# Oblique corrections when $m_{W}\neq m_{Z}\cos{\theta_{W}}$ at tree level

Simonas Draukšas,(1) Vytautas Dūdėnas,(1) and Luís Lavoura(2)
${}^{(1)}\\!$ Institute of Theoretical Physics and Astronomy, Faculty of
Physics, Vilnius University,
9 Saulėtekio, LT-10222 Vilnius, Lithuania
${}^{(2)}\\!$ CFTP, Instituto Superior Técnico, Universidade de Lisboa,
Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal  E-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The parametrization of the oblique corrections through $S$, $T$, and $U$—later
extended by $V$, $W$, and $X$—is a convenient way of comparing the predictions
for various electroweak observables at the one-loop level between the Standard
Model and its extensions. That parametrization assumes that the extensions
under consideration have ${SU(2)\times U(1)}$ gauge symmetry _and_ the tree-
level relation $m_{W}=m_{Z}\cos{\theta_{W}}$ between the Weinberg angle and
the gauge-boson masses. In models where that relation does not hold at the
Lagrangian level, the parameter $T$ is not ultraviolet-finite, making the
parametrization inadequate. We present expressions that parametrize the
difference of the various predictions of two models with $m_{W}\neq
m_{Z}\cos{\theta_{W}}$ in terms of oblique parameters. The parameter $T$ does
not play a role in those expressions. Conveniently, they may be reached, from
the ones that were derived for models with tree-level
$m_{W}=m_{Z}\cos{\theta_{W}}$, by performing a simple substitution for $T$. We
also discuss the difficulties in using oblique parameters when comparing a
model with $m_{W}\neq m_{Z}\cos{\theta_{W}}$ to the Standard Model. Finally,
we compute the relevant five oblique parameters in the SM extended by scalars
in both $Y=0$ and $Y=1$ triplets.

###### Contents

  1. 1 Introduction
    1. 1.1 Main definitions in $SU(2)\times U(1)$ gauge theory
    2. 1.2 Definitions of the oblique parameters
    3. 1.3 Plan of the paper
  2. 2 Observables in terms of the oblique parameters
    1. 2.1 Observables vs. bare parameters
    2. 2.2 $\rho$
    3. 2.3 $\rho_{\star}$ and the substitution
    4. 2.4 Summary
  3. 3 Extensions of the SM with scalar triplets
    1. 3.1 The BBM and the BM
    2. 3.2 Oblique parameters
  4. 4 Oblique corrections to the SM in a model with $\widehat{\rho}\neq 1$
  5. 5 Conclusions
  6. A The list of observables
  7. B Derivation of the oblique corrections for an NPM vs. the SM
    1. B.1 Derivation of the corrections to $\bar{\rho}$
    2. B.2 Derivation of the corrections to $s^{2}_{0}$ and $s^{2}_{Z}$
    3. B.3 Derivation of the corrections to the decay widths
  8. C Derivation of the oblique corrections for a BBM vs. a BM
    1. C.1 Derivation of the corrections to $\rho$
    2. C.2 Derivation of the corrections to $s^{2}_{0}$ and $s^{2}_{Z}$
    3. C.3 Decay widths
  9. D Passarino–Veltman functions
  10. E UV finiteness of the oblique parameters

##  1 Introduction

There have been many proposals for parametrizing the oblique corrections to
electroweak (EW) observables [1, 2, 3, 4, 5, 6]. Nowadays the most widely used
parametrization seems to be through the parameters $S$, $T$, and $U$ [7],
sometimes extended by $V$, $W$, and $X$ [8]. An important property of the
oblique parameters, which makes them very convenient, is that they do not
depend on the renormalization scheme utilized; they may be calculated from the
bare self-energies, i.e. without including any counterterms.

The six oblique parameters $S,\ldots,X$ consider the difference between two
models: the Standard Model (SM) and an extension thereof with extra matter
(i.e. fermions and/or scalars) that we call a New Physics Model (NPM). Both
the SM and the NPM have $SU(2)\times U(1)$ gauge symmetry. Moreover, they both
have a so-called ‘custodial symmetry’ that makes the $W$-boson and $Z$-boson
masses $m_{W}$ and $m_{Z}$, respectively, related at Lagrangian level. This is
often expressed in terms of the Veltman $\rho$ parameter [9] as

$\widehat{\rho}\equiv\frac{\widehat{m}_{W}^{2}}{\widehat{c}^{2}\widehat{m}_{Z}^{2}}=\begin{cases}1&\text{with
custodial symmetry},\\\ \text{not fixed}&\text{without custodial
symmetry},\end{cases}$ (1)

where $\widehat{c}\equiv\cos{\widehat{\theta}_{W}}$ is the cosine of the
Weinberg angle $\widehat{\theta}_{W}$. In eq. (1) and hereinafter, we use hats
to denote tree-level (bare) parameters, and we drop the hats in (one-)loop-
level parameters. Equation (1) gets loop corrections and the exact value of
the Veltman $\rho$ parameter depends on the renormalization scheme (or,
equivalently, the choice of input observables). In models with custodial
symmetry, only three observables are needed to describe the EW sector. We take
those observables to be the fine structure constant $\alpha$, the Fermi
coupling constant $G_{F(\mathrm{charged})}$ that is measured in muon decay,
and the mass $m_{Z}$ of the $Z$ boson.

In light of the new measurement of the $W$-boson mass by the CDF collaboration
[10], which disagrees with the SM prediction by seven standard deviations, it
is important to consider models without custodial symmetry, i.e. with
arbitrary $\widehat{\rho}$ . Then four, instead of just three, EW input
observables are needed to renormalize the EW sector of the model. In this
paper we take those four observables to be the former three plus the mass
$m_{W}$ of the $W$ boson, which in that case is not determined by eq. (1) any
more. Since there is one more input observable in the renormalization
procedure, one of the six parameters $S,\ldots,X$, is not calculable any more
from the bare self-energies only, i.e. it becomes renormalization scheme-
dependent. Indeed, when $\widehat{\rho}\neq 1$ the parameter $T$ is
ultraviolet (UV)-divergent, as has been affirmed before us by multiple authors
[6, 11, 12, 13, 14, 15, 16, 17, 18, 19] (see also appendix E). So, now there
are only five oblique parameters, viz. $S$, $U$, $V$, $W$, and $X$. We
emphasize that these implications of the non-existence of custodial symmetry
were noticed already long ago, for instance in refs. [20, 6, 7].

The essence of oblique parameters is that they make it easy to compare
predictions of two custodially symmetric models, i.e. an NPM vs. the SM.
However, if some base model (BM) is compared with a model beyond it (the BBM),
and neither the BM nor the BBM have $\widehat{\rho}$ fixed, then the
expressions that were derived to compare the SM with an NPM no longer hold,
since $T$ is UV-divergent. In order to clearly separate these cases,
throughout this paper we will use the terms SM, NPM, BM, and BBM as:

  * •

The SM and the NPM have $\widehat{\rho}=1$; the NPM is identical to the SM
plus some extra matter.

  * •

The BM and the BBM do not have fixed $\widehat{\rho}$; the BBM is identical to
the BM plus some extra matter.

For comparing the predictions of the EW sector between two models of interest
there are three cases that could be relevant and that need to be studied
separately:

  1. 1.

SM vs. NPM, i.e. two models that have $\widehat{\rho}=1$.

  2. 2.

BM vs. BBM, i.e. neither model has $\widehat{\rho}$ fixed.

  3. 3.

SM vs. BBM, i.e. one model has $\widehat{\rho}=1$ and the other one has
$\widehat{\rho}$ free.

The first case was studied in [7], where the oblique parameters were initially
introduced. To the best of our knowledge, a formalism of oblique parameters
has not yet been developed for the two remaining cases. In this paper, we
generalize the formalism of the oblique parameters to the second case. We also
outline the issues that one faces when trying to deal with the third case,
which we leave as an open problem. While the third case would be the most
useful generalization of the formalism, because the current base model is the
SM that has $\widehat{\rho}=1$, the comparison of BM vs. BBM is also
significant: if one calculates the full one-loop EW predictions of some BM,
then the possibility to relate them to the predictions of any BBM through
oblique parameters allows skipping the full calculation of those predictions
in the BBM. Moreover, it is not unthinkable that, in the future, the standard
model of particle physics could have $\widehat{\rho}$ free, and then the
second case would become immediately relevant.

###  1.1 Main definitions in $SU(2)\times U(1)$ gauge theory

All EW models considered in this paper have $SU(2)\times U(1)$ gauge symmetry.
We shortly review the main tree-level conventions and definitions for those
models.

The covariant derivative is written

$D_{\mu}=\partial_{\mu}-i\,\widehat{g}\,\sum_{a=1}^{3}T^{a}W_{\mu}^{a}-i\,\widehat{g}^{\,\prime}\,YB_{\mu},$
(2)

where the gauge coupling constants $\widehat{g}$ and $\widehat{g}^{\,\prime}$
of $SU(2)$ and $U(1)$, respectively, are positive, the $T^{a}$ are the
generators of $SU(2)$, and $Y$ is the (weak) hypercharge, viz. the generator
of $U(1)$.444For the sake of brevity, we omit hats over the gauge fields, viz.
we write $B_{\mu}$ instead of $\widehat{B}_{\mu}$, and so on. For matter
(i.e., either fermions or scalars) placed in doublets of $SU(2)$,
$T^{a}=\sigma^{a}/2$, where the $\sigma^{a}$ are the Pauli matrices.

After EW symmetry breaking, the photon field $A_{\mu}$ and the $Z$-boson field
$Z_{\mu}$ are related to the gauge-eigenstate fields $W^{3}_{\mu}$ and
$B_{\mu}$ by

$\begin{array}[]{rcl}A_{\mu}&=&\widehat{c}\,B_{\mu}+\widehat{s}\,W^{3}_{\mu}\\\\[2.84526pt]
Z_{\mu}&=&\widehat{c}\,W^{3}_{\mu}-\widehat{s}\,B_{\mu}\end{array}\quad\Leftrightarrow\quad\begin{array}[]{rcl}B_{\mu}&=&\widehat{c}\,A_{\mu}-\widehat{s}\,Z_{\mu}\\\\[2.84526pt]
W^{3}_{\mu}&=&\widehat{c}\,Z_{\mu}+\widehat{s}\,A_{\mu}\end{array},$ (3)

where $\widehat{s}$ and $\widehat{c}$ are the sine and the cosine,
respectively, of the Weinberg angle $\widehat{\theta}_{W}$. The $W$-boson
field is

$W_{\mu}^{\pm}=\frac{W_{\mu}^{1}\mp iW_{\mu}^{2}}{\sqrt{2}}.$ (4)

The absolute value of the electric charge $\widehat{e}$ is related to
$\widehat{g}$, $\widehat{g}^{\prime}$, and the fine structure constant
$\widehat{\alpha}$ by

$\widehat{e}=\widehat{g}\,\widehat{s}=\widehat{g}^{\,\prime}\,\widehat{c}=\sqrt{4\pi\widehat{\alpha}}.$
(5)

In the SM and in an NPM only scalar fields that have weak isospin $J=1/2$,
i.e. doublets of $SU(2)$, and have weak hypercharge $Y=\pm 1/2$ get vacuum
expectation values (VEVs).555Possibly, scalar fields with $J=Y=0$ acquire VEVs
too, but those VEVs neither break $SU(2)\times U(1)$ nor contribute to $m_{W}$
and $m_{Z}$. As a consequence, at tree level the masses of the gauge bosons
are related through

$\widehat{m}_{W}=\widehat{c}\,\widehat{m}_{Z}.$ (6)

This is due to the accidental ‘custodial’ $SU(2)$ symmetry of the scalar
potential [6] when all the scalars with VEVs are in $SU(2$) doublets. At loop
level eq. (6) gets corrections, because both the hypercharge gauge
interactions and the Yukawa couplings break the custodial symmetry.

An important equation relates the tree-level Fermi coupling constant of the
effective charged-current interaction, viz.
$\widehat{G}_{F(\mathrm{charged})}$, to the Weinberg angle:

$\widehat{s}^{\,2}=\frac{\pi\widehat{\alpha}}{\sqrt{2}\,\widehat{G}_{F(\mathrm{charged})}\,\widehat{m}_{W}^{2}}\,.$
(7)

When there is custodial symmetry, viz. when eq. (6) holds, one may use instead

$\widehat{c}^{\,2}\,\widehat{s}^{\,2}=\frac{\pi\widehat{\alpha}}{\sqrt{2}\,\widehat{G}_{F(\mathrm{charged})}\,\widehat{m}_{Z}^{2}}\,.$
(8)

Any of the three eqs. (6), (7), and (8) may be used as the _definition_ of the
Weinberg angle in models with $\widehat{\rho}=1$; which definition one uses
depends on which input observables one choses. However, eqs. (6) and (8) do
not hold when $\widehat{\rho}\neq 1$; in that case, only eq. (7) may be used
to define $\widehat{\theta}_{W}$.

###  1.2 Definitions of the oblique parameters

The oblique corrections originate in the self-energies of the gauge bosons.
Those self-energies are decomposed as

$\Pi^{\mu\nu}_{VV^{\prime}}\left(p\right)=g^{\mu\nu}\,\Pi_{VV^{\prime}}\left(p^{2}\right)+p^{\mu}p^{\nu}\,\text{terms},$
(9)

where $V$ and $V^{\prime}$ are the vector bosons, $\mu$ and $\nu$ are their
polarizations, and $p$ is their momentum. The oblique parameters do not depend
on the coefficients of the $p^{\mu}p^{\nu}$ terms; all the oblique parameters
are expressed in terms of the functions $\Pi_{VV^{\prime}}\left(p^{2}\right)$
and/or their derivatives relative to $p^{2}$, evaluated at various energy
scales.

Starting from $\Pi_{VV^{\prime}}\left(p^{2}\right)$, we define two other
functions:

$\displaystyle\widetilde{\Pi}_{VV^{\prime}}\left(p^{2}\right)$
$\displaystyle\equiv$
$\displaystyle\frac{\Pi_{VV^{\prime}}\left(p^{2}\right)-\Pi_{VV^{\prime}}\left(0\right)}{p^{2}},$
(10a) $\displaystyle\Pi_{VV^{\prime}}^{\prime}\left(p^{2}\right)$
$\displaystyle\equiv$
$\displaystyle\frac{\mathrm{d}\Pi_{VV^{\prime}}\left(p^{2}\right)}{\mathrm{d}p^{2}}.$
(10b)

The functions $\Pi_{VV^{\prime}}\left(p^{2}\right)$,
$\widetilde{\Pi}_{VV^{\prime}}\left(p^{2}\right)$, and
$\Pi_{VV^{\prime}}^{\prime}\left(p^{2}\right)$ depend on the matter content of
each model M. We put a superscript M on those functions to identify the model
M in which they are computed. We define the ‘model parameters’
$S^{\mathrm{M}},\ldots,X^{\mathrm{M}}$ as [8]

$\displaystyle S^{\mathrm{M}}$ $\displaystyle=$
$\displaystyle\frac{4s^{2}c^{2}}{\alpha}\left[\widetilde{\Pi}^{\mathrm{M}}_{ZZ}\left(m_{Z}^{2}\right)+\frac{s^{2}-c^{2}}{sc}\,\Pi_{ZA}^{\prime\,\mathrm{M}}\left(0\right)-\Pi_{AA}^{\prime\,\mathrm{M}}\left(0\right)\right],$
(11a) $\displaystyle T^{\mathrm{M}}$ $\displaystyle=$
$\displaystyle\frac{1}{\alpha}\left[\frac{\Pi^{\mathrm{M}}_{WW}\left(0\right)}{m_{W}^{2}}-\frac{\Pi_{ZZ}^{\mathrm{M}}\left(0\right)}{m_{Z}^{2}}\right],$
(11b) $\displaystyle U^{\mathrm{M}}$ $\displaystyle=$
$\displaystyle\frac{4s^{2}}{\alpha}\left[\widetilde{\Pi}^{\mathrm{M}}_{WW}\left(m_{W}^{2}\right)-c^{2}\,\widetilde{\Pi}^{\mathrm{M}}_{ZZ}\left(m_{Z}^{2}\right)-2sc\,\Pi_{ZA}^{\prime\,\mathrm{M}}\left(0\right)-s^{2}\,\Pi_{AA}^{\prime\,\mathrm{M}}\left(0\right)\right],$
(11c) $\displaystyle V^{\mathrm{M}}$ $\displaystyle=$
$\displaystyle\frac{1}{\alpha}\left[\Pi_{ZZ}^{\prime\,\mathrm{M}}\left(m_{Z}^{2}\right)-\widetilde{\Pi}^{\mathrm{M}}_{ZZ}\left(m_{Z}^{2}\right)\right],$
(11d) $\displaystyle W^{\mathrm{M}}$ $\displaystyle=$
$\displaystyle\frac{1}{\alpha}\left[\Pi_{WW}^{\prime\,\mathrm{M}}\left(m_{W}^{2}\right)-\widetilde{\Pi}^{\mathrm{M}}_{WW}\left(m_{W}^{2}\right)\right],$
(11e) $\displaystyle X^{\mathrm{M}}$ $\displaystyle=$
$\displaystyle\frac{sc}{\alpha}\left[\Pi_{ZA}^{\prime\,\mathrm{M}}\left(0\right)-\widetilde{\Pi}^{\mathrm{M}}_{ZA}\left(m_{Z}^{2}\right)\right].$
(11f)

Recall that non-hatted parameters are loop-level parameters. We emphasize that
the model parameters in eqs. (11) are in general neither gauge-
independent666This is because the functions
$\Pi_{VV^{\prime}}^{\mathrm{M}}\left(p^{2}\right)$ contain contributions from
loops of gauge bosons, Goldstone bosons, and ghosts. nor finite. Note that

$S^{\mathrm{M}}+U^{\mathrm{M}}=\frac{4s^{2}}{\alpha}\left[\widetilde{\Pi}^{\mathrm{M}}_{WW}\left(m_{W}^{2}\right)-\frac{c}{s}\,\Pi_{ZA}^{\prime\,\mathrm{M}}\left(0\right)-\Pi_{AA}^{\prime\,\mathrm{M}}\left(0\right)\right]$
(12)

is usually easier to compute than $U^{\mathrm{M}}$ itself. For this reason, in
Sec. 3 we write firstly $S^{\mathrm{M}}$ and then
$U^{\mathrm{M}}=-S^{\mathrm{M}}+\left(S^{\mathrm{M}}+U^{\mathrm{M}}\right)$
instead of writing the full expression for $U^{\mathrm{M}}$.

We introduce an abbreviation for the often-appearing linear combination

$\displaystyle K^{\mathrm{M}}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2c^{2}}\,S^{\mathrm{M}}+\frac{s^{2}-c^{2}}{4s^{2}c^{2}}\,U^{\mathrm{M}}$
(13a) $\displaystyle=$
$\displaystyle\frac{1}{\alpha}\left[\frac{s^{2}-c^{2}}{c^{2}}\,\widetilde{\Pi}^{\mathrm{M}}_{WW}\left(m_{W}^{2}\right)+\widetilde{\Pi}^{\mathrm{M}}_{ZZ}\left(m_{Z}^{2}\right)-\frac{s^{2}}{c^{2}}\,\Pi_{AA}^{\prime\,\mathrm{M}}\left(0\right)\right].$
(13b)

The oblique parameters are defined as _subtractions_

$O=O^{\mathrm{M}^{\prime}}-O^{\mathrm{M}}$ (14)

where $O$ may be either $S\ \mathrm{or}\ T\ldots\mathrm{or}\ X$. In eq. (14),
either M is the SM and M′ is an NPM, or M is the BM and M′ is the BBM; in both
cases, the models $M$ and $M^{\prime}$ are renormalized by utilizing the same
input observables. The oblique parameters are defined as _subtractions_ of
model parameters in eq. (14); contrary to the model parameters, the oblique
parameters are both gauge-independent and finite.

###  1.3 Plan of the paper

In section 2 we derive the prescription for substituting $T$ when comparing
the BBM with the BM. In section 3 we give formulas for all the oblique
parameters, except for $T$, when the BBM is the SM plus _two_ triplets of
scalars, one of them with $Y=0$ and the other one with $Y=1$, that both have
VEVs in their neutral components. Section 4 discusses the difficulties in
using oblique parameters for comparing the SM with a model where
$\widehat{\rho}$ is free. Section 5 summarizes the conclusions of our
research. Afterwards there is a set of appendices: appendix A lists the
observables that one compares between M′ and M; appendix B derives the
formulas for the comparison of the observables between the SM and an NPM;
appendix C does the same job for the comparison between the BM and the BBM;
appendix D defines the Passarino–Veltman functions utilized in section 3; and
appendix E discusses the finiteness of the oblique parameters.

##  2 Observables in terms of the oblique parameters

###  2.1 Observables vs. bare parameters

Observables are calculated from bare parameters at a given order in
perturbation theory by computing the loop corrections to them. One then
expresses the bare parameters in terms of the chosen input observables, and
one plugs them into the expressions for the observables that one wants to
predict. In that way, the procedure of defining counterterms is circumvented,
because everything is expressed in terms of observable quantities and loop
functions that are renormalization scheme-independent, as was originally
proposed in ref. [20]. We note, however, that the final expressions depend on
the choice of input observables.

Consider the following four observables, which at one-loop level are expressed
as

$\displaystyle m_{W}^{2}$ $\displaystyle=$
$\displaystyle\widehat{m}_{W}^{2}\left[1+\frac{\Pi_{WW}\left(m_{W}^{2}\right)}{m_{W}^{2}}\right],$
(15a) $\displaystyle m_{Z}^{2}$ $\displaystyle=$
$\displaystyle\widehat{m}_{Z}^{2}\left[1+\frac{\Pi_{ZZ}\left(m_{Z}^{2}\right)}{m_{Z}^{2}}\right],$
(15b) $\displaystyle G_{F(\mathrm{charged})}$ $\displaystyle=$
$\displaystyle\widehat{G}_{F(\mathrm{charged})}\left[1-\frac{\Pi_{WW}\left(0\right)}{m_{W}^{2}}+\delta_{Gc}\right],$
(15c) $\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\widehat{\alpha}\left[1+\Pi^{\prime}_{AA}\left(0\right)+2\,\frac{s}{c}\,\frac{\Pi_{ZA}\left(0\right)}{m_{Z}^{2}}\right].$
(15d)

$G_{F(\mathrm{charged})}$ in eq. (15c) is extracted from the decay rate of the
muon; also in that equation, $\delta_{Gc}$ stands for the one-loop vertex,
fermion line, and box corrections to that decay.

In the SM and in an NPM just three observables, out of the four in eqs. (15),
are needed as input, because $\widehat{\rho}=1$. We choose the

$\text{input observables for the SM and for the NPM:}\quad m_{Z},\
G_{F(\mathrm{charged})},\ \text{and}\ \alpha.$ (16)

The value of $m_{W}$ is _predicted_ by using eqs. (6) and (8). Alternatively,
one may say that it is the value of $\rho$ that is predicted.

Since the BM and the BBM do not have fixed $\widehat{\rho}$, they have
additional freedom and all four observables in eq. (15) are needed as input,
i.e. we utilize the

$\text{input observables for the BM and for the BBM:}\quad m_{W},\ m_{Z},\
G_{F(\mathrm{charged})},\ \text{and}\ \alpha.$ (17)

Other EW observables, that are related to these four by tree-level relations,
may be predicted. In this case $m_{W}$ _cannot_ be predicted.

Since many expressions depend on the Weinberg angle, it is convenient to
define that angle in terms of the input observables. If one considers the BM
and the BBM with eq. (17), then the Weinberg angle may be _defined_ from the
input as

$s^{2}\equiv\frac{\pi\alpha}{\sqrt{2}\,G_{F(\mathrm{charged})}\,m_{W}^{2}},$
(18)

which is equivalent to promoting the tree-level expression in eq. (7) and
making it hold at loop level. In the case with $\widehat{\rho}=1$ and the
input observables in eq. (16), one _defines_ the Weinberg angle
$\bar{\theta}_{W}$ through

$\bar{c}^{2}\bar{s}^{2}\equiv\frac{\pi\alpha}{\sqrt{2}\,G_{F(\mathrm{charged})}\,m_{Z}^{2}},$
(19)

where $\bar{c}\equiv\cos{\bar{\theta}_{W}}$ and
$\bar{s}\equiv\sin{\bar{\theta}_{W}}$. Note that $\bar{\theta}_{W}$ defined by
eq. (19) is different from $\theta_{W}$ defined by eq. (18) already at one-
loop level in models with $\widehat{\rho}=1$. For models with free
$\widehat{\rho}$ the definition in eq. (19) is just not correct.

A third definition is possible when $\widehat{\rho}=1$, viz. one _defines_ the
Weinberg angle $\tilde{\theta}_{W}$ through

$\tilde{c}^{2}=\frac{m_{W}^{2}}{m_{Z}^{2}},$ (20)

where $\tilde{c}\equiv\cos{\tilde{\theta}_{W}}$. This definition is used in
the on-shell (OS) renormalization scheme, for instance in refs. [21, 22],
where $m_{W}$, $m_{Z}$, and $\alpha$ are used as input observables. However,
as far as we know, this definition is not often employed in the context of
oblique parameters and therefore we consider here only the two definitions in
eqs. (18) and (19).

###  2.2 $\rho$

The definition of the Veltman $\rho$ parameter, just as the definition of the
Weinberg angle, depends on the input observables that one uses. Thus, using
either eq. (18) or eq. (19) one obtains two different definitions:

$\displaystyle\bar{\rho}$ $\displaystyle\equiv$
$\displaystyle\frac{m_{W}^{2}}{\bar{c}^{2}m_{Z}^{2}},$ (21a)
$\displaystyle\rho$ $\displaystyle\equiv$
$\displaystyle\frac{m_{W}^{2}}{c^{2}m_{Z}^{2}}.$ (21b)

The definition in eq. (21a) is only possible when $\widehat{\rho}=1$, while
the definition in eq. (21b) is possible for any $\widehat{\rho}$.

If one has a model M with $\widehat{\rho}=1$, one may plug the expressions of
eqs. (15) into the definition of $\bar{\rho}$ in eq. (21a). One then groups
the self-energies into model parameters by using eqs. (11) and eq. (13). The
procedure is explicitly carried out in appendix B, cf. eq. (B15). One gets

$\bar{\rho}^{\mathrm{M}}=\widehat{\rho}\left\\{1+\frac{\bar{c}^{2}}{\bar{c}^{2}-\bar{s}^{2}}\left(\alpha
T^{\mathrm{M}}-\alpha
K^{\mathrm{M}}\right)+\frac{\bar{s}^{2}}{\bar{s}^{2}-\bar{c}^{2}}\left[\delta_{Gc}^{\mathrm{M}}-2\,\frac{\bar{s}}{\bar{c}}\,\frac{\Pi_{ZA}^{\mathrm{M}}\left(0\right)}{m_{Z}^{2}}\right]\right\\},$
(22)

where $\delta_{Gc}^{\mathrm{M}}$ is the value of $\delta_{Gc}$ computed in the
model M and $\widehat{\rho}=1$ should be understood. Dividing eq. (22) for an
NPM by the same equation for the SM, we obtain

$\frac{\bar{\rho}^{\,\mathrm{NPM}}}{\bar{\rho}^{\,\mathrm{SM}}}=1+\frac{\bar{c}^{2}}{\bar{c}^{2}-\bar{s}^{2}}\left(\alpha
T-\alpha
K\right)+\frac{\bar{s}^{2}}{\bar{s}^{2}-\bar{c}^{2}}\left[\delta_{Gc}^{\mathrm{NPM}}-\delta_{Gc}^{\mathrm{SM}}-2\,\frac{\bar{s}}{\bar{c}}\,\frac{\Pi_{ZA}^{\mathrm{NPM}}\left(0\right)-\Pi_{ZA}^{\mathrm{SM}}\left(0\right)}{m_{Z}^{2}}\right].$
(23)

Now, for extra fermions and extra scalars one always has777Note that
$\Pi_{ZA}^{\mathrm{M}}\left(0\right)$ itself is nonzero, because of the
contributions to it from loops of gauge bosons, Goldstone bosons, and ghosts.
It is just the _new_ contributions to $\Pi_{ZA}\left(0\right)$ that vanish.

$\Pi_{ZA}^{\mathrm{M^{\prime}}}\left(0\right)=\Pi_{ZA}^{\mathrm{M}}\left(0\right).$
(24)

Moreover, we _assume_ that [7]

$\delta_{Gc}^{\mathrm{M^{\prime}}}-\delta_{Gc}^{\mathrm{M}}\ \mathrm{is\
negligible}.$ (25)

This is the crucial assumption in the philosophy of oblique parameters. We
thus keep only the _oblique_ corrections in eq. (23):

$\bar{\rho}^{\,\mathrm{NPM}}=\bar{\rho}^{\,\mathrm{SM}}\left[1+\frac{\bar{c}^{2}}{\bar{c}^{2}-\bar{s}^{2}}\left(\alpha
T-\alpha K\right)\right].$ (26)

We emphasize that eq. (26) holds because
$\widehat{\rho}^{\,\mathrm{SM}}=\widehat{\rho}^{\,\mathrm{NPM}}=1$.

If one has a model M with $\widehat{\rho}\neq 1$, plugging eqs. (15) into eq.
(21b) one gets

$\rho^{\mathrm{M}}=\widehat{\rho}\left\\{1+\alpha T^{\mathrm{M}}-\alpha
K^{\mathrm{M}}-\frac{s^{2}}{c^{2}}\left[\delta_{Gc}^{\mathrm{M}}-2\,\frac{s}{c}\,\frac{\Pi_{ZA}^{\mathrm{M}}\left(0\right)}{m_{Z}^{2}}\right]\right\\}$
(27)

instead of eq. (22), cf. eq. (C9f). Hence, using eq. (24) and the assumption
in eq. (25),

$\frac{\rho^{\mathrm{BBM}}}{\rho^{\mathrm{BM}}}=\frac{\widehat{\rho}^{\,\mathrm{BBM}}}{\widehat{\rho}^{\,\mathrm{BM}}}\left(1+\alpha
T-\alpha K\right).$ (28)

Now, the parameter $\rho$ in eq. (21b) is already expressed in terms of the
input observables of eq. (17), which are the same for both the BM and the BBM.
Therefore,

$\rho^{\mathrm{BM}}=\rho^{\mathrm{BBM}}\quad\Leftrightarrow\quad\frac{\widehat{\rho}^{\,\mathrm{BBM}}}{\widehat{\rho}^{\,\mathrm{BM}}}=1-\alpha
T+\alpha K.$ (29)

On its own, the second equality in eq. (29) does not have much physical
meaning, as it is renormalization scheme-dependent: both $T$ and
$\widehat{\rho}$ contain UV divergences. However, as we will shortly see, that
relation will allow us to cancel the occurrences of divergent T parameter from
the observable quantities.

###  2.3 $\rho_{\star}$ and the substitution

The parameter $\rho_{\star}$ is, by definition, the ratio between the charged
and the neutral effective Fermi constants [7]:

$\rho_{\star}\equiv\frac{G_{F(\mathrm{neutral})}}{G_{F(\mathrm{charged})}},$
(30)

where $G_{F(\mathrm{neutral})}$ is measured in low-energy neutrino scattering
via an intermediate $Z$ boson. The relation between the bare and the one-loop
charged Fermi constant is given by eq. (15c). The neutral Fermi constant is
similarly expressed as

$G_{F(\mathrm{neutral})}=\widehat{G}_{F(\mathrm{neutral})}\left[1-\frac{\Pi_{ZZ}\left(0\right)}{m_{Z}^{2}}+\delta_{Gn}\right]\,,$
(31)

where $\delta_{Gn}$ stands for the vertex, fermion line, and box corrections
in neutral-current neutrino scattering. Using this, we get

$\rho_{\star}^{\mathrm{M}}=\widehat{\rho}\left(1+\alpha
T^{\mathrm{M}}+\delta_{Gn}^{\mathrm{M}}-\delta_{Gc}^{\mathrm{M}}\right),$ (32)

because

$\widehat{\rho}\equiv\frac{\widehat{m}_{W}^{2}}{\widehat{c}^{2}\widehat{m}_{Z}^{2}}=\frac{\widehat{G}_{F(\mathrm{neutral})}}{\widehat{G}_{F(\mathrm{charged})}}.$
(33)

Using $\widehat{\rho}=1$ in eq. (32) for both the SM and an NPM, and assuming
that

$\delta_{Gn}^{\mathrm{M^{\prime}}}-\delta_{Gn}^{\mathrm{M}}\ \mathrm{is\
negligible}$ (34)

together with eq. (25), one obtains

$\frac{\rho_{\star}^{\mathrm{NPM}}}{\rho_{\star}^{\mathrm{SM}}}=1+\alpha T\,,$
(35)

which is a well-known relation, already given in ref. [7].

Taking exactly the same steps for the the BM and the BBM, while leaving the
parameter $\widehat{\rho}$ free, gives

$\frac{\rho_{\star}^{\mathrm{BBM}}}{\rho_{\star}^{\mathrm{BM}}}=\frac{\widehat{\rho}^{\,\mathrm{BBM}}}{\widehat{\rho}^{\,\mathrm{BM}}}\left(1+\alpha
T\right).$ (36)

Plugging
$\widehat{\rho}^{\mathrm{BBM}}\\!\left/\widehat{\rho}^{\mathrm{BM}}\right.$
from eq. (29) into eq. (36) one gets

$\frac{\rho_{\star}^{\mathrm{BBM}}}{\rho_{\star}^{\mathrm{BM}}}=1+\alpha K.$
(37)

Equation (37) is gauge-invariant, finite, and is expressed only in terms of
observable quantities and bare self-energies. Note that by using eq. (29) to
get rid of the dependence on
$\widehat{\rho}^{\mathrm{BBM}}\\!\left/\widehat{\rho}^{\mathrm{BM}}\right.$ we
effectively replaced the divergent quantity $T$ by the finite quantity $K$,
cf. eqs. (35) and (37). This is a general substitution rule that holds for all
the EW observables when one uses as input $m_{W}$ instead of a value of
$\widehat{\rho}$.888In fact, one may trade the oblique parameter $T$ for
$m_{W}$ in models with $\widehat{\rho}=1$ too. However, this is not really
practical, because instead of calculating $T$ one would then have to calculate
the difference in the predictions for $m_{W}$ between the SM and an NPM.

###  2.4 Summary

The main difference between the cases $\widehat{\rho}=1$ and $\widehat{\rho}$
free is the number of input observables. When $\widehat{\rho}$ is free one
uses the additional input observable $m_{W}$ to cancel the appearances of
$\widehat{\rho}$ in every other observable via eq. (29). By doing this, in the
expression for every observable quantity listed in appendix A, one effectively
replaces

$T\ \to\ K\equiv\frac{1}{2c^{2}}\,S+\frac{s^{2}-c^{2}}{4s^{2}c^{2}}\,U,$ (38)

in the relations that hold for the comparison between an NPM and the SM in
refs. [7, 8]. Those relations are given in appendix B, while the ones for BBM
vs. BM are written in appendix C.

##  3 Extensions of the SM with scalar triplets

###  3.1 The BBM and the BM

We consider an $SU(2)\times U(1)$ gauge model with the following scalar
multiplets of $SU(2)$:

  * •

One doublet with $Y=1/2$, viz.

$\Phi=\left(\begin{array}[]{c}\phi^{+}\\\
\phi\end{array}\right),\qquad\phi=\widehat{v}+\frac{R_{1}+iI_{1}}{\sqrt{2}},$
(39)

where $v$ is a real and positive VEV.

  * •

One triplet with $Y=1$, viz.

$\Theta=\left(\begin{array}[]{c}\theta^{++}\\\ \theta^{+}\\\
\theta\end{array}\right),\qquad\theta=\widehat{t}+\frac{R_{2}+iI_{2}}{\sqrt{2}},$
(40)

where $t$ is a real and positive VEV. The charge-2 field $\theta^{++}$ is an
eigenstate of mass with mass $m_{++}$.

  * •

One real triplet with $Y=0$, viz.

$\Delta=\left(\begin{array}[]{c}\delta^{+}\\\ \delta\\\
-\delta^{-}\end{array}\right),\qquad\delta=\widehat{u}+R_{3},$ (41)

where $\delta^{-}={\delta^{+}}^{\ast}$, $\delta=\delta^{\ast}$, and $u$ is a
real and positive VEV.

In eqs. (39)–(41), the fields $R_{1}$, $R_{2}$, $R_{3}$, $I_{1}$, and $I_{2}$
are real.

The gauge-boson masses are

$\displaystyle\widehat{m}_{W}$ $\displaystyle=$
$\displaystyle\frac{\widehat{g}}{\sqrt{2}}\,\widehat{\varrho}_{W},\quad\mbox{where}\quad\widehat{\varrho}_{W}\equiv\sqrt{\widehat{v}^{2}+2\widehat{t}^{2}+2\widehat{u}^{2}},$
(42a) $\displaystyle\widehat{m}_{Z}$ $\displaystyle=$
$\displaystyle\frac{\widehat{g}}{\sqrt{2}\,\widehat{c}}\,\widehat{\varrho}_{Z},\quad\mbox{where}\quad\widehat{\varrho}_{Z}\equiv\sqrt{\widehat{v}^{2}+4\widehat{t}^{2}}.$
(42b)

This leads to

$\frac{1}{2}\leq\widehat{\rho}=\frac{\widehat{\varrho}^{2}_{W}}{\widehat{\varrho}^{2}_{Z}}\leq\infty,$
(43)

viz. $\widehat{\rho}$ is free.

The mixing of the charge-1 scalars is given by

$\left(\begin{array}[]{c}G^{+}\\\ H_{2}^{+}\\\
H_{3}^{+}\end{array}\right)=\mathcal{O}\left(\begin{array}[]{c}\phi^{+}\\\
\delta^{+}\\\ \theta^{+}\end{array}\right),$ (44)

where $G^{+}$ is the charged Goldstone boson, the $H_{j}^{+}$ ($j=2,3$) are
physical charged scalars with masses $m_{j+}$, and

$\mathcal{O}=\left(\begin{array}[]{ccc}\widehat{v}\left/\widehat{\varrho}_{W}\right.&\sqrt{2}\,\widehat{u}\left/\widehat{\varrho}_{W}\right.&\sqrt{2}\,\widehat{t}\left/\widehat{\varrho}_{W}\right.\\\
\mathcal{O}_{21}&\mathcal{O}_{22}&\mathcal{O}_{23}\\\
\mathcal{O}_{31}&\mathcal{O}_{32}&\mathcal{O}_{33}\end{array}\right)$ (45)

is a $3\times 3$ orthogonal matrix. The orthogonality relations

$\sum_{k=1}^{3}\mathcal{O}_{mk}\mathcal{O}_{nk}=\delta_{mn},$ (46)

together with the fixed first row, leave only one free parameter in
$\mathcal{O}$.

The mixing of the pseudoscalars is given by

$\left(\begin{array}[]{c}G\\\
A\end{array}\right)=\frac{1}{\widehat{\varrho}_{Z}}\left(\begin{array}[]{cc}\widehat{v}&2\widehat{t}\\\
-2\widehat{t}&\widehat{v}\end{array}\right)\left(\begin{array}[]{c}I_{1}\\\
I_{2}\end{array}\right),$ (47)

where $G$ is the neutral Goldstone boson and $A$ is a physical pseudoscalar
with mass $m_{A}$.

The mixing of the neutral scalars is given by

$\left(\begin{array}[]{c}S_{1}\\\ S_{2}\\\
S_{3}\end{array}\right)=O\left(\begin{array}[]{c}R_{1}\\\ R_{3}\\\
R_{2}\end{array}\right),$ (48)

where $O$ is a $3\times 3$ real orthogonal matrix and the $S_{k}$ ($k=1,2,3$)
are physical scalars with mass $m_{k}$.

We take the model exposed above to be the BBM. The BM may then be

  * •

either identical to the BBM without $\Delta$, which we call BM1;

  * •

or identical to the BBM without $\Theta$, and we call this BM2.

Both the BM1 and the BM2 have free $\widehat{\rho}$ and therefore their
predictions may be compared with those of the BBM through oblique parameters.
One should however note that the BM1 has $u=0$, hence $\widehat{\rho}<1$,
while the BM2 has $t=0$, hence $\widehat{\rho}>1$, and therefore one may
compare each of them to the BBM only for well-defined (and non-intersecting)
ranges of $\widehat{\rho}^{\mathrm{BBM}}$. One must also take into
consideration that, while the input values of $m_{W}$ and $m_{Z}$ are the same
for both the BBM and the BMs, the values of $v$, $t$, and $u$ will turn out
different in the BBM and the corresponding BM.

###  3.2 Oblique parameters

In general, one may write each model parameter $O^{\mathrm{BBM}}$ in the form

$O^{\mathrm{BBM}}=O^{\mathrm{BBM}}_{\mathrm{scalars}}+O^{\mathrm{BBM}}_{\mathrm{fermions}}+O^{\mathrm{BBM}}_{\mathrm{gauge}},$
(49)

where

  * •

$O^{\mathrm{BBM}}_{\mathrm{scalars}}$ is the contribution to
$O^{\mathrm{BBM}}$ from diagrams that have _at least one physical_ scalar in
the loop (hence, one includes diagrams where either a gauge or a Goldstone
boson is in one internal line of the loop and a physical scalar is in the
other internal line).

  * •

$O^{\mathrm{BBM}}_{\mathrm{fermions}}$ is the contribution to
$O^{\mathrm{BBM}}$ from diagrams that have two fermions in the loop;

  * •

$O^{\mathrm{BBM}}_{\mathrm{gauge}}$ is the contribution to $O^{\mathrm{BBM}}$
from diagrams that have _only_ gauge bosons, Goldstone bosons, and/or ghosts
in the loop.

$O^{\mathrm{BBM}}_{\mathrm{fermions}}$ is gauge-independent because it has
neither gauge nor Goldstone bosons in the loops. We have found that
$O^{\mathrm{BBM}}_{\mathrm{scalars}}$ is also gauge-independent. On the other
hand, $O^{\mathrm{BBM}}_{\mathrm{gauge}}$ depends on the gauge, and that
gauge-dependence only disappears in the subtraction of the analogous
contribution in the BM. This means that the gauge-dependence of
$O^{\mathrm{M}}_{\mathrm{gauge}}$ is the same when M is either the BBM or the
BM1 or the BM2, provided the same value of $\rho$ is utilized and the input
observables in the renormalization procedure of the EW sector are the same.
Thus, gauge-dependence cancels out in the subtracted oblique parameters, viz.
in $O^{\mathrm{BBM}}_{\mathrm{gauge}}-O^{\mathrm{BM}}_{\mathrm{gauge}}$. We
have explicitly checked this fact for the triplet extensions of the SM
introduced above.

The contribution $O^{\mathrm{BBM}}_{\mathrm{scalars}}$ is not necessarily
finite before the subtraction of the same quantity in the base model. Indeed,
both $S^{\mathrm{BBM}}_{\mathrm{scalars}}$ and
$U^{\mathrm{BBM}}_{\mathrm{scalars}}$ are divergent; but their divergences
only depend on the _two_ quantities $\varrho_{W}$ and $\varrho_{Z}$—which are
functions of the input observables—instead of depending on all _three_ VEVs
$v$, $t$, and $u$; therefore they cancel out in the subtraction of
$S^{\mathrm{BM}}_{\mathrm{scalars}}$ and $U^{\mathrm{BM}}_{\mathrm{scalars}}$,
respectively. On the other hand, $V^{\mathrm{BBM}}_{\mathrm{scalars}}$,
$W^{\mathrm{BBM}}_{\mathrm{scalars}}$, and
$X^{\mathrm{BBM}}_{\mathrm{scalars}}$ are finite even without the need to
subtract from them the same parameters in the base model.

It is interesting to note that in the gauge $\xi_{W}=\xi_{Z}=\xi_{A}\equiv\xi$
the UV divergences of $S^{\mathrm{BBM}}_{\mathrm{scalars}}$ and
$U^{\mathrm{BBM}}_{\mathrm{scalars}}$ exactly cancel out the UV divergences
appearing in $S^{\mathrm{BBM}}_{\mathrm{gauge}}$ and
$U^{\mathrm{BBM}}_{\mathrm{gauge}}$, respectively (see Appendix E for an
explanation of this fact), leading to finite model parameters.

We give in the following analytical expressions for all the
$O^{\mathrm{BBM}}_{\mathrm{scalars}}$. They are expressed in terms of
Passarino–Veltman functions, defined in Appendix D. One has

$\displaystyle S^{\mathrm{BBM}}_{\mathrm{scalars}}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\left\\{\vphantom{\sum_{j=2}^{3}}4\left(s^{2}-c^{2}\right)^{2}\bar{B}_{00}\left(m^{2}_{Z},m^{2}_{++},m^{2}_{++}\right)\right.$
(50h)
$\displaystyle-8\left(s^{4}+c^{4}\right)B^{\prime}_{00}\left(0,m^{2}_{++},m^{2}_{++}\right)$
$\displaystyle+4\,\sum_{k=1}^{3}\frac{\left(vO_{k3}-tO_{k1}\right)^{2}}{\varrho_{Z}^{2}}\,\bar{B}_{00}\left(m^{2}_{Z},m^{2}_{A},m^{2}_{k}\right)$
$\displaystyle+\sum_{k=1}^{3}\frac{\left(vO_{k1}+4tO_{k3}\right)^{2}}{\varrho_{Z}^{2}}\
m^{2}_{Z}\,\bar{F}\left(m^{2}_{Z},m^{2}_{Z},m^{2}_{k}\right)$
$\displaystyle+4\,\sum_{j=2}^{3}\frac{\left(u\mathcal{O}_{j2}-t\mathcal{O}_{j3}\right)^{2}}{\varrho_{W}^{2}}\
m^{2}_{W}\,\bar{F}\left(m^{2}_{Z},m^{2}_{W},m^{2}_{j+}\right)$
$\displaystyle+2\left(\mathcal{O}_{22}\mathcal{O}_{32}-\mathcal{O}_{23}\mathcal{O}_{33}\right)^{2}\bar{B}_{00}\left(m^{2}_{Z},m^{2}_{2+},m^{2}_{3+}\right)$
$\displaystyle+\sum_{j=2}^{3}\left(s^{2}-c^{2}+\mathcal{O}_{j3}^{2}-\mathcal{O}_{j2}^{2}\right)^{2}\bar{B}_{00}\left(m^{2}_{Z},m^{2}_{j+},m^{2}_{j+}\right)$
$\displaystyle\left.-2\sum_{j=2}^{3}\left[(s^{4}+c^{4})+\left(s^{2}-c^{2}\right)\left(\mathcal{O}_{j3}^{2}-\mathcal{O}_{j2}^{2}\right)\right]B_{00}^{\prime}\left(0,m^{2}_{j+},m^{2}_{j+}\right)\right\\};$
$\displaystyle U^{\mathrm{BBM}}_{\mathrm{scalars}}$ $\displaystyle=$
$\displaystyle-S^{\mathrm{BBM}}_{\mathrm{scalars}}+\frac{1}{\pi}\left\\{\vphantom{+2\sum_{j=2}^{3}\frac{\left(u\mathcal{O}_{j2}-t\mathcal{O}_{j3}\right)^{2}}{\varrho_{Z}^{2}}}-8B^{\prime}_{00}\left(0,m_{++}^{2},m_{++}^{2}\right)\right.$
(51h)
$\displaystyle+2\sum_{j=2}^{3}\left(\mathcal{O}_{j3}^{2}-\mathcal{O}_{j2}^{2}-1\right)B^{\prime}_{00}\left(0,m_{j+}^{2},m_{j+}^{2}\right)$
$\displaystyle+4\sum_{j=2}^{3}\mathcal{O}_{j3}^{2}\bar{B}_{00}\left(m_{W}^{2},m_{++}^{2},m_{j+}^{2}\right)$
$\displaystyle+2\,\sum_{j=2}^{3}\frac{\left(v\mathcal{O}_{j3}-\sqrt{2}t\mathcal{O}_{j1}\right)^{2}}{\varrho_{Z}^{2}}\,\bar{B}_{00}\left(m_{W}^{2},m_{A}^{2},m_{j+}^{2}\right)$
$\displaystyle+\sum_{k=1}^{3}\sum_{j=2}^{3}\left(O_{k1}\mathcal{O}_{j1}+\sqrt{2}\,O_{k3}\mathcal{O}_{j3}+2\,O_{k2}\mathcal{O}_{j2}\right)^{2}\bar{B}_{00}\left(m_{W}^{2},m_{k}^{2},m_{j+}^{2}\right)$
$\displaystyle+\frac{8t^{2}}{\varrho_{W}^{2}}\,m_{W}^{2}\,\bar{F}\left(m_{W}^{2},m_{W}^{2},m_{++}^{2}\right)$
$\displaystyle+\sum_{k=1}^{3}\frac{\left(vO_{k1}+2tO_{k3}+2\sqrt{2}uO_{k2}\right)^{2}}{\varrho_{W}^{2}}\
m_{W}^{2}\,\bar{F}\left(m_{W}^{2},m_{W}^{2},m_{k}^{2}\right)$
$\displaystyle\left.+2\sum_{j=2}^{3}\frac{\left(u\mathcal{O}_{j2}-t\mathcal{O}_{j3}\right)^{2}}{\varrho_{Z}^{2}}\
m^{2}_{Z}\,\bar{F}\left(m^{2}_{W},m^{2}_{Z},m^{2}_{j+}\right)\right\\};$
$\displaystyle V^{\mathrm{BBM}}_{\mathrm{scalars}}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi
s^{2}c^{2}}\left\\{\vphantom{\frac{\left(\mathcal{O}_{22}\mathcal{O}_{32}-\mathcal{O}_{23}\mathcal{O}_{33}\right)^{2}}{2}}\left(s^{2}-c^{2}\right)^{2}\left[B^{\prime}_{00}\left(m_{Z}^{2},m_{++}^{2},m_{++}^{2}\right)-\bar{B}_{00}\left(m_{Z}^{2},m_{++}^{2},m_{++}^{2}\right)\right]\right.$
(52f)
$\displaystyle+\sum_{k=1}^{3}\frac{\left(vO_{k3}-tO_{k2}\right)^{2}}{\varrho_{Z}^{2}}\left[B^{\prime}_{00}\left(m_{Z}^{2},m_{A}^{2},m_{k}^{2}\right)-\bar{B}_{00}\left(m_{Z}^{2},m_{A}^{2},m_{k}^{2}\right)\right]$
$\displaystyle+\frac{m_{Z}^{2}}{4}\,\sum_{k=1}^{3}\frac{\left(vO_{k2}+4tO_{k3}\right)^{2}}{\varrho_{Z}^{2}}\left[F^{\prime}_{00}\left(m_{Z}^{2},m_{Z}^{2},m_{k}^{2}\right)-\bar{F}\left(m_{Z}^{2},m_{Z}^{2},m_{k}^{2}\right)\right]$
$\displaystyle+m_{W}^{2}\sum_{j=2}^{3}\frac{\left(u\mathcal{O}_{j2}-t\mathcal{O}_{j3}\right)^{2}}{\varrho_{W}^{2}}\left[F^{\prime}_{00}\left(m_{Z}^{2},m_{W}^{2},m_{j+}^{2}\right)-\bar{F}\left(m_{Z}^{2},m_{W}^{2},m_{j+}^{2}\right)\right]$
$\displaystyle+\sum_{j=2}^{3}\frac{\left(s^{2}-c^{2}+\mathcal{O}_{j3}^{2}-\mathcal{O}_{j2}^{2}\right)^{2}}{4}\left[B^{\prime}_{00}\left(m_{Z}^{2},m_{j+}^{2},m_{j+}^{2}\right)-\bar{B}_{00}\left(m_{Z}^{2},m_{j+}^{2},m_{j+}^{2}\right)\right]\hskip
19.91692pt$
$\displaystyle\left.+\frac{\left(\mathcal{O}_{22}\mathcal{O}_{32}-\mathcal{O}_{23}\mathcal{O}_{33}\right)^{2}}{2}\left[B^{\prime}_{00}\left(m_{Z}^{2},m_{2+}^{2},m_{3+}^{2}\right)-\bar{B}_{00}\left(m_{Z}^{2},m_{2+}^{2},m_{3+}^{2}\right)\right]\right\\};$
$\displaystyle W^{\mathrm{BBM}}_{\mathrm{scalars}}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi
s^{2}}\left\\{\sum_{j=2}^{3}\mathcal{O}_{j3}^{2}\left[B^{\prime}_{00}\left(m_{W}^{2},m_{++}^{2},m_{j+}^{2}\right)-\bar{B}_{00}\left(m_{W}^{2},m_{++}^{2},m_{j+}^{2}\right)\right]\right.$
(53f)
$\displaystyle+\sum_{j=2}^{3}\frac{\left(v\mathcal{O}_{j3}-\sqrt{2}t\mathcal{O}_{j1}\right)^{2}}{2\varrho_{Z}^{2}}\left[B^{\prime}_{00}\left(m_{W}^{2},m_{A}^{2},m_{j+}^{2}\right)-\bar{B}_{00}\left(m_{W}^{2},m_{A}^{2},m_{j+}^{2}\right)\right]$
$\displaystyle+\sum_{k=1}^{3}\sum_{j=2}^{3}\frac{\left(O_{k1}\mathcal{O}_{j1}+\sqrt{2}\,O_{k3}\mathcal{O}_{j3}+2\,O_{k2}\mathcal{O}_{j2}\right)^{2}}{4}\left[B^{\prime}_{00}\left(m_{W}^{2},m_{k}^{2},m_{j+}^{2}\right)\right.$
$\displaystyle\left.-\bar{B}_{00}\left(m_{W}^{2},m_{k}^{2},m_{j+}^{2}\right)\right]$
$\displaystyle+\frac{2m_{W}^{2}t^{2}}{\varrho_{W}^{2}}\left[F^{\prime}\left(m_{W}^{2},m_{W}^{2},m_{++}^{2}\right)-\bar{F}\left(m_{W}^{2},m_{W}^{2},m_{++}^{2}\right)\right]$
$\displaystyle+\frac{m_{W}^{2}}{4}\,\sum_{k=1}^{3}\frac{\left(vO_{k1}+2tO_{k3}+2\sqrt{2}uO_{k2}\right)^{2}}{\varrho_{W}^{2}}\left[F^{\prime}\left(m_{W}^{2},m_{W}^{2},m_{k}^{2}\right)\right.$
$\displaystyle\left.-\bar{F}\left(m_{W}^{2},m_{W}^{2},m_{k}^{2}\right)\right]$
$\displaystyle\left.+\frac{m_{Z}^{2}}{2}\,\sum_{j=2}^{3}\frac{\left(u\mathcal{O}_{j2}-t\mathcal{O}_{j3}\right)^{2}}{\varrho_{Z}^{2}}\left[F^{\prime}\left(m_{W}^{2},m_{Z}^{2},m_{j+}^{2}\right)-\bar{F}\left(m_{W}^{2},m_{Z}^{2},m_{j+}^{2}\right)\right]\right\\};\hskip
19.91692pt$ $\displaystyle X^{\mathrm{BBM}}_{\mathrm{scalars}}$
$\displaystyle=$
$\displaystyle\frac{2\left(s^{2}-c^{2}\right)}{\pi}\left[B^{\prime}_{00}\left(0,m_{++}^{2},m_{++}^{2}\right)-\bar{B}_{00}\left(m_{Z}^{2},m_{++}^{2},m_{++}^{2}\right)\right]$
(54b)
$\displaystyle+\sum_{j=2}^{3}\frac{s^{2}-c^{2}+\mathcal{O}_{j3}^{2}-\mathcal{O}_{j2}^{2}}{2\pi}\left[B^{\prime}_{00}\left(0,m_{j+}^{2},m_{j+}^{2}\right)-\bar{B}_{00}\left(m_{Z}^{2},m_{j+}^{2},m_{j+}^{2}\right)\right].\hskip
19.91692pt$

##  4 Oblique corrections to the SM in a model with $\widehat{\rho}\neq 1$

When one compares the SM to a model M where $\widehat{\rho}\neq 1$, one faces
complications due to the renormalization of the EW sector requiring different
numbers of input observables in the two models. This mismatch causes problems
when one attempts to use an input observable like $m_{W}$ in M. Indeed, in the
SM $m_{W}$ is fixed to a predicted value $m_{W}^{\mathrm{SM}}$. If one uses
$m_{W}\neq\left(m_{W}\right)_{\mathrm{SM}}$ as an input of M, then the gauge
dependences in the subtracted oblique parameters do not cancel out, and this
makes it necessary to take into account the non-oblique contributions. More
specifically, if one takes, for instance, $O^{\mathrm{BBM}}_{\mathrm{gauge}}$
from the previous section and the analogous
$O^{\mathrm{SM}}_{\mathrm{gauge}}$, they are not equal and thus the
subtraction of one from the other gives a gauge-dependent result.

Kennedy and Lynn [20, 23, 6] have discussed the issue of gauge dependence in
the context of oblique corrections. They have proposed a method to define
gauge-invariant self-energies by including vertex and box ‘universal’
contributions in the propagators. This approach may allow to define both the
(un-subtracted) model parameters and the (subtracted) oblique parameters in a
gauge-independent way. However, it is unclear whether the increased complexity
of such generalized parameters would justify the loss of accuracy, when
compared to a full calculation.

It remains an open question whether it is at all possible to construct a
convenient parametrization, similar to the oblique parameters presented in
this paper, when comparing a model with $\widehat{\rho}=1$ to a model with
free $\widehat{\rho}$.

##  5 Conclusions

The predictive power of models with free $\widehat{\rho}$ is smaller than the
one of models where $\widehat{\rho}=1$. An additional input observable, viz.
$m_{W}$, is needed in the renormalization of models with free
$\widehat{\rho}$. Still, should such a model be promoted in the future to
(new) standard model, oblique parameters could still serve as a powerful tool
for parametrizing the effects of physics beyond it. In this paper we have
provided analytical expressions for various electroweak observables in this
scenario and we have compared them to the known expressions for the case where
$\widehat{\rho}=1$. The new oblique corrections are parametrized by just five
oblique parameters $S$, $U$, $V$, $W$, and $X$; the parameter $T$ is absent
from the new expressions due to the additional input $m_{W}$ used, and is
replaced in all the expressions by a linear combination of $S$ and $U$ shown
in eq. (38). We have presented analytical expressions for the five oblique
parameters in a two-scalar-triplet extension of the SM, where both triplets
(one with hypercharge 1 and the other one with zero hypercharge) have nonzero
vacuum expectation values.

#### Acknowledgements:

The authors thank Duarte Fontes for reading and commenting on a draft of this
paper. S.D. and V.D. thank the Lithuanian Academy of Sciences for funding
received through the project DaFi2021. L.L. thanks the Portuguese Foundation
for Science and Technology for support through the projects UIDB/00777/2020,
UIDP/00777/2020, CERN/FIS-PAR/0002/2021, and CERN/FIS-PAR/0019/2021.

##  Appendix A The list of observables

####  $s_{0}$ and $s_{Z}$:

We define

$s^{2}_{0}\equiv s^{2}_{\mathrm{effective}}\left(p^{2}\approx 0\right),\quad
s^{2}_{Z}\equiv s^{2}_{\mathrm{effective}}\left(p^{2}\approx
m_{Z}^{2}\right),$ (A1)

where $s^{2}_{\mathrm{effective}}$ is the effective sine-squared of the
Weinberg angle.

#### Observables at low energies:

At $p^{2}\approx 0$ one measures the left-handed coupling and the right-handed
coupling in deep-inelastic neutrino scattering. They are given by

$\displaystyle g^{2}_{L}$ $\displaystyle=$
$\displaystyle\rho_{\star}^{2}\left[\left(\frac{1}{2}-\frac{2}{3}\,s^{2}_{0}\right)^{2}+\left(-\frac{1}{2}+\frac{1}{3}\,s^{2}_{0}\right)^{2}\right]$
(A2a) $\displaystyle=$
$\displaystyle\frac{1}{2}\,\rho_{\star}^{2}\left(1-2s^{2}_{0}+\frac{10}{9}\,s^{4}_{0}\right),$
(A2b) $\displaystyle g^{2}_{R}$ $\displaystyle=$
$\displaystyle\rho_{\star}^{2}\left[\left(-\frac{2}{3}\,s^{2}_{0}\right)^{2}+\left(\frac{1}{3}\,s^{2}_{0}\right)^{2}\right]$
(A2c) $\displaystyle=$ $\displaystyle\frac{5}{9}\
\rho_{\star}^{2}\,s^{4}_{0},$ (A2d)

respectively. Low-energy neutrino–electron scattering gives us the vector and
axial-vector couplings

$\displaystyle g_{V}$ $\displaystyle=$ $\displaystyle
2\rho_{\star}\left(s^{2}_{0}-\frac{1}{4}\right),$ (A3a) $\displaystyle g_{A}$
$\displaystyle=$ $\displaystyle-\frac{1}{2}\,\rho_{\star},$ (A3b)

respectively. The nuclear weak charge relevant for atomic parity violation is

$Q\left(Z,N\right)=-2\rho_{\star}\left[\left(s^{2}_{0}-\frac{1}{2}\right)Z+\frac{N}{2}\right],$
(A4)

where $Z$ is the number of protons and $N$ is the number of neutrons in the
nucleus. In a cesium-133 nucleus there are 55 protons and 78 neutrons, hence

$\displaystyle Q\left(55,78\right)$ $\displaystyle=$
$\displaystyle-2\rho_{\star}\left[55\left(s^{2}_{0}-\frac{1}{2}\right)+39\right]$
(A5a) $\displaystyle=$
$\displaystyle-110\rho_{\star}\left(s^{2}_{0}+\frac{23}{110}\right).$ (A5b)

#### Observables at the $Z$ pole:

At $p^{2}\approx m_{Z}^{2}$ one measures the left–right asymmetry

$\displaystyle A_{\mathrm{LR}}$ $\displaystyle=$ $\displaystyle
2\,\frac{1-4s^{2}_{Z}}{1+\left(1-4s^{2}_{Z}\right)^{2}}$ (A6a)
$\displaystyle=$ $\displaystyle\frac{1-4s^{2}_{Z}}{1-4s^{2}_{Z}+8s^{4}_{Z}}.$
(A6b)

One also measures

  * •

the forward–backward asymmetry of bottom quarks

$\displaystyle A_{\mathrm{FB}}\left(b\right)$ $\displaystyle=$
$\displaystyle\frac{3}{4}\left(\frac{1}{2}-\frac{2}{3}\,s^{2}_{Z}\right)\left(\frac{1}{2}-2s^{2}_{Z}\right)$
(A7a)
$\displaystyle\times\left[\frac{1}{4}+\left(\frac{1}{2}-\frac{2}{3}\,s^{2}_{Z}\right)^{2}\right]^{-1}\left[\frac{1}{4}+\left(\frac{1}{2}-2s^{2}_{Z}\right)^{2}\right]^{-1}$
$\displaystyle=$ $\displaystyle\frac{9}{4}\
\frac{3-16s^{2}_{Z}+16s^{4}_{Z}}{\left(9-12s^{2}_{Z}+8s^{4}_{Z}\right)\left(1-4s^{2}_{Z}+8s^{4}_{Z}\right)};$
(A7b)

  * •

the forward–backward asymmetry of charm quarks

$\displaystyle A_{\mathrm{FB}}\left(c\right)$ $\displaystyle=$
$\displaystyle\frac{3}{4}\left(\frac{1}{2}-\frac{4}{3}\,s^{2}_{Z}\right)\left(\frac{1}{2}-2s^{2}_{Z}\right)$
(A8a)
$\displaystyle\times\left[\frac{1}{4}+\left(\frac{1}{2}-\frac{4}{3}\,s^{2}_{Z}\right)^{2}\right]^{-1}\left[\frac{1}{4}+\left(\frac{1}{2}-2s^{2}_{Z}\right)^{2}\right]^{-1}$
$\displaystyle=$ $\displaystyle\frac{9}{4}\
\frac{3-20s^{2}_{Z}+32s^{4}_{Z}}{\left(9-24s^{2}_{Z}+32s^{4}_{Z}\right)\left(1-4s^{2}_{Z}+8s^{4}_{Z}\right)};$
(A8b)

  * •

the forward–backward asymmetry of muons

$\displaystyle A_{\mathrm{FB}}\left(\mu\right)$ $\displaystyle=$
$\displaystyle\frac{3}{4}\
\left(\frac{1}{2}-2s^{2}_{Z}\right)^{2}\left[\frac{1}{4}+\left(\frac{1}{2}-2s^{2}_{Z}\right)^{2}\right]^{-2}$
(A9a) $\displaystyle=$ $\displaystyle\frac{3}{4}\
\frac{1-8s^{2}_{Z}+16s^{4}_{Z}}{\left(1-4s^{2}_{Z}+8s^{4}_{Z}\right)^{2}}.$
(A9b)

#### Other observables:

Other observables that we consider are

  * •

the partial decay width $\Gamma\left(W^{-}\to e^{-}\bar{\nu}_{e}\right)$ of
the $W$ boson to leptons;

  * •

the partial decay width $\Gamma\left(Z\to u\bar{u}\right)$ of the $Z$ boson to
quark–antiquark pairs of charge $\pm 2/3$;

  * •

the partial decay width $\Gamma\left(Z\to d\bar{d}\right)$ of the $Z$ boson to
quark–antiquark pairs of charge $\mp 1/3$;

  * •

the partial decay width $\Gamma\left(Z\to\ell\bar{\ell}\right)$ of the $Z$
boson to pairs of charged leptons;

  * •

the partial decay width $\Gamma\left(Z\to\nu\bar{\nu}\right)$ of the $Z$ boson
to neutrino–antineutrino pairs;

  * •

the total decay width of the $Z$ boson

$\Gamma_{\mathrm{total}}\left(Z\right)=2\,\Gamma\left(Z\to
u\bar{u}\right)+3\left[\Gamma\left(Z\to
d\bar{d}\right)+\Gamma\left(Z\to\ell\bar{\ell}\right)+\Gamma\left(Z\to\nu\bar{\nu}\right)\right].$
(A10)

##  Appendix B Derivation of the oblique corrections for an NPM vs. the SM

We make here the explicit derivation of the observables listed in Appendix A,
for the case of a New Physics Model (NPM) compared to the Standard Model (SM),
both models featuring $m_{W}=m_{Z}\cos{\theta_{W}}$ at tree level.

We write the loop corrections as

$\displaystyle m_{W}^{2}$ $\displaystyle=$
$\displaystyle\widehat{m}_{W}^{2}\left(1+R_{W}\right),$ (B1a) $\displaystyle
m_{Z}^{2}$ $\displaystyle=$
$\displaystyle\widehat{m}_{Z}^{2}\left(1+R_{Z}\right),$ (B1b)
$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\widehat{\alpha}\left(1+R_{\alpha}\right),$ (B1c) $\displaystyle
G_{F(\mathrm{charged})}$ $\displaystyle=$
$\displaystyle\widehat{G}_{F(\mathrm{charged})}\left(1+R_{G}\right).$ (B1d)

We restrict ourselves to renormalization at the one-loop level. Then, only
terms _linear_ in $R_{W}$, $R_{Z}$, $R_{\alpha}$, and $R_{G}$ must be taken
into account. One has, according to eqs. (15),

$\displaystyle R_{W}$ $\displaystyle=$
$\displaystyle\frac{\Pi_{WW}\left(m_{W}^{2}\right)}{m_{W}^{2}},$ (B2a)
$\displaystyle R_{Z}$ $\displaystyle=$
$\displaystyle\frac{\Pi_{ZZ}\left(m_{Z}^{2}\right)}{m_{Z}^{2}},$ (B2b)
$\displaystyle R_{\alpha}$ $\displaystyle=$ $\displaystyle 2\
\frac{\Pi_{ZA}\left(0\right)}{m_{Z}^{2}}\,\tan{\theta_{W}}+\Pi_{AA}^{\prime}\left(0\right),$
(B2c) $\displaystyle R_{G}$ $\displaystyle=$
$\displaystyle-\frac{\Pi_{WW}\left(0\right)}{m_{W}^{2}}+\delta_{Gc},$ (B2d)

where $\delta_{Gc}$ is the correction to $G_{F(\mathrm{charged})}$ originating
from box diagrams and from the one-particle-irreducible diagrams that correct
the vertices and the external-fermion legs in muon decay—i.e., everything
except the oblique corrections. Notice that eq. (B2c) results from the use of
a Ward Identity.

###  B.1 Derivation of the corrections to $\bar{\rho}$

In both the SM and the NPM one uses the measured $\alpha$,
$G_{F(\mathrm{charged})}$, and $m_{Z}$ as the input of renormalization.
Following eq. (19), the Weinberg angle $\bar{\theta}_{W}$ is defined by

$A\equiv\frac{\pi\alpha}{\sqrt{2}\,G_{F(\mathrm{charged})}\,m_{Z}^{2}}=\bar{c}^{2}\bar{s}^{2},$
(B3)

where $\bar{c}\equiv\cos{\bar{\theta}_{W}}$ and
$\bar{s}\equiv\sin{\bar{\theta}_{W}}$. Equation (B3) means that

$\bar{c}^{2}=\frac{1+\sqrt{1-4A}}{2},\quad\bar{s}^{2}=\frac{1-\sqrt{1-4A}}{2}.$
(B4)

It follows from eqs. (B4) that

$\frac{\mathrm{d}\bar{c}^{2}}{\mathrm{d}A}=\frac{1}{\bar{s}^{2}-\bar{c}^{2}}.$
(B5)

We write the loop corrections as

$\displaystyle\bar{c}^{2}$ $\displaystyle=$
$\displaystyle\widehat{c}^{\,2}\left(1+R_{\bar{C}}\right),$ (B6a)
$\displaystyle\bar{s}^{2}$ $\displaystyle=$
$\displaystyle\widehat{s}^{\,2}\left(1+R_{\bar{S}}\right),$ (B6b)
$\displaystyle A$ $\displaystyle=$
$\displaystyle\widehat{A}\left(1+R_{A}\right),$ (B6c)
$\displaystyle\bar{\rho}$ $\displaystyle=$
$\displaystyle\widehat{\rho}\left(1+R_{\bar{\rho}}\right),$ (B6d)

where $\bar{\rho}$ was defined in eq. (21a). Clearly,

$R_{\bar{\rho}}=R_{W}-R_{\bar{C}}-R_{Z}.$ (B7)

We proceed to calculate $R_{\bar{C}}$. We have

$\bar{c}^{2}=\widehat{c}^{\,2}\left(1+R_{\bar{C}}\right)=\widehat{c}^{\,2}+\frac{\mathrm{d}\bar{c}^{\,2}}{\mathrm{d}A}\,\mathrm{d}A=\widehat{c}^{\,2}+\frac{1}{\bar{s}^{2}-\bar{c}^{2}}\,\widehat{A}R_{A}=\widehat{c}^{\,2}+\frac{\widehat{c}^{\,2}\widehat{s}^{\,2}R_{A}}{\bar{s}^{2}-\bar{c}^{2}}.$
(B8)

Therefore,

$R_{\bar{C}}=\frac{\bar{s}^{2}}{\bar{s}^{2}-\bar{c}^{2}}\,R_{A}.$ (B9)

Using the definition of $A$ in eq. (B3) one finally has

$R_{\bar{C}}=\frac{\bar{s}^{2}}{\bar{s}^{2}-\bar{c}^{2}}\left(R_{\alpha}-R_{G}-R_{Z}\right).$
(B10)

From eqs. (B6a), (B6b), and
$\bar{c}^{2}+\bar{s}^{2}=\widehat{c}^{\,2}+\widehat{s}^{\,2}=1$ one has

$\bar{c}^{2}R_{\bar{C}}+\bar{s}^{2}R_{\bar{S}}=0.$ (B11)

Thus,

$R_{\bar{S}}=\frac{\bar{c}^{2}}{\bar{c}^{2}-\bar{s}^{2}}\left(R_{\alpha}-R_{G}-R_{Z}\right).$
(B12)

Therefore, inverting eq. (B6b) and utilizing eq. (B12),

$\widehat{s}^{\,2}=\bar{s}^{2}\left(1-R_{\bar{S}}\right)=\bar{s}^{2}\left[1+\frac{\bar{c}^{2}}{\bar{s}^{2}-\bar{c}^{2}}\left(R_{\alpha}-R_{G}-R_{Z}\right)\right].$
(B13)

We now return to eq. (B7). Employing eqs. (B2) and (B10),

$\displaystyle R_{\bar{\rho}}$ $\displaystyle=$ $\displaystyle
R_{W}+\frac{\bar{c}^{2}}{\bar{s}^{2}-\bar{c}^{2}}\,R_{Z}+\frac{\bar{s}^{2}}{\bar{s}^{2}-\bar{c}^{2}}\left(R_{G}-R_{\alpha}\right)$
(B14c) $\displaystyle=$
$\displaystyle\frac{\Pi_{WW}\left(m_{W}^{2}\right)}{m_{W}^{2}}+\frac{\bar{c}^{2}}{\bar{s}^{2}-\bar{c}^{2}}\,\frac{\Pi_{ZZ}\left(m_{Z}^{2}\right)}{m_{Z}^{2}}$
$\displaystyle+\frac{\bar{s}^{2}}{\bar{s}^{2}-\bar{c}^{2}}\left[-\frac{\Pi_{WW}\left(0\right)}{m_{W}^{2}}+\delta_{Gc}-2\,\frac{\bar{s}}{\bar{c}}\
\frac{\Pi_{ZA}\left(0\right)}{m_{Z}^{2}}-\Pi_{AA}^{\prime}\left(0\right)\right]$
$\displaystyle=$
$\displaystyle\widetilde{\Pi}_{WW}\left(m_{W}^{2}\right)+\frac{\bar{c}^{2}}{\bar{s}^{2}-\bar{c}^{2}}\left[-\frac{\Pi_{WW}\left(0\right)}{m_{W}^{2}}+\widetilde{\Pi}_{ZZ}\left(m_{Z}^{2}\right)+\frac{\Pi_{ZZ}\left(0\right)}{m_{Z}^{2}}\right]$
$\displaystyle+\frac{\bar{s}^{2}}{\bar{s}^{2}-\bar{c}^{2}}\left[\delta_{Gc}-2\,\frac{\bar{s}}{\bar{c}}\
\frac{\Pi_{ZA}\left(0\right)}{m_{Z}^{2}}-\Pi_{AA}^{\prime}\left(0\right)\right].$

Up to now, all the equations in this subsection hold for both the SM and the
NPM. Subtracting eq. (B14) in the SM from the same equation in the NPM we get

$R_{\bar{\rho}}^{\mathrm{NPM}}-R_{\bar{\rho}}^{\mathrm{SM}}=\frac{\bar{c}^{2}}{\bar{c}^{2}-\bar{s}^{2}}\left(\alpha
T-\alpha
K\right)+\frac{\bar{s}^{2}}{\bar{s}^{2}-\bar{c}^{2}}\left[\delta_{Gc}^{\mathrm{NPM}}-\delta_{Gc}^{\mathrm{SM}}-2\,\frac{\bar{s}}{\bar{c}}\
\frac{\Pi_{ZA}^{\mathrm{NPM}}\left(0\right)-\Pi_{ZA}^{\mathrm{SM}}\left(0\right)}{m_{Z}^{2}}\right].$
(B15)

Equation (23) follows from eq. (B15).

###  B.2 Derivation of the corrections to $s^{2}_{0}$ and $s^{2}_{Z}$

One starts from

$\displaystyle s^{2}_{0}$ $\displaystyle=$
$\displaystyle\widehat{s}^{\,2}-\bar{s}\bar{c}\,\Pi^{\prime}_{ZA}\left(0\right),$
(B16a) $\displaystyle s^{2}_{Z}$ $\displaystyle=$
$\displaystyle\widehat{s}^{\,2}-\bar{s}\bar{c}\
\frac{\Pi_{ZA}\left(m_{Z}^{2}\right)}{m_{Z}^{2}}.$ (B16b)

Using eqs. (B13) and (B2),

$\displaystyle s^{2}_{0}$ $\displaystyle=$
$\displaystyle\bar{s}^{2}\left[1+\frac{\bar{c}^{2}}{\bar{s}^{2}-\bar{c}^{2}}\left(R_{\alpha}-R_{G}-R_{Z}\right)-\frac{\bar{c}}{\bar{s}}\
\Pi^{\prime}_{ZA}\left(0\right)\right]$ (B17c) $\displaystyle=$
$\displaystyle\bar{s}^{2}\left\\{1+\frac{\bar{c}^{2}}{\bar{s}^{2}-\bar{c}^{2}}\left[2\,\frac{\bar{s}}{\bar{c}}\
\frac{\Pi_{ZA}\left(0\right)}{m_{Z}^{2}}+\Pi^{\prime}_{AA}\left(0\right)+\frac{\Pi_{WW}\left(0\right)}{m_{W}^{2}}-\delta_{Gc}-\frac{\Pi_{ZZ}\left(m_{Z}^{2}\right)}{m_{Z}^{2}}\right]\right.$
$\displaystyle\left.-\frac{\bar{c}}{\bar{s}}\
\Pi^{\prime}_{ZA}\left(0\right)\vphantom{\frac{\bar{c}^{2}}{\bar{s}^{2}-\bar{c}^{2}}\left[2\,\frac{\bar{s}}{\bar{c}}\
\frac{\Pi_{ZA}\left(0\right)}{m_{Z}^{2}}\right.}\right\\}$ $\displaystyle=$
$\displaystyle\bar{s}^{2}\left\\{1+\frac{\bar{c}^{2}}{\bar{s}^{2}-\bar{c}^{2}}\left[-\delta_{Gc}+\frac{\Pi_{WW}\left(0\right)}{m_{W}^{2}}-\frac{\Pi_{ZZ}\left(0\right)}{m_{Z}^{2}}-\widetilde{\Pi}_{ZZ}\left(m_{Z}^{2}\right)\right.\right.$
$\displaystyle\left.\left.+2\,\frac{\bar{s}}{\bar{c}}\
\frac{\Pi_{ZA}\left(0\right)}{m_{Z}^{2}}+\Pi^{\prime}_{AA}\left(0\right)\right]-\frac{\bar{c}}{\bar{s}}\
\Pi^{\prime}_{ZA}\left(0\right)\vphantom{\frac{\bar{c}^{2}}{\bar{s}^{2}-\bar{c}^{2}}}\right\\},$

and

$\displaystyle s^{2}_{Z}$ $\displaystyle=$ $\displaystyle
s^{2}_{0}-\bar{s}\bar{c}\left[\frac{\Pi_{ZA}\left(m_{Z}^{2}\right)}{m_{Z}^{2}}-\Pi^{\prime}_{ZA}\left(0\right)\right]$
(B18a) $\displaystyle=$ $\displaystyle
s^{2}_{0}-\bar{s}\bar{c}\left[\frac{\Pi_{ZA}\left(0\right)}{m_{Z}^{2}}+\widetilde{\Pi}_{ZA}\left(m_{Z}^{2}\right)-\Pi^{\prime}_{ZA}\left(0\right)\right].$
(B18b)

Equations (B17) and (B18) hold in both SM and NPM. Subtracting eq. (B17) in
the SM from the same equation in the NPM and using eqs. (24) and (25), one has

$\frac{\left(s^{2}_{0}\right)^{\mathrm{NPM}}}{\left(s^{2}_{0}\right)^{\mathrm{SM}}}=1+\frac{\bar{c}^{2}}{\bar{s}^{2}-\bar{c}^{2}}\,\alpha
T+\frac{1}{4\bar{s}^{2}\left(\bar{c}^{2}-\bar{s}^{2}\right)}\,\alpha S.$ (B19)

Subtracting eq. (B18) in the SM from the same equation in the NPM, one obtains

$\frac{\left(s^{2}_{Z}\right)^{\mathrm{NPM}}}{\left(s^{2}_{Z}\right)^{\mathrm{SM}}}=1+\frac{F_{Z}}{4s^{2}},$
(B20)

where

$F_{Z}\equiv\frac{4\bar{s}^{2}\bar{c}^{2}}{\bar{s}^{2}-\bar{c}^{2}}\,\alpha
T+\frac{1}{\bar{c}^{2}-\bar{s}^{2}}\,\alpha S+4\alpha X.$ (B21)

The observables in eqs. (A2)–(A9) only involve $\rho_{\star}$, $s^{2}_{0}$,
and $s_{Z}^{2}$. Using eqs. (35), (B19), and (B20) one obtains

$\displaystyle\frac{g_{L}^{\mathrm{NPM}}}{g_{L}^{\mathrm{SM}}}$
$\displaystyle=$ $\displaystyle 1+\alpha T\
\frac{9-27\bar{s}^{2}+27\bar{s}^{4}-10\bar{s}^{6}}{9-36\bar{s}^{2}+46\bar{s}^{4}-20\bar{s}^{6}}-\alpha
S\
\frac{9-10\bar{s}^{2}}{4\left(9-36\bar{s}^{2}+46\bar{s}^{4}-20\bar{s}^{6}\right)},\hskip
28.45274pt$ (B22a)
$\displaystyle\frac{g_{R}^{\mathrm{NPM}}}{g_{R}^{\mathrm{SM}}}$
$\displaystyle=$ $\displaystyle 1-\alpha T\
\frac{\bar{s}^{2}}{1-2\bar{s}^{2}}+\alpha
S\,\frac{1}{4\bar{s}^{2}\left(1-2\bar{s}^{2}\right)},$ (B22b)
$\displaystyle\frac{g_{V}^{\mathrm{NPM}}}{g_{V}^{\mathrm{SM}}}$
$\displaystyle=$ $\displaystyle 1+\alpha T\
\frac{1-2\bar{s}^{2}+4\bar{s}^{4}}{1-6\bar{s}^{2}+8\bar{s}^{4}}-\alpha
S\,\frac{1}{1-6\bar{s}^{2}+8\bar{s}^{4}},$ (B22c)
$\displaystyle\frac{g_{A}^{\mathrm{NPM}}}{g_{A}^{\mathrm{SM}}}$
$\displaystyle=$ $\displaystyle 1+\alpha T,$ (B22d)
$\displaystyle\frac{Q^{\mathrm{NPM}}\left(55,78\right)}{Q^{\mathrm{SM}}\left(55,78\right)}$
$\displaystyle=$ $\displaystyle 1+\alpha T\
\frac{23-46\bar{s}^{2}-110\bar{s}^{4}}{23+64\bar{s}^{2}-220\bar{s}^{4}}+\alpha
S\
\frac{110\bar{s}^{4}}{4\bar{s}^{2}\left(23+64\bar{s}^{2}-220\bar{s}^{4}\right)},$
(B22e)

and

$\displaystyle\frac{A_{\mathrm{LR}}^{\mathrm{NPM}}}{A_{\mathrm{LR}}^{\mathrm{SM}}}$
$\displaystyle=$ $\displaystyle
1+\left(\frac{1-4\bar{s}^{2}}{1-4\bar{s}^{2}+8\bar{s}^{4}}-\frac{1}{1-4\bar{s}^{2}}\right)F_{Z},$
(B23a)
$\displaystyle\frac{A_{\mathrm{FB}}^{\mathrm{NPM}}\left(b\right)}{A_{\mathrm{FB}}^{\mathrm{SM}}\left(b\right)}$
$\displaystyle=$ $\displaystyle
1+\left(\frac{1-4\bar{s}^{2}}{1-4\bar{s}^{2}+8\bar{s}^{4}}+\frac{3-4\bar{s}^{2}}{9-12\bar{s}^{2}+8\bar{s}^{4}}-\frac{4-8\bar{s}^{2}}{3-16\bar{s}^{2}+16\bar{s}^{4}}\right)F_{Z},$
(B23b)
$\displaystyle\frac{A_{\mathrm{FB}}^{\mathrm{NPM}}\left(c\right)}{A_{\mathrm{FB}}^{\mathrm{SM}}\left(c\right)}$
$\displaystyle=$ $\displaystyle
1+\left(\frac{1-4\bar{s}^{2}}{1-4\bar{s}^{2}+8\bar{s}^{4}}+\frac{6-16\bar{s}^{2}}{9-24\bar{s}^{2}+32\bar{s}^{4}}-\frac{5-16\bar{s}^{2}}{3-20\bar{s}^{2}+32\bar{s}^{4}}\right)F_{Z},$
(B23c)
$\displaystyle\frac{A_{\mathrm{FB}}^{\mathrm{NPM}}\left(\mu\right)}{A_{\mathrm{FB}}^{\mathrm{SM}}\left(\mu\right)}$
$\displaystyle=$ $\displaystyle
1-8\bar{s}^{2}\left(1-4\bar{s}^{2}\right)\left(1-2\bar{s}^{2}\right)F_{Z},$
(B23d)

where $F_{Z}$ is defined in eq. (B21).

###  B.3 Derivation of the corrections to the decay widths

The partial decay width of a $W^{-}$ into an electron and an anti-neutrino is

$\Gamma\left(W^{-}\to
e^{-}\bar{\nu}_{e}\right)=\frac{\widehat{\alpha}\,m_{W}}{12\,\widehat{s}^{\,2}}\left[1+\Pi^{\prime}_{WW}\left(m_{W}^{2}\right)\right].$
(B24)

In eq. (B24), $\widehat{\alpha}$ and $\widehat{s}^{\,2}$ are bare parameters,
while $m_{W}$ is _not_ a bare parameter: it is the physical mass of the
$W^{-}$ that arises from phase-space integration. Moreover,
$\Pi^{\prime}_{WW}\left(m_{W}^{2}\right)$ originates in the field
renormalization. Inserting the expressions for $\widehat{\alpha}$ and
$\widehat{s}^{\,2}$, one obtains

$\displaystyle\frac{\widehat{\alpha}}{\widehat{s}^{2}}\left[1+\Pi^{\prime}_{WW}\left(m_{W}^{2}\right)\right]$
$\displaystyle=$
$\displaystyle\frac{\alpha}{\bar{s}^{2}}\left[1-R_{\alpha}+R_{\bar{S}}+\Pi^{\prime}_{WW}\left(m_{W}^{2}\right)\right]$
(B25a) $\displaystyle=$
$\displaystyle\frac{\alpha}{\bar{s}^{2}}\left[1-R_{\alpha}-\frac{\bar{c}^{2}}{\bar{s}^{2}-\bar{c}^{2}}\left(R_{\alpha}-R_{G}-R_{Z}\right)+\Pi^{\prime}_{WW}\left(m_{W}^{2}\right)\right]\hskip
19.91692pt$ (B25d) $\displaystyle=$
$\displaystyle\frac{\alpha}{\bar{s}^{2}}\left\\{1+\frac{\bar{s}^{2}}{\bar{c}^{2}-\bar{s}^{2}}\left[2\,\frac{\bar{s}}{\bar{c}}\
\frac{\Pi_{ZA}\left(0\right)}{m_{Z}^{2}}+\Pi^{\prime}_{AA}\left(0\right)\right]\right.$
$\displaystyle+\frac{\bar{c}^{2}}{\bar{s}^{2}-\bar{c}^{2}}\left[-\frac{\Pi_{WW}\left(0\right)}{m_{W}^{2}}+\delta_{Gc}+\frac{\Pi_{ZZ}\left(m_{Z}^{2}\right)}{m_{Z}^{2}}\right]$
$\displaystyle\left.+\Pi^{\prime}_{WW}\left(m_{W}^{2}\right)\right\\}$
$\displaystyle=$
$\displaystyle\frac{\alpha}{\bar{s}^{2}}\left\\{1+\frac{\bar{s}^{2}}{\bar{c}^{2}-\bar{s}^{2}}\left[2\,\frac{\bar{s}}{\bar{c}}\
\frac{\Pi_{ZA}\left(0\right)}{m_{Z}^{2}}+\Pi^{\prime}_{AA}\left(0\right)\right]\right.$
$\displaystyle+\frac{\bar{c}^{2}}{\bar{s}^{2}-\bar{c}^{2}}\left[-\frac{\Pi_{WW}\left(0\right)}{m_{W}^{2}}+\frac{\Pi_{ZZ}\left(0\right)}{m_{Z}^{2}}+\delta_{Gc}+\widetilde{\Pi}_{ZZ}\left(m_{Z}^{2}\right)\right]$
$\displaystyle\left.+\Pi^{\prime}_{WW}\left(m_{W}^{2}\right)\vphantom{\frac{\bar{c}^{2}}{\bar{s}^{2}-\bar{c}^{2}}}\right\\}.$

Using eqs. (24) and (25) we obtain

$\displaystyle\frac{\Gamma^{\mathrm{NPM}}\left(W^{-}\to
e^{-}\bar{\nu}_{e}\right)}{\Gamma^{\mathrm{SM}}\left(W^{-}\to
e^{-}\bar{\nu}_{e}\right)}$ $\displaystyle=$ $\displaystyle
1+\frac{\bar{s}^{2}}{\bar{c}^{2}-\bar{s}^{2}}\
\Pi^{\prime}_{AA}\left(0\right)+\frac{\bar{c}^{2}}{\bar{s}^{2}-\bar{c}^{2}}\left[-\alpha
T+\widetilde{\Pi}_{ZZ}\left(m_{Z}^{2}\right)\right]$ (B26a)
$\displaystyle+\alpha W+\widetilde{\Pi}_{WW}\left(m_{W}^{2}\right)$
$\displaystyle=$ $\displaystyle
1+\frac{\bar{c}^{2}}{\bar{c}^{2}-\bar{s}^{2}}\,\alpha
T+\frac{1}{2\left(\bar{s}^{2}-\bar{c}^{2}\right)}\,\alpha
S+\frac{1}{4\bar{s}^{2}}\,\alpha U+\alpha W.$ (B26b)

Analogously, the partial decay widths of a $Z$ into pairs of down-type quarks,
up-type quarks, charged leptons, and neutrinos are

$\displaystyle\Gamma\left(Z\to d\bar{d}\right)$ $\displaystyle=$
$\displaystyle\frac{\widehat{\alpha}\,m_{Z}}{2\,\widehat{c}^{\,2}\,\widehat{s}^{\,2}}\left[\left(-\frac{1}{2}+\frac{1}{3}\,s^{2}_{Z}\right)^{2}+\left(\frac{1}{3}\,s^{2}_{Z}\right)^{2}\right]\left[1+\Pi^{\prime}_{ZZ}\left(m_{Z}^{2}\right)\right]$
(B27a) $\displaystyle=$
$\displaystyle\frac{\widehat{\alpha}\,m_{Z}}{8\,\widehat{c}^{\,2}\,\widehat{s}^{\,2}}\left(1-\frac{4}{3}\,s^{2}_{Z}+\frac{8}{9}\,s^{4}_{Z}\right)\left[1+\Pi^{\prime}_{ZZ}\left(m_{Z}^{2}\right)\right],$
(B27b) $\displaystyle\Gamma\left(Z\to u\bar{u}\right)$ $\displaystyle=$
$\displaystyle\frac{\widehat{\alpha}\,m_{Z}}{2\,\widehat{c}^{\,2}\,\widehat{s}^{\,2}}\left[\left(\frac{1}{2}-\frac{2}{3}\,s^{2}_{Z}\right)^{2}+\left(\frac{2}{3}\,s^{2}_{Z}\right)^{2}\right]\left[1+\Pi^{\prime}_{ZZ}\left(m_{Z}^{2}\right)\right]$
(B27c) $\displaystyle=$
$\displaystyle\frac{\widehat{\alpha}\,m_{Z}}{8\,\widehat{c}^{\,2}\,\widehat{s}^{\,2}}\left(1-\frac{8}{3}\,s^{2}_{Z}+\frac{32}{9}\,s^{4}_{Z}\right)\left[1+\Pi^{\prime}_{ZZ}\left(m_{Z}^{2}\right)\right],$
(B27d) $\displaystyle\Gamma\left(Z\to\ell\bar{\ell}\right)$ $\displaystyle=$
$\displaystyle\frac{\widehat{\alpha}\,m_{Z}}{6\,\widehat{c}^{\,2}\,\widehat{s}^{\,2}}\left[\left(\frac{1}{2}-s^{2}_{Z}\right)^{2}+s^{4}_{Z}\right]\left[1+\Pi^{\prime}_{ZZ}\left(m_{Z}^{2}\right)\right]$
(B27e) $\displaystyle=$
$\displaystyle\frac{\widehat{\alpha}\,m_{Z}}{24\,\widehat{c}^{\,2}\,\widehat{s}^{\,2}}\left(1-4s^{2}_{Z}+8s^{4}_{Z}\right)\left[1+\Pi^{\prime}_{ZZ}\left(m_{Z}^{2}\right)\right],$
(B27f) $\displaystyle\Gamma\left(Z\to\nu\bar{\nu}\right)$ $\displaystyle=$
$\displaystyle\frac{\widehat{\alpha}\,m_{Z}}{24\,\widehat{c}^{\,2}\,\widehat{s}^{\,2}}\left[1+\Pi^{\prime}_{ZZ}\left(m_{Z}^{2}\right)\right],$
(B27g)

respectively. The total decay width of the $Z$ boson is

$\Gamma_{\mathrm{total}}\left(Z\right)=\frac{\widehat{\alpha}\,m_{Z}}{72\,\widehat{c}^{\,2}\,\widehat{s}^{\,2}}\left(63-120s^{2}_{Z}+160s^{4}_{Z}\right)\left[1+\Pi^{\prime}_{ZZ}\left(m_{Z}^{2}\right)\right].$
(B28)

We use

$\displaystyle\frac{\widehat{\alpha}}{\widehat{c}^{\,2}\,\widehat{s}^{\,2}}\left[1+\Pi^{\prime}_{ZZ}\left(m_{Z}^{2}\right)\right]$
$\displaystyle=$
$\displaystyle\frac{\alpha\left(1-R_{\alpha}\right)}{\bar{c}^{2}\left(1-R_{\bar{C}}\right)\bar{s}^{2}\left(1-R_{\bar{S}}\right)}\left[1+\Pi^{\prime}_{ZZ}\left(m_{Z}^{2}\right)\right]$
(B29a) $\displaystyle=$
$\displaystyle\frac{\alpha}{\bar{c}^{2}\bar{s}^{2}}\left[1-R_{\alpha}+R_{\bar{C}}+R_{\bar{S}}+\Pi^{\prime}_{ZZ}\left(m_{Z}^{2}\right)\right]$
(B29b) $\displaystyle=$
$\displaystyle\frac{\alpha}{\bar{c}^{2}\bar{s}^{2}}\left[1-R_{G}-R_{Z}+\Pi^{\prime}_{ZZ}\left(m_{Z}^{2}\right)\right]$
(B29c) $\displaystyle=$
$\displaystyle\frac{\alpha}{\bar{c}^{2}\bar{s}^{2}}\left[1+\frac{\Pi_{WW}\left(0\right)}{m_{W}^{2}}-\delta_{Gc}-\frac{\Pi_{ZZ}\left(m_{Z}^{2}\right)}{m_{Z}^{2}}+\Pi^{\prime}_{ZZ}\left(m_{Z}^{2}\right)\right]\hskip
19.91692pt$ (B29e) $\displaystyle=$
$\displaystyle\frac{\alpha}{\bar{c}^{2}\bar{s}^{2}}\left[1+\frac{\Pi_{WW}\left(0\right)}{m_{W}^{2}}-\frac{\Pi_{ZZ}\left(0\right)}{m_{Z}^{2}}-\delta_{Gc}\right.$
$\displaystyle\left.-\widetilde{\Pi}_{ZZ}\left(m_{Z}^{2}\right)+\Pi^{\prime}_{ZZ}\left(m_{Z}^{2}\right)\right].$

Neglecting $\delta^{\mathrm{NPM}}_{Gc}-\delta^{\mathrm{SM}}_{Gc}$, one then
gets

$\displaystyle\frac{\Gamma^{\mathrm{NPM}}\left(Z\to
d\bar{d}\right)}{\Gamma^{\mathrm{SM}}\left(Z\to d\bar{d}\right)}$
$\displaystyle=$ $\displaystyle 1+\alpha T+\alpha
V-\frac{3-4\bar{s}^{2}}{9-12\bar{s}^{2}+8\bar{s}^{4}}\,F_{Z},$ (B30a)
$\displaystyle\frac{\Gamma^{\mathrm{NPM}}\left(Z\to
u\bar{u}\right)}{\Gamma^{\mathrm{SM}}\left(Z\to u\bar{u}\right)}$
$\displaystyle=$ $\displaystyle 1+\alpha T+\alpha
V-\frac{6-16\bar{s}^{2}}{9-24\bar{s}^{2}+32\bar{s}^{4}}\,F_{Z},$ (B30b)
$\displaystyle\frac{\Gamma^{\mathrm{NPM}}\left(Z\to\ell\bar{\ell}\right)}{\Gamma^{\mathrm{SM}}\left(Z\to\ell\bar{\ell}\right)}$
$\displaystyle=$ $\displaystyle 1+\alpha T+\alpha
V-\frac{1-4\bar{s}^{2}}{1-4\bar{s}^{2}+8\bar{s}^{4}}\,F_{Z},$ (B30c)
$\displaystyle\frac{\Gamma^{\mathrm{NPM}}\left(Z\to\nu\bar{\nu}\right)}{\Gamma^{\mathrm{SM}}\left(Z\to\nu\bar{\nu}\right)}$
$\displaystyle=$ $\displaystyle 1+\alpha T+\alpha V,$ (B30d)
$\displaystyle\frac{\Gamma_{\mathrm{total}}^{\mathrm{NPM}}\left(Z\right)}{\Gamma_{\mathrm{total}}^{\mathrm{SM}}\left(Z\right)}$
$\displaystyle=$ $\displaystyle 1+\alpha T+\alpha
V-\frac{30-80\bar{s}^{2}}{63-120\bar{s}^{2}+160\bar{s}^{4}}\,F_{Z},$ (B30e)

where $F_{Z}$ is defined in eq. (B21).

##  Appendix C Derivation of the oblique corrections for a BBM vs. a BM

We now consider the case of a Base Model (BM) and a model Beyond the Base
Model (BBM), i.e. the BM plus extra fermions and/or extra scalars, assuming
that neither the BM nor the BBM have fixed $\rho$ at Lagrangian level. The
input observables of renormalization are chosen to be $\alpha$,
$G_{F(\mathrm{charged})}$, $m_{Z}$, and $m_{W}$. The Weinberg angle is defined
through eq. (18). Then,

$\rho=\frac{m_{W}^{2}}{\left(1-s^{2}\right)m_{Z}^{2}}=\frac{\sqrt{2}\,G_{F(\mathrm{charged})}\,m_{W}^{4}}{\left(\sqrt{2}\,G_{F(\mathrm{charged})}\,m_{W}^{2}-\pi\alpha\right)m_{Z}^{2}}$
(C1)

is fully given in terms of the input observables and therefore

$\rho^{\mathrm{BM}}=\rho^{\mathrm{BBM}},$ (C2)

since the input is the same for both models.

###  C.1 Derivation of the corrections to $\rho$

We now relate $\rho$, defined by the input, to the bare $\widehat{\rho}$. We
utilize once again eqs. (B1) and (B2).

From

$s^{2}=\frac{\pi\alpha}{\sqrt{2}\,G_{F(\mathrm{charged})}\,m_{W}^{2}}\quad\mbox{and}\quad\widehat{s}^{\,2}=\frac{\pi\widehat{\alpha}}{\sqrt{2}\,\widehat{G}_{F(\mathrm{charged})}\,\widehat{m}_{W}^{2}}$
(C3)

it follows that

$\displaystyle s^{2}$ $\displaystyle=$
$\displaystyle\frac{\pi\,\widehat{\alpha}\left(1+R_{\alpha}\right)}{\sqrt{2}\,\widehat{G}_{F(\mathrm{charged})}\left(1+R_{G}\right)\widehat{m}_{W}^{2}\left(1+R_{W}\right)}$
(C4a) $\displaystyle=$
$\displaystyle\widehat{s}^{\,2}\left(1+R_{\alpha}-R_{G}-R_{W}\right)$ (C4b)
$\displaystyle=$ $\displaystyle\widehat{s}^{\,2}\left[1+2\,\frac{s}{c}\
\frac{\Pi_{ZA}\left(0\right)}{m_{Z}^{2}}+\Pi^{\prime}_{AA}\left(0\right)-\widetilde{\Pi}_{WW}\left(m_{W}^{2}\right)-\delta_{Gc}\right].$
(C4c)

We write, just as in eqs. (B6),

$\displaystyle c^{2}$ $\displaystyle=$
$\displaystyle\widehat{c}^{\,2}\left(1+R_{C}\right),$ (C5a) $\displaystyle
s^{2}$ $\displaystyle=$ $\displaystyle\widehat{s}^{\,2}\left(1+R_{S}\right).$
(C5b)

The quantities $R_{C}$ and $R_{S}$ obey, just as in eq. (B11),

$c^{2}R_{C}+s^{2}R_{S}=0.$ (C6)

Comparing eqs. (C4) and (C5b) we see that

$R_{S}=2\,\frac{s}{c}\
\frac{\Pi_{ZA}\left(0\right)}{m_{Z}^{2}}+\Pi^{\prime}_{AA}\left(0\right)-\widetilde{\Pi}_{WW}\left(m_{W}^{2}\right)-\delta_{Gc}.$
(C7)

Therefore, from eq. (C6),

$R_{C}=-\frac{s^{2}}{c^{2}}\left[2\,\frac{s}{c}\
\frac{\Pi_{ZA}\left(0\right)}{m_{Z}^{2}}+\Pi^{\prime}_{AA}\left(0\right)-\widetilde{\Pi}_{WW}\left(m_{W}^{2}\right)-\delta_{Gc}\right].$
(C8)

Then,

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{m_{W}^{2}}{c^{2}m_{Z}^{2}}$ (C9a) $\displaystyle=$
$\displaystyle\frac{\widehat{m}_{W}^{2}\left(1+R_{W}\right)}{\widehat{c}^{\,2}\left(1+R_{C}\right)\widehat{m}_{Z}^{2}\left(1+R_{Z}\right)}$
(C9b) $\displaystyle=$
$\displaystyle\widehat{\rho}\left(1+R_{W}-R_{C}-R_{Z}\right)$ (C9f)
$\displaystyle=$
$\displaystyle\widehat{\rho}\left\\{1+\frac{\Pi_{WW}\left(m_{W}^{2}\right)}{m_{W}^{2}}+\frac{s^{2}}{c^{2}}\left[2\,\frac{s}{c}\
\frac{\Pi_{ZA}\left(0\right)}{m_{Z}^{2}}+\Pi^{\prime}_{AA}\left(0\right)-\widetilde{\Pi}_{WW}\left(m_{W}^{2}\right)-\delta_{Gc}\right]\right.$
$\displaystyle\left.-\frac{\Pi_{ZZ}\left(m_{Z}^{2}\right)}{m_{Z}^{2}}\right\\}$
$\displaystyle=$
$\displaystyle\widehat{\rho}\left\\{1+\frac{\Pi_{WW}\left(0\right)}{m_{W}^{2}}+\widetilde{\Pi}_{WW}\left(m_{W}^{2}\right)+\frac{s^{2}}{c^{2}}\left[2\,\frac{s}{c}\
\frac{\Pi_{ZA}\left(0\right)}{m_{Z}^{2}}+\Pi^{\prime}_{AA}\left(0\right)-\widetilde{\Pi}_{WW}\left(m_{W}^{2}\right)-\delta_{Gc}\right]\right.$
$\displaystyle\left.-\frac{\Pi_{ZZ}\left(0\right)}{m_{Z}^{2}}-\widetilde{\Pi}_{ZZ}\left(m_{Z}^{2}\right)\right\\}$
$\displaystyle=$
$\displaystyle\widehat{\rho}\left\\{1+\frac{\Pi_{WW}\left(0\right)}{m_{W}^{2}}-\frac{\Pi_{ZZ}\left(0\right)}{m_{Z}^{2}}+\frac{c^{2}-s^{2}}{c^{2}}\
\widetilde{\Pi}_{WW}\left(m_{W}^{2}\right)-\widetilde{\Pi}_{ZZ}\left(m_{Z}^{2}\right)+\frac{s^{2}}{c^{2}}\
\Pi^{\prime}_{AA}\left(0\right)\right.$
$\displaystyle\left.+\frac{s^{2}}{c^{2}}\left[2\,\frac{s}{c}\
\frac{\Pi_{ZA}\left(0\right)}{m_{Z}^{2}}-\delta_{Gc}\right]\right\\}.$

Using the definitions of the model parameters in eqs. (11) we get at eq. (27).

###  C.2 Derivation of the corrections to $s^{2}_{0}$ and $s^{2}_{Z}$

We make $\bar{s}\to s$ and $\bar{c}\to c$ in eq. (B16a) to obtain

$\displaystyle s^{2}_{0}$ $\displaystyle=$
$\displaystyle\widehat{s}^{\,2}-sc\,\Pi^{\prime}_{ZA}\left(0\right)$ (C10a)
$\displaystyle=$ $\displaystyle
s^{2}\left[1-R_{S}-\frac{c}{s}\,\Pi^{\prime}_{ZA}\left(0\right)\right]$ (C10b)
$\displaystyle=$ $\displaystyle s^{2}\left[1-2\,\frac{s}{c}\
\frac{\Pi_{ZA}\left(0\right)}{m_{Z}^{2}}-\Pi^{\prime}_{AA}\left(0\right)+\widetilde{\Pi}_{WW}\left(m_{W}^{2}\right)+\delta_{Gc}-\frac{c}{s}\
\Pi^{\prime}_{ZA}\left(0\right)\right],$ (C10c)

where we have employed eq. (C7). Utilizing eq. (24) and neglecting
$\delta_{Gc}^{\mathrm{BBM}}-\delta_{Gc}^{\mathrm{BM}}$,

$\frac{\left(s^{2}_{0}\right)^{\mathrm{BBM}}}{\left(s^{2}_{0}\right)^{\mathrm{BM}}}=1+\frac{1}{4s^{2}}\,\alpha
S+\frac{1}{4s^{2}}\,\alpha U.$ (C11)

Now returning to eq. (B18) (with $\bar{s}\to s$ and $\bar{c}\to c$) and
employing eq. (24), we get

$\displaystyle\frac{\left(s^{2}_{Z}\right)^{\mathrm{BBM}}}{\left(s^{2}_{Z}\right)^{\mathrm{BM}}}$
$\displaystyle=$ $\displaystyle 1+\frac{F_{Y}}{4s^{2}},$ (C12a) $\displaystyle
F_{Y}$ $\displaystyle\equiv$ $\displaystyle\alpha S+\alpha U+4\alpha X.$
(C12b)

We compare eqs. (26) and (C2), viz.

$\frac{\bar{\rho}^{\,\mathrm{NPM}}}{\bar{\rho}^{\,\mathrm{SM}}}=1+\frac{\bar{c}^{2}}{\bar{c}^{2}-\bar{s}^{2}}\left(\alpha
T-\alpha K\right),\qquad\frac{\rho^{\mathrm{BBM}}}{\rho^{\mathrm{BM}}}=1.$
(C13)

We also compare eqs. (B19)–(B21) to eqs. (C11) and (C12). We see that one
passes from the comparison NPM vs. SM to the comparison BBM vs. BM by making

$\bar{c}\to c,\quad\bar{s}\to s,\quad\alpha T\to\frac{1}{2c^{2}}\,\alpha
S+\frac{s^{2}-c^{2}}{4s^{2}c^{2}}\,\alpha U.$ (C14)

One should apply the transformation in eq. (C14) to eqs. (B22) and (B23) in
order to obtain the equations for the comparison of the observables between
the BBM and the BM.

###  C.3 Decay widths

Instead of eq. (B25) we now have

$\displaystyle\frac{\widehat{\alpha}}{\widehat{s}^{\,2}}\left[1+\Pi^{\prime}_{WW}\left(m_{W}^{2}\right)\right]$
$\displaystyle=$
$\displaystyle\frac{\alpha}{s^{2}}\left[1-R_{\alpha}+R_{S}+\Pi^{\prime}_{WW}\left(m_{W}^{2}\right)\right]$
(C15a) $\displaystyle=$
$\displaystyle\frac{\alpha}{s^{2}}\left[1-\widetilde{\Pi}_{WW}\left(m_{W}^{2}\right)-\delta_{Gc}+\Pi^{\prime}_{WW}\left(m_{W}^{2}\right)\right].$
(C15b)

Therefore, neglecting $\delta_{Gc}^{\mathrm{BBM}}-\delta_{Gc}^{\mathrm{BM}}$,

$\frac{\Gamma^{\mathrm{BBM}}\left(W^{-}\to
e^{-}\bar{\nu}_{e}\right)}{\Gamma^{\mathrm{BM}}\left(W^{-}\to
e^{-}\bar{\nu}_{e}\right)}=1+\alpha W.$ (C16)

Similarly, instead of eq. (B29) we now have

$\displaystyle\frac{\widehat{\alpha}}{\widehat{c}^{\,2}\,\widehat{s}^{\,2}}\left[1+\Pi^{\prime}_{ZZ}\left(m_{Z}^{2}\right)\right]$
$\displaystyle=$
$\displaystyle\frac{\alpha\left(1-R_{\alpha}\right)}{c^{2}\left(1-R_{C}\right)s^{2}\left(1-R_{S}\right)}\left[1+\Pi^{\prime}_{ZZ}\left(m_{Z}^{2}\right)\right]$
(C17a) $\displaystyle=$
$\displaystyle\frac{\alpha}{c^{2}s^{2}}\left[1-R_{\alpha}+R_{C}+R_{S}+\Pi^{\prime}_{ZZ}\left(m_{Z}^{2}\right)\right]$
(C17b) $\displaystyle=$
$\displaystyle\frac{\alpha}{c^{2}s^{2}}\left[1-R_{\alpha}+\left(1-\frac{s^{2}}{c^{2}}\right)R_{S}+\Pi^{\prime}_{ZZ}\left(m_{Z}^{2}\right)\right]$
(C17e) $\displaystyle=$
$\displaystyle\frac{\alpha}{c^{2}s^{2}}\left\\{1-\frac{s^{2}}{c^{2}}\left[2\,\frac{s}{c}\
\frac{\Pi_{ZA}\left(0\right)}{m_{Z}^{2}}+\Pi_{AA}^{\prime}\left(0\right)\right]\right.$
$\displaystyle\left.+\frac{s^{2}-c^{2}}{c^{2}}\left[\widetilde{\Pi}_{WW}\left(m_{W}^{2}\right)+\delta_{Gc}\right]+\Pi^{\prime}_{ZZ}\left(m_{Z}^{2}\right)\right\\}$
$\displaystyle=$
$\displaystyle\frac{\alpha}{c^{2}s^{2}}\left\\{1-2\,\frac{s^{3}}{c^{3}}\
\frac{\Pi_{ZA}\left(0\right)}{m_{Z}^{2}}+\frac{s^{2}-c^{2}}{c^{2}}\,\delta_{Gc}\right.$
$\displaystyle-\frac{s^{2}}{c^{2}}\
\Pi_{AA}^{\prime}\left(0\right)+\frac{s^{2}-c^{2}}{c^{2}}\
\widetilde{\Pi}_{WW}\left(m_{W}^{2}\right)+\widetilde{\Pi}_{ZZ}\left(m_{Z}^{2}\right)$
$\displaystyle\left.+\left[\Pi^{\prime}_{ZZ}\left(m_{Z}^{2}\right)-\widetilde{\Pi}_{ZZ}\left(m_{Z}^{2}\right)\right]\right\\}.$

Since

$\displaystyle-\frac{s^{2}}{c^{2}}\
\Pi^{\prime\,\mathrm{M}}_{AA}\left(0\right)+\frac{s^{2}-c^{2}}{c^{2}}\
\widetilde{\Pi}^{\mathrm{M}}_{WW}\left(m_{W}^{2}\right)+\widetilde{\Pi}^{\mathrm{M}}_{ZZ}\left(m_{Z}^{2}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{2c^{2}}\,\alpha
S^{\mathrm{M}}+\frac{s^{2}-c^{2}}{4c^{2}s^{2}}\,\alpha U^{\mathrm{M}},\hskip
19.91692pt$ (C18a)
$\displaystyle\Pi^{\prime\,\mathrm{M}}_{ZZ}\left(m_{Z}^{2}\right)-\widetilde{\Pi}_{ZZ}^{\mathrm{M}}\left(m_{Z}^{2}\right)$
$\displaystyle=$ $\displaystyle\alpha V^{\mathrm{M}},$ (C18b)

we conclude that

$\frac{\Gamma^{\mathrm{BBM}}\left(Z\to\nu\bar{\nu}\right)}{\Gamma^{\mathrm{BM}}\left(Z\to\nu\bar{\nu}\right)}=1+\frac{1}{2c^{2}}\,\alpha
S+\frac{s^{2}-c^{2}}{4s^{2}c^{2}}\,\alpha U+\alpha V.$ (C19)

Analogously,

$\displaystyle\frac{\Gamma^{\mathrm{BBM}}\left(Z\to
d\bar{d}\right)}{\Gamma^{\mathrm{BM}}\left(Z\to d\bar{d}\right)}$
$\displaystyle=$ $\displaystyle 1+\frac{1}{2c^{2}}\,\alpha
S+\frac{s^{2}-c^{2}}{4s^{2}c^{2}}\,\alpha U+\alpha
V-\frac{3-4s^{2}}{9-12s^{2}+8s^{4}}\,F_{Y},$ (C20a)
$\displaystyle\frac{\Gamma^{\mathrm{BBM}}\left(Z\to
u\bar{u}\right)}{\Gamma^{\mathrm{BM}}\left(Z\to u\bar{u}\right)}$
$\displaystyle=$ $\displaystyle 1+\frac{1}{2c^{2}}\,\alpha
S+\frac{s^{2}-c^{2}}{4s^{2}c^{2}}\,\alpha U+\alpha
V-\frac{6-16s^{2}}{9-24s^{2}+32s^{4}}\,F_{Y},$ (C20b)
$\displaystyle\frac{\Gamma^{\mathrm{BBM}}\left(Z\to\ell\bar{\ell}\right)}{\Gamma^{\mathrm{BM}}\left(Z\to\ell\bar{\ell}\right)}$
$\displaystyle=$ $\displaystyle 1+\frac{1}{2c^{2}}\,\alpha
S+\frac{s^{2}-c^{2}}{4s^{2}c^{2}}\,\alpha U+\alpha
V-\frac{1-4s^{2}}{1-4s^{2}+8s^{4}}\,F_{Y},$ (C20c)
$\displaystyle\frac{\Gamma_{\mathrm{total}}^{\mathrm{BBM}}\left(Z\right)}{\Gamma_{\mathrm{total}}^{\mathrm{BM}}\left(Z\right)}$
$\displaystyle=$ $\displaystyle 1+\frac{1}{2c^{2}}\,\alpha
S+\frac{s^{2}-c^{2}}{4s^{2}c^{2}}\,\alpha U+\alpha
V-\frac{30-80s^{2}}{63-120s^{2}+160s^{4}}\,F_{Y},\hskip 19.91692pt$ (C20d)

where $F_{Y}$ is defined in eq. (C12b). Comparing eq. (B26b) to eq. (C16), and
eqs. (B30) to eqs. (C19) and (C20), we see once again the transformation (C14)
operating.

##  Appendix D Passarino–Veltman functions

Our notation for the Passarino–Veltman functions is

$\displaystyle\mu^{\epsilon}\int\frac{\mathrm{d}^{4-\epsilon}\,k}{\left(2\pi\right)^{4-\epsilon}}\
\frac{1}{k^{2}-I}\ \frac{1}{\left(k+p\right)^{2}-J}$ $\displaystyle=$
$\displaystyle\frac{i}{16\pi^{2}}\ B_{0}\left(P,I,J\right),$ (D1a)
$\displaystyle\mu^{\epsilon}\int\frac{\mathrm{d}^{4-\epsilon}\,k}{\left(2\pi\right)^{4-\epsilon}}\
k^{\alpha}k^{\beta}\,\frac{1}{k^{2}-I}\ \frac{1}{\left(k+p\right)^{2}-J}$
$\displaystyle=$
$\displaystyle\frac{i}{16\pi^{2}}\left[g^{\alpha\beta}\,B_{00}\left(P,I,J\right)\right.$
(D1b)
$\displaystyle\left.+p^{\alpha}p^{\beta}\,B_{11}\left(P,I,J\right)\right],$

where $P\equiv p^{2}$ and in the limit $\epsilon\to 0^{+}$. The function
$B_{11}\left(P,I,J\right)$ in eq. (D1b) is irrelevant for our computations.
The function $B_{0}\left(P,I,J\right)$ of eq. (D1a) only appears in our
computations through the combination

$F\left(P,I,J\right)\equiv-
B_{0}\left(P,I,J\right)+\frac{B_{00}\left(P,I,J\right)}{I}.$ (D2)

Note that $B_{0}\left(P,I,J\right)$, $B_{00}\left(P,I,J\right)$, and
$F\left(P,I,J\right)$ are symmetric under $I\leftrightarrow J$.

We define

$\displaystyle B_{00}^{\prime}\left(P,I,J\right)$ $\displaystyle\equiv$
$\displaystyle\frac{\partial B_{00}\left(P,I,J\right)}{\partial P},$ (D3a)
$\displaystyle\bar{B}_{00}\left(P,I,J\right)$ $\displaystyle\equiv$
$\displaystyle\frac{B_{00}\left(P,I,J\right)-B_{00}\left(0,I,J\right)}{P},$
(D3b) $\displaystyle F^{\prime}\left(P,I,J\right)$ $\displaystyle\equiv$
$\displaystyle\frac{\partial F\left(P,I,J\right)}{\partial P},$ (D3c)
$\displaystyle\bar{F}\left(P,I,J\right)$ $\displaystyle\equiv$
$\displaystyle\frac{F\left(P,I,J\right)-F\left(0,I,J\right)}{P}.$ (D3d)

We regularize the Feynman integrals through dimensional regularization. The
divergent quantity

$\mathrm{div}\equiv\frac{2}{\epsilon}-\gamma+\ln{\left(4\pi\mu^{2}\right)}$
(D4)

($\gamma$ is the Euler–Mascheroni constant) appears in the Feynman integrals
in the limit $\epsilon\to 0^{+}$ as

$\displaystyle B_{00}\left(P,I,J\right)$ $\displaystyle=$
$\displaystyle\left(\frac{I+J}{4}-\frac{P}{12}\right)\mathrm{div}+\mathrm{finite\
terms},$ (D5a) $\displaystyle F\left(P,I,J\right)$ $\displaystyle=$
$\displaystyle\left(-\frac{3}{4}+\frac{J}{4I}-\frac{P}{12I}\right)\mathrm{div}+\mathrm{finite\
terms}.$ (D5b)

Therefore,

$\displaystyle B_{00}^{\prime}\left(P,I,J\right)$ $\displaystyle=$
$\displaystyle-\frac{\mathrm{div}}{12}+\mathrm{finite\ terms},$ (D6a)
$\displaystyle\bar{B}_{00}\left(P,I,J\right)$ $\displaystyle=$
$\displaystyle-\frac{\mathrm{div}}{12}+\mathrm{finite\ terms},$ (D6b)
$\displaystyle F^{\prime}\left(P,I,J\right)$ $\displaystyle=$
$\displaystyle-\frac{\mathrm{div}}{12I}+\mathrm{finite\ terms},$ (D6c)
$\displaystyle\bar{F}\left(P,I,J\right)$ $\displaystyle=$
$\displaystyle-\frac{\mathrm{div}}{12I}+\mathrm{finite\ terms}.$ (D6d)

Thus, all four functions $B_{00}^{\prime}\left(P,I,J\right)$,
$\bar{B}_{00}\left(P,I,J\right)$, $I\,F^{\prime}\left(P,I,J\right)$, and
$I\,\bar{F}\left(P,I,J\right)$ have identical divergent terms, and those terms
depend neither on $P$ nor on $I$ nor on $J$.

##  Appendix E UV finiteness of the oblique parameters

The model parameters defined in eqs. (11), i.e. the oblique parameters without
the subtraction between two models, are UV-divergent in a general gauge. In
the comparison between the SM and a NPM, $SU(2)\times U(1)$ gauge symmetry and
the condition $\widehat{\rho}=1$ ensure that the subtracted oblique parameters
are finite and gauge-invariant. In the case of BBM vs. BM, we have used
$m_{W}$ as an additional input observable, effectively eschewing the oblique
parameter $T$. While many authors have pointed out that $T$ is divergent when
$\widehat{\rho}$ is free, it is interesting to check the structure of its
divergence.

We find it insightful to check the UV divergences of the model (i.e.  _non-
subtracted_) parameters in the specific $R_{\xi}$ gauge that has

$\xi_{W}=\xi_{Z}=\xi_{A}\equiv\xi.$ (E1)

In this case the theory may be embedded into the background field formalism
(see for instance refs. [24], [25], and [26]), which preserves the global
gauge symmetry. Ultraviolet divergences may then be more easily tracked, since
UV-divergent operators must preserve the global gauge symmetry. As a result,
all the UV divergences are solely absorbed by the field, background field, and
parameter redefinitions. The renormalization of the VEVs is interpreted as the
renormalization of the Higgs background field [27].

When checking the divergences of the oblique parameters, it makes no
difference whether one uses the function
$\widetilde{\Pi}_{VV^{\prime}}\left(p^{2}\right)$ or the function
$\Pi^{\prime}_{VV^{\prime}}\left(p^{2}\right)$—defined in eqs. (10)—in the
expressions for the oblique parameters, since both collect the same divergent
terms, linear in $p^{2}$, of the function
${\Pi}_{VV^{\prime}}\left(p^{2}\right)$. Having this in mind, it is clear that
the model parameters $V^{\mathrm{M}}$, $W^{\mathrm{M}}$, and $X^{\mathrm{M}}$
are finite. Thus, one must check only the UV structure of $S^{\mathrm{M}}$,
$T^{\mathrm{M}}$, and $U^{\mathrm{M}}$ in the restricted gauge of eq. (E1).

For checking the UV divergences it is more convenient to write the oblique
parameters in terms of the gauge eigenstates, i.e. in the basis of the gauge
fields $B_{\mu}$ and $W_{\mu}^{a}$ ($a=1,2,3$). This is also the form utilized
in ref. [7]. We note, however, that the oblique parameters in ref. [7] differ
from the definitions in eqs. (11) not only by the replacement
$\widetilde{\Pi}\to\Pi^{\prime}$, but also by a term
$\propto\Pi^{\mathrm{M}}_{ZA}(0)$. Because of eq. (24), that term cancels out
in the subtracted oblique parameters, and so does the divergence coming from
this term. Having said that, to check the divergences we will use the form as
in ref. [7],

$\displaystyle-\frac{1}{4sc}\left(\alpha S^{\mathrm{M}}\right)$
$\displaystyle=$ $\displaystyle\Pi_{3B}^{\prime\,\mathrm{M}}\left(0\right),$
(E2a) $\displaystyle m_{W}^{2}\left(\alpha T^{\mathrm{M}}\right)$
$\displaystyle=$
$\displaystyle\Pi^{\mathrm{M}}_{11}\left(0\right)-\widehat{\rho}\,\Pi^{\mathrm{M}}_{33}\left(0\right),$
(E2b) $\displaystyle\frac{1}{4s^{2}}\left(\alpha U^{\mathrm{M}}\right)$
$\displaystyle=$
$\displaystyle\Pi_{11}^{\prime\,\mathrm{M}}\left(0\right)-\Pi_{33}^{\prime\,\mathrm{M}}\left(0\right).$
(E2c)

We consider the model parameters of the model described in section 3.1. The
renormalization transformation in the gauge-eigenstate basis is

$\displaystyle\widehat{B}_{\mu}=\left(1+\delta_{B}^{\mathrm{M}}\right)^{\frac{1}{2}}B_{\mu},\quad\widehat{W}^{a}_{\mu}=\left(1+\delta_{W}^{\mathrm{M}}\right)^{\frac{1}{2}}W^{a}_{\mu},\quad\widehat{g}=\left(1+\delta_{g}^{\mathrm{M}}\right)g,$
(E3a)
$\displaystyle\widehat{v}=\left(1+\delta_{v}^{\mathrm{M}}\right)v,\quad\widehat{t}=\left(1+\delta_{t}^{\mathrm{M}}\right)t,\quad\widehat{u}=\left(1+\delta_{u}^{\mathrm{M}}\right)u.$
(E3b)

Notice that there is a single $\delta_{W}$ for all three $W^{a}_{\mu}$.
Furthermore, there is no mixed $B_{\mu}$–$W^{3}_{\mu}$ renormalization
constant, because of the background-field gauge symmetry. Inserting the
renormalization transformation of eq. (E3) in the Lagrangian, one gets the
one-loop counterterms for the renormalized self-energies

$\displaystyle\delta\Pi_{11,22}^{\mathrm{M}}\left(p^{2}\right)$
$\displaystyle=$
$\displaystyle-p^{2}\delta_{W}^{\mathrm{M}}+\frac{g^{2}}{2}\,\left(\delta\varrho_{W}^{\mathrm{M}}\right)^{2}+\varrho_{W}^{2}\delta_{g}^{\mathrm{M}}g^{2},$
(E4a) $\displaystyle\delta\Pi_{33}^{\mathrm{M}}\left(p^{2}\right)$
$\displaystyle=$
$\displaystyle-p^{2}\delta_{W}^{\mathrm{M}}+\frac{g^{2}}{2}\,\left(\delta\varrho_{Z}^{\mathrm{M}}\right)^{2}+\varrho_{Z}^{2}\delta_{g}^{\mathrm{M}}g^{2},$
(E4b)

where $\delta\varrho_{W}^{2}$ and $\delta\varrho_{Z}^{2}$ are the variations
with respect to the renormalization transformations in eq. (E3) of the
parameters $\varrho_{W}$ and $\varrho_{Z}$, respectively, defined in eqs.
(42). Since the divergences in $S^{\mathrm{M}}$ and $U^{\mathrm{M}}$ come from
the divergences in the self-energies that are proportional to $p^{2}$, one
immediately sees that $S^{\mathrm{M}}$ and $U^{\mathrm{M}}$ are finite in the
gauge defined by eq. (E1). This holds for all models with $SU(2)\times U(1)$
gauge symmetry, independently of the value of $\widehat{\rho}$.

Plugging eq. (E4) into eq. (E2b), we gather that the UV divergences of
$T^{\mathrm{M}}$ must be of the form

$\displaystyle m_{W}^{2}\left(\alpha\,\delta T^{\mathrm{M}}\right)$
$\displaystyle=$
$\displaystyle\frac{g^{2}}{2}\left(\left(\delta\varrho_{W}^{\mathrm{M}}\right)^{2}-\frac{\varrho_{W}^{2}}{\varrho_{Z}^{2}}\left(\delta\varrho_{Z}^{\mathrm{M}}\right)^{2}\right)$
(E5a) $\displaystyle=$
$\displaystyle\frac{2g^{2}}{\varrho_{Z}^{2}}\left[\left(t^{2}-u^{2}\right)v^{2}\,\delta_{v}^{\mathrm{M}}+\left(4t^{2}+v^{2}\right)u^{2}\,\delta_{u}^{\mathrm{M}}-\left(4u^{2}+v^{2}\right)t^{2}\,\delta_{t}^{\mathrm{M}}\right].$
(E5b)

Setting both $u=0$ and $t=0$ in the right-hand side of eq. (E5b) makes it
vanish, recovering the null result that holds for the SM (see e.g. refs. [28]
and [24]). However, $T^{\mathrm{M}}$ is UV-divergent for either $u\neq 0$ or
$t\neq 0$. Note that $\delta^{\mathrm{M}}_{v}$ is different for M$=$BBM,
M$=$BM1, and M$=$BM2; therefore, in order to have the subtracted $T$ finite
one must have $u=t=0$ even if the new physics is purely fermionic. As a
result, the oblique (i.e., subtracted) parameter $T$ is also UV-divergent,
with divergences proportional to the VEVs $u$ and $t$ that break the custodial
symmetry. This explains the results previously obtained by many authors, e.g.
refs. [6, 11, 12, 13, 14, 15, 16, 17, 18, 19].

## References

  * [1] M.E. Peskin and T. Takeuchi, _New Constraint on a Strongly Interacting Higgs Sector_ , _Physical Review Letters_ 65 (1990) 964.
  * [2] W.J. Marciano and J.L. Rosner, _Atomic parity violation as a probe of new physics_ , _Phys. Rev. Lett._ 65 (1990) 2963.
  * [3] D.C. Kennedy and P. Langacker, _Precision Electroweak Experiments and Heavy Physics: A Global Analysis_ , _Physical Review Letters_ 65 (1990) 2967.
  * [4] G. Altarelli and R. Barbieri, _Vacuum Polarization Effects of New Physics on Electroweak Processes_ , _Physics Letters B_ 253 (1991) 161.
  * [5] G. Altarelli, R. Barbieri and S. Jadach, _Toward a Model-Independent Analysis of Electroweak Data_ , _Nuclear Physics B_ 369 (1992) 3.
  * [6] B.W. Lynn and E. Nardi, _Radiative Corrections in Unconstrained SU(2) $\times$ U(1) Models and the Top-Mass Problem_, _Nuclear Physics B_ 381 (1992) 467.
  * [7] M.E. Peskin and T. Takeuchi, _Estimation of Oblique Electroweak Corrections_ , _Physical Review D_ 46 (1992) 381.
  * [8] I. Maksymyk, C.P. Burgess and D. London, _Beyond S, T and U_ , _Physical Review D_ 50 (1994) 529 [hep-ph/9306267].
  * [9] M. Veltman, _Limit on mass differences in the Weinberg model_ , _Nuclear Physics B_ 123 (1977) 89.
  * [10] CDF Collaboration$\dagger\ddagger$, T. Aaltonen, S. Amerio, D. Amidei, A. Anastassov, A. Annovi et al., _High-precision measurement of the _W_ boson mass with the CDF II detector_, _Science_ 376 (2022) 170.
  * [11] T. Blank and W. Hollik, _Precision Observables in SU(2) $\times$ U(1) Models with an Additional Higgs Triplet_, _Nuclear Physics B_ 514 (1998) 113 [hep-ph/9703392].
  * [12] M. Czakon, J. Gluza, F. Jegerlehner and M. Zrałek, _Confronting electroweak precision measurements with New Physics models_ , _Eur. Phys. J. C_ 13 (2000) 275.
  * [13] J.R. Forshaw, D.A. Ross and B.E. White, _Higgs mass bounds in a Triplet Model_ , _J. High Energy Phys._ 2001 (2001) 007 [hep-ph/0107232].
  * [14] J.R. Forshaw, A.S. Vera and B.E. White, _Mass bounds in a model with a Triplet Higgs_ , _J. High Energy Phys._ 2003 (2003) 059 [hep-ph/0302256].
  * [15] M.-C. Chen, S. Dawson and T. Krupovnickas, _Constraining new models with precision electroweak data_ , _Int. J. Mod. Phys. A_ 21 (2006) 4045.
  * [16] P. Chankowski, S. Pokorski and J. Wagner, _(Non)decoupling of the Higgs triplet effects_ , _Eur. Phys. J. C_ 50 (2007) 919.
  * [17] M.-C. Chen, S. Dawson and C.B. Jackson, _Higgs triplets, decoupling, and precision measurements_ , _Phys. Rev. D_ 78 (2008) 093001.
  * [18] F. Albergaria and L. Lavoura, _Prescription for finite oblique parameters S and U in extensions of the SM with $m_{W}\neq m_{Z}\cos\theta_{W}$_, _J. Phys. G: Nucl. Part. Phys._ 49 (2022) 085005.
  * [19] T.G. Rizzo, _Kinetic Mixing, Dark Higgs Triplets, $M_{W}$ and All That_, Aug., 2022.  10.48550/arXiv.2206.09814.
  * [20] D. Kennedy and B. Lynn, _Electroweak radiative corrections with an effective lagrangian: Four-fermions processes_ , _Nuclear Physics B_ 322 (1989) 1.
  * [21] K.-i. Aoki, Z. Hioki, R. Kawabe, M. Konuma and T. Muta, _Electroweak Theory: Framework of On-Shell Renormalization and Study of Higher-Order Effects_ , _Prog. Theor. Phys. Suppl._ 73 (1982) 1.
  * [22] A. Denner, _Techniques for the Calculation of Electroweak Radiative Corrections at the One-Loop Level and Results forW-physics at LEP 200_ , _Fortschr. Phys._ 41 (1993) 307.
  * [23] B.W. Lynn, _GAUGE INVARIANCE OF RUNNING COUPLINGS IN SU(2)-L x U(1)_ , .
  * [24] A. Denner, S. Dittmaier and G. Weiglein, _Gauge invariance, gauge-parameter independence and properties of Green functions_ , .
  * [25] A. Denner, G. Weiglein and S. Dittmaier, _Application of the background-field method to the electroweak standard model_ , _Nuclear Physics B_ 440 (1995) 95.
  * [26] A. Denner and S. Dittmaier, _Electroweak radiative corrections for collider physics_ , _Physics Reports_ 864 (2020) 1 [1912.06823].
  * [27] M. Sperling, D. Stöckinger and A. Voigt, _Renormalization of Vacuum Expectation Values in Spontaneously Broken Gauge Theories_ , _Journal of High Energy Physics_ 2013 (2013) 132 [1305.1548].
  * [28] G. Degrassi, B.A. Kniehl and A. Sirlin, _Gauge-Invariant Formulation of the $S$, $T$, and $U$ Parameters_, _Physical Review D_ 48 (1993) R3963.