disposition [toc]toc.1toc.1ContentsContentstoc.1

DESY 21-030

TTK-21-07

Expectation management

Johannes Braathen${}^{\,a}$ 111e-mail<EMAIL_ADDRESS>Mark D.
Goodsell${}^{\,b}$ 222e-mail<EMAIL_ADDRESS>Sebastian
Paßehr${}^{\,c}$ 333e-mail<EMAIL_ADDRESS>and Emanuelle
Pinsard${}^{\,d}$ 444e-mail<EMAIL_ADDRESS>
a Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D-22607 Hamburg,
Germany.
bLaboratoire de Physique Théorique et Hautes Energies (LPTHE),
UMR 7589, Sorbonne Université et CNRS, 4 place Jussieu, 75252 Paris Cedex 05,
France.

cInstitute for Theoretical Particle Physics and Cosmology,
RWTH Aachen University, Sommerfeldstraße 16, 52074 Aachen, Germany.
dLaboratoire de Physique de Clermont (UMR 6533), CNRS/IN2P3,
Univ. Clermont Auvergne, 4 Av. Blaise Pascal, F-63178 Aubière Cedex, France

##  1 Introduction

The mass of the SM-like Higgs boson, discovered by ATLAS and CMS [1, 2, 3], is
now an electroweak precision observable, thanks to its outstandingly accurate
determination at the LHC [4, 5, 6], and it plays an important role in
constraining the allowed parameter space of Beyond-the-Standard-Model (BSM)
theories. On the one hand, the Higgs mass is a prediction in supersymmetric
theories (see Ref. [7] and references therein for a recent review) and
interestingly it depends most heavily on the electroweak couplings and scale –
quantities that are already known from other observations – while it is only
at loop level that a dependence on the scale of supersymmetric particles
appears. This property has spurred significant developments in precision
scalar-mass calculations, advanced in recent years by the KUTS initiative [8,
9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47,
48, 49, 50, 51, 52, 53, 54] as described in the report [7]. On the other hand,
in non-supersymmetric theories, the Higgs mass is not a prediction by itself,
but it can be used to extract the Higgs quartic coupling and, in turn,
investigate the stability of the electroweak vacuum. In this context, a
precise calculation is essential to produce reliable results on vacuum
stability (see Refs. [55, 56, 57, 58, 59] for works in the SM) and to
correctly appreciate the potential impact of new particles [60, 61, 62, 63,
64, 65].

We refer the interested reader to Ref. [7] and references therein for an in-
depth review of Higgs-mass computations, and we only recall here the main
steps involved (applicable for any BSM theory). The standard calculational
technique begins with the extraction of SM-like parameters – namely the
electroweak and strong gauge couplings, the quark and lepton Yukawa couplings,
and the Higgs vacuum expectation value (vev) – from observables. Adding then
the BSM parameters to these, the Higgs (and other particle) masses can be
calculated, along with any other desired predictions. The relevant observables
for the electroweak sector are typically, as in calculations in the SM, either
$M_{Z},M_{W},\alpha(0)$ or $M_{Z},G_{F},\alpha(0)$ where $M_{Z,W}$ are the $Z$
and $W$ boson masses, $\alpha(0)$ is the fine-structure constant extracted in
the Thompson limit, and $G_{F}$ is the Fermi constant. This latter quantity is
extracted from muon three-body decays, whereas the others are related
essentially to self-energies. In general, this extraction of the SM-like
couplings and the Higgs vev can be performed at one-loop for any theory, but
the two-loop relationships are only known for the SM and a small subset of
other models in certain limits.

At the tree level, the expectation value $v$ of the Higgs boson is related to
the other parameters in the theory by the requirement that the theory be at
the minimum of the potential. To be concrete, consider the Higgs potential of
the SM, $V=\mu^{2}\,\lvert H\rvert^{2}+\lambda\,\lvert H\rvert^{4}$; then the
minimisation condition gives

$\displaystyle 0$ $\displaystyle=\mu^{2}+\lambda\,v^{2}\,.$ (1.1)

Since we do not have an observable for $\mu^{2}$ we typically use this
equation to eliminate it, giving the Higgs mass to be

$\displaystyle m_{h}^{2}$
$\displaystyle=\mu^{2}+3\,\lambda\,v^{2}=2\,\lambda\,v^{2}\,.$ (1.2)

However, once we go beyond tree level, there are several possible choices. The
approach typically taken in BSM theories, and in the SM in Ref. [66], is to
insist that the expectation value $v$ is a fixed “observable”, and instead
keep solving for $\mu^{2}$ order-by-order in perturbation theory. In this way,

$\displaystyle\mu^{2}$
$\displaystyle=-\lambda\,v^{2}-\frac{1}{v}\,\frac{\partial\Delta V}{\partial
h}\bigg{|}_{h=0}\equiv-\lambda\,v^{2}-\frac{1}{v}\,t_{h}\,,$ (1.3)

where $\Delta V$ are the loop corrections to the effective potential, and then
the Higgs pole mass $M_{h}$ reads

$\displaystyle M_{h}^{2}$
$\displaystyle=2\,\lambda\,v^{2}-\frac{1}{v}\,t_{h}+\Pi_{hh}{\left(M_{h}^{2}\right)}\equiv
2\,\lambda\,v^{2}+\Delta M_{h}^{2}\,,$ (1.4)

where $\Pi_{hh}{\left(M_{h}^{2}\right)}$ is the Higgs self-energy evaluated
on-shell. One of the chief advantages of this approach is that tadpole
diagrams do not appear in any processes, since they vanish by construction.

On the other hand, while this is in principle a straightforward procedure to
follow, it is complicated by the fact that the self-energies and effective
potential implicitly depend on $\mu^{2}$. In Landau gauge, or the gaugeless
limit, this leads to the “Goldstone Boson Catastrophe” at two loops [67, 68,
69, 70] – its solution appears by consistently solving the above equation
order by order [27, 33]. Indeed, one way to formalise this is as a finite (or
possibly IR-divergent) counterterm for $\mu^{2}$:

$\displaystyle\mathcal{L}$
$\displaystyle\supset-\left(\mu^{2}+\delta\mu^{2}+\lambda\,v^{2}\right)v\,h-\frac{1}{2}\left(\mu^{2}+\delta\mu^{2}+3\,\lambda\,v^{2}\right)h^{2}+\ldots\,,$
(1.5)

where $\delta\mu^{2}=-\frac{1}{v}\,t_{h}.$ Another drawback is that it
manifestly breaks gauge invariance, since the loop corrections above depend on
the gauge; and it also means that the expectation value $v$ is not an
$\overline{\text{MS}}$ parameter, so the renormalisation-group equations for
the expectation value are no longer just given by those of $\mu^{2}$ and
$\lambda$, but have extra contributions [71, 72].

However, there is a further drawback to the above procedure which we wish to
highlight in this paper. When considering a BSM theory with additional scalars
that may have an expectation value, it is typical to take the same approach as
for the scalar field in the SM and fix their expectation values, solving the
additional tadpole equations for other dimensionful parameters – for example,
their mass-squared parameters, or sometimes a cubic scalar coupling. To take
the example of a real singlet $S$ with mass-squared _Lagrangian parameter_
$m_{S}^{2}$ – not to be confused with the pole mass, which we denote $M_{S}$ –
and expectation value $v_{S}$, this means that analagously to eq. (1.3),

$\displaystyle m_{S}^{2}$
$\displaystyle=\left(m_{S}^{2}\right)^{\mathrm{t}ree}-\frac{1}{v_{S}}\,\frac{\partial\Delta
V}{\partial S}.$ (1.6)

If the loop corrections are not large, and $v_{S}$ is not small, this is
completely acceptable – so for models such as the NMSSM there is generally no
problem. However, if we consider a different theory or regions of the
parameter space where $v_{S}$ is small, for example if $m_{S}\gg v$ and
$v_{S}\propto v^{2}$ (as may be found in examples of EFT matching [41]) then
we can easily find the case that $\delta
m_{S}^{2}>\left(m_{S}^{2}\right)^{\mathrm{t}ree}$. This makes the calculation
unreliable.

The archetypal example of this problem is the case where the neutral scalar
obtaining an expectation value actually comes from an $SU(2)$ triplet
$\mathbf{T}$ with expectation value $v_{T}$ and mass-squared $m_{T}^{2}$ – for
example in Dirac-gaugino models [73, 74, 75, 76]. In that case, $v_{T}\propto
v^{2}/m_{T}^{2}$ multiplied by other dimensionful parameters of the theory.
Moreover, we require that $v_{T}\lesssim 4$ $\mathrm{GeV}$ from electroweak-
precision constraints, generally requiring $m_{T}\gtrsim 1$ TeV. So then

$\displaystyle\delta m_{T}^{2}$ $\displaystyle\sim\frac{1}{4\
\mathrm{GeV}}\times\frac{1}{16\pi^{2}}\times\mathcal{O}{\left(\mathrm{TeV}^{3}\right)}\sim
2.5\times\mathcal{O}{\left(\mathrm{TeV}^{2}\right)}\,,$ (1.7)

i. e. we see that there is a severe problem whenever $v_{T}/m_{T}$ is of the
order of a loop factor.

Moreover, for such cases where $v_{S}$ is small, this procedure works in the
opposite way to that which we would desire. In BSM theories the scalar
expectation values beyond $v$ are not top-down inputs or tied closely to some
observables, whereas we may typically want to define the masses and couplings
as fixed by some high-energy boundary conditions (for example constrained or
minimal SUGRA conditions where soft masses have a common origin). In this case
we would like to solve the tadpole equations for $v_{S}$; even if this would
typically lead to coupled cubic equations, nowadays it is almost trivial to
solve them numerically, or start from an approximation.

In this paper we will instead examine an alternative procedure, proposed by
Fleischer and Jegerlehner in examining Higgs decays in the SM [77], which has
the potential to solve both of these issues. Instead of taking the expectation
values as fixed, we take them to be the tree-level solutions of the tadpole
equations. This means that we do not work at the “true” minimum of the
potential and must include tadpole diagrams in all processes. While this
implies the addition of some new Feynman diagrams in the Higgs mass
calculation, it is not technically more complicated than including finite
counterterm insertions for $\mu^{2}$. This approach has the additional
advantages that, since the Lagrangian is specified in terms of
$\overline{\text{MS}}$ parameters only, the result is manifestly gauge
independent, and the expectation values are just the solutions to the tree-
level tadpole equations. For these reasons, it has been used and advocated in
the SM, in particular at two loops in Ref. [78, 79, 80, 81, 82, 57]; and
applied to certain extensions of the Two Higgs Doublet Model (THDM) when
considering decays [83, 84, 85, 86]. We also note that this approach is
closely related to the various on-shell renormalisations used in e. g. Refs.
[87, 88, 89, 90] in the THDM and the Minimal Supersymmetric Standard Model
(MSSM).

In the example of the SM at the one-loop order, this would mean

$\displaystyle M_{h}^{2}$
$\displaystyle=2\,\lambda\,v^{2}-\frac{6\,\lambda\,v}{m_{h}^{2}}\,t_{h}^{(1)}+\Pi_{hh}^{(1)}{\left(m_{h}^{2}\right)}\,,$
(1.8)

where the superscripts in brackets indicate the loop order, and we put the
momentum in the self-energy at the tree-level Higgs mass in order to respect
the order of perturbation theory. In other words, the tadpole contribution is
suppressed by the mass-squared of the Higgs, although – since
$m_{h}^{2}=2\,\lambda\,v^{2}$ – here we find that they have a very similar
form to the previous approach. On the other hand, in the case of a heavy
singlet or triplet the contributions to the singlet self-energy would be
similarly suppressed by $m_{S}^{2}$, and we can have $m_{S}$ much greater than
the triplet coupling – so the corrections to the singlet mass would be well
under control.

On the other hand, in the BSM context this approach was proposed by Ref. [91]
for the following very different reason: by no-longer forcing the electroweak
expectation value to have its observed value, we allow new physics to disturb
the electroweak hierarchy. In the above approach, the contribution
$-\frac{6\,\lambda\,v}{m_{h}^{2}}\,t_{h}^{(1)}=-\frac{3}{v}\,t_{h}^{(1)}$ is
effectively the contribution _from a shift in $v$_. We can view the
calculation as equivalent to counterterms for the expectation value
$\delta^{(1)}v$, where

$\displaystyle\mathcal{L}$
$\displaystyle\supset-\left(\mu^{2}+\lambda\,v^{2}\right)v\,h-\left(\mu^{2}+3\,\lambda\,v^{2}\right)\delta^{(1)}v\,h-\ldots$
(1.9)

so that now

$\displaystyle\delta^{(1)}v$
$\displaystyle=-\frac{1}{m_{h}^{2}}\,t_{h}^{(1)}\,.$ (1.10)

In this case, if there is heavy new physics at a scale $\Lambda\gg m_{h}$,
then we shift the Higgs expectation value up to that new scale suppressed only
by a loop factor. Indeed in Ref. [91] the proposal was to use

$\displaystyle\frac{\delta m_{h}^{2}}{m_{h}^{2}}$
$\displaystyle\equiv\frac{1}{m_{h}^{2}}\left[-\frac{3}{v}\,t_{h}^{(1)}+\Pi_{hh}^{(1)}{\left(m_{h}^{2}\right)}\right]$
(1.11)

as _as a measure of fine-tuning of the theory_.

Another perspective on the difference between the two approaches is given by
viewing the SM as an EFT. In this case, in the EFT the SM receives corrections
to both $\mu^{2}$ and $\lambda$ at the matching scale from integrating out
heavy states which can be done with $v=0$. As discussed in Ref. [27], when
expanding in $v$, in order to respect gauge invariance we must have:

$\displaystyle\Delta V$ $\displaystyle=\Delta V_{0}+\frac{1}{2}\left.\Delta
V_{hh}\right\rvert_{v=0}\,v^{2}+\mathcal{O}{\left(v^{4}\right)}+\ldots\,,$
$\displaystyle\Pi_{hh}{\left(m_{h}^{2}\right)}$ $\displaystyle=\left.\Delta
V_{hh}\right\rvert_{v=0}+\mathcal{O}{\left(v^{2}\right)}$ (1.12)

and therefore $t_{h}=v\left.\Delta V_{hh}\right\rvert_{v=0}+\ldots$ This shows
that the EFT-matching correction to $\mu^{2}$, which is $\left.\Delta
V_{hh}\right\rvert_{v=0}$, and the origin of the hierarchy problem, correspond
to $t_{h}/v$ to lowest order in $v$. Hence in the “standard” approach of eq.
(1.4) this cancels out and leaves only corrections proportional to $v^{2}$ –
whereas in the modified approach it remains and gives a large shift to the
Higgs mass.

However, the reappearance of the hierarchy is a problem for the _light_ Higgs
mass, whereas the problem we wished to solve actually appeared in new, _heavy_
states! If we wish to explore theories which may remain natural while having
heavy states, such as those in Ref. [91], then the modified tadpole approach
should work best. There must consequently be some trade-off between losing
control of the light Higgs and losing control of the heavier states (and
losing gauge invariance too). In section 2 we will set up the necessary
general formalism and explore this in detail for a toy model.

However, there are _two_ potential solutions to allow us to have the best of
both worlds:

  1. 1.

Retain counterterms for $\mu^{2}$ as in eq. (1.5) for the SM Higgs, but _only_
for them. This is somewhat tricky to automate, since we must make a special
case of the electroweak sector, and we also lose gauge invariance.

  2. 2.

For cases where the tuning of the hierarchy becomes large, use EFT pole
matching [26] with the modified treatment of tadpoles. This way, the heavy
states remain entirely under control, we keep the heavy masses and couplings
as top-down inputs (that remain genuinely $\overline{\text{MS}}$ or
$\overline{\text{DR}}^{\prime}$), and we have gauge invariance built-in.

In section 4 we will adopt the second approach for the example of the general
NMSSM (and apply it specifically to the variant known as the $\mu$NMSSM [63]).
We establish the necessary formalism for the matching and give a detailed
examination, via implementing the computation in a modified SPheno [92, 93]
code generated from SARAH [94, 95, 96, 97, 98, 13, 16, 33].

##  2 Treatment of tadpoles for theories with heavy scalars

For a general renormalisable field theory, once we have solved the vacuum
minimisation conditions and diagonalised the mass matrices, we can write the
potential in terms of real scalar fields $\\{\phi_{i}\\}$ as

$\displaystyle V$
$\displaystyle=\text{const}+\frac{1}{2}\,m_{i}^{2}\,\phi_{i}^{2}+\frac{1}{6}\,a_{ijk}\,\phi_{i}\,\phi_{j}\,\phi_{k}+\frac{1}{24}\,\lambda_{ijkl}\,\phi_{i}\,\phi_{j}\,\phi_{k}\,\phi_{l}\,.$
(2.1)

If we take the standard approach and fix the expectation values, adjusting the
mass parameters order by order in perturbation theory, then as described in
Ref. [27] we can write the pole masses as

$\displaystyle\left(M_{i}^{2}\right)^{(1)}$
$\displaystyle=m_{i}^{2}+\Delta_{ii}+\Pi_{ii}^{(1)}{\left(m_{i}^{2}\right)}\equiv
m_{i}^{2}+\Delta M_{i}^{2}\,.$ (2.2)

To define the shifts $\Delta_{ii}$ in a general way, we must start from some
basis of fields $\big{\\{}\phi_{i}^{0}\big{\\}}$ split into expectation values
and fluctuations so that $\phi_{i}^{0}\equiv v_{i}+\hat{\phi}_{i}^{0}$ and
then diagonalise the fields via $\hat{\phi}_{i}^{0}=R_{ij}\,\phi_{i}$. In the
simplest case where we solve the tadpole equations for some mass-squared
parameters in the original basis and where we ignore pseudoscalars, we can
then write

$\displaystyle\Delta_{ii}=-\sum_{k}R_{ki}^{2}\,\frac{1}{v_{k}}\left.\frac{\partial\Delta
V}{\partial\hat{\phi}_{k}^{0}}\right|_{\hat{\phi}_{k}^{0}=0}=-\sum_{k,l}R_{ki}^{2}\,R_{lk}\,\frac{1}{v_{k}}\,t_{l}^{(1)}\,.$
(2.3)

The generalisation to solving for other variables (such as cubic scalar
couplings) and to include pseudoscalar mass shifts is given in Ref. [33].

On the other hand, taking the modified approach and including the tadpole
diagrams, the pole masses up to one loop are simply

$\displaystyle\left(M_{i}^{2}\right)^{(1)}$
$\displaystyle=\hat{m}_{i}^{2}-\frac{1}{\hat{m}_{j}^{2}}\,a_{iij}\,t_{j}^{(1)}+\Pi_{ii}^{(1)}{\left(\hat{m}_{i}^{2}\right)}\equiv\hat{m}_{i}^{2}+\hat{\Pi}_{ii}^{(1)}{\left(\hat{m}_{i}^{2}\right)}\,,$
(2.4)

where we have defined $\hat{m}_{i}^{2}$ to be the tree-level mass when we are
using the modified scheme (we will later drop the distinction between $m_{i}$
and $\hat{m}_{i}$, see below) and $\hat{\Pi}_{ij}\big{(}p^{2}\big{)}$ for
later use to be the self-energies including the tadpoles. The expressions for
the tadpoles and self-energies at one loop can be found e. g. in Refs. [99,
27]; this calculation is therefore more straightforward to automate, being
purely diagrammatic in nature. An _explicitly_ gauge-invariant expression for
this (i. e. one where there are no gauge-fixing parameters present) will be
given in future work.

At this point the reader may object that, no matter what technique we use to
calculate masses, the result for a given theory should be the same up to
higher-loop corrections. Unfortunately this is made obscure by the
difficulties in general in defining the parameters of our theory. To compare
the two calculations _for the same parameter point_ , in the standard approach
we are invited to treat the expectation values as fundamental, so if we start
from a theory defined in this way, we must:

  1. 1.

Calculate loop-level masses in the standard approach for a given choice of
expectation values (with the associated problems when those expectation values
are small).

  2. 2.

Extract the Lagrangian parameters from the loop-corrected tadpole equations.

  3. 3.

Solve the tree-level vacuum stability equations with these new parameters,
obtaining the expectation values for use in the alternative approach.

  4. 4.

Compute the new tree-level spectrum using these expectation values

  5. 5.

Compute the loop-corrected masses in the alternative approach.

Let us denote the tree-level masses and expectation values in the alternative
approach as $\hat{m}_{i}$ and $\hat{v}_{i}$, and for simplicity assume that we
solve the tadpole equations for some mass-squared parameters (rather than
cubic couplings, say). Then, by passing back to the basis in which the fields
are not diagonalised, where the Lagrangian mass parameters are
$m_{0,ij}^{2}=\hat{m}_{0,ij}^{2}+\delta m_{0,ij}^{2}$ and the Lagrangian
couplings are $a_{0,ijk},$ $\lambda_{0,ijkl}$, we can carry out the above
steps and solve perturbatively for the expectation values $\hat{v}_{i}$ in the
modified scheme:

$\displaystyle 0$ $\displaystyle=\big{(}m_{0,ij}^{2}+\delta
m_{0,ij}^{2}\big{)}\,v_{j}+\frac{1}{2}\,a_{0,ijk}\,v_{j}\,v_{k}+\frac{1}{6}\,\lambda_{0,ijkl}\,v_{j}\,v_{k}\,v_{l}+t_{0,i}$
$\displaystyle=\big{(}m_{0,ij}^{2}+\delta
m_{0,ij}^{2}\big{)}\,\hat{v}_{j}+\frac{1}{2}\,a_{0,ijk}\,\hat{v}_{j}\,\hat{v}_{k}+\frac{1}{6}\,\lambda_{0,ijkl}\,\hat{v}_{j}\,\hat{v}_{k}\,\hat{v}_{l}\,.$
(2.5)

We have written $t_{0,i}$ for the one-loop tadpole to emphasise that it is in
the undiagonalised basis; to go to the mass-diagonal basis we need to rotate
by the matrix $R_{ij}$ as above. Writing $\hat{v}_{i}=v_{i}+\delta v_{i}$ we
obtain

$\displaystyle 0$ $\displaystyle=-t_{0,i}+\mathcal{M}_{0,ij}^{2}\,\delta
v_{j},$ (2.6)

where $\mathcal{M}_{0,ij}^{2}$ is the tree-level mass matrix of scalars in the
standard scheme. This can be trivially solved by rotating to the mass-diagonal
basis. We then write the tree-level mass matrix in the alternative scheme as

$\displaystyle\hat{\mathcal{M}}_{0,ij}^{2}$
$\displaystyle=\mathcal{M}_{0,ij}^{2}+\delta m_{0,ij}^{2}+a_{0,ijk}\,\delta
v_{k}+\lambda_{0,ijkl}\,v_{k}\,\delta v_{l}\,.$ (2.7)

Using the _same_ matrix $R_{ij}$ we can rotate this to obtain111Recall that
$a_{ijk}=(a_{0,i^{\prime}j^{\prime}k^{\prime}}+\lambda_{0,i^{\prime}j^{\prime}k^{\prime}l^{\prime}}\,v_{l^{\prime}})\,R_{i^{\prime}i}\,R_{j^{\prime}j}\,R_{k^{\prime}k}$

$\displaystyle\hat{m}_{i}^{2}=\big{(}R^{T}\,\hat{\mathcal{M}}_{0}^{2}\,R\big{)}_{ii}$
$\displaystyle=m_{i}^{2}+\Delta_{ii}+a_{iik}\,\frac{t_{k}^{(1)}}{m_{k}^{2}}+\mathcal{O}(\text{2-loop}).$
(2.8)

Inserting this into (2.4) gives (2.2).

Of course, this comes with the associated problems of defining the theory in
the standard approach: if we have a small expectation value, then (as we shall
illustrate below) the loop corrections in $\Delta_{ii}$ can be very large, so
the mass of the heavy scalar may differ greatly from the tree-level one.
Making a conversion in this way just ensures that we see the same problem in
the alternative treatment. Instead, for such points we should start with a
theory defined in the _alternative_ manner.

Then, to compare the same point for the standard calculation one should:

  1. 1.

Calculate loop-level masses in the alternative approach for a given choice of
masses and couplings.

  2. 2.

Iteratively solve the loop-level vacuum stability equations to obtain the
loop-corrected expectation values $v_{i}$ for use in the standard scheme.

  3. 3.

Use these expectation values to compute the tree-level spectrum for use in the
standard scheme (if we are using the approach with “consistent tadpoles”)222In
principle it is possible, and simpler, to just use the “true” input masses in
the standard approach. This would vitiate the problem to a large extent, but
would then lead to the well-known infra-red issues at two loops, or
uncancelled logarithms in EFT matching, etc.

  4. 4.

Compute the loop-corrected masses in the standard approach.

In this way, we should obtain the same result (up to higher-order differences)
for our desired point as in the alternative scheme. However, the key
complicating factor is step 2: it assumes that we can efficiently and
accurately find the true minimum of the potential. This can only be done by
iteration of the tadpole equations; this involves _recomputing the masses and
couplings of the theory at each step_ and is therefore often numerically
expensive (especially at higher loop orders). On the other hand, if we do this
perturbatively, then we are effectively using the alternative scheme!

### Disclaimer

While the above discussion is reassuring for the consistency of our
calculations, in the following we will _not_ (for the most part) compare
masses at the same parameter point, for the obvious reason that the results
would be almost the same. Instead, what we want to illustrate is the
difficulty in even defining our theory: in the standard approach, since we are
required to choose a vacuum-expectation value for the heavy singlet fields
(which are not physical parameters), the phenomenologist will often use a
guess or a tree-level-approximate solution for this, rather than iteratively
solve the tadpole equations (which, in any case, would lead to a different
input value depending on the chosen loop order). We shall take this naive
approach below, and compare (in most cases) theories _with the same tree-level
spectrum_ by taking the expecation values to be the same in both the standard
and modified schemes. Of course, according to the discussion above, these are
not the same parameter points: we are instead illustrating the differences in
methods of defining the theory, and will show how the alternative scheme gives
a much more stable and efficient definition (at least in cases where the
hierarchy problem for the light Higgs does not become severe).

###  2.1 A toy model

Let us now apply the above general expressions to the simplest toy model that
can illustrate the differences of prescriptions for dealing with radiative
corrections to tadpoles. This consists of the abelian Goldstone model coupled
to a real singlet $S$, and has scalar potential

$\displaystyle V$ $\displaystyle=\mu^{2}\,\lvert
H\rvert^{2}+\frac{1}{4}\,\lambda\,\lvert
H\rvert^{4}+\frac{1}{2}\,m_{S}^{2}\,S^{2}+a_{SH}\,S\,\lvert
H\rvert^{2}+\lambda_{SH}\,S^{2}\,\lvert
H\rvert^{2}+a_{S}\,S^{3}+\lambda_{S}\,S^{4}$ (2.9)

with the fields

$\displaystyle H$
$\displaystyle\equiv\frac{1}{\sqrt{2}}\left(v+h+i\,G\right),\quad S\equiv
v_{S}+\hat{S}\,,$ (2.10)

$v$ and $v_{S}$ denoting the Higgs and singlet vacuum expectation values
(vevs), respectively. The minimisation conditions at the tree level yield the
equations

$\displaystyle-\mu^{2}$
$\displaystyle=\frac{1}{4}\,\lambda\,v^{2}+a_{SH}\,v_{S}+\lambda_{SH}\,v_{S}^{2}\,,$
(2.11a) $\displaystyle\left(m_{S}^{2}+\lambda_{SH}\,v^{2}\right)v_{S}$
$\displaystyle=-\frac{1}{2}\,a_{SH}\,v^{2}-3\,a_{S}\,v_{S}^{2}-4\,\lambda_{S}\,v_{S}^{3}$
(2.11b)

that lead to the tree-level (squared) mass matrix for the scalars (which do
not mix with the massless pseudoscalar):

$\displaystyle\mathcal{M}^{2}_{\text{tree}}$
$\displaystyle=\begin{pmatrix}\frac{1}{2}\,\lambda\,v^{2}&a_{SH}\,v+2\,\lambda_{SH}\,v\,v_{S}\\\
a_{SH}\,v+2\,\lambda_{SH}\,v\,v_{S}&m_{S}^{2}+\lambda_{SH}\,v^{2}+6\,a_{S}\,v_{S}+12\,\lambda_{S}\,v_{S}^{2}\end{pmatrix}.$
(2.12)

|  |
---|---|---
tadpole topologies | self-energy topologies | connected tadpole topologies
Figure 1: left: one-loop tadpole diagrams; middle: one-loop self-energy
diagrams appearing in standard and modified calculation; right: additional
self-energy diagrams in the modified approach.

The one-particle irreducible one-loop contributions to the one- and two-point
functions (see figure 1) of this toy model are given by

$\displaystyle t_{i}^{(1)}$
$\displaystyle=-\,\frac{\kappa}{2}\,a_{ijj}\,A{\left(m_{j}^{2}\right)}\,,$
(2.13a) $\displaystyle\Pi_{ij}^{(1)}{\left(p^{2}\right)}$
$\displaystyle=\kappa\left[\frac{1}{2}\,\lambda_{ijkk}\,A{\left(m_{k}^{2}\right)}-\frac{1}{2}\,a_{ikl}\,a_{jkl}\,B{\left(p^{2},m_{k}^{2},m_{l}^{2}\right)}\right]$
(2.13b)

with $A$ and $B$ denoting the scalar one-point and two-point one-loop
integrals in the conventions of e. g. Refs. [99, 27],
$\kappa\equiv(16\pi^{2})^{-1}$ and $p^{2}$ denoting the external momentum. In
the approach of keeping the vevs fixed, we find for the one-loop pole masses:

$\displaystyle\left(M_{i}^{2}\right)^{(1)}$
$\displaystyle=m_{i}^{2}-R_{i1}^{2}\,\frac{1}{v}\,t_{h}^{(1)}-R_{i2}^{2}\,\frac{1}{v_{S}}\,t_{S}^{(1)}+\Pi_{ii}{\left(m_{i}^{2}\right)}\,,$
(2.14)

where $t_{h}^{(1)}=\partial\Delta V\big{/}\partial h\big{|}_{h,\hat{S}=0}$ ,
$t_{S}^{(1)}=\partial\Delta V\big{/}\partial S\big{|}_{h,\hat{S}=0}$ . Thus
the tadpole corrections suffer from the division by the vev; in particular,
the mass predictions can become numerically unstable in scenarios with a small
singlet vev. Let us see this in practice for our example when $m_{S}^{2}$ is
large; in this case

$\displaystyle v_{S}\sim-\frac{a_{SH}\,v^{2}}{2\,m_{S}^{2}},\qquad
R\sim\begin{pmatrix}1&-\frac{a_{SH}\,v}{m_{S}^{2}}\\\
\frac{a_{SH}\,v}{m_{S}^{2}}&1\end{pmatrix}.$ (2.15)

If we take $v$ small and just look at the singlet mass in the limit
$p^{2}\rightarrow 0$ for simplicity,333This limit is not implemented in our
code and serves only the more lucid presentation. In fact, an off-shell
evaluation of the self-energies implies unphysical behaviour of Higgs-mass
predictions [100]. we have

$\displaystyle\Delta
M_{S}^{2}\approx\Pi_{SS}(0)-\frac{1}{v_{S}}\,t_{S}\supset-\frac{3\,a_{S}\,m_{S}^{2}\,\kappa}{v_{S}}\left(\,\overline{\log}\,m_{S}^{2}-1\right)+\ldots$
(2.16)

where $\overline{\log}m_{S}^{2}\equiv\log m_{S}^{2}/Q^{2}$ for renormalisation
scale $Q$. When the system is really decoupled and $v=0$, then $v_{S}\sim
m_{S}^{2}\big{/}(6a_{S})$ and this expression remains well-controlled, but
when $0<v\ll m_{S}$ – which is the case we are interested in – we instead have

$\displaystyle\Delta M_{S}^{2}$
$\displaystyle\propto\frac{6\,a_{S}\,m_{S}^{4}}{16\,\pi^{2}\,a_{SH}\,v^{2}}\
\overline{\log}\,m_{S}^{2}$ (2.17)

which can be very large compared to $m_{S}^{2}$.

If we take the modified approach to tadpoles, then the relevant generic
expression for the self-energy is

$\displaystyle\hat{\Pi}_{ij}^{(1)}{\left(p^{2}\right)}$
$\displaystyle=\frac{1}{16\,\pi^{2}}\left[\frac{1}{2}\,\lambda_{ijkk}\,A{\left(m_{k}^{2}\right)}-\frac{1}{2}\,a_{ikl}\,a_{jkl}\,B{\left(p^{2},m_{k}^{2},m_{l}^{2}\right)}-\frac{1}{2\,m_{k}^{2}}\,a_{ijk}\,a_{kll}\,A{\left(m_{l}^{2}\right)}\right];$
(2.18)

and for our example

$\displaystyle\hat{\Pi}_{SS}^{(1)}{\left(m_{S}^{2}\right)}$
$\displaystyle\approx\Pi_{SS}(0)-\frac{a_{SH}^{2}\,\kappa}{2\,m_{h}^{2}}\,A{\left(m_{S}^{2}\right)}-\frac{3\,a_{S}^{2}\,\kappa}{m_{S}^{2}}\,A{\left(m_{S}^{2}\right)}+\ldots\sim-\frac{\kappa}{2}\left(\frac{a_{SH}^{2}}{m_{h}^{2}}-24\,\lambda_{S}\right)m_{S}^{2}\
\overline{\log}\,m_{S}^{2}\,.\hskip 30.00005pt\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (2.19)

Provided that $a_{SH}\lesssim m_{h}$ this is well under control, in contrast
to the previous “standard” approach.

###  2.2 Numerical examples

In this section we shall illustrate the different behaviours of the two
approaches to tadpoles in the toy model defined in eq. (2.9) through numerical
examples. For this purpose, we present results for the one-loop pole masses
$M_{h}$ and $M_{S}$ computed diagrammatically both in the standard approach –
following eq. (2.2) – and in the modified approach of equation (2.4). We shall
consider points defined to have _the same tree-level spectrum_ , but whose
loop-corrected masses differ according to the scheme used. As described in the
disclaimer above, these are not therefore the same points in parameter space:
this illustrates _the difficulty in defining the model_.

For all the following figures, we set $\lambda=0.52$, to reproduce a light
“Higgs” (noting that there are no gauge fields) near $125$ GeV, and we also
fix $\lambda_{SH}=0$ and $\lambda_{S}=1/24$. In each case, we shall fix the
$\overline{\mathrm{MS}}$ parameter $m_{S}$ and solve the tree-level tadpole
equations numerically to obtain $v_{S}$ and fix $v=246\mathrm{GeV}$. Then the
calculation in the modified scheme gives the correct value for the scalar
masses. For comparison, in each of the figures 4, 4, 4 and 5 we use these same
values as inputs for the conventional scheme, where we treat the derived value
for $v_{S}$ as the “all orders” expectation value; this means that, in the
standard scheme, $(m_{S}^{2})^{\mathrm{m}od.}=(m_{S}^{2})^{\mathrm{t}ree}$,
the tree-level value, and is not actually the $\overline{\mathrm{MS}}$ mass-
squared parameter any more. Hence, as mentioned above, these represent
different parameter points now; only the tree-level spectra are the same. To
avoid ambiguity, we shall therefore use $(m_{S}^{2})^{\mathrm{t}ree}$ since it
is the input value for both schemes. In this way we see that two ways of
defining the theory at tree-level can give, at times, drastically different
results. In section 2.3 we provide as a consistency check a comparison of the
approaches with a conversion of the parameters.

Figure 2: $M_{h}$ (left) and $M_{S}$ (right) as a function of $a_{SH}$.
$m_{S}^{\mathrm{t}ree}=Q=2000\,\mathrm{GeV}$, $a_{S}=100~{}\mathrm{GeV}$,
$\lambda=0.52$, $\lambda_{SH}=0$, $\lambda_{S}=1/24$. The tree-level values
are shown with the green curves, while the red and blue curves correspond to
the one-loop results using respectively the standard (eq. (2.2)) and modified
(eq. (2.4)) treatments of tadpoles.

Figure 3: $M_{h}$ (left) and $M_{S}$ (right) as a function of
$m_{S}^{\mathrm{t}ree}$. $Q=m_{S}^{\mathrm{t}ree}$,
$a_{SH}=150~{}\mathrm{GeV}$, $a_{S}=100~{}\mathrm{GeV}$, $\lambda=0.52$,
$\lambda_{SH}=0$, $\lambda_{S}=1/24$. The colours for the different curves are
the same as in figure 4.

Figure 4: $M_{h}$ (left) and $M_{S}$ (right) as a function of $a_{SH}$.
$m_{S}^{\mathrm{t}ree}=1000~{}\mathrm{GeV}$, $Q=5000~{}\mathrm{GeV}$,
$a_{S}=0~{}\mathrm{GeV}$, $\lambda=0.52$, $\lambda_{SH}=0$,
$\lambda_{S}=1/24$. The colours for the different curves are the same as in
figure 4.

In figure 4, we show first $M_{h}$ (left side) and $M_{S}$ (right side) as a
function of the trilinear coupling $a_{SH}$, at tree level (green curves) and
at one loop in the standard (red curves) and modified (blue curves) schemes
for the tadpoles. We choose here a scenario with a large Lagrangian mass term
$m_{S}^{\mathrm{t}ree}=2000~{}\mathrm{GeV}$ and a non-zero trilinear self-
coupling $a_{S}=100~{}\mathrm{GeV}$ for the singlet (and we also fix the
renormalisation scale to be $Q=2000~{}\mathrm{GeV}$). Consequently, we find
ourselves exactly in the dangerous region $0<v\ll m_{S}$, c. f. eq. (2.17),
and as expected from our theoretical discussion, we find that the standard
treatment of the tadpoles breaks down. On the one hand, for $M_{h}$ one can
observe that the radiative corrections are larger in the standard approach and
lead to larger variations of the loop-corrected mass than in the modified
tadpole scheme. On the other hand, more strikingly, the results for $M_{S}$ in
the standard approach are manifestly spurious. Indeed, while the loop
corrections in the modified scheme remain very small (the green tree-level and
blue one-loop curves are almost superimposed), in the standard scheme the
corrections are huge: for large $a_{SH}\gtrsim v$ – meaning not too small
values of the singlet vev $v_{S}$ – they already amount to several hundred
GeV, and if one decreases $a_{SH}$ (thereby increasing $\Delta M_{S}^{2}$, c.
f. eq. (2.17)) the singlet pole mass becomes tachyonic below $a_{SH}=v$.

Next, in figure 4, we fix the trilinear coupling $a_{SH}=150~{}\mathrm{GeV}$
and now consider $M_{h}$ (left) and $M_{S}$ (right) as a function of the
Lagrangian mass term $m_{S}^{\mathrm{t}ree}$. We also set
$Q=m_{S}^{\text{tree}}$ and $a_{S}=100~{}\mathrm{GeV}$. Once again, with our
choice of a non-zero singlet trilinear self-coupling $a_{S}$ and relatively
small $a_{SH}$ – hence also a small singlet vev – we expect the standard
approach to exhibit instabilities. For $M_{h}$ (left side of figure 4) both
approaches behave relatively well and no instability seems to occur, although
the radiative corrections are significantly larger in the standard scheme.
However, for $M_{S}$ the calculation in the standard approach (red curve) once
again breaks down when $m_{S}^{\mathrm{t}ree}$ is increased – equivalently for
small $v_{S}$ – while the loop corrections to $M_{S}$ in the modified approach
(blue curve) remain minute.

In figure 4, we illustrate the behaviour of eq. (2.19). We plot once more
$M_{h}$ (left) and $M_{S}$ (right) as a function of the trilinear coupling
$a_{SH}$, but now for a scenario where $a_{S}=0$ (in order to avoid large
corrections $\Delta M_{S}^{2}$ in the standard scheme), and with
$m_{S}^{\mathrm{t}ree}=1000~{}\mathrm{GeV}$ and $Q=5000~{}\mathrm{GeV}$ so as
to increase the size of the logarithms
$\overline{\log}\,(m_{S}^{2})^{\mathrm{t}ree}$ . For small values of $a_{SH}$,
both schemes (red and blue curves) produce very similar results, however, as
$a_{SH}$ becomes larger the radiative corrections to $M_{h}$ as well as
$M_{S}$ increase significantly in the modified tadpole scheme, leading to less
reliable predictions (especially for $a_{SH}\gtrsim
300$–$400~{}\mathrm{GeV}$).

Finally, we present in figure 5 an example of scenario in which both ways to
treat the tadpole contributions give reliable results. We take a small singlet
mass parameter $m_{S}=500~{}\mathrm{GeV}$, set $a_{S}=0$ and maintain
$a_{SH}<200~{}\mathrm{GeV}$. We observe here that the radiative corrections to
$M_{h}$ and $M_{S}$ remain well behaved in both approaches.

Figure 5: $M_{h}$ (left) and $M_{S}$ (right) as a function of $a_{SH}$.
$m_{S}^{\mathrm{t}ree}=Q=500\,\mathrm{GeV}$, $a_{S}=0~{}\mathrm{GeV}$,
$\lambda_{SH}=0$, $\lambda=0.52$, $\lambda_{S}=1/24$. The colours for the
different curves are the same as in figure 4.

###  2.3 Comparisons at the same point

Here, for clarity (and as a consistency check) we shall follow the (first)
prescription in section 2 and compare the two schemes for computing the one-
loop masses in our toy model at the same parameter point. We consider the same
input parameters as in figure 4, except that now we scan over the true
$\overline{\mathrm{MS}}$ mass $m_{S}$ in both schemes. The calculation in the
modified scheme is therefore identical to those in figure 4, but we then solve
the tadpole equations for $\mu^{2}$ and $m_{S}^{2}$ at the one-loop order to
find the values of $v,v_{S}$; while the value for $v$ changes little, the
equation for $v_{S}$ becomes

$\displaystyle 0$
$\displaystyle=\left(m_{S}^{2}+\lambda_{SH}\,v^{2}\right)v_{S}+\frac{1}{2}\,a_{SH}\,v^{2}+3\,a_{S}\,v_{S}^{2}+4\,\lambda_{S}\,v_{S}^{3}+t_{S}\big{(}m_{S}^{2}\big{)}\,.$
(2.20)

We then use this new value for $v_{S}$ to compute the tree and loop-level
spectra in the standard scheme. In figure 7 we employ consistent tadpoles, so
that we obtain a value for $(m_{S}^{2})^{\mathrm{t}ree}$ which satisfies eq.
(2.11b) and use this to compute the tree-level spectrum, and as input for the
loop computation with the appropriate perturbative shifts to the loop mass;
neglecting mixing between the light and heavy scalars we have

$\displaystyle(M_{S}^{2})^{\mathrm{t}ree}$
$\displaystyle\simeq-\frac{a_{SH}\,v^{2}}{2\,v_{S}}+v_{S}\left(3\,a_{S}+8\,v_{S}\,\lambda_{S}\right),$
(2.21a) $\displaystyle M_{S}^{2}$
$\displaystyle\simeq(M_{S}^{2})^{\mathrm{t}ree}-\frac{1}{v_{S}}\,t_{S}\big{(}(m_{S}^{2})^{\mathrm{t}ree}\big{)}+\Pi_{SS}\big{(}(M_{S}^{2})^{\mathrm{t}ree};(m_{S}^{2})^{\mathrm{t}ree}\big{)}\,.$
(2.21b)

We have written $(m_{S}^{2})^{\mathrm{t}ree}$ in the arguments of the tadpoles
and self-energies to show the explicit dependence in the loop functions. In
the left and right-hand plots of figure 7 we therefore see that the shift
between $m_{S}^{2}$ and $(m_{S}^{2})^{\mathrm{t}ree}$ becomes very large, and
this leads to a breakdown of the (primitive) iterative algorithm that we use
to solve for $v_{S}$, hence the standard scheme curves end near $m_{S}=1250$
GeV, while the modified scheme has no such issue and the difference between
loop-corrected and tree-level masses is negligible. This gives a different
perspective on the general problem of calculating masses in such models. On
the other hand, we see that, while the tree-level masses can differ
significantly (even for the light “Higgs”) the loop masses agree to a high
precision, as they should.

Figure 6: $M_{h}$ (left) and $M_{S}$ (right) as a function of the true
$\overline{\mathrm{MS}}$ parameter $m_{S}$ in both the standard and modified
schemes, where the standard scheme is performed according to the “consistent
tadpole” prescription. Other parameters as in figure 4.

Figure 7: $M_{h}$ (left) and $M_{S}$ (right) as a function of the true
$\overline{\mathrm{MS}}$ parameter $m_{S}$ in both the standard and modified
schemes, where the standard scheme does not involve “consistent tadpoles” but
the true $\overline{\mathrm{MS}}$ mass $m_{S}$ is used everywhere. Other
parameters as in figure 4.

For a final comparison, we give in figure 7 the same computation but where,
instead of “consistent tadpoles” we use the true $\overline{\mathrm{MS}}$ mass
$m_{S}^{2}$ obtained from eq. (2.20) in all of the loop functions so that

$\displaystyle(M_{S}^{2})^{\mathrm{t}ree}$
$\displaystyle\simeq-\frac{a_{SH}v^{2}}{2\,v_{S}}+v_{S}\left(3\,a_{S}+8\,v_{S}\,\lambda_{S}\right)-\frac{1}{v_{S}}\,t_{S}\big{(}m_{S}^{2}\big{)}\,,$
(2.22a) $\displaystyle M_{S}^{2}$
$\displaystyle\simeq(M_{S}^{2})^{\mathrm{t}ree}+\Pi_{SS}\big{(}(M_{S}^{2})^{\mathrm{t}ree};m_{S}^{2}\big{)}\,.$
(2.22b)

Aside from a shuffling of the tadpole term between the “tree-level” mass in
the standard scheme, which now ensures that all of the curves on the right-
hand side of figure 7 lie on top of each other (modulo the same proviso that
the algorithm for finding $v_{S}$ breaks down) the differences between these
two versions of the standard scheme then only exist at two loops. From figure
7 it would seem that avoiding the consistent tadpoles would be preferable in
these cases, but of course then the above equations mix tree-level and loop-
level quantities, so we have problems with EFT matching at one loop (because
subleading logarithms do not cancel) and infra-red issues at two loops.

##  3 Pole mass matching with tadpole insertions

When matching two theories via pole masses, care must be taken that subleading
logarithms are correctly subtracted. The best way to do this is to expand the
expressions on both sides of the matching relation in terms of the same
parameters; the most efficient way to do this is to use those of the high-
energy theory (HET) even though this adds a layer of complication because it
is the SM parameters that we know from the bottom-up observations. To this end
we require the shifts in the vacuum expectation value as well as gauge, Yukawa
and of course quartic couplings.

The most straightforward way to match the vacuum-expectation value of the
Higgs is via matching the $Z$ mass, which gives (see e. g. Refs. [26, 29,
41]):

$\displaystyle v_{\mathrm{SM}}^{2}$
$\displaystyle=v_{\mathrm{HET}}^{2}+\frac{4}{g_{Y}^{2}+g_{2}^{2}}\left[\hat{\Pi}_{ZZ}^{\mathrm{HET}}(0)-\hat{\Pi}_{ZZ}^{\mathrm{SM}}(0)\right]+\mathcal{O}{\left(v^{4}\right)}\,.$
(3.1)

If we match the one-loop Higgs mass in the SM to the HET, where the light
Higgs mass at tree level is $m_{0}$, then we have

$\displaystyle
2\,\lambda_{\mathrm{SM}}\,v^{2}_{\mathrm{SM}}+\hat{\Pi}_{hh}^{\mathrm{SM}}{\left(2\,\lambda_{\mathrm{SM}}\,v^{2}_{\mathrm{SM}}\right)}=m_{0}^{2}+\hat{\Pi}_{hh}^{\mathrm{HET}}{\left(m_{0}^{2}\right)}$
(3.2a)
$\displaystyle\lambda_{\mathrm{SM}}=\frac{1}{2\,v_{\mathrm{HET}}^{2}}\bigg{\\{}m_{0}^{2}+\hat{\Pi}_{hh}^{\mathrm{HET}}{\left(m_{0}^{2}\right)}-\hat{\Pi}_{hh}^{\mathrm{SM}}{\left(m_{0}^{2}\right)}-\frac{4\,m_{0}^{2}}{v_{\mathrm{HET}}^{2}\left(g_{Y}^{2}+g_{2}^{2}\right)}\Big{[}\hat{\Pi}_{ZZ}^{\mathrm{HET}}(0)-\hat{\Pi}_{ZZ}^{\mathrm{SM}}(0)\Big{]}\bigg{\\}}.$
(3.2b)

It should be noted that – in order to preserve gauge invariance, and cancel
large logarithms exactly without introducing spurious subleading ones – the
matching of the quartic coupling should be performed according to this
equation, as opposed to performing some iteration, matching eigenvalues of the
mass matrices, or separately matching the expectation values and Higgs mass
(as performed in some codes) [32, 51]. With the prescription of including
tadpole diagrams, this leads to

$\displaystyle\hat{\Pi}_{hh}$
$\displaystyle\equiv\Pi_{hh}-a^{hhk}\,\frac{1}{m_{k}^{2}}\,t_{k}\,,\qquad\hat{\Pi}_{ZZ}\equiv\Pi_{ZZ}-g^{ZZk}\,\frac{1}{m_{k}^{2}}\,t_{k}\,.$
(3.3)

In the SM with $\mathcal{L}\supset-\lambda_{\mathrm{SM}}\,\lvert H\rvert^{4}$
we have

$\displaystyle\hat{\Pi}_{hh}^{\mathrm{SM}}$
$\displaystyle\equiv\Pi_{hh}^{\mathrm{SM}}-\frac{6\,\lambda\,v}{m_{h}^{2}}\,t_{h}^{\mathrm{SM}}=\Pi_{hh}^{\mathrm{SM}}-\frac{3}{v}\,t_{h}^{\mathrm{SM}}\,,\qquad\hat{\Pi}_{ZZ}^{\mathrm{SM}}\equiv\Pi_{ZZ}^{\mathrm{SM}}-\frac{2\,M_{Z}^{2}}{v\,m_{h}^{2}}\,t_{k}^{\mathrm{SM}}\,,$
(3.4) and so
$\displaystyle\hat{\Pi}_{hh}^{\mathrm{SM}}-\frac{m_{h}^{2}}{M_{Z}^{2}}\,\hat{\Pi}_{ZZ}^{\mathrm{SM}}$
$\displaystyle=\Pi_{hh}^{\mathrm{SM}}-\frac{m_{h}^{2}}{M_{Z}^{2}}\,\Pi_{ZZ}^{\mathrm{SM}}-\frac{3}{v}\,t_{h}^{\mathrm{SM}}+\frac{2}{v}\,t_{h}^{\mathrm{SM}}=\Delta
M^{2}_{\mathrm{SM}}-\frac{m_{h}^{2}}{M_{Z}^{2}}\,\Pi_{ZZ}^{\mathrm{SM}}\,,$
(3.5)

where the $\Delta M^{2}_{\mathrm{SM}}$ is now just the standard set of vacuum
conditions as in eqs. (1.4) or (2.2). So what we have shown is that the
modified treament of tadpoles cancels out exactly in the matching of the light
Higgs, _for the SM part_. Of course, the shift in the matching condition
should only depend on the Lagrangian parameters, which are not affected by the
treatment of tadpoles, so the same is true for the matching in the HET part
_up to terms of higher order in $v$_.

We have already implicitly shown how the change in scheme affects the matching
of the gauge bosons; now for fermions we have

$\displaystyle\Gamma_{F_{i}F_{j}}(p)$
$\displaystyle=i\left(\not{p}-m_{F}\right)\delta_{ij}+i\left[\not{p}\left(P_{L}\,\hat{\Sigma}_{ij}^{L}{\left(p^{2}\right)}+P_{R}\,\hat{\Sigma}_{ij}^{R}{\left(p^{2}\right)}\right)+P_{L}\,\hat{\Sigma}_{ij}^{SL}{\left(p^{2}\right)}+P_{R}\,\hat{\Sigma}_{ij}^{SR}{\left(p^{2}\right)}\right].$
(3.6)

For fermions at one loop we can write the mass-matrix corrections as

$\displaystyle\delta m_{F}$
$\displaystyle=-\Sigma^{SL}-\frac{1}{2}\left(\Sigma^{R}\,m+m\,\Sigma^{L}\right).$
(3.7)

This means that our tadpole shift just affects

$\displaystyle\delta\Sigma^{SL}$
$\displaystyle=\delta\Sigma^{SR}=\frac{1}{m_{k}^{2}}\,y^{ijk}\,\frac{\partial
V}{\partial\phi_{k}}\,,$ (3.8)

where $y^{ijk}$ are the Yukawa couplings, that can be written in terms of Weyl
spinors $\\{\psi_{i}\\}$ as

$\displaystyle\mathcal{L}\supset-\frac{1}{2}\,y^{ijk}\,\psi_{i}\,\psi_{j}\,\phi_{k}\,.$
(3.9)

To match the Yukawa couplings via the pole masses of the quarks, the matching
of the electroweak expectation value must also be included; working in the
basis with diagonalised Yukawa couplings, we can match the diagonal elements
as (using $Y^{F}\equiv y^{FFh}$ for $h$ the SM Higgs and a general fermion
$F$)

$\displaystyle M_{F}$
$\displaystyle=v\,Y^{F}-\Sigma^{SL}-\frac{1}{2}\left(\Sigma^{R}\,m+m\,\Sigma^{L}\right),$
(3.10a) $\displaystyle Y^{F}_{\mathrm{SM}}$
$\displaystyle=Y^{F}_{\mathrm{HET}}+\frac{1}{v_{\mathrm{HET}}}\left[(\delta
m_{F})^{\mathrm{HET}}-(\delta
m_{F})^{\mathrm{SM}}-\frac{1}{m_{k}^{2}}\,y^{FFk}_{\mathrm{HET}}\,t_{k}+\frac{1}{m_{h}^{2}}\,Y^{F}_{\mathrm{SM}}\,t_{h}^{\mathrm{SM}}\right]$
$\displaystyle\quad-\frac{Y^{F}_{\mathrm{HET}}}{2\,M_{Z}^{2}}\left[\hat{\Pi}_{ZZ}^{\mathrm{HET}}(0)-\hat{\Pi}_{ZZ}^{\mathrm{SM}}(0)\right]$
$\displaystyle=Y^{F}_{\mathrm{HET}}+\frac{1}{v_{\mathrm{HET}}}\left[(\delta
m_{F})^{\mathrm{HET}}-(\delta
m_{F})^{\mathrm{SM}}-\frac{1}{m_{k}^{2}}\,y^{FFk}_{\mathrm{HET}}\,t_{k}\right]-\frac{Y^{F}_{\mathrm{HET}}}{2\,M_{Z}^{2}}\left[\hat{\Pi}_{ZZ}^{\mathrm{HET}}(0)-\Pi_{ZZ}^{\mathrm{SM}}(0)\right],$
(3.10b)

where we once again see that the shift in the tadpole scheme cancels out
exactly in the SM part. This procedure is particularly important since the
shift to the expectation value arising in eq. (3.1) is very large, as
discussed in the introduction. In this case, since the corrections to
$\mu^{2}$ – and therefore also to $v^{2}$ – are very large, it becomes
impractical in an implementation to actually use the “correct” value of
$v^{2}$ in the high-energy theory. Indeed, this can even become impossible, if
$\delta\mu^{2}$ is such that $\mu^{2}$ would become positive in the SM!
Instead, provided we take $v$ much less than the matching scale, we can just
treat it as perturbation parameter to extract the SM values. In our numerical
calculation in the next section we do exactly this: we just use the SM value
of $v$ in both high- and low-energy theories, but use the correct shifts of
the expectation values in the matching of the parameters. This is very similar
to a standard EFT calculation, which assumes e. g. in split supersymmetry that
the heavy Higgs masses are tuned according to the mixing angle given as an
input, and takes $v=0$ explicitly, since we are not interested in corrections
to Lagrangian parameters of order $v^{2}\big{/}M^{2}$ where $M$ is the
matching scale.

##  4 Application in the $\mu$NMSSM

In the introduction, we explained that the modified treatment of tadpoles can
be useful for stability under perturbation theory of heavy scalar masses when
they are associated with a small expectation value. In section 2 we showed how
it worked in practice in a toy model. In section 3 we described how, for
theories where the new scalars are substantially above the electroweak scale,
it can be practically applied via EFT matching of the pole masses. Here, we
shall apply this technique to a real test case, the $\mu$NMSSM.

###  4.1 NMSSM, $\mu$NMSSM and GNMSSM

The superpotential of the most general form of the NMSSM – the GNMSSM – is
[101, 102]

$\displaystyle W_{\mathrm{GNMSSM}}$ $\displaystyle=Y_{u}\,Q\cdot
H_{u}\,U-Y_{d}\,Q\cdot H_{d}\,D-Y_{e}\,L\cdot
H_{d}\,E+\frac{1}{3}\,\kappa\,S^{3}+(\mu+\lambda\,S)H_{u}\cdot
H_{d}+\xi\,S+\frac{1}{2}\,\mu_{S}\,S^{2}$

and the supersymmetry-breaking terms in the Higgs sector are

$\displaystyle V_{\mathrm{s}oft}$ $\displaystyle\supset m_{S}^{2}\,\lvert
S\rvert^{2}+m_{H_{u}}^{2}\,\lvert H_{u}\rvert^{2}+m_{H_{d}}^{2}\lvert
H_{d}\rvert^{2}$ $\displaystyle\quad+\bigg{(}B_{\mu}\,H_{u}\cdot
H_{d}+T_{\lambda}\,S\,H_{u}\cdot
H_{d}+\frac{1}{3}\,T_{\kappa}\,S^{3}+\frac{1}{2}\,B_{S}\,S^{2}+\xi_{S}\,S+\text{h.\,c.}\bigg{)}\,.$
(4.1)

Once the singlet develops an expectation value, we can write effective terms

$\displaystyle\mu_{\mathrm{e}ff}$
$\displaystyle\equiv\mu+\frac{1}{\sqrt{2}}\,\lambda\,v_{S}\,,$ $\displaystyle
B_{\mathrm{e}ff}$ $\displaystyle\equiv
B_{\mu}+\frac{1}{\sqrt{2}}\,T_{\lambda}\,v_{S}+\lambda\left(\xi+\frac{1}{\sqrt{2}}\,\mu_{S}\,v_{S}+\frac{1}{2}\,\kappa\,v_{S}^{2}\right)$
(4.2)

and the tadpole equations become

$\displaystyle 0$
$\displaystyle=-B_{\mathrm{e}ff}\cot\beta+m_{H_{u}}^{2}+\mu_{\mathrm{e}ff}^{2}-\frac{M_{Z}^{2}}{2}\,c_{2\beta}+\frac{1}{2}\,\lambda\,c_{\beta}^{2}\,,$
(4.3a) $\displaystyle 0$
$\displaystyle=-B_{\mathrm{e}ff}\tan\beta+m_{H_{d}}^{2}+\mu_{\mathrm{e}ff}^{2}+\frac{M_{Z}^{2}}{2}\,c_{2\beta}+\frac{1}{2}\,\lambda\,s_{\beta}^{2}\,,$
(4.3b) $\displaystyle 0$
$\displaystyle=v_{S}\left(B_{S}+m_{S}^{2}+\mu_{S}^{2}+2\,\kappa\,\xi\right)+\frac{1}{\sqrt{2}}\,v_{S}^{2}\left(T_{\kappa}+3\,\kappa\,\mu_{S}\right)+\kappa^{2}\,v_{S}^{3}$
$\displaystyle\quad+\sqrt{2}\,\mu_{S}\,\xi+\sqrt{2}\,\xi_{S}+\frac{1}{2\,\sqrt{2}}\,v^{2}\Big{(}2\,\lambda\,\mu_{\mathrm{e}ff}-\left(T_{\lambda}+2\,\kappa\,\lambda\,v_{S}+\mu_{S}\,\lambda\right)s_{2\beta}\Big{)}\,.$
(4.3c)

The first two lines are essentially modified versions of the MSSM tadpole
equations with an extra term from the $\lambda$ coupling. The third line,
however, is the crucial one for our discussion. In a general non-
supersymmetric theory, we can redefine singlet fields to remove their tadpole
terms. However, in the GNMSSM, which has tadpole parameters $\xi$ in the
superpotential and $\xi_{S}$ in the soft-breaking terms, we can only remove
one of these, or the combination $\sqrt{2}\,\mu_{S}\,\xi+\sqrt{2}\,\xi_{S}$.

Clearly in the GNMSSM, it is most logical to choose a linear combination of
the singlet tadpole terms $\xi$ and $\xi_{S}$ (or just one) as the variable to
be eliminated by the tadpole equations. However, this is not possible in the
NMSSM or $\mu$NMSSM, since these terms vanish by the assumption of (at least
partial) $\mathbb{Z}_{3}$ symmetry. Then aside from
$(m_{H_{u}}^{2},m_{H_{d}}^{2})$ or $(\mu,B_{\mu})$, the dimensionful parameter
that we can now choose for elimination via the singlet tadpole equation is one
of $\big{\\{}m_{S}^{2},\mu_{\mathrm{e}ff},T_{\lambda},T_{\kappa}\big{\\}}$.

We are interested in the case that the singlet is rather heavier than the SM-
like Higgs, so that $v^{2}\big{/}m_{S}^{2}\ll 1$. This is clearly at best
problematic in the NMSSM, since $\mu_{\mathrm{e}ff},B_{\mathrm{e}ff}\propto
v_{S}$ so if we imagine $v_{S}\sim$ GeV we will have very light higgsinos,
pseudoscalar/charged Higgs and difficulties solving the tadpole equations.
Hence we turn to the $\mu$NMSSM, where we neglect all terms that break the
$\mathbb{Z}_{3}$ symmetry except for $\mu$ and $B_{\mu}$, and find

$\displaystyle v_{S}$ $\displaystyle\simeq-v^{2}\left(\frac{2\,\lambda\,\mu-
T_{\lambda}\,s_{2\beta}}{2\,\sqrt{2}\,m_{S}^{2}}\right),$ (4.4)

where the true value can be found numerically.

The logical choice for this case is to solve for $T_{\lambda}$. In this case
we have

$\displaystyle\Delta T_{\lambda}$
$\displaystyle=-\frac{2\,\sqrt{2}}{v^{2}\,s_{2\beta}}\,\frac{\partial\Delta
V}{\partial v_{S}}\,,$ (4.5)

and the terms in the mass matrix become

$\displaystyle\mathcal{M}^{2}_{h_{u}^{0}h_{u}^{0}}$
$\displaystyle\supset-\frac{v_{S}\,t_{S}^{(1)}}{v^{2}\,s_{\beta}\,c_{\beta}}+\ldots\propto\frac{t_{S}^{(1)}}{m_{S}^{2}}\,,$
$\displaystyle\mathcal{M}^{2}_{h_{u}^{0}s_{R}}$
$\displaystyle=-\frac{m_{S}^{2}\,v_{S}+t_{S}^{(1)}}{v\,s_{\beta}}+\ldots\,.$
(4.6)

Note that this is in the “flavour basis” before we diagonalise the fields at
tree level, so the contributions to the light Higgs and heavy singlet masses
are $\propto t_{S}^{(1)}\big{/}m_{S}^{2}$ .

On the other hand, this choice leads to a (potentially very) large quantum
correction to $T_{\lambda}$. Suppose we want to investigate gauge-mediation
scenarios where trilinears are small (nearly vanishing), or are otherwise
specified by the top-down inputs – this would be completely inappropriate.
Furthermore, we have to not only take into account shifts in the masses but
also the _couplings_ – this is moderately cumbersome to implement at one loop,
but much more so if we want to compute the two-loop corrections. Indeed, it is
not included in the algorithm to generate “consistent vacuum equations” of
Refs. [70, 27], which assumes that the parameters that we solve the tadpole
equations for only affect scalar masses.

To solve both of these issues the simplest choice is to solve for $m_{S}^{2}$,
and this leads to exactly the same problem as in the toy model, that the
corrections to the singlet mass scale as $t_{S}^{(1)}\big{/}v_{S}$ leading to
numerical instabilities for tiny $v_{S}$. Hence this model is an excellent
prototype for comparing the different approaches to solving the tadpole
equations.

###  4.2 Numerical comparison of tadpole schemes

In the $\mu$NMSSM and GNMSSM, we not only have a Higgs sector, but also
squarks, sleptons, a gluino and electroweakinos. In particular the colourful
states have a large impact on the mass of the light Higgs, and, when they are
heavy enough to be safe from current collider searches, they cause the “little
hierarchy problem” to manifest itself. If we try to apply our modified tadpole
scheme directly to these models, then we find all of the problems associated
with this little hierarchy in our Higgs-mass calculation. Therefore it is only
sensible to use EFT matching for the light Higgs mass. In this section we
shall endeavour to show that with such an approach we can solve the technical
difficulties with computing the masses of both light and heavy Higgs bosons.

We shall present here numerical investigations of several scenarios of the
$\mu$NMSSM and GNMSSM illustrating the differences between the two approaches
to the treatment of tadpoles, both using EFT matching. For this, we compare
results obtained using the original version of SPheno code obtained directly
from SARAH (for the model SMSSM), as well as with a version of the Fortran
output extensively modified according to the prescriptions described in
section 3.444This private code is not intended for public release, although it
is available on request from the authors. The new functionality should
eventually be made available in a future release of SARAH.

In these calculations we must refer the reader again to our disclaimer, that
we shall compare parameter points that generate the same _tree-level spectrum_
in the two schemes, but that differ from each other at higher order; because
this provides the clearest illustration of the problems faced (namely how to
even define the parameter point). In contrast to the toy model, we will give
no examples with a complete conversion of parameters, i. e. a comparison of
both calculations at the same point, since the actual procedure of converting
between the schemes is too onerous for technical reasons. In the SARAH/SPheno
code, while a numerical solution of the tadpole equations (required for
providing $\overline{\mathrm{MS}}$ input to the standard scheme) is in
principle possible, it is labourious and not implemented for loop computations
where the variable to solve for is the vacuum expectation value.555This
development in SARAH is envisioned in the future. Therefore, again we take the
tree-level value of $v_{S}$ as input for the modified scheme, and treat it as
the “all-orders” expectation value in the standard scheme (with consistent
tadpoles) thus ensuring the same tree-level spectrum, but potentially vastly
different results at one loop due to the large corrections to $m_{S}^{2}$ in
the standard scheme. Again we stress that this is typical of the ambiguity in
defining a parameter point that the phenomenologist is invited to suffer,
thanks to the expedient in the standard scheme of hiding loop corrections in
the definition of the expectation values.

In section 4.2.1 we give an example of the above reasoning in the $\mu$NMSSM.
For illustration in section 4.2.2 we also give examples in the GNMSSM where we
solve the tadpoles for the same variables ($m_{H_{u}}^{2}$, $m_{H_{d}}^{2}$,
$m_{S}^{2}$) which allows us to compare several different scenarios.

####  4.2.1 $\mu$NMSSM

Figure 8: $M_{h_{1}}$ (left) and $M_{h_{2}}$ and $M_{h_{3}}$ (right) as a
function of $v_{S}$ in a scenario of the $\mu$NMSSM. The other inputs are
taken as follows: $\lambda=\kappa=0.1$, $T_{\lambda}=200\text{ GeV}$,
$T_{\kappa}=-10\text{ GeV}$, $\mu=100\text{ GeV}$, $B_{\mu}=6\cdot
10^{5}\text{ GeV}^{2}$. Tree-level values are shown with green curves, while
the red and blue curves correspond to the pole masses computed at one loop,
respectively with the standard and modified approaches to the tadpoles. The
colour coding of the lines remains the same for all figures in this section.

In figure 8, we present the behaviour of the three CP-even mass eigenvalues –
i. e. the lightest Higgs mass $M_{h_{1}}$ (left side) and the masses of the
additional CP-even states $M_{h_{2}}$ and $M_{h_{3}}$ (right side) – as a
function of the singlet vev $v_{S}$ in a $\mu$NMSSM scenario, where the
underlying parameters are given in the caption. The tree-level values are
shown in green, while the one-loop results using the standard and the modified
treatments of tadpoles are in red and blue respectively. We consider here a
low range of values for $v_{S}$, so that, following our discussion in the
previous section, we expect the standard approach to perform poorly for the
singlet-like mass eigenstate. This is indeed what we observe if we turn to the
right-side plot: for lower $v_{S}$ ($\lesssim 0.1$–$1\text{ GeV}$) the
singlet-like scalar is the heaviest eigenstate $h_{3}$, while after level
crossing it is $h_{2}$ for larger $v_{S}$. For the entire range of $v_{S}$ the
mass corrections in the standard approach are huge, and they grow as large as
50 TeV for $v_{S}=0.001\text{ GeV}$ – i. e. 250% of the tree-level result! On
the other hand, if we look instead at the lightest Higgs boson $h_{1}$, we
find that the radiative corrections are somewhat larger with the modified
treatment of the tadpole diagrams, and increase significantly with $v_{S}$ in
this scenario – due to the contributions from the tadpole diagram with a
relatively large value of $T_{\lambda}=200\text{ GeV}$ and a relatively small
tree-level mass of the singlet-like state.

####  4.2.2 GNMSSM

While the $\mu$NMSSM provides an excellent prototype for the case of a heavy
singlet with a small expectation value, where we cannot hide the loop
corrections in a tadpole term, since it is a subset of the GNMSSM we can find
more varied scenarios exhibiting the same behaviour. Of course, this is with
the proviso that (with less justification in general) we restrict ourselves to
solving the tadpole equations for $m_{S}^{2}$.

We have devised three types of scenarios:

  * •

Scenario 1: large singlet vev and intermediate $\lambda$;

  * •

Scenario 2: small singlet vev and small $\lambda$;

  * •

Scenario 3: small singlet vev but large $\lambda$.

Table 1 summarises the values taken for the BSM input parameters relevant for
SPheno – note that we have adjusted the soft terms $m_{0}$ (scalar mass) and
$A_{0}$ (scalar trilinear coupling) in order to obtain a mass for the lightest
Higgs boson within the interval $[123~{}\mathrm{GeV},127~{}\mathrm{GeV}]$. We
should also emphasise that the numbers in table 1 are given to SPheno as high-
scale inputs (as this only requires a limited set of values). We then convert
these into low-scale input parameters using the standard version of the
$\mu$NMSSM SPheno code, and the plots presented in the following are obtained
by varying one of the low-scale inputs. In light of the analytic expressions
in the previous section, we can expect the two approaches to the tadpoles to
give relatively similar results in scenario 1, where the singlet vev is large.
However, in scenarios 2 and 3, the singlet vev is taken to be small, so that
the differences between the two schemes should be more pronounced. Scenario 3
furthermore allows us to investigate the effect of increasing the coupling
$\lambda$.

Scenario | 1 | 2 | 3
---|---|---|---
$m_{0}$ | [GeV] | 2000 | 1500 | 1500
$\lambda$ |  | $0.1^{\dagger}$ | 0.01 | 0.15
$\kappa$ |  | 0.005 | 0.05 | 0.05
$T_{\lambda}$ | [GeV] | 1000 | $1000^{\dagger}$ | $7500^{\dagger}$
$v_{S}$ | [GeV] | 3000 | $1.0^{\dagger}$ | $1.0^{\dagger}$
$\mu$ | [GeV] | 500 | 200 | 200
$\mu_{S}$ | [GeV] | 0 | –200 | –200
$\xi$ | $\big{[}\mathrm{GeV}^{2}\big{]}$ | $1.0\cdot 10^{8}$ | $1.7\cdot 10^{6}$ | $5.0\cdot 10^{4}$
$B_{\mu}$ | $\big{[}\mathrm{GeV}^{2}\big{]}$ | $2.0\cdot 10^{5}$ | $1.0\cdot 10^{6}$ | $4.0\cdot 10^{5}$
Table 1:  Definitions of the input parameters in the considered $\mu$NMSSM
scenarios. Some of the BSM parameters are not modified, and remain the same
for the three scenario. Namely, we take: $\tan\beta=10$,
$m_{12}=2~{}\mathrm{TeV}$, $A_{0}=3~{}\mathrm{TeV}$, $B_{0}=0$,
$m_{A}=500~{}\mathrm{GeV}$, $T_{\kappa}=-0.5~{}\mathrm{GeV}$. The
renormalisation scale is kept at $Q=3~{}\text{TeV}$ for all computations.
Finally, the numbers marked with a “$\dagger$” are varied for some of the
parameter scans.

Figure 9: $M_{h_{1}}$ (left) and $M_{h_{2}}$ and $M_{h_{3}}$ (right) as a
function of $\lambda$, in scenario 1. The other inputs are taken as in table
1. Tree-level values are shown with green curves, while the red and blue
curves correspond to the pole masses computed at one loop, respectively with
the standard and modified approaches to the tadpoles.

Figure 10: $M_{h_{1}}$ (left) and $M_{h_{2}}$ and $M_{h_{3}}$ (right) as a
function of the soft trilinear coupling $T_{\lambda}$, in scenario 2. The
values of the other BSM parameters are taken as in table 1.

Figure 11: $M_{h_{1}}$ (left) and $M_{h_{2}}$ and $M_{h_{3}}$ (right) as a
function of $v_{S}$, in scenario 2. Input values for the other BSM parameters
are given in table 1.

We show first in figure 11 the behaviour of the lightest Higgs mass
$M_{h_{1}}$ (left side) and of the additional CP-even Higgs-boson masses
$M_{h_{2}}$ and $M_{h_{3}}$ (right side) as a function of the superpotential
coupling $\lambda$. Among the two BSM states $h_{2}$ and $h_{3}$, the former
is singlet-like while the latter is doublet-like, in this figure. As can be
seen in the right-hand side plot of figure 11, the heavy Higgs bosons receive
only minute mass corrections in either of the approaches for tadpoles. For the
lightest scalar mass $M_{h_{1}}$, the results in the two schemes are also in
excellent agreement. However, we have cut off the plot before $\lambda=0.19$
because beyond this value perturbativity is lost: in the _standard approach_
the singlet-like pseudoscalar Higgs becomes tachyonic at one loop (from a
tree-level mass of $750$ GeV!). If we continued the plot into this regime we
would see the predictions diverging, with the standard approach predicting
ever decreasing masses and the modified approach increasing ones for larger
$\lambda$ (compare $104$ GeV and $138$ GeV respectively for $\lambda=0.3$).

Next, we turn to scenario 2, i. e. we consider a small $\lambda=0.01$ and
small singlet vev $v_{S}=1~{}\mathrm{GeV}$. Figure 11 shows the behaviour of
the CP-even masses as a function of the soft trilinear coupling $T_{\lambda}$,
at tree level and one loop (the colouring of the curves is the same as
previously explained). We should emphasise that we have made sure to fulfill
constraints from vacuum stability (and the absence of a charge-breaking
minimum) on $T_{\lambda}$ – see Ref. [63] – and the tree-level mass of the
charged Higgs boson remains positive for the entire range of $T_{\lambda}$
investigated here. While for $M_{h_{1}}$ (left side) and $M_{h_{2}}$ (lower
curves of the right-side plot) it seems essentially impossible to distinguish
the two approaches to the tadpole treatment, the radiative corrections to
$M_{h_{3}}$ – the mass of the singlet-like scalar – are clearly much larger
with the standard method, and the result of the modified scheme is certainly
more reliable. As a concrete comparison, we have for the intermediate value
$T_{\lambda}=2~{}\text{TeV}$ a one-loop correction to $M_{h_{3}}$ of 2752 GeV
(i. e. 24% of the tree-level result) in the standard approach, but only of
–4.5 GeV in the modified scheme.

We can confirm that the large difference between the two treatments of the
tadpoles arises from the small value of the singlet vev $v_{S}$. Indeed, in
figure 11, we present the same three CP-even scalar masses for $v_{S}$ varying
between 0.5 and 100 GeV. One can observe that the results using both
approaches for all three masses are in good agreement for large values of the
singlet vev. A short comment should be made for $M_{h_{1}}$: indeed, as
$v_{S}$ increases the results from the two schemes seem to grow apart, and it
is somewhat difficult to determine which one should be trusted more in this
case. We note that the radiative corrections to $M_{h_{1}}$ keep increasing
with $v_{S}$ in the standard approach while their size remains relatively
stable in the modified scheme. On the other hand, if we consider the situation
for $v_{S}\gtrsim 0.5~{}\mathrm{GeV}$, the breakdown of the standard approach
for small singlet vevs becomes obvious. Indeed, considering the different
results for the mass $M_{h_{3}}$ of the CP-even singlet-like scalar at
$v_{S}=0.5~{}\mathrm{GeV}$, the one-loop corrections in the standard scheme
amount to 6.5 TeV – in other words, 40% of the tree-level result – compared to
only –3.3 GeV (–0.02% of the tree-level mass) in the modified scheme.

Figure 12:  $M_{h_{1}}$ (left) and $M_{h_{2}}$ and $M_{h_{3}}$ (right) as a
function of $T_{\lambda}$, in scenario 3. The other BSM inputs are taken as in
table 1.

Figure 13:  $M_{h_{1}}$ (left) and $M_{h_{2}}$ and $M_{h_{3}}$ (right) as a
function of $v_{S}$, in scenario 3. The values of the other relevant inputs
are given in table 1.

Lastly, we consider the type of scenario 3, i. e. what happens if we keep a
small singlet vev $v_{S}=1~{}\mathrm{GeV}$ but increase the coupling $\lambda$
to 0.15. In figure 13, we present the CP-even scalar masses as a function of
$T_{\lambda}$ – having once again made sure to maintain vacuum stability [63].
Considering first the masses of the two doublet-like scalars $h_{1}$ and
$h_{2}$, we observe an excellent agreement of the results from the two tadpole
schemes for low to intermediate values of $T_{\lambda}$ – for $0\leq
T_{\lambda}\lesssim 4~{}\text{TeV}$. However, as $T_{\lambda}$ becomes larger,
the corrections to $M_{h_{1}}$ and $M_{h_{2}}$ in the modified approach start
growing out of control. This appears similar to the loss of accuracy of the
modified scheme that we encountered in the toy model of section 2 when
increasing the trilinear coupling $a_{SH}$, which plays the same role as
$T_{\lambda}$ – see eq. (2.19) and figure 4. Turning however to the singlet-
like mass $M_{h_{3}}$ we find (as in figure 11 for scenario 2) that the
radiative corrections are huge with the standard treatment of tadpoles, but
remain well-behaved with the modified one. Interestingly, having increased the
value of $\lambda$ has not made the breakdown of the standard calculation for
the singlet-like mass more severe than in scenario 2. Nevertheless, the one-
loop result $M_{h_{3}}$ using the modified tadpole scheme is undoubtedly more
reliable here.

Finally, we present in figure 13 the behaviour of the CP-even scalar masses as
a function of the singlet vev $v_{S}$ – restricting our attention to the low
range $0.5~{}\mathrm{GeV}\leq v_{S}\leq 5~{}\mathrm{GeV}$. As can be read from
table 1, we have chosen for this figure a large value of the soft trilinear
coupling $T_{\lambda}=7.5~{}\text{TeV}$, which corresponds to the right parts
of the plots in figure 13. Therefore, it is not surprising that we observe
some discrepancy between the results of the two tadpole schemes for all three
masses, as discussed above. More interestingly, we can compare the size of the
loop corrections to $M_{h_{3}}$ in the two approaches, as we vary $v_{S}$. On
the one hand, in the standard approach, the one-loop corrections increase from
2.3 TeV (19% of the tree-level result) for $v_{S}=2.5~{}\mathrm{GeV}$ to as
much as 9 TeV (40% of the tree-level mass) for $v_{S}=0.75~{}\mathrm{GeV}$,
for instance. On the other hand, in the modified scheme, the effects remain
minute and vary from –46 GeV for $v_{S}=2.5~{}\mathrm{GeV}$ to –3.6 GeV for
$v_{S}=0.75~{}\mathrm{GeV}$ (this amounts to –0.38% and –0.02% of the results
at tree level, respectively).

##  5 Conclusions

We have shown the advantages and limitations of taking a different
prescription for the solution of tadpole equations. In contrast to previous
applications of this technique, in the SM or as a measure of fine-tuning, we
have shown that it can be very useful when new scalars having a small
expectation value are present in the theory, and in the case that they are
much heavier than the electroweak scale, it is best employed via the matching
of pole masses in an EFT approach. While this technique offers the advantages
of perturbative stability for the heavy scalar masses, easy generalisability
(the corrections are simply computed diagramatically rather than via taking
derivatives of the tadpole equations) and gauge invariance, it can also lead
to numerical instabilities in extracting the _light_ Higgs mass, and the loss
of the ability to match the electroweak expectation value.

In future work, other than a general numerical implementation in SARAH, it
would be interesting to explore a hybrid approach (along the lines of option 1
described at the end of the introduction), where only the electroweak
expectation value is fixed by appropriate counterterms. On the other hand, we
intend to consider the corrections at two loops in this approach, and we shall
also provide general expressions for the one-loop self-energies which are
explicitly gauge independent.

## Acknowledgements

We would like to thank Sven Heinemeyer and Pietro Slavich for interesting
discussions. MDG acknowledges support from the grant “HiggsAutomator” of the
Agence Nationale de la Recherche (ANR) (ANR-15-CE31-0002). JB is supported by
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy – EXC 2121 “Quantum Universe” – 390833306. The
work of SP is supported by the BMBF Grant No. 05H18PACC2. This project
received support from the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement No
860881-HIDDeN.

## References

  * [1] ATLAS Collaboration,  “Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC”,  Phys. Lett. B716, 1 (2012), arXiv:1207.7214.
  * [2] CMS Collaboration,  “Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC”,  Phys. Lett. B716, 30 (2012), arXiv:1207.7235.
  * [3] ATLAS Collaboration, CMS Collaboration,  “Combined Measurement of the Higgs Boson Mass in $pp$ Collisions at $\sqrt{s}=7$ and 8 TeV with the ATLAS and CMS Experiments”,  Phys. Rev. Lett. 114, 191803 (2015), arXiv:1503.07589.
  * [4] ATLAS Collaboration, CMS Collaboration,  “Measurements of the Higgs boson production and decay rates and constraints on its couplings from a combined ATLAS and CMS analysis of the LHC pp collision data at $\sqrt{s}=7$ and 8 TeV”,  JHEP 08, 045 (2016), arXiv:1606.02266.
  * [5] CMS Collaboration, A. M. Sirunyan, _et al._ ,  “Combined measurements of Higgs boson couplings in proton–proton collisions at $\sqrt{s}=13\,\text{Te}\text{V}$”,  Eur. Phys. J. C79, 421 (2019), arXiv:1809.10733.
  * [6] ATLAS, G. Aad, _et al._ ,  “Combined measurements of Higgs boson production and decay using up to $80$ fb-1 of proton-proton collision data at $\sqrt{s}=$ 13 TeV collected with the ATLAS experiment”,  Phys. Rev. D 101, 012002 (2020), arXiv:1909.02845.
  * [7] P. Slavich, S. Heinemeyer (eds.), E. Bagnaschi, _et al._ ,  “Higgs-mass predictions in the MSSM and beyond”,  (2020), arXiv:2012.15629.
  * [8] W. Hollik, S. Paßehr,  “Two-loop top-Yukawa-coupling corrections to the Higgs boson masses in the complex MSSM”,  Phys. Lett. B733, 144 (2014), arXiv:1401.8275.
  * [9] S. Borowka, T. Hahn, S. Heinemeyer, G. Heinrich, W. Hollik,  “Momentum-dependent two-loop QCD corrections to the neutral Higgs-boson masses in the MSSM”,  Eur. Phys. J. C74, 2994 (2014), arXiv:1404.7074.
  * [10] E. Bagnaschi, G. F. Giudice, P. Slavich, A. Strumia,  “Higgs Mass and Unnatural Supersymmetry”,  JHEP 09, 092 (2014), arXiv:1407.4081.
  * [11] W. Hollik, S. Paßehr,  “Higgs boson masses and mixings in the complex MSSM with two-loop top-Yukawa-coupling corrections”,  JHEP 10, 171 (2014), arXiv:1409.1687.
  * [12] G. Degrassi, S. Di Vita, P. Slavich,  “Two-loop QCD corrections to the MSSM Higgs masses beyond the effective-potential approximation”,  Eur. Phys. J. C75, 61 (2015), arXiv:1410.3432.
  * [13] M. Goodsell, K. Nickel, F. Staub,  “Two-Loop Higgs mass calculations in supersymmetric models beyond the MSSM with SARAH and SPheno”,  Eur. Phys. J. C75, 32 (2015), arXiv:1411.0675.
  * [14] M. D. Goodsell, K. Nickel, F. Staub,  “Two-loop corrections to the Higgs masses in the NMSSM”,  Phys. Rev. D91, 035021 (2015), arXiv:1411.4665.
  * [15] M. Mühlleitner, D. T. Nhung, H. Rzehak, K. Walz,  “Two-loop contributions of the order $\mathcal{O}\left({\alpha}_{t}{\alpha}_{s}\right)$ to the masses of the Higgs bosons in the CP-violating NMSSM”,  JHEP 05, 128 (2015), arXiv:1412.0918.
  * [16] M. Goodsell, K. Nickel, F. Staub,  “Generic two-loop Higgs mass calculation from a diagrammatic approach”,  Eur. Phys. J. C75, 290 (2015), arXiv:1503.03098.
  * [17] S. Borowka, T. Hahn, S. Heinemeyer, G. Heinrich, W. Hollik,  “Renormalization scheme dependence of the two-loop QCD corrections to the neutral Higgs-boson masses in the MSSM”,  Eur. Phys. J. C75, 424 (2015), arXiv:1505.03133.
  * [18] F. Staub, P. Athron, U. Ellwanger, R. Gröber, M. Mühlleitner, P. Slavich, A. Voigt,  “Higgs mass predictions of public NMSSM spectrum generators”,  Comput. Phys. Commun. 202, 113 (2016), arXiv:1507.05093.
  * [19] T. Hahn, S. Paßehr,  “Implementation of the $\mathcal{O}{\left(\alpha_{t}^{2}\right)}$ MSSM Higgs-mass corrections in $\mathtt{FeynHiggs}$”,  Comput. Phys. Commun. 214, 91 (2017), arXiv:1508.00562.
  * [20] G. Lee, C. E. M. Wagner,  “Higgs bosons in heavy supersymmetry with an intermediate mA”,  Phys. Rev. D92, 075032 (2015), arXiv:1508.00576.
  * [21] M. D. Goodsell, K. Nickel, F. Staub,  “The Higgs Mass in the MSSM at two-loop order beyond minimal flavour violation”,  Phys. Lett. B758, 18 (2016), arXiv:1511.01904.
  * [22] P. Drechsel, L. Galeta, S. Heinemeyer, G. Weiglein,  “Precise Predictions for the Higgs-Boson Masses in the NMSSM”,  Eur. Phys. J. C77, 42 (2017), arXiv:1601.08100.
  * [23] M. D. Goodsell, F. Staub,  “The Higgs mass in the CP violating MSSM, NMSSM, and beyond”,  Eur. Phys. J. C77, 46 (2017), arXiv:1604.05335.
  * [24] J. Braathen, M. D. Goodsell, P. Slavich,  “Leading two-loop corrections to the Higgs boson masses in SUSY models with Dirac gauginos”,  JHEP 09, 045 (2016), arXiv:1606.09213.
  * [25] H. Bahl, W. Hollik,  “Precise prediction for the light MSSM Higgs boson mass combining effective field theory and fixed-order calculations”,  Eur. Phys. J. C76, 499 (2016), arXiv:1608.01880.
  * [26] P. Athron, J.-h. Park, T. Steudtner, D. Stöckinger, A. Voigt,  “Precise Higgs mass calculations in (non-)minimal supersymmetry at both high and low scales”,  JHEP 01, 079 (2017), arXiv:1609.00371.
  * [27] J. Braathen, M. D. Goodsell,  “Avoiding the Goldstone Boson Catastrophe in general renormalisable field theories at two loops”,  JHEP 12, 056 (2016), arXiv:1609.06977.
  * [28] P. Drechsel, R. Gröber, S. Heinemeyer, M. Mühlleitner, H. Rzehak, G. Weiglein,  “Higgs-Boson Masses and Mixing Matrices in the NMSSM: Analysis of On-Shell Calculations”,  Eur. Phys. J. C77, 366 (2017), arXiv:1612.07681.
  * [29] F. Staub, W. Porod,  “Improved predictions for intermediate and heavy Supersymmetry in the MSSM and beyond”,  Eur. Phys. J. C77, 338 (2017), arXiv:1703.03267.
  * [30] E. Bagnaschi, J. Pardo Vega, P. Slavich,  “Improved determination of the Higgs mass in the MSSM with heavy superpartners”,  Eur. Phys. J. C77, 334 (2017), arXiv:1703.08166.
  * [31] S. Paßehr, G. Weiglein,  “Two-loop top and bottom Yukawa corrections to the Higgs-boson masses in the complex MSSM”,  Eur. Phys. J. C78, 222 (2018), arXiv:1705.07909.
  * [32] H. Bahl, S. Heinemeyer, W. Hollik, G. Weiglein,  “Reconciling EFT and hybrid calculations of the light MSSM Higgs-boson mass”,  Eur. Phys. J. C78, 57 (2018), arXiv:1706.00346.
  * [33] J. Braathen, M. D. Goodsell, F. Staub,  “Supersymmetric and non-supersymmetric models without catastrophic Goldstone bosons”,  Eur. Phys. J. C77, 757 (2017), arXiv:1706.05372.
  * [34] R. V. Harlander, J. Klappert, A. Voigt,  “Higgs mass prediction in the MSSM at three-loop level in a pure $\overline{{\text{DR}}}$ context”,  Eur. Phys. J. C77, 814 (2017), arXiv:1708.05720.
  * [35] P. Athron, M. Bach, D. Harries, T. Kwasnitza, J.-h. Park, D. Stöckinger, A. Voigt, J. Ziebell,  “FlexibleSUSY 2.0: Extensions to investigate the phenomenology of SUSY and non-SUSY models”,  Comput. Phys. Commun. 230, 145 (2018), arXiv:1710.03760.
  * [36] T. Biekötter, S. Heinemeyer, C. Muñoz,  “Precise prediction for the Higgs-boson masses in the $\mu\nu$ SSM”,  Eur. Phys. J. C 78, 504 (2018), arXiv:1712.07475.
  * [37] S. Borowka, S. Paßehr, G. Weiglein,  “Complete two-loop QCD contributions to the lightest Higgs-boson mass in the MSSM with complex parameters”,  Eur. Phys. J. C78, 576 (2018), arXiv:1802.09886.
  * [38] D. Stöckinger, J. Unger,  “Three-loop MSSM Higgs-boson mass predictions and regularization by dimensional reduction”,  Nucl. Phys. B 935, 1 (2018), arXiv:1804.05619.
  * [39] H. Bahl, W. Hollik,  “Precise prediction of the MSSM Higgs boson masses for low MA”,  JHEP 07, 182 (2018), arXiv:1805.00867.
  * [40] R. V. Harlander, J. Klappert, A. D. Ochoa Franco, A. Voigt,  “The light CP-even MSSM Higgs mass resummed to fourth logarithmic order”,  Eur. Phys. J. C78, 874 (2018), arXiv:1807.03509.
  * [41] J. Braathen, M. D. Goodsell, P. Slavich,  “Matching renormalisable couplings: simple schemes and a plot”,  Eur. Phys. J. C79, 669 (2019), arXiv:1810.09388.
  * [42] M. Gabelmann, M. Mühlleitner, F. Staub,  “Automatised matching between two scalar sectors at the one-loop level”,  Eur. Phys. J. C79, 163 (2019), arXiv:1810.12326.
  * [43] H. Bahl, T. Hahn, S. Heinemeyer, W. Hollik, S. Paßehr, H. Rzehak, G. Weiglein,  “Precision calculations in the MSSM Higgs-boson sector with FeynHiggs 2.14”,  Comput. Phys. Commun. 249, 107099 (2020), arXiv:1811.09073.
  * [44] H. Bahl,  “Pole mass determination in presence of heavy particles”,  JHEP 02, 121 (2019), arXiv:1812.06452.
  * [45] T. N. Dao, R. Gröber, M. Krause, M. Mühlleitner, H. Rzehak,  “Two-loop $\mathcal{O}$ ( ${\alpha}_{t}^{2}$ ) corrections to the neutral Higgs boson masses in the CP-violating NMSSM”,  JHEP 08, 114 (2019), arXiv:1903.11358.
  * [46] E. Bagnaschi, G. Degrassi, S. Paßehr, P. Slavich,  “Full two-loop QCD corrections to the Higgs mass in the MSSM with heavy superpartners”,  Eur. Phys. J. C 79, 910 (2019), arXiv:1908.01670.
  * [47] M. D. Goodsell, S. Paßehr,  “All two-loop scalar self-energies and tadpoles in general renormalisable field theories”,  Eur. Phys. J. C 80, 417 (2020), arXiv:1910.02094.
  * [48] R. V. Harlander, J. Klappert, A. Voigt,  “The light CP-even MSSM Higgs mass including N3LO+N3LL QCD corrections”,  Eur. Phys. J. C 80, 186 (2020), arXiv:1910.03595.
  * [49] H. Bahl, S. Heinemeyer, W. Hollik, G. Weiglein,  “Theoretical uncertainties in the MSSM Higgs boson mass calculation”,  Eur. Phys. J. C 80, 497 (2020), arXiv:1912.04199.
  * [50] H. Bahl, I. Sobolev, G. Weiglein,  “Precise prediction for the mass of the light MSSM Higgs boson for the case of a heavy gluino”,  Phys. Lett. B 808, 135644 (2020), arXiv:1912.10002.
  * [51] T. Kwasnitza, D. Stöckinger, A. Voigt,  “Improved MSSM Higgs mass calculation using the 3-loop FlexibleEFTHiggs approach including $x_{t}$-resummation”,  JHEP 07, 197 (2020), arXiv:2003.04639.
  * [52] H. Bahl, I. Sobolev, G. Weiglein,  “The light MSSM Higgs boson mass for large $\tan\beta$ and complex input parameters”,  Eur. Phys. J. C 80, 1063 (2020), arXiv:2009.07572.
  * [53] H. Bahl, I. Sobolev,  “Two-loop matching of renormalizable operators: general considerations and applications”,  JHEP 03, 286 (2021), arXiv:2010.01989.
  * [54] H. Bahl, N. Murphy, H. Rzehak,  “Hybrid calculation of the MSSM Higgs boson masses using the complex THDM as EFT”,  Eur. Phys. J. C 81, 128 (2021), arXiv:2010.04711.
  * [55] G. Degrassi, S. Di Vita, J. Elias-Miro, J. R. Espinosa, G. F. Giudice, G. Isidori, A. Strumia,  “Higgs mass and vacuum stability in the Standard Model at NNLO”,  JHEP 08, 098 (2012), arXiv:1205.6497.
  * [56] D. Buttazzo, G. Degrassi, P. P. Giardino, G. F. Giudice, F. Sala, A. Salvio, A. Strumia,  “Investigating the near-criticality of the Higgs boson”,  JHEP 12, 089 (2013), arXiv:1307.3536.
  * [57] B. A. Kniehl, A. F. Pikelner, O. L. Veretin,  “Two-loop electroweak threshold corrections in the Standard Model”,  Nucl. Phys. B896, 19 (2015), arXiv:1503.02138.
  * [58] B. A. Kniehl, A. F. Pikelner, O. L. Veretin,  “mr: a C++ library for the matching and running of the Standard Model parameters”,  Comput. Phys. Commun. 206, 84 (2016), arXiv:1601.08143.
  * [59] S. P. Martin, D. G. Robertson,  “Standard Model parameters in the tadpole-free pure $\overline{\mathrm{MS}}$ scheme”,  (2019), arXiv:1907.02500.
  * [60] C. Coriano, L. Delle Rose, C. Marzo,  “Constraints on abelian extensions of the Standard Model from two-loop vacuum stability and $U(1)_{B-L}$”,  JHEP 02, 135 (2016), arXiv:1510.02379.
  * [61] J. Braathen, M. D. Goodsell, M. E. Krauss, T. Opferkuch, F. Staub,  “$N$-loop running should be combined with $N$-loop matching”,  Phys. Rev. D97, 015011 (2018), arXiv:1711.08460.
  * [62] M. E. Krauss, T. Opferkuch, F. Staub,  “The Ultraviolet Landscape of Two-Higgs Doublet Models”,  Eur. Phys. J. C 78, 1020 (2018), arXiv:1807.07581.
  * [63] W. G. Hollik, S. Liebler, G. Moortgat-Pick, S. Paßehr, G. Weiglein,  “Phenomenology of the inflation-inspired NMSSM at the electroweak scale”,  Eur. Phys. J. C79, 75 (2019), arXiv:1809.07371.
  * [64] J.-W. Wang, X.-J. Bi, P.-F. Yin, Z.-H. Yu,  “Impact of Fermionic Electroweak Multiplet Dark Matter on Vacuum Stability with One-loop Matching”,  Phys. Rev. D 99, 055009 (2019), arXiv:1811.08743.
  * [65] W. G. Hollik, G. Weiglein, J. Wittbrodt,  “Impact of Vacuum Stability Constraints on the Phenomenology of Supersymmetric Models”,  JHEP 03, 109 (2019), arXiv:1812.04644.
  * [66] S. P. Martin, D. G. Robertson,  “Higgs boson mass in the Standard Model at two-loop order and beyond”,  Phys. Rev. D90, 073010 (2014), arXiv:1407.4336.
  * [67] S. P. Martin,  “Complete Two Loop Effective Potential Approximation to the Lightest Higgs Scalar Boson Mass in Supersymmetry”,  Phys. Rev. D67, 095012 (2003), arXiv:hep-ph/0211366.
  * [68] S. P. Martin,  “Taming the Goldstone contributions to the effective potential”,  Phys. Rev. D90, 016013 (2014), arXiv:1406.2355.
  * [69] J. Elias-Miro, J. R. Espinosa, T. Konstandin,  “Taming Infrared Divergences in the Effective Potential”,  JHEP 08, 034 (2014), arXiv:1406.2652.
  * [70] N. Kumar, S. P. Martin,  “Resummation of Goldstone boson contributions to the MSSM effective potential”,  Phys. Rev. D94, 014013 (2016), arXiv:1605.02059.
  * [71] M. Sperling, D. Stöckinger, A. Voigt,  “Renormalization of vacuum expectation values in spontaneously broken gauge theories”,  JHEP 07, 132 (2013), arXiv:1305.1548.
  * [72] M. Sperling, D. Stöckinger, A. Voigt,  “Renormalization of vacuum expectation values in spontaneously broken gauge theories: Two-loop results”,  JHEP 01, 068 (2014), arXiv:1310.7629.
  * [73] G. Belanger, K. Benakli, M. Goodsell, C. Moura, A. Pukhov,  “Dark Matter with Dirac and Majorana Gaugino Masses”,  JCAP 08, 027 (2009), arXiv:0905.1043.
  * [74] K. Benakli, M. D. Goodsell, A.-K. Maier,  “Generating mu and Bmu in models with Dirac Gauginos”,  Nucl. Phys. B 851, 445 (2011), arXiv:1104.2695.
  * [75] K. Benakli, M. D. Goodsell, F. Staub,  “Dirac Gauginos and the 125 GeV Higgs”,  JHEP 06, 073 (2013), arXiv:1211.0552.
  * [76] M. D. Goodsell, S. Kraml, H. Reyes-González, S. L. Williamson,  “Constraining Electroweakinos in the Minimal Dirac Gaugino Model”,  SciPost Phys. 9, 047 (2020), arXiv:2007.08498.
  * [77] J. Fleischer, F. Jegerlehner,  “Radiative Corrections to Higgs Decays in the Extended Weinberg-Salam Model”,  Phys. Rev. D 23, 2001 (1981).
  * [78] F. Jegerlehner, M. Yu. Kalmykov, O. Veretin,  “MS versus pole masses of gauge bosons: Electroweak bosonic two loop corrections”,  Nucl. Phys. B641, 285 (2002), arXiv:hep-ph/0105304.
  * [79] F. Jegerlehner, M. Yu. Kalmykov, O. Veretin,  “Full two loop electroweak corrections to the pole masses of gauge bosons”,  Nucl. Phys. Proc. Suppl. 116, 382 (2003), arXiv:hep-ph/0212003,  [,382(2002)].
  * [80] F. Jegerlehner, M. Yu. Kalmykov, O. Veretin,  “MS-bar versus pole masses of gauge bosons. 2. Two loop electroweak fermion corrections”,  Nucl. Phys. B658, 49 (2003), arXiv:hep-ph/0212319.
  * [81] F. Jegerlehner, M. Y. Kalmykov,  “O(alpha alpha(s)) correction to the pole mass of the t quark within the standard model”,  Nucl. Phys. B 676, 365 (2004), arXiv:hep-ph/0308216.
  * [82] F. Bezrukov, M. Yu. Kalmykov, B. A. Kniehl, M. Shaposhnikov,  “Higgs Boson Mass and New Physics”,  JHEP 10, 140 (2012), arXiv:1205.2893,  [,275(2012)].
  * [83] M. Krause, R. Lorenz, M. Mühlleitner, R. Santos, H. Ziesche,  “Gauge-independent Renormalization of the 2-Higgs-Doublet Model”,  JHEP 09, 143 (2016), arXiv:1605.04853.
  * [84] A. Denner, L. Jenniches, J.-N. Lang, C. Sturm,  “Gauge-independent $\overline{MS}$ renormalization in the 2HDM”,  JHEP 09, 115 (2016), arXiv:1607.07352.
  * [85] L. Altenkamp, S. Dittmaier, H. Rzehak,  “Renormalization schemes for the Two-Higgs-Doublet Model and applications to h $\rightarrow$ WW/ZZ $\rightarrow$ 4 fermions”,  JHEP 09, 134 (2017), arXiv:1704.02645.
  * [86] M. Krause, M. Mühlleitner,  “Impact of Electroweak Corrections on Neutral Higgs Boson Decays in Extended Higgs Sectors”,  JHEP 04, 083 (2020), arXiv:1912.03948.
  * [87] P. Chankowski, S. Pokorski, J. Rosiek,  “Complete on-shell renormalization scheme for the minimal supersymmetric Higgs sector”,  Nucl. Phys. B423, 437 (1994), arXiv:hep-ph/9303309.
  * [88] A. Dabelstein,  “The One loop renormalization of the MSSM Higgs sector and its application to the neutral scalar Higgs masses”,  Z. Phys. C67, 495 (1995), arXiv:hep-ph/9409375.
  * [89] A. Freitas, D. Stöckinger,  “Gauge dependence and renormalization of tan beta in the MSSM”,  Phys. Rev. D 66, 095014 (2002), arXiv:hep-ph/0205281.
  * [90] S. Kanemura, Y. Okada, E. Senaha, C. P. Yuan,  “Higgs coupling constants as a probe of new physics”,  Phys. Rev. D 70, 115002 (2004), arXiv:hep-ph/0408364.
  * [91] M. Farina, D. Pappadopulo, A. Strumia,  “A modified naturalness principle and its experimental tests”,  JHEP 08, 022 (2013), arXiv:1303.7244.
  * [92] W. Porod,  “SPheno, a program for calculating supersymmetric spectra, SUSY particle decays and SUSY particle production at e+ e- colliders”,  Comput. Phys. Commun. 153, 275 (2003), arXiv:hep-ph/0301101.
  * [93] W. Porod, F. Staub,  “SPheno 3.1: Extensions including flavour, CP-phases and models beyond the MSSM”,  Comput. Phys. Commun. 183, 2458 (2012), arXiv:1104.1573.
  * [94] F. Staub,  “SARAH”,  (2008), arXiv:0806.0538.
  * [95] F. Staub,  “From Superpotential to Model Files for FeynArts and CalcHep/CompHep”,  Comput. Phys. Commun. 181, 1077 (2010), arXiv:0909.2863.
  * [96] F. Staub,  “Automatic Calculation of supersymmetric Renormalization Group Equations and Self Energies”,  Comput. Phys. Commun. 182, 808 (2011), arXiv:1002.0840.
  * [97] F. Staub,  “SARAH 3.2: Dirac Gauginos, UFO output, and more”,  Comput. Phys. Commun. 184, 1792 (2013), arXiv:1207.0906.
  * [98] F. Staub,  “SARAH 4: A tool for (not only SUSY) model builders”,  Comput. Phys. Commun. 185, 1773 (2014), arXiv:1309.7223.
  * [99] S. P. Martin,  “Two loop scalar self energies in a general renormalizable theory at leading order in gauge couplings”,  Phys. Rev. D70, 016005 (2004), arXiv:hep-ph/0312092.
  * [100] F. Domingo, S. Paßehr,  “Towards Higgs masses and decay widths satisfying the symmetries in the (N)MSSM”,  Eur. Phys. J. C 80, 1124 (2020), arXiv:2007.11010.
  * [101] U. Ellwanger, C. Hugonie, A. Teixeira,  “The Next-to-Minimal Supersymmetric Standard Model”,  Phys. Rept. 496, 1 (2010), arXiv:0910.1785.
  * [102] G. G. Ross, K. Schmidt-Hoberg, F. Staub,  “The Generalised NMSSM at One Loop: Fine Tuning and Phenomenology”,  JHEP 08, 074 (2012), arXiv:1205.1509.