scenario” as it has been defined in Ref. [135]. For the Standard Model
parameters, we use the values given in Eqs. (124). For the MSSM parameters, we
set

$\displaystyle t_{\beta}={}$ $\displaystyle 5,$ $\displaystyle m_{A}={}$
$\displaystyle 90\;\text{GeV},$ (125) $\displaystyle M_{\tilde{q}_{1}}^{2}={}$
$\displaystyle
M_{\tilde{u}_{1}}^{2}=M_{\tilde{d}_{1}}^{2}=(2\;\text{TeV})^{2},$
$\displaystyle M_{\tilde{q}_{2}}^{2}={}$ $\displaystyle
M_{\tilde{u}_{2}}^{2}=M_{\tilde{d}_{2}}^{2}=(2\;\text{TeV})^{2},$
$\displaystyle M_{\tilde{q}_{3}}^{2}={}$ $\displaystyle
M_{\tilde{u}_{3}}^{2}=M_{\tilde{d}_{3}}^{2}=(1.5\;\text{TeV})^{2},$
$\displaystyle\mu={}$ $\displaystyle 1\;\text{TeV},$ $\displaystyle A_{t}={}$
$\displaystyle A_{q},$ $\displaystyle A_{b}={}$ $\displaystyle A_{q}.$

The “$M_{h}^{125}$ scenario” was designed such that with the theoretical
prediction at that time one obtained a Higgs boson mass that was compatible
with the experimental value within the theoretical uncertainties over a wide
range of $m_{A}$ and $t_{\beta}$. In this original version, the stop-mixing
parameter $X_{t}$ is fixed and hence determines the trilinear coupling
$A_{q}$.

Figure 10: The dependence of the $\mathcal{CP}$-even Higgs boson masses,
$M_{h}$ and $M_{H}$, on the trilinear coupling $A_{q}$ in an illustrative
scenario that is not phenomenologically viable (see text). We plot the masses
against the stop-mixing parameter $X_{t}=A_{t}-\mu/t_{\beta}$. The cyan curves
include the tree-level prediction as well as the one-loop contributions of
$\order{N_{c}}$ and the two-loop contributions of $\order{N_{c}^{2}}$. The
black curves show the prediction the Higgs boson masses omitting the two-loop
contributions. The purple lines indicate the tree-level predictions. All
curves have been generated by including all considered contributions in a
strict fixed-order approach (see Eqs. (123)).

We pursue here a different approach; we impose values for $t_{\beta}$ and
$m_{A}$, allowing us to investigate the dependence of the Higgs boson masses
on the trilinear coupling $A_{q}$. We emphasise that a scenario with two light
$\mathcal{CP}$-even states and a similarly light $\mathcal{CP}$-odd state is
excluded by experimental measurements and searches. This scenario is,
nevertheless, useful to showcase some interesting features of our newly
calculated contributions. These features will be present in a similar manner
for the mixing between the nearly mass-degenerate two heavy neutral Higgs
bosons $H$ and $A$ in a $\mathcal{CP}$-violating scenario.

For our choice of parameters, the $\mathcal{CP}$-even tree-level masses are
$m_{h}\approx 71\;\text{GeV}$ and $m_{H}\approx 107\;\text{GeV}$, shown by the
purple lines in Fig. 10. The difference between these tree-level values is
small enough for large resonance effects to spoil the perturbative ansatz, as
we can see from the loop-corrected masses that are obtained using the fixed-
order method according to Eqs. (123) (black and cyan curves in Fig. 10). The
one-loop corrections shift $M_{h}$ by up to $40\;\text{GeV}$, $M_{H}$ is
increased by up to $30\;\text{GeV}$. The two-loop corrections, on the other
hand, lower $M_{h}$ by more than $50\;\text{GeV}$ around $X_{t}\approx
3\;\text{TeV}$; the two-loop prediction for $M_{H}$ is more than
$30\;\text{GeV}$ larger than the one-loop result for the same value of
$X_{t}$. As a result, in this scenario the perturbative series is no longer
well-behaved if the pole masses are determined by a strict fixed-order
approach. We therefore determine the pole masses numerically using a fixed-
point iteration incorporating the momentum dependence of the
$\order{(\alpha_{t}^{2}+\alpha_{b}^{2})N_{c}^{2}}$ Yukawa terms, which has not
been available before.

(a)

(b)

Figure 11: The dependence of the $\mathcal{CP}$-even Higgs boson masses,
$M_{h}$ and $M_{H}$, on $A_{q}$ for the same scenario as in Fig. 10 but using
a fixed-point iteration for determining the pole masses. We plot the masses
against the stop-mixing parameter $X_{t}=A_{t}-\mu/t_{\beta}$. The cyan curves
include the tree-level prediction as well as the one-loop contributions of
$\order{N_{c}}$ and the two-loop contributions of $\order{N_{c}^{2}}$. The
black curves show the prediction of the Higgs boson masses omitting the two-
loop contributions. The solid curves have been generated by including all
considered contributions. The dashed curves include only contributions from
the third generation of quarks and squarks.

The results of the fixed-point iteration are shown in Fig. 11 for $M_{h}$
(upper plot) and $M_{H}$ (lower plot). The plots contain gaps in the region
$X_{t}>4\;\text{TeV}$ because, for these parameter points, the fixed-point
iteration did not converge within the desired relative precision ($10^{-5}$)
after a designated number of steps ($\sim 1000$).

In both plots, the two-loop predictions are much closer to the one-loop
results than they were in Fig. 10. In contrast to the fixed-order method, for
which the hierarchy between the tree-level mass eigenstates $h$ and $H$ got
inverted at around $X_{t}\approx 5\;\text{TeV}$, we now also have a clear
separation between the masses. With the iterated procedure, the lighter mass
$M_{h}$ is always lower than $90\;\text{GeV}$ (Fig. 11(a)) while the heavier
mass exceeds $120\;\text{GeV}$ for $X_{t}<4\;\text{TeV}$ (Fig. 11(b)).

Figure 12: Two-loop contributions to the light Higgs boson mass $M_{h}$ for
the same scenario as in Fig. 10 but using a fixed-point iteration for
determining the pole masses. The solid cyan curve includes all two-loop
contributions at $\order{N_{c}^{2}}$, the dashed cyan curve incorporates only
contributions from the third generation. The red curve shows the contributions
that stem from the inclusion of the first and second generation of squarks as
well as generation mixing. These contributions vanish in the gaugeless limit.

A remarkable feature of the iterated results is the large difference between
the two-loop predictions which include either all (s)quarks (solid cyan) or
only the third generation (dashed cyan). For the light mass $M_{h}$, for which
the two-loop contributions are shown in Fig. 12, the inclusion of the first
and second generation as well as generation mixing lowers the Higgs mass
prediction by around $2\;\text{GeV}$ across the whole considered range of
$X_{t}$. The effect of these contributions exceeds the experimental
uncertainty by one order of magnitude and is also responsible for the bulk of
the two-loop corrections for $X_{t}>-3\;\text{TeV}$. We can also see that they
are two to three times larger than the contributions which stem from the third
generation alone (dotted red curve vs dashed cyan curve in Fig. 12). The
numerical impact of the inclusion of the first and second generation as well
as generation mixing is even larger for the heavier mass $M_{H}$, exceeding
$10\;\text{GeV}$, see Fig. 11(b).

In this scenario, we have showcased that a strict fixed-order method can lead
to unreliable predictions for pole masses if the states that mix with each
other have sufficiently similar tree-level masses. A fixed-point iteration,
which determines the exact location of the propagator pole, remedies the
issues of the fixed-order approach at the cost of mixing different loop orders
and contributions. In our case, the inclusion of gauge and gauge-Yukawa-mixing
contributions, which we calculated for the first time in this work, leads to a
large shift of the Higgs boson masses in such a scenario (which is related to
the suppression of the top-Yukawa contributions by $\tan\beta$). We stress
again, however, that a scenario with two light $\mathcal{CP}$-even Higgs
bosons is phenomenologically not viable while similar mixing scenarios can
appear between the heavy $\mathcal{CP}$-even $H$ and $\mathcal{CP}$-odd $A$
bosons if $\mathcal{CP}$-violating phases are non-zero.

##  6 Conclusions and outlook

The MSSM is a well-motivated model for physics beyond the SM. The probably
most striking feature of the MSSM (which holds also for non-minimal
supersymmetric models) is that it relates the masses of the various Higgs
bosons to other parameters. Therefore, predictions for Higgs boson masses can
be obtained within the model. Different approaches have been pursued in order
to obtain such predictions with sufficient accuracy, one of which is a
perturbative pole mass calculation in terms of self-energy Feynman diagrams.
So far, all one-loop contributions, a variety of two-loop terms, and leading
three-loop contributions have been calculated for the MSSM Higgs boson masses.
In this paper, we focused on so-far undetermined two-loop terms of
$\order{(\alpha_{\text{em}}+\alpha_{q})^{2}N_{c}^{2}}$, which are expected to
constitute the dominant part of those two-loop electroweak corrections that
had not been known up to now. From this class of contributions, only the pure
Yukawa subpart of $\order{\alpha_{q}^{2}N_{c}^{2}}$ was known so far, albeit
restricted to the limit of vanishing external momentum.

The inclusion of pure gauge and gauge-Yukawa-mixing contributions required us
to generalise a relation between two-loop mass counterterms of Higgs, (would-
be) Goldstone, and gauge bosons in order to obtain a finite result for the
Higgs boson self-energies. Specifically, in the case of complex parameters
giving rise to $\mathcal{CP}$ violation, the generalised relation for the
counterterms derived in this work is needed for the neutral Higgs boson self-
energies, while already in the $\mathcal{CP}$-conserving case this relation is
required in order to obtain a finite result for the masses of the charged
Higgs bosons. To the best of our knowledge, this new relation had not been
known in the literature up to now; the additional terms do not contribute at
the order of perturbation theory analysed in the existing literature and,
therefore, were not taken into account. Additionally, we generalised the
modified two-loop relation to an expression that holds in all orders of
perturbation theory (Eq. (19b)).

In our calculation, we employed a mixed OS–${\overline{\text{DR}}}$
renormalisation scheme at the two-loop level. We investigated in this context
how the choice of renormalisation scheme is related to the possible appearance
of so-called evanescent terms, arising in particular from the
$\order{\varepsilon}$ parts of one-loop counterterms, in the final prediction
for the Higgs boson masses. We have explicitly demonstrated that, in a fixed-
order prediction with on-shell self-energies, both a full OS renormalisation
and a full ${\overline{\text{DR}}}$ renormalisation lead to a total
cancellation of evanescent $\order{\varepsilon}$ parts of the loop integrals.
In case a mixed renormalisation scheme is used, in general all two-loop
parameter counterterms need to be defined in a momentum-subtraction scheme for
this cancellation to take place. We stress that the dependence of the
prediction for a physical mass on $\order{\varepsilon}$ contributions of
counterterms is a spurious one in the sense that these terms always drop out
in relations between physical observables. This can be viewed as a theoretical
limitation of such predictions, in particular since the $\order{\varepsilon}$
terms may give rise to numerical instabilities or gauge-dependent
contributions. Furthermore, a scheme involving uncancelled
$\order{\varepsilon}$ contributions of counterterms cannot be translated into
a different scheme via the usual kind of reparameterisation. Our results
clearly demonstrate that particular care is necessary from two-loop order
onwards in the application of mixed renormalisation schemes, which are widely
used in the literature (both for calculations in the SM and beyond).

As we opted for a mixed renormalisation scheme, we required an OS definition
at the one- and two-loop level for the parameter that is given by the ratio of
the vacuum expectation values of the two Higgs doublets, $t_{\beta}$.101010In
the literature, this parameter is more commonly defined as a
${\overline{\text{DR}}}$ quantity. We have performed the OS renormalisation
via the decay $A\to\tau^{-}\tau^{+}$ by requiring that the absolute square of
the associated physical amplitude should not receive any higher order
corrections. If this decay were to be observed, a numerical value for
$t_{\beta}^{\text{OS}}$ could be extracted from its measurement. We explicitly
checked that our definition of the one-loop (Eq. (83)) and the two-loop (Eq.
(92)) counterterm does not depend on the choice of the field renormalisations.
As expected, this definition of $t_{\beta}$ leads to a total cancellation of
the evanescent $\order{\varepsilon}$ terms of loop integrals in the Higgs
boson pole mass prediction at the two-loop order.

In our numerical analysis we have investigated the effects of our newly
calculated contributions on the predictions for the neutral MSSM Higgs boson
masses in four different $\mathcal{CP}$-conserving scenarios. We have compared
their size with the already known pure Yukawa contributions of
$\order{N_{c}^{2}}$ and also with the experimental uncertainty of the mass
measurement of the detected Higgs boson. While expectedly smaller than the
pure Yukawa contributions (by typically an order of magnitude), the pure gauge
and gauge-Yukawa-mixing terms turned out to be larger than or at least of a
similar size as the experimental uncertainty. Their inclusion in MSSM Higgs
mass predictions will therefore be important for improving the theoretical
uncertainty. In the fourth scenario, we have also showcased the fact that a
strict fixed-order treatment becomes insufficient for cases where there is
large mixing between particles that are nearly mass-degenerate at lowest
order. We showed that only the fixed-point iteration leads to a reliable
prediction in this case, but at the cost of mixing different orders of
perturbation theory. Depending on the investigated scenario, the appropriate
method has to be chosen and the drawbacks of each approach have to be taken
into account.

In the future, the newly calculated corrections are planned to be combined
with the other known higher-order corrections and a resummation of large
logarithms. This combined result is planned to be implemented into the code
FeynHiggs [181, 32, 182, 90, 69, 92, 130, 131, 183]. Based on this
implementation and the existing routines in FeynHiggs [133], the remaining
theoretical uncertainties will be estimated. In addition, the derived fixed-
order result can be used to derive so far unknown electroweak two-loop
threshold corrections for the EFT approach, which will further reduce the
theoretical uncertainties.

We, moreover, emphasise that the renormalisation of the MSSM Higgs–gauge
sector that we have carried out in detail at the two-loop level includes full
electroweak effects and goes significantly beyond the renormalisation of
electroweak two-loop contributions in the MSSM performed elsewhere. The
obtained results should hence be useful for any future prediction heading in a
similar direction. In particular, the presented results for the contributions
at $\order{N_{c}^{2}}$ can readily be extended to various Higgs production and
decay processes at the two-loop level, as it was demonstrated in this work for
the decay process $A\to\tau^{-}\tau^{+}$. We furthermore expect that the
ingredients of the analyses in this work can directly be transferred to a more
general $n$-loop order calculation of electroweak $\order{N_{c}^{n}}$ terms as
well; these contributions will similarly decompose into products of one-loop
integrals for the Higgs and vector boson self-energies. It remains to be seen
whether the renormalisation of the quark-squark sector, which for the two-loop
predictions of the Higgs boson masses carried out in this work was needed only
at the one-loop order, can as easily be extended to the two-loop case.

Since the renormalisation described in this work was performed allowing for
$\mathcal{CP}$-violating phases of MSSM parameters, the study of
$\mathcal{CP}$-violating scenarios requires only little additional work. The
on-shell renormalisation of $t_{\beta}$ in terms of a charged Higgs boson
decay like $H^{+}\to\tau^{+}\nu_{\tau}$ (instead of $A\to\tau^{-}\tau^{+}$) is
expected to be straightforward. The relevant Slavnov-Taylor identities
involving charged particles are included in App. D alongside the ones for the
neutral particles.

Electroweak two-loop contributions of $\order{N_{c}^{2}}$ also appear in
extensions of the MSSM, like e.g. the Next-to-Minimal Supersymmetric Standard
Model (NMSSM). Our results can therefore serve as a building block for similar
calculations in extended SUSY models as well.

Besides the particular relevance for supersymmetric models, our work has more
generally led to new insights about the renormalisation in spontaneously
broken gauge theories at two-loop order and beyond. In particular, the
features of unstable particles and particle mixing are present in a wide
variety of physical models and their proper treatment is paramount in order to
provide accurate theoretical predictions.

## Acknowledgements

We thank Johannes Braathen for useful discussions. H. B. acknowledges support
from the Alexander von Humboldt foundation. D. M. and G. W. acknowledge
support by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy — EXC 2121 “Quantum Universe”
— 390833306. This work has been partially funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) — 491245950.

##  Appendix A The ${\overline{\text{DR}}}$ scheme at the two-loop level

It has been noted that—starting at the two-loop level—the definition of
modified minimal subtraction is no longer unique [184, 185]. This can be seen
as follows. For dimensional reasons, all one-loop integrals are proportional
to $\mu_{\text{D}}^{2\varepsilon}$, owing to the way the regularisation scale
is introduced. The factor of $(4\pi)^{\varepsilon}$ always appears in the same
fashion as well, as it stems from a combination of the prefactor $C$ (see Eq.
(193a)) of the integrals and the angular integral in $D=4-2\varepsilon$
dimensions. However, different combinations of Gamma functions occur depending
on the considered integral. Therefore, different definitions of the modified
regularisation scale $\bar{\mu}_{\text{D}}$ and, hence, the
${\overline{\text{DR}}}$ scheme exist. The following conventions, among
others, are found in the literature:

$\displaystyle\bar{\mu}_{\text{D}}^{2\varepsilon}={}$ $\displaystyle\left(4\pi
e^{-\gamma_{E}}\right)^{\varepsilon}\mu_{\text{D}}^{2\varepsilon}\text{\cite[cite]{[\@@bibref{}{Martin:2003qz}{}{}]}},$
(126a) $\displaystyle\bar{\mu}_{\text{D}}^{2\varepsilon}={}$
$\displaystyle\frac{(4\pi)^{\varepsilon}}{\Gamma(1-\varepsilon)}\mu_{\text{D}}^{2\varepsilon}\text{\cite[cite]{[\@@bibref{}{Collins:2011zzd,Sperling:2013fwy}{}{}]}},$
(126b) $\displaystyle\bar{\mu}_{\text{D}}^{2\varepsilon}={}$
$\displaystyle\frac{(4\pi)^{\varepsilon}\Gamma^{2}(1-\varepsilon)\Gamma(1+\varepsilon)}{\Gamma(1-2\varepsilon)}\mu_{\text{D}}^{2\varepsilon}\text{\cite[cite]{[\@@bibref{}{Hahn:LT215}{}{}]}}.$
(126c)

All these conventions agree at $\order{\varepsilon}$. The second and third
conventions agree at $\order{\varepsilon^{2}}$, but differ from the first one
at that order. At $\order{\varepsilon^{3}}$, all conventions differ. While all
conventions are able to get rid of any $\log(4\pi)$ or $\gamma_{E}$ terms,
other irrational constants, which appear at higher orders, cannot be removed
simultaneously by any choice. It can be shown, however, that differences of
$\order{\varepsilon^{2}}$ in the definition of $\bar{\mu}_{\text{D}}$ do not
alter the value of a renormalised Green function after taking the limit
$\varepsilon\to 0$ [185]. Therefore, the exact choice of how to define
$\bar{\mu}_{\text{D}}$ matters only for technical reasons.

We work with the first of the aforementioned conventions,

$\bar{\mu}_{\text{D}}^{2}=4\pi e^{-\gamma_{E}}\mu_{\text{D}}^{2}.$ (127)

This replacement “hides” all appearances of $\log(4\pi)$ and $\gamma_{E}$ in
the newly defined regularisation scale $\bar{\mu}_{\text{D}}$.

In the DR renormalisation scheme, the counterterms contain only divergent
parts without any irrational constants. At the one- and two-loop level, they
can be cast into the form

$\displaystyle\delta^{(1)}p^{\text{DR}}={}$
$\displaystyle\frac{A}{\varepsilon},$ (128a)
$\displaystyle\delta^{(2)}p^{\text{DR}}={}$
$\displaystyle\frac{B}{\varepsilon^{2}}+\frac{C}{\varepsilon},$ (128b)

respectively, with the coefficients $A$, $B$ and $C$. With our convention for
$\bar{\mu}_{\text{D}}$, we write the one- and two-loop
${\overline{\text{DR}}}$ counterterms as

$\displaystyle\begin{split}\delta^{(1)}p^{{\overline{\text{DR}}}}\equiv{}&\left(4\pi
e^{-\gamma_{E}}\right)^{\varepsilon}\frac{\overline{A}}{\varepsilon}\\\
={}&A\left(\frac{1}{\varepsilon}+\log(4\pi)-\gamma_{E}+\frac{\varepsilon}{2}[\log(4\pi)-\gamma_{E}]^{2}+\order{\varepsilon^{2}}\right),\end{split}$
(129a)
$\displaystyle\begin{split}\delta^{(2)}p^{{\overline{\text{DR}}}}\equiv{}&\left(4\pi
e^{-\gamma_{E}}\right)^{2\varepsilon}\left(\frac{\overline{B}}{\varepsilon^{2}}+\frac{\overline{C}}{\varepsilon}\right)\\\
={}&B\left(\frac{1}{\varepsilon^{2}}-2[\log(4\pi)-\gamma_{E}]^{2}+\order{\varepsilon}\right)\\\
&+C\left(\frac{1}{\varepsilon}+2[\log(4\pi)-\gamma_{E}]+\order{\varepsilon}\right).\end{split}$
(129b)

The coefficients $\overline{A}$, $\overline{B}$, and $\overline{C}$ were
determined by comparing the divergent parts of DR and ${\overline{\text{DR}}}$
counterterms, which have to agree in order to obtain finite results in either
renormalisation scheme. It should be noted that, in this definition of the
${\overline{\text{DR}}}$ scheme, the one-loop counterterm contains terms of
$\order{\varepsilon}$. These terms are important for a cancellation of the
irrational constants at the two-loop level.

The ${\overline{\text{DR}}}$ coefficients $\overline{A}$, $\overline{B}$, and
$\overline{C}$ do not contain the irrational constants $\log(4\pi)$ and
$\gamma_{E}$. This prescription agrees with the one found in Ref. [185] and
was derived independently. In Ref. [185], the idea of adding one factor
$S_{\varepsilon}=(4\pi e^{-\gamma_{E}})^{\varepsilon}$ for each loop in the
counterterms is presented as well.111111In Ref. [185], the different but
equivalent convention
$S_{\varepsilon}=(4\pi)^{\varepsilon}/\Gamma(1-\varepsilon)$ is used.

##  Appendix B The quark and squark sector

In this section, we fix the notation for the quark and squark sector in the
MSSM. We give the renormalisation transformations and the resulting
expressions for the renormalised squark self-energy diagrams up to the one-
loop order. We present three different renormalisation schemes for the case of
massive quarks and we illustrate how the renormalisation has to be modified in
the massless case.

Throughout the whole paper, we assume that mass terms do not mix quarks and
squarks of different generations. This implies a unit CKM matrix and squark
mass matrices which are diagonal in flavour space. The generalisation of our
results to the case of non-zero mixing between the generations would easily be
possible but is expected to not yield any new insights. Quartic interaction
terms between squark flavours of different generations nevertheless lead to
Higgs self-energy diagrams with generation mixing.

The quark sector requires no renormalisation at our considered order of
perturbation theory, so we focus solely on the squark sector of the MSSM.

###  B.1 Tree-level

The bilinear squark Lagrangian reads

$\mathcal{L}_{\text{squark}}^{\text{bil.}}=\sum_{\tilde{q}}\left(\partial_{\mu}\tilde{q}_{L}^{*}\partial^{\mu}\tilde{q}_{L}+\partial_{\mu}\tilde{q}_{R}^{*}\partial^{\mu}\tilde{q}_{R}-\begin{pmatrix}\tilde{q}_{L}^{*}&\tilde{q}_{R}^{*}\end{pmatrix}\textbf{M}_{\tilde{q}}^{2}\begin{pmatrix}\tilde{q}_{L}\\\
\tilde{q}_{R}\end{pmatrix}\right),$ (130)

where the sum runs over the squark flavors
$\tilde{t},\tilde{b},\tilde{c},\tilde{s},\tilde{u},\tilde{d}$. As we do not
consider generation mixing, the squark mass matrices are flavour diagonal. We
denote the elements of a squark mass matrix by

$\textbf{M}_{\tilde{q}}^{2}=\begin{pmatrix}\left(\textbf{M}_{\tilde{q}}^{2}\right)_{LL}&\left(\textbf{M}_{\tilde{q}}^{2}\right)_{LR}\\\
\left(\textbf{M}_{\tilde{q}}^{2}\right)_{RL}&\left(\textbf{M}_{\tilde{q}}^{2}\right)_{RR}\end{pmatrix}.$
(131)

The squark fields carry non-vanishing quantum numbers and are thus complex
scalar fields; the mass matrices are, in the most general case, not symmetric
but hermitian so that their eigenvalues—the physical squark masses—are real.
The mass matrices for up- and down-type squarks read

$\displaystyle\textbf{M}_{\tilde{u}_{g}}^{2}={}$
$\displaystyle\begin{pmatrix}M_{\tilde{q}_{g}}^{2}+m_{u_{g}}^{2}+M_{Z}^{2}\cos(2\beta)(\frac{1}{2}-\frac{2}{3}s_{\text{w}}^{2})&m_{u_{g}}X_{u_{g}}^{*}\\\
m_{u_{g}}X_{u_{g}}&M_{\tilde{u}_{g}}^{2}+m_{u_{g}}^{2}+\frac{2}{3}M_{Z}^{2}\cos(2\beta)s_{\text{w}}^{2}\end{pmatrix},$
(132a) $\displaystyle\textbf{M}_{\tilde{d}_{g}}^{2}={}$
$\displaystyle\begin{pmatrix}M_{\tilde{q}_{g}}^{2}+m_{d_{g}}^{2}+M_{Z}^{2}\cos(2\beta)(-\frac{1}{2}+\frac{1}{3}s_{\text{w}}^{2})&m_{d_{g}}X_{d_{g}}^{*}\\\
m_{d_{g}}X_{d_{g}}&M_{\tilde{d}_{g}}^{2}+m_{d_{g}}^{2}-\frac{1}{3}M_{Z}^{2}\cos(2\beta)s_{\text{w}}^{2}\end{pmatrix}.$
(132b)

The index $g$ labels the three generations of matter such that
$m_{u_{3}}=m_{t}$ and $X_{d_{2}}=X_{s}$, for instance. We do not use this
convention for the soft SUSY-breaking masses $M_{\tilde{q}_{g}}^{2}$,
$M_{\tilde{u}_{g}}^{2}$, and $M_{\tilde{d}_{g}}^{2}$, i.e. there is no
parameter $M_{\tilde{t}}^{2}$, so that we can distinguish between the
respective left- ($M_{\tilde{q}_{3}}^{2}$) and right-handed
($M_{\tilde{u}_{3}}^{2}$) mass terms. Moreover, we introduced the common
abbreviations

$\displaystyle X_{u_{g}}={}$ $\displaystyle A_{u_{g}}-\mu^{*}\cot(\beta),$
(133a) $\displaystyle X_{d_{g}}={}$ $\displaystyle
A_{d_{g}}-\mu^{*}\tan(\beta).$ (133b)

The parameters
$M_{\tilde{q}_{g}}^{2},M_{\tilde{u}_{g}}^{2},M_{\tilde{d}_{g}}^{2},A_{u_{g}},A_{d_{g}}$
break supersymmetry softly. In the most general scenario of SUSY breaking,
they would be $3\times 3$ matrices in generation space. In our calculation, as
mentioned above, we neglect this mixing between generations and assume these
matrices to be diagonal.

To change from the gauge eigenbasis to the mass eigenbasis, we introduce the
unitary transformation

$\begin{pmatrix}\tilde{q}_{1}\\\
\tilde{q}_{2}\end{pmatrix}=\begin{pmatrix}c_{\tilde{q}}&-s_{\tilde{q}}e^{-\text{i}\phi_{\tilde{q}}}\\\
s_{\tilde{q}}e^{\text{i}\phi_{\tilde{q}}}&c_{\tilde{q}}\end{pmatrix}\begin{pmatrix}\tilde{q}_{L}\\\
\tilde{q}_{R}\end{pmatrix}\equiv U_{\tilde{q}}\begin{pmatrix}\tilde{q}_{L}\\\
\tilde{q}_{R}\end{pmatrix}$ (134)

for each squark flavor $\tilde{q}$. Here we introduced the abbreviations
$c_{\tilde{q}}=\cos(\theta_{\tilde{q}})$ and
$s_{\tilde{q}}=\sin(\theta_{\tilde{q}})$. The bilinear squark Lagrangian in
terms of the mass eigenbasis reads

$\mathcal{L}_{\text{squark}}^{\text{bil.}}=\sum_{\tilde{q}}\left(\partial_{\mu}\tilde{q}_{1}^{*}\partial^{\mu}\tilde{q}_{1}+\partial_{\mu}\tilde{q}_{2}^{*}\partial^{\mu}\tilde{q}_{2}-\begin{pmatrix}\tilde{q}_{1}^{*}&\tilde{q}_{2}^{*}\end{pmatrix}\textbf{D}_{\tilde{q}}^{2}\begin{pmatrix}\tilde{q}_{1}\\\
\tilde{q}_{2}\end{pmatrix}\right),$ (135)

where

$\textbf{D}_{\tilde{q}}^{2}=U_{\tilde{q}}\textbf{M}_{\tilde{q}}^{2}U_{\tilde{q}}^{\dagger}\equiv\begin{pmatrix}m_{\tilde{q}_{1}}^{2}&m_{\tilde{q}_{12}}^{2}\\\
m_{\tilde{q}_{21}}^{2}&m_{\tilde{q}_{2}}^{2}\end{pmatrix},\quad
m_{\tilde{q}_{21}}^{2}=m_{\tilde{q}_{12}}^{2*}.$ (136)

The angles $\theta_{\tilde{q}}\in[0,\tfrac{\pi}{2}]$ and
$\phi_{\tilde{q}}\in(-\pi,\pi]$ are then determined by the conditions

$m_{\tilde{q}_{12}}^{2}=0\quad\wedge\quad m_{\tilde{q}_{1}}^{2}\leq
m_{\tilde{q}_{2}}^{2}.$ (137)

To give explicit expressions for $\theta_{\tilde{q}}$ and $\phi_{\tilde{q}}$,
we have to distinguish between the degenerate and the non-degenerate case. The
two squark mass eigenvalues are degenerate if and only if the matrix
$\textbf{M}_{\tilde{q}}^{2}$ is proportional to the identity matrix, in which
case no rotation is needed, and $U_{\tilde{q}}$ can simply be chosen as unity.
This happens if both $m_{q}X_{q}=0$ and
$\left(\textbf{M}_{\tilde{q}}^{2}\right)_{LL}=\left(\textbf{M}_{\tilde{q}}^{2}\right)_{RR}$
are fulfilled simultaneously. When working in the gaugeless limit, assuming a
vanishing quark mass and a universal SUSY scale
$M_{\text{SUSY}}^{2}=M_{\tilde{q}_{g}}^{2}=M_{\tilde{u}_{g}}^{2}=M_{\tilde{d}_{g}}^{2}$,
this is always the case.

In the case of non-degenerate masses, the mass ordering ensures
$m_{\tilde{q}_{1}}^{2}<m_{\tilde{q}_{2}}^{2}$, and we can write

$\displaystyle\exp(\text{i}\phi_{\tilde{q}})={}$
$\displaystyle\frac{X_{q}}{|X_{q}|},$ (138a)
$\displaystyle\cos(2\theta_{\tilde{q}})={}$
$\displaystyle\frac{\left(\textbf{M}_{\tilde{q}}^{2}\right)_{RR}-\left(\textbf{M}_{\tilde{q}}^{2}\right)_{LL}}{m_{\tilde{q}_{2}}^{2}-m_{\tilde{q}_{1}}^{2}},$
(138b) $\displaystyle\sin(2\theta_{\tilde{q}})={}$
$\displaystyle\frac{2m_{q}|X_{q}|}{m_{\tilde{q}_{2}}^{2}-m_{\tilde{q}_{1}}^{2}}.$
(138c)

The angles can then uniquely be determined from

$\displaystyle\phi_{\tilde{q}}={}$ $\displaystyle\text{Arg}(X_{q}),\quad$
$\displaystyle-\pi<{}$ $\displaystyle\phi_{\tilde{q}}\leq\pi,$ (139a)
$\displaystyle\theta_{\tilde{q}}={}$
$\displaystyle\frac{1}{2}\arccos\frac{\left(\textbf{M}_{\tilde{q}}^{2}\right)_{RR}-\left(\textbf{M}_{\tilde{q}}^{2}\right)_{LL}}{m_{\tilde{q}_{2}}^{2}-m_{\tilde{q}_{1}}^{2}},\quad$
$\displaystyle 0\leq{}$ $\displaystyle\theta_{\tilde{q}}\leq\tfrac{\pi}{2}.$
(139b)

Mathematically, $\phi_{\tilde{q}}$ is undefined if $X_{q}$ vanishes, and we
set it to 0 for simplicity in this case.

###  B.2 Renormalisation transformations

For a full two-loop prediction of the Higgs boson masses, the one-loop
renormalisation of the squark sector is needed. As we are only interested in
electroweak corrections of
$\order{(\alpha_{\text{em}}+\alpha_{q})^{2}N_{c}^{2}}$, no
lepton/slepton/quark renormalisation constants are needed.

We renormalise the squark mass matrices and fields via

$\displaystyle\textbf{D}_{\tilde{q}}^{2}\to{}$
$\displaystyle\textbf{D}_{\tilde{q}}^{2}+\begin{pmatrix}\delta
m_{\tilde{q}_{1}}^{2}&\delta m_{\tilde{q}_{12}}^{2}\\\ \delta
m_{\tilde{q}_{21}}^{2}&\delta m_{\tilde{q}_{2}}^{2}\end{pmatrix},$ (140a)
$\displaystyle\begin{pmatrix}\tilde{q}_{1}\\\ \tilde{q}_{2}\end{pmatrix}\to{}$
$\displaystyle\begin{pmatrix}\sqrt{1+\delta
Z_{\tilde{q}_{11}}}&\frac{1}{2}\delta Z_{\tilde{q}_{12}}\\\ \frac{1}{2}\delta
Z_{\tilde{q}_{21}}&\sqrt{1+\delta
Z_{\tilde{q}_{22}}}\end{pmatrix}\begin{pmatrix}\tilde{q}_{1}\\\
\tilde{q}_{2}\end{pmatrix},$ (140b)
$\displaystyle\begin{pmatrix}\tilde{q}_{1}^{*}\\\
\tilde{q}_{2}^{*}\end{pmatrix}\to{}$
$\displaystyle\begin{pmatrix}\sqrt{1+\delta
Z_{\tilde{q}_{11}}}&\frac{1}{2}\delta\bar{Z}_{\tilde{q}_{21}}\\\
\frac{1}{2}\delta\bar{Z}_{\tilde{q}_{12}}&\sqrt{1+\delta
Z_{\tilde{q}_{22}}}\end{pmatrix}\begin{pmatrix}\tilde{q}_{1}^{*}\\\
\tilde{q}_{2}^{*}\end{pmatrix}.$ (140c)

It should be noted that we introduce separate off-diagonal field counterterms
for the squark and anti-squark fields. This follows the convention of Refs.
[51, 52], where it enables the inclusion of absorptive contributions into the
field counterterms. If the absorptive parts are left out, the off-diagonal
field counterterms are related by $\delta\bar{Z}_{\tilde{q}_{ij}}=\delta
Z_{\tilde{q}_{ji}}^{*}$.

###  B.3 Renormalisation at the one-loop level

Figure 13: Unrenormalised one-loop squark self-energy
$\Sigma^{(1)}_{\tilde{q}_{g,i}\tilde{q}_{g,j}}$. The indices $g$ and $h$ are
flavour labels, which we make explicit here since the squark in the loop,
$\tilde{q}_{h}$, can have a different flavour than the external squark. The
indices $i$, $j$, and $k$ label the two mass eigenstates. We note the
convention for naming the self-energy, where the first label corresponds to
the outgoing squark in the diagram.

In this section, we give an overview over the most important renormalisation
schemes for the squark sector. We present expressions for the relevant one-
loop renormalisation constants; they enter the prediction for the two-loop
Higgs boson masses through the sub-loop part of two-loop Higgs self-energies.

With the renormalisation transformations given in the previous section, the
renormalised one-loop squark self-energies read

$\displaystyle\hat{\Sigma}^{(1)}_{\tilde{q}_{1}\tilde{q}_{1}}(p^{2})={}$
$\displaystyle\Sigma^{(1)}_{\tilde{q}_{1}\tilde{q}_{1}}(p^{2})+\delta^{(1)}Z_{\tilde{q}_{11}}(p^{2}-m_{\tilde{q}_{1}}^{2})-\delta^{(1)}m_{\tilde{q}_{1}}^{2},$
(141a)
$\displaystyle\begin{split}\hat{\Sigma}^{(1)}_{\tilde{q}_{1}\tilde{q}_{2}}(p^{2})={}&\Sigma^{(1)}_{\tilde{q}_{1}\tilde{q}_{2}}(p^{2})+\frac{1}{2}\delta^{(1)}Z_{\tilde{q}_{12}}(p^{2}-m_{\tilde{q}_{1}}^{2})\\\
&+\frac{1}{2}\delta^{(1)}\bar{Z}_{\tilde{q}_{12}}(p^{2}-m_{\tilde{q}_{2}}^{2})-\delta^{(1)}m_{\tilde{q}_{12}}^{2},\end{split}$
(141b)
$\displaystyle\begin{split}\hat{\Sigma}^{(1)}_{\tilde{q}_{2}\tilde{q}_{1}}(p^{2})={}&\Sigma^{(1)}_{\tilde{q}_{2}\tilde{q}_{1}}(p^{2})+\frac{1}{2}\delta^{(1)}Z_{\tilde{q}_{21}}(p^{2}-m_{\tilde{q}_{2}}^{2})\\\
&+\frac{1}{2}\delta^{(1)}\bar{Z}_{\tilde{q}_{21}}(p^{2}-m_{\tilde{q}_{1}}^{2})-\delta^{(1)}m_{\tilde{q}_{21}}^{2},\end{split}$
(141c)
$\displaystyle\hat{\Sigma}^{(1)}_{\tilde{q}_{2}\tilde{q}_{2}}(p^{2})={}$
$\displaystyle\Sigma^{(1)}_{\tilde{q}_{2}\tilde{q}_{2}}(p^{2})+\delta^{(1)}Z_{\tilde{q}_{22}}(p^{2}-m_{\tilde{q}_{2}}^{2})-\delta^{(1)}m_{\tilde{q}_{2}}^{2}.$
(141d)

The only topology contributing to the
$\Sigma^{(1)}_{\tilde{q}_{i}\tilde{q}_{j}}$ self-energy at $\order{N_{c}}$ is
shown in Fig. 13. These diagrams are independent of the external momentum, and
no field renormalisation constants are needed to yield finite renormalised
one-loop self-energies. Since in our calculation squarks appear as internal
particles only, any finite part of their field renormalisation constants
cancels in the sub-loop renormalisation of the two-loop Higgs (and vector
boson) self-energies, and we will set them to zero for simplicity:

$\displaystyle\delta^{(1)}Z_{\tilde{q}_{ij}}={}$ $\displaystyle 0,$ (142a)
$\displaystyle\delta^{(1)}\bar{Z}_{\tilde{q}_{ij}}={}$ $\displaystyle 0.$
(142b)

While this choice is not necessary, it simplifies the algebraic expressions
and, if an on-shell renormalisation scheme is chosen for the squark masses, it
implies that the squark self-energies vanish for arbitrary external momenta.

By virtue of the momentum independence, the self-energies are free from
absorptive contributions and they can only be complex because of the involved
couplings. Consequently, the off-diagonal unrenormalised squark self-energies
are related via

$\Sigma^{(1)*}_{\tilde{q}_{1}\tilde{q}_{2}}\overset{\order{N_{c}}}{=}\Sigma^{(1)}_{\tilde{q}_{2}\tilde{q}_{1}},$
(143)

which holds for the renormalised self-energies as well. Due to the absence of
absorptive contributions, the diagonal self-energies are real.

####  B.3.1 Renormalisation conditions and counterterms for the massive case

In this paper, unless explicitly stated otherwise, we work with a massive
third generation of quarks, while the first two generations are treated as
massless. For the third generation squark sector, we consider several
different renormalisation schemes; an on-shell scheme (OS), a
${\overline{\text{DR}}}$ scheme, and a mixed scheme. In each of these schemes,
we allow for either sbottom mass $m_{\tilde{b}_{i}}^{2}$ to be used as input
parameter, which amounts to a total of six different renormalisation schemes
for the third generation squark sector.

Regarding the choice of renormalisation conditions, it is useful to count the
number of independent parameters first. A set of independent parameters in the
stop-sbottom sector is for example given by
$\\{M_{\tilde{q}_{3}}^{2},M_{\tilde{u}_{3}}^{2},M_{\tilde{d}_{3}}^{2},A_{t},A_{b}\\}$,
of which the trilinear couplings can be complex. This requires us to impose
seven (real) renormalisation conditions. Independent of the chosen
renormalisation scheme for the squark sector, we require that

$\mu$, $A_{b}$ are renormalised in the ${\overline{\text{DR}}}$ scheme. (144)

We derive the ${\overline{\text{DR}}}$ expressions for $\mu$ and $A_{b}$ in
Sect. B.4.

In all schemes, the stop masses and one of the sbottom masses are used as
independent input parameters. We label the independent sbottom with $n$ and
the dependent sbottom with $f=3-n$. In the case of the
$\order{(\alpha_{\text{em}}+\alpha_{q})^{2}N_{c}^{2}}$ contributions that are
of interest to us, the quark sector is not renormalised. For the sake of
completeness, we include the vanishing quark mass counterterms in the
expressions below.

##### (i) OS scheme.

We use on-shell definitions for the stop parameters and the $n$th sbottom
mass:

$\displaystyle\delta^{(1)}m_{\tilde{t}_{i}}^{2}={}$
$\displaystyle\real\Sigma^{(1)}_{\tilde{t}_{i}\tilde{t}_{i}}$
$\displaystyle\overset{\order{N_{c}}}{=}\Sigma^{(1)}_{\tilde{t}_{i}\tilde{t}_{i}},\quad
i\in\\{1,2\\},$ (145a) $\displaystyle\delta^{(1)}m_{\tilde{b}_{n}}^{2}={}$
$\displaystyle\real\Sigma^{(1)}_{\tilde{b}_{n}\tilde{b}_{n}}$
$\displaystyle\overset{\order{N_{c}}}{=}\Sigma^{(1)}_{\tilde{b}_{n}\tilde{b}_{n}},$
(145b) $\displaystyle\delta^{(1)}m_{\tilde{t}_{12}}^{2}={}$
$\displaystyle\,\widetilde{\text{Re}}\
\Sigma^{(1)}_{\tilde{t}_{1}\tilde{t}_{2}}$
$\displaystyle\overset{\order{N_{c}}}{=}\Sigma^{(1)}_{\tilde{t}_{1}\tilde{t}_{2}},$
(145c) $\displaystyle\delta^{(1)}m_{\tilde{t}_{21}}^{2}={}$
$\displaystyle\delta^{(1)}m_{\tilde{t}_{12}}^{2*}$
$\displaystyle\overset{\order{N_{c}}}{=}\Sigma^{(1)}_{\tilde{t}_{2}\tilde{t}_{1}}.$
(145d)

The operator $\widetilde{\text{Re}}$ takes the real part of the loop integrals
but leaves complex couplings unaffected. In our calculation, the squark self-
energies are momentum independent and so we do not specify any momentum at
which the self-energies are to be evaluated. It should be noted that the
renormalisation condition for the off-diagonal stop mass terms implies that
the whole renormalised self-energy vanishes only in our calculation of $N_{c}$
contributions (because we chose $\delta
Z_{\tilde{q}_{ij}}=\delta\bar{Z}_{\tilde{q}_{ij}}=0$); in a setting where the
squark self-energies are momentum-dependent, an unphysical MOM scheme is often
used instead, see Refs. [52, 50, 70, 187, 166].

In this scheme, the $A_{t}$ counterterm is a dependent quantity:

$\begin{split}\delta^{(1)}A_{t}={}&\frac{1}{m_{t}}\left[U_{\tilde{t}_{11}}U_{\tilde{t}_{12}}^{*}\left(\delta^{(1)}m_{\tilde{t}_{1}}^{2}-\delta^{(1)}m_{\tilde{t}_{2}}^{2}\right)+U_{\tilde{t}_{21}}U_{\tilde{t}_{12}}^{*}\delta^{(1)}m_{\tilde{t}_{12}}^{2}+U_{\tilde{t}_{11}}U_{\tilde{t}_{22}}^{*}\delta^{(1)}m_{\tilde{t}_{21}}^{2}\right]\\\
&-\frac{X_{t}\,\delta^{(1)}m_{t}}{m_{t}}+\frac{\delta^{(1)}\mu^{*}}{t_{\beta}}-\frac{\mu^{*}\delta^{(1)}t_{\beta}}{t_{\beta}^{2}}.\end{split}$
(146)

##### (ii) ${\overline{\text{DR}}}$ scheme.

In this scheme, we use $A_{t}$ to formulate a renormalisation condition
instead of $m_{\tilde{t}_{12}}^{2}$. Now, all the input counterterms are
defined in the ${\overline{\text{DR}}}$ scheme:

$m_{\tilde{t}_{1}}^{2}$, $m_{\tilde{t}_{2}}^{2}$, $m_{\tilde{b}_{n}}^{2}$, and
$A_{t}$ are renormalised in the ${\overline{\text{DR}}}$ scheme. (147)

The ${\overline{\text{DR}}}$ counterterms for the masses are obtained by
simply discarding the finite parts of the OS counterterms. The
${\overline{\text{DR}}}$ expression for $\delta^{(1)}A_{t}$ is derived in
Sect. B.4. Finally, the $m_{\tilde{t}_{12}}^{2}$ counterterm is a dependent
quantity now:

$\begin{split}\delta^{(1)}m_{\tilde{t}_{12}}^{2}={}&\frac{1}{|U_{\tilde{t}_{11}}|^{2}-|U_{\tilde{t}_{12}}|^{2}}\Big{[}U_{\tilde{t}_{11}}U_{\tilde{t}_{21}}^{*}\left(\delta^{(1)}m_{\tilde{t}_{1}}^{2}-\delta^{(1)}m_{\tilde{t}_{2}}^{2}\right)\\\
&+U_{\tilde{t}_{11}}U_{\tilde{t}_{22}}^{*}\left(m_{t}\,\delta^{(1)}X_{t}^{*}+X_{t}^{*}\,\delta^{(1)}m_{t}\right)-U_{\tilde{t}_{12}}U_{\tilde{t}_{21}}^{*}\left(m_{t}\,\delta^{(1)}X_{t}+X_{t}\,\delta^{(1)}m_{t}\right)\Big{]}.\end{split}$
(148)

##### (iii) Mixed scheme.

The input counterterms are the same as in the ${\overline{\text{DR}}}$ scheme,
but now

$m_{\tilde{t}_{1}}^{2}$, $m_{\tilde{t}_{2}}^{2}$, and $m_{\tilde{b}_{n}}^{2}$
are renormalised on-shell. $A_{t}$ is renormalised ${\overline{\text{DR}}}$.
(149)

$\delta^{(1)}m_{\tilde{t}_{12}}^{2}$ is calculated by the same expression as
in the ${\overline{\text{DR}}}$ scheme. In this scheme, just as in the
${\overline{\text{DR}}}$ scheme, $\mu$, $t_{\beta}$, and both $A_{t}$ and
$A_{b}$ are ${\overline{\text{DR}}}$ quantities. Consequently, $X_{t}$ and
$X_{b}$ are ${\overline{\text{DR}}}$ quantities as well.

##### Scheme-independent relations.

In all three schemes, the counterterms for the sbottom masses
$m_{\tilde{b}_{f}}^{2}$ and $m_{\tilde{b}_{12}}^{2}$ are dependent quantities
and as such they have to be expressed in terms of the input counterterms. To
find the expression for the remaining sbottom mass counterterm, we make use of
a relation between the $LL$ entries (see Eq. (131)) of the stop and sbottom
mass matrix. Both entries contain the SUSY-breaking parameter
$M_{\tilde{q}_{3}}^{2}$, yielding

$\displaystyle\begin{split}M_{\tilde{q}_{3}}^{2}={}&\left(\textbf{M}_{\tilde{t}}^{2}\right)_{LL}-m_{t}^{2}-M_{Z}^{2}c_{2\beta}\left(\frac{1}{2}-\frac{2}{3}s_{\text{w}}^{2}\right)\\\
={}&\left(\textbf{M}_{\tilde{b}}^{2}\right)_{LL}-m_{b}^{2}+M_{Z}^{2}c_{2\beta}\left(\frac{1}{2}-\frac{1}{3}s_{\text{w}}^{2}\right)\end{split}$
(150a)
$\displaystyle\Leftrightarrow\left(\textbf{M}_{\tilde{b}}^{2}\right)_{LL}={}$
$\displaystyle\left(\textbf{M}_{\tilde{t}}^{2}\right)_{LL}-m_{t}^{2}+m_{b}^{2}-M_{W}^{2}c_{2\beta}.$
(150b)

In order to simplify the notation, we introduce the auxiliary renormalisation
constant

$\begin{split}\delta^{(1)}\\!\left(\textbf{M}_{\tilde{b}}^{2}\right)_{LL}={}&|U_{\tilde{t}_{11}}|^{2}\delta^{(1)}m_{\tilde{t}_{1}}^{2}+|U_{\tilde{t}_{12}}|^{2}\delta^{(1)}m_{\tilde{t}_{2}}^{2}-2\,\real\left\\{U_{\tilde{t}_{22}}U_{\tilde{t}_{12}}^{*}\delta^{(1)}m_{\tilde{t}_{12}}^{2}\right\\}\\\
&-2m_{t}\delta^{(1)}m_{t}+2m_{b}\delta^{(1)}m_{b}-c_{2\beta}\delta^{(1)}M_{W}^{2}+4M_{W}^{2}s_{\beta}c_{\beta}^{3}\delta^{(1)}t_{\beta}.\end{split}$
(151)

This constant allows us to write the dependent sbottom mass counterterm as

$\begin{split}\delta^{(1)}m_{\tilde{b}_{f}}^{2}={}&\frac{1}{|U_{\tilde{b}_{1f}}|^{2}}\Big{[}|U_{\tilde{b}_{1n}}|^{2}\delta^{(1)}m_{\tilde{b}_{n}}^{2}+(n-f)\Big{(}2\,\real\left\\{U_{\tilde{b}_{11}}U_{\tilde{b}_{12}}^{*}\left(m_{b}\,\delta^{(1)}X_{b}^{*}+X_{b}^{*}\,\delta^{(1)}m_{b}\right)\right\\}\\\
&+\left(|U_{\tilde{b}_{11}}|^{2}-|U_{\tilde{b}_{12}}|^{2}\right)\delta^{(1)}\\!\left(\textbf{M}_{\tilde{b}}^{2}\right)_{LL}\Big{)}\Big{]}.\end{split}$
(152)

Lastly, the counterterm for $m_{\tilde{b}_{12}}^{2}$ reads

$\begin{split}\delta^{(1)}m_{\tilde{b}_{12}}^{2}={}&\frac{1}{|U_{\tilde{b}_{11}}|^{2}-|U_{\tilde{b}_{12}}|^{2}}\Big{[}U_{\tilde{b}_{11}}U_{\tilde{b}_{21}}^{*}\left(\delta^{(1)}m_{\tilde{b}_{1}}^{2}-\delta^{(1)}m_{\tilde{b}_{2}}^{2}\right)\\\
&+U_{\tilde{b}_{11}}U_{\tilde{b}_{22}}^{*}\left(m_{b}\,\delta^{(1)}X_{b}^{*}+X_{b}^{*}\,\delta^{(1)}m_{b}\right)-U_{\tilde{b}_{12}}U_{\tilde{b}_{21}}^{*}\left(m_{b}\,\delta^{(1)}X_{b}+X_{b}\,\delta^{(1)}m_{b}\right)\Big{]}.\end{split}$
(153)

####  B.3.2 Renormalisation in the massless case

To extract all terms of order
$\order{(\alpha_{\text{em}}+\alpha_{q})^{2}N_{c}^{2}}$ in the Higgs boson mass
prediction at the two-loop level, the first and second generation of quarks
and squarks need to be taken into account as well. These generations
contribute even if their quarks are assumed to be massless. As there are also
two-loop Higgs self-energies with both a third generation squark in one loop
and a first/second generation squark in the other, those contributions cannot
simply be obtained by taking the results for the third generation and applying
the massless limit. Instead, the whole calculation has to be done anew. To
this end, we assume both the first and second generation of quarks to be
massless and again a diagonal CKM matrix.

As for the third generation of squarks, seven independent real parameters
appear in the squark mass matrices of each of the first two generations. In
the massless limit, however, the trilinear counterterms $\delta^{(1)}A_{q}$
and correspondingly the off-diagonal mass counterterms
$\delta^{(1)}m_{\tilde{q}_{12}}^{2}$ do not contribute in the sub-loop
renormalisation of the Higgs self-energies. This leaves us with three
independent parameters in each generation. If we assume generation $g$ to be
massless, these are $M_{\tilde{q}_{g}}^{2}$, $M_{\tilde{u}_{g}}^{2}$, and
$M_{\tilde{d}_{g}}^{2}$. As before, we fix these parameters by imposing
renormalisation conditions on the diagonal squark self-energies. When working
with the massive third generation, we used both the stop and one of the
sbottom masses as independent input parameters. The remaining sbottom mass was
then fixed by virtue of the $SU(2)$ symmetry of the SUSY breaking parameter
$M_{\tilde{q}_{3}}^{2}$. For the massless first two generations, this
procedure needs to be adapted; the squark mass matrices are diagonal in the
massless limit and so the corresponding rotation matrices $U_{\tilde{q}}$ are
either purely diagonal or purely off-diagonal. One of the mass eigenstates
thus corresponds to the left-handed gauge eigenstate $\tilde{q}_{L}$, the
other one to the right-handed gauge eigenstate $\tilde{q}_{R}$. The $SU(2)$
symmetry fixes the mass of the left-handed down-type squark in terms of the
left-handed up-type squark, and its mass counterterm cannot be chosen
independently. This issue becomes clear when looking at Eq. (152); this
expression for the dependent mass counterterm
$\delta^{(1)}m_{\tilde{b}_{f}}^{2}$ is undefined if $U_{\tilde{b}_{1f}}=0$.
This can be circumvented by removing the freedom of choice as for which mass
is treated independently. We demonstrate the procedure for the second
generation and an on-shell renormalisation of the input mass (the first
generation and ${\overline{\text{DR}}}$ renormalisation are treated in
analogous fashion). The scalar charm quark mass counterterms are

$\delta^{(1)}m_{\tilde{c}_{i}}^{2}=\real\Sigma^{(1)}_{\tilde{c}_{i}\tilde{c}_{i}}=\Sigma^{(1)}_{\tilde{c}_{i}\tilde{c}_{i}},\quad
i\in\\{1,2\\}.$ (154)

We cannot freely choose which of the two scalar strange quark masses is used
as input. Instead, we fix our choice by the form of $U_{\tilde{s}}$ in order
to avoid divergent and thus meaningless expressions:

#####  $U_{\tilde{s}}$ is diagonal.

This means that $U_{\tilde{s}_{12}}=U_{\tilde{s}_{21}}=0$ and the second
generation analogue of Eq. (152) is only meaningful if $f=1,n=2$ is chosen. We
arrive at

$\displaystyle\delta^{(1)}m_{\tilde{s}_{1}}^{2}={}$
$\displaystyle\delta^{(1)}\\!\left(\textbf{M}_{\tilde{s}}^{2}\right)_{LL},$
(155a) $\displaystyle\delta^{(1)}m_{\tilde{s}_{2}}^{2}={}$
$\displaystyle\Sigma^{(1)}_{\tilde{s}_{2}\tilde{s}_{2}}.$ (155b)

#####  $U_{\tilde{s}}$ is purely off-diagonal.

This means that $U_{\tilde{s}_{11}}=U_{\tilde{s}_{22}}=0$ and the second
generation analogue of Eq. (152) is only meaningful if $f=2,n=1$ is chosen. We
arrive at

$\displaystyle\delta^{(1)}m_{\tilde{s}_{1}}^{2}={}$
$\displaystyle\Sigma^{(1)}_{\tilde{s}_{1}\tilde{s}_{1}},$ (156a)
$\displaystyle\delta^{(1)}m_{\tilde{s}_{2}}^{2}={}$
$\displaystyle\delta^{(1)}\\!\left(\textbf{M}_{\tilde{s}}^{2}\right)_{LL}.$
(156b)

We can combine both cases in the formulae

$\displaystyle\delta^{(1)}m_{\tilde{s}_{1}}^{2}={}$
$\displaystyle|U_{\tilde{s}_{12}}|^{2}\Sigma^{(1)}_{\tilde{s}_{1}\tilde{s}_{1}}+|U_{\tilde{s}_{11}}|^{2}\delta^{(1)}\\!\left(\textbf{M}_{\tilde{s}}^{2}\right)_{LL},$
(157a) $\displaystyle\delta^{(1)}m_{\tilde{s}_{2}}^{2}={}$
$\displaystyle|U_{\tilde{s}_{11}}|^{2}\Sigma^{(1)}_{\tilde{s}_{2}\tilde{s}_{2}}+|U_{\tilde{s}_{12}}|^{2}\delta^{(1)}\\!\left(\textbf{M}_{\tilde{s}}^{2}\right)_{LL}.$
(157b)

$\delta^{(1)}\\!\left(\textbf{M}_{\tilde{s}}^{2}\right)_{LL}$ is obtained from
Eq. (151) by replacing the third generation labels by second generation
labels.

###  B.4 ${\overline{\text{DR}}}$ renormalisation of $\mu$ and $A_{q}$

For a full renormalisation of the squark sector, the counterterms
$\delta^{(1)}\mu$, $\delta^{(1)}A_{t}$, and $\delta^{(1)}A_{b}$ need to be
fixed. The higgsino mass parameter $\mu$ is typically defined via the
chargino-neutralino sector (see e.g. Ref. [52]). As that sector is otherwise
irrelevant to our calculation, we choose a ${\overline{\text{DR}}}$
renormalisation for $\mu$. As can be seen in Ref. [52], the OS expression for
the $\mu$ counterterm in a CCN scheme involves elements of both chargino
rotation matrices, which transform the gauge eigenstates into mass
eigenstates. Taking the divergent part of the OS counterterm yields an
expression which is still rather complicated as the rotation matrix elements
do not easily cancel out algebraically.

As the higgsino mass parameter enters the $h\tilde{t}\tilde{t}^{*}$ vertex, an
expression for its counterterm can also be obtained from the renormalisation
of this vertex. This approach naturally leads to an expression for
$\delta^{(1)}A_{t}$ as well and avoids the need to deal with the chargino
rotation matrices. Therefore, we calculate the amplitudes
$h\to\tilde{t}_{1}\tilde{t}_{2}^{*}$ and $H\to\tilde{t}_{1}\tilde{t}_{2}^{*}$
at the one-loop level and determine $\delta^{(1)}\mu^{\overline{\text{DR}}}$
and $\delta^{(1)}A_{t}^{\overline{\text{DR}}}$ from requiring both amplitudes
to be finite.

As both amplitudes $\hat{\Gamma}^{(1)}_{h\tilde{t}_{1}\tilde{t}_{2}^{*}}$ and
$\hat{\Gamma}^{(1)}_{H\tilde{t}_{1}\tilde{t}_{2}^{*}}$ involve both
counterterms $\delta^{(1)}\mu$ and $\delta^{(1)}A_{t}$, we have to form
appropriate combinations of both amplitudes:

$\displaystyle
c_{\alpha}\hat{\Gamma}^{(1)}_{H\tilde{t}_{1}\tilde{t}_{2}^{*}}-s_{\alpha}\hat{\Gamma}^{(1)}_{h\tilde{t}_{1}\tilde{t}_{2}^{*}}:\quad\text{$\delta^{(1)}A_{t}$
drops out},$ (158a) $\displaystyle
c_{\alpha}\hat{\Gamma}^{(1)}_{h\tilde{t}_{1}\tilde{t}_{2}^{*}}+s_{\alpha}\hat{\Gamma}^{(1)}_{H\tilde{t}_{1}\tilde{t}_{2}^{*}}:\quad\text{$\delta^{(1)}\mu$
drops out}.$ (158b)

Now the first expression is used to determine $\delta^{(1)}\mu$ and the second
one to determine $\delta^{(1)}A_{t}$. The same procedure works for
$\delta^{(1)}A_{b}$ as well:

$\displaystyle
c_{\alpha}\hat{\Gamma}^{(1)}_{H\tilde{b}_{1}\tilde{b}_{2}^{*}}-s_{\alpha}\hat{\Gamma}^{(1)}_{h\tilde{b}_{1}\tilde{b}_{2}^{*}}:\quad\text{$\delta^{(1)}\mu$
drops out},$ (159a) $\displaystyle
c_{\alpha}\hat{\Gamma}^{(1)}_{h\tilde{b}_{1}\tilde{b}_{2}^{*}}+s_{\alpha}\hat{\Gamma}^{(1)}_{H\tilde{b}_{1}\tilde{b}_{2}^{*}}:\quad\text{$\delta^{(1)}A_{b}$
drops out}.$ (159b)

Taking into account also the other counterterms contributing to the
Higgs–sfermion amplitudes, we obtain

$\displaystyle\frac{\delta^{(1)}\mu^{\overline{\text{DR}}}}{\mu}={}$
$\displaystyle\frac{\alpha_{\text{em}}N_{c}}{16\pi
M_{W}^{2}s_{\text{w}}^{2}}\Big{(}\frac{m_{t}^{2}}{s_{\beta}^{2}}+\frac{m_{b}^{2}}{c_{\beta}^{2}}\Big{)}\frac{1}{\varepsilon},$
(160a)
$\displaystyle\frac{\delta^{(1)}A_{t}^{\overline{\text{DR}}}}{A_{t}}={}$
$\displaystyle\frac{\alpha_{\text{em}}N_{c}}{8\pi
M_{W}^{2}s_{\text{w}}^{2}}\frac{m_{t}^{2}}{s_{\beta}^{2}}\frac{1}{\varepsilon},$
(160b)
$\displaystyle\frac{\delta^{(1)}A_{b}^{\overline{\text{DR}}}}{A_{b}}={}$
$\displaystyle\frac{\alpha_{\text{em}}N_{c}}{8\pi
M_{W}^{2}s_{\text{w}}^{2}}\frac{m_{b}^{2}}{c_{\beta}^{2}}\frac{1}{\varepsilon}.$
(160c)

##  Appendix C One- and two-loop field and parameter counterterms

In this appendix, we list the one- and two-loop expressions for the
renormalised tadpoles and self-energies in the Higgs–gauge sector that appear
in our calculation. They are obtained by applying the renormalisation
transformations given in Sec. 2 to the Higgs–gauge Lagrangian.

We also give the one- and two-loop expressions for the counterterms of all
Higgs boson mass parameters appearing in Eq. (8). The counterterms will be
expressed as combinations of the one- and two-loop counterterms of the input
parameters. The renormalisation of the input parameters is explained in Sects.
2.3 and 2.4. We also relate the renormalisation constants of the Higgs fields
to the Higgs doublet counterterms.

###  C.1 Renormalised one-loop tadpoles and self-energies

The one-loop one-point vertex functions are renormalised by the tadpole
counterterms:

$\displaystyle\hat{\Gamma}^{(1)}_{h}={}$
$\displaystyle\Gamma^{(1)}_{h}+\delta^{(1)}T_{h},$ (161a)
$\displaystyle\hat{\Gamma}^{(1)}_{H}={}$
$\displaystyle\Gamma^{(1)}_{H}+\delta^{(1)}T_{H},$ (161b)
$\displaystyle\hat{\Gamma}^{(1)}_{A}={}$
$\displaystyle\Gamma^{(1)}_{A}+\delta^{(1)}T_{A},$ (161c)
$\displaystyle\hat{\Gamma}^{(1)}_{G}={}$
$\displaystyle\Gamma^{(1)}_{G}+\delta^{(1)}T_{G}.$ (161d)

The neutral $\mathcal{CP}$-even self-energies are

$\displaystyle\hat{\Sigma}^{(1)}_{hh}(p^{2})={}$
$\displaystyle\Sigma^{(1)}_{hh}(p^{2})+\delta^{(1)}Z_{hh}(p^{2}-m_{h}^{2})-\delta^{(1)}m_{h}^{2},$
(162a) $\displaystyle\hat{\Sigma}^{(1)}_{hH}(p^{2})={}$
$\displaystyle\Sigma^{(1)}_{hH}(p^{2})+\delta^{(1)}Z_{hH}\left(p^{2}-\frac{m_{h}^{2}+m_{H}^{2}}{2}\right)-\delta^{(1)}m_{hH}^{2},$
(162b) $\displaystyle\hat{\Sigma}^{(1)}_{HH}(p^{2})={}$
$\displaystyle\Sigma^{(1)}_{HH}(p^{2})+\delta^{(1)}Z_{HH}(p^{2}-m_{H}^{2})-\delta^{(1)}m_{H}^{2},$
(162c)

while the $\mathcal{CP}$-odd self-energies read

$\displaystyle\hat{\Sigma}^{(1)}_{AA}(p^{2})={}$
$\displaystyle\Sigma^{(1)}_{AA}(p^{2})+\delta^{(1)}Z_{AA}(p^{2}-m_{A}^{2})-\delta^{(1)}m_{A}^{2},$
(163a) $\displaystyle\hat{\Sigma}^{(1)}_{AG}(p^{2})={}$
$\displaystyle\Sigma^{(1)}_{AG}(p^{2})+\delta^{(1)}Z_{AG}\left(p^{2}-\frac{m_{A}^{2}}{2}\right)-\delta^{(1)}m_{AG}^{2},$
(163b) $\displaystyle\hat{\Sigma}^{(1)}_{GG}(p^{2})={}$
$\displaystyle\Sigma^{(1)}_{GG}(p^{2})+\delta^{(1)}Z_{GG}\,p^{2}-\delta^{(1)}m_{G}^{2}.$
(163c)

In the case of $\mathcal{CP}$ violation, the self-energies

$\displaystyle\hat{\Sigma}^{(1)}_{hA}(p^{2})={}$
$\displaystyle\Sigma^{(1)}_{hA}(p^{2})-\delta^{(1)}m_{hA}^{2},$ (164a)
$\displaystyle\hat{\Sigma}^{(1)}_{hG}(p^{2})={}$
$\displaystyle\Sigma^{(1)}_{hG}(p^{2})-\delta^{(1)}m_{hG}^{2},$ (164b)
$\displaystyle\hat{\Sigma}^{(1)}_{HA}(p^{2})={}$
$\displaystyle\Sigma^{(1)}_{HA}(p^{2})-\delta^{(1)}m_{HA}^{2},$ (164c)
$\displaystyle\hat{\Sigma}^{(1)}_{HG}(p^{2})={}$
$\displaystyle\Sigma^{(1)}_{HG}(p^{2})-\delta^{(1)}m_{HG}^{2}$ (164d)

do not vanish. The neutral self-energies are symmetric such that for instance
${\hat{\Sigma}^{(1)}_{Hh}=\hat{\Sigma}^{(1)}_{hH}}$.

The charged Higgs self-energies are

$\displaystyle\hat{\Sigma}^{(1)}_{H^{-}H^{+}}(p^{2})={}$
$\displaystyle\Sigma^{(1)}_{H^{-}H^{+}}(p^{2})+\delta^{(1)}Z_{H^{-}H^{+}}\left(p^{2}-m_{H^{\pm}}^{2}\right)-\delta^{(1)}m_{H^{\pm}}^{2},$
(165a) $\displaystyle\hat{\Sigma}^{(1)}_{H^{-}G^{+}}(p^{2})={}$
$\displaystyle\Sigma^{(1)}_{H^{-}G^{+}}(p^{2})+\delta^{(1)}Z_{H^{-}G^{+}}\left(p^{2}-\frac{m_{H^{\pm}}^{2}}{2}\right)-\delta^{(1)}m_{H^{-}G^{+}}^{2},$
(165b) $\displaystyle\hat{\Sigma}^{(1)}_{G^{-}H^{+}}(p^{2})={}$
$\displaystyle\Sigma^{(1)}_{G^{-}H^{+}}(p^{2})+\delta^{(1)}Z_{G^{-}H^{+}}\left(p^{2}-\frac{m_{H^{\pm}}^{2}}{2}\right)-\delta^{(1)}m_{G^{-}H^{+}}^{2},$
(165c) $\displaystyle\hat{\Sigma}^{(1)}_{G^{-}G^{+}}(p^{2})={}$
$\displaystyle\Sigma^{(1)}_{G^{-}G^{+}}(p^{2})+\delta^{(1)}Z_{G^{-}G^{+}}\,p^{2}-\delta^{(1)}m_{G^{\pm}}^{2}.$
(165d)

They are symmetric in the sense that

$\hat{\Sigma}^{(1)}_{G^{-}H^{+}}=\hat{\Sigma}^{(1)}_{H^{+}G^{-}}$ (166)

but in general

$\big{(}\hat{\Sigma}^{(1)}_{G^{-}H^{+}}(p^{2})\big{)}^{*}\neq\hat{\Sigma}^{(1)}_{H^{-}G^{+}}(p^{2}).$
(167)

Instead, we have

$\big{(}\hat{\Sigma}^{(1)}_{G^{-}H^{+}}(p^{2})\big{)}^{*}=\widetilde{\text{Co}}\
\hat{\Sigma}^{(1)}_{H^{-}G^{+}}(p^{2}),$ (168)

where $\widetilde{\text{Co}}$ takes the complex conjugate of loop integrals
only and leaves complex couplings unaffected. Loop integrals are complex
quantities for sufficiently large external momenta and so the
$\widetilde{\text{Co}}$ must not be left out.

Vector boson self-energies are Lorentz tensors of rank two. We decompose them
into a transverse and into a longitudinal component

$\Sigma^{\mu\nu}(p)=\left(-g^{\mu\nu}+\frac{p^{\mu}p^{\nu}}{p^{2}}\right)\Sigma^{T}(p^{2})-\frac{p^{\mu}p^{\nu}}{p^{2}}\Sigma^{L}(p^{2}),$
(169)

using the same convention as in Ref. [173].

The renormalised transverse parts of the gauge boson self-energies are

$\displaystyle\hat{\Sigma}^{T,(1)}_{\gamma\gamma}(p^{2})={}$
$\displaystyle\Sigma^{T,(1)}_{\gamma\gamma}(p^{2})+\delta^{(1)}Z_{\gamma\gamma}\,p^{2},$
(170a) $\displaystyle\begin{split}\hat{\Sigma}^{T,(1)}_{\gamma
Z}(p^{2})={}&\Sigma^{T,(1)}_{\gamma
Z}(p^{2})+\tfrac{1}{2}\delta^{(1)}Z_{\gamma
Z}\,p^{2}+\tfrac{1}{2}\delta^{(1)}Z_{Z\gamma}(p^{2}-M_{Z}^{2})\\\
={}&\hat{\Sigma}^{T,(1)}_{Z\gamma}(p^{2}),\end{split}$ (170b)
$\displaystyle\hat{\Sigma}^{T,(1)}_{ZZ}(p^{2})={}$
$\displaystyle\Sigma^{T,(1)}_{ZZ}(p^{2})+\delta^{(1)}Z_{ZZ}(p^{2}-M_{Z}^{2})-\delta^{(1)}M_{Z}^{2},$
(170c)
$\displaystyle\begin{split}\hat{\Sigma}^{T,(1)}_{W^{-}W^{+}}(p^{2})={}&\Sigma^{T,(1)}_{W^{-}W^{+}}(p^{2})+\delta^{(1)}Z_{WW}(p^{2}-M_{W}^{2})-\delta^{(1)}M_{W}^{2}\\\
={}&\hat{\Sigma}^{T,(1)}_{W^{+}W^{-}}(p^{2}).\end{split}$ (170d)

The transverse parts of vector self-energies are relevant in our calculation
already at the one-loop level, as they are used to determine the mass and
field counterterms of the gauge bosons. The longitudinal vector boson self-
energies enter a Higgs boson mass prediction at the three-loop order and
higher, and will not be discussed here.

For a two-loop calculation, we also need self-energies which mix scalars and
vectors. Their Lorentz decomposition reads

$\displaystyle\Sigma^{\mu}_{SV}(p)=p^{\mu}\Sigma^{L}_{SV}(p^{2}),$ (171a)
$\displaystyle\Sigma^{\mu}_{VS}(p)=p^{\mu}\Sigma^{L}_{VS}(p^{2}),$ (171b)

where $\Sigma^{\mu}_{SV}(p)$ denotes a self-energy with incoming vector
$V^{\dagger}$ and outgoing scalar $S$, and $\Sigma^{\mu}_{VS}(p)$ denotes a
self-energy with incoming scalar $S^{\dagger}$ and outgoing vector $V$. We
have four neutral scalar–vector self-energies

$\displaystyle\begin{split}\hat{\Sigma}^{L,(1)}_{A\gamma}(p^{2})={}&\Sigma^{L,(1)}_{A\gamma}(p^{2})\\\
={}&-\hat{\Sigma}^{L,(1)}_{\gamma
A}(p^{2})\overset{\order{N_{c}}}{=}0,\end{split}$ (172a)
$\displaystyle\begin{split}\hat{\Sigma}^{L,(1)}_{AZ}(p^{2})={}&\Sigma^{L,(1)}_{AZ}(p^{2})-\text{i}M_{Z}\left(\tfrac{1}{2}\delta^{(1)}Z_{AG}+c_{\beta}^{2}\delta^{(1)}t_{\beta}\right)\\\
={}&-\hat{\Sigma}^{L,(1)}_{ZA}(p^{2}),\end{split}$ (172b)
$\displaystyle\begin{split}\hat{\Sigma}^{L,(1)}_{G\gamma}(p^{2})={}&\Sigma^{L,(1)}_{G\gamma}(p^{2})-\frac{\text{i}M_{Z}}{2}\delta^{(1)}Z_{Z\gamma}\\\
={}&-\hat{\Sigma}^{L,(1)}_{\gamma
G}(p^{2})\overset{\order{N_{c}}}{=}0,\end{split}$ (172c)
$\displaystyle\begin{split}\hat{\Sigma}^{L,(1)}_{GZ}(p^{2})={}&\Sigma^{L,(1)}_{GZ}(p^{2})-\frac{\text{i}M_{Z}}{2}\left(\frac{\delta^{(1)}M_{Z}^{2}}{M_{Z}^{2}}+\delta^{(1)}Z_{ZZ}+\delta^{(1)}Z_{GG}\right)\\\
={}&-\hat{\Sigma}^{L,(1)}_{ZG}(p^{2}).\end{split}$ (172d)

The charged self-energies are

$\displaystyle\begin{split}\hat{\Sigma}^{L,(1)}_{H^{-}W^{+}}(p^{2})={}&\Sigma^{L,(1)}_{H^{-}W^{+}}(p^{2})+M_{W}\left(\tfrac{1}{2}\delta^{(1)}Z_{H^{-}G^{+}}+c_{\beta}^{2}\delta^{(1)}t_{\beta}\right)\\\
={}&-\hat{\Sigma}^{L,(1)}_{W^{+}H^{-}}(p^{2}),\end{split}$ (173a)
$\displaystyle\begin{split}\hat{\Sigma}^{L,(1)}_{W^{-}H^{+}}(p^{2})={}&\Sigma^{L,(1)}_{W^{-}H^{+}}(p^{2})+M_{W}\left(\tfrac{1}{2}\delta^{(1)}Z_{G^{-}H^{+}}+c_{\beta}^{2}\delta^{(1)}t_{\beta}\right)\\\
={}&-\hat{\Sigma}^{L,(1)}_{H^{+}W^{-}}(p^{2}).\end{split}$ (173b)
$\displaystyle\begin{split}\hat{\Sigma}^{L,(1)}_{G^{-}W^{+}}(p^{2})={}&\Sigma^{L,(1)}_{G^{-}W^{+}}(p^{2})+\frac{M_{W}}{2}\left(\frac{\delta^{(1)}M_{W}^{2}}{M_{W}^{2}}+\delta^{(1)}Z_{WW}+\delta^{(1)}Z_{G^{-}G^{+}}\right)\\\
={}&-\hat{\Sigma}^{L,(1)}_{W^{+}G^{-}}(p^{2}),\end{split}$ (173c)
$\displaystyle\begin{split}\hat{\Sigma}^{L,(1)}_{W^{-}G^{+}}(p^{2})={}&\Sigma^{L,(1)}_{W^{-}G^{+}}(p^{2})+\frac{M_{W}}{2}\left(\frac{\delta^{(1)}M_{W}^{2}}{M_{W}^{2}}+\delta^{(1)}Z_{WW}+\delta^{(1)}Z_{G^{-}G^{+}}\right)\\\
={}&-\hat{\Sigma}^{L,(1)}_{G^{+}W^{-}}(p^{2}).\end{split}$ (173d)

Again, conjugated diagrams are related via

$\big{(}\hat{\Sigma}^{L,(1)}_{W^{-}H^{+}}(p^{2})\big{)}^{*}=\widetilde{\text{Co}}\
\hat{\Sigma}^{L,(1)}_{H^{-}W^{+}}(p^{2}).$ (174)

Not all of the self-energies presented in this section are actually needed for
our calculation. We have numerically shown the finiteness of all given one-
loop self-energies as a cross-check and provide the derived expressions for
the sake of completeness and for future reference.

###  C.2 One-loop counterterms

The one-loop mass counterterms read

$\displaystyle\begin{split}\delta^{(1)}m_{h}^{2}={}&\delta^{(1)}m_{A}^{2}c_{\alpha-\beta}^{2}+\delta^{(1)}M_{Z}^{2}s_{\alpha+\beta}^{2}\\\
&+\delta^{(1)}t_{\beta}c_{\beta}^{2}\big{(}m_{A}^{2}s_{2(\alpha-\beta)}+M_{Z}^{2}s_{2(\alpha+\beta)}\big{)}\\\
&+\frac{es_{\alpha-\beta}}{2M_{W}s_{\text{w}}}\big{(}\delta^{(1)}T_{h}(1+c_{\alpha-\beta}^{2})+\delta^{(1)}T_{H}c_{\alpha-\beta}s_{\alpha-\beta}\big{)},\end{split}$
(175a)
$\displaystyle\begin{split}\delta^{(1)}m_{hH}^{2}={}&\delta^{(1)}m_{A}^{2}c_{\alpha-\beta}s_{\alpha-\beta}-\delta^{(1)}M_{Z}^{2}c_{\alpha+\beta}s_{\alpha+\beta}\\\
&-\delta^{(1)}t_{\beta}c_{\beta}^{2}\big{(}m_{A}^{2}c_{2(\alpha-\beta)}+M_{Z}^{2}c_{2(\alpha+\beta)}\big{)}\\\
&-\frac{e}{2M_{W}s_{\text{w}}}\big{(}\delta^{(1)}T_{h}c_{\alpha-\beta}^{3}-\delta^{(1)}T_{H}s_{\alpha-\beta}^{3}\big{)},\end{split}$
(175b)
$\displaystyle\begin{split}\delta^{(1)}m_{H}^{2}={}&\delta^{(1)}m_{A}^{2}s_{\alpha-\beta}^{2}+\delta^{(1)}M_{Z}^{2}c_{\alpha+\beta}^{2}\\\
&-\delta^{(1)}t_{\beta}c_{\beta}^{2}\big{(}m_{A}^{2}s_{2(\alpha-\beta)}+M_{Z}^{2}s_{2(\alpha+\beta)}\big{)}\\\
&-\frac{ec_{\alpha-\beta}}{2M_{W}s_{\text{w}}}\big{(}\delta^{(1)}T_{h}c_{\alpha-\beta}s_{\alpha-\beta}+\delta^{(1)}T_{H}(1+s_{\alpha-\beta}^{2})\big{)},\end{split}$
(175c)
$\displaystyle\begin{split}\delta^{(1)}m_{AG}^{2}={}&-\delta^{(1)}t_{\beta}c_{\beta}^{2}m_{A}^{2}\\\
&-\frac{e}{2M_{W}s_{\text{w}}}\big{(}\delta^{(1)}T_{h}c_{\alpha-\beta}+\delta^{(1)}T_{H}s_{\alpha-\beta}\big{)},\end{split}$
(175d) $\displaystyle\delta^{(1)}m_{G}^{2}={}$
$\displaystyle\frac{e}{2M_{W}s_{\text{w}}}\big{(}\delta^{(1)}T_{h}s_{\alpha-\beta}-\delta^{(1)}T_{H}c_{\alpha-\beta}\big{)},$
(175e) $\displaystyle\delta^{(1)}m_{hA}^{2}={}$
$\displaystyle\frac{es_{\alpha-\beta}}{2M_{W}s_{\text{w}}}\delta^{(1)}T_{A},$
(175f) $\displaystyle\delta^{(1)}m_{HA}^{2}={}$
$\displaystyle-\frac{ec_{\alpha-\beta}}{2M_{W}s_{\text{w}}}\delta^{(1)}T_{A},$
(175g) $\displaystyle\delta^{(1)}m_{hG}^{2}={}$
$\displaystyle\frac{ec_{\alpha-\beta}}{2M_{W}s_{\text{w}}}\delta^{(1)}T_{A}=-\delta^{(1)}m_{HA}^{2},$
(175h) $\displaystyle\delta^{(1)}m_{HG}^{2}={}$
$\displaystyle\frac{es_{\alpha-\beta}}{2M_{W}s_{\text{w}}}\delta^{(1)}T_{A}=\delta^{(1)}m_{hA}^{2},$
(175i)
$\displaystyle\begin{split}\delta^{(1)}m_{H^{-}G^{+}}^{2}={}&-\delta^{(1)}t_{\beta}c_{\beta}^{2}m_{H^{\pm}}^{2}\\\
&-\frac{e}{2M_{W}s_{\text{w}}}\big{(}\delta^{(1)}T_{h}c_{\alpha-\beta}+\delta^{(1)}T_{H}s_{\alpha-\beta}-\text{i}\delta^{(1)}T_{A}\big{)},\end{split}$
(175j)
$\displaystyle\begin{split}\delta^{(1)}m_{G^{-}H^{+}}^{2}={}&-\delta^{(1)}t_{\beta}c_{\beta}^{2}m_{H^{\pm}}^{2}\\\
&-\frac{e}{2M_{W}s_{\text{w}}}\big{(}\delta^{(1)}T_{h}c_{\alpha-\beta}+\delta^{(1)}T_{H}s_{\alpha-\beta}+\text{i}\delta^{(1)}T_{A}\big{)}\\\
={}&\big{(}\delta^{(1)}m_{H^{-}G^{+}}^{2}\big{)}^{*},\end{split}$ (175k)
$\displaystyle\delta^{(1)}m_{G^{\pm}}^{2}={}$
$\displaystyle\frac{e}{2M_{W}s_{\text{w}}}\big{(}\delta^{(1)}T_{h}s_{\alpha-\beta}-\delta^{(1)}T_{H}c_{\alpha-\beta}\big{)}=\delta^{(1)}m_{G}^{2}.$
(175l)

The one-loop field counterterms are given by

$\displaystyle\delta^{(1)}Z_{hh}={}$ $\displaystyle
s_{\alpha}^{2}\delta^{(1)}Z_{\mathcal{H}_{1}}+c_{\alpha}^{2}\delta^{(1)}Z_{\mathcal{H}_{2}},$
(176a) $\displaystyle\delta^{(1)}Z_{hH}={}$ $\displaystyle
s_{\alpha}c_{\alpha}\big{(}\delta^{(1)}Z_{\mathcal{H}_{2}}-\delta^{(1)}Z_{\mathcal{H}_{1}}\big{)},$
(176b) $\displaystyle\delta^{(1)}Z_{HH}={}$ $\displaystyle
c_{\alpha}^{2}\delta^{(1)}Z_{\mathcal{H}_{1}}+s_{\alpha}^{2}\delta^{(1)}Z_{\mathcal{H}_{2}},$
(176c) $\displaystyle\delta^{(1)}Z_{AA}={}$ $\displaystyle
s_{\beta}^{2}\delta^{(1)}Z_{\mathcal{H}_{1}}+c_{\beta}^{2}\delta^{(1)}Z_{\mathcal{H}_{2}},$
(176d) $\displaystyle\delta^{(1)}Z_{AG}={}$ $\displaystyle
s_{\beta}c_{\beta}\big{(}\delta^{(1)}Z_{\mathcal{H}_{2}}-\delta^{(1)}Z_{\mathcal{H}_{1}}\big{)},$
(176e) $\displaystyle\delta^{(1)}Z_{GG}={}$ $\displaystyle
c_{\beta}^{2}\delta^{(1)}Z_{\mathcal{H}_{1}}+s_{\beta}^{2}\delta^{(1)}Z_{\mathcal{H}_{2}},$
(176f) $\displaystyle\delta^{(1)}Z_{H^{-}H^{+}}={}$ $\displaystyle
s_{\beta}^{2}\delta^{(1)}Z_{\mathcal{H}_{1}}+c_{\beta}^{2}\delta^{(1)}Z_{\mathcal{H}_{2}},$
(176g) $\displaystyle\delta^{(1)}Z_{H^{-}G^{+}}={}$ $\displaystyle
s_{\beta}c_{\beta}\big{(}\delta^{(1)}Z_{\mathcal{H}_{2}}-\delta^{(1)}Z_{\mathcal{H}_{1}}\big{)},$
(176h) $\displaystyle\delta^{(1)}Z_{G^{-}H^{+}}={}$ $\displaystyle
s_{\beta}c_{\beta}\big{(}\delta^{(1)}Z_{\mathcal{H}_{2}}-\delta^{(1)}Z_{\mathcal{H}_{1}}\big{)},$
(176i) $\displaystyle\delta^{(1)}Z_{G^{-}G^{+}}={}$ $\displaystyle
c_{\beta}^{2}\delta^{(1)}Z_{\mathcal{H}_{1}}+s_{\beta}^{2}\delta^{(1)}Z_{\mathcal{H}_{2}}.$
(176j)

###  C.3 Renormalised two-loop tadpoles and self-energies

At the two-loop level, the renormalised one-point vertex functions read

$\displaystyle\hat{\Gamma}^{(2)}_{h}={}$
$\displaystyle\Gamma^{(2)}_{h}+\delta^{(2)}T_{h}+\tfrac{1}{2}\delta^{(1)}Z_{hh}\delta^{(1)}T_{h}+\tfrac{1}{2}\delta^{(1)}Z_{hH}\delta^{(1)}T_{H},$
(177a) $\displaystyle\hat{\Gamma}^{(2)}_{H}={}$
$\displaystyle\Gamma^{(2)}_{H}+\delta^{(2)}T_{H}+\tfrac{1}{2}\delta^{(1)}Z_{HH}\delta^{(1)}T_{H}+\tfrac{1}{2}\delta^{(1)}Z_{hH}\delta^{(1)}T_{h},$
(177b) $\displaystyle\hat{\Gamma}^{(2)}_{A}={}$
$\displaystyle\Gamma^{(2)}_{A}+\delta^{(2)}T_{A}+\tfrac{1}{2}\delta^{(1)}Z_{AA}\delta^{(1)}T_{A}+\tfrac{1}{2}\delta^{(1)}Z_{AG}\delta^{(1)}T_{G},$
(177c) $\displaystyle\hat{\Gamma}^{(2)}_{G}={}$
$\displaystyle\Gamma^{(2)}_{G}+\delta^{(2)}T_{G}+\tfrac{1}{2}\delta^{(1)}Z_{GG}\delta^{(1)}T_{G}+\tfrac{1}{2}\delta^{(1)}Z_{AG}\delta^{(1)}T_{A}.$
(177d)

The renormalised $\mathcal{CP}$-even two-loop self-energies read

$\displaystyle\begin{split}\hat{\Sigma}^{(2)}_{hh}(p^{2})={}&\Sigma^{(2)}_{hh}(p^{2})+\delta^{(2)}Z_{hh}(p^{2}-m_{h}^{2})-\delta^{(2)}m_{h}^{2}\\\
&+\tfrac{1}{4}\left(\delta^{(1)}Z_{hH}\right)^{2}(p^{2}-m_{H}^{2})-\delta^{(1)}Z_{hh}\delta^{(1)}m_{h}^{2}-\delta^{(1)}Z_{hH}\delta^{(1)}m_{hH}^{2},\end{split}$
(178a)
$\displaystyle\begin{split}\hat{\Sigma}^{(2)}_{hH}(p^{2})={}&\Sigma^{(2)}_{hH}(p^{2})+\delta^{(2)}Z_{hH}\left(p^{2}-\frac{m_{h}^{2}+m_{H}^{2}}{2}\right)-\delta^{(2)}m_{hH}^{2}\\\
&+\tfrac{1}{4}\delta^{(1)}Z_{hH}\delta^{(1)}Z_{hh}(p^{2}-m_{h}^{2})+\tfrac{1}{4}\delta^{(1)}Z_{hH}\delta^{(1)}Z_{HH}(p^{2}-m_{H}^{2})\\\
&-\delta^{(1)}Z_{hH}\frac{\delta^{(1)}m_{h}^{2}+\delta^{(1)}m_{H}^{2}}{2}-\frac{\delta^{(1)}Z_{hh}+\delta^{(1)}Z_{HH}}{2}\delta^{(1)}m_{hH}^{2},\end{split}$
(178b)
$\displaystyle\begin{split}\hat{\Sigma}^{(2)}_{HH}(p^{2})={}&\Sigma^{(2)}_{HH}(p^{2})+\delta^{(2)}Z_{HH}(p^{2}-m_{H}^{2})-\delta^{(2)}m_{H}^{2}\\\
&+\tfrac{1}{4}\left(\delta^{(1)}Z_{hH}\right)^{2}(p^{2}-m_{h}^{2})-\delta^{(1)}Z_{HH}\delta^{(1)}m_{H}^{2}-\delta^{(1)}Z_{hH}\delta^{(1)}m_{hH}^{2}.\end{split}$
(178c)

Similarly, the $\mathcal{CP}$-odd self-energies are

$\displaystyle\begin{split}\hat{\Sigma}^{(2)}_{AA}(p^{2})={}&\Sigma^{(2)}_{AA}(p^{2})+\delta^{(2)}Z_{AA}(p^{2}-m_{A}^{2})-\delta^{(2)}m_{A}^{2}\\\
&+\tfrac{1}{4}\left(\delta^{(1)}Z_{AG}\right)^{2}p^{2}-\delta^{(1)}Z_{AA}\delta^{(1)}m_{A}^{2}-\delta^{(1)}Z_{AG}\delta^{(1)}m_{AG}^{2},\end{split}$
(179a)
$\displaystyle\begin{split}\hat{\Sigma}^{(2)}_{AG}(p^{2})={}&\Sigma^{(2)}_{AG}(p^{2})+\delta^{(2)}Z_{AG}\left(p^{2}-\frac{m_{A}^{2}}{2}\right)-\delta^{(2)}m_{AG}^{2}\\\
&+\tfrac{1}{4}\delta^{(1)}Z_{AG}\delta^{(1)}Z_{AA}(p^{2}-m_{A}^{2})+\tfrac{1}{4}\delta^{(1)}Z_{AG}\delta^{(1)}Z_{GG}\,p^{2}\\\
&-\delta^{(1)}Z_{AG}\frac{\delta^{(1)}m_{A}^{2}+\delta^{(1)}m_{G}^{2}}{2}-\frac{\delta^{(1)}Z_{AA}+\delta^{(1)}Z_{GG}}{2}\delta^{(1)}m_{AG}^{2},\end{split}$
(179b)
$\displaystyle\begin{split}\hat{\Sigma}^{(2)}_{GG}(p^{2})={}&\Sigma^{(2)}_{GG}(p^{2})+\delta^{(2)}Z_{GG}\,p^{2}-\delta^{(2)}m_{G}^{2}\\\
&+\tfrac{1}{4}\left(\delta^{(1)}Z_{AG}\right)^{2}(p^{2}-m_{A}^{2})-\delta^{(1)}Z_{GG}\delta^{(1)}m_{G}^{2}-\delta^{(1)}Z_{AG}\delta^{(1)}m_{AG}^{2}.\end{split}$
(179c)

Finally, the $\mathcal{CP}$-mixing self-energies read

$\displaystyle\begin{split}\hat{\Sigma}^{(2)}_{hA}(p^{2})={}&\Sigma^{(2)}_{hA}(p^{2})-\delta^{(2)}m_{hA}^{2}\\\
&-\frac{\delta^{(1)}Z_{hh}+\delta^{(1)}Z_{AA}}{2}\delta^{(1)}m_{hA}^{2}-\tfrac{1}{2}Z_{hH}m_{HA}^{2}-\tfrac{1}{2}Z_{AG}m_{hG}^{2},\end{split}$
(180a)
$\displaystyle\begin{split}\hat{\Sigma}^{(2)}_{hG}(p^{2})={}&\Sigma^{(2)}_{hG}(p^{2})-\delta^{(2)}m_{hG}^{2}\\\
&-\frac{\delta^{(1)}Z_{hh}+\delta^{(1)}Z_{GG}}{2}\delta^{(1)}m_{hG}^{2}-\tfrac{1}{2}Z_{hH}m_{HG}^{2}-\tfrac{1}{2}Z_{AG}m_{hA}^{2},\end{split}$
(180b)
$\displaystyle\begin{split}\hat{\Sigma}^{(2)}_{HA}(p^{2})={}&\Sigma^{(2)}_{HA}(p^{2})-\delta^{(2)}m_{HA}^{2}\\\
&-\frac{\delta^{(1)}Z_{HH}+\delta^{(1)}Z_{AA}}{2}\delta^{(1)}m_{HA}^{2}-\tfrac{1}{2}Z_{hH}m_{hA}^{2}-\tfrac{1}{2}Z_{AG}m_{HG}^{2},\end{split}$
(180c)
$\displaystyle\begin{split}\hat{\Sigma}^{(2)}_{HG}(p^{2})={}&\Sigma^{(2)}_{HG}(p^{2})-\delta^{(2)}m_{HG}^{2}\\\
&-\frac{\delta^{(1)}Z_{HH}+\delta^{(1)}Z_{GG}}{2}\delta^{(1)}m_{HG}^{2}-\tfrac{1}{2}Z_{hH}m_{hG}^{2}-\tfrac{1}{2}Z_{AG}m_{HA}^{2}.\end{split}$
(180d)

The renormalised charged self-energies are

$\displaystyle\begin{split}\hat{\Sigma}^{(2)}_{H^{-}H^{+}}(p^{2})={}&\Sigma^{(2)}_{H^{-}H^{+}}(p^{2})+\delta^{(2)}Z_{H^{-}H^{+}}(p^{2}-m_{H^{\pm}}^{2})-\delta^{(2)}m_{H^{\pm}}^{2}\\\
&+\frac{\delta^{(1)}Z_{H^{-}G^{+}}\delta^{(1)}Z_{G^{-}H^{+}}}{4}p^{2}-\delta^{(1)}Z_{H^{-}H^{+}}\delta^{(1)}m_{H^{\pm}}^{2}\\\
&-\tfrac{1}{2}\delta^{(1)}Z_{H^{-}G^{+}}\delta^{(1)}m_{G^{-}H^{+}}^{2}-\tfrac{1}{2}\delta^{(1)}Z_{G^{-}H^{+}}\delta^{(1)}m_{H^{-}G^{+}}^{2},\end{split}$
(181a)
$\displaystyle\begin{split}\hat{\Sigma}^{(2)}_{H^{-}G^{+}}(p^{2})={}&\Sigma^{(2)}_{H^{-}G^{+}}(p^{2})+\delta^{(2)}Z_{H^{-}G^{+}}\left(p^{2}-\frac{m_{H^{\pm}}^{2}}{2}\right)-\delta^{(2)}m_{H^{-}G^{+}}^{2}\\\
&+\tfrac{1}{4}\delta^{(1)}Z_{H^{-}G^{+}}\delta^{(1)}Z_{H^{-}H^{+}}(p^{2}-m_{H^{\pm}}^{2})\\\
&+\tfrac{1}{4}\delta^{(1)}Z_{H^{-}G^{+}}\delta^{(1)}Z_{G^{-}G^{+}}\,p^{2}\\\
&-\delta^{(1)}Z_{H^{-}G^{+}}\frac{\delta^{(1)}m_{H^{\pm}}^{2}+\delta^{(1)}m_{G^{\pm}}^{2}}{2}\\\
&-\frac{\delta^{(1)}Z_{H^{-}H^{+}}+\delta^{(1)}Z_{G^{-}G^{+}}}{2}\delta^{(1)}m_{H^{-}G^{+}}^{2},\end{split}$
(181b)
$\displaystyle\begin{split}\hat{\Sigma}^{(2)}_{G^{-}H^{+}}(p^{2})={}&\Sigma^{(2)}_{G^{-}H^{+}}(p^{2})+\delta^{(2)}Z_{G^{-}H^{+}}\left(p^{2}-\frac{m_{H^{\pm}}^{2}}{2}\right)-\delta^{(2)}m_{G^{-}H^{+}}^{2}\\\
&+\tfrac{1}{4}\delta^{(1)}Z_{G^{-}H^{+}}\delta^{(1)}Z_{H^{-}H^{+}}(p^{2}-m_{H^{\pm}}^{2})\\\
&+\tfrac{1}{4}\delta^{(1)}Z_{G^{-}H^{+}}\delta^{(1)}Z_{G^{-}G^{+}}\,p^{2}\\\
&-\delta^{(1)}Z_{G^{-}H^{+}}\frac{\delta^{(1)}m_{H^{\pm}}^{2}+\delta^{(1)}m_{G^{\pm}}^{2}}{2}\\\
&-\frac{\delta^{(1)}Z_{H^{-}H^{+}}+\delta^{(1)}Z_{G^{-}G^{+}}}{2}\delta^{(1)}m_{G^{-}H^{+}}^{2},\end{split}$
(181c)
$\displaystyle\begin{split}\hat{\Sigma}^{(2)}_{G^{-}G^{+}}(p^{2})={}&\Sigma^{(2)}_{G^{-}G^{+}}(p^{2})+\delta^{(2)}Z_{G^{-}G^{+}}\,p^{2}-\delta^{(2)}m_{G^{\pm}}^{2}\\\
&+\frac{\delta^{(1)}Z_{H^{-}G^{+}}\delta^{(1)}Z_{G^{-}H^{+}}}{4}(p^{2}-m_{H^{\pm}}^{2})-\delta^{(1)}Z_{G^{-}G^{+}}\delta^{(1)}m_{G^{\pm}}^{2}\\\
&-\tfrac{1}{2}\delta^{(1)}Z_{H^{-}G^{+}}\delta^{(1)}m_{G^{-}H^{+}}^{2}-\tfrac{1}{2}\delta^{(1)}Z_{G^{-}H^{+}}\delta^{(1)}m_{H^{-}G^{+}}^{2}.\end{split}$
(181d)

The renormalised transverse parts of the two-loop gauge boson self-energies
are

$\displaystyle\hat{\Sigma}^{T,(2)}_{\gamma\gamma}(p^{2})={}$
$\displaystyle\Sigma^{T,(2)}_{\gamma\gamma}(p^{2})+\delta^{(2)}Z_{\gamma\gamma}\,p^{2}+\tfrac{1}{4}\left(\delta^{(1)}Z_{Z\gamma}\right)^{2}(p^{2}-M_{Z}^{2}),$
(182a) $\displaystyle\begin{split}\hat{\Sigma}^{T,(2)}_{\gamma
Z}(p^{2})={}&\Sigma^{T,(2)}_{\gamma
Z}(p^{2})+\tfrac{1}{2}\left(\delta^{(2)}Z_{\gamma
Z}+\tfrac{1}{2}\delta^{(1)}Z_{\gamma
Z}\delta^{(1)}Z_{\gamma\gamma}\right)p^{2}\\\
&+\tfrac{1}{2}\left(\delta^{(2)}Z_{Z\gamma}+\tfrac{1}{2}\delta^{(1)}Z_{Z\gamma}\delta^{(1)}Z_{ZZ}\right)(p^{2}-M_{Z}^{2})\\\
&-\tfrac{1}{2}\delta^{(1)}Z_{Z\gamma}\delta^{(1)}M_{Z}^{2}\\\
={}&\hat{\Sigma}^{T,(2)}_{Z\gamma}(p^{2}),\end{split}$ (182b)
$\displaystyle\begin{split}\hat{\Sigma}^{T,(2)}_{ZZ}(p^{2})={}&\Sigma^{T,(2)}_{ZZ}(p^{2})+\delta^{(2)}Z_{ZZ}(p^{2}-M_{Z}^{2})-\delta^{(2)}M_{Z}^{2}\\\
&+\tfrac{1}{4}\left(\delta^{(1)}Z_{\gamma
Z}\right)^{2}p^{2}-\delta^{(1)}Z_{ZZ}\delta^{(1)}M_{Z}^{2},\end{split}$ (182c)
$\displaystyle\begin{split}\hat{\Sigma}^{T,(2)}_{W^{-}W^{+}}(p^{2})={}&\Sigma^{T,(2)}_{W^{-}W^{+}}(p^{2})+\delta^{(2)}Z_{WW}(p^{2}-M_{W}^{2})-\delta^{(2)}M_{W}^{2}\\\
&-\delta^{(1)}Z_{WW}\delta^{(1)}M_{W}^{2}\\\
={}&\hat{\Sigma}^{T,(2)}_{W^{+}W^{-}}(p^{2}).\end{split}$ (182d)

At the two-loop level, the renormalised transverse part of the photon self-
energy receives a non-vanishing contribution at $p^{2}=0$ from the mixing with
the $Z$ boson. Another non-vanishing contribution stems from the sub-loop part
of the unrenormalised self-energy. To demonstrate this, we write the
unrenormalised self-energy as

$\Sigma^{T,(2)}_{\gamma\gamma}(p^{2})=\widetilde{\Sigma}^{T,(2)}_{\gamma\gamma}(p^{2})+\delta^{(1)}Z_{\gamma\gamma}\Sigma^{T,(1)}_{\gamma\gamma}(p^{2})+\delta^{(1)}Z_{Z\gamma}\Sigma^{T,(1)}_{\gamma
Z}(p^{2}),$ (183)

where $\widetilde{\Sigma}^{T,(2)}_{\gamma\gamma}(p^{2})$ does not contain any
field renormalisation constants. While the second term on the right-hand side
vanishes at zero momentum due to a Slavnov-Taylor identity, the third term
will usually give a non-vanishing contribution.121212For the set of
contributions considered in this paper, $\Sigma^{T,(1)}_{\gamma Z}(0)=0$. The
third term of Eq. (182a) and the third term of Eq. (183) will drop out once
the effective two-loop self-energy, which we defined in Eq. (49a), is
considered.

The renormalised two-loop self-energy vanishes at zero momentum if an on-shell
renormalisation is chosen for $\delta^{(1)}Z_{Z\gamma}$. In our calculation,
the on-shell condition leads to $\delta Z_{Z\gamma}=0$.

The two-loop Higgs–vector mixing self-energies

$\displaystyle\begin{split}\hat{\Sigma}^{L,(2)}_{AZ}(p^{2})={}&\Sigma^{L,(2)}_{AZ}(p^{2})-\text{i}M_{Z}\Big{(}c_{\beta}^{2}\delta^{(2)}t_{\beta}-c_{\beta}^{3}s_{\beta}(\delta^{(1)}t_{\beta})^{2}\\\
&+\tfrac{1}{2}\delta^{(2)}Z_{AG}+\tfrac{1}{2}c_{\beta}^{2}\delta^{(1)}t_{\beta}\delta^{(1)}Z_{AA}\Big{)}\\\
&-\frac{\text{i}M_{Z}}{2}\left(c_{\beta}^{2}\delta^{(1)}t_{\beta}+\tfrac{1}{2}\delta^{(1)}Z_{AG}\right)\left(\frac{\delta^{(1)}M_{Z}^{2}}{M_{Z}^{2}}+\delta^{(1)}Z_{ZZ}\right)\\\
={}&-\hat{\Sigma}^{L,(2)}_{ZA}(p^{2}),\end{split}$ (184a)
$\displaystyle\begin{split}\hat{\Sigma}^{L,(2)}_{H^{-}W^{+}}(p^{2})={}&\Sigma^{L,(2)}_{H^{-}W^{+}}(p^{2})+M_{W}\Big{(}c_{\beta}^{2}\delta^{(2)}t_{\beta}-c_{\beta}^{3}s_{\beta}(\delta^{(1)}t_{\beta})^{2}\\\
&+\tfrac{1}{2}\delta^{(2)}Z_{H^{-}G^{+}}+\tfrac{1}{2}c_{\beta}^{2}\delta^{(1)}t_{\beta}\delta^{(1)}Z_{H^{-}H^{+}}\Big{)}\\\
&+\frac{M_{W}}{2}\left(c_{\beta}^{2}\delta^{(1)}t_{\beta}+\tfrac{1}{2}\delta^{(1)}Z_{H^{-}G^{+}}\right)\left(\frac{\delta^{(1)}M_{W}^{2}}{M_{W}^{2}}+\delta^{(1)}Z_{WW}\right)\\\
={}&-\hat{\Sigma}^{L,(2)}_{W^{+}H^{-}}(p^{2})\end{split}$ (184b)

are used in some schemes to determine the two-loop counterterm for
$t_{\beta}$, see Sect. 2.5.

###  C.4 Two-loop counterterms

The two-loop mass counterterms are given by

$\displaystyle\begin{split}\delta^{(2)}m_{h}^{2}={}&\delta^{(2)}m_{A}^{2}c_{\alpha-\beta}^{2}+\delta^{(2)}M_{Z}^{2}s_{\alpha+\beta}^{2}\\\
&+\delta^{(2)}t_{\beta}c_{\beta}^{2}\big{(}m_{A}^{2}s_{2(\alpha-\beta)}+M_{Z}^{2}s_{2(\alpha+\beta)}\big{)}\\\
&+\delta^{(1)}t_{\beta}c_{\beta}^{2}\big{(}\delta^{(1)}m_{A}^{2}s_{2(\alpha-\beta)}+\delta^{(1)}M_{Z}^{2}s_{2(\alpha+\beta)}\big{)}\\\
&+\tfrac{1}{2}\big{(}\delta^{(1)}t_{\beta}\big{)}^{2}c_{\beta}^{3}\Big{[}m_{A}^{2}s_{\alpha-\beta}(3s_{\alpha-2\beta}-s_{\alpha})+2M_{Z}^{2}c_{2\alpha+3\beta}\Big{]}\\\
&+\frac{es_{\alpha-\beta}}{2M_{W}s_{\text{w}}}\Big{[}\big{(}\delta^{(2)}T_{h}+\delta^{(1)}Z_{\text{w}}\delta^{(1)}T_{h}\big{)}(1+c_{\alpha-\beta}^{2})\\\
&\hphantom{{}+\frac{es_{\alpha-\beta}}{2M_{W}s_{\text{w}}}\Big{[}{}}+\big{(}\delta^{(2)}T_{H}+\delta^{(1)}Z_{\text{w}}\delta^{(1)}T_{H}\big{)}c_{\alpha-\beta}s_{\alpha-\beta}\\\
&\hphantom{{}+\frac{es_{\alpha-\beta}}{2M_{W}s_{\text{w}}}\Big{[}{}}+\delta^{(1)}t_{\beta}c_{\beta}^{2}\big{(}\delta^{(1)}T_{h}c_{\alpha-\beta}+\delta^{(1)}T_{H}s_{\alpha-\beta}\big{)}s_{\alpha-\beta}\Big{]},\end{split}$
(185a)
$\displaystyle\begin{split}\delta^{(2)}m_{hH}^{2}={}&\delta^{(2)}m_{A}^{2}c_{\alpha-\beta}s_{\alpha-\beta}-\delta^{(2)}M_{Z}^{2}c_{\alpha+\beta}s_{\alpha+\beta}\\\
&-\delta^{(2)}t_{\beta}c_{\beta}^{2}\big{(}m_{A}^{2}c_{2(\alpha-\beta)}+M_{Z}^{2}c_{2(\alpha+\beta)}\big{)}\\\
&-\delta^{(1)}t_{\beta}c_{\beta}^{2}\big{(}\delta^{(1)}m_{A}^{2}c_{2(\alpha-\beta)}+\delta^{(1)}M_{Z}^{2}c_{2(\alpha+\beta)}\big{)}\\\
&-\tfrac{1}{2}\big{(}\delta^{(1)}t_{\beta}\big{)}^{2}c_{\beta}^{3}\Big{[}m_{A}^{2}\frac{c_{\alpha-\beta}(3s_{\alpha-2\beta}-s_{\alpha})+s_{\alpha-\beta}(3c_{\alpha-2\beta}-c_{\alpha})}{2}\\\
&\hphantom{{}-\tfrac{1}{4}\big{(}\delta^{(1)}t_{\beta}\big{)}^{2}c_{\beta}^{3}\Big{[}{}}-2M_{Z}^{2}s_{2\alpha+3\beta}\Big{]}\\\
&-\frac{e}{2M_{W}s_{\text{w}}}\Big{[}\big{(}\delta^{(2)}T_{h}+\delta^{(1)}Z_{\text{w}}\delta^{(1)}T_{h}\big{)}c_{\alpha-\beta}^{3}\\\
&\hphantom{{}-\frac{e}{2M_{W}s_{\text{w}}}\Big{[}{}}-\big{(}\delta^{(2)}T_{H}+\delta^{(1)}Z_{\text{w}}\delta^{(1)}T_{H}\big{)}s_{\alpha-\beta}^{3}\\\
&\hphantom{{}-\frac{e}{2M_{W}s_{\text{w}}}\Big{[}{}}+\delta^{(1)}t_{\beta}c_{\beta}^{2}\big{(}\delta^{(1)}T_{h}c_{\alpha-\beta}+\delta^{(1)}T_{H}s_{\alpha-\beta}\big{)}c_{\alpha-\beta}s_{\alpha-\beta}\Big{]},\end{split}$
(185b)
$\displaystyle\begin{split}\delta^{(2)}m_{H}^{2}={}&\delta^{(2)}m_{A}^{2}s_{\alpha-\beta}^{2}+\delta^{(2)}M_{Z}^{2}c_{\alpha+\beta}^{2}\\\
&-\delta^{(2)}t_{\beta}c_{\beta}^{2}\big{(}m_{A}^{2}s_{2(\alpha-\beta)}+M_{Z}^{2}s_{2(\alpha+\beta)}\big{)}\\\
&-\delta^{(1)}t_{\beta}c_{\beta}^{2}\big{(}\delta^{(1)}m_{A}^{2}s_{2(\alpha-\beta)}+\delta^{(1)}M_{Z}^{2}s_{2(\alpha+\beta)}\big{)}\\\
&+\tfrac{1}{2}\big{(}\delta^{(1)}t_{\beta}\big{)}^{2}c_{\beta}^{3}\Big{[}m_{A}^{2}c_{\alpha-\beta}(3c_{\alpha-2\beta}-c_{\alpha})-2M_{Z}^{2}c_{2\alpha+3\beta}\Big{]}\\\
&-\frac{ec_{\alpha-\beta}}{2M_{W}s_{\text{w}}}\Big{[}\big{(}\delta^{(2)}T_{h}+\delta^{(1)}Z_{\text{w}}\delta^{(1)}T_{h}\big{)}c_{\alpha-\beta}s_{\alpha-\beta}\\\
&\hphantom{{}-\frac{ec_{\alpha-\beta}}{2M_{W}s_{\text{w}}}\Big{[}{}}+\big{(}\delta^{(2)}T_{H}+\delta^{(1)}Z_{\text{w}}\delta^{(1)}T_{H}\big{)}(1+s_{\alpha-\beta}^{2})\\\
&\hphantom{{}-\frac{ec_{\alpha-\beta}}{2M_{W}s_{\text{w}}}\Big{[}{}}-\delta^{(1)}t_{\beta}c_{\beta}^{2}\big{(}\delta^{(1)}T_{h}c_{\alpha-\beta}+\delta^{(1)}T_{H}s_{\alpha-\beta}\big{)}c_{\alpha-\beta}\Big{]},\end{split}$
(185c)
$\displaystyle\begin{split}\delta^{(2)}m_{AG}^{2}={}&-\delta^{(2)}t_{\beta}c_{\beta}^{2}m_{A}^{2}-\delta^{(1)}t_{\beta}c_{\beta}^{2}\delta^{(1)}m_{A}^{2}+\big{(}\delta^{(1)}t_{\beta}\big{)}^{2}c_{\beta}^{3}s_{\beta}m_{A}^{2}\\\
&-\frac{e}{2M_{W}s_{\text{w}}}\Big{[}\big{(}\delta^{(2)}T_{h}+\delta^{(1)}Z_{\text{w}}\delta^{(1)}T_{h}\big{)}c_{\alpha-\beta}\\\
&\hphantom{{}-\frac{e}{2M_{W}s_{\text{w}}}\Big{[}{}}+\big{(}\delta^{(2)}T_{H}+\delta^{(1)}Z_{\text{w}}\delta^{(1)}T_{H}\big{)}s_{\alpha-\beta}\Big{]},\end{split}$
(185d)
$\displaystyle\begin{split}\delta^{(2)}m_{G}^{2}={}&\big{(}\delta^{(1)}t_{\beta}\big{)}^{2}c_{\beta}^{4}m_{A}^{2}\\\
&+\frac{e}{2M_{W}s_{\text{w}}}\Big{[}\big{(}\delta^{(2)}T_{h}+\delta^{(1)}Z_{\text{w}}\delta^{(1)}T_{h}\big{)}s_{\alpha-\beta}\\\
&\hphantom{{}+\frac{e}{2M_{W}s_{\text{w}}}\Big{[}{}}-\big{(}\delta^{(2)}T_{H}+\delta^{(1)}Z_{\text{w}}\delta^{(1)}T_{H}\big{)}c_{\alpha-\beta}\\\
&\hphantom{{}+\frac{e}{2M_{W}s_{\text{w}}}\Big{[}{}}+\delta^{(1)}t_{\beta}c_{\beta}^{2}\big{(}\delta^{(1)}T_{h}c_{\alpha-\beta}+\delta^{(1)}T_{H}s_{\alpha-\beta}\big{)}\Big{]},\end{split}$
(185e) $\displaystyle\delta^{(2)}m_{hA}^{2}={}$
$\displaystyle\frac{es_{\alpha-\beta}}{2M_{W}s_{\text{w}}}\big{(}\delta^{(2)}T_{A}+\delta^{(1)}Z_{\text{w}}\delta^{(1)}T_{A}\big{)},$
(185f) $\displaystyle\delta^{(2)}m_{HA}^{2}={}$
$\displaystyle-\frac{ec_{\alpha-\beta}}{2M_{W}s_{\text{w}}}\big{(}\delta^{(2)}T_{A}+\delta^{(1)}Z_{\text{w}}\delta^{(1)}T_{A}\big{)},$
(185g) $\displaystyle\delta^{(2)}m_{hG}^{2}={}$
$\displaystyle\frac{ec_{\alpha-\beta}}{2M_{W}s_{\text{w}}}\big{(}\delta^{(2)}T_{A}+\delta^{(1)}Z_{\text{w}}\delta^{(1)}T_{A}\big{)}=-\delta^{(2)}m_{HA}^{2},$
(185h) $\displaystyle\delta^{(2)}m_{HG}^{2}={}$
$\displaystyle\frac{es_{\alpha-\beta}}{2M_{W}s_{\text{w}}}\big{(}\delta^{(2)}T_{A}+\delta^{(1)}Z_{\text{w}}\delta^{(1)}T_{A}\big{)}=\delta^{(2)}m_{hA}^{2},$
(185i)
$\displaystyle\begin{split}\delta^{(2)}m_{H^{-}G^{+}}^{2}={}&-\delta^{(2)}t_{\beta}c_{\beta}^{2}m_{H^{\pm}}^{2}-\delta^{(1)}t_{\beta}c_{\beta}^{2}\delta^{(1)}m_{H^{\pm}}^{2}+\big{(}\delta^{(1)}t_{\beta}\big{)}^{2}c_{\beta}^{3}s_{\beta}m_{H^{\pm}}^{2}\\\
&-\frac{e}{2M_{W}s_{\text{w}}}\Big{[}\big{(}\delta^{(2)}T_{h}+\delta^{(1)}Z_{\text{w}}\delta^{(1)}T_{h}\big{)}c_{\alpha-\beta}\\\
&\hphantom{{}-\frac{e}{2M_{W}s_{\text{w}}}\Big{[}{}}+\big{(}\delta^{(2)}T_{H}+\delta^{(1)}Z_{\text{w}}\delta^{(1)}T_{H}\big{)}s_{\alpha-\beta}\\\
&\hphantom{{}-\frac{e}{2M_{W}s_{\text{w}}}\Big{[}{}}-\text{i}\big{(}\delta^{(2)}T_{A}+\delta^{(1)}Z_{\text{w}}\delta^{(1)}T_{A}\big{)}\Big{]},\end{split}$
(185j)
$\displaystyle\begin{split}\delta^{(2)}m_{G^{-}H^{+}}^{2}={}&-\delta^{(2)}t_{\beta}c_{\beta}^{2}m_{H^{\pm}}^{2}-\delta^{(1)}t_{\beta}c_{\beta}^{2}\delta^{(1)}m_{H^{\pm}}^{2}+\big{(}\delta^{(1)}t_{\beta}\big{)}^{2}c_{\beta}^{3}s_{\beta}m_{H^{\pm}}^{2}\\\
&-\frac{e}{2M_{W}s_{\text{w}}}\Big{[}\big{(}\delta^{(2)}T_{h}+\delta^{(1)}Z_{\text{w}}\delta^{(1)}T_{h}\big{)}c_{\alpha-\beta}\\\
&\hphantom{{}-\frac{e}{2M_{W}s_{\text{w}}}\Big{[}{}}+\big{(}\delta^{(2)}T_{H}+\delta^{(1)}Z_{\text{w}}\delta^{(1)}T_{H}\big{)}s_{\alpha-\beta}\\\
&\hphantom{{}-\frac{e}{2M_{W}s_{\text{w}}}\Big{[}{}}+\text{i}\big{(}\delta^{(2)}T_{A}+\delta^{(1)}Z_{\text{w}}\delta^{(1)}T_{A}\big{)}\Big{]}\\\
={}&\big{(}\delta^{(2)}m_{H^{-}G^{+}}^{2}\big{)}^{*},\end{split}$ (185k)
$\displaystyle\begin{split}\delta^{(2)}m_{G^{\pm}}^{2}={}&\big{(}\delta^{(1)}t_{\beta}\big{)}^{2}c_{\beta}^{4}m_{H^{\pm}}^{2}\\\
&+\frac{e}{2M_{W}s_{\text{w}}}\Big{[}\big{(}\delta^{(2)}T_{h}+\delta^{(1)}Z_{\text{w}}\delta^{(1)}T_{h}\big{)}s_{\alpha-\beta}\\\
&\hphantom{{}+\frac{e}{2M_{W}s_{\text{w}}}\Big{[}{}}-\big{(}\delta^{(2)}T_{H}+\delta^{(1)}Z_{\text{w}}\delta^{(1)}T_{H}\big{)}c_{\alpha-\beta}\\\
&\hphantom{{}+\frac{e}{2M_{W}s_{\text{w}}}\Big{[}{}}+\delta^{(1)}t_{\beta}c_{\beta}^{2}\big{(}\delta^{(1)}T_{h}c_{\alpha-\beta}+\delta^{(1)}T_{H}s_{\alpha-\beta}\big{)}\Big{]}.\end{split}$
(185l)

The two-loop field counterterms read

$\displaystyle\begin{split}\delta^{(2)}Z_{hh}={}&s_{\alpha}^{2}\left(\delta^{(2)}Z_{\mathcal{H}_{1}}-\frac{1}{4}\left(\delta^{(1)}Z_{\mathcal{H}_{1}}\right)^{2}\right)+c_{\alpha}^{2}\left(\delta^{(2)}Z_{\mathcal{H}_{2}}-\frac{1}{4}\left(\delta^{(1)}Z_{\mathcal{H}_{2}}\right)^{2}\right)\\\
&+\frac{1}{4}\left(\delta^{(1)}Z_{hh}\right)^{2},\end{split}$ (186a)
$\displaystyle\delta^{(2)}Z_{hH}={}$ $\displaystyle
s_{\alpha}c_{\alpha}\left[\delta^{(2)}Z_{\mathcal{H}_{2}}-\frac{1}{4}\left(\delta^{(1)}Z_{\mathcal{H}_{2}}\right)^{2}-\delta^{(2)}Z_{\mathcal{H}_{1}}+\frac{1}{4}\left(\delta^{(1)}Z_{\mathcal{H}_{1}}\right)^{2}\right],$
(186b)
$\displaystyle\begin{split}\delta^{(2)}Z_{HH}={}&c_{\alpha}^{2}\left(\delta^{(2)}Z_{\mathcal{H}_{1}}-\frac{1}{4}\left(\delta^{(1)}Z_{\mathcal{H}_{1}}\right)^{2}\right)+s_{\alpha}^{2}\left(\delta^{(2)}Z_{\mathcal{H}_{2}}-\frac{1}{4}\left(\delta^{(1)}Z_{\mathcal{H}_{2}}\right)^{2}\right)\\\
&+\frac{1}{4}\left(\delta^{(1)}Z_{HH}\right)^{2},\end{split}$ (186c)
$\displaystyle\begin{split}\delta^{(2)}Z_{AA}={}&s_{\beta}^{2}\left(\delta^{(2)}Z_{\mathcal{H}_{1}}-\frac{1}{4}\left(\delta^{(1)}Z_{\mathcal{H}_{1}}\right)^{2}\right)+c_{\beta}^{2}\left(\delta^{(2)}Z_{\mathcal{H}_{2}}-\frac{1}{4}\left(\delta^{(1)}Z_{\mathcal{H}_{2}}\right)^{2}\right)\\\
&+\frac{1}{4}\left(\delta^{(1)}Z_{AA}\right)^{2},\end{split}$ (186d)
$\displaystyle\delta^{(2)}Z_{AG}={}$ $\displaystyle
s_{\beta}c_{\beta}\left[\delta^{(2)}Z_{\mathcal{H}_{2}}-\frac{1}{4}\left(\delta^{(1)}Z_{\mathcal{H}_{2}}\right)^{2}-\delta^{(2)}Z_{\mathcal{H}_{1}}+\frac{1}{4}\left(\delta^{(1)}Z_{\mathcal{H}_{1}}\right)^{2}\right],$
(186e)
$\displaystyle\begin{split}\delta^{(2)}Z_{GG}={}&c_{\beta}^{2}\left(\delta^{(2)}Z_{\mathcal{H}_{1}}-\frac{1}{4}\left(\delta^{(1)}Z_{\mathcal{H}_{1}}\right)^{2}\right)+s_{\beta}^{2}\left(\delta^{(2)}Z_{\mathcal{H}_{2}}-\frac{1}{4}\left(\delta^{(1)}Z_{\mathcal{H}_{2}}\right)^{2}\right)\\\
&+\frac{1}{4}\left(\delta^{(1)}Z_{GG}\right)^{2},\end{split}$ (186f)
$\displaystyle\begin{split}\delta^{(2)}Z_{H^{-}H^{+}}={}&s_{\beta}^{2}\left(\delta^{(2)}Z_{\mathcal{H}_{1}}-\frac{1}{4}\left(\delta^{(1)}Z_{\mathcal{H}_{1}}\right)^{2}\right)+c_{\beta}^{2}\left(\delta^{(2)}Z_{\mathcal{H}_{2}}-\frac{1}{4}\left(\delta^{(1)}Z_{\mathcal{H}_{2}}\right)^{2}\right)\\\
&+\frac{1}{4}\left(\delta^{(1)}Z_{H^{-}H^{+}}\right)^{2},\end{split}$ (186g)
$\displaystyle\delta^{(2)}Z_{H^{-}G^{+}}={}$ $\displaystyle
s_{\beta}c_{\beta}\left[\delta^{(2)}Z_{\mathcal{H}_{2}}-\frac{1}{4}\left(\delta^{(1)}Z_{\mathcal{H}_{2}}\right)^{2}-\delta^{(2)}Z_{\mathcal{H}_{1}}+\frac{1}{4}\left(\delta^{(1)}Z_{\mathcal{H}_{1}}\right)^{2}\right],$
(186h) $\displaystyle\delta^{(2)}Z_{G^{-}H^{+}}={}$ $\displaystyle
s_{\beta}c_{\beta}\left[\delta^{(2)}Z_{\mathcal{H}_{2}}-\frac{1}{4}\left(\delta^{(1)}Z_{\mathcal{H}_{2}}\right)^{2}-\delta^{(2)}Z_{\mathcal{H}_{1}}+\frac{1}{4}\left(\delta^{(1)}Z_{\mathcal{H}_{1}}\right)^{2}\right],$
(186i)
$\displaystyle\begin{split}\delta^{(2)}Z_{G^{-}G^{+}}={}&c_{\beta}^{2}\left(\delta^{(2)}Z_{\mathcal{H}_{1}}-\frac{1}{4}\left(\delta^{(1)}Z_{\mathcal{H}_{1}}\right)^{2}\right)+s_{\beta}^{2}\left(\delta^{(2)}Z_{\mathcal{H}_{2}}-\frac{1}{4}\left(\delta^{(1)}Z_{\mathcal{H}_{2}}\right)^{2}\right)\\\
&+\frac{1}{4}\left(\delta^{(1)}Z_{GG}\right)^{2}.\end{split}$ (186j)

##  Appendix D Slavnov-Taylor identities for scalar-vector mixing

Slavnov-Taylor (ST) identities are the generalisation of the abelian Ward-
Takahashi (WT) identities to non-abelian gauge theories. While WT identities
follow from gauge symmetry, ST identities are a consequence of the Becchi-
Rouet-Stora-Tyutin (BRST) symmetry, which is an extension of gauge symmetry
after gauge fixing. For the present discussion, we will not derive ST
identities from BRST invariance but simply check relations between the
relevant self-energies algebraically or numerically.

In Refs. [188, 189], MSSM Slavnov-Taylor identities for self-energies in the
$AGZ$ and the $H^{\pm}G^{\pm}W^{\pm}$ system are given. As was pointed out in
Ref. [164], the identities given in Refs. [188, 189] hold only in a linear
gauge and on-shell. Refs. [164, 190] give ST identities also for off-shell
momenta.

In our analysis, we only consider self-energy contributions of $\order{N_{c}}$
and in a general $R_{\xi}$ gauge. Therefore, no diagrams with electroweak
particles in the loops appear, and the self-energies are gauge-parameter
independent.

Our starting point are the equations

$\displaystyle\Sigma^{(1),\text{notad}}_{AG}(p^{2})-\text{i}\frac{p^{2}}{M_{Z}}\Sigma^{L,(1),\text{notad}}_{AZ}(p^{2})\overset{\order{N_{c}}}{=}{}$
$\displaystyle\frac{e}{2s_{\text{w}}M_{W}}\big{(}\Gamma^{(1)}_{h}c_{\alpha-\beta}+\Gamma^{(1)}_{H}s_{\alpha-\beta}\big{)},$
(187a)
$\displaystyle\Sigma^{(1),\text{notad}}_{H^{-}G^{+}}(p^{2})-\frac{p^{2}}{M_{W}}\Sigma^{L,(1),\text{notad}}_{H^{-}W^{+}}(p^{2})\overset{\order{N_{c}}}{=}{}$
$\displaystyle\frac{e}{2s_{\text{w}}M_{W}}\big{(}\Gamma^{(1)}_{h}c_{\alpha-\beta}+\Gamma^{(1)}_{H}s_{\alpha-\beta}-\text{i}\Gamma^{(1)}_{A}\big{)},$
(187b)
$\displaystyle\Sigma^{(1),\text{notad}}_{G^{-}H^{+}}(p^{2})-\frac{p^{2}}{M_{W}}\Sigma^{L,(1),\text{notad}}_{W^{-}H^{+}}(p^{2})\overset{\order{N_{c}}}{=}{}$
$\displaystyle\frac{e}{2s_{\text{w}}M_{W}}\big{(}\Gamma^{(1)}_{h}c_{\alpha-\beta}+\Gamma^{(1)}_{H}s_{\alpha-\beta}+\text{i}\Gamma^{(1)}_{A}\big{)},$
(187c)

which we have explicitly verified. The superscript ‘notad’ denotes that we do
not include tadpole contributions in the respective self-energy. Instead, the
tadpole contributions appear explicitly in the form of unrenormalised one-
point functions on the right-hand side of the equations.

Using on-shell definitions for the one-loop tadpole counterterms,
$\delta^{(1)}T_{i}=-\Gamma^{(1)}_{i}$, we can write

$\displaystyle\Sigma^{(1),\text{notad}}_{AG}(p^{2})-\text{i}\frac{p^{2}}{M_{Z}}\Sigma^{L,(1),\text{notad}}_{AZ}(p^{2})\overset{\order{N_{c}}}{=}{}$
$\displaystyle\delta^{(1)}m_{AG}^{2}+\delta^{(1)}t_{\beta}c_{\beta}^{2}m_{A}^{2},$
(188a)
$\displaystyle\Sigma^{(1),\text{notad}}_{H^{-}G^{+}}(p^{2})-\frac{p^{2}}{M_{W}}\Sigma^{L,(1),\text{notad}}_{H^{-}W^{+}}(p^{2})\overset{\order{N_{c}}}{=}{}$
$\displaystyle\delta^{(1)}m_{H^{-}G^{+}}^{2}+\delta^{(1)}t_{\beta}c_{\beta}^{2}m_{H^{\pm}}^{2},$
(188b)
$\displaystyle\Sigma^{(1),\text{notad}}_{G^{-}H^{+}}(p^{2})-\frac{p^{2}}{M_{W}}\Sigma^{L,(1),\text{notad}}_{W^{-}H^{+}}(p^{2})\overset{\order{N_{c}}}{=}{}$
$\displaystyle\delta^{(1)}m_{G^{-}H^{+}}^{2}+\delta^{(1)}t_{\beta}c_{\beta}^{2}m_{H^{\pm}}^{2},$
(188c)

where he have used the expressions for the one-loop Higgs mass counterterms
which were introduced in App. C.

For the renormalised self-energies (the definitions are given in Sect. C.1),
we arrive at the following off-shell Slavnov-Taylor identities

$\displaystyle\hat{\Sigma}^{(1)}_{AG}(p^{2})-\text{i}\frac{p^{2}}{M_{Z}}\hat{\Sigma}^{L,(1)}_{AZ}(p^{2})\overset{\order{N_{c}}}{=}{}$
$\displaystyle(p^{2}-m_{A}^{2})\big{(}\tfrac{1}{2}\delta^{(1)}Z_{AG}-\delta^{(1)}t_{\beta}c_{\beta}^{2}\big{)},$
(189a)
$\displaystyle\hat{\Sigma}^{(1)}_{H^{-}G^{+}}(p^{2})-\frac{p^{2}}{M_{W}}\hat{\Sigma}^{L,(1)}_{H^{-}W^{+}}(p^{2})\overset{\order{N_{c}}}{=}{}$
$\displaystyle(p^{2}-m_{H^{\pm}}^{2})\big{(}\tfrac{1}{2}\delta^{(1)}Z_{H^{-}G^{+}}-\delta^{(1)}t_{\beta}c_{\beta}^{2}\big{)},$
(189b)
$\displaystyle\hat{\Sigma}^{(1)}_{G^{-}H^{+}}(p^{2})-\frac{p^{2}}{M_{W}}\hat{\Sigma}^{L,(1)}_{W^{-}H^{+}}(p^{2})\overset{\order{N_{c}}}{=}{}$
$\displaystyle(p^{2}-m_{H^{\pm}}^{2})\big{(}\tfrac{1}{2}\delta^{(1)}Z_{G^{-}H^{+}}-\delta^{(1)}t_{\beta}c_{\beta}^{2}\big{)},$
(189c)

which are in agreement with the ones given in Ref. [164].

The right-hand side of Eqs. (189) vanishes in the ${\overline{\text{DR}}}$
scheme for any value of $p^{2}$, see Sect. 2.5. This is in agreement with Ref.
[190], which employs the ${\overline{\text{DR}}}$ version. As we allow for an
on-shell renormalisation of $t_{\beta}$ while keeping the field counterterms
defined in the minimal ${\overline{\text{DR}}}$ scheme, the right-hand side
will not vanish in general. The on-shell ST identities

$\displaystyle\hat{\Sigma}^{(1)}_{AG}(m_{A}^{2})-\text{i}\frac{m_{A}^{2}}{M_{Z}}\hat{\Sigma}^{L,(1)}_{AZ}(m_{A}^{2})={}$
$\displaystyle 0,$ (190a)
$\displaystyle\hat{\Sigma}^{(1)}_{H^{-}G^{+}}(m_{H^{\pm}}^{2})-\frac{m_{H^{\pm}}^{2}}{M_{W}}\hat{\Sigma}^{L,(1)}_{H^{-}W^{+}}(m_{H^{\pm}}^{2})={}$
$\displaystyle 0,$ (190b)
$\displaystyle\hat{\Sigma}^{(1)}_{G^{-}H^{+}}(m_{H^{\pm}}^{2})-\frac{m_{H^{\pm}}^{2}}{M_{W}}\hat{\Sigma}^{L,(1)}_{W^{-}H^{+}}(m_{H^{\pm}}^{2})={}$
$\displaystyle 0$ (190c)

hold independently of the renormalisation chosen for $t_{\beta}$ and the Higgs
field counterterms. In a linear gauge, they are also valid if terms of
$\order{N_{c}^{0}}$ are taken into account [164, 190].

##  Appendix E One-loop integrals

For the definitions of the one-loop integrals, we follow the conventions used
in Ref. [153]. Here, we provide additional details on the $B_{0}^{\prime}$
integral, which is used less often than the $A_{0}$ and $B_{0}$ integrals. It
is defined as

$B_{0}^{\prime}(p^{2},m_{1}^{2},m_{2}^{2})\equiv\frac{\partial}{\partial
m_{1}^{2}}B_{0}(p^{2},m_{1}^{2},m_{2}^{2}).$ (191)

To derive an expression for the $B_{0}^{\prime}$ integral, we insert a factor
$1=D^{-1}\frac{\partial q^{\mu}}{\partial q^{\mu}}$ into the definition of the
$B_{0}$ integral, and we integrate by parts:

$B_{0}(p^{2},m_{1}^{2},m_{2}^{2})=-\frac{C}{D}\int\differential[D]{q}q^{\mu}\frac{\partial}{\partial
q^{\mu}}\frac{1}{[q^{2}-m_{1}^{2}][(q+p)^{2}-m_{2}^{2}]},$ (192)

where we introduced the abbreviations

$\displaystyle
C=\frac{16\pi^{2}}{\text{i}}\frac{\mu_{\text{D}}^{2\varepsilon}}{(2\pi)^{D}},$
(193a) $\displaystyle D=4-2\varepsilon.$ (193b)

Solving the resulting set of equations for
$B_{0}^{\prime}(p^{2},m_{1}^{2},m_{2}^{2})$, we obtain

$\displaystyle B_{0}^{\prime}(p^{2},m_{1}^{2},m_{2}^{2})={}$
$\displaystyle\frac{-1}{\lambda(p^{2},m_{1}^{2},m_{2}^{2})}\bigg{[}(p^{2}-m_{1}^{2}+m_{2}^{2})(1-2\varepsilon)B_{0}(p^{2},m_{1}^{2},m_{2}^{2})$
(194)
$\displaystyle-2(1-\varepsilon)A_{0}(m_{2}^{2})-\frac{p^{2}-m_{1}^{2}-m_{2}^{2}}{m_{1}^{2}}(1-\varepsilon)A_{0}(m_{1}^{2})\bigg{]},$

where $\lambda$ is the Källén function,
$\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2ab-2ac-2bc$. From Eq. (194), a few special
cases are readily derived:

$\displaystyle B_{0}^{\prime}(p^{2},m^{2},m^{2})={}$
$\displaystyle\frac{(1-2\varepsilon)B_{0}(p^{2},m^{2},m^{2})-B_{0}(0,m^{2},m^{2})}{4m^{2}-p^{2}},$
(195a) $\displaystyle B_{0}^{\prime}(p^{2},m^{2},0)={}$
$\displaystyle\frac{(1-2\varepsilon)B_{0}(p^{2},m^{2},0)-B_{0}(0,m^{2},m^{2})}{m^{2}-p^{2}},$
(195b) $\displaystyle B_{0}^{\prime}(0,m^{2},m^{2})={}$
$\displaystyle\frac{-\varepsilon}{2m^{2}}B_{0}(0,m^{2},m^{2}).$ (195c)

As we explained above, for identical mass arguments we take the derivative
with respect to the first mass argument before setting the masses equal.

Setting $m_{1}^{2}=0$ and $m_{2}^{2}=m^{2}$ in Eq. (194), we obtain the
relation

$B_{0}(0,0,0)-(p^{2}-m^{2})B_{0}^{\prime}(p^{2},0,m^{2})=\frac{(p^{2}+m^{2})(1-2\varepsilon)B_{0}(p^{2},m^{2},0)-2(1-\varepsilon)A_{0}(m^{2})}{p^{2}-m^{2}}.$
(196)

$B_{0}(0,0,0)$ and $B_{0}^{\prime}(p^{2},0,m^{2})$ are both IR-divergent,
while the integrals on the right-hand side are IR-finite.

##  Appendix F Generation of plots

In this appendix, we explain how the different plots shown in Sec. 5 were
obtained. For any given point in parameter space, we always calculate the
Higgs boson mass square at the one- ($1L$) and two-loop order ($2L$) including
different generations of quarks and squarks:

  * •

Only the third generation quarks ($t$ and $b$) and squarks are included
($3g$).

  * •

Quarks and squarks of all generations are included ($ag$).

We perform these calculations in several different limits:

  * •

Full prediction at $\order{N_{c}}$/$\order{N_{c}^{2}}$ ($full$).

  * •

Gaugeless limit, $\alpha_{\text{em}}=0$ ($gl$). We numerically take this limit
by replacing $M_{Z}\to\widehat{M}_{Z}$ (where $\widehat{M}_{Z}$ is a dummy
variable), $M_{W}\to M_{W}(\widehat{M}_{Z}/M_{Z})$, and
$\alpha_{\text{em}}\to\alpha_{\text{em}}(\widehat{M}_{Z}/M_{Z})^{2}$, where
$\widehat{M}_{Z}\ll M_{Z}$. In scenarios 1 and 2 we use
$\widehat{M}_{Z}=M_{S}/1000$, in scenario 3
$\widehat{M}_{Z}=\max\\{|A_{q}|/1000,1\;\text{GeV}\\}$, and in scenario 4 we
set $\widehat{M}_{Z}=M_{S}/500$.

  * •

Limit of vanishing bottom mass, $m_{b}=0$ ($bl$). We employ this numerically
by replacing $m_{b}\to m_{b}/10=0.418\;\text{GeV}$.

  * •

$\alpha_{\text{em}}=0$ and $m_{b}=0$ ($gl+bl$). For this limit, we simply
combine the aforementioned prescriptions for the gaugeless limit and the limit
of vanishing bottom mass.

We use the symbol $(M_{h_{i}}^{2})^{l,g,c}$ for each of the different
predictions for the squared Higgs boson masses, where

$\displaystyle h_{i}\in{}$ $\displaystyle\\{h,H\\},$ (197) $\displaystyle
l\in{}$ $\displaystyle\\{1L,2L\\},$ $\displaystyle g\in{}$
$\displaystyle\\{3g,ag\\},$ $\displaystyle c\in{}$
$\displaystyle\\{full,gl,bl,gl+bl\\}.$

As the first and second generation are assumed to be massless, there is no
difference between working with the third or all generations in the gaugeless
limit:

$\displaystyle(M_{h_{i}}^{2})^{l,3g,gl}={}$
$\displaystyle(M_{h_{i}}^{2})^{l,ag,gl},$ (198a)
$\displaystyle(M_{h_{i}}^{2})^{l,3g,gl+bl}={}$
$\displaystyle(M_{h_{i}}^{2})^{l,ag,gl+bl}.$ (198b)

We have used these identities to validate our implementation of the gaugeless
limit. Nevertheless, we occasionally use gaugeless results with either one or
three generations. This leaves us with six predictions for the Higgs boson
mass at one-loop order1313133g/full, 3g/gl, 3g/bl, 3g/gl+bl, ag/full, and
ag/bl. and the same number of predictions for the two-loop masses per
parameter point. We combine them to obtain all contributions we are interested
in.

In the plots for the Higgs boson masses in Sec. 5 (these are Figs. 10, 11(a),
and11(b)), up to five curves are shown:

$\displaystyle\text{cyan, solid: }M_{h_{i}}={}$
$\displaystyle\sqrt{(M_{h_{i}}^{2})^{2L,ag,full}},$ (199a)
$\displaystyle\text{cyan, dashed: }M_{h_{i}}={}$
$\displaystyle\sqrt{(M_{h_{i}}^{2})^{2L,3g,full}-(M_{h_{i}}^{2})^{1L,3g,full}+(M_{h_{i}}^{2})^{1L,ag,full}},$
(199b) $\displaystyle\text{green, solid: }M_{h_{i}}={}$
$\displaystyle\sqrt{(M_{h_{i}}^{2})^{2L,ag,gl}-(M_{h_{i}}^{2})^{1L,3g,gl}+(M_{h_{i}}^{2})^{1L,ag,full}},$
(199c) $\displaystyle\text{green, dashed: }M_{h_{i}}={}$
$\displaystyle\sqrt{(M_{h_{i}}^{2})^{2L,3g,gl+bl}-(M_{h_{i}}^{2})^{1L,3g,gl+bl}+(M_{h_{i}}^{2})^{1L,ag,full}},$
(199d) $\displaystyle\text{black, solid: }M_{h_{i}}={}$
$\displaystyle\sqrt{(M_{h_{i}}^{2})^{1L,ag,full}}.$ (199e)

When we make a prediction for the two-loop mass in any given limit and for any
number of fermion generations, the same properties are also applied to the
tree-level and one-loop contribution. As we are interested in estimating the
size of the newly calculated two-loop corrections, we subtract the appropriate
one-loop prediction from the two-loop value and add the full $\order{N_{c}}$
one-loop result, which includes all generations of quarks and squarks.

In the plots for the two-loop contributions to the Higgs boson mass (these are
Figs. 5, 6, 7, 8, 9, and 12), twelve combinations are possible:

$\displaystyle\text{cyan, solid: }\Delta^{(2)}M_{h_{i}}={}$
$\displaystyle\sqrt{(M_{h_{i}}^{2})^{2L,ag,full}}-\sqrt{(M_{h_{i}}^{2})^{1L,ag,full}},$
(200a) $\displaystyle\begin{split}\text{cyan, dashed:
}\Delta^{(2)}M_{h_{i}}={}&\sqrt{(M_{h_{i}}^{2})^{2L,3g,full}-(M_{h_{i}}^{2})^{1L,3g,full}+(M_{h_{i}}^{2})^{1L,ag,full}}\\\
&-\sqrt{(M_{h_{i}}^{2})^{1L,ag,full}},\end{split}$ (200b)
$\displaystyle\begin{split}\text{magenta, solid:
}\Delta^{(2)}M_{h_{i}}={}&\sqrt{(M_{h_{i}}^{2})^{2L,ag,bl}-(M_{h_{i}}^{2})^{1L,ag,bl}+(M_{h_{i}}^{2})^{1L,ag,full}}\\\
&-\sqrt{(M_{h_{i}}^{2})^{1L,ag,full}},\end{split}$ (200c)
$\displaystyle\begin{split}\text{magenta, dashed:
}\Delta^{(2)}M_{h_{i}}={}&\sqrt{(M_{h_{i}}^{2})^{2L,3g,bl}-(M_{h_{i}}^{2})^{1L,3g,bl}+(M_{h_{i}}^{2})^{1L,ag,full}}\\\
&-\sqrt{(M_{h_{i}}^{2})^{1L,ag,full}},\end{split}$ (200d)
$\displaystyle\begin{split}\text{green, solid:
}\Delta^{(2)}M_{h_{i}}={}&\sqrt{(M_{h_{i}}^{2})^{2L,ag,gl}-(M_{h_{i}}^{2})^{1L,ag,gl}+(M_{h_{i}}^{2})^{1L,ag,full}}\\\
&-\sqrt{(M_{h_{i}}^{2})^{1L,ag,full}},\end{split}$ (200e)
$\displaystyle\begin{split}\text{green, dashed:
}\Delta^{(2)}M_{h_{i}}={}&\sqrt{(M_{h_{i}}^{2})^{2L,3g,gl+bl}-(M_{h_{i}}^{2})^{1L,3g,gl+bl}+(M_{h_{i}}^{2})^{1L,ag,full}}\\\
&-\sqrt{(M_{h_{i}}^{2})^{1L,ag,full}},\end{split}$ (200f)
$\displaystyle\begin{split}\text{green, dotted:
}\Delta^{(2)}M_{h_{i}}={}&\sqrt{(M_{h_{i}}^{2})^{2L,ag,gl}-(M_{h_{i}}^{2})^{1L,ag,gl}+(M_{h_{i}}^{2})^{1L,ag,full}}\\\
&-\sqrt{(M_{h_{i}}^{2})^{2L,3g,gl+bl}-(M_{h_{i}}^{2})^{1L,3g,gl+bl}+(M_{h_{i}}^{2})^{1L,ag,full}},\end{split}$
(200g) $\displaystyle\begin{split}\text{red, solid:
}\Delta^{(2)}M_{h_{i}}={}&\sqrt{(M_{h_{i}}^{2})^{2L,ag,full}}\\\
&-\sqrt{(M_{h_{i}}^{2})^{2L,ag,gl}-(M_{h_{i}}^{2})^{1L,ag,gl}+(M_{h_{i}}^{2})^{1L,ag,full}},\end{split}$
(200h) $\displaystyle\begin{split}\text{red, dashed:
}\Delta^{(2)}M_{h_{i}}={}&\sqrt{(M_{h_{i}}^{2})^{2L,3g,full}-(M_{h_{i}}^{2})^{1L,3g,full}+(M_{h_{i}}^{2})^{1L,ag,full}}\\\
&-\sqrt{(M_{h_{i}}^{2})^{2L,3g,gl}-(M_{h_{i}}^{2})^{1L,3g,gl}+(M_{h_{i}}^{2})^{1L,ag,full}},\end{split}$
(200i) $\displaystyle\begin{split}\text{red, dotted:
}\Delta^{(2)}M_{h_{i}}={}&\sqrt{(M_{h_{i}}^{2})^{2L,ag,full}}\\\
&-\sqrt{(M_{h_{i}}^{2})^{2L,3g,full}-(M_{h_{i}}^{2})^{1L,3g,full}+(M_{h_{i}}^{2})^{1L,ag,full}},\end{split}$
(200j) $\displaystyle\begin{split}\text{orange, solid:
}\Delta^{(2)}M_{h_{i}}={}&\sqrt{(M_{h_{i}}^{2})^{2L,ag,full}}\\\
&-\sqrt{(M_{h_{i}}^{2})^{2L,ag,bl}-(M_{h_{i}}^{2})^{1L,ag,bl}+(M_{h_{i}}^{2})^{1L,ag,full}},\end{split}$
(200k) $\displaystyle\begin{split}\text{orange, dashed:
}\Delta^{(2)}M_{h_{i}}={}&\sqrt{(M_{h_{i}}^{2})^{2L,3g,full}}\\\
&-\sqrt{(M_{h_{i}}^{2})^{2L,3g,bl}-(M_{h_{i}}^{2})^{1L,3g,bl}+(M_{h_{i}}^{2})^{1L,ag,full}}.\end{split}$
(200l)

## References

  * [1] ATLAS collaboration, _Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC_ , _Phys. Lett. B_ 716 (2012) 1 [1207.7214].
  * [2] CMS collaboration, _Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC_ , _Phys. Lett. B_ 716 (2012) 30 [1207.7235].
  * [3] ATLAS, CMS collaboration, _Combined Measurement of the Higgs Boson Mass in $pp$ Collisions at $\sqrt{s}=7$ and 8 TeV with the ATLAS and CMS Experiments_, _Phys. Rev. Lett._ 114 (2015) 191803 [1503.07589].
  * [4] ATLAS, CMS collaboration, _Measurements of the Higgs boson production and decay rates and constraints on its couplings from a combined ATLAS and CMS analysis of the LHC pp collision data at $\sqrt{s}=7$ and 8 TeV_, _JHEP_ 08 (2016) 045 [1606.02266].
  * [5] ATLAS collaboration, _A detailed map of Higgs boson interactions by the ATLAS experiment ten years after the discovery_ , _Nature_ 607 (2022) 52 [2207.00092].
  * [6] CMS collaboration, _A portrait of the Higgs boson by the CMS experiment ten years after the discovery_ , _Nature_ 607 (2022) 60 [2207.00043].
  * [7] ATLAS collaboration, _Combined measurement of the Higgs boson mass from the $H\to\gamma\gamma$ and $H\to ZZ^{*}\to 4\ell$ decay channels with the ATLAS detector using $\sqrt{s}$ = 7, 8 and 13 TeV $pp$ collision data_, 2308.04775.
  * [8] ATLAS collaboration, _Measurement of the Higgs boson mass with $H\rightarrow\gamma\gamma$ decays in 140 fb-1 of $\sqrt{s}=13$ TeV $pp$ collisions with the ATLAS detector_, 2308.07216.
  * [9] CMS collaboration, _Measurement of the Higgs boson mass and width using the four leptons final state_ , CMS PAS HIG-21-019 (2023).
  * [10] H.P. Nilles, _Supersymmetry, Supergravity and Particle Physics_ , _Phys. Rept._ 110 (1984) 1.
  * [11] H.E. Haber and G.L. Kane, _The Search for Supersymmetry: Probing Physics Beyond the Standard Model_ , _Phys. Rept._ 117 (1985) 75.
  * [12] K. Inoue, A. Kakuto, H. Komatsu and S. Takeshita, _Low-Energy Parameters and Particle Masses in a Supersymmetric Grand Unified Model_ , _Prog. Theor. Phys._ 67 (1982) 1889.
  * [13] S.P. Li and M. Sher, _Upper Limit to the Lightest Higgs Mass in Supersymmetric Models_ , _Phys. Lett. B_ 140 (1984) 339.
  * [14] J.F. Gunion and A. Turski, _Corrections to Higgs Boson Mass Sum Rules from the Sfermion Sector of a Supersymmetric Model_ , _Phys. Rev. D_ 40 (1989) 2333.
  * [15] M.S. Berger, _Radiative Corrections to Higgs Boson Mass Sum Rules in the Minimal Supersymmetric Extension to the Standard Model_ , _Phys. Rev. D_ 41 (1990) 225.
  * [16] Y. Okada, M. Yamaguchi and T. Yanagida, _Upper bound of the lightest Higgs boson mass in the minimal supersymmetric standard model_ , _Prog. Theor. Phys._ 85 (1991) 1.
  * [17] J.R. Ellis, G. Ridolfi and F. Zwirner, _Radiative corrections to the masses of supersymmetric Higgs bosons_ , _Phys. Lett. B_ 257 (1991) 83.
  * [18] H.E. Haber and R. Hempfling, _Can the mass of the lightest Higgs boson of the minimal supersymmetric model be larger than m(Z)?_ , _Phys. Rev. Lett._ 66 (1991) 1815.
  * [19] R. Barbieri and M. Frigeni, _The Supersymmetric Higgs searches at LEP after radiative corrections_ , _Phys. Lett. B_ 258 (1991) 395.
  * [20] J.R. Ellis, G. Ridolfi and F. Zwirner, _On radiative corrections to supersymmetric Higgs boson masses and their implications for LEP searches_ , _Phys. Lett. B_ 262 (1991) 477.
  * [21] A. Brignole, J.R. Ellis, G. Ridolfi and F. Zwirner, _The supersymmetric charged Higgs boson mass and LEP phenomenology_ , _Phys. Lett. B_ 271 (1991) 123.
  * [22] P.H. Chankowski, S. Pokorski and J. Rosiek, _Charged and neutral supersymmetric Higgs boson masses: Complete one loop analysis_ , _Phys. Lett. B_ 274 (1992) 191.
  * [23] A. Brignole, _Radiative corrections to the supersymmetric charged Higgs boson mass_ , _Phys. Lett. B_ 277 (1992) 313.
  * [24] A. Brignole, _Radiative corrections to the supersymmetric neutral Higgs boson masses_ , _Phys. Lett. B_ 281 (1992) 284.
  * [25] P.H. Chankowski, S. Pokorski and J. Rosiek, _Complete on-shell renormalization scheme for the minimal supersymmetric Higgs sector_ , _Nucl. Phys. B_ 423 (1994) 437 [hep-ph/9303309].
  * [26] A. Dabelstein, _The One loop renormalization of the MSSM Higgs sector and its application to the neutral scalar Higgs masses_ , _Z. Phys. C_ 67 (1995) 495 [hep-ph/9409375].
  * [27] D.M. Pierce, J.A. Bagger, K.T. Matchev and R.-j. Zhang, _Precision corrections in the minimal supersymmetric standard model_ , _Nucl. Phys. B_ 491 (1997) 3 [hep-ph/9606211].
  * [28] R. Hempfling and A.H. Hoang, _Two loop radiative corrections to the upper limit of the lightest Higgs boson mass in the minimal supersymmetric model_ , _Phys. Lett. B_ 331 (1994) 99 [hep-ph/9401219].
  * [29] S. Heinemeyer, W. Hollik and G. Weiglein, _QCD corrections to the masses of the neutral CP - even Higgs bosons in the MSSM_ , _Phys. Rev. D_ 58 (1998) 091701 [hep-ph/9803277].
  * [30] S. Heinemeyer, W. Hollik and G. Weiglein, _Precise prediction for the mass of the lightest Higgs boson in the MSSM_ , _Phys. Lett. B_ 440 (1998) 296 [hep-ph/9807423].
  * [31] R.-J. Zhang, _Two loop effective potential calculation of the lightest CP even Higgs boson mass in the MSSM_ , _Phys. Lett. B_ 447 (1999) 89 [hep-ph/9808299].
  * [32] S. Heinemeyer, W. Hollik and G. Weiglein, _The Masses of the neutral CP - even Higgs bosons in the MSSM: Accurate analysis at the two loop level_ , _Eur. Phys. J. C_ 9 (1999) 343 [hep-ph/9812472].
  * [33] J.R. Espinosa and R.-J. Zhang, _MSSM lightest CP even Higgs boson mass to O(alpha(s) alpha(t)): The Effective potential approach_ , _JHEP_ 03 (2000) 026 [hep-ph/9912236].
  * [34] M. Carena, H.E. Haber, S. Heinemeyer, W. Hollik, C.E.M. Wagner and G. Weiglein, _Reconciling the two loop diagrammatic and effective field theory computations of the mass of the lightest CP - even Higgs boson in the MSSM_ , _Nucl. Phys. B_ 580 (2000) 29 [hep-ph/0001002].
  * [35] J.R. Espinosa and R.-J. Zhang, _Complete two loop dominant corrections to the mass of the lightest CP even Higgs boson in the minimal supersymmetric standard model_ , _Nucl. Phys. B_ 586 (2000) 3 [hep-ph/0003246].
  * [36] G. Degrassi, P. Slavich and F. Zwirner, _On the neutral Higgs boson masses in the MSSM for arbitrary stop mixing_ , _Nucl. Phys. B_ 611 (2001) 403 [hep-ph/0105096].
  * [37] A. Brignole, G. Degrassi, P. Slavich and F. Zwirner, _On the O(alpha(t)**2) two loop corrections to the neutral Higgs boson masses in the MSSM_ , _Nucl. Phys. B_ 631 (2002) 195 [hep-ph/0112177].
  * [38] A. Brignole, G. Degrassi, P. Slavich and F. Zwirner, _On the two loop sbottom corrections to the neutral Higgs boson masses in the MSSM_ , _Nucl. Phys. B_ 643 (2002) 79 [hep-ph/0206101].
  * [39] A. Dedes and P. Slavich, _Two loop corrections to radiative electroweak symmetry breaking in the MSSM_ , _Nucl. Phys. B_ 657 (2003) 333 [hep-ph/0212132].
  * [40] A. Dedes, G. Degrassi and P. Slavich, _On the two loop Yukawa corrections to the MSSM Higgs boson masses at large tan beta_ , _Nucl. Phys. B_ 672 (2003) 144 [hep-ph/0305127].
  * [41] B.C. Allanach, A. Djouadi, J.L. Kneur, W. Porod and P. Slavich, _Precise determination of the neutral Higgs boson masses in the MSSM_ , _JHEP_ 09 (2004) 044 [hep-ph/0406166].
  * [42] S. Heinemeyer, W. Hollik, H. Rzehak and G. Weiglein, _High-precision predictions for the MSSM Higgs sector at O(alpha(b) alpha(s))_ , _Eur. Phys. J. C_ 39 (2005) 465 [hep-ph/0411114].
  * [43] M. Frank, L. Galeta, T. Hahn, S. Heinemeyer, W. Hollik, H. Rzehak et al., _Charged Higgs Boson Mass of the MSSM in the Feynman Diagrammatic Approach_ , _Phys. Rev. D_ 88 (2013) 055013 [1306.1156].
  * [44] W. Hollik and S. Paßehr, _Two-loop top-Yukawa-coupling corrections to the charged Higgs-boson mass in the MSSM_ , _Eur. Phys. J. C_ 75 (2015) 336 [1502.02394].
  * [45] S.P. Martin, _Complete Two Loop Effective Potential Approximation to the Lightest Higgs Scalar Boson Mass in Supersymmetry_ , _Phys. Rev. D_ 67 (2003) 095012 [hep-ph/0211366].
  * [46] S.P. Martin, _Two Loop Effective Potential for the Minimal Supersymmetric Standard Model_ , _Phys. Rev. D_ 66 (2002) 096001 [hep-ph/0206136].
  * [47] S.P. Martin, _Strong and Yukawa two-loop contributions to Higgs scalar boson self-energies and pole masses in supersymmetry_ , _Phys. Rev. D_ 71 (2005) 016012 [hep-ph/0405022].
  * [48] S.P. Martin, _Evaluation of two loop selfenergy basis integrals using differential equations_ , _Phys. Rev. D_ 68 (2003) 075002 [hep-ph/0307101].
  * [49] S.P. Martin and D.G. Robertson, _TSIL: A Program for the calculation of two-loop self-energy integrals_ , _Comput. Phys. Commun._ 174 (2006) 133 [hep-ph/0501132].
  * [50] S. Heinemeyer, H. Rzehak and C. Schappacher, _Proposals for Bottom Quark/Squark Renormalization in the Complex MSSM_ , _Phys. Rev. D_ 82 (2010) 075010 [1007.0689].
  * [51] T. Fritzsche, S. Heinemeyer, H. Rzehak and C. Schappacher, _Heavy Scalar Top Quark Decays in the Complex MSSM: A Full One-Loop Analysis_ , _Phys. Rev. D_ 86 (2012) 035014 [1111.7289].
  * [52] T. Fritzsche, T. Hahn, S. Heinemeyer, F. von der Pahlen, H. Rzehak and C. Schappacher, _The Implementation of the Renormalized Complex MSSM in FeynArts and FormCalc_ , _Comput. Phys. Commun._ 185 (2014) 1529 [1309.1692].
  * [53] S. Borowka, T. Hahn, S. Heinemeyer, G. Heinrich and W. Hollik, _Momentum-dependent two-loop QCD corrections to the neutral Higgs-boson masses in the MSSM_ , _Eur. Phys. J. C_ 74 (2014) 2994 [1404.7074].
  * [54] G. Degrassi, S. Di Vita and P. Slavich, _Two-loop QCD corrections to the MSSM Higgs masses beyond the effective-potential approximation_ , _Eur. Phys. J. C_ 75 (2015) 61 [1410.3432].
  * [55] J. Carter and G. Heinrich, _SecDec: A general program for sector decomposition_ , _Comput. Phys. Commun._ 182 (2011) 1566 [1011.5493].
  * [56] S. Borowka, J. Carter and G. Heinrich, _Numerical Evaluation of Multi-Loop Integrals for Arbitrary Kinematics with SecDec 2.0_ , _Comput. Phys. Commun._ 184 (2013) 396 [1204.4152].
  * [57] S. Borowka, T. Hahn, S. Heinemeyer, G. Heinrich and W. Hollik, _Renormalization scheme dependence of the two-loop QCD corrections to the neutral Higgs-boson masses in the MSSM_ , _Eur. Phys. J. C_ 75 (2015) 424 [1505.03133].
  * [58] S. Borowka, S. Paßehr and G. Weiglein, _Complete two-loop QCD contributions to the lightest Higgs-boson mass in the MSSM with complex parameters_ , _Eur. Phys. J. C_ 78 (2018) 576 [1802.09886].
  * [59] A. Pilaftsis, _Higgs scalar - pseudoscalar mixing in the minimal supersymmetric standard model_ , _Phys. Lett. B_ 435 (1998) 88 [hep-ph/9805373].
  * [60] D.A. Demir, _Effects of the supersymmetric phases on the neutral Higgs sector_ , _Phys. Rev. D_ 60 (1999) 055006 [hep-ph/9901389].
  * [61] A. Pilaftsis and C.E.M. Wagner, _Higgs bosons in the minimal supersymmetric standard model with explicit CP violation_ , _Nucl. Phys. B_ 553 (1999) 3 [hep-ph/9902371].
  * [62] S.Y. Choi, M. Drees and J.S. Lee, _Loop corrections to the neutral Higgs boson sector of the MSSM with explicit CP violation_ , _Phys. Lett. B_ 481 (2000) 57 [hep-ph/0002287].
  * [63] M. Carena, J.R. Ellis, A. Pilaftsis and C.E.M. Wagner, _Renormalization group improved effective potential for the MSSM Higgs sector with explicit CP violation_ , _Nucl. Phys. B_ 586 (2000) 92 [hep-ph/0003180].
  * [64] T. Ibrahim and P. Nath, _Corrections to the Higgs boson masses and mixings from chargino, W and charged Higgs exchange loops and large CP phases_ , _Phys. Rev. D_ 63 (2001) 035009 [hep-ph/0008237].
  * [65] S. Heinemeyer, _The Higgs boson sector of the complex MSSM in the Feynman diagrammatic approach_ , _Eur. Phys. J. C_ 22 (2001) 521 [hep-ph/0108059].
  * [66] M. Carena, J.R. Ellis, A. Pilaftsis and C.E.M. Wagner, _Higgs Boson Pole Masses in the MSSM with Explicit CP Violation_ , _Nucl. Phys. B_ 625 (2002) 345 [hep-ph/0111245].
  * [67] T. Ibrahim and P. Nath, _Neutralino exchange corrections to the Higgs boson mixings with explicit CP violation_ , _Phys. Rev. D_ 66 (2002) 015005 [hep-ph/0204092].
  * [68] J.R. Ellis, J.S. Lee and A. Pilaftsis, _CERN LHC signatures of resonant CP violation in a minimal supersymmetric Higgs sector_ , _Phys. Rev. D_ 70 (2004) 075010 [hep-ph/0404167].
  * [69] M. Frank, T. Hahn, S. Heinemeyer, W. Hollik, H. Rzehak and G. Weiglein, _The Higgs Boson Masses and Mixings of the Complex MSSM in the Feynman-Diagrammatic Approach_ , _JHEP_ 02 (2007) 047 [hep-ph/0611326].
  * [70] S. Heinemeyer, W. Hollik, H. Rzehak and G. Weiglein, _The Higgs sector of the complex MSSM at two-loop order: QCD contributions_ , _Phys. Lett. B_ 652 (2007) 300 [0705.0746].
  * [71] S. Heinemeyer, W. Hollik, F. Merz and S. Penaranda, _Electroweak precision observables in the MSSM with nonminimal flavor violation_ , _Eur. Phys. J. C_ 37 (2004) 481 [hep-ph/0403228].
  * [72] J. Cao, G. Eilam, K.-i. Hikasa and J.M. Yang, _Experimental constraints on stop-scharm flavor mixing and implications in top-quark FCNC processes_ , _Phys. Rev. D_ 74 (2006) 031701 [hep-ph/0604163].
  * [73] A. Brignole, _The supersymmetric Higgs boson with flavoured A-terms_ , _Nucl. Phys. B_ 898 (2015) 644 [1504.03273].
  * [74] M. Arana-Catania, S. Heinemeyer, M.J. Herrero and S. Penaranda, _Higgs Boson masses and B-Physics Constraints in Non-Minimal Flavor Violating SUSY scenarios_ , _JHEP_ 05 (2012) 015 [1109.6232].
  * [75] M.E. Gómez, T. Hahn, S. Heinemeyer and M. Rehman, _Higgs masses and Electroweak Precision Observables in the Lepton-Flavor-Violating MSSM_ , _Phys. Rev. D_ 90 (2014) 074016 [1408.0663].
  * [76] W. Hollik and S. Paßehr, _Two-loop top-Yukawa-coupling corrections to the Higgs boson masses in the complex MSSM_ , _Phys. Lett. B_ 733 (2014) 144 [1401.8275].
  * [77] W. Hollik and S. Paßehr, _Higgs boson masses and mixings in the complex MSSM with two-loop top-Yukawa-coupling corrections_ , _JHEP_ 10 (2014) 171 [1409.1687].
  * [78] T. Hahn and S. Paßehr, _Implementation of the $\mathcal{O}{\left(\alpha_{t}^{2}\right)}$ MSSM Higgs-mass corrections in FeynHiggs_, _Comput. Phys. Commun._ 214 (2017) 91 [1508.00562].
  * [79] S. Paßehr and G. Weiglein, _Two-loop top and bottom Yukawa corrections to the Higgs-boson masses in the complex MSSM_ , _Eur. Phys. J. C_ 78 (2018) 222 [1705.07909].
  * [80] M.D. Goodsell and S. Paßehr, _All two-loop scalar self-energies and tadpoles in general renormalisable field theories_ , _Eur. Phys. J. C_ 80 (2020) 417 [1910.02094].
  * [81] F. Domingo and S. Paßehr, _Fighting off field dependence in MSSM Higgs-mass corrections of order $\alpha_{t}\,\alpha_{s}$ and $\alpha_{t}^{2}$_, _Eur. Phys. J. C_ 81 (2021) 661 [2105.01139].
  * [82] E.A. Reyes R. and R. Fazio, _High-Precision Calculations of the Higgs Boson Mass_ , _Particles_ 5 (2022) 53 [2112.15295].
  * [83] R. Barbieri, M. Frigeni and F. Caravaglios, _The Supersymmetric Higgs for heavy superpartners_ , _Phys. Lett. B_ 258 (1991) 167.
  * [84] J.R. Espinosa and M. Quiros, _Two loop radiative corrections to the mass of the lightest Higgs boson in supersymmetric standard models_ , _Phys. Lett. B_ 266 (1991) 389.
  * [85] J.A. Casas, J.R. Espinosa, M. Quiros and A. Riotto, _The Lightest Higgs boson mass in the minimal supersymmetric standard model_ , _Nucl. Phys. B_ 436 (1995) 3 [hep-ph/9407389].
  * [86] H.E. Haber and R. Hempfling, _The Renormalization group improved Higgs sector of the minimal supersymmetric model_ , _Phys. Rev. D_ 48 (1993) 4280 [hep-ph/9307201].
  * [87] M. Carena, J.R. Espinosa, M. Quiros and C.E.M. Wagner, _Analytical expressions for radiatively corrected Higgs masses and couplings in the MSSM_ , _Phys. Lett. B_ 355 (1995) 209 [hep-ph/9504316].
  * [88] M. Carena, M. Quiros and C.E.M. Wagner, _Effective potential methods and the Higgs mass spectrum in the MSSM_ , _Nucl. Phys. B_ 461 (1996) 407 [hep-ph/9508343].
  * [89] H.E. Haber, R. Hempfling and A.H. Hoang, _Approximating the radiatively corrected Higgs mass in the minimal supersymmetric model_ , _Z. Phys. C_ 75 (1997) 539 [hep-ph/9609331].
  * [90] G. Degrassi, S. Heinemeyer, W. Hollik, P. Slavich and G. Weiglein, _Towards high precision predictions for the MSSM Higgs sector_ , _Eur. Phys. J. C_ 28 (2003) 133 [hep-ph/0212020].
  * [91] S.P. Martin, _Three-loop corrections to the lightest Higgs scalar boson mass in supersymmetry_ , _Phys. Rev. D_ 75 (2007) 055005 [hep-ph/0701051].
  * [92] T. Hahn, S. Heinemeyer, W. Hollik, H. Rzehak and G. Weiglein, _High-Precision Predictions for the Light CP -Even Higgs Boson Mass of the Minimal Supersymmetric Standard Model_ , _Phys. Rev. Lett._ 112 (2014) 141801 [1312.4937].
  * [93] P. Draper, G. Lee and C.E.M. Wagner, _Precise estimates of the Higgs mass in heavy supersymmetry_ , _Phys. Rev. D_ 89 (2014) 055023 [1312.5743].
  * [94] N. Arkani-Hamed and S. Dimopoulos, _Supersymmetric unification without low energy supersymmetry and signatures for fine-tuning at the LHC_ , _JHEP_ 06 (2005) 073 [hep-th/0405159].
  * [95] G.F. Giudice and A. Romanino, _Split supersymmetry_ , _Nucl. Phys. B_ 699 (2004) 65 [hep-ph/0406088].
  * [96] M. Carena, G. Nardini, M. Quiros and C.E.M. Wagner, _The Effective Theory of the Light Stop Scenario_ , _JHEP_ 10 (2008) 062 [0806.4297].
  * [97] M. Binger, _Higgs boson mass in split supersymmetry at two-loops_ , _Phys. Rev. D_ 73 (2006) 095001 [hep-ph/0408240].
  * [98] N. Bernal, A. Djouadi and P. Slavich, _The MSSM with heavy scalars_ , _JHEP_ 07 (2007) 016 [0705.1496].
  * [99] P.P. Giardino and P. Lodone, _Threshold Corrections to Hard Supersymmetric Relations_ , _Mod. Phys. Lett. A_ 29 (2014) 1450101 [1112.2635].
  * [100] G.F. Giudice and A. Strumia, _Probing High-Scale and Split Supersymmetry with Higgs Mass Measurements_ , _Nucl. Phys. B_ 858 (2012) 63 [1108.6077].
  * [101] E. Bagnaschi, G.F. Giudice, P. Slavich and A. Strumia, _Higgs Mass and Unnatural Supersymmetry_ , _JHEP_ 09 (2014) 092 [1407.4081].
  * [102] C. Tamarit, _Decoupling heavy sparticles in hierarchical SUSY scenarios: Two-loop Renormalization Group equations_ , 1204.2292.
  * [103] K. Benakli, L. Darmé, M.D. Goodsell and P. Slavich, _A Fake Split Supersymmetry Model for the 126 GeV Higgs_ , _JHEP_ 05 (2014) 113 [1312.5220].
  * [104] P.J. Fox, D.E. Kaplan, E. Katz, E. Poppitz, V. Sanz, M. Schmaltz et al., _Supersplit supersymmetry_ , hep-th/0503249.
  * [105] L.J. Hall and Y. Nomura, _A Finely-Predicted Higgs Boson Mass from A Finely-Tuned Weak Scale_ , _JHEP_ 03 (2010) 076 [0910.2235].
  * [106] M.E. Cabrera, J.A. Casas and A. Delgado, _Upper Bounds on Superpartner Masses from Upper Bounds on the Higgs Boson Mass_ , _Phys. Rev. Lett._ 108 (2012) 021802 [1108.3867].
  * [107] G. Degrassi, S. Di Vita, J. Elias-Miro, J.R. Espinosa, G.F. Giudice, G. Isidori et al., _Higgs mass and vacuum stability in the Standard Model at NNLO_ , _JHEP_ 08 (2012) 098 [1205.6497].
  * [108] J. Pardo Vega and G. Villadoro, _SusyHD: Higgs mass Determination in Supersymmetry_ , _JHEP_ 07 (2015) 159 [1504.05200].
  * [109] E. Bagnaschi, J. Pardo Vega and P. Slavich, _Improved determination of the Higgs mass in the MSSM with heavy superpartners_ , _Eur. Phys. J. C_ 77 (2017) 334 [1703.08166].
  * [110] R.V. Harlander, J. Klappert, A.D. Ochoa Franco and A. Voigt, _The light CP-even MSSM Higgs mass resummed to fourth logarithmic order_ , _Eur. Phys. J. C_ 78 (2018) 874 [1807.03509].
  * [111] E. Bagnaschi, G. Degrassi, S. Paßehr and P. Slavich, _Full two-loop QCD corrections to the Higgs mass in the MSSM with heavy superpartners_ , _Eur. Phys. J. C_ 79 (2019) 910 [1908.01670].
  * [112] H. Bahl, I. Sobolev and G. Weiglein, _Precise prediction for the mass of the light MSSM Higgs boson for the case of a heavy gluino_ , _Phys. Lett. B_ 808 (2020) 135644 [1912.10002].
  * [113] H. Bahl, I. Sobolev and G. Weiglein, _The light MSSM Higgs boson mass for large $\tan\beta$ and complex input parameters_, _Eur. Phys. J. C_ 80 (2020) 1063 [2009.07572].
  * [114] M. Carena, J. Ellis, J.S. Lee, A. Pilaftsis and C.E.M. Wagner, _CP Violation in Heavy MSSM Higgs Scenarios_ , _JHEP_ 02 (2016) 123 [1512.00437].
  * [115] N. Murphy and H. Rzehak, _Higgs-boson masses and mixings in the MSSM with CP violation and heavy SUSY particles_ , _Eur. Phys. J. C_ 82 (2022) 1093 [1909.00726].
  * [116] M. Gorbahn, S. Jager, U. Nierste and S. Trine, _The supersymmetric Higgs sector and $B-\bar{B}$ mixing for large tan $\beta$_, _Phys. Rev. D_ 84 (2011) 034030 [0901.2065].
  * [117] H. Bahl and W. Hollik, _Precise prediction of the MSSM Higgs boson masses for low M A_, _JHEP_ 07 (2018) 182 [1805.00867].
  * [118] G. Lee and C.E.M. Wagner, _Higgs bosons in heavy supersymmetry with an intermediate m A_, _Phys. Rev. D_ 92 (2015) 075032 [1508.00576].
  * [119] H. Bahl and I. Sobolev, _Two-loop matching of renormalizable operators: general considerations and applications_ , _JHEP_ 03 (2021) 286 [2010.01989].
  * [120] P.S. Bhupal Dev and A. Pilaftsis, _Maximally Symmetric Two Higgs Doublet Model with Natural Standard Model Alignment_ , _JHEP_ 12 (2014) 024 [1408.3405].
  * [121] A.V. Bednyakov, _On three-loop RGE for the Higgs sector of 2HDM_ , _JHEP_ 11 (2018) 154 [1809.04527].
  * [122] I. Schienbein, F. Staub, T. Steudtner and K. Svirina, _Revisiting RGEs for general gauge theories_ , _Nucl. Phys. B_ 939 (2019) 1 [1809.06797].
  * [123] J. Oredsson and J. Rathsman, _$\mathbb{Z}_{2}$ breaking effects in 2-loop RG evolution of 2HDM_, _JHEP_ 02 (2019) 152 [1810.02588].
  * [124] F. Herren, L. Mihaila and M. Steinhauser, _Gauge and Yukawa coupling beta functions of two-Higgs-doublet models to three-loop order_ , _Phys. Rev. D_ 97 (2018) 015016 [1712.06614].
  * [125] E. Bagnaschi, F. Brümmer, W. Buchmüller, A. Voigt and G. Weiglein, _Vacuum stability and supersymmetry at high scales with two Higgs doublets_ , _JHEP_ 03 (2016) 158 [1512.07761].
  * [126] E. Bagnaschi et al., _Benchmark scenarios for low $\tan\beta$ in the MSSM_, LHCHXSWG-2015-002 (2015).
  * [127] H. Bahl, S. Liebler and T. Stefaniak, _MSSM Higgs benchmark scenarios for Run 2 and beyond: the low $\tan\beta$ region_, _Eur. Phys. J. C_ 79 (2019) 279 [1901.05933].
  * [128] K. Cheung, R. Huo, J.S. Lee and Y.-L. Sming Tsai, _Dark Matter in Split SUSY with Intermediate Higgses_ , _JHEP_ 04 (2015) 151 [1411.7329].
  * [129] T. Kwasnitza and D. Stöckinger, _Resummation of terms enhanced by trilinear squark-Higgs couplings in the MSSM_ , _JHEP_ 08 (2021) 070 [2103.08616].
  * [130] H. Bahl and W. Hollik, _Precise prediction for the light MSSM Higgs boson mass combining effective field theory and fixed-order calculations_ , _Eur. Phys. J. C_ 76 (2016) 499 [1608.01880].
  * [131] H. Bahl, S. Heinemeyer, W. Hollik and G. Weiglein, _Reconciling EFT and hybrid calculations of the light MSSM Higgs-boson mass_ , _Eur. Phys. J. C_ 78 (2018) 57 [1706.00346].
  * [132] H. Bahl, _Pole mass determination in presence of heavy particles_ , _JHEP_ 02 (2019) 121 [1812.06452].
  * [133] H. Bahl, S. Heinemeyer, W. Hollik and G. Weiglein, _Theoretical uncertainties in the MSSM Higgs boson mass calculation_ , _Eur. Phys. J. C_ 80 (2020) 497 [1912.04199].
  * [134] H. Bahl, N. Murphy and H. Rzehak, _Hybrid calculation of the MSSM Higgs boson masses using the complex THDM as EFT_ , _Eur. Phys. J. C_ 81 (2021) 128 [2010.04711].
  * [135] E. Bagnaschi et al., _MSSM Higgs Boson Searches at the LHC: Benchmark Scenarios for Run 2 and Beyond_ , _Eur. Phys. J. C_ 79 (2019) 617 [1808.07542].
  * [136] I. Sobolev, _Precise predictions for Higgs physics in supersymmetric models_ , Ph.D. thesis, Hamburg U., 2020.  10.3204/PUBDB-2020-02962.
  * [137] P. Athron, J.-h. Park, T. Steudtner, D. Stöckinger and A. Voigt, _Precise Higgs mass calculations in (non-)minimal supersymmetry at both high and low scales_ , _JHEP_ 01 (2017) 079 [1609.00371].
  * [138] F. Staub and W. Porod, _Improved predictions for intermediate and heavy Supersymmetry in the MSSM and beyond_ , _Eur. Phys. J. C_ 77 (2017) 338 [1703.03267].
  * [139] P. Athron, M. Bach, D. Harries, T. Kwasnitza, J.-h. Park, D. Stöckinger et al., _FlexibleSUSY 2.0: Extensions to investigate the phenomenology of SUSY and non-SUSY models_ , _Comput. Phys. Commun._ 230 (2018) 145 [1710.03760].
  * [140] T. Kwasnitza, D. Stöckinger and A. Voigt, _Improved MSSM Higgs mass calculation using the 3-loop FlexibleEFTHiggs approach including $x_{t}$-resummation_, _JHEP_ 07 (2020) 197 [2003.04639].
  * [141] R.V. Harlander, J. Klappert and A. Voigt, _The light CP-even MSSM Higgs mass including N 3LO+N3LL QCD corrections_, _Eur. Phys. J. C_ 80 (2020) 186 [1910.03595].
  * [142] P. Slavich et al., _Higgs-mass predictions in the MSSM and beyond_ , _Eur. Phys. J. C_ 81 (2021) 450 [2012.15629].
  * [143] G. Weiglein, _Results for precision observables in the electroweak standard model at two loop order and beyond_ , _Acta Phys. Polon. B_ 29 (1998) 2735 [hep-ph/9807222].
  * [144] A. Stremplat, “diploma thesis.” Univ. of Karlsruhe, 1998.
  * [145] L. Chen and A. Freitas, _Leading fermionic three-loop corrections to electroweak precision observables_ , _JHEP_ 07 (2020) 210 [2002.05845].
  * [146] L. Chen and A. Freitas, _Mixed EW-QCD leading fermionic three-loop corrections at $\mathcal{O}(\alpha_{s}\alpha^{2})$ to electroweak precision observables_, _JHEP_ 03 (2021) 215 [2012.08605].
  * [147] D. Meuser, _Complete $\mathcal{O}(N_{c}^{2})$ two-loop contributions to the Higgs boson masses in the MSSM and aspects of two-loop renormalisation_, Ph.D. thesis, University of Hamburg, 2023.
  * [148] R.D. Peccei and H.R. Quinn, _CP Conservation in the Presence of Instantons_ , _Phys. Rev. Lett._ 38 (1977) 1440.
  * [149] R.D. Peccei and H.R. Quinn, _Constraints Imposed by CP Conservation in the Presence of Instantons_ , _Phys. Rev. D_ 16 (1977) 1791.
  * [150] S. Dimopoulos and S.D. Thomas, _Dynamical relaxation of the supersymmetric CP violating phases_ , _Nucl. Phys. B_ 465 (1996) 23 [hep-ph/9510220].