On the other hand, if one intends to obtain finite Green's functions (GFs) and not only finite $S$-matrix elements, fields must be renormalized as well. This can be done in the symmetric form of the theory [229, 245]—i.e. before the SSB of the symmetry—, in which case it is enough to consider one field counterterm for each multiplet [119]. Alternatively, one can renormalize the fields in the mass basis [228, 137, 119]. Here, counterterms for the mixing of fields (mixed fields counterterms) must in general be considered whenever fields mix at loop-level.

In the present section, we start by identifying the bare quantities of the theory with the sum of renormalized ones and counterterms (we do not do this for gauge parameters, nor ghost fields, since it would be irrelevant for the calculation of $S$-matrix elements [119]).

Actually, in linear gauges (such as the Feynman gauge), it can be shown that one can simply refrain from applying any renormalization transformation whatsoever to the Gauge Fixing Lagrangian $\mathcal{L}_{\text{GF}}$
[246, 247, 248].
That is, all the parameters and fields therein are assumed to be already renormalized.

Then, in section <ref>, we review the need to consider a general parameterization, and show that the renormalized parameters can be described by a simpler parameterization.

§.§ Expansion of the bare quantities

In eq. <ref> above, we considered generic expansions. We now need to apply them to the specific content of the C2HDM.
Concerning the parameters, we include not only those of eqs. <ref>, but also some dependent parameters, for convenience (recall section <ref>).
Some renormalized parameters will be set equal to non-obvious quantities, which will be justified below.
As for the fields, we choose those in the mass basis.

We start with the gauge boson masses and fields. For the masses, we have:

m_W(0)^2 = m_W^2 + δm_W^2,
m_Z(0)^2 = m_Z^2 + δm_Z^2,

while for the fields:

W_μ(0)^+ =

( 1 + 12 δZ_W ) W_μ^+,

1 + 12 δZ_AA     12 δZ_AZ

12 δZ_ZA     1 + 12 δZ_ZZ

In the fermions, the masses obey

m_f,i(0) = m_f,i + δm_f,i,

in such a way that $m_{f,i}$ are real, and we define:

For details, cf. appendix <ref>.

δm_f,i^R ≡δm_f,i,
δm_f,i^L ≡δm_f,i^*.

The fields obey:

f_i(0)^L =
∑_j ( δ_ij + 12 δZ_ij^f,L ) f_j^L,

f_i(0)^R =

∑_j ( δ_ij + 12 δZ_ij^f,R ) f_j^R.

Considering now the scalar sector, we have for the masses:

m_H^+(0)^2 = m_H^+^2 + δm_H^+^2,
m_1(0)^2 = m_1^2 + δm_1^2,
m_2(0)^2 = m_2^2 + δm_2^2.

We renormalize $m_{3(0)}^2$ for convenience:

m_3(0)^2 = m_3R^2 + δm_3^2.

The renormalized squared mass in this case explicitly includes an index $\mathrm{R}$ (from `renormalized').

Not to be confused with the index $\mathrm{R}$ expressing `real', as in eq. <ref>.

The reason why this is necessary in this case is that, since $m_{3(0)}$ is a dependent parameter, it is fixed by other parameters. As a consequence, when the bare quantities are identified with a renormalized quantity plus a counterterm, the renormalized squared mass $m_{3\mathrm{R}}^2$ will also be fixed by other parameters, which means that it cannot be set equal to the pole squared mass—which we identify in the following as $m_{3\mathrm{P}}^2$. Moreover, $\delta m_{3}^2$ will be a dependent counterterm.

All this implies that $h_3$ plays a special role in theory, which is described in detail in appendix <ref>.

For the scalar fields,

1 + 12 δZ_G^+G^+     12 δZ_G^+H^+

12 δZ_H^+G^+     1 + 12 δZ_H^+H^+

1 + 1/2 δZ_h_1h_1     1/2 δZ_h_1h_2     1/2 δZ_h_1h_3     1/2 δZ_h_1G_0

1/2 δZ_h_2h_1     1 + 1/2 δZ_h_2h_2     1/2 δZ_h_2h_3     1/2 δZ_h_2G_0

1/2 δZ_h_3h_1     1/2 δZ_h_3h_2     1 + 1/2 δZ_h_3h_3      1/2 δZ_h_3G_0

1/2 δZ_G_0h_1     1/2 δZ_G_0h_2     1/2 δZ_G_0h_3     1 + 1/2 δZ_G_0G_0

The mixing parameters are such that:

Recall that $\alpha_{0(0)}$, $\alpha_{4(0)}$ and $\alpha_{5(0)}$ will be independent according to the combination $C_i$ chosen.

\begin{gather}
\label{Chap-Reno:eq:my-dza}
\beta_{(0)} = \beta + \delta \beta,
\qquad
\zeta_{a(0)} = \delta \zeta_a,
\\
\alpha_{1(0)} = \alpha_1 + \delta \alpha_{1},
\qquad
\alpha_{2(0)} = \alpha_2 + \delta \alpha_{2},
\qquad
\alpha_{3(0)} = \alpha_3 + \delta \alpha_{3},
\\
\alpha_{0(0)} = \beta + \delta \alpha_{0},
\qquad
\alpha_{4(0)} = \delta \alpha_{4},
\qquad
\alpha_{5(0)} = \delta \alpha_{5}.
\label{Chap-Reno:eq:consoante}
\end{gather}

The electric charge and the CKM matrix obey:

= e + δZ_e e,

V_ij(0) = V_ij + δV_ij.

As for the remaining parameters that are taken as independent ($\zeta_{1(0)}$ will be independent in the combination $C_4$), we have:

μ_(0)^2 = μ^2 + δμ^2,
ζ_1(0) = δζ_1,
ζ_2(0) = δζ_2.

For convenience, we also renormalize several dependent parameters:

In the combinations $C_i$ for which the parameters $\alpha_{0(0)}$, $\alpha_{4(0)}$, $\alpha_{5(0)}$ and $\zeta_{1(0)}$ are dependent, we also renormalize them for convenience (according to eqs. <ref> and <ref>).

We do not renormalize $\chi$, $\zeta_b$, $g_1$, $g_2$ or $v$. This means that, before renormalization, they must be replaced by the corresponding expressions (cf. eqs. <ref> and <ref>), and the latter must be renormalized. For details, cf. section <ref>.

\begin{gather}
s_{{\text{w}}(0)} = s_{\text{w}} + \delta s_{\text{w}},
\quad
c_{{\text{w}}(0)} = c_{\text{w}} + \delta c_{\text{w}},
\nonumber
\\
m_{11(0)}^2 = m_{11}^2 + \delta m_{11}^2,
\quad
m_{22(0)}^2 = m_{22}^2 + \delta m_{22}^2,
\qquad
m_{12\text{I}(0)}^2 = m_{12\text{I}}^2 + \delta m_{12\text{I}}^2,
\nonumber
\\
\lambda_{i(0)} = \lambda_i + \delta \lambda_i,
\qquad
X^{\dagger}_{ij(0)} = X^{\dagger}_{ij} + \delta X^{\dagger}_{ij},
\qquad
Q_{ij(0)} = Q_{ij} + \delta Q_{ij}.
\label{Chap-Reno:eq:unnec}
\end{gather}

§.§ General vs. simple parameterization

When studying the theory at tree-level, we considered the most general parameterization of the Higgs doublets, as well as of the diagonalization of the scalar states—recall section <ref>.
This general information was required in order to have general counterterms (which are necessary to absorb the UV divergences at one-loop). But it is not required to describe the renormalized parameters. Indeed, if the only reason we considered such a general parameterization was to account for the generality of the counterterms, there is no need to apply such general description to the renormalized parameters as well.

To better understand the claim we are making, consider for example the phase $\zeta_a$. At tree-level, we had the freedom to rephase the fields $G^+$ and $H^+$ to absorb $\zeta_a$. Had we used that freedom, though, we would have run into problems; for when we were to consider the theory up to one-loop level, we would have no counterterm $\delta \zeta_a$ (as there would have been no bare parameter $\zeta_{a(0)}$ to start with). It turns out that such counterterm is necessary for the absorption of the divergences of the theory, so that it cannot be discarded.

That is to say, we could not use the freedom to rephase $\zeta_a$ away at tree-level.

But once in the up-to-one-loop theory, and given that $\delta \zeta_a$ was already generated, there is nothing stopping us from using the rephasing freedom. We can thus rephase the renormalized parameter $\zeta_a$ away in the up-to-one-loop theory; and when we do so, the counterterm $\delta \zeta_a$ (that was meanwhile generated because we did not rephase $\zeta_a$ away at tree-level) will not vanish.

To see this explicitly, we may consider a simplified version of eq. <ref>, where we take $\zeta_b=\chi=0$ (which are irrelevant for the argument):

e^i ζ_a     0

0     e^-i ζ_a

It is obvious that, in this tree-level relation, one is free to rephase $\zeta_a$ away through $G^+ \to e^{-i \zeta_a} G^+$, $H^+ \to e^{i \zeta_a} H^+$. Yet, as we want to generate a counterterm $\delta \zeta_a$, we do not use that rephasing freedom. So, when considering the theory up to one-loop level, we take the bare version of eq. <ref>;

then, expanding the charged scalar states in the mass basis and $\zeta_{a(0)}$ into renormalized quantities and counterterms, according to eq. <ref> and

ζ_a(0) = ζ_a + δζ_a,

we obtain, to first order,

e^i ζ_a ( i δζ_a + δZ_G^+G^+ + 1 )
e^i ζ_a δZ_G^+H^+

e^-i ζ_a δZ_H^+G^+
e^-i ζ_a ( -i δζ_a + δZ_H^+H^+ + 1 )

Since $\delta \zeta_a$ was already generated, the renormalized parameter $\zeta_a$ can be rephased away through
$G^+ \to e^{-i \zeta_a} G^+$,

$H^+ \to e^{i \zeta_a} H^+$,

$Z_{G^{+}H^{+}} = e^{2 i \zeta_a} Z_{G^{+}H^{+}}^{\prime}$,

$Z_{H^{+}G^{+}} = e^{-2 i \zeta_a} Z_{H^{+}G^{+}}^{\prime}$,

i δζ_a + δZ_G^+G^+ + 1

-i δζ_a + δZ_H^+H^+ + 1

This simple example illustrates several relevant points.

First, it shows that one must start with a general description including the tree-level parameter $\zeta_a$, for otherwise the counterterm $\delta \zeta_a$ will not be generated.

Second, after $\delta \zeta_a$ was generated, the renormalized parameter $\zeta_a$ can be rephased away, and in such a way that  $\delta \zeta_a$ does not vanish.

Third, instead of starting by splitting $\zeta_{a(0)}$ into a non-zero $\zeta_a$ plus a counterterm (as in eq. <ref>), and afterwards rephasing $\zeta_a$ away, we can assume beforehand that we will work in a basis for the renormalized parameters where $\zeta_a=0$, which lets us equate $\zeta_{a(0)}$ immediately with the counterterm only. This is what we did in eq. <ref> above.

Finally, what we obtained for the renormalized parameters looks just the same as that which we would have obtained at tree-level should we not have been aiming at renormalizing the theory.
As a matter of fact, if we were not concerned with the generation of counterterms, but rather in a mere tree-level description (as in chapter <ref>), we would not have needed $\zeta_a$, so that we could have rephased it away in eq. <ref>. In that case, we would have obtained the same which we obtained in eq. <ref> for the renormalized terms (i.e. excluding counterterms).

Care should be taken with the notion `the same', because in one case what is at stake are renormalized quantities in the context of the up-to-one-theory, whereas in the other case there are only tree-level quantities in a tree-level theory. Hence, they are formally different. Nonetheless, the physical description in both cases is equivalent.

This also holds for the other parameters that were not rephased away for the sake of renormalization only. Therefore, the basis for the renormalized parameters (i.e. the basis that we can and shall use for the renormalized parameters) is considerably simpler than the general one used in section <ref>, and is just the same as the one usually employed when considering the theory solely at tree-level—as in chapter <ref>. Comparing the parameterization used in that chapter with the general parameterization used in section <ref>, we conclude that, for the renormalized parameters in the up-to-one-loop theory, all the fermion masses are real, and

The reason for identifying the renormalized phases $\zeta_{1\mathrm{R}}$ and $\zeta_{2\mathrm{R}}$ with the index R is given in note <ref>.

ζ_1R =
ζ_2R =
ζ_a =
α_4 =
α_5 =

α_0 = β.

As a consequence, the relations one obtains for the renormalized parameters in the context of the up-to-one-loop theory are precisely those that were derived in chapter <ref>. We checked this explicitly for different expressions: the minimum equations (eqs. <ref>), the relation for $m_{3\mathrm{R}}^2$ (eq. <ref>), as well as the relations for the $\lambda_i$ (eqs. <ref>).
Moreover, given eq. <ref>, the renormalized matrix $Q$ becomes:

Q_321   Q_β,

Q_3   Q_2   Q_1

R        0

0   0   0    1

1   0   0   0

0   1   0   0

0   0   - s_β     c_β

0   0   c_β     s_β

with $R$ given in eq. <ref>.

The original parameterization of $Q$ in eqs. <ref> and <ref> thus allows a simple connection between the matrix $Q$ for the renormalized parameters and the matrix $R$.

The matrix $X$ becomes orthogonal, and is simply:

c_β     s_β

- s_β     c_β

which is just matrix $R_H^T$ of eq. <ref>.

Finally, the total independent renormalized parameters become:

m_f, V

e,   m_W,   m_Z,
α_1,   α_2,   α_3,
β,   m_1,   m_2,   m_H^+,   μ^2

These parameters are precisely those which are usually taken as independent in the theory considered solely at tree-level (recall eq. <ref>).
Moreover, the set of renormalized parameters of the Higgs sector, $\{p^{\mathrm{H}}_{\text{reno.}}\}$, is the same whichever the set of bare parameters $\{p^{\mathrm{H}}_{C_i(0)}\}$ taken as the independent (recall eqs. <ref>).

Actually, not only it is the same, but it contains less parameters than each of the sets $\{p^{\mathrm{H}}_{C_i(0)}\}$, since the renormalized versions of the last three independent bare parameters of each set were rephased away.
On the other hand, for each bare independent parameter, there is an independent counterterm.
Thus, whichever the combination of independent bare parameters chosen, there will be more independent counterterms than independent renormalized parameters.

In fact, identifying $\{\delta p^{\mathrm{H}}_{C_i}\}$ with the set of independent counterterms of the Higgs sector in the combination $C_i$, we have:

{δe, δm_W, δm_Z,
δα_1, δα_2, δα_3,
δβ, δm_1, δm_2, δm_H^+, δμ^2, δζ_2, δζ_a,

Something similar happens in the Yukawa sector, as the phases of the renormalized fermion masses were rephased away. This means that, while the counterterms $\delta m_f$ are complex, the renormalized masses $m_f$ are real.
In summary, the presence of more independent counterterms than independent renormalized parameters is a consequence of CP violation, which forces the introduction of several parameters; these are required for the one-loop renormalization of the model, but their renormalized versions can be rephased away.

§ SELECTION OF THE TRUE VEV

We studied in detail the selection of the true vev in chapter <ref>. We realized that the most convenient way to perform such selection—the so-called Fleischer-Jegerlehner tadpole scheme, or FJTS—turns out to be equivalent to an approach where the tree-level vev is selected. We verified this explicitly for the SM.

In this section, we perform the selection of the true vev in the C2HDM using the FJTS. The complexity over the SM is increased not only by the existence of more fields, but also due to the presence of phases. Nonetheless, we will reach the same conclusion as before, namely: the FJTS can be mimicked by selecting the bare vev and by resorting to connected GFs.

We start by considering the bare version of eq. <ref>, using the notation of bars introduced in chapter <ref>:

Φ_1(0) =

1√(2) (v̅_1   e^i ζ̅_1+ ρ_1(0) + i η_1(0))

Φ_2(0) =

1√(2) (v̅_2   e^i ζ̅_2+ ρ_2(0) + i η_2(0))

As we are interested in selecting the true vev, the quantities with a bar correspond to true quantities, in the sense that they guarantee that no proper tadpoles (i.e. 1-point GFs) show up in the up-to-one-loop theory. In the FJTS, they are split as follows:

v̅_1 = v_1(0) + Δv_1,
v̅_2 = v_2(0) + Δv_2,
ζ̅_1 = ζ_1(0) + Δζ_1,
ζ̅_2 = ζ_2(0) + Δζ_2.

Thus, each quantity with a bar is split into two terms: the bare one and an extra quantity. These extra quantities are identified in what follows as the $\Delta$ quantities and are responsible for the elimination of proper tadpoles (they are introduced with this purpose). To see this, it is convenient to write eq. <ref> in an alternative formulation:

Φ_1(0) =

1√(2) (v̅_ρ_1 + ρ_1(0) + i v̅_η_1 + i η_1(0))

Φ_2(0) =

1√(2) (v̅_ρ_2 + ρ_2(0) + i v̅_η_2 + i η_2(0))

with $\bar{v}_{\rho_1}$, $\bar{v}_{\rho_2}$, $\bar{v}_{\eta_1}$, $\bar{v}_{\eta_2}$ real parameters, such that:

v̅_ρ_1 = v_ρ_1(0) + Δv_ρ_1,
v̅_η_1 = v_η_1(0) + Δv_η_1,

v̅_ρ_2 = v_ρ_2(0) + Δv_ρ_2,
v̅_η_2 = v_η_2(0) + Δv_η_2.

Comparing with eqs. <ref> and <ref>, we obtain:

Δv_ρ_1 = Δv_1   cosζ_1(0) - v_1(0)   Δζ_1 sinζ_1(0),

Δv_η_1 = Δv_1   sinζ_1(0) + v_1(0)   Δζ_1   cosζ_1(0),

Δv_ρ_2 = Δv_2   cosζ_2(0) - v_2(0)   Δζ_2 sinζ_2(0),

Δv_η_2 = Δv_2   sinζ_2(0) + v_2(0)   Δζ_2   cosζ_2(0).

We neglected terms of order $\Delta^2$, since the $\Delta$ quantities are of one-loop order (as we are about to show). This also means that the bare quantities can be replaced by their renormalized versions.

We shall use this argument in the equations that follow.

Hence, using the fact that we choose a basis for the renormalized parameters such that the renormalized phases $\zeta_{1\mathrm{R}}$ and $\zeta_{2\mathrm{R}}$ vanish (eq. <ref>), we get:

The phases $\zeta_{1\mathrm{R}}$ and $\zeta_{2\mathrm{R}}$ are the renormalized phases, such that $\zeta_{1(0)} = \zeta_{1\mathrm{R}} + \delta \zeta_1$
$\zeta_{2(0)} = \zeta_{2\mathrm{R}} + \delta \zeta_2$.
We use the index $\mathrm{R}$ to distinguish them from the true phases,
just as we did with the vev in chapter <ref>.

Finally, note that we keep the bare vevs for convenience in eq. <ref>.

v_1(0)   Δζ_1

v_2(0)   Δζ_2

Now, when the Lagrangian is expanded with eq. <ref>, the $\Delta v_{\phi_{n}}$ of eq. <ref> must ensure that proper tadpoles vanish. In order for this to happen, the terms with $\Delta v_{\phi_{n,j}}$ that contribute to the 1-point function of a certain neutral scalar field $\phi_{n,i}$ must precisely cancel the one-loop 1-point function in that field. That is, if $T_{\phi_{n,i}}$ represents the one-loop tadpole for $\phi_{n,i}$ (i.e. the one-loop contribution to the 1-point function in $\phi_{n,i}$), we must have:

T_ϕ_n,i = - ∑_j

∂^2 ℒ∂ϕ_n,i   ∂Δv_ϕ_n,j |_ϕ=0,   Δv = 0

Using the fact that every $\Delta v_{\phi_{n,j}}$ always shows up together with the corresponding field $\phi_{n,j}$ (cf. eqs. <ref> and <ref>), we can write:

∑_j ∂^2 V∂ϕ_n,i   ∂ϕ_n,j |_ϕ=0,   Δv = 0 Δv_ϕ_n,j
∑_j (M_n^2)_ij Δv_ϕ_n,j ,

where the mass matrix ${\cal M}_n^{2}$ obeys eq. <ref>.
Then, going to the diagonal basis,

which implies:

Δv_h_1 = T_h_1m_1^2,
Δv_h_2 = T_h_2m_2^2,
Δv_h_3 = T_h_3m_3R^2,
Δv_G_0 = 0,

where we used the fact that $T_{G_0} = 0$, which is a consequence of the Goldstone theorem. Some comments are in order.

First, just as in the SM, by obeying eq. <ref> the $\Delta$ quantities—in the physical basis, $\Delta v_{h_1}$, $\Delta v_{h_2}$, $\Delta v_{h_3}$, $\Delta v_{G_0}$—ensure that no proper tadpoles (1-point GFs) show up in the theory. In other words, they guarantee that the true vev up to one-loop level is selected. Yet, just as in the SM, since they are contained inside the scalar doublets, they will contribute to other GFs of theory. For example, although no 1-point GF for $h_2$ remains in the theory (that is, no proper tadpole for $h_2$ remains), $\Delta v_{h_2}$ will contribute to some 2-point and 3-point functions; and given eq. <ref>, the one-loop tadpole $T_{h_2}$ will contribute to those GFs; just as in chapter <ref>, we dub these one-loop tadpole structures contributing to GFs other than the 1-point GFs broad tadpoles.

In this way, $\Delta v_{h_2}$ ensures that no proper tadpole for $h_2$ exists, but introduces broad tadpoles in the theory. Broad tadpoles are thus a consequence of the selection of the true vev and must be considered for consistency.

Second, also as in the description of the FJTS in chapter <ref>, all broad tadpoles can be accounted for by considering all possible one-loop tadpole insertions in all possible GFs. Indeed,

as suggested above, each renormalized scalar field will have the corresponding $\Delta$ quantity added to it: $\Delta v_{h_1} + h_1$, $\Delta v_{h_2} + h_2$, and so on. This means that, for every term in the Lagrangian with (say) $\Delta v_{h_2}$, there will be a term with $h_2$ instead. As a consequence, for every $n$-point function with a $\Delta v_{h_2}$ contribution, there is a $(n+1)$-point function with a $h_2$ field contribution instead; and with that field, a reducible diagram with a one-loop tadpole can be formed. But it is easy to see that such reducible diagram precisely equals the $n$-point function with $\Delta v_{h_2}$: one just needs to consider the definition of the one-loop tadpole for $h_2$,

i   T_h_2,

and the zero-momentum $h_2$ propagator,

dashes,label=$\hspace{12mm} p=0$,label.side=left,tension=3,label.dist=2thicknJ1,nJ1nJ2

i 1/- m_2^2,

and compare with eq. <ref>.

The same argument obviously applies to the remaining broad tadpoles. In sum, although broad tadpoles must be considered for consistency, in the FJTS they can be accounted for by including all possible one-loop tadpole insertions in all possible GFs.

Third, the alternative just proposed (calculating reducible diagrams with one-loop tadpoles instead of tracing down the terms with $\Delta$ showing up in the Lagrangian) will be the one adopted in the following. The reason is simply that, as argued in chapter <ref>, the calculation of reducible diagrams with one-loop tadpoles is trivial with software such as ;

For details, cf. appendix <ref>.

by contrast, the task of expanding the Lagrangian and tracing down all the terms with $\Delta$ is quite cumbersome; and since the two methods are equivalent, we opt for the former.

But it is clear that this option (calculating reducible diagrams with one-loop tadpoles) is precisely the same as the option that a) selects the bare vev instead of the true up-to-one-loop vev and b) considers connected GFs. In fact, by opting for calculating reducible diagrams with one-loop tadpoles in the FJTS, all GFs that may receive an insertion of a one-loop tadpole must receive such insertion.

In general, then, non-renormalized 2-point functions in the FJTS will get contributions from two different types of terms:

where the full lines represent any type of particle and the dashed lines represent $h_1$, $h_2$, $h_3$. The same will happen for non-renormalized 3-point functions, which then include:

These two equations are nothing but the non-renormalized contributions for 2-point and 3-point connected GFs in an approch where the bare vev is selected (i.e. where all the $\Delta$ quantities are simply ignored).

The conclusion is thus the same as in the SM: although the $\Delta$ quantities are used in the FJTS to select the true vev, they do not contribute to connected GFs (with more than one external leg). Hence, if one decides to calculate the 2-point and 3-point functions as in eqs. <ref> and <ref>, all the discussion between eqs. <ref> and <ref> concerning the $\Delta$ quantities is not necessary and can be ignored.

In other words, one does not need to take the trouble of selecting the true vev up to one-loop level: by ignoring $\Delta$ quantities and by including the reducible diagrams with one-loop tadpoles, one obtains the same predictions as in the FJTS. This was already noted with generality in ref. [132], but was now explicitly verified in a non-trivial case with CP violation.

It is worth mentioning that nowhere in the present section did we need to mention tadpole counterterms. As discussed in section <ref>, this notion can lead to misunderstandings, so that it is preferable to avoid it.

§ COUNTERTERMS

As mentioned in section <ref>, the renormalization of the theory is completed by fixing the counterterms.
Independent counterterms are a priori not fixed; rather, they must be fixed through independent renormalization conditions.

Now, counterterms in general contain a divergent part and a finite part; whichever the renormalization condition used to fix a certain counterterm, its divergent part will always be the same. But the same is not true for the finite part, which in general depends on the prescription used. The different prescriptions that can be used to define the finite part of counterterms are called subtraction schemes.

In what follows, most counterterms will be fixed either through the modified minimal subtraction ($\overline{\text{MS}}$) scheme or through the on-shell subtraction (OSS) scheme.

The former is simpler: one selects a process that depends on the counterterm to be calculated; then, the renormalization condition simply states that the terms of the renormalized one-loop amplitude proportional to $\Delta_{\varepsilon}$ are zero. Here, $\Delta_{\varepsilon}$ is

Δ_ε = 2ε - γ_E + ln4 π,

where $\varepsilon = 4-d$ ($d$ being the dimension) and $\gamma_{\text{E}}$ is the Euler-Mascheroni constant. The counterterm at stake will then be proportional to $\Delta_{\varepsilon}$ only. When $\overline{\text{MS}}$ is used to calculate a parameter counterterm,
the corresponding renormalized parameter will have no clear physical meaning.
By contrast, OSS is defined precisely so that all renormalized parameters have a direct physical meaning [119]; the renormalization conditions in this case are less direct, though, and are discussed in detail in appendix <ref>.

In what follows, we adopt the conventions therein defined.

In section <ref>, we fix all of the counterterms of the theory, proceeding by sectors.

Finally, in section <ref>, we discuss the gauge dependence of the counterterms.

§.§ Calculation of counterterms

§.§.§ Gauge masses and fields

For gauge fields, the  renormalized up-to-one-loop 2-point functions are, in the Feynman gauge,

Although 2-point functions configure two momentum arguments, we exploit momentum conservation to write only one argument, always positive.

Note also that our notation for signs is different from that of ref. [119]: where we use a plus sign (in eqs. <ref>), they use a minus.
Moreover, ref. [119] also uses the relative sign between the non-renormalized transverse one-loop functions and the corresponding counterterms (in eqs. <ref> below) opposite to ours. These differences led us to introduce the variable $\eta_g$ in appendix <ref> (cf. eqs. <ref> and <ref>), so that ref. [119] uses $\eta_g=-1$, whereas $\eta_g=1$ is used in this thesis. Finally, note that ref. [12] implicitly uses the $\eta_g=-1$.

Γ̂^W^+ W^-_μν (k)    =    -g_μν (k^2 - m_W^2) + Σ̂_μν^W^+ W^- (k),

Γ̂^V V^'_μν (k)    =    -g_μν   δ_V V^' (k^2 - m_V^2) + Σ̂_μν^V V^' (k),

where $V, V^{\prime}=\{\gamma, Z\}$. They can be decomposed into transversal (T) and longitudinal (L) components:

Γ̂_μν^W^+W^-(k)    =    (g_μν-k_μ k_ν/k^2) Γ̂_T^W^+W^-(k^2)+k_μ k_ν/k^2 Γ̂_L^W^+W^-(k^2),

Γ̂_μν^V V^'(k)    =    (g_μν-k_μ k_ν/k^2) Γ̂_T^V V^'(k^2)+k_μ k_ν/k^2 Γ̂_L^V V^'(k^2),

which can also be applied to the particular case of the renormalized one-loop GFs:

Eqs. <ref> and <ref> also hold for non-renormalized GFs.

Σ̂_μν^W^+W^-(k)    =    (g_μν-k_μ k_ν/k^2) Σ̂_T^W^+W^-(k^2)+k_μ k_ν/k^2 Σ̂_L^W^+W^-(k^2),

Σ̂_μν^V V^'(k)    =    (g_μν-k_μ k_ν/k^2) Σ̂_T^V V^'(k^2) + k_μ k_ν/k^2 Σ̂_L^V V^'(k^2).

Now, expanding the Lagrangian using eqs. <ref> and <ref>, we find:

Σ̂_T^W^+W^- (k^2)    =    Σ_T^W^+W^-(k^2) - (k^2-m_W^2) δZ_W + δm_W^2,

Σ̂_T^V V^'(k^2)    =    Σ_T^V V^'(k^2) - 1/2(k^2-m_V^2) δZ_V V^'   - 1/2(k^2-m_V^'^2) δZ_V^' V + δ_V V^' δm_V^2.

Having seen this, we start by considering the charged fields. The OSS renormalization conditions are:
Note that the residue of the propagator of a gauge field (of mass $m$, projected onto the state $\varepsilon^{\sigma}$) is defined with a plus sign:

$\hat{R}^{\mu} =  \Lim{k^2 \to m^2} i \, g^{\mu \nu} (k^2 - m^2) \, \widetilde{\operatorname{Re}} \left[ i \hat{\Gamma}_{\nu \sigma}^{-1}(k) \right] \varepsilon^{\sigma}$ (compare with eq. <ref>).

.Re [Γ̂_μν^W^+W^-(k)] ε^ν(k)|_k^2=m_W^2    =    0,

lim_k^2 →m_W^2 (-1)/k^2-m_W^2 Re [Γ̂_μν^W^+W^-(k)] ε^ν(k)    =    ε_μ(k).

Inserting eq. <ref>, we get respectively:In this way, the longitudinal components disappear in OSS. This is consistent with the fact that they are non-physical, in the sense that they do not contribute to the 2-point function in the OS limit.

Re   Γ̂_T^W^+W^-(m_W^2)    =    0,

.  Re   ∂Γ̂_T^W^+W^-(k^2)/∂k^2|_k^2 = m_W^2    =    -1.

Then, using eqs.
<ref>, <ref> in eqs. <ref> and <ref>, we get respectively:

δm_W^2    =    - Re   Σ_T^W^+W^-(m_W^2),

δZ_W    =    .Re ∂Σ_T^W^+W^-(k^2)/∂k^2|_k^2=m_W^2.

In a similar way for the neutral fields, the OSS renormalization conditions read:

We thus write in a compact form the OSS conditions for the mass counterterms (eq. <ref> for $V^{\prime} = V$), for the off-diagonal field counterterms (eq. <ref> for $V^{\prime} \neq V$) and for the diagonal field counterterms (eq. <ref>).
Moreover, note that $\widetilde{\operatorname{Re}} \, \hat{\Gamma}_{\mu \nu}^{V^{\prime} V}(k)$ $=$ $\widetilde{\operatorname{Re}} \, \hat{\Gamma}_{\mu \nu}^{V V^{\prime}}(k)$, as a consequence of the Lorentz invariance of the renormalized up-to-one-loop action.

.Re [Γ̂_μν^V V^'(k)] ε^ν(k)|_k^2=m_V^2    =    0,

lim_k^2 →m_V^2 (-1)/k^2-m_V^2 Re [Γ̂_μν^V V(k)]   ε^ν(k)    =    ε_μ(k).

Inserting eq. <ref>, we get respectively:

Re   Γ̂_T^V V^'(m_V^2)    =    0,

.  Re   ∂Γ̂_T^V V(k^2)/∂k^2|_k^2 = m_V^2    =    -1.

Using eqs.
<ref>, <ref>, we obtain:

δm_Z^2    =    - Re   Σ_T^Z Z(m_Z^2),

δZ_A A     δZ_A Z

δZ_Z A     δZ_Z Z

.Re ∂Σ_T^A A(k^2)∂k^2|_k^2=0

2 Re Σ_T^A Z(m_Z^2)m_Z^2

-2 Σ_T^A Z(0)m_Z^2

.Re ∂Σ_T^Z Z(k^2)∂k^2|_k^2=m_Z^2

Finally, if one neglects the absorptive parts of loop integrals (through the operator $\widetilde{\operatorname{Re}}$), the one-loop processes introduce no new imaginary parts.

For details, cf. appendix <ref>.

It goes without saying that the absorptive parts of loop integrals are neglected in the calculation of counterterms only; the loop integrals of non-renormalized one-loop diagrams contributing to a certain process have in general absorptive parts.

In that case, the hermiticity of the Lagrangian implies the hermiticity of the up-to-one-loop action. Specifically in the case of up-to-one-loop 2-point functions of the gauge bosons of the C2HDM, this means:

Re   Γ_μν^W^+W^-(k)
Re   Γ_μν^W^+W^-(k)

Re   Γ_μν^V V^'(k)
[Re   Γ_μν^V V^'(k)]^*,

which implies, in particular,

As the different orders are formally independent, the hermiticity of the up-to-one-loop 2-point function implies the hermiticity of the one-loop 2-point function, which in turn leads to the hermiticity of its independent components (transversal and longitudinal).

Re   Σ_T^W^+W^-(k^2)
Re   Σ_T^W^+W^-(k^2)

Re   Σ_T^V V^'(k^2)
[Re   Σ_T^V V^'(k^2)]^*.

As a consequence, all the counterterms for gauge boson masses and fields are real.

§.§.§ Fermion masses and fields

In the case of fermionic 2-point functions, we have:

Γ̂^f̅ f_i j (p) = δ_i j (p - m_f,i) + Σ̂^f̅ f_i j (p),

and we can use the parameterization [119]:

Γ̂_i j^f̅ f(p) = p 1-γ_5/2 Γ̂_i j^f, L(p^2)+ p 1+γ_5/2 Γ̂_i j^f, R(p^2)+1-γ_5/2 Γ̂_i j^f, l(p^2)+1+γ_5/2 Γ̂_i j^f, r(p^2),

which, in the particular case of the renormalized one-loop GFs, reads:

Σ̂_i j^f̅ f(p) = p 1-γ_5/2 Σ̂_i j^f, L(p^2)+ p 1+γ_5/2 Σ̂_i j^f, R(p^2)+1-γ_5/2 Σ̂_i j^f, l(p^2)+1+γ_5/2 Σ̂_i j^f, r(p^2).

Expanding the Lagrangian using eqs. <ref> to <ref>, we find

Σ̂_i j^f, L(p^2)    =    Σ_i j^f, L(p^2)+1/2(δZ_i j^f, L+δZ_i j^f, L^†),

Σ̂_i j^f, R(p^2)    =    Σ_i j^f, R(p^2)+1/2(δZ_i j^f, R+δZ_i j^f, R^†),

Σ̂_i j^f, l(p^2)    =    Σ_i j^f, l(p^2)-1/2(m_f, i δZ_i j^f, L+m_f, j δZ_i j^f, R^†)-δ_i j   δm_f, i^L,

Σ̂_i j^f, r(p^2)    =    Σ_i j^f, r(p^2)-1/2(m_f, i δZ_i j^f, R+m_f, j δZ_i j^f, L †)-δ_i j   δm_f, i^R,

so that, using eqs. <ref>, <ref> and <ref>, we obtain:

Γ̂_i j^f, L(p^2)    =    δ_ij + Σ_i j^f, L(p^2)+1/2(δZ_i j^f, L+δZ_i j^f, L^†),

Γ̂_i j^f, R(p^2)    =    δ_ij + Σ_i j^f, R(p^2)+1/2(δZ_i j^f, R+δZ_i j^f, R^†),

Γ̂_i j^f, l(p^2)    =    - δ_ij m_f,i + Σ_i j^f, l(p^2)-1/2(m_f, i δZ_i j^f, L+m_f, j δZ_i j^f, R^†)-δ_i j   δm_f, i^L,

Γ̂_i j^f, r(p^2)    =    - δ_ij m_f,i + Σ_i j^f, r(p^2)-1/2(m_f, i δZ_i j^f, R+m_f, j δZ_i j^f, L †)-δ_i j   δm_f, i^R.

The OSS renormalization conditions are (no summation in the indices is implicit):

The conditions for the spinor $\bar{u}(p)$ do not need to be considered, since they end up containing no new information if the hermiticity of the up-to-one-loop action is used.

Re {Γ̂_i j^f̅ f(p)} u_j(p)
|_p   u_j(p) = m_f, j u_j(p)    =    0,

lim_p   u_i(p) →m_f, i u_i(p)

p+m_f,i/p^2-m_f, i^2Re [Γ̂_i i^f̅ f(p)] u_i(p)    =    u_i(p).

Inserting eq. <ref>, we have respectively:

The derivation of eq. <ref> is subtle and requires eqs. <ref>. We thank Ansgar Denner for clarifications on this aspect.

m_f, j   Re   Γ̂_i j^f, L(m_f, j^2)+Re   Γ̂_i j^f, r(m_f, j^2)=0,

m_f, j   Re   Γ̂_i j^f, R(m_f, j^2)+Re   Γ̂_i j^f, l(m_f, j^2)=0,

Re   {Γ̂_i i^f, R(m_f, i^2)+Γ̂_i i^f, L(m_f, i^2)+2 ∂/∂p^2 [m_f, i^2(Γ̂_i i^f, R(p^2)+Γ̂_i i^f, L(p^2))

55mm +m_f, i(Γ̂_i i^f, r(p^2)+Γ̂_i i^f, l(p^2))]|_p^2=m_f, i^2}=2.

In this case, however, the equations leave some freedom. We show in detail in appendix <ref> how to derive the counterterms. The results are:

12 Re
m_f,i   Σ_ii^f, L(m_f,i^2)
+ m_f,i   Σ_ii^f, R(m_f,i^2)
+ 2   Σ_ii^f, l(m_f,i^2)

12 Re
m_f,i   Σ_ii^f, L(m_f,i^2)
+ m_f,i   Σ_ii^f, R(m_f,i^2)
+ 2   Σ_ii^f, r(m_f,i^2)

δZ_i i^f, L
-Re   Σ_i i^f, L(m_f, i^2) - m_f, i ∂/∂p^2 Re[m_f, i(Σ_i i^f, L(p^2)+Σ_i i^f, R(p^2))

80mm+ Σ_i i^f, l(p^2)+Σ_i i^f, r(p^2)]|_p^2=m_f, i^2,

δZ_i i^f, R
-Re   Σ_i i^f, R(m_f, i^2) - m_f, i ∂/∂p^2 Re[m_f, i(Σ_i i^f, L(p^2)+Σ_i i^f, R(p^2))

80mm+Σ_i i^f, l(p^2)+Σ_i i^f, r(p^2)]|_p^2=m_f, i^2,

δZ_i j^f, L
i ≠j=
2/m_f, i^2-m_f, j^2 Re[m_f, j^2 Σ_i j^f, L(m_f, j^2)+m_f, i m_f, j Σ_i j^f, R(m_f, j^2)

60mm+m_f, i Σ_i j^f, l(m_f, j^2)+m_f, j Σ_i j^f, r(m_f, j^2)],

δZ_i j^f, R

i ≠j=
2/m_f, i^2-m_f, j^2 Re[m_f, j^2 Σ_i j^f, R(m_f, j^2)+m_f, i m_f, j Σ_i j^f, L(m_f, j^2)

60mm+m_f, j Σ_i j^f, l(m_f, j^2)+m_f, i Σ_i j^f, r(m_f, j^2)].

Finally, in the case of fermions, the hermiticity of the up-to-one-loop action implies:

Re  Γ_ij^f̅ f(p) = γ^0  [ Re   Γ_ij^f̅ f (p) ]^† γ^0,

so that, in particular,

\begin{gather}
\widetilde{\operatorname{Re}} \,\Sigma_{ij}^{f,\mathrm{L}}(p^2) =
\left[
\widetilde{\operatorname{Re}} \,\Sigma_{ji}^{f,\mathrm{L}}(p^2)
\right]^*,
\qquad
\widetilde{\operatorname{Re}} \,\Sigma_{ij}^{f,\mathrm{R}}(p^2)
\left[
\widetilde{\operatorname{Re}} \,\Sigma_{ji}^{f,\mathrm{R}}(p^2)
\right]^*,
\no
\widetilde{\operatorname{Re}} \,\Sigma_{ij}^{f,\mathrm{l}}(p^2)
\left[
\widetilde{\operatorname{Re}} \,\Sigma_{ji}^{f,\mathrm{r}}(p^2)
\right]^*.
\label{Chap-Reno:eq:apD:30c}
\end{gather}

It follows that $\delta m_{f,i}^{\mathrm{L}}$, $\delta m_{f,i}^{\mathrm{R}}$, $\delta Z_{i j}^{f, \mathrm{L}}$, $\delta Z_{i j}^{f, \mathrm{R}}$ (for $i \neq j$) are in general complex, whereas $\delta Z_{i i}^{f, \mathrm{L}}$ and $\delta Z_{i i}^{f, \mathrm{R}}$ are real.

§.§.§ Scalar masses and fields

In this case, we simply have:

For the neutral scalar fields, $\hat{\Gamma}^{S_n S_n^{\prime}} (k^2) = \hat{\Gamma}^{S_n^{\prime} S_n} (k^2)$ by construction.

Γ̂^S_c S_c^' (k^2)    =    δ_S_c S_c^' (k^2 - m_S_c^2) + Σ̂^S_c S_c^' (k^2),

Γ̂^S_n S_n^' (k^2)    =    δ_S_n S_n^' (k^2 - m_S_n^2) + Σ̂^S_n S_n^' (k^2),

where $S_c, S_c^{\prime} = \{H^+, G^+\}$, $S_n, S_n^{\prime} = \{h_1, h_2, h_3, G_0\}$, and $m_{S_n}^2 = m_{3\mathrm{R}}^2$ for $S_n = h_3$.

Moreover, we are taking $m_{G_0} = m_{G^{+}} = 0$ (and also $\delta m_{G_0}^{2} = \delta m_{G^{+}}^{2} = 0$ in eq. <ref>).
The would-be Goldstone boson fields are not physical, so that they cannot be OS; this means that, strictly speaking, one cannot apply the OSS renormalization conditions to GFs with external would-be Goldstone boson fields. Yet, since it will prove convenient, we extend these conditions by supposing that those fields are physical, with masses $m_{G_0} = m_{G^{+}} = 0$.

Expanding the Lagrangian with eqs. <ref> to <ref>, we find:

Σ̂^S_c S_c^'(k^2)    =    Σ^S_c S_c^'(k^2)  +1/2(k^2-m_S_c^'^2) δZ_S_c^' S_c

35mm + 1/2(k^2-m_S_c^2) δZ_S_c S_c^'^* -δ_S_c S_c^'   δm_S_c^2,

Σ̂^S_n S_n^'(k^2)    =    Σ^S_n S_n^'(k^2)  +1/2(k^2-m_S_n^'^2) δZ_S_n^' S_n

35mm + 1/2(k^2-m_S_n^2) δZ_S_n S_n^'
-δ_S_n S_n^'   δm_S_n^2.

We start by considering the charged fields. The OSS renormalization conditions are:

.Re   Γ̂^S_c^* S_c^'(k)|_k^2=m_S_c^2    =    0,

lim_k^2 →m_S_c^2 1/k^2-m_S_c^2 Re   Γ̂^S_c^* S_c(k)    =    1,

Inserting eq. <ref>, we get respectively:

Re   Σ̂^S_c^* S_c^'(m_S_c^2)    =    0,

.  Re   ∂Σ̂^S_c^* S_c(k^2)/∂k^2|_k^2 = m_S_c^2    =    0,

and, using eq. <ref>, we obtain:

δm_H^+^2    =    Re   Σ^H^+H^-(m_H^+^2),

δZ_G^+G^+     δZ_G^+H^+

δZ_H^+G^+     δZ_H^+H^+

-.Re   ∂Σ^G^+G^+(k^2)∂k^2|_k^2=0

- 2  Re  Σ^H^+G^+(m_H^+^2)m_H^+^2

2   Re  Σ^G^+H^+(0)m_H^+^2

-.Re   ∂Σ^H^+H^+(k^2)∂k^2|_k^2=m_H^+^2

The case of the neutral fields is more subtle, and is considered in detail in appendix <ref>. The independent counterterms are:

\begin{gather}
\delta m_1^2 = \widetilde{\operatorname{Re}} \, \Sigma^{h_1h_1}(m_1^2), \qquad
\delta m_2^2 = \widetilde{\operatorname{Re}} \, \Sigma^{h_2h_2}(m_2^2),
\\[5mm]
\delta Z_{S_n S_n} = -\left.\widetilde{\operatorname{Re}} \, \frac{\partial \Sigma^{S_nS_n}\left(k^{2}\right)}{\partial k^{2}}\right|_{k^{2}=m_{S_n}^{2}},
\qquad
\delta Z_{S_n^{\prime} S_n} \stackrel{S_n^{\prime} \neq S_n}{=}  2 \, \frac{\widetilde{\operatorname{Re}}  \, \Sigma^{S_n S_n^{\prime}}\left(m_{S_n}^{2}\right)}{m_{S_n^{\prime}}^2 - m_{S_n}^2},
\label{Chap-Reno:eq:CT-scalar-neutral}
\end{gather}

and the pole squared mass $m_{3\mathrm{P}}^2$ is given by:

m_3P^2 = m_3R^2 - Re   Σ^h_3h_3(m_3R^2) + δm_3^2.

The hermiticity of the up-to-one-loop action implies in this case:

Re   Γ^S_c^' S_c(k)
[Re   Γ^S_c S_c^'(k)]^*,

Re   Γ^S_n S_n^'(k)
[Re   Γ^S_n S_n^'(k)]^*.

so that, in particular,

Re    Σ^S_c^' S_c(k^2)
[Re    Σ^S_c S_c^'(k^2)]^*,

Re    Σ^S_n S_n^'(k^2)
[Re    Σ^S_n S_n^'(k^2)]^*.

Therefore, the counterterms $\delta Z_{G^{+}H^{+}}$ and $\delta Z_{H^{+}G^{+}}$ are in general complex, while the remaining counterterms of this subsection are real.

§.§.§ Electric charge

The counterterm for the electric charge is also fixed through OSS; this involves considering the interaction between electrons and photon, on the one hand, and requiring that, for OS electrons and for the Thomson limit (i.e. for vanishing photon momentum), the full (renormalized) interaction is given by the tree-level one, on the other.

Or, which is equivalent, the OSS scheme imposes that higher orders do not contribute to the interaction $A_{\mu}\bar{e}e$ in the particular limit just mentioned.

Besides the vertex (or 1PI) functions, the full interaction in general involves LSZ factors—c.f. appendix <ref>. However, as discussed in that appendix, the field counterterms in OSS are chosen in such a way that those factors become trivial, and can be ignored.

The OSS condition then becomes:

i Γ̂^Ae̅e_μ(0,-p,p) |_1L
u(p) = 0,

where 1L indicates the one-loop contribution.
The reason why this condition allows to fix $\delta Z_e$ is that $\hat{\Gamma}^{A\bar{f}f}_{\mu}(0,-p,p) \Big|_{\text{1L}}$, which is a renormalized GF, depends on
that counterterm.

Cf. eqs. <ref> and <ref>.

Hence, $\delta Z_e$  becomes fixed (i.e. it is fixed precisely so that eq. <ref> is verified). However, that results in a quite complicated expression. It turns out that there is a Ward identity that substantially simplifies the result, as it leads to a simple expression that relates $\delta Z_e$ to other counterterms. A detailed derivation of both the Ward identity and the expression for $\delta Z_e$ are provided in appendix <ref>. The final result is:

δZ_e = -1/2 δZ_A A + s_w/c_w 1/2 δZ_Z A=.1/2 ∂Σ_T^A A(k^2)/∂k^2|_k^2=0 + s_w/c_w Σ_T^A Z(0)/m_Z^2.

§.§.§ CKM matrix

For the CKM matrix, we follow the prescription originally presented in ref. [249]. We fix the expression in the Feynman gauge, which will be justified in section <ref> below. We then have:

δV_i j=1/4 .[(δZ_i k^u, L-δZ_i k^u, L †) V_k j-V_i k(δZ_k j^d, L-δZ_k j^d, L †)]|_ξ=1,

where $\xi=1$ represents the Feynman gauge.

§.§.§ Mixing parameters

The renormalization of mixing parameters has been recurrently discussed in recent years [232, 233, 121, 125, 132, 134, 128, 124], especially in ref. [128].

Mixing parameters usually reduce to mixing angles. Yet, in the present model (and due to CP violation), there are not only mixing angles (like $\beta$), but also mixing phases (like $\zeta_a$). Hence, we promote the designation `mixing angles' to `mixing parameters'.

In the latter, several methods for fixing counterterms for mixing parameters are proposed and analysed.

In the present work, we consider one of those methods, which involves fixing the independent counterterms for mixing parameters using symmetry relations. This method is simple and leads to well behaved results, as shall be discussed below.

A description of the procedure, as well as the derivation of the expressions for the counterterms for mixing parameters of the C2HDM, can be found in appendix <ref>.

In this section, we present the results therein derived. For the charged sector,

=    14
.[δZ_G^+H^+ - δZ_H^+G^+]

-12 (2 β) Im [δZ_G^+H^+ ]|_ξ=1.

For the neutral sector, the set of independent counterterms for mixing parameters depends on the combination $C_i$ (recall eq. <ref>). There are three counterterms which are independent in all the combinations:

14 (α_2) . [  cos(α_3) ( δZ_h_1h_2 - δZ_h_2h_1 )  +  sin(α_3) ( δZ_h_3h_1 -δZ_h_1h_3 )  ] |_ξ=1,

14  sin(α_3) [ δZ_h_1h_2 - δZ_h_2h_1 + (α_3) ( δZ_h_1h_3 - δZ_h_3h_1 )  ] |_ξ=1,

14 [δZ_h_2h_3 - δZ_h_3h_2 -   cos(α_3) tan(α_2)  ( δZ_h_1h_2 - δZ_h_2h_1 )

50mm + sin(α_3) tan(α_2) ( δZ_h_1h_3 - δZ_h_3h_1 )  ] |_ξ=1.

Then, the combinations $C_1$, $C_2$ and $C_3$ have $\delta \alpha_5$, $\delta \alpha_4$ and $\delta \alpha_0$ as independent, respectively, given by:

The notation $\stackrel{C_i}{=}$ clarifies which combination $C_i$ the expression is applicable to. For example, eq. <ref> is only valid if $C_1$ is chosen; in other combinations, $\delta \alpha_5$ will be a dependent counterterm, so that eq. <ref> is not valid.
Finally, note that $\delta \alpha_5$, $\delta \alpha_4$ and $\delta \alpha_0$ are all dependent counterterms in the combination $C_4$.

14 [δZ_h_2G_0 -δZ_G_0h_2 + tan(α_3) ( δZ_G_0h_3 - δZ_h_3G_0 )

14 [δZ_h_1G_0 -δZ_G_0h_1 + (α_3) tan(α_2) ( δZ_G_0h_3 - δZ_h_3G_0 )  ]

14 (α_2) (α_3)  ( δZ_G_0h_3 - δZ_h_3G_0 )

§.§.§ Remaining parameters

By now, all the field counterterms are determined; as for the independent parameter counterterms, considering eq. <ref> we see that we still need to fix $\delta \mu^2$ and $\delta \zeta_2$ (both required for all combinations $C_i$) and, in the specific case of the combination $C_4$, also $\delta \zeta_1$.

We calculate the latter and $\delta \mu^2$ in the $\overline{\text{MS}}$ scheme.

As described above, this requires selecting a process where the counterterm to be fixed intervenes.

For $\delta \mu^2$, we choose $h_3 \to H^+H^-$. Here, the total counterterm is such that:

i e2 m_W s_w c_β s_β
(s_β R_31 + c_β R_32) δμ^2

Then, by requiring the renormalized 3-point function to be such that the terms proportional to $\Delta_{\varepsilon}$ are zero,

i Γ̂^h_3 H^+ H^-|_Δ_ε

= 0 ,

we fix $\delta \mu^2$. Concerning $\delta \zeta_1$ in $C_4$, we choose $h_3 \to  \bar{\tau} \tau$. Knowing that:

i   δζ_1 (e   m_τ   R_33 tanβ2 m_W s_w + ...),

where the ellipses represent terms proportional to $\gamma_5$,
we fix $\delta \zeta_1$ in $C_4$ by imposing:

i Γ̂^h_3  τ̅ τ|_Δ_ε

C_4= 0 .

As for $\delta \zeta_2$, we checked that its divergent part is related to that of $\delta \zeta_1$ via:

The divergent part of $\delta \zeta_2$ can be easily calculated by requiring e.g. the process $h_3 \to \bar{u} u$ to be finite.

δζ_2 |_Δ_ε = -^2(β)    δζ_1 |_Δ_ε.

Then, similarly to what can be done with symmetry relations (cf. appendix <ref>), we fix $\delta \zeta_2$ as a whole by requiring that eq. <ref> also holds for the finite parts. That is, we fix $\delta \zeta_2$ by imposing:

Eq. <ref> thus provides a simple prescription to fix $\delta \zeta_2$. This simplicity is the reason why we chose this counterterm as independent in all the four combinations discussed above.

δζ_2 = -^2(β)   δζ_1 |_ξ=1.

Note that this condition applies to all four combinations $C_i$. Accordingly, $\delta \zeta_2$ will have a general finite part in the first three combinations, whereas in $C_4$ it will be proportional to $\Delta_{\varepsilon}$ only.

§.§ Gauge dependence

Physical observables cannot depend on the gauge chosen, which implies that $S$-matrix elements must be gauge independent.

In a renormalized up-to-one-loop process, both the (non-renormalized) one-loop diagrams and the counterterms are in general gauge dependent.
Now, it can be shown that, if parameter counterterms are gauge independent, the gauge dependences of the one-loop diagrams and those of the field counterterms precisely cancel, thus rendering the $S$-matrix elements gauge independent [140].

Conversely, gauge dependent parameter counterterms in general lead to gauge dependent $S$-matrix elements.

Motivated by this, several authors have insisted in recent years on the importance of finding gauge independent parameter counterterms [121, 124, 132, 123].

However, as noted in ref. [128], the circumstance that parameter counterterms are gauge dependent does not by itself ruin the gauge independence of $S$-matrix elements. In fact, all that is required for $S$-matrix elements to be gauge independent is that parameter counterterms are considered gauge independent. In order to ensure the gauge independence of $S$-matrix elements, indeed, it does not really matter whether the parameter counterterms are truly gauge independent or not; all what is required is that they are considered gauge independent, i.e. that they do not change when the gauge is changed [128, 140].

We thank Ansgar Denner for clarifications on this point.

But this can be easily implemented by calculating a gauge dependent counterterm in a particular gauge. When calculating observables in other gauges, one must stick to the value of the parameter counterterm in the gauge it was calculated in. Then, the gauge dependence of field counterterms cancels the gauge dependence of the one-loop diagrams, and $S$-matrix elements will be gauge independent.

In face of this, there does not seem to be a clear advantage of true gauge independence in parameter counterterms over considered gauge independence. In both scenarios, $S$-matrix elements end up being gauge-independent, and both lead to correct relations between physical observables.

It goes without saying that, when using considered gauge independent counterterms, one has to make sure that no inconsistencies are introduced (i.e. one must make sure not to change the gauge in the gauge dependent parameter counterterms). In this sense, truly gauge independent counterterms may be preferable.
It has been claimed that true gauge independence is a desirable property, as it allows a “more physical interpretation” of non-observable parameters [250]. However, a non-observable parameter is always non-physical anyway; the fact that it is truly gauge independent does not render it any more physical (in whatever physically meaningful way).

Now, this procedure is obviously irrelevant when parameter counterterms are truly gauge independent. In the FJTS, this happens for parameter counterterms fixed in either the OSS or the $\overline{\text{MS}}$ schemes (recall section <ref>).

However, the method described in the previous paragraph is very convenient for the remaining independent parameter counterterms, which in this model (and according to our choices) consist of those for the CKM matrix elements, those for the independent mixing parameters, and $\delta \zeta_2$ (sections <ref>, <ref> and <ref> above, respectively).

We thus applied the method in those cases, following the suggestion proposed in ref. [128] of selecting the Feynman gauge as the gauge in which the counterterms are calculated, which avoids artificially large parameters.

§ RESULTS

We now study the behaviour of the different combinations $C_i$ introduced above. This agenda is quite novel in studies of renormalization, due to the mere existence of combinations. In fact, there is usually no such thing as different combinations of independent parameters, as the number of independent renormalized parameters usually equals the number of independent counterterms. The studies of the renormalization of models thus tend to be focused either on the choice of different subtraction schemes for the counterterms, or on the choice of different renormalization scales.

In our case, the fact that there are more independent counterterms than independent renormalized parameters means that, for the same set of the latter, there can be different combinations of the former. Our goal, then, is to compare those combinations, ascertaining how the choice of one or the other combination (i.e. the choice of one or the other set of independent counterterms) affects the predictions of the theory at NLO.

In this paper, we restrict ourselves to the four combinations $C_i$ implied in eq. <ref> and, for a certain combination, to the subtraction schemes proposed in section <ref>.

To do that, we consider three specific NLO processes: the decays of $h_2$ to $ZZ$, $h_1 Z$ and $h_1 h_1$.

In the present section, we start by describing both the computational tools and the simulation procedure used to study these decays. Then, we discuss how the decays are influenced by the different combinations $C_i$. Finally, after presenting the expressions for the one-loop decay widths, we show numerical results that allow to compare the different combinations.

§.§ Computational tools and simulation procedure

The generation of Feynman rules for both the tree-level and the counterterm interactions, the drawing of the Feynman rules and diagrams, and the calculation of one-loop amplitudes, counterterms and one-loop decay widths were all performed with . These different tasks are discussed in detail in appendix <ref>.
was also used for two additional tasks.
The first was a confirmation that the model is renormalized; by considering processes of the different sectors, we numerically checked that all the counterterms calculated in section <ref> lead to finite results.

The second task was a conversion of the results to Fortran, which were then numerically evaluated using LoopTools [105].

This conversion is highly non-trivial, involving millions of lines of code; for details, see appendix <ref>.

For the scatter plots of section <ref> below, we
essentially followed the procedure outlined in section <ref>, with four differences.

First, we assumed that $h_1$ corresponds to the SM-like Higgs boson, such that $m_1 = 125 \, \, \textrm{GeV}$.

Second, for the experiments concerning the different signals for the SM-like Higgs boson, we used the more recent reference [47].

Third, we used a more recent version of HiggsBounds, namely HiggsBounds5 [178].

Finally, for the EDM, we considered the most recent result from the ACME collaboration, $|d_e| < 1.1 \times 10^{-29} e \mbox{ cm}$ ($90\%$ confidence level) [222].

§.§ The influence of the different combinations

The renormalized NLO amplitude for the process $j$ (with $j = \{ h_2 \to ZZ, h_2 \to h_1Z, h_2 \to h_1h_1\}$),
which we represent as $\hat{\mathcal{M}}_j$, can generically be written as:

ℳ̂_j = ℳ^tree_j + ℳ̂^loop_j,

where $\mathcal{M}^{\mathrm{tree}}_j$ and $\hat{\mathcal{M}}^{\mathrm{loop}}_j$ respectively represent the tree-level amplitude and the renormalized one-loop amplitude, the latter of which can in turn be split according to:

ℳ̂^loop_j =
ℳ^loop_j + ℳ^CT_j,

where the terms on the right-hand side represent the non-renormalized one-loop amplitude and the counterterms, respectively.

Here, $\mathcal{M}^{\mathrm{CT}}_j$ includes all the counterterms that, after expanding the bare quantities into renormalized plus counterterm, end up contributing to the process $j$. Now, the set of counterterms that constitute $\mathcal{M}^{\mathrm{CT}}_j$ is just a consequence of such an expansion, and is the same for all the combinations $C_i$. The differences between the combinations begin
when the counterterms are to be given an expression,

since the latter in general depend on the combination. For example, suppose that $\delta \zeta_1$ contributes to $\mathcal{M}^{\mathrm{CT}}_j$; recalling eq. <ref>, it is clear that the expression for this counterterm depends on $C_i$. Indeed, in $C_4$, $\delta \zeta_1$ is an independent counterterm, so that it is determined by eq. <ref>. In the remaining combinations, by contrast, $\delta \zeta_1$ is dependent, which means that it is determined by a function (which in general depends on the particular combination at stake) of independent counterterms. As a consequence, $\mathcal{M}^{\mathrm{CT}}_j$ will in general have different values according to the combination $C_i$ chosen. In this way, the different $C_i$ in general lead to different predictions for the renormalized NLO amplitudes.

At this point, it is worth noting that, in a renormalized function such that all its counterterms are fixed through OSS or other momentum-dependent subtraction schemes, the dependence on the renormalization scale $\mu_{\mathrm{R}}$ drops out.

A simple way to see this is to note that, in a one-loop integral calculated through dimensional regularization, the UV divergent term ($\frac{2}{\varepsilon}$) and the renormalization scale dependent term ($\ln\mu_{\mathrm{R}}^2$) always come together ($\frac{2}{\varepsilon} + \ln\mu_{\mathrm{R}}^2$). Now, a momentum-dependent subtraction scheme is such that the counterterms are defined as a function of a one-loop integral evaluated at a specific momentum. Thus, since neither $\frac{2}{\varepsilon}$ nor $\ln\mu_{\mathrm{R}}^2$ depend on the momentum, the counterterms will contain both. And just as they (i.e. the counterterms) ensure that the renormalized function is UV finite (i.e. ensure that the $\frac{2}{\varepsilon}$ of the counterterms cancel against that of the original one-loop integral), they also ensure that it does not depend on the renormalization scale (i.e. the $\ln\mu_{\mathrm{R}}^2$ cancel).
A final note: the renormalization scale $\mu_{\mathrm{R}}$ should not be confused with the parameter $\mu^2$ introduced in eq. <ref>.

Conversely, if there is at least a counterterm which is fixed in $\overline{\text{MS}}$, such counterterm will not cancel the dependence of the non-renormalized function on the renormalization scale, so that the renormalized function at stake will depend on $\mu_{\mathrm{R}}$.

In section <ref>, all independent counterterms were fixed through a momentum-dependent subtraction scheme, except for $\delta \mu^2$ and (whenever it is independent) $\delta \zeta_1$.

Whenever $\delta \zeta_1$ is fixed in $\overline{\text{MS}}$, $\delta \zeta_2$ will also be (cf. eq. <ref>).

These exceptions will be quite relevant for the impact of the different combinations on the three processes $j$ introduced above.

In order to better illustrate such impact, we show in table <ref> all the independent counterterms contributing to the different $\mathcal{M}^{\mathrm{CT}}_{j}$, for the four combinations $C_i$.

These counterterms are fixed according to the prescriptions presented in section <ref>. We should mention that $\delta \zeta_2$ contributes to all processes, in all combinations; and since it is an independent counterterm, it should in principle be included in table <ref>. However, we fixed it by reference to $\delta \zeta_1$ (recall eq. <ref>), which is only independent in $C_4$. In the remaining combinations, the counterterms which $\delta \zeta_1$ depends on are already accounted for in the table. Hence, for the sake of clarity, we do not show $\delta \zeta_2$.

Process $j$     Common to all the $C_i$     $C_1$     $C_2$     $C_3$     $C_4$

$h_2 \to ZZ$
{\delta Z_e},  \,  {\delta m_{\mathrm{W}}^2},  \,  {\delta m_{\mathrm{Z}}^2},  \,  {\delta \alpha_1},  \,  {\delta \alpha_2}, \, {\delta \alpha_3}, \, {\delta \beta},

{\delta Z_{ZZ}},  \,   {\delta Z_{h_1h_2}},  \,  {\delta Z_{h_2h_2}},  \,  {\delta Z_{h_3h_2}}

$\delta \alpha_5$     $\delta \alpha_4$     $\delta \alpha_0$     $\delta \zeta_1$

$h_2 \to h_1 Z$
{\delta Z_e},  \,  {\delta m_{\mathrm{W}}^2},  \,  {\delta m_{\mathrm{Z}}^2},  \,  {\delta \alpha_1},  \,  {\delta \alpha_2},  \,  {\delta \alpha_3}, \, {\delta \beta}, \, {\delta Z_{ZZ}},$

${\delta Z_{h_1h_1}},  \,  {\delta Z_{h_2h_2}},  \,  {\delta Z_{h_3h_1}},  \,  {\delta Z_{h_3h_2}},  \,  {\delta Z_{G_0h_1}},  \,  {\delta Z_{G_0h_2}}
$\delta \alpha_5$     $\delta \alpha_4$     $\delta \alpha_0$
$\delta \zeta_1$

$h_2 \to h_1 h_1$

${\delta Z_e},  \,  {\delta m_{\mathrm{W}}^2},  \,  {\delta m_{\mathrm{Z}}^2},  \,  {\delta \alpha_1},  \,  {\delta \alpha_2},  \,  {\delta \alpha_3}, \, {\delta \beta}, \, {\delta m_1^2}, $

$ {\delta m_2^2}, \,  {\delta \mu^2}, \, {\delta Z_{h_1h_1}},  \,  {\delta Z_{h_1h_2}},  \,  {\delta Z_{h_2h_1}}, \, {\delta Z_{h_2h_2}}, $

${\delta Z_{h_3h_1}},  \,  {\delta Z_{h_3h_2}},  \,  {\delta Z_{G_0h_1}},  \,  {\delta Z_{G_0h_2}}$

$\delta \alpha_5$     $\delta \alpha_4$     $\delta \alpha_0$     $\delta \zeta_1$

Total set of independent counterterms contributing to $\mathcal{M}^{\mathrm{CT}}_{j}$ for the combinations $C_i$. For each process $j$, the counterterms are separated in two groups: those that are common to all the four combinations (second column), and those that are specific to each combination (last four columns).

It is clear that, in all three processes, the complete set of independent counterterms always depends on the combination $C_i$ (note that the last four columns of the table reflect eq. <ref>).
Moreover, processes $h_1 \to ZZ$ and $h_2 \to h_1 Z$ should have no scale dependence in $C_1$, $C_2$ and $C_3$, since they do not involve any counterterm fixed in $\overline{\text{MS}}$ in those combinations. By contrast, the combination $C_4$ in all processes is expected to lead to scale dependent results, as it includes the $\overline{\text{MS}}$-fixed $\delta \zeta_1$.
Finally, process $h_2 \to h_1 h_1$ depends on $\delta \mu^2$, so that we expect scale dependence in all combinations of that process.

§.§ Decay widths

Using , the expressions for the renormalized NLO decay width $\Gamma^{\mathrm{NLO}}_j$ for the process $j$ can be easily calculated from the renormalized NLO amplitude $\hat{\mathcal{M}}_j$. We assume that all external particles are OS. The final expressions are considerably clear, as long as they are written in terms of form factors.

In what follows, we follow the conventions of ; we omit the Feynman diagrams, as well as the expressions for the form factors.

We start with the decay $h_2 \to ZZ$. Defining the momenta and Lorentz indices such that $h_2(p_1) \to Z^{\nu}(q_1) Z^{\sigma}(q_2)$, we have:

ℳ_h_2 →ZZ^tree
= ε^*_ν(q_1)   ε^*_σ(q_2)      f_3^tree   g^νσ,

ℳ̂_h_2 →ZZ^loop
= ε^*_ν(q_1)   ε^*_σ(q_2)
f_3   g^νσ +
f_6   p_1^ν   p_1^σ +
f_9   p_1^ν   q_1^σ +
f_24   p_1^σ   q_2^ν +
f_27   q_1^σ   q_2^ν

+ f_33   p_1^ω   q_1^υ   ϵ^νσωυ +
f_15   p_1^σ   q_1^ν +
f_18   q_1^ν   q_1^σ +
f_21   q_1^ν   q_2^σ
), 8mm

where the different $f_k$

correspond to form factors.

We identify tree-level form factors with the superscript `tree'.
The form factor $f_{33}$ corresponds to the CP-violating component predicted in ref. [82]. It turns out that it does not contribute to $\Gamma^{\mathrm{NLO}}_{h_2 \to ZZ}$, as can be seen in eq. <ref>.

These in general contain contributions from the counterterms, so that they will in general have different values in the different combinations.
The form factors $f_{15}$, $f_{18}$ and $f_{21}$ do not contribute, since the fact that the $Z$ bosons are OS implies $\varepsilon^*_{\nu}(q_1) \, q_1^{\nu} = \varepsilon^*_{\sigma}(q_2) \, q_2^{\sigma} = 0$.
The decay width is:

Γ^NLO_h_2 →ZZ
√(m_2^4 - 4   m_2^2   m_Z^2)128  π  m_2^3   m_Z^4 f_3^tree
{ ( m_2^4 - 4   m_2^2   m_Z^2 + 12   m_Z^4 ) ( f_3^tree  + 2   Re [f_3] )

+ m_2^2    ( m_2^4 - 6   m_2^2   m_Z^2 + 8   m_Z^4 )    Re[f_6 + f_9 + f_24 + f_27] }.

As for the decay $h_2 \to h_1 Z$, we define the momenta and Lorentz indices such that $h_2(p_1) \to h_1(q_1) Z^{\sigma}(q_2)$, so that

ℳ_h_2 →h_1 Z^tree
f_10^tree   q_1^σ + f_12^tree   p_1^σ

ℳ̂_h_2 →h_1 Z^loop
= ε^*_σ(q_2)
f_10   q_1^σ +
f_11   q_2^σ +
f_12   p_1^σ

but, as the $Z$ boson is OS, $f_{11}$ does not contribute. The decay width is:

Γ^NLO_h_2 →h_1 Z
164   π  m_2^3   m_Z^2
m_1^4 + m_2^4 + m_Z^4 - 2   m_1^2   m_2^2 - 2   m_1^2   m_Z^2 - 2   m_2^2   m_Z^2

20mm ×{
|f_10^tree + f_12^tree|^2
2   Re
(f_10^tree + f_12^tree)
(f_10^* + f_12^*)

Finally, the decay $h_2 \to h_1 h_1$ is such that the renormalized NLO amplitude is simply given by:

ℳ̂_h_2 →h_1 h_1
ℳ_h_2 →h_1 h_1^tree
ℳ̂_h_2 →h_1 h_1^loop
f_1^tree + f_1,

and the decay width is:

Γ^NLO_h_2 →h_1 h_1
√(m_2^4 -4   m_1^2   m_2^2)32   π  m_2^3   f_1^tree (f_1^tree + 2   Re[f_1]).

§.§ Numerical results

At last, we investigate the behaviour of the different combinations on the decay widths using numerical results.

First and foremost, we checked that all combinations in all processes lead to gauge independent decay widths.

This is in agreement with the discussion presented in section <ref>, and validates the simple prescription for the renormalization of mixing parameters proposed in section <ref>.

We compare the different combinations by ascertaining how their results affect the numerical stability of the perturbative expansion. If perturbation theory is to be trusted, higher orders should contribute with smaller corrections. In particular, the NLO results should generate small corrections when compared to the LO ones. This leads us to define the quantity [122, 124]:

ΔΓ_j ≡Γ^NLO_j - Γ^LO_jΓ^LO_j,

where $\Gamma^{\mathrm{LO}}_j$ represents the decay width at LO for the process $j$.

$\Delta \Gamma_{h_2 \to ZZ}$ in percentage as a function of $\Gamma^{\mathrm{LO}}_{h_2 \to ZZ}$, for the combination $C_1$.

Only the interval $ 0.1 \, \, \textrm{GeV} < \Gamma^{\mathrm{LO}} < 1 \, \, \textrm{GeV}$ is shown.

In red, points passing all constraints except HiggsBounds5; in green, points passing all constraints.

In fig. <ref>, we show $\Delta \Gamma_{h_2 \to ZZ}$ in percentage as a function of $\Gamma^{\mathrm{LO}}_{h_2 \to ZZ}$ for the combination $C_1$. Several aspects should be stressed here.
First if all, we checked numerically that the results are invariant under the renormalization scale, as expected from the discussion of section <ref>.

Both scale invariance and finiteness were checked with the subroutines setmudim and setdelta from LoopTools.

Besides, the plot shows that HiggsBounds5 hinders points with large values of $\Gamma^{\mathrm{LO}}$;

this is consistent with the fact that, had $h_2 \to ZZ$ a large value of $\Gamma^{\mathrm{LO}}$, $h_2$ would probably already have been discovered by now.

The majority of points passing all constraints thus obey $\Gamma^{\mathrm{LO}} < 0.7 \, \, \textrm{GeV}$; points with $\Gamma^{\mathrm{LO}}$ almost up to $0.8 \, \, \textrm{GeV}$ are still allowed, albeit very scarcely (a stringent fine tuning is required).

All in all, the points describe a smooth and well-behaved pattern; in particular, there is no region of parameter space generating singularities in $\Delta \Gamma$.

Finally, although there are points with small $\Delta \Gamma$ for all allowed values of $\Gamma^{\mathrm{LO}}$—indicating that, for those points, perturbation theory is valid—, larger values of $\Delta \Gamma$ are also possible for smaller values of  $\Gamma^{\mathrm{LO}}$.

From eq. <ref>, larger values of $\Delta \Gamma$ are expected for smaller values of $\Gamma^{\mathrm{LO}}$.

For those points, one would in principle need to calculate the following order in perturbation theory to check whether or not a perturbative description is possible.

Similar to fig. <ref> are the two panels of fig. <ref>, where we show equivalent results for the combinations $C_2$ and $C_3$.

$\Delta \Gamma_{h_2 \to ZZ}$ in percentage as a function of $\Gamma^{\mathrm{LO}}_{h_2 \to ZZ}$, for the combinations $C_2$ (left) and $C_3$ (right).

Only the interval $ 0.1 \, \, \textrm{GeV} < \Gamma^{\mathrm{LO}} < 1 \, \, \textrm{GeV}$ is shown.

The color conventions are those of fig. <ref>.

We checked that, as before, there is no scale dependence.
The values of $\Delta \Gamma_{h_2 \to ZZ}$ are essentially the same as in fig. <ref>; moreover, we find the same smooth pattern in the points. This allows us to attest the quality of our prescription for the determination of the counterterms for the mixing parameters, proposed in section <ref>.

Except for $\delta \zeta_a$, which we do not investigate in this paper.

Indeed, from table <ref>, the combinations $C_1$, $C_2$ and $C_3$ for the process $h_2 \to ZZ$, besides depending all on $\delta \beta$, $\delta \alpha_1$, $\delta \alpha_2$ and $\delta \alpha_3$, depend individually on $\delta \alpha_5$, $\delta \alpha_4$ and $\delta \alpha_0$, respectively. Then, by considering fig.s <ref> and <ref>, we conclude that all these counterterms lead to well-behaved results. Actually, not only are the results all well-behaved, but they are also very similar.
This is consistent with the circumstance that $\delta \alpha_5$, $\delta \alpha_4$ and $\delta \alpha_0$ were all determined via symmetry relations, so that one would expect similar behaviours from the different lines of eqs. <ref>.

All in all, then,
the combinations $C_1$, $C_2$ and $C_3$ are essentially equivalent for the process $h_2 \to ZZ$.

The results for $C_4$ are shown in fig. <ref>.

$\Delta \Gamma_{h_2 \to ZZ}$ in percentage as a function of $\Gamma^{\mathrm{LO}}_{h_2 \to ZZ}$, for the combination $C_4$.

Only the interval $ 0.1 \, \, \textrm{GeV} < \Gamma^{\mathrm{LO}} < 1 \, \, \textrm{GeV}$ is shown.

In red, points with $\mu_{\mathrm{R}}= 350$ GeV passing all constraints except HiggsBounds5. The remaining colors represent different values of the renormalization scale for points passing all the constraints: in cyan, $\mu_{\mathrm{R}} = 125$ GeV; in green, $\mu_{\mathrm{R}}= 350$ GeV; in blue, $\mu_{\mathrm{R}}= 700$ GeV.

Here, as expected from the discussion of section <ref>, the results depend on the renormalization scale $\mu_{\mathrm{R}}$. We plot three different scales, for points passing all constraints. It is clear that the lighter scale leads to a big dispersion in the values of $\Delta \Gamma$, and that the intermediate scale leads to the most stable results.
For $\Gamma^{\mathrm{LO}}_{h_2 \to ZZ} \gtrsim 0.4$ GeV, these turn out to be quite similar to those of the points passing all constraints in the remaining combinations: the ranges of values of $\Delta \Gamma$ are essentially the same, and the same well-behaved pattern is observed. This could hardly be expected from the expressions for the different counterterms; not only is the specific counterterm in $C_4$ ($\delta \zeta_1$) fixed through a method completely different from the one used to fix the specific counterterms of the combinations $C_1$, $C_2$ and $C_3$ ($\delta \alpha_5$, $\delta \alpha_4$ and $\delta \alpha_0$, respectively), but the role of counterterms is also different: $\delta \zeta_1$ is a counterterm for a phase of the potential, whereas $\delta \alpha_5$, $\delta \alpha_4$ and $\delta \alpha_0$ are counterterms for mixing parameters. For smaller values of $\Gamma^{\mathrm{LO}}_{h_2 \to ZZ}$, however, and whichever the scale chosen, $C_4$ leads to more unstable results than the remaining combinations.

In fig. <ref>, we show the results for the decay $h_2 \to h_1Z$, for the four combinations.

$\Delta \Gamma_{h_2 \to h_1Z}$ in percentage as a function of $\Gamma^{\mathrm{LO}}_{h_2 \to h_1Z}$, for the combinations $C_1$ (top left), $C_2$ (top right), $C_3$ (down left) and $C_4$ (down right).
Only the interval $ 0.1 \, \, \textrm{GeV} < \Gamma^{\mathrm{LO}} < 2 \, \, \textrm{GeV}$ is shown.
The color conventions for $C_1$, $C_2$ and $C_3$ are those of fig. <ref>, whereas for $C_4$ are those of fig. <ref>.

The conclusions are similar to the ones for $h_2 \to ZZ$, which we just discussed.
In particular, the combinations $C_1$, $C_2$ and $C_3$ are scale independent, whereas $C_4$ is scale dependent (in which case the intermediate scale is again the most stable one); HiggsBounds5 hampers points with large values of $\Gamma^{\mathrm{LO}}$, and all points passing all constraints in the four combinations show a similar and well-behaved pattern. Broadly speaking, these points are similar to the equivalent ones in $h_2 \to ZZ$ (except that, for small values of $\Gamma^{\mathrm{LO}}_{h_2 \to h_1Z}$, there are valid points with larger values—in modulus—of $\Delta \Gamma$, especially for $C_4$).

This similarity is expected from table <ref>, which shows that the only difference in the set of independent counterterms contributing to the two decays concerns field counterterms; but since all field counterterms are fixed through the same method (namely, OSS), it is not surprising that the two processes turn out to lead to similar NLO corrections.

In fig. <ref>, we show the results for $h_2 \to h_1 h_1$, for the four combinations.

$\Delta \Gamma_{h_2 \to h_1h_1}$ in percentage as a function of $\Gamma^{\mathrm{LO}}_{h_2 \to h_1h_1}$, for the combinations $C_1$ (top left), $C_2$ (top right), $C_3$ (down left) and $C_4$ (down right).

Only the interval $ 0.1 \, \, \textrm{GeV} < \Gamma^{\mathrm{LO}} < 2.4 \, \, \textrm{GeV}$ is shown.
All points correspond to $\mu_{\mathrm{R}}= 350$ GeV; the color conventions are those of fig. <ref>.

Here, as predicted in section <ref> above, all combinations are scale dependent, due to the contribution of the counterterm $\delta \mu^2$; we plot the intermediate scale only, which we checked is again the most stable one.

The four combinations lead to similar results; a well-behaved pattern is once again to be observed in all of them. In contrast to what happened previously, HiggsBounds5 does not preclude points with large values of $\Gamma^{\mathrm{LO}}_{h_2 \to h_1h_1}$; we checked that these points lead to small values of $\Gamma^{\mathrm{LO}}_{h_2 \to ZZ}$ and  $\Gamma^{\mathrm{LO}}_{h_2 \to h_1Z}$, and that only large values of these would be excluded by the experimental results.

Moreover, values of $\Delta \Gamma$ as high as $40\%$ are allowed for all the region of allowed points in all combinations. We found that more stable results can be obtained by requiring smaller values for the different $\lambda_i$ (parameters of the potential), as can be seen in fig. <ref> for the combination $C_1$, where we constrained $|\lambda_i| < 2$ for all $\lambda_i$.

Identical to fig. <ref> top left, except that blue/black points are included, satisfying all constraints and, additionally, $|\lambda_i|<2$ (see text for details).

This can be explained by the circumstance that smaller values for the $\lambda_i$ correspond to a more stringent constraint regarding the tree-level perturbative unitarity. Hence, it is not surprising that such values lead to a more stable perturbative expansion [122, 132].

§ SUMMARY

We presented the renormalization of the C2HDM, which is one of the simplest models containing an extended scalar sector with CP violation. We showed in detail how the presence of CP violation in the scalar sector leads to a rather peculiar program of renormalization, due to two main reasons.

First, several non-physical parameters must be introduced in order to ensure the generation of the necessary counterterms. This happens in such a way that, once the counterterms are obtained, the corresponding renormalized parameters are not needed and can be rephased away.

Second, for the same set of independent physical parameters, multiple combinations $C_i$ of independent counterterms can be chosen. It thus becomes relevant to study the impact of different combinations on observables.

We considered four combinations and applied them to three specific one-loop decays of $h_2$; the first three combinations take as independent a counterterm for a mixing parameter, and they fix it through symmetry relations, whereas $C_4$ takes a parameter of the potential as independent, and fixes it through $\overline{\text{MS}}$—thus yielding a dependence on the renormalization scale $\mu_{\mathrm{R}}$.

We checked that all combinations in all processes lead to gauge-independent and well-behaved results, which validates the prescriptions used to fix the counterterms.

For all processes, we found that the combination $C_4$ leads (at least for some values of $\mu_{\mathrm{R}}$ and of the LO decay width) to results very similar to those of the remaining combinations.

Among the points passing all constraints in all processes, one finds regions in the four combinations that lead to a numerical stability of the perturbative expansion; points leading to large NLO corrections are also allowed, in which case the results of higher orders should be investigated.

We stressed several aspects related to an adequate treatment of the C2HDM when considered up to one-loop level.

We used the Fleischner-Jegerlehner tadpole scheme to ensure the selection of the true vev; this was the first time that this scheme was applied to a model with CP violation in scalar sector (and thus to a model with generally complex vevs); we confirmed that this scheme is equivalent to an approach where the tree-level vevs are used and where the (connected) GFs include one-loop tadpole insertions.

We discussed the renormalization of a field whose mass is a dependent parameter.

We showed that the OSS conditions in the fermion sector leave some freedom, which can be used to define complex counterterms for the fermion masses.

We proposed a simple and easily generalizable prescription to calculate counterterms for mixing parameters, which ensures that observables are gauge independent.

Finally, proved to be an ideal tool for renormalizing models such as the C2HDM and study them at NLO, since it calculates a multiplicity of crucial elements—Feynman rules, counterterms, one-loop amplitudes, decay widths, etc.—in a simultaneously automatic and flexible way. The model file for C2HDM can be found in the webpage of the program, and a detailed description of the advantages of using to perform the renormalization of the model can be found in appendix <ref>.

CHAPTER: NLO CORRECTIONS TO $Z \TO B\BAR{B}$ IN MHM

Multi-Higgs Models (MHM) can be investigated in a generic fashion; this is what we propose to do in this chapter, by considering a general model with an arbitrary number of scalar particles.

While light scalar particles may be detected directly through their production, heavy ones may be detected indirectly through their impact on the radiative corrections. Here, we investigate the second approach, focusing on one-loop corrections to $Z \to b\bar{b}$.

Such corrections can be connected to experiment through e.g. the forward–backward asymmetry measured in the process $e^- e^+ \to b \overline{b}$. The present values for these quantities are within 1$\sigma$ of the SM predictions [251]; therefore,
studying the one-loop corrections to  $Z \to b\bar{b}$ can be used to constrain New Physics.

Haber and Logan [208] have already addressed this case in models with extra scalars in any representation of the gauge group $\mathrm{SU(2)_L}$.

Their work has been used to constrain a multiplicity of models, e.g.,

2HDMs [252, 253, 122, 254, 255, 5],

the Georgi–Machacek model[256, 257, 258, 259, 260],

scotogenic models [261] and

models with $\mathrm{SU(2)_L}$ singlet scalars [262, 263];

it was also used in fitting programs[264, 265].

In this chapter, we improve their analysis
in several aspects: first, we consider CP-violating scalar sectors; second, we provide explicit expressions that a model should verify in order for the divergences of the one-loop $Z \to b\bar{b}$ to cancel; third, we introduce a practical formalism to write the results in models with singlets and doublets, which is possible due to a convenient parameterization that was introduced in refs. [266, 267, 268, 269]; finally, we apply the results to a particular CP-violating model (the C2HDM), where we ascertain the validity of common approximations—in particular, neglecting both $m_{\mathrm{Z}}$ and the contributions of the neutral scalars.

§ THE ONE-LOOP CALCULATION

We start by considering a generic MHM with $n$ complex charged scalars $H_a^\pm$ and $m$ real neutral scalars $S_l^0$, being in everything else equal to the SM.

In this notation, $H_1^{\pm}$ and $S_1^0$ are the would-be Goldstone bosons $G^{\pm}$ and $G^0$, respectively.

We approximate the CKM matrix such that $V_{tb} = 1$,
which leads us to consider only the quarks bottom with mass $m_b$ and
top with mass $m_t$. Besides, we neglect $m_b$ in the propagators and loop
functions, but not in the couplings (which will be justified below).

As usual in this thesis, we use the conventions of ref. [138],
with all $\eta$'s positive.

§.§ Couplings

The relevant interactions between gauge bosons and quarks can be parameterized as:

ℒ_Zbb    =    -
e/s_w c_w  Z_μ  b̅  γ^μ( g_Lb γ_L + g_Rb γ_R ) b,

ℒ_Ztt    =    -
e/s_w c_w  Z_μ  t̅  γ^μ( g_Lt γ_L + g_Rt γ_R ) t,

ℒ_Wtb    =     -
e/√(2) s_w
( t̅ γ^μγ_L b  W_μ^+
+ b̅ γ^μγ_L t  W_μ^-

in such a way that the couplings $g_{\mathrm{L}b}$, $g_{\mathrm{R}b}$, $g_{\mathrm{L}t}$ and $g_{\mathrm{R}t}$ read:

g_Lb = s_w^2/3 - 1/2,
g_Rb = s_w^2/3,
g_Lt = 1/2 - 2 s_w^2/3,
g_Rt = - 2 s_w^2/3.

To parameterize the interaction between scalars and quarks, we introduce the $n$-dimensional complex vector $c_a$, the $n$-dimensional complex vector $d_a$ and the $m$-dimensional complex vector $r_l$, such that:

ℒ_Htb    =
∑_a=1^n [
H_a^+  t̅ ( c_a^∗γ_L - d_a γ_R ) b + H_a^-  b̅ ( c_a γ_R - d_a^∗γ_L ) t

ℒ_Sbb    =    ∑_l=1^m S_l^0
b̅ ( r_l γ_R + r_l^∗γ_L ) b.

Finally, we parameterize the interaction between scalars and the $Z$ gauge boson by introducing the $n \times n$ hermitian matrix $X$, the $m \times m$ real and antisymmetric matrix $Y$ and the $m$-dimensional real vector $y$, such that:

ℒ_ZHH    =     -
e/s_w c_w  Z_μ∑_a, a^'= 1^n X_a a^'
( H_a^+  i ∂^μH_a^'^-
- H_a^'^-  i ∂^μH_a^+ ),

ℒ_ZSS    =
i e/s_w c_w  Z_μ∑_l, l^'= 1^m Y_l l^'
( S_l^0  i ∂^μS_l^'^0
- S_l^'^0  i ∂^μS_l^0 ),

ℒ_ZZS    =
e   m_Z/2 s_w c_w  Z_μZ^μ∑_l= 1^m y_l S_l^0.

§.§ One-loop contribution: overview

We are looking for corrections—arising from scalar particles—that modify the tree-level couplings $g_{\mathrm{L}b}$ and $g_{\mathrm{R}b}$. Hence, we define the quantities $\Delta g_{\mathrm{L}b}$ and $\Delta g_{\mathrm{R}b}$ as the renormalized one-loop corrections that involve scalar particles, including the would-be Goldstone bosons and the already-observed neutral scalar with mass 125 GeV. We thus write the renormalized effective couplings as:

g_Lb^eff = g_Lb + Δg_Lb,
g_Rb^eff = g_Rb + Δg_Rb.

In Figs. <ref>, <ref> and <ref>, we represent all the (non-renormalized) one-loop diagrams involving scalar particles.
We follow the terminology of Haber and Logan [208] closely. In particular, we identify the first diagram of figures <ref> and <ref> as type a) and the second one as  type b). Haber and Logan also classify the diagrams of fig. <ref> as type d), but neglected them.

a)     5mm b)

Two diagrams with charged scalars contributing to
$Z \to b\bar{b}$.

a)     5mm b)

Two diagrams with neutral scalars contributing to
$Z \to b\bar{b}$.

i)     5mm ii)     5mm iii)     5mm iv)

[Diagrams referred as to “type d)”.]Diagrams referred as to “type d)” in  ref. [208].

Here, we shall ignore the diagrams <ref> i) and ii), for two reasons: first, they are convergent; second, in models with only scalar singlets and doublets (which we are primarly interested in), they do not give new contributions beyond the SM, as there are no $Z W^\pm H_a^\mp$ couplings in those models (other than the $Z W^\pm G^\mp$ already present in the SM).

As for the diagrams <ref> iii) and iv), they are proportional to $m_b$;

This can be explained as follows. The coupling of the $Z$ to the bottom quarks conserves chirality (i.e. the $\bar{b}$ and the $b$ in eq. <ref> necessarily have the same chirality, due to the matrix $\gamma^{\mu}$); the same happens in the one-loop corrections to $Z \to b\bar{b}$, so that the incoming and the outgoing $b$ quarks have the same chirality. Now, in the diagrams in fig. <ref> iii) and iv), besides the chirality-conserving coupling $Z \bar{b} b$, there is the coupling $S \bar{b} b$, which violates chirality (as there is no $\gamma^{\mu}$ in eq. <ref>). So, there must be a mass insertion in the internal $b$ propagator in order to change the chirality again.

hence, and since they are also convergent, one may neglect them by taking $m_b=0$ (which is what was done in ref. [208]).

Nevertheless, since $m_b$ could appear multiplied by a large coefficient,

Such as $\tan{\beta}$ in the $\mathbbm{Z}_2$-symmetric 2HDM,
see for instance table 2 in ref. [59].

we will also present their calculation in order to
check the validity of this approximation.

For the renormalization, we follow the usual procedure, which involves identifying the original quantities of the theory (parameters and fields) as bare quantities, and separating them into renormalized quantities and counterterms.

For details, see chapters <ref> and <ref>.

In the end, several counterterms contribute to the process $Z \to b\bar{b}$. However, in order to ensure the finiteness of the diagrams that interest us here, it is enough to consider the fermionic field counterterms; more specifically, it is enough to consider the contributions of the scalar particles to those counterterms.

We follow ref. [208] in neglecting finite corrections coming from other counterterms.

We calculate the fermionic field counterterms using OSS, which fixes them in terms of one-loop fermionic 2-point functions.

For details, cf. section <ref>.

As we just mentioned, we consider only the contributions of the scalar particles (both charged and neutral), shown in fig. <ref>.

i)        ii)

Contribution of the charged and neutral scalars
to the self-energy of the bottom quark,
leading to “type c)” contributions to the vertex.

We follow once again the classification of Haber and Logan, and identify the contribution of these diagrams to the effective vertex (via fermionic fields counterterms) as type c).
Using the conventions of section <ref>, we find:

The second term on the r.h.s. of eqs. <ref> and <ref> is finite; therefore, and since it is proportional to $m_b$, we ignore it. Note, moreover, that we do not need to consider the operator $\widetilde{\operatorname{Re}}$ in this case, as there are no absorptive parts in the diagrams of fig. <ref>.
Finally, although the quantities $\Delta g_{\aleph b}$ (with $\aleph = \{\mathrm{L},\mathrm{R}\}$) are finite (as they are the renormalized corrections), their different components ($\Delta g_{\aleph b}(i)$, for the diagrams of the generic type $i$) are in general divergent.

Δg_Lb ( c )
- g_Lb  Σ^f,L_bb (m_b^2 ),

Δg_Rb ( c )
- g_Rb  Σ^f,R_bb (m_b^2 ).

Our calculations have been performed with , by considering a model with a sufficiently large number of scalar particles, and by deriving general expressions from it. In this way, is a very useful tool also for investigating generic models.

§.§ Calculation of the diagrams
involving charged scalars

The diagrams in fig. <ref>a) lead to:

We have set $m_b=0$ inside all the Passarino–Veltman functions;
when evaluating them numerically, however, it is convenient to keep $m_b \neq 0$ to avoid numerical instabilities.

Note, moreover, that the sums in eqs. <ref>
start at $a=1$ (i.e. they include the charged would-be Goldstone bosons $G^\pm$). However, one may show that $X_{1a}=X_{a1}=0$ for $a \neq 1$, so that the sum in eq. (<ref>) may start at $a, a^\prime = 2$, while the term with $a = a^\prime = 1$ is separately included in the SM contribution.

Finally, the one-loop results for $\Delta g_{\mathrm{L}b}$ and $\Delta g_{\mathrm{R}b}$ have absorptive (i.e. imaginary) parts. If there are no scalars with mass below $m_{\mathrm{Z}}/2$, the imaginary parts only appear through cuts of the internal bottom-quark lines of fig. <ref>b), thus affecting only the
contributions with neutral scalars. Yet, although those imaginary parts
may be of the same order of magnitude as the real parts, they
can be ignored here; indeed, the observables will depend on, for example,

| g_Lb |^2
| g_Lb + Δg_Lb |^2
| g_Lb |^2
+ 2  Re ( g_Lb  Δg_Lb^∗)
+ 𝒪 ( Δg_Lb^2 )
| g_Lb |^2
+ 2  g_Lb  Re( Δg_Lb )
+ 𝒪 ( Δg_Lb^2 ),

where the last line follows from the fact that $g_{\mathrm{L}b}$ is real.  As a result, the impact of an imaginary $\Delta g_{\mathrm{L}b}$ on the observables effectively appears only at higher order.

Δg_Lb ( a )    =
∑_a,a'=1^n c_a X_aa' c_a'^∗
C_00 ( m_Z^2, 0, 0, m_a'^2, m_a^2, m_t^2 ),

Δg_Rb ( a )    =
Δg_Lb ( a ) ( c_a →d_a^∗),

where $m_a$ denotes the mass of $H_a^\pm$.
The diagrams in fig. <ref>b) lead to:

We took into account the fact that $\left( d - 2 \right) C_{00} \left( \dots \right) = 2\, C_{00} \left( \dots \right) - 1/2$, with
$d$ being the dimension of space-time.

Δg_Lb ( b )    =
1/16 π^2 ∑_a=1^n | c_a |^2
to 14 pt
- m_t^2  g_Lt  C_0 ( 0, m_Z^2, 0, m_a^2, m_t^2, m_t^2 )

+ g_Rt [
to 12 pt
2  C_00 ( 0, m_Z^2, 0, m_a^2, m_t^2, m_t^2 )
- 1/2 - m_Z^2  C_12 ( 0, m_Z^2, 0, m_a^2, m_t^2, m_t^2 )
to 12 pt
to 14 pt
} ,

Δg_Rb ( b )    =    Δg_Lb ( b )
( c_a →d_a, g_Lt ↔g_Rt ).

As for the type c) contributions,
arising from fig. <ref> i), we find:

Δg_Lb ( c )    =
g_Lb/16 π^2
∑_a=1^n | c_a |^2
B_1 ( 0, m_t^2, m_a^2 ),

Δg_Rb ( c )    =
Δg_Lb ( c ) ( c_a →d_a,g_Lb →g_Rb ).

As mentioned before, we do not consider the contributions from the diagrams <ref> i) and ii).

In the CP-conserving limit,
eqs. <ref>–<ref> agree with eqs. (4.1)
of ref. [208].

The functions $B_1$ and $C_{00}$ are divergent;
all the other Passarino–Veltman functions appearing in this section are finite. Defining $\Delta_{\varepsilon}$ as in eq. <ref>, one has [270]:

B_1 ( ... ) = - Δ_ε/2 +
finite terms,

C_00 ( ... ) = Δ_ε/4 + finite terms.

We can thus conclude that the terms in eqs. <ref>–<ref> proportional to $\Delta_{\varepsilon}$ are:

\begin{gather}
\Big\{
\Delta g_{\mathrm{L}b} \left( a \right) + \Delta g_{\mathrm{L}b} \left( b \right) + \Delta g_{\mathrm{L}b} \left( c \right)
\Big\}\Big|_{\Delta_{\varepsilon}}
\frac{\Delta_{\varepsilon}}{32 \pi^2}
\left[
\sum_{a,a'=1}^{n} c_a X_{aa'} c_{a'}^\ast
+ \left( g_{\mathrm{R}t} - g_{\mathrm{L}b} \right) \sum_{a=1}^{n} \left| c_a \right|^2
\right], \\
\Big\{
\Delta g_{\mathrm{R}b} \left( a \right) + \Delta g_{\mathrm{R}b} \left( b \right) + \Delta g_{\mathrm{R}b} \left( c \right)
\Big\}\Big|_{\Delta_{\varepsilon}}
\frac{\Delta_{\varepsilon}}{32 \pi^2} \left[
\sum_{a,a'=1}^{n} d_a^\ast X_{aa'} d_{a'}
+ \left( g_{\mathrm{L}t} - g_{\mathrm{R}b} \right) \sum_{a=1}^{n} \left| d_a \right|^2
\right].
\end{gather}

Therefore, and using eq. <ref>, the divergences cancel if and only if:

∑_a,a' c_a X_aa' c_a'^∗   =
s_w^2 - c_w^2/2 ∑_a | c_a |^2,

∑_a,a' d_a^∗X_aa' d_a'
s_w^2 - c_w^2/2 ∑_a | d_a |^2.

§.§ Calculation the diagrams
involving neutral scalars

The diagrams in fig. <ref>a) lead to:

Δg_Lb ( a )    =
∑_l,l'=1^m r_l Y_ll' r_l'^∗
C_00 ( 0, m_Z^2, 0, 0, m_l'^2, m_l^2 ),

Δg_Rb ( a )    =
Δg_Lb ( a ) ( r_l →r_l^∗),

where $m_l$ denotes the mass of $S^0_l$.
The diagrams in fig. <ref>b) lead to:

\begin{align}
\Delta g_{\mathrm{L}b} \left( b \right) =&\, \frac{g_{\mathrm{R}b}}{16 \pi^2}
\sum_{l=1}^{m} \left| r_l \right|^2
\bigg[ 2\, C_{00} \left( 0, m_{\mathrm{Z}}^2, 0, m_l^2, 0, 0 \right)  - \frac{1}{2} - m_{\mathrm{Z}}^2\, C_{12} \left( 0, m_{\mathrm{Z}}^2, 0, m_l^2, 0, 0 \right)
\bigg],
\label{Chap-Lavou:eq:155b}
\\
\Delta g_{\mathrm{R}b} \left( b \right) =&\,
\Delta g_{\mathrm{L}b} \left( b \right) \left( g_{\mathrm{R}b} \to g_{\mathrm{L}b} \right).
\label{Chap-Lavou:eq:156b}
\end{align}

As for the type c) contributions, arising from fig. <ref> ii), we find:

Δg_Lb ( c )    =
g_Lb/16 π^2
∑_l=1^m | r_l |^2  B_1 ( 0, 0, m_l^2 ),

Δg_Rb ( c )    =
Δg_Lb ( c ) ( g_Lb →g_Rb ).

In the CP-conserving limit,
eqs. <ref>–<ref> agree with eqs. (5.1)
of ref. [208].

Collecting all the divergent terms in eqs. <ref>,
<ref> and <ref>, we find:

Δg_Lb ( a )
+ Δg_Lb ( b )
+ Δg_Lb ( c )
Δ_ε/32 π^2
[ 2 i ∑_l,l'=1^m r_l Y_ll' r_l'^∗+ ( g_Rb - g_Lb ) ∑_l=1^m | r_l |^2

Hence, given eq. <ref>, we conclude that a consistent theory requires:

The same condition can also be obtained by collecting all the
divergent terms in eqs. <ref>, <ref>,
and <ref>.

∑_l,l'=1^m r_l Y_ll' r_l'^*
∑_l |r_l|^2.

Finally, we calculate the diagrams in fig. <ref> iii) and iv). As mentioned above, although they are suppressed by $m_b$, they might be enhanced when the coupling of neutral scalars to the bottom quark gets enhanced. They lead to:

Δg_Lb ( d )    =
e   m_b   m_Z/8 π^2 s_w c_w
∑_l=1^m y_l  Re [r_l]
to 14 pt

g_Lb [ C_0 ( m_Z^2, 0, 0, m_Z^2, m_l^2, 0 )
- C_1 ( m_Z^2, 0, 0, m_Z^2, m_l^2, 0 ) ]

36mm + g_Rb  C_1 ( m_Z^2, 0, 0, m_l^2, m_Z^2, 0 )
to 14 pt },

Δg_Rb ( d )    =
Δg_Lb ( d ) ( g_Lb ↔g_Rb ).

§ MODELS WITH DOUBLET AND SINGLET SCALARS

We now slightly restrict the spectrum of the previous section, and consider the more specific case of extensions of the SM with $n_d$ scalar doublets, $n_c$ singly-charged scalar $\mathrm{SU(2)_L}$ singlets and $n_n$ real scalar
gauge-invariant fields.

The particle content is then $n \equiv n_d + n_c$ (complex) charged scalars $H_a^\pm$ and $m \equiv 2 n_d + n_n$ (real) neutral scalars $S_l^0$. As before, this counting includes the would-be Goldstone bosons $H_1^\pm = G^\pm$ and $S_1^0 = G^0$.  Without loss of generality, one may assume that the scalar with mass 125 GeV found at the LHC is $S^0_2$; the generality is lost if one makes the further assumption that the masses are ordered, since there might be massive scalars below 125 GeV.

§.§ Formalism

We use the formalism of refs. [267, 268, 269, 266] (cf. also ref. [10]).

We thus parameterize the $n_d$ doublets as:

Φ_k = (  φ_k^+
Φ̃_k ≡i σ_2 Φ_k^∗= (  φ_k^0^∗
- φ_k^-

and we assume that the fields $\varphi_k^0$ have vevs $v_k \left/ \sqrt{2} \right.$, where the $v_k$ may be complex.

In order to write $\varphi_k^+$ and $\varphi_k^0$ as superpositions of the physical (i.e. eigenstates of mass) fields $H^+_a$ and $S_l^0$, we introduce the $n_d \times n$ matrix $\mathcal{U}$ and the $n_d \times m$ matrix $\mathcal{V}$, such that:

Since $H_1^\pm$ and $S_1^0$ are would-be Goldstone bosons,
the first columns of $\mathcal{U}$ and $\mathcal{V}$ are fixed and given by
$\mathcal{U}_{k1} = v_k/v$ and $\mathcal{V}_{k1} = i v_k/v$, respectively, with $v^2 \equiv \sum_{k=1}^{n_d} |v_k|^2$
($v$ is real and positive by definition).

φ_k^+ = ∑_a=1^n 𝒰_ka H_a^+,

φ_k^0 = 1/√(2)
( v_k + ∑_l=1^m 𝒱_kl S_l^0 ).

Besides, we define an $n \times n$ unitary matrix $\tilde{\mathcal{U}}$,

𝒰̃ =
(  𝒰
𝒯  ),

where the matrix $\mathcal{T}$ only exists when $n_c \neq 0$.

We also define an $m \times m$ real and orthogonal matrix $\tilde{\mathcal{V}}$,

𝒱̃ =
(  Re  𝒱

Im  𝒱

ℛ  ),

where the matrix $\mathcal{R}$ only exists when $n_n \neq 0$.

Finally, to parameterize the Yukawa Lagrangian, we introduce the $n_d$-dimensional complex vectors $e$ and $f$, such that:

It goes without saying that charged and neutral scalar singlets have no Yukawa couplings.

ℒ_Yukawa = -
(  t̅_L     b̅_L  )
∑_k=1^n_d [
(  φ_k^+

φ_k^0  ) b_R
+ e_k
(  φ_k^0^∗

- φ_k^-
) t_R ] + h.c..

§.§ Connection with the general description

We can compare the new formalism to the one introduced in section <ref> above.

For the Yukawa couplings, we can use eqs. <ref>,
<ref>, and <ref>–<ref> to conclude that:

c_a = ∑_k=1^n_d 𝒰_ka^∗e_k,

d_a = ∑_k=1^n_d 𝒰_ka f_k,

r_l = - 1/√(2) ∑_k=1^n_d 𝒱_kl f_k.

As for the couplings between $Z$ and scalars, one can show that, in this class of models [268],

The last relation leads to $y_{l=1} = 0$, because $\mathcal{V}^\dagger \mathcal{V}$ is hermitian, which implies $\textrm{Im} \left( \mathcal{V}^\dagger \mathcal{V} \right)_{11} = 0$. Therefore, the sum in eq. <ref> really starts at $l=2$, as there is no vertex $Z Z G^0$.

X_a a^'    =     s_w^2 δ_a a^' -
( 𝒰^T 𝒰^∗)_a a^'/2
s_w^2 - c_w^2/2  δ_a a^'
+ ( 𝒯^T 𝒯^∗)_a a^'/2,

Y_l l^'    =     -
Im ( 𝒱^†𝒱 )_l l^',

y_l    =    - Im ( 𝒱^†𝒱 )_1l.

§ CONNECTION WITH EXPERIMENT

Before we apply the new formalism to a particular case, we discuss how to relate the theoretical calculations with experimental observations (for details, cf. ref. [251] and references therein).

The effective couplings $g_{\mathrm{L}b}^{\mathrm{eff}}$ and $g_{\mathrm{R}b}^{\mathrm{eff}}$ in eq. <ref>
may be determined experimentally from:

* The rate $R_b$, given by:

R_b =
Γ( Z →b b̅ )/Γ( Z →hadrons ).

* Several asymmetries, including:

* the $Z$-pole forward–backward asymmetry measured at LEP1,

σ( e^- →b_F )
- σ( e^- →b_B )/σ( e^-
→b_F ) + σ( e^- →b_B )
= 3/4 A_e  A_b,

where $b_F$ ($b_B$) stands for final-state bottom quarks
moving in the forward (backward) direction
with respect to the direction of the initial-state electron;

* the left–right forward–backward asymmetry
measured by the SLD Collaboration,

A_LR^FB ( b ) =  σ (e_L^-  → b_F )
- σ(e_L^-  → b_B )
- σ(e_R^-  → b_F )
+ σ(e_R^-  → b_B )/
σ(e_L^-  → b_F )
+ σ(e_L^-  → b_B )
+ σ(e_R^-  → b_F )
+ σ(e_R^-  → b_B )
= 3/4 A_b,

where $e_L^-$ ($e_R^-$) are initial-state left-handed (right-handed) electrons.

The recent fit of $R_b$ and $A_b$ to the electroweak data
by Erler and Freitas in ref. [251] finds:

R_b^fit = 0.21629 ±0.00066,
A_b^fit = 0.923 ±0.020,

to be compared with the SM values:

R_b^SM = 0.21582 ±0.00002,
A_b^SM = 0.9347.

Thus, the experimental $R_b$
is about $0.7\sigma$ above the SM value, while $A_b$ is about
$0.6\sigma$ below the SM value.  However, this good agreement only
applies to the overall fit of many observables producing
eqs. <ref>; the measured values of the bottom-quark
asymmetries by themselves alone reveal a discrepancy close to $3\sigma$.

As pointed out in ref. [251], extracting $A_b$ from
$A_{\mathrm{FB}}^{(0,b)}$ and using $A_e = 0.1501 \pm 0.0016$ leads to a result which is $3.1 \sigma$ below the SM (the precise value of $A_b$ depends on which observables $A_e$ is extracted from), while combining
$A_{\mathrm{FB}}^{(0,b)}$ with $A_{\mathrm{LR}}^{\mathrm{FB}}$ leads to $A_b = 0.899 \pm 0.013$, which deviates from the SM value by $2.8 \sigma$.

This suggests two possible approaches.

The first one consists in accepting the values in eq. <ref> and using them as constraints on New Physics (NP).

The second approach, by contrast, seeks NP that might explain an $R_b$ just slightly above the SM, together with an $A_b$ that undershoots the SM by around $3\sigma$.

For what follows, it is convenient to switch from the parameterization defined in eq. <ref> (which separates the tree-level contribution from the one-loop one) to a parameterization which separates the SM contribution from the BSM (or NP) one:

g_L b^eff = g_L b^SM + δg_L b,

g_R b^eff = g_R b^SM + δg_R b.

The quantities in $\delta$ thus represent the NP contributions. This notation can also be applied to $R_b$ and $A_b$, thus yielding $\delta R_b$ and $\delta A_b$, which turn out to be related with $\delta g_{\mathrm{L}b}$ and $\delta g_{\mathrm{R}b}$ according to [208]:

δR_b = - 0.7785  δg_L b + 0.1409  δg_R b,

δA_b = - 0.2984  δg_L b - 1.6234  δg_R b.

Hence, if one wishes to follow the second approach mentioned above—using NP to keep $R_b$ close to its SM value and to reduce $A_b$ significantly—, one needs a small $\delta g_{\mathrm{L} b}$, together with a large (and positive) $\delta g_{\mathrm{R} b}$.

§ A SIMPLE PARTICULAR CASE: THE C2HDM

We now investigate the application of the calculation described in section <ref> to a particular case: the C2HDM, studied in detail in the previous chapters.

Since the renormalization described in section <ref> is simple, there is no need to resort to the unphysical parameters introduced in chapter <ref>.

In order to apply the one-loop corrections of sections <ref> and <ref>, we must determine the values that the quantities introduced in section <ref> take in the C2HDM. Such task can be simplified by determining first the values of the quantities of section <ref> (which is the reason why the formalism of that section is so useful). This we do in section <ref> below; then, in section <ref>, we present our numerical results.

§.§ Expressions

The first thing to notice is that, since there are no scalar singlets in the C2HDM, the description of section <ref> becomes even more simplified, as the matrices $\mathcal{T}$ and $\mathcal{R}$ do not exist.

The matrix $\mathcal{U}$ can be easily obtained by comparing eqs. <ref>, <ref>, <ref> and <ref>. To obtain $\mathcal{V}$, eq. <ref> also needs to be taken into account. The results are:

𝒰    =    (
c_β     - s_β
s_β     c_β

𝒱    =    (
i c_β    R_11 - i s_βR_13     R_21 - i s_βR_23
R_31 - i s_βR_33

i s_β    R_12 + i c_βR_13     R_22 + i c_βR_23
R_32 + i c_βR_33

To determine the Yukawa couplings $e$ and $f$, we assume the Type-II C2HDM.
Therefore, comparing eqs. <ref> and <ref>, we directly find:

e_1 = 0,
f_2 = 0,
e_2 =  √(2) m_t/v_2,
f_1 =  √(2) m_b/v_1.

Now that we have all the parameters of section <ref>, we can trivially determine the general parameters of section <ref> using the relations of section <ref>, and thus calculate the one-loop corrections $\Delta g_{\mathrm{L}b}$ and $\Delta g_{\mathrm{R}b}$ in the C2HDM.

Particularly relevant for what follows are the values $c_2$ and $d_2$, given by:

c_2 =  √(2) m_t/v β,     d_2 = - √(2) m_b/v tanβ.

It is thus clear that $\left| c_2 \right|$ and $\left| d_2 \right|$ may be of vastly different orders of magnitude—in particular,
$\left| d_2 \right| \ll \left| c_2 \right|$ for $\tan{\beta} \sim 1$.
However, when $\tan{\beta} \gtrsim \sqrt{m_t / m_b} \approx 6$,
$\left| d_2 \right|$ becomes larger than $\left| c_2 \right|$,
which is the regime that we will be mostly interested in.

§.§ Results

From the results for $\Delta g_{\mathrm{L}b}$ and $\Delta g_{\mathrm{R}b}$, we can easily obtain the NP contributions $\delta g_{\mathrm{L}b}$ and $\delta g_{\mathrm{R}b}$ introduced in eq. <ref>; as already seen, these allow a simple comparison with experiment, and are thus preferred in the results that are presented below.

We use superscripts $c$ and $n$ to denote the NP contributions to $\delta g_{\mathrm{L}b}$ and $\delta g_{\mathrm{R}b}$ coming from the charged scalar (in the C2HDM, just $H^{\pm}$) and neutral scalars, respectively.

The calculations were performed with and converted to LoopTools [105] for numerical evaluation.

The numerical simulation follows the strategy described in section <ref>—which selects, in particular, the range $0.8 \le \tan \beta  \le 35$.

We combine the results obtained from that run with those from a new run, characterized by the range $0 \le \tan \beta \le 100$. Such extreme (very low and very high) values of $\tan{\beta}$ may be in contradiction with certain Flavour Physics observables, notably $Z \rightarrow b \bar{b}$.

[Cf. fig. <ref> below. Those extreme values of $\tan{\beta}$ (both high and low) also violate perturbative unitarity.]

Nevertheless, we will consider them in order to stress that the details of such a bound may require both the charged-scalar and the neutral-scalar contributions.

We start by exploring $\delta g_{\mathrm{L}b}^c$ and $\delta g_{\mathrm{R}b}^c$, i.e. the contributions from the charged scalar.
More precisely, we want to ascertain the impact of the approximation $m_{\mathrm{Z}}=0$ on those contributions, since this approximation is usually employed in the literature.

The function $m_t^2\, C_0 \left( 0, 0, 0, m_a^2, m_t^2, m_t^2 \right)
= \frac{x}{1 - x} \left( 1 + \frac{\ln{x}}{1 - x} \right)$, with $x = \frac{m_t^2}{m_a^2}$, which shows up in the limit $m_{\mathrm{Z}}=0$,
has been given in eq. (4.5) of ref. [208] and used by many authors (cf. e.g. ref. [271] and references therein).

It is easy to prove that, with such approximation, we have:

On the left panel of fig. <ref>, we show the exact contributions (i.e. without $m_{\mathrm{Z}}=0$) from the charged scalar to $\delta g_{\mathrm{L}b}$ and $\delta g_{\mathrm{R}b}$. We see that eq. <ref> holds very well.

Left panel: contribution of $H^{\pm}$
to $\delta g_{\mathrm{L}b}$ (red curve) and to $- \delta g_{\mathrm{R}b}$ (blue curve). Right panel: the assymetries $R_{g_{\mathrm{L}}}^c$ (red curve) and $R_{g_{\mathrm{R}}}^c$ (blue curve), defined in eq. <ref>.

This result can also be ascertained by the asymmetries $R_{g_{\mathrm{L,R}}}$, defined in terms of the $\delta$ quantities evaluated with $m_{\mathrm{Z}} \neq 0$ and with $m_{\mathrm{Z}} = 0$:

R_g_L^c = δg_Lb^c ( m_Z ) -
δg_Lb^c ( 0 )/δg_Lb^c ( m_Z ) + δg_Lb^c ( 0 ),

R_g_R^c = δg_Rb^c ( m_Z ) -
δg_Rb^c ( 0 )/δg_Rb^c ( m_Z ) + δg_Rb^c ( 0 ).

We observe on the right panel of fig. <ref> that both asymmetries are very small, which thus confirms the validity of the approximation that neglects the $Z$ boson mass in the charged scalar contributions to NP.

Note that $\delta g_{\mathrm{L}b}^c$ is positive, while $\delta g_{\mathrm{R}b}^c$ is negative. From eq. <ref>, we see that a positive $\delta g_{\mathrm{L}b}$ tends to make $R_b$ smaller—whereas we expected it to become larger, recall eq. <ref>.
From eq. <ref>, we also conclude that the correction $\delta g_{\mathrm{R}b}^c$ is too small to have an impact on $R_b$.

In principle, it could have a substantial impact on $A_b$
going in the wrong direction when compared with the
experimental measurements (cf. eq. <ref>).
However, we will see below (on the right panel of fig. <ref>) that this only happens for large values of $\tan\beta$, not allowed by perturbativity.

Comparison of $\delta g_{\mathrm{L}b}^n$ with $\delta g_{\mathrm{L}b}^c$ (left) and of $\delta g_{\mathrm{R}b}^n$ with $\delta g_{\mathrm{R}b}^c$ (right).
Blue points correspond to $0 \le \tan\beta \le 10$,
green points to $10 \le \tan\beta \le 30$ and
red ones to $30 \le \tan\beta \le 100$.

We now compare the contributions from the charged scalar with those from the neutral scalars. The latter are frequently disconsidered in the literature (cf. e.g. refs. [202, 271]), which motivates us to investigate under which circumstances they can be large.

On the left panel of fig. <ref>,
we display $\delta g_{\mathrm{L}b}^n$ versus $\delta g_{\mathrm{L}b}^c$;
on the right panel, $-\delta g_{\mathrm{R}b}^n$ is displayed against $-\delta g_{\mathrm{R}b}^c$ (both $\delta g_{\mathrm{R}b}^n$ and $\delta g_{\mathrm{R}b}^c$ are negative).
We consider separately three different ranges of $\tan \beta$ and we represent each one by a different color.
We see in the right panel that $\left| \delta g_{\mathrm{R}b}^n \right|$
generally is of order $\left. \left| \delta g_{\mathrm{R}b}^c \right| \right/ 10$, but they may be comparable in the low-$\tan{\beta}$ regime.
The left panel shows a very different behaviour, namely: $\delta g_{\mathrm{L}b}^n \ll \delta g_{\mathrm{L}b}^c$ for low $\tan{\beta}$, $\delta g_{\mathrm{L}b}^n \sim \delta g_{\mathrm{L}b}^c$ for $\tan{\beta} \sim 30$ and $\delta g_{\mathrm{L}b}^n \gg \delta g_{\mathrm{L}b}^c$ for high $\tan{\beta}$. So,

one really cannot neglect the contributions from neutral scalars when $\tan{\beta} \gtrsim 10$. For low $\tb \sim 1$, $\delta g_{\mathrm{L}b}^c$ is much larger than $\delta g_{\mathrm{L}b}^n$, but $\delta g_{\mathrm{R}b}^n$ may not be much smaller than $\delta g_{\mathrm{R}b}^c$.

In fig. <ref>, we provide a more explicit comparison between the contributions to NP arising from the charged and from the neutral scalars.

Left panel: comparison of $|\delta g_{\mathrm{L}b}^n|$ (in blue)
and $|\delta g_{\mathrm{L}b}^c|$ (in green) with $|\delta g_{\mathrm{L}b}^n + \delta g_{\mathrm{L}b}^c|$ (in pink).

Right panel: comparison of $|\delta g_{\mathrm{R}b}^n|$ (in blue) and $|\delta g_{\mathrm{R}b}^c|$ (in green) with $|\delta g_{\mathrm{R}b}^n + \delta g_{\mathrm{R}b}^c|$ (in pink).
Also displayed (in red) are the contributions of the neutral type d) diagrams—Fig. <ref> iii) and iv)—to both $\delta g_{\mathrm{L}b}^n$ (on the left panel) and $\delta g_{\mathrm{R}b}^n$ (on the right panel).

On the left panel, we see that the contribution from neutral scalars to $\delta g_{\mathrm{L}b}$ becomes larger than the contribution from the charged scalar—eventually by many orders of magnitude—as soon as $\tan{\beta} > 30$. We thus reinforce the same conclusion derived above, namely: one cannot neglect the contribution of the neutral scalars to $\delta g_{\mathrm{L}b}$.

In the same figure, we ascertain (in red) the importance of contributions from the diagrams <ref> iii) and iv), calculated in section <ref>.

We see that, when $\tb$ is low, they may constitute a substantial part of the NP contribution; yet, that is precisely the range in which $\delta g_{\mathrm{L} b}^n$ and $\delta g_{\mathrm{R} b}^n$ are too small to be of practical relevance (as can be seen by the blue points).

We conclude that, at least in the particular case of the C2HDM, it is correct to neglect the diagrams in fig. <ref> iii) and iv), as was done in ref. [208].

The pink points in fig. <ref> allow us to infer that a significant impact on $A_b$ and $R_b$
can only occur for either very low or very high values of $\tan\beta$;
specifically, for $\tb \lesssim 1$, $\delta g_{\mathrm{L}b}^c + \delta g_{\mathrm{L}b}^n \sim 10^{-3}$ and for $\tb \gtrsim 50$, $- \delta g_{\mathrm{R}b}^c - \delta g_{\mathrm{R}b}^n \gtrsim 10^{-3}$.
The impact on $A_b$ and $R_b$ is explored in more detail in fig. <ref>.

$A_b$ versus $R_b$ in the C2HDM for all values of
$0<\tan\beta <100$.
The charged-scalar contribution is shown in
red for low $\tb$ and in orange for large $\tb$.
The contribution of the neutral scalars is shown in blue and lies very
close to the SM point.
In green (in background) the sum of the contributions.

There, we see that points deviate significantly from the experimental value of $R_b$ in the low $\tan\beta$ regime; we checked that points only lie within the 2$\sigma$ interval for $\tb > 0.8$, which is why this lower bound was imposed in chapter <ref>. The contribution of the neutral scalars (in blue) is always very small; we verified that, for the neutral scalars to have meaningful impact in the C2HDM, one would have to consider values of $\tan\beta>250$, which would violate perturbativity. This is because the neutral contributions in the C2HDM are related to a ratio of vevs ($v_2/v_1 = \tan{\beta}$), which is limited by perturbative unitarity. However, such relation is not necessary in more general models, in which case the contribution from the neutral scalars may very well be important.

It is a matter of course that, when studying any model, all the theoretical and experimental constraints need to be included, which may curtail a large part of the phase space for such extreme couplings. This will have to be evaluated in a case-by-case basis.

§ SUMMARY

We studied the one-loop contributions to $Z \rightarrow b \bar b$
in generic models with extra scalars. We started by performing the one-loop calculations for general couplings and we derived the conditions that must hold for the divergences to cancel.
We then concentrated on models with a generic number of scalar doublets and singlets, using the formalism of refs. [267, 268, 269, 266], which is very useful to connect the general one-loop results to particular cases. We verified this explicitly in the case of the C2HDM; here, we also highlighted the possible importance of contributions arising from neutral scalars, which may in some cases be much larger than the contributions of the charged scalars. This imbalance must be considered when evaluating the limits on $A_b$ and $R_b$.

CHAPTER: SIMPLEST LINEAR SEESAW MODEL

lepton flavour violation
lepton number violating
$\mathrm{SU(3)_c \otimes SU(2)_L \otimes U(1)_Y}$

In this final chapter, we investigate the phenomenology of a MHM motivated by non-zero neutrino masses. As is well known, neutrino masses constitute one of the most robust evidences for new physics. Ever since the discovery [272, 273] and confirmation [274, 275] of neutrino oscillations, the efforts to understand the mechanism of neutrino mass have been intense.

A popular approach is the type-I seesaw mechanism, in which neutrinos get mass due to the exchange of heavy singlet mediators.

In its standard high-scale realization, the seesaw mechanism hardly leads to any phenomenological implication besides those associated to the neutrino masses themselves. However, the seesaw can arise from low-scale physics [276, 277], e.g. through the linear seesaw mechanism [278, 279, 280].

Here, we examine the scalar sector of the simplest variant of such mechanism, where the seesaw mechanism is realized using just the SM gauge structure associated to the symmetry [281, 282, 283].

In addition to the SM Higgs doublet, the model contains a second Higgs doublet, as well as a complex singlet scalar, both carrying lepton number charges. Spontaneous breaking of the lepton number (global) symmetry hence implies the existence of a Nambu-Golstone boson, a variant of the so-called majoron [282, 283].

We start by presenting the scalar sector of the model in section
<ref>, after which we provide our results.
The latter illustrate the phenomenology of the model,
giving special emphasis to the invisible Higgs decay channel with majoron emission [276]. This has implications for collider experiments [284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294], and has indeed been searched by LEP [295] and LHC collaborations [296, 297].
The new signals can be studied in proton-proton collisions such as at the High-Luminosity LHC setup, as well as in the next generation of lepton collider experiments such as CEPC, FCC-ee, ILC and CLIC [298, 299, 300, 301].

§ THE SCALAR SECTOR

As we just mentioned, besides the SM doublet $\Phi$, we consider an additional doublet $\chi_L$ and a (complex) singlet $\sigma$. We parameterize all these multiplets according to:

\begin{equation}
\label{Chap-Valle:eq:1}
\Phi=
\left(
\begin{array}{c}
\phi^+\\[+2mm]
\frac{1}{\sqrt{2}} \left( v_{\phi} + R_1 + i\, I_1\right)
\end{array}
\right),
\left(
\begin{array}{c}
\chi_L^+\\[+2mm]
\frac{1}{\sqrt{2}} \left( v_L + R_2 + i\, I_2\right)
\end{array}
\right),
~~~\sigma =     \frac{1}{\sqrt{2}} \left( v_\sigma + R_3 + i\, I_3\right),
\end{equation}

with $v_{\phi}$, $v_L$ and $v_\sigma$ real parameters, $\phi^+$ and $\chi_L^+$ complex fields, and $R_i$ and $I_i$ real fields.

We choose the following lepton number assignments,

\begin{equation}
\label{Chap-Valle:eq:2}
L[\Phi] =0,\quad L[\chi_L] =-2,\quad L[\sigma]=1.
\end{equation}

Therefore, the most general potential is:

\begin{align}
\label{Chap-Valle:eq:7}
V = &-\mu^2\, \Phi^\dagger \Phi + \lambda\, \left(\Phi^\dagger
\Phi\right)^2 -\mu_L^2\, \chi_L^\dagger \chi_L + \lambda_L\,
\left(\chi_L^\dagger   \chi_L\right)^2
- \mu_\sigma^2 \sigma^\dagger \sigma
+ \lambda_\sigma  \left(\sigma^\dagger \sigma\right)^2\nonumber\\[+2mm]
+ \beta_1\, \Phi^\dagger \Phi\, \chi_L^\dagger \chi_L
+ \beta_2\, \Phi^\dagger \Phi\,  \sigma^\dagger \sigma
+ \beta_3\,  \chi_L^\dagger \chi_L\,  \sigma^\dagger \sigma
+  \beta_5\,  \Phi^\dagger \chi_L\,  \chi_L^\dagger \Phi
- \left(\beta_4 \Phi \chi_L \sigma^2 + \text{h.c.}\right).
\end{align}

Although $\beta_4$ is in general complex, we will consider the region of parameter space where it takes real values only.

$\beta_4$ cannot be rendered real by a basis choice; more precisely, there is no basis where $v_{\phi}$, $v_L$, $v_\sigma$ and $\beta_4$ are all real. In principle, these quantities could be set real by imposing the symmetry CP on the theory; however, this is not a sound option, because this symmetry is violated by quark interactions. In other words, if $v_{\phi}$, $v_L$, $v_\sigma$ and $\beta_4$ are all taken to be real parameters, the model will be as theoretically unsound as the real 2HDM (recall the discussion in chapter <ref>). Strictly speaking, then, and if we choose a basis where $v_{\phi}$, $v_L$, $v_\sigma$ are real, $\beta_4$ should be allowed to be complex, which would imply CP violation in the scalar sector, and the diagonalization of the neutral scalar states should be done accordingly. However, if we focus on the particular region of the parameter space of the (CP-violating) model where $\beta_4$ takes real values, then, in that region, CP can be well defined in the potential, so that CP-even states and CP-odd states can be separately diagonalized. This is what we do in the following.

After writing the minimization equations,

we can derive the following charged scalar mass matrix in the basis $(\phi^+,\chi^+_L)$:

\begin{equation}
\label{Chap-Valle:eq:9}
M^2_{\rm ch}=
\begin{pmatrix}
\dfrac{\beta_4 v_L v_{\sigma}^2-\beta_5 v_L^2 v_\phi}{2 v}
& \dfrac{\beta_5 v_L v_\phi - \beta_4 v_{\sigma}^2}{2} \\[+2mm]
\dfrac{\beta_5 v_L v_\phi -\beta_4 v_{\sigma}^2}{2}
& \dfrac{\beta_4 v_{\phi} v_{\sigma}^2-\beta_5 v_L v_\phi^2}{2 v_L}
\end{pmatrix}.
\end{equation}

Diagonalizing this matrix, we find two mass states: the charged (massless) would-be Goldstone boson, $G^+$, as well as a massive state, $H^+$. The squared mass of the latter is:

\begin{equation}
\label{Chap-Valle:eq:11}
= \frac{(\beta_4 v_{\sigma}^2 -\beta_5 v_L v_\phi) \left(v_{\phi}^2+v_L^2\right)}{2 v_{\phi} v_L} =
\frac{\beta_4 v_{\sigma}^2}{\sin 2\beta} - \frac{1}{2} \beta_5 v^2,
\end{equation}

where we defined:

\begin{equation}
\label{Chap-Valle:eq:16}
\qquad
\tan\beta= \frac{v_L}{v_{\phi}}.
\end{equation}

The neutral scalar (CP-even) mass matrix in the basis $(R_1, R_2, R_3)$ is given by:

\begin{equation}
\label{Chap-Valle:eq:12}
M^2_{\rm ns} =
\begin{pmatrix}
2 \lambda v_{\phi}^2+\dfrac{\beta_4 v_L v_{\sigma}^2}{2 v_{\phi}} & \beta_1 v_{\phi}
v_L-\dfrac{\beta_4 v_{\sigma}^2}{2} + \beta_5 v_L v_\phi & (\beta_2 v_{\phi}-\beta_4
v_L) v_{\sigma} \\[+2mm]
\beta_1 v_{\phi} v_L-\dfrac{\beta_4 v_{\sigma}^2}{2}+ \beta_5 v_L v_\phi & 2 \lambda_L
v_L^2+\dfrac{\beta_4 v_{\phi} v_{\sigma}^2}{2 v_L} & \beta_3 v_L
v_{\sigma}-\beta_4 v_{\phi} v_{\sigma} \\[+2mm]
(\beta_2 v-\beta_4 v_L) v_{\sigma} & \beta_3 v_L v_{\sigma}-\beta_4 v_{\phi} v_{\sigma} & 2
\lambda_\sigma v_{\sigma}^2
\end{pmatrix}.
\end{equation}

We can thus define the scalar mass eigenstates $h_1$,
$h_2$ and $h_3$, as well as the orthogonal matrix $\mathcal{O}^{R}$, such that:

\begin{equation}
\label{Chap-Valle:eq:20}
\begin{pmatrix}
\end{pmatrix}
= \mathcal{O}^{R}
\begin{pmatrix}
\end{pmatrix},
\qquad
\mathrm{with}
\qquad
\mathcal{O}^{R}  \cdot M_{\rm ns}^2\cdot \mathcal{O}^{R}{}^{\mathrm{T}} =
\text{diag}(m_{1}^2,m_{2}^2,m_{3}^2),
\end{equation}

where $m_1^2$, $m_2^2$ and $m_3^2$ are the squared masses of $h_1$,
$h_2$ and $h_3$, respectively, with $m_1 < m_2 < m_3$.

The matrix $\mathcal{O}^{R}$ can
be parameterized precisely as the matrix $R$ of the C2HDM (eq. <ref>):

\begin{equation}
\label{Chap-Valle:eq:21}
\mathcal{O}^{R}
%=\mathcal{O}^{R}_3 \cdot \mathcal{O}^{R}_2 \cdot \mathcal{O}^{R}_1
\begin{pmatrix}
c_1c_2 & s_1 c_2 & s_2\\
\end{pmatrix},
\end{equation}

where $c_i=\cos\alpha_i,s_i=\sin\alpha_i$.

The neutral pseudoscalar (CP-odd) mass matrix in the basis $(I_1, I_2, I_3)$ is given by:

\begin{equation}
\label{Chap-Valle:eq:13}
M^2_{\rm nps} =
\begin{pmatrix}
\dfrac{\beta_4 v_L v_{\sigma}^2}{2 v_{\phi}} & -\dfrac{\beta_4 v_{\sigma}^2}{2} & -\beta_4
v_L v_{\sigma} \\[+2mm]
-\dfrac{\beta_4 v_{\sigma}^2}{2} & \dfrac{\beta_4 v_{\phi} v_{\sigma}^2}{2 v_L} & \beta_4 v_{\phi} v_{\sigma}
\\[+2mm]
-\beta_4 v_L v_{\sigma} & \beta_4 v_{\phi} v_{\sigma} & 2 \beta_4 v_{\phi} v_L
\end{pmatrix}.
\end{equation}

Similarly to what we did for the scalar states, we can define the pseudoscalar mass eigenstates $G_0$, $J$ and $A$, as well as the orthogonal matrix $\mathcal{O}^{I}$, such that:

\begin{equation}
\label{Chap-Valle:eq:24}
\begin{pmatrix}
\end{pmatrix}
\begin{pmatrix}
\end{pmatrix},
\qquad
\mathrm{with}
\qquad
\text{diag}(0,0,m_{A}^2) =
\mathcal{O}^{I} \cdot M_{\rm ns}^2\cdot\mathcal{O}^{I}{}^{\mathrm{T}}.
\end{equation}

$G_0$ is the neutral (massless) would-be Goldstone boson and $J$ is the majoron (which is also massless). The squared mass of $A$ is given by:

\begin{equation}
\label{Chap-Valle:eq:14}
m_A^2= \frac{\beta_4 \left(4 v_{\phi}^2 v_L^2+v_{\phi}^2 v_{\sigma}^2+v_L^2
v_{\sigma}^2\right)}{2 v_{\phi} v_L}= \beta_4 \left[ v^2
\sin 2\beta  + \frac{v_{\sigma}^2}{\sin2\beta} \right].
\end{equation}

Finally, we can use the relations of the model to rewrite the quartic parameters of the potential in terms of other parameters. Particularly relevant for what follows is the expression for $\lambda_L$, given by:

\begin{align}
\lambda_L=&-\frac{1}{4 v_{L}^3}\left[\vb{14}
\beta_{4} c_{1}^4 v_{\phi} v_{\sigma}^2+2
c_{1}^2 \left(\beta_{4} c_{3}^4
s_{1}^2 v_{\phi} v_{\sigma}^2+c_{3}^2 \left(2 \beta_{4}
s_{1}^2 s_{3}^2
v_{\phi} v_{\sigma}^2-m_{2}^2 v_{L}\right)+\beta_{4} s_{1}^2
v_{\phi} v_{\sigma}^2
\right.\right.
\nonumber\\
-m_{3}^2 s_{3}^2
v_{L}\right)+s_{1}^2 \left(\beta_{4}
c_{2}^4 s_{1}^2 v_{\phi} v_{\sigma}^2-2 c_{2}^2 \left(m_{1}^2
v_{L}-\beta_{4} s_{1}^2 s_{2}^2 v_{\phi}
\left(\beta_{4} c_{3}^4 s_{1}^2 s_{2}^2 v_{\phi}
\right.\right.\right.
\nonumber\\
+c_{3}^2 \left(2
\beta_{4} s_{1}^2 s_{2}^2 s_{3}^2 v_{\phi} v_{\sigma}^2-2 m_{3}^2
v_{L}\right)+\beta_{4} s_{1}^2 s_{2}^2 s_{3}^4
v_{\phi} v_{\sigma}^2-2
m_{2}^2 s_{3}^2 v_{L}\right)\right)
+4 c_{1}
c_{3} s_{1}
s_{2} s_{3} v_{L} (m_{2}^2-m_{3}^2) \vb{14}\right].
\label{Chap-Valle:eq:23b}
\end{align}

§ RESULTS

§.§ Simulation procedure and constraints

The generation of Feynman rules and the calculation of leading order (LO) decay widths was performed with .
For the scatter plots below, we generated points in the parameter space of the model with $m_1$ fixed to 125 GeV, and we varied the other parameters in the following way:

\begin{gather}
\label{Chap-Valle:eq:59}
m_{2}, m_{3},m_{A},m_{\mathrm{H}^+} \in [125, 800] \, \text{GeV},
\qquad
\alpha_1,\alpha_2,\alpha_3 \in [-\frac{\pi}{2},\frac{\pi}{2}],
\qquad
v_L \in [10^{-6},10^2] \, \text{GeV},
\nonumber\\
v_{\sigma} \in [10^3,1.2 \times 10^4] \, \text{GeV}.
\end{gather}

We imposed both theoretical and experimental constraints on the generated points; details can be found in ref. [8].

On the theoretical side, we considered BFB [302], unitarity [303] and the oblique parameters S, T and U [268, 59].

On the experimental side, we took into account both astrophysical constraints and contraints from the LHC. The first
affect the majoron $J$, which would be copiously produced in stars, leading to new mechanisms of stellar cooling.

In our model, that leads to the constraint [304, 305]:

\begin{equation}
\label{Chap-Valle:eq:60}
|\vev{ J|\Phi}|=
\frac{2 v_{\phi} v_L^2}{\sqrt{(v_{\phi}^2+v_L^2) \big(v_{\phi}^2 (4 v_L^2 +
v_{\sigma}^2) + v_L^2 v_{\sigma}^2\big)}}
\lesssim10^{-7},
\end{equation}

with $\vev{ J|\Phi}$ denoting the majoron projection into the SM doublet.

As for the LHC constraints, these concern both the experiments related to the 125 GeV scalar Higgs boson, $h_{1}$, as well as those related to additional scalar bosons. For the former, we used combined ATLAS and CMS analyses for 8 TeV [306], and ATLAS results for 13 TeV [307]; by default, we imposed these constraints at the $3 \sigma$ level. The information relative to other scalars was included using the HiggsBounds4 package [177].

§.§ The impact of the astrophysical constraints on $v_L$

We start by ascertaining the impact of the astrophysical constraint, eq. (<ref>).

In order to more easily comply with this constraint, we have sampled
$v_\sigma$ values above 1 TeV, aware that lower values could be possible.

The allowed region in the ($v_\sigma,v_L$) plane is shown in fig. <ref>.

$v_\sigma$ versus $v_L$. The yellow points satisfy eq. (<ref>) without any other constraint. The other points satisfy all the other theoretical and experimental constraints: in dark green, $v_L> 0.1$ GeV; in green, $v_L\in[0.01,01]$GeV; in yellow green, $v_L < 0.01$ GeV. Besides, a cut $v_\sigma > 1$ TeV was imposed.

The yellow region is the set of values of
$(v_\sigma,v_L)$ that satisfy eq. (<ref>), ignoring all the remaining contraints. The green regions, in contrast, satisfy all constraints. We explicitly associate different shades of green to different ranges of $v_L$: points with $v_L > 0.1 \, \text{GeV}$ are shown in dark green, points with $v_L \in [0.01,0.1] \, \text{GeV}$ in green and points with $v_L < 0.01 \, \text{GeV}$ in yellow green. The figure shows that all points (in particular, the yellow ones) obey $v_L < 0.5$ GeV, which represents a much smaller range than that which we started with.
In other words, the astrophysical contraints by themselves force $v_L$ to be quite constrained.

Note that, besides eq. (<ref>), there is another expression relating the vevs, $m_{\mathrm{W}}=\dfrac{1}{2}\, \dfrac{e}{s_{\mathrm{w}}} \, \sqrt{v_L^2+v_\phi^2}$, which explains the boundary in the plane ($v_\sigma,v_L$).

§.§ Mixing angles

In fig. <ref>, the mixing angles in the neutral CP-even rotation matrix are shown.

$\alpha_2$ (left) and $\alpha_3$ (right) versus $\alpha_1$. Points obey to all theoretical and experimental constraints: in dark green, $v_L> 0.1$ GeV; in green, $v_L\in[0.01,01]$GeV; in yellow green, $v_L < 0.01$ GeV.

We see that $\alpha_1$ is close to zero for very small values of $v_L$.

The reason can be traced back to the perturbative unitarity constraints on the quartic coulings, in particular to those on $\lambda_L$, eq. (<ref>). In fact, as $\lambda_L$ is inversely proportional to the third power of $v_L$, small values of $v_L$ demand a very small numerator, in order to prevent $\lambda_L$ violating the perturbative unitarity constraints.
One can verify that the smallness of the numerator occurs for $\alpha_1$  close to zero, which thus explains the observed behaviour.

It turns out that it is not only $\alpha_1$ that almost vanishes for small values of $v_L$; the same should happen with $\beta$, according to eq. <ref>. This is what is found in fig. <ref>.

Correlation between $\beta$ and $v_L$. The color
code is that of fig. <ref>.

§.§ Higgs masses

The argument of perturbative unitarity which we have just used to justify the smallness of $\alpha_1$ also leads to a compressed spectrum of the Higgs masses. Indeed, small values of $v_L$ require almost degenerate scalar masses in order for the unitarity constraints to be verified. This behaviour is illustrated in the plots in fig. <ref>.

Correlation between $m_{A}/m_{2}$ (left), $m_{\mathrm{H}^+}/m_{2}$ (middle), $m_{3}/m_{A}$ (right) and $v_L$. The color code is that of fig. <ref>.

We should stress that this compression is much stronger than that usually imposed by the oblique parameters. On the other hand, the figures also show that, for larger values of $v_L$, the points satisfying all the constraints (including those from the oblique parameters) can have a sizeable splitting.

§.§ Invisible Higgs decays

We now turn to the invisible decays of the CP-even neutral scalars to the majoron, $h_{i}\to JJ$ and $h_{i}\to 2h_{j} \to 4J$.

For the latter, we must have $m_{i}> 2 m_{j}$.

We investigate the branching ratios (BRs) corresponding to the total invisible decay of the scalar $h_i$ into $J$, $\textrm{BR}_{\textrm{Inv}}(h_{i})$.
We ascertain the impact of the LHC constraints on the 125 GeV Higgs boson by considering separately points complying with $3 \sigma$ (in green) and with $2 \sigma$ (in yellow green).

In fig. <ref>, we plot $\textrm{BR}_{\textrm{Inv}}(h_{1})$ as a function of $v_L$ on the left panel, and as a function of $m_2$ on the right panel.

Left panel: $\textrm{BR}_{\textrm{Inv}}(h_{1})$ versus $v_L$. Right panel: $\textrm{BR}_{\textrm{Inv}}(h_{1})$ versus $m_2$. The LHC constraints on the 125 GeV Higgs boson are imposed at 3$\sigma$ in green and at 2$\sigma$ in yellow green.

We see that an invisible branching ratio around 20%—close to the present upper bound [296, 297]—is possible for 3$\sigma$ constraints.
If 2$\sigma$ limits are applied, however, the ratio reduces to a maximum of 10%. This is also illustrated in fig. <ref>, where we plot $\textrm{BR}_{\textrm{Inv}}(h_{1})$ versus the signal strength of the Higgs boson produced via gluon fusion and decaying in the $ZZ$ final state, $\mu_{zz}^{ggF}$.

$\textrm{BR}_{\textrm{Inv}}(h_{1})$  versus $\mu_{zz}^{ggF}$. The color code is that of fig. <ref>.

We see that, if this signal strength becomes closer to one, $\textrm{BR}_{\textrm{Inv}}(h_{1})$ will be smaller, which can be important for LHC searches.

Finally, we show in fig. <ref> the correlation between the invisible branching ratios of different scalars.

In the right panel, which shows $\textrm{BR}_{\textrm{Inv}}(h_{3})$ versus $\textrm{BR}_{\textrm{Inv}}(h_{2})$, it is clear that both BRs can have a wide range of values, from very small (hence visible) to close to 100%.

Left panel: $\textrm{BR}_{\textrm{Inv}}(h_{2})$ versus $\textrm{BR}_{\textrm{Inv}}(h_{1})$. Right panel: $\textrm{BR}_{\textrm{Inv}}(h_{3})$ versus $\textrm{BR}_{\textrm{Inv}}(h_{2})$. The color code is that of fig. <ref>.

§ SUMMARY

We investigated the phenomenology of the scalar sector of the simplest realization of the linear seesaw mechanism within the context of the SM gauge symmetry. In addition to the SM scalar doublet, it features two scalar multiplets charged under lepton-number: an extra doublet and a complex singlet. The physical content of the model includes a majoron $J$, which leads to severe constraints from astrophysical experiments.

This is reflected in the parameter space of the model, which nonetheless still allows a rich phenomenology.

In particular, values of an invisible branching ratio up to $20\%$ are allowed for the current experimental precision. However, future
lepton colliders may play a decisive role here; in fact, $\textrm{BR}_{\textrm{Inv}}(h_{i})$ is
expected to be measured with precision better than 1% level [299], which will impose severe constraints on these decay modes.

CHAPTER: CONCLUSION

In this thesis, we discussed several aspects of Multi-Higgs models (MHM).

As a toolkit to more practically address the different topics, we presented , a multi-tasking software for particle physics studies. By making use of already existing programs (, , ), automatically generates Feynman rules, generates and draws Feynman diagrams, generates amplitudes, performs both loop and algebraic calculations, and fully renormalizes models. In parallel with this automatic character, allows the user to manipulate the generated results in Mathematica notebooks in a flexible and consistent way.

Throughout the thesis, the versatility of the software became manifest.

It proved suitable for building and testing simple models, as was shown with the toy model discussed in chapter <ref>; there, was used to generate Feynman rules and one-loop amplitudes, and to quickly check that the counterterms were not able to absorb all divergences.

The program also revealed quite useful to investigate the tree-level phenomenology of non-trivial models such as the one presented in chapter <ref>, by generating the Feynman rules and by calculating amplitudes and decay widths in a user-friendly way.

It also proved very convenient to ascertain more complex problems, such as the verification of both Ward identities in non-abelian theories (appendix <ref>) and particular cases of the theorem proposed in appendix <ref>, or the cancellations of UV divergences in generic models (chapter <ref>).

It is fair to say that it even went beyond our initial expectations, by allowing to calculate the three-loop amplitudes involved in chapter <ref>, which corresponds to a highly complex calculation with hundreds of Feynman diagrams.