diagram

# Oblique corrections from leptoquarks

Francisco Albergaria and Luís Lavoura
Universidade de Lisboa, Instituto Superior Técnico, CFTP,
Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present general formulas for the oblique-correction parameters $S$, $T$,
$U$, $V$, $W$, and $X$ in a model of New Physics having arbitrary numbers of
scalar leptoquarks of the five permissible types. We allow for a general
mixing among the scalars of the various electric charges, viz. $-4/3$, $-1/3$,
$2/3$, and $5/3$. We then extend the formulas to the case of a New Physics
model with additional scalars in any representations of the gauge group
$SU(2)\times U(1)$, mixing arbitrarily among themselves.

##  1 Introduction and notation

#### Leptoquarks:

In this paper we consider a model of New Physics (NP), i.e. an extension of
the Standard Model (SM), that has arbitrary numbers of scalars placed in
triplets of colour-$SU(3)$ which are

$SU(2)$ singlets with weak hypercharge555We use the normalization $Y=Q-T_{3}$,
where $Y$ is the weak hypercharge, $Q$ is the electric charge, and $T_{3}$ is
the third component of weak isospin. $-1/3$

$\sigma_{0,-2};$ (1)

$SU(2)$ singlets with weak hypercharge $-4/3$

$\sigma_{0,-8};$ (2)

$SU(2)$ doublets with weak hypercharge $7/6$

$\left(\begin{array}[]{c}\delta_{1,7}\\\ \delta_{-1,7}\end{array}\right);$ (3)

$SU(2)$ doublets with weak hypercharge $1/6$

$\left(\begin{array}[]{c}\delta_{1,1}\\\ \delta_{-1,1}\end{array}\right);$ (4)

$SU(2)$ triplets with weak hypercharge $-1/3$

$\left(\begin{array}[]{c}\tau_{2,-2}\\\ \tau_{0,-2}\\\
\tau_{-2,-2}\end{array}\right).$ (5)

In Eqs. (1)–(5),

the letter $\sigma$ denotes singlets of gauge $SU(2)$, the letter $\delta$
stands for doublets, and the letter $\tau$ means triplets;

the first number in the subscript is two times the third component of weak
isospin $T_{3}$;

the second number in the subscript is six times the weak hypercharge $Y$.

In the notation of the recent Ref. [1], which refers to the original Ref. [2],

the scalars in Eq. (1) are leptoquarks of type $\Phi_{1}$, viz. scalars placed
in the representation $\left(\mathbf{3},\mathbf{1},-1/3\right)$ of the gauge
group $SU(3)\times SU(2)\times U(1)$;

the scalars in Eq. (2) are leptoquarks of type $\Phi_{\tilde{1}}$, viz.
scalars placed in the representation $\left(\mathbf{3},\mathbf{1},-4/3\right)$
of $SU(3)\times SU(2)\times U(1)$;

the scalars in Eq. (3) are leptoquarks of type $\Phi_{2}$, viz. scalars placed
in the representation $\left(\mathbf{3},\mathbf{2},7/6\right)$ of the gauge
group;

the scalars in Eq. (4) are leptoquarks of type $\Phi_{\tilde{2}}$, viz.
scalars placed in the representation $\left(\mathbf{3},\mathbf{2},1/6\right)$;

the scalars in Eq. (5) are leptoquarks of type $\Phi_{3}$, viz. scalars placed
in the representation $\left(\mathbf{3},\mathbf{3},-1/3\right)$ of
$SU(3)\times SU(2)\times U(1)$.

Leptoquarks666In this paper we focus exclusively on _scalar_ leptoquarks.
Various authors use the term ‘leptoquarks’ to mean some vector, i.e. spin-one,
fields. are scalars that are in multiplets of $SU(3)\times SU(2)\times U(1)$
such that they have (renormalizable) Yukawa couplings to one lepton multiplet
and one quark multiplet of the SM. Leptoquarks have recently been much used in
models that seek to explain one or more unexpected experimental results like
those on the muon magnetic moment, the $W^{\pm}$ mass, the decays $b\to
c\tau\nu$, and the decays $b\to s\ell^{+}\ell^{-}$; see for instance Refs. [3,
4, 5, 6, 7, 8, 9, 10].

#### Oblique parameters:

The oblique parameters are defined as [11]777We use the sign conventions in
Ref. [12]. Those conventions differ from the ones used in many other papers,
viz. in Ref. [11]. For a resource paper on sign conventions, see Ref. [13];
using the notation of that paper, our convention has $\eta_{e}=\eta_{Z}=1$ and
$\eta=-1$.,888The definitions (6) build on, and generalize, previous work in
Refs. [14, 15, 16]. They are appropriate for the case where the functions
$A_{VV^{\prime}}\left(q^{2}\right)$ are not linear in the range
$0<q^{2}<m_{Z}^{2}$. This means that, using those definitions, NP does not
need to be much above the Fermi scale.

$\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{16\pi
c_{W}^{2}}{g^{2}}\left[\frac{A_{ZZ}\left(m_{Z}^{2}\right)-A_{ZZ}\left(0\right)}{m_{Z}^{2}}-\left.\frac{\partial
A_{\gamma\gamma}\left(q^{2}\right)}{\partial q^{2}}\right|_{q^{2}=0}\right.$
(6a)
$\displaystyle\left.+\frac{c_{W}^{2}-s_{W}^{2}}{c_{W}s_{W}}\,\left.\frac{\partial
A_{\gamma Z}\left(q^{2}\right)}{\partial q^{2}}\right|_{q^{2}=0}\right],$
$\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{4\pi}{g^{2}s_{W}^{2}}\left[\frac{A_{WW}\left(0\right)}{m_{W}^{2}}-\frac{A_{ZZ}\left(0\right)}{m_{Z}^{2}}\right],$
(6b) $\displaystyle U$ $\displaystyle=$
$\displaystyle\frac{16\pi}{g^{2}}\left[\frac{A_{WW}\left(m_{W}^{2}\right)-A_{WW}\left(0\right)}{m_{W}^{2}}-c_{W}^{2}\,\frac{A_{ZZ}\left(m_{Z}^{2}\right)-A_{ZZ}\left(0\right)}{m_{Z}^{2}}\right.$
(6c) $\displaystyle\left.-s_{W}^{2}\left.\frac{\partial
A_{\gamma\gamma}\left(q^{2}\right)}{\partial
q^{2}}\right|_{q^{2}=0}+2c_{W}s_{W}\left.\frac{\partial A_{\gamma
Z}\left(q^{2}\right)}{\partial q^{2}}\right|_{q^{2}=0}\right],$ $\displaystyle
V$ $\displaystyle=$
$\displaystyle\frac{4\pi}{g^{2}s_{W}^{2}}\left[\left.\frac{\partial
A_{ZZ}\left(q^{2}\right)}{\partial
q^{2}}\right|_{q^{2}=m_{Z}^{2}}-\frac{A_{ZZ}\left(m_{Z}^{2}\right)-A_{ZZ}\left(0\right)}{m_{Z}^{2}}\right],$
(6d) $\displaystyle W$ $\displaystyle=$
$\displaystyle\frac{4\pi}{g^{2}s_{W}^{2}}\left[\left.\frac{\partial
A_{WW}\left(q^{2}\right)}{\partial
q^{2}}\right|_{q^{2}=m_{W}^{2}}-\frac{A_{WW}\left(m_{W}^{2}\right)-A_{WW}\left(0\right)}{m_{W}^{2}}\right],$
(6e) $\displaystyle X$ $\displaystyle=$ $\displaystyle\frac{4\pi
c_{W}}{g^{2}s_{W}}\left[\left.\frac{\partial A_{\gamma
Z}\left(q^{2}\right)}{\partial q^{2}}\right|_{q^{2}=0}-\frac{A_{\gamma
Z}\left(m_{Z}^{2}\right)-A_{\gamma Z}\left(0\right)}{m_{Z}^{2}}\right],$ (6f)

where $g$ is the $SU(2)$ gauge coupling constant and $c_{W}$ and $s_{W}$ are
the cosine and the sine, respectively, of the Weinberg angle. The functions
$A_{VV^{\prime}}\left(q^{2}\right)$ are the coefficients of the metric tensor
$g^{\mu\nu}$ in the vacuum-polarization tensor

$\Pi^{\mu\nu}_{VV^{\prime}}\left(q^{2}\right)=g^{\mu\nu}\,A_{VV^{\prime}}\left(q^{2}\right)+q^{\mu}q^{\nu}\,B_{VV^{\prime}}\left(q^{2}\right)$
(7)

between gauge bosons $V_{\mu}$ and $V^{\prime}_{\nu}$ carrying four-momentum
$q$. In $A_{VV^{\prime}}\left(q^{2}\right)$

one only takes into account the dispersive part—one discards the absorptive
part;

one subtracts the SM contribution from the full result.999In this paper we do
not need to perform this subtraction, since we are dealing with NP scalars
that have non-integer electric charges and, therefore, do not mix with the SM
scalars.

Note that

$S+U=\frac{16\pi}{g^{2}}\left[\frac{A_{WW}\left(m_{W}^{2}\right)-A_{WW}\left(0\right)}{m_{W}^{2}}-\left.\frac{\partial
A_{\gamma\gamma}\left(q^{2}\right)}{\partial
q^{2}}\right|_{q^{2}=0}+\frac{c_{W}}{s_{W}}\left.\frac{\partial A_{\gamma
Z}\left(q^{2}\right)}{\partial q^{2}}\right|_{q^{2}=0}\right]$ (8)

has a somewhat simpler expression than $U$.

It is convenient to separate the parameters $S$ and $U$ in two parts:

$S=S^{\prime}+S^{\prime\prime},\quad U=U^{\prime}+U^{\prime\prime}.$ (9)

The parameters $S^{\prime}$ and $U^{\prime}$ are identical to the original $S$
and $U$, respectively, as they were defined in Ref. [14]:

$\displaystyle S^{\prime}$ $\displaystyle=$ $\displaystyle\frac{16\pi
c_{W}^{2}}{g^{2}}\left[\left.\frac{\partial A_{ZZ}\left(q^{2}\right)}{\partial
q^{2}}\right|_{q^{2}=0}-\left.\frac{\partial
A_{\gamma\gamma}\left(q^{2}\right)}{\partial
q^{2}}\right|_{q^{2}=0}+\frac{c_{W}^{2}-s_{W}^{2}}{c_{W}s_{W}}\left.\frac{\partial
A_{\gamma Z}\left(q^{2}\right)}{\partial q^{2}}\right|_{q^{2}=0}\right],\hskip
19.91692pt$ (10a) $\displaystyle U^{\prime}$ $\displaystyle=$
$\displaystyle-S^{\prime}+\frac{16\pi}{g^{2}}\left[\left.\frac{\partial
A_{WW}\left(q^{2}\right)}{\partial
q^{2}}\right|_{q^{2}=0}-\left.\frac{\partial
A_{\gamma\gamma}\left(q^{2}\right)}{\partial
q^{2}}\right|_{q^{2}=0}+\frac{c_{W}}{s_{W}}\left.\frac{\partial A_{\gamma
Z}\left(q^{2}\right)}{\partial q^{2}}\right|_{q^{2}=0}\right].$ (10b)

Clearly then,

$\displaystyle S^{\prime\prime}$ $\displaystyle=$ $\displaystyle\frac{16\pi
c_{W}^{2}}{g^{2}}\left[\frac{A_{ZZ}\left(m_{Z}^{2}\right)-A_{ZZ}\left(0\right)}{m_{Z}^{2}}-\left.\frac{\partial
A_{ZZ}\left(q^{2}\right)}{\partial q^{2}}\right|_{q^{2}=0}\right],$ (11a)
$\displaystyle U^{\prime\prime}$ $\displaystyle=$
$\displaystyle-S^{\prime\prime}+\frac{16\pi}{g^{2}}\left[\frac{A_{WW}\left(m_{W}^{2}\right)-A_{WW}\left(0\right)}{m_{W}^{2}}-\left.\frac{\partial
A_{WW}\left(q^{2}\right)}{\partial q^{2}}\right|_{q^{2}=0}\right].$ (11b)

#### Purpose of this paper:

In this paper we compute the oblique parameters for a NP model with arbitrary
numbers of leptoquarks, mixing arbitrarily among themselves. Our work
generalizes earlier partial results in Refs. [17, 18, 19, 20, 21, 22]. At the
end of this paper we present a generalization wherein the oblique parameters
are computed for any NP model solely with extra scalars, whatever the
representations of $SU(2)\times U(1)$ those new scalars are in—just assuming
that no new scalar either develops a vacuum expectation value or mixes with
the scalars of the SM.

#### Plan of the paper:

In the next section we write the Lagrangian for the gauge interactions of the
scalars, carefully defining the mixing matrices that appear in that
Lagrangian. Section 3 performs the computation of the relevant diagrams and
explicitly gives various functions that appear in the formulas for the oblique
parameters. Those formulas are then given in Section 4. Section 5 presents a
few illustrative examples of application of our formulas to simple models with
leptoquarks. Section 6 generalizes our work to scalars in any representations
of $SU(2)\times U(1)$. Section 7 summarizes our main results. In Appendices A
and B we deal, respectively, with the counting of mixing parameters and with
the parameterization of the mixing in models with leptoquarks. In Appendix C
we explicitly demonstrate that $S^{\prime}$ and $U^{\prime}$ are finite as
they should be.

##  2 Interactions

#### Numbers of scalars:

There are in our NP model $n_{\sigma,-2}$ multiplets of type (1),
$n_{\sigma,-8}$ multiplets of type (2), $n_{\delta,7}$ multiplets of type (3),
$n_{\delta,1}$ multiplets of type (4), and $n_{\tau,-2}$ multiplets of type
(5). All these five numbers $n_{\sigma,-2}$, $n_{\sigma,-8}$, $n_{\delta,7}$,
$n_{\delta,1}$, and $n_{\tau,-2}$ must be multiples of three, since all the
scalars are in triplets of $SU(3)$; otherwise, those numbers are free.

Our model has fractionary-charge scalars that we call

$h$-type scalars, viz. the ones that have electric charge $Q_{h}=5/3$;

$u$-type scalars, viz. the ones that have electric charge $Q_{u}=2/3$;

$d$-type scalars, viz. the ones that have electric charge $Q_{d}=-1/3$;

$l$-type scalars, viz. the ones that have electric charge $Q_{l}=-4/3$.

The total numbers of $h$-type scalars, $u$-type scalars, $d$-type scalars, and
$l$-type scalars are

$\displaystyle n_{h}$ $\displaystyle=$ $\displaystyle n_{\delta,7},$ (12a)
$\displaystyle n_{u}$ $\displaystyle=$ $\displaystyle
n_{\delta,7}+n_{\delta,1}+n_{\tau,-2},$ (12b) $\displaystyle n_{d}$
$\displaystyle=$ $\displaystyle n_{\sigma,-2}+n_{\delta,1}+n_{\tau,-2},$ (12c)
$\displaystyle n_{l}$ $\displaystyle=$ $\displaystyle
n_{\sigma,-8}+n_{\tau,-2},$ (12d)

respectively. Notice that

$n_{d}\geq n_{u}-n_{h}\geq 0.$ (13)

#### Scalar mixing:

One bi-diagonalizes the leptoquark mass matrices by making

$\displaystyle\delta_{1,7}$ $\displaystyle=$ $\displaystyle H_{1}h,$ (14a)
$\displaystyle\left(\begin{array}[]{c}\delta_{-1,7}\\\ \delta_{1,1}\\\
\tau_{2,-2}\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}U_{1}\\\ U_{2}\\\
U_{3}\end{array}\right)u,$ (14h)
$\displaystyle\left(\begin{array}[]{c}\sigma_{0,-2}\\\ \delta_{-1,1}\\\
\tau_{0,-2}\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}D_{1}\\\ D_{2}\\\
D_{3}\end{array}\right)d,$ (14o)
$\displaystyle\left(\begin{array}[]{c}\sigma_{0,-8}\\\
\tau_{-2,-2}\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}L_{1}\\\ L_{2}\end{array}\right)l,$
(14t)

where $h$ in Eq. (14a) stands for a column matrix containing the $n_{h}$
$h$-type scalars; and analogously for $u$, $d$, and $l$ in the other three
Eqs. (14). The matrices $H_{1}$, $U_{1}$, $U_{2}$, $U_{3}$, $D_{1}$, $D_{2}$,
$D_{3}$, $L_{1}$, and $L_{2}$ have dimensions $n_{\delta,7}\times n_{h}$,
$n_{\delta,7}\times n_{u}$, $n_{\delta,1}\times n_{u}$, $n_{\tau,-2}\times
n_{u}$, $n_{\sigma,-2}\times n_{d}$, $n_{\delta,1}\times n_{d}$,
$n_{\tau,-2}\times n_{d}$, $n_{\sigma,-8}\times n_{l}$, and $n_{\tau,-2}\times
n_{l}$, respectively. The matrices

$H_{1},\quad\left(\begin{array}[]{c}U_{1}\\\ U_{2}\\\
U_{3}\end{array}\right),\quad\left(\begin{array}[]{c}D_{1}\\\ D_{2}\\\
D_{3}\end{array}\right),\quad\mbox{and}\ \left(\begin{array}[]{c}L_{1}\\\
L_{2}\end{array}\right)$ (15)

are unitary.101010Without loss of generality, one may chose a basis where
$H_{1}$ is equal to the unit matrix. We refrain from doing that, in order to
keep the notation as general as possible.

#### Mixing matrices:

We next define the mixing matrices that appear in the charged-current
interactions of the scalars. They are

$\displaystyle\mathcal{N}$ $\displaystyle=$ $\displaystyle
H_{1}^{\dagger}U_{1},$ (16a) $\displaystyle\mathcal{V}$ $\displaystyle=$
$\displaystyle U_{2}^{\dagger}D_{2}+\sqrt{2}\,U_{3}^{\dagger}D_{3},$ (16b)
$\displaystyle\mathcal{Q}$ $\displaystyle=$
$\displaystyle\sqrt{2}\,D_{3}^{\dagger}L_{2},$ (16c)

respectively. The factors $\sqrt{2}$ in Eqs. (16) arise because the charged
gauge interactions of the triplets are $\sqrt{2}$ times stronger than those of
the doublets. The mixing matrices that appear in the neutral-current
interactions of the scalars are

$\displaystyle\bar{H}$ $\displaystyle=$
$\displaystyle\mathcal{H}-2Q_{h}s_{W}^{2}\times\mathbbm{1}_{n_{h}},$ (17a)
$\displaystyle\bar{U}$ $\displaystyle=$
$\displaystyle\mathcal{U}-2Q_{u}s_{W}^{2}\times\mathbbm{1}_{n_{u}},$ (17b)
$\displaystyle\bar{D}$ $\displaystyle=$
$\displaystyle\mathcal{D}+2Q_{d}s_{W}^{2}\times\mathbbm{1}_{n_{d}},$ (17c)
$\displaystyle\bar{L}$ $\displaystyle=$
$\displaystyle\mathcal{L}+2Q_{l}s_{W}^{2}\times\mathbbm{1}_{n_{l}},$ (17d)

where $\mathbbm{1}_{m}$ denotes the $m\times m$ unit matrix and the matrices
$\mathcal{H}$, $\mathcal{U}$, $\mathcal{D}$, and $\mathcal{L}$ are related to
the mixing matrices in Eqs. (16) through

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\mathcal{N}\mathcal{N}^{\dagger},$ (18a)
$\displaystyle\mathcal{U}$ $\displaystyle=$
$\displaystyle\mathcal{V}\mathcal{V}^{\dagger}-\mathcal{N}^{\dagger}\mathcal{N},$
(18b) $\displaystyle\mathcal{D}$ $\displaystyle=$
$\displaystyle\mathcal{V}^{\dagger}\mathcal{V}-\mathcal{Q}\mathcal{Q}^{\dagger},$
(18c) $\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\mathcal{Q}^{\dagger}\mathcal{Q},$ (18d)

respectively.

#### Gauge interactions:

The gauge interactions of the scalars are given by the following pieces of the
Lagrangian:

$\displaystyle\mathcal{L}_{ASS}$ $\displaystyle=$ $\displaystyle-
igs_{W}A_{\theta}\left[Q_{h}\sum_{h}\left(h^{\ast}\partial^{\theta}h-h\partial^{\theta}h^{\ast}\right)+Q_{u}\sum_{u}\left(u^{\ast}\partial^{\theta}u-u\partial^{\theta}u^{\ast}\right)\right.$
(19a)
$\displaystyle\left.+Q_{d}\sum_{d}\left(d^{\ast}\partial^{\theta}d-d\partial^{\theta}d^{\ast}\right)+Q_{l}\sum_{l}\left(l^{\ast}\partial^{\theta}l-l\partial^{\theta}l^{\ast}\right)\right],$
$\displaystyle\mathcal{L}_{AASS}$ $\displaystyle=$ $\displaystyle
g^{2}s_{W}^{2}A_{\theta}A^{\theta}\left(Q_{h}^{2}\,\sum_{h}hh^{\ast}+Q_{u}^{2}\,\sum_{u}uu^{\ast}+Q_{d}^{2}\,\sum_{d}dd^{\ast}+Q_{l}^{2}\,\sum_{l}ll^{\ast}\right),\hskip
8.53581pt$ (19b) $\displaystyle\mathcal{L}_{WSS}$ $\displaystyle=$
$\displaystyle
i\,\frac{g}{\sqrt{2}}\,W_{\theta}^{+}\left[\sum_{h,u}\mathcal{N}_{hu}\left(h^{\ast}\partial^{\theta}u-u\partial^{\theta}h^{\ast}\right)+\sum_{u,d}\mathcal{V}_{ud}\left(u^{\ast}\partial^{\theta}d-d\partial^{\theta}u^{\ast}\right)\right.$
(19c)
$\displaystyle\left.+\sum_{d,l}\mathcal{Q}_{dl}\left(d^{\ast}\partial^{\theta}l-l\partial^{\theta}d^{\ast}\right)\right]+\mathrm{H.c.},$
$\displaystyle\mathcal{L}_{WWSS}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{2}\,W^{+}_{\theta}W^{-\theta}\left[\sum_{h,h^{\prime}}\left(\mathcal{N}\mathcal{N}^{\dagger}\right)_{hh^{\prime}}h^{\ast}h^{\prime}+\sum_{u,u^{\prime}}\left(\mathcal{N}^{\dagger}\mathcal{N}+\mathcal{V}\mathcal{V}^{\dagger}\right)_{uu^{\prime}}u^{\ast}u^{\prime}\right.$
(19d)
$\displaystyle\left.+\sum_{d,d^{\prime}}\left(\mathcal{V}^{\dagger}\mathcal{V}+\mathcal{Q}\mathcal{Q}^{\dagger}\right)_{dd^{\prime}}d^{\ast}d^{\prime}+\sum_{l,l^{\prime}}\left(\mathcal{Q}^{\dagger}\mathcal{Q}\right)_{ll^{\prime}}l^{\ast}l^{\prime}\right],$
$\displaystyle\mathcal{L}_{ZSS}$ $\displaystyle=$ $\displaystyle
i\,\frac{g}{2c_{W}}\,Z_{\theta}\left[\sum_{h}\bar{H}_{hh^{\prime}}\left(h^{\ast}\partial^{\theta}h^{\prime}-h^{\prime}\partial^{\theta}h^{\ast}\right)+\sum_{u,u^{\prime}}\bar{U}_{uu^{\prime}}\left(u^{\ast}\,\partial^{\theta}u^{\prime}-u^{\prime}\,\partial^{\theta}u^{\ast}\right)\right.$
(19e)
$\displaystyle\left.-\sum_{d,d^{\prime}}\bar{D}_{dd^{\prime}}\left(d^{\ast}\,\partial^{\theta}d^{\prime}-d^{\prime}\,\partial^{\theta}d^{\ast}\right)-\sum_{l,l^{\prime}}\bar{L}_{ll^{\prime}}\left(l^{\ast}\,\partial^{\theta}l^{\prime}-l^{\prime}\,\partial^{\theta}l^{\ast}\right)\right],$
$\displaystyle\mathcal{L}_{ZZSS}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{4c_{W}^{2}}\,Z_{\theta}Z^{\theta}\left[\sum_{h,h^{\prime}}\left(\bar{H}^{2}\right)_{hh^{\prime}}h^{\ast}h^{\prime}+\sum_{u,u^{\prime}}\left(\bar{U}^{2}\right)_{uu^{\prime}}u^{\ast}u^{\prime}\right.$
(19f)
$\displaystyle\left.+\sum_{d,d^{\prime}}\left(\bar{D}^{2}\right)_{dd^{\prime}}d^{\ast}d^{\prime}+\sum_{l,l^{\prime}}\left(\bar{L}^{2}\right)_{ll^{\prime}}l^{\ast}l^{\prime}\right],$
$\displaystyle\mathcal{L}_{AZSS}$ $\displaystyle=$
$\displaystyle-\frac{g^{2}s_{W}}{c_{W}}\,A_{\theta}Z^{\theta}\left(Q_{h}\,\sum_{h,h^{\prime}}\bar{H}_{hh^{\prime}}\,h^{\ast}h^{\prime}+Q_{u}\,\sum_{u,u^{\prime}}\bar{U}_{uu^{\prime}}\,u^{\ast}u^{\prime}\right.$
(19g)
$\displaystyle\left.-Q_{d}\,\sum_{d,d^{\prime}}\bar{D}_{dd^{\prime}}\,d^{\ast}d^{\prime}-Q_{l}\,\sum_{l,l^{\prime}}\bar{L}_{ll^{\prime}}\,l^{\ast}l^{\prime}\right).$

Notice the presence in Eqs. (19c) and (19d) of the matrices defined in Eqs.
(16) and the presence in Eqs. (19e)–(19g) of the matrices defined in Eqs.
(17). Also notice that, out of the four matrices in Eq. (19d), only
$\mathcal{N}\mathcal{N}^{\dagger}=\mathcal{H}$ and
$\mathcal{Q}^{\dagger}\mathcal{Q}=\mathcal{L}$ coincide with matrices in Eqs.
(18).

##  3 Tools for the computation

#### PV functions:

The Passarino–Veltman (PV) functions [23]
$B_{00}\left(q^{2},m_{1}^{2},m_{2}^{2}\right)$ and
$A_{0}\left(m_{1}^{2}\right)$ are defined by111111We use the definitions of
Ref. [24] for the PV functions.

$\displaystyle\mu^{\epsilon}\int\frac{\mathrm{d}^{4-\epsilon}k}{\left(2\pi\right)^{4-\epsilon}}\
\,k^{\theta}k^{\psi}\,\frac{1}{k^{2}-m_{1}^{2}}\
\frac{1}{\left(k+q\right)^{2}-m_{2}^{2}}$ $\displaystyle=$
$\displaystyle\frac{i}{16\pi^{2}}\,\left[g^{\theta\psi}\,B_{00}\left(q^{2},m_{1}^{2},m_{2}^{2}\right)\right.$
(20a)
$\displaystyle\left.+q^{\theta}q^{\psi}\,B_{11}\left(q^{2},m_{1}^{2},m_{2}^{2}\right)\right],\hskip
19.91692pt$
$\displaystyle\mu^{\epsilon}\int\frac{\mathrm{d}^{4-\epsilon}k}{\left(2\pi\right)^{4-\epsilon}}\
\frac{1}{k^{2}-m_{1}^{2}}\ $ $\displaystyle=$
$\displaystyle\frac{i}{16\pi^{2}}\ A_{0}\left(m_{1}^{2}\right),$ (20b)

respectively, where $\mu$ is an arbitrary quantity with mass dimension. The
quantities $q^{2}$, $m_{1}^{2}$, and $m_{2}^{2}$ are assumed to be non-
negative. The PV function $B_{11}\left(q^{2},m_{1}^{2},m_{2}^{2}\right)$ in
Eq. (20a) is not needed in this paper; on the other hand, we need

$\displaystyle B_{00}^{\prime}\left(q^{2},m_{1}^{2},m_{2}^{2}\right)$
$\displaystyle\equiv$ $\displaystyle\frac{\partial
B_{00}\left(q^{2},m_{1}^{2},m_{2}^{2}\right)}{\partial q^{2}},$ (21a)
$\displaystyle\bar{B}_{00}\left(q^{2},m_{1}^{2},m_{2}^{2}\right)$
$\displaystyle\equiv$
$\displaystyle\frac{B_{00}\left(q^{2},m_{1}^{2},m_{2}^{2}\right)-B_{00}\left(0,m_{1}^{2},m_{2}^{2}\right)}{q^{2}}.$
(21b)

The PV functions may be numerically evaluated by using softwares like
LoopTools [24] and COLLIER [25].121212For the present purposes, in the results
given by those codes one should take only the _real_ , viz. dispersive, parts
of the PV functions, while dropping the absorptive parts, which have no
relevance for the oblique parameters. They may also be evaluated through
analytic formulas given in Ref. [26].

#### The function $g$:

One has [26]

$B^{\prime}_{00}\left(0,m_{1}^{2},m_{2}^{2}\right)=-\frac{\mathrm{div}}{12}+\frac{1}{24}\left[\ln{\frac{m_{1}^{2}}{\mu^{2}}}+\ln{\frac{m_{2}^{2}}{\mu^{2}}}+g\left(\frac{m_{1}^{2}}{m_{2}^{2}}\right)\right],$
(22)

where $\mathrm{div}$ is a divergent quantity which is defined in Eq. (C3) and
cancels out in the final results, and

$g\left(x\right)=\left\\{\begin{array}[]{lcl}{\displaystyle\frac{x^{3}-3x^{2}-3x+1}{\left(x-1\right)^{3}}\,\ln{x}-\frac{5x^{2}-22x+5}{3\left(x-1\right)^{2}}}&\Leftarrow&x\neq
1,\\\\[5.69054pt] 0&\Leftarrow&x=1\end{array}\right.$ (23)

is a function that obeys $g\left(1\right)=0$ and
$g\left(x\right)=g\left(1/x\right)$. This function is depicted in Fig. 1.

Figure 1: Plot of the function $g\left(x\right)$. The scale in the horizontal
axis is logarithmic.

#### The function $\theta_{+}$:

From the formulas in Ref. [26] one gathers that

$B_{00}\left(0,m_{1}^{2},m_{2}^{2}\right)=\frac{A_{0}\left(m_{1}^{2}\right)+A_{0}\left(m_{2}^{2}\right)}{4}+\frac{\theta_{+}\left(m_{1}^{2},m_{2}^{2}\right)}{8},$
(24)

where [27]

$\theta_{+}\left(m_{1}^{2},m_{2}^{2}\right)\equiv\left\\{\begin{array}[]{lcl}\displaystyle{m_{1}^{2}+m_{2}^{2}-\frac{2m_{1}^{2}m_{2}^{2}}{m_{1}^{2}-m_{2}^{2}}\,\ln{\frac{m_{1}^{2}}{m_{2}^{2}}}}&\Leftarrow&m_{1}^{2}\neq
m_{2}^{2},\\\\[5.69054pt] 0&\Leftarrow&m_{1}^{2}=m_{2}^{2}.\end{array}\right.$
(25)

#### The function $\rho$:

We define the function $\rho\left(x,y\right)$—where $x$ and $y$ are
positive—through

$\rho\left(x,y\right)\equiv
2\left[B_{00}^{\prime}\left(q^{2},q^{2}x,q^{2}y\right)-\bar{B}_{00}\left(q^{2},q^{2}x,q^{2}y\right)\right],$
(26)

which is $q^{2}$-independent. Using the analytic formulas in ref. [26] one
finds that

$\rho\left(x,y\right)=\left\\{\begin{array}[]{lcl}\displaystyle{\frac{1}{6}-\frac{3\left(x+y\right)}{4}+\frac{\left(x-y\right)^{2}}{2}+\left[\frac{\left(y-x\right)^{3}}{4}+\frac{x^{2}+y^{2}}{4\left(y-x\right)}\right.}&&\\\\[5.69054pt]
\displaystyle{\left.+\frac{x^{2}-y^{2}}{2}\right]\ln{\frac{x}{y}}+\frac{\left(x-y\right)^{2}-x-y}{4}\
f\left(x,y\right)}&\Leftarrow&x\neq y,\\\\[11.38109pt]
\displaystyle{\frac{1}{6}-2x-\frac{x}{2}\
f\left(x,x\right)}&\Leftarrow&x=y.\end{array}\right.$ (27)

In Eq. (27),

$f\left(x,y\right)=\left\\{\begin{array}[]{lcl}\displaystyle{\sqrt{\Delta\left(x,y\right)}\
\ln{\frac{x+y-1+\sqrt{\Delta\left(x,y\right)}}{x+y-1-\sqrt{\Delta\left(x,y\right)}}}}&\Leftarrow&\Delta\left(x,y\right)\geq
0,\\\\[11.38109pt]
\displaystyle{-2\sqrt{-\Delta\left(x,y\right)}\left[\arctan{\frac{x-y+1}{\sqrt{-\Delta\left(x,y\right)}}}+\left(x\leftrightarrow
y\right)\right]}&\Leftarrow&\Delta\left(x,y\right)<0,\end{array}\right.$ (28)

where

$\Delta\left(x,y\right)=1-2\left(x+y\right)+\left(x-y\right)^{2}.$ (29)

#### The function $\zeta$:

The function $\zeta\left(x,y\right)$—where $x$ and $y$ are positive—is defined
through

$\zeta\left(x,y\right)\equiv
2\left[B_{00}^{\prime}\left(0,q^{2}x,q^{2}y\right)-\bar{B}_{00}\left(q^{2},q^{2}x,q^{2}y\right)\right],$
(30)

which is $q^{2}$-independent. Using the analytic formulas in ref. [26] one
finds that

$\zeta\left(x,y\right)=\left\\{\begin{array}[]{lcl}\displaystyle{\frac{11}{36}-\frac{5\left(x+y\right)}{12}+\frac{xy}{3\left(x-y\right)^{2}}+\frac{\left(x-y\right)^{2}}{6}+\left[\frac{x^{2}-y^{2}}{4}+\frac{\left(y-x\right)^{3}}{12}\right.}&&\\\\[5.69054pt]
\displaystyle{\left.+\frac{x^{2}+y^{2}}{4\left(y-x\right)}+\frac{x+y}{12\left(x-y\right)}+\frac{xy\left(x+y\right)}{6\left(y-x\right)^{3}}\right]\ln{\frac{x}{y}}}&&\\\\[5.69054pt]
\displaystyle{+\frac{\Delta\left(x,y\right)}{12}\
f\left(x,y\right)}&\Leftarrow&x\neq y,\\\\[8.53581pt]
\displaystyle{\frac{4}{9}-\frac{4x}{3}+\frac{\Delta\left(x,x\right)}{12}\
f\left(x,x\right)}&\Leftarrow&x=y.\end{array}\right.$ (31)

The functions $\rho\left(x,y\right)$ and $\zeta\left(x,y\right)$ are
illustrated in Fig. 2. They are both very small when $x\gtrsim 1$ and
$y\gtrsim 1$.

Figure 2: Plots of the functions $\rho\left(x,y\right)$ (left panel) and
$\zeta\left(x,y\right)$ (right panel).

#### Computation of the diagrams:

Suppose the vertices $V_{\mu}S_{1}S_{2}$ and
$V_{\mu}V^{\prime}_{\nu}S_{1}S_{2}$ have Feynman rules

$\displaystyle=iX\left(k-p\right)_{\mu},$ (32a) (75,75) i1,i2 o1,o2
dashesi1,w1 dashesi2,w1 photonw1,o1 photonw1,o2 $S_{2}$i1 $S_{1}$i2
$V_{\mu}$o2 $V_{\nu}^{\prime}$o1 $\displaystyle=iYg_{\mu\nu},$ (32b)

respectively, where $V$ and $V^{\prime}$ are gauge bosons (v.g. either $A$,
$Z$, $W^{+}$, or $W^{-}$) and $S_{1}$ and $S_{2}$ are scalars. Using those
Feynman rules we have

$A^{VV^{\prime}S_{1}S_{2}}\left(q^{2}\right)=\frac{X_{1}X_{2}}{4\pi^{2}}\,B_{00}\left(q^{2},m_{1}^{2},m_{2}^{2}\right)$
(32ag)

for diagrams of the form in Fig. 3,

(100,100) i1 o1 boson,label.side=lefti1,w1 boson,label.side=leftw2,o1
dashes,right,tension=.3,label=$S_{2}$w1,w2
dashes,left,tension=.3,label=$S_{1}$w1,w2 $V$i1 $V^{\prime}$o1

Figure 3: One type of diagram for vacuum polarization. $V$ and $V^{\prime}$
are gauge bosons, $S_{1}$ is a scalar with mass $m_{1}$, and $S_{2}$ is a
scalar with mass $m_{2}$.

and

$A^{VV^{\prime}S_{1}}\left(q^{2}\right)=-\frac{Y}{16\pi^{2}}\,A_{0}\left(m_{1}^{2}\right)$
(32ah)

for diagrams of the form in Fig. 4.

(100,100) i1 o1 boson,label.side=lefti1,w1 boson,label.side=leftw1,o1
dashes,label=$S_{1}$w1,w1 $V$i1 $V^{\prime}$o1

Figure 4: Another type of diagram for vacuum polarization. $V$ and
$V^{\prime}$ are gauge bosons and $S_{1}$ is a scalar with mass $m_{1}$.

In Eqs. (32ag) and (32ah), $q$ is the four-momentum of the gauge bosons.

##  4 Results for the oblique parameters

####  $T$:

For the oblique parameter $T$ we have

$\displaystyle T$ $\displaystyle=$ $\displaystyle\frac{1}{8\pi
s_{W}^{2}m_{W}^{2}}\left\\{4\left[\sum_{h,u}\left|\mathcal{N}_{hu}\right|^{2}B_{00}(0,m_{h}^{2},m_{u}^{2})+\sum_{u,d}\left|\mathcal{V}_{ud}\right|^{2}B_{00}(0,m_{u}^{2},m_{d}^{2})\right.\right.$
(32aid)
$\displaystyle\left.+\sum_{d,l}\left|\mathcal{Q}_{dl}\right|^{2}B_{00}(0,m_{d}^{2},m_{l}^{2})\right]$
$\displaystyle-\sum_{h}\left(\mathcal{N}\mathcal{N}^{\dagger}\right)_{hh}\,A_{0}\left(m_{h}^{2}\right)-\sum_{u}\left(\mathcal{N}^{\dagger}\mathcal{N}+\mathcal{V}\mathcal{V}^{\dagger}\right)_{uu}\,A_{0}\left(m_{u}^{2}\right)$
$\displaystyle-\sum_{d}\left(\mathcal{V}^{\dagger}\mathcal{V}+\mathcal{Q}\mathcal{Q}^{\dagger}\right)_{dd}\,A_{0}\left(m_{d}^{2}\right)-\sum_{l}\left(\mathcal{Q}^{\dagger}\mathcal{Q}\right)_{ll}\,A_{0}\left(m_{l}^{2}\right)$
$\displaystyle-2\left[\sum_{h,h^{\prime}}\left|\bar{H}_{hh^{\prime}}\right|^{2}B_{00}\left(0,m_{h}^{2},m_{h^{\prime}}^{2}\right)+\sum_{u,u^{\prime}}\left|\bar{U}_{uu^{\prime}}\right|^{2}B_{00}\left(0,m_{u}^{2},m_{u^{\prime}}^{2}\right)\right.$
$\displaystyle\left.+\sum_{d,d^{\prime}}\left|\bar{D}_{dd^{\prime}}\right|^{2}B_{00}\left(0,m_{d}^{2},m_{d^{\prime}}^{2}\right)+\sum_{l,l^{\prime}}\left|\bar{L}_{ll^{\prime}}\right|^{2}B_{00}\left(0,m_{l}^{2},m_{l^{\prime}}^{2}\right)\right]$
$\displaystyle+\sum_{h}\left(\bar{H}^{2}\right)_{hh}\,A_{0}\left(m_{h}^{2}\right)+\sum_{u}\left(\bar{U}^{2}\right)_{uu}\,A_{0}\left(m_{u}^{2}\right)$
$\displaystyle\left.+\sum_{d}\left(\bar{D}^{2}\right)_{dd}\,A_{0}\left(m_{d}^{2}\right)+\sum_{l}\left(\bar{L}^{2}\right)_{ll}\,A_{0}\left(m_{l}^{2}\right)\right\\},$

where lines (32aid) and (32aid) originate in $A_{WW}\left(0\right)$ while
lines (32aid) and (32aid) originate in $A_{ZZ}\left(0\right)$. Utilizing Eq.
(24) on Eq. (32ai), we obtain

$\displaystyle T$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi
s_{W}^{2}m_{W}^{2}}\left[\sum_{h,u}\left|\mathcal{N}_{hu}\right|^{2}\theta_{+}\left(m_{h}^{2},m_{u}^{2}\right)\right.$
(32aj)
$\displaystyle+\sum_{u,d}\left|\mathcal{V}_{ud}\right|^{2}\theta_{+}\left(m_{u}^{2},m_{d}^{2}\right)+\sum_{d,l}\left|\mathcal{Q}_{dl}\right|^{2}\theta_{+}\left(m_{d}^{2},m_{l}^{2}\right)$
$\displaystyle-\sum_{h<h^{\prime}}\left|\mathcal{H}_{hh^{\prime}}\right|^{2}\theta_{+}\left(m_{h}^{2},m_{h^{\prime}}^{2}\right)-\sum_{u<u^{\prime}}\left|\mathcal{U}_{uu^{\prime}}\right|^{2}\theta_{+}\left(m_{u}^{2},m_{u^{\prime}}^{2}\right)$
$\displaystyle\left.-\sum_{d<d^{\prime}}\left|\mathcal{D}_{dd^{\prime}}\right|^{2}\theta_{+}\left(m_{d}^{2},m_{d^{\prime}}^{2}\right)-\sum_{l<l^{\prime}}\left|\mathcal{L}_{ll^{\prime}}\right|^{2}\theta_{+}\left(m_{l}^{2},m_{l^{\prime}}^{2}\right)\right],$

The function $\theta_{+}\left(m_{1}^{2},m_{2}^{2}\right)$ is given in Eq.
(25). It is a finite function; thus, in Eq. (32aj) there are no divergences,
contrary to what happens in Eq. (32ai), wherein the functions
$B_{00}\left(0,m_{1}^{2},m_{2}^{2}\right)$ and $A_{0}\left(m_{3}^{2}\right)$
are divergent. The function $\theta_{+}\left(m_{1}^{2},m_{2}^{2}\right)$ is
moreover positive definite; thus, the two first lines of Eq. (32aj) constitute
a positive contribution to $T$, which always exists in a model with
leptoquarks, while the two last lines of that equation are a negative
contribution, which only exists if at least one of the matrices $\mathcal{H}$,
$\mathcal{U}$, $\mathcal{D}$, and $\mathcal{L}$ has nonzero off-diagonal
matrix elements.

####  $V$ and $W$:

For the oblique parameters $V$ and $W$ we obtain the results

$\displaystyle V$ $\displaystyle=$ $\displaystyle\frac{1}{8\pi
c_{W}^{2}s_{W}^{2}}\left[\sum_{h,h^{\prime}}\left|\bar{H}_{hh^{\prime}}\right|^{2}\rho\left(\frac{m_{h}^{2}}{m_{Z}^{2}},\frac{m_{h^{\prime}}^{2}}{m_{Z}^{2}}\right)+\sum_{u,u^{\prime}}\left|\bar{U}_{uu^{\prime}}\right|^{2}\rho\left(\frac{m_{u}^{2}}{m_{Z}^{2}},\frac{m_{u^{\prime}}^{2}}{m_{Z}^{2}}\right)\right.$
(32aka)
$\displaystyle\left.+\sum_{d,d^{\prime}}\left|\bar{D}_{dd^{\prime}}\right|^{2}\rho\left(\frac{m_{d}^{2}}{m_{Z}^{2}},\frac{m_{d^{\prime}}^{2}}{m_{Z}^{2}}\right)+\sum_{l,l^{\prime}}\left|\bar{L}_{ll^{\prime}}\right|^{2}\rho\left(\frac{m_{l}^{2}}{m_{Z}^{2}},\frac{m_{l^{\prime}}^{2}}{m_{Z}^{2}}\right)\right],$
$\displaystyle W$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi
s_{W}^{2}}\left[\sum_{h,u}\left|\mathcal{N}_{hu}\right|^{2}\rho\left(\frac{m_{h}^{2}}{m_{W}^{2}},\frac{m_{u}^{2}}{m_{W}^{2}}\right)+\sum_{u,d}\left|\mathcal{V}_{ud}\right|^{2}\rho\left(\frac{m_{u}^{2}}{m_{W}^{2}},\frac{m_{d}^{2}}{m_{W}^{2}}\right)\right.$
(32akb)
$\displaystyle\left.+\sum_{d,l}\left|\mathcal{Q}_{dl}\right|^{2}\rho\left(\frac{m_{d}^{2}}{m_{W}^{2}},\frac{m_{l}^{2}}{m_{W}^{2}}\right)\right],$

where the function $\rho$ is given in Eq. (27).

####  $X$:

For the oblique parameter $X$ one has

$\displaystyle X$ $\displaystyle=$
$\displaystyle-\frac{1}{4\pi}\left[Q_{h}\sum_{h}\bar{H}_{hh}\
\zeta\left(\frac{m_{h}^{2}}{m_{Z}^{2}},\frac{m_{h}^{2}}{m_{Z}^{2}}\right)+Q_{u}\sum_{u}\bar{U}_{uu}\
\zeta\left(\frac{m_{u}^{2}}{m_{Z}^{2}},\frac{m_{u}^{2}}{m_{Z}^{2}}\right)\right.$
(32al) $\displaystyle\left.-Q_{d}\sum_{d}\bar{D}_{dd}\
\zeta\left(\frac{m_{d}^{2}}{m_{Z}^{2}},\frac{m_{d}^{2}}{m_{Z}^{2}}\right)-Q_{l}\sum_{l}\bar{L}_{ll}\
\zeta\left(\frac{m_{l}^{2}}{m_{Z}^{2}},\frac{m_{l}^{2}}{m_{Z}^{2}}\right)\right],$

where the function $\zeta$ is given in Eq. (31).

####  $S$ and $U$:

Following the decomposition in Eqs. (9)–(11), for the oblique parameters $S$
and $U$ we have

$\displaystyle S^{\prime\prime}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\pi}\left[\sum_{h,h^{\prime}}\left|\bar{H}_{hh^{\prime}}\right|^{2}\zeta\left(\frac{m_{h}^{2}}{m_{Z}^{2}},\frac{m_{h^{\prime}}^{2}}{m_{Z}^{2}}\right)+\sum_{u,u^{\prime}}\left|\bar{H}_{uu^{\prime}}\right|^{2}\zeta\left(\frac{m_{u}^{2}}{m_{Z}^{2}},\frac{m_{u^{\prime}}^{2}}{m_{Z}^{2}}\right)\right.$
(32ama)
$\displaystyle\left.+\sum_{d,d^{\prime}}\left|\bar{H}_{dd^{\prime}}\right|^{2}\zeta\left(\frac{m_{d}^{2}}{m_{Z}^{2}},\frac{m_{d^{\prime}}^{2}}{m_{Z}^{2}}\right)+\sum_{l,l^{\prime}}\left|\bar{H}_{ll^{\prime}}\right|^{2}\zeta\left(\frac{m_{l}^{2}}{m_{Z}^{2}},\frac{m_{l^{\prime}}^{2}}{m_{Z}^{2}}\right)\right],$
$\displaystyle U^{\prime\prime}$ $\displaystyle=$
$\displaystyle-S^{\prime\prime}-\frac{1}{\pi}\left[\sum_{h,u}\left|\mathcal{N}_{hu}\right|^{2}\zeta\left(\frac{m_{h}^{2}}{m_{W}^{2}},\frac{m_{u}^{2}}{m_{W}^{2}}\right)+\sum_{u,d}\left|\mathcal{V}_{ud}\right|^{2}\zeta\left(\frac{m_{u}^{2}}{m_{W}^{2}},\frac{m_{d}^{2}}{m_{W}^{2}}\right)\right.$
(32amb)
$\displaystyle\left.+\sum_{d,l}\left|\mathcal{Q}_{dl}\right|^{2}\zeta\left(\frac{m_{d}^{2}}{m_{W}^{2}},\frac{m_{l}^{2}}{m_{W}^{2}}\right)\right],$

where the function $\zeta$ is given in Eq. (31); and

$\displaystyle S^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\left\\{\sum_{h,h^{\prime}}\left|\mathcal{H}_{hh^{\prime}}\right|^{2}B_{00}^{\prime}\left(0,m_{h}^{2},m_{h^{\prime}}^{2}\right)+\sum_{u,u^{\prime}}\left|\mathcal{U}_{uu^{\prime}}\right|^{2}B_{00}^{\prime}\left(0,m_{u}^{2},m_{u^{\prime}}^{2}\right)\right.$
(32ana)
$\displaystyle+\sum_{d,d^{\prime}}\left|\mathcal{D}_{dd^{\prime}}\right|^{2}B_{00}^{\prime}\left(0,m_{d}^{2},m_{d^{\prime}}^{2}\right)+\sum_{l,l^{\prime}}\left|\mathcal{L}_{ll^{\prime}}\right|^{2}B_{00}^{\prime}\left(0,m_{l}^{2},m_{l^{\prime}}^{2}\right)$
$\displaystyle-2\left[Q_{h}\,\sum_{h}\mathcal{H}_{hh}\,B_{00}^{\prime}\left(0,m_{h}^{2},m_{h}^{2}\right)+Q_{u}\,\sum_{u}\mathcal{U}_{uu}\,B_{00}^{\prime}\left(0,m_{u}^{2},m_{u}^{2}\right)\right]$
$\displaystyle\left.+2\left[Q_{d}\,\sum_{d}\mathcal{D}_{dd}\,B_{00}^{\prime}\left(0,m_{d}^{2},m_{d}^{2}\right)+Q_{l}\,\sum_{l}\mathcal{L}_{ll}\,B_{00}^{\prime}\left(0,m_{l}^{2},m_{l}^{2}\right)\right]\right\\},$
$\displaystyle U^{\prime}$ $\displaystyle=$
$\displaystyle-S^{\prime}+\frac{2}{\pi}\left[\sum_{h,u}\left|\mathcal{N}_{hu}\right|^{2}B_{00}^{\prime}\left(0,m_{h}^{2},m_{u}^{2}\right)+\sum_{u,d}\left|\mathcal{V}_{ud}\right|^{2}B_{00}^{\prime}\left(0,m_{u}^{2},m_{d}^{2}\right)\right.$
(32anb)
$\displaystyle+\sum_{d,l}\left|\mathcal{Q}_{dl}\right|^{2}B_{00}^{\prime}\left(0,m_{d}^{2},m_{l}^{2}\right)$
$\displaystyle-
Q_{h}\,\sum_{h}\mathcal{H}_{hh}\,B_{00}^{\prime}\left(0,m_{h}^{2},m_{h}^{2}\right)-Q_{u}\,\sum_{u}\mathcal{U}_{uu}\,B_{00}^{\prime}\left(0,m_{u}^{2},m_{u}^{2}\right)$
$\displaystyle\left.+Q_{d}\,\sum_{d}\mathcal{D}_{dd}\,B_{00}^{\prime}\left(0,m_{d}^{2},m_{d}^{2}\right)+Q_{l}\,\sum_{l}\mathcal{L}_{ll}\,B_{00}^{\prime}\left(0,m_{l}^{2},m_{l}^{2}\right)\right].$

Utilizing Eq. (22), and the cancellation of the divergences proved in Appendix
C, one simplifies Eqs. (32an) to

$\displaystyle S^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{12\pi}\left\\{\sum_{h<h^{\prime}}\left|\mathcal{H}_{hh^{\prime}}\right|^{2}g\left(\frac{m_{h}^{2}}{m_{h^{\prime}}^{2}}\right)+\sum_{u<u^{\prime}}\left|\mathcal{U}_{uu^{\prime}}\right|^{2}g\left(\frac{m_{u}^{2}}{m_{u^{\prime}}^{2}}\right)\right.$
(32aod)
$\displaystyle+\sum_{d<d^{\prime}}\left|\mathcal{D}_{dd^{\prime}}\right|^{2}g\left(\frac{m_{d}^{2}}{m_{d^{\prime}}^{2}}\right)+\sum_{l<l^{\prime}}\left|\mathcal{L}_{ll^{\prime}}\right|^{2}g\left(\frac{m_{l}^{2}}{m_{l^{\prime}}^{2}}\right)$
$\displaystyle+\sum_{h}\left[\left(\mathcal{H}^{2}\right)_{hh}-2Q_{h}\mathcal{H}_{hh}\right]\ln{\frac{m_{h}^{2}}{\mu^{2}}}+\sum_{u}\left[\left(\mathcal{U}^{2}\right)_{uu}-2Q_{u}\mathcal{U}_{uu}\right]\ln{\frac{m_{u}^{2}}{\mu^{2}}}$
$\displaystyle\left.+\sum_{d}\left[\left(\mathcal{D}^{2}\right)_{dd}+2Q_{d}\mathcal{D}_{dd}\right]\ln{\frac{m_{d}^{2}}{\mu^{2}}}+\sum_{l}\left[\left(\mathcal{L}^{2}\right)_{ll}+2Q_{l}\mathcal{L}_{ll}\right]\ln{\frac{m_{l}^{2}}{\mu^{2}}}\right\\},$
$\displaystyle U^{\prime}$ $\displaystyle=$
$\displaystyle-S^{\prime}+\frac{1}{12\pi}\left\\{\sum_{h,u}\left|\mathcal{N}_{hu}\right|^{2}g\left(\frac{m_{h}^{2}}{m_{u}^{2}}\right)+\sum_{u,d}\left|\mathcal{V}_{ud}\right|^{2}g\left(\frac{m_{u}^{2}}{m_{d}^{2}}\right)+\sum_{d,l}\left|\mathcal{Q}_{dl}\right|^{2}g\left(\frac{m_{d}^{2}}{m_{l}^{2}}\right)\right.\hskip
19.91692pt$ (32aog)
$\displaystyle+\left(1-2Q_{h}\right)\sum_{h}\mathcal{H}_{hh}\ln{\frac{m_{h}^{2}}{\mu^{2}}}+\sum_{u}\left[\left(\mathcal{N}^{\dagger}\mathcal{N}+\mathcal{V}\mathcal{V}^{\dagger}\right)_{uu}-2Q_{u}\mathcal{U}_{uu}\right]\ln{\frac{m_{u}^{2}}{\mu^{2}}}$
$\displaystyle\left.+\sum_{d}\left[\left(\mathcal{V}^{\dagger}\mathcal{V}+\mathcal{Q}\mathcal{Q}^{\dagger}\right)_{dd}+2Q_{d}\mathcal{D}_{dd}\right]\ln{\frac{m_{d}^{2}}{\mu^{2}}}+\left(1+2Q_{l}\right)\sum_{l}\mathcal{L}_{ll}\ln{\frac{m_{l}^{2}}{\mu^{2}}}\right\\}.$

The function $g$ in lines (32aod), (32aod), and (32aog) is given in Eq. (23).
In lines (32aod), (32aod), (32aog), and (32aog) $\mu$ is the arbitrary mass
introduced in Eqs. (20); $S^{\prime}$ and $U^{\prime}$ do not depend on the
value of $\mu$, just as they do not depend on the divergent quantity of Eq.
(C3).

##  5 Models with few parameters

We present in this section three examples of leptoquark models, each of them
with two leptoquark multiplets and one mixing angle. For the sake of
simplicity, for each model we only write down the expressions for the three
original oblique parameters $T$, $S^{\prime}$, and $U^{\prime}$. Note that we
multiply the original Eqs. (32aj) and (32ao) by a factor 3 because of the
three possible colours of each leptoquark.

###  5.1
$\left(n_{\sigma,-2},n_{\sigma,-8},n_{\delta,7},n_{\delta,1},n_{\tau,-2}\right)=\left(1,0,0,1,0\right)$

There is one $u$-type scalar with mass $m_{u}$ and there are two $d$-type
scalars with masses $m_{d_{1}}$ and $m_{d_{2}}$. The matrices $U_{2}=1$,
$D_{1}=\left(c_{\theta},\ -s_{\theta}\right)$, and $D_{2}=\left(s_{\theta},\
c_{\theta}\right)$, where $\theta$ is a mixing angle. Therefore, the matrices
$\mathcal{N}$, $\mathcal{Q}$, $\mathcal{H}$, and $\mathcal{L}$ do not exist,
$\mathcal{V}=\left(s_{\theta},\ c_{\theta}\right)$, $\mathcal{U}=1$, and
$\mathcal{D}=\begin{pmatrix}s_{\theta}^{2}&c_{\theta}s_{\theta}\\\
c_{\theta}s_{\theta}&c_{\theta}^{2}\end{pmatrix}$. So,

$\displaystyle T$ $\displaystyle=$ $\displaystyle\frac{3}{16\pi
s_{W}^{2}m_{W}^{2}}\left[s_{\theta}^{2}\
\theta_{+}\left(m_{u}^{2},m_{d_{1}}^{2}\right)+c_{\theta}^{2}\
\theta_{+}\left(m_{u}^{2},m_{d_{2}}^{2}\right)-c_{\theta}^{2}s_{\theta}^{2}\
\theta_{+}\left(m_{d_{1}}^{2},m_{d_{2}}^{2}\right)\right],$ (32apa)
$\displaystyle S^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{12\pi}\left[3\,c_{\theta}^{2}s_{\theta}^{2}\
g\left(\frac{m_{d_{1}}^{2}}{m_{d_{2}}^{2}}\right)+s_{\theta}^{2}\,\ln{\frac{m_{d_{1}}^{2}}{m_{u}^{2}}}+c_{\theta}^{2}\,\ln{\frac{m_{d_{2}}^{2}}{m_{u}^{2}}}\right],$
(32apb) $\displaystyle U^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\left[s_{\theta}^{2}\,g\left(\frac{m_{u}^{2}}{m_{d_{1}}^{2}}\right)+c_{\theta}^{2}\,g\left(\frac{m_{u}^{2}}{m_{d_{2}}^{2}}\right)-c_{\theta}^{2}s_{\theta}^{2}\
g\left(\frac{m_{d_{1}}^{2}}{m_{d_{2}}^{2}}\right)\right].$ (32apc)

###  5.2
$\left(n_{\sigma,-2},n_{\sigma,-8},n_{\delta,7},n_{\delta,1},n_{\tau,-2}\right)=\left(0,0,1,1,0\right)$

The is one $h$-type leptoquark with mass $m_{h}$, one $d$-type leptoquark with
mass $m_{d}$, and two up-type leptoquarks with masses $m_{u1}$ and $m_{u2}$.
The matrices $U_{3}$, $D_{1}$, $D_{3}$, $L_{1}$, and $L_{2}$ do not exist; the
matrices $H_{1}=D_{2}=1$ are just numbers. We have
$\mathcal{N}=U_{1}=\left(c_{\theta},\ -s_{\theta}\right)$ and
$\mathcal{V}^{\dagger}=U_{2}=\left(s_{\theta},\ c_{\theta}\right)$; the
matrices $\mathcal{Q}$ and $\mathcal{L}$ do not exist,
$\mathcal{H}=\mathcal{D}=1$, and
$\mathcal{U}=\left(\begin{array}[]{cc}s_{\theta}^{2}-c_{\theta}^{2}&2s_{\theta}c_{\theta}\\\
2s_{\theta}c_{\theta}&c_{\theta}^{2}-s_{\theta}^{2}\end{array}\right)$. So,

$\displaystyle T$ $\displaystyle=$ $\displaystyle\frac{3}{16\pi
s_{W}^{2}m_{W}^{2}}\left\\{c_{\theta}^{2}\left[\theta_{+}\left(m_{h}^{2},m_{u1}^{2}\right)+\theta_{+}\left(m_{u2}^{2},m_{d}^{2}\right)\right]+s_{\theta}^{2}\left[\theta_{+}\left(m_{h}^{2},m_{u2}^{2}\right)+\theta_{+}\left(m_{u1}^{2},m_{d}^{2}\right)\right]\right.$
(32asa)
$\displaystyle\left.-4c_{\theta}^{2}s_{\theta}^{2}\,\theta_{+}\left(m_{u1}^{2},m_{u2}^{2}\right)\right\\},$
$\displaystyle S^{\prime}$ $\displaystyle=$
$\displaystyle\frac{c_{\theta}^{2}s_{\theta}^{2}}{\pi}\,g\left(\frac{m_{u1}^{2}}{m_{u2}^{2}}\right)+\frac{1}{12\pi}\left(7\ln{\frac{m_{u1}^{2}}{m_{h}^{2}}}+\ln{\frac{m_{d}^{2}}{m_{u2}^{2}}}+8s_{\theta}^{2}\,\ln{\frac{m_{u2}^{2}}{m_{u1}^{2}}}\right),$
(32asb) $\displaystyle U^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\left\\{c_{\theta}^{2}\left[g\left(\frac{m_{h}^{2}}{m_{u1}^{2}}\right)+g\left(\frac{m_{u2}^{2}}{m_{d}^{2}}\right)\right]+s_{\theta}^{2}\left[g\left(\frac{m_{h}^{2}}{m_{u2}^{2}}\right)+g\left(\frac{m_{u1}^{2}}{m_{d}^{2}}\right)\right]-4c_{\theta}^{2}s_{\theta}^{2}\,g\left(\frac{m_{u1}^{2}}{m_{u2}^{2}}\right)\right\\}.$

###  5.3
$\left(n_{\sigma,-2},n_{\sigma,-8},n_{\delta,7},n_{\delta,1},n_{\tau,-2}\right)=\left(0,0,1,0,1\right)$

There are one $h$-type leptoquark with mass $m_{h}$, two up-type leptoquarks
with masses $m_{u1}$ and $m_{u2}$, respectively, one $d$-type leptoquark with
mass $m_{d}$, and one $l$-type leptoquark with mass $m_{l}$. The matrices
$U_{2}$, $D_{1}$, $D_{2}$, and $L_{1}$ do not exist; the matrices
$H_{1}=D_{3}=L_{2}=1$ are just numbers, while
$\mathcal{N}=U_{1}=\left(c_{\theta},\ -s_{\theta}\right)$ and
$\mathcal{V}^{\dagger}\left/\sqrt{2}\right.=U_{3}=\left(s_{\theta},\
c_{\theta}\right)$. The matrix $\mathcal{Q}=\sqrt{2}$ is a number.
Furthermore, $\mathcal{H}=1$, $\mathcal{D}=0$, $\mathcal{L}=2$, and
$\mathcal{U}=\left(\begin{array}[]{cc}3s_{\theta}^{2}-1&3s_{\theta}c_{\theta}\\\
3s_{\theta}c_{\theta}&3c_{\theta}^{2}-1\end{array}\right)$. So,

$\displaystyle T$ $\displaystyle=$ $\displaystyle\frac{3}{16\pi
s_{W}^{2}m_{W}^{2}}\left\\{c_{\theta}^{2}\left[\theta_{+}\left(m_{h}^{2},m_{u1}^{2}\right)+2\,\theta_{+}\left(m_{u2}^{2},m_{d}^{2}\right)\right]\right.$
(32ava)
$\displaystyle+s_{\theta}^{2}\left[\theta_{+}\left(m_{h}^{2},m_{u2}^{2}\right)+2\,\theta_{+}\left(m_{u1}^{2},m_{d}^{2}\right)\right]$
$\displaystyle\left.+2\,\theta_{+}\left(m_{d}^{2},m_{l}^{2}\right)-9c_{\theta}^{2}s_{\theta}^{2}\,\theta_{+}\left(m_{u1}^{2},m_{u2}^{2}\right)\right\\},$
$\displaystyle S^{\prime}$ $\displaystyle=$
$\displaystyle\frac{9c_{\theta}^{2}s_{\theta}^{2}}{4\pi}\,g\left(\frac{m_{u1}^{2}}{m_{u2}^{2}}\right)+\frac{1}{12\pi}\left(7\ln{\frac{m_{u1}^{2}}{m_{h}^{2}}}+4\ln{\frac{m_{u2}^{2}}{m_{l}^{2}}}+3s_{\theta}^{2}\,\ln{\frac{m_{u2}^{2}}{m_{u1}^{2}}}\right),$
(32avb) $\displaystyle U^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\left\\{c_{\theta}^{2}\left[g\left(\frac{m_{h}^{2}}{m_{u1}^{2}}\right)+2g\left(\frac{m_{u2}^{2}}{m_{d}^{2}}\right)\right]+s_{\theta}^{2}\left[g\left(\frac{m_{h}^{2}}{m_{u2}^{2}}\right)+2g\left(\frac{m_{u1}^{2}}{m_{d}^{2}}\right)\right]\right.$
(32avc)
$\displaystyle\left.+2g\left(\frac{m_{l}^{2}}{m_{d}^{2}}\right)-9c_{\theta}^{2}s_{\theta}^{2}\,g\left(\frac{m_{u1}^{2}}{m_{u2}^{2}}\right)\right\\}+\frac{1}{2\pi}\left(s_{\theta}^{2}\ln{\frac{m_{d}^{2}}{m_{u1}^{2}}}+c_{\theta}^{2}\ln{\frac{m_{d}^{2}}{m_{u2}^{2}}}+\ln{\frac{m_{d}^{2}}{m_{l}^{2}}}\right).$

##  6 Generalization

In this section we generalize the formulas that we derived for leptoquarks. We
consider a set of NP physical scalars with any electric charges, making the
provisos that none of the new scalars has vacuum expectation value and that
they do not mix with the scalars of the SM. Also, we assume that all the new
scalar fields are complex.

We consider a set of physical New Physics scalars $s$ that have electric
charges $Q_{s}$ (i.e. each $s$ has a corresponding $Q_{s}$) which differ among
themselves by integers.131313If in the NP model under consideration there are
physical scalars with electric charges that differ among themselves by non-
integer values, then they must be treated separately. For instance, a set of
NP scalars with electric charges $5/3$, $2/3$, $-1/3$, and $-4/3$ (like the
leptoquarks) must be treated separately from a set of NP scalars with electric
charges $3$, $2$, and $1$. The charges $Q_{s}$ may be either positive or
negative.

We place all the scalars of each set in a column vector, ordering the scalars
by their decreasing electric charges; this vector generalizes the vector

$\left(\begin{array}[]{c}h\\\ u\\\ d\\\ l\end{array}\right)$ (32ba)

wherein sit the $n_{h}+n_{u}+n_{d}+n_{l}$ leptoquarks of Section 2.

The column vector of the previous paragraph must constitute a—in general,
reducible—representation of the gauge group $SU(2)$. Therefore, the following
operators, represented by matrices, act on it:

  1. 1.

The operator $M=\sqrt{2}\,T_{+}$, where $T_{+}$ is the raising operator of the
gauge group $SU(2)$.

  2. 2.

The operator $M^{\dagger}=\sqrt{2}\,T_{-}$, where $T_{-}$ is the lowering
operator of $SU(2)$.

  3. 3.

The Hermitian operator $\bar{M}\equiv 2T_{3}$, where $T_{3}$ is the third
generator of $SU(2)$.

Since the algebra of $SU(2)$ has $T_{3}=\left[T_{+},T_{-}\right]$,

$\bar{M}=MM^{\dagger}-M^{\dagger}M.$ (32bb)

The three operators are, of course, represented by matrices that act on the
column vector. The matrix $M$ is off-diagonal: it only has nonzero matrix
elements $M_{ss^{\prime}}$ for $Q_{s^{\prime}}=Q_{s}-1$. Conversely, the
nonzero matrix elements $\left(M^{\dagger}\right)_{ss^{\prime}}$ only occur
for $Q_{s}=Q_{s^{\prime}}-1$. The matrix $\bar{M}$ only has nonzero matrix
elements $\bar{M}_{ss^{\prime}}$ when $Q_{s^{\prime}}=Q_{s}$. Since the
scalars in the column vector are ordered by decreasing electric charges, one
has

$M=\left(\begin{array}[]{ccccccc}0&M_{1}&0&0&\cdots&0&0\\\
0&0&M_{2}&0&\cdots&0&0\\\ 0&0&0&M_{3}&\cdots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&0&0&\cdots&0&M_{n}\\\
0&0&0&0&\cdots&0&0\end{array}\right).$ (32bi)

For instance, in our leptoquark model we had $M_{1}=\mathcal{N}$,
$M_{2}=\mathcal{V}$, and $M_{3}=\mathcal{Q}$. Clearly then,

$\bar{M}=\left(\begin{array}[]{cccccc}M_{1}M_{1}^{\dagger}&0&0&\cdots&0&0\\\
0&M_{2}M_{2}^{\dagger}-M_{1}^{\dagger}M_{1}&0&\cdots&0&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&0&\cdots&M_{n}M_{n}^{\dagger}-M_{n-1}^{\dagger}M_{n-1}&0\\\
0&0&0&\cdots&0&-M_{n}^{\dagger}M_{n}\end{array}\right).$ (32bo)

The electric-charge operator is given in this basis by

$Q=\left(\begin{array}[]{cccccc}Q_{1}\times\mathbbm{1}&0&0&\cdots&0&0\\\
0&Q_{2}\times\mathbbm{1}&0&\cdots&0&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&0&\cdots&Q_{n}\times\mathbbm{1}&0\\\
0&0&0&\cdots&0&Q_{n+1}\times\mathbbm{1}\end{array}\right),$ (32bu)

where the unit matrices $\mathbbm{1}$ have the appropriate dimensions, which
may not all be equal. Note that

$Q_{1}=Q_{2}+1=\cdots=Q_{n}+\left(n-1\right)=Q_{n+1}+n.$ (32bv)

One must also take into account the $SU(2)$ commutation relation
$\left[T_{3},T_{+}\right]=T_{+}$. With $T_{3}=\bar{M}/2$ and
$T_{+}=M\left/\sqrt{2}\right.$, it yields $\left[\bar{M},M\right]=2M$. Hence,
the submatrices in Eq. (32bi) are related by

$\displaystyle 2M_{1}$ $\displaystyle=$ $\displaystyle
2M_{1}M_{1}^{\dagger}M_{1}-M_{1}M_{2}M_{2}^{\dagger},$ (32bwa) $\displaystyle
2M_{2}$ $\displaystyle=$ $\displaystyle
2M_{2}M_{2}^{\dagger}M_{2}-M_{1}^{\dagger}M_{1}M_{2}-M_{2}M_{3}M_{3}^{\dagger},$
(32bwb) $\displaystyle\vdots$ $\displaystyle\vdots$ $\displaystyle 2M_{n-1}$
$\displaystyle=$ $\displaystyle
2M_{n-1}M_{n-1}^{\dagger}M_{n-1}-M_{n-2}^{\dagger}M_{n-2}M_{n-1}-M_{n-1}M_{n}M_{n}^{\dagger},$
(32bwc) $\displaystyle 2M_{n}$ $\displaystyle=$ $\displaystyle
2M_{n}M_{n}^{\dagger}M_{n}-M_{n-1}^{\dagger}M_{n-1}M_{n}.$ (32bwd)

The gauge-kinetic Lagrangian for the NP scalars is

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\sum_{s}\left(\partial_{\mu}s+igs_{W}A_{\mu}Q_{s}s-i\,\frac{g}{2c_{W}}\,Z_{\mu}\sum_{s^{\prime}}F_{ss^{\prime}}s^{\prime}\right.$
(32bxb)
$\displaystyle\left.-i\,\frac{g}{\sqrt{2}}\,W_{\mu}^{+}\sum_{s^{\prime}}M_{ss^{\prime}}s^{\prime}-i\,\frac{g}{\sqrt{2}}\,W_{\mu}^{-}\sum_{s^{\prime}}M^{\dagger}_{ss^{\prime}}s^{\prime}\right)$
$\displaystyle\times\left(\partial^{\mu}s^{\ast}-igs_{W}A^{\mu}Q_{s}s^{\ast}+i\,\frac{g}{2c_{W}}\,Z^{\mu}\sum_{s^{\prime\prime}}F^{\ast}_{ss^{\prime\prime}}{s^{\prime\prime}}^{\ast}\right.$
$\displaystyle\left.+i\,\frac{g}{\sqrt{2}}\,W^{\mu-}\sum_{s^{\prime\prime}}M^{\ast}_{ss^{\prime\prime}}{s^{\prime\prime}}^{\ast}+i\,\frac{g}{\sqrt{2}}\,W^{\mu+}\sum_{s^{\prime\prime}}M^{T}_{ss^{\prime\prime}}{s^{\prime\prime}}^{\ast}\right),$

where

$F_{ss^{\prime}}\equiv\bar{M}_{ss^{\prime}}-2s_{W}^{2}Q_{s}\delta_{ss^{\prime}}.$
(32by)

The Lagrangian (32bx) produces the following interactions of the scalars with
the photons:

$\displaystyle\mathcal{L}_{ASS}$ $\displaystyle=$ $\displaystyle
igs_{W}A_{\theta}\sum_{s}Q_{s}\left(s\partial^{\theta}s^{\ast}-s^{\ast}\partial^{\theta}s\right)$
(32bza) $\displaystyle\mathcal{L}_{AASS}$ $\displaystyle=$ $\displaystyle
g^{2}s_{W}^{2}A_{\theta}A^{\theta}\sum_{s}Q_{s}^{2}\,ss^{\ast}.$ (32bzb)

It also produces interactions of the scalars with the $W$ gauge bosons, given
by

$\displaystyle\mathcal{L}_{WSS}$ $\displaystyle=$ $\displaystyle
i\,\frac{g}{\sqrt{2}}\,\sum_{s,s^{\prime}}\left[\vphantom{\left(M^{\dagger}\right)_{ss^{\prime}}}W_{\theta}^{+}M_{ss^{\prime}}\left(s^{\ast}\partial^{\theta}s^{\prime}-s^{\prime}\partial^{\theta}s^{\ast}\right)+W_{\theta}^{-}\left(M^{\dagger}\right)_{ss^{\prime}}\left(s^{\ast}\partial^{\theta}s^{\prime}-s^{\prime}\partial^{\theta}s^{\ast}\right)\right],\hskip
19.91692pt$ (32caa) $\displaystyle\mathcal{L}_{WWSS}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{2}\,W_{\theta}^{+}W^{\theta-}\sum_{s,s^{\prime}}\left(MM^{\dagger}+M^{\dagger}M\right)_{ss^{\prime}}s^{\ast}s^{\prime}.$
(32cab)

The interactions of the scalars with the $Z$ are given by

$\displaystyle\mathcal{L}_{ZSS}$ $\displaystyle=$ $\displaystyle
i\,\frac{g}{2c_{W}}\,Z_{\theta}\sum_{s,s^{\prime}}F_{ss^{\prime}}\left(s^{\ast}\partial^{\theta}s^{\prime}-s^{\prime}\partial^{\theta}s^{\ast}\right),$
(32cba) $\displaystyle\mathcal{L}_{ZZSS}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{4c_{W}^{2}}\,Z_{\theta}Z^{\theta}\sum_{s,s^{\prime}}\left(F^{2}\right)_{ss^{\prime}}s^{\ast}s^{\prime}.$
(32cbb)

Last but not least, there is the ‘seagull’ interaction of a photon and a $Z$
with two scalars, given by

$\mathcal{L}_{AZSS}=-\frac{g^{2}s_{W}}{c_{W}}\,A_{\theta}Z^{\theta}\sum_{s,s^{\prime}}Q_{s}F_{ss^{\prime}}s^{\ast}s^{\prime}.$
(32cc)

Therefore,

$\displaystyle A_{\gamma\gamma}\left(q^{2}\right)$ $\displaystyle=$
$\displaystyle\frac{g^{2}s_{W}^{2}}{4\pi^{2}}\,\sum_{s}\,Q_{s}^{2}\left[B_{00}\left(q^{2},m_{s}^{2},m_{s}^{2}\right)-\frac{A_{0}\left(m_{s}^{2}\right)}{2}\right],$
(32cda) $\displaystyle A_{\gamma Z}\left(q^{2}\right)$ $\displaystyle=$
$\displaystyle-\frac{g^{2}s_{W}}{8\pi^{2}c_{W}}\,\sum_{s}\,Q_{s}F_{ss}\left[B_{00}\left(q^{2},m_{s}^{2},m_{s}^{2}\right)-\frac{A_{0}\left(m_{s}^{2}\right)}{2}\right],$
(32cdb) $\displaystyle A_{ZZ}\left(q^{2}\right)$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{16\pi^{2}c_{W}^{2}}\left[\sum_{s,s^{\prime}}\left|F_{ss^{\prime}}\right|^{2}B_{00}\left(q^{2},m_{s}^{2},m_{s^{\prime}}^{2}\right)-\sum_{s}\left(F^{2}\right)_{ss}\frac{A_{0}\left(m_{s}^{2}\right)}{2}\right],$
(32cdc) $\displaystyle A_{WW}\left(q^{2}\right)$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{8\pi^{2}}\left[\sum_{ss^{\prime}}\left|M_{ss^{\prime}}\right|^{2}B_{00}\left(q^{2},m_{s}^{2},m_{s^{\prime}}^{2}\right)-\sum_{s}\left(MM^{\dagger}+M^{\dagger}M\right)_{ss}\frac{A_{0}\left(m_{s}^{2}\right)}{4}\right].$

It immediately follows that

$\displaystyle T$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi
s_{W}^{2}m_{W}^{2}}\left[\sum_{s,s^{\prime}}\left|M_{ss^{\prime}}\right|^{2}B_{00}\left(0,m_{s}^{2},m_{s^{\prime}}^{2}\right)-\sum_{s}\left(MM^{\dagger}+M^{\dagger}M\right)_{ss}\frac{A_{0}\left(m_{s}^{2}\right)}{4}\right.$
(32ce)
$\displaystyle\left.-\frac{1}{2}\,\sum_{s,s^{\prime}}\left|F_{ss^{\prime}}\right|^{2}B_{00}\left(0,m_{s}^{2},m_{s^{\prime}}^{2}\right)+\sum_{s}\left(F^{2}\right)_{ss}\frac{A_{0}\left(m_{s}^{2}\right)}{4}\right],$

where we have used $c_{W}^{2}m_{Z}^{2}=m_{W}^{2}$. Employing Eq. (24), one
obtains

$T=\frac{1}{16\pi
s_{W}^{2}m_{W}^{2}}\left[\sum_{s,s^{\prime}}\left|M_{ss^{\prime}}\right|^{2}\theta_{+}\left(m_{s}^{2},m_{s^{\prime}}^{2}\right)-\sum_{s<s^{\prime}}\left|\bar{M}_{ss^{\prime}}\right|^{2}\theta_{+}\left(m_{s}^{2},m_{s^{\prime}}^{2}\right)\right].$
(32cf)

One furthermore obtains

$\displaystyle V$ $\displaystyle=$ $\displaystyle\frac{1}{8\pi
c_{W}^{2}s_{W}^{2}}\,\sum_{s,s^{\prime}}\left|F_{ss^{\prime}}\right|^{2}\rho\left(\frac{m_{s}^{2}}{m_{Z}^{2}},\frac{m_{s^{\prime}}^{2}}{m_{Z}^{2}}\right),$
(32cga) $\displaystyle W$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi
s_{W}^{2}}\,\sum_{s,s^{\prime}}\left|M_{ss^{\prime}}\right|^{2}\rho\left(\frac{m_{s}^{2}}{m_{W}^{2}},\frac{m_{s^{\prime}}^{2}}{m_{W}^{2}}\right),$
(32cgb)

and

$\displaystyle S^{\prime\prime}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\pi}\,\sum_{s,s^{\prime}}\left|F_{ss^{\prime}}\right|^{2}\,\zeta\left(\frac{m_{s}^{2}}{m_{Z}^{2}},\frac{m_{s^{\prime}}^{2}}{m_{Z}^{2}}\right),$
(32cha) $\displaystyle U^{\prime\prime}$ $\displaystyle=$
$\displaystyle-S^{\prime\prime}-\frac{1}{\pi}\,\sum_{s,s^{\prime}}\left|M_{ss^{\prime}}\right|^{2}\,\zeta\left(\frac{m_{s}^{2}}{m_{W}^{2}},\frac{m_{s^{\prime}}^{2}}{m_{W}^{2}}\right),$
(32chb) $\displaystyle X$ $\displaystyle=$
$\displaystyle-\frac{1}{4\pi}\,\sum_{s}Q_{s}F_{ss}\
\zeta\left(\frac{m_{s}^{2}}{m_{Z}^{2}},\frac{m_{s}^{2}}{m_{Z}^{2}}\right).$
(32chc)

On the other hand,

$S^{\prime}=\frac{1}{\pi}\,\sum_{s,s^{\prime}}\left|\bar{M}_{ss^{\prime}}\right|^{2}B_{00}^{\prime}\left(0,m_{s}^{2},m_{s^{\prime}}^{2}\right)-\frac{2}{\pi}\,\sum_{s}Q_{s}\bar{M}_{ss}\,B_{00}^{\prime}\left(0,m_{s}^{2},m_{s}^{2}\right)$
(32ci)

and

$U^{\prime}=-S^{\prime}+\frac{2}{\pi}\left[\sum_{s,s^{\prime}}\left|M_{ss^{\prime}}\right|^{2}B_{00}^{\prime}\left(0,m_{s}^{2},m_{s^{\prime}}^{2}\right)-\sum_{s}Q_{s}\bar{M}_{ss}B_{00}^{\prime}\left(0,m_{s}^{2},m_{s}^{2}\right)\right].$
(32cj)

In order to establish the finiteness of $S^{\prime}$ and of
$S^{\prime}+U^{\prime}$ we have to make sure that

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\sum_{s,s^{\prime}}\left|\bar{M}_{ss^{\prime}}\right|^{2}-2\,\sum_{s}Q_{s}\bar{M}_{ss},$
(32cka) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\sum_{d,s^{\prime}}\left|M_{ss^{\prime}}\right|^{2}-\sum_{s}Q_{s}\bar{M}_{ss},$
(32ckb)

respectively. Now,

$\displaystyle\sum_{s,s^{\prime}}\left|M_{ss^{\prime}}\right|^{2}$
$\displaystyle=$ $\displaystyle\mathrm{tr}\left(MM^{\dagger}\right)$ (32cla)
$\displaystyle=$
$\displaystyle\mathrm{tr}\left(M_{1}M_{1}^{\dagger}+M_{2}M_{2}^{\dagger}+\cdots+M_{n}M_{n}^{\dagger}\right),$
$\displaystyle\sum_{s}Q_{s}\bar{M}_{ss}$ $\displaystyle=$
$\displaystyle\mathrm{tr}\left(Q\bar{M}\right)$ (32clb) $\displaystyle=$
$\displaystyle\mathrm{tr}\left[Q_{1}M_{1}M_{1}^{\dagger}+Q_{2}\left(M_{2}M_{2}^{\dagger}-M_{1}^{\dagger}M_{1}\right)+\cdots\right.$
$\displaystyle\left.+Q_{n}\left(M_{n}M_{n}^{\dagger}-M_{n-1}^{\dagger}M_{n-1}\right)-Q_{n+1}M_{n}^{\dagger}M_{n}\right]$
$\displaystyle=$
$\displaystyle\mathrm{tr}\left[\left(Q_{1}-Q_{2}\right)M_{1}M_{1}^{\dagger}+\left(Q_{2}-Q_{3}\right)M_{2}M_{2}^{\dagger}+\cdots\right.$
$\displaystyle\left.+\left(Q_{n}-Q_{n+1}\right)M_{n}M_{n}^{\dagger}\right],$

which proves Eq. (32ckb), because
$Q_{1}-Q_{2}=Q_{2}-Q_{3}=\cdots=Q_{n}-Q_{n+1}=1$. Moreover,

$\displaystyle\sum_{s,s^{\prime}}\left|\bar{M}_{ss^{\prime}}\right|^{2}$
$\displaystyle=$ $\displaystyle\mathrm{tr}\left(\bar{M}^{2}\right)$ (32cmb)
$\displaystyle=$
$\displaystyle\mathrm{tr}\left[\left(M_{1}M_{1}^{\dagger}\right)^{2}+\left(M_{2}M_{2}^{\dagger}-M_{1}^{\dagger}M_{1}\right)^{2}+\left(M_{3}M_{3}^{\dagger}-M_{2}^{\dagger}M_{2}\right)^{2}+\cdots\right.$
$\displaystyle\left.+\left(M_{n}M_{n}^{\dagger}-M_{n-1}^{\dagger}M_{n-1}\right)^{2}+\left(-M_{n}^{\dagger}M_{n}\right)^{2}\right]$
$\displaystyle=$ $\displaystyle
2\,\mathrm{tr}\left(M_{1}M_{1}^{\dagger}+M_{2}M_{2}^{\dagger}+M_{3}M_{3}^{\dagger}+\cdots+M_{n-1}M_{n-1}^{\dagger}+M_{n}M_{n}^{\dagger}\right),$
(32cmc)

because of Eqs. (32bw). Therefore, using Eq. (32clb), we see that Eq. (32cka)
is also true.

Employing Eq. (22) on Eqs. (32ci) and (32cj), one at last obtains

$\displaystyle S^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{24\pi}\left\\{\sum_{s,s^{\prime}}\left|\bar{M}_{ss^{\prime}}\right|^{2}g\left(\frac{m_{s}^{2}}{m_{s^{\prime}}^{2}}\right)+2\,\sum_{s}\left[\left(\bar{M}^{2}\right)_{ss}-2Q_{s}\bar{M}_{ss}\right]\ln{m_{s}^{2}}\right\\},$
(32cna) $\displaystyle U^{\prime}$ $\displaystyle=$
$\displaystyle-S^{\prime}+\frac{1}{12\pi}\left\\{\sum_{s,s^{\prime}}\left|M_{ss^{\prime}}\right|^{2}g\left(\frac{m_{s}^{2}}{m_{s^{\prime}}^{2}}\right)+\sum_{s}\left[\left(MM^{\dagger}+M^{\dagger}M\right)_{ss}-2Q_{s}\bar{M}_{ss}\right]\ln{m_{s}^{2}}\right\\}.$

##  7 Conclusions

In this paper we have derived general formulas for the oblique parameters in a
model with an arbitrary number of scalar leptoquarks. Those formulas include
the mixing matrices $\mathcal{N}$, $\mathcal{V}$, and $\mathcal{Q}$ that
appear in the charged current of Eq. (19c); and also the matrices $\bar{H}$,
$\bar{U}$, $\bar{D}$, and $\bar{L}$ that are derived from $\mathcal{N}$,
$\mathcal{V}$, and $\mathcal{Q}$ through Eqs. (17) and (18). The final
formulas for the oblique parameters are $T$ given by Eq. (32aj), $V$ and $W$
given by Eqs. (32ak), $X$ given by Eq. (32al), and $S$ and $U$ given by Eqs.
(32am) and (32ao). The relevant functions $g$, $\theta_{+}$, $\rho$, and
$\zeta$ are given by Eqs. (23), (25), (27), and (31), respectively.

We have then considered the more general situation where the New Physics is
constituted by (complex) physical scalars $s$ that originate in arbitrary
representations of the gauge group; those scalars are allowed to freely mix
among themselves, but not to mix with the scalar doublet of the SM. In this
case the fundamental mixing matrix is $M$ that appears in Eq. (32ca);
therefrom one derives $\bar{M}$ in Eq. (32bb) and $F$ in Eq. (32by). The final
results for the oblique parameters are then given in Eqs. (32cf), (32cg),
(32ch), and (32cn).

#### Acknowledgements:

The authors thank the Portuguese Foundation for Science and Technology for
support through the projects UIDB/00777/2020, UIDP/00777/2020, and CERN/FIS-
PAR/0002/2021. The work of F.A. was furthermore supported by grant
UI/BD/153763/2022. The work of L.L. was furthermore supported by project
CERN/FIS-PAR/0019/2021.

##  Appendix A Parameter counting

In our leptoquark models the total number of parameters is
$n_{\mathrm{total}}=n_{\mathrm{masses}}+n_{\mathrm{mixings}}$. Here,

$\displaystyle n_{\mathrm{masses}}$ $\displaystyle=$ $\displaystyle
n_{h}+n_{u}+n_{d}+n_{l}$ (A1) $\displaystyle=$ $\displaystyle
n_{\delta,7}+\left(n_{\delta,7}+n_{\delta,1}+n_{\tau,-2}\right)+\left(n_{\sigma,-2}+n_{\delta,1}+n_{\tau,-2}\right)+\left(n_{\sigma,-8}+n_{\tau,-2}\right)\hskip
14.22636pt$

is the total number of distinct leptoquark masses; $n_{\mathrm{mixings}}$ is
the number of independent parameters in the mixing matrices. The latter may be
computed as follows.

Since the matrices $\mathcal{H}$, $\mathcal{U}$, $\mathcal{D}$, and
$\mathcal{L}$ may be derived fom the matrices $\mathcal{N}$, $\mathcal{V}$,
and $\mathcal{Q}$ through Eqs. (17) and (18), in order to know
$n_{\mathrm{mixings}}$ one must count the parameters in the latter matrices.
They are derived from $H_{1}$, $U_{1}$, $U_{2}$, $U_{3}$, $D_{2}$, $D_{3}$,
and $L_{2}$ through Eqs. (16). Let us assume, for the sake of simplicity, the
latter matrices to be real. We use the fact that the first $m$ rows of an
$n\times n$ real orthogonal matrix may be parameterized by
$\left.m\left(2n-m-1\right)\right/2$ parameters. Therefore,

$\displaystyle H_{1}$ has
$\displaystyle\frac{n_{\delta,7}\left(n_{\delta,7}-1\right)}{2}\,\
\mbox{parameters},$ (A2a) $\displaystyle\left(\begin{array}[]{c}U_{1}\\\
U_{2}\\\ U_{3}\end{array}\right)$ has
$\displaystyle\frac{\left(n_{\delta,7}+n_{\delta,1}+n_{\tau,-2}\right)\left(n_{\delta,7}+n_{\delta,1}+n_{\tau,-2}-1\right)}{2}\,\
\mbox{parameters},$ (A2e) $\displaystyle\left(\begin{array}[]{c}D_{2}\\\
D_{3}\end{array}\right)$ has
$\displaystyle\frac{\left(n_{\delta,1}+n_{\tau,-2}\right)\left(2n_{\sigma,-2}+n_{\delta,1}+n_{\tau,-2}-1\right)}{2}\,\
\mbox{parameters},$ (A2h) $\displaystyle L_{2}$ has
$\displaystyle\frac{n_{\tau,-2}\left(2n_{\sigma,-8}+n_{\tau,-2}-1\right)}{2}\,\
\mbox{parameters}.$ (A2i)

However, one must take into account the fact that, in Eqs. (16), the
transformations

$\displaystyle H_{1}\to AH_{1},\ U_{1}\to AU_{1},$ (A3a) $\displaystyle
U_{2}\to BU_{2},\ D_{2}\to BD_{2},$ (A3b) $\displaystyle U_{3}\to CU_{3},\
D_{3}\to CD_{3},\ L_{2}\to CL_{2},$ (A3c)

where $A$, $B$, and $C$ are arbitrary $n_{\delta,7}\times n_{\delta,7}$,
$n_{\delta,1}\times n_{\delta,1}$, and $n_{\tau,-2}\times n_{\tau,-2}$ real
orthogonal matrices, respectively, leave $\mathcal{N}$, $\mathcal{V}$, and
$\mathcal{Q}$ invariant. Therefore the total number of parameters in
$\mathcal{N}$, $\mathcal{V}$, and $\mathcal{Q}$ is

$\displaystyle n_{\mathrm{mixings}}$ $\displaystyle=$
$\displaystyle\frac{n_{\delta,7}\left(n_{\delta,7}-1\right)}{2}+\frac{\left(n_{\delta,7}+n_{\delta,1}+n_{\tau,-2}\right)\left(n_{\delta,7}+n_{\delta,1}+n_{\tau,-2}-1\right)}{2}$
(A4)
$\displaystyle+\frac{\left(n_{\delta,1}+n_{\tau,-2}\right)\left(2n_{\sigma,-2}+n_{\delta,1}+n_{\tau,-2}-1\right)}{2}+\frac{n_{\tau,-2}\left(2n_{\sigma,-8}+n_{\tau,-2}-1\right)}{2}$
$\displaystyle-\frac{n_{\delta,7}\left(n_{\delta,7}-1\right)}{2}-\frac{n_{\delta,1}\left(n_{\delta,1}-1\right)}{2}-\frac{n_{\tau,-2}\left(n_{\tau,-2}-1\right)}{2}.$

Utilizing both Eqs. (A1) and (A4), we find that the leptoquark models with the
smallest numbers of parameters are the ones in Table 1.

$n_{\sigma,-2}$ | $n_{\sigma,-8}$ | $n_{\delta,1}$ | $n_{\delta,7}$ | $n_{\tau,-2}$ | $n_{h}$ | $n_{u}$ | $n_{d}$ | $n_{l}$ | $n_{\mathrm{masses}}$ | $n_{\mathrm{mixings}}$ | $n_{\mathrm{total}}$
---|---|---|---|---|---|---|---|---|---|---|---
0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 2 | 0 | 2
0 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 2 | 0 | 2
0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 3 | 0 | 3
1 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 3 | 0 | 3
0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 3 | 0 | 3
0 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 3 | 0 | 3
1 | 0 | 0 | 1 | 0 | 0 | 1 | 2 | 0 | 3 | 1 | 4
2 | 0 | 1 | 0 | 0 | 1 | 1 | 2 | 0 | 4 | 0 | 4
0 | 2 | 1 | 0 | 0 | 1 | 1 | 0 | 2 | 4 | 0 | 4
1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 4 | 0 | 4
0 | 2 | 0 | 1 | 0 | 0 | 1 | 1 | 2 | 4 | 0 | 4
0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 2 | 4 | 1 | 5
1 | 0 | 0 | 0 | 1 | 0 | 1 | 2 | 1 | 4 | 1 | 5
0 | 0 | 1 | 1 | 0 | 1 | 2 | 1 | 0 | 4 | 1 | 5
1 | 1 | 0 | 1 | 0 | 0 | 1 | 2 | 1 | 4 | 1 | 5
0 | 0 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 4 | 1 | 5
0 | 0 | 0 | 2 | 0 | 0 | 2 | 2 | 0 | 4 | 1 | 5
3 | 0 | 1 | 0 | 0 | 1 | 1 | 3 | 0 | 5 | 0 | 5
0 | 3 | 1 | 0 | 0 | 1 | 1 | 0 | 3 | 5 | 0 | 5
2 | 1 | 1 | 0 | 0 | 1 | 1 | 2 | 1 | 5 | 0 | 5
1 | 2 | 1 | 0 | 0 | 1 | 1 | 1 | 2 | 5 | 0 | 5
0 | 3 | 0 | 1 | 0 | 0 | 1 | 1 | 3 | 5 | 0 | 5
Table 1: The leptoquark models with the smallest numbers of parameters. Models
with $n_{\mathrm{total}}>5$ and (trivial) models with
$n_{\delta,1}=n_{\delta,7}=n_{\tau,-2}=0$ are not included.

##  Appendix B Parameterization of the mixing

One may choose to parameterize the mixing matrices in the following way:

  1. 1.

One sets $H_{1}=\mathbbm{1}_{n_{\delta,7}}$.

  2. 2.

One leaves the matrix

$\left(\begin{array}[]{c}U_{1}\\\ U_{2}\\\ U_{3}\end{array}\right)$ (B4)

fully free, i.e. with all its parameters.

  3. 3.

One parameterizes the matrix

$\left(\begin{array}[]{c}L_{1}\\\ L_{2}\end{array}\right)$ (B7)

through $\left.n_{\tau,-2}\left(n_{\tau,-2}-1\right)\right/2$ rotations among
the rows of $L_{2}$ followed by $n_{\sigma,-8}n_{\tau,-2}$ rotations between
the rows of $L_{1}$ and the ones of $L_{2}$. For instance, if
$n_{\sigma,-8}=n_{\tau,-2}=2$, one parameterizes the matrix (B7) as
$O_{34}O_{13}O_{14}O_{23}O_{24}$.

  4. 4.

One parameterizes the matrix

$\left(\begin{array}[]{c}D_{1}\\\ D_{2}\\\ D_{3}\end{array}\right)$ (B11)

through $n_{\sigma,-2}n_{\delta,1}$ rotations between the rows of $D_{1}$ and
the ones of $D_{2}$, followed by $n_{\sigma,-2}n_{\tau,-2}$ rotations between
the rows of $D_{1}$ and the ones of $D_{3}$ and by $n_{\delta,1}n_{\tau,-2}$
rotations between the rows of $D_{2}$ and the ones of $D_{3}$. For instance,
if $n_{\sigma,-2}=n_{\delta,1}=2$ and $n_{\tau,-2}=1$, one parameterizes the
matrix (B11) as $O_{13}O_{14}O_{23}O_{24}O_{15}O_{25}O_{35}O_{45}$.

##  Appendix C Cancellation of the divergences in $S^{\prime}$ and
$U^{\prime}$

Because of Eqs. (14),

$\displaystyle n_{h}$ $\displaystyle=$ $\displaystyle n_{\delta,7},$ (C1a)
$\displaystyle\mathrm{tr}\left(U_{1}^{\dagger}U_{1}\right)$ $\displaystyle=$
$\displaystyle n_{\delta,7},$ (C1b)
$\displaystyle\mathrm{tr}\left(U_{2}^{\dagger}U_{2}\right)$ $\displaystyle=$
$\displaystyle n_{\delta,1},$ (C1c)
$\displaystyle\mathrm{tr}\left(U_{3}^{\dagger}U_{3}\right)$ $\displaystyle=$
$\displaystyle n_{\tau,-2},$ (C1d)
$\displaystyle\mathrm{tr}\left(D_{2}^{\dagger}D_{2}\right)$ $\displaystyle=$
$\displaystyle n_{\delta,1},$ (C1e)
$\displaystyle\mathrm{tr}\left(D_{3}^{\dagger}D_{3}\right)$ $\displaystyle=$
$\displaystyle n_{\tau,-2},$ (C1f)
$\displaystyle\mathrm{tr}\left(L_{2}^{\dagger}L_{2}\right)$ $\displaystyle=$
$\displaystyle n_{\tau,-2}.$ (C1g)

Using Eqs. (16) and the unitarity of the matrices in Eq. (15), the matrices
(18) may be rewritten

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\mathbbm{1}_{n_{h}},$ (C2a) $\displaystyle\mathcal{U}$
$\displaystyle=$ $\displaystyle-
U_{1}^{\dagger}U_{1}+U_{2}^{\dagger}U_{2}+2\,U_{3}^{\dagger}U_{3},$ (C2b)
$\displaystyle\mathcal{D}$ $\displaystyle=$ $\displaystyle
D_{2}^{\dagger}D_{2},$ (C2c) $\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle 2\,L_{2}^{\dagger}L_{2}.$ (C2d)

#### Divergences:

When $\epsilon\to 0^{+}$, the quantity

$\mathrm{div}\equiv\frac{2}{\epsilon}-\gamma+\ln{\left(4\pi\right)}$ (C3)

diverges. In Eq. (C3), $\gamma$ is Euler’s constant. The PV function

$B_{00}\left(q^{2},m_{1}^{2},m_{2}^{2}\right)=\left(\frac{m_{1}^{2}+m_{2}^{2}}{4}-\frac{q^{2}}{12}\right)\mathrm{div}+\mathrm{convergent\
terms}.$ (C4)

Therefore,

$\displaystyle B_{00}^{\prime}\left(q^{2},m_{1}^{2},m_{2}^{2}\right)$
$\displaystyle=$ $\displaystyle-\frac{\mathrm{div}}{12}+\mathrm{convergent\
terms},$ (C5a)
$\displaystyle\bar{B}_{00}\left(q^{2},m_{1}^{2},m_{2}^{2}\right)$
$\displaystyle=$ $\displaystyle-\frac{\mathrm{div}}{12}+\mathrm{convergent\
terms}.$ (C5b)

####  $S^{\prime}$ divergence:

Taking into account Eqs. (C5), the divergent part of $S^{\prime}$ in Eq.
(32ana) is proportional to

$\displaystyle\mathrm{tr}\,\mathcal{H}^{2}-2Q_{h}\,\mathrm{tr}\,\mathcal{H}+\mathrm{tr}\,\mathcal{U}^{2}-2Q_{u}\,\mathrm{tr}\,\mathcal{U}+\mathrm{tr}\,\mathcal{D}^{2}+2Q_{d}\,\mathrm{tr}\,\mathcal{D}+\mathrm{tr}\,\mathcal{L}^{2}+2Q_{l}\,\mathrm{tr}\,\mathcal{L}$
(C6) $\displaystyle=$
$\displaystyle\mathrm{tr}\left(\mathbbm{1}_{n_{h}}\right)-2Q_{h}\,\mathrm{tr}\left(\mathbbm{1}_{n_{h}}\right)$
$\displaystyle+\mathrm{tr}\left(U_{1}^{\dagger}U_{1}+U_{2}^{\dagger}U_{2}+4U_{3}^{\dagger}U_{3}\right)-2Q_{u}\,\mathrm{tr}\left(-U_{1}^{\dagger}U_{1}+U_{2}^{\dagger}U_{2}+2U_{3}^{\dagger}U_{3}\right)$
$\displaystyle+\mathrm{tr}\left(D_{2}^{\dagger}D_{2}\right)+2Q_{d}\,\mathrm{tr}\left(D_{2}^{\dagger}D_{2}\right)$
$\displaystyle+4\,\mathrm{tr}\left(L_{2}^{\dagger}L_{2}\right)+4Q_{l}\,\mathrm{tr}\left(L_{2}^{\dagger}L_{2}\right)$
$\displaystyle=$ $\displaystyle n_{\delta,7}-2Q_{h}n_{\delta,7}$
$\displaystyle+n_{\delta,7}+n_{\delta,1}+4n_{\tau,-2}-2Q_{u}\left(-n_{\delta,7}+n_{\delta,1}+2n_{\tau,-2}\right)$
$\displaystyle+n_{\delta,1}+2Q_{d}n_{\delta,1}$
$\displaystyle+4n_{\tau,-2}+4Q_{l}n_{\tau,-2}$ $\displaystyle=$ $\displaystyle
n_{\delta,7}\left(2-2Q_{h}+2Q_{u}\right)+n_{\delta,1}\left(2-2Q_{u}+2Q_{d}\right)+n_{\tau,-2}\left(8-4Q_{u}+4Q_{l}\right)$
$\displaystyle=$ $\displaystyle 0,$

where we have used Eqs. (C1). Thus, the parameter $S^{\prime}$ is finite.

####  $S^{\prime}+U^{\prime}$ divergence:

The divergent part of $S^{\prime}+U^{\prime}$ in Eq. (32anb) is proportional
to

$\displaystyle\mathrm{tr}\left(\mathcal{N}\mathcal{N}^{\dagger}+\mathcal{V}\mathcal{V}^{\dagger}+\mathcal{Q}\mathcal{Q}^{\dagger}\right)-\left(Q_{h}\,\mathrm{tr}\,\mathcal{H}+Q_{u}\,\mathrm{tr}\,\mathcal{U}\right)+\left(Q_{d}\,\mathrm{tr}\,\mathcal{D}+Q_{l}\,\mathrm{tr}\,\mathcal{L}\right)$
(C7) $\displaystyle=$
$\displaystyle\mathrm{tr}\left(\mathcal{N}\mathcal{N}^{\dagger}+\mathcal{V}\mathcal{V}^{\dagger}+\mathcal{Q}\mathcal{Q}^{\dagger}\right)-Q_{h}\,\mathrm{tr}\left(\mathcal{N}\mathcal{N}^{\dagger}\right)-Q_{u}\,\mathrm{tr}\left(\mathcal{V}\mathcal{V}^{\dagger}-\mathcal{N}\mathcal{N}^{\dagger}\right)$
$\displaystyle+Q_{d}\,\mathrm{tr}\left(\mathcal{V}\mathcal{V}^{\dagger}-\mathcal{Q}\mathcal{Q}^{\dagger}\right)+Q_{l}\,\mathrm{tr}\left(\mathcal{Q}\mathcal{Q}^{\dagger}\right)$
$\displaystyle=$
$\displaystyle\left(1-Q_{h}+Q_{u}\right)\mathrm{tr}\left(\mathcal{N}\mathcal{N}^{\dagger}\right)+\left(1-Q_{u}+Q_{d}\right)\mathrm{tr}\left(\mathcal{V}\mathcal{V}^{\dagger}\right)+\left(1-Q_{d}+Q_{l}\right)\mathrm{tr}\left(\mathcal{Q}\mathcal{Q}^{\dagger}\right)$
$\displaystyle=$ $\displaystyle 0,$

where we have used $Q_{h}-Q_{u}=Q_{u}-Q_{d}=Q_{d}-Q_{l}=1$. Thus,
$S^{\prime}+U^{\prime}$ is finite.

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