculations by including all relevant SMEFT operators
systematically, explicitly verifying the appearance of the constructed
invariants. In addition, considering higher orders in the Yukawa couplings
would also prove interesting. Another possible extension of our work is to
consider the effect of higher-dimensional effective operators (e.g.
considering the double insertions of dimension-six SMEFT operators, or a
single insertion of the dimension-eight SMEFT operators). Since these
operators could allow for different topologies than those of Fig. 1, their
effect might not be in general trivially extrapolated from our results. While
we estimated that a higher-dimensional effect would be suppressed, as
expected, in the case of the double insertion of ${\cal
O}^{(1)}_{\mathrm{quqd}}$ in Appendix D.1, a more systematic study of higher-
dimensional operators could be relevant.

Furthermore, we have solely focused on contributions to the linear term in the
axion potential in this work. However, similar computations could be performed
to estimate the small instanton effects on the axion mass term using CP-even
SMEFT invariants. Having more reliable estimates of observables to probe SMEFT
operators can help to better direct experimental searches.

## Acknowledgments

We thank Quentin Bonnefoy, Pablo Quílez, Jasper Roosmale Nepveu and Maximilian
Ruhdorfer for useful discussions. The work of R.B. and T.G. is supported in
part by the Department of Energy under Grant No. DE-SC0011842 at the
University of Minnesota. The work of C.G., G.G., J.K. and P.N.H.V. is
supported by the Deutsche Forschungsgemeinschaft under Germany’s Excellence
Strategy EXC 2121 “Quantum Universe” – 390833306, as well as by the grant
491245950. This project also has received funding from the European Union’s
Horizon Europe research and innovation programme under the Marie Skłodowska-
Curie Staff Exchange grant agreement No 101086085 - ASYMMETRY. T.G. and C.G.
acknowledge the Aspen Center for Physics, which is supported by National
Science Foundation grant PHY-2210452, where this work was initiated.

##  Appendix A SMEFT conventions

In this work, we consider the SMEFT up to dimension-six terms and assume
dimensionless Wilson coefficients $C_{{\rm a},\,i_{1}...i_{n}}$ defined as

$\mathcal{L}^{\mathrm{SMEFT}}=\mathcal{L}^{\mathrm{SM}}+\frac{1}{\Lambda_{\cancel{\rm
CP}}^{2}}\sum_{\rm a}C_{{\rm a},\,{i_{1}...i_{n}}}\mathcal{O}_{\rm
a}^{i_{1}...i_{n}}\,,$ (A.1)

where $\Lambda_{\cancel{\rm CP}}$ is the cutoff scale associated with the CP-
violating operator, $\rm a$ labels the type of dimension-six operator and
$i_{1}...i_{n}$ correspond to the $n$ flavor indices of fermionic operators.
Above the electroweak breaking scale, the SM Lagrangian is given by

$\displaystyle\mathcal{L}^{\rm SM}$
$\displaystyle=-\dfrac{1}{4}G_{\mu\nu}^{A}G^{A,\mu\nu}-\dfrac{1}{4}W_{\mu\nu}^{I}W^{I,\mu\nu}-\dfrac{1}{4}B_{\mu\nu}B^{\mu\nu}+\theta_{\text{QCD}}\dfrac{g^{2}}{32\pi^{2}}G^{A}_{\mu\nu}\tilde{G}^{A,\,\mu\nu}$
$\displaystyle+i\big{(}\bar{Q}\not{D}Q+\bar{u}\not{D}u+\bar{d}\not{D}d+\bar{L}\not{D}L+\bar{e}\not{D}e\big{)}-\big{(}\bar{Q}Y_{\rm
u}\tilde{H}u+\bar{Q}Y_{\rm d}Hd+\bar{L}Y_{\rm e}He+\text{h.c.}\big{)}$
$\displaystyle+(D^{\mu}H)^{\dagger}(D_{\mu}H)+m_{H}^{2}(H^{\dagger}H)-\dfrac{\lambda}{2}(H^{\dagger}H)^{2}\,,$
(A.2)

where $Y_{\rm u,d,e}$ are the Yukawa coupling matrices and
$\tilde{H}^{i}=\epsilon^{ij}H_{j}^{*}$. Our sign convention for covariant
derivatives is mostly positive, e.g. the covariant derivative acting on a
field $\phi$ reads

$D_{\mu}\phi=\big{(}\partial_{\mu}+igT_{C}^{A}G^{A}_{\mu}+ig_{2}T^{I}W^{I}_{\mu}+ig_{1}\mathbf{Y}_{\phi}B_{\mu}\big{)}\phi\,,$
(A.3)

where $g,~{}g_{2},~{}g_{1}$ are dimensionless gauge coupling constants and
$T_{C}^{A},~{}T^{I}$ are the $SU(3)$ and $SU(2)$ generators in the
representations of $\phi$ respectively; $\mathbf{Y}_{\phi}$ stands for the
hypercharge of $\phi$. Here, it is convenient to define the covariant
derivatives acting to the field and its Hermitian conjugate,

$H^{\dagger}i\overleftrightarrow{D}_{\mu}H\equiv
H^{\dagger}(iD_{\mu}H)-(iD_{\mu}H^{\dagger})H\,,\quad
H^{\dagger}i\overleftrightarrow{D}^{I}_{\mu}H\equiv
H^{\dagger}\tau^{I}(iD_{\mu}H)-(iD_{\mu}H^{\dagger})\tau^{I}H\,,$ (A.4)

where $\tau^{I}$ are the Pauli matrices. We adopt the Warsaw basis Grzadkowski
_et al._ (2010) conventions for the definitions of the effective operators.
The fermionic operators are given for completeness in Tables 2 and 3. We only
consider fermionic operators and neglect fully bosonic ones in our analysis
because only the former have zero modes projected out resulting in
determinant-like structures, as explored in detail in Section 3.

Label | Operator | # phases | # primary phases
---|---|---|---
$\mathcal{O}_{\rm eH}$ | $(H^{\dagger}H)(\bar{L}_{i}e_{j}H)+\text{h.c.}$ | 9 | 3
$\mathcal{O}_{\rm uH}$ | $(H^{\dagger}H)(\bar{Q}_{i}u_{j}\widetilde{H})+\text{h.c.}$ | 9 | 9
$\mathcal{O}_{\rm dH}$ | $(H^{\dagger}H)(\bar{Q}_{i}d_{j}H)+\text{h.c.}$ | 9 | 9
$\mathcal{O}_{\rm eW}$ | $(\bar{L}_{i}\sigma^{\mu\nu}e_{j})\tau^{I}HW_{\mu\nu}^{I}+\text{h.c.}$ | 9 | 3
$\mathcal{O}_{\rm eB}$ | $(\bar{L}_{i}\sigma^{\mu\nu}e_{j})HB_{\mu\nu}+\text{h.c.}$ | 9 | 3
$\mathcal{O}_{\rm uG}$ | $(\bar{Q}_{i}\sigma^{\mu\nu}T^{A}u_{j})\widetilde{H}\,G_{\mu\nu}^{A}+\text{h.c.}$ | 9 | 9
$\mathcal{O}_{\rm uW}$ | $(\bar{Q}_{i}\sigma^{\mu\nu}u_{j})\tau^{I}\widetilde{H}\,W_{\mu\nu}^{I}+\text{h.c.}$ | 9 | 9
$\mathcal{O}_{\rm uB}$ | $(\bar{Q}_{i}\sigma^{\mu\nu}u_{j})\widetilde{H}\,B_{\mu\nu}+\text{h.c.}$ | 9 | 9
$\mathcal{O}_{\rm dG}$ | $(\bar{Q}_{i}\sigma^{\mu\nu}T^{A}d_{j})H\,G_{\mu\nu}^{A}+\text{h.c.}$ | 9 | 9
$\mathcal{O}_{\rm dW}$ | $(\bar{Q}_{i}\sigma^{\mu\nu}d_{j})\tau^{I}H\,W_{\mu\nu}^{I}+\text{h.c.}$ | 9 | 9
$\mathcal{O}_{\rm dB}$ | $(\bar{Q}_{i}\sigma^{\mu\nu}d_{j})H\,B_{\mu\nu}+\text{h.c.}$ | 9 | 9
$\mathcal{O}_{\rm Hl}^{(1)}$ | $(H^{\dagger}i\overleftrightarrow{D}_{\mu}H)(\bar{L}_{i}\gamma^{\mu}L_{j})$ | 3 | 0
$\mathcal{O}_{\rm Hl}^{(3)}$ | $(H^{\dagger}i\overleftrightarrow{D}^{I}_{\mu}H)(\bar{L}_{i}\tau^{I}\gamma^{\mu}L_{j})$ | 3 | 0
$\mathcal{O}_{\rm He}$ | $(H^{\dagger}i\overleftrightarrow{D}_{\mu}H)(\bar{e}_{i}\gamma^{\mu}e_{j})$ | 3 | 0
$\mathcal{O}_{\rm Hq}^{(1)}$ | $(H^{\dagger}i\overleftrightarrow{D}_{\mu}H)(\bar{Q}_{i}\gamma^{\mu}Q_{j})$ | 3 | 3
$\mathcal{O}_{\rm Hq}^{(3)}$ | $(H^{\dagger}i\overleftrightarrow{D}^{I}_{\mu}H)(\bar{Q}_{i}\tau^{I}\gamma^{\mu}Q_{j})$ | 3 | 3
$\mathcal{O}_{\rm Hu}$ | $(H^{\dagger}i\overleftrightarrow{D}_{\mu}H)(\bar{u}_{i}\gamma^{\mu}u_{j})$ | 3 | 3
$\mathcal{O}_{\rm Hd}$ | $(H^{\dagger}i\overleftrightarrow{D}_{\mu}H)(\bar{d}_{i}\gamma^{\mu}d_{j})$ | 3 | 3
$\mathcal{O}_{\rm Hud}$ | $(\widetilde{H}^{\dagger}iD_{\mu}H)(\bar{u}_{i}\gamma^{\mu}d_{j})+\text{h.c.}$ | 9 | 9

Table 2:  Table of bilinear fermionic operators in the Warsaw basis
Grzadkowski _et al._ (2010). For each operator, we also indicate the number of
phases and the number of primary phases Bonnefoy _et al._ (2022), i.e. the
number of flavor-invariant CP-odd quantities capturing the interference with
the SM. The lower-case indices $i,j,k,l$ denote flavor indices, while the
uppercase indices $I$ and $A$ denote the indices of the adjoint representation
of the $SU(2)_{L}$ and $SU(3)_{c}$ gauge groups, respectively.

Label | Operator | # phases | # primary phases
---|---|---|---
${\cal O}_{\rm ll}$ | $(\bar{L}_{i}\gamma_{\mu}L_{j})(\bar{L}_{k}\gamma^{\mu}L_{l})$ | 18 | 0
${\cal O}_{\rm qq}^{(1)}$ | $(\bar{Q}_{i}\gamma_{\mu}Q_{j})(\bar{Q}_{k}\gamma^{\mu}Q_{l})$ | 18 | 18
${\cal O}_{\rm qq}^{(3)}$ | $(\bar{Q}_{i}\gamma_{\mu}\tau^{I}Q_{j})(\bar{Q}_{k}\gamma^{\mu}\tau^{I}Q_{l})$ | 18 | 18
${\cal O}_{\rm lq}^{(1)}$ | $(\bar{L}_{i}\gamma_{\mu}L_{j})(\bar{Q}_{k}\gamma^{\mu}Q_{l})$ | 36 | 9
${\cal O}_{\rm lq}^{(3)}$ | $(\bar{L}_{i}\gamma_{\mu}\tau^{I}L_{j})(\bar{Q}_{k}\gamma^{\mu}\tau^{I}Q_{l})$ | 36 | 9
${\cal O}_{\rm ee}$ | $(\bar{e}_{i}\gamma_{\mu}e_{j})(\bar{e}_{k}\gamma^{\mu}e_{l})$ | 15 | 0
${\cal O}_{\rm uu}$ | $(\bar{u}_{i}\gamma_{\mu}u_{j})(\bar{u}_{k}\gamma^{\mu}u_{l})$ | 18 | 18
${\cal O}_{\rm dd}$ | $(\bar{d}_{i}\gamma_{\mu}d_{j})(\bar{d}_{k}\gamma^{\mu}d_{l})$ | 18 | 18
${\cal O}_{\rm eu}$ | $(\bar{e}_{i}\gamma_{\mu}e_{j})(\bar{u}_{k}\gamma^{\mu}u_{l})$ | 36 | 9
${\cal O}_{\rm ed}$ | $(\bar{e}_{i}\gamma_{\mu}e_{j})(\bar{d}_{k}\gamma^{\mu}d_{l})$ | 36 | 9
${\cal O}_{\rm ud}^{(1)}$ | $(\bar{u}_{i}\gamma_{\mu}u_{j})(\bar{d}_{k}\gamma^{\mu}d_{l})$ | 36 | 36
${\cal O}_{\rm ud}^{(8)}$ | $(\bar{u}_{i}\gamma_{\mu}T^{A}u_{j})(\bar{d}_{k}\gamma^{\mu}T^{A}d_{l})$ | 36 | 36
${\cal O}_{\rm le}$ | $(\bar{L}_{i}\gamma_{\mu}L_{j})(\bar{e}_{k}\gamma^{\mu}e_{l})$ | 36 | 3
${\cal O}_{\rm lu}$ | $(\bar{L}_{i}\gamma_{\mu}L_{j})(\bar{u}_{k}\gamma^{\mu}u_{l})$ | 36 | 9
${\cal O}_{\rm ld}$ | $(\bar{L}_{i}\gamma_{\mu}L_{j})(\bar{d}_{k}\gamma^{\mu}d_{l})$ | 36 | 9
${\cal O}_{\rm qe}$ | $(\bar{Q}_{i}\gamma_{\mu}Q_{j})(\bar{e}_{k}\gamma^{\mu}e_{l})$ | 36 | 9
${\cal O}_{\rm qu}^{(1)}$ | $(\bar{Q}_{i}\gamma_{\mu}Q_{j})(\bar{u}_{k}\gamma^{\mu}u_{l})$ | 36 | 36
${\cal O}_{\rm qu}^{(8)}$ | $(\bar{Q}_{i}\gamma_{\mu}T^{A}Q_{j})(\bar{u}_{k}\gamma^{\mu}T^{A}u_{l})$ | 36 | 36
${\cal O}_{\rm qd}^{(1)}$ | $(\bar{Q}_{i}\gamma_{\mu}Q_{j})(\bar{d}_{k}\gamma^{\mu}d_{l})$ | 36 | 36
${\cal O}_{\rm qd}^{(8)}$ | $(\bar{Q}_{i}\gamma_{\mu}T^{A}Q_{j})(\bar{d}_{k}\gamma^{\mu}T^{A}d_{l})$ | 36 | 36
${\cal O}_{\rm ledq}$ | $(\bar{L}_{i}^{a}e_{j})(\bar{d}_{k}Q_{la})+\text{h.c.}$ | 81 | 27
${\cal O}_{\rm quqd}^{(1)}$ | $(\bar{Q}_{i}^{a}u_{j})(\bar{Q}_{k}^{b}d_{l})+\text{h.c.}$ | 81 | 81
${\cal O}_{\rm quqd}^{(8)}$ | $(\bar{Q}_{i}^{a}T^{A}u_{j})(\bar{Q}_{k}^{b}T^{A}d_{l})+\text{h.c.}$ | 81 | 81
${\cal O}_{\rm lequ}^{(1)}$ | $(\bar{L}_{i}^{a}e_{j})(\bar{Q}_{k}^{b}u_{l})+\text{h.c.}$ | 81 | 27
${\cal O}_{\rm lequ}^{(3)}$ | $(\bar{L}_{i}^{a}\sigma_{\mu\nu}e_{j})(\bar{Q}_{s}^{k}\sigma^{\mu\nu}u_{t})+\text{h.c.}$ | 81 | 27

Table 3:  Table of four-fermion operators in the Warsaw basis Grzadkowski _et
al._ (2010). For each operator, we also indicate the number of phases and the
number of primary phases Bonnefoy _et al._ (2022), i.e. the number of flavor-
invariant CP-odd objects capturing the interference with the SM.

##  Appendix B Details on determinant-like invariants and their relation to
trace invariants

###  B.1 Complete set of flavor invariants featuring $\theta_{\text{QCD}}$ for
all SMEFT operators

In this Appendix, we present a complete set of flavor invariants, featuring
$\theta_{\text{QCD}}$ and are linear in the Wilson coefficient, for all
dimension-six SMEFT operators, which were not shown in the main text due to
their length. We begin with the operators that are used as examples in the
main text. For the operator ${\cal O}_{\rm quqd}^{(1,8)}$, a complete set can
be built with the following 81 invariants

$\displaystyle{\cal A}^{0000}_{0000}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{0000}_{1000}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{1000}_{0000}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{1000}_{1000}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{0000}_{0100}\left(C_{\rm quqd}^{(1,8)}\right),\,$ $\displaystyle{\cal
A}^{0100}_{0000}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{0000}_{1100}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{0000}_{0110}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{0100}_{1000}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{1000}_{0100}\left(C_{\rm quqd}^{(1,8)}\right),\,$ $\displaystyle{\cal
A}^{1100}_{0000}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{0110}_{0000}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{1000}_{1100}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{1000}_{0110}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{1100}_{1000}\left(C_{\rm quqd}^{(1,8)}\right),\,$ $\displaystyle{\cal
A}^{0100}_{0100}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{0100}_{1100}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{0100}_{0110}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{0110}_{0100}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{0000}_{2200}\left(C_{\rm quqd}^{(1,8)}\right),\,$ $\displaystyle{\cal
A}^{0000}_{0220}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{0200}_{2000}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{1100}_{1100}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{1100}_{0110}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{2000}_{0200}\left(C_{\rm quqd}^{(1,8)}\right),\,$ $\displaystyle{\cal
A}^{2100}_{0100}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{0110}_{1100}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{0110}_{0110}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{0210}_{1000}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{0000}_{1220}\left(C_{\rm quqd}^{(1,8)}\right),\,$ $\displaystyle{\cal
A}^{1200}_{2000}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{0000}_{0122}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{0100}_{1220}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{1000}_{0122}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{1100}_{2200}\left(C_{\rm quqd}^{(1,8)}\right),\,$ $\displaystyle{\cal
A}^{1100}_{0220}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{1200}_{2100}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{2100}_{1200}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{2100}_{0210}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{2200}_{0110}\left(C_{\rm quqd}^{(1,8)}\right),\,$ $\displaystyle{\cal
A}^{0110}_{2200}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{0110}_{0220}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{0112}_{2000}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{1100}_{1220}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{2100}_{0112}\left(C_{\rm quqd}^{(1,8)}\right),\,$ (B.1)
$\displaystyle{\cal A}^{1200}_{1220}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{2200}_{2200}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{0110}_{1122}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{0122}_{2100}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
A}^{0220}_{0220}\left(C_{\rm quqd}^{(1,8)}\right),\,$ $\displaystyle{\cal
B}^{0000}_{0000}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
B}^{0000}_{0100}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
B}^{0000}_{1000}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
B}^{0000}_{1100}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
B}^{0000}_{2200}\left(C_{\rm quqd}^{(1,8)}\right),\,$ $\displaystyle{\cal
B}^{0000}_{0110}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
B}^{0000}_{0122}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
B}^{0000}_{0220}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
B}^{0100}_{0000}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
B}^{0100}_{1000}\left(C_{\rm quqd}^{(1,8)}\right),\,$ $\displaystyle{\cal
B}^{0100}_{1100}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
B}^{0100}_{2100}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
B}^{0100}_{0120}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
B}^{0100}_{1220}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
B}^{0200}_{1120}\left(C_{\rm quqd}^{(1,8)}\right),\,$ $\displaystyle{\cal
B}^{1000}_{0000}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
B}^{1000}_{0100}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
B}^{1000}_{1200}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
B}^{1000}_{0110}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
B}^{1000}_{0122}\left(C_{\rm quqd}^{(1,8)}\right),\,$ $\displaystyle{\cal
B}^{1000}_{0210}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
B}^{1100}_{0000}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
B}^{1100}_{1100}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
B}^{1100}_{2200}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
B}^{1100}_{0110}\left(C_{\rm quqd}^{(1,8)}\right),\,$ $\displaystyle{\cal
B}^{1100}_{0220}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
B}^{1100}_{1122}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
B}^{1200}_{2100}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
B}^{2100}_{0122}\left(C_{\rm quqd}^{(1,8)}\right),\,{\cal
B}^{2200}_{0000}\left(C_{\rm quqd}^{(1,8)}\right),\,$ $\displaystyle{\cal
A}^{2200}_{1122}\left(C_{\rm quqd}^{(1,8)}\right),\,$

where the two structures ${\cal A}(C_{\rm quqd}^{(1,8)})$ and ${\cal B}(C_{\rm
quqd}^{(1,8)})$ are defined in Eq. (2.11).

We have also used the operator ${\cal O}_{\rm lequ}^{(1,3)}$ in Section 2.1
for which the full list of 27 invariants reads

$\begin{split}&{\cal I}^{0}_{0000}\left(C_{\rm lequ}^{(1,3)}\right),\,{\cal
I}^{1}_{0000}\left(C_{\rm lequ}^{(1,3)}\right),\,{\cal
I}^{2}_{0000}\left(C_{\rm lequ}^{(1,3)}\right),\,{\cal
I}^{0}_{1000}\left(C_{\rm lequ}^{(1,3)}\right),\,{\cal
I}^{1}_{1000}\left(C_{\rm lequ}^{(1,3)}\right),\\\ &{\cal
I}^{2}_{1000}\left(C_{\rm lequ}^{(1,3)}\right),\,{\cal
I}^{0}_{0100}\left(C_{\rm lequ}^{(1,3)}\right),\,{\cal
I}^{1}_{0100}\left(C_{\rm lequ}^{(1,3)}\right),\,{\cal
I}^{2}_{0100}\left(C_{\rm lequ}^{(1,3)}\right),\,{\cal
I}^{0}_{1100}\left(C_{\rm lequ}^{(1,3)}\right),\\\ &{\cal
I}^{1}_{1100}\left(C_{\rm lequ}^{(1,3)}\right),\,{\cal
I}^{2}_{1100}\left(C_{\rm lequ}^{(1,3)}\right),\,{\cal
I}^{0}_{0110}\left(C_{\rm lequ}^{(1,3)}\right),\,{\cal
I}^{1}_{0110}\left(C_{\rm lequ}^{(1,3)}\right),\,{\cal
I}^{2}_{0110}\left(C_{\rm lequ}^{(1,3)}\right),\\\ &{\cal
I}^{0}_{2200}\left(C_{\rm lequ}^{(1,3)}\right),\,{\cal
I}^{1}_{2200}\left(C_{\rm lequ}^{(1,3)}\right),\,{\cal
I}^{2}_{2200}\left(C_{\rm lequ}^{(1,3)}\right),\,{\cal
I}^{0}_{0220}\left(C_{\rm lequ}^{(1,3)}\right),\,{\cal
I}^{1}_{0220}\left(C_{\rm lequ}^{(1,3)}\right),\\\ &{\cal
I}^{2}_{0220}\left(C_{\rm lequ}^{(1,3)}\right),\,{\cal
I}^{0}_{1220}\left(C_{\rm lequ}^{(1,3)}\right),\,{\cal
I}^{1}_{1220}\left(C_{\rm lequ}^{(1,3)}\right),\,{\cal
I}^{2}_{1220}\left(C_{\rm lequ}^{(1,3)}\right),\,{\cal
I}^{0}_{0122}\left(C_{\rm lequ}^{(1,3)}\right),\\\ &{\cal
I}^{1}_{0122}\left(C_{\rm lequ}^{(1,3)}\right),\,{\cal
I}^{2}_{0122}\left(C_{\rm lequ}^{(1,3)}\right)\end{split}$ (B.2)

where ${\cal I}(C_{\rm lequ}^{(1,3)})$ is defined in Eq. (2.12).

We next continue with all other invariants in the Warsaw basis of the SMEFT.
Only the form of the invariants will be given and we refer to Ref. Bonnefoy
_et al._ (2022) for the index assignments that are needed to obtain a complete
set of invariants which will be exactly the same as those that are needed for
the determinant-like invariants. Consider first the fermion bilinears where we
have already defined the invariants for ${\cal O}_{\rm uH}$ in Eq. (2.10) and
the invariant for ${\cal O}_{\rm dH}$ can be defined in a similar way.
Furthermore, the dipole operators ${\cal O}_{\rm uB},{\cal O}_{\rm uW},{\cal
O}_{\rm uG}$ and ${\cal O}_{\rm dB},{\cal O}_{\rm dW},{\cal O}_{\rm dG}$, fall
into the same class of operators and the invariants have exactly the same form
as those for ${\cal O}_{\rm uH}$ and ${\cal O}_{\rm dH}$. For ${\cal O}_{\rm
eH}$, the form of the invariants is

${\cal I}_{a}(C_{\rm
eH})\equiv\text{Im}\left[e^{-i\theta_{\text{QCD}}}\text{Tr}\left(X_{\rm
e}^{a}C_{\rm eH}\right)\det\left(Y_{\rm u}Y_{\rm d}\right)\right]\,,$ (B.3)

where the index assignments for $a$, for this operator and all other operators
below, are the same as for the trace invariants in Ref. Bonnefoy _et al._
(2022). ${\cal O}_{\rm eB}$ and ${\cal O}_{\rm eW}$ fall again into the same
class of operators.

For the bilinear current-current operators, the Hermitian leptonic operators
${\cal O}_{\rm Hl}^{(1,3)}$ and ${\cal O}_{\rm He}$ do not have any phases
interfering with the SM at the leading order and consequently there exist no
flavor invariants. For the operators containing quarks we have already defined
the invariants for ${\cal O}_{\rm Hq}^{(1,3)}$ in Eq. (2.13). The phases
introduced by the operators ${\cal O}_{\rm Hu}$ and ${\cal O}_{\rm Hd}$ can be
obtained from the invariants of ${\cal O}_{\rm Hq}^{(1,3)}$ by replacing
$C_{\rm Hq}^{(1,3)}\to Y_{\rm u}^{\vphantom{\dagger}}C_{\rm
Hu}^{\vphantom{\dagger}}Y_{\rm u}^{\dagger},\,Y_{\rm
d}^{\vphantom{\dagger}}C_{\rm Hd}^{\vphantom{\dagger}}Y_{\rm d}^{\dagger}$,
while the phases in ${\cal O}_{\rm Hud}$ can be obtained from the invariants
of ${\cal O}_{\rm uH}$ by replacing $C_{\rm uH}^{\vphantom{\dagger}}\to C_{\rm
Hud}^{\vphantom{\dagger}}Y_{\rm d}^{\dagger}$.

Only the four-fermion operators remain to be treated. Again, for the purely
leptonic “Hermitian” four-fermion operators ${\cal O}_{\rm ll}$ and ${\cal
O}_{\rm ee}$, whose Wilson coefficients satisfy the identity
$C_{ijkl}^{*}=C_{jilk}$, no CP-odd invariants arise at leading order in the
Wilson coefficients. For ${\cal O}_{\rm le}$, we can define the following
invariants

${\cal I}_{a}(C_{\rm le})\equiv\text{Im}\left[e^{-i\theta_{\text{QCD}}}(X_{\rm
e}^{a})_{ij}^{\vphantom{\dagger}}Y_{{\rm e},kl}^{\dagger}C_{{\rm
le},jlmi}^{\vphantom{\dagger}}Y_{{\rm
e},mk}^{\vphantom{\dagger}}\det\left(Y_{\rm u}Y_{\rm d}\right)\right]\,.$
(B.4)

The remaining semi-leptonic four-fermion operators can be divided into two
classes. In the first class ${\cal O}_{\rm qe},{\cal O}_{\rm ed},{\cal O}_{\rm
eu}$, can be captured by the following invariant forms

$\begin{split}&{\cal I}_{abcd}^{f}(C_{\rm
qe})\equiv\text{Im}\left[e^{-i\theta_{\text{QCD}}}(X_{\rm u}^{a}X_{\rm
d}^{b}X_{\rm u}^{c}X_{\rm d}^{d})_{ij}^{\vphantom{\dagger}}(X_{\rm
e}^{f}Y_{\rm e}^{\vphantom{f}})_{kl}^{\vphantom{\dagger}}C_{{\rm
qe},jilm}^{\vphantom{\dagger}}Y_{{\rm e},mk}^{\dagger}\det\left(Y_{\rm
u}Y_{\rm d}\right)\right]\,,\\\ &{\cal I}_{abcd}^{f}(C_{\rm
ed})\equiv\text{Im}\left[e^{-i\theta_{\text{QCD}}}(X_{\rm e}^{f}Y_{\rm
e}^{\vphantom{f}})_{ij}^{\vphantom{\dagger}}(X_{\rm u}^{a}X_{\rm d}^{b}X_{\rm
u}^{c}X_{\rm d}^{d}Y_{\rm d})_{lm}^{\vphantom{\dagger}}C_{{\rm
ed},jkmn}^{\vphantom{\dagger}}Y_{{\rm e},ki}^{\dagger}Y_{{\rm
d},nl}^{\dagger}\det\left(Y_{\rm u}Y_{\rm d}\right)\right]\,,\end{split}$
(B.5)

and the ${\cal O}_{\rm eu}$ invariants are obtained by $Y_{{\rm
d},lm}^{\vphantom{\dagger}}C_{{\rm ed},jkmn}^{\vphantom{\dagger}}Y_{{\rm
d},nl}^{\dagger}\to Y_{{\rm u},lm}^{\vphantom{\dagger}}C_{{\rm
eu},jkmn}^{\vphantom{\dagger}}Y_{{\rm u},nl}^{\dagger}$.

The second class ${\cal O}_{\rm lq}^{(1,3)},{\cal O}_{\rm
ld}^{\vphantom{\dagger}},{\cal O}_{\rm lu}^{\vphantom{\dagger}}$ is captured
by the invariants of the following form

$\begin{split}&{\cal I}_{abcd}^{f}(C_{\rm
lq}^{(1,3)})\equiv\text{Im}\left[e^{-i\theta_{\text{QCD}}}(X_{\rm
e}^{f})_{ij}^{\vphantom{\dagger}}(X_{\rm u}^{a}X_{\rm d}^{b}X_{\rm
u}^{c}X_{\rm d}^{d})_{kl}^{\vphantom{\dagger}}C_{{\rm
lq},jilk}^{(1,3)}\det\left(Y_{\rm u}Y_{\rm d}\right)\right]\,,\\\ &{\cal
I}_{abcd}^{f}(C_{\rm ld})\equiv\text{Im}\left[e^{-i\theta_{\text{QCD}}}(X_{\rm
e}^{f})_{ij}^{\vphantom{\dagger}}(X_{\rm u}^{a}X_{\rm d}^{b}X_{\rm
u}^{c}X_{\rm d}^{d}Y_{\rm d})_{kl}^{\vphantom{\dagger}}C_{{\rm
ld},jilm}^{\vphantom{\dagger}}Y_{{\rm d},mk}^{\dagger}\det\left(Y_{\rm
u}Y_{\rm d}\right)\right]\,,\end{split}$ (B.6)

and the ${\cal O}_{\rm lu}$ invariants are obtained by $Y_{{\rm
d},kl}^{\vphantom{\dagger}}C_{{\rm ld},jilm}^{\vphantom{\dagger}}Y_{{\rm
d},mk}^{\dagger}\to Y_{{\rm u},kl}^{\vphantom{\dagger}}C_{{\rm
lu},jilm}^{\vphantom{\dagger}}Y_{{\rm u},mk}^{\dagger}$.

The phases introduced by the operator ${\cal O}_{\rm ledq}$ are captured by
the following invariant

${\cal I}_{abcd}^{f}(C_{\rm
ledq})\equiv\text{Im}\left[e^{-i\theta_{\text{QCD}}}(Y_{\rm e}^{\dagger}X_{\rm
e}^{f})_{ij}^{\vphantom{\dagger}}(Y_{\rm d}X_{\rm u}^{a}X_{\rm d}^{b}X_{\rm
u}^{c}X_{\rm d}^{d})_{kl}^{\vphantom{\dagger}}C_{{\rm
ledq},jilk}^{(1,3)}\det\left(Y_{\rm u}Y_{\rm d}\right)\right]\,.$ (B.7)

The operators ${\cal O}_{\rm qq}^{(1,3)},{\cal O}_{\rm
uu}^{\vphantom{\dagger}},{\cal O}_{\rm dd}^{\vphantom{\dagger}}$ can all be
described by invariants of the form

$\begin{split}&{\cal A}_{a_{1}b_{1}c_{1}d_{1}}^{a_{2}b_{2}c_{2}d_{2}}(C_{\rm
qq}^{(1,3)})\equiv\text{Im}\left[e^{-i\theta_{\text{QCD}}}(X_{\rm
u}^{a_{1}}X_{\rm d}^{b_{1}}X_{\rm u}^{c_{1}}X_{\rm
d}^{d_{1}})_{ij}^{\vphantom{\dagger}}(X_{\rm u}^{a_{2}}X_{\rm d}^{b_{2}}X_{\rm
u}^{c_{2}}X_{\rm d}^{d_{2}})_{kl}^{\vphantom{\dagger}}C_{{\rm
qq},jilk}^{(1,3)}\det\left(Y_{\rm u}Y_{\rm d}\right)\right]\,,\\\ &{\cal
B}_{a_{1}b_{1}c_{1}d_{1}}^{a_{2}b_{2}c_{2}d_{2}}(C_{\rm
qq}^{(1,3)})\equiv\text{Im}\left[e^{-i\theta_{\text{QCD}}}(X_{\rm
u}^{a_{1}}X_{\rm d}^{b_{1}}X_{\rm u}^{c_{1}}X_{\rm
d}^{d_{1}})_{ij}^{\vphantom{\dagger}}(X_{\rm u}^{a_{2}}X_{\rm d}^{b_{2}}X_{\rm
u}^{c_{2}}X_{\rm d}^{d_{2}})_{kl}^{\vphantom{\dagger}}C_{{\rm
qq},lijk}^{(1,3)}\det\left(Y_{\rm u}Y_{\rm d}\right)\right]\,,\end{split}$
(B.8)

where the following replacements have to be made for the latter two operators
$C_{{\rm qq},ijkl}^{(1,3)}\to Y_{{\rm u},im}^{\vphantom{\dagger}}Y_{{\rm
u},nj}^{\dagger}C_{\vphantom{d}{\rm uu},mnop}^{\vphantom{\dagger}}Y_{{\rm
u},ko}^{\vphantom{\dagger}}Y_{{\rm u},pl}^{\dagger}\,,Y_{{\rm
d},im}^{\vphantom{\dagger}}Y_{{\rm d},nj}^{\dagger}C_{{\rm
dd},mnop}^{\vphantom{\dagger}}Y_{{\rm d},ko}^{\vphantom{\dagger}}Y_{{\rm
d},pl}^{\dagger}$.777Note, that we have chosen the same form for all
invariants here while Ref. Bonnefoy _et al._ (2022) chooses a form with fewer
insertions of Yukawa couplings, whenever no insertions of $X_{u,d}$ are made
in one of the bilinears of right-handed quarks. We chose to do this here to
present the invariants in a more compact form. We will also do this for the
following four-fermion operators containing only quark fields.

For the operators ${\cal O}_{\vphantom{d}\rm qu}^{(1,8)},{\cal O}_{\rm
qd}^{(1,8)}$, a complete set of invariants can be found by considering the
following forms of invariants

$\begin{split}{\cal A}_{a_{1}b_{1}c_{1}d_{1}}^{a_{2}b_{2}c_{2}d_{2}}(C_{\rm
qu}^{(1,8)})\equiv&~{}\text{Im}\left[e^{-i\theta_{\text{QCD}}}(X_{\rm
u}^{a_{1}}X_{\rm d}^{b_{1}}X_{\rm u}^{c_{1}}X_{\rm
d}^{d_{1}})_{ij}^{\vphantom{\dagger}}(X_{\rm u}^{a_{2}}X_{\rm d}^{b_{2}}X_{\rm
u}^{c_{2}}X_{\rm d}^{d_{2}})_{kl}^{\vphantom{\dagger}}\right.\\\
&\qquad\left.\times Y_{{\rm u},lm}^{\vphantom{\dagger}}C_{{\rm
qu},jimn}^{(1,8)}Y_{{\rm u},nk}^{\dagger}\det\left(Y_{\rm u}Y_{\rm
d}\right)\right]\,,\\\ {\cal
B}_{a_{1}b_{1}c_{1}d_{1}}^{a_{2}b_{2}c_{2}d_{2}}(C_{\rm
qu}^{(1,8)})\equiv&~{}\text{Im}\left[e^{-i\theta_{\text{QCD}}}(X_{\rm
u}^{a_{1}}X_{\rm d}^{b_{1}}X_{\rm u}^{c_{1}}X_{\rm
d}^{d_{1}})_{ij}^{\vphantom{\dagger}}(X_{\rm u}^{a_{2}}X_{\rm d}^{b_{2}}X_{\rm
u}^{c_{2}}X_{\rm d}^{d_{2}})_{kl}^{\vphantom{\dagger}}\right.\\\
&\qquad\left.\times Y_{{\rm u},lm}^{\vphantom{\dagger}}C_{{\rm
qu},mijn}^{(1,8)}Y_{{\rm u},nk}^{\dagger}\det\left(Y_{\rm u}Y_{\rm
d}\right)\right]\,,\end{split}$ (B.9)

where for ${\cal O}_{\rm qd}^{(1,8)}$ the replacement $Y_{{\rm
u},lm}^{\vphantom{\dagger}}C_{{\rm qu},mijn}^{(1,8)}Y_{{\rm
u},nk}^{\dagger}\to Y_{{\rm d},lm}^{\vphantom{\dagger}}C_{{\rm
qd},mijn}^{(1,8)}Y_{{\rm d},nk}^{\dagger}$ has to be made.

Finally, the CP violation introduced by the operator ${\cal O}_{\rm
ud}^{(1,8)}$ can be captured by invariants of the form

$\begin{split}{\cal A}_{a_{1}b_{1}c_{1}d_{1}}^{a_{2}b_{2}c_{2}d_{2}}(C_{\rm
ud}^{(1,8)})\equiv&~{}\text{Im}\left[e^{-i\theta_{\text{QCD}}}(X_{\rm
u}^{a_{1}}X_{\rm d}^{b_{1}}X_{\rm u}^{c_{1}}X_{\rm
d}^{d_{1}})_{ij}^{\vphantom{\dagger}}(X_{\rm u}^{a_{2}}X_{\rm d}^{b_{2}}X_{\rm
u}^{c_{2}}X_{\rm d}^{d_{2}})_{kl}^{\vphantom{\dagger}}\right.\\\
&\qquad\left.\times Y_{{\rm u},jm}^{\vphantom{\dagger}}Y_{{\rm
d},ln}^{\vphantom{\dagger}}C_{{\rm ud},monp}^{(1,8)}Y_{{\rm
u},oi}^{\dagger}Y_{{\rm d},pk}^{\dagger}\det\left(Y_{\rm u}Y_{\rm
d}\right)\right]\,,\\\ {\cal
B}_{a_{1}b_{1}c_{1}d_{1}}^{a_{2}b_{2}c_{2}d_{2}}(C_{\rm
ud}^{(1,8)})\equiv&~{}\text{Im}\left[e^{-i\theta_{\text{QCD}}}(X_{\rm
u}^{a_{1}}X_{\rm d}^{b_{1}}X_{\rm u}^{c_{1}}X_{\rm
d}^{d_{1}})_{ij}^{\vphantom{\dagger}}(X_{\rm u}^{a_{2}}X_{\rm d}^{b_{2}}X_{\rm
u}^{c_{2}}X_{\rm d}^{d_{2}})_{kl}^{\vphantom{\dagger}}\right.\\\
&\qquad\left.\times Y_{{\rm u},jm}^{\vphantom{\dagger}}Y_{{\rm
d},ln}^{\vphantom{\dagger}}C_{{\rm ud},nomp}^{(1,8)}Y_{{\rm
u},oi}^{\dagger}Y_{{\rm d},pk}^{\dagger}\det\left(Y_{\rm u}Y_{\rm
d}\right)\right]\,.\end{split}$ (B.10)

###  B.2 Conversion of invariants with negative powers of Yukawa couplings

In Section 2.1, all determinant-like invariants were related to the basis of
trace invariants of Ref. Bonnefoy _et al._ (2022), except for invariants of
the form ${\cal I}_{0bcd}(C_{\rm uH})$ which are mapped to the trace
invariants $L(C)$ and $R(C)$ defined below Eq. (2.15)

${\cal I}_{0bcd}(C_{\rm uH})=2\,\left(J_{\theta}\,R_{(-1)bcd}(C_{\rm
uH}^{\vphantom{\dagger}}Y_{\vphantom{H}\rm
u}^{\dagger})+K_{\theta}\,L_{(-1)bcd}(C_{\rm
uH}^{\vphantom{\dagger}}Y_{\vphantom{H}\rm u}^{\dagger})\right)\,,$ (B.11)

that contain inverse powers of Yukawa couplings, which is clearly not in the
basis of Ref. Bonnefoy _et al._ (2022). To convert the right-hand side of Eq.
(B.11) to the basis we will use the following relation

$A^{-1}=\frac{1}{\det
A}\left[A^{2}-A\,\text{Tr}\,A+\frac{1}{2}\left((\text{Tr}A)^{2}-\text{Tr}(A^{2})\right)\mathbb{1}\right]\,,$
(B.12)

which directly follows from the Cayley–Hamilton theorem. Making use of this
identity, we can write

$\begin{split}L_{(-1)bcd}(C_{\rm uH}^{\vphantom{\dagger}}Y_{\vphantom{H}\rm
u}^{\dagger})&=\text{Im}\,\text{Tr}\left(X_{\rm u}^{-1}X_{\rm d}^{b}X_{\rm
u}^{c}X_{\rm d}^{d}C_{\rm uH}^{\vphantom{\dagger}}Y_{\vphantom{H}\rm
u}^{\dagger}\right)\\\ &=\frac{1}{\det X_{\rm u}}\left(L_{2bcd}(C_{\rm
uH}^{\vphantom{\dagger}}Y_{\vphantom{H}\rm u}^{\dagger})-\text{Tr}(X_{\rm
u})\,L_{1bcd}(C_{\rm uH}^{\vphantom{\dagger}}Y_{\vphantom{H}\rm
u}^{\dagger})\right.\\\ &\qquad\left.+\frac{1}{2}\left((\text{Tr}X_{\rm
u})^{2}-\text{Tr}(X_{\rm u}^{2})\right)L_{0bcd}(C_{\rm
uH}^{\vphantom{\dagger}}Y_{\vphantom{H}\rm u}^{\dagger})\right)\,,\end{split}$
(B.13)

given that $\det X_{\rm u}\neq 0$. Repeating the same for $R_{(-1)000}(C_{\rm