#  Two-loop analysis of classically scale-invariant models with extended Higgs
sectors

Johannes Braathen<EMAIL_ADDRESS>Department of Physics, Osaka
University, Toyonaka, Osaka 560-0043, Japan  Deutsches Elektronen-Synchrotron
DESY, Notkestraße 85, D-22607 Hamburg, Germany  Shinya Kanemura
<EMAIL_ADDRESS>Department of Physics, Osaka University,
Toyonaka, Osaka 560-0043, Japan  Makoto Shimoda
<EMAIL_ADDRESS>Department of Physics, Osaka University,
Toyonaka, Osaka 560-0043, Japan

###### Abstract

We present the first explicit calculation of leading two-loop corrections to
the Higgs trilinear coupling $\lambda_{hhh}$ in models with classical scale
invariance (CSI), using the effective-potential approximation. Furthermore, we
also study – for the first time at two loops – the relation that appears
between the masses of all states in CSI theories, due to the requirement of
reproducing correctly the 125-GeV Higgs-boson mass. In addition to obtaining
analytic results for general CSI models, we consider two particular examples
of Beyond-the-Standard-Model theories with extended Higgs sectors, namely an
$N$-scalar model (endowed with a global $O(N)$ symmetry) and a CSI version of
the Two-Higgs-Doublet Model, and we perform detailed numerical studies of
these scenarios. While at one loop the value of the Higgs trilinear coupling
is identical in all CSI models, and deviates by approximately $82\%$ from the
(one-loop) SM prediction, we find that the inclusion of two-loop corrections
lifts this universality and allows distinguishing different BSM scenarios with
CSI. Taking into account constraints from perturbative unitarity and the
relation among masses, we find for both types of scenarios we consider that at
two loops $\lambda_{hhh}$ deviates from its SM prediction by $100\pm 10\%$ –
$i.e.$ a quite significant further deviation with respect to the one-loop
result of $\sim 82\%$.

††preprint: OU-HET-1077††preprint: DESY 20-192

##  I Introduction

The discovery of a Higgs boson of mass 125 GeV at the CERN Large Hadron
Collider Chatrchyan:2012xdj ; Aad:2012tfa  has completed the particle spectrum
of the Standard Model (SM) of particle physics, and has established the Higgs
sector as the origin of electroweak symmetry breaking (EWSB). Nevertheless,
numerous open questions remain unsolved by the SM, and point to the need for
Beyond-the-Standard-Model (BSM) physics. Among these issues, one can mention,
on the theory side, for instance the hierarchy problem, the lack of a
description of quantum gravity, or inflation. Additionally, a number of
experimental results have contradicted prior assumptions – upon which the SM
was built – such as for example (to cite only a few) the need to realise
baryogenesis, the evidence for dark matter and dark energy, or the discovery
that neutrinos have tiny masses.

There is no doubt that the Higgs boson plays a central role in new physics, as
well as in searches for it. Indeed, many (if not most) problems of the SM can
be related to Higgs physics. In order to address these, BSM theories commonly
feature a Higgs sector with extended particle content or modified dynamics –
as there is no compelling argument for it to be minimal. Therefore, shedding
light on the structure of the Higgs sector is one of the most pressing tasks
for particle physicists. Unfortunately, for the time being, all measurements
of Higgs properties seem to be in excellent agreement with SM predictions
Aad:2019mbh ; Sirunyan:2018koj , and this forces us to investigate how new
physics can appear to solve problems of the SM while simultaneously evading
detection. A first, but unexciting, possibility is decoupling – $i.e.$ that
all new states are well beyond our experimental reach and their contributions
to SM(-like) observables are vanishingly small due to the decoupling theorem
Appelquist:1974tg . Another more interesting option is alignment without
decoupling Gunion:2002zf , which occurs111Alignment is usually defined in
models with several Higgs doublets, and is then the limit in which one of the
CP-even mass eigenstates is colinear in field space with the total electroweak
vacuum expectation value.  when the couplings of the 125-GeV Higgs boson to
fermions and gauge bosons are SM-like at tree level, possibly because of some
mechanism or symmetry of the Lagrangian – see for instance Refs. Dev:2014yca ;
Deshpande:1977rw ; Silveira:1985rk ; Barbieri:2006dq ; Benakli:2018vqz ;
Benakli:2018vjk ; Coyle:2019exn ; Aiko:2020atr . We will return to this latter
case in the following.222Slightly non-alignment cases will be almost
completely surveyed and, if nothing appears, will be excluded by the synergy
of the HL-LHC and future precision measurements of the Higgs boson couplings
such as at the ILC, FCCee, CEPC, CLIC, etc. Aiko:2020ksl .

Among the issues of the SM, one that has received considerable attention is
the (gauge) hierarchy problem Veltman:1976rt , or in other words the need for
a mechanism to stabilise the mass of the Higgs boson $M_{h}$ at the
electroweak (EW) scale. Because the Higgs boson is a scalar, its mass is in
the SM not protected by any symmetry, meaning that it typically receives huge
– quadratically divergent – radiative corrections. In turn, this implies that
a cancellation of tremendous – and unrealistic – accuracy would have to occur
between tree-level and loop-level contributions to $M_{h}$. Solutions to the
hierarchy problem usually involve some new symmetry or mechanism that protects
the Higgs mass. For example, supersymmetry Fayet:1977yc  introduces a new
symmetry between bosons and fermions, so that scalar masses become protected
by the chiral symmetry. Meanwhile, in composite Higgs models Kaplan:1983fs ;
Kaplan:1983sm , there are no fundamental scalars and the Higgs boson appears
as a composite state of elementary fermions. But because new physics has so
far eluded discovery, experimental constraints on most popular BSM scenarios
have become more stringent (see for instance Refs. ATLAS:2020uqv ;
ATLAS:2020ujn  and references therein) and the need to explore new avenues of
research is increasingly evident.

Another interesting possibility to address the hierarchy problem, which we
will consider in this paper, comes from the concept of classical scale
invariance (CSI), first proposed by Bardeen in Ref. Bardeen:1995kv . The idea
of CSI is to forbid all mass terms in the Lagrangian, so that the theory is
scale invariant at the tree ($i.e.$ classical) level. Of course, as this is
only imposed at tree level, scale invariance is explicitly broken by radiative
corrections, and EWSB can occur radiatively, via the Coleman-Weinberg
mechanism Coleman:1973jx  – which was extended to multi-scalar cases by
Gildener and Weinberg in Ref. Gildener:1976ih . The Higgs vacuum expectation
value (VEV) $v$ and all particle masses (proportional to $v$) are then
generated via dimensional transmutation. If the CSI model is taken to be UV
complete, or in other words if scale invariance is assumed at the Planck
scale, the absence of a tree-level Higgs mass parameter together with the
generation of the Higgs mass entirely at loop level can ensure that there is
no hierarchy problem – see for instance Refs. Bardeen:1995kv ; Foot:2007as ;
Foot:2007iy ; Meissner:2007xv ; Iso:2009ss ; Iso:2012jn . On the other hand,
models in which scale invariance is imposed at the EW scale can often not be
extended up to the Planck scale, because of Landau poles appearing in the
running of dimensionless parameters. However, even if a Landau pole appears at
some intermediate scale $\Lambda$ and thus requires the existence of some
further new physics beyond $\Lambda$, tree-level scalar masses are prohibited
up to this scale thanks to the CSI. Therefore there is no hierarchy problem up
to the scale $\Lambda$, and a problem only remains between $\Lambda$ and the
Planck scale – arguably a less severe issue than in the SM. The hierarchy
problem is thus not entirely solved in such theories, although it can be
somewhat alleviated.

Nonetheless, motivated by a number of phenomenological considerations, we
restrict our attention in this paper on the latter type of CSI models, defined
with extended scalar sectors at the EW scale. A first interesting feature of
such models is that, in the absence of any (BSM) mass term, new states cannot
be decoupled, nor very heavy, and hence should be found in a foreseeable
future – if they exist. Additionally, in CSI models, the 125-GeV Higgs boson,
which corresponds in the language of Ref. Gildener:1976ih  to the “scalon”
direction along which EWSB is radiatively realised, is automatically aligned
at tree level – as pointed out in Ref. Lane:2019dbc  for a CSI variant of the
Two-Higgs-Doublet Model (2HDM) – thereby explaining current experimental
results.

A further discriminative characteristic of CSI models comes from the Higgs
trilinear coupling $\lambda_{hhh}$. This coupling is of great importance as it
controls the shape of the Higgs potential, and in turn, the nature and
strength of the EW phase transition (EWPT). In particular, one of the
requirements to make the scenario of electroweak baryogenesis work
Sakharov:1967dj ; Kuzmin:1985mm ; Cohen:1993nk  is for the EWPT to be of
strong first order. In turn, a large deviation of the Higgs trilinear coupling
from its SM prediction is one of the main signs of a strong first-order EWPT –
for massive models, a deviation of at least 20% was found to be necessary in
Refs. Grojean:2004xa ; Kanemura:2004ch  – together with the special spectrum
of gravitational waves that would be produced by bubble collisions during the
EWPT Grojean:2006bp .

Couplings of the Higgs boson – and especially its trilinear coupling – can
potentially differ significantly from their SM values in BSM theories with
extended scalar sectors, because of non-decoupling effects from loop
corrections involving the BSM states – this was found at one loop, first in
the 2HDM in Refs. Kanemura:2002vm ; Kanemura:2004mg , and later in various BSM
models with extra singlets Kanemura:2015fra ; Kanemura:2016lkz ; He:2016sqr ;
Kanemura:2017wtm , doublets Kanemura:2015mxa ; Arhrib:2015hoa ;
Kanemura:2016sos ; Kanemura:2017wtm , or triplets Aoki:2012jj . The limit when
these BSM corrections are maximal is when the BSM scalars acquire large masses
exclusively from the Higgs VEV (rather from some BSM mass parameter).
Incidently, this is exactly what happens in CSI models, and as a matter of
fact, the Higgs trilinear coupling has been found Hill:2014mqa ;
Hashino:2015nxa ; Agrawal:2019bpm  to be at one loop
$(\lambda_{hhh}^{\text{CSI}})^{(1)}=5/3(\lambda_{hhh}^{\text{SM}})^{(0)}$ in
any CSI model (without mixing), where $(\lambda_{hhh}^{\text{SM}})^{(0)}$ is
the tree-level SM prediction for $\lambda_{hhh}$ – in other words
$(\lambda_{hhh}^{\text{CSI}})^{(1)}$ deviates by $\sim 67\%$ from
$(\lambda_{hhh}^{\text{SM}})^{(0)}$. The universality of this prediction is a
particularly strong and unique property of CSI models, and follows (as we will
see in section II) from the especially simple form that the effective
potential takes at one-loop order along the “scalon” direction. This value of
$(\lambda_{hhh}^{\text{CSI}})^{(1)}$ also means that the EWPT is always of
strong first order in CSI theories – this, together with the synergy between
the measurements of gravitational waves and of the Higgs trilinear coupling
was investigated in Ref. Hashino:2016rvx  (for the example of $O(N)$-symmetric
CSI models).

One may then naturally wonder what is the current experimental status for the
determination of $\lambda_{hhh}$ as well as future perspectives. It turns out
that large deviations of $\lambda_{hhh}$ from its SM prediction are still
allowed by experiments: indeed, at the present time the most stringent limits
on the trilinear coupling come from the ATLAS collaboration using single-Higgs
production searches with LHC Run 2 data and give
$-3.2<\lambda_{hhh}/\lambda_{hhh}^{\text{SM}}<11.9$ at 95% confidence level
(CL) ATL-PHYS-PUB-2019-009  (see also Refs. Ferrari:2018akh ; Aaboud:2018ftw
and Sirunyan:2018iwt ; Sirunyan:2018two  respectively for ATLAS and CMS
results using double-Higgs production). However, these limits will be greatly
improved at future colliders, thus making $\lambda_{hhh}$ an ideal target to
search for BSM effects. We refer the interested reader to Ref. deBlas:2019rxi
for a detailed discussion of the current prospects for the determination of
$\lambda_{hhh}$ at upcoming experiments, and we only summarise here some of
the main results. First of all, the high-luminosity upgrade of the LHC (HL-
LHC) is expected to reach an accuracy of about $50-60\%$ with $3\text{
ab}^{-1}$ of data Cepeda:2019klc  (see also Ref. Chang:2019ncg ) while a high-
energy extension (HE-LHC) might – optimistically333Note that this result
depends on the treatment of backgrounds – see the discussions in Refs.
Homiller:2018dgu ; Goncalves:2018yva  – be able to attain an
$\mathcal{O}(15\%)$ precision Goncalves:2018yva  at 68% CL. Turning next to
potential lepton colliders, the high-energy stages of the ILC can access the
value of $\lambda_{hhh}$ respectively to 27% accuracy at 500 GeV (with
$4\text{ ab}^{-1}$ of data) and 10% at 1 TeV ($8\text{ ab}^{-1}$ of data)
Fujii:2015jha  – both values being once again at 68% CL. Independently, CLIC
could achieve a precision of approximately 10% (at 68% CL) by combining the
data from all phases (up to 3 TeV) Abramowicz:2016zbo ; Charles:2018vfv ;
Roloff:2019crr . Finally, a possible 100-TeV hadron collider would be able to
determine $\lambda_{hhh}$ to a level of accuracy of 5-7% at 68% CL using
$30\text{ ab}^{-1}$ of data Goncalves:2018yva ; Cepeda:2019klc ; Chang:2018uwu
.

The universality of the value of $\lambda_{hhh}$ in CSI models at one loop
also raises the question of the impact of two-loop corrections. Indeed, as the
form of the effective potential is changed at two loops – with new squared-
logarithmic terms appearing, see $e.g.$ Refs. vanderBij:1983bw ; Ford:1992pn ;
Martin:2001vx  – one may expect the effective Higgs trilinear coupling derived
from it to be modified as well. Moreover, two-loop corrections to
$\lambda_{hhh}$ have already been investigated in a number of extension of the
SM with enlarged scalar sectors,444For completeness, we must also mention
Refs. Brucherseifer:2013qva ; Muhlleitner:2015dua  in which the leading two-
loop $\mathcal{O}(\alpha_{s}\alpha_{t})$ corrections to $\lambda_{hhh}$ are
computed, respectively for the Minimal-Supersymmetric-SM and Next-to-Minimal-
Supersymmetric-SM, for the purpose of matching the precision at which the
Higgs mass is calculated in these models.  namely in the Inert Doublet Model
Senaha:2018xek ; Braathen:2019pxr ; Braathen:2019zoh , in the 2HDM
Braathen:2019pxr ; Braathen:2019zoh , and in a real-singlet extension of the
SM Braathen:2019zoh  (this last model actually corresponds to the $N$-scalar
theory we will consider in this work for $N=1$). It is therefore also
important to compute two-loop corrections to $\lambda_{hhh}$ in the CSI
theories, to match the accuracy achieved in non-CSI scenarios and allow
precise comparisons, as the computational method differs between the two types
of models.

In this paper, we have performed the first explicit555 A study including two-
loop corrections to the effective potential, and its derivatives, via
renormalisation-group improvement were performed in Ref. Hill:2014mqa .
calculation of dominant two-loop corrections to $\lambda_{hhh}$ in theories
with CSI: we extend the known one-loop (effective-potential) calculation to
two loops, so as to include the same level of corrections as in our previous
works on non-CSI models Braathen:2019pxr ; Braathen:2019zoh . We investigate
in particular two types of BSM scenarios: first, $O(N)$-symmetric CSI models,
and second, a CSI variant of the 2HDM Lee:2012jn . Moreover, we include the
renormalisation scheme conversion, from the $\overline{\rm MS}$ scheme (in
which the effective potential is obtained) to the OS scheme, in order to
express our results in terms of physical quantities – note that we will be
referring to the $\overline{\rm MS}$ and OS results respectively as
$\lambda_{hhh}$ and $\hat{\lambda}_{hhh}$ to avoid confusions. We also take
into account the requirement of generating the 125-GeV mass of the Higgs boson
correctly, at the two-loop order. This yields a relation between the masses of
all the states in the CSI models, which we use to constrain the BSM parameters
(masses and couplings) appearing in our calculations. We also ensure that
perturbative unitarity is maintained and that the EW vacuum remains the true
vacuum of the effective potential for the parameter points we consider. As
mentioned already, the Higgs trilinear coupling has a universal prediction at
one loop in all CSI models –
$(\lambda_{hhh}^{\text{CSI}})^{(1)}=5/3(\lambda_{hhh}^{\text{SM}})^{(0)}$ –
and our main result in this paper is finding that the inclusion of two-loop
corrections breaks this universality. Through our detailed numerical studies,
we observe that the different theoretical constraints we impose on the
considered scenarios limit the allowed range of values for $\lambda_{hhh}$ at
two loops – the strongest constraint coming from the requirement of correctly
reproducing the 125-GeV Higgs mass. Nevertheless, the two-loop Higgs trilinear
coupling retains dependence on the parameters of the BSM scalar sectors, and
may provide an opportunity to distinguish different BSM scenarios with CSI.

This paper is organised as follows: in section II we present general results,
applicable to a wide range of CSI models, and we illustrate our calculational
scheme with the example of the CSI Standard Model. Next, in sections III and
IV, we study the Higgs trilinear coupling and the relation between masses in
scenarios of $N$-scalar models with $O(N)$ global symmetries and also in a CSI
variant of the 2HDM. We discuss implications of our present in section V,
before concluding in section VI. A number of appendices provide details about
our computations and some results we deemed too long for the main text.
Definitions of one- and two-loop functions are given in appendix A,
complementary expressions for the $O(N)$-symmetric models are provided in
appendix B, and results in the CSI-2HDM for general masses are presented in
appendix C. Finally, we provide in appendix D generic results for the
coefficients in the effective potential that can be applied for any CSI model
without mixing.

##  II Generalities about classically scale-invariant theories

We begin this paper by recalling known features of CSI models, before deriving
some new results applicable for all such models.

###  II.1 Classically scale-invariant theories

Classically scale-invariant models arise from the assumption that at some
energy scale – which we will in this paper consider to be at or near the EW
scale – the scalar potential exhibits scale invariance, by which we mean that
all mass-dimensionful quantities vanish at the classical ($i.e.$ tree) level.
Radiative corrections explicitly violate the CSI and allow the EW gauge
symmetry to be broken dynamically, giving rise to a mass scale via dimensional
transmutation. This mechanism to realise spontaneous EWSB radiatively was
first studied for a model of a single scalar by S. Coleman and E. Weinberg
Coleman:1973jx . However, it is known that it is impossible to explain the
Higgs boson mass in this simplest single-scalar case: neglecting the top-quark
effects, the original paper of Coleman and Weinberg Coleman:1973jx  predicted
a scalar of mass less than 10 GeV, and including the top quark renders the
(one-loop) potential unstable. In order to reproduce the 125-GeV Higgs mass in
models with CSI, it is therefore necessary to consider extended scalar
sectors. The treatment of this general case of multi-scalar theories requires
more care as one must also identify the direction in field space along which
the EW gauge symmetry is broken. This is the purpose of the method devised by
E. Gildener and S. Weinberg Gildener:1976ih , which we will review briefly
now.

###  II.2 The Gildener-Weinberg method

The tree-level scalar potential of a generic CSI theory with scalars
$\\{\varphi_{i}\\}$ takes the form

$\displaystyle
V^{(0)}(\\{\varphi_{i}\\})=\Lambda_{ijkl}\varphi_{i}\varphi_{j}\varphi_{k}\varphi_{l}\,,$
(II.1)

where the $\Lambda_{ijkl}$ are (dimensionless) quartic couplings – tadpole,
mass, and trilinear terms all have a non-zero mass dimension and must
therefore vanish. Clearly, this potential does not admit any other minimum
than the origin $\\{\varphi_{i}\\}=0$ and in order for the EW gauge symmetry
to be broken spontaneously using the Coleman-Weinberg mechanism, it is
necessary that the tree-level potential vanish along a ray in field space.
This particular flat direction corresponds to the SM-like Higgs boson
$\varphi_{h}$ (called scalon in Ref. Gildener:1976ih ) which carries the EW
vacuum expectation value (VEV) $v$ and acquires a mass of 125 GeV at loop
level.

Along the flat direction, the scalar potential is generated at one-loop order,
and it can be written as a function of the order parameter ($i.e.$ the Higgs
field) $h\equiv\varphi_{h}-v$ using the supertrace formula Jackiw:1974cv

$\displaystyle V_{\text{eff}}(h)=\frac{1}{64\pi^{2}}\bigg{\\{}$
$\displaystyle\operatorname{\text{tr}}\left[M_{S}^{4}(h)\left(\log\frac{M_{S}^{2}(h)}{Q^{2}}-\frac{3}{2}\right)\right]-4\operatorname{\text{tr}}\left[M_{f}^{4}(h)\left(\log\frac{M_{f}^{2}(h)}{Q^{2}}-\frac{3}{2}\right)\right]$
$\displaystyle+3\operatorname{\text{tr}}\left[M_{V}^{4}(h)\left(\log\frac{M_{V}^{2}(h)}{Q^{2}}-\frac{5}{6}\right)\right]\bigg{\\}}\,,$
(II.2)

where $M_{S,f,V}(h)$ denote respectively the scalar, fermion, and gauge-boson
field-dependent mass matrices (at tree level), and $Q$ is the renormalisation
scale. Note that, this expression is given in the $\overline{\rm MS}$
renormalisation scheme. Additionally, compared to the paper of Gildener and
Weinberg Gildener:1976ih , we have kept here all the terms of the supertrace
formula, rather than just the logarithmic terms. As we will see in the next
section, this has no impact on any physical quantity or discussion.

Due to the absence of any mass term in CSI models, the field-dependent masses
of all particles in theory – scalars, fermions, gauge bosons – can be related
to the field-independent masses as

$m_{X}^{2}(h)=m_{X}^{2}\left(1+\frac{h}{v}\right)^{2}\,.$ (II.3)

We can exploit this particularity of CSI models to write the one-loop
effective potential under the simple form

$V_{\text{eff}}(h)=\kappa\left[A^{(1)}\ (v+h)^{4}+B^{(1)}\
(v+h)^{4}\log\frac{(v+h)^{2}}{Q^{2}}\right]\,,$ (II.4)

where $\kappa$ is the loop factor defined in eq. (A.1) and

$\displaystyle A^{(1)}$
$\displaystyle\equiv\frac{1}{4v^{4}}\left\\{\operatorname{\text{tr}}\left[M_{S}^{4}\left(\log\frac{M_{S}^{2}}{v^{2}}-\frac{3}{2}\right)\right]-4\operatorname{\text{tr}}\left[M_{f}^{4}\left(\log\frac{M_{f}^{2}}{v^{2}}-\frac{3}{2}\right)\right]+3\operatorname{\text{tr}}\left[M_{V}^{4}\left(\log\frac{M_{V}^{2}}{v^{2}}-\frac{5}{6}\right)\right]\right\\}\,,$
$\displaystyle B^{(1)}$
$\displaystyle\equiv\frac{1}{4v^{4}}\left(\operatorname{\text{tr}}\left[M_{S}^{4}\right]-4\operatorname{\text{tr}}\left[M_{f}^{4}\right]+3\operatorname{\text{tr}}\left[M_{V}^{4}\right]\right)\,.$
(II.5)

We will discuss in the following how two-loop corrections modify this form of
$V_{\text{eff}}$, however, first we will review the one-loop result for the
effective Higgs trilinear coupling that is computed with this potential.

###  II.3 The Higgs trilinear coupling at one-loop order

We study in this paper corrections to an effective Higgs trilinear coupling,
calculated as the third derivative of the effective potential with respect to
the Higgs field, and evaluated at its minimum

$\lambda_{hhh}\equiv\frac{\partial^{3}V_{\text{eff}}}{\partial
h^{3}}\bigg{|}_{h=0}\,.$ (II.6)

In such a computation based on the effective potential, lower derivatives of
the potential also play an important role. First of all, from the first
derivative of $V_{\text{eff}}$ we obtain the tadpole equation

$\displaystyle\frac{\partial V_{\text{eff}}}{\partial h}\bigg{|}_{h=0}=2\kappa
v^{3}\bigg{(}2A^{(1)}+B^{(1)}+2B^{(1)}\log\frac{v^{2}}{Q^{2}}\bigg{)}=0\,,$
(II.7)

which allows us – as $v\neq 0$ – to eliminate666This confirms our earlier
statement: a finite shift to the quantity $A^{(1)}$ plays no role physically,
as $A^{(1)}$ is anyway eliminated using the tadpole equation.  the quantity
$A^{(1)}$

$\displaystyle
A^{(1)}=-B^{(1)}\left(\frac{1}{2}+\log\frac{v^{2}}{Q^{2}}\right)\,.$ (II.8)

Next, the second derivative of $V_{\text{eff}}$ defines the effective-
potential, or curvature, mass of the Higgs boson, which we denote
$[M_{h}^{2}]_{V_{\text{eff}}}$. We have that

$[M_{h}^{2}]_{V_{\text{eff}}}\equiv\frac{\partial^{2}V_{\text{eff}}}{\partial
h^{2}}\bigg{|}_{h=0}=2\kappa
v^{2}\left(6A^{(1)}+7B^{(1)}+6B^{(1)}\log\frac{v^{2}}{Q^{2}}\right)=8\kappa
v^{2}B^{(1)}\,.$ (II.9)

This equation has a very important consequence, unique to models with
classical scale invariance: it relates on the left-hand side the Higgs
curvature mass, close to its known pole mass of 125 GeV, to the tree-level
masses of all particles in the theory through the quantity $B^{(1)}$ on the
right-hand side. If the model we consider contains no extra fermions and gauge
bosons but only additional scalars, we obtain Gildener:1976ih ; Lee:2012jn ;
Hashino:2015nxa ; Lane:2018ycs  using eq. (II.2)

$4v^{4}B^{(1)}=8\pi^{2}v^{2}[M_{h}^{2}]_{V_{\text{eff}}}=\operatorname{\text{tr}}[M_{S}^{4}]-12m_{t}^{4}+6m_{W}^{4}+3m_{Z}^{4}\,,$
(II.10)

or equivalently

$\operatorname{\text{tr}}[M_{S}^{4}]=8\pi^{2}v^{2}[M_{h}^{2}]_{V_{\text{eff}}}+12m_{t}^{4}-6m_{W}^{4}-3m_{Z}^{4}\,,$
(II.11)

having taken into account the $N_{c}=3$ colour factor for the top quark and
with $m_{t}$, $m_{W}$, and $m_{Z}$ the tree-level masses of the top quark and
of the $W$ and $Z$ bosons respectively. Once one considers specific scenarios
for the particle contents of BSM CSI theories, it becomes possible to obtain
constraints on the masses of the new states – as studied for example in Refs.
Hashino:2015nxa ; Lane:2019dbc ; Brooijmans:2020yij . We will return to this
type of relations among masses in the following, when we consider particular
BSM models.

Returning now to the one-loop potential ($c.f.$ eq. (II.4)) we find that the
only two parameters on which it depends – $A^{(1)}$ and $B^{(1)}$ – are fixed
by the tadpole equation and the knowledge of the Higgs mass. This implies that
if we take further derivatives of the one-loop potential, the computed
quantities are also fixed in terms of the Higgs curvature mass. Indeed, the
effective Higgs trilinear coupling is at one-loop order

$\displaystyle\lambda_{hhh}\equiv\frac{\partial^{3}V_{\text{eff}}}{\partial
h^{3}}\bigg{|}_{h=0}=4\kappa
v\left(6A^{(1)}+13B^{(1)}+6B^{(1)}\log\frac{v^{2}}{Q^{2}}\right)\,,$ (II.12)

and using equations (II.7) and (II.9), we can eliminate $A^{(1)}$ and
$B^{(1)}$, and rewrite $\lambda_{hhh}$ in terms of the tree-level result for
$\lambda_{hhh}$ in the Standard Model as Hashino:2015nxa

$\displaystyle\lambda_{hhh}=40\kappa
vB^{(1)}=\frac{5[M_{h}^{2}]_{V_{\text{eff}}}}{v}=\frac{5}{3}(\lambda_{hhh}^{(0)})^{\text{SM}}\,.$
(II.13)

Similarly, we can compute a one-loop (effective) quartic coupling
$\lambda_{hhhh}$

$\displaystyle\lambda_{hhhh}=\frac{\partial^{4}V_{\text{eff}}}{\partial
h^{4}}\bigg{|}_{h=0}=$ $\displaystyle\
4\kappa\left(6A^{(1)}+25B^{(1)}+6B^{(1)}\log\frac{v^{2}}{Q^{2}}\right)=88\kappa
B^{(1)}$ $\displaystyle=$ $\displaystyle\
\frac{11[M_{h}^{2}]_{V_{\text{eff}}}}{v^{2}}=\frac{11}{3}(\lambda_{hhhh}^{(0)})^{\text{SM}}\,.$
(II.14)

Note that for these two last equations, we made use of the known tree-level
results for the Standard Model

$\displaystyle(\lambda_{hhh}^{(0)})^{\text{SM}}=\frac{3m_{h}^{2}}{v}\,,\qquad\text{and}\qquad(\lambda_{hhhh}^{(0)})^{\text{SM}}=\frac{3m_{h}^{2}}{v^{2}}\,.$
(II.15)

The predictions in equations (II.13) and (II.3) deviate significantly from the
usual SM values – for the case of $\lambda_{hhh}$ this effect is sufficiently
large to be observed already at the HL-LHC. A further crucial aspect of these
results is that, in CSI models, the Higgs trilinear and quartic coupling are,
at one loop, completely independent of the particle content of the theory.
However, as we will see now, this universality is lost when two-loop
corrections are included in the effective potential.

###  II.4 Two-loop corrections to the Higgs trilinear coupling

At two loops, corrections to the effective potential are obtained by computing
one-particle irreducible vacuum bubble diagrams. In addition to logarithmic
and non-logarithmic terms – as are present at one loop – such two-loop
diagrams also contain squared logarithms, meaning that the effective potential
takes the form

$\displaystyle V_{\text{eff}}(h)=A\ (v+h)^{4}+B\
(v+h)^{4}\log\frac{(v+h)^{2}}{Q^{2}}+C\
(v+h)^{4}\log^{2}\frac{(v+h)^{2}}{Q^{2}}\,,$ (II.16)

where the coefficient $C$ appears only at two-loop order. Expanding this
expression in terms of one- and two-loop quantities, it becomes

$\displaystyle V_{\text{eff}}(h)\equiv$ $\displaystyle\ \kappa
V^{(1)}(h)+\kappa^{2}V^{(2)}(h)+\mathcal{O}(\kappa^{3})$ $\displaystyle=$
$\displaystyle\ \kappa\bigg{[}A^{(1)}\ (v+h)^{4}+B^{(1)}\
(v+h)^{4}\log\frac{(v+h)^{2}}{Q^{2}}\bigg{]}$ (II.17)
$\displaystyle+\kappa^{2}\bigg{[}A^{(2)}\ (v+h)^{4}+B^{(2)}\
(v+h)^{4}\log\frac{(v+h)^{2}}{Q^{2}}+C^{(2)}\
(v+h)^{4}\log^{2}\frac{(v+h)^{2}}{Q^{2}}\bigg{]}+\mathcal{O}(\kappa^{3})\,.$

While $A^{(1)}$ and $B^{(1)}$ are exactly the quantities given in eq. (II.2),
$A^{(2)}$, $B^{(2)}$, and $C^{(2)}$ must be derived from the two-loop diagrams
contributing to the effective potential – and will depend on the model one is
considering.

Before investigating specific models, it is enlighting to repeat the procedure
of the previous section, and derive general expressions for the derivatives of
$V_{\text{eff}}$, using now the two-loop form in eq. (II.16). First, the
tadpole equation reads

$\frac{\partial V_{\text{eff}}}{\partial
h}\bigg{|}_{h=0}=2v^{3}\bigg{[}2A+B+2(B+C)\log\frac{v^{2}}{Q^{2}}+2C\log^{2}\frac{v^{2}}{Q^{2}}\bigg{]}=0\,,$
(II.18)

from which we can isolate $A$ as

$\displaystyle
A=-\frac{1}{2}B-(B+C)\log\frac{v^{2}}{Q^{2}}-C\log^{2}\frac{v^{2}}{Q^{2}}\,.$
(II.19)

Next, the Higgs curvature mass is, after eliminating $A$

$\displaystyle[M_{h}^{2}]_{V_{\text{eff}}}=\frac{\partial^{2}V_{\text{eff}}}{\partial
h^{2}}\bigg{|}_{h=0}$
$\displaystyle=8v^{2}\bigg{[}B+C+2C\log\frac{v^{2}}{Q^{2}}\bigg{]}\,,$ (II.20)

so that $B$ can be rewritten in terms of $[M_{h}^{2}]_{V_{\text{eff}}}$ and
$C$

$B=\frac{[M_{h}^{2}]_{V_{\text{eff}}}}{8v^{2}}-C\bigg{(}1+2\log\frac{v^{2}}{Q^{2}}\bigg{)}\,.$
(II.21)

Equation (II.20) is also important as it constitutes the two-loop version of
the relation between masses in CSI models – which we first encountered at one
loop in the previous section. Once we consider particular BSM models in
sections III and IV, we will see that this relation places stringent
constraints on the allowed BSM mass and parameter ranges.

Now, if we compute the effective Higgs trilinear coupling, we obtain at two
loops

$\displaystyle\lambda_{hhh}=\frac{\partial^{3}V_{\text{eff}}}{\partial
h^{3}}\bigg{|}_{h=0}$
$\displaystyle=\frac{5[M_{h}^{2}]_{V_{\text{eff}}}}{v}+32Cv\,,$ (II.22)

once $A$ and $B$ are replaced using eqs. (II.19) and (II.21). We see here that
because of the new term $C$, appearing from two loops in $V_{\text{eff}}$,
non-universal, model-dependent corrections arise in the expression of
$\lambda_{hhh}$. In practice we will be computing $C$ from the two-loop
diagrams contributing to $V_{\text{eff}}$ and use this result to obtain
$\lambda_{hhh}$ at two loops. However, we note that we can formally use the
last equation to determine $C$ in terms of $[M_{h}^{2}]_{V_{\text{eff}}}$ and
$\lambda_{hhh}$, as

$\displaystyle
C=\frac{1}{32v}\bigg{(}\lambda_{hhh}-\frac{5[M_{h}^{2}]_{V_{\text{eff}}}}{v}\bigg{)}\,,$
(II.23)

We can finally use this expression of $C$ to compute $\lambda_{hhhh}$ at two
loops in terms of $[M_{h}^{2}]_{V_{\text{eff}}}$ and $\lambda_{hhh}$ only

$\displaystyle\lambda_{hhhh}=\frac{\partial^{4}V_{\text{eff}}}{\partial
h^{4}}\bigg{|}_{h=0}=\frac{11[M_{h}^{2}]_{V_{\text{eff}}}}{v^{2}}+192C=\frac{6\lambda_{hhh}}{v}-\frac{19[M_{h}^{2}]_{V_{\text{eff}}}}{v^{2}}\,.$
(II.24)

This equation implies that once $\lambda_{hhh}$ is measured, the Higgs quartic
coupling can be predicted in a universal way for all CSI theories, up to two
loops. This being said, once three-loop corrections are included, we can
expect this universality to be lost because of $\log^{3}$ terms appearing in
$V_{\text{eff}}$ – similarly to what happens with $\lambda_{hhh}$ once two-
loop effects are taken into account.

###  II.5 Results for the classically scale-invariant Standard Model

Figure 1: Dominant two-loop diagrams contributing to the SM effective
potential.

We end this section by calculating the leading two-loop contributions to
$V_{\text{eff}}$ and $\lambda_{hhh}$ for the classically scale-invariant
version of the SM, which will also provide an occasion to present the
renormalisation scheme conversion that we perform to obtain expressions in
terms of physical quantities. We must emphasise that the CSI-SM is already
known not to be a valid theory of Nature as it cannot reproduce the Higgs mass
correctly Coleman:1973jx , however, its simplicity makes it the ideal setting
to explain our calculational setup.

The tree-level scalar potential of the CSI-SM does not contain a mass term,
and reads simply

$\displaystyle V^{(0)}=\lambda|\Phi|^{4}\,,$ (II.25)

where the Higgs doublet $\Phi$ is defined as

$\Phi=\begin{pmatrix}G^{+}\\\ \frac{1}{\sqrt{2}}(v+h+iG)\end{pmatrix}\,.$
(II.26)

$G$ and $G^{\pm}$ are respectively the neutral and charged would-be Goldstone
bosons, while $h$ is the 125-GeV Higgs boson. At tree level, the field-
dependent masses of the Higgs and Goldstone bosons read

$m_{h}^{2}(h)=3\lambda(v+h)^{2}\,,\qquad\text{and}\qquad
m_{G}^{2}(h)=\lambda(v+h)^{2}\,,$ (II.27)

while the top-quark mass is (like for the usual SM)

$\displaystyle m_{t}(h)=\frac{y_{t}}{\sqrt{2}}(v+h)\,.$ (II.28)

The need for a flat direction in the tree-level potential imposes $\lambda=0$,
which is also the result of solving the tree-level tadpole equation.

The dominant contributions to the CSI-SM two-loop effective potential arise
from the diagrams shown in figure 1 and involving the $SU(3)_{c}$ gauge
coupling $g_{3}$ and the top Yukawa coupling $y_{t}$ – the two EW gauge
couplings $g_{2}$ and $g_{Y}$, and the light fermion Yukawa couplings are
negligible in comparison. These contributions read (in the $\overline{\rm MS}$
scheme)

$\displaystyle V^{(2)}(h)=$
$\displaystyle-4g_{3}^{2}m_{t}^{2}(h)\bigg{(}4J(m_{t}^{2}(h))-8m_{t}^{2}(h)-\frac{6J(m_{t}^{2}(h))^{2}}{m_{t}^{2}(h)}\bigg{)}\,,$
$\displaystyle+3y_{t}^{2}\bigg{[}2m_{t}^{2}(h)I(m_{t}^{2}(h),m_{t}^{2}(h),0)+m_{t}^{2}(h)I(m_{t}^{2}(h),0,0)+J(m_{t}^{2}(h))^{2}\bigg{]}\,,$
(II.29)

where $J$ and $I$ are respectively one- and two-loop functions, for which
definitions777Note also that the function we denote as $J$ is the same as the
Passarino-Veltman function $A$. We choose the notation $J$ as $e.g.$ in Refs.
Ford:1992pn ; Martin:2001vx  to avoid confusion with the quantity $A$ in the
effective potential – $c.f.$ eq. (II.16).  and useful limits are given in
appendix A.

In the previous section, we found that to obtain the two-loop corrections to
$\lambda_{hhh}$, we need to compute $C$, $i.e.$ the coefficient of the
$\log^{2}$ term in $V_{\text{eff}}$. For the CSI-SM, after expanding the
expression in equation (II.5), we find

$\displaystyle
C^{(2)}=\frac{24g_{3}^{2}m_{t}^{4}}{v^{4}}-\frac{9m_{t}^{6}}{v^{6}}\,.$
(II.30)

Using then equation (II.22), we obtain for the effective Higgs trilinear
coupling in the CSI-SM, to leading two-loop order,

$\displaystyle\lambda_{hhh}=\frac{5[M_{h}^{2}]_{V_{\text{eff}}}}{v}+\frac{1}{(16\pi^{2})^{2}}\bigg{[}\frac{768g_{3}^{2}m_{t}^{4}}{v^{3}}-\frac{288m_{t}^{6}}{v^{5}}\bigg{]}\,.$
(II.31)

At this point, we should also discuss the choice of renormalisation scheme for
the parameters appearing in the results presented in this work. As the
effective potential, from which we derive $\lambda_{hhh}$, is calculated in
the $\overline{\rm MS}$ scheme, the expression in equation (II.31) should be
understood as being written in terms of $\overline{\rm MS}$ parameters. In the
following, we will however choose to convert $\overline{\rm MS}$ expressions
into on-shell (OS) ones – $i.e.$ written in terms of physical quantities and
including also finite corrections from wave-function renormalisation (WFR). An
OS version of the Higgs trilinear coupling can be obtained as

$\displaystyle\hat{\lambda}_{hhh}=\left(\frac{Z_{h}^{\text{OS}}}{Z_{h}^{\overline{\rm
MS}}}\right)^{\frac{3}{2}}\lambda_{hhh}=\left(1+\frac{3}{2}\kappa\frac{d\Pi_{hh}^{(1)}}{dp^{2}}\bigg{|}_{p^{2}=0}+\mathcal{O}(\kappa^{2})\right)\lambda_{hhh}\,,$
(II.32)

where $Z_{h}^{\text{OS}}$ and $Z_{h}^{\overline{\rm MS}}$ are respectively the
on-shell and $\overline{\rm MS}$ scheme Higgs WFR constants, $\Pi^{(1)}_{hh}$
is the finite part of the one-loop Higgs boson self-energy, and with
$\lambda_{hhh}$ expressed in terms of physical quantities. Note that it
suffices to compute the WFR constants to one-loop order, because
$\lambda_{hhh}$ is generated at one loop.

In the two-loop result for $\lambda_{hhh}$ in eq. (II.31), $m_{t}$ and $g_{3}$
only enter at two loops, and therefore they require no scheme conversion here
– this would only produce corrections at three-loop order. However, it is
necessary to include a one-loop scheme conversion for the Higgs $\overline{\rm
MS}$ VEV $v$ and curvature mass $[M_{h}^{2}]_{V_{\text{eff}}}$, both appearing
from one loop. First, the $\overline{\rm MS}$ VEV can be related to the Fermi
constant at (leading) one loop as

$\displaystyle
v^{2}=\frac{1}{\sqrt{2}G_{F}}+\frac{3M_{t}^{2}}{16\pi^{2}}\big{(}2\operatorname{\overline{\text{log}}}M_{t}^{2}-1\big{)}\,.$
(II.33)

For notational convenience, we also define an OS Higgs VEV as
$v_{\text{OS}}^{2}\equiv(\sqrt{2}G_{F})^{-1}$. Next, we recall that the Higgs
pole ($i.e.$ physical) and curvature masses are related by the following
formula

$\displaystyle
M_{h}^{2}=[M_{h}^{2}]_{V_{\text{eff}}}+\Pi_{hh}(p^{2}=M_{h}^{2})-\Pi_{hh}(p^{2}=0)\,.$
(II.34)

In the CSI-SM, the Higgs mass is generated at loop level, and thus this
relation simplifies into

$\displaystyle
M_{h}^{2}=[M_{h}^{2}]_{V_{\text{eff}}}+\frac{M_{h}^{2}}{16\pi^{2}}\bigg{[}\frac{2M_{t}^{2}}{v_{\text{OS}}^{2}}\big{(}2+3\operatorname{\overline{\text{log}}}M_{t}^{2}\big{)}\bigg{]}\,.$
(II.35)

Combining all these intermediate results together, we obtain finally

$\displaystyle\hat{\lambda}_{hhh}=\frac{5M_{h}^{2}}{v_{\text{OS}}}+\frac{1}{16\pi^{2}}\frac{35}{2}\frac{M_{h}^{2}M_{t}^{2}}{v_{\text{OS}}^{3}}+\frac{1}{(16\pi^{2})^{2}}\bigg{[}\frac{768g_{3}^{2}M_{t}^{4}}{v_{\text{OS}}^{3}}-\frac{288M_{t}^{6}}{v_{\text{OS}}^{5}}\bigg{]}\,.$
(II.36)

It is important to note that, because $M_{h}^{2}$ is generated at loop level,
the second term, which is proportional to $\kappa
M_{h}^{2}M_{t}^{2}/v_{\text{OS}}^{3}$, is formally of two-loop order – only
the first term in this expression is of one-loop order.

Taking from the PDG Zyla:2020zbs , the following values for the physical
inputs

$\displaystyle M_{h}=125.10\text{ GeV}\,,\quad M_{t}=172.4\text{ GeV}\,,\quad
G_{F}=1.1663787\cdot 10^{-5}\text{
GeV}^{-2}\,,\quad\alpha_{s}(M_{Z})=0.1179\,,$ (II.37)

we find numerically

$\displaystyle\frac{5M_{h}^{2}}{v_{\text{OS}}}=$ $\displaystyle\ 317.8\text{
GeV}$
$\displaystyle\frac{5M_{h}^{2}}{v_{\text{OS}}}+\frac{1}{16\pi^{2}}\frac{35}{2}\frac{M_{h}^{2}M_{t}^{2}}{v_{\text{OS}}^{3}}+\frac{1}{(16\pi^{2})^{2}}\bigg{[}\frac{768g_{3}^{2}M_{t}^{4}}{v_{\text{OS}}^{3}}-\frac{288M_{t}^{6}}{v_{\text{OS}}^{5}}\bigg{]}=$
$\displaystyle\ 323.6\text{ GeV}\,,$ (II.38)

or in other words, the two-loop corrections to $\lambda_{hhh}$ yield a
positive shift of the order of $2\%$.

##  III $N$-scalar models

###  III.1 Model definitions

The first class of BSM CSI theories that we consider are extensions of the SM
with $N$ additional real singlet scalar states. We will furthermore assume
that these models are endowed with a global $O(N)$ symmetry, under which only
the singlet scalars are charged while the SM particles do not carry any
charge. Although this assumption greatly simplifies the expressions we obtain
– by reducing the number of different masses and coupling constants – it is
important to note that it does not significantly affect our discussion about
corrections to the Higgs trilinear coupling. Such $O(N)$-symmetric CSI models
have been discussed in many references – see Refs. Endo:2015ifa ; Endo:2015nba
; Endo:2016koi ; Helmboldt:2016mpi ; Fujitani:2017gma  – from various points
of view. In particular, the exact $O(N)$ symmetry provides a symmetry to
stabilise dark matter candidates in these models. It was found in Ref.
Endo:2015nba  that there is an upper bound on the value of $N$ if one
considers direct dark matter search results. However, in this paper, we choose
to study $O(N)$ models exclusively from the point of view of Higgs physics and
corrections to $\lambda_{hhh}$, and we do not consider potential dark matter
aspects of these models. In other words, we consider here $O(N)$ as a
phenomenological symmetry – which greatly simplifies the study of Higgs
physics in $N$-scalar theories – rather than a symmetry to stabilise dark
matter.

We can write the scalar potential of $O(N)$-symmetric models with CSI as

$\displaystyle V^{(0)}=$ $\displaystyle\ \lambda|\Phi|^{4}+\lambda_{\Phi
S}\vec{S}^{2}|\Phi|^{2}+\frac{1}{4}\lambda_{S}(\vec{S}^{2})^{2}\,,$ (III.1)

where $\Phi$ is the SM-like Higgs doublet – same as that shown in eq. (II.26)
– and $\vec{S}=(S_{1}\,,S_{2}\,,\cdots S_{N})$ is a real scalar, singlet under
the SM gauge groups and belonging to an $N$-dimensional representation of the
global $O(N)$ symmetry group – another way to see this is to think of $N$ real
scalar singlets. Only the neutral component of the Higgs doublet acquires a
non-zero VEV – the singlets do not as this would break the global $O(N)$
symmetry.

The scalar masses at tree-level are

$\displaystyle m_{h}^{2}(h)=3\lambda(v+h)^{2}\,,\qquad
m_{G}^{2}(h)=\lambda(v+h)^{2}\,,\qquad m_{S_{i}}^{2}(h)=\lambda_{\Phi
S}(v+h)^{2}\equiv m_{S}^{2}(h)\,.$ (III.2)

and the tree-level tadpole condition simply gives $\lambda=0$. Finally, it
follows immediately from this equation that we can replace $\lambda_{\Phi S}$
by the tree-level BSM scalar mass $m_{S}$, as $\lambda_{\Phi
S}=m_{S}^{2}/v^{2}$.

###  III.2 Dominant two-loop corrections to the Higgs trilinear coupling

Figure 2: Leading two-loop BSM diagrams contributing to the effective
potential of the $O(N)$-symmetric CSI model – in addition to the leading SM
diagrams of figure 1.

We can obtain for the leading BSM contributions to $V^{(2)}$ by considering
the new diagrams involving $\lambda_{\Phi S}$ and $\lambda_{S}$, shown in
figure 2. Their expressions are

$\displaystyle V^{(2)}=$ $\displaystyle\ V^{(2)}_{hSS}+V^{(2)}_{SS}\,,$
$\displaystyle\text{with}\qquad V^{(2)}_{hSS}=$ $\displaystyle-N\lambda_{\Phi
S}^{2}(v+h)^{2}I(m_{S}^{2}(h),m_{S}^{2}(h),0)\,,$ $\displaystyle
V^{(2)}_{SS}=$ $\displaystyle\
\frac{1}{4}N(N+2)\lambda_{S}J(m_{S}^{2}(h))^{2}\,.$ (III.3)

The squared-logarithm terms in these two-loop contributions are, in turn

$\displaystyle C^{(2)}_{hSS}=$ $\displaystyle\ \frac{Nm_{S}^{6}}{v^{6}}\,,$
$\displaystyle C^{(2)}_{SS}=$ $\displaystyle\
\frac{1}{4}N(N+2)\frac{\lambda_{S}m_{S}^{4}}{v^{4}}\,.$ (III.4)

Using once again equation (II.22), we can obtain the leading two-loop order
$\overline{\rm MS}$ result for $\lambda_{hhh}$

$\displaystyle\lambda_{hhh}=\frac{5[M_{h}^{2}]_{V_{\text{eff}}}}{v}+\frac{1}{(16\pi^{2})^{2}}\bigg{[}\frac{768g_{3}^{2}m_{t}^{4}}{v^{3}}-\frac{288m_{t}^{6}}{v^{5}}+\frac{32Nm_{S}^{6}}{v^{5}}+8N(N+2)\frac{\lambda_{S}m_{S}^{4}}{v^{3}}\bigg{]}\,.$
(III.5)

Among the terms in the brackets, the first two (those involving $m_{t}$) are
the SM-like contributions, while the last two are the new BSM effects. Note
that, like in the SM case, $m_{S}$ and $\lambda_{S}$ only enter the expression
of $\lambda_{hhh}$ at two loops, and hence we do not need to specify their
renormalisation scheme. If we want to express the one-loop result in terms of
$M_{h}^{2}$ and $v_{\text{OS}}$ we must however include scheme conversions for
both $[M_{h}^{2}]_{V_{\text{eff}}}$ and $v$ and take into account WFR effects
– as discussed already in section II.5. The only difference we must consider
arises from the new momentum-dependent term in the Higgs self-energy in the
$O(N)$-symmetric model, which reads

$\displaystyle\Pi^{(1)}_{hh}(p^{2})\supset-\frac{2Nm_{S}^{4}}{v^{2}}B_{0}(p^{2},m_{S}^{2},m_{S}^{2})\,,$
(III.6)

where the one-loop Passarino-Veltman function Passarino:1978jh  $B_{0}$ is
defined in appendix A. Adding this additional term, we find the OS expression
for $\hat{\lambda}_{hhh}$ at leading two loops to be

$\displaystyle\hat{\lambda}_{hhh}=$ $\displaystyle\
\frac{5M_{h}^{2}}{v_{\text{OS}}}+\frac{1}{16\pi^{2}}\frac{5M_{h}^{2}}{v_{\text{OS}}^{3}}\bigg{[}\frac{7}{2}M_{t}^{2}-\frac{1}{6}NM_{S}^{2}\bigg{]}$
$\displaystyle+\frac{1}{(16\pi^{2})^{2}}\bigg{[}\frac{768g_{3}^{2}M_{t}^{4}}{v_{\text{OS}}^{3}}-\frac{288M_{t}^{6}}{v_{\text{OS}}^{5}}+\frac{32NM_{S}^{6}}{v_{\text{OS}}^{5}}+8N(N+2)\frac{\lambda_{S}M_{S}^{4}}{v_{\text{OS}}^{3}}\bigg{]}\,.$
(III.7)

Once again, we emphasise that the terms in the first pair of brackets –
proportional to $\kappa M_{h}^{2}/v_{\text{OS}}^{3}$ – are formally of two-
loop order.

In principle the above result for $\hat{\lambda}_{hhh}$ involves 3 BSM
parameters, namely $M_{S}$, $\lambda_{S}$, and $N$. However, the classical
scale invariance produces a further constraint on the BSM parameters, by
relating the masses of the particles in the theory, as we will see in the
following section.

###  III.3 Mass relation

When investigating derivatives of the effective potential in general CSI
models in section II.4, we had found that the curvature mass of the Higgs
boson is related to the $B$ and $C$ coefficients in $V_{\text{eff}}$. In terms
of one- and two-loop quantities, this relation reads

$\displaystyle
8\pi^{2}[M_{h}^{2}]_{V_{\text{eff}}}=4v^{2}B^{(1)}+\frac{4v^{2}}{16\pi^{2}}\left[B^{(2)}+C^{(2)}\left(1+2\log\frac{v^{2}}{Q^{2}}\right)\right]\,.$
(III.8)

As we have computed the effective potential in $O(N)$-symmetric CSI models, it
is a simple matter to derive all the necessary ingredients to apply this
relation to this class of models. In addition to $C^{(2)}$, which was given in
the previous section, we also need the logarithmic terms at one and two loops
– $B^{(1)}$ and $B^{(2)}$. Including both SM-like and BSM terms, these are

$\displaystyle B^{(1)}=$ $\displaystyle\
\frac{1}{4v^{4}}\left(Nm_{S}^{4}-12m_{t}^{4}+6m_{W}^{4}+3m_{Z}^{4}\right)\,,$
$\displaystyle B^{(2)}=$ $\displaystyle\
\frac{4g_{3}^{2}m_{t}^{4}}{v^{4}}\left[12\log\frac{m_{t}^{2}}{v^{2}}-16\right]+\frac{6m_{t}^{6}}{v^{6}}\left[-3\log\frac{m_{t}^{2}}{v^{2}}+8\right]$
$\displaystyle+\frac{Nm_{S}^{6}}{v^{6}}\bigg{[}2\log\frac{m_{S}^{2}}{v^{2}}-4\bigg{]}+\frac{1}{2}N(N+2)\frac{\lambda_{S}m_{S}^{4}}{v^{4}}\bigg{[}\log\frac{m_{S}^{2}}{v^{2}}-1\bigg{]}\,.$
(III.9)

Inserting these into equation (III.8), we obtain in terms of $\overline{\rm
MS}$-renormalised parameters

$\displaystyle 8\pi^{2}v^{2}[M_{h}^{2}]_{V_{\text{eff}}}=$ $\displaystyle\
Nm_{S}^{4}-12m_{t}^{4}+6m_{W}^{4}+3m_{Z}^{4}$
$\displaystyle+\frac{4m_{t}^{4}}{16\pi^{2}}\bigg{[}g_{3}^{2}\left(48\log\frac{m_{t}^{2}}{Q^{2}}-40\right)+\frac{m_{t}^{2}}{v^{2}}\left(-18\log\frac{m_{t}^{2}}{Q^{2}}+39\right)\bigg{]}$
$\displaystyle+\frac{4m_{S}^{4}}{16\pi^{2}}\bigg{[}\frac{Nm_{S}^{2}}{v^{2}}\bigg{(}2\log\frac{m_{S}^{2}}{Q^{2}}-3\bigg{)}+\frac{1}{2}N(N+2)\lambda_{S}\bigg{(}\log\frac{m_{S}^{2}}{Q^{2}}-\frac{1}{2}\bigg{)}\bigg{]}\,.$
(III.10)

The first line is simply the one-loop result, while the second and third lines
are respectively the SM-like and BSM two-loop corrections to the relation
between masses.

In order to convert this result in terms of physical masses for the SM and BSM
states, we require the Higgs and BSM scalar self-energies – provided in
appendix B.1 – as well as equation (II.33) for the Higgs VEV. We obtain
finally

$\displaystyle\frac{4\sqrt{2}\pi^{2}}{G_{F}}M_{h}^{2}=$ $\displaystyle\
NM_{S}^{4}-12M_{t}^{4}+6M_{W}^{4}+3M_{Z}^{4}$ (III.11)
$\displaystyle+M_{h}^{2}\bigg{[}\frac{7}{2}M_{t}^{2}-\frac{1}{6}NM_{S}^{2}\bigg{]}+\frac{3M_{t}^{4}}{8\pi^{2}}\bigg{[}16g_{3}^{2}-\frac{6M_{t}^{2}}{v_{\text{OS}}^{2}}\bigg{]}+\frac{NM_{S}^{4}}{16\pi^{2}}\bigg{[}(N+2)\lambda_{S}+\frac{4M_{S}^{2}}{v_{\text{OS}}^{2}}\bigg{]}\,.$

If one considers this relation from the point of view of computing $M_{S}$ in
terms of $\lambda_{S}$, $N$, and the known SM inputs, the two-loop corrections
decrease the extracted value of $M_{S}$. However, in our numerical
investigations, we will instead use this relation to compute $\lambda_{S}$ as
a function of $M_{S}$ and $N$.

###  III.4 Numerical study

At this point, we have derived all the analytical results necessary to study
the possible magnitude of the leading two-loop corrections to the Higgs
trilinear coupling in the $O(N)$-symmetric CSI model. Rather than examining
the absolute value of $\hat{\lambda}_{hhh}$, we prefer to investigate by how
much it deviates from the prediction in the (non-CSI) Standard Model, at the
same leading two-loop order. Therefore, we present in the following results
for the BSM deviation $\delta R$, defined as

$\displaystyle\delta R$
$\displaystyle\equiv\frac{\hat{\lambda}_{hhh}^{\text{BSM}}-\hat{\lambda}_{hhh}^{\text{SM}}}{\hat{\lambda}_{hhh}^{\text{SM}}}\,,$
(III.12)

where $\hat{\lambda}_{hhh}$ is the leading two-loop result for the Higgs
trilinear coupling in the OS scheme and the SM – its analytical expression can
be found for instance in Ref. Braathen:2019zoh  (see also Ref. Senaha:2018xek
; Braathen:2019pxr ). Taking the same input values as in eq. (II.37), we
obtain numerically $\hat{\lambda}_{hhh}^{\text{SM}}\simeq 177.4\text{ GeV}$.

Figure 3: BSM deviation $\delta R$ – defined in eq. (III.12) – of the Higgs
trilinear coupling $\hat{\lambda}_{hhh}$ computed in the $O(N)$-symmetric CSI
model with respect to its prediction in the usual ($i.e.$ non-CSI) SM. The
blue (green) curves show the results at two loops for $N=1$ ($N=4$). Solid and
dashed curves are plotted respectively for $\lambda_{S}=0$ and
$\lambda_{S}=2$. The red dot-dashed line shows the one-loop result – $i.e.$
comparing the one-loop CSI result of $5M_{h}^{2}/v_{\text{OS}}$ to the one-
loop (non-CSI) SM result. The black dotted line corresponds to the comparison
of the one-loop CSI result for $\lambda_{hhh}$ to the SM result at tree level
– $i.e.$ the 67% deviation mentioned in the introduction.

We present in figure 3 our results for $\delta R$ using only the expressions
from section III.2 – in other words, we do not impose the relation between
masses from eq. (III.11) yet. The red dot-dashed line shows the one-loop
result, comparing the one-loop CSI result
$(\hat{\lambda}_{hhh})^{(1)}=5M_{h}^{2}/v_{\text{OS}}$ to the (leading) one-
loop expression in the SM – see for instance Ref. Kanemura:2004mg . We should
point out here that this result differs from the $\sim 67\%$ mentioned in the
introduction, and shown with the black dotted curve: we have $(\delta
R)^{(1)}\simeq 82\%$. This is because the value of $\sim 67\%$ is obtained by
comparing the one-loop $(\hat{\lambda}_{hhh}^{\text{CSI}})^{(1)}$ to the tree-
level SM result for $(\hat{\lambda}_{hhh}^{\text{SM}})^{(0)}$, but once we
include the one-loop contribution from the top quark in the SM expression we
obtain a smaller value for
$(\hat{\lambda}_{hhh}^{\text{SM}})^{(1)}<(\hat{\lambda}_{hhh}^{\text{SM}})^{(0)}$,
and hence a larger deviation. We must emphasise that with our definition of
$(\delta R)^{(1)}$ we are comparing one-loop effective-potential results,
which do not include any dependence on external momenta. It should however be
noted that, as studied in Ref. Kanemura:2004mg , the momentum dependence in
the SM result can be quite significant, and could modify significantly the
value of the BSM deviation. In Ref. Fujitani:2017gma , momentum-dependent
corrections were also included in the study of the Higgs trilinear coupling in
$O(N)$-symmetric CSI models, and were found to be non-negligible.
Nevertheless, we still choose to define $(\delta R)^{(1)}$ in terms of
effective Higgs trilinear couplings, as this allows simpler comparisons with
our new two-loop results, calculated with the effective potential. The result
we find for $(\delta
R)^{(1)}=(\hat{\lambda}_{hhh}^{\text{CSI}})^{(1)}/(\hat{\lambda}_{hhh}^{\text{SM}})^{(1)}-1$
remains totally independent of the particle content of the CSI model.

Next, the blue and green curves in figure 3 are the values of $\delta R$ at
two loops, for different choices of $N$ and $\lambda_{S}$: blue and green
curves correspond respectively to $N=1$ and $N=4$; solid curves are made for
$\lambda_{S}=0$ while the dashed ones are for $\lambda_{S}=2$. For the lowest
values of $M_{S}$ – $i.e.$ where the BSM effects are the smallest – the two-
loop deviation $(\delta R)^{(2)}$ is only slightly larger than at one loop,
however, as one increases $M_{S}$, the BSM contributions grow rapidly. This
growth is stronger for larger values of $N$ and $\lambda_{S}$, and, crucially,
the universality of the BSM shift found at one loop is entirely lost at two
loops. Looking at concrete numerical values of $(\delta R)^{(2)}$, for example
for $M_{S}=500\text{ GeV}$, we have on the one hand for $N=1$ and
$\lambda_{S}=0$ a deviation of $\sim 94\%$, while on the other hand for $N=4$
and $\lambda_{S}=2$ we find $(\delta R)^{(2)}\sim 168\%$. Anticipating a
little on the discussion in the next pages, we note that all parameter points
in this figure 3 fulfill the criterion of tree-level perturbative unitarity
Lee:1977eg ; Hashino:2016rvx .

Figure 4: Comparison of the BSM deviations $\delta R$ computed in
$O(N)$-symmetric models, with (solid curves) and without (dashed curves) CSI,
as a function of the pole mass $M_{S}$ of the BSM scalars. Results for $N=1$
and $N=4$ are shown respectively in the left- and right-hand plots. Red curves
are one-loop values, blue and green curves are two-loop results for
$\lambda_{S}=0$ (respectively for $N=1$ and $N=4$), and black and brown curves
are two-loop results for $\lambda_{S}=2$ (respectively for $N=1$ and $N=4$).

Another interesting comparison we can perform is to investigate the difference
in the BSM deviation in $\hat{\lambda}_{hhh}$ between CSI and non-CSI versions
of $O(N)$-symmetric models. For this purpose, we derive in appendix B.2 the
dominant two-loop corrections to $\hat{\lambda}_{hhh}$ in a non-CSI variant of
the $O(N)$-symmetric model, and for generic values of $N$ – this calculation
follows the procedure developed in Refs. Braathen:2019pxr ; Braathen:2019zoh
(note also that the case of $N=1$ was already treated in Ref. Braathen:2019zoh
, under the name of “Higgs singlet model”). We present in figure 4 a
comparison of our results for the BSM deviation $\delta R$ computed with
respect to the SM result in both CSI and non-CSI $O(N)$-symmetric models, for
$N=1$ (left side) and for $N=4$ (right side). Note that, as there is no mass
term in the CSI $O(N)$-symmetric model, we must set $\tilde{\mu}_{S}=0$ in the
result for the non-CSI version of the model in order to be able to compare
both variants consistently. The solid curves in fig. 4 are obtained for the
versions of the models with CSI, while the results for the non-CSI variants
are given by the dashed curves. Additionally, in both plots, the red curves
show one-loop values, while the blue and black (green and brown) curves are
the two-loop results for $N=1$ ($N=4$) for $\lambda_{S}=0$ and
$\lambda_{S}=2$.

One clearly notices that CSI and non-CSI variants of this BSM model behave
quite differently. First, for low BSM masses, the BSM deviation in the non-CSI
case is minute, while in the CSI case the deviation is already large ($80\%$
or so) – this is because of the difference between the normal SM and CSI SM.
Then, when $M_{S}$ increases, the non-CSI corrections grow much faster than
their CSI counterparts, as in the non-CSI models the one-loop corrections are
proportional to $M_{S}^{4}$ – $c.f.$ equation (B.9).

Figure 5: Regions of the $M_{S}$ and $\lambda_{S}$ parameter space of the
$O(N)$-symmetric CSI model allowed (light green) and excluded (light red) by
the requirement of tree-level perturbative unitarity – following Ref.
Hashino:2016rvx . Additionally, the blue and red curves give the values of
$\lambda_{S}$ computed at respectively one and two loops using equation
(III.11) as a function of $M_{S}$. The left pane is for $N=1$, while the right
one is for $N=4$.

Until this point, we have not taken into account the mass relation – as
derived in section III.3 – nor the theoretical constraint from unitarity. For
the latter, we choose to use as our criterion tree-level888We want to
constrain the values of $\lambda_{S}$ and $M_{S}$, both of which appearing in
$\hat{\lambda}_{hhh}$ only at two-loop order. Therefore, the difference from
including higher-order (instead of tree-level) perturbative unitarity
constraints should be formally a three-loop effect, and thus within the
expected theoretical uncertainty of our result. perturbative unitarity
Lee:1977eg , and we can for the $O(N)$-symmetric model employ the results
given in Ref. Hashino:2016rvx , in particular equation (22) therein. We show
in figure 5 the regions of parameter space – in the $M_{S}$-$\lambda_{S}$
plane – allowed (light green) and excluded (light red) under this criterion of
tree-level perturbative unitarity, for both $N=1$ (left) and $N=4$ (right).
Moreover, we superimpose on top of this plane the curves obtained when
computing $\lambda_{S}$ as a function of $M_{S}$ (and $N$) using the mass
relation we derived in eq. (III.11) – the blue line is for $N=1$ while the red
one is for $N=4$. We recall also that negative values of $\lambda_{S}$ are
forbidden, otherwise the potential would not be bounded from below. Although a
wide range of BSM masses seems to be allowed by unitarity, once one takes into
account the relation between masses in the CSI model – or in other words once
one imposes that $M_{h}=125\text{ GeV}$ – only a narrow interval remains.
Concretely, we find for $N=1$ that $509\text{ GeV}\lesssim M_{S}\lesssim
524\text{ GeV}$, and for $N=4$ that $365\text{ GeV}\lesssim M_{S}\lesssim
375\text{ GeV}$. In fact, at one-loop order – see the first line of eq.
(III.11) – only a single value of $M_{S}$ is allowed for a given $N$, however
once two-loop contributions are included the presence of a new parameter
$\lambda_{S}$ allows moderate variations of $M_{S}$. Conversely, as it only
appears from two loops, $\lambda_{S}$ is not constrained very severely: indeed
values as large as $\lambda_{S}=7.4$ (3.7) are possible for $N=1$ ($N=4$).

Another important criterion that we use to verify the validity of the
parameter points we consider is the true-vacuum condition Politzer:1978ic  –
$i.e.$ the requirement that the EW vacuum is the true minimum of the (two-
loop) potential and corresponds to a lower value of $V_{\text{eff}}$ than the
origin of the potential (at $\varphi_{h}=v+h=0$). In CSI theories, as all
states acquire their masses entirely from their coupling to the Higgs boson,
all masses vanish at the origin $v+h=0$, and hence
$V_{\text{eff}}(\varphi_{h}=0)=0$ and we only need to make sure that
$V_{\text{eff}}(\varphi_{h}=v)$ is negative. The effective potential depends
not only on the field $\varphi_{h}$, but also on the renormalisation scale $Q$
and therefore we examine the value of $V_{\text{eff}}(\varphi_{h}=v)$ for
$Q=v$ and $Q=M_{S}$ – the two choices that we believe to be the most natural.
We find for $N=1$ (where $M_{S}\sim 515\text{ GeV}$) that the true-vacuum
condition is fulfilled for $Q=M_{S}$ but not for $Q=v$, while for $N=4$ (where
$M_{S}\sim 370\text{ GeV})$ the condition is verified for both values of $Q$.
As it is unclear which value of $Q$ would be optimal (both values being around
the EW scale anyway), the case of $N=1$ is not conclusively excluded by this
criterion.

Figure 6: BSM deviation $\delta R$ – defined in eq. (III.12) – of the Higgs
trilinear coupling $\hat{\lambda}_{hhh}$ computed at two loops in the
$O(N)$-symmetric CSI model with respect to its usual ($i.e.$ non-CSI) SM
prediction. The blue curve (left side) is the two-loop result obtained for
$N=1$, while the green curve (right side) is the two-loop result for $N=4$.
Moreover, for this figure, $\lambda_{S}$ is calculated as a function of
$M_{S}$ (for $N=1$ on the left side and $N=4$ on the right side) using
equation (III.11). The red dot-dashed line shows the one-loop result for
$\delta R$, while the black dotted line is the comparison of the one-loop CSI
result with the tree-level SM one. Finally, the light-red shaded regions are
excluded by tree-level perturbative unitarity.

Finally, we present in figure 6 the BSM deviations $\delta R$ we obtain once
we take into account the mass relation and the constraint from perturbative
unitarity. As for previous figures, the left and right panes are for $N=1$ and
$N=4$ respectively, and the red dot-dashed lines show the one-loop result for
$\delta R$ – independent of $M_{S}$ and $N$. The blue and green curves give
the two-loop results for $N=1$ and $N=4$. We emphasise that there is now only
a single result for $\delta R$ at two loops because once $M_{S}$ and $N$ are
fixed, the only remaining parameter – $\lambda_{S}$ – is fixed via equation
(III.11). The ranges of $M_{S}$ for these plots correspond to $\lambda_{S}$
varying between $0$ and $4\pi$, but the light-red shaded regions are excluded
by the criterion of tree-level perturbative unitarity, as discussed above.

Taking into consideration all these theoretical constraints, we find that the
two-loop BSM deviation to $\hat{\lambda}_{hhh}$ must be within the intervals
from $\sim 99\%$ to $\sim 114\%$ and from $\sim 93\%$ to $\sim 109\%$
respectively for $N=1$ and $N=4$. This means that while the universality found
at one loop for $(\delta R)^{(1)}$ is lost at two loops, only a limited range
of values of $(\delta R)^{(2)}$ is possible. Moreover, our computation finds
the Higgs trilinear coupling in CSI $O(N)$-symmetric model to be about twice
as large as in the SM – $(\delta R)^{(2)}\sim 100\%\pm 10\%$ – and is larger
than what is obtained at one loop, which makes the BSM deviation easier to
access in experimental searches.

Before ending our discussion of the CSI $O(N)$-symmetric models, it is also
important to comment on the experimental status of the parameter points we
have considered. We have verified using HiggsBounds Bechtle:2008jh ;
Bechtle:2011sb ; Bechtle:2013wla ; Bechtle:2015pma ; Bechtle:2020pkv  that the
points allowed by under the theoretical criteria discussed above are also
allowed from the point of view of collider searches. The input values for
HiggsBounds are produced by using the Mathematica package SARAH Staub:2008uz ;
Staub:2009bi ; Staub:2010jh ; Staub:2012pb ; Staub:2013tta  to create a
SPheno-based Porod:2003um ; Porod:2011nf  spectrum generator for the CSI
$O(N)$-symmetric models, with $N=1$ as well as $N=4$.

##  IV CSI Two-Higgs-Doublet Model

###  IV.1 Model definition

We now turn to the case of a CSI version of the Two-Higgs-Doublet Model, as
devised in Ref. Lee:2012jn  (see also Ref. Funakubo:1993jg ). We will be
assuming here that CP is conserved in the Higgs sector, and thus we can write
the tree-level potential of the model in terms of two $SU(2)_{L}$ doublets
$\Phi_{1}$ and $\Phi_{2}$ of hypercharge $1/2$ as

$\displaystyle V^{(0)}=$ $\displaystyle\
\frac{1}{2}\lambda_{1}|\Phi_{1}|^{4}+\frac{1}{2}\lambda_{2}|\Phi_{2}|^{4}+\lambda_{3}|\Phi_{1}|^{2}|\Phi_{2}|^{2}+\lambda_{4}|\Phi_{1}^{\dagger}\Phi_{2}|^{2}+\frac{1}{2}\lambda_{5}\left((\Phi_{1}^{\dagger}\Phi_{2})^{2}+(\Phi_{2}^{\dagger}\Phi_{1})^{2}\right)\,,$
(IV.1)

where all the quartic couplings $\lambda_{i}$ are real. To prevent the
occurrence of tree-level flavour-changing neutral currents, we impose a
discrete $\mathbb{Z}_{2}$ symmetry Glashow:1976nt ; Paschos:1976ay  under
which the scalar doublets transform as

$\Phi_{1}\xrightarrow{\mathbb{Z}_{2}}\Phi_{1}\,,\qquad\Phi_{2}\xrightarrow{\mathbb{Z}_{2}}-\Phi_{2}\,,$
(IV.2)

while the fermions remain unaffected.

We expand the two doublets as

$\displaystyle\Phi_{i}=\begin{pmatrix}w_{i}^{+}\\\
\frac{1}{\sqrt{2}}(v_{i}+h_{i}+iz_{i})\end{pmatrix}\,,\qquad i=1,2\,,$ (IV.3)

where $v_{i}$ ($i=1,2$) are the vacuum expectation values of the neutral
components of the doublets. These can be taken to be real – owing to the
assumption of CP conservation – and furthermore verify the relation
$v_{1}^{2}+v_{2}^{2}=v^{2}\simeq(246\text{ GeV})^{2}$. From the ratio of
$v_{1}$ and $v_{2}$, we can also define the usual quantity $\tan\beta\equiv
v_{2}/v_{1}$.

Requiring both VEVs to be non-vanishing, we find for the tree-level tadpole
equations

$\displaystyle\lambda_{1}v_{1}^{2}+\lambda_{345}v_{2}^{2}$
$\displaystyle=0\,,$
$\displaystyle\lambda_{2}v_{2}^{2}+\lambda_{345}v_{1}^{2}$
$\displaystyle=0\,,$ (IV.4)

with the shorthand notation
$\lambda_{345}\equiv\lambda_{3}+\lambda_{4}+\lambda_{5}$. These equations can
be solved as

$\displaystyle\lambda_{1}=-\lambda_{345}\tan^{2}\beta\,,\quad\text{and}\quad\lambda_{2}=-\lambda_{345}\cot^{2}\beta\,,$
(IV.5)

or equivalently as

$\displaystyle\frac{\lambda_{1}}{\lambda_{2}}=\tan^{4}\beta\,,\quad\text{and}\quad\lambda_{1}\lambda_{2}=\lambda_{345}^{2}\,.$
(IV.6)

Next, mass eigenstates are obtained by rotating the gauge eigenstate fields
with the angle $\beta$

$\displaystyle\begin{pmatrix}w_{1}^{\pm}\\\
w_{2}^{\pm}\end{pmatrix}=R(\beta)\begin{pmatrix}G^{\pm}\\\
H^{\pm}\end{pmatrix}\,,\quad\begin{pmatrix}z_{1}\\\
z_{2}\end{pmatrix}=R(\beta)\begin{pmatrix}G\\\
A\end{pmatrix}\,,\quad\begin{pmatrix}h_{1}\\\
h_{2}\end{pmatrix}=R(\beta)\begin{pmatrix}h\\\ H\end{pmatrix}\,,$ (IV.7)

having defined

$\displaystyle R(x)\equiv\begin{pmatrix}\cos x&-\sin x\\\ \sin x&\cos
x\end{pmatrix}\,.$ (IV.8)

Among these fields, $h,H$ are CP-even Higgs boson, $A$ is a CP-odd Higgs
boson, and $H^{\pm}$ is the charged Higgs boson. As in the SM, $G$ and
$G^{\pm}$ are respectively the neutral and charged would-be Goldstone bosons.

In this new basis, the tree-level scalar mass eigenvalues can be found to be

$\displaystyle m_{G}^{2}$ $\displaystyle=m_{G^{\pm}}^{2}=m_{h}^{2}=0\,,$
$\displaystyle m_{H}^{2}=-\lambda_{345}v^{2}\,,\qquad m_{A}^{2}$
$\displaystyle=-\lambda_{5}v^{2}\,,\qquad
m_{H^{\pm}}^{2}=-\frac{1}{2}(\lambda_{4}+\lambda_{5})v^{2}\,.$ (IV.9)

Interestingly, and to the difference of the usual (non-CSI) 2HDM, the CP-even
mass matrix is already diagonal after the rotation of angle $\beta$. In other
words, no additional rotation of angle $\alpha-\beta$ is necessary, and the
CSI-2HDM is naturally aligned (at tree level). Moreover, finding $m_{h}=0$
also tells us that $h$ is indeed the flat direction along which we want to
work in what follows.

We also follow the common choice of trading the free Lagrangian quartic
couplings $\lambda_{3},\,\lambda_{4},\,\lambda_{5}$ for the tree-level masses
of the scalars. From eq. (IV.1), one has

$\displaystyle\lambda_{3}=-\frac{m_{H}^{2}-2m_{H^{\pm}}^{2}}{v^{2}}\,,\qquad\lambda_{4}=\frac{m_{A}^{2}-2m_{H^{\pm}}^{2}}{v^{2}}\,,\qquad\lambda_{5}=-\frac{m_{A}^{2}}{v^{2}}\,.$
(IV.10)

We should note also that in the following, we neglect loop-induced deviations
from the alignment found at tree level. Such effects have been studied for
instance in Ref. Lane:2019dbc  in the CSI 2HDM (see also Ref. Braathen:2017izn
for the usual 2HDM) and were found to be relatively small for most of the
parameter space, so that we consider them subleading before the two-loop
corrections from the BSM scalar states. This enables us to write the effective
potential under the form of equation (II.16) and thus to follow the
calculational procedure described in section II.

As in the usual 2HDM, the two scalar doublets in the CSI-2HDM can couple to
quarks and leptons. Under the $\mathbb{Z}_{2}$ symmetry, four types of
coupling assignments are possible, corresponding to the four types of 2HDMs
Barger:1989fj ; Grossman:1994jb ; Aoki:2009ha . We note however that, for the
major part of the discussion in this section, we do not need to specify a type
as we only consider effects from the top quark – only when verifying
experimental constraints do we assume interactions of type I in order to have
less severe limits from flavour physics than for instance for types II and Y
Misiak:2017bgg .

###  IV.2 Dominant two-loop corrections to the Higgs trilinear coupling

At one loop, the result for $\lambda_{hhh}$ is model-independent and was
discussed in section II. We can now consider the dominant two-loop
contributions to the effective potential, and in turn to the Higgs trilinear
coupling. We choose in this section to present results for the case of
degenerate BSM scalar masses $M_{H}=M_{A}=M_{H^{\pm}}\equiv M_{\Phi}$, because
the expressions for general values of the BSM scalar masses are quite long and
cumbersome, and moreover because this will not affect the physics we discuss
here. We provide the complete expressions for $V^{(2)}$, $\lambda_{hhh}$, and
$\hat{\lambda}_{hhh}$ in appendix C.1. We should also note that as the BSM
scalar masses will only appear in the Higgs trilinear coupling starting from
the two-loop order, it does not make a difference whether we use pole or tree-
level masses for the BSM scalars – the difference in $\lambda_{hhh}$ from
using one or the other is formally of three-loop order.

Figure 7: New dominant two-loop BSM diagrams contributing to the effective
potential of the CSI-2HDM – in addition to the leading SM diagrams of figure
1. The diagrams in the second line all vanish in the limit of degenerate BSM
scalar masses.

The BSM diagrams contributing to leading order to $V^{(2)}$ are shown in
figure 7. In the limit of degenerate masses, their expressions read

$\displaystyle V^{(2)}_{SSS}(h)=$
$\displaystyle-\frac{4m_{\Phi}^{4}(v+h)^{2}}{v^{4}}I(m_{\Phi}^{2}(h),m_{\Phi}^{2}(h),0)-\frac{6m_{\Phi}^{4}\cot^{2}2\beta(v+h)^{2}}{v^{4}}I(m_{\Phi}^{2}(h),m_{\Phi}^{2}(h),m_{\Phi}^{2}(h))\,,$
$\displaystyle V^{(2)}_{SS}(h)=$ $\displaystyle\
\frac{12m_{\Phi}^{2}\cot^{2}2\beta}{v^{2}}J(m_{\Phi}^{2}(h))^{2}\,,$
$\displaystyle V^{(2)}_{FFS}(h)=$ $\displaystyle\
\frac{6m_{t}^{2}\cot^{2}\beta}{v^{2}}\Big{[}J(m_{t}^{2}(h))^{2}-3J(m_{\Phi}^{2}(h))J(m_{t}^{2}(h))+(m_{t}^{2}(h)-m_{\Phi}^{2}(h))I(0,m_{\Phi}^{2}(h),m_{t}^{2}(h))$
$\displaystyle\hskip
56.9055pt+(2m_{t}^{2}(h)-m_{\Phi}^{2}(h))I(m_{\Phi}^{2}(h),m_{t}^{2}(h),m_{t}^{2}(h))\Big{]}\,.$
(IV.11)

From these equations we can extract the BSM logarithm-squared terms, and
including also the SM-like contributions, we find in total for $C^{(2)}$

$\displaystyle C^{(2)}=$ $\displaystyle\
\frac{24g_{3}^{2}m_{t}^{4}}{v^{4}}-\frac{9m_{t}^{6}}{v^{6}}+\frac{m_{\Phi}^{6}}{v^{6}}(4+21\cot^{2}2\beta)+\frac{6m_{t}^{2}\cot^{2}\beta}{v^{6}}\bigg{(}m_{\Phi}^{4}-3m_{\Phi}^{2}m_{t}^{2}-\frac{3}{2}m_{t}^{4}\bigg{)}\,.$
(IV.12)

Applying equation (II.22), we obtain the dominant two-loop corrections to
$\lambda_{hhh}$, involving $g_{3}$, $m_{t}$, $m_{\Phi}$, and $\tan\beta$ as

$\displaystyle\lambda_{hhh}=$ $\displaystyle\
\frac{5[M_{h}^{2}]_{V_{\text{eff}}}}{v}+\frac{1}{(16\pi^{2})^{2}}\bigg{[}\frac{768g_{3}^{2}m_{t}^{4}}{v^{3}}-\frac{288m_{t}^{6}}{v^{5}}+\frac{32m_{\Phi}^{6}}{v^{5}}(4+21\cot^{2}2\beta)$
$\displaystyle\hskip
99.58464pt+\frac{192m_{t}^{2}m_{\Phi}^{4}\cot^{2}\beta}{v^{5}}\left(1-\frac{3m_{t}^{2}}{m_{\Phi}^{2}}-\frac{3m_{t}^{4}}{2m_{\Phi}^{4}}\right)\bigg{]}\,.$
(IV.13)

Lastly, we can – like in the previous sections – express the one-loop result
$5[M_{h}^{2}]_{V_{\text{eff}}}/v$ in terms of the Higgs pole mass and the
physical Higgs VEV, and take into account finite WFR effects, to obtain an OS
result. The only 2HDM-specific corrections come from the momentum-dependent
BSM scalar contributions to the Higgs self-energy. These can be found to be

$\displaystyle\Pi_{hh}^{(1)}(p^{2})\supset-\sum_{\mathclap{\Phi=H,A,H^{\pm}}}\frac{2n_{\Phi}m_{\Phi}^{4}}{v^{2}}B_{0}(p^{2},m_{\Phi}^{2},m_{\Phi}^{2})\,,$
(IV.14)

where $n_{H,A}=1$ and $n_{H^{\pm}}=2$. From this, we have

$\displaystyle\frac{d\Pi_{hh}^{(1)}}{dp^{2}}\bigg{|}_{p^{2}=0}\supset-\frac{1}{3}\sum_{\mathclap{\Phi=H,A,H^{\pm}}}\frac{n_{\Phi}m_{\Phi}^{2}}{v^{2}}\,.$
(IV.15)

Finally, we obtain in terms of OS-renormalised parameters

$\displaystyle\hat{\lambda}_{hhh}=$ $\displaystyle\
\frac{5M_{h}^{2}}{v_{\text{OS}}}+\frac{1}{16\pi^{2}}\frac{5M_{h}^{2}}{v_{\text{OS}}}\bigg{[}\frac{7}{2}\frac{M_{t}^{2}}{v_{\text{OS}}^{2}}-\frac{2}{3}\frac{M_{\Phi}^{2}}{v_{\text{OS}}^{2}}\bigg{]}$
$\displaystyle+\frac{1}{(16\pi^{2})^{2}}\bigg{[}\frac{768g_{3}^{2}M_{t}^{4}}{v_{\text{OS}}^{3}}-\frac{288M_{t}^{6}}{v_{\text{OS}}^{5}}+\frac{32M_{\Phi}^{6}}{v_{\text{OS}}^{5}}(4+21\cot^{2}2\beta)$
$\displaystyle\hskip
56.9055pt+\frac{192M_{t}^{2}M_{\Phi}^{4}\cot^{2}\beta}{v_{\text{OS}}^{5}}\left(1-\frac{3M_{t}^{2}}{M_{\Phi}^{2}}-\frac{3M_{t}^{4}}{2M_{\Phi}^{4}}\right)\bigg{]}\,.$
(IV.16)

###  IV.3 Mass relation

Next, we also consider the mass relation for the case of the CSI-2HDM, which
will serve to relate the BSM scalar masses $M_{H},\,M_{A},\,M_{H^{\pm}}$ and
$\tan\beta$. As in the previous section, we present here only expressions in
the limit of degenerate scalar masses, and leave the complete results for
appendix C.2. To derive corrections to $\lambda_{hhh}$, we have already used
the expression of $C^{(2)}$ given in eq. (IV.12). We must now also derive
results for $B$ at one- and two-loop orders. First, at one loop it is
straightforward to calculate

$\displaystyle B^{(1)}=$ $\displaystyle\
\frac{1}{4v^{4}}\big{(}4m_{\Phi}^{4}-12m_{t}^{4}+6m_{W}^{4}+3m_{Z}^{4}\big{)}\,.$
(IV.17)

At two loops, we have

$\displaystyle B^{(2)}=$ $\displaystyle\
\frac{4g_{3}^{2}m_{t}^{4}}{v^{4}}\left[12\log\frac{m_{t}^{2}}{v^{2}}-16\right]+\frac{6m_{t}^{6}}{v^{6}}\left[-3\log\frac{m_{t}^{2}}{v^{2}}+8\right]+B^{(2)}_{SS}+B^{(2)}_{SSS}+B^{(2)}_{F\bar{F}S}\,,$
(IV.18)

where the first two terms are SM-like contributions and the latter three –
$B^{(2)}_{SS},\,B^{(2)}_{SSS},\,B^{(2)}_{F\bar{F}S}$ – are the BSM ones. These
can be found to be in the limit of degenerate BSM masses

$\displaystyle B^{(2)}_{SS}=$ $\displaystyle\
\frac{24m_{\Phi}^{6}\cot^{2}2\beta}{v^{6}}\left(\log\frac{m_{\Phi}^{2}}{v^{2}}-1\right)\,,$
$\displaystyle B^{(2)}_{SSS}=$ $\displaystyle\
\frac{2m_{\Phi}^{6}}{v^{6}}(4+9\cot^{2}2\beta)\left(\log\frac{m_{\Phi}^{2}}{v^{2}}-2\right)\,,$
$\displaystyle B^{(2)}_{F\bar{F}S}=$ $\displaystyle\
\frac{6m_{t}^{2}\cot^{2}\beta}{v^{6}}\bigg{\\{}2m_{\Phi}^{4}\left(\log\frac{m_{\Phi}^{2}}{v^{2}}-2\right)-6m_{t}^{2}m_{\Phi}^{2}\left(\log\frac{m_{\Phi}^{2}}{v^{2}}-1\right)-m_{t}^{4}\left(3\log\frac{m_{t}^{2}}{v^{2}}-8\right)\bigg{\\}}\,.$
(IV.19)

Applying equation (III.8), the mass relation at two loops is, in terms of
$\overline{\rm MS}$-renormalised quantities,

$\displaystyle 8\pi^{2}v^{2}[M_{h}^{2}]_{V_{\text{eff}}}=$ $\displaystyle\
4m_{\Phi}^{4}-12m_{t}^{4}+6m_{W}^{4}+3m_{Z}^{4}$
$\displaystyle+\frac{m_{t}^{4}}{4\pi^{2}}\bigg{[}g_{3}^{2}\left(48\log\frac{m_{t}^{2}}{Q^{2}}-40\right)+\frac{m_{t}^{2}}{v^{2}}\left(-18\log\frac{m_{t}^{2}}{Q^{2}}+39\right)\bigg{]}$
$\displaystyle+\frac{1}{4\pi^{2}}\Bigg{\\{}\frac{24m_{\Phi}^{6}\cot^{2}2\beta}{v^{2}}\left(\log\frac{m_{\Phi}^{2}}{Q^{2}}-\frac{1}{2}\right)+\frac{2m_{\Phi}^{6}}{v^{2}}[4+9\cot^{2}2\beta]\left(\log\frac{m_{\Phi}^{2}}{Q^{2}}-\frac{3}{2}\right)$
$\displaystyle\hskip
42.67912pt+\frac{6m_{t}^{2}\cot^{2}\beta}{v^{2}}\bigg{[}2m_{\Phi}^{4}\left(\log\frac{m_{\Phi}^{2}}{Q^{2}}-\frac{3}{2}\right)-6m_{t}^{2}m_{\Phi}^{2}\left(2\log\frac{m_{\Phi}^{2}}{Q^{2}}-\frac{1}{2}\right)$
$\displaystyle\hskip 113.81102pt-
m_{t}^{4}\left(3\log\frac{m_{t}^{2}}{Q^{2}}-\frac{13}{2}\right)\bigg{]}\Bigg{\\}}\,.$
(IV.20)

The first line in this equation is the one-loop result, while the additional
lines are the two-loop corrections. Among these, the first (square) brackets
are the SM-like contributions, and the remains terms (in the curly brackets)
are the BSM terms, proper to the CSI-2HDM.

To convert equation (IV.3) from the $\overline{\rm MS}$ to the OS scheme, we
require as previously the correction to the Higgs VEV – $c.f$ equation (II.33)
and the top-quark and Higgs-boson one-loop self-energies, as well as the self-
energies for the BSM scalars $H,\,A,\,H^{\pm}$ (also at one loop). All
necessary expressions are provided in appendix C.2. With these, we are able to
obtain the following result, in terms only of physical parameters

$\displaystyle\frac{4\sqrt{2}\pi^{2}}{G_{F}}M_{h}^{2}=$ $\displaystyle\
4M_{\Phi}^{4}-12M_{t}^{4}+6M_{W}^{4}+3M_{Z}^{4}+M_{h}^{2}\left[\frac{7}{2}M_{t}^{2}-\frac{2}{3}M_{\Phi}^{2}\right]+\frac{3M_{t}^{4}}{8\pi^{2}}\left[16g_{3}^{2}-\frac{6M_{t}^{2}}{v_{\text{OS}}^{2}}\right]$
$\displaystyle+\frac{M_{\Phi}^{6}}{4\pi^{2}v_{\text{OS}}^{2}}\bigg{[}4+3\bigg{(}7-2\sqrt{3}\pi\bigg{)}\cot^{2}2\beta\bigg{]}+\frac{3M_{t}^{2}\cot^{2}\beta}{2\pi^{2}v_{\text{OS}}^{2}}\Bigg{\\{}-M_{\Phi}^{4}-3M_{t}^{2}M_{\Phi}^{2}+\frac{13}{2}M_{t}^{4}$
$\displaystyle\hskip
14.22636pt-2M_{t}^{2}(M_{\Phi}^{2}-2M_{t}^{2})\mathfrak{Re}\bigg{[}f_{B}\left(\frac{M_{\Phi}^{2}}{2M_{t}^{2}}-\sqrt{\frac{M_{\Phi}^{2}}{4M_{t}^{4}}-1}\right)+f_{B}\left(\frac{M_{\Phi}^{2}}{2M_{t}^{2}}+\sqrt{\frac{M_{\Phi}^{2}}{4M_{t}^{4}}-1}\right)\bigg{]}$
$\displaystyle\hskip 14.22636pt-
M_{\Phi}^{2}(M_{\Phi}^{2}-2M_{t}^{2})\mathfrak{Re}\bigg{[}f_{B}\left(\frac{1}{2}-\sqrt{\frac{1}{4}-\frac{M_{t}^{2}}{M_{\Phi}^{2}}}\right)+f_{B}\left(\frac{1}{2}+\sqrt{\frac{1}{4}-\frac{M_{t}^{2}}{M_{\Phi}^{2}}}\right)\bigg{]}\Bigg{\\}}\,,$
(IV.21)

where the function $f_{B}$ is defined in equation (A.1), and
$\mathfrak{Re}[x]$ denotes the real part of a given quantity $x$.

###  IV.4 Numerical study

Figure 8: BSM deviation $\delta R$ – defined in eq. (III.12) – of the Higgs
trilinear coupling $\hat{\lambda}_{hhh}$ computed at one loop (dot-dashed
line) and at two loops (solid and dashed curves) in the CSI-2HDM with respect
to its SM prediction as a function of the degenerate mass of the BSM scalars
$M_{\Phi}=M_{H}=M_{A}=M_{H^{\pm}}$. The different grey, teal, blue, and green
two-loop curves correspond respectively to $\tan\beta$ equal to 1, 1.1, 1.2,
and 1.4. As in the previous section, the black dotted line is obtained by
comparing the one-loop CSI result for $\lambda_{hhh}$ to the SM tree-level
result.

We have now derived all the results we require to study numerically the
leading two-loop corrections to the Higgs trilinear coupling in the CSI-2HDM.
We follow here the same outline as we did for the $O(N)$-symmetric model, and
we provide results for the same type of BSM deviation $\delta R$ – as defined
in eq. (III.12) – comparing the prediction for the Higgs trilinear coupling in
the CSI-2HDM to that in the usual (non-CSI) SM. Moreover, we will consider the
three BSM scalars $H,\,A,\,H^{\pm}$ to be degenerate in mass – this ensures
that the $\rho$ parameter remains close to one, and furthermore this
conveniently reduces the number of BSM mass scales to one.

We present first in figure 8 the deviation $\delta R$ at one loop (red line)
and two loops (grey, teal, blue, and green curves) for different values of
$\tan\beta$. We employ here the expression in equation (IV.2), without for now
taking into account the relation between masses. As we had seen for the
$N$-scalar model, while at one loop the BSM deviation in $\hat{\lambda}_{hhh}$
has a universal value of $\delta R\sim 82\%$, the two-loop corrections
introduce new dependences on both $M_{\Phi}$ and $\tan\beta$, thereby spoiling
the universality of $\delta R$. We consider in this plot different values of
$\tan\beta$, ranging from $\tan\beta=1$ to $\tan\beta=1.4$, which we find to
be the largest value of $\tan\beta$ for which tree-level perturbative
unitarity Lee:1977eg ; Kanemura:1993hm  is maintained all the way to
$M_{\Phi}=500\text{ GeV}$ – see figure 10 and the discussion in the following.
Considering for instance $M_{\Phi}=500\text{ GeV}$, we find $(\delta
R)^{(2)}=134\%$ for $\tan\beta=1$, and $(\delta R)^{(2)}=162\%$ for
$\tan\beta=1.4$, in stark contrast to the one-loop result.

At this point, we should comment briefly on our interpretation of the quantity
$\tan\beta$. While $\tan\beta$ is originally defined as the ratio of the VEVs
of the two scalar doublets in the model, it appears in two-loop corrections to
the Higgs trilinear coupling via the dependence of four-point scalar
interactions at tree level and of the coupling between BSM scalars and the top
quark. It should be emphasised however that because $\tan\beta$ appears only
from the two-loop order, any scheme difference in how its value could be
extracted from some experimental data (in the event a 2HDM Higgs sector is
discovered) only has an impact from three-loop order and beyond – $i.e.$ below
the level of accuracy to which we are working.

Figure 9: Comparison of the BSM deviations $\delta R$ computed in versions of
the 2HDM, with (solid curves) and without (dashed curves) CSI, as a function
of the degenerate pole mass $M_{\Phi}$ of the BSM scalars. Red curves present
one-loop values, while grey and green curves are two-loop results for
$\tan\beta=1$ and $\tan\beta=1.4$ respectively.

Next, it is also interesting to compare predictions for the Higgs trilinear
coupling in variants of the 2HDM with or without CSI. For this purpose, we
present in figure 9 the BSM deviations in $\hat{\lambda}_{hhh}$ computed with
respect to the usual-SM prediction in both variants of the 2HDM. For the usual
2HDM, we use the analytic expression for $\hat{\lambda}_{hhh}$ derived in Ref.
Braathen:2019pxr ; Braathen:2019zoh  – see specifically equation (5.15) in
Ref. Braathen:2019zoh  – in which we set the mass term $\tilde{M}$
(representing the scale of the soft breaking of the $\mathbb{Z}_{2}$ symmetry
acting on the Higgs doublets) to zero to allow the comparison with the CSI
scenario.

We can observe that the Higgs trilinear coupling behaves very differently in
CSI and non-CSI versions of the 2HDM. The main discrepancy occurs at one-loop
level: indeed, while in the CSI-2HDM $(\delta R)^{(1)}$ is constant, in the
usual (non-CSI) 2HDM, one-loop corrections involving the BSM scalars are
minute in the low-mass range, but grow rapidly (as $M_{\Phi}^{4}$) and reach
$\simeq 240\%$ for $M_{\Phi}=500\text{ GeV}$. At two loops, the radiative
corrections to $\hat{\lambda}_{hhh}$ behave quite similarly in the CSI and
non-CSI scenarios – the main difference arising from effects due to the scheme
conversion of the one-loop BSM corrections in the non-CSI 2HDM case ($c.f.$
Ref. Braathen:2019zoh ).

Figure 10: Regions of the $\tan\beta$ and $M_{\Phi}$ parameter space of the
CSI-2HDM allowed (light green) and excluded (light red) by the requirement of
tree-level perturbative unitarity Kanemura:1993hm . Additionally, the red and
blue curves give the values of $M_{\Phi}$ computed at respectively one and two
loops using equation (IV.3) as a function of $\tan\beta$. The right-hand side
plot is an enlargement of the left-hand one.

At this point, we should discuss the theoretical constraints on the BSM
parameters – $M_{\Phi}$ and $\tan\beta$ – coming from unitarity and from the
mass relation in eq. (IV.3). For the former, we choose to take tree-level
perturbative unitarity Lee:1977eg  as our criterion, and we employ results
from Ref. Kanemura:1993hm  (see also Ref. Akeroyd:2000wc ). Figure 10 shows
the allowed (light green) and excluded (light red) regions of the CSI-2HDM
parameter space, in the $M_{\Phi}-\tan\beta$ plane. Additionally, the values
of $M_{\Phi}$ that are extracted from the equation (IV.3) are given by the
dashed red (at one loop) and solid blue (at two loops) curves. On the one
hand, at one loop $\tan\beta$ does not appear in the mass relation and hence
at a constant value is found for $M_{\Phi}$. On the other hand, at two loops,
the result for $M_{\Phi}$ obtained from eq. (IV.3) becomes a function of
$\tan\beta$. Along the two-loop (blue) curve, we find that the maximal
possible value of $\tan\beta$ is 1.855, which corresponds to a maximal value
of the degenerate mass of the BSM scalars $M_{\Phi}\leq 382.2\text{ GeV}$. As
for the $O(N)$-symmetric models, we have also verified that the true-vacuum
condition is respected in the CSI-2HDM: we have found that the value of the
two-loop potential is, at the EW minimum, indeed lower than the value of the
potential at the origin, and this is true both for $Q=v$ and $Q=M_{\Phi}\sim
378\text{ GeV}$.

Figure 11: BSM deviation $\delta R$ – defined in eq. (III.12) – of the Higgs
trilinear coupling $\hat{\lambda}_{hhh}$ computed at two loops in the CSI-2HDM
with respect to its SM prediction as a function of the degenerate mass of the
BSM scalars $M_{\Phi}=M_{H}=M_{A}=M_{H^{\pm}}$. The value of $\tan\beta$ in
the point in this figure is computed as a function of $M_{\Phi}$ using
equation (IV.3). Finally, the maximal value of $M_{\Phi}$ (equivalently the
maximal value of $\tan\beta$) in this plot is constrained by the requirement
of tree-level perturbative unitarity – $c.f.$ figure 10. The red dot-dashed
and black dotted lines show the comparison of the one-loop CSI value of
$\lambda_{hhh}$ with respectively the one-loop effective-potential and tree-
level results in the SM.

Having considered the constraints on $M_{\Phi}$ and $\tan\beta$, we can
finally investigate the theoretically-allowed values of the Higgs trilinear
coupling. Figure 11 therefore presents our results for the deviation $\delta
R$ at one loop (red dot-dashed line) and at two loops (blue solid curve), once
$\tan\beta$ is extracted as a function of $M_{\Phi}$ from the mass relation in
eq. (IV.3). As can be seen in figure 10, the possible values of $M_{\Phi}$ are
all within the narrow interval from $373.9\text{ GeV}$ to $382.1\text{ GeV}$.
We find that $\delta R$ varies between $89.9\%$ at the lowest, and $112.5\%$
at most. In other words, as for the $O(N)$-symmetric model earlier, we obtain
a larger deviation of $\hat{\lambda}_{hhh}$ from its deviation at two loops,
and the total BSM deviation is (approximately) in the range $100\pm 10\%$.

We have verified that the corresponding parameter points are not excluded by
experimental searches for BSM scalars. Assuming Yukawa interactions of type I
(to benefit from less severe constraints from flavour physics than for types
II or Y, see $e.g.$ Ref. Misiak:2017bgg ), we create a SPheno-based spectrum
generator for the CSI-2HDM using SARAH, which we in turn used to provide
inputs for HiggsBounds to verify the experimental status of the parameter
choices we took. We ensured that the Lagrangian inputs provided to SPheno
yielded the corrected spectrum of scalar masses – a 125-GeV SM-like Higgs
boson and BSM scalars at 374 GeV, with an allowed error of 1 GeV – using the
Mathematica package SSP Staub:2011dp .

##  V Discussion

The computations presented in this paper allow studying the Higgs trilinear
coupling in models with classical scale invariance to the same level of
accuracy as for non-CSI extensions of the SM – as done for instance in Refs.
Braathen:2019pxr ; Braathen:2019zoh ; Senaha:2018xek  (and appendix B.2 of
this work). Classical scale invariance is an attractive concept for model
building, and may relate to the solution to the hierarchy problem. It is
therefore of paramount importance to find new ways to investigate CSI models,
in complement of collider and dark matter searches – see $e.g.$ Refs
Endo:2015ifa ; Endo:2015nba ; Endo:2016koi ; Helmboldt:2016mpi ;
Fujitani:2017gma ; Lane:2019dbc ; Brooijmans:2020yij .

Due to the simple form of the potential at one loop, containing only two free
quantities – recall equation (II.4) – $\hat{\lambda}_{hhh}$ is found to be
predicted universally at the one-loop level, in all CSI models. This
universality of $(\hat{\lambda}_{hhh})^{(1)}$ is lifted by the inclusion of
two-loop effects, which thus allows distinguishing different scenarios of CSI
models. We found that the new corrections at two loops are quite significant,
leading to a further 5-30% deviation of $\hat{\lambda}_{hhh}$ from the (non-
CSI) SM prediction, compared to the one-loop CSI result. We should emphasise
that this is by no means a problem from the point of view of the validity of
the perturbative loop expansion: indeed the two-loop corrections to
$\hat{\lambda}_{hhh}$ involve new parameters that are not present at one loop
– namely $M_{S},\,\lambda_{S},\,N$ for the $O(N)$-symmetric models and
$M_{H},\,M_{A},\,M_{H^{\pm}},\,\tan\beta$ in the CSI-2HDM. Furthermore, as
there are numerous arguments in favour of extended scalar sectors, it is
interesting to note that the two-loop contributions to the Higgs trilinear
coupling receives from scalars are always positive, which makes it deviate
further from the SM value so that it is more easily accessible in collider
searches than what is expected from the one-loop result.

We have also paid particular attention to the relation among masses arising in
theories with CSI, which is often over-looked in the literature. More than
being simply bounded from above, the masses of BSM states in CSI models are
actually strongly constrained by the requirement of reproducing the correct
125-GeV mass of the Higgs boson – which is entirely generated at loop level.
In the scenarios considered in the main text of this paper, we imposed
additional constraints – a global $O(N)$ symmetry for the $N$-scalar model,
and mass degeneracy of the BSM scalars in the CSI-2HDM – which resulted in
severe bounds on the allowed BSM masses.999Admittedly, if these restrictions
were to be loosened (as $e.g.$ in appendix C), there would be more freedom to
vary the BSM scalar masses. Nevertheless, the masses would remain related so
as to ensure the correctness of the Higgs mass.  In turn, this allows us to
find predictions for $\hat{\lambda}_{hhh}$ within a narrow range: in both
types of BSM models, we obtain $90\%\lesssim(\delta R)^{(2)}\lesssim 115\%$.
Looking back at figures 4 and 9, it is worth noting that the values obtained
for $(\delta R)^{(2)}$ for the allowed BSM mass ranges are very close to those
found for the same masses in the non-CSI counterparts of the scenarios. Taking
the example of the 2HDM, while in the CSI version of the model we have
$90\%\lesssim(\delta R)^{(2)}\lesssim 110\%$, in the non-CSI variant and
taking the non-decoupling limit ($i.e.$ $\tilde{M}=0$) the value of $(\delta
R)^{(2)}$ computed at $M_{\Phi}\simeq 374\text{ GeV}$ varies from $\sim 90\%$
(for $\tan\beta=1$) to $105\%$ (for $\tan\beta=1.9$, close to the upper limit
from perturbative unitarity). This implies that the magnitude of a potential
BSM deviation in the Higgs trilinear coupling may not provide a clear
indication of whether the associated new physics exhibits classical scale
invariance – instead, a better criterion might come from whether masses and
couplings fulfill the mass relations imposed by CSI.

Rather than considering only a single type of observable, a powerful strategy
to investigate CSI theories, and to discriminate theories with or without CSI,
would be to study the correlation between several observables. In particular,
synergies between measurements of the Higgs trilinear coupling and direct
searches of new particles at colliders may prove crucial to ascertain whether
Nature exhibits classical scale invariance or not. Indeed, we should recall
that due to the relation among masses in CSI models, the new BSM states cannot
be arbitrarily heavy (even in non-degenerate scenarios) and will be within
reach of experiments in a foreseeable future (see for instance the discussions
for the CSI-2HDM in Refs. Lane:2019dbc ; Brooijmans:2020yij ). In the
eventuality that a new scalar state is found in direct searches, determining
the value of the Higgs trilinear coupling and verifying whether the relation
among masses is fulfilled or not can potentially provide clear evidence of the
CSI nature of the underlying theory. Furthermore, if one considers a model
with charged scalars ($e.g.$ $H^{\pm}$ in the CSI-2HDM), sizeable one-loop
corrections to the Higgs-to-two-photon ($h\gamma\gamma$) coupling can also be
expected, as studied in Ref. Hashino:2015nxa . Such effects could also help
confirm the existence (or absence) of CSI in BSM models. In this context,
depending on the achieved accuracy of the determination of the $h\gamma\gamma$
coupling at future colliders – possibly down to percent level, see $e.g.$
Refs. Fujii:2015jha ; Fujii:2017vwa  for prospects at the ILC – it may become
important (if not unavoidable) to compute theory predictions for this coupling
at two-loop level in order to match the precision of experimental results and
allow consistent comparisons between theory and experiment. We leave this
endeavour for future work.

A caveat that should be mentioned is that the scenarios considered in this
work require relatively large scalar quartic couplings – once again, this
comes from the need to generate the correct mass of the Higgs boson, in the
absence of any BSM mass term in the Lagrangian. Consequently, under
renormalisation-group running, we can expect these couplings to grow rapidly
and encounter Landau poles well before the Planck scale – in a similar way as
what occurs in the non-decoupling limit of (non-CSI) extensions of the SM.
This is consistent with our expectation that the assumption of CSI at the
electroweak scale can a priori not be extended up to the Planck scale, and
therefore does not solve the hierarchy problem by itself.

Finally, we should also point out the importance of computing the Higgs
trilinear coupling to high precision when studying the behaviour of a number
of BSM phenomena occuring in models with extended scalar sectors – in
particular the possibility of a strong-first order electroweak phase
transition and the spectrum of gravitational waves that may be produced during
the former. In complementarity with collider searches, the future measurement
of primordial gravitational waves (GW) at the LISA and DECIGO space-based
interferometers will provide a new way to probe the value of the Higgs
trilinear coupling and the shape of the Higgs potential. The synergy between
the two – collider and GW measurements – was studied at one loop for
$O(N)$-symmetric CSI models in Ref. Hashino:2016rvx  (see also Refs.
Kakizaki:2015wua ; Hashino:2016xoj ; Hashino:2018wee  for similar studies in
non-CSI theories). Given the large deviation from the SM in the one-loop
predictions for $\lambda_{hhh}$, one expects in CSI models the EWPT to be of
strong first order ($c.f.$ Ref Kanemura:2004ch ). However, two-loop
corrections to the Higgs trilinear coupling certainly affect the strength of
the EWPT to some extent, although for the time being this type of effects have
only been studied for a non-CSI IDM in Ref. Senaha:2018xek  – the EWPT was
then found to be slightly weaker once two-loop corrections to
$\hat{\lambda}_{hhh}$ where included. In turn, this also modifies the spectrum
of gravitational waves produced during the EWPT if it is of strong first-order
nature. There is therefore strong motivation to consider two-loop corrections
to the Higgs trilinear coupling at finite temperatures in CSI theories.

##  VI Summary

In this paper, we have performed the first explicit calculation of leading
two-loop corrections to the Higgs trilinear coupling in models with classical
scale invariance, using the effective-potential approximation. For the wide
range of CSI models in which the scalon/Higgs direction does not mix with
other states – so that we can compute $V_{\text{eff}}$ as in eq. (II.16) – we
have derived general two-loop results relating the coefficients of the
effective potential to the corrections to the Higgs trilinear coupling.
Importantly, we find that while at one loop the prediction for
$\hat{\lambda}_{hhh}$ is the same in all CSI theories, this is not any more
the case at two loops.

We focused our investigations on two particular types of CSI theories: first
an $N$-scalar model with a global $O(N)$ symmetry (where we took either $N=1$
or $N=4$ for numerical applications), and next a CSI-2HDM. For both models, we
computed the leading corrections at two loops involving the BSM scalars – and
in the CSI-2HDM, the top quark as well. Because these expressions are derived
from $V^{(2)}$, which is computed in the $\overline{\rm MS}$ scheme, results
are obtained at first in the $\overline{\rm MS}$ scheme. Therefore, we have
included the necessary scheme conversion to rewrite the corrections to the
Higgs trilinear coupling in terms of physical parameters (pole masses and
physical Higgs VEV) and to take into account the effects from finite WFR. Once
we turn to the numerical study of these models, we observe that the two-loop
corrections clearly lift the degeneracy in $\hat{\lambda}_{hhh}$ among
different BSM masses and parameters. Furthermore, the results for
$\hat{\lambda}_{hhh}$ in CSI and non-CSI variants of the same model strongly
differ in their theoretical behaviours, mainly because of the difference at
one loop – constant (mass-independent) value in the CSI case, and corrections
proportional to the fourth power of the BSM mass(es) in the non-CSI case.

However, when we also consider the constraints on the allowed ranges of the
BSM parameters coming from the criterion of tree-level perturbative unitarity
and from the mass relation in CSI theories, the possible values of
$\hat{\lambda}_{hhh}$ become quite restricted, namely $90\%\lesssim(\delta
R)^{(2)}\lesssim 110-115\%$. In other words, while the Higgs trilinear
coupling at two loops is not universally predicted in all CSI models (without
mixing), its allowed values are still severely limited by the classical scale
invariance. Nevertheless, the additional positive deviation of order $5-30\%$
of the Higgs trilinear coupling from its SM prediction at two loops can make
the BSM discrepancy easier to find at future experiments, and is of the same
order of magnitude as the expected accuracy of the determination of
$\lambda_{hhh}$ at future colliders (as discussed in the introduction).
Therefore, two-loop calculations will in the future prove necessary for CSI
theories in order to consistently compare theoretical predictions with
experimental results.

While it may be difficult to distinguish between CSI scenarios and non-CSI
scenarios (in the non-decoupling limit) simply from the size of the Higgs
trilinear coupling, our work demonstrates the important role of the mass
relation that strongly constrains masses and parameters in CSI theories, and
may prove to be one of the most powerful tools to probe the CSI nature of a
potential BSM discovery. Additionally, as we have mentioned in the previous
section, the synergy of the measurement of the spectrum of primordial
gravitational waves produced during the EWPT together with that of the Higgs
trilinear coupling can also help distinguish scenarios with or without CSI
Hashino:2016rvx .

Finally, in appendix D we have provided generic $\overline{\rm MS}$ results
for the coefficients $B^{(2)}$ and $C^{(2)}$ (see eq. (II.16)) applicable to
all CSI models without mixing, and which served to cross-check some of our
calculations. We hope these can be of use for the community to study further
CSI scenarios.

## Acknowledgements

This work is supported by JSPS, Grant-in-Aid for Scientific Research, No.
16H06492, 18F18022, 18F18321 and 20H00160. This work is also partly supported
by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy – EXC 2121 “Quantum Universe” – 390833306.

##  Appendix A Loop functions

This appendix summarises notations and definitions of loop functions used
throughout this paper.

First of all, the loop factor is defined as

$\kappa=\frac{1}{16\pi^{2}}\,.$ (A.1)

Next, we denote the regularisation and renormalisation scales respectively
$\mu$ and $Q$. These two scales are related as

$\displaystyle Q^{2}=4\pi e^{-\gamma_{E}}\mu^{2}\,,$ (A.2)

where $\gamma_{E}\simeq 0.577$ is the Euler constant. For convenience, we also
make extensive use of the notation

$\operatorname{\overline{\text{log}}}x\equiv\log\frac{x}{Q^{2}}\,.$ (A.3)

###  A.1 One-loop functions

At one-loop order, we employ the following two integrals

$\displaystyle\mathbf{J}(x)$
$\displaystyle\equiv-16\pi^{2}\frac{\mu^{2\epsilon}}{i(2\pi)^{d}}\int_{k}\frac{d^{d}k}{k^{2}-x}\,,$
$\displaystyle\mathbf{B}(p^{2},x,y)$ $\displaystyle\equiv
16\pi^{2}\frac{\mu^{2\epsilon}}{i(2\pi)^{d}}\int_{k}\frac{d^{d}k}{(k^{2}-x)((p-k)^{2}-y)}\,,$
(A.4)

defined in dimensional regularisation, with $d=4-2\epsilon$. Their finite (and
$\epsilon$-independent) part yield the usual Passarino-Veltmann functions
Passarino:1978jh

$\displaystyle J(x)$ $\displaystyle\equiv\lim_{\epsilon\to
0}\bigg{[}\mathbf{J}(x)+\frac{x}{\epsilon}\bigg{]}=x(\operatorname{\overline{\text{log}}}x-1)\,,$
$\displaystyle B_{0}(p^{2},x,y)$ $\displaystyle\equiv\lim_{\epsilon\to
0}\bigg{[}\mathbf{B}(p^{2},x,y)-\frac{1}{\epsilon}\bigg{]}=-\operatorname{\overline{\text{log}}}p^{2}-f_{B}(x_{+})-f_{B}(x_{-}),$
(A.5)

where

$\displaystyle f_{B}(x)$
$\displaystyle=\log(1-x)-x\log\left(1-\frac{1}{x}\right)-1\,,$ $\displaystyle
x_{\pm}$
$\displaystyle=\frac{p^{2}+x-y\pm\sqrt{(p^{2}+x-y)^{2}-4p^{2}x}}{2p^{2}}\,.$
(A.6)

Some simple limits of particular interest include

$\displaystyle B_{0}(x,0,0)$
$\displaystyle=B_{0}(x,0,x)=2-\operatorname{\overline{\text{log}}}x\,,$
$\displaystyle B_{0}(x,x,x)$
$\displaystyle=2-\frac{\pi}{\sqrt{3}}-\operatorname{\overline{\text{log}}}x\,,$
$\displaystyle B_{0}(x,0,y)$
$\displaystyle=2-\operatorname{\overline{\text{log}}}y+\left(\frac{y}{x}-1\right)\log\left(1-\frac{x}{y}\right)\,.$
(A.7)

Finally, when computing contributions from finite WFR, it is useful to expand
the $B_{0}$ function for low momentum $p^{2}$ with the relation

$\displaystyle B_{0}(p^{2},x,x)\underset{p^{2}\ll
x}{=}-\operatorname{\overline{\text{log}}}x+\frac{p^{2}}{6x}+\mathcal{O}\left(\frac{p^{4}}{x^{2}}\right)\,.$
(A.8)

###  A.2 Two-loop functions

When considering two-loop corrections to the effective potential, we also come
to encounter the sunrise integral $\mathbf{I}$ (see $e.g.$ Ref. Ford:1992pn )
defined as

$\displaystyle\mathbf{I}(x,y,z)\equiv(16\pi^{2})^{2}\frac{\mu^{4\epsilon}}{(2\pi)^{2d}}\int_{k_{1}}\int_{k_{2}}\frac{d^{d}k_{1}d^{d}k_{2}}{(k_{1}^{2}-x)(k_{2}^{2}-y)((k_{1}+k_{2})^{2}-z)}\,.$
(A.9)

Its finite part is

$\displaystyle I(x,y,z)\equiv\lim_{\epsilon\to
0}\left[\mathbf{I}(x,y,z)-\frac{\mathbf{J}(x)+\mathbf{J}(y)+\mathbf{J}(z)}{\epsilon}-\frac{1}{2}\left(\frac{1}{\epsilon^{2}}-\frac{1}{\epsilon}\right)(x+y+z)\right]\,,$
(A.10)

with $\mathbf{J}$ the one-loop integral from eq. (A.1). Expressions for $I$,
equivalent up to relations among dilogarithms can be found in many references
(see for instance Refs. Ford:1992pn ; Martin:2001vx ; Martin:2003qz ;
Degrassi:2009yq ). Ref. Degrassi:2009yq  provides the result in a convenient
way, which we reproduce here

$\displaystyle I(x,y,z)=$ $\displaystyle\
\frac{1}{2}\big{[}(x-y-z)\operatorname{\overline{\text{log}}}y\operatorname{\overline{\text{log}}}z+(y-z-x)\operatorname{\overline{\text{log}}}x\operatorname{\overline{\text{log}}}z+(z-x-y)\operatorname{\overline{\text{log}}}x\operatorname{\overline{\text{log}}}y\big{]}$
$\displaystyle+2\big{[}x\operatorname{\overline{\text{log}}}x+y\operatorname{\overline{\text{log}}}y+z\operatorname{\overline{\text{log}}}z\big{]}-\frac{5}{2}(x+y+z)-\frac{\Delta(x,y,z)}{2z}\Phi(x,y,z)\,,$
(A.11)

with

$\displaystyle\Delta(x,y,z)=$ $\displaystyle\
x^{2}+y^{2}+z^{2}-2(xy+xz+yz)\,,$ (A.12) $\displaystyle\Phi(x,y,z)=$
$\displaystyle\ \frac{1}{\lambda(x,y,z)}\bigg{[}2\log X_{+}\log
X_{-}-\log\frac{x}{z}\log\frac{y}{z}-2\big{(}\mathrm{Li}_{2}(X_{+})+\mathrm{Li}_{2}(X_{-})\big{)}+\frac{\pi^{2}}{3}\bigg{]}\,.$
(A.13)

$\mathrm{Li}_{2}(z)\equiv-\int_{0}^{z}\log(1-t)/tdt$ is the dilogarithm
function, and

$\displaystyle\lambda(x,y,z)=$ $\displaystyle\
\frac{1}{z}\big{[}\big{(}z-x-y\big{)}^{2}-4xy\big{]}^{1/2}\,,$ $\displaystyle
X_{\pm}=$ $\displaystyle\
\frac{1}{2}\bigg{[}1\pm\frac{x-y}{z}-\lambda(x,y,z)\bigg{]}\,.$ (A.14)

Note that the definition of the function $\Phi$ in eq. (A.13) is only valid
for $x,y<z$, and the other branches of the function can be found with the
relations

$\displaystyle\Phi(x,y,z)=\Phi(y,x,z)\,,\qquad\text{and}\qquad x\
\Phi(x,y,z)=z\ \Phi(z,y,x)\,.$ (A.15)

Furthermore, a number of useful limits are included in Refs. Martin:2003qz ;
Braathen:2016cqe , among which

$\displaystyle I(x,x,x)=$ $\displaystyle\
\frac{3}{2}x\left[-\operatorname{\overline{\text{log}}}^{2}x+4\operatorname{\overline{\text{log}}}x-5-\frac{i}{\sqrt{3}}\bigg{(}\frac{\pi^{2}}{9}-4\
\mathrm{Li}_{2}\left(\frac{1}{2}-\frac{i\sqrt{3}}{2}\right)\bigg{)}\right]\,,$
$\displaystyle I(x,x,0)=$ $\displaystyle\
x\big{[}-\operatorname{\overline{\text{log}}}^{2}x+4\operatorname{\overline{\text{log}}}x-5\big{]}\,,$
$\displaystyle I(x,0,0)=$ $\displaystyle\
\frac{1}{2}x\left[-\operatorname{\overline{\text{log}}}^{2}x+4\operatorname{\overline{\text{log}}}x-5-\frac{\pi^{2}}{3}\right]\,.$
(A.16)

##  Appendix B Detailed results for the $N$-scalar model

###  B.1 With classical scale invariance

We present in this appendix intermediate results that we employed for the
$\overline{\rm MS}$ to OS scheme conversion of our expressions for the CSI
$O(N)$-symmetric model. The different parameters we need conversions for are
$v$, $m_{t}$, $[M_{h}^{2}]_{V_{\text{eff}}}$, and $m_{S}$. First of all, for
the Higgs VEV, we can use the same result as in the SM, $c.f.$ equation
(II.33).

Next, the one-loop self-energy of the top quark is the same as in the SM (the
BSM singlet scalars do not couple to it) and it reads

$\displaystyle\Pi_{tt}^{(1)}(p^{2}=m_{t}^{2})=$ $\displaystyle\
\frac{4}{3}g_{3}^{2}m_{t}^{2}(8-6\operatorname{\overline{\text{log}}}m_{t}^{2})+\frac{m_{t}^{4}}{v^{2}}(-8+3\operatorname{\overline{\text{log}}}m_{t}^{2})\,.$
(B.1)

For the conversion from the Higgs curvature to pole mass and the Higgs WFR, we
only require the momentum-dependent contributions to the one-loop Higgs self-
energy – momentum-independent terms cancel out when taking the difference
$\Pi_{hh}(p^{2}=m_{h}^{2})-\Pi_{hh}(p^{2}=0)$ and moreover their derivative
with respect to $p^{2}$ is also zero of course. The momentum-dependent
corrections are

$\displaystyle\Pi^{(1)}_{hh}(p^{2})\bigg{|}_{p^{2}\text{-dep.}}=\frac{6m_{t}^{2}}{v^{2}}\bigg{[}(4m_{t}^{2}-p^{2})B_{0}(p^{2},m_{t}^{2},m_{t}^{2})\bigg{]}-\frac{2Nm_{S}^{4}}{v^{2}}B_{0}(p^{2},m_{S}^{2},m_{S}^{2})\,,$
(B.2)

where the first term is the SM-like correction from the top quark and the
second term is the BSM contribution from the scalars $S_{i}$.

Finally, we find for the one-loop self-energy of the BSM scalars

$\displaystyle\Pi^{(1)}_{SS}(p^{2})=(N+2)\lambda_{S}J(m_{S}^{2})-4\lambda_{\Phi
S}^{2}v^{2}B_{0}(p^{2},0,m_{S}^{2})\,.$ (B.3)

###  B.2 Without classical scale invariance

In order to compare predictions for the Higgs trilinear coupling in
$O(N)$-symmetric models with and without CSI, we derive in this appendix the
leading two-loop contributions to $\hat{\lambda}_{hhh}$ in the non-CSI version
of these models. Note that the case for $N=1$ was already considered in Ref.
Braathen:2019zoh  – where it was referred to as the ”Higgs-Singlet Model”
(HSM).

If we do not impose classical scale invariance, the tree-level scalar
potential in a theory invariant under a global $O(N)$ symmetry reads

$\displaystyle V^{(0)}=$ $\displaystyle\
\mu^{2}|\Phi|^{2}+\frac{1}{2}\mu_{S}^{2}\vec{S}^{2}+\lambda|\Phi|^{4}+\lambda_{\Phi
S}\vec{S}^{2}|\Phi|^{2}+\frac{1}{4}\lambda_{S}(\vec{S}^{2})^{2}\,.$ (B.4)

Due to the unbroken $O(N)$ symmetry, the singlets $S_{i}$ do not acquire VEVs,
and the only tadpole equation gives at tree level

$\mu^{2}=-\lambda v^{2}\,.$ (B.5)

The field-dependent tree-level masses read

$\displaystyle m_{h}^{2}(h)=\mu^{2}+3\lambda(v+h)^{2}\,,\quad
m_{G}^{2}(h)=m_{G^{\pm}}^{2}(h)=\mu^{2}+\lambda(v+h)^{2}\,,\quad
m_{S_{i}}^{2}(h)=\mu_{S}^{2}+\lambda_{\Phi S}(v+h)^{2}\,.$ (B.6)

The dominant two-loop contributions to the effective potential in this model
are the same as shown in figure 2, and the expressions are (like for the CSI
version of the model):

$\displaystyle V^{(2)}=$ $\displaystyle\ V^{(2)}_{hSS}+V^{(2)}_{SS}$
$\displaystyle V^{(2)}_{hSS}=$ $\displaystyle-\sum_{i=1}^{N}\lambda_{\Phi
S}^{2}(v+h)^{2}I(0,m_{S_{i}}^{2}(h),m_{S_{i}}^{2}(h))$ $\displaystyle=$
$\displaystyle-N\lambda_{\Phi S}^{2}(v+h)^{2}I(m_{S}^{2}(h),m_{S}^{2}(h),0)$
$\displaystyle V^{(2)}_{SS}=$ $\displaystyle\
\sum_{i=1}^{N}\frac{1}{8}\lambda_{S_{i}S_{i}S_{i}S_{i}}J(m_{S_{i}}^{2}(h))^{2}+\sum_{i=1}^{N}\sum_{j=1,j\neq
i}^{N}\frac{1}{8}\lambda_{S_{i}S_{i}S_{j}S_{j}}J(m_{S_{i}}^{2}(h))J(m_{S_{j}}^{2}(h))$
$\displaystyle=$ $\displaystyle\
\frac{1}{4}N(N+2)\lambda_{S}J(m_{S}^{2}(h))^{2}$

Following the calculation method explained in detail in Ref. Braathen:2019zoh
and using the differential operator $\mathcal{D}_{3}$ defined therein, we find
for the dominant two-loop BSM corrections to the Higgs trilinear coupling
$\lambda_{hhh}$

$\displaystyle\delta^{(2)}\lambda_{hhh}=$ $\displaystyle\
\frac{1}{(16\pi^{2})^{2}}\mathcal{D}_{3}V^{(2)}\big{|}_{\text{min.}}$
$\displaystyle=$ $\displaystyle\
\frac{1}{(16\pi^{2})^{2}}\bigg{\\{}\frac{16Nm_{S}^{4}}{v^{5}}\left(1-\frac{\mu_{S}^{2}}{m_{S}^{2}}\right)^{4}\left[-m_{S}^{2}-2\mu_{S}^{2}+(2m_{S}^{2}+\mu_{S}^{2})\operatorname{\overline{\text{log}}}m_{S}^{2}\right]$
$\displaystyle\hskip
56.9055pt+\frac{4N(N+2)\lambda_{S}m_{S}^{4}}{v^{3}}\left(1-\frac{\mu_{S}^{2}}{m_{S}^{2}}\right)^{3}[1+2\operatorname{\overline{\text{log}}}m_{S}^{2}]\bigg{\\}}\,.$
(B.8)

For the leading one-loop BSM corrections, we have simply

$\displaystyle\delta^{(1)}\lambda_{hhh}=$ $\displaystyle\
\frac{1}{16\pi^{2}}\mathcal{D}_{3}V^{(1)}\big{|}_{\text{min.}}=\frac{1}{16\pi^{2}}\frac{4Nm_{S}^{4}}{v^{3}}\left(1-\frac{\mu_{S}^{2}}{m_{S}^{2}}\right)^{3}\,.$
(B.9)

To convert this result to the on-shell scheme, we need the one-loop self-
energy of the scalars $S_{i}$, given in eq. (B.3), as well as the finite
counter-term for $\mu_{S}^{2}$ (to ensure proper decoupling of the OS result
when taking $\tilde{\mu}_{S}^{2}\to\infty$ as discussed at length in Ref.
Braathen:2019zoh ). The latter can be obtained straightforwardly by adapting
the result in eq. (5.29) of Ref. Braathen:2019zoh  (originally for the HSM) to
our case. We find

$\displaystyle\delta^{\text{OS}}\mu_{S}^{2}=\frac{1}{16\pi^{2}}(N+2)\lambda_{S}\tilde{\mu}_{S}^{2}(\operatorname{\overline{\text{log}}}M_{S}^{2}-1)\,.$
(B.10)

Combining all these results, we obtain for the dominant one-loop and two-loop
BSM contributions to the Higgs trilinear coupling, expressed in the OS scheme

$\displaystyle(16\pi^{2})\,\delta^{(1)}\hat{\lambda}_{hhh}=$ $\displaystyle\
\frac{4NM_{S}^{4}}{v_{\text{OS}}^{3}}\left(1-\frac{\tilde{\mu}_{S}^{2}}{M_{S}^{2}}\right)^{3}-\frac{NM_{S}^{2}M_{h}^{2}}{2v_{\text{OS}}^{3}}\left(1-\frac{\tilde{\mu}_{S}^{2}}{M_{S}^{2}}\right)^{2}\,,$
$\displaystyle(16\pi^{2})^{2}\delta^{(2)}\hat{\lambda}_{hhh}=$ $\displaystyle\
\frac{48NM_{S}^{6}}{v_{\text{OS}}^{5}}\left(1-\frac{\tilde{\mu}_{S}^{2}}{M_{S}^{2}}\right)^{4}+\frac{12N(N+2)\lambda_{S}M_{S}^{4}}{v_{\text{OS}}^{3}}\left(1-\frac{\tilde{\mu}_{S}^{2}}{M_{S}^{2}}\right)^{3}$
$\displaystyle+\frac{42NM_{S}^{4}M_{t}^{2}}{v_{\text{OS}}^{5}}\left(1-\frac{\tilde{\mu}_{S}^{2}}{M_{S}^{2}}\right)^{3}+\frac{24NM_{S}^{2}M_{t}^{4}}{v_{\text{OS}}^{5}}\left(1-\frac{\tilde{\mu}_{S}^{2}}{M_{S}^{2}}\right)^{2}-\frac{2N^{2}M_{S}^{6}}{v_{\text{OS}}^{5}}\left(1-\frac{\tilde{\mu}_{S}^{2}}{M_{S}^{2}}\right)^{5}\,.$
(B.11)

The last three terms ($i.e.$ the second line) come from WF and VEV
renormalisations. Setting $N=1$ and taking into account the slightly different
convention for $\lambda_{S}$, we do recover the same result as in equation
(5.31) of Ref. Braathen:2019zoh .

##  Appendix C Detailed results for the CSI-2HDM

In this appendix, we generalise the results of the section IV for the case
where the masses of the three BSM scalars are not degenerate.

###  C.1 Results for $V^{(2)}$ and $\hat{\lambda}_{hhh}$

First of all, the leading two-loop BSM corrections to $V_{\text{eff}}$ of the
CSI-2HDM read

$\displaystyle V^{(2)}_{SSS}(h)=$
$\displaystyle-\sum_{\Phi=H,A,H^{\pm}}\frac{n_{\Phi}m_{\Phi}^{4}(v+h)^{2}}{v^{4}}I(0,m_{\Phi}^{2}(h),m_{\Phi}^{2}(h))$
$\displaystyle-\sum_{\Phi=A,H^{\pm}}\frac{n_{\Phi}m_{H}^{4}\cot^{2}2\beta(v+h)^{2}}{v^{4}}I(m_{H}^{2}(h),m_{\Phi}^{2}(h),m_{\Phi}^{2}(h))$
$\displaystyle-\frac{3m_{H}^{4}\cot^{2}2\beta(v+h)^{2}}{v^{4}}I(m_{H}^{2}(h),m_{H}^{2}(h),m_{H}^{2}(h))$
$\displaystyle-\frac{(v+h)^{2}}{2v^{4}}\bigg{[}(m_{H}^{2}-m_{A}^{2})^{2}I(m_{H}^{2}(h),m_{A}^{2}(h),0)+2(m_{H}^{2}-m_{H^{\pm}}^{2})^{2}I(m_{H}^{2}(h),m_{H^{\pm}}^{2}(h),0)$
$\displaystyle\hskip
56.9055pt+2(m_{A}^{2}-m_{H^{\pm}}^{2})^{2}I(m_{A}^{2}(h),m_{H^{\pm}}^{2}(h),0)\bigg{]}$
(C.1) $\displaystyle V^{(2)}_{SS}(h)=$ $\displaystyle\
\frac{m_{H}^{2}\cot^{2}2\beta}{v^{2}}\bigg{[}\frac{3}{2}J(m_{H}^{2}(h))^{2}+\frac{3}{2}J(m_{A}^{2}(h))^{2}+4J(m_{H^{\pm}}^{2}(h))^{2}+J(m_{A}^{2}(h))J(m_{H}^{2}(h))$
$\displaystyle\hskip
113.81102pt+2J(m_{H}^{2}(h))J(m_{H^{\pm}}^{2}(h))+2J(m_{A}^{2}(h))J(m_{H^{\pm}}^{2}(h))\bigg{]}$
(C.2) $\displaystyle V^{(2)}_{FFS}(h)=$
$\displaystyle-\frac{3}{2}y_{t}^{2}c_{\beta}^{2}\bigg{[}2J(m_{t}^{2}(h))J(m_{H}^{2}(h))-J(m_{t}^{2}(h))^{2}-(4m_{t}^{2}(h)-m_{H}^{2}(h))I(m_{H}^{2}(h),m_{t}^{2}(h),m_{t}^{2}(h))\bigg{]}$
$\displaystyle-\frac{3}{2}y_{t}^{2}c_{\beta}^{2}\bigg{[}2J(m_{t}^{2}(h))J(m_{A}^{2}(h))-J(m_{t}^{2}(h))^{2}+m_{A}^{2}(h)I(m_{A}^{2}(h),m_{t}^{2}(h),m_{t}^{2}(h))\bigg{]}$
$\displaystyle-3y_{t}^{2}c_{\beta}^{2}\bigg{[}J(m_{t}^{2}(h))J(m_{H^{\pm}}^{2}(h))-(m_{t}^{2}(h)-m_{H^{\pm}}^{2}(h))I(m_{H^{\pm}}^{2}(h),m_{t}^{2}(h),0)\bigg{]}$
(C.3)

Turning next to the $\log^{2}$ terms $C^{(2)}$, we have first for the scalar
sunrise diagrams

$\displaystyle C^{(2)}_{SSS}=$ $\displaystyle\
\frac{m_{H}^{6}+m_{A}^{6}+2m_{H^{\pm}}^{6}}{v^{6}}+\frac{m_{H}^{4}\cot^{2}2\beta}{v^{6}}\big{[}6m_{H}^{2}+m_{A}^{2}+2m_{H^{\pm}}^{2}\big{]}$
$\displaystyle+\frac{1}{4}\frac{(m_{H}^{2}-m_{A}^{2})^{2}(m_{H}^{2}+m_{A}^{2})}{v^{6}}+\frac{1}{2}\frac{(m_{H}^{2}-m_{H^{\pm}}^{2})^{2}(m_{H}^{2}+m_{H^{\pm}}^{2})}{v^{6}}$
$\displaystyle+\frac{1}{2}\frac{(m_{A}^{2}-m_{H^{\pm}}^{2})^{2}(m_{A}^{2}+m_{H^{\pm}}^{2})}{v^{6}}$
(C.4)

Second, for the eight-shaped diagrams we have

$\displaystyle C^{(2)}_{SS}$ $\displaystyle=\
\frac{m_{H}^{2}\cot^{2}2\beta}{2v^{6}}\left[3m_{H}^{4}+3m_{A}^{4}+8m_{H^{\pm}}^{4}+2m_{H}^{2}m_{A}^{2}+4m_{H^{\pm}}^{2}(m_{H}^{2}+m_{A}^{2})\right]\,.$
(C.5)

Finally, for the diagrams with top quarks, we find

$\displaystyle C^{(2)}_{F\bar{F}S}$
$\displaystyle=\frac{3m_{t}^{2}\cot^{2}\beta}{2v^{6}}\big{[}m_{H}^{4}+m_{A}^{4}+2m_{H^{\pm}}^{4}-2m_{t}^{2}\big{(}3m_{H}^{2}+m_{A}^{2}+2m_{H^{\pm}}^{2}\big{)}-6m_{t}^{4}\big{]}\,.$
(C.6)

Grouping all these results together, and using eq. (II.22), we find

$\displaystyle\lambda_{hhh}=\frac{5[M_{h}^{2}]_{V_{\text{eff}}}}{v}+\frac{1}{(16\pi^{2})^{2}}\bigg{[}$
$\displaystyle\frac{768g_{3}^{2}m_{t}^{4}}{v^{3}}-\frac{288m_{t}^{6}}{v^{5}}+\frac{56m_{H}^{6}+56m_{A}^{6}+96m_{H^{\pm}}^{6}}{v^{5}}$
$\displaystyle-\frac{8m_{H}^{4}m_{A}^{2}+8m_{H}^{2}m_{A}^{4}+16m_{H}^{4}m_{H^{\pm}}^{2}+16m_{H}^{2}m_{H^{\pm}}^{4}+16m_{A}^{4}m_{H^{\pm}}^{2}+16m_{A}^{2}m_{H^{\pm}}^{4}}{v^{5}}$
$\displaystyle+\frac{32m_{H}^{4}\cot^{2}2\beta}{v^{5}}\big{(}6m_{H}^{2}+m_{A}^{2}+2m_{H^{\pm}}^{2}\big{)}$
$\displaystyle+\frac{16m_{H}^{2}\cot^{2}2\beta}{v^{5}}\left(3m_{H}^{4}+3m_{A}^{4}+8m_{H^{\pm}}^{4}+2m_{H}^{2}m_{A}^{2}+4m_{H^{\pm}}^{2}(m_{H}^{2}+m_{A}^{2})\right)$
$\displaystyle+\frac{48m_{t}^{2}\cot^{2}\beta}{v^{5}}\big{(}m_{H}^{4}+m_{A}^{4}+2m_{H^{\pm}}^{4}-2m_{t}^{2}\big{(}3m_{H}^{2}+m_{A}^{2}+2m_{H^{\pm}}^{2}\big{)}-6m_{t}^{4}\big{)}\bigg{]}\,.$
(C.7)

By taking $m_{H}=m_{A}=m_{H^{\pm}}=m_{\Phi}$, we can straightforwardly recover
eq. (IV.2).

In order to convert this expression from the $\overline{\rm MS}$ to the OS
scheme, we need to take into account the finite renormalisation of the Higgs
VEV – as given in equation (II.33), as well as the correction to the Higgs
mass and the effect of finite WFR. The latter two are calculated from the
momentum-dependent part of the Higgs self-energy, which in the CSI-2HDM
receives new contributions from each of the BSM scalars. In total, it reads

$\displaystyle\Pi_{hh}^{(1)}(p^{2})\big{|}_{p^{2}\text{-dep.}}=$
$\displaystyle\
\frac{6m_{t}^{2}}{v^{2}}\bigg{[}(4m_{t}^{2}-p^{2})B_{0}(p^{2},m_{t}^{2},m_{t}^{2})\bigg{]}$
$\displaystyle-\frac{2m_{H}^{4}}{v^{2}}B_{0}(p^{2},m_{H}^{2},m_{H}^{2})-\frac{2m_{A}^{4}}{v^{2}}B_{0}(p^{2},m_{A}^{2},m_{A}^{2})-\frac{4m_{H^{\pm}}^{4}}{v^{2}}B_{0}(p^{2},m_{H^{\pm}}^{2},m_{H^{\pm}}^{2})\,.$
(C.8)

This enables us to obtain finally in the OS scheme

$\displaystyle\hat{\lambda}_{hhh}=$ $\displaystyle\
\frac{5M_{h}^{2}}{v_{\text{OS}}}+\frac{1}{16\pi^{2}}\frac{5M_{h}^{2}}{v_{\text{OS}}}\bigg{[}\frac{7}{2}\frac{M_{t}^{2}}{v_{\text{OS}}^{2}}-\frac{1}{6}\frac{M_{H}^{2}+M_{A}^{2}+2M_{H^{\pm}}^{2}}{v_{\text{OS}}^{2}}\bigg{]}$
$\displaystyle+\frac{1}{(16\pi^{2})^{2}}\bigg{[}\frac{768g_{3}^{2}M_{t}^{4}}{v_{\text{OS}}^{3}}-\frac{288M_{t}^{6}}{v_{\text{OS}}^{5}}+\frac{56M_{H}^{6}+56M_{A}^{6}+96M_{H^{\pm}}^{6}}{v_{\text{OS}}^{5}}$
$\displaystyle\hskip
49.79231pt-\frac{8(M_{H}^{4}M_{A}^{2}+M_{H}^{2}M_{A}^{4}+2M_{H}^{4}M_{H^{\pm}}^{2}+2M_{H}^{2}M_{H^{\pm}}^{4}+2M_{A}^{4}M_{H^{\pm}}^{2}+2M_{A}^{2}M_{H^{\pm}}^{4})}{v_{\text{OS}}^{5}}$
$\displaystyle\hskip
49.79231pt+\frac{32M_{H}^{4}\cot^{2}2\beta}{v_{\text{OS}}^{5}}\big{(}6M_{H}^{2}+M_{A}^{2}+2M_{H^{\pm}}^{2}\big{)}$
$\displaystyle\hskip
49.79231pt+\frac{16M_{H}^{2}\cot^{2}2\beta}{v_{\text{OS}}^{5}}\left(3M_{H}^{4}+3M_{A}^{4}+8M_{H^{\pm}}^{4}+2M_{H}^{2}M_{A}^{2}+4M_{H^{\pm}}^{2}(M_{H}^{2}+M_{A}^{2})\right)$
$\displaystyle\hskip
49.79231pt+\frac{48M_{t}^{2}\cot^{2}\beta}{v_{\text{OS}}^{5}}\big{(}M_{H}^{4}+M_{A}^{4}+2M_{H^{\pm}}^{4}-2M_{t}^{2}\big{(}3M_{H}^{2}+M_{A}^{2}+2M_{H^{\pm}}^{2}\big{)}-6M_{t}^{4}\big{)}\bigg{]}\,.$
(C.9)

###  C.2 Results for the mass relation

####  C.2.1 Expression in the $\overline{\rm MS}$ scheme

Finally, we also consider the mass relation for the CSI-2HDM with general BSM
scalar masses, and therefore we start by deriving expressions for the
quantities $C^{(2)},\,B^{(1)},\,B^{(2)}$. First of all, we have that

$\displaystyle
C^{(2)}=\frac{24g_{3}^{2}m_{t}^{4}}{v^{4}}-\frac{9m_{t}^{6}}{v^{6}}+C^{(2)}_{SS}+C^{(2)}_{SSS}+C^{(2)}_{F\bar{F}S}\,,$
(C.10)

with $C^{(2)}_{SS}$, $C^{(2)}_{SSS}$, and $C^{(2)}_{F\bar{F}S}$ given
respectively in equations (C.1), (C.5), and (C.6). Next, $B^{(1)}$ is simply

$\displaystyle B^{(1)}=$ $\displaystyle\
\frac{1}{4v^{4}}\big{(}m_{H}^{4}+m_{A}^{4}+2m_{H^{\pm}}^{4}-12m_{t}^{4}+6m_{W}^{4}+3m_{Z}^{4}\big{)}\,.$
(C.11)

The expression for $B^{(2)}$ is significantly longer, and can be written as

$\displaystyle B^{(2)}=$ $\displaystyle\
\frac{4g_{3}^{2}m_{t}^{4}}{v^{4}}\left[12\log\frac{m_{t}^{2}}{v^{2}}-16\right]+\frac{6m_{t}^{6}}{v^{6}}\left[-3\log\frac{m_{t}^{2}}{v^{2}}+8\right]+B^{(2)}_{SS}+B^{(2)}_{SSS}+B^{(2)}_{F\bar{F}S}\,,$
(C.12)

with

$\displaystyle B^{(2)}_{SS}=$ $\displaystyle\
\frac{3m_{H}^{6}\cot^{2}2\beta}{v^{6}}\left(\log\frac{m_{H}^{2}}{v^{2}}-1\right)+\frac{3m_{H}^{2}m_{A}^{4}\cot^{2}2\beta}{v^{6}}\left(\log\frac{m_{A}^{2}}{v^{2}}-1\right)$
$\displaystyle+\frac{8m_{H}^{2}m_{H^{\pm}}^{4}\cot^{2}2\beta}{v^{6}}\left(\log\frac{m_{H^{\pm}}^{2}}{v^{2}}-1\right)+\frac{m_{H}^{4}m_{A}^{2}\cot^{2}2\beta}{v^{6}}\left(\log\frac{m_{H}^{2}}{v^{2}}+\log\frac{m_{A}^{2}}{v^{2}}-2\right)$
$\displaystyle+\frac{2m_{H}^{4}m_{H^{\pm}}^{2}\cot^{2}2\beta}{v^{6}}\left(\log\frac{m_{H}^{2}}{v^{2}}+\log\frac{m_{H^{\pm}}^{2}}{v^{2}}-2\right)$
$\displaystyle+\frac{2m_{H}^{2}m_{A}^{2}m_{H^{\pm}}^{2}\cot^{2}2\beta}{v^{6}}\left(\log\frac{m_{A}^{2}}{v^{2}}+\log\frac{m_{H^{\pm}}^{2}}{v^{2}}-2\right)\,,$
$\displaystyle B^{(2)}_{SSS}=$ $\displaystyle\
\frac{2m_{H}^{6}}{v^{6}}\left(\log\frac{m_{H}^{2}}{v^{2}}-2\right)+\frac{2m_{A}^{6}}{v^{6}}\left(\log\frac{m_{A}^{2}}{v^{2}}-2\right)+\frac{4m_{H^{\pm}}^{6}}{v^{6}}\left(\log\frac{m_{H^{\pm}}^{2}}{v^{2}}-2\right)$
$\displaystyle+\frac{12m_{H}^{6}\cot^{2}2\beta}{v^{6}}\left(\log\frac{m_{H}^{2}}{v^{2}}-2\right)+\frac{2m_{H}^{4}m_{A}^{2}\cot^{2}2\beta}{v^{6}}\left(\log\frac{m_{A}^{2}}{v^{2}}-2\right)$
$\displaystyle+\frac{4m_{H}^{4}m_{H^{\pm}}^{2}\cot^{2}2\beta}{v^{6}}\left(\log\frac{m_{H^{\pm}}^{2}}{v^{2}}-2\right)$
$\displaystyle+\frac{1}{2v^{6}}\bigg{\\{}(m_{H}^{2}-m_{A}^{2})^{2}\bigg{[}m_{H}^{2}\left(\log\frac{m_{H}^{2}}{v^{2}}-2\right)+m_{A}^{2}\left(\log\frac{m_{A}^{2}}{v^{2}}-2\right)\bigg{]}$
$\displaystyle\hskip
36.98866pt+2(m_{H}^{2}-m_{H^{\pm}}^{2})^{2}\bigg{[}m_{H}^{2}\left(\log\frac{m_{H}^{2}}{v^{2}}-2\right)+m_{H^{\pm}}^{2}\left(\log\frac{m_{H^{\pm}}^{2}}{v^{2}}-2\right)\bigg{]}$
$\displaystyle\hskip
36.98866pt+2(m_{A}^{2}-m_{H^{\pm}}^{2})^{2}\bigg{[}m_{A}^{2}\left(\log\frac{m_{A}^{2}}{v^{2}}-2\right)+m_{H^{\pm}}^{2}\left(\log\frac{m_{H^{\pm}}^{2}}{v^{2}}-2\right)\bigg{]}\bigg{\\}}\,,$
(C.13)

and

$\displaystyle B^{(2)}_{F\bar{F}S}=$ $\displaystyle\
B^{(2)}_{ttH}+B^{(2)}_{ttA}+B^{(2)}_{tbH^{\pm}}\,,\text{ with}$ $\displaystyle
B^{(2)}_{ttH}=$ $\displaystyle\
\frac{3m_{t}^{2}\cot^{2}\beta}{v^{6}}\bigg{\\{}(4m_{t}^{2}-m_{H}^{2})\bigg{[}2m_{t}^{2}\left(2-\log\frac{m_{t}^{2}}{v^{2}}\right)+m_{H}^{2}\left(2-\log\frac{m_{H}^{2}}{v^{2}}\right)\bigg{]}$
$\displaystyle\hskip
71.13188pt+2m_{t}^{4}\left(\log\frac{m_{t}^{2}}{v^{2}}-1\right)-2m_{t}^{2}m_{H}^{2}\left(\log\frac{m_{t}^{2}}{v^{2}}+\log\frac{m_{H}^{2}}{v^{2}}-2\right)\bigg{\\}}$
$\displaystyle=$ $\displaystyle\
\frac{3m_{t}^{2}\cot^{2}\beta}{v^{6}}\bigg{\\{}m_{H}^{4}\left(\log\frac{m_{H}^{2}}{v^{2}}-2\right)-2m_{t}^{2}m_{H}^{2}\left(3\log\frac{m_{H}^{2}}{v^{2}}-4\right)-2m_{t}^{4}\left(3\log\frac{m_{t}^{2}}{v^{2}}-7\right)\bigg{\\}}\,,$
$\displaystyle B^{(2)}_{ttA}=$ $\displaystyle\
\frac{3m_{t}^{2}\cot^{2}\beta}{v^{6}}\bigg{\\{}-m_{A}^{2}\bigg{[}2m_{t}^{2}\left(2-\log\frac{m_{t}^{2}}{v^{2}}\right)+m_{A}^{2}\left(2-\log\frac{m_{A}^{2}}{v^{2}}\right)\bigg{]}$
$\displaystyle\hskip
71.13188pt+2m_{t}^{4}\left(\log\frac{m_{t}^{2}}{v^{2}}-1\right)-2m_{t}^{2}m_{A}^{2}\left(\log\frac{m_{t}^{2}}{v^{2}}+\log\frac{m_{A}^{2}}{v^{2}}-2\right)\bigg{\\}}$
$\displaystyle=$ $\displaystyle\
\frac{3m_{t}^{2}\cot^{2}\beta}{v^{6}}\bigg{[}m_{A}^{4}\left(\log\frac{m_{A}^{2}}{v^{2}}-2\right)-2m_{t}^{2}m_{A}^{2}\log\frac{m_{A}^{2}}{v^{2}}+2m_{t}^{4}\left(\log\frac{m_{t}^{2}}{v^{2}}-1\right)\bigg{]}\,,$
$\displaystyle B^{(2)}_{tbH^{\pm}}=$ $\displaystyle\
\frac{6m_{t}^{2}\cot^{2}\beta}{v^{6}}\bigg{\\{}m_{H^{\pm}}^{4}\left(\log\frac{m_{H^{\pm}}^{2}}{v^{2}}-2\right)-2m_{t}^{2}m_{H^{\pm}}^{2}\left(\log\frac{m_{H^{\pm}}^{2}}{v^{2}}-1\right)-m_{t}^{4}\left(\log\frac{m_{t}^{2}}{v^{2}}-2\right)\bigg{\\}}\,.$
(C.14)

As an intermediate step, we next calculate the sum
$B^{(2)}+C^{(2)}(1+2\log{v^{2}}/{Q^{2}})$ for the different contributions –
$SS$, $SSS$, $ttH$, $ttA$, and $tbH^{\pm}$ – and we denote this quantity as
$D^{(2)}$. We obtain

$\displaystyle D^{(2)}_{SS}\equiv$ $\displaystyle\
B^{(2)}_{SS}+C^{(2)}_{SS}\left(1+2\log\frac{v^{2}}{Q^{2}}\right)$
$\displaystyle=$ $\displaystyle\
\frac{3m_{H}^{6}\cot^{2}2\beta}{v^{6}}\left(\log\frac{m_{H}^{2}}{Q^{2}}-\frac{1}{2}\right)+\frac{3m_{H}^{2}m_{A}^{4}\cot^{2}2\beta}{v^{6}}\left(\log\frac{m_{A}^{2}}{Q^{2}}-\frac{1}{2}\right)$
$\displaystyle+\frac{8m_{H}^{2}m_{H^{\pm}}^{4}\cot^{2}2\beta}{v^{6}}\left(\log\frac{m_{H^{\pm}}^{2}}{Q^{2}}-\frac{1}{2}\right)+\frac{m_{H}^{4}m_{A}^{2}\cot^{2}2\beta}{v^{6}}\left(\log\frac{m_{H}^{2}}{Q^{2}}+\log\frac{m_{A}^{2}}{Q^{2}}-1\right)$
$\displaystyle+\frac{2m_{H}^{4}m_{H^{\pm}}^{2}\cot^{2}2\beta}{v^{6}}\left(\log\frac{m_{H}^{2}}{Q^{2}}+\log\frac{m_{H^{\pm}}^{2}}{Q^{2}}-1\right)$
$\displaystyle+\frac{2m_{H}^{2}m_{A}^{2}m_{H^{\pm}}^{2}\cot^{2}2\beta}{v^{6}}\left(\log\frac{m_{A}^{2}}{Q^{2}}+\log\frac{m_{H^{\pm}}^{2}}{Q^{2}}-1\right)\,,$
(C.15)

$\displaystyle D^{(2)}_{SSS}\equiv$ $\displaystyle\
B^{(2)}_{SSS}+C^{(2)}_{SSS}\left(1+2\log\frac{v^{2}}{Q^{2}}\right)$
$\displaystyle=$ $\displaystyle\
\frac{2m_{H}^{6}}{v^{6}}\left(\log\frac{m_{H}^{2}}{Q^{2}}-\frac{3}{2}\right)+\frac{2m_{A}^{6}}{v^{6}}\left(\log\frac{m_{A}^{2}}{Q^{2}}-\frac{3}{2}\right)+\frac{4m_{H^{\pm}}^{6}}{v^{6}}\left(\log\frac{m_{H^{\pm}}^{2}}{Q^{2}}-\frac{3}{2}\right)$
$\displaystyle+\frac{12m_{H}^{6}\cot^{2}2\beta}{v^{6}}\left(\log\frac{m_{H}^{2}}{Q^{2}}-\frac{3}{2}\right)+\frac{2m_{H}^{4}m_{A}^{2}\cot^{2}2\beta}{v^{6}}\left(\log\frac{m_{A}^{2}}{Q^{2}}-\frac{3}{2}\right)$
$\displaystyle+\frac{4m_{H}^{4}m_{H^{\pm}}^{2}\cot^{2}2\beta}{v^{6}}\left(\log\frac{m_{H^{\pm}}^{2}}{Q^{2}}-\frac{3}{2}\right)$
$\displaystyle+\frac{1}{2v^{6}}\bigg{\\{}(m_{H}^{2}-m_{A}^{2})^{2}\bigg{[}m_{H}^{2}\left(\log\frac{m_{H}^{2}}{Q^{2}}-\frac{3}{2}\right)+m_{A}^{2}\left(\log\frac{m_{A}^{2}}{Q^{2}}-\frac{3}{2}\right)\bigg{]}$
$\displaystyle\hskip
42.67912pt+2(m_{H}^{2}-m_{H^{\pm}}^{2})^{2}\bigg{[}m_{H}^{2}\left(\log\frac{m_{H}^{2}}{Q^{2}}-\frac{3}{2}\right)+m_{H^{\pm}}^{2}\left(\log\frac{m_{H^{\pm}}^{2}}{Q^{2}}-\frac{3}{2}\right)\bigg{]}$
$\displaystyle\hskip
42.67912pt+2(m_{A}^{2}-m_{H^{\pm}}^{2})^{2}\bigg{[}m_{A}^{2}\left(\log\frac{m_{A}^{2}}{Q^{2}}-\frac{3}{2}\right)+m_{H^{\pm}}^{2}\left(\log\frac{m_{H^{\pm}}^{2}}{Q^{2}}-\frac{3}{2}\right)\bigg{]}\bigg{\\}}\,,$
(C.16)

$\displaystyle D^{(2)}_{ttH}\equiv$ $\displaystyle\
B^{(2)}_{ttH}+C^{(2)}_{ttH}\left(1+2\log\frac{v^{2}}{Q^{2}}\right)$
$\displaystyle=$ $\displaystyle\
\frac{3m_{t}^{2}\cot^{2}\beta}{v^{6}}\bigg{[}m_{H}^{4}\left(\log\frac{m_{H}^{2}}{Q^{2}}-\frac{3}{2}\right)-6m_{t}^{2}m_{H}^{2}\left(\log\frac{m_{H}^{2}}{Q^{2}}-\frac{5}{6}\right)-6m_{t}^{4}\left(\log\frac{m_{t}^{2}}{Q^{2}}-\frac{11}{6}\right)\bigg{]}\,,$
(C.17)

$\displaystyle D^{(2)}_{ttA}\equiv$ $\displaystyle\
B^{(2)}_{ttA}+C^{(2)}_{ttA}\left(1+2\log\frac{v^{2}}{Q^{2}}\right)$
$\displaystyle=$ $\displaystyle\
\frac{3m_{t}^{2}\cot^{2}\beta}{v^{6}}\bigg{[}m_{A}^{4}\left(\log\frac{m_{A}^{2}}{Q^{2}}-\frac{3}{2}\right)-2m_{t}^{2}m_{A}^{2}\left(\log\frac{m_{A}^{2}}{Q^{2}}+\frac{1}{2}\right)+2m_{t}^{4}\left(\log\frac{m_{t}^{2}}{Q^{2}}-\frac{1}{2}\right)\bigg{]}\,,$
(C.18)

$\displaystyle D^{(2)}_{tbH^{\pm}}\equiv$ $\displaystyle\
B^{(2)}_{tbH^{\pm}}+C^{(2)}_{tbH^{\pm}}\left(1+2\log\frac{v^{2}}{Q^{2}}\right)$
$\displaystyle=$ $\displaystyle\
\frac{6m_{t}^{2}\cot^{2}\beta}{v^{6}}\bigg{[}m_{H^{\pm}}^{4}\left(\log\frac{m_{H^{\pm}}^{2}}{Q^{2}}-\frac{3}{2}\right)-2m_{t}^{2}m_{H^{\pm}}^{2}\left(\log\frac{m_{H^{\pm}}^{2}}{Q^{2}}-\frac{1}{2}\right)-m_{t}^{4}\left(\log\frac{m_{t}^{2}}{Q^{2}}-\frac{3}{2}\right)\bigg{]}\,.$
(C.19)

The mass relation at two loops, for general masses and in terms of
$\overline{\rm MS}$ parameters, is then

$\displaystyle 8\pi^{2}v^{2}[M_{h}^{2}]_{V_{\text{eff}}}=$ $\displaystyle\
m_{H}^{4}+m_{A}^{4}+2m_{H^{\pm}}^{4}-12m_{t}^{4}+6m_{W}^{4}+3m_{Z}^{4}$