Light vectors coupled to anomalous currents
with harmless Wess-Zumino terms

Luca Di Luzioa,b, Marco Nardecchiac, Claudio Tonic

aDipartimento di Fisica e Astronomia ‘G. Galilei’, Università di Padova, Italy
bIstituto Nazionale Fisica Nucleare, Sezione di Padova, Italy
cPhysics Department and INFN Sezione di Roma La Sapienza,
Piazzale Aldo Moro 5, 00185 Roma, Italy

> We reconsider the case of light vectors coupled to anomalous fermionic
> currents, focussing on the interplay between UV and IR dynamics. Taking as a
> general framework the gauging of the Standard Model accidental symmetries,
> we show that it is possible to devise an anomaly-free UV completion with
> mostly chiral heavy fermions such that anomalous Wess-Zumino terms are
> suppressed in the IR, thus relaxing would-be strong bounds from the
> longitudinal emission of light vectors coupled to non-conserved currents. We
> classify such scenarios and show that they will be extensively probed at the
> high-luminosity phase of the LHC via the measurement of the $h\to Z\gamma$
> rate and the direct search for non-decoupling charged leptons.

###### Contents

  1. 1 Introduction
  2. 2 Gauging the Standard Model accidental symmetries
    1. 2.1 UV model
      1. 2.1.1 Anomaly cancellation
      2. 2.1.2 Renormalizable operators
      3. 2.1.3 Spectrum
    2. 2.2 EFT of a light vector and decoupling of WZ terms
    3. 2.3 Neutrino masses
  3. 3 Electroweak anomalons phenomenology
    1. 3.1 Electroweak precision tests
    2. 3.2 Higgs physics
    3. 3.3 Direct searches
      1. 3.3.1 Stable charged leptons
      2. 3.3.2 Unstable charged leptons
  4. 4 Conclusions
  5. A Calculation of the Wess-Zumino terms
    1. A.1 Toy model
    2. A.2 $\gamma_{5}$ in dimensional regularization
    3. A.3 1-loop matching
    4. A.4 Reproducing the chiral anomaly in the EFT
    5. A.5 WZ terms for massive vector bosons and Goldstone bosons
    6. A.6 Properties of the WZ coefficients

##  1 Introduction

The physics of light spin-1 dark bosons has witnessed a growing amount of
interest in the recent years, from both a theoretical and phenomenological
standpoint. A standard benchmark is that of a secluded ${\rm U}(1)$ gauge
boson, kinetically mixed with the photon [1] and hence universally coupled to
the Standard Model (SM) sector via the electromagnetic current. This
framework, although elegant and predictive, can be too restrictive for
phenomenological applications and hence more general forms of the light vector
boson interactions with the SM fields can be envisaged. Going beyond the
kinetic mixing framework, a theoretically motivated option is provided by the
gauging of the accidental global symmetries of the SM, that is baryon number
${\rm U}(1)_{B}$ and family lepton number ${\rm U}(1)_{L_{i}}$ (with
$i=e,\mu,\tau$) in the limit of massless neutrinos. Within the SM field
content, only the $L_{i}-L_{j}$ combinations turn out to be anomaly free [2,
3, 4]. Hence, in order to consistently gauge a general linear combination

$X=\alpha_{B}B+\sum_{i=e,\,\mu,\,\tau}\alpha_{i}L_{i}\,,$ (1.1)

one requires new fermions, also known as _anomalons_ , which cancel the
anomalies of the new ${\rm U}(1)_{X}$ factor, also in combination with the
electroweak gauge group. Note that Eq. (1.1) is the most general linear
combination of abelian global symmetries of the SM that can be gauged, under
the assumption that all SM Yukawa operators are allowed in the quark sector at
the renormalizable level. Barring the cases of $B/3-L_{i}$ and linear
combinations thereof, the anomalons need to be charged under the electroweak
gauge group (henceforth indicated more precisely as electroweak
anomalons).111This is not the case if some of the quark Yukawa operators arise
at the non-renormalizable level, as e.g. discussed recently in Ref. [5]. In
the latter case, in order to evade detection at high-energy particle
colliders, the new fermions need to be heavier than the electroweak scale.
Consequently, their effects on the physics of the light vector boson
associated with the ${\rm U}(1)_{X}$ gauge symmetry, here denoted as
$\mathcal{X}$, can be described within an effective field theory (EFT)
approch. In particular, after integrating out the new heavy fermions at one
loop, one generates dimension-4 Wess-Zumino (WZ) terms, schematically of the
form $\mathcal{X}(W\partial W+WWW)$ and $\mathcal{X}B\partial B$ (with $W$ and
$B$ denoting ${\rm SU}(2)_{L}$ and ${\rm U}(1)_{Y}$ gauge bosons). These
contact interactions have the role of compensating, in the EFT without
electroweak anomalons, the anomalous shift of the effective action due to the
anomalous SM fermion current coupled to $\mathcal{X}$ (see e.g. [6, 7, 8, 9]).

As it was emphasized more recently in Refs. [10, 11], WZ terms display an
axion-like behaviour (as can be understood by applying the equivalence theorem
to the longitudinal component of $\mathcal{X}$) and lead to amplitudes that
grow with the energy. The anomalous $\mathcal{X}W\partial W$ vertex can be
dressed with SM flavour-violating interactions leading to loop-induced flavour
changing neutral current (FCNC) processes, while the anomalous
$\mathcal{X}B\partial B$ vertex is responsible for $Z\to\gamma\mathcal{X}$
decays at the tree level (see also [12, 13, 14, 15]). In both cases these
processes are enhanced as $(\text{energy}/m_{\mathcal{X}})^{2}$, thus
resulting into the typically most stringent bounds on light vectors with no
direct couplings to electrons, as e.g. in the case of gauged baryon number.

It is known (see e.g. [11]) that in the limit where the mass of the anomalons
stems from a SM-preserving vacuum expectation value (VEV), the low-energy
coefficients of the WZ terms are entirely fixed by the requirement of
cancelling the ${\rm SU}(2)_{L}^{2}{\rm U}(1)_{X}$ and ${\rm U}(1)_{Y}^{2}{\rm
U}(1)_{X}$ anomalies of the SM sector. On the other hand, if the anomalons
pick up a mass contribution from the electroweak VEV then the coefficients of
the WZ terms become model-dependent. In particular, in the limit where the
anomalons mass is completely due to electroweak symmetry breaking sources, the
anomalous couplings of the longitudinal component of $\mathcal{X}$ with SM
electroweak gauge bosons goes to zero, thus relaxing the above mentioned
strong bounds on light vectors.

In this paper, we revisit the argument why WZ terms become harmless in the
limit where the electroweak anomalons obtain their mass solely from the Higgs,
classify the structure of UV completions that allow for such a pattern and
discuss their electroweak-scale phenomenology. Due to its non-decoupling
nature, the phenomenology of the electroweak anomalons is tightly constrained
(but not yet ruled out) by Higgs couplings measurements and direct searches,
thus making the whole setup testable at the high-luminosity phase of the LHC
(HL-LHC). Chiral fermionic extensions of the SM, sharing some similarities
with our setup, were previously discussed in a different context in Refs. [16,
17]. Here, the main phenomenological interest consists in the physics of light
(i.e. sub-GeV) vector bosons coupled to anomalous SM currents and the
possibility of re-opening a large portion of parameter space, which might be
probed by several low-energy experiments or help in explaining current
experimental anomalies, such as e.g. that of the muon $g$-2 [18].

The paper is structured as follows. Sect. 2 is the core of the work, in which
we provide the general setup for the gauging of the generic linear combination
of U(1) factors in Eq. (1.1). We discuss in particular the heavy anomalons
sector leading to the cancellation of gauge anomalies and compute the
resulting WZ terms in the EFT. In passing, we also deal with the issue of
neutrino masses when lepton family generators are gauged. Sect. 3 is devoted
instead to the phenomenology of the electroweak anomalons, in the limit where
their mass dominantly stems from the Higgs VEV. We conclude in Sect. 4, while
in App. A we collect a series of technical results about the calculation of WZ
terms.

##  2 Gauging the Standard Model accidental symmetries

In this Section, we provide an explicit UV completion for the gauging of the
most general combination of the SM global symmetries in Eq. (1.1), discuss the
the conditions for the cancellation of gauge anomalies, compute the spectrum
and the EFT below the scale of the heavy fermions (anomalons) assuming that
the only new physics light state is the vector boson $\mathcal{X}$.

###  2.1 UV model

The field content of the model is displayed in Table 1, where the anomalon
fields are highlighted in color and we also extended the scalar sector of the
SM in order to spontaneously break the ${\rm U}(1)_{X}$ symmetry. Similar
setups for anomaly cancellation were considered e.g. in Refs. [19, 20, 21, 22,
23, 24].222For other anomalon configurations leading to anomaly cancellation
when baryon and/or lepton number generators are gauged see e.g. [25, 26].
Here, the more general SM charges of the electroweak anomalon fields $({\cal
L},{\cal N},{\cal E})$ are needed to evade LHC constraints on purely-chiral
fermions for ${\cal Y}\approx 2,-1$ [16, 17], as it will be reviewed in Sect.
3. In fact, as already anticipated in the Introduction, we will be interested
in exploring the limit in which the electroweak anomalon masses are dominantly
due to the Higgs, so that the strong bounds stemming from the anomalous WZ
couplings of the light vector with SM gauge bosons are relaxed. We have also
included $N$ copies of chiral SM-singlet fermions $\nu^{\alpha}_{R}$
($\alpha=1,\ldots,N$) which allow to have more freedom for the cancellation of
${\rm U}(1)_{X}$ and ${\rm U}(1)^{3}_{X}$ anomalies (as well as provide a
seesaw setup for neutrino masses), but whose presence does not impact the
calculation of the electroweak WZ terms.

Field | Lorentz | ${\rm SU}(3)_{C}$ | ${\rm SU}(2)_{L}$ | ${\rm U}(1)_{Y}$ | ${\rm U}(1)_{X}$
---|---|---|---|---|---
$q^{i}_{L}$ | $(\tfrac{1}{2},0)$ | 3 | 2 | 1/6 | $\alpha_{B}/3$
$u^{i}_{R}$ | $(0,\tfrac{1}{2})$ | 3 | 1 | 2/3 | $\alpha_{B}/3$
$d^{i}_{R}$ | $(0,\tfrac{1}{2})$ | 3 | 1 | $-1/3$ | $\alpha_{B}/3$
$\ell^{i}_{L}$ | $(\tfrac{1}{2},0)$ | 1 | 2 | $-1/2$ | $\alpha_{i}$
$e^{i}_{R}$ | $(0,\tfrac{1}{2})$ | 1 | 1 | $-1$ | $\alpha_{i}$
$H$ | $(0,0)$ | 1 | 2 | 1/2 | $0$
${\cal L}_{L}$ | $(\tfrac{1}{2},0)$ | 1 | 2 | ${\cal Y}-1/2$ | $X_{{\cal L}_{L}}$
${\cal L}_{R}$ | $(0,\tfrac{1}{2})$ | 1 | 2 | ${\cal Y}-1/2$ | $X_{{\cal L}_{R}}$
${\cal E}_{L}$ | $(\tfrac{1}{2},0)$ | 1 | 1 | ${\cal Y}-1$ | $X_{{\cal E}_{L}}$
${\cal E}_{R}$ | $(0,\tfrac{1}{2})$ | 1 | 1 | ${\cal Y}-1$ | $X_{{\cal E}_{R}}$
${\cal N}_{L}$ | $(\tfrac{1}{2},0)$ | 1 | 1 | ${\cal Y}$ | $X_{{\cal N}_{L}}$
${\cal N}_{R}$ | $(0,\tfrac{1}{2})$ | 1 | 1 | ${\cal Y}$ | $X_{{\cal N}_{R}}$
$\nu_{R}^{\alpha}$ | $(0,\tfrac{1}{2})$ | 1 | 1 | $0$ | $X^{\alpha}_{\nu_{R}}$
$\mathcal{S}$ | $(0,0)$ | 1 | 1 | 0 | $X_{\mathcal{S}}$
Table 1:  Anomaly-free field content for a general ${\rm SU}(3)_{C}\times{\rm
SU}(2)_{L}\times{\rm U}(1)_{Y}\times{\rm U}(1)_{X}$ gauge theory, with
$X=\alpha_{B}B+\sum_{i=e,\mu,\tau}\alpha_{i}L_{i}$. The conditions on the
${\rm U}(1)_{X}$ charges fulfilling the cancellation of gauge anomalies are
reported in the text.

####  2.1.1 Anomaly cancellation

The ${\rm U}(1)_{X}$ charges are required to cancel all gauge anomalies. This
corresponds to the following five conditions:

$\displaystyle\text{Gravity}\times{\rm U}(1)_{X}:$ $\displaystyle\quad
2(X_{{\cal L}_{L}}-X_{{\cal L}_{R}})+(X_{{\cal E}_{L}}-X_{{\cal
E}_{R}})+(X_{{\cal N}_{L}}-X_{{\cal
N}_{R}})-\sum_{\alpha=1}^{N}X^{\alpha}_{\nu_{R}}$
$\displaystyle\quad+\alpha_{e}+\alpha_{\mu}+\alpha_{\tau}=0\,,$ (2.1)
$\displaystyle{\rm U}(1)^{3}_{X}:$ $\displaystyle\quad 2(X_{{\cal
L}_{L}}^{3}-X_{{\cal L}_{R}}^{3})+(X_{{\cal E}_{L}}^{3}-X_{{\cal
E}_{R}}^{3})+(X_{{\cal N}_{L}}^{3}-X_{{\cal
N}_{R}}^{3})-\sum_{\alpha=1}^{N}\left(X^{\alpha}_{\nu_{R}}\right)^{3}$
$\displaystyle\quad+\alpha^{3}_{e}+\alpha^{3}_{\mu}+\alpha^{3}_{\tau}=0\,,$
(2.2) $\displaystyle{\rm SU}(2)^{2}_{L}\times{\rm U}(1)_{X}:$
$\displaystyle\quad\frac{1}{2}(X_{{\cal L}_{L}}-X_{{\cal
L}_{R}})+\frac{1}{2}(3\alpha_{B}+\alpha_{e}+\alpha_{\mu}+\alpha_{\tau})=0\,,$
(2.3) $\displaystyle{\rm U}(1)^{2}_{Y}\times{\rm U}(1)_{X}:$
$\displaystyle\quad 2({\cal Y}-\frac{1}{2})^{2}(X_{{\cal L}_{L}}-X_{{\cal
L}_{R}})+({\cal Y}-1)^{2}(X_{{\cal E}_{L}}-X_{{\cal E}_{R}})+{\cal
Y}^{2}(X_{{\cal N}_{L}}-X_{{\cal N}_{R}})$
$\displaystyle\quad-\frac{1}{2}(3\alpha_{B}+\alpha_{e}+\alpha_{\mu}+\alpha_{\tau})=0\,,$
(2.4) $\displaystyle{\rm U}(1)_{Y}\times{\rm U}(1)^{2}_{X}:$
$\displaystyle\quad 2({\cal Y}-\frac{1}{2})(X_{{\cal L}_{L}}^{2}-X_{{\cal
L}_{R}}^{2})+({\cal Y}-1)(X_{{\cal E}_{L}}^{2}-X_{{\cal E}_{R}}^{2})+{\cal
Y}(X_{{\cal N}_{L}}^{2}-X_{{\cal N}_{R}}^{2})=0\,.$ (2.5)

####  2.1.2 Renormalizable operators

Further constraints on the ${\rm U}(1)_{X}$ charges are obtained by the
requirement that the electroweak anomalons pick up their mass from the VEV of
$H$. Hence, the Yukawa Lagrangian involving the electroweak anomalon fields is
(the discussion of neutrino masses is postponed to Sect. 2.3)

$\displaystyle-\mathscr{L}_{Y}$ $\displaystyle=y_{1}\overline{{\cal
L}}_{L}{\cal E}_{R}H+y_{2}\overline{{\cal L}}_{R}{\cal
E}_{L}H+y_{3}\overline{{\cal L}}_{L}{\cal N}_{R}\tilde{H}+y_{4}\overline{{\cal
L}}_{R}{\cal N}_{L}\tilde{H}+\text{h.c.}\,,$ (2.6)

with $\tilde{H}=i\sigma_{2}H^{*}$. The extra conditions on the ${\rm
U}(1)_{X}$ charges stemming from Eq. (2.6) read

$\displaystyle X_{{\cal E}_{R}}=X_{{\cal L}_{L}}\,,$ (2.7) $\displaystyle
X_{{\cal E}_{L}}=X_{{\cal L}_{R}}\,,$ (2.8) $\displaystyle X_{{\cal
N}_{R}}=X_{{\cal L}_{L}}\,,$ (2.9) $\displaystyle X_{{\cal N}_{L}}=X_{{\cal
L}_{R}}\,,$ (2.10)

thus reducing the number of independent charges to two. By substituting Eqs.
(2.7)–(2.10) into Eqs. (2.1.1)–(2.5), we obtain the following non-trivial
conditions

$\displaystyle X_{{\cal L}_{R}}-X_{{\cal L}_{L}}$
$\displaystyle=3\alpha_{B}+\alpha_{e}+\alpha_{\mu}+\alpha_{\tau}\equiv
3\alpha_{B+L}\,,$ (2.11)
$\displaystyle\sum_{\alpha=1}^{N}X^{\alpha}_{\nu_{R}}$
$\displaystyle=\alpha_{e}+\alpha_{\mu}+\alpha_{\tau}\,,$ (2.12)
$\displaystyle\sum_{\alpha=1}^{N}\left(X^{\alpha}_{\nu_{R}}\right)^{3}$
$\displaystyle=\alpha^{3}_{e}+\alpha^{3}_{\mu}+\alpha^{3}_{\tau}\,,$ (2.13)

where we have introduced the shorthand $\alpha_{B+L}$ defined in Eq. (2.11).
Note that the condition of cancellation of electroweak anomalies fixes only
the difference $X_{{\cal L}_{R}}-X_{{\cal L}_{R}}$, leaving us with one free
charge that we choose to be $X_{{\cal L}_{L}}$. This redundancy is related to
the electroweak anomalon number ${\rm U}(1)_{A}$, corresponding to a common
re-phasing of the electroweak anomalon fields.

Other renormalizable operators, which are allowed by the SM gauge symmetry,
may or may not be allowed by ${\rm U}(1)_{X}$ invariance. For instance, extra
Yukawas of the type333The case $\mathcal{S}\to\mathcal{S}^{*}$ is trivially
obtained by replacing $X_{\mathcal{S}}\to-X_{\mathcal{S}}$.

$-\Delta\mathscr{L}_{Y}=y_{\cal L}\overline{{\cal L}}_{L}{\cal
L}_{R}\mathcal{S}^{*}+y_{\cal E}\overline{{\cal E}}_{L}{\cal
E}_{R}\mathcal{S}+y_{\cal N}\overline{{\cal N}}_{L}{\cal
N}_{R}\mathcal{S}+\text{h.c.}\,,$ (2.14)

are only permitted for $X_{\mathcal{S}}=X_{{\cal L}_{R}}-X_{{\cal
L}_{L}}=3\alpha_{B+L}$. These terms would yield an additional vector-like mass
to the anomalons after ${\rm U}(1)_{X}$ symmetry breaking. Finally, for
specific values of ${\rm U}(1)_{Y}$ and ${\rm U}(1)_{X}$ charges, the
electroweak anomalons can mix with the SM leptons at the renormalizable level.
The classification of $d=4$ mixing operators is provided in Table 2, where we
emphasized the phenomenologically relevant case ${\cal Y}=2,-1$ (see Sect.
3.2). Note that in the presence of mixing operators the electroweak anomalon
number is explicitly broken, and hence $X_{{\cal L}_{L}}$ gets fixed in terms
of the coefficients of the $X$ generator in Eq. (1.1).

Mixing operator | ${\rm U}(1)_{Y}$ | ${\rm U}(1)_{X}$
---|---|---
$\overline{\ell}^{i}_{L}{\cal E}_{R}H$ | ${\cal Y}=0$ | $X_{{\cal L}_{L}}=\alpha_{i}$
$\overline{\ell}^{i}_{L}{\cal E}_{R}\tilde{H}$ | ${\cal Y}=1$ | $X_{{\cal L}_{L}}=\alpha_{i}$
$\overline{\ell}^{i}_{L}({\cal E}_{L})^{c}H$ | ${\cal Y}=2$ | $X_{{\cal L}_{L}}=-\alpha_{i}-3\alpha_{B+L}$
$\overline{\ell}^{i}_{L}({\cal E}_{L})^{c}\tilde{H}$ | ${\cal Y}=1$ | $X_{{\cal L}_{L}}=-\alpha_{i}-3\alpha_{B+L}$
$\overline{\ell}^{i}_{L}{\cal N}_{R}H$ | ${\cal Y}=-1$ | $X_{{\cal L}_{L}}=\alpha_{i}$
$\overline{\ell}^{i}_{L}{\cal N}_{R}\tilde{H}$ | ${\cal Y}=0$ | $X_{{\cal L}_{L}}=\alpha_{i}$
$\overline{\ell}^{i}_{L}({\cal N}_{L})^{c}H$ | ${\cal Y}=1$ | $X_{{\cal L}_{L}}=-\alpha_{i}-3\alpha_{B+L}$
$\overline{\ell}^{i}_{L}({\cal N}_{L})^{c}\tilde{H}$ | ${\cal Y}=0$ | $X_{{\cal L}_{L}}=-\alpha_{i}-3\alpha_{B+L}$
$\overline{\cal L}_{L}e^{i}_{R}H$ | ${\cal Y}=0$ | $X_{{\cal L}_{L}}=\alpha_{i}$
$\overline{\cal L}_{L}e^{i}_{R}\tilde{H}$ | ${\cal Y}=-1$ | $X_{{\cal L}_{L}}=\alpha_{i}$
$\overline{\cal L}_{R}(e^{i}_{R})^{c}H$ | ${\cal Y}=2$ | $X_{{\cal L}_{L}}=-\alpha_{i}-3\alpha_{B+L}$
$\overline{\cal L}_{R}(e^{i}_{R})^{c}\tilde{H}$ | ${\cal Y}=1$ | $X_{{\cal L}_{L}}=-\alpha_{i}-3\alpha_{B+L}$
$\overline{\cal L}_{L}\nu^{\alpha}_{R}H$ | ${\cal Y}=1$ | $X_{{\cal L}_{L}}=X^{\alpha}_{\nu_{R}}$
$\overline{\cal L}_{L}\nu^{\alpha}_{R}\tilde{H}$ | ${\cal Y}=0$ | $X_{{\cal L}_{L}}=X^{\alpha}_{\nu_{R}}$
$\overline{\cal L}_{R}(\nu^{\alpha}_{R})^{c}H$ | ${\cal Y}=1$ | $X_{{\cal L}_{L}}=-X^{\alpha}_{\nu_{R}}-3\alpha_{B+L}$
$\overline{\cal L}_{R}(\nu^{\alpha}_{R})^{c}\tilde{H}$ | ${\cal Y}=0$ | $X_{{\cal L}_{L}}=-X^{\alpha}_{\nu_{R}}-3\alpha_{B+L}$
$\overline{\cal L}_{R}\ell^{i}_{L}\mathcal{S}$ | ${\cal Y}=0$ | $X_{{\cal L}_{L}}=\alpha_{i}+X_{\mathcal{S}}-3\alpha_{B+L}$
$\overline{\cal L}_{L}(\ell^{i}_{L})^{c}\mathcal{S}$ | ${\cal Y}=1$ | $X_{{\cal L}_{L}}=-\alpha_{i}+X_{\mathcal{S}}$
$\overline{\cal E}_{L}e^{i}_{R}\mathcal{S}$ | ${\cal Y}=0$ | $X_{{\cal L}_{L}}=\alpha_{i}+X_{\mathcal{S}}-3\alpha_{B+L}$
$\overline{\cal E}_{L}\nu^{\alpha}_{R}\mathcal{S}$ | ${\cal Y}=1$ | $X_{{\cal L}_{L}}=X_{\nu_{R}}^{\alpha}+X_{\mathcal{S}}-3\alpha_{B+L}$
$\overline{\cal E}_{R}(e^{i}_{R})^{c}\mathcal{S}$ | ${\cal Y}=2$ | $X_{{\cal L}_{L}}=-\alpha_{i}+X_{\mathcal{S}}$
$\overline{\cal E}_{R}(\nu^{\alpha}_{R})^{c}\mathcal{S}$ | ${\cal Y}=1$ | $X_{{\cal L}_{L}}=-X_{\nu_{R}}^{\alpha}+X_{\mathcal{S}}$
$\overline{\cal N}_{L}e^{i}_{R}\mathcal{S}$ | ${\cal Y}=-1$ | $X_{{\cal L}_{L}}=\alpha_{i}+X_{\mathcal{S}}-3\alpha_{B+L}$
$\overline{\cal N}_{L}\nu^{\alpha}_{R}\mathcal{S}$ | ${\cal Y}=0$ | $X_{{\cal L}_{L}}=X_{\nu_{R}}^{\alpha}+X_{\mathcal{S}}-3\alpha_{B+L}$
$\overline{\cal N}_{R}(e^{i}_{R})^{c}\mathcal{S}$ | ${\cal Y}=1$ | $X_{{\cal L}_{L}}=-\alpha_{i}+X_{\mathcal{S}}$
$\overline{\cal N}_{R}(\nu^{\alpha}_{R})^{c}\mathcal{S}$ | ${\cal Y}=0$ | $X_{{\cal L}_{L}}=-X_{\nu_{R}}^{\alpha}+X_{\mathcal{S}}$
Table 2:  Renormalizable operators leading to a mixing between electroweak
anomalons and SM leptons (first column) and required conditions on ${\rm
U}(1)_{Y}$ and ${\rm U}(1)_{X}$ charges (second and third columns). For
completeness, we also include mixings via RH neutrinos and/or $\mathcal{S}$,
whose ${\rm U}(1)_{X}$ charges depend on the mechanism giving mass to
neutrinos (see Sect. 2.3). Mixing operators via $\mathcal{S}^{*}$ are
trivially obtained by flipping the sign of $X_{\mathcal{S}}$ in the third
column.

####  2.1.3 Spectrum

By adding a proper term in the scalar potential, $\Delta V(H,\mathcal{S})$,
the following VEV configurations are generated

$\left\langle H\right\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\
v\end{pmatrix}\,,\qquad\left\langle\mathcal{S}\right\rangle=\frac{v_{X}}{\sqrt{2}}\,,$
(2.15)

with $v\simeq 246$ GeV and $v_{X}$ being the order parameter of ${\rm
U}(1)_{X}$ breaking. The latter is responsible for the mass of the ${\rm
U}(1)_{X}$ gauge boson, $\mathcal{X}^{\mu}$, that is

$m_{\mathcal{X}}=X_{\mathcal{S}}g_{X}v_{X}\,,$ (2.16)

where $g_{X}$ is the ${\rm U}(1)_{X}$ gauge coupling entering the covariant
derivative, i.e.
$D^{\mu}\mathcal{S}=(\partial^{\mu}+ig_{X}X_{\mathcal{S}}\mathcal{X}^{\mu})\mathcal{S}$.
The scalar field can be expanded around the vacuum as
$\mathcal{S}=\frac{v_{X}}{\sqrt{2}}e^{i\xi/v_{X}}+\ldots$, where $\xi$ is the
Goldstone boson associated with the massive state $\mathcal{X}$ and we
neglected the radial mode. After ${\rm U}(1)_{X}$ and electroweak symmetry
breaking the Yukawa terms in $\mathscr{L}_{Y}+\Delta\mathscr{L}_{Y}$ (see Eq.
(2.6) and Eq. (2.14)) give mass to the electroweak anomalons (neglecting for
simplicity possible mixings with the SM sector)

$\displaystyle-\mathscr{L}_{\rm mass}$ $\displaystyle=\overline{\Psi}^{{\cal
E}}_{L}\mathcal{M}_{{\cal E}}\,\Psi^{{\cal E}}_{R}+\overline{\Psi}^{{\cal
N}}_{L}\mathcal{M}_{{\cal N}}\,\Psi^{{\cal N}}_{R}+\text{h.c.}\,,$ (2.17)

which can be cast into 2-flavour Dirac fermions, $\Psi^{{\cal E}}_{L,R}=({\cal
E}_{{\cal L}_{L,R}},{\cal E}_{L,R})$ and $\Psi^{{\cal N}}_{L,R}=({\cal
N}_{{\cal L}_{L,R}},{\cal N}_{L,R})$, with

$\mathcal{M}_{{\cal E}}=\begin{pmatrix}m_{{\cal L}}&m_{1}\\\
m_{2}^{*}&m_{{\cal E}}\\\ \end{pmatrix}\,,\qquad\mathcal{M}_{{\cal
N}}=\begin{pmatrix}m_{{\cal L}}&m_{3}\\\ m_{4}^{*}&m_{{\cal N}}\\\
\end{pmatrix}\,,$ (2.18)

and

$m_{{\cal L},\,{\cal E},\,{\cal N}}=\frac{y_{{\cal L},\,{\cal E},\,{\cal
N}}}{\sqrt{2}}v_{X}\,,\qquad
m_{1,\,2,\,3,\,4}=\frac{y_{1,\,2,\,3,\,4}}{\sqrt{2}}v\,.$ (2.19)

The mass matrices are diagonalized via the bi-unitary transformations
$\Psi^{{\cal E},\,{\cal N}}_{R}\to U_{{\cal E},\,{\cal N}}\,\Psi^{{\cal
E},\,{\cal N}}_{R}$ and $\Psi^{{\cal E},\,{\cal N}}_{L}\to V_{{\cal E},\,{\cal
N}}\,\Psi^{{\cal E},\,{\cal N}}_{L}$, with the unitary matrices entering non-
trivially into the gauge currents in the mass basis. In the limit $y_{{\cal
E}}=y_{{\cal N}}$, $y_{1}=y_{3}$, $y_{2}=y_{4}$ (and hence $\mathcal{M}_{{\cal
E}}=\mathcal{M}_{{\cal N}}$), the Yukawa Lagrangian features a custodial
symmetry which helps in taming corrections to electroweak precision
observables (see Sect. 3.1). In the following, we will stick to the custodial
limit, while for the calculations in App. A we will consider the more general
case.

###  2.2 EFT of a light vector and decoupling of WZ terms

We are interested in the limit where the electroweak anomalons are heavier the
electroweak scale, while the vector $\mathcal{X}$ is much lighter than the
electroweak scale. Parametrically (see Eq. (2.16)), this can be obtained in
two ways: $i)$ $v_{X}\gtrsim v$ and $g_{X}\ll 1$ or $ii)$ $v_{X}\ll v$ and
$g_{X}\lesssim 1$. In case $ii)$ or if the $\Delta\mathscr{L}_{Y}$ operators
in Eq. (2.14) are absent due to charge assignment (i.e. $X_{\mathcal{S}}\neq
3\alpha_{B+L}$), this requires $y_{1,2,3,4}\sim\sqrt{4\pi}$ in order for the
anomalons to be heavier than the electroweak scale. Upon integrating out the
electroweak anomalons at one loop one finds in the EFT given by the SM and the
light vector $\mathcal{X}$ (also keeping the Goldstone mode $\xi$, see App. A
for details)

$\displaystyle\mathscr{L}_{\rm EFT}^{{\rm U}(1)_{X}}$ $\displaystyle\supset
g_{X}g^{\prime
2}\frac{C_{BB}}{24\pi^{2}}\epsilon^{\alpha\mu\nu\beta}\mathcal{X}_{\alpha}B_{\mu}\partial_{\beta}B_{\nu}+g_{X}g^{2}\frac{C_{ab}}{24\pi^{2}}\epsilon^{\alpha\mu\nu\beta}\mathcal{X}_{\alpha}W_{\mu}^{a}\partial_{\beta}W_{\nu}^{b}$
$\displaystyle+g_{X}gg^{\prime}\frac{C_{aB}}{24\pi^{2}}\epsilon^{\alpha\mu\nu\beta}\mathcal{X}_{\alpha}W_{\mu}^{a}\partial_{\beta}B_{\nu}+g_{X}gg^{\prime}\frac{C_{Ba}}{24\pi^{2}}\epsilon^{\alpha\mu\nu\beta}\mathcal{X}_{\alpha}B_{\mu}\partial_{\beta}W_{\nu}^{a}\,$
$\displaystyle+g_{X}g^{2}\frac{D_{ab}}{48\pi^{2}}\frac{\xi}{m_{\mathcal{X}}}\epsilon^{\alpha\mu\beta\nu}(\partial_{\alpha}W_{\mu}^{a})(\partial_{\beta}W_{\nu}^{b})+g_{X}g^{\prime
2}\frac{D_{BB}}{48\pi^{2}}\frac{\xi}{m_{\mathcal{X}}}\epsilon^{\alpha\mu\beta\nu}(\partial_{\alpha}B_{\mu})(\partial_{\beta}B_{\nu})\,$
$\displaystyle+g_{X}gg^{\prime}\frac{D_{aB}}{24\pi^{2}}\frac{\xi}{m_{\mathcal{X}}}\epsilon^{\alpha\mu\beta\nu}(\partial_{\alpha}W_{\mu}^{a})(\partial_{\beta}B_{\nu})\,,$
(2.20)

where $a,b=1,2,3$ and we neglected non-abelian $W$ terms scaling with an extra
gauge coupling $g$. In general, from the requirement that the electromagnetic
group remains unbroken, one obtains

$\displaystyle C_{ab}$ $\displaystyle=\begin{pmatrix}C_{11}&C_{12}&0\\\
-C_{12}&C_{11}&0\\\ 0&0&C_{33}\end{pmatrix}\,,\quad
C_{aB}=\begin{pmatrix}0,&0,&C_{3B}\end{pmatrix}\,,\quad
C_{Ba}=\begin{pmatrix}0,&0,&C_{B3}\end{pmatrix}\,,$ (2.21) $\displaystyle
D_{ab}$ $\displaystyle=\begin{pmatrix}D_{11}&0&0\\\ 0&D_{11}&0\\\
0&0&D_{33}\end{pmatrix}\,,\quad
D_{aB}=\begin{pmatrix}0,&0,&D_{3B}\end{pmatrix}\,,$ (2.22)

together with the sum-rules

$\displaystyle C_{33}+C_{3B}+C_{B3}+C_{BB}=0\,,$ (2.23) $\displaystyle
D_{33}+2D_{3B}+D_{BB}=0\,.$ (2.24)

A relatively simple case is given in the limit where the masses of the
anomalon fields stem completely from the VEV of $\mathcal{S}$, yielding

$\displaystyle C_{11}$ $\displaystyle=C_{33}=-C_{BB}=3\alpha_{B+L}\,,$ (2.25)
$\displaystyle C_{B3}$ $\displaystyle=-C_{3B}=D_{3B}=C_{12}=0\,,$ (2.26)
$\displaystyle D_{11}$ $\displaystyle=D_{33}=-D_{BB}=-9\alpha_{B+L}\,,$ (2.27)

where the effective coefficients are set by the anomalous trace of the SM
current (see Eqs. (A.46)–(A.47)). Here, instead, we focus on the more general
case where the anomalon masses have both a SM-singlet and an electroweak
symmetry breaking source. Although we were not able to cast explicit
expressions for the EFT coefficients into a simple analytical form (see Eqs.
(A.28)–(A.29)), we will present them here under the simplified (but
phenomenologically motivated) hypothesis in which the anomalon masses are
degenerate, that is

$\mathcal{M}_{{\cal E}}^{{\dagger}}\mathcal{M}_{{\cal E}}=\mathcal{M}_{{\cal
N}}^{{\dagger}}\mathcal{M}_{{\cal N}}=m_{\Psi}^{2}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\,,$ (2.28)

with $m_{\Psi}$ denoting the degenerate anomalons mass. Eq. (2.28) enforces
the mass matrices to be

$\mathcal{M}_{{\cal E}}=\mathcal{M}_{{\cal
N}}=\frac{1}{\sqrt{2}}\begin{pmatrix}y_{\mathcal{S}}v_{X}&iy_{H}v\\\
iy_{H}v&y_{\mathcal{S}}v_{X}\end{pmatrix}=m_{\Psi}\begin{pmatrix}\cos\theta&i\sin\theta\\\
i\sin\theta&\cos\theta\end{pmatrix}\,,$ (2.29)

with $y_{\mathcal{S},H}$ real parameters,444The accidental global ${\rm
U}(1)^{6}$ symmetry corresponding to the re-phasing of the each electroweak
anomalon field is broken by the $\mathscr{L}_{Y}+\Delta\mathscr{L}_{Y}$,
leaving an unbroken electroweak anomalon number ${\rm U}(1)_{A}$, that is the
subgroup corresponding to the common re-phasing of all the anomalon fields.
Hence, of the seven complex parameters introduced in
$\mathscr{L}_{Y}+\Delta\mathscr{L}_{Y}$, $6-1=5$ phases are unphysical and can
be rotated away. One possible choice is to set $\text{Arg}(y_{{\cal
L}})=\text{Arg}(y_{{\cal E}})=\text{Arg}(y_{{\cal N}})=0$,
$\text{Arg}(y_{1})=-\text{Arg}(y_{2})$, and
$\text{Arg}(y_{3})=-\text{Arg}(y_{4})$. while

$m^{2}_{\Psi}=\frac{1}{2}\left((y_{H}v)^{2}+(y_{S}v_{X})^{2}\right)\,,$ (2.30)

and

$\tan\theta=\frac{y_{H}v}{y_{S}v_{X}}\,.$ (2.31)

Specializing the general expressions in Eqs. (A.28)–(A.29) to the degenerate
case above, we find

$\displaystyle C_{11}$
$\displaystyle=C_{33}=-C_{BB}=\frac{3}{4}\alpha_{B+L}(1+3\cos 2\theta)\,,$
(2.32) $\displaystyle C_{B3}$
$\displaystyle=-C_{3B}=\frac{9}{4}\alpha_{B+L}(1-\cos
2\theta),\,\,\,\,\,\,\,\,\,C_{12}=0\,,$ (2.33) $\displaystyle D_{11}$
$\displaystyle=D_{33}=-D_{BB}=-\frac{9}{2}\alpha_{B+L}(1+\cos
2\theta),\,\,\,\,\,\,\,D_{3B}=0\,.$ (2.34)

The important point to be noted is that when the anomalons pick-up a mass from
both electroweak preserving and breaking sources, the low-energy WZ
coefficients acquire a model dependence through the angle $\theta$.

In order to understand the phenomenological implications of this model
dependence, we briefly recall here the argument of Refs. [10, 11] regarding
the energy-enhanced emission of the longitudinal modes of $\mathcal{X}$
stemming from the WZ operators. Taking the limit $g_{X}\rightarrow 0$ and
$m_{\mathcal{X}}\rightarrow 0$, while keeping fixed the ratio
$m_{\mathcal{X}}/g_{X}\propto v_{X}$, the transverse modes of $\mathcal{X}$
decouple, while the longitudinal mode is enhanced as $E/m_{\mathcal{X}}$. In
this regime, the equivalence theorem states that the longitudinally polarized
vectors are equivalent to the corresponding scalar Goldstone bosons. This is
readily seen by working in the so-called “Equivalent Gauge” of Ref. [27],
where the longitudinally polarized state, $|\mathcal{X}_{L}\rangle$, is
represented as

$\langle
0|\mathcal{X}_{\mu}(x)|\mathcal{X}_{L}(\vec{p})\rangle=\epsilon_{\mu}^{L}(\vec{p})e^{-ipx}\,,\qquad\langle
0|\xi(x)|\mathcal{X}_{L}(\vec{p})\rangle=-ie^{-ipx}\,,$ (2.35)

with the polarization vector

$\epsilon_{\mu}^{L}(\vec{p})=-\frac{m_{\mathcal{X}}}{E_{\vec{p}}+|\vec{p}|}\left\\{1,\frac{\vec{p}}{|\vec{p}|}\right\\}\,,$
(2.36)

vanishing in the $m_{\mathcal{X}}\rightarrow 0$ limit. The advantage of this
representation is that it makes the equivalence theorem explicit, since in the
high-energy limit (or equivalently $m_{\mathcal{X}}\rightarrow 0$) only the
Goldstone mode survives. Hence, adopting the above prescription, only the
diagrams with one external $\xi$ contribute to physical processes in the
$m_{\mathcal{X}}\rightarrow 0$ limit. For instance, upon integrating out the
$W$ boson, the axion-like operator $\xi W^{-}\tilde{W}^{+}$ proportional to
$D_{11}$ in Eq. (2.2) yields the effective interaction

$g_{\xi
d_{i}d_{j}}\overline{d}_{j}\gamma^{\mu}P_{L}d_{i}\,(\partial_{\mu}\xi/m_{\mathcal{X}})\,+\text{h.c.}\,,$
(2.37)

in terms of the effective coupling [28]

$g_{\xi
d_{i}d_{j}}=-\frac{g_{X}g^{4}}{(4\pi)^{4}}D_{11}\sum_{\alpha=u,c,t}V_{\alpha
i}V_{\alpha j}^{*}F(m_{\alpha}^{2}/m_{W}^{2})\,,$ (2.38)

with $D_{11}\propto(1+\cos 2\theta)$ given in Eq. (2.34), $V$ denoting the CKM
matrix and the loop function

$F(x)=\frac{x(1+x(\ln x-1))}{(1-x)^{2}}\,.$ (2.39)

This leads to FCNC processes, such as $K\to\pi\mathcal{X}_{L}$, $B\to
K\mathcal{X}_{L}$, etc, whose rate is enhanced as $(E/m_{\mathcal{X}})^{2}$,
where $E$ is the decay energy (cf. the derivative operator in Eq. (2.37)),
thus implying strong bounds on light vector bosons coupled to anomalous
currents [10, 11].

On the other hand, the above constraints from energy-enhanced
$\mathcal{X}_{L}$ emission disappear for $D_{11}=0$, that is when the ${\rm
U}(1)_{X}$ Goldstone decouples from the electroweak anomalons. This
corresponds to $\theta=\pi/2$, which implies that the anomalon masses are
entirely due to the Higgs VEV (cf. Eq. (2.31)). From a top-down perspective
this condition can be neatly obtained in terms of ${\rm U}(1)_{X}$ charges
($X_{\mathcal{S}}\neq 3\alpha_{B+L}$) which forbid the operators of
$\Delta\mathscr{L}_{Y}$ in Eq. (2.14). Alternatively, it can be parametrically
obtained by taking $v_{X}\ll v$ or $y_{\mathcal{S}}\approx 0$. Note that the
latter condition is radiatively stable, since it corresponds to an enhanced
U(1) global symmetry of the Lagrangian in which LH and RH anomalons fields are
rotated with an opposite phase.

In conclusion, we have shown that the bounds of Refs. [10, 11] can be relaxed
by assuming that the anomalon fields are mostly chiral (namely, their mass
mostly stems from the Higgs VEV). This possibility, however, leads to non-
decoupling signatures in Higgs observables and direct searches, to be
discussed in Sect. 3.

###  2.3 Neutrino masses

If the $X$ generator has a non-trivial projection on family lepton numbers,
$\alpha_{i}\neq 0$ ($i=e,\mu,\tau$), then we need RH neutrinos,
$\nu_{R}^{\alpha}$ ($\alpha=1,\ldots,N$), in order to cancel ${\rm U}(1)_{X}$
and ${\rm U}(1)^{3}_{X}$ anomalies (cf. conditions in Eqs. (2.12)–(2.13)). The
simplest solution is to introduce one RH neutrino for each $\alpha_{i}\neq 0$
and set $X^{i}_{\nu_{R}}=\alpha_{i}$. Another possibility is to have universal
charges $X_{\nu_{R}}^{\alpha}=X_{\nu_{R}}$, so that the anomaly-free
conditions are

$\displaystyle
X_{\nu_{R}}=\left(\frac{\alpha^{3}_{e}+\alpha^{3}_{\mu}+\alpha^{3}_{\tau}}{\alpha_{e}+\alpha_{\mu}+\alpha_{\tau}}\right)^{1/2}\,,\qquad
N=\left(\frac{(\alpha_{e}+\alpha_{\mu}+\alpha_{\tau})^{3}}{\alpha^{3}_{e}+\alpha^{3}_{\mu}+\alpha^{3}_{\tau}}\right)^{1/2}\,.$
(2.40)

The SM-singlet states $\nu_{R}^{\alpha}$ can be used to give mass to light
neutrinos via the seesaw mechanism. In fact, SM gauge invariance would allow
the operators555We neglect here bare Majorana mass terms, since in that case
RH neutrinos would not contribute to the cancellation of ${\rm U}(1)_{X}$ and
${\rm U}(1)^{3}_{X}$ anomalies. Instead, possible mixings between RH neutrinos
and electroweak anomalons have been classified in Table 2.

$-\mathscr{L}_{Y}^{\nu_{R}}=y^{i\beta}_{D}\overline{\ell}^{i}_{L}\nu_{R}^{\beta}\tilde{H}+\frac{1}{2}y^{\alpha\beta}_{\nu_{R}}\nu_{R}^{\alpha}\nu_{R}^{\beta}\mathcal{S}^{*}+\text{h.c.}\quad\longrightarrow\quad
m_{D}^{i\beta}\,\overline{\ell}^{i}_{L}\nu_{R}^{\beta}+\frac{1}{2}M_{R}^{\alpha\beta}\nu_{R}^{\alpha}\nu_{R}^{\beta}+\text{h.c.}\,,$
(2.41)

with $m_{D}=y_{D}v/\sqrt{2}$ and $M_{R}=y_{R}v_{X}/\sqrt{2}$, leading to light
neutrino masses via the seesaw mechanism

$m_{\nu}=m_{D}M_{R}^{-1}m_{D}^{T}\,.$ (2.42)

However, in order for the operators in Eq. (2.41) to be ${\rm U}(1)_{X}$
invariant, the following constraints on ${\rm U}(1)_{X}$ charges need to be
satisfied

$\displaystyle-\alpha_{i}+X^{\beta}_{\nu_{R}}$ $\displaystyle=0\,,$ (2.43)
$\displaystyle X^{\alpha}_{\nu_{R}}+X^{\beta}_{\nu_{R}}-X_{\mathcal{S}}$
$\displaystyle=0\,.$ (2.44)

While the first condition can be easily fulfilled (since it also ensures the
cancellation of ${\rm U}(1)_{X}$ and ${\rm U}(1)^{3}_{X}$ anomalies), the
second one could imply texture zeros in $M_{R}$ if some leptonic generators
are non-universal $\alpha_{i}\neq\alpha_{j}$. Consistency with light neutrino
data might then require the introduction of extra scalars in order to
reproduce realistic low-energy textures (see e.g. [29, 30, 31]).

##  3 Electroweak anomalons phenomenology

In the previous Section we have seen that mostly chiral electroweak anomalons
(i.e. which take their mass mostly from the Higgs VEV) allow to decouple
dangerous WZ terms, which would otherwise lead to the energy-enhanced
longitudinal emission of light vectors coupled to anomalous currents. We are
hence interested in studying the electroweak phenomenology of the exotic
leptons ${\cal L}+{\cal N}+{\cal E}$, whose quantum numbers are displayed in
Table 1. In particular, following the analysis of Refs. [16, 17], we will
argue that phenomenology requires ${\cal Y}\approx 2,-1$.

###  3.1 Electroweak precision tests

The contribution of the electroweak anomalons in terms of mass eigenstates
(cf. Table 1 and Eq. (2.17)) to the $S$ and $T$ parameters is [16]

$\displaystyle S$ $\displaystyle=\frac{1}{6\pi}\left[\left(1-2({\cal
Y}-\frac{1}{2})\log\frac{m^{2}_{{\cal N}_{1}}}{m^{2}_{{\cal
E}_{1}}}\right)+\left(1+2({\cal Y}-\frac{1}{2})\log\frac{m^{2}_{{\cal
N}_{2}}}{m^{2}_{{\cal
E}_{2}}}\right)+\mathcal{O}\left(\frac{m^{2}_{Z}}{m^{2}_{{\cal N},\,{\cal
E}}}\right)\right]\approx\frac{1}{3\pi}\,,$ (3.1) $\displaystyle T$
$\displaystyle=\frac{1}{16\pi c^{2}_{W}s^{2}_{W}m^{2}_{Z}}\left(m^{2}_{{\cal
N}_{1}}+m^{2}_{{\cal E}_{1}}-2\frac{m^{2}_{{\cal N}_{1}}m^{2}_{{\cal
E}_{1}}}{m^{2}_{{\cal N}_{1}}-m^{2}_{{\cal E}_{1}}}\log\frac{m^{2}_{{\cal
N}_{1}}}{m^{2}_{{\cal E}_{1}}}\right)$ $\displaystyle+\frac{1}{16\pi
c^{2}_{W}s^{2}_{W}m^{2}_{Z}}\left(m^{2}_{{\cal N}_{2}}+m^{2}_{{\cal
E}_{2}}-2\frac{m^{2}_{{\cal N}_{2}}m^{2}_{{\cal E}_{2}}}{m^{2}_{{\cal
N}_{2}}-m^{2}_{{\cal E}_{2}}}\log\frac{m^{2}_{{\cal N}_{2}}}{m^{2}_{{\cal
E}_{2}}}\right)\approx 0\,,$ (3.2)

where the approximation in the last steps holds in the custodial limit
$m_{{\cal N}_{1,2}}\approx m_{{\cal E}_{1,2}}$. Recent fits for oblique
parameters, e.g. from Gfitter [32], yield

$S=0.05\pm 0.11\,,\qquad T=0.09\pm 0.13\,,$ (3.3)

which are easily satisfied in the custodial limit, although a mass splitting
might play a role to explain the recent $M_{W}$ anomaly [33].

###  3.2 Higgs physics

We now consider the constraints from Higgs couplings measurements. In
particular, we assess the impact of the new heavy fermions on the decay rate
of the Higgs boson to two photons, or to a photon and a $Z$ boson. Taking a
fermion $\psi$ of mass $m_{\psi}$ the interaction Lagrangian is given by

${\cal L}_{\psi}^{\rm
int}=-\frac{m_{\psi}}{v}h\overline{\psi}\psi+eQ_{\psi}\overline{\psi}\gamma^{\mu}\psi
A_{\mu}+\frac{e}{c_{W}s_{W}}\overline{\psi}\gamma^{\mu}\left(\frac{T^{3}_{\psi}}{2}-Q_{\psi}s_{W}^{2}-\frac{T^{3}_{\psi}}{2}\gamma_{5}\right)\psi
Z_{\mu}\ ,$ (3.4)

where $h$ is the $125$ GeV Higgs, $A_{\mu}$ and $Z_{\mu}$ the photon and $Z$
boson fields, $T^{3}_{\psi}$ is the eigenvalue of the third generator of ${\rm
SU}(2)_{L}$ when it acts on the left-handed component of $\psi$, so that
$T^{3}_{\psi}=\pm\frac{1}{2}$ when $\psi_{L}$ arises from a doublet in the
fundamental of ${\rm SU}(2)_{L}$. Its one-loop contributions to the amplitudes
$h\to\gamma\gamma$ and $h\to\gamma Z$ are [34]

${\cal A}^{\psi}_{\gamma\gamma}\approx\frac{4}{3}Q_{\psi}^{2}\ ,\quad{\cal
A}^{\psi}_{Z\gamma}\approx-\frac{1}{3}Q_{\psi}\frac{T^{3}_{\psi}-2Q_{\psi}s_{W}^{2}}{c_{W}}\
,$ (3.5)

where we assumed that $\psi$ is much heavier than the Higgs and the $Z$ boson,
which holds for the heavy fermions we consider here. In the SM, these
amplitudes are dominated by the loop of the $W$ gauge boson interfering
negatively with the loop of the top quark and they amount to
$A_{\gamma\gamma}^{\rm SM}\approx-6.5$ and $\mathcal{A}_{\gamma Z}^{\rm
SM}\approx 5.7$ at leading order. In the presence of a single Higgs doublet,
the new physics contribution yields $\mathcal{A_{\gamma\gamma}^{\rm
NP}}\approx\frac{8}{3}(1-2{\cal Y}+2{\cal Y}^{2})$. Writing the modified Higgs
width to photons as

$R_{\gamma\gamma}=\frac{|\mathcal{A_{\gamma\gamma}^{\rm
SM}}+\mathcal{A_{\gamma\gamma}^{\rm NP}|}^{2}}{\mathcal{|A_{\gamma\gamma}^{\rm
SM}|}^{2}}\,,$ (3.6)

a recent ATLAS analysis found $R_{\gamma\gamma}=1.00\pm 0.12$ [35]. There is
in fact the possibility that the new physics contribution interferes
negatively with the SM amplitude, namely $\mathcal{A_{\gamma\gamma}^{\rm
NP}}\approx-2\mathcal{A}_{\gamma\gamma}^{\rm SM}\approx 13.0$. This is
obtained either for ${\cal Y}\approx 2$ ($1.93\lesssim{\cal Y}\lesssim 2.03$
[2$\sigma$ range]) or ${\cal Y}\approx-1$ ($-1.03\lesssim{\cal
Y}\lesssim-0.93$ [2$\sigma$ range]), both yielding
$\mathcal{A_{\gamma\gamma}^{\rm NP}}({\cal
Y}=2)=\mathcal{A_{\gamma\gamma}^{\rm NP}}({\cal Y}=-1)\approx 13.3$. A
correlated signal in the $\gamma Z$ channel yields $\mathcal{A}_{\gamma
Z}^{\rm NP}\approx-\frac{2}{3}c_{W}[1-(3-8{\cal Y}+8{\cal Y}^{2})t^{2}_{W}]$,
leading to a large deviation $\mathcal{A}_{\gamma Z}^{\rm NP}({\cal
Y}=2)=\mathcal{A}_{\gamma Z}^{\rm NP}({\cal Y}=-1)\approx 2.33$ in the region
where the value of ${\cal Y}$ is compatible with the di-photon channel. The
$\gamma Z$ decay channel of the Higgs has not been observed yet and HL-LHC is
expected to measure $\kappa_{\gamma Z}$ within $10\%$ precision [36]. Thus the
model with a single Higgs doublet predicts a strong departure of $R_{Z\gamma}$
from its SM value, although extended Higgs sectors can help to tame
modifications of Higgs signals (see e.g. [17]).

###  3.3 Direct searches

Direct searches at high-energy particle colliders depend on whether the exotic
leptons mix with the SM leptons. In fact, this is possible only for the values
${\cal Y}=0,\pm 1,2$ (see Table 2), including the phenomenologically favoured
case ${\cal Y}=2,-1$. We discuss in turn the two different scenarios
corresponding to ${\cal Y}\neq 2,-1$ (stable charged leptons) and ${\cal
Y}=2,-1$ (unstable charged leptons).

####  3.3.1 Stable charged leptons

For ${\cal Y}\approx 2,-1$ (but ${\cal Y}\neq 2,-1$) the exotic leptons do not
mix the SM ones and the lightest state of the spectrum is electrically charged
and stable due to exotic lepton number. Charged relics are cosmologically
dangerous and largely excluded. To avoid cosmological problems one has to
invoke low-scale inflation, such that charged relics are either diluted by
inflation or never thermally produced. On the other hand, stable charged
particles yield striking signatures at colliders in the form of charged track,
anomalous energy loss in calorimeters, longer times of flight, etc. Applying
the experimental limits of [37] at 13 TeV LHC with the leading-order Drell-
Yann cross-sections rescaled for $|Q|=2$ (see also [38]), Ref. [17] obtained
$m_{{\cal N},\,{\cal E}}\gtrsim 800$ GeV. Since $m_{{\cal N},\,{\cal
E}}=y_{{\cal N},\,{\cal E}}\,v/\sqrt{2}$, direct searches imply Yukawa
couplings, $y_{{\cal N},\,{\cal E}}\approx 4.6$, at the boundary of
perturbative unitarity (see e.g. [39, 40]).666Large higher-order corrections
(starting at two loops) are expected for Higgs decays and they might slightly
change the solutions ${\cal Y}\approx 2,-1$.

####  3.3.2 Unstable charged leptons

For ${\cal Y}=2,-1$ the electroweak anomalons have electric charge $Q=2,-1$
(${\cal N}$ components) and $Q=1,-2$ (${\cal E}$ components). The $|Q|=2$
states can decay into a $W$ and a $|Q|=1$ fermion, while the latter can mix
with SM leptons and decay into $Z\ell$ or $h\ell$. Signatures of this type
were previously studied in Ref. [41], which estimated a mass reach at the LHC
14 up to $m_{{\cal N},\,{\cal E}}\sim 800$ GeV (depending on the integrated
luminosity). To our knowledge, however, such an analysis has never been
performed by the experimental collaborations. Anyhow, the bounds appear to be
of the same order of those obtained in the case of stable charged leptons.

##  4 Conclusions

In this work we have revisited the case of light vector bosons coupled to
anomalous currents which are UV completed by new anomaly-cancelling heavy
fermions (anomalons). After the latter have been integrated out, WZ terms of
the type in Eq. (2.2) are generated. On the one hand, they take care of
anomaly cancellation in the IR and, on the other, they source the energy-
enhanced emission of longitudinally polarized vectors, $\mathcal{X}$, which
typically results in very strong bounds on $g_{X}/m_{\mathcal{X}}\propto
1/v_{X}$ whenever the decay channels $Z\to\gamma\mathcal{X}$, $B\to
K\mathcal{X}$, $K\to\pi\mathcal{X}$, etc, are kinematically open [10, 11].
Here, we have studied the model-dependence of such bounds, considering as a
paradigmatic framework the gauging of the most general (anomalous) linear
combination of SM global symmetries, ${\rm U}(1)_{X}$, with the generator $X$
given in Eq. (1.1). To this end, we provided a UV completion including
electroweak anomalons ${\cal L}+{\cal E}+{\cal N}$ (cf. Table 1) to cancel
${\rm U}(1)_{X}$ anomalies in combination with electroweak gauge factors and
RH neutrinos to take care of ${\rm U}(1)_{X}$ anomalies in isolation when the
lepton number generators are gauged. An extra scalar $\mathcal{S}$ provides
the spontaneous breaking of the ${\rm U}(1)_{X}$ factor and gives mass to the
vector $\mathcal{X}$. Then, we have computed the EFT of a light $\mathcal{X}$
when the heavy anomalons are integrated, keeping in general both electroweak
symmetry breaking and preserving sources for the mass of the anomalons (see
App. A for details). This allowed us to conclude (cf. e.g. Eq. (2.38)) that
the bounds mentioned above on light $\mathcal{X}$ can be evaded in the limit
where the mass of the electroweak anomalons comes mostly from the Higgs VEV.
This condition can be neatly imposed in terms of ${\rm U}(1)_{X}$ gauge
charges (so that the operators in Eq. (2.6) are allowed while those in Eq.
(2.14) are forbidden) or parametrically by decoupling the vector-like masses
of the exotic leptons by taking a small yukawa and/or a small VEV for
$\mathcal{S}$. On the other hand, mostly chiral exotic leptons (receiving
their mass mostly from the Higgs VEV) are strongly constrained due to their
non-decoupling nature by electroweak-scale phenomenology, in particular Higgs
couplings and direct searches. We have reviewed in Sect. 3 those constraints,
based on the previous analyses in [16, 17], and argued that it is possible to
evade $h\to\gamma\gamma$ bounds for ${\cal Y}\approx 2,-1$ (including the
exact cases ${\cal Y}=2,-1$ allowing for mixings between anomalons and SM
leptons, cf. Table 2). For ${\cal Y}\approx 2,-1$, however, the $h\to\gamma Z$
channel differs $\mathcal{O}(1)$ from the SM and it will be possible to test
this scenario at the HL-LHC. Direct searches, whose signatures depend on
whether the electroweak anomalons mix or not with the SM leptons, are also
very stringent and they practically push the Yukawas of the exotic leptons to
the boundary of perturbativity.

Since, somewhat surprisingly, mostly chiral exotic leptons are not yet ruled-
out, it would be interesting to see the potential implications of the setup
discussed in this paper on the physics of light vectors coupled to anomalous
currents. The most relevant application turns out to be for the case of
electro-phobic light vectors ($\alpha_{e}=0$ in Eq. (1.1)), for which the
bounds stemming from WZ terms are typically the most important ones. These
include, for instance, baryonic forces ($\alpha_{B}\neq 0$) which have a rich
accelerator and collider phenomenology (see e.g. [42, 43]), while a case
motivated by the $g$-2 anomaly [18] is that of purely muonic forces
($\alpha_{\mu}\neq 0$) which are in principle distinguishable from more
standard scenarios based on ${\rm U}(1)_{\mu-\tau}$ (see e.g. [44]). The
general framework discussed in Sect. 2 provides a consistent UV completion for
light vectors coupled to anomalous SM fermionic currents, allowing to free
large portions of parameter space from the bounds of Ref. [10, 11], until the
HL-LHC will give the final word on the existence of mostly chiral exotic
leptons.

## Acknowledgments

The work of LDL is partially supported by the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No 860881 - HIDDEN. The work of MN and CT was supported in part by
MIUR under contract PRIN 2017L5W2PT

##  Appendix A Calculation of the Wess-Zumino terms

In this Appendix we present the calculation of the effective WZ terms
involving gauge and Goldstone bosons that arise after integrating out heavy
fermionic degrees of freedom. In particular, we focus on the 3-point vertices
involving the epsilon tensor $\epsilon^{\alpha\beta\mu\nu}$ (with
$\epsilon^{0123}=1$) which are related to anomaly cancellation in the EFT when
the heavy fermions are integrated out.

###  A.1 Toy model

We assume a toy model with a set of gauge bosons $G_{\mu}^{A}$ related to the
generators $Q^{A}$ of the gauge symmetry group $\mathcal{G}$ (that can be in
general semi-simple). The model contains a fermionic sector, whose fields are
labeled as $\psi_{i}$, that acquire a mass term $\mathcal{M}_{ij}$ after a
spontaneously symmetry breaking (SSB) mechanism. The $(\frac{1}{2},0)$ and
$(0,\frac{1}{2})$ Lorentz components of the $\psi$ field are separately
(reducible) representations of $\mathcal{G}$ and the generators act on them as

$Q^{A}\psi_{i}=\sum_{j}(Q_{L}^{A})_{ij}\psi_{jL}+\sum_{j}(Q_{R}^{A})_{ij}\psi_{jR}\,,$
(A.1)

where $(Q_{L,R}^{A})_{ij}$ are the matrix rapresentation of the gauge
multiplets $\psi_{L,R}\equiv P_{L,R}\psi$. We restrict ourselves to models
with a ${\rm U}(1)_{\psi}$ symmetry corresponding to the fermionic number of
the $\psi$ fields ($\psi_{i}\to e^{i\phi}\psi_{i}$). The real scalar Higgs
fields, responsible for the SSB mechanism, are labeled as $H_{a}=(H_{a})^{*}$
and belong to a (reducible) representation of the gauge group $\mathcal{G}$.
By performing an infinitesimal transformation of angle $\alpha_{A}$ along the
$Q^{A}$ generator, the $H_{a}$ fields transform like

$\delta H_{a}=\sum_{b}g_{A}\alpha_{A}(iQ_{H}^{A})_{ab}H_{b}\,,$ (A.2)

where $(iQ_{H}^{A})_{ab}$ is a real and antisymmetric matrix. Hence,

$\mathcal{L}_{\text{toy}\,\text{model}}\supset\sum_{i}\overline{\psi}_{i}i\not{\partial}\psi_{i}-\sum_{a,i,j}H_{a}(\overline{\psi}_{iL}\mathcal{Y}_{ij}^{a}\psi_{jR}+\text{h.c.})-\sum_{A}g_{A}G_{\mu}^{A}J^{\mu
A}\,,$ (A.3)

with

$J^{\mu
A}=\sum_{i,j}\Bigl{[}\overline{\psi}_{iL}\gamma^{\mu}(Q_{L}^{A})_{ij}\psi_{jL}+\overline{\psi}_{iR}\gamma^{\mu}(Q_{R}^{A})_{ij}\psi_{jR}\Bigr{]}\,.$
(A.4)

The Yukawa couplings must preserve gauge invariance and hence they satisfy

$\sum_{k}\mathcal{Y}_{ik}^{a}(Q_{R}^{A})_{kj}-\sum_{k}(Q_{L}^{A})_{ik}\mathcal{Y}_{kj}^{a}+\sum_{b}\mathcal{Y}_{ij}^{b}(Q_{H}^{A})_{ba}=0\,.$
(A.5)

The Higgs fields acquire the VEVs $\langle H_{a}\rangle=v_{a}$ which break the
gauge group, leaving an unbroken subgroup $\mathcal{G}_{0}$. Then, the mass
matrix of the $\psi$ fields is given by

$\mathcal{M}_{ij}=\sum_{a}\mathcal{Y}_{ij}^{a}v_{a}\,,$ (A.6)

leading to

$\begin{split}\mathcal{L}_{\text{toy}\,\text{model}}\supset&\sum_{i}\overline{\psi}_{i}i\not{\partial}\psi_{i}-\sum_{i,j}(\overline{\psi}_{iL}\mathcal{M}_{ij}\psi_{jR}+\text{h.c.})\\\
&-\sum_{a,i,j}\tilde{H}_{a}(\overline{\psi}_{iL}\mathcal{Y}_{ij}^{a}\psi_{jR}+\text{h.c.})-\sum_{A}g_{A}G_{\mu}^{A}J^{\mu
A}\,,\end{split}$ (A.7)

where $\tilde{H}_{a}=H_{a}\\!-\\!v_{a}$ are the Higgs fluctuations around the
vacuum.

In order to go in the mass basis, the mass matrix $\mathcal{M}$ is
diagonalized via the bi-unitary transformations $\psi_{R}\to U_{R}\,\psi_{R}$
and $\psi_{L}\to U_{L}\,\psi_{L}$, which by construction satisfy
$U_{L}^{{\dagger}}\mathcal{M}U_{R}=\text{diag}(m_{1},m_{2},...)$. This yields

$\mathcal{L}_{\text{toy}\,\text{model}}\supset\sum_{i}\overline{\psi}_{i}(i\not{\partial}-m_{i})\psi_{i}-\sum_{A}g_{A}G_{\mu}^{A}J_{U}^{\mu
A}-\sum_{a,i,j}\tilde{H}_{a}\overline{\psi}_{i}(\hat{\mathcal{Y}}_{R}^{a}P_{R}+\hat{\mathcal{Y}}_{L}^{a}P_{L})_{ij}\psi_{j}\,,$
(A.8)

where
$\hat{\mathcal{Y}}_{R}^{a}=U_{L}^{{\dagger}}\mathcal{Y}^{a}U_{R}=(\hat{\mathcal{Y}}_{L}^{a})^{{\dagger}}$,
while the gauge currents in the mass basis are equal to

$J_{U}^{\mu
A}=\sum_{i,j}\Bigl{[}\overline{\psi}_{iL}\gamma^{\mu}(U_{L}^{{\dagger}}Q_{L}^{A}U_{L})_{ij}\psi_{jL}+\overline{\psi}_{iR}\gamma^{\mu}(U_{R}^{{\dagger}}Q_{R}^{A}U_{R})_{ij}\psi_{jR}\Bigr{]}\,.$
(A.9)

After integrating out the heavy fermion fields, we get EFT operators of the
type

$\begin{split}\mathcal{L}_{\text{toy}\,\text{model}}\supset&\sum_{A,B,C}\frac{g_{A}g_{B}g_{C}}{48\pi^{2}}C^{ABC}\epsilon^{\alpha\mu\nu\beta}G_{\alpha}^{A}G_{\mu}^{B}\partial_{\beta}G_{\nu}^{C}\\\
-&\sum_{a,B,C}\frac{g_{B}g_{C}}{48\pi^{2}}D^{aBC}\epsilon^{\mu\nu\alpha\beta}\tilde{H}_{a}\partial_{\alpha}G_{\mu}^{B}\partial_{\beta}G_{\nu}^{C}\,,\end{split}$
(A.10)

in terms of the EFT coefficients $C^{ABC}=-C^{BAC}$ and $D^{aBC}=D^{aCB}$ that
we want to compute. Moreover, integrating by parts the term on the first line
of Eq. (A.10), one also obtains

$C^{ABC}+C^{CAB}+C^{BCA}=0\,.$ (A.11)

###  A.2 $\gamma_{5}$ in dimensional regularization

Dimensional regularization allows to regularize the divergences arising from
loop calculations in 4 dimensions, while explicitly preserving Lorentz
covariance and gauge invariance. In the $d$-dimensional spacetime, the mass
dimensions of the quantum fields are equal to

$[\psi]=\frac{d-1}{2}\,,\qquad[H]=[G_{\mu}]=\frac{d-2}{2}\,.$ (A.12)

Hence, in order to keep the gauge couplings dimensionless, one introduces the
renormalization scale $\mu$ by the substitution

$g\rightarrow\mu^{\frac{4-d}{2}}\,g$ (A.13)

in the Lagrangian. The use of dimensional regularization poses some potential
problems in calculations where the $\gamma^{5}$ matrix is involved. In fact,
$\gamma^{5}$ (or equivalently the antisymmetric tensor
$\epsilon^{\alpha\beta\mu\nu}$) is a quantity whose definition is strictly
connected to the fact that space-time is four dimensional, and a definition in
$d$ dimensions requires special care. Here, we adopt the Breitenlohner-
Maison-’t Hooft-Veltman (BMHV) scheme, which is able to reproduce the chiral
anomaly (see [45] for a recent review).

We decompose all matrices into a four-dimensional (denoted by bars) and an
extra-dimensional (also called “evanescent”, denoted by hats) component:

$\gamma^{\mu}=\overline{\gamma}^{\mu}+\hat{\gamma}^{\mu}\,,$ (A.14)

where $\overline{\gamma}^{\mu}$ is non-zero only when $\mu$ takes the ordinary
values $0,1,2,3$ and $\hat{\gamma}^{\mu}$ vanishes for $\mu=0,1,2,3$.
Correspondingly, the matrix tensor $g_{\mu\nu}$ has a four-dimensional and an
extra-dimensional part,

$g_{\mu\nu}=\overline{g}_{\mu\nu}+\hat{g}_{\mu\nu}\,,$ (A.15)

while mixed components vanish. The gamma matrices satisfy

$\\{\overline{\gamma}^{\mu},\overline{\gamma}^{\nu}\\}=2\overline{g}^{\mu\nu}\,,\qquad\\{\hat{\gamma}^{\mu},\hat{\gamma}^{\nu}\\}=2\hat{g}^{\mu\nu}\,,\qquad\\{\overline{\gamma}^{\mu},\hat{\gamma}^{\nu}\\}=0\,.$
(A.16)

Then, we simply define $\gamma^{5}$ as in four dimensions, that is

$\gamma^{5}=i\overline{\gamma}^{0}\overline{\gamma}^{1}\overline{\gamma}^{2}\overline{\gamma}^{3}\,.$
(A.17)

It is easy to check that the definition in Eq. (A.17) implies

$\displaystyle\\{\gamma^{5},\overline{\gamma}^{\mu}\\}=0\,,\qquad[\gamma^{5},\hat{\gamma}^{\mu}]=0\,,$
(A.18)
$\displaystyle\text{Tr}\,\gamma^{5}\gamma^{\alpha}\gamma^{\beta}\gamma^{\mu}\gamma^{\nu}=\text{Tr}\,\gamma^{5}\overline{\gamma}^{\alpha}\overline{\gamma}^{\beta}\overline{\gamma}^{\mu}\overline{\gamma}^{\nu}=4i\epsilon^{\alpha\beta\mu\nu}\,,$
(A.19)

which is the correct four-dimensional result.

In a general chiral gauge theory, the fermion fields are introduced as Weyl
spinors whose formalism is intrinsically tied to 4-dimensional space. In $d$
dimensions, we replace the Weyl spinors by projections of Dirac spinors, which
can be generalized to arbitrary dimensions. The right and left projections are
$P_{R,L}=\frac{1}{2}(1\pm\gamma_{5})$, as in the 4-dimensional space. Then,
there are three possible inequivalent choices for the $d$-dimensional
extension of the right-handed chiral current
$\overline{\psi}_{iR}\gamma^{\mu}\psi_{jR}$ coupled to gauge bosons, which are

$\overline{\psi}_{i}P_{L}\gamma^{\mu}\psi_{j}\,,\qquad\overline{\psi}_{i}\gamma^{\mu}P_{R}\psi_{j}\,,\qquad\overline{\psi}_{i}P_{L}\gamma^{\mu}P_{R}\psi_{j}\,.$
(A.20)

They are different because $P_{L}\gamma^{\mu}\neq\gamma^{\mu}P_{R}$ in $d$
dimensions. Each of these does lead to valid $d$-dimensional extensions of the
model that are perfectly renormalizable using dimensional regularization and
the BMHV scheme. However, the intermediate calculations and the final
$d$-dimensional results will differ, depending on the choice for this
interaction term. Our choice for this work is to use the third option, that is

$\overline{\psi}_{i}P_{L}\gamma^{\mu}P_{R}\psi_{j}=\overline{\psi}_{iR}\overline{\gamma}^{\mu}\psi_{jR}\,,$
(A.21)

is the most symmetric one, and leads to the simplest expressions. Similar
considerations hold for the left-handed chiral current
$\overline{\psi}_{iL}\gamma^{\mu}\psi_{jL}$. A different choice has to be
taken instead for the kinetic terms
$\overline{\psi}_{iR}i\not{\partial}\psi_{iR}$ and
$\overline{\psi}_{iL}i\not{\partial}\psi_{iL}$. Indeed, in order to properly
regularize the theory, we need to consider the full Dirac fermion kinetic term
$\overline{\psi}_{i}i\not{\partial}\psi_{i}$, including the evanescent terms.

Once the regulated amplitude is well-defined, we can perform all the necessary
subtractions of the divergences of its sub-diagrams and the resulting finite
expression is interpreted in the physical 4-dimensional space by setting all
quantities to their 4-dimensional values, i.e. first taking the $d\to 4$ limit
and then, setting all remaining evanescent terms to zero.

###  A.3 1-loop matching

The epsilon tensor structure occurs in the 3-point functions
$\Gamma_{ABC}^{\alpha\mu\nu}(x,y,z)$ and $\Gamma_{aBC}^{\mu\nu}(x,y,z)$ at
1-loop through fermionic triangle diagrams (see Fig. 1). The amplitudes in
momentum space are defined via

$\int\\!\text{d}^{4}\\!x\,\text{d}^{4}\\!y\,\text{d}^{4}\\!z\,e^{i(xq_{1}+yq_{2}+zq_{3})}\,\Gamma_{ABC}^{\alpha\mu\nu}(x,y,z)|_{\text{1-loop}}=(2\pi)^{4}\delta^{(4)}\\!(q_{1}+q_{2}+q_{3})\,\mu^{\frac{4-d}{2}}\,iM_{ABC}^{\alpha\mu\nu}(q_{1},q_{2},q_{3})\,,$
(A.22)

and

$\int\\!\text{d}^{4}\\!x\,\text{d}^{4}\\!y\,\text{d}^{4}\\!z\,e^{i(xq_{1}+yq_{2}+zq_{3})}\,\Gamma_{aBC}^{\mu\nu}(x,y,z)|_{\text{1-loop}}=(2\pi)^{4}\delta^{(4)}\\!(q_{1}+q_{2}+q_{3})\,\mu^{\frac{4-d}{2}}\,iM_{aBC}^{\mu\nu}(q_{1},q_{2},q_{3})\,,$
(A.23)

Figure 1: Feynman diagrams relative to the 3-point functions in Eqs.
(A.22)–(A.23).

which yield

$\begin{split}M_{ABC}^{\alpha\mu\nu}=\sum_{\begin{subarray}{c}i,j,k\\\
\chi_{1},\chi_{2},\chi_{3}\end{subarray}}\\!&g_{A}g_{B}g_{C}(U_{\chi_{1}}^{{\dagger}}Q_{\chi_{1}}^{A}U_{\chi_{1}})_{jk}(U_{\chi_{2}}^{{\dagger}}Q_{\chi_{2}}^{B}U_{\chi_{2}})_{ki}(U_{\chi_{3}}^{{\dagger}}Q_{\chi_{3}}^{C}U_{\chi_{3}})_{ij}\\\
&\times
i\mu^{4-d}\\!\int\\!\frac{\text{d}^{d}\\!k}{(2\pi)^{d}}\,\frac{\text{Tr}_{\text{D}}[\overline{\gamma}^{\mu}\\!P_{\chi_{2}}\\!(\not{k}+m_{i})\overline{\gamma}^{\nu}\\!P_{\chi_{3}}\\!(\not{k}+\not{q}_{3}+m_{j})\overline{\gamma}^{\alpha}\\!P_{\chi_{1}}\\!(\not{k}-\not{q}_{2}+m_{k})]}{[k^{2}-m_{i}^{2}]\,[(k+q_{3})^{2}-m_{j}^{2}]\,[(k-q_{2})^{2}-m_{k}^{2}]}\\\
+\sum_{\begin{subarray}{c}i,j,k\\\
\chi_{1},\chi_{2},\chi_{3}\end{subarray}}\\!&g_{A}g_{C}g_{B}(U_{\chi_{1}}^{{\dagger}}Q_{\chi_{1}}^{A}U_{\chi_{1}})_{kj}(U_{\chi_{3}}^{{\dagger}}Q_{\chi_{3}}^{C}U_{\chi_{3}})_{ji}(U_{\chi_{2}}^{{\dagger}}Q_{\chi_{2}}^{B}U_{\chi_{2}})_{ik}\\\
&\times
i\mu^{4-d}\\!\int\\!\frac{\text{d}^{d}\\!k}{(2\pi)^{d}}\,\frac{\text{Tr}_{\text{D}}[\overline{\gamma}^{\nu}\\!P_{\chi_{3}}\\!(\not{k}+m_{i})\overline{\gamma}^{\mu}\\!P_{\chi_{2}}\\!(\not{k}+\not{q}_{2}+m_{k})\overline{\gamma}^{\alpha}\\!P_{\chi_{1}}\\!(\not{k}-\not{q}_{3}+m_{j})]}{[k^{2}-m_{i}^{2}]\,[(k+q_{2})^{2}-m_{k}^{2}]\,[(k-q_{3})^{2}-m_{j}^{2}]}\,,\end{split}$
(A.24)

and

$\begin{split}M_{aBC}^{\mu\nu}=\sum_{\begin{subarray}{c}i,j,k\\\
\chi_{1},\chi_{2},\chi_{3}\end{subarray}}\\!&g_{B}g_{C}(\hat{\mathcal{Y}}_{\chi_{1}}^{a})_{jk}(U_{\chi_{2}}^{{\dagger}}Q_{\chi_{2}}^{B}U_{\chi_{2}})_{ki}(U_{\chi_{3}}^{{\dagger}}Q_{\chi_{3}}^{C}U_{\chi_{3}})_{ij}\\\
&\times
i\mu^{4-d}\\!\int\\!\frac{\text{d}^{d}\\!k}{(2\pi)^{d}}\,\frac{\text{Tr}_{\text{D}}[\overline{\gamma}^{\mu}\\!P_{\chi_{2}}\\!(\not{k}+m_{i})\overline{\gamma}^{\nu}\\!P_{\chi_{3}}\\!(\not{k}+\not{q}_{3}+m_{j})P_{\chi_{1}}\\!(\not{k}-\not{q}_{2}+m_{k})]}{[k^{2}-m_{i}^{2}]\,[(k+q_{3})^{2}-m_{j}^{2}]\,[(k-q_{2})^{2}-m_{k}^{2}]}\\\
+\sum_{\begin{subarray}{c}i,j,k\\\
\chi_{1},\chi_{2},\chi_{3}\end{subarray}}\\!&g_{B}g_{C}(\hat{\mathcal{Y}}_{\chi_{1}}^{a})_{kj}(U_{\chi_{3}}^{{\dagger}}Q_{\chi_{3}}^{C}U_{\chi_{3}})_{ji}(U_{\chi_{2}}^{{\dagger}}Q_{\chi_{2}}^{B}U_{\chi_{2}})_{ik}\\\
&\times
i\mu^{4-d}\\!\int\\!\frac{\text{d}^{d}\\!k}{(2\pi)^{d}}\,\frac{\text{Tr}_{\text{D}}[\overline{\gamma}^{\nu}\\!P_{\chi_{3}}\\!(\not{k}+m_{i})\overline{\gamma}^{\mu}\\!P_{\chi_{2}}\\!(\not{k}+\not{q}_{2}+m_{k})P_{\chi_{1}}\\!(\not{k}-\not{q}_{3}+m_{j})]}{[k^{2}-m_{i}^{2}]\,[(k+q_{2})^{2}-m_{k}^{2}]\,[(k-q_{3})^{2}-m_{j}^{2}]}\,.\end{split}$
(A.25)

Since we have regularized the theory, the loop integrals over momentum $k$ are
convergent and can be evaluated with the usual well-known techniques. Next, we
perform the traces over the Dirac indices to extract the terms involving the
epsilon tensor structure we are interested on. One finds that such terms are
finite, i.e. they do not contain $1/(d\\!-\\!4)$ poles, and are independent
from the renormalization scale $\mu$. Hence, we can send $d\rightarrow 4$ and
set the evanescent components to zero.

In order to obtain the EFT coefficients in Eq. (A.10), we have to match the
expressions that we have calculated above to the EFT matrix elements in the
limit of heavy fermion masses, i.e.

$\lim_{\begin{subarray}{c}m_{i,j,k}^{2}\gg\\\
q_{2}^{2},q_{3}^{2},q_{2}\\!\cdot\\!q_{3}\end{subarray}}M_{ABC}^{\alpha\mu\nu}|_{\epsilon-
tensor}=\frac{g_{A}g_{B}g_{C}}{24\pi^{2}}\epsilon^{\alpha\mu\nu\beta}(C^{ABC}iq_{3}+C^{CAB}iq_{2}+C^{BCA}iq_{1})_{\beta}\,,$
(A.26)

and

$\lim_{\begin{subarray}{c}m_{i,j,k}^{2}\gg\\\
q_{2}^{2},q_{3}^{2},q_{2}\\!\cdot\\!q_{3}\end{subarray}}M_{aBC}^{\mu\nu}|_{\epsilon-
tensor}=\frac{g_{B}g_{C}}{24\pi^{2}}D^{aBC}\epsilon^{\mu\nu\alpha\beta}q_{2\alpha}q_{3\beta}\,.$
(A.27)

Thus we get

$\begin{split}C^{ABC}=\int_{0}^{+\infty}\\!\\!\\!\text{d}s\\!\int_{0}^{1}\\!\text{d}x\\!&\int_{0}^{1}\\!\text{d}y\\!\int_{0}^{1}\\!\text{d}z\,2\,\delta(1\\!-\\!x\\!-\\!y\\!-\\!z)\times\\\
\times\,\text{Re}\Biggl{\\{}3y\,&\text{Tr}\\!\left[e^{-sy\mathcal{M}^{{\dagger}}\mathcal{M}}Q_{R}^{A}\mathcal{M}^{{\dagger}}e^{-sz\mathcal{M}\mathcal{M}^{{\dagger}}}Q_{L}^{B}e^{-sx\mathcal{M}\mathcal{M}^{{\dagger}}}Q_{L}^{C}\mathcal{M}\right]\\\
-3y\,&\text{Tr}\\!\left[e^{-sy\mathcal{M}^{{\dagger}}\mathcal{M}}Q_{R}^{B}\mathcal{M}^{{\dagger}}e^{-sz\mathcal{M}\mathcal{M}^{{\dagger}}}Q_{L}^{A}e^{-sx\mathcal{M}\mathcal{M}^{{\dagger}}}Q_{L}^{C}\mathcal{M}\right]\\\
+3y\,&\text{Tr}\\!\left[e^{-sy\mathcal{M}\mathcal{M}^{{\dagger}}}Q_{L}^{B}\mathcal{M}e^{-sz\mathcal{M}^{{\dagger}}\mathcal{M}}Q_{R}^{A}e^{-sx\mathcal{M}^{{\dagger}}\mathcal{M}}Q_{R}^{C}\mathcal{M}^{{\dagger}}\right]\\\
-3y\,&\text{Tr}\\!\left[e^{-sy\mathcal{M}\mathcal{M}^{{\dagger}}}Q_{L}^{A}\mathcal{M}e^{-sz\mathcal{M}^{{\dagger}}\mathcal{M}}Q_{R}^{B}e^{-sx\mathcal{M}^{{\dagger}}\mathcal{M}}Q_{R}^{C}\mathcal{M}^{{\dagger}}\right]\\\
+y\,&\text{Tr}\\!\left[e^{-sy\mathcal{M}^{{\dagger}}\mathcal{M}}\mathcal{M}^{{\dagger}}\mathcal{M}Q_{R}^{A}e^{-sz\mathcal{M}^{{\dagger}}\mathcal{M}}Q_{R}^{B}e^{-sx\mathcal{M}^{{\dagger}}\mathcal{M}}Q_{R}^{C}\right]\\\
-x\,&\text{Tr}\\!\left[e^{-sy\mathcal{M}^{{\dagger}}\mathcal{M}}Q_{R}^{A}e^{-sz\mathcal{M}^{{\dagger}}\mathcal{M}}Q_{R}^{B}e^{-sx\mathcal{M}^{{\dagger}}\mathcal{M}}\mathcal{M}^{{\dagger}}\mathcal{M}Q_{R}^{C}\right]\\\
+x\,&\text{Tr}\\!\left[e^{-sy\mathcal{M}\mathcal{M}^{{\dagger}}}Q_{L}^{A}e^{-sz\mathcal{M}\mathcal{M}^{{\dagger}}}Q_{L}^{B}e^{-sx\mathcal{M}\mathcal{M}^{{\dagger}}}\mathcal{M}\mathcal{M}^{{\dagger}}Q_{L}^{C}\right]\\\
-y\,&\text{Tr}\\!\left[e^{-sy\mathcal{M}\mathcal{M}^{{\dagger}}}\mathcal{M}\mathcal{M}^{{\dagger}}Q_{L}^{A}e^{-sz\mathcal{M}\mathcal{M}^{{\dagger}}}Q_{L}^{B}e^{-sx\mathcal{M}\mathcal{M}^{{\dagger}}}Q_{L}^{C}\right]\Biggr{\\}}\,,\end{split}$
(A.28)

and

$\begin{split}D^{aBC}=\int_{0}^{+\infty}\\!\\!\\!\\!\\!\\!\text{d}s\\!\int_{0}^{1}\\!\text{d}x\\!\int_{0}^{1}\\!&\text{d}y\\!\int_{0}^{1}\\!\text{d}z\,6\,\delta(1\\!-\\!x\\!-\\!y\\!-\\!z)\times\\\
\times\,\text{Im}\Biggl{\\{}x\,&\text{Tr}\\!\left[e^{-sz\mathcal{M}^{{\dagger}}\mathcal{M}}Q_{R}^{B}\mathcal{M}^{{\dagger}}e^{-sx\mathcal{M}\mathcal{M}^{{\dagger}}}Q_{L}^{C}e^{-sy\mathcal{M}\mathcal{M}^{{\dagger}}}\mathcal{Y}^{a}\right]\\\
+x\,&\text{Tr}\\!\left[e^{-sz\mathcal{M}^{{\dagger}}\mathcal{M}}Q_{R}^{C}\mathcal{M}^{{\dagger}}e^{-sx\mathcal{M}\mathcal{M}^{{\dagger}}}Q_{L}^{B}e^{-sy\mathcal{M}\mathcal{M}^{{\dagger}}}\mathcal{Y}^{a}\right]\\\
+y\,&\text{Tr}\\!\left[e^{-sy\mathcal{M}^{{\dagger}}\mathcal{M}}\mathcal{M}^{{\dagger}}Q_{L}^{C}e^{-sx\mathcal{M}\mathcal{M}^{{\dagger}}}Q_{L}^{B}e^{-sz\mathcal{M}\mathcal{M}^{{\dagger}}}\mathcal{Y}^{a}\right]\\\
+y\,&\text{Tr}\\!\left[e^{-sy\mathcal{M}^{{\dagger}}\mathcal{M}}\mathcal{M}^{{\dagger}}Q_{L}^{B}e^{-sx\mathcal{M}\mathcal{M}^{{\dagger}}}Q_{L}^{C}e^{-sz\mathcal{M}\mathcal{M}^{{\dagger}}}\mathcal{Y}^{a}\right]\\\
+y\,&\text{Tr}\\!\left[e^{-sz\mathcal{M}^{{\dagger}}\mathcal{M}}Q_{R}^{B}e^{-sx\mathcal{M}^{{\dagger}}\mathcal{M}}Q_{R}^{C}\mathcal{M}^{{\dagger}}e^{-sy\mathcal{M}\mathcal{M}^{{\dagger}}}\mathcal{Y}^{a}\right]\\\
+y\,&\text{Tr}\\!\left[e^{-sz\mathcal{M}^{{\dagger}}\mathcal{M}}Q_{R}^{C}e^{-sx\mathcal{M}^{{\dagger}}\mathcal{M}}Q_{R}^{B}\mathcal{M}^{{\dagger}}e^{-sy\mathcal{M}\mathcal{M}^{{\dagger}}}\mathcal{Y}^{a}\right]\Biggr{\\}}\,.\end{split}$
(A.29)

###  A.4 Reproducing the chiral anomaly in the EFT

A consistent gauge theory must be anomaly free and hence the chiral anomaly
needs to cancel when we sum over all the fermion fields of the theory. If we
integrate out a heavy fermionic sector of the complete UV model, the
corresponding chiral anomaly is reproduced in the EFT action
$\mathcal{S}_{eff}$ thanks to the WZ effective operators in Eq. (A.10). To
show this, we make an infinitesimal transformation of angle $\alpha_{A}$ along
the $Q^{A}$ generator. The gauge fields $G_{\mu}^{B}$ and the Higgs fields
$\tilde{H}_{a}$ transform like

$\displaystyle\delta\tilde{H}_{a}=\sum_{b}\alpha_{A}(iQ_{H}^{A})_{ab}v_{b}\,+\,\text{linear\,terms}\,,$
(A.30) $\displaystyle\delta
G_{\mu}^{B}=-\delta_{AB}\,(\partial_{\mu}\alpha_{B})/g_{B}\,+\,\text{linear\,terms}\,,$
(A.31)

which, from the variation of the effective Lagrangian in Eq. (A.10), yields

$\begin{split}\delta\mathcal{S}_{eff}=&\sum_{BC}\frac{g_{B}g_{C}}{48\pi^{2}}\left[C^{ABC}+C^{ACB}+D^{aBC}(iQ_{H}^{A})_{ab}v_{b}\right]\int\\!\text{d}^{4}x\,\alpha_{A}\partial_{\alpha}G_{\mu}^{B}\partial_{\beta}G_{\nu}^{C}\epsilon^{\mu\nu\alpha\beta}\\\
=&\sum_{BC}\frac{g_{B}g_{C}}{48\pi^{2}}\left[\text{Tr}\,Q_{R}^{A}\\{Q_{R}^{B},Q_{R}^{C}\\}-\text{Tr}\,Q_{L}^{A}\\{Q_{L}^{B},Q_{L}^{C}\\}\right]\int\\!\text{d}^{4}x\,\alpha_{A}\partial_{\alpha}G_{\mu}^{B}\partial_{\beta}G_{\nu}^{C}\epsilon^{\alpha\mu\beta\nu}\,.\end{split}$
(A.32)

###  A.5 WZ terms for massive vector bosons and Goldstone bosons

The VEVs of the Higgs fields contribute to the mass matrix
$\mathcal{M}_{gauge}^{2}$ for the gauge bosons $G_{\mu}^{A}$ with elements

$(\mathcal{M}_{gauge}^{2})_{AB}=\sum_{a,b,c}g_{A}(iQ_{H}^{A})_{ca}v_{a}g_{B}(iQ_{H}^{B})_{cb}v_{b}\,.$
(A.33)

Since the matrix is real and symmetric, we can diagonalize it through an
orthogonal matrix $O_{AB}$ such that

$\sum_{A,B}\,O_{DB}O_{CA}(\mathcal{M}_{gauge}^{2})_{AB}=m_{C}^{2}\delta_{CD}\,.$
(A.34)

The massive eigenstates are then defined by

$Z_{\mu}^{A}=\sum_{B}\,O_{AB}\,G_{\mu}^{B}\,,$ (A.35)

with corresponding symmetry generator

$\tilde{g}_{A}T^{A}=\sum_{B}\,O_{AB}\,g_{B}Q^{B}\,.$ (A.36)

There are two scenarios for each generator $T^{A}$:

  * •

$(iT_{H}^{A})_{ab}v_{b}=0$, if $T^{A}$ belongs to the unbroken subgroup
$\mathcal{G}_{0}$, such that the corresponding vector boson $Z_{\mu}^{A}$ is
then massless, i.e. $m_{A}=0$;

  * •

$(iT_{H}^{A})_{ab}v_{b}\neq 0$, if $T^{A}$ is spontaneously broken by the
Higgs VEVs. The corresponding Nambu-Goldstone boson $\eta^{A}$ is given by

$\eta^{A}=\sum_{a}\,t_{a}^{A}\tilde{H}_{a}\,,$ (A.37)

where $t_{a}^{A}=\tilde{g}_{A}(iT_{H}^{A})_{ab}v_{b}/m_{A}$ are a (incomplete)
set of orthogonal vectors in the Higgs space, i.e.
$\sum_{a}t_{a}^{A}t_{a}^{B}=\delta_{AB}$. The Goldstone field is then eaten by
the vector boson $Z_{\mu}^{A}$ which acquires a mass $m_{A}$.

The $\tilde{H}_{a}$ contains the Goldstone modes along the $t_{a}^{A}$
directions while the remaining modes, orthogonal to the Goldstones, are all
physical, i.e.

$\tilde{H}_{a}=\sum_{\text{NG\,modes}}t_{a}^{A}\eta^{A}+\dots\,.$ (A.38)

The interaction terms between the $\psi$ fields and the Goldstone bosons are
given by

$\sum_{a}\mathcal{Y}_{ij}^{a}t_{a}^{A}=\frac{i\tilde{g}_{A}}{m_{A}}\sum_{k}\left[(T_{L}^{A})_{ik}\mathcal{M}_{kj}-\mathcal{M}_{ik}(T_{R}^{A})_{kj}\right]\,,$
(A.39)

because of the gauge invariance of the Yukawa couplings. Upon an infinitesimal
transformation of angle $\alpha_{A}$ along a broken $T^{A}$ generator, the
Goldstone field $\eta^{A}$ transform like

$\delta\eta^{A}=\alpha_{A}m_{A}+\text{linear\,terms}\,.$ (A.40)

Finally, the effective operators in Eq. (A.10) written in terms of the
$Z_{\mu}^{A}$ and $\eta^{A}$ fields are

$\sum_{A,B,C}\frac{\tilde{g}_{A}\tilde{g}_{B}\tilde{g}_{C}}{48\pi^{2}}C_{Z}^{ABC}\epsilon^{\alpha\mu\nu\beta}Z_{\alpha}^{A}Z_{\mu}^{B}\partial_{\beta}Z_{\nu}^{C}-\sum_{A,B,C}\frac{\tilde{g}_{A}\tilde{g}_{B}\tilde{g}_{C}}{48\pi^{2}}D_{\eta}^{ABC}\epsilon^{\mu\nu\alpha\beta}\frac{\eta^{A}}{m_{A}}\partial_{\alpha}Z_{\mu}^{B}\partial_{\beta}Z_{\nu}^{C}+\dots,$
(A.41)

where the dots contain the interaction terms with the Higgs physical modes.
The rotated EFT coefficients are equal to

$\begin{split}C_{Z}^{ABC}=\int_{0}^{+\infty}\\!\\!\\!\text{d}s\\!\int_{0}^{1}\\!\text{d}x\\!&\int_{0}^{1}\\!\text{d}y\\!\int_{0}^{1}\\!\text{d}z\,2\,\delta(1\\!-\\!x\\!-\\!y\\!-\\!z)\times\\\
\times\,\text{Re}\Biggl{\\{}3y\,&\text{Tr}\\!\left[e^{-sy\mathcal{M}^{{\dagger}}\mathcal{M}}T_{R}^{A}\mathcal{M}^{{\dagger}}e^{-sz\mathcal{M}\mathcal{M}^{{\dagger}}}T_{L}^{B}e^{-sx\mathcal{M}\mathcal{M}^{{\dagger}}}T_{L}^{C}\mathcal{M}\right]\\\
-3y\,&\text{Tr}\\!\left[e^{-sy\mathcal{M}^{{\dagger}}\mathcal{M}}T_{R}^{B}\mathcal{M}^{{\dagger}}e^{-sz\mathcal{M}\mathcal{M}^{{\dagger}}}T_{L}^{A}e^{-sx\mathcal{M}\mathcal{M}^{{\dagger}}}T_{L}^{C}\mathcal{M}\right]\\\
+3y\,&\text{Tr}\\!\left[e^{-sy\mathcal{M}\mathcal{M}^{{\dagger}}}T_{L}^{B}\mathcal{M}e^{-sz\mathcal{M}^{{\dagger}}\mathcal{M}}T_{R}^{A}e^{-sx\mathcal{M}^{{\dagger}}\mathcal{M}}T_{R}^{C}\mathcal{M}^{{\dagger}}\right]\\\
-3y\,&\text{Tr}\\!\left[e^{-sy\mathcal{M}\mathcal{M}^{{\dagger}}}T_{L}^{A}\mathcal{M}e^{-sz\mathcal{M}^{{\dagger}}\mathcal{M}}T_{R}^{B}e^{-sx\mathcal{M}^{{\dagger}}\mathcal{M}}T_{R}^{C}\mathcal{M}^{{\dagger}}\right]\\\
+y\,&\text{Tr}\\!\left[e^{-sy\mathcal{M}^{{\dagger}}\mathcal{M}}\mathcal{M}^{{\dagger}}\mathcal{M}T_{R}^{A}e^{-sz\mathcal{M}^{{\dagger}}\mathcal{M}}T_{R}^{B}e^{-sx\mathcal{M}^{{\dagger}}\mathcal{M}}T_{R}^{C}\right]\\\
-x\,&\text{Tr}\\!\left[e^{-sy\mathcal{M}^{{\dagger}}\mathcal{M}}T_{R}^{A}e^{-sz\mathcal{M}^{{\dagger}}\mathcal{M}}T_{R}^{B}e^{-sx\mathcal{M}^{{\dagger}}\mathcal{M}}\mathcal{M}^{{\dagger}}\mathcal{M}T_{R}^{C}\right]\\\
+x\,&\text{Tr}\\!\left[e^{-sy\mathcal{M}\mathcal{M}^{{\dagger}}}T_{L}^{A}e^{-sz\mathcal{M}\mathcal{M}^{{\dagger}}}T_{L}^{B}e^{-sx\mathcal{M}\mathcal{M}^{{\dagger}}}\mathcal{M}\mathcal{M}^{{\dagger}}T_{L}^{C}\right]\\\
-y\,&\text{Tr}\\!\left[e^{-sy\mathcal{M}\mathcal{M}^{{\dagger}}}\mathcal{M}\mathcal{M}^{{\dagger}}T_{L}^{A}e^{-sz\mathcal{M}\mathcal{M}^{{\dagger}}}T_{L}^{B}e^{-sx\mathcal{M}\mathcal{M}^{{\dagger}}}T_{L}^{C}\right]\Biggr{\\}}\,,\end{split}$
(A.42)

and

$\begin{split}D_{\eta}^{ABC}=\int_{0}^{+\infty}\\!\\!\\!\\!\\!\\!\text{d}s\\!\int_{0}^{1}\\!\text{d}x\\!\int_{0}^{1}\\!&\text{d}y\\!\int_{0}^{1}\\!\text{d}z\,6\,\delta(1\\!-\\!x\\!-\\!y\\!-\\!z)\times\\\
\times\,\text{Re}\Biggl{\\{}x\,&\text{Tr}\\!\left[e^{-sz\mathcal{M}^{{\dagger}}\mathcal{M}}T_{R}^{B}\mathcal{M}^{{\dagger}}e^{-sx\mathcal{M}\mathcal{M}^{{\dagger}}}T_{L}^{C}e^{-sy\mathcal{M}\mathcal{M}^{{\dagger}}}(T_{L}^{A}\mathcal{M}-\mathcal{M}T_{R}^{A})\right]\\\
+x\,&\text{Tr}\\!\left[e^{-sz\mathcal{M}^{{\dagger}}\mathcal{M}}T_{R}^{C}\mathcal{M}^{{\dagger}}e^{-sx\mathcal{M}\mathcal{M}^{{\dagger}}}T_{L}^{B}e^{-sy\mathcal{M}\mathcal{M}^{{\dagger}}}(T_{L}^{A}\mathcal{M}-\mathcal{M}T_{R}^{A})\right]\\\
+y\,&\text{Tr}\\!\left[e^{-sy\mathcal{M}^{{\dagger}}\mathcal{M}}\mathcal{M}^{{\dagger}}T_{L}^{C}e^{-sx\mathcal{M}\mathcal{M}^{{\dagger}}}T_{L}^{B}e^{-sz\mathcal{M}\mathcal{M}^{{\dagger}}}(T_{L}^{A}\mathcal{M}-\mathcal{M}T_{R}^{A})\right]\\\
+y\,&\text{Tr}\\!\left[e^{-sy\mathcal{M}^{{\dagger}}\mathcal{M}}\mathcal{M}^{{\dagger}}T_{L}^{B}e^{-sx\mathcal{M}\mathcal{M}^{{\dagger}}}T_{L}^{C}e^{-sz\mathcal{M}\mathcal{M}^{{\dagger}}}(T_{L}^{A}\mathcal{M}-\mathcal{M}T_{R}^{A})\right]\\\
+y\,&\text{Tr}\\!\left[e^{-sz\mathcal{M}^{{\dagger}}\mathcal{M}}T_{R}^{B}e^{-sx\mathcal{M}^{{\dagger}}\mathcal{M}}T_{R}^{C}\mathcal{M}^{{\dagger}}e^{-sy\mathcal{M}\mathcal{M}^{{\dagger}}}(T_{L}^{A}\mathcal{M}-\mathcal{M}T_{R}^{A})\right]\\\
+y\,&\text{Tr}\\!\left[e^{-sz\mathcal{M}^{{\dagger}}\mathcal{M}}T_{R}^{C}e^{-sx\mathcal{M}^{{\dagger}}\mathcal{M}}T_{R}^{B}\mathcal{M}^{{\dagger}}e^{-sy\mathcal{M}\mathcal{M}^{{\dagger}}}(T_{L}^{A}\mathcal{M}-\mathcal{M}T_{R}^{A})\right]\Biggr{\\}}\,.\end{split}$
(A.43)

###  A.6 Properties of the WZ coefficients

In general, the expressions (A.42) and (A.43) involve non-trivial integrations
which are difficult to compute. Special simplifications occur if the fermion
mass term $\overline{\psi}_{Li}\mathcal{M}_{ij}\psi_{Rj}$ is invariant under
any of the symmetry generators $T^{A}$. If so, the mass matrix $\mathcal{M}$
satisfies

$\sum_{k}\mathcal{M}_{ik}(T_{R}^{A})_{kj}-\sum_{k}(T_{L}^{A})_{ik}\mathcal{M}_{kj}=0\,.$
(A.44)

Alternatively, the invariance of the mass term reads

$\sum_{a,b}\mathcal{Y}_{ij}^{a}(T_{H}^{A})_{ab}v_{b}=0\,,$ (A.45)

which could occur if $T^{A}$ belongs to $\mathcal{G}_{0}$ or some Yukawa
coupling vanishes. Then, one finds

$C_{Z}^{ABC}=\begin{cases}\text{Tr}\,T_{R}^{A}\\{T_{R}^{B},T_{R}^{C}\\}-\text{Tr}\,T_{L}^{A}\\{T_{L}^{B},T_{L}^{C}\\}&\text{if
\quad$\sum_{a,b}\mathcal{Y}_{ij}^{a}(T_{H}^{B,C})_{ab}v_{b}=0$\, ,}\\\
\text{Tr}\,T_{L}^{A}\\{T_{L}^{B},T_{L}^{C}\\}-\text{Tr}\,T_{R}^{A}\\{T_{R}^{B},T_{R}^{C}\\}&\text{if
\quad$\sum_{a,b}\mathcal{Y}_{ij}^{a}(T_{H}^{A,C})_{ab}v_{b}=0$\, ,}\\\
0&\text{if \quad$\sum_{a,b}\mathcal{Y}_{ij}^{a}(T_{H}^{A,B})_{ab}v_{b}=0$\,
.}\end{cases}$ (A.46)

and

$D_{\eta}^{ABC}=3\left[\text{Tr}\,T_{L}^{A}\\{T_{L}^{B},T_{L}^{C}\\}-\text{Tr}\,T_{R}^{A}\\{T_{R}^{B},T_{R}^{C}\\}\right]\,\,\,\,\,\,\,\,\,\,\,\text{if
\quad$\sum_{a,b}\mathcal{Y}_{ij}^{a}(T_{H}^{B,C})_{ab}v_{b}=0$}\,.$ (A.47)

Note that $D_{\eta}^{ABC}$ vanishes if

$\sum_{a,b}\mathcal{Y}_{ij}^{a}(T_{H}^{A})_{ab}v_{b}=0\,,$ (A.48)

i.e. if the fermion mass term is invariant under symmetry generator $T^{A}$.

Consider now an unbroken generator $Q\in\mathcal{G}_{0}$, hence satisfying

$\sum_{k}\mathcal{M}_{ik}(Q_{R})_{kj}-\sum_{k}(Q_{L})_{ik}\mathcal{M}_{kj}=0\,,$
(A.49)

with the commutation rules

$[Q,T^{A}]=\sum_{B}q_{AB}T^{B}\,.$ (A.50)

Thanks to (A.49) and the cyclic property of the trace, the expression

$C_{Z}^{ABC}|_{T^{A}\rightarrow[Q,T^{A}]}\,+\,C_{Z}^{ABC}|_{T^{B}\rightarrow[Q,T^{B}]}\,+\,C_{Z}^{ABC}|_{T^{C}\rightarrow[Q,T^{C}]}\,=0$
(A.51)

is identically zero. Then, we find that the EFT coefficients satisfy

$\sum_{D}(q_{AD}C_{Z}^{DBC}+q_{BD}C_{Z}^{ADC}+q_{CD}C_{Z}^{ABD})=0\,.$ (A.52)

The same argument yields

$\sum_{D}(q_{AD}D_{\eta}^{DBC}+q_{BD}D_{\eta}^{ADC}+q_{CD}D_{\eta}^{ABD})=0\,.$
(A.53)

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