*[arabic]figure *[arabic]table [toc]toc.1toc.1ContentsContentstoc.1

DESY–17–067

IFT–UAM/CSIC–17–042

On-Shell neutral Higgs bosons in

the NMSSM with complex parameters

Florian Domingo1,2***email<EMAIL_ADDRESS>Peter Drechsel3†††email:
<EMAIL_ADDRESS>and Sebastian Paßehr3‡‡‡email:
<EMAIL_ADDRESS>
1Instituto de Física Teórica (UAM/CSIC), Universidad Autónoma de Madrid,
Cantoblanco, E-28049 Madrid, Spain

2Instituto de Física de Cantabria (CSIC-UC), E-39005 Santander, Spain

3Deutsches Elektronensynchrotron DESY, Notkestraße 85, D–22607 Hamburg,
Germany

###### Abstract

The Next-to-Minimal Supersymmetric Standard model (NMSSM) appears as an
interesting candidate for the interpretation of the Higgs-measurement at the
LHC and as a rich framework embedding physics beyond the Standard Model. We
consider the renormalization of the Higgs sector of this model in its ${\cal
CP}$-violating version, and propose a renormalization scheme for the
calculation of on-shell Higgs masses. Moreover, the connection between the
physical states and the tree-level ones is no longer trivial at the radiative
level: a proper description of the corresponding transition thus proves
necessary in order to calculate Higgs production and decays at a consistent
loop order. After discussing these formal aspects, we compare the results of
our mass calculation to the output of existing tools. We also study the
relevance of the on-shell transition-matrix in the example of the
$h_{i}\to\tau^{+}\tau^{-}$ width. We find deviations between our full
prescription and popular approximations that can exceed $10\%$.

###### Contents

  1. 1 Introduction
  2. 2 Higgs masses and mixing in the ${\cal CP}$-violating NMSSM
    1. 2.1 Conventions and relations at the tree level
    2. 2.2 Renormalization of the Higgs potential
    3. 2.3 Renormalization conditions at the one-loop order
    4. 2.4 Quark Yukawa couplings
    5. 2.5 Higgs masses at higher orders
    6. 2.6 Wave function normalization factors: the matrix $\mathbf{Z}^{\mbox{\tiny mix}}$
  3. 3 Numerical Analysis
    1. 3.1 Comparison with FeynHiggs in the MSSM-limit
    2. 3.2 Comparison in the ${\cal CP}$-conserving limit
    3. 3.3 Comparison with NMSSMCALC
    4. 3.4 The matrix $\mathbf{Z}^{\mbox{\tiny mix}}$ and the $h_{i}\to\tau^{+}\tau^{-}$ decays
  4. 4 Conclusions

##  1 Introduction

Since the discovery of a Higgs-like particle with a mass around
$125\text{\,}\mathrm{GeV}$ by the ATLAS and CMS experiments Aad:2012tfa ;
Chatrchyan:2012ufa  at CERN, a lot of effort has been invested to reveal its
nature as the particle responsible for electroweak symmetry breaking. While
within the present experimental uncertainties the properties of the observed
state are compatible with the predictions of the Standard Model (SM)
Khachatryan:2016vau  many other interpretations are possible as well, in
particular as a Higgs boson of an extended Higgs sector.

One of the prime candidates for physics beyond the SM is softly-broken
supersymmetry (SUSY), which doubles the particle degrees of freedom by
predicting two scalar partners for each SM fermion, as well as fermionic
partners for all bosons—for reviews see Nilles:1983ge ; Haber:1984rc . The
Next-to-Minimal Supersymmetric Standard Model (NMSSM) Ellwanger:2009dp ;
Maniatis:2009re  is a well-motivated extension of the SM. In particular it
provides a solution for the “$\mu$ problem” Kim:1983dt  of the Minimal
Supersymmetric Standard Model (MSSM), by naturally relating the $\mu$
parameter to a dynamical scale of the Higgs potential Ellis:1988er ;
Miller:2003ay .

In contrast to the single Higgs doublet in the SM, the Higgs sector of the
NMSSM contains two Higgs doublets (like the MSSM) and one Higgs singlet. After
electroweak symmetry breaking the physical spectrum consists of five neutral
Higgs bosons, $h_{i}$ ($i\in[1,5]$), and the charged Higgs boson pair
$H^{\pm}$. Ever since the Higgs discovery, the possibility to interpret this
signal in terms of an NMSSM (mostly) ${\cal CP}$-even Higgs boson has been
emphasized in Refs. Gunion:2012zd ; Ellwanger:2012ke ; Gunion:2012gc ;
Kowalska:2012gs ; Bae:2012am ; Beskidt:2013gia ; Moretti:2013lya ;
Moretti:2015bua ; Domingo:2015eea ; Carena:2015moc . In particular, it has
been argued that such a solution came with improved naturalness compared to
the MSSM interpretation Hall:2011aa ; Arvanitaki:2011ck ; Kang:2012sy ;
Cao:2012yn ; Kyae:2012rv ; Jeong:2012ma ; Agashe:2012zq ; Gherghetta:2012gb ;
Barbieri:2013hxa . Moreover, several works have pointed out the possibility to
accommodate deviations from a strict SM behavior in the diphoton rate, in
Higgs-pair production or in associated production Ellwanger:2011aa ;
Cao:2012fz ; Benbrik:2012rm ; Heng:2012at ; Choi:2012he ; King:2012tr ;
Vasquez:2012hn ; Jia-Wei:2013eea ; Munir:2013wka ; Cao:2013si ; Nhung:2013lpa
; Christensen:2013dra ; Ellwanger:2013ova ; Han:2013sga ; Cao:2014kya ;
Wu:2015nba ; Badziak:2016tzl ; Das:2017tob . Admittedly, the viability of the
extended NMSSM Higgs sector would be comforted by the detection of additional
Higgs states. To this end, several search channels have been suggested,
especially for states lighter than $125$ GeV Belanger:2012tt ;
Almarashi:2012ri ; Rathsman:2012dp ; Cerdeno:2013cz ; Barbieri:2013nka ;
Badziak:2013bda ; Kang:2013rj ; Cao:2013gba ; King:2014xwa ; Bomark:2014gya ;
Bomark:2015fga ; Buttazzo:2015bka ; Guchait:2015owa ; Ellwanger:2015uaz ;
Conte:2016zjp ; Guchait:2016pes ; Das:2016eob ; Cao:2016uwt . Another feature
of the NMSSM phenomenology is the extended neutralino sector, due to the
singlino.

In contrast to the situation in the MSSM, ${\cal CP}$-violation can already
occur at the tree-level in the NMSSM Higgs sector Garisto:1993ms ;
Matsuda:1995ta ; Haba:1995aw ; Asatrian:1996wn ; Haba:1996bg ; Ham:1999zs ;
Branco:2000dq ; Davies:2001uv ; Ham:2001wt ; Ham:2001kf ; Hugonie:2003yu ;
Miller:2003ay ; Ham:2003jf ; Funakubo:2004ka ; Ham:2007mt ; Ham:2008cg ;
Cheung:2010ba ; Moretti:2013lya ; Munir:2013dya . While low-energy observables
place limits on such ${\cal CP}$-violating scenarios King:2015oxa , especially
on MSSM-like phases Arbey:2014msa , ${\cal CP}$-violation beyond the SM
appears as a well-motivated requirement for a successful baryogenesis
Sakharov:1967dj . Correspondingly, several computer tools have been proposed
in the past few years to promote the study of the ${\cal CP}$-violating NMSSM:
`SPHENO` Porod:2003um ; Porod:2011nf ; Goodsell:2014bna ; spheno-www  and
`FlexibleSUSY` Athron:2014yba ; FlexibleSUSY-www —which employ `SARAH`
Staub:2009bi ; Staub:2010jh ; Staub:2012pb ; Staub:2013tta  in order to
produce their modelfiles; `FlexibleSUSY` contains components from `SoftSUSY`
Allanach:2001kg ; Allanach:2013kza  and only the ${\cal CP}$-conserving case
is explicitly mentioned for both—as well as `NMSSMCALC` Baglio:2013iia ;
NMSSMCALC-www  and `NMSSMTools` Ellwanger:2004xm ; Ellwanger:2005dv ;
Domingo:2015qaa ; NMSSMTOOLS-www .

In this work, we specialize in the $Z_{3}$-conserving version of the NMSSM,
characterized by a scale-invariant superpotential. The main effort of our
project consists in analyzing radiative corrections in the Higgs sector of the
${\cal CP}$-violating NMSSM. To serve this purpose, we elaborated a `FeynArts`
Kublbeck:1990xc ; Hahn:2000kx  model file and a set of Mathematica routines
for the evaluation of the Higgs masses and wave-function normalization matrix
at full one-loop order and beyond. These should serve as a basis for a future
inclusion of the ${\cal CP}$-violating NMSSM in the `FeynHiggs`
Heinemeyer:1998np ; Heinemeyer:1998yj ; Degrassi:2002fi ; Frank:2006yh ;
Hahn:2010te ; Bahl:2016brp ; FH-www  package—originally designed for precise
calculations of the masses, decays, and other properties of the Higgs bosons
in the ${\cal CP}$-conserving or -violating MSSM. A first step in this
direction is represented by Ref. Drechsel:2016jdg , centering on the ${\cal
CP}$-conserving NMSSM. In the current paper, we expand this project further.
We follow the general methodology of `FeynHiggs`, relying on a Feynman-
diagrammatic calculation of radiative corrections, which employs `FeynArts`
Kublbeck:1990xc ; Hahn:2000kx , `FormCalc` Hahn:1998yk  and `LoopTools`
Hahn:1998yk . Our chosen renormalization scheme differs somewhat from earlier
proposals Graf:2012hh ; Goodsell:2014bna ; Drechsel:2016jdg . In particular,
in our renormalization scheme, the electromagnetic coupling $e$—which is
related to the fine-structure constant $\alpha=e^{2}/(4\pi)$—is defined in
terms of the Fermi constant $G_{F}$ measured in muon decays.

In section 2, we shall introduce relevant notations and describe the
renormalization procedure underpinning our model file for the ${\cal
CP}$-violating NMSSM. In this section we also describe our implementation of
higher-order corrections in the Higgs sector. A numerical evaluation of our
results follows in section 3, where we will validate our calculation by a
comparison with public codes. We will also insist on the relevance of the
field-renormalization matrix for a consistent evaluation of the Higgs decays
at the one-loop level, before a short conclusion in section 4.

##  2 Higgs masses and mixing in the ${\cal CP}$-violating NMSSM

After a few general remarks concerning our notations and conventions, we
present the renormalization conditions that we employ in our calculation.
There, we focus on effects beyond the MSSM in the Higgs and higgsino sectors,
since we otherwise align with the conventions of FeynHiggs, described in
Frank:2006yh . Then we discuss how to formally extract the loop-corrected
Higgs masses and the wave function normalization factors.

###  2.1 Conventions and relations at the tree level

In the following, we consider the $Z_{3}$-conserving version of the NMSSM and
neglect flavor-mixing. The superpotential of the NMSSM (showing only one
generation of fermions/sfermions) reads

$\displaystyle
W=Y_{u}\,\hat{u}\left(\hat{H}_{2}\cdot\hat{Q}\right)-Y_{d}\,\hat{d}\left(\hat{H}_{1}\cdot\hat{Q}\right)-Y_{e}\,\hat{e}\left(\hat{H}_{1}\cdot\hat{L}\right)+\lambda\,\hat{S}\left(\hat{H}_{2}\cdot\hat{H}_{1}\right)+\frac{1}{3}\,\kappa\,\hat{S}^{3},$
(1)

where $\hat{Q}$, $\hat{u}$, $\hat{d}$, $\hat{L}$, $\hat{e}$, $\hat{H}_{1}$,
$\hat{H}_{2}$, $\hat{S}$ denote the quark, lepton and Higgs superfields. The
dot $\cdot$ stands for the $SU(2)_{\text{L}}$-invariant product. The Yukawa
couplings in Eq. (1) can be complex in general. However, their phases can be
absorbed in a redefinition of the quark and lepton superfields. We may write
the scalar fields in $\hat{H}_{1}$, $\hat{H}_{2}$ and $\hat{S}$ explicitly in
terms of their (real and positive) vacuum expectation values (vevs), $v_{1}$,
$v_{2}$ and $v_{s}$, respectively, as well as their ${\cal CP}$-even, ${\cal
CP}$-odd, and charged components, $\phi_{i}$, $\chi_{i}$, and
$\phi^{\pm}_{i}$,

$\displaystyle\mathcal{H}_{1}$ $\displaystyle=\mathrm{e}^{{i\mkern
1.0mu}\,\xi_{1}}\begin{pmatrix}v_{1}+\frac{\left(\phi_{1}+{i\mkern
1.0mu}\chi_{1}\right)}{\sqrt{2}}\\\ \phi_{1}^{-}\end{pmatrix},$
$\displaystyle\mathcal{H}_{2}$ $\displaystyle=\mathrm{e}^{{i\mkern
1.0mu}\,\xi_{2}}\begin{pmatrix}\phi_{2}^{+}\\\
v_{2}+\frac{\left(\phi_{2}+{i\mkern
1.0mu}\chi_{2}\right)}{\sqrt{2}}\end{pmatrix},$ $\displaystyle\mathcal{S}$
$\displaystyle=\mathrm{e}^{{i\mkern
1.0mu}\,\xi_{s}}\left[v_{s}+\tfrac{\left(\phi_{s}+i\chi_{s}\right)}{\sqrt{2}}\right].$
(2)

Here $\xi_{1}$, $\xi_{2}$ and $\xi_{s}$ are the phases of the two Higgs
doublets and the Higgs singlet, respectively. It is convenient to define the
ratio $\tan{\beta}=v_{2}/v_{1}$, the geometric mean of the doublet vevs
$v=\sqrt{v_{1}^{2}+v_{2}^{2}}$, as well as the sum $\xi=\xi_{1}+\xi_{2}$ of
the doublet phases. Since $\hat{S}$ transforms as a singlet under the SM-gauge
transformations, the $D$-terms of the scalar potential are unchanged with
respect to the MSSM. On the other hand, as compared to the MSSM, additional
dimensionless, complex parameters
$\lambda=\lvert\lambda\rvert\,\mathrm{e}^{{i\mkern 1.0mu}\,\phi_{\lambda}}$
and $\kappa=\lvert\kappa\rvert\,\mathrm{e}^{{i\mkern 1.0mu}\,\phi_{\kappa}}$
appear while the complex $\mu$-term is absent. The latter is dynamically
generated as an effective $\mu$-term when the singlet field takes its vev,

$\displaystyle\mu_{\text{eff}}=\lvert\mu_{\text{eff}}\rvert\,\mathrm{e}^{{i\mkern
1.0mu}\,\phi_{\mu}}=\lvert\lambda\rvert\,v_{s}\,\mathrm{e}^{{i\mkern
1.0mu}\left(\phi_{\lambda}+\xi_{s}\right)}.$ (3)

In the NMSSM the phases $\xi$ and $\xi_{s}$ only appear in the combinations
$\phi_{\lambda}+\xi_{s}+\xi$ and $\phi_{\kappa}+3\,\xi_{s}$, so that they
could be absorbed in a re-definition of $\phi_{\lambda}$ and $\phi_{\kappa}$.
Nevertheless, we will keep the dependence on all phases of the Higgs sector
explicitly, in order to allow for more flexibility on the choice of input.

Soft SUSY-breaking in the NMSSM is parametrized by the complex trilinear soft-
breaking parameters $A_{\lambda}=\lvert
A_{\lambda}\rvert\,\mathrm{e}^{{i\mkern 1.0mu}\,\phi_{A_{\lambda}}}$,
$A_{\kappa}=\lvert A_{\kappa}\rvert\,\mathrm{e}^{{i\mkern
1.0mu}\,\phi_{A_{\kappa}}}$, $A_{u}=\lvert A_{u}\rvert\,\mathrm{e}^{{i\mkern
1.0mu}\,\phi_{A_{u}}}$, $A_{d}=\lvert A_{d}\rvert\,\mathrm{e}^{{i\mkern
1.0mu}\,\phi_{A_{d}}}$, and $A_{e}=\lvert A_{e}\rvert\,\mathrm{e}^{{i\mkern
1.0mu}\,\phi_{A_{e}}}$, as well as the real soft-breaking mass terms
$m_{1,2}^{2}$ and $m_{S}^{2}$ for the Higgs fields, and $m^{2}_{\tilde{Q}}$,
$m^{2}_{\tilde{U}}$, $m^{2}_{\tilde{D}}$, $m^{2}_{\tilde{L}}$ and
$m^{2}_{\tilde{E}}$ for the sfermions,

$\displaystyle\begin{split}\mathcal{L}_{\text{soft}}&=-m_{1}^{2}\,\lvert\mathcal{H}_{1}\rvert^{2}-m_{2}^{2}\,\lvert\mathcal{H}_{2}\rvert^{2}-m_{S}^{2}\,\lvert\mathcal{S}\rvert^{2}-\left[\lambda\,A_{\lambda}\,\mathcal{S}\left(\mathcal{H}_{2}\cdot\mathcal{H}_{1}\right)+\tfrac{1}{3}\,\kappa\,A_{\kappa}\,\mathcal{S}^{3}+\text{h.c.}\right]\\\
&\quad-
m_{\tilde{Q}}^{2}\,\lvert\tilde{Q}\rvert^{2}-m_{\tilde{U}}^{2}\,\lvert\tilde{u}\rvert^{2}-m_{\tilde{D}}^{2}\,\lvert\tilde{d}\rvert^{2}-m_{\tilde{L}}^{2}\,\lvert\tilde{L}\rvert^{2}-m_{\tilde{E}}^{2}\,\lvert\tilde{e}\rvert^{2}\\\
&\quad-\left[-Y_{u}\,A_{u}\,\tilde{u}\left(\mathcal{H}_{2}\cdot\tilde{Q}\right)+Y_{d}\,A_{d}\,\tilde{d}\left(\mathcal{H}_{1}\cdot\tilde{Q}\right)+Y_{e}\,A_{e}\,\tilde{e}\left(\mathcal{H}_{1}\cdot\tilde{L}\right)+\text{h.c.}\right]\,.\end{split}$
(4)

Expanding the Higgs potential in terms of the charged Higgs fields
$(\phi_{1}^{+},\phi_{2}^{+})=(\phi_{1}^{-},\phi_{2}^{-})^{*}$, and neutral
Higgs fields
$\left(\phi,\chi\right)=\left(\phi_{1},\phi_{2},\phi_{s},\chi_{1},\chi_{2},\chi_{s}\right)$
yields

$\displaystyle
V_{\text{H}}=-\mathbf{T}\left(\phi,\chi\right)^{\text{T}}+\frac{1}{2}\left(\phi,\chi\right)\mathbf{M}^{2}\left(\phi,\chi\right)^{\text{T}}+\begin{pmatrix}\phi^{-}_{1},\phi^{-}_{2}\end{pmatrix}\mathbf{M}_{\phi^{\pm}}^{2}\begin{pmatrix}\phi^{+}_{1}\\\
\phi^{+}_{2}\end{pmatrix}+\cdots.$ (5)

Here
$\mathbf{T}=\left(T_{\phi_{1}},T_{\phi_{2}},T_{\phi_{s}},T_{\chi_{1}},T_{\chi_{2}},T_{\chi_{s}}\right)$
denotes the tadpole coefficients of the neutral Higgs fields, and
$\mathbf{M}^{2}$ and $\mathbf{M}_{\phi^{\pm}}^{2}$ denote the mass matrices of
the neutral and charged Higgs bosons, respectively.

Since $\mathbf{M}^{2}$ and $\mathbf{M}^{2}_{\phi^{\pm}}$ are symmetric and
hermitian matrices, respectively, we diagonalize them by an orthogonal
$(6\times 6)$ matrix $\mathbf{U}_{n}$ and a unitary $(2\times 2)$ matrix
$\mathbf{U}_{c}$, respectively,

$\displaystyle\mathbf{D}_{hG}=\text{diag}{\left(m_{h_{1}}^{2},\,m_{h_{2}}^{2},\,m_{h_{3}}^{2},\,m_{h_{4}}^{2},\,m_{h_{5}}^{2},\,0\right)}$
$\displaystyle=\mathbf{U}_{n}\,\mathbf{M}^{2}\,\mathbf{U}_{n}^{\textrm{T}}\,,$
(6a)
$\displaystyle\mathbf{D}_{h^{\pm}G^{\pm}}=\text{diag}{\left(M_{H^{\pm}}^{2},\,0\right)}$
$\displaystyle=\mathbf{U}_{c}\,\mathbf{M}^{2}_{\phi^{\pm}}\,\mathbf{U}_{c}^{\dagger}\,.$
(6b)

These transformations define the five neutral Higgs boson mass eigenstates,
$h_{i}$, ($i=1,\ldots,5$), and the (would-be) Goldstone boson $G$, as well as
the charged Higgs and (would-be) Goldstone states, $H^{\pm}$ and $G^{\pm}$, at
the tree level,

$\displaystyle\left(h,G\right)^{\textrm{T}}\equiv\left(h_{1},h_{2},h_{3},h_{4},h_{5},G\right)^{\textrm{T}}$
$\displaystyle=\mathbf{U}_{n}\left(\phi,\chi\right)^{\textrm{T}},$ (7a)
$\displaystyle\left(H^{\pm},G^{\pm}\right)^{\textrm{T}}$
$\displaystyle=\mathbf{U}_{c}\left(\phi_{1}^{\pm},\phi_{2}^{\pm}\right)^{\textrm{T}}.$
(7b)

It is convenient to decompose $\mathbf{U}_{n}$ into two matrices
$\mathbf{U}_{n}^{G}$ and $\mathbf{U}_{n}^{5}$, where $\mathbf{U}_{n}^{G}$
singularizes out the neutral Goldstone boson,

$\displaystyle\mathbf{U}_{n}\left(\phi,\chi\right)^{\textrm{T}}=\mathbf{U}_{n}^{5}\mathbf{U}_{n}^{G}\left(\phi,\chi\right)^{\textrm{T}}=\mathbf{U}_{n}^{5}\left(\phi_{1},\phi_{2},\phi_{s},A,\chi_{s},G\right)^{\textrm{T}}=\left(h,G\right)^{\textrm{T}}.$
(8)

In the ${\cal CP}$-violating NMSSM the five fields $h_{i}$ are in general
superpositions of the ${\cal CP}$-even and -odd components $\phi_{i}$ and
$\chi_{j}$. In the special case of

$\displaystyle\sin{\left(\xi-2\,\xi_{s}+\phi_{\lambda}-\phi_{\kappa}\right)}$
$\displaystyle=0$ (9)

${\cal CP}$-conservation is restored in the Higgs sector at the tree level,
and the neutral mass matrix $\mathbf{M}^{2}$ becomes block-diagonal with two
$(3\times 3)$ sub-matrices for the ${\cal CP}$-even and -odd entries.

The five linearly independent tadpole coefficients are related to soft-
breaking terms and combinations of phases as

$\displaystyle\begin{split}m_{1}^{2}&=-\lvert\mu_{\text{eff}}\rvert^{2}-\frac{1}{2}\,M_{Z}^{2}\,\cos{\left(2\beta\right)}-\left(\lvert\lambda\rvert\,v\,\sin{\beta}\right)^{2}-\frac{T_{\phi_{1}}}{\sqrt{2}\,v\,\cos{\beta}}\\\
&\quad+\lvert\mu_{\text{eff}}\rvert\,\tan{\beta}\left(\lvert
A_{\lambda}\rvert\,\cos{\zeta_{2}}+\frac{\lvert\kappa\rvert\,\lvert\mu_{\text{eff}}\rvert}{\lvert\lambda\rvert}\cos{\zeta_{1}}\right)\,,\end{split}$
(10a)
$\displaystyle\begin{split}m_{2}^{2}&=-\lvert\mu_{\text{eff}}\rvert^{2}+\frac{1}{2}\,M_{Z}^{2}\,\cos{\left(2\beta\right)}-\left(\lvert\lambda\rvert\,v\,\cos{\beta}\right)^{2}-\frac{T_{\phi_{2}}}{\sqrt{2}\,v\,\sin{\beta}}\\\
&\quad+\frac{\lvert\mu_{\text{eff}}\rvert}{\tan{\beta}}\left(\lvert
A_{\lambda}\rvert\,\cos{\zeta_{2}}+\frac{\lvert\kappa\rvert\,\lvert\mu_{\text{eff}}\rvert}{\lvert\lambda\rvert}\cos{\zeta_{1}}\right)\,,\end{split}$
(10b)
$\displaystyle\begin{split}m_{s}^{2}&=-\lvert\lambda\rvert^{2}\,v^{2}-\frac{T_{\phi_{s}}\,\lvert\lambda\rvert}{\sqrt{2}\,\lvert\mu_{\text{eff}}\rvert}-\frac{\lvert\kappa\rvert\,\lvert\mu_{\text{eff}}\rvert}{\lvert\lambda\rvert}\left(\lvert
A_{\kappa}\rvert\,\cos{\zeta_{3}}+2\frac{\lvert\kappa\rvert\,\lvert\mu_{\text{eff}}\rvert}{\lvert\lambda\rvert}\right)\\\
&\quad+\frac{1}{2}\frac{\lvert\lambda\rvert^{2}\,v^{2}}{\lvert\mu_{\text{eff}}\rvert}\sin{\left(2\beta\right)}\left(\lvert
A_{\lambda}\rvert\,\cos{\zeta_{2}}+2\frac{\lvert\kappa\rvert\,\lvert\mu_{\text{eff}}\rvert}{\lvert\lambda\rvert}\cos{\zeta_{1}}\right)\end{split}$
(10c) $\displaystyle\sin{\zeta_{2}}$ $\displaystyle=\frac{1}{\lvert
A_{\lambda}\rvert}\left(-\frac{\lvert\kappa\rvert\,\lvert\mu_{\text{eff}}\rvert}{\lvert\lambda\rvert}\sin{\zeta_{1}}-\frac{T_{\chi_{1}}}{\sqrt{2}\,\lvert\mu_{\text{eff}}\rvert\,v\,\sin{\beta}}\right)\,,$
(10d) $\displaystyle\sin{\zeta_{3}}$
$\displaystyle=\frac{\lvert\lambda\rvert^{2}}{\lvert\kappa\rvert\,\lvert\mu_{\text{eff}}\rvert^{2}\,\lvert
A_{\kappa}\rvert}\left(\frac{\lvert\lambda\rvert\,v^{2}}{2}\sin{\left(2\beta\right)}\left(\lvert
A_{\lambda}\rvert\,\sin{\zeta_{2}}-2\frac{\lvert\kappa\rvert\,\lvert\mu_{\text{eff}}\rvert}{\lvert\lambda\rvert}\sin{\zeta_{1}}\right)+\frac{T_{\chi_{s}}}{\sqrt{2}}\right)\,,$
(10e)

where the masses of the $W$ and $Z$ bosons are denoted by $M_{W}$ and $M_{Z}$,
respectively, and the phases combine to
$\zeta_{1}=\xi-2\,\xi_{s}+\phi_{\lambda}-\phi_{\kappa}$,
$\zeta_{2}=\xi+\xi_{s}+\phi_{A_{\lambda}}+\phi_{\lambda}$ and
$\zeta_{3}=3\,\xi_{s}+\phi_{A_{\kappa}}+\phi_{\kappa}$. The expressions of Eq.
(10) make plain that the tadpole coefficients can substitute the five
parameters $m_{1}^{2}$, $m_{2}^{2}$, $m_{S}^{2}$, $\phi_{A_{\lambda}}$ and
$\phi_{A_{\kappa}}$, so that the latter will not be regarded as free
parameters in the following. Finally, the tadpole coefficients in the (tree-
level) mass basis,
$\mathbf{T}_{h}=(T_{h_{1}},T_{h_{2}},T_{h_{3}},T_{h_{4}},T_{h_{5}},0)$, where
the zero denotes the vanishing tadpole coefficient of the Goldstone mode, are
obtained by $\mathbf{T}_{h}=\mathbf{U}_{n}\mathbf{T}$. The minimization of
$V_{\text{H}}$ at the chosen Higgs vevs is guaranteed through the condition
that all tadpole coefficients $\mathbf{T}$ vanish at the tree level.

The trilinear parameter $\left|A_{\lambda}\right|$ can be expressed in terms
of the charged Higgs mass $M_{H^{\pm}}$ as

$\displaystyle\lvert A_{\lambda}\rvert\,\cos{\zeta_{2}}$
$\displaystyle=-\frac{\lvert\kappa\rvert\,\lvert\mu_{\text{eff}}\rvert}{\lvert\lambda\rvert}\cos{\zeta_{1}}+\left(M_{H^{\pm}}^{2}-M_{W}^{2}+\lvert\lambda\rvert^{2}\,v^{2}\right)\frac{\sin{\left(2\beta\right)}}{2\,\lvert\mu_{\text{eff}}\rvert}\,.$
(11)

Our renormalization scheme will also involve the fermionic superpartners of
the Higgs bosons, known as the higgsinos. We thus introduce here the Dirac
spinors $\tilde{H}^{\pm}$ of the charged higgsino fields, as well as the
Majorana spinor of the singlino $\tilde{S}$. In turn, these higgsino gauge
eigenstates mix with the gauginos to form the mass states known as the
neutralinos and charginos—see e. g. Eq. (11) in Ref. Drechsel:2016jdg , where
the NMSSM parameters $\lambda$, $\kappa$, $M_{1,2}$ and $\mu_{\text{eff}}$
should be promoted to complex values. Yet these mass states will play no role
in the discussion below.

###  2.2 Renormalization of the Higgs potential

In the past, radiative corrections to the Higgs masses of the ${\cal
CP}$-conserving NMSSM have been considered in the effective potential
approach, see e. g. Refs. Ellwanger:1992jp ; Ellwanger:1993hn ; Elliott:1993ex
; Elliott:1993uc ; Elliott:1993bs ; Pandita:1993hx ; Pandita:1993tg ;
Ham:1996mi ; Ellwanger:1999ji . This topic has also been analyzed from the
perspective of a diagrammatic expansion, including radiative corrections from
part or the full set of the particle content of the NMSSM: see Refs.
Degrassi:2009yq ; Belanger:2017rgu ; Drechsel:2016jdg . Both procedures have
also been employed for the ${\cal CP}$-violating case: contributions to the
effective potential have been discussed in Refs. Matsuda:1995ta ; Haba:1995aw
; Asatrian:1996wn ; Haba:1996bg ; Ham:1999zs ; Branco:2000dq ; Davies:2001uv ;
Ham:2001wt ; Ham:2001kf ; Hugonie:2003yu ; Ham:2003jf ; Funakubo:2004ka ;
Ham:2007mt ; Ham:2008cg ; Cheung:2010ba ; Moretti:2013lya ; Munir:2013dya ;
Domingo:2015qaa , while contributions using the diagrammatic approach have
been presented in Refs. Graf:2012hh ; Goodsell:2014bna .

In the present work, the radiative corrections to the Higgs sector are
calculated in the diagrammatic approach. To this end, we first establish a
list of the independent parameters appearing in the linear and bilinear terms
of the Higgs potential in Eq. (5):

$\displaystyle T_{h_{1,\ldots,5}},\ M_{H^{\pm}}^{2},\ M_{W}^{2},\ M_{Z}^{2},\
e,\ \tan{\beta},\ \lvert\mu_{\text{eff}}\rvert,\ \lvert\lambda\rvert,\
\phi_{\lambda},\ \lvert\kappa\rvert,\ \phi_{\kappa},\ \lvert
A_{\kappa}\rvert,\ \xi,\ \xi_{s},$ (12)

where the electromagnetic coupling $e$ is related to the fine structure
constant $\alpha$ by $e=\sqrt{4\,\pi\,\alpha}$. Compared to Ref.
Drechsel:2016jdg , we use $e$ as an independent parameter instead of the vev
$v$. This choice allows to renormalize $e$ to its value derived from the Fermi
constant $G_{F}$ and does not require the reparametrization procedure employed
in Ref. Drechsel:2016jdg . The difference between the renormalization employed
here, and the renormalization and reparametrization procedure described in
Ref. Drechsel:2016jdg  is a sub-leading effect of two-loop order, however.
Other proposals in the literature consist in fixing $e$ from
$\alpha\left(M_{Z}\right)$ Graf:2012hh ; Goodsell:2014bna .

To all real and complex independent parameters, $g_{r}$ and $g_{c}$,
respectively, that are given in Eq. (12), we apply the renormalization
transformations

$\displaystyle g_{r}$ $\displaystyle\rightarrow g_{r}\left(1+\delta
Z_{r}\right)=g_{r}+\delta{g_{r}}\ ,$ $\displaystyle g_{c}$
$\displaystyle\rightarrow g_{c}\left(1+\delta
Z_{c}\right)=g_{c}+\delta{g_{c}}=g_{c}+\delta{\left|g_{c}\right|}\,\mathrm{e}^{{i\mkern
1.0mu}\,\phi_{c}}+{i\mkern 1.0mu}\,g_{c}\,\delta{\phi_{c}}\ .$ (13)

The renormalization transformations for the Higgs, singlino and charged
higgsino fields read

$\displaystyle\mathcal{H}_{1,2}$
$\displaystyle\rightarrow\left(1+\tfrac{1}{2}\,\delta
Z_{\mathcal{H}_{1,2}}\right)\mathcal{H}_{1,2},$ $\displaystyle\mathcal{S}$
$\displaystyle\rightarrow\left(1+\tfrac{1}{2}\,\delta
Z_{\mathcal{S}}\right)\mathcal{S},$ (14a) $\displaystyle\tilde{H}^{\pm}$
$\displaystyle\rightarrow\left(1+\tfrac{1}{2}\,\delta{Z_{\tilde{H}^{\pm}}^{\textrm{L}}}\,P_{\textrm{L}}+\tfrac{1}{2}\,\delta{Z_{\tilde{H}^{\pm}}^{\textrm{R}}}\,P_{\textrm{R}}\right)\tilde{H}^{\pm},$
$\displaystyle\tilde{S}$
$\displaystyle\rightarrow\left(1+\tfrac{1}{2}\,\delta{Z_{\tilde{S}}^{\textrm{L}}}\,P_{\textrm{L}}+\tfrac{1}{2}\,\delta{Z_{\tilde{S}}^{\textrm{R}}}\,P_{\textrm{R}}\right)\tilde{S},$
(14b)

with $P_{\textrm{L}}$ and $P_{\textrm{R}}$ denoting the left- and right-handed
projectors, respectively. Since the singlino is a Majorana field, the
corresponding wave-function counterterms $\delta{Z_{\tilde{S}}^{\textrm{L}}}$
and $\delta{Z_{\tilde{S}}^{\textrm{R}}}$ are complex conjugates of one
another.

###  2.3 Renormalization conditions at the one-loop order

For the parameters $M_{H^{\pm}}$, $M_{W}$, $M_{Z}$ and $\tan{\beta}$, which
enter the one-loop calculation of the Higgs masses in the MSSM as well, we
follow the renormalization prescription outlined in Ref. Frank:2006yh : the
on-shell renormalization scheme is employed for the gauge boson masses,
$M_{Z}$ and $M_{W}$, and the charged Higgs mass $M_{H^{\pm}}$, while the
parameter $\tan{\beta}$ is renormalized $\overline{\mathrm{DR}}$.

We apply the minimization conditions in order to fix the tadpole counterterms:

$\displaystyle T^{(1)}_{h_{i}}+\delta{T_{h_{i}}}$ $\displaystyle=0,$ (15)

where the $T^{(1)}_{h_{i}}$ correspond to the one-loop contributions to the
tadpole parameters.

The counterterm of the electromagnetic coupling $e$ is fixed by

$\displaystyle\delta{Z_{e}}$
$\displaystyle=\delta{Z_{e}^{\text{Th}}}-\frac{1}{2}\,\Delta{r}^{\textrm{NMSSM}},$
(16a) $\displaystyle\delta Z_{e}^{\text{Th}}$
$\displaystyle=\frac{1}{2}\,\Pi^{\gamma\gamma}(0)+\frac{s_{\mathrm{w}}}{c_{\mathrm{w}}}\frac{\Sigma_{T}^{\gamma
Z}(0)}{M_{Z}^{2}}\ .$ (16b)

Here $\delta Z_{e}^{\text{Th}}$ is the counterterm of the charge
renormalization within the NMSSM according to the static (Thomson) limit. The
quantities $\Pi^{\gamma\gamma}(0)$ and $\Sigma_{T}^{\gamma Z}(0)$ are
respectively the derivative of the transverse part of the photon self-energy
and the transverse part of the photon–$Z$ self-energy at zero momentum
transfer. For the quantity $\Delta r^{\textrm{NMSSM}}$, relating the
elementary charge to the Fermi constant $G_{F}$ measured in muon decays, we
use the result of Ref. Stal:2015zca  (see also Ref. Domingo:2011uf ). The
numerical value for the electromagnetic coupling $e$ in this parametrization
is obtained from the Fermi constant in the usual way as
$e=2\,M_{W}\,s_{\mathrm{w}}\,\sqrt{\sqrt{2}\,G_{F}}$. This choice differs from
previous works, where either the charge renormalization condition was
determined in terms of $\alpha(M_{Z})$ Graf:2012hh , or instead $v$ was
renormalized $\overline{\mathrm{DR}}$ and the result was subsequently
reparametrized to use the value of $e$ derived from the Fermi constant
Drechsel:2016jdg .

The remaining independent parameters and the field renormalization constants
are renormalized $\overline{\mathrm{DR}}$. We present a detailed description
of the $\overline{\mathrm{DR}}$ renormalization conditions that we apply. The
actual cancellation of UV-divergences, that we recover at the diagrammatic
level, represents a non-trivial check for the validity of the FeynArts model-
file employed for our calculation.

The $\overline{\mathrm{DR}}$ field renormalization constants for the Higgs
fields are obtained as

$\displaystyle\delta Z_{\mathcal{H}_{1}}$
$\displaystyle=-\Re\mathfrak{e}{\left[\frac{d\Sigma_{\phi_{1}\phi_{1}}^{(1)}}{dp^{2}}\right]}_{\textrm{div}},$
$\displaystyle\delta Z_{\mathcal{H}_{2}}$
$\displaystyle=-\Re\mathfrak{e}{\left[\frac{d\Sigma_{\phi_{2}\phi_{2}}^{(1)}}{dp^{2}}\right]}_{\textrm{div}},$
$\displaystyle\delta Z_{\mathcal{S}}$
$\displaystyle=-\Re\mathfrak{e}{\left[\frac{d\Sigma_{\phi_{s}\phi_{s}}^{(1)}}{dp^{2}}\right]}_{\textrm{div}},$
(17)

where $\Sigma_{ii}^{(1)}$ denotes the self-energy of field $i$ at the one-loop
order, and the subscript ’div’ denotes the UV-divergent piece (along with the
universal finite pieces that are associated in the $\overline{\mathrm{DR}}$
scheme) of the quantity that it follows. The result does not depend on the
momentum $p^{2}$.

The field renormalization constants for the charged higgsino and the singlino
fields are defined by the following conditions (the momentum $p^{2}$ again
does not matter)

$\displaystyle\delta Z_{\tilde{H}^{\pm}}^{\text{L}}$
$\displaystyle=-\left.\Sigma_{\tilde{H}^{\pm}\tilde{H}^{\pm}}^{\text{vec\,L}\,(1)}\right|_{\textrm{div}}\,,$
$\displaystyle\delta Z_{\tilde{H}^{\pm}}^{\text{R}}$
$\displaystyle=-\left.\Sigma_{\tilde{H}^{\pm}\tilde{H}^{\pm}}^{\text{vec\,R}\,(1)}\right|_{\textrm{div}}\,,$
$\displaystyle\delta Z_{\tilde{S}}^{\text{L}}$
$\displaystyle=-\left.\Sigma_{\tilde{S}\tilde{S}}^{\text{vec\,L}\,(1)}\right|_{\textrm{div}}\,,$
(18)

where the self-energies of the fermion fields are decomposed into the left and
right vector and the scalar contributions,

$\displaystyle\Sigma_{ff}^{(1)}{\left(p^{2}\right)}$
$\displaystyle=\Sigma_{ff}^{\text{scal}\,(1)}{\left(p^{2}\right)}+p_{\mu}\gamma^{\mu}\left[P_{\textrm{L}}\,\Sigma_{ff}^{\text{vec\,L}\,(1)}{\left(p^{2}\right)}+P_{\textrm{R}}\,\Sigma_{ff}^{\text{vec\,R}\,(1)}{\left(p^{2}\right)}\right].$
(19)

The renormalization constants $\delta{\lvert\lambda\rvert}$,
$\delta{\lvert\kappa\rvert}$, $\delta{\phi_{\lambda}}$,
$\delta{\phi_{\kappa}}$, $\delta\xi$, $\delta\xi_{s}$ and $\delta{\lvert
A_{\kappa}\rvert}$ are fixed by $\overline{\mathrm{DR}}$ conditions imposed on
trilinear vertices involving scalar, ${\cal CP}$-even Higgs fields
$\phi_{1,2,s}$, the singlino $\tilde{S}$, and the charged higgsino fields
$\tilde{H}^{\pm}$, in the interaction basis in analogy to the procedure
outlined in Drechsel:2016ukp . The renormalization condition imposed on the
renormalized three-point function $\hat{\Gamma}_{ijk}$ for three arbitrary
fields $i$, $j$ and $k$ reads

$\displaystyle\hat{\Gamma}_{ijk}$
$\displaystyle=\Gamma^{(0)}_{ijk}+\Gamma^{(1)}_{ijk}+\delta{\Gamma}_{ijk}\overset{!}{=}\text{finite}\,,$
$\displaystyle\Leftarrow$ $\displaystyle\delta{\Gamma}_{ijk}$
$\displaystyle=-\left.\Gamma^{(1)}_{ijk}\right|_{\text{div}}\,,$ (20)

where $\Gamma_{ijk}^{(0)}$ and $\Gamma_{ijk}^{(1)}$ denote the vertex function
at the tree-level and one-loop order, respectively, and $\delta{\Gamma_{ijk}}$
denotes the counterterm. The counterterm $\delta{\Gamma_{ijk}}$ is thus fixed
by the divergent part of the vertex function. The renormalization constants of
the independent parameters are subsequently fixed by linear relations to
$\delta{\Gamma_{ijk}}$.

  * •

For $\delta{\left|\lambda\right|}$ and $\delta{\phi_{\lambda}}$ we impose the
$\overline{\mathrm{DR}}$ renormalization condition of Eq. (20) on the vertices
${\Gamma_{\tilde{S}\tilde{H}^{-}\phi_{1}^{+}}^{(0)}=\lambda}$ and
${\Gamma_{\tilde{S}\tilde{H}^{+}\phi_{1}^{-}}^{(0)}=\lambda^{\ast}}$, which
yields

$\displaystyle\frac{\delta\left|\lambda\right|}{\left|\lambda\right|}$
$\displaystyle=-\frac{1}{2}\left\\{\left.\frac{\Gamma_{\tilde{S}\tilde{H}^{-}\phi_{1}^{+}}^{(1)}}{\Gamma_{\tilde{S}\tilde{H}^{-}\phi_{1}^{+}}^{(0)}}\right|_{\text{div}}+\frac{1}{2}\left(\delta
Z_{\tilde{S}}+\delta Z_{\tilde{H}^{\pm}}^{\text{R}}+\delta
Z_{\mathcal{H}_{1}}\right)\right\\}+\text{c.\,c.}\,,$ (21a)
$\displaystyle\delta\phi_{\lambda}$ $\displaystyle=-\frac{1}{2{i\mkern
1.0mu}}\left\\{\left.\frac{\Gamma_{\tilde{S}\tilde{H}^{-}\phi_{1}^{+}}^{(1)}}{\Gamma_{\tilde{S}\tilde{H}^{-}\phi_{1}^{+}}^{(0)}}\right|_{\text{div}}+\frac{1}{2}\left(\delta
Z_{\tilde{S}}+\delta Z_{\tilde{H}^{\pm}}^{\text{R}}+\delta
Z_{\mathcal{H}_{1}}\right)\right\\}+\text{c.\,c.}$ (21b)

  * •

For $\delta\xi$ we impose the renormalization condition of Eq. (20) on the
vertices
${\Gamma_{\tilde{S}\tilde{H}^{+}\phi_{2}^{-}}^{(0)}=\lambda\,\mathrm{e}^{{i\mkern
1.0mu}\,\xi}}$ and
${\Gamma_{\tilde{S}\tilde{H}^{-}\phi_{2}^{+}}^{(0)}=\lambda^{\ast}\,\mathrm{e}^{-{i\mkern
1.0mu}\,\xi}}$. The counterterm reads

$\displaystyle\delta\xi$ $\displaystyle=\left\\{-\frac{1}{2{i\mkern
1.0mu}}\left[\left.\frac{\Gamma_{\tilde{S}\tilde{H}^{+}\phi_{2}^{-}}^{(1)}}{\Gamma_{\tilde{S}\tilde{H}^{+}\phi_{2}^{-}}^{(0)}}\right|_{\text{div}}+\frac{1}{2}\left(\delta
Z_{\tilde{S}}+\delta Z_{\tilde{H}^{\pm}}^{\text{L}}+\delta
Z_{\mathcal{H}_{2}}\right)\right]+\text{c.\,c.}\right\\}-\delta\phi_{\lambda}\,.$
(22)

  * •

We fix the renormalization constant $\delta\xi_{s}$ for the phase $\xi_{s}$ by
applying Eq. (20) on the vertices
${\Gamma_{\tilde{H}^{-},\tilde{H}^{+}\phi_{s}}^{(0)}=\lambda\,\mathrm{e}^{{i\mkern
1.0mu}\,\xi_{s}}/\sqrt{2}}$ and
${\Gamma_{\tilde{H}^{+},\tilde{H}^{-}\phi_{s}}^{(0)}=\lambda^{\ast}\,\mathrm{e}^{-{i\mkern
1.0mu}\,\xi_{s}}/\sqrt{2}}$, which yields

$\displaystyle\delta\xi_{s}$ $\displaystyle=\left\\{-\frac{1}{2{i\mkern
1.0mu}}\left[\left.\frac{\Gamma_{\tilde{H}^{-},\tilde{H}^{+}\phi_{s}}^{(1)}}{\Gamma_{\tilde{H}^{-},\tilde{H}^{+}\phi_{s}}^{(0)}}\right|_{\text{div}}+\frac{1}{2}\left(\delta
Z_{\tilde{H}^{\pm}}^{\text{L}}+\delta Z_{\tilde{H}^{\pm}}^{\text{R}}+\delta
Z_{\phi_{s}}\right)\right]+\text{c.\,c.}\right\\}-\delta\phi_{\lambda}\,.$
(23)

  * •

The absolute value and phase of $\kappa$ are renormalized by
$\delta{\left|\kappa\right|}$ and $\delta{\phi_{\kappa}}$. We fix both
renormalization constants by applying Eq. (20) on the vertex
${\Gamma_{\tilde{S}\tilde{S}\phi_{s}}^{(0)}=\sqrt{2}\,\kappa\,\mathrm{e}^{{i\mkern
1.0mu}\,\xi_{s}}}$, which yields

$\displaystyle\frac{\delta\left|\kappa\right|}{\left|\kappa\right|}$
$\displaystyle=-\frac{1}{2}\left\\{\left.\frac{\Gamma_{\tilde{S}\tilde{S}\phi_{s}}^{(1)}}{\Gamma_{\tilde{S}\tilde{S}\phi_{s}}^{(0)}}\right|_{\text{div}}+\frac{1}{2}\left(2\delta
Z_{\tilde{S}}+\delta Z_{\phi_{s}}\right)\right\\}+\text{c.\,c.}\,,$ (24a)
$\displaystyle\delta\phi_{\kappa}$ $\displaystyle=\left\\{-\frac{1}{2{i\mkern
1.0mu}}\left[\left.\frac{\Gamma_{\tilde{S}\tilde{S}\phi_{s}}^{(1)}}{\Gamma_{\tilde{S}\tilde{S}\phi_{s}}^{(0)}}\right|_{\text{div}}+\frac{1}{2}\left(2\delta
Z_{\tilde{S}}+\delta
Z_{\phi_{s}}\right)\right]+\text{c.\,c.}\right\\}-\delta\xi_{s}\,.$ (24b)

  * •

The parameter $\lvert\mu_{\text{eff}}\rvert$ could be renormalized in the on-
shell scheme for one of the charginos or neutralinos Fritzsche:2011nr ;
Heinemeyer:2011gk ; Bharucha:2012re . However, such schemes cannot be
stabilized over the whole parameter space (due to mass-crossings). We thus
prefer to apply the $\overline{\mathrm{DR}}$ condition

$\displaystyle\delta\lvert\mu_{\text{eff}}\rvert$
$\displaystyle=\lvert\mu_{\text{eff}}\rvert\left(\frac{\delta\lvert\lambda\rvert}{\lvert\lambda\rvert}+\frac{1}{2}\delta
Z_{S}\right)\,.$ (25)

  * •

In order to fix $\delta{\left|A_{\kappa}\right|}$, we impose the
renormalization condition of Eq. (20) on the vertex
$\Gamma_{\phi_{s}\phi_{s}\phi_{s}}^{(0)}=-\sqrt{2}\,\lvert\kappa\rvert\left(\frac{6\,\lvert\kappa\rvert\,\lvert\mu_{\text{eff}}\rvert}{\lvert\lambda\rvert}+\lvert
A_{\kappa}\rvert\cos{\zeta_{3}}\right)$, ($\zeta_{3}$ was defined after Eq.
(10)). It reads

$\displaystyle\begin{split}\delta\lvert A_{\kappa}\rvert&=-\lvert
A_{\kappa}\rvert\left(\frac{\delta{\lvert\kappa\rvert}}{\lvert\kappa\rvert}+\frac{\delta{\cos{\zeta_{3}}}}{\cos{\zeta_{3}}}\right)-\frac{6\,\lvert\kappa\rvert\,\lvert\mu_{\text{eff}}\rvert}{\lvert\lambda\rvert\,\cos{\zeta_{3}}}\left(\frac{\delta\lvert\mu_{\text{eff}}\rvert}{\lvert\mu_{\text{eff}}\rvert}-\frac{\delta\lvert\lambda\rvert}{\lvert\lambda\rvert}+\frac{2\,\delta\lvert\kappa\rvert}{\lvert\kappa\rvert}\right)\\\
&\quad-\frac{\delta\Gamma_{\phi_{s}\phi_{s}\phi_{s}}}{\sqrt{2}\,\lvert\kappa\rvert\cos{\zeta_{3}}}\,,\end{split}$
(26a)
$\displaystyle\begin{split}\delta\zeta_{3}&=\left(\frac{\delta\lvert\kappa\rvert}{\lvert\kappa\rvert}+\frac{2\,\delta\lvert\lambda\rvert}{\lvert\lambda\rvert}-\frac{\delta\lvert\mu_{\text{eff}}\rvert}{\mu_{\text{eff}}}+\frac{\delta\sin{\left(2\beta\right)}}{\sin{\left(2\beta\right)}}+\frac{\delta\sin{\zeta_{1}}}{\sin{\zeta_{1}}}+\left.\frac{2\,\delta
v}{v}\right|_{\text{div}}\right)\cos{\zeta_{3}}\sin{\zeta_{3}}\\\
&\quad+\left[\frac{\delta\Gamma_{\phi_{s}\phi_{s}\phi_{s}}}{\sqrt{2}\,\lvert\kappa\rvert}+\frac{6\,\lvert\kappa\rvert\,\lvert\mu_{\text{eff}}\rvert}{\lvert\lambda\rvert}\left(\frac{\delta\lvert\mu_{\text{eff}}\rvert}{\lvert\mu_{\text{eff}}\rvert}-\frac{\delta\lvert\lambda\rvert}{\lvert\lambda\rvert}+\frac{2\,\delta\lvert\kappa\rvert}{\lvert\kappa\rvert}\right)\right]\frac{\sin{\zeta_{3}}}{\lvert
A_{\kappa}\rvert}\\\ &\quad+\left(\left.\delta
T_{\chi_{s}}\right|_{\text{div}}-\frac{\lvert\lambda\rvert\,v\,\cos{\beta}}{\lvert\mu_{\text{eff}}\rvert}\left.\delta
T_{\chi_{1}}\right|_{\text{div}}\right)\frac{\lvert\lambda\rvert^{2}\,\cos{\zeta_{3}}}{\sqrt{2}\,\lvert\kappa\rvert\,\lvert
A_{\kappa}\rvert\,\lvert\mu_{\text{eff}}\rvert^{2}}\,.\end{split}$ (26b)

where we used the one-loop relation
$\delta{\cos{\zeta_{3}}}=-\sin{\zeta_{3}}\,\delta{\zeta_{3}}$, and $\delta v$
is not an independent counterterm, but a quantity depending on the
counterterms to the electroweak parameters

$\displaystyle\delta v$ $\displaystyle=v\left(\frac{\delta
M_{W}}{M_{W}}+\frac{\delta s_{\text{w}}}{s_{\text{w}}}-\delta Z_{e}\right)\,,$
$\displaystyle\delta s_{\text{w}}$
$\displaystyle=\frac{c_{\text{w}}^{2}}{s_{\text{w}}}\left(\frac{\delta
M_{Z}}{M_{Z}}-\frac{\delta M_{W}}{M_{W}}\right)\,.$ (27)

We performed various consistency checks of our model file at the one-loop
order:

  * •

all the renormalized Higgs self-energies are UV-finite, for arbitrary values
of the momentum,

  * •

all the vertex-diagram amplitudes of a Higgs state decaying to SM-particles or
a pair of charginos/neutralinos are UV-finite,

  * •

the UV-divergences of the counterterms to gauge couplings, superpotential
parameters or soft terms are consistent with the corresponding one-loop beta
functions (see e. g. Refs. Ellwanger:2009dp ; Martin:1993zk ),

  * •

in the ${\cal CP}$-conserving limit, our parameters and couplings are
identical to the findings of a previously developed model file
Drechsel:2016jdg ,

  * •

in the MSSM limit, we have found agreement of the values of all our couplings
with their counterparts in the complex MSSM, obtained with the model file of
Fritzsche:2013fta .

  * •

we checked that $\phi_{\lambda}+\xi+\xi_{s}$ and $\phi_{\kappa}+3\,\xi_{s}$
were the only relevant combinations of the phases $\phi_{\lambda}$,
$\phi_{\kappa}$, $\xi$ and $\xi_{s}$ at the level of amplitudes,

  * •

finally, we checked explicitly, that the counterterms
$\delta{\phi_{\lambda}}$, $\delta{\phi_{\kappa}}$, $\delta\xi$ and
$\delta\xi_{s}$ vanish when all NMSSM contributions are included, as pointed
out in Ref. Graf:2012hh . This can also be placed in the perspective of the
$\beta$-functions Cheng:1973nv ; Machacek:1983fi ; Machacek:1984zw ;
Jones:1984cx ; West:1984dg ; Martin:1993zk : the phases from the
superpotential parameters have no scale-dependence (at least up to two-loop
order); since $\xi$ and $\xi_{s}$ are spurious degrees of freedom, we could
expect their counterterms to present the same vanishing behaviors as
$\delta{\phi_{\lambda}}$ and $\delta{\phi_{\kappa}}$.

###  2.4 Quark Yukawa couplings

The Yukawa couplings of the top and bottom quarks, $Y_{t}$ and $Y_{b}$, have a
sizable impact on radiative corrections to the Higgs masses. We present our
prescriptions in this subsection.

The top Yukawa coupling $Y_{t}=\sqrt{2\sqrt{2}\,G_{F}}\,m_{t}/\sin\beta$ is
defined by the on-shell top mass $m_{t}$.

For the bottom quark, we employ the running $\overline{\mathrm{DR}}$ bottom-
mass of the SM (containing one-loop QCD corrections), $\overline{m}_{b}$, at
the scale $m_{t}$ Williams:2011bu . Additionally, we subtract the possibly
large $\tan\beta$-enhanced one-loop contributions to
$\overline{m}_{b}$—induced by gaugino–squark and higgsino–squark loops—from
the numerical definition of $Y_{b}$ at the tree level:
$Y_{b}=\sqrt{2\sqrt{2}\,G_{F}}\,\overline{m}_{b}/[\cos\beta\,\lvert
1+\Delta_{b}\rvert]$, where $\Delta_{b}$ is discussed in e. g. Refs.
Banks:1987iu ; Hall:1993gn ; Hempfling:1993kv ; Carena:1994bv ; Carena:1999py
; Eberl:1999he ; Williams:2011bu ; Baglio:2013iia .

###  2.5 Higgs masses at higher orders

The masses of the Higgs bosons are obtained from the complex poles of the full
propagator matrix. After rotating out the Goldstone mode111Besides Higgs–$G$
mixing, we neglect the kinetic Higgs–$Z$ and Higgs–photon mixing, since they
are sub-leading effects of two- and three-loop order, respectively. the
inverse propagator matrix for the five Higgs fields $h_{i}$ is a $(5\times 5)$
matrix that reads

$\displaystyle\mathbf{\hat{\Delta}}_{hh}^{-1}{\left(k^{2}\right)}=i\left[k^{2}\mathbf{1}-\mathbf{D}_{hh}+\mathbf{\hat{\Sigma}}_{hh}{\left(k^{2}\right)}\right].$
(28)

Here
$\mathbf{D}_{hh}=\text{diag}\\{m^{2}_{h_{1}},\,m^{2}_{h_{2}},\,m^{2}_{h_{3}},\,m^{2}_{h_{4}},\,m^{2}_{h_{5}}\\}$
denotes the diagonalized mass matrix of the Higgs fields without the Goldstone
at the tree level, and $\mathbf{\hat{\Sigma}}_{hh}$ denotes the matrix of the
renormalized self-energy corrections of the neutral Higgs fields.

The five complex poles of the propagator are given by the values of the
squared external momentum $k^{2}$ for which the determinant of the inverse
propagator matrix vanishes,

$\displaystyle\det{\left[\mathbf{\hat{\Delta}}^{-1}_{hh}{\left(k^{2}\right)}\right]_{k^{2}=\mathcal{M}^{2}_{i}}}$
$\displaystyle\overset{!}{=}0\,,$ $\displaystyle\mathcal{M}_{i}^{2}$
$\displaystyle\overset{!}{=}M_{h_{i}}^{2}+{i\mkern
1.0mu}\,\Gamma_{h_{i}}\,M_{h_{i}}\,,$ $\displaystyle i$
$\displaystyle\in\\{1,\ldots,5\\}\,,$ (29)

where we have explicitly stated the connection between the pole
$\mathcal{M}_{i}^{2}$, the Higgs mass $M_{h_{i}}$ and the total width
$\Gamma_{h_{i}}$ for each Higgs field $h_{i}$.

In order to account for the imaginary parts of the poles of the propagator
matrix, we perform an expansion of the self-energies in terms of the imaginary
part of the momentum, which is assumed to be small (also see section 4.3.5 of
Williams:2008 ),

$\displaystyle\mathbf{\hat{\Sigma}}_{hh}{\left(k^{2}\right)}\approx\mathbf{\hat{\Sigma}}_{hh}{\left(\Re\mathfrak{e}{\left[k^{2}\right]}\right)}+{i\mkern
1.0mu}\,\Im\mathfrak{m}{\left[k^{2}\right]}\,\frac{\mathrm{d}}{\mathrm{d}k^{2}}\mathbf{\hat{\Sigma}}_{hh}{\left(\Re\mathfrak{e}{\left[k^{2}\right]}\right)}$
(30)

In this work, the renormalized self-energy $\mathbf{\hat{\Sigma}}_{hh}$,

$\displaystyle\mathbf{\hat{\Sigma}}_{hh}{\left(k^{2}\right)}\approx\left.\mathbf{\hat{\Sigma}}^{(\text{1L})}_{hh}{\left(k^{2}\right)}\right|^{\text{NMSSM}}+\left.\mathbf{\hat{\Sigma}}^{(\text{2L})}_{hh}{\left(k^{2}\right)}\right|_{k^{2}\;=\;0}^{\text{MSSM}}.$
(31)

is evaluated by taking into account the full contributions from the ${\cal
CP}$-violating NMSSM at one-loop order and, as an approximation, the MSSM-like
contributions at two-loop order of
$\mathcal{O}{\left(\alpha_{t}\alpha_{s}\right)}$ Heinemeyer:2007aq  and
$\mathcal{O}{\left(\alpha_{t}^{2}\right)}$ Hollik:2014wea ; Hollik:2014bua  at
vanishing external momentum as implemented in FeynHiggs.222Additional MSSM-
like contributions, at two-loop order or beyond—e. g. resummation of large
logarithms for heavy sfermions Bahl:2016brp —could be incorporated as well
(see Drechsel:2016jdg ). However, we will confine our discussion in this paper
to the leading two-loop contributions.

We note that the two-loop $O(\alpha_{b}\alpha_{s})$ contributions to the Higgs
self-energies are not included in our calculation. Still, as we employ the
running bottom mass in the definition of $Y_{b}$ entering
$\left.\mathbf{\hat{\Sigma}}^{(\text{1L})}_{hh}{\left(k^{2}\right)}\right|^{\text{NMSSM}}$,
we expect that the missing two-loop piece is numerically subleading
Dedes:2003km ; Heinemeyer:2004xw ; Heinemeyer:2010mm .

###  2.6 Wave function normalization factors: the matrix
$\mathbf{Z}^{\mbox{\tiny mix}}$

In the Feynman-diagrammatic approach physical processes with external Higgs
fields are defined in terms of the tree-level mass states $h_{i}$. When
higher-order contributions are considered, however, the tree-level mass states
are not physical states. Indeed, radiative corrections induce additional mass
and kinematic mixing among the fields $h_{i}$, and the poles of the tree-level
propagators do not coincide with $\mathcal{M}_{i}^{2}$. A relation between the
amplitudes with an external tree-level Higgs mass state and those with an
external physical Higgs state is necessary (though this relation is trivial if
the fields are renormalized on-shell). For example, for a Higgs decaying into
two fermions $f$ this relation is given by the LSZ reduction formula,

$\displaystyle{\cal A}[h_{i}^{\mbox{\tiny phys}}\to f\bar{f}]=Z^{\mbox{\tiny
mix}}_{ij}\,{\cal A}[h_{j}\to f\bar{f}].$ (32)

Here the superscript ’phys’ denotes the amplitude with an external physical
field. The coefficients $Z^{\mbox{\tiny mix}}_{ij}$ can be expressed
explicitly in terms of the full propagator matrix (see Refs. Chankowski:1992er
; Heinemeyer:2001iy ; Fuchs:2016swt ; Fuchs:2017wkq  and also section $5.3$ of
Ref. Fuchs:2015jwa ), as

$\displaystyle Z^{\mbox{\tiny mix}}_{ij}=\left.\left[{i\mkern
1.0mu}\,\frac{\mathrm{d}}{\mathrm{d}k^{2}}\left(\mathbf{\hat{\Delta}}_{hh}^{-1}(k^{2})\right)_{ii}\right]^{\frac{1}{2}}\frac{\left(\mathbf{\hat{\Delta}}_{hh}(k^{2})\right)_{ij}}{\left(\mathbf{\hat{\Delta}}_{hh}(k^{2})\right)_{ii}}\right|_{k^{2}\to\mathcal{M}^{2}_{i}}.$
(33)

However, the analytical inversion becomes time-consuming in the case of a
$(5\times 5)$ propagator matrix. Additionally, $\mathbf{\hat{\Delta}}_{hh}$
needs to be evaluated at (or close to) its singular points
$\mathcal{M}_{i}^{2}$, which can lead to numerical instabilities on the right-
hand side of Eq. (33) (only the ratio of propagator matrix elements
$(\mathbf{\hat{\Delta}}_{hh})_{ij}$ is finite). In order to avoid these issues
we employed an equivalent formulation of the coefficients $Z^{\mbox{\tiny
mix}}_{ij}$, which is outlined below.

We consider the following effective Lagrangian for the tree-level mass states
$h_{i}^{\mbox{\tiny(tree)}}$,

$\displaystyle{\cal L}_{\mbox{\tiny eff}}=\frac{1}{2{i\mkern
1.0mu}}\left[\mathbf{\hat{\Delta}}^{-1}_{hh}{\left(k^{2}\right)}\right]_{ij}h_{i}^{\mbox{\tiny(tree)}}h_{j}^{\mbox{\tiny(tree)}}$
(34)

and set $\mathbf{Z}^{\mbox{\tiny mix}}=(Z^{\mbox{\tiny mix}}_{ij})$ as the
transition matrix to the physical Higgs fields:
$h_{i}^{\mbox{\tiny(loop)}}=Z^{\mbox{\tiny
mix}}_{ij}h_{j}^{\mbox{\tiny(tree)}}$. The states $h_{i}^{\mbox{\tiny(loop)}}$
are defined such that

$\displaystyle\mathcal{L}_{\mbox{\tiny
eff}}=\frac{1}{2}\left[k^{2}-\mathcal{M}_{h_{i}}^{2}\right](h_{i}^{\mbox{\tiny(loop)}})^{2}+\mathcal{O}{\left(\left[k^{2}-\mathcal{M}_{h_{i}}^{2}\right]^{2}\right)}\,.$
(35)

In other terms, the $h_{i}^{\mbox{\tiny(loop)}}$ should appear as on-shell
fields with standard kinetic terms close to their mass-pole. Thus the
coefficients $Z^{\mbox{\tiny mix}}_{ij}$ should satisfy the ‘eigenvalue’
conditions333Note that the system is non-linear due to the momentum dependence
of $\mathbf{\hat{\Sigma}}_{hG}$. However,
$\left\\{\mathcal{M}_{h_{i}}^{2},\left(\mathbf{Z}^{\mbox{\tiny
mix}}\right)_{i}\right\\}$ is a genuine eigenstate of
$\mathbf{D}_{hG}-\mathbf{\hat{\Sigma}}_{hG}{\left(\mathcal{M}_{h_{i}}^{2}\right)}$.

$\displaystyle\left[\mathbf{D}_{hG}-\mathbf{\hat{\Sigma}}_{hG}{\left(\mathcal{M}_{h_{i}}^{2}\right)}\right]_{kl}Z^{\mbox{\tiny
mix}}_{il}$ $\displaystyle=\mathcal{M}_{h_{i}}^{2}Z^{\mbox{\tiny
mix}}_{ik}\,.$ (36)

Once the roots of
$\det{\left[\mathbf{\hat{\Delta}}^{-1}_{hh}{\left(k^{2}\right)}\right]}$ are
known, the $i$-th line of $\mathbf{Z}^{\mbox{\tiny mix}}$ is thus determined
as the eigenvector of $\mathbf{D}_{hG}-\mathbf{\hat{\Sigma}}_{hG}$ for the
eigenvalue $\mathcal{M}_{h_{i}}^{2}$.

Finally, the normalization of the $i$-th line of $\mathbf{Z}^{\mbox{\tiny
mix}}$ is specified by the following condition on the kinetic term,

$\displaystyle\left[\frac{\mathrm{d}\mathbf{\hat{\Delta}}^{-1}_{hh}}{\mathrm{d}k^{2}}{\left(\mathcal{M}_{i}^{2}\right)}\right]_{kl}Z^{\mbox{\tiny
mix}}_{ik}\,Z^{\mbox{\tiny
mix}}_{il}=\left[\mathbb{I}+\frac{\mathrm{d}\mathbf{\hat{\Sigma}}_{hG}}{\mathrm{d}k^{2}}{\left(\mathcal{M}_{i}^{2}\right)}\right]_{kl}Z^{\mbox{\tiny
mix}}_{ik}\,Z^{\mbox{\tiny mix}}_{il}=1,$ (37)

such that the coefficients $Z^{\mbox{\tiny mix}}_{ij}$ are uniquely specified
(up to a sign without physical meaning) by Eq. (35). For the study of effects
from the normalization of $\mathbf{Z}^{\mbox{\tiny mix}}$, it is convenient to
define the (squared) norm $\left|Z_{i}\right|^{2}$ of its rows,

$\displaystyle\lvert Z_{i}\rvert^{2}=\sum_{j\;=\;1}^{5}{\left|Z^{\mbox{\tiny
mix}}_{ij}\right|^{2}}\,,$ (38)

i. e. $\left|Z_{i}\right|$ correspond to the norm of the eigenvectors, that
are associated to the complex pole $\mathcal{M}_{h_{i}}$, see Eqs. (35) and
(37).

The determination of $\mathbf{Z}^{\mbox{\tiny mix}}$ in terms of the
eigenstates of $\mathbf{\hat{\Delta}}^{-1}_{hh}{\left(k^{2}\right)}$ is
numerically easier to handle than its determination via Eq. (33). Applying the
two defining conditions Eqs. (36) and (37) to the expression of Eq. (33), one
can verify that both definitions of $\mathbf{Z}^{\mbox{\tiny mix}}$ are
identical.

As we discussed, the components of the matrix $\mathbf{Z}^{\mbox{\tiny mix}}$
establish the connection between the physical fields and the tree-level
external legs. In the literature this matrix $\mathbf{Z}^{\mbox{\tiny mix}}$
is often replaced by simplified versions neglecting the momentum dependence of
the self-energies. With the aim of performing numerical comparisons in the
following section, we introduce two such approximate definitions of the
relation between loop-corrected and tree-level fields:

  * •

The first approach consists in freezing the momentum to $k^{2}=0$ in the self-
energy of Eq. (28). This assumption is known as the effective potential
approximation. In this approach the inverse propagator matrix
$\mathbf{\Delta}_{hh}^{-1}{(k^{2}=0)}$, as given by Eq. (28), is diagonalized
by a simple orthogonal matrix $\mathbf{U}^{0}$, which approximates
$\mathbf{Z}^{\mbox{\tiny mix}}$.

  * •

Another choice consists in replacing the momentum dependence of the self-
energy in Eq. (28) by
$\left[\mathbf{\hat{\Sigma}}_{hh}{\left(k^{2}\right)}\right]_{ij}\rightarrow\left[\mathbf{\hat{\Sigma}}_{hh}{\left((m_{h_{i}}^{2}+m_{h_{j}}^{2})/2\right)}\right]_{ij}$
(given in the basis of the tree-level mass states). This procedure aims at
more closely mimicking the actual values of the self-energies involved in the
mass calculation. In this approach the inverse propagator is also diagonalized
by an orthogonal matrix $\mathbf{U}^{m}$.

While these procedures capture the mixing effects induced by radiative
corrections, at least partially, it is nevertheless obvious that they miss the
normalization of the fields outlined in Eq. (37). This means in particular
that the norm as defined in Eq. (38) will always be identical to 1 if the
matrix $\mathbf{Z}^{\mbox{\tiny mix}}$ is approximated by either
$\mathbf{U}^{0}$ or $\mathbf{U}^{m}$. We will discuss the impact of these
approximations in the following section.

##  3 Numerical Analysis

In this section we present the results of our Higgs mass calculation and
compare them with the output of public tools for several ${\cal CP}$-violating
scenarios. We also investigate the relevance of the matrix
$\mathbf{Z}^{\mbox{\tiny mix}}$ for transition amplitudes in the example of
the one-loop corrected decays of one Higgs field into a tau/anti-tau pair,
$h_{i}\to\tau^{+}\tau^{-}$.

The choice for the top-quark mass is $m_{t}=173.2$ GeV. Throughout this
section all $\overline{\mathrm{DR}}$ parameters are defined at $m_{t}$, and
all stop-parameters are on-shell parameters.

From the point of view of the Higgs phenomenology we test the scenarios
presented in this section with the full set of experimental constraints and
signals implemented in the public tools HiggsBounds-4.3.1 Bechtle:2008jh ;
Bechtle:2011sb ; Bechtle:2013gu ; Bechtle:2013wla ; Bechtle:2015pma ; HB-www
and HiggsSignals-1.3.1 Bechtle:2013xfa ; HB-www .

###  3.1 Comparison with FeynHiggs in the MSSM-limit

In the limit of vanishing $\lambda$ and $\kappa$, the singlet superfield
decouples from the MSSM sector and one is left with an effective MSSM—the
$\mu_{\text{eff}}$ term persists as long as $\kappa\sim\lambda$. We may then
compare our results for the Higgs masses and the matrix
$\mathbf{Z}^{\mbox{\tiny mix}}$ in this limit to those of `FeynHiggs-2.12.0`.
For this to be meaningful, we adjust the settings of `FeynHiggs`, so that they
match the higher-order contributions and renormalization scheme of our NMSSM
calculation. In particular, we impose that the one-loop field-renormalization
constants and $\tan\beta$ are $\overline{\mathrm{DR}}$-renormalized and select
full one-loop and leading two-loop MSSM contributions of
$\mathcal{O}{(\alpha_{t}\alpha_{s}+\alpha_{t}^{2})}$. We also require that
`FeynHiggs` takes the $\tan\beta$-enhanced contributions to the down-type
Yukawa couplings into account. The corresponding FeynHiggs input flags read
FHSetFlags[4,0,0,3,0,2,0,0,1,1].

Figure 1:  Masses of the light, SM-like state $h_{1}$ (left plot) and the two
heavy states $h_{2}$ and $h_{3}$ (right plot) as a function of $\phi_{A_{t}}$.
The solid lines denote the masses obtained with our calculation, while the
squares denote the masses obtained with FeynHiggs with the options indicated
in the text. The scenario is representative of the MSSM-limit of the NMSSM and
we employ the following parameters: $\lambda=\kappa=10^{-5}$, $\tan\beta=10$,
$m_{H^{\pm}}=500$ GeV, $\mu_{\text{eff}}=250$ GeV, $A_{\kappa}=-100$ GeV,
$m_{\tilde{F}}=1.5$ TeV, $|A_{t}|=A_{b}=2.5$ TeV, $2\,M_{1}=M_{2}=M_{3}/5=0.5$
TeV. Figure 2: Modules for the nine elements of the matrix
$\mathbf{Z}^{\mbox{\tiny mix}}$, obtained with our calculation (solid line)
and FeynHiggs (squares). The colors follow the convention of Fig. 1: red for
the coefficients defining the wave-function normalization of the light, SM-
like state $h_{1}$, blue and green for the coefficients corresponding to the
heavy states $h_{2}$ and $h_{3}$, respectively. The parameters are chosen as
in Fig. 1.

We consider a region in the parameter space of the NMSSM with the following
characteristics: $\lambda=\kappa=10^{-5}$, $\tan\beta=10$, $m_{H^{\pm}}=500$
GeV, $\mu_{\text{eff}}=250$ GeV, $A_{\kappa}=-100$ GeV; the sfermion soft
masses are set to the universal value of $1.5$ TeV and the sfermion trilinear
couplings to a value of $0.5$ TeV, with the exception of the third generation
parameters $|A_{t}|=A_{b}=2.5$ TeV; the gaugino masses are chosen as follows:
$2M_{1}=M_{2}=M_{3}/5=0.5$ TeV. We then vary the phase $\phi_{A_{t}}$. A
variation of $\phi_{A_{t}}$ (or of any MSSM-like phase) in such a naive
direction is of limited phenomenological interest, since in this case limits
from Electric Dipole Moments (EDM) are violated almost as soon as ${\cal CP}$,
see e.g. Ref. Arbey:2014msa . In the following we dismiss this issue, however,
and allow $\phi_{A_{t}}$ to vary over its full range. Indeed, we are only
interested in comparing our results with those of `FeynHiggs`. Due to the
largely SM-like properties of the state with a mass close to $125$ GeV, our
scenario appears to retain characteristics that are compatible with the
experimental data implemented in HiggsBounds and HiggsSignals, over the whole
range of $\phi_{A_{t}}$. The results for the Higgs masses are displayed in
Fig. 1. We observe a near perfect agreement between our results (solid curves)
and those of `FeynHiggs` (squares) with differences of order MeV. This
agreement is expected, since we closely follow the procedure for the
renormalization and processing of the MSSM-like input of FeynHiggs. Moreover,
due to the small values for $\lambda$ and $\kappa$, deviations induced by
genuine NMSSM effects remain negligible. The results for the elements of the
matrix $\mathbf{Z}^{\mbox{\tiny mix}}$ are displayed in Fig. 2. Again, we find
a very good agreement between our results and FeynHiggs with differences below
1‰ for the modules.

###  3.2 Comparison in the ${\cal CP}$-conserving limit

We now turn away from the MSSM limit. Our mass calculation can be confronted
to the routines presented in Drechsel:2016jdg  in the ${\cal CP}$-conserving
case. Both approaches employ an identical renormalization scheme in this
limit, with the exception of the electroweak vev, which receives a
$\overline{\mathrm{DR}}$ renormalization in Drechsel:2016jdg  while we
parametrize $v$ in terms of $M_{W}$, $M_{Z}$ and $e$ (see Eq. (27)). However,
in Drechsel:2016jdg  the input for $v$ is obtained via a reparametrization
from our scheme (the scheme using $\alpha(M_{Z})$ as input is also
considered), as explained in section 2.3 of that reference. Therefore, both
mass predictions are directly comparable and the mismatch between them should
be understood as an effect of two-loop electroweak order, due to the
approximations in the reparametrization used by Drechsel:2016jdg .

We consider the following region in the parameter space of the NMSSM:
$\kappa=\lambda/2$, $\tan\beta=10$, $M_{H^{\pm}}=1$ TeV,
$\mu_{\text{eff}}=125$ GeV, $A_{\kappa}=-70$ GeV; the soft masses are taken as
in the previous subsection while the trilinear soft sfermion couplings are all
set to $0.5$ TeV, with the exception of $A_{t}=1.2$ TeV. We scan over
$\lambda$ from $\sim 0$ (the MSSM-limit) to $0.5$. The masses of the three
lightest Higgs states are displayed in the plot on the left-hand side of Fig.
3. The dominantly SM-like state is the heaviest of the three (green curve).
The two lighter states are dominantly singlet, ${\cal CP}$-even (red curve) or
${\cal CP}$-odd (blue curve). In the MSSM-limit, these three states are
significantly lighter than $125$ GeV: this results in an unsatisfactory
phenomenological situation in view of the LHC measurements. With increasing
$\lambda$, the ${\cal CP}$-even singlet-doublet mixing uplifts the mass of the
dominantly SM-like state, leading to phenomenologically viable
characteristics—as tested with HiggsSignals—for $\lambda\sim 0.2$. There, we
observe that $h_{1}$ possesses a sizable doublet component and a mass
$M_{h_{1}}\simeq 100$ GeV, so that this state could offer an interpretation of
the local excess observed at LEP in $e^{+}e^{-}\to Z+(H\to b\bar{b})$ searches
Barate:2003sz .

Figure 3: Masses of the three lightest Higgs states $h_{1}$ (red), $h_{2}$
(blue), $h_{3}$ (green) in the ${\cal CP}$-conserving limit for varying
$\lambda=2\,\kappa$ and the following input: $\tan\beta=10$, $M_{H^{\pm}}=1$
TeV, $\mu_{\text{eff}}=125$ GeV, $A_{\kappa}=-70$ GeV, $m_{\tilde{F}}=1.5$
TeV, $A_{t}=2$ TeV, $A_{f\;\neq\;t}=0.5$ TeV, $2\,M_{1}=M_{2}=M_{3}/5=0.5$
TeV. On the left-hand side, we present our result (solid line) and that of
Drechsel:2016jdg  (squares). On the right-hand side, we plot the mass-
differences between these two codes, due to the scheme applied to the electric
coupling.

We then focus on the comparison with the masses predicted by Drechsel:2016jdg
. The plot on the left-hand side of Fig. 3 illustrates a general agreement
between our calculation (solid curves) and the results of Drechsel:2016jdg
(squares). On the right-hand side of Fig. 3, we display the mass differences
between the two procedures, which are due to differences of two-loop order
induced by the reparametrization used by Drechsel:2016jdg . We observe
vanishing effects in the MSSM-limit while the mass differences eventually
reach $\mathcal{O}{\left(40\,\text{MeV}\right)}$ for $\lambda\simeq 0.16$.
This can be understood in the following fashion: the leading effect originates
in the Higgs mass matrix at the tree level, where an explicit dependence on
$v$ appears only444We remind the reader that both in Drechsel:2016jdg  and in
our calculation $M_{W}^{2}$ and $M_{Z}^{2}$ are chosen as independent, on-
shell parameters. Therefore, the corresponding terms in the Higgs mass-matrix
are not affected by the differences in the renormalization/reparametrization
discussed here. through terms of the form $\lambda\,v$ and $\kappa\,v$
(quadratically for the doublet and singlet mass entries, and linearly for the
doublet–singlet mixing). These terms are processed differently in both
approaches: in Drechsel:2016jdg  $v$ is regarded as an independent
$\overline{\mathrm{DR}}$ parameter, while in our calculation $v$ is a
dependent quantity that is expressed in terms of the independent parameters
$M_{W}$, $M_{Z}$ and $e$. While the reparametrization of Drechsel:2016jdg
should restore the agreement between the two procedures, neglected effects of
two-loop electroweak order in this reparametrization result in a small
mismatch. Since the terms that convey this mismatch come with prefactors
$\lambda$ or $\kappa$, the difference vanishes in the MSSM limit
($\lambda,\kappa\to 0$). Moreover, in the regime under consideration, where
$\tan\beta\gg 1$, it is possible to understand why the mass of the ${\cal
CP}$-odd singlet (blue curve) is largely insensitive to the mismatch: terms
$\propto(\lambda\,v)^{2}$ in the ${\cal CP}$-odd singlet mass entry are
suppressed as $1/\tan\beta$. Additionally, leading one-loop radiative
corrections of $\mathcal{O}{\left(\alpha_{t}\right)}$ induce further
dependence on the processing of $v$. However, these corrections are suppressed
for the points of Fig. 3, as the stops are relatively light.

On the whole, the numerical mismatch with the procedure of Drechsel:2016jdg
is very minor, which places our current code in the direct continuity of this
earlier work.

###  3.3 Comparison with NMSSMCALC

NMSSMCALC is particularly suitable for a comparison with our calculation,
since its mixed $\overline{\mathrm{DR}}$/on-shell renormalization scheme is
relatively close to the one that we use.555Note that it is somewhat more
involved to compare our results quantitatively with RGE-based tools, as the
input requires a conversion to the appropriate scheme (usually
$\overline{\mathrm{DR}}$) and a running to the correct input scale
Staub:2015aea . For this reason, we shall confine our discussion to
comparisons with NMSSMCALC, which shares closer characteristics with our
approach. A similar comparison for real parameters has been presented in
Drechsel:2016htw . Yet, we note several differences between the prescriptions
implemented by NMSSMCALC and the procedure that we have outlined in section 2
(defining our “default” calculation). First, NMSSMCALC applies a
renormalization scheme for the electric charge employing $\alpha(M_{Z})$ as
input—whereas we decided to define $\alpha$ via its relation to $G_{F}$. Then
the input parameters in the stop sector are defined in the
$\overline{\mathrm{DR}}$ scheme in NMSSMCALC—while we employ on-shell
definitions. Additionally, we resum large $\tan\beta$ effects from our
definition of the bottom Yukawa, contrarily to the Higgs-mass calculation of
NMSSMCALC. Finally, NMSSMCALC includes only
$\mathcal{O}{\left(\alpha_{t}\alpha_{s}\right)}$ corrections at the two-loop
order—where we consider $\mathcal{O}{\left(\alpha_{t}^{2}\right)}$ effects as
well. However, the two-loop $\mathcal{O}{\left(\alpha_{t}\alpha_{s}\right)}$
contributions of NMSSMCALC are exhaustive in the NMSSM (including corrections
for the self-energies with at least one external singlet field)—whereas ours
are obtained in the MSSM approximation.

These observations mean that our mass-calculation is not directly (at least,
not quantitatively) comparable to the predictions of NMSSMCALC, since, of the
items listed above, the first few produce a deviation relative to the scheme,
while the later ones generate a mismatch of higher orders. Consequently,
several adjustments need to be performed in order to make a comparison
meaningful and control the sources of deviations. Thus, NMSSMCALC has been
adjusted in view of accepting on-shell input in the stop sector.666We thank K.
Walz for providing a modified version of NMSSMCALC for this feature. Moreover,
we also establish a “modified” version of our routines that attempts to mimic
the choices of NMSSMCALC—i. e. employing $\alpha(M_{Z})$, discarding
large-$\tan\beta$ effects for $Y_{b}$ and subtracting
$\mathcal{O}{\left(\alpha_{t}^{2}\right)}$ corrections—although we cannot
currently include $\mathcal{O}{\left(\alpha_{t}\alpha_{s}\right)}$ corrections
beyond the MSSM, so that this effect should control the difference of our
modified version with NMSSMCALC. Beyond this comparison with NMSSMCALC, we
will also try to quantify the magnitude of the other higher-order effects that
distinguish our “default” result from NMSSMCALC.

First, we consider the regime of the NMSSM with low $\tan\beta$ and large
$\lambda$. This region in parameter space is well-known for maximizing the
specific NMSSM tree-level contributions to the mass of the SM-like Higgs state
as well as for stimulating singlet–doublet mixing effects and other genuine
aspects of the NMSSM phenomenology. We employ the following parameters:
$\lambda=0.7$, $|\kappa|=0.1$, $\tan\beta=2$, $M_{H^{\pm}}=1170$ GeV,
$\mu_{\text{eff}}=500$ GeV, $A_{\kappa}=-70$ GeV; the soft masses are taken as
in Fig. 1 with the exception of the squarks of the third generation, for which
the soft masses and trilinear couplings are set to $500$ GeV and $100$ GeV,
respectively. In the regime under consideration genuine NMSSM effects are
indeed sufficient to produce a SM-like state in the observed mass-range
without requiring large top/stop corrections. We vary the phase
$\phi_{\kappa}$ (we restrict to a range where the tree-level squared Higgs
masses remain positive). We note that, contrarily to MSSM-like phases, the
phases from the singlet sector are allowed a wide range of variation without
conflicting with the measured EDM King:2015oxa ; Cheung:2011wn .

The results for the mass prediction are presented in Fig. 4. At vanishing
$\phi_{\kappa}$ the mass of the SM-like state (in blue) is somewhat low,
$m_{h_{2}}\sim 120$ GeV, so that this point in parameter space has a very
marginal agreement with the observed characteristics of the Higgs state. For
non-vanishing $\phi_{\kappa}$, however, a ${\cal CP}$-violating mixing with
the lighter pseudoscalar singlet (in red) develops: this effect increases the
mass of the light mostly ${\cal CP}$-even state $h_{2}$ but affects its
otherwise SM-like properties only in a subleading way. Consequently, we
recover an excellent agreement with the LHC results—as tested by
`HiggsSignals` and HiggsBounds—for e. g. $\phi_{\kappa}\simeq-0.11$.
Additionally, the dominantly ${\cal CP}$-odd singlet $h_{1}$ then has a mass
close to $100$ GeV. As it acquires a doublet ${\cal CP}$-even component via
mixing, it could explain the LEP local excess in $b\bar{b}$ final states
Barate:2003sz . The mostly ${\cal CP}$-even singlet $h_{3}$ (in green), with
mass at $\sim 210$ GeV plays no significant role. The masses of the heavier
doublet-like fields $h_{4}$ and $h_{5}$ are approximately constant and close
to $M_{H^{\pm}}$.

In Fig. 4, we observe a good agreement between our results (solid lines),
computed as described in section 2, and the predictions of NMSSMCALC
(squares), although the corresponding masses are defined in different schemes
and at different orders. For a more quantitative comparison, we turn to our
“modified” scheme for the mass calculation. On the left-hand side of Fig. 5,
we plot the deviation between the corresponding results and the predictions of
NMSSMCALC for the three lightest Higgs states. We checked that the one-loop
results are virtually identical, so that the differences between NMSSMCALC and
our calculation are entirely controlled by two-loop effects. We observe
typical deviations of order $0.5$–$1$ GeV that should be interpreted as the
impact of $\mathcal{O}{\left(\alpha_{t}\alpha_{s}\right)}$ corrections beyond
the MSSM-approximation. As could be expected, the masses of the mostly singlet
states (red and green lines) tend to exhibit the largest effect, though the
mass-predictions for the SM-like state may still differ by $\sim 0.5$ GeV (for
$\phi_{\kappa}\simeq 0$). The plot on the right-hand side of Fig. 5 depicts
the magnitude of $\mathcal{O}{\left(\alpha_{t}^{2}\right)}$ effects, which is
quantified in our “default” scheme. Here again, the typical impact on the
masses is of order $1$ GeV. Expectedly, the masses of the almost pure singlet
states (red curve at $\phi_{\kappa}\simeq 0$ or green curve) are insensitive
to the corrections implemented in the MSSM-approximation. The mass of the
mostly ${\cal CP}$-odd singlet (red curve) is only affected when the
corresponding state acquires a non-vanishing doublet component
($\phi_{\kappa}\neq 0$).

In Fig. 6, we compare the $\mathbf{U}^{0}$ matrix elements that are delivered
by NMSSMCALC (squares) with ours (solid line; NMSSMCALC does not provide
$\mathbf{Z}^{\mbox{\tiny mix}}$). The results show a satisfactory agreement
also at this level.

Figure 4:  Masses of the three lighter Higgs fields as a function of
$\phi_{\kappa}$ for the scenario $\lambda=0.7$, $|\kappa|=0.1$, $\tan\beta=2$,
$M_{H^{\pm}}=1170$ GeV, $\mu_{\text{eff}}=500$ GeV, $A_{\kappa}=-70$ GeV,
$m_{\tilde{Q}_{3},\tilde{T},\tilde{B}}=0.5$ TeV, $A_{t}=A_{b}=0.1$ TeV,
$2\,M_{1}=M_{2}=M_{3}/5=0.5$ TeV. The red color depicts the mass of the mostly
${\cal CP}$-odd, singlet-like state $h_{1}$; blue is associated to the
essentially SM-like state $h_{2}$ and green corresponds to the mostly ${\cal
CP}$-even singlet-like state $h_{3}$. We display our “default” result for the
masses (solid curves) as well as the predictions of NMSSMCALC (squares).
Figure 5:  Impact of two-loop contributions in the scenario of Fig. 4. On the
left-hand side, $\Delta{M_{h_{i}}}=M_{h_{i}}-M_{h_{i}}^{\texttt{NC}}$
correspond to the mass-differences (for each of the three lightest Higgs
states) between the predictions of our “modified” scheme and NMSSMCALC:
$\mathcal{O}{\left(\alpha_{t}\alpha_{s}\right)}$ should dominate these
deviations. On the right-hand side, $\Delta{M_{h_{i}}}$ corresponds to the
mass-shifts associated to $\mathcal{O}{\left(\alpha_{t}^{2}\right)}$
contributions, which are calculated in our “default” scheme. The color of the
curves match the convention of Fig. 4.

Figure 6: The matrix-elements $|U^{0}_{ij}|$ in our calculation (full curves)
and in NMSSMCALC (squares) for the scenario of Fig. 4.

Subsequently, we present our results in another region of the parameter space:
$\lambda=0.2$, $|\kappa|=0.6$, $\tan\beta=25$, $m_{H^{\pm}}=1$ TeV,
$\mu_{\text{eff}}=200$ GeV, $A_{\kappa}=-750$ GeV, the gaugino soft masses as
well as the soft masses for the sfermions of first and second generations are
chosen as before; for the third generation, the soft sfermion mass is set to
$1.1$ TeV; the trilinear soft terms are set to $-2$ TeV. With this choice of
parameters, the singlet-like ${\cal CP}$-even state and the heavy ${\cal
CP}$-even and ${\cal CP}$-odd doublet-like states receive comparable masses of
the order of $1$ TeV. This results in a sizable mixing for the corresponding
fields $h_{2}$, $h_{3}$ and $h_{4}$, which includes both singlet–doublet
admixture as well as ${\cal CP}$-violation (for non-vanishing
$\phi_{\kappa}$). The SM-like Higgs state has a mass close to $\sim 124$ GeV
on the whole range of $\phi_{\kappa}$, which leads to a good agreement with
the Higgs properties measured at the LHC (as tested with `HiggsSignals`). The
heaviest state $h_{5}$ has a mass of $\sim 1.2$ TeV, which we will not comment
further below.

On the left-hand side of Fig. 7, we show our prediction for the mass of the
lightest (SM-like) Higgs state (full curve). The mass delivered by NMSSMCALC
is represented by the squares at about $118$ GeV, which is substantially
smaller than ours (by $\approx 6$ GeV). If we mimic the settings of NMSSMCALC
(our “modified result”, dotted curve), this discrepancy is considerably
reduced. In fact, the difference between our full result and NMSSMCALC’s is
largely driven by the $\mathcal{O}{\left(\alpha_{t}^{2}\right)}$ two-loop
contributions, missing in NMSSMCALC. Again, both results are virtually
identical at the one-loop order.

Figure 7:  Mass predictions as a function of $\phi_{\kappa}$ for the lightest,
mostly SM-like Higgs state $h_{1}$ on the left-hand side, and the states
$h_{2}$ (blue), $h_{3}$ (green) and $h_{4}$ (orange) on the right-hand side.
The latter involve essentially the ${\cal CP}$-even singlet and the two ${\cal
CP}$-even and ${\cal CP}$-odd heavy-doublet degrees of freedom that mix
substantially. The full curves correspond to our default result; the squares
are obtained with NMSSMCALC; the dotted line represents our modified result.
The parameters are chosen as follows: $\lambda=0.2$, $|\kappa|=0.6$,
$\tan\beta=25$, $m_{H^{\pm}}=1$ TeV, $\mu_{\text{eff}}=200$ GeV,
$A_{\kappa}=-750$ GeV, $m_{\tilde{Q}_{3},\tilde{T},\tilde{B}}=1.1$ TeV,
$A_{t}=A_{b}=-2$ TeV, $2\,M_{1}=M_{2}=M_{3}/5=0.5$ TeV.

On the right-hand side of Fig. 7, we turn to the heavier states $h_{2}$,
$h_{3}$ and $h_{4}$ of this scenario. Our default results (full curves) are
compatible with the predictions of NMSSMCALC (squares). The discrepancies are
of order $1$–$3$ GeV only, which should be considered from both the
perspective of the different renormalization scheme of the electric coupling
$e$ and the different two-loop contributions. Actually, the mass predictions
match almost exactly when comparing NMSSMCALC with our modified scheme (dotted
curves). The corresponding deviations are shown on the left-hand side of Fig.
8 and fall in the range of $100$ MeV. In this precise case, the difference
between our results and NMSSMCALC is essentially driven by the resummation of
large-$\tan\beta$ effects in the $b$-quark Yukawa coupling. On the right-hand
side of Fig. 8, we quantify the associated mass-shift and find an impact of a
few GeV.

Finally, we turn to the $\mathbf{U}^{0}$ matrix elements for $h_{2}$, $h_{3}$
and $h_{4}$ in Fig. 9. There, we observe sizable deviations between our
default result (solid curves) and NMSSMCALC (squares), which, however, have no
deep-reason to agree in view of the diverging options. If we keep in mind that
the main difference between our full scheme and NMSSMCALC is controlled by the
large-$\tan\beta$ corrections to $Y_{b}$ in this precise scenario, it is not
surprising to observe large shifts, as mixing angles are indeed very sensitive
to small deviations in the mass-matrix for states that are very close in mass.
These differences largely vanish when we identify the output of NMSSMCALC with
our modified results (dotted lines), which is better equipped for comparisons
with this code.

In summary, the results of our mass calculation are largely compatible with
the predictions of NMSSMCALC. Deviations with a magnitude of
$\mathcal{O}{\left(\text{GeV}\right)}$ are indeed within the expected range if
we allow for the different renormalization scheme of $e$ and higher-order
contributions. Such discrepancies tend to be reduced sizable when we modify
our routines to adopt the assumptions of NMSSMCALC. The impact of two-loop
$\mathcal{O}{\left(\alpha_{t}\alpha_{s}\right)}$ contributions beyond the
MSSM, two-loop $\mathcal{O}{\left(\alpha_{t}^{2}\right)}$ and the resummation
of large-$\tan\beta$ effects in the Yukawa couplings can be clearly identified
from this comparison. As a final remark, let us emphasize that the uncertainty
on the Higgs-mass calculation from unknown higher-order contributions and
parametric errors may reach several GeV Degrassi:2002fi ; Staub:2015aea ;
Drechsel:2016htw .

Figure 8:  Impact of the higher-order effects in the mass-predictions for
$h_{2}$, $h_{3}$ and $h_{4}$ in the scenario of Fig. 7. On the left-hand side,
we show the deviation in mass between NMSSMCALC and our modified scheme
(controlled by $\mathcal{O}{\left(\alpha_{t}\alpha_{s}\right)}$ corrections
beyond the MSSM-approximation). The plot on the right-hand side illustrates
the impact of large-$\tan\beta$ effects in $Y_{b}$: the considered mass-
difference is that induced in our default scheme by the $\Delta_{b}$ term. The
colors follow the conventions of Fig. 7. The discontinuities at
$\phi_{\kappa}=\pm\frac{\pi}{2}$ originate from the mass calculation in
NMSSMCALC.

Figure 9:  The matrix-elements $\lvert U^{0}_{ij}\rvert$ in our calculation
(solid curves), in NMSSMCALC (squares), and in our modified calculation closer
to the options of NMSSMCALC (dotted) for the scenario of Fig. 7.

###  3.4 The matrix $\mathbf{Z}^{\mbox{\tiny mix}}$ and the
$h_{i}\to\tau^{+}\tau^{-}$ decays

The matrix $\mathbf{Z}^{\mbox{\tiny mix}}$ is not an observable quantity in
itself. It is a renormalization-scheme dependent object relating the tree-
level mass states of the Higgs sector to the physical Higgs fields. For on-
shell renormalized fields $\mathbf{Z}^{\mbox{\tiny mix}}$ is trivial. In any
other renormalization scheme, however, it is mandatory to include this
transition to the physical fields for a proper description of external legs in
Feynman diagrams at higher orders.

A remarkable aspect of $\mathbf{Z}^{\mbox{\tiny mix}}$ is that the
eigenvectors that it contains do not preserve unitarity with respect to the
tree-level fields. Instead, they satisfy the normalization condition given in
Eq. (37). This is a feature that the approximations $\mathbf{U}^{0}$ and
$\mathbf{U}^{m}$ are unable to capture (by construction). In a first step, we
will show that the norms in $\mathbf{Z}^{\mbox{\tiny mix}}$ can differ from
$1$ by a few percent in the scheme that we have described in section 2. Beyond
the normalization of the fields, $\mathbf{U}^{0}$ and $\mathbf{U}^{m}$ also
differ from $\mathbf{Z}^{\mbox{\tiny mix}}$ in that they diagonalize the mass-
matrix away from the poles of the propagator.

However, as we wrote above, $\mathbf{Z}^{\mbox{\tiny mix}}$ is a scheme-
dependent object and we should not pay excessive attention to its actual
structure. In order to characterize its role in an observable quantity, we
will consider the $h_{i}\to\tau^{+}\tau^{-}$ decays at the one-loop level. We
have chosen this particular channel as it is one of the main fermionic Higgs
decays and proves technically simple to implement in a predictive way.
Moreover, one-loop corrections are of purely electroweak nature—QCD
contributions occur only at three-loop order and beyond—so that radiative
corrections are expected to be moderate. This allows for a clean
appreciation—free of large higher-order uncertainties—of the impact of the
wave-function normalization matrix $\mathbf{Z}^{\mbox{\tiny mix}}$. Radiative
corrections777Details on the calculation of the decays at the one-loop level
will be presented in a future publication. are computed with our model file,
except for the QED contributions, which are included according to the
prescriptions of Refs. Braaten:1980yq ; Drees:1990dq . There,
$\mathbf{Z}^{\mbox{\tiny mix}}$ intervenes in the decay amplitudes of the
physical fields according to Eq. (32) (we dismiss the superscript ’phys’
throughout this section). We will show that the substitution of
$\mathbf{Z}^{\mbox{\tiny mix}}$ by the approximations $\mathbf{U}^{0}$ and
$\mathbf{U}^{m}$ may lead to sizable deviations in certain regions of the the
NMSSM parameter space. This result confirms the outcome of similar studies in
the MSSM Frank:2006yh .

Figure 10:  On the left-hand side, the masses of the three lighter Higgs
states are depicted as a function of $\phi_{\kappa}$, including all available
two-loop contributions. On the right-hand side, the squared norms
$\left|Z_{i}\right|^{2}$ (as defined in Eq. (38)) of the five eigenvectors
defining $\mathbf{Z}^{\mbox{\tiny mix}}$ are shown. The scenario is
characterized by $\lambda=0.7$, $|\kappa|=0.1$, $\tan\beta=2$,
$m_{H^{\pm}}=1.2$ TeV, $\mu_{\text{eff}}=500$ GeV, $A_{\kappa}=-100$ GeV,
$m_{\tilde{Q}_{3},\tilde{T},\tilde{B}}=0.5$ TeV, $A_{t}=A_{b}=0.1$ TeV,
$2\,M_{1}=M_{2}=M_{3}/5=0.5$ TeV. Figure 11:  In the left column, we show the
decay widths $\Gamma(h_{i}\to\tau^{+}\tau^{-})$ in the scenario of Fig. 10 for
the five neutral Higgs states. The widths are computed at the one-loop level,
and the mixing of the external, physical Higgs fields is expressed in terms of
the matrix $\mathbf{Z}^{\mbox{\tiny mix}}$ (solid), or approximated by the
matrices $\mathbf{U}^{m}$ (dashed) or $\mathbf{U}^{0}$ (dotted). In the right
column, the differences $\Delta{\Gamma}=\Gamma-\Gamma_{\textrm{appr}}$ between
the widths obtained with $\mathbf{Z}^{\mbox{\tiny mix}}$ and its approximate
treatments $\mathbf{U}^{m}$ (dashed) and $\mathbf{U}^{0}$ (dotted) are
depicted, normalized to the width $\Gamma$ obtained with
$\mathbf{Z}^{\mbox{\tiny mix}}$.

We turn to the following NMSSM input: the parameters are chosen as in Fig. 4,
except for $M_{H^{\pm}}=1.2$ TeV and $A_{\kappa}=-100$ GeV. We have plotted
the masses of the three lighter Higgs fields as a function of $\phi_{\kappa}$
in the plot on the left-hand side of Fig. 10. For vanishing $\phi_{\kappa}$
the lightest Higgs state $h_{1}$ is SM-like and we checked with `HiggsBounds`
and `HiggsSignals` that this point is consistent with the experimental data.
The dominantly ${\cal CP}$-odd, singlet-like state $h_{2}$ is only slightly
heavier than the state $h_{1}$ in this case. For increasing values of
$\phi_{\kappa}$ the mixing of the states $h_{1}$ and $h_{2}$ tends to lower
the mass of the SM-like state $h_{1}$, which eventually becomes too light to
accommodate the experimental data. The dominantly ${\cal CP}$-even, singlet-
like state $h_{3}$ has a near constant mass of $\sim 210$ GeV for all depicted
values of $\phi_{\kappa}$. The two heavier, ${\cal CP}$-even and ${\cal
CP}$-odd doublet-like states have masses close to $\sim 1.2$ TeV.

The results for the squared norms $\lvert Z_{i}\rvert^{2}$ of the
eigenvectors—see Eq. (38)—in this scenario are shown in the plot on the right-
hand side of Fig. 10. We observe a departure from the value $1$—which would
correspond to a unitary transition, as modeled by the approximations
$\mathbf{U}^{0}$ and $\mathbf{U}^{m}$—by a few percent. The local extrema at
$\phi_{\kappa}\simeq 0$ for $\lvert Z_{1}\rvert^{2}$ (red curve) and $\lvert
Z_{2}\rvert^{2}$ (blue curve) are associated to the sudden disappearance of
the mixing between the light ${\cal CP}$-odd singlet and the SM-like states at
$\phi_{\kappa}=0$ (${\cal CP}$-conserving limit). The discontinuities of
$\lvert Z_{2}\rvert^{2}$ and $\lvert Z_{3}\rvert^{2}$ (green curve) at
$\phi_{\kappa}\simeq\pm 0.5$ and $\pm 0.7$ correspond to the crossing of decay
thresholds ($h_{2}\to W^{+}W^{-}$, $h_{2}\to 2\,Z$, $h_{3}\to h_{1}\,h_{2}$).
These “spikes” are associated to the singularities of the first derivatives of
the loop functions involved in the determination of $\mathbf{Z}^{\mbox{\tiny
mix}}$—the apparent singularities actually come with a finite height due to
the imaginary parts of the poles. A proper description of these threshold
regions would require that the interactions among the daughter particles (of
the decays at threshold) are properly taken into account, which would result
in e. g. interactions between the Higgs state and bound-states or $s$-waves of
the daughter particles. This, however goes beyond the scope of the present
work.

We now turn to the decay widths $\Gamma{(h_{i}\rightarrow\tau^{+}\tau^{-})}$
in the scenario of Fig. 10. The widths are displayed in the left column of
Fig. 11 in the exhaustive description of the Higgs external leg (i. e.
employing $\mathbf{Z}^{\mbox{\tiny mix}}$; solid curves), in the
$\mathbf{U}^{m}$ approximation (dashed lines) and in the $\mathbf{U}^{0}$
approximation (dotted lines), for the five Higgs mass-eigenstates. We observe
a sharp variation close to $\phi_{\kappa}=0$ for the decays of $h_{1}$ and
$h_{2}$, both in the full and approximate descriptions. It is associated to
the mixing that develops between the SM-like state $h_{1}$ and the dominantly
${\cal CP}$-odd, singlet-like state $h_{2}$: this effect transfers part of the
doublet component of $h_{1}$ to $h_{2}$, so that the second state acquires a
non-vanishing coupling to SM fermions at the expense of the first. The sum of
the decay widths for both these states remains approximately constant in the
vicinity of $\phi_{\kappa}=0$. The width $\Gamma(h_{3}\to\tau^{+}\tau^{-})$
appears to be an order of magnitude smaller than the corresponding widths for
$h_{1}$ and $h_{2}$, an effect that is associated to the dominantly ${\cal
CP}$-even, singlet-like nature of $h_{3}$. Still,
$\Gamma(h_{3}\to\tau^{+}\tau^{-})$ nearly doubles in the considered interval
of $\phi_{\kappa}$, while the mass of $h_{3}$ is fairly stable: we can
understand this fact in terms of the acquisition of a larger doublet
component, which is channeled by the increased proximity of the masses of
$h_{2}$ and $h_{3}$. The widths of $h_{4}$ and $h_{5}$ are essentially
constant with only small relative changes. The general
$\phi_{\kappa}$-dependency of the approximated and the full results are very
similar. Yet, a systematic shift can be observed, especially in the case of
$\mathbf{U}^{0}$. This is consistent with the findings of similar studies in
the context of the MSSM Frank:2006yh .

On the right-hand side of Fig. 11, we show the difference between the full and
the approximate results, $\Delta{\Gamma}=\Gamma-\Gamma_{\textrm{appr}}$,
normalized to the more accurate one obtained with $\mathbf{Z}^{\mbox{\tiny
mix}}$. When $\mathbf{Z}^{\mbox{\tiny mix}}$ is approximated by
$\mathbf{U}^{0}$ (dotted lines), the typical discrepancy averages $4\%$,
although the deviation reaches beyond $10\%$ in the case of the decays of the
two lightest Higgs states in the vicinity of, but not at, $\phi_{\kappa}\simeq
0$. We stress that this interval close to $\phi_{\kappa}=0$ corresponds to the
phenomenologically relevant region from the perspective of the measured Higgs
properties. The approximation of $\mathbf{Z}^{\mbox{\tiny mix}}$ by
$\mathbf{U}^{m}$ tends to provide better estimates of the full result, though
deviations reach up to $\sim 7\%$. For both approximations the largest
deviations from the more complete result employing $\mathbf{Z}^{\mbox{\tiny
mix}}$ are intimately related to the proximity in mass of the SM-like and
dominantly ${\cal CP}$-odd, singlet-like states: as the approximations capture
the dependence on the external momentum either partially ($\mathbf{U}^{m}$) or
not at all ($\mathbf{U}^{0}$), the gap between the diagonal elements of the
Higgs mass-matrix, hence the mixing between the two states, is not quantified
properly. While this precise configuration might appear somewhat anecdotal, we
wish to point out the popularity of NMSSM scenarios with a sizable
singlet–doublet mixing. Dismissing this extreme case, the approximations of
$\mathbf{Z}^{\mbox{\tiny mix}}$ by $\mathbf{U}^{0}$, and to a lesser extent by
$\mathbf{U}^{m}$, still generate errors of the order of a few percent at the
level of the decay widths. In view of the precision of the measurements
achievable at the LHC CMS-HL ; ATLAS-HL ; Khachatryan:2016vau ; Zenz:2016lia ;
Slawinska:2016zeh , such discrepancies may appear of secondary importance. In
the long run, however, if the Higgs couplings are studied more closely, e. g.
at a linear collider Baer:2013cma ; Fujii:2015jha ; Fujii:2017ekh ; Moortgat-
Picka:2015yla , one would have to try and keep such sources of error to a
minimum.

##  4 Conclusions

In this paper, we have discussed the renormalization of the NMSSM Higgs
sector, including complex parameters. Radiative contributions to the Higgs
self-energies have been included up to the leading two-loop MSSM-like effects
of $\mathcal{O}{\left(\alpha_{t}\alpha_{s}+\alpha_{t}^{2}\right)}$. Beyond the
calculation of on-shell Higgs masses in this scheme, we were interested in
determining the transition matrix $\mathbf{Z}^{\mbox{\tiny mix}}$ between the
mass- and tree-level states. The latter plays an essential role in the proper
description of external Higgs legs in physical processes at the radiative
level.

Our predictions for the Higgs masses have been compared to the calculations of
existing tools in several NMSSM scenarios. In the MSSM-limit of the model, we
have recovered an excellent agreement with `FeynHiggs`. For non-vanishing
$\lambda$ and $\kappa$, we first compared our Higgs-mass prediction with the
findings of a previous extension of `FeynHiggs` to the NMSSM in the case of
real parameters, and found nearly identical results. Second, we compared our
calculation in the case of complex parameters with `NMSSMCalc` and found
values of the Higgs masses which are compatible, although small differences
emerge as a result of different processing of the two-loop pieces, both for
low and high $\tan\beta$.

Finally, we investigated the impact of the transition matrix
$\mathbf{Z}^{\mbox{\tiny mix}}$ on the $h_{i}\to\tau^{+}\tau^{-}$ width in a
scenario with low $\tan\beta$ and large $\lambda$, where the SM-like Higgs
state may have a sizable mixing with the ${\cal CP}$-odd singlet. We compared
the full one-loop calculation of the width—i. e. including
$\mathbf{Z}^{\mbox{\tiny mix}}$—with the popular approximations
$\mathbf{U}^{0}$ and $\mathbf{U}^{m}$—which are determined for fixed,
unphysical momenta. We found typical deviations at the percent level, although
larger effects can develop in the presence of almost-degenerate states,
especially in the $\mathbf{U}^{0}$ approximation. Such precision effects will
matter when the measurement of fermionic Higgs couplings reaches comparable
accuracy.

In its current form, our mass-computing tool is contained within a Mathematica
package. In time, our routines should be incorporated in an extension of
`FeynHiggs` to the NMSSM.

## Acknowledgments

We thank S. Heinemeyer for helpful comments on the manuscript. The work of F.
Domingo is supported in part by CICYT (Grant FPA 2013-40715-P), in part by the
MEINCOP Spain under contract FPA2016-78022-P, and in part by the Spanish
“Agencia Estatal de Investigación” (AEI) and the EU “Fondo Europeo de
Desarrollo Regional” (FEDER) through the project FPA2016-78645-P. The work of
P. Drechsel and S. Paßehr is supported by the Collaborative Research Center
SFB676 of the DFG, “Particles, Strings and the early Universe.”

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