aainstitutetext: Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607
Hamburg, Germanybbinstitutetext: Institut für Physik, Humboldt-Universität zu
Berlin, 12489 Berlin, Germanyccinstitutetext: School of Physics, Nankai
University, Tianjin 300071, China

#  Hilbert series for ALP EFTs

Christophe Grojean<EMAIL_ADDRESS>a,b Jonathan Kley
<EMAIL_ADDRESS>a,c Chang-Yuan Yao<EMAIL_ADDRESS>
###### Abstract

Axions and axion-like particles (ALPs) are ubiquitous in popular attempts to
solve supercalifragilisticexpialidocious puzzles of Nature. A widespread and
vivid experimental programme spanning a vast range of mass scales and decades
of couplings strives to find evidence for these elusive but theoretically
well-motivated particles. In the absence of clear guiding principle, effective
field theories (EFTs) prove to be an efficient tool in this experimental
quest. Hilbert series technologies are a privileged instrument of the EFT
toolbox to enumerate and classify operators. In this work, we compute
explicitly the Hilbert series capturing the interactions of a generic ALP to
the Standard Model particles above and below the electroweak symmetry scale,
which allow us to build bases of operators up to dimension-8. In particular,
we revealed a remarkable structure of the Hilbert series that isolates the
shift-symmetry breaking and preserving interactions. Furthermore, we provide
an ancillary file of the Hilbert series up to dimension 15 to supplement our
findings, which can be used for further analysis and exploration.

DESY-23-098

HU-EP-23/39

##  1 Introduction

Axions and axionlike particles (ALPs) are well-motivated new physics
candidates appearing in many models beyond the Standard Model (SM). The axion
was first introduced as a – by now the most well-known – solution to the
strong CP problem of QCD Peccei and Quinn (1977a, b); Weinberg (1978); Wilczek
(1978); Kim (1979); Shifman _et al._ (1980); Dine _et al._ (1981); Zhitnitsky
(1980) in the Peccei–Quinn (PQ) mechanism. In addition, it can also act as a
candidate for dark matter Preskill _et al._ (1983); Abbott and Sikivie (1983);
Dine and Fischler (1983); Marsh (2016); Di Luzio _et al._ (2020) solving two
shortcomings of the SM at once. One of the most important properties of the
axion is that it benefits from a shift symmetry which is due to its Goldstone
nature under the spontaneously broken $U(1)_{\text{PQ}}$ symmetry in the PQ
mechanism. Thanks to this shift symmetry, the axion acquires special
properties. It is, for instance, protected from receiving a potential which
ensures in the case of the QCD axion that the strong CP problem will still be
solved by going to the minimum of the potential as long as any shift-breaking
effects beyond QCD instantons are small enough. As the shift symmetry is a
vital property of ALPs, it is important to study how explicit breaking effects
can introduce deviations to the ALP construction from the exact shift
symmetric case. Above the scale of spontaneous PQ-breaking, the shift-breaking
terms can be understood as terms which break the global $U(1)_{\text{PQ}}$
symmetry. Henceforth, we will therefore also refer to these effects as PQ-
breaking effects.

Explicit breaking effects to the PQ symmetry appear in many well-motivated new
physics scenarios. First of all, quantum gravity does not allow for exact
global symmetries Hawking (1987); Giddings and Strominger (1988); Kamionkowski
and March-Russell (1992); Banks and Seiberg (2011) leading to Planck scale
suppressed operators111These effects can become much larger than naively
expected when heavy particles are present in the UV above the scale of
spontaneous PQ-breaking. Integrating them out can significantly lower the
scale of gravity-induced breaking effects Bonnefoy (2022) worsening the so-
called axion quality problem. that break the shift symmetry of the ALP.
Besides that, it can also be interesting from a model building point of view
to allow for some breaking of the shift symmetry. Examples are the relaxion
Graham _et al._ (2015); Espinosa _et al._ (2015), where the breaking term is
needed to scan the Higgs mass parameter, or model building for collider
anomalies Franceschini _et al._ (2016). Therefore, it is important to
understand how these effects can be captured in a generic way. Due to the vast
landscape of theories in which axions can appear, it is convenient to work
with an effective field theory (EFT) of the axion coupled to the SM degrees of
freedom. In this language, one can capture the interactions between axions and
the SM particles in a bottom-up approach while being mostly agnostic about the
details of the UV physics Srednicki (1985); Georgi _et al._ (1986).

A first step towards understanding PQ-breaking effects in an EFT language has
been laid out in Ref. Bonnefoy _et al._ (2023), where the leading shift-
breaking effects have been captured in Jarlskog-like flavor invariants. Here,
we want to carry on this analysis further and explicitly check how these PQ-
breaking effects are encoded in operators at higher mass dimensions. One of
the main points of Ref. Bonnefoy _et al._ (2023) is that, in the presence of
shift-breaking effects, the same operators coupling the ALP to fermions
describe both shift-breaking and shift-preserving interactions. Therefore, it
is difficult to give the interactions their appropriate power counting which
can be very different as the scales of spontaneous and explicit PQ-breaking
are usually well-separated. In this paper, we want to study if a similar
mixing between the shift-breaking and shift-preserving sector also appears in
the fermionic sector at higher orders in the EFT expansion. For that we build
an operator basis for a pseudoscalar with and without a shift symmetry coupled
to the SM degrees of freedom.

The operator basis we derive could prove useful for phenomenological as well
as theoretical studies. Most analyses study the leading dimension-5
interactions of the ALPs, while some also consider effects from higher
dimensional operators at dimension 6 Draper and McKeen (2012); Bauer _et al._
(2017, 2019); Davoudiasl _et al._ (2021); Brivio _et al._ (2021) and dimension
7 Bauer _et al._ (2016, 2017, 2019). In particular, the analyses at dimension
7 use an incomplete basis, which may lead to the omission of contributions
from other operators that could alter the results of the study. On the more
theoretical side, the operators at dimension 8 are of interest for discussion
of positivity in the context of the ALP EFT. Furthermore, it could be of
interest for matching calculations Quevillon _et al._ (2022) to have a
complete set of operators beyond the leading interactions. An important probe
for new physics are low-energy experiments looking for small corrections in
high-precision experiments and exotic decays of mesons involving ALPs (see
e.g. Refs. Björkeroth _et al._ (2018); Bauer _et al._ (2021a); Martin Camalich
_et al._ (2020); Calibbi _et al._ (2021); Bauer _et al._ (2022)). In order to
have a complete effective description of such effects below the electroweak
(EW) scale, we will also derive an operator basis for the so called low-energy
effective field theory (LEFT) extended with an ALP.

To simplify the procedure of building an operator basis, we will borrow the
so-called Hilbert series from representation theory that counts all
combinations of objects transforming as singlets under a given group. Here, we
will use it to count the singlets under the Lorentz and gauge group as well as
to address equation of motion (EOM) and integration by parts (IBP)
redundancies that are a big nuisance when building an operator basis. In the
context of EFTs, these tools have been developed in Refs. Lehman and Martin
(2015, 2016); Henning _et al._ (2016, 2017a, 2017b) to build operator bases
for EFTs addressing EOM and IBP redundancies with the help of ideas from
conformal representation theory. Since then, these tools have proven helpful
in many analyses of different EFTs Lehman and Martin (2015, 2016); Henning _et
al._ (2016, 2017a); Kobach and Pal (2017); Henning _et al._ (2017b); Ruhdorfer
_et al._ (2020); Marinissen _et al._ (2020); Graf _et al._ (2021); Kondo _et
al._ (2023); Gráf _et al._ (2023); Sun _et al._ (2022); Bijnens _et al._
(2022) to build operator bases and to study different aspects of these EFTs
like their behavior under CP.

The main focus of this paper is to demonstrate how the Hilbert series can
provide a clear and concise understanding of the separation of the shift-
symmetric and shift-breaking sectors in the ALP EFTs. We highlight that these
two sectors can be distinctly categorized above mass dimension 5, without any
observed mixing between them. We will furthermore show that, making a change
of basis that is often considered in the literature and convenient to work in
in the presence of shift-breaking effects, one has to consider more seemingly
shift-breaking operators with completely constrained Wilson coefficients.

The paper is organized as follows. In Section 2, we briefly introduce the
Hilbert series in the context of constructing an EFT operator basis, and
explain how to implement a shift symmetry in the calculation. In Section 3, we
extend the SMEFT with both shift-symmetric and non-shift-symmetric axions, the
Hilbert series and the operator counting are present, we identify a remarkable
Peccei–Quinn breaking isolation property of the Hilbert series, and the
complete and non-redundant operator bases up to dimension 8 are constructed.
Furthermore, we discuss the shift-symmetric limit at mass dimension 5 and
beyond, and discuss conditions beyond dimension 5 that have to be considered
if one changes the operator basis from the derivatively coupled to the Yukawa-
like ALP interactions at dimension-5. In Section 4, we investigate the axion-
extended LEFT with and without a shift symmetry, we calculate the Hilbert
series and construct the operator basis up to dimension 8. Finally, we
conclude the paper in Section 5 and outline potential future directions for
our research. In order to provide more information, we show the complete
operator bases up to dimension 8 for both axion-extended SMEFT and LEFT in
App. A and App. B respectively. In App. C, we show additional results for the
Hilbert series, and operator counting is also performed for higher dimensions.
Finally, in App. D, we give details about the change of operator basis from
the derivatively coupled to the Yukawa basis, and list all the relevant
operators along their constrained Wilson coefficient that have to be
considered in the Yukawa basis up to dimension 8.

##  2 Hilbert series for EFT operators

The Hilbert series serves the purpose of systematically counting operators
based on their order in fields and derivatives, although it does not provide
an explicit construction of these operators. Knowing the number of independent
operators is extremely helpful for the construction of an operator basis. In
this section, we will briefly review the tools for the Hilbert series that
will be used throughout this paper and have been developed for EFTs in Refs.
Lehman and Martin (2015, 2016); Henning _et al._ (2016, 2017a, 2017b).

###  2.1 Hilbert series and operator redundancies

The Hilbert series is a mathematical tool that allows one to determine the
number of independent invariants in a theory by considering the power series
representation. In the context of operator basis construction, it is a
generating function designed to count the number of gauge and Lorentz
invariant operators associated with a specific field content $\\{\phi_{i}\\}$
and derivatives ${\cal D}$, referred to as spurions. The general form of the
Hilbert series is given by

${\cal H}({\cal
D},\\{\phi_{i}\\})=\sum_{r_{1},\dots,r_{n}}\sum_{k}c_{\mathbf{r}\,k}\
\phi_{1}^{r_{1}}\dots\phi_{n}^{r_{n}}{\cal D}^{k},$ (2.1)

where $c_{\mathbf{r}\,k}\equiv c_{r_{1},\dots,r_{n},k}$ counts the number of
independent operators with $k$ derivatives ${\cal D}$ and $r_{i}$ fields
$\phi_{i}$. In order to construct an operator basis at given mass dimension,
it is helpful to obtain the Hilbert series and interpret the number
$c_{\mathbf{r},k}$ as a guiding factor. The calculation of the Hilbert series
of the operator basis can be accomplished by utilizing the group characters.
When dealing with compact Lie groups, the group characters are orthonormal
when integrate over the group’s Haar measure, i.e.,

$\int
d\mu_{G}(g)\,\chi_{\mathbf{R}}(g)\,\chi_{\mathbf{R^{\prime}}}^{*}(g)=\delta_{\mathbf{R},\mathbf{R^{\prime}}}\,,$
(2.2)

where $\chi_{\mathbf{R}}(g)$ is the character of representation $\mathbf{R}$
of a group $G$ with $g\in G$, and $d\mu_{G}$ is the Haar measure. Therefore,
by considering all possible tensor products of the spurions, multiplying their
characters, the orthonormality of the group characters allows one to project
these products onto the singlets of the group, which results in a complete set
of group invariants. The generating function that yields all possible tensor
products of spurions is called the plethystic exponential (PE) Feng _et al._
(2007); Henning _et al._ (2017b)

$\text{PE}\left[\phi_{\mathbf{R}}\,\chi_{\mathbf{R}}(z)\right]=\exp\left(\sum_{r=1}^{\infty}\frac{1}{r}(\pm
1)^{r+1}\phi_{\mathbf{R}}^{r}\,\chi_{\mathbf{R}}(z^{r})\right)\,,$ (2.3)

where $z=\\{z_{1},\dots,z_{n}\\}$ are the complex parameters of the maximal
torus $U(1)^{n}\subset G$ with $n=\text{rank}\,G$, and
$z^{r}=\\{z_{1}^{r},\dots,z_{n}^{r}\\}$, the plus and minus signs correspond
to the bosonic and fermionic spurions respectively. For operator basis
construction, the PE should include the complete field content of the theory,
and the full PE is defined as
$\text{PE}[\\{\phi_{i}\\}]=\prod_{i}\text{PE}[\phi_{i}]$ without showing
characters explicitly. The Hilbert series will be obtained after the group
integration, projecting out all of the singlets under the group $G$

${\cal H}(\\{\phi_{i}\\})=\int d\mu_{G}\,\text{PE}[\\{\phi_{i}\\}]\,.$ (2.4)

Using the procedure described above, it is possible to calculate the Hilbert
series for all potential invariants associated with the symmetry group of a
theory. However, it is important to note that these invariants do not form an
independent operator basis because the EOM Arzt (1995) and IBP redundancies
are not directly taken into account during the calculation of the Hilbert
series. In Ref. Henning _et al._ (2017b), a comprehensive exploration is
carried out to examine the structure and implications of the conformal group.
It turns out that an operator basis is well organized by the conformal group,
which in turn enables one to effectively address redundancies arising from EOM
and IBP. To capture the correct degrees of freedom, the so-called single
particle module is introduced as a building block which reads as follows for a
scalar field $a$

$R_{a}=\begin{pmatrix}a\\\ \partial_{\mu_{1}}a\\\
\partial_{\\{\mu_{1}}\partial_{\mu_{2}\\}}a\\\ \vdots\end{pmatrix}\,,$ (2.5)

where only the symmetrized combinations of the derivatives are included, as
the other combinations just yield a field strength. To avoid EOM redundancies,
we have to remove the terms of the form
$\left(\partial^{2}a,\partial_{\mu_{1}}\partial^{2}a,\partial_{\mu_{1}}\partial_{\mu_{2}}\partial^{2}a,\dots\right)$
from the module. This is done by imposing a so-called shortening condition
after which only the traceless part of the module remains. We can obtain the
character for this single particle module that captures the right degrees of
freedom by summing over the characters of its symmetrized and traceless
components, i.e. Henning _et al._ (2017b)

$\chi_{a}\left({\cal D},x\right)=\sum_{n=0}^{\infty}{\cal
D}^{n+d_{a}}\chi_{\text{Sym}^{n}\left(\frac{1}{2},\frac{1}{2}\right)}(x)-\sum_{n=2}^{\infty}{\cal
D}^{n+d_{a}}\chi_{\text{Sym}^{n-2}\left(\frac{1}{2},\frac{1}{2}\right)}(x)={\cal
D}\left(1-{\cal D}^{2}\right)P({\cal D},x)\,,$ (2.6)

where $\chi_{\text{Sym}^{n}\left(\frac{1}{2},\frac{1}{2}\right)}$ is the
character of the $n$th tensor product of the symmetrized
$\left(\frac{1}{2},\frac{1}{2}\right)$ representation of the Lorentz
group222Note that the the Lorentz group is non-compact, and its characters are
not orthonormal, which make it difficult to project out the Lorentz invariant
by using group integral method in calculation of Hilbert series. However, we
can conveniently work in Euclidean space, the Lorentz group $SO(4)$ is
isomorphic to $[SU(2)_{L}\otimes SU(2)_{R}]/Z_{2}$, which is a compact group.
Then the covariant derivative ${\cal D}$ transforms in the fundemental
$\left(\frac{1}{2},\frac{1}{2}\right)$ representation of the Lorentz group.,
that the covariant derivative, counted by the spurion ${\cal D}$, lives in.
$d_{a}=1$ is the scaling dimension of the $a$, which accounts for the
additional factor of ${\cal D}$ in Eq. (2.6). We have defined the generating
function of symmetric products of the vector representation

$P({\cal D},x)=\sum_{n=0}^{\infty}{\cal
D}^{n}\chi_{\text{Sym}^{n}\left(\frac{1}{2},\frac{1}{2}\right)}(x)\,,$ (2.7)

which will also appear later in the Hilbert series. $x$ are the maximal torus
coordinates of the Lorentz group that will be integrated over in the Hilbert
series. The same procedure can be applied to the SM fermions and field
strengths to remove EOM redundancies (for details see Ref. Henning _et al._
(2017b)).

Furthermore, when dealing with IBP redundancy, it has been discovered that the
single particle modules align perfectly with unitary conformal representations
of free fields. The local operators can then be constructed by combining
single particle modules through tensor products of the unitary conformal
representations, which can further be decomposed into irreducible conformal
representations. The irreducible representations in the tensor product form a
set of independent operators with both the IBP and EOM redundancies
eliminated. By performing an additional integral over the conformal group, the
independent Lorentz and gauge invariant operators will be obtained, and thus
an operator basis can be constructed Henning _et al._ (2017b). The final
expression of the Hilbert series can be organized as follows

${\cal H}({\cal D},\\{\phi_{i}\\})=\int d\mu_{\text{Lorentz}}\int
d\mu_{\text{gauge}}\frac{1}{P}\prod_{i}\text{PE}\left[\frac{\phi_{i}}{{\cal
D}^{d_{i}}}\chi_{i}\right]+\Delta{\cal H}({\cal D},\\{\phi_{i}\\})\,,$ (2.8)

where $\\{\phi_{i}\\}$ corresponds to all spurions in the theory, and the
character $\chi_{i}$ should be understood as the character of the single
particle module $R_{\phi_{i}}$. The calculation of the conformal character of
$R_{\phi_{i}}$ is weighted with the scaling dimension $d_{i}$, therefore, each
spurion $\phi_{i}$ in the PE is weighted by ${\cal D}^{-d_{i}}$. The factor
$1/P$ as well as the $d\mu_{\text{Lorentz}}$ are the remnants of the Haar
measure for the conformal group after the integral of the dilatations, and $P$
is the generating function for symmetric products of the vector representation
that we have defined above, which plays an important role in eliminating IBP
redundancies.

One subtlety arises when using the shortened conformal characters to remove
the EOM redundancies. Conformal characters are only unitary if their scaling
dimensions satisfies a lower bound, its unitarity bound. The characters we use
saturate this bound leading to problems with orthogonality of these characters
Henning _et al._ (2017b); Ruhdorfer _et al._ (2020). The additional terms in
the Hilbert series arising due to these issues are removed by $\Delta{\cal H}$
in Eq. (2.8), its specific form for a general EFT can be found in Ref. Henning
_et al._ (2017b). These terms can be interpreted with tools from differential
geometry Graf _et al._ (2021); Bijnens _et al._ (2022) and usually only appear
at mass dimension four and less. We will discuss this in details in the next
section where we calculate the Hilbert series.

In order to calculate the Hilbert series defined in Eq. (2.8), the conformal
characters corresponding to various single particle modules and the Haar
measures for different groups are needed. They have been extensively discussed
in previous papers Henning _et al._ (2017b, a). This paper does not aim to
replicate the aforementioned results. Instead, our calculations rely on the
formulas presented in Ref. Henning _et al._ (2017a), specifically addressing
the characters of typical fields and Haar measures.

###  2.2 Implementing the ALP shift symmetry

After introducing how we can obtain the Hilbert series for operators removing
IBP and EOM redundancies, we will now discuss how to implement the shift-
symmetric character of the axion. The Hilbert series for a shift-symmetric
theory was first discussed in Ref. Henning _et al._ (2017b), where a general
treatment was introduced within the framework of non-linear realizations. This
approach has been further applied to construct operator bases, such as the
operator basis for the shift-symmetric scalar coupled to gravity Ruhdorfer _et
al._ (2020), and for the ${\cal O}(N)$ nonlinear sigma model Bijnens _et al._
(2022). This paper will provide a detailed discussion of the Hilbert series
for the SMEFT and LEFT extended with a shift-symmetric axion. The Hilbert
series will serve as a guiding tool in constructing the operator basis for
axion EFTs. In this section, we will briefly summarize how to implement a
shift-symmetric scalar in the Hilbert series using conformal characters.

As the axion arises as the Goldstone boson of the spontaneously broken
$U(1)_{\text{PQ}}$, we can adopt the machinery for non-linearly realized
symmetries à la CCWZ Coleman _et al._ (1969); Callan _et al._ (1969) developed
in Ref. Henning _et al._ (2017b) to impose its properties in the Hilbert
series. For a spontaneously broken symmetry $G\to H\subset G$, the Goldstone
degrees of freedom $\pi^{i}(x)$ can be parameterized using the following
matrix field

$\xi(x)=e^{\frac{i\pi^{i}(x)X^{i}}{f_{\pi}}}\,,$ (2.9)

where $X^{i}$ are the broken generators living in the coset space $G/H$ and
$f_{\pi}$ is the pion decay constant. To write down the EFT of the pions of
the spontaneously broken symmetry, one usually defines the Cartan form

$w_{\mu}\equiv\xi^{-1}\partial_{\mu}\xi=u_{\mu}^{i}X^{i}+v_{\mu}^{a}T^{a}\equiv
u_{\mu}+v_{\mu}$ (2.10)

decomposing the degrees of freedom along the broken generators $X^{i}$ and the
unbroken generators $T^{a}$. Due to the simplicity of the symmetry breaking
pattern $U(1)_{\text{PQ}}\to\not{0}$, the discussion simplifies drastically
for us. There exists only one broken generator and we can simply write

$\xi=e^{i\frac{a}{f}},\qquad w_{\mu}=u_{\mu}=i\frac{\partial_{\mu}a}{f}\,.$
(2.11)

In the following, instead of working with the Cartan form $u_{\mu}$, we will
work with the simplified expression $u_{\mu}\sim\partial_{\mu}a$ for the ALP.
In order to implement this derivative coupling, we have to remove the scalar
itself as a building block from the Hilbert series, amounting to removing the
first entry from the single particle module in Eq. (2.5). This yields

$R_{\partial a}=\begin{pmatrix}\partial_{\mu_{1}}a\\\
\partial_{\\{\mu_{1}}\partial_{\mu_{2}\\}}a\\\
\partial_{\\{\mu_{1}}\partial_{\mu_{2}}\partial_{\mu_{3}\\}}a\\\
\vdots\end{pmatrix}\,.$ (2.12)

To remove the first entry in the single particle module in Eq. (2.5) with the
help of characters, we have to apply another shortening condition on top of
the previous one that eliminates EOM redundancy. By summing over the
characters of the remaining elements of the scalar single particle module, we
obtain the character of a shift-symmetric singlet scalar Henning _et al._
(2017b)

$\begin{split}\chi_{\partial a}\left({\cal
D},x\right)&=\sum_{n=1}^{\infty}{\cal
D}^{n+d_{a}}\chi_{\text{Sym}^{n}\left(\frac{1}{2},\frac{1}{2}\right)}(x)-\sum_{n=2}^{\infty}{\cal
D}^{n+d_{a}}\chi_{\text{Sym}^{n-2}\left(\frac{1}{2},\frac{1}{2}\right)}(x)\\\
&={\cal D}^{d_{a}}\left(-1+\sum_{n=0}^{\infty}{\cal
D}^{n}\chi_{\text{Sym}^{n}\left(\frac{1}{2},\frac{1}{2}\right)}(x)-\sum_{n=2}^{\infty}{\cal
D}^{n}\chi_{\text{Sym}^{n-2}\left(\frac{1}{2},\frac{1}{2}\right)}(x)\right)\\\
&={\cal D}\left(\left(1-{\cal D}^{2}\right)P\left({\cal
D},x\right)-1\right)\,.\end{split}$ (2.13)

Together with the characters for all other building blocks in the EFT (see
Ref. Henning _et al._ (2017b)) this completes the discussion of the
ingredients for the Hilbert series. We will fix the exact spurion content and
some other conventions in the following section.

###  2.3 Conventions

This paper will discuss axion EFTs at different energy scales, we consider
both SMEFT and LEFT extended with axion, and they are refereed to as aSMEFT
and aLEFT respectively. Depending on whether there is a shift symmetry for the
axion, four types of EFTs are shown explicitly along with corresponding
spurions as follows

  * •

$\text{aSMEFT}_{\text{PQ}}$: SMEFT extended with a shift-symmetric axion

$\\{{\cal D},\partial
a,Q,Q^{\dagger},L,L^{\dagger},H,H^{\dagger},u,u^{\dagger},d,d^{\dagger},e,e^{\dagger},B_{L},B_{R},W_{L},W_{R},G_{L},G_{R}\\}$
,

  * •

$\text{aSMEFT}_{\cancel{\text{PQ}}}$: SMEFT extended with a non-shift-
symmetric axion

$\\{{\cal
D},a,Q,Q^{\dagger},L,L^{\dagger},H,H^{\dagger},u,u^{\dagger},d,d^{\dagger},e,e^{\dagger},B_{L},B_{R},W_{L},W_{R},G_{L},G_{R}\\}$
,

  * •

$\text{aLEFT}_{\text{PQ}}$: LEFT extended with a shift-symmetric axion

$\\{{\cal D},\partial
a,u_{L}^{\vphantom{\dagger}},u_{L}^{\dagger},u_{R}^{\vphantom{\dagger}},u_{R}^{\dagger},d_{L}^{\vphantom{\dagger}},d_{L}^{\dagger},d_{R}^{\vphantom{\dagger}},d_{R}^{\dagger},e_{L}^{\vphantom{\dagger}},e_{L}^{\dagger},e_{R}^{\vphantom{\dagger}},e_{R}^{\dagger},\nu_{L}^{\vphantom{\dagger}},\nu_{L}^{\dagger},F_{L},F_{R},G_{L},G_{R}\\}$
,

  * •

$\text{aLEFT}_{\cancel{\text{PQ}}}$: LEFT extended with a non-shift-symmetric
axion

$\\{{\cal
D},a,u_{L}^{\vphantom{\dagger}},u_{L}^{\dagger},u_{R}^{\vphantom{\dagger}},u_{R}^{\dagger},d_{L}^{\vphantom{\dagger}},d_{L}^{\dagger},d_{R}^{\vphantom{\dagger}},d_{R}^{\dagger},e_{L}^{\vphantom{\dagger}},e_{L}^{\dagger},e_{R}^{\vphantom{\dagger}},e_{R}^{\dagger},\nu_{L}^{\vphantom{\dagger}},\nu_{L}^{\dagger},F_{L},F_{R},G_{L},G_{R}\\}$
,

where we have followed the convention in Ref. Henning _et al._ (2017a) to only
use the $\left(\frac{1}{2},0\right)$ representation left-handed fermions to
form the Hilbert series, and the superscript “$c$” for the right-handed
conjugate fields $u^{c}_{(R)},d^{c}_{(R)}$ and $e^{c}_{(R)}$ are omitted when
we calculate the Hilbert series. The field strength $X=F,B,W,G$ is redefined
to extract the chiral components $X^{\mu\nu}_{L,R}=\frac{1}{2}(X^{\mu\nu}\pm
i\tilde{X}^{\mu\nu})$ that transform as $(1,0)$ and $(0,1)$ under the Lorentz
group respectively. The representations and charges of these SM fields under
Lorentz group $SO(4)$ and gauge groups $SU(3)_{c}\otimes SU(2)_{W}\otimes
U(1)_{Y}$ are shown explicitly in Ref. Henning _et al._ (2017a), we will not
show them here. The axion field transforms as a singlet under all groups.

With the spurions shown for $\text{aSMEFT}_{\cancel{\text{PQ}}}$, as we
mentioned previously, we can use the Haar measures and characters in Ref.
Henning _et al._ (2017a) to calculate the Hilbert series easily based on Eq.
(2.8). The only difference compared to the SMEFT lies in the inclusion of the
axion field $a$, which can be easily handled by adding additional PE of the
spurion $a$ in the calculation of the Hilbert series. For
$\text{aSMEFT}_{\text{PQ}}$, we only need to change the spurion “$a$” to
“$\partial a$”, and adopt the character in Eq. (2.13), the calculation of
Hilbert series will also be straightforward.

For the axion in LEFT, all left-handed and right-handed fermion fields are
completely independent below the electroweak (EW) scale as a UV completion
could be chiral (LEFT captures both SMEFT- and HEFT-like UV completions). The
$SU(2)_{W}$ symmetry is broken, and $U(1)_{\text{em}}$ is generated with the
charge given by $Q=Y+T_{3}$. The calculation of the Hilbert series of
$\text{aLEFT}_{\text{PQ}}$ and $\text{aLEFT}_{\cancel{\text{PQ}}}$ follows the
same fashion, except that we don’t need to integrate over the $SU(2)$ group,
which makes the calculation even easier. Similarly, in the calculation, we
need to take care of the spurions “$\partial a$” and “$a$”, and use the
corresponding shift and non-shift-symmetric character respectively.

Throughout this paper we will use a grading in the mass dimension of the
fields to obtain the Hilbert series at each order in the EFT expansion. For
this, we will rescale the spurions with their mass dimensions
$\phi\to\epsilon\phi$ for scalars, $\psi\to\epsilon^{3/2}\psi$ for fermions,
$X\to\epsilon^{2}X$ for field strengths, $\partial a\to\epsilon^{2}\partial a$
and ${\cal D}\to\epsilon{\cal D}$ for the covariant derivative. We define the
graded Hilbert series as ${\cal H}(\epsilon)=\sum_{i}\epsilon^{i}{\cal
H}_{i}$. It should be noted that the calculation of the full Hilbert series is
impossible. However, in the construction of the operator basis, it suffices to
focus only on a specific mass dimension.

The calculation of the Hilbert series can be simplified by working only with
the corresponding mass dimension of the integrand in Eq. (2.8). This means
that we can expand $\text{PE}/P$ to the desired mass dimension first and then
perform the integration. Nevertheless, expanding the integrand itself becomes
challenging at higher mass dimensions. To address this, a FORM code called ECO
(Efficient Counting of Operators) Marinissen _et al._ (2020) has been
developed specifically for efficient Hilbert series calculation. For this
project, we developed our own Mathematica code to generate the Hilbert series
that can be used efficiently at higher mass dimensions. It allowed us to
compute the Hilbert series up to dimension 15 for all axion EFTs. Our code is
designed with a broader scope in mind and we intend to publish it in a
forthcoming paper Grojean _et al._ , making it readily available whenever the
explicit form of the Hilbert series is needed.

##  3 aSMEFT

We are now ready to calculate the Hilbert series and construct all operators
of the SMEFT extended with a light pseudoscalar. We will start with an ALP,
i.e., a pseudoscalar with a shift symmetry, stemming from its Goldstone boson
nature under the spontaneously broken global $U(1)_{\text{PQ}}$ symmetry.

###  3.1 $\text{aSMEFT}_{\text{PQ}}$

Using the tools we have introduced in the last section, we can calculate the
Hilbert series for the given spurions that define the EFT at low energies. For
the explicit shift symmetry we will work with a derivatively coupled
pseudoscalar $\partial a$ as mentioned in the last section. Evaluating Eq.
(2.8) for the given spurions, we find the Hilbert series for one generation of
fermions up to mass dimension 8 to be333Note that ${\cal H}_{5}^{\text{PQ}}$
here only corresponds to the first term in Eq. (2.8) and we still have to add
$\Delta{\cal H}$, which is non-trivial here, to get the correct full result.
We also remove the pure SMEFT operators from our Hilbert series to only
capture the axion coupled operators.

$\displaystyle{\cal H}_{5}^{\text{PQ}}\,=\,$ $\displaystyle\partial
a\,QQ^{\dagger}+\partial a\,uu^{\dagger}+\partial a\,dd^{\dagger}+\partial
a\,LL^{\dagger}+\partial a\,ee^{\dagger}+\partial a\,HH^{\dagger}{\cal D}$
$\displaystyle-\partial a\,B_{L}{\cal D}-\partial a\,B_{R}{\cal D}-\partial
a\,{\cal D}^{3}\,,$ $\displaystyle{\cal H}_{6}^{\text{PQ}}\,=\,$
$\displaystyle(\partial a)^{2}HH^{\dagger}\,,$ $\displaystyle{\cal
H}_{7}^{\text{PQ}}\,=\,$ $\displaystyle\partial a\,QQ^{\dagger}B_{L}+\partial
a\,QQ^{\dagger}B_{R}+\partial a\,QQ^{\dagger}G_{L}+\partial
a\,QQ^{\dagger}G_{R}+\partial a\,QQ^{\dagger}W_{L}+\partial
a\,QQ^{\dagger}W_{R}$ $\displaystyle+\partial a\,uu^{\dagger}B_{L}+\partial
a\,uu^{\dagger}B_{R}+\partial a\,uu^{\dagger}G_{L}+\partial
a\,uu^{\dagger}G_{R}+\partial a\,dd^{\dagger}B_{L}+\partial
a\,dd^{\dagger}B_{R}$ $\displaystyle+\partial a\,dd^{\dagger}G_{L}+\partial
a\,dd^{\dagger}G_{R}+\partial a\,LL^{\dagger}B_{L}+\partial
a\,LL^{\dagger}B_{R}+\partial a\,LL^{\dagger}W_{L}+\partial
a\,LL^{\dagger}W_{R}$ $\displaystyle+\partial a\,ee^{\dagger}B_{L}+\partial
a\,ee^{\dagger}B_{R}+2\partial a\,QQ^{\dagger}HH^{\dagger}+\partial
a\,uu^{\dagger}HH^{\dagger}+\partial a\,dd^{\dagger}HH^{\dagger}$
$\displaystyle+2\partial a\,LL^{\dagger}HH^{\dagger}+\partial
a\,ee^{\dagger}HH^{\dagger}+\partial a\,B_{L}HH^{\dagger}{\cal D}+\partial
a\,B_{R}HH^{\dagger}{\cal D}+\partial a\,W_{L}HH^{\dagger}{\cal D}$
$\displaystyle+\partial a\,W_{R}HH^{\dagger}{\cal D}+\partial
a\,H^{2}H^{\dagger 2}{\cal D}+2\partial a\,QuH{\cal D}+2\partial
a\,Q^{\dagger}u^{\dagger}H^{\dagger}{\cal D}+2\partial a\,QdH^{\dagger}{\cal
D}$ (3.1) $\displaystyle+2\partial a\,Q^{\dagger}d^{\dagger}H{\cal
D}+2\partial a\,LeH^{\dagger}{\cal D}+2\partial
a\,L^{\dagger}e^{\dagger}H{\cal D}\,,$ $\displaystyle{\cal
H}_{8}^{\text{PQ}}\,=\,$ $\displaystyle(\partial a)^{4}+(\partial
a)^{2}QQ^{\dagger}{\cal D}+(\partial a)^{2}uu^{\dagger}{\cal D}+(\partial
a)^{2}dd^{\dagger}{\cal D}+(\partial a)^{2}LL^{\dagger}{\cal D}+(\partial
a)^{2}ee^{\dagger}{\cal D}$ $\displaystyle+(\partial a)^{2}B_{L}^{2}+(\partial
a)^{2}B_{L}B_{R}+(\partial a)^{2}B_{R}^{2}+(\partial a)^{2}G_{L}^{2}+(\partial
a)^{2}G_{L}G_{R}+(\partial a)^{2}G_{R}^{2}$ $\displaystyle+(\partial
a)^{2}W_{L}^{2}+(\partial a)^{2}W_{L}W_{R}+(\partial a)^{2}W_{R}^{2}+\partial
a\,Qd^{\dagger 2}L^{\dagger}+\partial a\,Q^{\dagger}d^{2}L+\partial
a\,ud^{\dagger}L^{\dagger 2}$ $\displaystyle+\partial
a\,u^{\dagger}dL^{2}+2(\partial a)^{2}HH^{\dagger}{\cal D}^{2}+2\partial
a\,L^{2}H^{2}{\cal D}+2\partial a\,L^{\dagger 2}H^{\dagger 2}{\cal
D}+(\partial a)^{2}H^{2}H^{\dagger 2}$ $\displaystyle+(\partial
a)^{2}QuH+(\partial a)^{2}Q^{\dagger}u^{\dagger}H^{\dagger}+(\partial
a)^{2}QdH^{\dagger}+(\partial a)^{2}Q^{\dagger}d^{\dagger}H+(\partial
a)^{2}LeH^{\dagger}$ $\displaystyle+(\partial
a)^{2}L^{\dagger}e^{\dagger}H\,.$

We can interpret the Hilbert series as follows. Every term corresponds to an
operator with the field content given by the spurions and the multiplicity
given by the prefactor. For the first term in ${\cal H}_{5}^{\text{PQ}}$, for
instance, we expect one operator with the field content
$\partial_{\mu}a,Q,Q^{\dagger}$. With this information, it is simple to build
a gauge and Lorentz invariant operator
$\partial_{\mu}a\,\bar{Q}\gamma^{\mu}Q$.

The negative terms appearing at dimension 5 do not correspond to non-redundant
operators and they are canceled exactly by other terms in $\Delta{\cal H}$ as
mentioned in the last section. They correspond to co-closed but not co-exact
forms and can be calculated immediately from the expression of $\Delta{\cal
H}$ given in Ref. Henning _et al._ (2017b). Evaluating the expression, we find
for the terms that are relevant for the ALP EFT444Note that the ${\cal D}^{4}$
term that usually appears in ${\cal H}_{0}$ and is canceled by a term in
$\Delta{\cal H}$ does not appear here because we only keep terms in the
Hilbert series which include at least on ALP field $a$.

$\Delta{\cal H}=\partial a\,B_{L}{\cal D}+\partial a\,B_{R}{\cal D}+\partial
a\,{\cal D}^{3}\,,$ (3.2)

which exactly cancel the negative terms in ${\cal H}_{0}$. The form of the
terms is similar to those found in the discussion of the QCD chiral Lagrangian
in Ref. Graf _et al._ (2021), where a more involved case of a non-linearly
realized symmetry is analyzed. The only difference is due to the different
counting of mass dimension of the spurions capturing the Goldstone degrees of
freedom. As mentioned in the last section, in the case of the ALP we have
simplified $u_{\mu}=i\frac{\partial_{\mu}a}{f}$ where we put the $1/f$
suppression into the Wilson coefficient. Hence, in our case $\left[\partial
a\right]=2$, whereas usually $\left[u_{\mu}\right]=1$. Therefore terms like
$u\,B_{L}{\cal D}\sim\partial a\,B_{L}{\cal D}$ appear at dimension 5 here.
Note that a term of the form of the last term in Eq. (3.2) only appears for
scalars that are singlets under the gauge group Henning _et al._ (2017b),
while the other terms can also appear for non-Abelian gauge groups if the
scalar transforms in the adjoint representation.

We should also note that the Hilbert series in Eq. (3.1) only takes one flavor
of fermions into account. Sometimes this is not enough for constructing an
operator basis with multiple flavors because some operator structures are only
non-vanishing for multiple flavors of fermions. In order to construct an
operator basis for a general number of flavors $N_{f}$, we need to capture the
dependence of the Hilbert series on $N_{f}$. This can be easily realized by
making $N_{f}$ copies of the corresponding fermions’ PE in the calculation of
the Hilbert series, which is equivalent to simply adding a factor of $N_{f}$
in front of the fermionic part of the PE (c.f. Eq. (2.3)). Indeed, looking at
the Hilbert series at dimension 8 for generic $N_{f}$, we find the following
terms

${\cal
H}_{8}^{\text{PQ}}\supset\frac{1}{3}N_{f}^{2}\left(N_{f}^{2}-1\right)\partial
a\,d^{\dagger 3}e+\frac{1}{3}N_{f}^{2}\left(N_{f}^{2}-1\right)\partial
a\,d^{3}e^{\dagger}\,,$ (3.3)

which evidently vanishes for $N_{f}=1$. The reason why these terms only appear
for $N_{f}>1$ is the antisymmetric color structure of the down quarks in that
operator which only gives rise to a non-vanishing operator if there are at
least two different flavors of down quarks (c.f. operator ${\cal O}_{\partial
aed}$ in Tab. 4).

The Hilbert series in Eq.(3.1) does not quite give the correct number of non-
redundant operators and some further adjustments have to be performed. First
of all, the redundant operator ${\cal O}_{\partial
aH}=\partial^{\mu}a\,\Bigl{(}H^{\dagger}i\overleftrightarrow{D}_{\mu}H\Bigr{)}$
corresponding to the term $\partial a\,HH^{\dagger}{\cal D}$ in ${\cal
H}_{5}^{\text{PQ}}$ is not removed automatically.555We can check that this
does not happen again at higher mass dimensions using Eq. (3.11). Without
imposing a shift symmetry any operator that exactly gives an EOM redundant
operator upon using IBP will be removed because one derivative is no longer
fixed to the ALP by demanding derivatively coupled ALP interactions. This is
because it can be removed by a global hypercharge transformation on the Higgs
field which is not captured in our Hilbert series approach. In general, all
derivative couplings of the ALP to SM particles are only defined up to
redefinitions by exact global symmetries Georgi _et al._ (1986). This is also
relevant for the conservation of baryon and lepton number. We did not impose
the condition $\partial_{\mu}j^{\mu}=0$ for conserved currents in our
approach. This applies to the operators of type
$\partial_{\mu}a\,\bar{\psi}\gamma^{\mu}\psi$ where flavor diagonal parts of
the Wilson coefficients can be removed by moving the derivative to the
fermions by IBP and using the conservation of baryon and lepton family number
$\partial_{\mu}j_{B}^{\mu}=\partial_{\mu}j_{L_{i}}^{\mu}=0$ Bonilla _et al._
(2021).666To be precise, baryon number and lepton family number are both only
conserved classically and are anomalous at loop level. Taking the anomalies
into account shifts the Wilson coefficients of the operators ${\cal
O}_{a\tilde{B}}$ and ${\cal O}_{a\tilde{W}}$ Bonilla _et al._ (2021).  In
principal we could also implement a shortening condition through conformal
characters that removes redundancies of the form $\partial_{\mu}j^{\mu}=0$
Henning _et al._ (2017b). Then, one would have to implement the conserved
currents explicitly into the Hilbert series, so it is easier to just remove
these redundancies by hand at the end, as we do here.

Secondly, there are operators of the form $aF\widetilde{F}$ at mass dimension
5 which do not appear in the Hilbert series. This is due to the fact that we
use $\partial a$ and $F$ as a building block because they have nice
transformation properties under gauge and Lorentz transformations. One can
easily show by moving the derivative from the first field strength by IBP to
the ALP that this operator is shift-symmetric. Then however, the gauge field
$A_{\mu}$ appears by itself and we would have to use the gauge fields
themselves as a building block which is practically unfeasible with the
Hilbert series, and would also be inconvenient from the perspective of
constructing the operators. Alternatively, as is well-known, after shifting
the ALP by a constant $a\to a+c$, chiral transformations can remove the shift
in front of this type of operator. It will prove useful that we also build an
operator basis for an ALP without a shift symmetry in Section 3.2, where we
use just $a$ as a building block and notice that after taking the shift-
symmetric limit the $aF\widetilde{F}$ type operators will remain. In this way
we can systematically include this type of operator in our construction.

For most terms in the Hilbert series it is straightforward to build Lorentz
and gauge invariant operators from the spurion content and get the correct
number of independent operators as indicated by the Hilbert series. There is
one exception that is a bit more involved, the operators of type $(\partial
a)^{2}X^{2}$ at dimension 8. From the Hilbert series in Eq. (3.1), we can read
of that we should expect three non-redundant operators $X_{L}^{2}(\partial
a)^{2}$, $X_{L}X_{R}(\partial a)^{2}$ and $X_{R}^{2}(\partial a)^{2}$.
However, one can naively build 4 operators

$\partial_{\mu}a\partial^{\mu}a\,B_{\nu\rho}B^{\nu\rho},\quad\partial_{\mu}a\partial^{\mu}a\,B_{\nu\rho}\widetilde{B}^{\nu\rho},\quad\partial_{\mu}a\partial^{\nu}a\,B^{\mu\rho}B_{\nu\rho},\quad\partial_{\mu}a\partial^{\nu}a\,B^{\mu\rho}\widetilde{B}_{\nu\rho}$
(3.4)

sharing a complicated relation that renders one of the operators redundant. We
can use the Schouten identity (see e.g. Ref. Chala _et al._ (2022))

$g_{\mu\nu}\epsilon_{\alpha\beta\gamma\delta}+g_{\mu\alpha}\epsilon_{\beta\gamma\delta\nu}+g_{\mu\beta}\epsilon_{\gamma\delta\nu\alpha}+g_{\mu\gamma}\epsilon_{\delta\nu\alpha\beta}+g_{\mu\delta}\epsilon_{\nu\alpha\beta\gamma}=0$
(3.5)

to relate the two operators with a dual field strength. Contracting the
indices in the identity with a generic rank-2 tensor $T^{\mu\nu}$ (which we
identify with $\partial^{\mu}a\partial^{\nu}a$) and an anti-symmetric rank-2
tensor $X_{\mu\nu}$ yields

$T_{\mu\nu}X^{\mu\rho}\widetilde{X}_{\hphantom{\nu}\rho}^{\nu}=\frac{1}{4}T_{\mu}^{\mu}X_{\nu\rho}\widetilde{X}^{\nu\rho}$
(3.6)

explaining the number of operators of that type in the Hilbert series. Our
complete basis up to mass dimension 8 can be found in Tabs. 1, 2, 3 and 4 in
App. A.1.

There are several ways to cross-check our results. As a sanity check for our
implementation of the Hilbert series we can use the $a^{0}$ terms to compare
our results for the Hilbert series with Ref. Henning _et al._ (2017a) and the
operators with the SMEFT operator basis up to dimension 8 Weinberg (1979);
Grzadkowski _et al._ (2010); Lehman (2014); Murphy (2020).

Furthermore, some results for the ALP EFT are available in the literature up
to dimension 7. Our results at dimension 5 are consistent with the usual basis
at dimension 5 (see e.g. Ref. Georgi _et al._ (1986)). The results at
dimension 6 are consistent with Ref. Bauer _et al._ (2017, 2019); Brivio _et
al._ (2021); Bonilla _et al._ (2021). Some results at dimension 7 can be found
in Ref. Bauer _et al._ (2016, 2017) and are consistent with our operator
basis. More results for higher-dimensional operators can be found in Ref.
Brivio _et al._ (2017) where the authors match some of the operators in the
chiral electroweak EFT extended with an axion to the one in a linear
realization. All operators they find are either in our basis or equivalent to
operators in our basis due to field redefinitions and IBP.

To get a feeling for how the number of operators $\\#\,{\cal
O}_{d_{i}}^{\text{PQ}}$ behaves as a function of the mass dimension $d_{i}$,
we set all field spurions in the Hilbert series with full flavor dependence to
unity such that only the dependence on $N_{f}$ remains. By rephasing all
fermionic spurions with lepton number and baryon number transformations
respectively, i.e. $\ell\to\epsilon_{L}\ell,\
\ell^{\dagger}\to\epsilon_{L}^{-1}\ell^{\dagger}$ and
$q\to\epsilon_{B}^{1/3}q,\ q^{\dagger}\to\epsilon_{B}^{-1/3}q^{\dagger}$, we
can in addition obtain the number of operators that break lepton and baryon
number at each mass dimension. Then for each power of $\epsilon_{B,L}$ in the
following expressions, baryon number and lepton number are violated by one
unit. After taking care of the caveats we have mentioned above by hand, we
find777After all terms have been simplified, we make the replacement
$\epsilon_{i}^{-n}\to\epsilon_{i}^{n}$, such that we count all operators in
the same way that break baryon and lepton number by $|n|$ units with respect
to the conserving case.

$\displaystyle\\#\,{\cal O}_{5}^{\text{PQ}}\,=\,2-N_{f}+5N_{f}^{2}\,,$
$\displaystyle\\#\,{\cal O}_{6}^{\text{PQ}}\,=\,1\,,$ $\displaystyle\\#\,{\cal
O}_{7}^{\text{PQ}}\,=\,5+39N_{f}^{2}\,,$ $\displaystyle\\#\,{\cal
O}_{8}^{\text{PQ}}\,=\,\left(13+11N_{f}^{2}\right)+\left(-\frac{2N_{f}^{2}}{3}+\frac{8N_{f}^{4}}{3}\right)\epsilon_{B}\epsilon_{L}+\left(4N_{f}^{2}+2N_{f}^{4}\right)\epsilon_{L}^{2}\,,$
(3.7) $\displaystyle\\#\,{\cal
O}_{9}^{\text{PQ}}\,=\,\left(74+\frac{1799N_{f}^{2}}{4}-\frac{N_{f}^{3}}{2}+\frac{847N_{f}^{4}}{4}\right)+46N_{f}^{4}\epsilon_{B}\epsilon_{L}+\left(N_{f}+N_{f}^{2}\right)\epsilon_{L}^{2}\,,$
$\displaystyle\\#\,{\cal
O}_{10}^{\text{PQ}}\,=\,\left(74+\frac{431N_{f}^{2}}{2}+\frac{91N_{f}^{4}}{2}\right)+\left(8N_{f}^{3}+106N_{f}^{4}\right)\epsilon_{B}\epsilon_{L}+\left(-N_{f}+75N_{f}^{2}+N_{f}^{3}+121N_{f}^{4}\right)\epsilon_{L}^{2}\,.$

One can also easily get the total number of operators by setting
$\epsilon_{B,L}\to 1$. Beyond mass dimension 10 the expressions become too
long to be presented here. We give more results in App. C.1 and the Hilbert
series up to mass dimension 15 with full spurion and flavor dependence in an
ancillary Mathematica notebook.888In the notebook, we have taken care of the
caveats mentioned above by hand and have also added $\Delta{\cal H}$ such that
the negative terms coming from co-closed but not co-exact forms are canceled,
so that the correct number of non-redundant operators can be obtained easily.
One notices that baryon and lepton number breaking operators only appear at
mass dimension 8. This is due to the derivative coupling of the ALP that only
allows for baryon and lepton number breaking terms through the coupling of
$\partial_{\mu}a$ to 4-fermion operators.

In Fig. 1, we have plotted the number of operators of the ALP EFT up to mass
dimension 15 for one and three flavors of fermions. One can see the usual
growth of operators with mass dimension where some unusual features appear for
the ALP EFT due to the derivative nature of the shift-symmetric ALP couplings.
The shift symmetry, for instance, only allows for one coupling at dimension 6.
At higher mass dimensions on the other hand, the multiplicity of operators
increases in the same manner as one is accustomed to from other EFTs.999There
is an interesting asymptotic scaling behavior Melia and Pal (2021) for the
number of operators in effective field theories that should also apply to the
plot in Fig. 1 for large mass dimensions beyond the drop at dimension 6.

Figure 1: The number of operators in the aSMEFT with and without a shift
symmetry for the ALP plotted against the mass dimension for $N_{f}=1$ and
$N_{f}=3$ number of flavors.

###  3.2 $\text{aSMEFT}_{\cancel{\text{PQ}}}$

In this section, we will analyze the difference between an ALP and a generic
pseudoscalar. Hence, we relax the assumption of a shift symmetry for the
pseudoscalar which no longer necessarily has to be connected to the
spontaneous breaking of a PQ symmetry. However, there can still be such a
connection by assuming that the spontaneously broken global symmetry is only
approximate. Then, all shift-breaking operators are understood as small
corrections to those that break the shift symmetry and it is important to
understand the limit of taking the shift symmetry.

As the shift symmetry is now relaxed, we can work with $a$ itself as a
building block for the Hilbert series. As in the previous section, we can
evaluate the Hilbert series from Eq. (2.8) using the appropriate characters
and Haar measures for the given field spurions and symmetries. For one
generation of fermions we obtain up to mass dimension 7

$\begin{split}{\cal
H}_{5}^{\cancel{\text{PQ}}}=&a^{5}+aB_{L}^{2}+aB_{R}^{2}+aG_{L}^{2}+aG_{R}^{2}+aW_{L}^{2}+aW_{R}^{2}+a^{3}HH^{\dagger}+aH^{2}H^{\dagger
2}+aQuH\\\
&+aQ^{\dagger}u^{\dagger}H^{\dagger}+aQdH^{\dagger}+aQ^{\dagger}d^{\dagger}H+aLeH^{\dagger}+aL^{\dagger}e^{\dagger}H\\\
=&a\,{\cal H}_{4}^{\text{SM}}+a^{5}+a^{3}HH^{\dagger}\,,\\\\[3.0pt] {\cal
H}_{6}^{\cancel{\text{PQ}}}=&a\,{\cal
H}_{5}^{\cancel{\text{PQ}}}+aH^{2}L^{2}+aH^{\dagger 2}L^{\dagger
2}+a^{2}HH^{\dagger}{\cal D}^{2}\,,\\\\[3.0pt] {\cal
H}_{7}^{\cancel{\text{PQ}}}=&a\,{\cal H}_{6}^{\cancel{\text{PQ}}}+a\,{\cal
H}_{6}^{\text{SMEFT}}+aQQ^{\dagger}B_{L}{\cal D}+aQQ^{\dagger}B_{R}{\cal
D}+aQQ^{\dagger}G_{L}{\cal D}+aQQ^{\dagger}G_{R}{\cal D}\\\
&+aQQ^{\dagger}W_{L}{\cal D}+aQQ^{\dagger}W_{R}{\cal
D}+auu^{\dagger}B_{L}{\cal D}+auu^{\dagger}B_{R}{\cal
D}+auu^{\dagger}G_{L}{\cal D}+auu^{\dagger}G_{R}{\cal D}\\\
&+add^{\dagger}B_{L}{\cal D}+add^{\dagger}B_{R}{\cal
D}+add^{\dagger}G_{L}{\cal D}+add^{\dagger}G_{R}{\cal
D}+aLL^{\dagger}B_{L}{\cal D}+aLL^{\dagger}B_{R}{\cal D}\\\
&+aLL^{\dagger}W_{L}{\cal D}+aLL^{\dagger}W_{R}{\cal
D}+aee^{\dagger}B_{L}{\cal D}+aee^{\dagger}B_{R}{\cal
D}+2aQQ^{\dagger}HH^{\dagger}{\cal D}\\\ &+auu^{\dagger}HH^{\dagger}{\cal
D}+add^{\dagger}HH^{\dagger}{\cal D}+2aLL^{\dagger}HH^{\dagger}{\cal
D}+aee^{\dagger}HH^{\dagger}{\cal D}+aB_{L}HH^{\dagger}{\cal D}^{2}\\\
&+aB_{R}HH^{\dagger}{\cal D}^{2}+aW_{L}HH^{\dagger}{\cal
D}^{2}+aW_{R}HH^{\dagger}{\cal D}^{2}+aH^{2}H^{\dagger 2}{\cal
D}^{2}+2aQuH{\cal D}^{2}\\\ &+2aQ^{\dagger}u^{\dagger}H^{\dagger}{\cal
D}^{2}+2aQdH^{\dagger}{\cal D}^{2}+2aQ^{\dagger}d^{\dagger}H{\cal
D}^{2}+2aLeH^{\dagger}{\cal D}^{2}+2aL^{\dagger}e^{\dagger}H{\cal
D}^{2}\,.\end{split}$ (3.8)

At dimension 5, ${\cal H}_{5}^{\text{PQ}}$ is absent and we find

${\cal H}_{5}^{\cancel{\text{PQ}}}=a{\cal H}_{4}^{\cancel{\text{PQ}}}+a{\cal
H}_{4}^{\text{SM}}\,,$ (3.9)

which is the well-known result that, at dimension 5, the fermionic operators
with the derivatively coupled ALP are redundant by the EOM Brivio _et al._
(2017); Bauer _et al._ (2021a); Chala _et al._ (2021); Bonilla _et al._
(2021). Allowing for lepton number breaking, we find the following relation at
the level of the Hilbert series

${\cal H}_{6}^{\cancel{\text{PQ}}}=a\,{\cal
H}_{5}^{\cancel{\text{PQ}}}+a\,{\cal H}_{5}^{\text{SMEFT}}+{\cal
H}_{6}^{\text{PQ}}\left(\partial a\to a{\cal D}\right)\,.$ (3.10)

Here, ${\cal H}_{i}^{\cancel{\text{PQ}}}$ is the Hilbert series of the ALP EFT
with $a$ as a building block, ${\cal H}_{i}^{\text{SMEFT}}$ is the Hilbert
series of the SMEFT and ${\cal H}_{6}^{\text{PQ}}$ is the Hilbert series of
the ALP EFT with $\partial a$ as a building block, each at mass dimension $i$.
The expression in the bracket is understood as replacing the spurion $\partial
a$ of the derivatively coupled ALP with the ALP spurion $a$ and the spurion of
the covariant derivative ${\cal D}$.

This relation at the level of the Hilbert series should hold true to any mass
dimension beyond dimension 5.101010We have checked that this separation
appears up to mass dimension 15 and believe that it holds also true for all
higher order operators. To get a mixing of the two sector one needs an EOM
relation between two of the operators which implies that the effects of one
operator is already captured by the other. In particular such an EOM relation
has to arise upon moving the derivative from the axion to the rest of the
operator using IBP. As the operators become more and more complicated for
higher mass dimensions, it is less and less likely that through this procedure
a structure is obtained which exactly resembles the EOM of an SM particle as
at dimension 5. In general, we conjecture that the Hilbert series fulfills the
following condition at mass dimension $n$

${\cal H}_{n}^{\cancel{\text{PQ}}}=a\,{\cal
H}_{n-1}^{\cancel{\text{PQ}}}+a\,{\cal H}_{n-1}^{\text{SMEFT}}+{\cal
H}_{n}^{\text{PQ}}(\partial a\to a{\cal D})\,,\qquad n>5$ (3.11)

which we have verified to hold true up to $n=15$. From now on we will refer to
this relation as the _Peccei–Quinn breaking isolation condition_ or _shift-
breaking isolation condition_. We want to emphasize the importance of this
equation. It states that above dimension 5 the EFT splits into a part
generated by simply multiplying the operators at the previous mass dimension –
which immediately follows from the singlet scalar nature of the ALP – and a
second part which is exactly the EFT built with a derivatively coupled, i.e.
explicitly shift-invariant ALP. This separation of the shift-breaking and
shift-symmetric sectors of the ALP EFT can be captured with the Hilbert series
in a concise way. We will explore the implications of this further in Section
3.3.

The operator construction for the shift-breaking case is trivial as the ALP is
a singlet both under the Lorentz and the gauge group. Then, one can multiply
any gauge and Lorentz invariant operator to receive a new Lorentz invariant
operator. Looking again at Eq. (3.11), it can be seen that, by this
construction, one can obtain the complete operator basis after adding the
derivatively coupled terms that we have constructed in Section 3.1.

Our complete basis for an ALP without a shift symmetry coupled to the SM
particles at mass dimension 5 can be found in Tab. 5. The operator bases up to
mass dimension 8 can be constructed easily with the shift-breaking isolation
condition, see App. A.2 for details. For completeness, we also give the
renormalizable part of the Lagrangian here, as a potential can be generated
for the ALP once the assumption of a shift symmetry for the ALP is loosened.
The renormalizable Lagrangian is given by

${\cal L}_{\leq
4}^{a}=\frac{1}{2}\partial_{\mu}a\,\partial^{\mu}a-\frac{m_{a,0}^{2}}{2}\,a^{2}+C_{a^{3}}\,a^{3}+C_{a^{4}}\,a^{4}+C_{aH^{2}}\,a\,|H|^{2}+C_{a^{2}H^{2}}\,a^{2}|H|^{2}\,.$
(3.12)

There are some results available in the literature. We have cross-checked our
operator basis with those of Refs. Gripaios and Sutherland (2016);
Franceschini _et al._ (2016) and find agreement.

As before, we can obtain a formula for the number of operators at each mass
dimension by setting all spurions to unity. We find

$\displaystyle\\#\,{\cal O}_{5}^{\cancel{\text{PQ}}}\,=\,$ $\displaystyle
9+6N_{f}^{2}\,,$ $\displaystyle\\#\,{\cal O}_{6}^{\cancel{\text{PQ}}}\,=\,$
$\displaystyle\left(10+6N_{f}^{2}\right)+\left(N_{f}+N_{f}^{2}\right)\epsilon_{L}^{2}\,,$
$\displaystyle\\#\,{\cal O}_{7}^{\cancel{\text{PQ}}}\,=\,$
$\displaystyle\left(30+\frac{315N_{f}^{2}}{4}+\frac{N_{f}^{3}}{2}+\frac{107N_{f}^{4}}{4}\right)+\left(\frac{2N_{f}^{2}}{3}+N_{f}^{3}+\frac{19N_{f}^{4}}{3}\right)\epsilon_{B}\epsilon_{L}+\left(N_{f}+N_{f}^{2}\right)\epsilon_{L}^{2}\,,$
$\displaystyle\\#\,{\cal O}_{8}^{\cancel{\text{PQ}}}\,=\,$
$\displaystyle\left(43+\frac{359N_{f}^{2}}{4}+\frac{N_{f}^{3}}{2}+\frac{107N_{f}^{4}}{4}\right)+\left(3N_{f}+\frac{41N_{f}^{2}}{3}+N_{f}^{3}+\frac{37N_{f}^{4}}{3}\right)\epsilon_{L}^{2}$
(3.13)
$\displaystyle+\left(2N_{f}^{3}+16N_{f}^{4}\right)\epsilon_{B}\epsilon_{L}\,.$

Note that lepton number violating terms already appear at dimension 6 for an
ALP without a shift symmetry, whereas for an ALP with a shift symmetry lepton
and baryon number violating terms only appear at dimension 8 because the ALP
no longer has to be derivatively coupled, and can for instance simply multiply
the Weinberg operator of the SMEFT to give a lepton number violating operator
at dimension 6. One can again find more results beyond dimension 8 in App. C.1
and the Hilbert series up to mass dimension 15 with full spurion content and
flavor dependence in the ancillary Mathematica notebook.

In Fig. 1, we have plotted the number of operators against the mass dimension
for the SMEFT extended with an ALP without a shift symmetry. When the shift
symmetry is relaxed, it is trivial to build new singlets under the gauge and
Lorentz group by just multiplying by $a$ as can also be seen in the structure
of the Hilbert series in Eq. (3.11). Comparing the number of operators at
dimension 5 between the explicitly shift-symmetric and non-shift-symmetric
Lagrangian in Fig 1, one can see that the difference $63-44=19$ corresponds
exactly to the number of shift-breaking invariants from Ref. Bonnefoy _et al._
(2023) (13) plus the number of conditions that have to be imposed on the
bosonic shift symmetry breaking operators (6) (c.f. Tab. 5). We will discuss
the shift-symmetric limit in more details in Section 3.3.

###  3.3 Taking the shift-symmetric limit

In the aSMEFT there is a subtlety in how to properly identify the shift-
symmetric and shift-breaking couplings of the ALP to the fermions at dimension
5. This is because, as we saw in Section 3.2, the dimension-5 operators
coupling the ALP derivatively to the SM particles become redundant due to the
fermion EOM once the shift symmetry is relaxed. After removing the EOM
redundancy, one can use the apparently non-shift-symmetric interactions
between the ALP and the SM particles to describe a shift-symmetric ALP given
that the couplings of the interactions follow a set of 13 relations Bonnefoy
_et al._ (2023) (see also Refs. Brivio _et al._ (2017); Bauer _et al._
(2021a); Chala _et al._ (2021); Bonilla _et al._ (2021) where the constraints
on the couplings were formulated as matrix relations in flavor space). This
leads to difficulties in the EFT picture because the same operator has to
capture physics corresponding to the shift-breaking and shift-symmetric sector
which usually arise at very different scales. Furthermore, taking the shift-
symmetric limit in the EFT where one uses $a$ as a building block requires
some care.

We will first present the results at dimension 5 known in the literature and
will then check if similar new relations arise at higher mass dimensions using
our results from the previous section. We consider the following Lagrangian

${\cal
L}^{a}=-\bar{L}Y_{e}He-\bar{Q}Y_{u}\tilde{H}u-\bar{Q}Y_{d}Hd+\frac{a}{f}\left(\bar{L}C_{ae}He+\bar{Q}C_{au}\tilde{H}u+\bar{Q}C_{ad}Hd\right)+\text{h.c.}\,.$
(3.14)

On a first glance, this Lagrangian does not look shift invariant. However,
after shifting the ALP $a\to a+c$, one can perform field redefinitions on the
fermion fields that allow to remove the shift at ${\cal
O}\left(\frac{1}{f}\right)$ from the Lagrangian by imposing the following
relations on the dimension-5 Wilson coefficients111111Another way to derive
these results can be found in App. D, where we start in the derivatively
coupled basis and make a change of basis to go to the Yukawa basis.

$C_{au}=i\left(c_{Q}Y_{u}-Y_{u}c_{u}\right),\
C_{ad}=i\left(c_{Q}Y_{d}-Y_{d}c_{d}\right),\
C_{ae}=i\left(c_{L}Y_{e}-Y_{e}c_{e}\right)\,.$ (3.15)

Then, the apparently non-shift-symmetric Lagrangian can be made shift-
symmetric. Furthermore, these matrix relations can be cast into order
parameters which allow to implement the different power countings of the
shift-breaking and shift-conserving sector in a straightforward way Bonnefoy
_et al._ (2023). From now on, we will refer to the Lagrangian in Eq. (3.14) as
the Yukawa basis given that the relations in Eq. (3.15) are fulfilled.

We will now explore if similar relations exist in the non-shift-symmetric EFT
at higher mass dimensions using the operator basis we have derived above. The
first observation we want to make is based on the Peccei–Quinn breaking
isolation condition in Eq. (3.11). For $n>5$ we have found previously

${\cal H}_{n}^{\cancel{\text{PQ}}}=a\,{\cal
H}_{n-1}^{\cancel{\text{PQ}}}+a\,{\cal H}_{n-1}^{\text{SMEFT}}+{\cal
H}_{n}^{\text{PQ}}\left(\partial a\to a{\cal D}\right)\,.$ (3.16)

The Hilbert series which is obtained by imposing the shift symmetry
explicitly, ${\cal H}_{n}^{\text{PQ}}$, appears fully in the Hilbert series of
the theory where $a$ itself is used as a spurion.

This implies that beyond dimension-5 no further EOM redundancies appear and
all operator structures stay non-redundant in the presence of shift-breaking
interactions. Therefore, if one decides to work in the operator basis with
derivatively coupled interactions at dimension-5, all shift-symmetric
couplings are exactly captured by the derivative interactions.

One has to be more careful when working in the Yukawa basis, which is the more
natural basis in the presence of shift-breaking effects as we will comment on
below. Here, one has to take care while removing the EOM redundancy at
dimension 5 when higher-order operators are considered in the EFT.121212We
thank Quentin Bonnefoy for pointing this out and Hoa Vuong for sharing
calculations with us convincing us of the importance of those terms. In
particular, one should use field redefinitions instead of simply plugging in
the SM EOM to remove the derivatively coupled operators at dimension 5 and
keep all terms that are generated by these field redefinitions up to the
considered order in the EFT. We have done this carefully in App. D and find
that the field redefinition removing the EOM redundancy at dimension 5 indeed
generates more (seemingly shift-breaking) operators with fully constrained
Wilson coefficients that restore the shift symmetry.

This is important as the spontaneous and explicit breaking of PQ usually arise
at very different scales. Working in the Yukawa basis, these effects are
captured by the same operators and it is not straightforward how to implement
the correct power counting for both sectors. As was pointed out in Ref.
Bonnefoy _et al._ (2023), one way around this is to consider flavor invariants
acting as order parameters for the ALP shift symmetry. In this language it is
possible to consistently implement the power counting of the theory.

One also has to keep these relations in mind while taking the shift-symmetric
limit going from the EFT without a shift-symmetry to the EFT with a shift-
symmetry in the Yukawa basis. Instead of setting all non-derivatively coupled
operators to zero, one should set them to the constrained form that is found
applying the appropriate field redefinitions. Note that the Yukawa basis is in
some sense the more natural basis to perform this limit because the EOM
redundancy at dimension-5 requires the derivatively coupled fermionic operator
to be absent from the operator basis in favor of the Yukawa-like operator. We
list all the relations that have to be imposed in App. D.3.

These additional constrained interactions are crucial for explicit
calculations in the Yukawa basis. If the additional terms are not included,
one will run into results in the shift-symmetric EFT in the Yukawa basis which
are not shift-invariant. E.g. if two insertions of the dimension-5 ALP-Yukawa
couplings are considered, one must also add the diagram with the constrained
interaction of the dimension-6 ALP-Yukawa coupling. Note that up to
dimension-7 only the ALP-Yukawa operators with higher powers of the ALPs have
to be considered, and only starting at dimension-8 more operators with
constrained Wilson coefficients are generated by the field redefinition if
additional operators generated by applying the field redefinitions to SMEFT
operators are ignored.

The analysis presented here can also be understood from an amplitudes point of
view Bertuzzo _et al._ by imposing the Adler zero condition Adler (1965a, b).
Here, special care has to be taken in how to impose the Adler zero condition
leading to the well-known conditions on the dimension-5 couplings and more
relations at higher mass dimensions, consistent with our analysis with the
Hilbert series and field redefinitions. In the amplitudes approach, these
relations can be understood from fundamental properties of amplitudes like
analyticity and regularity of the amplitude in the limit of soft ALP momenta
(for details see Ref. Bertuzzo _et al._ ).

For the aLEFT, that we will construct in the next section, one can perform a
similar analysis which works in the same way. Therefore, we will skip the
discussion of the shift symmetric limit there.

##  4 aLEFT

With the Hilbert series implemented for the SMEFT extended with an ALP, it is
fairly straightforward to also construct it for the EFT below the electroweak
scale. As discussed in Section 2, the main difference is the different
particle content where the heavy particles of the SM, the $W,Z,t$ and $h$, now
have been integrated out and the fact that the left-handed fermions are no
longer related through their appearance in $SU(2)$ doublets. Since the gauge
group below the EW scale is only $SU(3)_{c}\times U(1)_{\text{em}}$, both a
linear and a non-linear realization of the EW symmetry can be captured in the
LEFT. Hence, we can capture effects of HEFT-like ALP couplings to the SM
particles Brivio _et al._ (2017); Bonnefoy _et al._ (2021); Altmannshofer _et
al._ (2022) in the LEFT extended with an ALP. Furthermore, the effective
description of these interactions is of importance for experiments operating
at these scales, for instance meson decays giving flavor constraints on ALP
couplings to fermions Bauer _et al._ (2022). Performing the construction and
matching to the QCD chiral Lagrangian Georgi _et al._ (1986); Grilli di
Cortona _et al._ (2016); Bauer _et al._ (2021a, b) beyond leading order would
complete a full EFT description beyond leading order at all scales.

###  4.1 $\text{aLEFT}_{\text{PQ}}$

We start again by calculating the Hilbert series for the effective operators
by mass dimension. We present here the Hilbert series up to mass dimension 7

$\displaystyle{\cal H}_{5}^{\text{aLEFT}_{\text{PQ}}}\,=\,$
$\displaystyle\partial a\,u_{L}u_{L}^{\dagger}+\partial
a\,u_{R}u_{R}^{\dagger}+\partial a\,d_{L}d_{L}^{\dagger}+\partial
a\,d_{R}d_{R}^{\dagger}+\partial a\,\nu_{L}\nu_{L}^{\dagger}+\partial
a\,e_{L}e_{L}^{\dagger}+\partial a\,e_{R}e_{R}^{\dagger}$
$\displaystyle-\partial a\,F_{L}{\cal D}-\partial a\,F_{R}{\cal D}-\partial
a\,{\cal D}^{3}\,,$ $\displaystyle{\cal
H}_{6}^{\text{aLEFT}_{\text{PQ}}}\,=\,$ $\displaystyle 0\,,$
$\displaystyle{\cal H}_{7}^{\text{aLEFT}_{\text{PQ}}}\,=\,$
$\displaystyle(\partial a)^{2}u_{L}u_{R}+(\partial
a)^{2}u_{L}^{\dagger}u_{R}^{\dagger}+(\partial a)^{2}d_{L}d_{R}+(\partial
a)^{2}d_{L}^{\dagger}d_{R}^{\dagger}+(\partial a)^{2}\nu_{L}^{2}+(\partial
a)^{2}\nu_{L}^{\dagger 2}$ $\displaystyle+(\partial a)^{2}e_{L}e_{R}+(\partial
a)^{2}e_{L}^{\dagger}e_{R}^{\dagger}+\partial
a\,u_{L}u_{L}^{\dagger}F_{L}+\partial a\,u_{L}u_{L}^{\dagger}F_{R}+\partial
a\,u_{L}u_{L}^{\dagger}G_{L}$ (4.1) $\displaystyle+\partial
a\,u_{L}u_{L}^{\dagger}G_{R}+\partial a\,u_{R}u_{R}^{\dagger}F_{L}+\partial
a\,u_{R}u_{R}^{\dagger}F_{R}+\partial a\,u_{R}u_{R}^{\dagger}G_{L}+\partial
a\,u_{R}u_{R}^{\dagger}G_{R}$ $\displaystyle+\partial
a\,d_{L}d_{L}^{\dagger}F_{L}+\partial a\,d_{R}d_{R}^{\dagger}F_{L}+\partial
a\,d_{L}d_{L}^{\dagger}F_{R}+\partial a\,d_{R}d_{R}^{\dagger}F_{R}+\partial
a\,d_{L}d_{L}^{\dagger}G_{L}$ $\displaystyle+\partial
a\,d_{R}d_{R}^{\dagger}G_{L}+\partial a\,d_{L}d_{L}^{\dagger}G_{R}+\partial
a\,d_{R}d_{R}^{\dagger}G_{R}+\partial
a\,\nu_{L}\nu_{L}^{\dagger}F_{L}+\partial a\,\nu_{L}\nu_{L}^{\dagger}F_{R}$
$\displaystyle+\partial a\,e_{L}e_{L}^{\dagger}F_{L}+\partial
a\,e_{R}e_{R}^{\dagger}F_{L}+\partial a\,e_{L}e_{L}^{\dagger}F_{R}+\partial
a\,e_{R}e_{R}^{\dagger}F_{R}\,,$

and more results can be found in App. C.2 and in the ancillary Mathematica
notebook.

As for the SMEFT extended with an axion, it is mostly straightforward to
identify the independent Lorentz and gauge invariant operator structures from
the spurions in the Hilbert series. Due to the increased number of independent
fermions below the electroweak scale the number of operators increases quickly
in the aLEFT. In particular, it can be challenging to identify the many
4-fermion operators coupled to $\partial a$ at dimension 8. Using Fierz
identities from Refs. Liao and Liu (2012); Liao and Ma (2017), one can show
that the operators in Tab. 8 are the only non-redundant ones.

Something else that one has to keep in mind is that the flavor symmetries
below the EW scale change with respect to what we have discussed before as the
left-handed fermions no longer come together in doublets. Therefore, all mass
terms can be diagonalized and the misalignment captured by the
Cabibbo–Kobayashi–Maskawa (CKM) matrix that one finds above the EW scale is
shifted to dimension 6 in the $1/m_{W}$ expansion allowing for more possible
rephasings for the quarks. Instead of only applying baryon number rephasings
on the quark fields, we can rephase each flavor of quarks by themselves. In
the lepton sector, the charged leptons keep their lepton family number
rephasing properties while the neutrinos do not benefit from any flavor
symmetries due to the lepton number breaking term
$\bar{\nu}_{L}\nu_{L}^{c}+\text{h.c.}$ that we allow for in the most generic
Lagrangian. In total we expect $N_{u}+N_{d}+N_{e}$ conserved currents that
enable us to remove operators via imposing $\partial_{\mu}j^{\mu}=0$ after
integrating by parts. As in the aSMEFT, we have to impose these conditions by
hand after calculating the Hilbert series. As discussed in the last section,
we have to add $aF\widetilde{F},aG\widetilde{G}$ to the operator basis by hand
as well.

Our complete basis up to mass dimension 8 for an ALP coupled derivatively to
all particles in the SM below the EW scale can be found in Tabs. 6, 7 and 8 in
App. B.1. The operator basis at mass dimension 5 in the aLEFT is consistent
with the operators used in Ref. Bauer _et al._ (2021a). As before, we can use
the ALP-independent terms in the Hilbert series as a sanity check for our
implementation of the Hilbert series and compare them with the known results
for the operator bases up to dimension 8 in the LEFT Jenkins _et al._ (2018);
Liao _et al._ (2020); Murphy (2021).

Figure 2: The number of operators in the aLEFT with and without a shift
symmetry for the ALP plotted against the mass dimension for
$N_{u}=N_{d}=N_{\nu}=N_{e}=1$ and $N_{u}=2,N_{d}=N_{\nu}=N_{e}=3$ number of
flavors. Note that the y-axis has a linear scaling between 0 and 1 to
accommodate for the 0 at mass dimension 6.

With the Hilbert series calculated, we will now repeat the analysis of the
flavor dependence we performed for the aSMEFT. As before, we count the number
of independent operators for generic number of flavors using lepton and baryon
number rephasings to single out the lepton and baryon number breaking
operators. Using $N_{\nu}=N_{e}=N_{d}$131313We keep $N_{u}$ independent here,
since we want to take the limit $N_{\nu}=N_{e}=N_{d}=3$, $N_{u}=2$ later
amounting to the usual flavor content of the EFT below the EW scale. to keep
the expressions more concise, we find

$\begin{split}\\#\,{\cal
O}_{5}^{\text{aLEFT}_{\text{PQ}}}\,=\,&2-2N_{d}+5N_{d}^{2}-N_{u}+2N_{u}^{2}\,,\\\
\\#\,{\cal O}_{6}^{\text{aLEFT}_{\text{PQ}}}\,=\,&0\,,\\\ \\#\,{\cal
O}_{7}^{\text{aLEFT}_{\text{PQ}}}\,=\,&\left(18N_{d}^{2}+10N_{u}^{2}\right)+\left(N_{d}+N_{d}^{2}\right)\epsilon_{L}^{2}\,,\\\
\\#\,{\cal
O}_{8}^{\text{aLEFT}_{\text{PQ}}}\,=\,&\left(7+23N_{d}^{2}+36N_{d}^{4}+16N_{d}^{3}N_{u}+14N_{u}^{2}+52N_{d}^{2}N_{u}^{2}+8N_{u}^{4}\right)\\\
&+\left(-\frac{4N_{d}^{2}}{3}+\frac{16N_{d}^{4}}{3}+16N_{d}^{3}N_{u}+16N_{d}^{2}N_{u}^{2}\right)\epsilon_{B}\epsilon_{L}\\\
&+\left(-N_{d}+\frac{7N_{d}^{2}}{3}+\frac{26N_{d}^{4}}{3}+16N_{d}^{3}N_{u}+4N_{d}^{2}N_{u}^{2}\right)\epsilon_{L}^{2}\,.\end{split}$
(4.2)

As before, shift symmetry protects from lepton number breaking. The lepton
number breaking, gauge-invariant neutrino mass term
$\bar{\nu}_{L}^{c}\nu_{L}+\text{h.c.}$ that can be written down below the EW
scale allows for lepton number breaking already at dimension 7.

###  4.2 $\text{aLEFT}_{\cancel{\text{PQ}}}$

We will now proceed by calculating the Hilbert series for the LEFT extended
with a generic scalar field $a$ that can but no longer necessarily has to be
connected to the spontaneous breaking of a PQ symmetry. The first two terms in
the expansion of the Hilbert series in the mass dimension of the operators are
given by

$\begin{split}{\cal
H}_{5}^{\text{aLEFT}_{\cancel{\text{PQ}}}}\,=\,&a^{5}+a^{2}u_{L}u_{R}+a^{2}u_{L}^{\dagger}u_{R}^{\dagger}+a^{2}d_{L}d_{R}+a^{2}d_{L}^{\dagger}d_{R}^{\dagger}+a^{2}\nu_{L}^{2}+a^{2}\nu_{L}^{\dagger
2}+a^{2}e_{L}e_{R}\\\
&+a^{2}e_{L}^{\dagger}e_{R}^{\dagger}+aF_{L}^{2}+aF_{R}^{2}+aG_{L}^{2}+aG_{R}^{2}\,,\\\
{\cal
H}_{6}^{\text{aLEFT}_{\cancel{\text{PQ}}}}\,=\,&a^{6}+a^{3}u_{L}u_{R}+a^{3}u_{L}^{\dagger}u_{R}^{\dagger}+a^{3}d_{L}d_{R}+a^{3}d_{L}^{\dagger}d_{R}^{\dagger}+a^{3}\nu_{L}^{2}+a^{3}\nu_{L}^{\dagger
2}+a^{3}e_{L}e_{R}\\\
&+a^{3}e_{L}^{\dagger}e_{R}^{\dagger}+a^{2}F_{L}^{2}+a^{2}F_{R}^{2}+a^{2}G_{L}^{2}+a^{2}G_{R}^{2}+au_{L}u_{R}F_{L}+au_{L}u_{R}G_{L}\\\
&+au_{L}^{\dagger}u_{R}^{\dagger}F_{R}+au_{L}^{\dagger}u_{R}^{\dagger}G_{R}+ad_{L}d_{R}F_{L}+ad_{L}^{\dagger}d_{R}^{\dagger}F_{R}+ad_{L}d_{R}G_{L}\\\
&+ad_{L}^{\dagger}d_{R}^{\dagger}G_{R}+ae_{L}e_{R}F_{L}+ae_{L}^{\dagger}e_{R}^{\dagger}F_{R}\,.\\\
\end{split}$ (4.3)

From these, one can already see again that the same structure appears as for
the SMEFT. Firstly, due to the ALP being a singlet pseudoscalar operators can
simply be built by multiplying operators at the previous mass dimension with
$a$. Secondly, beyond dimension 5 we find a similar PQ-breaking isolation
condition that we have found for the aSMEFT

${\cal H}_{n}^{\text{aLEFT}_{\cancel{\text{PQ}}}}=a\,{\cal
H}_{n-1}^{\text{aLEFT}_{\cancel{\text{PQ}}}}+a\,{\cal
H}_{n-1}^{\text{LEFT}}+{\cal H}_{n}^{\text{aLEFT}_{\text{PQ}}}\left(\partial
a\to a{\cal D}\right)\,$ (4.4)

Therefore the same discussion about shift symmetry that we will present in the
following section also applies to the aLEFT.

In the aLEFT, there exists a subtlety in how to count the couplings of a
single ALP to the fermion mass terms. This can be seen by studying the
renormalizable part of the Lagrangian

$\begin{split}{\cal L}_{\leq
4}^{a}=&\frac{1}{2}\partial_{\mu}a\partial^{\mu}a-\frac{m_{a,0}^{2}}{2}a^{2}+C_{a^{3}}a^{3}+C_{a^{4}}a^{4}+\frac{a}{f}\left(\bar{u}_{L}C_{au}^{SR}u_{R}+\bar{d}_{L}C_{ad}^{SR}d_{R}+\bar{e}_{L}C_{ae}^{SR}e_{R}\right.\\\
&\left.+\bar{\nu}_{L}C_{a\nu}^{SR}\nu_{L}^{c}+\text{h.c.}\right)\,,\end{split}$
(4.5)

that we have neglected up to this point. Here, one can see that the operators
that couple one ALP field to the fermions naively already appear at dimension
4, instead of dimension 5 in the SMEFT analysis. If one however performs a
matching of the aSMEFT to the aLEFT one obtains

$\\{C_{au}^{SR},C_{ad}^{SR},C_{ae}^{SR}\\}=\frac{v}{\sqrt{2}}\\{C_{au},C_{ad},C_{ae}\\}\quad\text{and}\quad
C_{a\nu}^{SR}=\frac{v^{2}}{2}C_{aLH}$ (4.6)

after expanding the Higgs around its vacuum expectation value $v$ and the
operators can be identified with dimension-5 and dimension-6 operators in the
aSMEFT.

The operator basis at dimension 5 can be found in Tab. 11. Based on the shift-
breaking isolation condition in Eq. (4.4), the higher dimensional operator
bases can be constructed easily, see App. B.2 for details.

We can once more compare the numbers in Fig. 2 to see if we can understand the
results in terms of the invariants from Ref. Bonnefoy _et al._ (2023). As in
the dimension-5 aSMEFT, the derivatively coupled ALP interactions with the
fermions become redundant in the presence of the dimension-5 ALP Yukawa
couplings. We have just discussed that the Yukawa ALP operators already
naively appear at mass dimension 4 in the aLEFT, so we have to be careful in
our discussion. Looking at Tab. 11, we can see that the same number of ALP-
fermion interactions appear at dimension 4 and dimension 5 in the aLEFT
because one operator can just be obtained by multiplying the other one by $a$.
Hence, we can rely on the numbers at dimension 5 to understand the counting.

The difference at dimension 5 in Fig. 2 can be explained as follows. According
to Ref. Bonnefoy _et al._ (2023), the 13 conditions for shift invariance
reduce to 8 invariants below the electroweak scale at leading order. The
remaining 4 invariants that are present due to the correlations induced by the
left-handed quark doublet get shifted to higher mass dimensions in the
$1/m_{W}^{2}$ expansion after integrating out the $W$ from the aSMEFT141414If
one starts from a HEFT-like scenario, these correlations will not be there in
the first place which is both captured in the aLEFT in higher-dimensional
operators upon matching to a HEFT-/SMEFT-like scenario. and one more invariant
is removed because the top quark is integrated out. With respect to Ref.
Bonnefoy _et al._ (2023) we also allow for lepton number breaking here which
allows for a neutrino mass term. This implies that there are an additional 3
relations in the fermionic sector. In total, we have to subtract the 11
conditions obtained from the fermionic sector at dimension 4 (which are
counted in the same way at dimension 5), the 1 condition obtained from
removing the operator $a^{5}$ and the 2 conditions obtained from removing
$aFF$ and $aGG$ from the 61 terms at dimension 5 which yields exactly the 47
terms in Fig. 2.

We have also once more performed the counting of operators for each mass
dimension by setting all spurions to unity and applying the same procedure as
before to single out the lepton and baryon number violating terms. We find

$\displaystyle\\#\,{\cal O}_{5}^{\text{aLEFT}_{\cancel{\text{PQ}}}}\,=\,$
$\displaystyle\left(5+4N_{d}^{2}+2N_{u}^{2}\right)+\left(N_{d}+N_{d}^{2}\right)\epsilon_{L}^{2}\,,$
$\displaystyle\\#\,{\cal O}_{6}^{\text{aLEFT}_{\cancel{\text{PQ}}}}\,=\,$
$\displaystyle\left(5+10N_{d}^{2}+6N_{u}^{2}\right)+2N_{d}^{2}\epsilon_{L}^{2}\,,$
$\displaystyle\\#\,{\cal O}_{7}^{\text{aLEFT}_{\cancel{\text{PQ}}}}\,=\,$
$\displaystyle\left(7+\frac{131N_{d}^{2}}{4}+\frac{3N_{d}^{3}}{2}+\frac{87N_{d}^{4}}{4}+10N_{d}^{3}N_{u}+19N_{u}^{2}+32N_{d}^{2}N_{u}^{2}+5N_{u}^{4}\right)$
$\displaystyle+\left(-\frac{4N_{d}^{2}}{3}-2N_{d}^{3}+\frac{10N_{d}^{4}}{3}-4N_{d}^{2}N_{u}+10N_{d}^{3}N_{u}+10N_{d}^{2}N_{u}^{2}\right)\epsilon_{B}\epsilon_{L}$
$\displaystyle+\left(N_{d}+3N_{d}^{2}+2N_{d}^{3}+6N_{d}^{4}+10N_{d}^{3}N_{u}+N_{d}N_{u}^{2}+3N_{d}^{2}N_{u}^{2}\right)\epsilon_{L}^{2}$
$\displaystyle+\left(-\frac{N_{d}^{2}}{6}+\frac{N_{d}^{4}}{6}\right)\epsilon_{L}^{4}\,,$
(4.7) $\displaystyle\\#\,{\cal
O}_{8}^{\text{aLEFT}_{\cancel{\text{PQ}}}}\,=\,$
$\displaystyle\left(14+\frac{335N_{d}^{2}}{4}-\frac{N_{d}^{3}}{2}+\frac{303N_{d}^{4}}{4}+34N_{d}^{3}N_{u}+53N_{u}^{2}+110N_{d}^{2}N_{u}^{2}+17N_{u}^{4}\right)$
$\displaystyle+\left(-\frac{4N_{d}^{2}}{3}+2N_{d}^{3}+\frac{34N_{d}^{4}}{3}+4N_{d}^{2}N_{u}+34N_{d}^{3}N_{u}+34N_{d}^{2}N_{u}^{2}\right)\epsilon_{B}\epsilon_{L}$
$\displaystyle+\left(4N_{d}+10N_{d}^{2}-3N_{d}^{3}+19N_{d}^{4}+34N_{d}^{3}N_{u}-N_{d}N_{u}^{2}+9N_{d}^{2}N_{u}^{2}\right)\epsilon_{L}^{2}$
$\displaystyle+\left(-\frac{N_{d}^{2}}{6}+\frac{N_{d}^{4}}{6}\right)\epsilon_{L}^{4}\,.$

Again, we only show the leading results here and the remaining results with
full spurion and flavor dependence can be found in the ancillary notebook.
Note that due to the operator $a\bar{\nu}\nu_{L}^{c}+\text{h.c.}$, a lepton-
number violating term can already be written down at dimension 4 (it has the
same dependence on the number of flavors as the corresponding term at mass
dimension 5 quoted at the end of $\\#\,{\cal
O}_{5}^{\text{aLEFT}_{\cancel{\text{PQ}}}}$ in Eq. (4.2)), which is the
operator that captures the effects of the derivatively coupled operator
$\partial_{\mu}a\,\bar{\nu}_{L}\gamma^{\mu}\nu_{L}$ that does not violate
lepton number. This only makes sense if the coefficient of
$a\bar{\nu}\nu_{L}^{c}+\text{h.c.}$ is proportional to the renormalizable
spurion of lepton number breaking $m_{\nu}$ which is indeed the case as one
can check from the usual relations one expects at dimension 5.

##  5 Conclusions

In this paper, we have investigated how shift-breaking effects in ALP EFTs
above and below the electroweak scale can be captured beyond the leading
interactions at dimension 5. For that we have constructed operator bases up to
dimension 8 using the Hilbert series and presented the results for the Hilbert
series up to dimension 15, making it possible to extend our operator bases to
higher mass dimensions if needed. We provide the full results of the Hilbert
series in an ancillary file including the full $N_{f}$ dependence. These
results can be used to perform the operator counting for any specific $N_{f}$.
We have furthermore identified the operators associated with violation of
baryon and lepton number in the full expression of the Hilbert series. The
Hilbert series is calculated using our own Mathematica code, which is
specifically designed to address general problems. This code will be made
publicly available in an upcoming publication Grojean _et al._ , allowing
researchers to use it for a wide range of applications beyond the scope of
this study.

At the level of the Hilbert series, we have found what we call the PQ-breaking
isolation condition stating that beyond mass dimension 5 the operators
describing shift-breaking couplings of the ALP to the SM are clearly isolated
from the shift-preserving couplings. This is in stark contrast with the
dimension-5 interactions where the two sectors mix due to the EOM redundancy
that relate the derivatively coupled operators with fermions to the ALP-Yukawa
couplings. We have discussed how to properly take care of the EOM redundancy
at dimension-5 when higher order operators are considered in the EFT. Then,
more relations beyond the well-known relations at dimension-5 should be
imposed when going from the derivatively coupled basis to the Yukawa basis. We
have constructed those relations and show them explicitly up to dimension-8.

There are several ways in which one could proceed with our results. First, one
can study how the higher-dimensional shift-symmetric operators influence
phenomenology. Since at dimension 5 the ALP already receives couplings to all
SM particles except the Higgs, and at dimension 7 the ALP is coupled to all SM
particles, we do not expect large corrections. The only exception could be
specific channels which do not get a contribution in the EFT at lower mass
order or which rely on intermediate particles from the SM implying that their
amplitudes do not grow (as fast) with energy as a pure contact interaction.
Another interesting possibility would be to study the interplay of shift-
symmetric and shift-breaking operators if the scale of explicit PQ-breaking is
not much larger than the scale of spontaneous breaking. In regards to low-
energy experiments, our complete basis for the LEFT extended with an ALP
should prove helpful to perform analyses beyond the leading order (c.f. for
instance Ref. Bauer _et al._ (2021a, 2022)). With the full operator basis at
dimension 8 one could study the implications of positivity on ALP EFTs,
extending the analysis of Ref. Kim _et al._ (2023) in the scalar sector.
Finally, with a complete basis one could extend the efforts of Refs. Chala _et
al._ (2021); Bauer _et al._ (2021a); Das Bakshi _et al._ (2023) and calculate
the renormalization group equations of operators of higher mass dimension and
their contributions to the renormalization group equations of operators at
dimension 5.

## Acknowledgments

We thank Enrico Bertuzzo and Gabriel Massoni Salla for enlightening
discussions on the axion scattering amplitudes that give an alternative way to
obtain a basis of the axion interactions to the SM particles. We thank
Guilherme Guedes for discussions on operator basis redundancies and Hoa Vuong
for discussions on shift-breaking effects in ALP EFTs. Furthermore, we thank
Emanuele Gendy for collaboration at early stages of this project and Quentin
Bonnefoy for numerous discussions on this topic and comments on the
manuscript. We thank Katharina Albrecht for cross-checking some of the results
in this paper.

This work 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. This
research was supported in part by Perimeter Institute for Theoretical Physics.
Research at Perimeter Institute is supported by the Government of Canada
through the Department of Innovation, Science and Economic Development and by
the Province of Ontario through the Ministry of Research and Innovation.
C.Y.Y. is supported in part by the Grants No. NSFC-11975130, No.
NSFC-12035008, No. NSFC-12047533, the National Key Research and Development
Program of China under Grant No. 2017YFA0402200, the China Postdoctoral
Science Foundation under Grant No. 2018M641621 and the Helmholtz-OCPC
International Postdoctoral Exchange Fellowship Program.

## Note added

During the final stages of this project, the papers Song _et al._ (2023a, b)
appeared on the arXiv. There, the EFT describing the interactions of a
pseudoscalar, with and without a shift symmetry, with the SM particles above
the electroweak symmetry breaking scale was also considered. The Young tensor
methods was used to find the Lorentz and SM gauge group singlets, which is an
orthogonal approach to the Hilbert series methods we have used. The main
result of our work is the identification of the particular structure of the
shift-symmetry of an ALP in the EFT picture. We have cross-checked our results
with those in Refs. Song _et al._ (2023a, b), and find agreement for the
number of operators for $N_{f}=1,3$, and the bases of operators seem to be
equivalent after applying appropriate transformations.

##  Appendix A Operator basis for the aSMEFT up to mass dimension 8

Using the Hilbert series as a guide, we have constructed independent operator
bases for aSMEFT, encompassing dimensions up to 8, for both shift-symmetric
and non-shift-symmetric theories, which are shown explicitly in the following
two subsections.

###  A.1 With shift symmetry

For aSMEFT with a shift symmetry, the operator bases from dimension 5 to
dimension 8 are constructed, they are grouped in Tabs. 1, 2, 3 and 4
respectively. The operators are classified according to the reduced Hilbert
series, we identify fermion with $\psi$, field strength with $X$, and scalar
with $H$, and also taking $\partial a$ as a building block, the Hilbert series
with $N_{f}=1$ for each mass dimension will be reduced as

$\begin{split}{\cal H}_{5}^{\text{PQ}}\,=\,&5\partial
a\,\psi^{2}+3aX^{2}\,,\qquad{\cal H}_{6}^{\text{PQ}}\,=\,(\partial
a)^{2}H^{2}\,,\\\ {\cal H}_{7}^{\text{PQ}}\,=\,&20\partial
a\,\psi^{2}X+4\partial a\,XH^{2}{\cal D}+7\partial a\,\psi^{2}H^{2}+\partial
a\,H^{4}{\cal D}+12\partial a\,\psi^{2}H{\cal D}\,,\\\ {\cal
H}_{8}^{\text{PQ}}\,=\,&(\partial a)^{4}+5(\partial a)^{2}\psi^{2}{\cal
D}+9(\partial a)^{2}X^{2}+4\partial a\,\psi^{4}+2(\partial a)^{2}H^{2}{\cal
D}^{2}+4\partial a\,\psi^{2}H^{2}{\cal D}\\\ &+(\partial
a)^{2}H^{4}+6(\partial a)^{2}\psi^{2}H+[2\partial a\,\psi^{4}]\,,\end{split}$
(A.1)

where for the Hilbert series at dimension 5, we have taken care of the caveats
discussed in Section 3.1, i.e., the Higgs coupled term and the negative terms
are removed, and the $aX^{2}$ terms are added by hand. As already mentioned in
Section 3.1, setting $N_{f}=1$ will lead to vanishing terms. In order to
construct an operator basis for general $N_{f}$, such vanishing terms should
also be taken into account, which are given as additional terms in the
brackets in Eq. (A.1) and the corresponding operators are marked with
$(\star)$ in the tables of operator bases. For instance, the additional terms
$[2\partial a\,\psi^{4}]$ correspond to the operator ${\cal O}_{\partial
aed}+\text{h.c.}$ in Tab. 4.

$\partial a\,\psi^{2}$ | $aX^{2}$
---|---
${\cal O}_{\partial aL}$ | $\partial_{\mu}a\,\left(\bar{L}\gamma^{\mu}L\right)$ | ${\cal O}_{a\tilde{B}}$ | $aB_{\mu\nu}\widetilde{B}^{\mu\nu}$
${\cal O}_{\partial ae}$ | $\partial_{\mu}a\,\left(\bar{e}\gamma^{\mu}e\right)$ | ${\cal O}_{a\tilde{W}}$ | $aW_{\mu\nu}^{I}\widetilde{W}^{I,\mu\nu}$
${\cal O}_{\partial aQ}$ | $\partial_{\mu}a\,\left(\bar{Q}\gamma^{\mu}Q\right)$ | ${\cal O}_{a\tilde{G}}$ | $aG_{\mu\nu}^{a}\widetilde{G}^{a,\mu\nu}$
${\cal O}_{\partial au}$ | $\partial_{\mu}a\,\left(\bar{u}\gamma^{\mu}u\right)$ |  |
${\cal O}_{\partial ad}$ | $\partial_{\mu}a\,\left(\bar{d}\gamma^{\mu}d\right)$ |  |
Table 1: Operators in the aSMEFT at mass dimension 5 with $\partial a$ as a
building block. Note that ${\cal O}_{\partial
aH}=\partial^{\mu}a\,\Bigl{(}H^{\dagger}i\overleftrightarrow{D}_{\mu}H\Bigr{)}$
is a redundant operator and can be removed via a global hypercharge
transformation Georgi _et al._ (1986); Bauer _et al._ (2021a). Imposing lepton
and baryon number conservation at the level of the renormalizable Lagrangian,
3 (1) flavor diagonal entries of the operators coupling the ALP to leptons
(quarks), for instance ${\cal O}_{\partial aL,ii}$ and ${\cal O}_{\partial
aQ,11}$, can be removed Bonilla _et al._ (2021). Furthermore, we have used
that the shift in the operators of class $aX^{2}$ can be removed using
anomalous chiral transformations on the fermion fields making the operators
shift-symmetric without an explicit derivative on the axion field. $(\partial
a)^{2}H^{2}$
---
${\cal O}_{\partial a^{2}H^{2}}$ | $\partial_{\mu}a\,\partial^{\mu}a\,|H|^{2}$
Table 2: Operators in the aSMEFT at mass dimension 6 with $\partial a$ as a building block. $\partial a\,\psi^{2}X$ | $\partial a\,XH^{2}D$
---|---
${\cal O}_{\partial aLB}$ | $\partial^{\mu}a\,\left(\bar{L}\gamma^{\nu}L\right)B_{\mu\nu}$ | ${\cal O}_{\partial aHB}$ | $\partial_{\mu}a\,\Bigl{(}H^{\dagger}i\overleftrightarrow{D}_{\nu}H\Bigr{)}B^{\mu\nu}$
${\cal O}_{\partial aL\tilde{B}}$ | $\partial^{\mu}a\,\left(\bar{L}\gamma^{\nu}L\right)\widetilde{B}_{\mu\nu}$ | ${\cal O}_{\partial aH\tilde{B}}$ | $\partial_{\mu}a\,\Bigl{(}H^{\dagger}i\overleftrightarrow{D}_{\nu}H\Bigr{)}\widetilde{B}^{\mu\nu}$
${\cal O}_{\partial aeB}$ | $\partial^{\mu}a\,\left(\bar{e}\gamma^{\nu}e\right)B_{\mu\nu}$ | ${\cal O}_{\partial aHW}$ | $\partial_{\mu}a\,\Bigl{(}H^{\dagger}i\overleftrightarrow{D}_{\nu}^{I}H\Bigr{)}W^{I,\mu\nu}$
${\cal O}_{\partial ae\tilde{B}}$ | $\partial^{\mu}a\,\left(\bar{e}\gamma^{\nu}e\right)\widetilde{B}_{\mu\nu}$ | ${\cal O}_{\partial aH\tilde{W}}$ | $\partial_{\mu}a\,\Bigl{(}H^{\dagger}i\overleftrightarrow{D}_{\nu}^{I}H\Bigr{)}\widetilde{W}^{I,\mu\nu}$
${\cal O}_{\partial aQB}$ | $\partial^{\mu}a\,\left(\bar{Q}\gamma^{\nu}Q\right)B_{\mu\nu}$ | $\partial a\,\psi^{2}H^{2}$
${\cal O}_{\partial aQ\tilde{B}}$ | $\partial^{\mu}a\,\left(\bar{Q}\gamma^{\nu}Q\right)\widetilde{B}_{\mu\nu}$ | ${\cal O}_{\partial aLH^{2}}^{(1)}$ | $\partial_{\mu}a\,\left(\bar{L}\gamma^{\mu}L\right)|H|^{2}$
${\cal O}_{\partial auB}$ | $\partial^{\mu}a\,\left(\bar{u}\gamma^{\nu}u\right)B_{\mu\nu}$ | ${\cal O}_{\partial aLH^{2}}^{(2)}$ | $\partial_{\mu}a\,\left(\bar{L}\gamma^{\mu}\sigma^{I}L\right)\left(H^{\dagger}\sigma^{I}H\right)$
${\cal O}_{\partial au\tilde{B}}$ | $\partial^{\mu}a\,\left(\bar{u}\gamma^{\nu}u\right)\widetilde{B}_{\mu\nu}$ | ${\cal O}_{\partial aeH^{2}}$ | $\partial_{\mu}a\,\left(\bar{e}\gamma^{\mu}e\right)|H|^{2}$
${\cal O}_{\partial adB}$ | $\partial^{\mu}a\,\left(\bar{d}\gamma^{\nu}d\right)B_{\mu\nu}$ | ${\cal O}_{\partial aQH^{2}}^{(1)}$ | $\partial_{\mu}a\,\left(\bar{Q}\gamma^{\mu}Q\right)|H|^{2}$
${\cal O}_{\partial ad\tilde{B}}$ | $\partial^{\mu}a\,\left(\bar{d}\gamma^{\nu}d\right)\widetilde{B}_{\mu\nu}$ | ${\cal O}_{\partial aQH^{2}}^{(2)}$ | $\partial_{\mu}a\,\left(\bar{Q}\gamma^{\mu}\sigma^{I}Q\right)\left(H^{\dagger}\sigma^{I}H\right)$
${\cal O}_{\partial aLW}$ | $\partial^{\mu}a\,\left(\bar{L}\gamma^{\nu}\sigma^{I}L\right)W_{\mu\nu}^{I}$ | ${\cal O}_{\partial auH^{2}}$ | $\partial_{\mu}a\,\left(\bar{u}\gamma^{\mu}u\right)|H|^{2}$
${\cal O}_{\partial aL\tilde{W}}$ | $\partial^{\mu}a\,\left(\bar{L}\gamma^{\nu}\sigma^{I}L\right)\widetilde{W}_{\mu\nu}^{I}$ | ${\cal O}_{\partial adH^{2}}$ | $\partial_{\mu}a\,\left(\bar{d}\gamma^{\mu}d\right)|H|^{2}$
${\cal O}_{\partial aQW}$ | $\partial^{\mu}a\,\left(\bar{Q}\gamma^{\nu}\sigma^{I}Q\right)W_{\mu\nu}^{I}$ | $\partial a\,\psi^{2}HD+\text{h.c.}$
${\cal O}_{\partial aQ\tilde{W}}$ | $\partial^{\mu}a\,\left(\bar{Q}\gamma^{\nu}\sigma^{I}Q\right)\widetilde{W}_{\mu\nu}^{I}$ | ${\cal O}_{\partial aeHD}^{(1)}$ | $\partial_{\mu}a\left(D^{\mu}\bar{L}\right)He$
${\cal O}_{\partial aQG}$ | $\partial^{\mu}a\,\left(\bar{Q}\gamma^{\nu}T^{a}Q\right)G_{\mu\nu}^{a}$ | ${\cal O}_{\partial aeHD}^{(2)}$ | $\partial_{\mu}a\,\bar{L}H\left(D^{\mu}e\right)$
${\cal O}_{\partial aQ\tilde{G}}$ | $\partial^{\mu}a\,\left(\bar{Q}\gamma^{\nu}T^{a}Q\right)\widetilde{G}_{\mu\nu}^{a}$ | ${\cal O}_{\partial auHD}^{(1)}$ | $\partial_{\mu}a\left(D^{\mu}\bar{Q}\right)\widetilde{H}u$
${\cal O}_{\partial auG}$ | $\partial^{\mu}a\,\left(\bar{u}\gamma^{\nu}T^{a}u\right)G_{\mu\nu}^{a}$ | ${\cal O}_{\partial auHD}^{(2)}$ | $\partial_{\mu}a\,\bar{Q}\widetilde{H}\left(D^{\mu}u\right)$
${\cal O}_{\partial au\tilde{G}}$ | $\partial^{\mu}a\,\left(\bar{u}\gamma^{\nu}T^{a}u\right)\widetilde{G}_{\mu\nu}^{a}$ | ${\cal O}_{\partial adHD}^{(1)}$ | $\partial_{\mu}a\left(D^{\mu}\bar{Q}\right)Hd$
${\cal O}_{\partial adG}$ | $\partial^{\mu}a\,\left(\bar{d}\gamma^{\nu}T^{a}d\right)G_{\mu\nu}^{a}$ | ${\cal O}_{\partial adHD}^{(2)}$ | $\partial_{\mu}a\,\bar{Q}H\left(D^{\mu}d\right)$
${\cal O}_{\partial ad\tilde{G}}$ | $\partial^{\mu}a\,\left(\bar{d}\gamma^{\nu}T^{a}d\right)\widetilde{G}_{\mu\nu}^{a}$ | $\partial a\,H^{4}D$
|  | ${\cal O}_{\partial aH^{4}}$ | $\partial^{\mu}a\,\Bigl{(}H^{\dagger}i\overleftrightarrow{D}_{\mu}H\Bigr{)}|H|^{2}$
Table 3: Operators in the aSMEFT at mass dimension 7 with $\partial a$ as a building block.  $(\partial a)^{2}X^{2}$ | $(\partial a)^{2}\psi^{2}D$
---|---
${\cal O}_{\partial a^{2}B}^{(1)}$ | $\partial_{\mu}a\partial^{\mu}a\,B_{\nu\rho}B^{\nu\rho}$ | ${\cal O}_{\partial a^{2}LD}$ | $\partial_{\mu}a\partial_{\nu}a\,\left(\bar{L}\gamma^{\mu}\overleftrightarrow{D}^{\nu}L\right)$
${\cal O}_{\partial a^{2}B}^{(2)}$ | $\partial_{\mu}a\partial^{\nu}a\,B^{\mu\rho}B_{\nu\rho}$ | ${\cal O}_{\partial a^{2}eD}$ | $\partial_{\mu}a\partial_{\nu}a\,\left(\bar{e}\gamma^{\mu}\overleftrightarrow{D}^{\nu}e\right)$
${\cal O}_{\partial a^{2}\tilde{B}}$ | $\partial_{\mu}a\partial^{\mu}a\,B_{\nu\rho}\widetilde{B}^{\nu\rho}$ | ${\cal O}_{\partial a^{2}QD}$ | $\partial_{\mu}a\partial_{\nu}a\,\left(\bar{Q}\gamma^{\mu}\overleftrightarrow{D}^{\nu}Q\right)$
${\cal O}_{\partial a^{2}W}^{(1)}$ | $\partial_{\mu}a\partial^{\mu}a\,W_{\nu\rho}^{I}W^{I,\nu\rho}$ | ${\cal O}_{\partial a^{2}uD}$ | $\partial_{\mu}a\partial_{\nu}a\,\left(\bar{u}\gamma^{\mu}\overleftrightarrow{D}^{\nu}u\right)$
${\cal O}_{\partial a^{2}W}^{(2)}$ | $\partial_{\mu}a\partial^{\nu}a\,W^{I,\mu\rho}W_{\nu\rho}^{I}$ | ${\cal O}_{\partial a^{2}dD}$ | $\partial_{\mu}a\partial_{\nu}a\,\left(\bar{d}\gamma^{\mu}\overleftrightarrow{D}^{\nu}d\right)$
${\cal O}_{\partial a^{2}\tilde{W}}^{(2)}$ | $\partial_{\mu}a\partial^{\mu}a\,W_{\nu\rho}^{I}\widetilde{W}^{I,\nu\rho}$ | $(\partial a)^{2}\psi^{2}H+\text{h.c.}$
${\cal O}_{\partial a^{2}G}^{(1)}$ | $\partial_{\mu}a\partial^{\mu}a\,G_{\nu\rho}^{a}G^{a,\nu\rho}$ | ${\cal O}_{\partial a^{2}eH}$ | $\partial_{\mu}a\partial^{\mu}a\,\bar{L}He$
${\cal O}_{\partial a^{2}G}^{(2)}$ | $\partial_{\mu}a\partial^{\nu}a\,G^{a,\mu\rho}G_{\nu\rho}^{a}$ | ${\cal O}_{\partial a^{2}uH}$ | $\partial_{\mu}a\partial^{\mu}a\,\bar{Q}Hu$
${\cal O}_{\partial a^{2}\tilde{G}}$ | $\partial_{\mu}a\partial^{\mu}a\,G_{\nu\rho}^{a}\widetilde{G}^{a,\nu\rho}$ | ${\cal O}_{\partial a^{2}dH}$ | $\partial_{\mu}a\partial^{\mu}a\,\bar{Q}Hd$
$(\partial a)^{4}$ | $(\partial a)^{2}H^{2}D^{2}$
${\cal O}_{\partial a^{4}}$ | $\partial_{\mu}a\partial^{\mu}a\partial_{\nu}a\partial^{\nu}a$ | ${\cal O}_{\partial a^{2}DH^{2}}^{(1)}$ | $\partial_{\mu}a\partial^{\mu}aD_{\nu}H^{\dagger}D^{\nu}H$
$(\partial a)^{2}H^{4}$ | ${\cal O}_{\partial a^{2}DH^{2}}^{(2)}$ | $\partial_{\mu}a\partial_{\nu}aD^{\mu}H^{\dagger}D^{\nu}H$
${\cal O}_{\partial a^{2}H^{4}}$ | $\partial_{\mu}a\partial^{\mu}a|H|^{4}$ |  |
$\not{B}$ and $\not{L}$ terms
$\partial a\,\psi^{4}+\text{h.c.}$ | $\partial a\,\psi^{2}H^{2}D+\text{h.c.}$
${\cal O}_{\partial aLdu}$ | $\partial_{\mu}a\,\left(\bar{L}^{c}L\right)\left(\bar{d}\gamma^{\mu}u\right)$ | ${\cal O}_{\partial aLHD}^{(1)}$ | $\partial_{\mu}a\,\left(\bar{L}^{c}H\right)\left(\tilde{H}^{\dagger}D^{\mu}L\right)$
${\cal O}_{\partial aLQd}$ | $\epsilon^{\alpha\beta\gamma}\partial_{\mu}a\,\left(\bar{L}d_{\alpha}\right)\left(\bar{Q}_{\beta}^{c}\gamma^{\mu}d_{\gamma}\right)$ | ${\cal O}_{\partial aLHD}^{(2)}$ | $\partial_{\mu}a\,\left(\bar{L}^{c}D^{\mu}H\right)\left(\tilde{H}^{\dagger}L\right)$
${\cal O}_{\partial aed}~{}(\star)$ | $\epsilon^{\alpha\beta\gamma}\partial_{\mu}a\,\left(\bar{d}_{\alpha}^{c}d_{\beta}\right)\left(\bar{e}\gamma^{\mu}d_{\gamma}\right)$ |  |
Table 4: Operators in the aSMEFT at mass dimension 8 with $\partial a$ as a
building block. Note that the operator ${\cal O}_{\partial aed}$ marked with
$(\star)$ only exists for $N_{f}\neq 1$ because otherwise all contractions of
the antisymmetric color structure will sum to zero (the first current in the
operator is symmetric under $\alpha\leftrightarrow\beta$ for one generation of
fermions).

###  A.2 Without shift symmetry

For aSMEFT without a shift symmetry, we can use the shift-breaking isolation
condition Eq. (3.11) to construct the operator basis easily. The operator
bases at higher dimensions can be constructed with

${\cal L}_{n}^{\cancel{\text{PQ}}}=a\,{\cal
L}_{n-1}^{\cancel{\text{PQ}}}+a\,{\cal L}_{n-1}^{\text{SMEFT}}+{\cal
L}_{n}^{\text{PQ}}$ (A.2)

for $n>5$. We start with the dimension-5 operator basis shown in Tab. 5, the
dimension-6, 7, and 8 operator bases can be constructed successively. For
instance, the operators at dimension 6 can be constructed with ${\cal
L}_{6}^{\cancel{\text{PQ}}}=a\,{\cal L}_{5}^{\cancel{\text{PQ}}}+a\,{\cal
L}_{5}^{\text{SMEFT}}+{\cal L}_{6}^{\text{PQ}}$, where the operators in ${\cal
L}_{5}^{\text{SMEFT}}$ have been shown in Ref. Weinberg (1979), and the
dimension-6 shift-symmetric operator basis associated with ${\cal
L}_{6}^{\text{PQ}}$ is given in Tab. 2. The construction of operator bases of
dimension 7 and 8 follows the same manner, and the SMEFT operator bases at
dimension 6 and 7 are needed Grzadkowski _et al._ (2010); Lehman (2014). For
completeness, the axion-dependent renormalizable operators can be found in Eq.
(3.12).

$V(a,H)$ | $aX^{2}$
---|---
${\cal O}_{a^{5}}$ | $a^{5}$ | ${\cal O}_{aB}$ | $aB_{\mu\nu}B^{\mu\nu}$
${\cal O}_{a^{3}H^{2}}$ | $a^{3}|H|^{2}$ | ${\cal O}_{a\tilde{B}}$ | $aB_{\mu\nu}\widetilde{B}^{\mu\nu}$
${\cal O}_{aH^{4}}$ | $a|H|^{4}$ | ${\cal O}_{aW}$ | $aW_{\mu\nu}^{I}W^{I,\mu\nu}$
$a\psi^{2}H+\text{h.c.}$ | ${\cal O}_{a\tilde{W}}$ | $aW_{\mu\nu}^{I}\widetilde{W}^{I,\mu\nu}$
${\cal O}_{ae}$ | $a\bar{L}He$ | ${\cal O}_{aG}$ | $aG_{\mu\nu}^{a}G^{a,\mu\nu}$
${\cal O}_{au}$ | $a\bar{Q}\tilde{H}u$ | ${\cal O}_{a\tilde{G}}$ | $aG_{\mu\nu}^{a}\widetilde{G}^{a,\mu\nu}$
${\cal O}_{ad}$ | $a\bar{Q}Hd$ |  |
Table 5: Operators in the aSMEFT at mass dimension 5 with $a$ as a building
block.

##  Appendix B Operator basis for the aLEFT up to mass dimension 8

###  B.1 With shift symmetry

By setting $N_{u,d,\nu,e}\to 1$ and restoring the vanishing terms, the reduced
Hilbert series for operator basis up to dimension 8 are given by

$\displaystyle{\cal H}_{5}^{\text{aLEFT}_{\text{PQ}}}\,=\,$ $\displaystyle
7\partial a\,\psi^{2}+2aX^{2}\,,\quad{\cal
H}_{6}^{\text{aLEFT}_{\text{PQ}}}\,=\,0\,,\quad{\cal
H}_{7}^{\text{aLEFT}_{\text{PQ}}}\,=\,8(\partial a)^{2}\psi^{2}+22\,\partial
a\,\psi^{2}X\,,$ $\displaystyle{\cal H}_{8}^{\text{aLEFT}_{\text{PQ}}}\,=\,$
$\displaystyle(\partial a)^{4}+7(\partial a)^{2}\psi^{2}{\cal D}+6(\partial
a)^{2}X^{2}+32\partial a\,\psi^{2}X{\cal D}+176\partial a\,\psi^{4}+[6\partial
a\,\psi^{4}]\,.$ (B.1)

The dimension-5 and dimension-7 operator bases are given in Tab. 6 and Tab. 7
respectively, and dimension-8 operator basis is collected in Tabs. 8, 9, 10.
We can see that the numbers of operators precisely match the numbers present
in the reduced Hilbert series.

$\partial a\,\psi^{2}$ |  $\partial a\,\psi^{2}$ (cont.)
---|---
${\cal O}_{\partial ae}^{VL}$ | $\partial_{\mu}a\,\left(\bar{e}_{L}\gamma^{\mu}e_{L}\right)$ | ${\cal O}_{\partial ad}^{VL}$ | $\partial_{\mu}a\,\left(\bar{d}_{L}\gamma^{\mu}d_{L}\right)$
${\cal O}_{\partial ae}^{VR}$ | $\partial_{\mu}a\,\left(\bar{e}_{R}\gamma^{\mu}e_{R}\right)$ | ${\cal O}_{\partial ad}^{VR}$ | $\partial^{\mu}a\,\left(\bar{d}_{R}\gamma^{\mu}d_{R}\right)$
${\cal O}_{\partial a\nu}^{VL}$ | $\partial_{\mu}a\,\left(\bar{\nu}_{L}\gamma^{\mu}\nu_{L}\right)$ | $aX^{2}$
${\cal O}_{\partial au}^{VL}$ | $\partial_{\mu}a\,\left(\bar{u}_{L}\gamma^{\mu}u_{L}\right)$ | ${\cal O}_{a\tilde{F}}$ | $aF_{\mu\nu}\widetilde{F}^{\mu\nu}$
${\cal O}_{\partial au}^{VR}$ | $\partial_{\mu}a\,\left(\bar{u}_{R}\gamma^{\mu}u_{R}\right)$ | ${\cal O}_{a\tilde{G}}$ | $aG_{\mu\nu}^{a}\widetilde{G}^{a,\mu\nu}$
Table 6: Operators in the aLEFT at mass dimension 5 with $\partial a$ as a building block. Note that imposing lepton and baryon number conservation, 3 flavor diagonal entries of the operators coupling the ALP to leptons and quarks, for instance ${\cal O}_{\partial ae,ii}^{VL}$, ${\cal O}_{\partial au,ii}^{VL}$ and ${\cal O}_{\partial ad,ii}^{VL}$, can be removed Bonilla _et al._ (2021). Furthermore, we have used that the shift in the operators of class $aX^{2}$ can be removed using anomalous chiral transformations on the fermion fields making the operators shift-symmetric without an explicit derivative on the axion field. $(\partial a)^{2}\psi^{2}+\text{h.c.}$ |  $\partial a\,X\psi^{2}$ (cont.)
---|---
${\cal O}_{\partial a^{2}e}^{SR}$ | $\partial_{\mu}a\partial^{\mu}a\,\left(\bar{e}_{L}e_{R}\right)$ | ${\cal O}_{\partial au\tilde{F}}^{VR}$ | $\partial^{\mu}a\,\left(\bar{u}_{R}\gamma^{\nu}u_{R}\right)\widetilde{F}_{\mu\nu}$
${\cal O}_{\partial a^{2}\nu}^{SR}\left(\not{L}\right)$ | $\partial_{\mu}a\partial^{\mu}a\,\left(\bar{\nu}_{L}\nu_{L}^{c}\right)$ | ${\cal O}_{\partial adF}^{VL}$ | $\partial^{\mu}a\,\left(\bar{d}_{L}\gamma^{\nu}d_{L}\right)F_{\mu\nu}$
${\cal O}_{\partial a^{2}u}^{SR}$ | $\partial_{\mu}a\partial^{\mu}a\,\left(\bar{u}_{L}u_{R}\right)$ | ${\cal O}_{\partial adF}^{VR}$ | $\partial^{\mu}a\,\left(\bar{d}_{R}\gamma^{\nu}d_{R}\right)F_{\mu\nu}$
${\cal O}_{\partial a^{2}d}^{SR}$ | $\partial_{\mu}a\partial^{\mu}a\,\left(\bar{d}_{L}d_{R}\right)$ | ${\cal O}_{\partial ad\tilde{F}}^{VL}$ | $\partial^{\mu}a\,\left(\bar{d}_{L}\gamma^{\nu}d_{L}\right)\widetilde{F}_{\mu\nu}$
$\partial a\,X\psi^{2}$ | ${\cal O}_{\partial ad\tilde{F}}^{VR}$ | $\partial^{\mu}a\,\left(\bar{d}_{R}\gamma^{\nu}d_{R}\right)\widetilde{F}_{\mu\nu}$
${\cal O}_{\partial aeF}^{VL}$ | $\partial^{\mu}a\,\left(\bar{e}_{L}\gamma^{\nu}e_{L}\right)F_{\mu\nu}$ | ${\cal O}_{\partial auG}^{VL}$ | $\partial^{\mu}a\,\left(\bar{u}_{L}\gamma^{\nu}T^{a}u_{L}\right)G_{\mu\nu}^{a}$
${\cal O}_{\partial aeF}^{VR}$ | $\partial^{\mu}a\,\left(\bar{e}_{R}\gamma^{\nu}e_{R}\right)F_{\mu\nu}$ | ${\cal O}_{\partial auG}^{VR}$ | $\partial^{\mu}a\,\left(\bar{u}_{R}\gamma^{\nu}T^{a}u_{R}\right)G_{\mu\nu}^{a}$
${\cal O}_{\partial ae\tilde{F}}^{VL}$ | $\partial^{\mu}a\,\left(\bar{e}_{L}\gamma^{\nu}e_{L}\right)\widetilde{F}_{\mu\nu}$ | ${\cal O}_{\partial au\tilde{G}}^{VL}$ | $\partial^{\mu}a\,\left(\bar{u}_{L}\gamma^{\nu}T^{a}u_{L}\right)\widetilde{G}_{\mu\nu}^{a}$
${\cal O}_{\partial ae\tilde{F}}^{VR}$ | $\partial^{\mu}a\,\left(\bar{e}_{R}\gamma^{\nu}e_{R}\right)\widetilde{F}_{\mu\nu}$ | ${\cal O}_{\partial au\tilde{G}}^{VR}$ | $\partial^{\mu}a\,\left(\bar{u}_{R}\gamma^{\nu}T^{a}u_{R}\right)\widetilde{G}_{\mu\nu}^{a}$
${\cal O}_{\partial a\nu F}^{VL}$ | $\partial^{\mu}a\,\left(\bar{\nu}_{L}\gamma^{\nu}\nu_{L}\right)F_{\mu\nu}$ | ${\cal O}_{\partial adG}^{VL}$ | $\partial^{\mu}a\,\left(\bar{d}_{L}\gamma^{\nu}T^{a}d_{L}\right)G_{\mu\nu}^{a}$
${\cal O}_{\partial a\nu\tilde{F}}^{VL}$ | $\partial^{\mu}a\,\left(\bar{\nu}_{L}\gamma^{\nu}\nu_{L}\right)\widetilde{F}_{\mu\nu}$ | ${\cal O}_{\partial adG}^{VR}$ | $\partial^{\mu}a\,\left(\bar{d}_{R}\gamma^{\nu}T^{a}d_{R}\right)G_{\mu\nu}^{a}$
${\cal O}_{\partial auF}^{VL}$ | $\partial^{\mu}a\,\left(\bar{u}_{L}\gamma^{\nu}u_{L}\right)F_{\mu\nu}$ | ${\cal O}_{\partial ad\tilde{G}}^{VL}$ | $\partial^{\mu}a\,\left(\bar{d}_{L}\gamma^{\nu}T^{a}d_{L}\right)\widetilde{G}_{\mu\nu}^{a}$
${\cal O}_{\partial auF}^{VR}$ | $\partial^{\mu}a\,\left(\bar{u}_{R}\gamma^{\nu}u_{R}\right)F_{\mu\nu}$ | ${\cal O}_{\partial ad\tilde{G}}^{VR}$ | $\partial^{\mu}a\,\left(\bar{d}_{R}\gamma^{\nu}T^{a}d_{R}\right)\widetilde{G}_{\mu\nu}^{a}$
${\cal O}_{\partial au\tilde{F}}^{VL}$ | $\partial^{\mu}a\,\left(\bar{u}_{L}\gamma^{\nu}u_{L}\right)\widetilde{F}_{\mu\nu}$ |  |
Table 7: Operators in the aLEFT at mass dimension 7 with $\partial a$ as a building block. The lepton number violating operator ${\cal O}_{\partial a^{2}\nu}^{LR}$ is marked with $(\not{L})$.  $\partial a\,\psi^{4}+\text{h.c.}$ |  $\partial a\,\psi^{4}+\textrm{h.c.}$ (cont.)
---|---
${\cal O}_{\partial aee}^{VL,SR}$ | $\partial_{\mu}a\,\left(\bar{e}_{L}\gamma^{\mu}e_{L}\right)\left(\bar{e}_{L}e_{R}\right)$ | ${\cal O}_{\partial aeu}^{VL,TR}$ | $\partial_{\mu}a\,\left(\bar{e}_{L}\gamma_{\nu}e_{L}\right)\left(\bar{u}_{L}\sigma^{\mu\nu}u_{R}\right)$
${\cal O}_{\partial aee}^{VR,SR}$ | $\partial_{\mu}a\,\left(\bar{e}_{R}\gamma^{\mu}e_{R}\right)\left(\bar{e}_{L}e_{R}\right)$ | ${\cal O}_{\partial aeu}^{VR,SR}$ | $\partial_{\mu}a\,\left(\bar{e}_{R}\gamma^{\mu}e_{R}\right)\left(\bar{u}_{L}u_{R}\right)$
${\cal O}_{\partial auu}^{VL1,SR}$ | $\partial_{\mu}a\,\left(\bar{u}_{L}\gamma^{\mu}u_{L}\right)\left(\bar{u}_{L}u_{R}\right)$ | ${\cal O}_{\partial aeu}^{VR,TR}$ | $\partial_{\mu}a\,\left(\bar{e}_{R}\gamma_{\nu}e_{R}\right)\left(\bar{u}_{L}\sigma^{\mu\nu}u_{R}\right)$
${\cal O}_{\partial auu}^{VL8,SR}$ | $\partial_{\mu}a\,\left(\bar{u}_{L}\gamma^{\mu}T^{a}u_{L}\right)\left(\bar{u}_{L}T^{a}u_{R}\right)$ | ${\cal O}_{\partial aed}^{VL,SR}$ | $\partial_{\mu}a\,\left(\bar{e}_{L}\gamma^{\mu}e_{L}\right)\left(\bar{d}_{L}d_{R}\right)$
${\cal O}_{\partial auu}^{VR1,SR}$ | $\partial_{\mu}a\,\left(\bar{u}_{R}\gamma^{\mu}u_{R}\right)\left(\bar{u}_{L}u_{R}\right)$ | ${\cal O}_{\partial aed}^{VL,TR}$ | $\partial_{\mu}a\,\left(\bar{e}_{L}\gamma_{\nu}e_{L}\right)\left(\bar{d}_{L}\sigma^{\mu\nu}d_{R}\right)$
${\cal O}_{\partial auu}^{VR8,SR}$ | $\partial_{\mu}a\,\left(\bar{u}_{R}\gamma^{\mu}T^{a}u_{R}\right)\left(\bar{u}_{L}T^{a}u_{R}\right)$ | ${\cal O}_{\partial aed}^{VR,SR}$ | $\partial_{\mu}a\,\left(\bar{e}_{R}\gamma^{\mu}e_{R}\right)\left(\bar{d}_{L}d_{R}\right)$
${\cal O}_{\partial add}^{VL1,SR}$ | $\partial_{\mu}a\,\left(\bar{d}_{L}\gamma^{\mu}d_{L}\right)\left(\bar{d}_{L}d_{R}\right)$ | ${\cal O}_{\partial aed}^{VR,TR}$ | $\partial_{\mu}a\,\left(\bar{e}_{R}\gamma_{\nu}e_{R}\right)\left(\bar{d}_{L}\sigma^{\mu\nu}d_{R}\right)$
${\cal O}_{\partial add}^{VL8,SR}$ | $\partial_{\mu}a\,\left(\bar{d}_{L}\gamma^{\mu}T^{a}d_{L}\right)\left(\bar{d}_{L}T^{a}d_{R}\right)$ | ${\cal O}_{\partial aue}^{VL,SR}$ | $\partial_{\mu}a\,\left(\bar{u}_{L}\gamma^{\mu}u_{L}\right)\left(\bar{e}_{L}e_{R}\right)$
${\cal O}_{\partial add}^{VR1,SR}$ | $\partial_{\mu}a\,\left(\bar{d}_{R}\gamma^{\mu}d_{R}\right)\left(\bar{d}_{L}d_{R}\right)$ | ${\cal O}_{\partial aue}^{VL,TR}$ | $\partial_{\mu}a\,\left(\bar{u}_{L}\gamma_{\nu}u_{L}\right)\left(\bar{e}_{L}\sigma^{\mu\nu}e_{R}\right)$
${\cal O}_{\partial add}^{VR8,SR}$ | $\partial_{\mu}a\,\left(\bar{d}_{R}\gamma^{\mu}T^{a}d_{R}\right)\left(\bar{d}_{L}T^{a}d_{R}\right)$ | ${\cal O}_{\partial aue}^{VR,SR}$ | $\partial_{\mu}a\,\left(\bar{u}_{R}\gamma^{\mu}u_{R}\right)\left(\bar{e}_{L}e_{R}\right)$
${\cal O}_{\partial a\nu e}^{VL,SR}$ | $\partial_{\mu}a\,\left(\bar{\nu}_{L}\gamma^{\mu}\nu_{L}\right)\left(\bar{e}_{L}e_{R}\right)$ | ${\cal O}_{\partial aue}^{VR,TR}$ | $\partial_{\mu}a\,\left(\bar{u}_{R}\gamma_{\nu}u_{R}\right)\left(\bar{e}_{L}\sigma^{\mu\nu}e_{R}\right)$
${\cal O}_{\partial a\nu e}^{VL,TR}$ | $\partial_{\mu}a\,\left(\bar{\nu}_{L}\gamma_{\nu}\nu_{L}\right)\left(\bar{e}_{L}\sigma^{\mu\nu}e_{R}\right)$ | ${\cal O}_{\partial ade}^{VL,SR}$ | $\partial_{\mu}a\,\left(\bar{d}_{L}\gamma^{\mu}d_{L}\right)\left(\bar{e}_{L}e_{R}\right)$
${\cal O}_{\partial aud}^{VL1,SR}$ | $\partial_{\mu}a\,\left(\bar{u}_{L}\gamma^{\mu}u_{L}\right)\left(\bar{d}_{L}d_{R}\right)$ | ${\cal O}_{\partial ade}^{VL,TR}$ | $\partial_{\mu}a\,\left(\bar{d}_{L}\gamma_{\nu}d_{L}\right)\left(\bar{e}_{L}\sigma^{\mu\nu}e_{R}\right)$
${\cal O}_{\partial aud}^{VL8,SR}$ | $\partial_{\mu}a\,\left(\bar{u}_{L}\gamma^{\mu}T^{a}u_{L}\right)\left(\bar{d}_{L}T^{a}d_{R}\right)$ | ${\cal O}_{\partial ade}^{VR,SR}$ | $\partial_{\mu}a\,\left(\bar{d}_{R}\gamma^{\mu}d_{R}\right)\left(\bar{e}_{L}e_{R}\right)$
${\cal O}_{\partial aud}^{VL1,TR}$ | $\partial_{\mu}a\,\left(\bar{u}_{L}\gamma_{\nu}u_{L}\right)\left(\bar{d}_{L}\sigma^{\mu\nu}d_{R}\right)$ | ${\cal O}_{\partial ade}^{VR,TR}$ | $\partial_{\mu}a\,\left(\bar{d}_{R}\gamma_{\nu}d_{R}\right)\left(\bar{e}_{L}\sigma^{\mu\nu}e_{R}\right)$
${\cal O}_{\partial aud}^{VL8,TR}$ | $\partial_{\mu}a\,\left(\bar{u}_{L}\gamma_{\nu}T^{a}u_{L}\right)\left(\bar{d}_{L}\sigma^{\mu\nu}T^{a}d_{R}\right)$ | ${\cal O}_{\partial a\nu u}^{VL,SR}$ | $\partial_{\mu}a\,\left(\bar{\nu}_{L}\gamma^{\mu}\nu_{L}\right)\left(\bar{u}_{L}u_{R}\right)$
${\cal O}_{\partial aud}^{VR1,SR}$ | $\partial_{\mu}a\,\left(\bar{u}_{R}\gamma^{\mu}u_{R}\right)\left(\bar{d}_{L}d_{R}\right)$ | ${\cal O}_{\partial a\nu u}^{VL,TR}$ | $\partial_{\mu}a\,\left(\bar{\nu}_{L}\gamma_{\nu}\nu_{L}\right)\left(\bar{u}_{L}\sigma^{\mu\nu}u_{R}\right)$
${\cal O}_{\partial aud}^{VR8,SR}$ | $\partial_{\mu}a\,\left(\bar{u}_{R}\gamma^{\mu}T^{a}u_{R}\right)\left(\bar{d}_{L}T^{a}d_{R}\right)$ | ${\cal O}_{\partial a\nu d}^{VL,SR}$ | $\partial_{\mu}a\,\left(\bar{\nu}_{L}\gamma^{\mu}\nu_{L}\right)\left(\bar{d}_{L}d_{R}\right)$
${\cal O}_{\partial aud}^{VR1,TR}$ | $\partial_{\mu}a\,\left(\bar{u}_{R}\gamma_{\nu}u_{R}\right)\left(\bar{d}_{L}\sigma^{\mu\nu}d_{R}\right)$ | ${\cal O}_{\partial a\nu d}^{VL,TR}$ | $\partial_{\mu}a\,\left(\bar{\nu}_{L}\gamma_{\nu}\nu_{L}\right)\left(\bar{d}_{L}\sigma^{\mu\nu}d_{R}\right)$
${\cal O}_{\partial aud}^{VR8,TR}$ | $\partial_{\mu}a\,\left(\bar{u}_{R}\gamma_{\nu}T^{a}u_{R}\right)\left(\bar{d}_{L}\sigma^{\mu\nu}T^{a}d_{R}\right)$ | $(\partial a)^{2}X^{2}$
${\cal O}_{\partial adu}^{VL1,SR}$ | $\partial_{\mu}a\,\left(\bar{d}_{L}\gamma^{\mu}d_{L}\right)\left(\bar{u}_{L}u_{R}\right)$ | ${\cal O}_{\partial a^{2}F}^{(1)}$ | $\partial_{\mu}a\partial^{\mu}a\,F_{\nu\rho}F^{\nu\rho}$
${\cal O}_{\partial adu}^{VL8,SR}$ | $\partial_{\mu}a\,\left(\bar{d}_{L}\gamma^{\mu}T^{a}d_{L}\right)\left(\bar{u}_{L}T^{a}u_{R}\right)$ | ${\cal O}_{\partial a^{2}F}^{(2)}$ | $\partial_{\mu}a\partial^{\nu}a\,F^{\mu\rho}F_{\nu\rho}$
${\cal O}_{\partial adu}^{VL1,TR}$ | $\partial_{\mu}a\,\left(\bar{d}_{L}\gamma_{\nu}d_{L}\right)\left(\bar{u}_{L}\sigma^{\mu\nu}u_{R}\right)$ | ${\cal O}_{\partial a^{2}\tilde{F}}$ | $\partial_{\mu}a\partial^{\mu}a\,F_{\nu\rho}\widetilde{F}^{\nu\rho}$
${\cal O}_{\partial adu}^{VL8,TR}$ | $\partial_{\mu}a\,\left(\bar{d}_{L}\gamma_{\nu}T^{a}d_{L}\right)\left(\bar{u}_{L}\sigma^{\mu\nu}T^{a}u_{R}\right)$ | ${\cal O}_{\partial a^{2}G}^{(1)}$ | $\partial_{\mu}a\partial^{\mu}a\,G_{\nu\rho}^{a}G^{a,\nu\rho}$
${\cal O}_{\partial adu}^{VR1,SR}$ | $\partial_{\mu}a\,\left(\bar{d}_{R}\gamma^{\mu}d_{R}\right)\left(\bar{u}_{L}u_{R}\right)$ | ${\cal O}_{\partial a^{2}G}^{(2)}$ | $\partial_{\mu}a\partial^{\nu}a\,G^{a,\mu\rho}G_{\nu\rho}^{a}$
${\cal O}_{\partial adu}^{VR8,SR}$ | $\partial_{\mu}a\,\left(\bar{d}_{R}\gamma^{\mu}T^{a}d_{R}\right)\left(\bar{u}_{L}T^{a}u_{R}\right)$ | ${\cal O}_{\partial a^{2}\tilde{G}}$ | $\partial_{\mu}a\partial^{\mu}a\,G_{\nu\rho}^{a}\widetilde{G}^{a,\nu\rho}$
${\cal O}_{\partial adu}^{VR1,TR}$ | $\partial_{\mu}a\,\left(\bar{d}_{R}\gamma_{\nu}d_{R}\right)\left(\bar{u}_{L}\sigma^{\mu\nu}u_{R}\right)$ | $(\partial a)^{4}$
${\cal O}_{\partial adu}^{VR8,TR}$ | $\partial_{\mu}a\,\left(\bar{d}_{R}\gamma_{\nu}T^{a}d_{R}\right)\left(\bar{u}_{L}\sigma^{\mu\nu}T^{a}u_{R}\right)$ | ${\cal O}_{\partial a^{4}}$ | $\partial_{\mu}a\partial^{\mu}a\partial_{\nu}a\partial^{\nu}a$
${\cal O}_{\partial aeu}^{VL,SR}$ | $\partial_{\mu}a\,\left(\bar{e}_{L}\gamma^{\mu}e_{L}\right)\left(\bar{u}_{L}u_{R}\right)$ |  |
Table 8: Operators in the aLEFT at mass dimension 8 with $\partial a$ as a building block.  $(\partial a)^{2}\psi^{2}D$ |  $\partial a\,\psi^{2}XD+\text{h.c.}$ (cont.)
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${\cal O}_{\partial a^{2}eD}^{VL}$ | $\partial_{\mu}a\partial_{\nu}a\,\left(\bar{e}_{L}\gamma^{\mu}\overleftrightarrow{D}^{\nu}e_{L}\right)$ | ${\cal O}_{\partial aFuD}^{TR}$ | $\partial_{\mu}a\,F_{\nu\rho}\left(\bar{u}_{L}\sigma^{\mu\nu}D^{\rho}u_{R}\right)$
${\cal O}_{\partial a^{2}eD}^{VR}$ | $\partial_{\mu}a\partial_{\nu}a\,\left(\bar{e}_{R}\gamma^{\mu}\overleftrightarrow{D}^{\nu}e_{R}\right)$ | ${\cal O}_{\partial a\tilde{F}uD}^{SR}$ | $\partial_{\mu}a\,\widetilde{F}^{\mu\nu}\left(\bar{u}_{L}D_{\nu}u_{R}\right)$
${\cal O}_{\partial a^{2}\nu D}^{VL}$ | $\partial_{\mu}a\partial_{\nu}a\,\left(\bar{\nu}_{L}\gamma^{\mu}\overleftrightarrow{D}^{\nu}\nu_{L}\right)$ | ${\cal O}_{\partial aFdD}^{SR}$ | $\partial_{\mu}a\,F^{\mu\nu}\left(\bar{d}_{L}D_{\nu}d_{R}\right)$
${\cal O}_{\partial a^{2}uD}^{VL}$ | $\partial_{\mu}a\partial_{\nu}a\,\left(\bar{u}_{L}\gamma^{\mu}\overleftrightarrow{D}^{\nu}u_{L}\right)$ | ${\cal O}_{\partial aFdD}^{TR}$ | $\partial_{\mu}a\,F_{\nu\rho}\left(\bar{d}_{L}\sigma^{\mu\nu}D^{\rho}d_{R}\right)$
${\cal O}_{\partial a^{2}uD}^{VR}$ | $\partial_{\mu}a\partial_{\nu}a\,\left(\bar{u}_{R}\gamma^{\mu}\overleftrightarrow{D}^{\nu}u_{R}\right)$ | ${\cal O}_{\partial a\tilde{F}dD}^{SR}$ | $\partial_{\mu}a\,\widetilde{F}^{\mu\nu}\left(\bar{d}_{L}D_{\nu}d_{R}\right)$
${\cal O}_{\partial a^{2}dD}^{VL}$ | $\partial_{\mu}a\partial_{\nu}a\,\left(\bar{d}_{L}\gamma^{\mu}\overleftrightarrow{D}^{\nu}d_{L}\right)$ | ${\cal O}_{\partial aGuD}^{SR}$ | $\partial_{\mu}a\,G^{a,\mu\nu}\left(\bar{u}_{L}T^{a}D_{\nu}u_{R}\right)$
${\cal O}_{\partial a^{2}dD}^{VR}$ | $\partial_{\mu}a\partial_{\nu}a\,\left(\bar{d}_{R}\gamma^{\mu}\overleftrightarrow{D}^{\nu}d_{R}\right)$ | ${\cal O}_{\partial aGuD}^{TR}$ | $\partial_{\mu}a\,G_{\nu\rho}^{a}\left(\bar{u}_{L}\sigma^{\mu\nu}T^{a}D^{\rho}u_{R}\right)$
$\partial a\,\psi^{2}XD+\text{h.c.}$ | ${\cal O}_{\partial a\tilde{G}uD}^{SR}$ | $\partial_{\mu}a\,\widetilde{G}^{a,\mu\nu}\left(\bar{u}_{L}T^{a}D_{\nu}u_{R}\right)$
${\cal O}_{\partial aFeD}^{SR}$ | $\partial_{\mu}a\,F^{\mu\nu}\left(\bar{e}_{L}D_{\nu}e_{R}\right)$ | ${\cal O}_{\partial aGdD}^{SR}$ | $\partial_{\mu}a\,G^{a,\mu\nu}\left(\bar{d}_{L}T^{a}D_{\nu}d_{R}\right)$
${\cal O}_{\partial aFeD}^{TR}$ | $\partial_{\mu}a\,F_{\nu\rho}\left(\bar{e}_{L}\sigma^{\mu\nu}D^{\rho}e_{R}\right)$ | ${\cal O}_{\partial aGdD}^{TR}$ | $\partial_{\mu}a\,G_{\nu\rho}^{a}\left(\bar{d}_{L}\sigma^{\mu\nu}T^{a}D^{\rho}d_{R}\right)$
${\cal O}_{\partial a\tilde{F}eD}^{SR}$ | $\partial_{\mu}a\,\widetilde{F}^{\mu\nu}\left(\bar{e}_{L}D_{\nu}e_{R}\right)$ | ${\cal O}_{\partial a\tilde{G}dD}^{SR}$ | $\partial_{\mu}a\,\widetilde{G}^{a,\mu\nu}\left(\bar{d}_{L}T^{a}D_{\nu}d_{R}\right)$
${\cal O}_{\partial aFuD}^{SR}$ | $\partial_{\mu}a\,F^{\mu\nu}\left(\bar{u}_{L}D_{\nu}u_{R}\right)$ |  |
Table 9: Operators in the aLEFT at mass dimension 8 with $\partial a$ as a
building block. [Tab. 8 continued.]  $\not{B}$ and $\not{L}$ terms
---
$\partial a\,\psi^{4}+\text{h.c.}$ |  $\partial a\,\psi^{4}+\text{h.c.}$ (cont.)
${\cal O}_{\partial ae\nu}^{VL,SR}$ | $\partial_{\mu}a\,\left(\bar{e}_{L}\gamma^{\mu}e_{L}\right)\left(\bar{\nu}_{L}\nu_{L}^{c}\right)$ | ${\cal O}_{\partial aeddd}^{VL,SR}$ | $\partial_{\mu}a\,\epsilon^{\alpha\beta\gamma}\left(\bar{e}_{L}\gamma^{\mu}d_{L,\alpha}\right)\left(\bar{d}_{R,\beta}^{c}d_{R,\gamma}\right)$
${\cal O}_{\partial ae\nu}^{VR,SR}$ | $\partial_{\mu}a\,\left(\bar{e}_{R}\gamma^{\mu}e_{R}\right)\left(\bar{\nu}_{L}\nu_{L}^{c}\right)$ | ${\cal O}_{\partial addde}^{VL,SR}$ | $\partial_{\mu}a\,\epsilon^{\alpha\beta\gamma}\left(\bar{d}_{L,\alpha}\gamma^{\mu}d_{R,\beta}^{c}\right)\left(\bar{d}_{L,\gamma}e_{R}\right)$
${\cal O}_{\partial au\nu}^{VL,SR}$ | $\partial_{\mu}a\,\left(\bar{u}_{L}\gamma^{\mu}u_{L}\right)\left(\bar{\nu}_{L}\nu_{L}^{c}\right)$ | ${\cal O}_{\partial aeduu}^{VR,SR}$ | $\partial_{\mu}a\,\epsilon^{\alpha\beta\gamma}\left(\bar{e}_{L}^{c}\gamma^{\mu}d_{R,\alpha}\right)\left(\bar{u}_{R,\beta}^{c}u_{R,\gamma}\right)$
${\cal O}_{\partial au\nu}^{VR,SR}$ | $\partial_{\mu}a\,\left(\bar{u}_{R}\gamma^{\mu}u_{R}\right)\left(\bar{\nu}_{L}\nu_{L}^{c}\right)$ | ${\cal O}_{\partial adeuu}^{VR,SR}$ | $\partial_{\mu}a\,\epsilon^{\alpha\beta\gamma}\left(\bar{d}_{L,\alpha}^{c}\gamma^{\mu}e_{R}\right)\left(\bar{u}_{R,\beta}^{c}u_{R,\gamma}\right)$
${\cal O}_{\partial ad\nu}^{VL,SR}$ | $\partial_{\mu}a\,\left(\bar{d}_{L}\gamma^{\mu}d_{L}\right)\left(\bar{\nu}_{L}\nu_{L}^{c}\right)$ | ${\cal O}_{\partial aeudu}^{VL,SR}$ | $\partial_{\mu}a\,\epsilon^{\alpha\beta\gamma}\left(\bar{e}_{R}^{c}\gamma^{\mu}u_{L,\alpha}\right)\left(\bar{d}_{R,\beta}^{c}u_{R,\gamma}\right)$
${\cal O}_{\partial ad\nu}^{VR,SR}$ | $\partial_{\mu}a\,\left(\bar{d}_{R}\gamma^{\mu}d_{R}\right)\left(\bar{\nu}_{L}\nu_{L}^{c}\right)$ | ${\cal O}_{\partial adueu}^{VL,SR}$ | $\partial_{\mu}a\,\epsilon^{\alpha\beta\gamma}\left(\bar{d}_{R,\alpha}^{c}\gamma^{\mu}u_{L,\beta}\right)\left(\bar{e}_{R}^{c}u_{R,\gamma}\right)$
${\cal O}_{\partial aude\nu}^{VL,SR}$ | $\partial_{\mu}a\,\left(\bar{u}_{L}\gamma^{\mu}d_{L}\right)\left(\bar{e}_{L}\nu_{L}^{c}\right)$ | ${\cal O}_{\partial auude}^{VL,SR}$ | $\partial_{\mu}a\,\epsilon^{\alpha\beta\gamma}\left(\bar{u}_{L,\alpha}\gamma^{\mu}u_{R,\beta}^{c}\right)\left(\bar{d}_{L,\beta}e_{L}^{c}\right)$
${\cal O}_{\partial aedu\nu}^{VL,SR}$ | $\partial_{\mu}a\,\left(\bar{e}_{L}\gamma^{\mu}d_{L}\right)\left(\bar{u}_{L}\nu_{L}^{c}\right)$ | ${\cal O}_{\partial aduue}^{VL,SR}$ | $\partial_{\mu}a\,\epsilon^{\alpha\beta\gamma}\left(\bar{d}_{L,\alpha}\gamma^{\mu}u_{R,\beta}^{c}\right)\left(\bar{u}_{L,\beta}e_{L}^{c}\right)$
${\cal O}_{\partial au\nu ed}^{VL,SR}$ | $\partial_{\mu}a\,\left(\bar{u}_{L}\gamma^{\mu}\nu_{L}\right)\left(\bar{e}_{L}d_{R}\right)$ | ${\cal O}_{\partial aueud}^{VL,SR}$ | $\partial_{\mu}a\,\epsilon^{\alpha\beta\gamma}\left(\bar{u}_{L,\alpha}\gamma^{\mu}e_{R}^{c}\right)\left(\bar{u}_{L,\beta}d_{L,\gamma}^{c}\right)$
${\cal O}_{\partial ae\nu ud}^{VL,SR}$ | $\partial_{\mu}a\,\left(\bar{e}_{L}\gamma^{\mu}\nu_{L}\right)\left(\bar{u}_{L}d_{R}\right)$ | ${\cal O}_{\partial audue}^{VL,SR}$ | $\partial_{\mu}a\,\epsilon^{\alpha\beta\gamma}\left(\bar{u}_{L,\alpha}\gamma^{\mu}d_{R,\beta}^{c}\right)\left(\bar{u}_{L,\beta}e_{L}^{c}\right)$
${\cal O}_{\partial ade\nu u}^{VL,SR}$ | $\partial_{\mu}a\,\left(\bar{d}_{L}\gamma^{\mu}e_{L}\right)\left(\bar{\nu}_{L}u_{R}\right)$ | ${\cal O}_{\partial a\nu ddu}^{VL,SR}$ | $\partial_{\mu}a\,\epsilon^{\alpha\beta\gamma}\left(\bar{\nu}_{L}\gamma^{\mu}d_{R,\alpha}^{c}\right)\left(\bar{d}_{L,\beta}u_{L,\gamma}^{c}\right)$
${\cal O}_{\partial a\nu edu}^{VL,SR}$ | $\partial_{\mu}a\,\left(\bar{\nu}_{L}\gamma^{\mu}e_{L}\right)\left(\bar{d}_{L}u_{R}\right)$ | ${\cal O}_{\partial add\nu u}^{VL,SR}$ | $\partial_{\mu}a\,\epsilon^{\alpha\beta\gamma}\left(\bar{d}_{L,\alpha}\gamma^{\mu}d_{R,\beta}^{c}\right)\left(\bar{\nu}_{L}u_{L,\gamma}^{c}\right)$
${\cal O}_{\partial aeu\nu d}^{VL,SR}$ | $\partial_{\mu}a\,\left(\bar{e}_{L}\gamma^{\mu}u_{R}^{c}\right)\left(\bar{\nu}_{L}d_{R}\right)$ | ${\cal O}_{\partial ad\nu ud}^{VL,SR}$ | $\partial_{\mu}a\,\epsilon^{\alpha\beta\gamma}\left(\bar{d}_{L,\alpha}\gamma^{\mu}\nu_{L}\right)\left(\bar{u}_{L,\beta}d_{L,\gamma}^{c}\right)$
${\cal O}_{\partial a\nu ued}^{VL,SR}$ | $\partial_{\mu}a\,\left(\bar{\nu}_{L}\gamma^{\mu}u_{R}^{c}\right)\left(\bar{e}_{L}d_{R}\right)$ | ${\cal O}_{\partial aud\nu d}^{VR,SR}$ | $\partial_{\mu}a\,\epsilon^{\alpha\beta\gamma}\left(\bar{u}_{L,\alpha}^{c}\gamma^{\mu}d_{R,\beta}\right)\left(\bar{\nu}_{L}d_{R,\gamma}\right)$
${\cal O}_{\partial a\nu eud}^{VL,SR}$ | $\partial_{\mu}a\,\left(\bar{\nu}_{L}\gamma^{\mu}e_{R}^{c}\right)\left(\bar{u}_{L}d_{R}\right)$ | ${\cal O}_{\partial ad\nu du}^{VL,SR}$ | $\partial_{\mu}a\,\epsilon^{\alpha\beta\gamma}\left(\bar{d}_{R,\alpha}^{c}\gamma^{\mu}\nu_{L}\right)\left(\bar{d}_{R,\beta}^{c}u_{R,\gamma}\right)$
${\cal O}_{\partial aue\nu d}^{VL,SR}$ | $\partial_{\mu}a\,\left(\bar{u}_{L}\gamma^{\mu}e_{R}^{c}\right)\left(\bar{\nu}_{L}d_{R}\right)$ | ${\cal O}_{\partial adud\nu}^{VL,SR}$ | $\partial_{\mu}a\,\epsilon^{\alpha\beta\gamma}\left(\bar{d}_{L,\alpha}\gamma^{\mu}u_{R,\beta}^{c}\right)\left(\bar{d}_{L,\gamma}\nu_{L}^{c}\right)$
${\cal O}_{\partial adu\nu e}^{VL,SR}$ | $\partial_{\mu}a\,\left(\bar{d}_{L}\gamma^{\mu}u_{L}\right)\left(\bar{\nu}_{L}e_{R}\right)$ | ${\cal O}_{\partial add\nu u}^{VR,SR}$ | $\partial_{\mu}a\,\epsilon^{\alpha\beta\gamma}\left(\bar{d}_{L,\alpha}^{c}\gamma^{\mu}d_{R,\beta}\right)\left(\bar{\nu}_{L}u_{R,\gamma}\right)$
${\cal O}_{\partial a\nu ude}^{VL,SR}$ | $\partial_{\mu}a\,\left(\bar{\nu}_{L}\gamma^{\mu}u_{L}\right)\left(\bar{d}_{L}e_{R}\right)$ | ${\cal O}_{\partial adu\nu d}^{VR,SR}$ | $\partial_{\mu}a\,\epsilon^{\alpha\beta\gamma}\left(\bar{d}_{L,\alpha}^{c}\gamma^{\mu}u_{R,\beta}\right)\left(\bar{\nu}_{L}d_{R,\gamma}\right)$
${\cal O}_{\partial adu\nu e}^{VR,SR}$ | $\partial_{\mu}a\left(\bar{d}_{R}\gamma^{\mu}u_{R}\right)\left(\bar{\nu}_{L}e_{R}\right)$ | ${\cal O}_{\partial adedd}^{VL,SR}~{}(\star)$ | $\partial_{\mu}a\,\epsilon^{\alpha\beta\gamma}\left(\bar{d}_{L,\alpha}^{c}\gamma^{\mu}e_{L}\right)\left(\bar{d}_{L,\beta}d_{L,\gamma}^{c}\right)$
${\cal O}_{\partial ade\nu u}^{VR,SR}$ | $\partial_{\mu}a\left(\bar{d}_{R}\gamma^{\mu}e_{R}\right)\left(\bar{\nu}_{L}u_{R}\right)$ | ${\cal O}_{\partial aeddd}^{VR,SR}~{}(\star)$ | $\partial_{\mu}a\,\epsilon^{\alpha\beta\gamma}\left(\bar{e}_{R}\gamma^{\mu}d_{R,\alpha}\right)\left(\bar{d}_{R,\beta}^{c}d_{R,\gamma}\right)$
${\cal O}_{\partial a\nu edu}^{VR,SR}$ | $\partial_{\mu}a\left(\bar{\nu}_{L}^{c}\gamma^{\mu}e_{R}\right)\left(\bar{d}_{L}u_{R}\right)$ | $\partial a\,\psi^{2}XD+\text{h.c.}$
${\cal O}_{\partial a\nu ude}^{VR,SR}$ | $\partial_{\mu}a\left(\bar{\nu}_{L}^{c}\gamma^{\mu}u_{R}\right)\left(\bar{d}_{L}e_{R}\right)$ | ${\cal O}_{\partial aF\nu D}^{SR}$ | $\partial_{\mu}a\,F^{\mu\nu}\left(\bar{\nu}_{L}\sigma_{\nu\rho}\partial^{\rho}\nu_{L}^{c}\right)$
${\cal O}_{\partial a\nu}^{VL,SR}~{}(\star)$ | $\partial_{\mu}a\,\left(\bar{\nu}_{L}\gamma^{\mu}\nu_{L}\right)\left(\bar{\nu}_{L}\nu_{L}^{c}\right)$ |  |
Table 10: B- and L-breaking operators in the aLEFT at mass dimension 8 with
$\partial a$ as a building block. Note that the operators marked with
$(\star)$ do not appear when fermion flavors are set to 1. [Tab. 9 continued].

###  B.2 Without shift symmetry

For aLEFT without a shift symmetry, once again, we can use the PQ-breaking
isolation condition Eq. (4.4) to construct the operator basis easily. The
operator bases at higher dimensions can be constructed with

${\cal L}_{n}^{\text{aLEFT}_{\cancel{\text{PQ}}}}=a\,{\cal
L}_{n-1}^{\text{aLEFT}_{\cancel{\text{PQ}}}}+a\,{\cal
L}_{n-1}^{\text{LEFT}}+{\cal L}_{n}^{\text{aLEFT}_{\text{PQ}}}$ (B.2)

for $n>5$. As a start point, we show the operator basis at dimension 5 in Tab.
11. The higher-dimensional operator bases can be easily constructed with the
LEFT operator bases Jenkins _et al._ (2018); Liao _et al._ (2020) and the
shift-symmetric operator bases. For completeness, the axion-dependent
renormalizable operators are shown in Eq. (4.5).

$V(a,H)$ | $a^{2}\psi^{2}+\text{h.c.}$
---|---
${\cal O}_{a^{5}}$ | $a^{5}$ | ${\cal O}_{a^{2}e}^{SR}$ | $a^{2}\bar{e}_{L}e_{R}$
$aX^{2}$ |  ${\cal O}_{a^{2}\nu}^{SR}$ $(\not{L})$ | $a^{2}\bar{\nu}_{L}\nu_{L}^{c}$
${\cal O}_{aF}$ | $aF_{\mu\nu}F^{\mu\nu}$ | ${\cal O}_{a^{2}u}^{SR}$ | $a^{2}\bar{u}_{L}u_{R}$
${\cal O}_{a\tilde{F}}$ | $aF_{\mu\nu}\widetilde{F}^{\mu\nu}$ | ${\cal O}_{a^{2}d}^{SR}$ | $a^{2}\bar{d}_{L}d_{R}$
${\cal O}_{aG}$ | $aG_{\mu\nu}^{a}G^{a,\mu\nu}$ |  |
${\cal O}_{a\tilde{G}}$ | $aG_{\mu\nu}^{a}\widetilde{G}^{a,\mu\nu}$ |  |
Table 11: Operators in the aLEFT at mass dimension 5 with $a$ as a building
block. It is worth noting that the dimension-5 ALP Yukawa couplings in the
aSMEFT in Tab. 5 become dimension-4 ALP-dependent mass terms in the aLEFT. The
lepton number violating operator ${\cal O}_{a^{2}\nu}$ is marked with
$(\not{L})$.

##  Appendix C Additional results for the Hilbert series and the operator
counting

###  C.1 aSMEFT

In Section 3, we have shown the Hilbert series for aSMEFT with (without) a
shift symmetry up to dimension 8 (7). Due to the excessive length of the
Hilbert series in higher dimensions, it cannot be included in this paper.
Instead, we provide an ancillary Mathematica notebook to present all the
Hilbert series from dimension 5 to dimension 15. By keeping the information of
the baryon and lepton number violation in each term of the Hilbert series, and
setting all spurions to unity, the number of operators at each mass dimension
can be expanded in powers of the $B$ and $L$ violating unit $\epsilon_{B,L}$.
The Hilbert series is calculated with $N_{f}$ flavors of the fermions, which
allow us to count the operators with flavor dependence. The full results of
the operator counting is also available in our ancillary notebook file. In
this section, we will only show the results for number counting up to
dimension 12.

For aSMEFT with a shift symmetry, the numbers of operators at each dimension
are given by

$\displaystyle\\#\,{\cal O}_{5}^{\text{PQ}}\,=\,$ $\displaystyle
2-N_{f}+5N_{f}^{2}\,,$ $\displaystyle\\#\,{\cal O}_{6}^{\text{PQ}}\,=\,$
$\displaystyle 1\,,$ $\displaystyle\\#\,{\cal O}_{7}^{\text{PQ}}\,=\,$
$\displaystyle 5+39N_{f}^{2}\,,$ $\displaystyle\\#\,{\cal
O}_{8}^{\text{PQ}}\,=\,$
$\displaystyle\left(13+11N_{f}^{2}\right)+\left(-\frac{2N_{f}^{2}}{3}+\frac{8N_{f}^{4}}{3}\right)\epsilon_{B}\epsilon_{L}+\left(4N_{f}^{2}+2N_{f}^{4}\right)\epsilon_{L}^{2}\,,$
$\displaystyle\\#\,{\cal O}_{9}^{\text{PQ}}\,=\,$
$\displaystyle\left(74+\frac{1799N_{f}^{2}}{4}-\frac{N_{f}^{3}}{2}+\frac{847N_{f}^{4}}{4}\right)+46N_{f}^{4}\epsilon_{B}\epsilon_{L}+\left(N_{f}+N_{f}^{2}\right)\epsilon_{L}^{2}\,,$
$\displaystyle\\#\,{\cal O}_{10}^{\text{PQ}}\,=\,$
$\displaystyle\left(74+\frac{431N_{f}^{2}}{2}+\frac{91N_{f}^{4}}{2}\right)+\left(8N_{f}^{3}+106N_{f}^{4}\right)\epsilon_{B}\epsilon_{L}+\left(-N_{f}+75N_{f}^{2}+N_{f}^{3}+121N_{f}^{4}\right)\epsilon_{L}^{2}\,,$
$\displaystyle\\#\,{\cal O}_{11}^{\text{PQ}}\,=\,$