#  The $Z$ Penguin in Generic Extensions of the Standard Model

$\text{Joachim Brod}^{a}$, $\text{Martin Gorbahn}^{b}$
aDepartment of Physics, University of Cincinnati, Cincinnati, OH 45221, USA
bDepartment of Mathematical Sciences, University of Liverpool, Liverpool, L69
7ZL, UK

###### Abstract

Precision flavour observables play an important role in the interpretation of
results at the LHC in terms of models of new physics. We present the result
for the one-loop $Z$ penguin in generic extensions of the standard model which
exhibit exact perturbative unitarity. We use Slavnov-Taylor identities to
study the implications of unitarity on the renormalisation of the $Z$ penguin,
and derive a manifestly finite result that depends on a reduced set of of
physical couplings.

##  1 Introduction

It is well known that precision flavour observables put strong constraints on
models of new physics. While these models typically predict new particles
which might be found by LHC experiments or at future colliders, first hints
could show up as anomalies in precision flavour observables. Their examination
could then lead to clues towards the nature of new physics. It is therefore of
interest to calculate these observables in a generic extension of the standard
model (SM).

We consider an arbitrary number of additional heavy degrees of freedom: gauge
bosons, fermions and scalars. Perturbative unitarity imposes important
constraints on such generic extensions. The required cancellation of unbounded
high-energy growth of scattering amplitudes leads to specific relations among
the coupling constants that are common to all models. These relations allow us
to understand and perform the renormalisation of the observables in a general
way. The feasibility of this approach is expected on general grounds since the
equations implied by perturbative unitarity uniquely reflect the spontaneously
broken gauge structure [1, 2, 3] and thus may as well be derived by means of
Slavnov-Taylor identities (STI). Here we advocate the practical implementation
of those simple relations in the calculation and renormalisation of generic
loop amplitudes. This goes beyond the typical application of perturbative
unitarity in which one derives upper bounds on yet unobserved mass spectra [4,
5, 6] and combinations of masses and/or couplings [7, 8, 9, 10, 11].

As an example we study the flavour-changing transition between two SM down-
type quarks $d$ and $d^{\prime}$ of different generations, induced by a heavy
neutral gauge boson at one-loop – the so-called $Z$ penguin. An example is the
FCNC $s\to d$ transition with the emission of a virtual $Z$ boson, leading to
processes like the rare $K\to\pi\nu\bar{\nu}$ decays. Just like the SM, many
models of new physics generate this transition first at the one-loop level,
since their neutral SM-fermion currents are flavour-conserving.111Since this
property drastically improves the potential agreement of a model with
experimental flavour constraints, especially from $\Delta S=2$ observables, it
is sometimes enforced by imposing an additional $Z_{2}$ parity on the particle
content (an example is $T$ parity in little Higgs models [12]). In this
article we present the general renormalised one-loop result for this process,
assuming the absence of tree-level contributions to the $d\to d^{\prime}$
transition. It is then straightforward to obtain the $Z$ penguin in any given
model by just inserting the specific couplings into our generic result. Due to
sum rules derived from the STI, only a reduced set of couplings needs to be
specified in practice. We illustrate this procedure in detail for several
examples.

Our method has several interesting applications. It provides explicit and
manifestly finite results for a very general class of extensions of the SM.
The strategy is not restricted to flavour observables, but might also be
applicable, for instance, to collider and dark-matter phenomenology.

This paper is organised as follows. After a definition of the generic
Lagragian in Sec. 2, we present in Sec. 3 the general analytic result which in
many models is as yet unknown, and elaborate on its renormalisation. In Sec. 4
we explicitly perform the renormalization of our result for the case of
charged heavy particles. As an illustration, we (re-)derive the $Z$ penguin in
various renormalisable models, to wit, the SM, the two-Higgs-doublet model, an
extension of the SM with vector-like quarks, and the minimal supersymmetric SM
(MSSM). In the appendices we give the results for the box diagrams in our
notation and provide the definitions and explicit expressions of the requisite
loop functions. Moreover, we provide the full list of Slavnov-Taylor
identities for four-point couplings.

##  2 The generic Lagrangian

In this work we consider an extension of the SM by an arbitrary number of
heavy scalar, fermion, and vector fields (in this context, “heavy” means that
the particle masses are of the order of the electroweak scale or larger). As
our starting point we define the parts of the generic Lagrangian which are
relevant to the calculation of box and penguin diagrams. The interaction terms
involving massless SM vector fields – photons and gluons – are fixed by QED
and QCD gauge invariance. In particular, the massive-massless interaction
terms are given in terms of the covariant derivative

$(D_{\mu})_{ij}=(\partial_{\mu}-ieQ_{F}A_{\mu})\delta_{ij}-ig_{s}G_{\mu}^{a}T^{a}_{F,ij}$
(2.1)

by the usual kinetic terms of the massive fields $F$. Here $T^{a}_{F,ij}$ and
$Q_{F}$ generate the action of the respective gauge group $SU(3)_{c}$ and
$U(1)_{em}$ on the field $F$.

The three-point interactions of massive fields

$\begin{split}\mathcal{L}_{3}=&\mkern
5.0mu{\textstyle\sum\limits_{f_{1}f_{2}s_{1}\sigma}}\mkern
5.0muy_{s_{1}\bar{f}_{1}f_{2}}^{\sigma,abc}h_{s_{1}}^{a}\bar{\psi}_{f_{1}}^{b}P_{\sigma}\psi_{f_{2}}^{c}+{\textstyle\sum\limits_{f_{1}f_{2}v_{1}\sigma}}\mkern
5.0mug_{v_{1}\bar{f}_{1}f_{2}}^{\sigma,abc}V_{v_{1},\mu}^{a}\bar{\psi}_{f_{1}}^{b}\gamma^{\mu}P_{\sigma}\psi_{f_{2}}^{c}\\\\[2.0pt]
&+\tfrac{i}{6}\mkern 5.0mu{\textstyle\sum\limits_{v_{1}v_{2}v_{3}}}\mkern
5.0mug_{v_{1}v_{2}v_{3}}^{abc}\Big{(}V_{v_{1},\mu}^{a}V_{v_{2},\nu}^{b}\,\partial^{[\mu}V_{v_{3}}^{c,\nu]}+V_{v_{3},\mu}^{c}V_{v_{1},\nu}^{a}\,\partial_{\vphantom{v_{2}}}^{[\mu}V_{v_{2}}^{b,\nu]}+V_{v_{2},\mu}^{b}V_{v_{3},\nu}^{c}\,\partial_{\vphantom{v_{1}}}^{[\mu}V_{v_{1}}^{a,\nu]}\Big{)}\\\\[2.0pt]
&+\tfrac{1}{2}\mkern 5.0mu{\textstyle\sum\limits_{v_{1}v_{2}s_{1}}}\mkern
5.0mug_{v_{1}v_{2}s_{1}}^{abc}\,V_{v_{1},\mu}^{a}V_{v_{2}}^{b,\mu}h_{s_{1}}^{c}-\tfrac{i}{2}\mkern
5.0mu{\textstyle\sum\limits_{v_{1}s_{1}s_{2}}}\mkern
5.0mug_{v_{1}s_{1}s_{2}}^{abc}\,V_{v_{1}}^{a,\mu}\Big{(}h_{s_{1}}^{b}\,\partial_{\mu}h_{s_{2}}^{c}-\big{(}\partial_{\mu}h_{s_{1}}^{b}\big{)}\,h_{s_{2}}^{c}\Big{)}\,.\end{split}$
(2.2)

involve physical scalars $h_{s_{i}}$, Dirac fermions $\psi_{f_{i}}$, and
vector fields $V_{v_{i}}$, with non-zero masses $M_{s_{i}}$, $m_{f_{i}}$ and
$M_{v_{i}}$, respectively. These fields are enumerated by the corresponding
indices $s_{i}$, $f_{i}$, $v_{i}$. The index $\sigma$ denotes the two
chiralities $\sigma=L,R$, via the chiral projectors
$P_{R,L}=(1\pm\gamma_{5})/2$. Square brackets denote antisymmetrization of the
enclosed Lorentz indices (no symmetry factors are implied).

The sums in Eq. (2.2) run over all particles in a given multiplet. Consider,
for instance, the last term in Eq. (2.2): if $v_{1}$ corresponds to the SM $Z$
boson and the scalar indices to a charged scalar multiplet, the sum runs of
both positively and negatively charged scalar particles. Alternatively, one
could sum over positive particle only and omit the factor $1/2$ in front of
the sum.

We assume that all vector fields obtain their mass by the spontaneous
breakdown of a local symmetry. The Lagrangian $\mathcal{L}_{3}$ comprises only
the model-dependent couplings; all remaining “unphysical” interactions, for
instance of the would-be Goldstone bosons associated with the spontaneous
symmetry breaking, can be inferred from the requirement of perturbative
unitarity, via the STIs which we discuss below.

Due to $SU(3)\times U(1)$ gauge invariance non-vanishing couplings may only
exist for index combinations which allow the fields to combine to form an
uncharged singlet. For instance, a nonvanishing coefficient
$y_{s_{1}\bar{f}_{1}f_{2}}^{\sigma,abc}$ implies the charge relation
$Q_{s_{1}}+Q_{f_{2}}=Q_{f_{1}}$, and

$\displaystyle
y^{\sigma,dbc}_{s_{1}\bar{f}_{1}f_{2}}\,T^{e}_{s_{1},da}+y^{\sigma,abd}_{s_{1}\bar{f}_{1}f_{2}}\,T^{e}_{f_{2},dc}=T^{e}_{f_{1},bd}\,y^{\sigma,adc}_{s_{1}\bar{f}_{1}f_{2}}\,.$
(2.3)

This last property is important for the calculation of QCD corrections in the
spirit of Refs. [13, 14, 15]. In the following, we will suppress the colour
indices. They can always be thought of as being subsumed in the field indices
$v_{i}$, $s_{i}$, and $f_{i}$, if necessary.

If one of the fermions (e.g., $\psi_{f_{2}}$) is uncharged, Schur’s lemma
implies that even $T_{s_{1}}=T_{f_{1}}$. Hermiticity puts further restrictions
on the couplings. For instance, we can express the couplings of negatively
charged Higgs and gauge bosons to fermions through the couplings of the
corresponding positively charged particles. In general we have

$\begin{gathered}y_{s_{1}\bar{f}_{2}f_{1}}^{\sigma}=\big{(}y_{\bar{s}_{1}\bar{f}_{1}f_{2}}^{\bar{\sigma}}\big{)}^{*}\,,\qquad
g_{v_{1}\bar{f}_{2}f_{1}}^{\sigma}=\big{(}g_{\bar{v}_{1}\bar{f}_{1}f_{2}}^{\sigma}\big{)}^{*}\,,\qquad
g_{v_{1}v_{2}s_{1}}=\big{(}g_{\bar{v}_{1}\bar{v}_{2}\bar{s}_{1}}\big{)}^{*}\,,\\\
g_{v_{1}s_{1}s_{2}}=-\big{(}g_{\bar{v}_{1}\bar{s}_{1}\bar{s}_{2}}\big{)}^{*}\,,\quad
g_{v_{1}v_{2}v_{3}}=-\big{(}g_{\bar{v}_{1}\bar{v}_{2}\bar{v}_{3}}\big{)}^{*}\,.\end{gathered}$
(2.4)

The bars over bosonic indices denote the exchange of indices within a pair of
oppositely charged particles (as in
$g_{\overline{W^{+}}\ldots}=g_{W^{-}\ldots}$) and have no effect for neutral
particles. The bars over the $\sigma$s denote the opposite chirality.

#### Slavnov-Taylor Identities for Feynman Rules

The constraints derived from perturbative unitarity reflect a spontaneously
broken gauge symmetry. To exploit these constraints for our generic Lagrangian
we use the STIs of an arbitrary fundamental spontaneously broken gauge theory.
The massive vector fields of (2.2) are the gauge bosons of the fundamental
theory supplemented by a standard $R_{\xi}$ gauge-fixing term. This has two
consequences. First, the couplings of Goldstone bosons can be linked directly
to the couplings of the corresponding vectors in the mass-eigenstate basis.
This use of STIs is well known and summarized in the Goldstone-boson
equivalence theorem [3, 4, 16, 17]. Second, we obtain certain sum rules, i.e.
equations that impose non-trivial constraints on the couplings of physical
fields and encode the full spontaneously broken gauge structure on the level
of Feynman rules222The couplings in (2.2) are defined such that the Feynman
rules are given, after multiplication by a factor of $i$, in terms of the
usual Lorentz structures in the conventions of FeynArts [18].. We will use
these sum rules later to renormalise the $Z$ penguin.

From a technical point of view it is easiest to derive the sum rules from the
vanishing Becchi-Rouet-Stora-Tyutin (BRST) [19, 20] transformation of suitable
vertex functions. To start, we note that throughout this work the gauge
freedom of (2.2) is fixed with a standard linear $R_{\xi}$ Lagrangian [21]

$\mathcal{L}_{\text{fix}}=-\sum_{v}(2\xi_{v})^{-1}F_{\bar{v}}F_{v}\,,\qquad
F_{v}=\partial_{\mu}V_{v}^{\mu}-\sigma_{v}\xi_{v}M_{v}\phi_{v}\,,$ (2.5)

for every vector field $V_{v}$ of mass $M_{v}$ and the corresponding pseudo-
Goldstone boson field $\phi_{v}$. The coefficients $\sigma_{v}$ can have the
values $\pm i$ for complex fields and $\pm 1$ for real fields. For the SM
fields they are given by $\sigma_{W^{\pm}}=\pm i$ and $\sigma_{Z}=1$, and we
choose this convention in general for all charged and neutral vector fields.

By applying a BRST transformation $s$ to a Green’s function

$G^{\bar{u}_{v}(\ldots)_{\text{ph}}}\equiv\left\langle
T\big{\\{}\bar{u}_{v}(\ldots)_{\text{ph}}\big{\\}}\right\rangle$ (2.6)

which involves an anti-ghost field $\bar{u}_{v}$, and using the transformation
property $s\bar{u}_{v}=-F_{v}/\xi_{v}$, we obtain a linear relation between
the connected, truncated Green’s functions
$G_{c}^{\underline{W_{v}}(\ldots)_{\text{ph}}}$ and
$G_{c}^{\underline{\phi_{v}}(\ldots)_{\text{ph}}}$. Here, the dots
$(\ldots)_{\text{ph}}$ stand for any combination of physical, asymptotic on-
shell fields, whose BRST variations vanish. The underlining of a field
indicates that the corresponding external leg has been amputated. In our
convention, labels on vertex functions denote outgoing fields, whereas all
momenta are incoming333The momentum configuration of the vectors and Goldstone
bosons appearing in the gauge-fixing function is not restricted any further,
in contrast to the procedure used, for instance, in applications of the
Goldstone-boson equivalence theorem [22].. Angle brackets denote a vacuum
expectation value, and $T\\{\ldots\\}$ the time-ordered product of fields.

The STIs lead to the following relation in momentum space:

$\begin{pmatrix}k^{\mu}\\\
i\sigma_{\bar{v}}\xi_{v}M_{v}\end{pmatrix}^{T}G^{\bar{v}}_{(\mu\nu)}\begin{pmatrix}\langle\underline{V}_{v}^{\nu}(\ldots)_{\text{ph}}\rangle_{c}\\\
\langle\underline{\phi}_{v}(\ldots)_{\text{ph}}\rangle_{c}\end{pmatrix}=0\,.$
(2.7)

$G^{\bar{v}}_{(\mu\nu)}$ denotes the propagator for a vector boson or its
Goldstone boson, and is given by the inverse of the two-point vertex function
$\Gamma^{v\,(\mu\nu)}$. These functions can be decomposed as

$\Gamma^{v}_{(\mu\nu)}(k,-k)=\begin{pmatrix}{\textstyle\sum\limits_{P=T,L}}g^{P}_{\mu\nu}\Gamma^{V_{v}V_{\bar{v}}}_{P}(k^{2})&k_{\mu}\Gamma^{V_{v}\phi_{\bar{v}}}_{L}(k^{2})\\\
k_{\nu}\Gamma^{\phi^{a}V_{\bar{v}}}_{L}(k^{2})&\Gamma^{\phi_{v}\phi_{\bar{v}}}(k^{2})\end{pmatrix}\,,\quad
G^{\bar{v}}_{(\mu\lambda)}\Gamma^{v(\lambda\nu)}=i\begin{pmatrix}\delta_{\mu}^{\nu}&0\\\
0&1\end{pmatrix}\,.$ (2.8)

where $g^{T}_{\mu\nu}\equiv g_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{k^{2}}$ and
$g^{L}_{\mu\nu}\equiv g_{\mu\nu}-g^{T}_{\mu\nu}$. It follows [23] that the
STIs are given by

$\langle
T\Big{\\{}k^{\mu}\underline{V_{v}^{\mu}}-i\sigma_{\bar{v}}M_{v}A_{v}(k^{2})\underline{\phi_{v}}\Big{\\}}(\ldots)_{\text{ph}}\rangle=0\,,\quad
A_{v}=\frac{\Gamma^{V_{v}V_{\bar{v}}}_{L}+\frac{k^{2}}{\xi_{v}}}{M_{v}\big{(}M_{v}-i\sigma_{v}\Gamma^{V_{v}\phi_{\bar{v}}}_{L}\big{)}}\,.$
(2.9)

In principle, one would have to account for the mixing of different bosons;
consider, for instance, $Z-A$ mixing in the SM. However, this only affects Eq.
(2.9) at loop level, while we use the equation to evaluate Feynman rules and
tree-level sum rules, where, in fact, $A_{v}(k^{2})=1$. Evaluating this
identity at tree level shows that the three-point couplings involving
Goldstone bosons $\phi_{v}$ are related to the couplings involving the
corresponding gauge bosons $V_{v}$ as follows:

$\begin{gathered}\begin{aligned}
&g_{v_{1}\phi_{2}\phi_{3}}=\sigma_{v_{2}}\sigma_{v_{3}}\,\tfrac{M_{v_{2}}^{2}+M_{v_{3}}^{2}-M_{v_{1}}^{2}}{2\,M_{v_{2}}M_{v_{3}}}\,g_{v_{1}v_{2}v_{3}}\,,&\;&g_{\phi_{1}\phi_{2}s_{1}}=-\sigma_{v_{1}}\sigma_{v_{2}}\,\tfrac{M_{s_{1}}^{2}}{2\,M_{v_{1}}M_{v_{2}}}\,g_{v_{1}v_{2}s_{1}}\,,\\\
&g_{v_{1}v_{2}\phi_{3}}=-i\sigma_{v_{3}}\,\tfrac{M_{v_{1}}^{2}-M_{v_{2}}^{2}}{M_{v_{3}}}\,g_{v_{1}v_{2}v_{3}}\,,&\;&g_{\phi_{1}s_{1}s_{2}}=i\sigma_{v_{1}}\,\tfrac{M_{s_{1}}^{2}-M_{s_{2}}^{2}}{M_{v_{1}}}\,g_{v_{1}s_{1}s_{2}}\,,\\\
&g_{v_{1}\phi_{2}s_{1}}=-i\sigma_{v_{2}}\,\tfrac{1}{2\,M_{v_{2}}}\,g_{v_{1}v_{2}s_{1}}\,,&\;&g_{\phi_{1}\phi_{2}\phi_{3}}=0\,,\\\
\end{aligned}\\\
y_{\phi_{1}\bar{f}_{1}f_{2}}^{\sigma}=-i\sigma_{v_{1}}\,\tfrac{1}{M_{v_{1}}}\,\big{(}m_{f_{1}}g_{v_{1}\bar{f}_{1}f_{2}}^{\sigma}-g_{v_{1}\bar{f}_{1}f_{2}}^{\bar{\sigma}}m_{f_{2}}\big{)}\,.\end{gathered}$
(2.10)

Here the subscripts $\phi_{i}$ correspond to Goldstone-boson indices and are
used in distinction to the subscripts $s_{i}$ corresponding to physical
scalars (for instance, Higgs bosons).

The STIs for four-point diagrams have two consequences. If the diagram
contains a four-point coupling, the resulting relation allows to express this
coupling in terms of three-point couplings. In this way, all four-point
couplings with at least one Goldstone or vector boson can be derived; they are
summarized in Appendix C. If the diagram does not contain a four-point
coupling, the STIs yield sum rules which imply additional relations among
three-point couplings. For instance, the Lie-algebra structure of the vector
and fermion couplings is reflected by the two sum rules

$\displaystyle{\textstyle\sum\limits_{v_{5}}}\,\big{(}g_{v_{1}v_{2}v_{5}}g_{v_{3}v_{4}\bar{v}_{5}}+g_{v_{2}v_{3}v_{5}}g_{v_{1}v_{4}\bar{v}_{5}}+g_{v_{3}v_{1}v_{5}}g_{v_{2}v_{4}\bar{v}_{5}}\big{)}=0\,,$
(2.11)
$\displaystyle{\textstyle\sum\limits_{v_{3}}}\,g_{v_{3}\bar{f}_{1}f_{2}}^{\sigma}g_{v_{1}v_{2}\bar{v}_{3}}={\textstyle\sum\limits_{f_{3}}}\,\big{(}g_{v_{1}\bar{f}_{1}f_{3}}^{\sigma}g_{v_{2}\bar{f}_{3}f_{2}}^{\sigma}-g_{v_{2}\bar{f}_{1}f_{3}}^{\sigma}g_{v_{1}\bar{f}_{3}f_{2}}^{\sigma}\big{)}\,.$
(2.12)

Eq. (2.11) is simply the Jacobi identity, and Eq. (2.12) relates the structure
constants of fermion and vector representations. It is interesting to note
that Eq. (2.12) implies the unitarity of the Cabibbo-Kobayashi-Maskawa (CKM)
matrix for an universal, diagonal $Z$-boson coupling to fermions. The
remaining sum rules provide non-trivial constraints on the unitarization
properties of the given couplings. Here we show only the two sum rules needed
for the renormalisation of the $Z$ penguin:

$\displaystyle{\textstyle\sum\limits_{s_{1}}}\,g_{v_{1}v_{2}\bar{s}_{1}}y_{s_{1}\bar{f}_{1}f_{2}}^{\sigma}={\textstyle\sum\limits_{v_{3}}}\,\tfrac{M_{v_{1}}^{2}-M_{v_{2}}^{2}}{M_{v_{3}}^{2}}\,g_{v_{1}v_{2}\bar{v}_{3}}\Big{(}m_{f_{1}}g_{v_{3}\bar{f}_{1}f_{2}}^{\sigma}-g_{v_{3}\bar{f}_{1}f_{2}}^{\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern 1.0mu}m_{f_{2}}\Big{)}$
$\displaystyle\mkern
135.0mu+{\textstyle\sum\limits_{f_{3}}}\Big{(}-m_{f_{1}}\big{(}g_{v_{2}\bar{f}_{1}f_{3}}^{\sigma}g_{v_{1}\bar{f}_{3}f_{2}}^{\sigma}+g_{v_{1}\bar{f}_{1}f_{3}}^{\sigma}g_{v_{2}\bar{f}_{3}f_{2}}^{\sigma}\big{)}$
$\displaystyle\mkern 187.0mu-m_{f_{2}}\big{(}g_{v_{2}\bar{f}_{1}f_{3}}^{\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern
1.0mu}g_{v_{1}\bar{f}_{3}f_{2}}^{\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern
1.0mu}+g_{v_{1}\bar{f}_{1}f_{3}}^{\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern
1.0mu}g_{v_{2}\bar{f}_{3}f_{2}}^{\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern 1.0mu}\big{)}$
$\displaystyle\mkern
187.0mu+2\,m_{f_{3}}\big{(}g_{v_{2}\bar{f}_{1}f_{3}}^{\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern
1.0mu}g_{v_{1}\bar{f}_{3}f_{2}}^{\sigma}+g_{v_{1}\bar{f}_{1}f_{3}}^{\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern
1.0mu}g_{v_{2}\bar{f}_{3}f_{2}}^{\sigma}\big{)}\Big{)}\,,$ (2.13)
$\displaystyle{\textstyle\sum\limits_{s_{1}}}\,g_{v_{1}s_{2}\bar{s}_{1}}y_{s_{1}\bar{f}_{1}f_{2}}^{\sigma}=-{\textstyle\sum\limits_{v_{3}}}\,\tfrac{1}{2M_{v_{3}}^{2}}g_{v_{1}\bar{v}_{3}s_{2}}\Big{(}m_{f_{1}}g_{v_{3}\bar{f}_{1}f_{2}}^{\sigma}-g_{v_{3}\bar{f}_{1}f_{2}}^{\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern 1.0mu}m_{f_{2}}\Big{)}$
$\displaystyle\mkern
133.0mu+{\textstyle\sum\limits_{f_{3}}}\Big{(}g_{v_{1}\bar{f}_{1}f_{3}}^{\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern
1.0mu}y_{s_{2}\bar{f}_{3}f_{2}}^{\sigma}-y_{s_{2}\bar{f}_{1}f_{3}}^{\sigma}g_{v_{1}\bar{f}_{3}f_{2}}^{\sigma}\Big{)}\,.$
(2.14)

We refer to Appendix C for the remaining rules.

##  3 Result for the $Z$ penguin

We present our result for the renormalised $Z$ penguin in the form of an
effective vertex. The penguin function $C^{\sigma}_{d_{j}d_{i}Z}$ describing
the transition between two light SM fermions $d_{j}\to d_{i}$ is defined in
terms of the amputated vertex function as

$\Gamma_{\mu\,\sigma}^{d_{j}d_{i}Z}=\gamma_{\mu}P_{\sigma}\times\frac{G_{F}}{\sqrt{2}}\frac{e}{\pi^{2}}M_{Z}^{2}\frac{c_{w}}{s_{w}}C^{\sigma}_{d_{j}d_{i}Z}\equiv\frac{\gamma_{\mu}P_{\sigma}}{(4\pi)^{2}}\hat{C}^{\sigma}_{d_{j}d_{i}Z}\,.$
(3.1)

The function $C^{\sigma}_{d_{j}d_{i}Z}$, or $\hat{C}^{\sigma}_{d_{j}d_{i}Z}$
respectively, depends on all masses and couplings that appear; $\sigma=L,R$
stands for the chiral projection. Setting the external momenta and light
fermion masses to zero in our results, we obtain in the ’t Hooft-Feynman gauge

$\begin{split}&\hat{C}^{\sigma}_{d_{j}d_{i}Z}=\Bigg{\\{}\sum_{f_{1}v_{1}}\sum_{f_{2}\notin\,\rm
SM}\bigg{[}\kappa^{\sigma}_{f_{1}f_{2}v_{1}}B_{0}\big{(}m_{f_{1}},m_{v_{1}}\big{)}+\kappa^{\prime\sigma}_{f_{1}f_{2}v_{1}}\bigg{]}+\sum_{f_{1}s_{1}}\sum_{f_{2}\notin\,\rm
SM}\kappa^{\sigma}_{f_{1}f_{2}s_{1}}B_{0}\big{(}m_{f_{1}},m_{s_{1}}\big{)}\\\
&\quad+\sum_{f_{1}f_{2}v_{1}}\bigg{[}\tilde{k}^{\sigma}_{f_{1}f_{2}v_{1}}\Big{(}\tilde{C_{0}}\big{(}m_{f_{1}},m_{f_{2}},M_{v_{1}}\big{)}-\tfrac{1}{2}\Big{)}+k^{\sigma}_{f_{1}f_{2}v_{1}}C_{0}\big{(}m_{f_{1}},m_{f_{2}},M_{v_{1}}\big{)}+k^{\prime\sigma}_{f_{1}f_{2}v_{1}}\bigg{]}\\\
&\quad+\sum_{f_{1}v_{1}v_{2}}\bigg{[}\tilde{k}^{\sigma}_{f_{1}v_{1}v_{2}}\Big{(}\tilde{C_{0}}\big{(}m_{f_{1}},M_{v_{1}},M_{v_{2}}\big{)}+\tfrac{1}{2}\Big{)}+k^{\sigma}_{f_{1}v_{1}v_{2}}C_{0}\big{(}m_{f_{1}},M_{v_{1}},M_{v_{2}}\big{)}+k^{\prime\sigma}_{f_{1}v_{1}v_{2}}\bigg{]}\\\
&\quad+\sum_{f_{1}v_{1}s_{1}}\bigg{[}\tilde{k}^{\sigma}_{f_{1}v_{1}s_{1}}\Big{(}\tilde{C_{0}}\big{(}M_{s_{1}},m_{f_{1}},M_{v_{1}}\big{)}+\tfrac{1}{2}\Big{)}+k^{\sigma}_{f_{1}v_{1}s_{1}}C_{0}\big{(}M_{s_{1}},m_{f_{1}},M_{v_{1}}\big{)}\bigg{]}\\\
&\quad+\sum_{f_{1}f_{2}s_{1}}\bigg{[}\tilde{k}^{\sigma}_{f_{1}f_{2}s_{1}}\Big{(}\tilde{C_{0}}\big{(}M_{s_{1}},m_{f_{1}},m_{f_{2}}\big{)}-\tfrac{1}{2}\Big{)}+k^{\sigma}_{f_{1}f_{2}s_{1}}C_{0}\big{(}M_{s_{1}},m_{f_{1}},m_{f_{2}}\big{)}\bigg{]}\\\
&\quad+\sum_{f_{1}s_{1}s_{2}}\tilde{k}^{\sigma}_{f_{1}s_{1}s_{2}}\Big{(}\tilde{C_{0}}\big{(}M_{s_{1}},M_{s_{2}},m_{f_{1}}\big{)}+\tfrac{1}{2}\Big{)}\Bigg{\\}}\,.\end{split}$
(3.2)

The couplings are contained in the $\kappa$ and $k$ factors that are defined
as follows:

$\begin{split}\kappa^{\sigma}_{f_{1}f_{2}v_{1}}&=-\frac{m_{f_{1}}}{m_{f_{2}}}\bigg{(}g_{Z\bar{d}_{i}f_{2}}^{\sigma}\bigg{[}\bigg{(}4-\frac{m_{f_{1}}^{2}}{M_{v_{1}}^{2}}\bigg{)}g_{v_{1}\bar{f}_{2}f_{1}}^{\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern
1.0mu}+\frac{m_{f_{1}}m_{f_{2}}}{M_{v_{1}}^{2}}g_{v_{1}\bar{f}_{2}f_{1}}^{\sigma}\bigg{]}g_{\bar{v}_{1}\bar{f}_{1}d_{j}}^{\sigma}\\\
&\mkern
60.0mu+g_{\bar{v}_{1}\bar{d}_{i}f_{1}}^{\sigma}\bigg{[}\bigg{(}4-\frac{m_{f_{1}}^{2}}{M_{v_{1}}^{2}}\bigg{)}g_{v_{1}\bar{f}_{1}f_{2}}^{\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern
1.0mu}+\frac{m_{f_{1}}m_{f_{2}}}{M_{v_{1}}^{2}}g_{v_{1}\bar{f}_{1}f_{2}}^{\sigma}\bigg{]}g_{Z\bar{f}_{2}d_{j}}^{\sigma}\bigg{)}\,,\\\
\kappa^{\prime\sigma}_{f_{1}f_{2}v_{1}}&=\frac{2m_{f_{1}}}{m_{f_{2}}}\Big{(}g_{Z\bar{d}_{i}f_{2}}^{\sigma}g_{v_{1}\bar{f}_{2}f_{1}}^{\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern
1.0mu}g_{\bar{v}_{1}\bar{f}_{1}d_{j}}^{\sigma}+g_{v_{1}\bar{d}_{i}f_{1}}^{\sigma}g_{\bar{v}_{1}\bar{f}_{1}f_{2}}^{\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern
1.0mu}g_{Z\bar{f}_{2}d_{j}}^{\sigma}\Big{)}\,,\\\ \end{split}$ (3.3)
$\begin{split}\kappa^{\sigma}_{f_{1}f_{2}s_{1}}&=\frac{m_{f_{1}}}{m_{f_{2}}}\Big{(}g_{Z\bar{d}_{i}f_{2}}^{\sigma}y_{s_{1}\bar{f}_{2}f_{1}}^{\sigma}y_{\bar{s}_{1}\bar{f}_{1}d_{j}}^{\sigma}+y_{s_{1}\bar{d}_{i}f_{1}}^{\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern
1.0mu}y_{\bar{s}_{1}\bar{f}_{1}f_{2}}^{\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern
1.0mu}g_{Z\bar{f}_{2}d_{j}}^{\sigma}\Big{)}\,,\\\ \end{split}$ (3.4)
$\begin{split}\tilde{k}^{\sigma}_{f_{1}f_{2}v_{1}}&=\bigg{(}g_{Z\bar{f}_{2}f_{1}}^{\sigma}+\frac{m_{f_{1}}m_{f_{2}}}{2M_{v_{1}}^{2}}g_{Z\bar{f}_{2}f_{1}}^{\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern
1.0mu}\bigg{)}\,g_{\bar{v}_{1}\bar{d}_{i}f_{2}}^{\sigma}g_{v_{1}\bar{f}_{1}d_{j}}^{\sigma}\,,\\\
k^{\sigma}_{f_{1}f_{2}v_{1}}&=-\frac{m_{f_{1}}m_{f_{2}}}{M_{v_{1}}^{2}}\,\big{(}m_{f_{1}}m_{f_{2}}g_{Z\bar{f}_{2}f_{1}}^{\sigma}+2M_{v_{1}}^{2}g_{Z\bar{f}_{2}f_{1}}^{\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern
1.0mu}\big{)}\,g_{\bar{v}_{1}\bar{d}_{i}f_{2}}^{\sigma}g_{v_{1}\bar{f}_{1}d_{j}}^{\sigma}\,,\\\
k^{\prime\sigma}_{f_{1}f_{2}v_{1}}&=-g_{Z\bar{f}_{2}f_{1}}^{\sigma}g_{\bar{v}_{1}\bar{d}_{i}f_{2}}^{\sigma}g_{v_{1}\bar{f}_{1}d_{j}}^{\sigma}+\frac{1}{2}\big{(}g_{Z\bar{d}_{i}d_{i}}^{\sigma}g_{v_{1}\bar{d}_{i}f_{1}}^{\sigma}g_{\bar{v}_{1}\bar{f}_{1}d_{j}}^{\sigma}+g_{\bar{v}_{1}\bar{d}_{i}f_{1}}^{\sigma}g_{v_{1}\bar{f}_{1}d_{j}}^{\sigma}g_{Z\bar{d}_{j}d_{j}}^{\sigma}\big{)}\delta_{f_{1}f_{2}}\,,\\\
\end{split}$ (3.5)
$\begin{split}\tilde{k}^{\sigma}_{f_{1}v_{1}v_{2}}&=-\bigg{(}3+\frac{m_{f_{1}}^{2}(M_{v_{1}}^{2}+M_{v_{2}}^{2}-M_{Z}^{2})}{4M_{v_{1}}^{2}M_{v_{2}}^{2}}\bigg{)}\,g_{Zv_{1}\bar{v}_{2}}g_{\bar{v}_{1}\bar{d}_{i}f_{1}}^{\sigma}g_{v_{2}\bar{f}_{1}d_{j}}^{\sigma}\\\
&\quad-\frac{1}{2}\bigg{(}1+\frac{m_{f_{1}}^{2}}{2M_{v_{1}}^{2}}\bigg{)}\big{(}g_{Z\bar{d}_{i}d_{i}}^{\sigma}g_{v_{1}\bar{d}_{i}f_{1}}^{\sigma}g_{\bar{v}_{1}\bar{f}_{1}d_{j}}^{\sigma}+g_{v_{1}\bar{d}_{i}f_{1}}^{\sigma}g_{\bar{v}_{1}\bar{f}_{1}d_{j}}^{\sigma}g_{Z\bar{d}_{j}d_{j}}^{\sigma}\big{)}\delta_{v_{1}v_{2}}\,,\\\
k^{\sigma}_{f_{1}v_{1}v_{2}}&=\frac{m_{f_{1}}^{2}\left(M_{v_{1}}^{4}+M_{v_{2}}^{4}-M_{Z}^{2}(M_{v_{1}}^{2}+M_{v_{2}}^{2})\right)}{M_{v_{1}}^{2}M_{v_{2}}^{2}}\,g_{Zv_{1}\bar{v}_{2}}g_{\bar{v}_{1}\bar{d}_{i}f_{1}}^{\sigma}g_{v_{2}\bar{f}_{1}d_{j}}^{\sigma}\,,\\\
k^{\prime\sigma}_{f_{1}v_{1}v_{2}}&=2\,g_{Zv_{1}\bar{v}_{2}}g_{\bar{v}_{1}\bar{d}_{i}f_{1}}^{\sigma}g_{v_{2}\bar{f}_{1}d_{j}}^{\sigma}\,,\\\
\end{split}$ (3.6)
$\begin{split}\tilde{k}^{\sigma}_{f_{1}v_{1}s_{1}}&=-\frac{k^{\sigma}_{f_{1}v_{1}s_{1}}}{4M_{v_{1}}^{2}}\,,\qquad
k^{\sigma}_{f_{1}v_{1}s_{1}}=m_{f_{1}}\left(g_{Z\bar{v}_{1}s_{1}}y_{\bar{s}_{1}\bar{d}_{i}f_{1}}^{\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern
1.0mu}g_{v_{1}\bar{f}_{1}d_{j}}^{\sigma}+g_{Zv_{1}\bar{s}_{1}}y_{s_{1}\bar{f}_{1}d_{j}}^{\sigma}g_{\bar{v}_{1}\bar{d}_{i}f_{1}}^{\sigma}\right)\,,\\\
\end{split}$ (3.7)
$\begin{split}\tilde{k}^{\sigma}_{f_{1}f_{2}s_{1}}&=\frac{1}{2}\,g_{Z\bar{f}_{2}f_{1}}^{\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern
1.0mu}y_{\bar{s}_{1}\bar{d}_{i}f_{2}}^{\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern
1.0mu}y_{s_{1}\bar{f}_{1}d_{j}}^{\sigma}\,,\qquad
k^{\sigma}_{f_{1}f_{2}s_{1}}=-m_{f_{1}}m_{f_{2}}g_{Z\bar{f}_{2}f_{1}}^{\sigma}y_{\bar{s}_{1}\bar{d}_{i}f_{2}}^{\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern
1.0mu}y_{s_{1}\bar{f}_{1}d_{j}}^{\sigma}\,,\\\ \end{split}$ (3.8)
$\begin{split}\tilde{k}^{\sigma}_{f_{1}s_{1}s_{2}}&=\frac{1}{2}\,g_{Z\bar{s}_{1}s_{2}}y_{\bar{s}_{2}\bar{d}_{i}f_{1}}^{\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern
1.0mu}y_{s_{1}\bar{f}_{1}d_{j}}^{\sigma}-\frac{1}{4}\Big{(}g_{Z\bar{d}_{i}d_{i}}^{\sigma}y_{\bar{s}_{1}\bar{d}_{i}f_{1}}^{\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern
1.0mu}y_{s_{1}\bar{f}_{1}d_{j}}^{\sigma}+y_{\bar{s}_{1}\bar{d}_{i}f_{1}}^{\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern
1.0mu}y_{s_{1}\bar{f}_{1}d_{j}}^{\sigma}g_{Z\bar{d}_{j}d_{j}}^{\sigma}\Big{)}\delta_{s_{1}s_{2}}\,.\end{split}$
(3.9)

The sums in (3.2) run over all components of the fields (e.g., explicitly over
$W^{+}$ and $W^{-}$ in the SM). Note that the scalar-index sums in (3.2) run
only over physical fields; the contributions of would-be Goldstone bosons have
been accounted for via the replacement of the Goldstone couplings in terms of
couplings to physical particles, using Eq. (2.10). The contribution of the $Z$
penguin to processes like rare $B$ and $K$ decays is not separately gauge-
invariant without including the box contributions. We present the explicit
generic results for completeness in Appendix A.

Our result (3.1) consists of several contributions. In general, we can write
($i\neq j$)

$\begin{split}\Gamma_{\mu\,\sigma}^{d_{j}d_{i}Z}=\Gamma_{\mu\,\sigma}^{d_{j}d_{i}Z,(1)}+\Big{[}\delta
Z_{Z\bar{d}_{i}d_{j}}^{g,\sigma}+\tfrac{1}{2}{\textstyle\sum\limits_{f_{1}}}\Big{(}g_{Z\bar{d}_{i}f_{1}}^{\sigma}\delta
Z_{\bar{f}_{1}d_{j}}^{\psi,\sigma}+\delta
Z_{\bar{d}_{i}f_{1}}^{\psi,\sigma\,\dagger}g_{Z\bar{f}_{1}d_{j}}^{\sigma}\Big{)}+\tfrac{1}{2}{\textstyle\sum\limits_{v_{1}}}\delta
Z_{Zv_{1}}^{V}g_{\bar{v}_{1}\bar{d}_{i}d_{j}}^{\sigma}\Big{]}P_{\sigma}\,.\end{split}$
(3.10)

Here $\Gamma_{\mu\,\sigma}^{Zd_{j}d_{i},(1)}$ denotes the sum of all
contributing one-loop diagrams, and the rest are the vertex, fermion field,
and gauge-boson field renormalisation constants, respectively. Note that in
this work we assume the absence of tree-level FCNC transitions between light
quarks, i.e. $g_{\bar{v}_{1}\bar{d}_{i}d_{j}}^{\sigma}=0$. In this case, we
can use the STIs (2.12), (2.13) and (2.14) to show that the fermion field
renormalisation constants are sufficient to cancel all divergences in (3.10)
(in other words, $\delta Z_{Z\bar{d}_{i}d_{j}}^{g,\sigma}$ can be chosen to
vanish). This is done explicitly in Sec. 4.

Figure 1: One-heavy-particle reducible contribution to the $Z$ penguin.

The two-point functions in the full theory lead to mixing among the fermions
beyond tree level. We perform an off-diagonal field renormalisation for all
fermions fields, with finite terms chosen to restore a diagonal and
canonically normalised kinetic and mass term. As one consequence, diagrams
with a reducible heavy-fermion line (of the form shown in Figure 1) are
exactly cancelled by corresponding counterterm diagrams.

The one-loop corrections to the renormalised fermion two-point function can be
written as

$\delta\Gamma_{\sigma}^{f_{i}f_{j}}=\Big{[}\big{(}\Sigma_{V,\bar{f}_{j}f_{i}}^{\psi,\sigma}+\delta
Z_{H,\bar{f}_{j}f_{i}}^{\psi,\sigma}\big{)}\not{p}+\Sigma_{S,\bar{f}_{j}f_{i}}^{\psi,\sigma}-\tfrac{1}{2}(m_{f_{j}}\delta
Z_{\bar{f}_{j}f_{i}}^{\psi,\sigma}+\delta Z_{\bar{f}_{j}f_{i}}^{\psi,\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern
1.0mu\,\dagger}m_{f_{i}})-\delta_{ji}\delta m_{f_{i}}\Big{]}P_{\sigma}\,.$
(3.11)

The finite parts of the field renormalisation constants for $j\neq i$ are
obtained from the requirement that the off-diagonal parts of the self energy
(3.11) vanish. They are given by

$\delta Z_{\bar{f}_{j}f_{i}}^{\psi,\sigma}\big{|}_{\rm
fin}=\frac{2}{m_{f_{j}}^{2}-m_{f_{i}}^{2}}\Big{(}m_{f_{i}}^{2}\Sigma_{V,\bar{f}_{j}f_{i}}^{\psi,\sigma}+m_{f_{j}}m_{f_{i}}\Sigma_{V,\bar{f}_{j}f_{i}}^{\psi,\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern
1.0mu}+m_{f_{j}}\Sigma_{S,\bar{f}_{j}f_{i}}^{\psi,\sigma}+m_{f_{i}}\Sigma_{S,\bar{f}_{j}f_{i}}^{\psi,\mkern
1.0mu\overline{\mkern-1.0mu\sigma\mkern-1.0mu}\mkern 1.0mu}\Big{)}\big{|}_{\rm
fin}\,.$ (3.12)

They enter our generic result after having been expanded in small mass ratios.
The diagonal field renormalization constants are not necessary in our case. If
we had allowed for the presence of tree-level $d_{j}\to d_{i}$ transitions,
one would have to fix these constants by renormalization conditions. Note
that, in this case, the one-loop result would correspond to a tiny correction
of an already suppressed tree-level amplitude. These consideration can be
important, however, in an analogous treatment of charged current couplings
[24].

We will show in the next section that all divergences in our generic result
completely cancel against the divergent terms in the field rotation, without
introducing additional counterterms. In order to make this cancellation
manifest, it is necessary to use the consequences of tree-level perturbative
unitarity derived in Sec. 2 in form of the STIs. This step is independent of
the specification of a model and can be applied without detailed knowledge
about the sector responsible for the spontaneous symmetry breaking.

##  4 Renormalized Results

The result (3.2) is valid for any number of heavy new scalars, vectors, and
Dirac fermions. It is, however, not very suitable for numerical evaluation:
the functions $\tilde{C_{0}}$ in Eq. (4.4) contain divergent terms (cf.
Appendix B). While the STI (via relations between the couplings appearing in
the coefficients (3.3) - (3.9)) ensure that the result is finite, the
cancellation of the divergences is not manifest. In this section we will
derive manifestly finite versions of our result that, in addition, depend on a
minimal number of physical couplings.

In order to make contact to phenomenological applications, we remind the
reader that the loop functions appearing in rare $K$ and $B$ meson decays are
given by the following combinations [25]

$\begin{split}X^{\sigma}(x)=C_{sdZ}^{\sigma}(x)-4B_{sd\nu\nu}^{\sigma
L}(x)\,,\quad
Y^{\sigma\sigma^{\prime}}(x)=C_{sdZ}^{\sigma}(x)-B(x)_{sd\ell\ell}^{\sigma\sigma^{\prime}}\,,\end{split}$
(4.1)

with $C_{sdZ}^{\sigma}$ and $B_{sd\ell_{k}\ell_{k}}^{\sigma\sigma^{\prime}}$
given in Eq. (3.1) and Eq. (A.2), respectively.

###  4.1 SM Fermions and charged scalars and vectors

We first consider the simplified case of theories with heavy scalars and
vectors – in other words, we drop all coefficients in Eq. (3.2) that involve
couplings to heavy new fermions. We then simplify the remaining expression by
repeatedly eliminating physical coupling constants via application of the STI.
The result obtained in this way is not only manifestly finite, but also
depends on fewer physical coupling constants than the original expression.
This procedure is not unique. While the STI ensure that one will finally
arrive at a finite result, there is some freedom of which couplings to
eliminate and which ones to retain.

For instance, continuing with our simplified case, we can first eliminate the
right-handed $Z$-fermion coupling by solving the “Yukawa sum rule” Eq. (2.14)
for $g_{Z\bar{q}q}^{R}$, and solving the “gauge boson mass sum rule” Eq.
(2.13) for $g_{Z\bar{t}t}^{R}$. We then apply the “unitarity sum rule”, Eq.
(2.12), together with the universality of the $Z$ coupling to fermions, to
eliminate all dependence on couplings involving light quarks (this step is a
generalization of the GIM mechanism). We can then solve Eq. (2.12) for
$g_{Z\bar{t}t}^{L}$. Applying the resulting relations to Eqs. (3.3)-(3.9)
eliminates all divergences, and we obtain the manifestly finite expression

$\begin{split}\hat{C}^{L}_{d_{j}d_{i}Z}=&\sum_{s_{1}s_{2}}f_{S}(m_{t},M_{s_{1}},M_{s_{2}})y_{s_{2}^{+}\bar{t}d_{j}}^{L}\left(\delta_{s_{1}s_{2}}y_{s_{2}^{-}\bar{d_{i}}t}^{R}\left(g_{Z\bar{d_{j}}d_{j}}^{L}-g_{Z\bar{t}t}^{L}\right)+g_{Zs_{1}^{+}s_{2}^{-}}y_{s_{1}^{-}\bar{d_{i}}t}^{R}\right)\\\
&+\sum_{v_{1}v_{2}}f_{V}(m_{t},M_{v_{1}},M_{v_{2}})g_{Zv_{2}^{+}v_{1}^{-}}g_{v_{1}^{+}\bar{t}d_{j}}^{L}g_{v_{2}^{-}\bar{d_{i}}t}^{L}\\\
&+\sum_{s_{1}v_{1}}f_{VS}(m_{t},M_{s_{1}},M_{v_{1}})y_{s_{1}^{+}\bar{t}d_{j}}^{L}g_{v_{1}^{-}\bar{d_{i}}t}^{L}g_{Zv_{1}^{+}s_{1}^{-}}\\\
&+\sum_{s_{1}v_{1}}f_{VS^{\prime}}(m_{t},M_{s_{1}},M_{v_{1}})y_{s_{1}^{-}\bar{d_{i}}t}^{R}g_{v_{1}^{+}\bar{t}d_{j}}^{L}g_{Zv_{1}^{-}s_{1}^{+}}\,.\end{split}$
(4.2)

The loop functions are given by

$\begin{split}f_{V}(m_{i},m_{j},m_{k})&=m_{i}^{2}C_{0}\left(m_{i},m_{k},m_{k}\right)-\frac{m_{i}^{2}\left(m_{j}^{2}+m_{k}^{2}-M_{Z}^{2}\right)}{4m_{j}^{2}m_{k}^{2}}\\\
&+\frac{m_{i}^{2}\left(-3m_{j}^{2}+m_{k}^{2}-M_{Z}^{2}\right)+4m_{k}^{2}\left(m_{j}^{2}-m_{k}^{2}+M_{Z}^{2}\right)}{4m_{j}^{2}m_{k}^{2}}m_{i}^{2}C_{0}\left(m_{i},m_{i},m_{k}\right)\\\
&+\frac{-M_{Z}^{2}\left(3m_{j}^{2}+4m_{k}^{2}\right)-13m_{j}^{2}m_{k}^{2}+3m_{j}^{4}+4m_{k}^{4}}{4m_{j}^{2}m_{k}^{2}}m_{i}^{2}C_{0}\left(m_{i},m_{j},m_{k}\right)\,,\\\
f_{S}(m_{i},m_{j},m_{k})&=\frac{1}{2}m_{i}^{2}C_{0}\left(m_{i},m_{i},m_{k}\right)-\frac{1}{2}m_{j}^{2}C_{0}\left(m_{i},m_{j},m_{k}\right)-\frac{1}{2}\,,\\\
f_{VS}(m_{i},m_{j},m_{k})&=m_{i}\left(1-\frac{m_{j}^{2}}{4m_{k}^{2}}\right)C_{0}\left(m_{i},m_{j},m_{k}\right)\\\
&+\left(\frac{m_{i}^{3}}{4m_{k}^{2}}-m_{i}\right)C_{0}\left(m_{i},m_{i},m_{k}\right)-\frac{m_{i}}{4m_{k}^{2}}\,,\\\
f_{VS^{\prime}}(m_{i},m_{j},m_{k})&=\frac{m_{i}^{3}C_{0}\left(m_{i},m_{i},m_{j}\right)}{4m_{k}^{2}}+\frac{3}{4}m_{i}C_{0}\left(m_{i},m_{j},m_{k}\right)-\frac{m_{i}}{4m_{k}^{2}}\,.\end{split}$
(4.3)

The function $C_{0}$ can be found in Eq. (B.2).

A comment is in order: The sums over charged vectors $v_{i}$ and charged
scalars $s_{i}$ in Eq. (4.2) run over the particle types, but not over the
different charges: for SM fermion content the charge of the internal vectors
and scalars is fixed by the external fermions (e.g., for a strange to down
transition, there are only up-type quarks in the loop, so the internal vectors
and scalars must have negative charge $-e$).

####  4.1.1 Standard Model

We illustrate the general procedure described above by considering the $s\to
d$ transition in the SM. Here, the only relevant heavy degrees of freedom are
the top quark, and the gauge bosons $Z$ and $W^{\pm}$. Starting with the
general result (3.2), we need to keep only the terms

$\begin{split}\hat{C}^{L,\text{SM}}_{sdZ}=&\sum_{f_{1}f_{2}v_{1}}\bigg{[}\tilde{k}^{\sigma}_{f_{1}f_{2}v_{1}}\Big{(}\tilde{C_{0}}\big{(}m_{f_{1}},m_{f_{2}},M_{v_{1}}\big{)}-\tfrac{1}{2}\Big{)}+k^{\sigma}_{f_{1}f_{2}v_{1}}C_{0}\big{(}m_{f_{1}},m_{f_{2}},M_{v_{1}}\big{)}+k^{\prime\sigma}_{f_{1}f_{2}v_{1}}\bigg{]}\\\
+&\sum_{f_{1}v_{1}v_{2}}\bigg{[}\tilde{k}^{\sigma}_{f_{1}v_{1}v_{2}}\Big{(}\tilde{C_{0}}\big{(}m_{f_{1}},M_{v_{1}},M_{v_{2}}\big{)}+\tfrac{1}{2}\Big{)}+k^{\sigma}_{f_{1}v_{1}v_{2}}C_{0}\big{(}m_{f_{1}},M_{v_{1}},M_{v_{2}}\big{)}+k^{\prime\sigma}_{f_{1}v_{1}v_{2}}\bigg{]}\,\end{split}$
(4.4)

Next, we insert the SM couplings of $W$ bosons

$g^{L}_{W^{+}\bar{u}_{j}d_{k}}=\frac{e}{s_{w}\sqrt{2}}V_{jk}\,,\qquad
g^{L}_{W^{+}\bar{\nu}_{j}\ell_{k}}=\frac{e}{s_{w}\sqrt{2}}\delta_{jk}\,,\qquad
g_{ZW^{+}W^{-}}=\frac{e}{t_{w}}$ (4.5)

and $Z$ bosons

$g^{L}_{Z\bar{f}_{j}f_{k}}=\frac{2e}{s_{2w}}\left(T_{3}^{f}-Q_{f}s_{w}^{2}\right)\delta_{jk}\,,\qquad
g^{R}_{Z\bar{f}_{j}f_{k}}=-\frac{2e}{s_{2w}}Q_{f}s_{w}^{2}\delta_{jk}\,.$
(4.6)

Here $T_{3}^{f}=\pm 1/2$ is the third component of the weak isospin of the
fermion $f$ and $Q_{f}$ is its electric charge in units of the positron charge
$e$. The sine of the weak mixing angle is denoted by $s_{w}=\sin(\theta_{w})$.
In addition we defined $s_{2w}=\sin(2\theta_{w})$ and
$t_{w}=\tan(\theta_{w})$. The CKM matrix elements are denoted by $V_{jk}$.

The $Z$-boson coupling to fermions in the SM is universal and diagonal; hence,
the sum rule (2.12) leads to the unitarity of the CKM matrix. Defining
$\lambda_{q}^{(d^{\prime}d)}=V_{qd^{\prime}}V_{qd}^{*}$, this allows us to
eliminate $\lambda_{u}^{sd}=-\lambda_{c}^{sd}-\lambda_{t}^{sd}$, and thus the
coefficient $\lambda_{t}^{sd}$ multiplies the difference of the diagrams with
massive top and massless up quark. This leads to a partial cancellation of UV
divergences – the well-known GIM (Glashow-Iliopoulos-Maiani) mechanism.

After this manipulation, the result still has a left-over divergence,
proportional to

$\frac{1}{M_{W}^{2}}-\frac{1}{c_{w}^{2}M_{Z}^{2}}\,.$ (4.7)

Evaluating the sum rule (2.13) for the choice $\sigma=L$, $v_{1}=Z$,
$v_{2}=W^{+}$, $f_{1}=t$, $f_{2}=b$, and inserting the SM couplings, leads
immediately to the relation

$M_{W}^{2}=c_{w}^{2}M_{Z}^{2}\,.$ (4.8)

Thus, the divergent term vanishes. The finite term exactly reproduces the
result of Inami and Lim [26]

$C^{L\,(\rm SM)}_{d^{\prime}dZ}=\lambda_{t}^{(d^{\prime}d)}\,C(x_{t})\,,\quad
C(x)=\frac{x}{8}\bigg{[}\frac{x-6}{x-1}+\frac{3x+2}{(x-1)^{2}}\log(x)\bigg{]}\,,$
(4.9)

for the top-quark contribution to the $d^{\prime}\to dZ$ vertex. The ratio of
the quark and $W$-boson masses squared is denoted by
$x_{q}=m_{q}^{2}/M_{W}^{2}$.

Of course, the same result is obtained in a much simpler way by directly using
the result (4.2). Both the GIM mechanism and “gauge boson mass relation” (4.8)
are already built in. Moreover, it suffices to specify the reduced set of SM
couplings (4.5) – the $Z$-fermion couplings (4.6) are then fixed via the STI.

####  4.1.2 2HDM

As an example with additional charged scalars we consider the contribution of
charged Higgs bosons in a two Higgs doublet model (2HDM) [27]. Using that the
couplings involving both charged gauge and Higgs bosons vanish,
$g_{ZW_{j}^{+}h_{i}^{-}}=g_{ZW_{j}^{-}h_{i}^{+}}=0$, we see that only the
first line in Eq. (4.2) contributes, with a prefactor given by

$y_{h_{j}^{+}\bar{t}d}^{L}\left(\delta_{ij}y_{h_{j}^{-}\bar{s}t}^{R}\left(g_{Z\bar{d}d}^{L}-g_{Z\bar{t}t}^{L}\right)+g_{Zh_{i}^{+}h_{j}^{-}}y_{h_{i}^{-}\bar{s}t}^{R}\right)\,.$
(4.10)

Specializing to the case of one charged Higgs, we need the following
additional Feynman rules [27]

$g_{Zh^{-}h^{+}}=-e\frac{c_{2w}}{2s_{w}c_{w}}\,,\quad
y_{h^{+}\bar{t}d_{i}}^{L}=\big{(}y_{h^{-}\bar{d_{i}}t}^{R}\big{)}^{*}=\frac{m_{t}}{t_{\beta}}\frac{V_{td}e}{\sqrt{2}s_{w}M_{W}}\,,\quad
y_{h^{+}\bar{t}d_{i}}^{R}=\big{(}y_{h^{-}\bar{d_{i}}t}^{L}\big{)}^{*}={\mathcal{O}}(m_{d_{i}})\,,$
(4.11)

with $t_{\beta}=\tan\beta$. We then find

$C(x_{t},y_{t})=-\frac{V_{td}V_{ts}^{*}}{8t_{\beta}^{2}}\,x_{t}\,f_{S}(m_{t},M_{h},M_{h})$
(4.12)

where

$f_{S}(m_{t},M_{h},M_{h})=\frac{y_{t}}{1-y_{t}}+\frac{y_{t}\log
y_{t}}{(y_{t}-1)^{2}}$ (4.13)

and we defined $x=m_{t}^{2}/M_{W}^{2}$, $y=m_{t}^{2}/M_{h}^{2}$. This
reproduces the function

$C_{H}(x_{t},y_{t})=-\frac{1}{8}|Y|^{2}x_{t}y_{t}\bigg{[}\frac{1}{1-y_{t}}+\frac{\log
y_{t}}{(y_{t}-1)^{2}}\bigg{]}$ (4.14)

from [28], where $Y=v_{1}/v_{2}=\cot\beta$.

###  4.2 Arbitrary charged Fermions, Scalars, and Vectors

We now consider the general case of theories with arbitrary numbers of heavy
charged fermions, scalars and vectors, and simplify the general result Eq.
(3.2) by repeatedly eliminating physical coupling constants via application of
the STI.

In analogy to our procedure described in the previous section, we can first
eliminate the diagonal right-handed $Z$-fermion coupling by solving the
“Yukawa sum rule” Eq. (2.14) for $g_{Z\bar{f}f}^{R}$, thus eliminating this
coupling in conjunction with couplings to heavy scalars. Next, we solve the
“gauge boson mass sum rule” Eq. (2.13) for $g_{Z\bar{f}f}^{R}$ and use the
resulting relation to eliminate all diagonal right-handed $Z$ couplings to
fermions in conjunction with vector couplings. We then repeatedly apply the
“unitarity sum rule”, Eq. (2.12), to eliminate all couplings of fermions to
the charged vectors and all couplings of the $Z$ boson to left-handed
fermions, looping over all heavy fermions. Together with our assumption of
universality and diagonality of the $Z$-boson couplings to down-type SM
quarks, this eliminates all divergences. Note that the “unitarity sum rule”,
Eq. (2.12), leads to a “generalized GIM mechanism”, effectively eliminating
some of the couplings of one (arbitrarily chosen) heavy fermions in the loop
(denoted below by the subscript $f^{\prime}$). The resulting, manifestly
finite expression for the $Z$ penguin for an arbitrary number of charged
fermions, scalars, and vectors is then

$\begin{split}\hat{C}_{d_{j}d_{i}Z}^{L}=&\quad\sum_{f_{1}f_{2}v_{1}}g_{Z\bar{f}_{2}f_{1}}^{L}g_{v_{1}\bar{f}_{1}d_{j}}^{L}g_{\bar{v}_{1}\bar{d_{i}}f_{2}}^{L}F_{V}\left(m_{f_{1}},m_{f_{2}},M_{v_{1}}\right)\\\
&+\sum_{f_{1}f_{2}v_{1}}g_{Z\bar{f}_{2}f_{1}}^{R}g_{v_{1}\bar{f}_{1}d_{j}}^{L}g_{\bar{v}_{1}\bar{d_{i}}f_{2}}^{L}F_{V^{\prime}}\left(m_{f_{1}},m_{f_{2}},M_{v_{1}}\right)\\\
&+\sum_{f_{1}v_{1}v_{2}}g_{Zv_{2}\bar{v}_{1}}g_{v_{1}\bar{f}_{1}d_{j}}^{L}g_{\bar{v}_{2}\bar{d_{i}}f_{1}}^{L}F_{V^{\prime\prime}}\left(m_{f^{\prime}},m_{f_{1}},M_{v_{1}},M_{v_{2}}\right)\\\
&+\sum_{f_{1}f_{2}s_{1}}g_{Z\bar{f}_{2}f_{1}}^{L}y_{s_{1}\bar{f}_{1}d_{j}}^{L}y_{\bar{s}_{1}\bar{d_{i}}f_{2}}^{R}F_{S}\left(m_{f_{1}},m_{f_{2}},M_{s_{1}}\right)\\\
&+\sum_{f_{1}s_{1}s_{2}}\left(g_{Zs_{2}\bar{s}_{1}}+\delta_{s_{1}s_{2}}g_{Z\bar{d}_{j}d_{j}}^{L}\right)y_{s_{1}\bar{f}_{1}d_{j}}^{L}y_{\bar{s}_{2}\bar{d_{i}}f_{1}}^{R}F_{S^{\prime}}\left(m_{f_{1}},M_{s_{1}},M_{s_{2}}\right)\\\
&+\sum_{f_{1}f_{2}s_{1}}g_{Z\bar{f}_{2}f_{1}}^{R}y_{s_{1}\bar{f}_{1}d_{j}}^{L}y_{\bar{s}_{1}\bar{d_{i}}f_{2}}^{R}F_{S^{\prime\prime}}\left(m_{f_{1}},m_{f_{2}},M_{s_{1}}\right)\\\
&+\sum_{f_{1}s_{1}v_{1}}g_{Zv_{1}\bar{s}_{1}}y_{s_{1}\bar{f}_{1}d_{j}}^{L}g_{\bar{v}_{1}\bar{d_{i}}f_{1}}^{L}F_{\text{SV}}\left(m_{f_{1}},M_{s_{1}},M_{v_{1}}\right)\\\
&+\sum_{f_{1}s_{1}v_{1}}g_{Z\bar{v}_{1}s_{1}}g_{v_{1}\bar{f}_{1}d_{j}}^{L}y_{\bar{s}_{1}\bar{d_{i}}f_{1}}^{R}F_{\text{SV}^{\prime}}\left(m_{f_{1}},M_{s_{1}},M_{v_{1}}\right)\,,\end{split}$
(4.15)

The respective loop functions can be written

$\begin{split}&F_{V}\left(m_{i},m_{j},m_{k}\right)=\tilde{C}_{\delta}\left(m_{i},m_{j},m_{k}\right)+\frac{m_{j}^{2}}{m_{k}^{2}}\left(m_{j}^{2}C_{0}\left(m_{j},m_{j},m_{k}\right)-m_{i}^{2}C_{0}\left(m_{i},m_{j},m_{k}\right)\right)\,,\\\
&F_{V^{\prime}}\left(m_{i},m_{j},m_{k}\right)=\frac{m_{i}m_{j}}{2m_{k}^{2}}\left(\tilde{C}_{\delta}\left(m_{i},m_{j},m_{k}\right)-4m_{k}^{2}C_{\delta}\left(m_{i},m_{j},m_{k}\right)\right)\,,\\\
&F_{V^{\prime\prime}}(m_{i},m_{j},m_{k},m_{l})=\tilde{C}_{\Delta}(m_{i},m_{j},m_{k},m_{l})+f_{V^{\prime\prime}}(m_{i},m_{k},m_{l})-f_{V^{\prime\prime}}(m_{j},m_{k},m_{l})\,,\\\
&F_{S}\left(m_{i},m_{j},m_{k}\right)=-m_{i}m_{j}C_{0}\left(m_{i},m_{j},m_{k}\right)\,,\\\
&F_{S^{\prime}}\left(m_{i},m_{j},m_{k}\right)=-\frac{1}{2}\left(\tilde{C}_{\delta}\left(m_{j},m_{i},m_{k}\right)+1\right)\,,\\\
&F_{S^{\prime\prime}}\left(m_{i},m_{j},m_{k}\right)=\frac{1}{2}\tilde{C}_{\delta}\left(m_{i},m_{j},m_{k}\right)\,,\\\
&F_{\text{SV}}\left(m_{i},m_{j},m_{k}\right)=\frac{m_{i}}{4m_{k}^{2}}\left(4m_{k}^{2}C_{\delta}\left(m_{j},m_{i},m_{k}\right)-\tilde{C}_{\delta}\left(m_{j},m_{i},m_{k}\right)-1\right)\,,\\\
&F_{\text{SV}^{\prime}}\left(m_{i},m_{j},m_{k}\right)=\frac{m_{i}}{4m_{k}^{2}}\left(4m_{k}^{2}C_{0}\left(m_{i},m_{j},m_{k}\right)-\tilde{C}_{\delta}\left(m_{k},m_{i},m_{j}\right)-1\right)\,,\\\
\end{split}$ (4.16)

where

$\begin{split}\frac{1}{m_{i}^{2}}f_{V^{\prime\prime}}(m_{i},m_{j},m_{k})&=\frac{m_{j}^{2}+m_{k}^{2}-M_{Z}^{2}}{4m_{j}^{2}m_{k}^{2}}\big{(}1+\tilde{C}_{\delta}(m_{j},m_{i},m_{k})\big{)}\\\
&+\left(\frac{m_{i}^{2}}{m_{k}^{2}}+\frac{m_{k}^{2}-M_{Z}^{2}}{m_{j}^{2}}+1\right)C_{0}(m_{i},m_{i},m_{k})\\\
&-\frac{m_{j}^{4}+m_{k}^{4}-M_{Z}^{2}\left(m_{j}^{2}+m_{k}^{2}\right)}{m_{j}^{2}m_{k}^{2}}C_{0}(m_{i},m_{j},m_{k})\end{split}$
(4.17)

and

$\begin{split}\tilde{C}_{\Delta}\left(m_{i},m_{j},m_{k},m_{l}\right)&=3\left(\tilde{C}_{0}\left(m_{i},m_{k},m_{l}\right)-\tilde{C}_{0}\left(m_{j},m_{k},m_{l}\right)\right)\\\
&\quad-\tilde{C}_{0}\left(m_{i},m_{i},m_{l}\right)+\tilde{C}_{0}\left(m_{j},m_{j},m_{l}\right)\\\
\tilde{C}_{\delta}(m_{i},m_{j},m_{k})&=\tilde{C}_{0}\left(m_{i},m_{j},m_{k}\right)-\tilde{C}_{0}\left(m_{j},m_{j},m_{k}\right)\\\
C_{\delta}(m_{i},m_{j},m_{k})&=C_{0}\left(m_{i},m_{j},m_{k}\right)-C_{0}\left(m_{j},m_{j},m_{k}\right)\,.\end{split}$
(4.18)

These loop functions are manifestly finite; it is interesting to note that one
could perform the whole calculation without regulator in four space-time
dimensions if one applies the sum rules first. Note also that several of the
loop functions vanish if the fermion masses are equal; namely, we have

$F_{V}\left(m,m,M\right)=F_{V^{\prime}}\left(m,m,M\right)=F_{V^{\prime\prime}}\left(m,m,M\right)=F_{S^{\prime\prime}}\left(m,m,M\right)=0\,.$
(4.19)

As a consequence, the result depends only the off-diagonal right-handed
couplings of the $Z$ boson to fermions. Furthermore, we have
$F_{S}\left(m,m,M\right)=F_{S^{\prime}}\left(m,M,M\right)$, and the only
contributions in the fourth line in Eq. (4.15) proportional to diagonal $Z$
couplings arise from the finite parts of the field renormalisation constants.

As before, given the charges of the internal fermions, the charges of the
internal scalars and vectors are determined by the charges of the external
particles, via $Q_{s,v}=1/3+Q_{f}$; so the sums in Eq. (4.15) effectively run
only over the particles with positive charge. Finally, we remind the reader
that we suppressed colour indices throughout; coloured particles can easily be
taken into account with the understanding that the sums also run over all
components of fields in multiplets of $SU(3)$.

####  4.2.1 Vector-like quarks

As an example, we consider vector-like quarks in a representation that does
not generate tree-level FCNC transitions in the down-quark sector [29]. For
simplicity, we treat the case of an up-type singlet vector-like quark, $U$,
with charge $2/3$, isospin zero, and mass $m_{U}$.

The additional quark $U$ mixes with the SM fermions via Yukawa term in the
Lagrangian, given by

$\mathcal{L}_{Y}=-\sum_{k}\overline{Q}_{L}^{k}Y_{k}^{U}U_{R}\tilde{H}+\text{h.c.}\,$
(4.20)

(note that hard mixing terms
$\mathcal{L}_{\text{mix}}=-\sum_{k=1}^{3}m_{uu}^{k}\,\overline{u}_{R}^{k}U_{L}+\text{h.c.}$
can be eliminated via a field rotation). Here $k$ is a SM generation index,
and the charge conjugate of the Higgs field $H$ is given by
$\tilde{H}=i\sigma^{2}H^{*}=i\sigma^{2}(H^{\dagger})^{T}$.

We obtain the contributions to the $Z$ penguin by inserting the appropriate
couplings into the general result (4.15). First, note that all RH $Z$
couplings as well as the LH tree-level $Z$ couplings to down-type quarks are
not changed from their SM values. Therefore, we need only the following
additional Feynman rules:

$\begin{split}g_{Z\bar{u}_{i}u_{j}}^{L}=\frac{e}{s_{w}c_{w}}\bigg{[}\frac{1}{2}(VV^{\dagger})_{ij}-\frac{2}{3}s_{w}^{2}\delta_{ij}\bigg{]}\,,\quad
g_{W^{+}\bar{u}_{i}d_{j}}^{L}=\frac{e}{\sqrt{2}s_{w}}V_{ij}\,,\quad
g_{ZW^{+}W^{-}}=\frac{ec_{w}}{s_{w}}\end{split}$ (4.21)

where $V$ is the generalized, non-unitary CKM matrix (see, e.g., [30, 31]).
(Note that the Higgs couplings contribute only indirectly to the $Z$ penguin,
via the mixing angles comprised by the matrix $V$.) The resulting expression
for the $Z$ penguin is

$\begin{split}C&=V_{td}V_{ts}^{*}f_{1}(x_{t})+V_{Ud}V_{Us}^{*}f_{1}(y_{U})+V_{Ud}V_{ts}^{*}(VV^{\dagger})_{tU}f_{4}(x_{t},y_{U})+V_{td}V_{Us}^{*}(VV^{\dagger})_{Ut}f_{4}(y_{U},x_{t})\\\
&+V_{td}\big{[}V_{cs}^{*}(VV^{\dagger})_{ct}+V_{us}^{*}(VV^{\dagger})_{ut}\big{]}f_{2}(x_{t})+V_{Ud}\big{[}V_{cs}^{*}(VV^{\dagger})_{cU}+V_{us}^{*}(VV^{\dagger})_{uU}\big{]}f_{2}(y_{U})\\\
&+V_{ts}^{*}\big{[}V_{cd}(VV^{\dagger})_{tc}+V_{ud}(VV^{\dagger})_{tu}\big{]}f_{3}(x_{t})+V_{Us}^{*}\big{[}V_{cd}(VV^{\dagger})_{Uc}+V_{ud}(VV^{\dagger})_{Uu}\big{]}f_{3}(y_{U})\end{split}$
(4.22)

where $x_{t}=m_{t}^{2}/M_{W}^{2}$, $y_{U}=m_{U}^{2}/M_{W}^{2}$, and the loop
functions are given by

$\begin{split}f_{1}(x)&=\frac{x(x-6)}{8(x-1)}+\frac{x(2+3x)\log
x}{8(x-1)^{2}}\,,\quad f_{2}(x)=\frac{x\log x}{8(1-x)}\,,\quad
f_{3}(x)=-\frac{x}{8}-f_{2}(x)\,,\\\\[5.69054pt]
f_{4}(x,y)&=\frac{x(x-2y+xy)\log x}{8(x-1)(x-y)}-\frac{(x-1)y^{2}\log
y}{8(y-1)(x-y)}-\frac{x}{8}\,.\end{split}$ (4.23)

The box contribution for external neutrinos (needed for the rare
$K\to\pi\nu\bar{\nu}$ decays) is

$B=V_{td}V_{ts}^{*}\,g(x_{t})+V_{Ud}V_{Us}^{*}\,g(y_{U})$ (4.24)

where

$g(x)=\frac{x}{x-1}-\frac{x\log x}{(x-1)^{2}}\,.$ (4.25)

#### The large-mass limit

As a simple application we study the limit of a vector-like quark with a mass
much larger than the electroweak symmetry-breaking scale. In general, we
expect the effects of the heavy quark to decouple – all contributions to
physical observables should be suppressed by a power of $m_{U}$. This is not
immediately obvious from the loop function; to see this, it is necessary to
expand also the mixing matrix $V$ in inverse powers of $m_{U}$. In a basis
where the SM up-quark Yukawas are diagonal, the leading contributions can be
written as (see also the discussion in Ref. [30])

$V=\begin{pmatrix}V_{\text{CKM}}\\\\[5.69054pt]
\tfrac{v}{\sqrt{2}m_{U}}Y^{U\dagger}V_{\text{CKM}}\end{pmatrix}\,,$ (4.26)

where $V_{\text{CKM}}$ is the SM CKM matrix. Using this explicit form, it can
easily be shown that the non-decoupling terms in (4.22) cancel.

The leading terms in the large-mass expansion are thus of order $1/m_{U}^{2}$,
and they can be captured by an effective theory description (cf. Ref. [32]).
We will treat here the special case $Y_{1}^{U}=Y_{2}^{U}=0$, so that only the
flavor-diagonal top-quark couplings will be modified, but no FCNC transitions
in the up-quark sector are generated. Working again in a basis where the up-
quark SM Yukawas are diagonal, only the following operators are generated at
order $1/m_{U}^{2}$ [29]:

$\begin{split}Q_{Hq,33}^{(3)}&\equiv(H^{\dagger}i\\!\stackrel{{\scriptstyle\leftrightarrow}}{{D_{\mu}^{a}}}H)(\bar{Q}_{L}^{3}\gamma^{\mu}\sigma^{a}Q_{L}^{3})\,,\\\
Q_{Hq,33}^{(1)}&\equiv(H^{\dagger}i\\!\stackrel{{\scriptstyle\leftrightarrow}}{{D}}_{\mu}H)(\bar{Q}_{L}^{3}\gamma^{\mu}Q_{L}^{3})\,,\\\
Q_{uH,33}&\equiv(H^{\dagger}H)(\bar{Q}_{L}^{3}\tilde{H}t_{R})\,.\end{split}$
(4.27)

These operators contain the Higgs doublet $H$ and its charge conjugate
$\tilde{H}$, the left-handed third-generation quark doublet $Q_{L}^{3}$, and
the right-handed top quark $t_{R}$. Moreover, $\sigma^{a}$ are the Pauli
matrices and $D_{\mu}$ is the SM gauge-covariant derivative and we defined

$\begin{split}(H^{\dagger}i\\!\stackrel{{\scriptstyle\leftrightarrow}}{{D_{\mu}}}H)&=iH^{\dagger}\big{(}D_{\mu}H\big{)}-i\big{(}D_{\mu}H\big{)}^{\dagger}H\,,\\\
(H^{\dagger}i\\!\stackrel{{\scriptstyle\leftrightarrow}}{{D_{\mu}^{a}}}H)&=iH^{\dagger}\sigma^{a}\big{(}D_{\mu}H\big{)}-i\big{(}D_{\mu}H\big{)}^{\dagger}\sigma^{a}H\,,\end{split}$
(4.28)

so that the operators $Q_{Hq,33}^{(3)}$ and $Q_{Hq,33}^{(1)}$ are manifestly
Hermitian.

Here, we are not interested in Higgs physics observables, so we will
concentrate on the $Z$-penguin operators $Q_{Hq,33}^{(3)}$ and
$Q_{Hq,33}^{(1)}$. The implications of constraints from rare meson decays for
anomalous $t\bar{t}Z$ couplings in the limit of heavy quark masses have been
treated in Ref. [33] in an effective theory framework. We want to compare
these results to those obtained in our concrete model.

The effective theory approach allows to calculate the leading-logarithmic
contribution to the rare meson decays via operator mixing (see Ref. [33] for
details). From Ref. [29] we can read off the Wilson coefficients

$C_{Hq,33}^{(3)}=-C_{Hq,33}^{(1)}=\tfrac{1}{4}V_{td}V_{ts}^{*}\big{|}Y_{3}^{U}\big{|}^{2}\,,$
(4.29)

while the results in Ref. [33] provide the logarithmic contribution to the
$X$and $Y$ functions

$\begin{split}\delta Y^{\rm NP}=\delta X^{\rm NP}=-\frac{3+2x_{t}}{2}C_{\phi
q,33}^{(1)}\frac{v^{2}}{\Lambda^{2}}\log\frac{\mu_{W}}{\Lambda}=-\frac{3+2x_{t}}{8}V_{td}V_{ts}^{*}\big{|}Y_{3}^{U}\big{|}^{2}\frac{v^{2}}{m_{U}^{2}}\log\frac{\mu_{W}}{m_{U}}\,,\end{split}$
(4.30)

where $x_{t}=m_{t}^{2}/M_{W}^{2}$ and we identified $\Lambda=m_{U}$. Inserting
the expansion (4.26) into the explicit result, we obtain the logarithmic term

$\begin{split}X\big{|}_{\log}=-\frac{3+2x_{t}}{8}V_{td}V_{ts}^{*}\big{|}Y_{3}^{U}\big{|}^{2}\frac{v^{2}}{m_{U}^{2}}\log\frac{M_{W}}{m_{U}}\end{split}$
(4.31)

which reproduces the EFT result.

In fact, the EFT approach at one-loop allows only to calculate the leading-
logarithmic term in the $1/m_{U}^{2}$ expansion. Having the full theory at
hand, we can now check whether this is a reasonable appoximation. Choosing, as
an example, a mass $m_{U}=1\,$TeV, and adding the box contribution to obtain a
gauge-invariant result, we find that the logarithmic term dominates the
remaining terms of order $1/m_{U}^{2}$ by a factor of seven for the rare
$K\to\pi\nu\bar{\nu}$ decays, and by a factor of four for
$B_{s}\to\mu^{+}\mu^{-}$. We see that the EFT result gives, in this instance,
a good estimate of the leading contributions.

Another interesting question is the contribution of dimension-eight operators
(corresponding to terms suppressed by $1/m_{U}^{4}$). We find that the
dimension-six terms dominate over the dimension-eight contributions for masses
$m_{U}\gtrsim 150\,$GeV – in particular, for all masses where an expansion in
$v/m_{U}$ is justified.

####  4.2.2 Charginos in the MSSM

As a further check of our formalism, we compare the result for additional
scalars and fermions with the chargino contributions to the $Z$ penguin in the
MSSM. The particles in the loop are the two charginos,
$(\tilde{\chi}_{1},\tilde{\chi}_{2})$, and the six up-type squarks,
$(\tilde{U}_{j})$, where we follow the notation of Ref. [34]. Using the charge
conjugated charginos $(\tilde{\chi}_{1}^{c},\tilde{\chi}_{2}^{c})$ inside the
loop, the relevant coupling constants read

$\begin{split}g_{ZU_{i}^{-}U_{j}^{+}}&=-\frac{e}{s_{2W}}\left(\sum_{I=1}^{3}Z_{U}^{Ii*}Z_{U}^{Ij}-\frac{4}{3}s_{W}^{2}\delta_{ij}\right)\,,\\\
g^{L}_{Z\overline{\chi^{c}_{i}}\chi^{c}_{j}}&=-\frac{e}{s_{2w}}\left(Z_{-}^{1i}Z_{-}^{1j*}+(c_{W}^{2}-s_{W}^{2})\delta_{ij}\right)\\\
g^{R}_{Z\overline{\chi^{c}_{i}}\chi^{c}_{j}}&=-\frac{e}{s_{2w}}\left(Z_{+}^{1i}Z_{+}^{1j*}+(c_{W}^{2}-s_{W}^{2})\delta_{ij}\right)\\\
y_{U^{-}_{i}\overline{\chi^{c}_{j}}d_{k}}^{L}&=\sum_{I=1}^{3}V_{Ik}\big{(}-\frac{e}{s_{W}}Z_{U}^{Ii*}Z_{+}^{1j}+Y_{u}^{I}Z_{U}^{(I+3)i*}Z_{+}^{2j}\big{)}\,,\\\
y_{U^{-}_{i}\overline{\chi^{c}_{j}}d_{k}}^{R}&=-\sum_{I=1}^{3}V_{Ik}\big{(}Y_{d}^{k}Z_{U}^{Ii*}Z_{-}^{2j}\big{)}\,,\\\
\end{split}$ (4.32)

Using these coupling constants and the standard model couplings of the Z-Boson
to the down quarks we reproduce the results of the chargino contributions
presented in Ref [35].

##  5 Conclusion

In this work we presented a manifestly finite result for the three-point
function involving two light SM fermions and the $Z$ boson (the “$Z$
penguin”), in generic extensions of the SM that satisfy the condition of
perturbative unitarity. The $Z$ penguin is the main ingredient for the
prediction of decay rates for rare meson decays.

We allow for an arbitrary number of heavy new scalar, fermionic, and vector
particles. The vector particles are assumed to obtain their mass via the
spontaneous breaking of a gauge symmetry. The specific form of the result is
independent of the symmetry-breaking mechanism.

Our results are not only finite, but depend also on a reduced set of physical
couplings (reflecting the structure of the underlying symmetry group). This
elimination of redundant couplings is important in particular when one
performs a fit to flavor or collider data. In this paper, we have focused on
the contributions of charged heavy particles. The study of neutral particles
is relegated to future work. Furthermore, we plan to apply our methods also in
the context of dark matter and collider physics to study simplified models
with a consistent UV behaviour.

## Acknowledgements

We thank Sandro Casagrande for collaboration at an earlier stage of this work,
and Andrzej Buras for comments on the manuscript. This work was performed in
part at the Aspen Center for Physics, which is supported by National Science
Foundation grant PHY-1066293. MG is supported in part by the UK STFC under
Consolidated Grant ST/L000431/1 and also acknowledges support from COST Action
CA16201 PARTICLEFACE.

##  Appendix A Result for the box diagrams

The box-diagram contribution to the effective Hamiltonian for $d_{j}\to
d_{i}\ell_{k}\bar{\ell}_{l}$ transitions reads

$\begin{split}\mathcal{H}_{\rm eff}^{\Delta F=1\,\rm
Box}=\frac{4G_{F}}{\sqrt{2}}\frac{\alpha}{2\pi
s_{w}^{2}}\bigg{\\{}&\sum_{\sigma_{1}\sigma_{2}}\Big{(}B^{S\sigma_{1}\sigma_{2}}_{d_{j}d_{i}\ell_{l}\ell_{k}}(\bar{d}_{i}P_{\sigma_{1}}d_{j})(\bar{\ell}_{k}P_{\sigma_{2}}\ell_{l})\\\
&\qquad+B^{V\sigma_{1}\sigma_{2}}_{d_{j}d_{i}\ell_{l}\ell_{k}}(\bar{d}_{i}\gamma^{\mu}P_{\sigma_{1}}d_{j})(\bar{\ell}_{k}\gamma_{\mu}P_{\sigma_{2}}\ell_{l})\Big{)}\\\
&+\sum_{\sigma}B^{T\sigma}_{d_{j}d_{i}\ell_{l}\ell_{k}}(\bar{d}_{i}\sigma^{\mu\nu}P_{\sigma}d_{j})(\bar{\ell}_{k}\sigma_{\mu\nu}P_{\sigma}\ell_{l})\bigg{\\}}\,,\end{split}$
(A.1)

with $\sigma_{\mu\nu}=i[\gamma^{\mu},\gamma^{\nu}]/2$. The use of chiral Fierz
identities (see e.g. [36]) shows that the tensor structures with mixed
chirality
$(\bar{d}_{i}\sigma^{\mu\nu}P_{\sigma}d_{j})(\bar{\ell}_{k}\sigma_{\mu\nu}P_{\bar{\sigma}}\,\ell_{l})$
vanish identically. Hence, they are omitted from (A.1). We decompose the box
functions $B^{M\sigma_{1}\sigma_{2}}_{d_{j}d_{i}\ell_{l}\ell_{k}}$ as

$\begin{split}B^{M\sigma_{1}\sigma_{2}}_{d_{j}d_{i}\ell_{l}\ell_{k}}=\frac{\sqrt{2}}{G_{F}}\frac{s_{w}^{2}}{\alpha}\frac{1}{32\pi}\times\bigg{\\{}&{\textstyle\sum\limits_{f_{1}f_{2}v_{1}v_{2}}}\Big{(}\tilde{c}_{f_{1}f_{2}v_{1}v_{2}}^{M\sigma_{1}\sigma_{2}}\,\tilde{D}_{0}\big{(}m_{f_{1}},m_{f_{2}},M_{v_{1}},M_{v_{2}}\big{)}\\\
&\qquad\qquad+c_{f_{1}f_{2}v_{1}v_{2}}^{M\sigma_{1}\sigma_{2}}D_{0}\big{(}m_{f_{1}},m_{f_{2}},M_{v_{1}},M_{v_{2}}\big{)}\Big{)}\\\
&\quad+{\textstyle\sum\limits_{f_{1}f_{2}v_{1}s_{1}}}\Big{(}\tilde{c}_{f_{1}f_{2}v_{1}s_{1}}^{M\sigma_{1}\sigma_{2}}\tilde{D}_{0}\big{(}M_{s_{1}},m_{f_{1}},m_{f_{2}},M_{v_{1}}\big{)}\\\
&\qquad\qquad+c_{f_{1}f_{2}v_{1}s_{1}}^{M\sigma_{1}\sigma_{2}}D_{0}\big{(}M_{s_{1}},m_{f_{1}},m_{f_{2}},M_{v_{1}}\big{)}\Big{)}\\\
&\quad+{\textstyle\sum\limits_{f_{1}f_{2}s_{1}s_{2}}}\Big{(}\tilde{c}_{f_{1}f_{2}s_{1}s_{2}}^{M\sigma_{1}\sigma_{2}}\tilde{D}_{0}\big{(}M_{s_{1}},M_{s_{2}},m_{f_{1}},m_{f_{2}}\big{)}\\\
&\qquad\qquad+c_{f_{1}f_{2}s_{1}s_{2}}^{M\sigma_{1}\sigma_{2}}D_{0}\big{(}M_{s_{1}},M_{s_{2}},m_{f_{1}},m_{f_{2}}\big{)}\Big{)}\bigg{\\}}\,,\end{split}$
(A.2)

where the superscript takes the values $M=S,V,T$. The coefficients for the
vector projection are given by

$\displaystyle c_{f_{1}f_{2}v_{1}v_{2}}^{V\sigma_{1}\sigma_{2}}$
$\displaystyle=m_{f_{1}}^{2}m_{f_{2}}^{2}\bigg{(}\frac{1}{M_{v_{1}}^{2}}+\frac{1}{M_{v_{2}}^{2}}\bigg{)}\,g_{\bar{v}_{2}\bar{d}_{i}f_{1}}^{\sigma_{1}}g_{v_{1}\bar{f}_{1}d_{j}}^{\sigma_{1}}\bigg{(}g_{\bar{v}_{1}\bar{\ell}_{k}f_{2}}^{\sigma_{2}}g_{v_{2}\bar{f}_{2}\ell_{l}}^{\sigma_{2}}-\text{flip}^{V}_{f_{2}}\bigg{)}\,,$
$\displaystyle\tilde{c}_{f_{1}f_{2}v_{1}v_{2}}^{V\sigma_{1}\sigma_{2}}$
$\displaystyle=-\frac{1}{4(M_{v_{1}}^{2}+M_{v_{2}}^{2})}c_{f_{1}f_{2}v_{1}v_{2}}^{V\sigma_{1}\sigma_{2}}$
(A.3)
$\displaystyle\quad+\bigg{(}g_{\bar{v}_{1}\bar{\ell}_{k}f_{2}}^{\sigma_{2}}g_{v_{2}\bar{f}_{2}\ell_{l}}^{\sigma_{2}}-4\,\text{flip}^{V}_{f_{2}}\bigg{)}\times\begin{cases}-g_{\bar{v}_{2}\bar{d}_{i}f_{1}}^{\sigma_{1}}g_{v_{1}\bar{f}_{1}d_{j}}^{\sigma_{1}},&\mkern-10.0mu\sigma_{1}=\sigma_{2}\\\
g_{\bar{v}_{1}\bar{d}_{i}f_{1}}^{\sigma_{1}}g_{v_{2}\bar{f}_{1}d_{j}}^{\sigma_{1}},&\mkern-10.0mu\sigma_{1}\neq\sigma_{2}\end{cases}\,,$
$\displaystyle c_{f_{1}f_{2}v_{1}s_{1}}^{V\sigma_{1}\sigma_{2}}$
$\displaystyle=m_{f_{1}}m_{f_{2}}\Big{(}g_{\bar{v}_{1}\bar{d}_{i}f_{1}}^{\sigma_{1}}y_{s_{1}\bar{f}_{1}d_{j}}^{\sigma_{1}}+\text{flip}^{V}_{f_{1}}\Big{)}\Big{(}y_{\bar{s}_{1}\bar{\ell}_{k}f_{2}}^{\bar{\sigma}_{2}}g_{v_{1}\bar{f}_{2}\ell_{l}}^{\sigma_{2}}+\text{flip}^{V}_{f_{2}}\Big{)}\,,$
(A.4) $\displaystyle\tilde{c}_{f_{1}f_{2}v_{1}s_{1}}^{V\sigma_{1}\sigma_{2}}$
$\displaystyle=-\frac{1}{4M_{v_{1}}^{2}}c_{f_{1}f_{2}v_{1}s_{1}}^{V\sigma_{1}\sigma_{2}}\,,\qquad
c_{f_{1}f_{2}s_{1}s_{2}}^{V\sigma_{1}\sigma_{2}}=0\,,$
$\displaystyle\tilde{c}_{f_{1}f_{2}s_{1}s_{2}}^{V\sigma_{1}\sigma_{2}}$
$\displaystyle=-\frac{1}{4}\,y_{\bar{s}_{2}\bar{d}_{i}f_{1}}^{\bar{\sigma}_{1}}y_{s_{1}\bar{f}_{1}d_{j}}^{\sigma_{1}}\bigg{(}y_{\bar{s}_{1}\bar{\ell}_{k}f_{2}}^{\bar{\sigma}_{2}}y_{s_{2}\bar{f}_{2}\ell_{l}}^{\sigma_{2}}-\text{flip}^{V}_{f_{2}}\bigg{)}\,.$

Here, $\text{flip}^{V}_{f_{1}}$ represents contributions for which we
interchange the coupling constants via
$g^{\sigma}_{\ldots\,\bar{d}_{i}f_{1}}\leftrightarrow
g^{\sigma}_{\ldots\,\bar{f}_{1}d_{j}}$ and
$y^{\sigma}_{\ldots\,\bar{d}_{i}f_{1}}\leftrightarrow
y^{\bar{\sigma}}_{\ldots\,\bar{f}_{1}d_{j}}$, and $\text{flip}^{V}_{f_{2}}$
acts analogously on $\ell_{l,k}$ and $f_{2}$. For the scalar projections we
obtain the coefficients

$\displaystyle\tilde{c}_{f_{1}f_{2}v_{1}v_{2}}^{S\sigma_{1}\sigma_{2}}$
$\displaystyle=m_{f_{1}}m_{f_{2}}\bigg{(}\frac{1}{M_{v_{1}}^{2}}+\frac{1}{M_{v_{2}}^{2}}\bigg{)}\,g_{\bar{v}_{2}\bar{d}_{i}f_{1}}^{\bar{\sigma}_{1}}g_{v_{1}\bar{f}_{1}d_{j}}^{\sigma_{1}}\Big{\\{}g_{\bar{v}_{1}\bar{\ell}_{k}f_{2}}^{\bar{\sigma}_{2}}g_{v_{2}\bar{f}_{2}\ell_{l}}^{\sigma_{2}}+\text{flip}^{S}_{f_{2}}\Big{\\}}\,,$
$\displaystyle c_{f_{1}f_{2}v_{1}v_{2}}^{S\sigma_{1}\sigma_{2}}$
$\displaystyle=-\frac{4M_{v_{1}}^{2}M_{v_{2}}^{2}+m_{f_{1}}^{2}m_{f_{2}}^{2}}{M_{v_{1}}^{2}+M_{v_{2}}^{2}}\tilde{c}_{f_{1}f_{2}v_{1}v_{2}}^{S\sigma_{1}\sigma_{2}}\,,$
$\displaystyle\tilde{c}_{f_{1}f_{2}v_{1}s_{1}}^{S\sigma_{1}\sigma_{2}}$
$\displaystyle=\Big{(}y_{\bar{s}_{1}\bar{d}_{i}f_{1}}^{\sigma_{1}}g_{v_{1}\bar{f}_{1}d_{j}}^{\sigma_{1}}-\text{flip}^{S}_{f_{1}}\Big{)}\Big{(}g_{\bar{v}_{1}\bar{\ell}_{k}^{\prime}f_{2}}^{\bar{\sigma}_{2}}y_{s_{1}\bar{f}_{2}l}^{\sigma_{2}}-\text{flip}^{S}_{f_{2}}\Big{)}\,,$
(A.5) $\displaystyle c_{f_{1}f_{2}v_{1}s_{1}}^{S\sigma_{1}\sigma_{2}}$
$\displaystyle=-\frac{m_{f_{1}}^{2}m_{f_{2}}^{2}}{M_{v_{1}}^{2}}\tilde{c}_{f_{1}f_{2}v_{1}s_{1}}^{S\sigma_{1}\sigma_{2}}\,,\qquad\tilde{c}_{f_{1}f_{2}s_{1}s_{2}}^{S\sigma_{1}\sigma_{2}}=0\,,$
$\displaystyle c_{f_{1}f_{2}s_{1}s_{2}}^{S\sigma_{1}\sigma_{2}}$
$\displaystyle=-m_{f_{1}}m_{f_{2}}\,y_{\bar{s}_{2}\bar{d}_{i}f_{1}}^{\sigma_{1}}y_{s_{1}\bar{f}_{1}d_{j}}^{\sigma_{1}}\Big{(}y_{\bar{s}_{1}\bar{\ell}_{k}f_{2}}^{\sigma_{2}}y_{s_{2}\bar{f}_{2}\ell_{l}}^{\sigma_{2}}+\text{flip}^{S}_{f_{2}}\Big{)}\,,$

and finally the tensor coefficients read

$\displaystyle c_{f_{1}f_{2}v_{1}v_{2}}^{T\sigma}$
$\displaystyle=-m_{f_{1}}m_{f_{2}}g_{\bar{v}_{2}\bar{d}_{i}f_{1}}^{\bar{\sigma}}g_{v_{1}\bar{f}_{1}d_{j}}^{\sigma}\Big{(}g_{\bar{v}_{1}\bar{\ell}_{k}f_{2}}^{\bar{\sigma}}g_{v_{2}\bar{f}_{2}\ell_{l}}^{\sigma}-\text{flip}^{S}_{f_{2}}\Big{)}\,,$
$\displaystyle\tilde{c}_{f_{1}f_{2}v_{1}v_{2}}^{T\sigma}$
$\displaystyle=-\frac{1}{4}\bigg{(}\frac{1}{M_{v_{1}}^{2}}+\frac{1}{M_{v_{2}}^{2}}\bigg{)}c_{f_{1}f_{2}v_{1}v_{2}}^{T\sigma}\,,$
(A.6) $\displaystyle\tilde{c}_{f_{1}f_{2}v_{1}s_{1}}^{T\sigma}$
$\displaystyle=\frac{1}{4}\Big{(}y_{\bar{s}_{1}\bar{d}_{i}f_{1}}^{\sigma}g_{v_{1}\bar{f}_{1}d_{j}}^{\sigma}+\text{flip}^{S}_{f_{1}}\Big{)}\Big{(}g_{\bar{v}_{1}\bar{\ell}_{k}f_{2}}^{\bar{\sigma}}y_{s_{1}\bar{f}_{2}\ell_{l}}^{\sigma}+\text{flip}^{S}_{f_{2}}\Big{)}\,,$
$\displaystyle c_{f_{1}f_{2}v_{1}s_{1}}^{T\sigma}$
$\displaystyle=c_{f_{1}f_{2}s_{1}s_{2}}^{T\sigma}=\tilde{c}_{f_{1}f_{2}s_{1}s_{2}}^{T\sigma}=0\,.$

Similar to above $\text{flip}^{S}_{f_{1}}$ stands for additional
contributions, here with the coupling constants
$y^{\sigma}_{\;\cdot\;\bar{d}_{i}f_{1}}\leftrightarrow
y^{\sigma}_{\;\cdot\;\bar{f}_{1}d_{j}}$ and
$g^{\sigma}_{\;\cdot\;\bar{d}_{i}f_{1}}\leftrightarrow
g^{\bar{\sigma}}_{\;\cdot\;\bar{f}_{1}d_{j}}$ interchanged, and
$\text{flip}^{S}_{f_{2}}$ acts analogously on $\ell_{l,k}$ and $f_{2}$.

It is possible to extract the generic effective Hamiltonian for the $|\Delta
F|=2$ boxes from the above results, if one keeps in mind that the external
particle – antiparticle pairs allow for additional Wick contractions of the
transition amplitude and that the conventional normalization of the effective
Hamiltonian is different. We use Fierz identities to express the result in
terms of the operator basis of [37], where

$\displaystyle\mathcal{H}_{\textrm{eff}}^{\Delta
F=2\;\textrm{Box}}=\frac{G_{F}^{2}}{4\pi^{2}}M_{W}^{2}\bigg{(}\sum_{\sigma}S^{V\sigma\sigma}_{1,d_{j}d_{i}}\,Q_{1,d_{i}d_{j}}^{V\sigma\sigma}+\sum_{n=1,2}\Big{(}S^{LR}_{n,d_{j}d_{i}}\,Q_{n,d_{i}d_{j}}^{LR}+\sum_{\sigma}S^{S\sigma\sigma}_{n,d_{j}d_{i}}\,Q_{n,d_{i}d_{j}}^{S\sigma\sigma}\Big{)}\bigg{)}\,,$
$\displaystyle\mkern 80.0mu\begin{aligned}
Q_{1}^{V\sigma\sigma}&=(\bar{d}_{j}\gamma_{\mu}P_{\sigma}d_{i})(\bar{d}_{j}\gamma^{\mu}P_{\sigma}d_{i})\,,\\\
Q_{1}^{LR}&=(\bar{d}_{j}\gamma_{\mu}P_{L}d_{i})(\bar{d}_{j}\gamma^{\mu}P_{R}d_{i})\,,\;&Q_{2}^{LR}&=(\bar{d}_{j}P_{L}d_{i})(\bar{d}_{j}P_{R}d_{i})\,,\\\
Q_{1}^{S\sigma\sigma}&=(\bar{d}_{j}P_{\sigma}d_{i})(\bar{d}_{j}P_{\sigma}d_{i})\,,\;&Q_{2}^{S\sigma\sigma}&=(\bar{d}_{j}^{\alpha}P_{\sigma}d_{i}^{\beta})(\bar{d}_{j}^{\beta}P_{\sigma}d_{i}^{\alpha})\,.\end{aligned}$
(A.7)

A summation over colour indices $\alpha,\beta$ is understood. In order to
obtain the coefficients $S$ from (A.4) – (A.6), the following prescription
holds:

$\displaystyle S^{V\sigma\sigma}_{1,d_{j}d_{i}}$
$\displaystyle=4\,B^{V\sigma\sigma}_{d_{j}d_{i}\ell_{l}\ell_{k}}|_{\ell_{k,l}\,\to\,d_{i,j}}\,,$
(A.8) $\displaystyle S^{LR}_{1,d_{j}d_{i}}$
$\displaystyle=8\,B^{VLR}_{d_{j}d_{i}\ell_{l}\ell_{k}}|_{\ell_{k,l}\,\to\,d_{i,j}}\,,\;$
$\displaystyle S^{LR}_{2,d_{j}d_{i}}$
$\displaystyle=8\,B^{SLR}_{d_{j}d_{i}\ell_{l}\ell_{k}}|_{\ell_{k,l}\,\to\,d_{i,j}}\,,$
$\displaystyle S^{S\sigma\sigma}_{1,d_{j}d_{i}}$
$\displaystyle=\big{(}4\,B^{S\sigma\sigma}_{d_{j}d_{i}\ell_{l}\ell_{k}}-16\,B^{T\sigma\sigma}_{d_{j}d_{i}\ell_{l}\ell_{k}}\big{)}|_{\ell_{k,l}\,\to\,d_{i,j}}\,,\;$
$\displaystyle S^{S\sigma\sigma}_{2,d_{j}d_{i}}$
$\displaystyle=-32\,B^{T\sigma\sigma}_{d_{j}d_{i}\ell_{l}\ell_{k}}|_{\ell_{k,l}\,\to\,d_{i,j}}\,.$

##  Appendix B Loop functions

In the UV-divergent loop functions we set $\epsilon=(4-D)/2$. The loop
functions are defined as (cf. [37])

$\begin{split}\frac{i}{(4\pi)^{2}}B_{0}\left(m_{1},m_{2}\right)\left(\frac{4\pi}{\mu^{2}}e^{-\gamma_{E}}\right)^{\epsilon}&=\int\frac{d^{D}q}{(2\pi)^{D}}\frac{1}{q^{2}-m_{1}^{2}}\frac{1}{q^{2}-m_{2}^{2}}\\\
\frac{i}{(4\pi)^{2}}C_{0}\left(m_{1},m_{2},m_{3}\right)\left(\frac{4\pi}{\mu^{2}}e^{-\gamma_{E}}\right)^{\epsilon}&=\int\frac{d^{D}q}{(2\pi)^{D}}\frac{1}{q^{2}-m_{1}^{2}}\frac{1}{q^{2}-m_{2}^{2}}\frac{1}{q^{2}-m_{3}^{2}}\\\
\frac{i}{(4\pi)^{2}}D_{0}\left(m_{1},m_{2},m_{3},m_{4}\right)\left(\frac{4\pi}{\mu^{2}}e^{-\gamma_{E}}\right)^{\epsilon}&=\int\frac{d^{D}q}{(2\pi)^{D}}\frac{1}{q^{2}-m_{1}^{2}}\frac{1}{q^{2}-m_{2}^{2}}\frac{1}{q^{2}-m_{3}^{2}}\frac{1}{q^{2}-m_{4}^{2}}\end{split}$
(B.1)

These functions read:

$\begin{split}B_{0}\left(m_{1},m_{2}\right)&=\frac{1}{\epsilon}+1+\frac{m_{1}^{2}\log\left(\frac{\mu^{2}}{m_{1}^{2}}\right)-m_{2}^{2}\log\left(\frac{\mu^{2}}{m_{2}^{2}}\right)}{m_{1}^{2}-m_{2}^{2}}\\\
C_{0}\left(m_{1},m_{2},m_{3}\right)&=\frac{m_{1}^{2}\,m_{2}^{2}\log\left(\frac{m_{2}^{2}}{m_{1}^{2}}\right)+m_{3}^{2}\,m_{2}^{2}\log\left(\frac{m_{3}^{2}}{m_{2}^{2}}\right)+m_{1}^{2}\,m_{3}^{2}\log\left(\frac{m_{1}^{2}}{m_{3}^{2}}\right)}{\left(m_{1}^{2}-m_{2}^{2}\right)\left(m_{1}^{2}-m_{3}^{2}\right)\left(m_{2}^{2}-m_{3}^{2}\right)}\\\
D_{0}\left(m_{1},m_{2},m_{3},m_{4}\right)&=\sum_{\textrm{cyclic
permutations}}\frac{m_{1}^{2}\log
m_{1}^{2}}{\big{(}m_{1}^{2}-m_{2}^{2}\big{)}\big{(}m_{1}^{2}-m_{3}^{2}\big{)}\big{(}m_{4}^{2}-m_{1}^{2}\big{)}}\end{split}$
(B.2)
$\begin{split}\tilde{C}_{0}(m_{1},m_{2},m_{3})=&B_{0}(m_{2},m_{3})+m_{1}^{2}C_{0}(m_{1},m_{2},m_{3})\\\
\tilde{D}_{0}(m_{1},m_{2},m_{3},m_{4})=&C_{0}(m_{2},m_{3},m_{4})+m_{1}^{2}D_{0}(m_{1},m_{2},m_{3},m_{4})\end{split}$
(B.3)

##  Appendix C Complete list of STIs for Feynman Rules

Here we list the full set of identities for the four-point couplings provided
by the tree-level analysis of STIs described in section 2. Their derivation
involves the following additional three- and four-point coupling

$\mathcal{L}_{\rm
3}\supset-{\textstyle\sum\limits_{v_{i},v_{j}}}\Big{\\{}\tfrac{ie}{2}\,\omega_{A,v_{i}v_{j}}F^{\mu\nu}V_{v_{i},\mu}^{a}V_{v_{j},\nu}^{a}+\tfrac{ig_{s}}{2}\,\omega_{G,v_{i}v_{j}}^{abc}G^{a,\mu\nu}V_{v_{i},\mu}^{b}V_{v_{j},\nu}^{c}\Big{\\}}\,.$
(C.1)

The kinetic term of the vector fields $V_{v_{i}}$ and the couplings to the
field strength tensors $\omega$ contribute to triple gauge boson vertices with
one photon or gluon. $U(1)_{\text{em}}\times SU(3)_{\text{colour}}$ gauge
invariance restricts these couplings to be of the form

$\omega_{A,ij}=\delta_{\mkern 1.5mu\mkern
1.0mu\overline{\mkern-1.0mui\mkern-1.0mu}\mkern
1.0muj}Q_{V_{j}}\,,\qquad\omega_{G,ij}^{abc}=\delta_{\mkern 1.5mu\mkern
1.0mu\overline{\mkern-1.0mui\mkern-1.0mu}\mkern 1.0muj}T^{a}_{V_{j},bc}\,.$
(C.2)

$\displaystyle
g_{{\phi}_{1}{\phi}_{2}{\phi}_{3}{\phi}_{4}}=\tfrac{-\sigma_{v_{1}}\sigma_{v_{2}}\sigma_{v_{3}}\sigma_{v_{4}}}{4m_{v_{1}}m_{v_{2}}m_{v_{3}}m_{v_{4}}}\,{\textstyle\sum\limits_{s_{5}}}\,m_{s_{5}}^{2}\Big{(}g_{v_{1}v_{4}s_{5}}g_{v_{2}v_{3}\bar{s}_{5}}+\text{symm}(v_{2},v_{3},v_{4})\Big{)}\,,$
(C.3) $\displaystyle
g_{v_{1}v_{2}{\phi}_{3}{\phi}_{4}}=\tfrac{\sigma_{v_{3}}\sigma_{v_{4}}}{2m_{v_{3}}m_{v_{4}}}\bigg{\\{}{\textstyle\sum\limits_{s_{5}}}\,g_{v_{1}v_{2}s_{5}}g_{v_{3}v_{4}\bar{s}_{5}}$
$\displaystyle\mkern
155.0mu+{\textstyle\sum\limits_{v_{5}}}\Big{(}\big{(}m_{v_{1}}^{2}+m_{v_{2}}^{2}-2m_{v_{5}}^{2}\big{)}\big{(}g_{v_{1}v_{4}v_{5}}g_{v_{2}v_{3}\bar{v}_{5}}+g_{v_{1}v_{3}v_{5}}g_{v_{2}v_{4}\bar{v}_{5}}\big{)}$
$\displaystyle\mkern
205.0mu-\tfrac{(m_{v_{1}}^{2}-m_{v_{2}}^{2})(m_{v_{3}}^{2}-m_{v_{4}}^{2})}{m_{v_{5}}^{2}}\,g_{v_{1}v_{2}v_{5}}g_{v_{3}v_{4}\bar{v}_{5}}\Big{)}\bigg{\\}}\,,$
$\displaystyle
g_{{\phi}_{1}{\phi}_{2}{\phi}_{3}s_{4}}=\tfrac{i\,\sigma_{v_{1}}\sigma_{v_{2}}\sigma_{v_{3}}}{6m_{v_{1}}m_{v_{2}}m_{v_{3}}}\bigg{\\{}{\textstyle\sum\limits_{s_{5}}}\,(2m_{s_{4}}^{2}-3m_{s_{5}}^{2})g_{v_{1}s_{4}s_{5}}g_{v_{2}v_{3}\bar{s}_{5}}$
$\displaystyle\mkern
180.0mu+m_{s_{4}}^{2}{\textstyle\sum\limits_{v_{5}}}\,\tfrac{m_{v_{2}}^{2}-m_{v_{3}}^{2}}{m_{v_{5}}^{2}}\,g_{v_{1}v_{5}s_{4}}g_{v_{2}v_{3}\bar{v}_{5}}+\text{symm}(v_{1},v_{2},v_{3})\bigg{\\}}\,,$
$\displaystyle
g_{v_{1}v_{2}{\phi}_{3}s_{4}}=\tfrac{i\,\sigma_{v_{3}}}{m_{v_{3}}}\bigg{\\{}{\textstyle\sum\limits_{v_{5}}}\Big{(}-g_{v_{1}v_{5}s_{4}}g_{v_{2}v_{3}\bar{v}_{5}}-g_{v_{2}v_{5}s_{4}}g_{v_{1}v_{3}\bar{v}_{5}}$
$\displaystyle\mkern
158.0mu+g_{v_{3}v_{5}s_{4}}g_{v_{1}v_{2}\bar{v}_{5}}\tfrac{m_{v_{1}}^{2}-m_{v_{2}}^{2}}{2m_{v_{5}}^{2}}\Big{)}+{\textstyle\sum\limits_{s_{5}}}\,g_{v_{3}s_{4}s_{5}}g_{v_{1}v_{2}\bar{s}_{5}}\bigg{\\}}\,,$
$\displaystyle
g_{{\phi}_{1}{\phi}_{2}s_{3}s_{4}}=\tfrac{\sigma_{v_{1}}\sigma_{v_{2}}}{2m_{v_{1}}m_{v_{2}}}\bigg{\\{}{\textstyle\sum\limits_{s_{5}}}\,\Big{(}g_{s_{3}s_{4}s_{5}}g_{v_{1}v_{2}\bar{s}_{5}}$
$\displaystyle\mkern
185.0mu+\Big{[}g_{v_{1}s_{3}s_{5}}g_{v_{2}s_{4}\bar{s}_{5}}\,(2m_{s_{5}}^{2}-m_{s_{3}}^{2}-m_{s_{4}}^{2})+\text{symm}(s_{3},s_{4})\Big{]}\Big{)}$
$\displaystyle\mkern
120.0mu+{\textstyle\sum\limits_{v_{5}}}\,\Big{(}g_{v_{5}s_{3}s_{4}}g_{v_{1}v_{2}\bar{v}_{5}}\,\big{(}m_{s_{3}}^{2}-m_{s_{4}}^{2}\big{)}\,\tfrac{m_{v_{1}}^{2}-m_{v_{2}}^{2}}{m_{v_{5}}^{2}}$
$\displaystyle\mkern
170.0mu+\Big{[}g_{v_{1}v_{5}s_{4}}g_{v_{2}\bar{v}_{5}s_{3}}\,\big{(}m_{s_{3}}^{2}+m_{s_{4}}^{2}\big{)}\,\tfrac{1}{4m_{v_{5}}^{2}}+\text{symm}(v_{1},v_{2})\Big{]}\Big{)}\bigg{\\}}\,,$
$\displaystyle
g_{{\phi}_{1}s_{2}s_{3}s_{4}}=\tfrac{i\,\sigma_{v_{1}}}{m_{v_{1}}}\bigg{\\{}{\textstyle\sum\limits_{s_{5}}}\,g_{s_{2}s_{3}s_{5}}g_{v_{1}s_{4}\bar{s}_{5}}-{\textstyle\sum\limits_{v_{5}}}\,g_{v_{5}s_{2}s_{3}}g_{v_{1}\bar{v}_{5}s_{4}}\tfrac{m_{s_{2}}^{2}-m_{s_{3}}^{2}}{2m_{v_{5}}^{2}}+\text{symm}(s_{2},s_{3},s_{4})\bigg{\\}}\,.$

Furthermore, all quartic interactions with vectors are fixed by the three-
point couplings

$\displaystyle
g_{v_{1}v_{2}v_{3}v_{4}}={\textstyle\sum\limits_{v_{5}}}\,\Big{(}g_{v_{1}v_{4}v_{5}}g_{v_{2}v_{3}\bar{v}_{5}}+g_{v_{1}v_{3}v_{5}}g_{v_{2}v_{4}\bar{v}_{5}}\Big{)}\,,$
(C.4) $\displaystyle
g_{v_{1}v_{2}s_{3}s_{4}}=\bigg{\\{}{\textstyle\sum\limits_{v_{5}}}\,g_{v_{1}v_{5}s_{4}}g_{v_{2}\bar{v}_{5}s_{3}}\tfrac{1}{4m_{v_{5}}^{2}}-{\textstyle\sum\limits_{s_{5}}}\,g_{v_{1}s_{3}s_{5}}g_{v_{2}s_{4}\bar{s}_{5}}+\text{symm}(v_{1},v_{2})\bigg{\\}}\,.$

The remaining, purely bosonic sum-rules are given by

$\displaystyle{\textstyle\sum\limits_{s_{5}}}\,\Big{(}g_{v_{1}v_{2}s_{5}}g_{v_{3}v_{4}\bar{s}_{5}}-g_{v_{1}v_{4}s_{5}}g_{v_{2}v_{3}\bar{s}_{5}}\Big{)}$
(C.5) $\displaystyle\mkern
100.0mu={\textstyle\sum\limits_{v_{5}}}\bigg{(}g_{v_{1}v_{3}v_{5}}g_{v_{2}v_{4}\bar{v}_{5}}\,\big{(}2m_{v_{5}}^{2}-m_{v_{1}}^{2}-m_{v_{2}}^{2}-m_{v_{3}}^{2}-m_{v_{4}}^{2}\big{)}$
$\displaystyle\mkern
160.0mu+g_{v_{1}v_{2}v_{5}}g_{v_{3}v_{4}\bar{v}_{5}}\Big{(}m_{v_{5}}^{2}+\tfrac{(m_{v_{1}}^{2}-m_{v_{2}}^{2})(m_{v_{3}}^{2}-m_{v_{4}}^{2})}{m_{v_{5}}^{2}}\Big{)}$
$\displaystyle\mkern
160.0mu+g_{v_{1}v_{4}v_{5}}g_{v_{2}v_{3}\bar{v}_{5}}\Big{(}m_{v_{5}}^{2}-\tfrac{(m_{v_{1}}^{2}-m_{v_{4}}^{2})(m_{v_{2}}^{2}-m_{v_{3}}^{2})}{m_{v_{5}}^{2}}\Big{)}\bigg{)}\,,$
$\displaystyle{\textstyle\sum\limits_{s_{5}}}\,\Big{(}g_{v_{1}v_{2}s_{5}}g_{v_{3}s_{4}\bar{s}_{5}}-g_{v_{2}v_{3}s_{5}}g_{v_{1}s_{4}\bar{s}_{5}}\Big{)}$
$\displaystyle\mkern
100.0mu={\textstyle\sum\limits_{v_{5}}}\bigg{(}g_{v_{2}v_{5}s_{4}}g_{v_{1}v_{3}\bar{v}_{5}}+g_{v_{3}v_{5}s_{4}}g_{v_{1}v_{2}\bar{v}_{5}}\,\tfrac{1}{2}\Big{(}1-\tfrac{m_{v_{1}}^{2}-m_{v_{2}}^{2}}{m_{v_{5}}^{2}}\Big{)}$
$\displaystyle\mkern
250.0mu+g_{v_{1}v_{5}s_{4}}g_{v_{2}v_{3}\bar{v}_{5}}\,\tfrac{1}{2}\Big{(}1+\tfrac{m_{v_{2}}^{2}-m_{v_{3}}^{2}}{m_{v_{5}}^{2}}\Big{)}\bigg{)}\,,$
$\displaystyle{\textstyle\sum\limits_{v_{5}}}\,g_{v_{1}v_{2}\bar{v}_{5}}g_{v_{5}s_{3}s_{4}}={\textstyle\sum\limits_{v_{5}}}\,\tfrac{1}{4m_{v_{5}}^{2}}\Big{(}g_{v_{1}v_{5}s_{4}}g_{v_{2}\bar{v}_{5}s_{3}}-g_{v_{1}v_{5}s_{3}}g_{v_{2}\bar{v}_{5}s_{4}}\Big{)}$
$\displaystyle\mkern
135.0mu+{\textstyle\sum\limits_{s_{5}}}\,\Big{(}g_{v_{1}s_{3}s_{5}}g_{v_{2}s_{4}\bar{s}_{5}}-g_{v_{2}s_{3}s_{5}}g_{v_{1}s_{4}\bar{s}_{5}}\Big{)}\,.$

All rules given here and in section 2 hold in the limiting case of any of the
particles being massless. The couplings of photons or gluons of course
simplify considerably and can be expressed by using the $U(1)$ and $SU(3)$
charges defined in (2.1). Moreover they can be separated into two classes:
Couplings involving $V$-$A$ or $V$-$G$ transitions (the generalization of
$Z$-$A$ transitions known from the SM) and those without. Couplings of the
latter class are either directly included in the covariant kinetic lagrangian
or zero. This includes444To derive the first line, one has to note that
$g_{Av_{1}s_{2}}$ vanishes. This coupling has to come from the kinetic term of
a multiplet $\Phi$ of the full gauge group, and is thus proportional to
$eQ_{\Phi}A_{\mu}V_{a}^{\mu}(\Phi^{\dagger}T_{V_{a}}\Phi)$. However the
Goldstone directions are precisely given by $T_{V_{a}}\langle\Phi\rangle$.

$\begin{gathered}g_{A{\phi}_{1}{\phi}_{2}}=g_{AA{\phi}_{1}{\phi}_{2}}=g_{A{\phi}_{1}s_{1}}=g_{AA{\phi}_{1}s_{1}}=0\,,\\\
g_{AA{\phi}_{1}{\phi}_{2}}=-2i(g_{Av_{1}\bar{v}_{1}})^{2}\delta_{v_{1}\bar{v}_{2}}\,.\end{gathered}$
(C.6)

The equations also hold with $G$ instead of $A$. The class of couplings with
$V$-$A$ or $V$-$G$ transitions has to be derived from the STIs. They are given
by

$\begin{split}&g_{Av_{1}{\phi}_{2}}=ie\,m_{v_{1}}\delta_{v_{1}\bar{v}_{2}}\,,\\\
&g_{Av_{1}{\phi}_{2}{\phi}_{3}}=e\,\sigma_{v_{2}}\sigma_{v_{3}}\big{(}Q_{v_{2}}-Q_{v_{3}}\big{)}\,\tfrac{m_{v_{2}}^{2}+m_{v_{3}}^{2}-m_{v_{1}}^{2}}{2m_{v_{2}}m_{v_{3}}}\,g_{v_{1}v_{2}v_{3}}\,,\\\
&g_{Av_{1}{\phi}_{2}s_{3}}=-ie\,\sigma_{v_{2}}\big{(}Q_{v_{2}}-Q_{s_{3}}\big{)}\,\tfrac{1}{2m_{v_{2}}}\,g_{v_{1}v_{2}s_{3}}\,,\\\
&g_{Av_{1}v_{2}v_{3}}=-e\,(Q_{v_{2}}-Q_{v_{3}})\,g_{v_{1}v_{2}v_{3}}\,,\\\
&g_{Av_{1}s_{2}s_{3}}=e\,(Q_{s_{2}}-Q_{s_{3}})\,g_{v_{1}s_{2}s_{3}}\,.\\\
\end{split}$ (C.7)

In the quartic coupling we used charge conservation to simplify the right hand
side. The corresponding equations with a gluon are

$\begin{split}&g_{Gv_{1}{\phi}_{2}}^{c_{3}c_{1}c_{2}}=i\sigma_{v_{2}}\big{(}T_{v_{2}}^{c_{3}}\big{)}_{c_{1}c_{2}}\,m_{v_{1}}\delta_{v_{1}\bar{v}_{2}}\,,\\\
&g_{Gv_{1}{\phi}_{2}{\phi}_{3}}^{c_{4}c_{1}c_{2}c_{3}}=\sigma_{v_{2}}\sigma_{v_{3}}\tfrac{m_{v_{2}}^{2}+m_{v_{3}}^{2}-m_{v_{1}}^{2}}{2m_{v_{2}}m_{v_{3}}}\,{\textstyle\sum\limits_{c_{5}}}\,\Big{(}\big{(}T_{v_{2}}^{c_{4}}\big{)}_{c_{2}c_{5}}\,g_{v_{1}v_{2}v_{3}}^{c_{1}c_{5}c_{3}}-\big{(}T_{V_{a_{3}}}^{c_{4}}\big{)}_{c_{3}c_{5}}\,g_{v_{1}v_{2}v_{3}}^{c_{1}c_{2}c_{5}}\Big{)}\,,\\\
&g_{Gv_{1}{\phi}_{2}s_{3}}^{c_{4}c_{1}c_{2}c_{3}}=-i\,\sigma_{v_{2}}\,\tfrac{1}{2m_{v_{2}}}\,{\textstyle\sum\limits_{c_{5}}}\,\Big{(}\big{(}T_{v_{2}}^{c_{4}}\big{)}_{c_{2}c_{5}}\,g_{v_{1}v_{2}s_{3}}^{c_{1}c_{5}c_{3}}-\big{(}T_{s_{3}}^{c_{4}}\big{)}_{c_{3}c_{5}}\,g_{v_{1}v_{2}s_{3}}^{c_{1}c_{2}c_{5}}\Big{)}\,,\\\
&g_{Gv_{1}v_{2}v_{3}}^{c_{4}c_{1}c_{2}c_{3}}=-{\textstyle\sum\limits_{c_{5}}}\,\Big{(}\big{(}T_{v_{2}}^{c_{4}}\big{)}_{c_{2}c_{5}}\,g_{v_{1}v_{2}v_{3}}^{c_{1}c_{5}c_{3}}-\big{(}T_{v_{3}}^{c_{4}}\big{)}_{c_{3}c_{5}}\,g_{v_{1}v_{2}v_{3}}^{c_{1}c_{2}c_{5}}\Big{)}\,,\\\
&g_{Gv_{1}s_{1}s_{2}}^{c_{4}c_{1}c_{2}c_{3}}={\textstyle\sum\limits_{c_{5}}}\,\Big{(}\big{(}T_{s_{1}}^{c_{4}}\big{)}_{c_{2}c_{5}}\,g_{v_{1}s_{1}s_{2}}^{c_{1}c_{5}c_{3}}-\big{(}T_{s_{2}}^{c_{4}}\big{)}_{c_{3}c_{5}}\,g_{v_{1}s_{1}s_{2}}^{c_{1}c_{2}c_{5}}\Big{)}\,.\\\
\end{split}$ (C.8)

We have checked explicitly that the STIs for five-point vertex functions do
not imply additional sum rules.

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