# Type-II two-Higgs-doublet model in noncommutative geometry

Fredy Jimenez111<EMAIL_ADDRESS>Instituto de Física, Universidad
de Antioquia,
Calle 70 No 52-21, Medellín, Colombia  Diego Restrepo222<EMAIL_ADDRESS>Instituto de Física, Universidad de Antioquia,
Calle 70 No 52-21, Medellín, Colombia  Instituto de Física Gleb Wataghin,
UNICAMP, 13083-859, Campinas, SP, Brazil  Andrés Rivera333
<EMAIL_ADDRESS>Instituto de Física, Universidad de Antioquia,
Calle 70 No 52-21, Medellín, Colombia

###### Abstract

In noncommutative geometry (NCG) the spectral action principle predicts the
standard model (SM) particle masses by constraining the scalar and Yukawa
couplings at some heavy scale, but gives an inconsistent value for the Higgs
mass. Nevertheless, the scalar sector in the NCG approach to the standard
model, is in general composed of two Higgs doublets and its phenomenology
remains unexplored. In this work, we present a type-II two-Higgs-doublet model
in NCG, with a SM-like Higgs mass compatible with the 125 GeV experimental
value and extra scalars within the alignment limit without decoupling with
masses from $350$ GeV.

##  1 Introduction

After the discovery of the Higgs scalar particle by the ATLAS and CMS
experiments at the Large Hadron Collider (LHC) [1, 2], the question remains
whether the discovered scalar is just one piece of an extended Higgs sector.
The two-Higgs-doublet model (2HDM) [3, 4] is one of the simplest standard
model (SM) extensions. It offers a rich phenomenology, which could explain
some of the facts pointing out the need for physics beyond the SM, like the
evidence of baryon asymmetry [5], the presence of dark matter in the universe
[6, 7] or the smallness of Dirac neutrino masses [8]. It demands the inclusion
of two complex doublet scalar fields, $\Phi_{1}$ and $\Phi_{2}$, sharing the
same hypercharge, and both acquiring vacuum expectation values (VEVs), $v_{1}$
and $v_{2}$, respectively. The current measurements of the Higgs boson
properties show no substantial deviations from SM expectations [9, 10], and
only one of the neutral Higgs mass eigenstates ‘aligns’ to the SM-like Higgs
boson. Despite this, searches for additional Higgs bosons in collider
experiments provide only weak absolute exclusion limits on the masses of the
scalars [11], which results in a large freedom in the choice of the 2HDM
parameters. With this in mind, it would be desirable to have mathematical
structures which could impose more rigid bounds on the parameter space of
these kind of models with extended Higgs sectors.

Connes’ noncommutative geometry (NCG) [12, 13, 14] is a branch of mathematics
whose main application to particle physics is to endow the SM with an internal
geometry. Such an underlying geometrical structure makes difficult to fit SM
extensions into the NCG context [15, 16, 17]. In consequence, it would be
desirable to use this framework to constraint the parameter space of models
with extended Higgs sectors. In NCG, the topological and the metric data of
manifolds are encoded by spectral triples $\\{A,H,D\\}$, which are defined by
an associative algebra $A$ represented on a Hilbert space $H$, together with
the action of the Hermitian (Dirac) operator $D$, which contains the masses
and mixing terms for the fermion fields. This algebra can be considered in
general as noncommutative (including finite-dimensional matrix algebras) so
that the Dirac operator allows to extrapolate differential geometry notions to
spaces with a finite number of points. In particular, the SM internal geometry
is characterized by the finite-dimensional (and noncommutative) matrix algebra
$\mathbb{C}\oplus\mathbb{H}\oplus M_{3}(\mathbb{C})$, where its automorphism
group contains the SM gauge group SU(3)$\times$SU(2)$\times$U(1). The faithful
representation of this algebra on a $30$-dimensional (complex) Hilbert space
accounts for the SM fermions and antifermions with their respective quantum
numbers. The action functional describing the dynamics of the bosonic fields
is given by the spectral action principle [18], which is based on the spectrum
of the Dirac operator mixed with the continuous (Riemannian) space-time. This
action constrains the Yukawa and scalar couplings at the grand unification
scale, so that by flowing the renormalization group equations (RGEs) down to
the electroweak scale it is possible to make phenomenological estimations on
the masses of the particle fields. In particular, NCG predicted the existence
of a Higgs boson (provided with a geometric interpretation) as the internal
analog of a gauge field, but with a wrong mass near to $170$ GeV [19]. Such
discordance can be solved by enforcing the appearance of a singlet scalar
field in the spectral action [20, 21], which can be turned on rigorously by
considering a larger algebra [22, 23], by relaxing the first-order axiom [24,
25], or from a $U(1)_{B-L}$ (baryon-lepton) gauge symmetry [26].

In this work, we try to get the proper SM-like Higgs boson mass but preserving
the minimal SM algebra and gauge symmetry. Instead of introducing a singlet
scalar field, we build a phenomenologically viable 2HDM in the context of NCG.
We observe that the Higgs sector (encoded into the inner fluctuations of the
Dirac operator) is composed of two complex doublet scalar fields when we use
the second-order axiom [27] rather than the massless photon condition [28].
For simplicity, we assume a $CP$-conserving 2HDM scalar potential given by
[29]

$\displaystyle V=$
$\displaystyle\mu_{1}^{2}\Phi_{1}^{\dagger}\Phi_{1}+\mu_{2}^{2}\Phi_{2}^{\dagger}\Phi_{2}-\left(\mu_{12}^{2}\Phi_{1}^{\dagger}\Phi_{2}+\text{h.c.}\right)+\frac{\lambda_{1}}{2}\left(\Phi_{1}^{\dagger}\Phi_{1}\right)^{2}+\frac{\lambda_{2}}{2}\left(\Phi_{2}^{\dagger}\Phi_{2}\right)^{2}$
$\displaystyle+\lambda_{3}\Phi_{1}^{\dagger}\Phi_{1}\Phi_{2}^{\dagger}\Phi_{2}+\lambda_{4}\Phi_{1}^{\dagger}\Phi_{2}\Phi_{2}^{\dagger}\Phi_{1}+\left[\frac{\lambda_{5}}{2}\left(\Phi_{1}^{\dagger}\Phi_{2}\right)^{2}+\text{h.c.}\right],$
(1)

where the coefficients $\mu_{1}^{2}$, $\mu_{2}^{2}$,
$\lambda_{1},\lambda_{2},\lambda_{3},$ and $\lambda_{4}$, are real by
hermiticity of the potential [30, 31], whereas $\mu_{12}^{2}$ and
$\lambda_{5}$ are assumed to be real444There are another quartic couplings,
not considered here, which are forbidden if one of the Higgs doublets is odd
under an extra $\mathbb{Z}_{2}$ symmetry softly violated by the $\mu_{12}^{2}$
bilinear term.. To prevent the emergence of spontaneous $CP$ violation, we
choose real and positive values for $v_{1}$ and $v_{2}$ [32] and define

$\displaystyle\tan\beta=\frac{v_{2}}{v_{1}}\,.$ (2)

The quartic couplings $\lambda_{1}(\Lambda),...,\lambda_{4}(\Lambda)$ become
constrained by the spectral action at the unification scale $\Lambda$, while
$\mu_{12}^{2}$ and $\lambda_{5}(\Lambda)$ together with $\tan\beta$ are the
free parameters of the model.

The main result of our work is that the type-II 2HDM in NCG with the scalar
potential in Eq. (1), accounts for the correct SM-like Higgs boson mass of
$125$ GeV, involving extra scalars compatible with the alignment limit without
decoupling from masses around $350\ \text{GeV}$, for a very constrained
parameter space encompassing $\tan\beta\approx 2.1$.

The rest of the paper is organized as follows. In Sec. 2, we introduce the
main notions in NCG to emphasize its underlying 2HDM structure. In Sec. 3, we
deduce the spectral action constrains of the model in order to get the
boundary conditions for the quartic couplings of the 2HDM scalar potential. In
Sec. 4, we review the mass formulas for the scalar fields in 2HDM in
preparation for the phenomenological analysis. In Sec. 5, we scan the
parameter space at the unification scale and evolve all the relevant Yukawa
couplings and quartic scalar parameters down to the electroweak scale. There,
we check if there are regions of the parameter space compatible with a SM-like
Higgs boson. Finally, we close with the conclusions of our work.

##  2 Noncommutative geometry: Higgs sector

In this section, we introduce the main NCG tools which are necessary to
describe the SM emphasizing their underlying 2HDM structure.

The study of noncommutative geometry has its roots in the Gelfand-Naimark
theorem, which establishes an equivalence between compact topological spaces
and commutative, C∗-algebras [33, 34]. In consequence, the topology of a
manifold $M$ is encoded by the algebra of (complex) continuously
differentiable functions defined over the manifold $C^{\infty}(M,\mathbb{C})$,
so that the points $x\in M$ correspond to irreducible representations of the
algebra.

In order to characterize the geometrical information of a Riemannian manifold,
we introduce the spectral triple formulation (for complete reviews see e.g.
[35, 36]). Let us suppose that $\mathcal{S}\rightarrow M$ is a spinor bundle
on a 4-dimensional manifold $M$, so that spinor fields on $M$ are given by
smooth sections $\psi$ in the Hilbert space of square-integrable sections
$L^{2}(M,\mathcal{S})$. Then, we can represent the algebra elements by bounded
operators acting on spinors as $(f\psi)(x)=f(x)\psi(x)$. The so-called
‘canonical spectral triple’ is defined by
$\\{C^{\infty}(M,\mathbb{C}),L^{2}(M,\mathcal{S}),\not{D},\gamma_{5},J_{M}\\}$,
where the Dirac operator is defined by

$\displaystyle\not{D}=-i\gamma^{\mu}\nabla^{\mathcal{S}}_{\mu}\,,$

with $\nabla^{\mathcal{S}}_{\mu}$ being the spin Levi-Civita connection on
$\mathcal{S}$ and $\gamma^{\mu}$ are the usual gamma matrices which generate
the Dirac algebra. Additionally, the (Hermitian) chirality operator
$\gamma_{5}$, and the anti-unitary charge conjugation $J_{M}$ are given after
fixing a Dirac algebra basis $\gamma^{\mu}$, such that

$\displaystyle\left[f,J_{M}g^{*}J_{M}^{-1}\right]$ $\displaystyle=0,$
$\displaystyle\gamma_{5}^{2}=$ $\displaystyle 1_{4},$ $\displaystyle
J_{M}^{2}$ $\displaystyle=-1_{4},$ $\displaystyle J_{M}\gamma_{5}=$
$\displaystyle\gamma_{5}J_{M}\,,$ (3)

for all $f,g\in C^{\infty}(M,\mathbb{C})$. The Dirac operator $\not{D}$
satisfies the Leibniz rule

$\displaystyle\nabla^{\mathcal{S}}_{\mu}({f}\psi)={f}\nabla^{\mathcal{S}}_{\mu}(\psi)+\partial_{\mu}({f})\psi\,,$

for all $f\in C^{\infty}(M,\mathbb{C})$ and $\psi\in L^{2}(M,\mathcal{S})$. It
also satisfies the following commutation properties

$\displaystyle J_{M}\not{D}$ $\displaystyle=\not{D}J_{M},$
$\displaystyle\not{D}\gamma_{5}=$ $\displaystyle-\gamma_{5}\not{D},$
$\displaystyle\left[\left[\not{D},f\right],J_{M}g^{*}J_{M}^{-1}\right]=0\,,$
(4)

for all $f,g\in C^{\infty}(M,\mathbb{C})$. Note that the one-forms are
bounded, because they are given by the commutator
$\left[\not{D},f\right]=-i\gamma^{\mu}(\partial_{\mu}f)$, which is necessary
to recover the metric information from $M$ [37, 38].

The next step in NCG is to inquire about the geometry of a noncommutative
$*$-algebra. In this case, the points are mimicked by the set of ‘states’ of
the algebra555The states of an algebra are defined as linear maps of the form
$\pi_{x}:A\rightarrow\mathbb{C}$, which are positive $\pi_{x}(f^{*}f)\geq
0\in\mathbb{R}$, for all $f$ in $A$ and so that $\pi_{x}(\mathbf{1}_{A})=1$,
for $\mathbf{1}_{A}$ the unit of $A$., which corresponds to its set of
characters in the case that the algebra is commutative [39]. For physical
motives, we concentrate in the finite-dimensional algebra
$A_{F}=\mathbb{C}\oplus\mathbb{H}\oplus M_{3}(\mathbb{C})$, where the second
and last summands stand for the quaternion algebra and the set of $3\times 3$
complex matrices, respectively. We introduce the ‘finite spectral triple’
$\\{A_{F},H_{F},D_{F},\gamma_{F},J_{F}\\}$, where the algebra elements
($D_{F},\gamma_{F}$ and $J_{F}$ will be defined below) can be represented as
matrices acting on the finite-dimensional (complex) Hilbert space [40]

$\displaystyle H_{F}=(H_{l}\oplus H_{q}\oplus H_{\overline{l}}\oplus
H_{\overline{q}})^{\oplus 3}\,,$

which accounts for the three SM generations (without right-handed neutrinos),
where the spaces of leptons $H_{l}=\mathbb{C}^{3}$ and the quarks
$H_{q}=\mathbb{C}^{4}\otimes\mathbb{C}^{3}$ are spanned by
$\\{e_{R},\nu_{L},e_{L}\\}$ and $\\{u_{R},d_{R},u_{L},d_{L}\\}$, whereas the
anti-lepton $H_{\overline{l}}=\mathbb{C}^{3}$ and the anti-quark
$H_{\overline{q}}=\mathbb{C}^{4}\otimes\mathbb{C}^{3}$ spaces are generated by
the basis $\\{\overline{e_{R}},\overline{\nu_{L}},\overline{e_{L}}\\}$ and
$\\{\overline{u_{R}},\overline{d_{R}},\overline{u_{L}},\overline{d_{L}}\\}$,
respectively. The action of an element $a=(\lambda,q,m)\in A_{F}$ (with
$q\in\mathbb{H}$) on the particle side of the Hilbert space is defined by

$\displaystyle\pi(a)|_{H_{l}}$
$\displaystyle=\begin{pmatrix}\overline{\lambda}&0&0\\\ 0&\alpha&\beta\\\
0&-\overline{\beta}&\overline{\alpha}\end{pmatrix},$ and
$\displaystyle\pi(a)|_{H_{q}}$
$\displaystyle=\begin{pmatrix}{\lambda}&0&0&0\\\ 0&\overline{\lambda}&0&0\\\
0&0&\alpha&{\beta}\\\
0&0&-\overline{\beta}&\overline{\alpha}\end{pmatrix}\otimes 1_{3}\,,$ (5)

where $m\in M_{3}(\mathbb{C})\ \text{and}\ \lambda,\alpha,\beta\in\mathbb{C}$,
and the bar over them stands for their complex conjugated. The action of $a$
on the anti-particles is so that $\pi(a)\overline{l}=\lambda
1_{3}\overline{l}$ and $\pi(a)\overline{q}=(1_{4}\otimes m)\overline{q}$. For
easy use, from now on we will drop the symbol $\pi$ to denote the
representation of an algebra element.

Regarding the other parts of the finite spectral triple, the finite charge
conjugation operator $J_{F}$ is defined so that it interchanges particles with
their respective antiparticles, whereas the finite chirality operator
$\gamma_{F}$ assigns the eigenvalue $+1$ to left-handed particles, and the
eigenvalue $-1$ to the right-handed ones. Finally, the finite self-adjoint
Dirac operator, $D_{F}$, involves the Yukawa matrices for leptons and quarks
as follows

$\displaystyle D_{F}=\begin{pmatrix}S&T^{\dagger}\\\
T&\overline{S}\end{pmatrix},$

where $S$ and $T$ are operators acting on the SM particles, while
$T^{\dagger}$ and $\overline{S}$ act on the anti-particles. The action of $S$
on the SM fermions is given by

$\displaystyle S|_{H_{l}}$
$\displaystyle={\begin{pmatrix}0&0&Y_{l,1}^{\dagger}&Y_{l,2}^{\dagger}\\\
Y_{l,1}&0&0&0\\\ Y_{l,2}&0&0&0\end{pmatrix}},$ $\displaystyle S|_{H_{q}}$
$\displaystyle=\begin{pmatrix}0&0&Y_{q,11}^{\dagger}&Y_{q,21}^{\dagger}\\\
0&0&Y_{q,12}^{\dagger}&Y_{q,22}^{\dagger}\\\ Y_{q,11}&Y_{q,12}&0&0\\\
Y_{q,21}&Y_{q,22}&0&0\end{pmatrix}\otimes 1_{3}\,,$ (6)

where $Y_{l,i}$ and $Y_{q,ij}$ are $3\times 3$ complex matrices. On the other
hand, $T$ contains some unwanted terms which are usually removed by
introducing the non-geometric massless photon condition [27, 28] or by making
use of the second order axiom [36]

$\displaystyle\left[\left[D_{F},a\right],J_{F}\left[D_{F},b\right]^{*}J_{F}^{-1}\right]=0\,,$
(7)

for all $a,b\in A_{F}$. In the absence of right-handed neutrinos, the
condition in Eq. (7) reduces $T$ to $0$. One advantage (which we will exploit
later) of using the second-order axiom instead of the massless photon
condition is that none of the $Y_{l,i}$ or $Y_{q,ij}$ in the Eq. (6) is
required to be zero.

In general, given an arbitrary spectral triple $\\{A,H,D,\gamma,J\\}$, the
last two relations in Eq. (3), together with the first one in Eq. (4) can be
shifted to

$\displaystyle J^{2}$ $\displaystyle=\epsilon\mathbb{I},$ $\displaystyle JD$
$\displaystyle=\epsilon^{\prime}DJ,$ $\displaystyle J\gamma$
$\displaystyle=\epsilon^{\prime\prime}\gamma J\,,$

where $\mathbb{I}$ stands for the identity operator on $H$, while the symbols
$\epsilon,\epsilon^{\prime},\epsilon^{\prime\prime}\in\\{1,-1\\}$ define the
KO-dimension666The name ‘KO’ comes from the Bott periodicity theorems for real
K-theory based on real vector bundles [41]. (or signature) modulo $8$ as shown
in Table 1. We choose $\epsilon=1$, $\epsilon^{\prime}=1$ and
$\epsilon^{\prime\prime}=-1$ so that it belongs to a KO-dimension $6$ modulo
$8$ (see Table 1), which is necessary to preserve the correct fermion degrees
of freedom [42, 43, 44].

| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---|---|---
$\epsilon$ | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1
$\epsilon^{\prime}$ | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1
$\epsilon^{\prime\prime}$ | 1 |  | -1 |  | 1 |  | -1 |
Table 1: Module 8 KO-dimension.

To have a complete picture involving all of the SM fields, we take the tensor
product between the canonical spectral triple and the finite one, which
defines the so-called almost-commutative manifold. In this case, the resulting
spectral triple is given by $\mathcal{A}=C^{\infty}(M,\mathbb{C})\otimes
A_{F}$ represented on the tensor product space
$\mathcal{H}=L^{2}(\mathcal{S},M)\otimes H_{F}$. The chirality and the charge
conjugation operators are

$\displaystyle\Gamma=$ $\displaystyle\gamma_{5}\otimes\gamma_{F}\,,$ and
$\displaystyle\mathcal{J}=$ $\displaystyle J_{M}\otimes J_{F}\,,$

respectively, whereas the total Dirac operator is defined by

$\displaystyle\mathcal{D}=\not{D}\otimes 1_{F}+\gamma_{5}\otimes D_{F}\,,$ (8)

where $1_{F}$ stands for the identity matrix on the finite space $H_{F}$.

The fermion fields $\Psi\in\mathcal{H}$ should satisfy $\Gamma\Psi=\Psi$ [43],
and are given by the tensor product between anti-commuting Weyl spinors and
finite vector basis elements, both denoted by the same symbol as follows

$\displaystyle\Psi\equiv$ $\displaystyle e_{L}\otimes e_{L}+e_{R}\otimes
e_{R}+\overline{e_{L}}\otimes\overline{e_{L}}+\overline{e_{R}}\otimes\overline{e_{R}}+\nu_{L}\otimes\nu_{L}+\overline{\nu_{L}}\otimes\overline{\nu_{L}}+u_{L}\otimes
u_{L}$ $\displaystyle+u_{R}\otimes
u_{R}+\overline{u_{L}}\otimes\overline{u_{L}}+\overline{u_{R}}\otimes\overline{u_{R}}+d_{L}\otimes
d_{L}+d_{R}\otimes
d_{R}+\overline{d_{L}}\otimes\overline{d_{L}}+\overline{d_{R}}\otimes\overline{d_{R}}\,,$
(9)

where for simplicity we omitted family and color indices.

The bosonic fields come along with the inner fluctuations of the Dirac
operator, which are given by linear combinations of terms of the form
$a[\mathcal{D},b]$, for $a,b\in\mathcal{A}$, which we write as
$a=(\lambda,q,m)\,$ and $b=(\lambda^{\prime},q^{\prime},m^{\prime})\,$ [40,
35]. By using Eq. eqrefeq:dirac, we can split such commutator into the
following two terms

$\displaystyle a[\not{D}\otimes 1_{F},b]$
$\displaystyle\equiv\gamma^{\mu}\otimes A_{\mu}\ ,$ $\displaystyle
a[\gamma_{5}\otimes D_{F},b]$ $\displaystyle\equiv\gamma_{5}\otimes\phi\,,$

where $A_{\mu}\equiv-ia\partial_{\mu}b\,$ and $\phi\equiv a[D_{F},b]$
determine the Hermitian gauge and scalar fields, respectively.

The gauge fields can be expressed as follows

$\displaystyle A_{\mu}|_{H_{l}}$
$\displaystyle=\begin{pmatrix}\Lambda_{\mu}&0\\\ 0&Q_{\mu}\end{pmatrix},$
$\displaystyle A_{\mu}|_{H_{\overline{l}}}$
$\displaystyle=\Lambda_{\mu}1_{3}\,,$ $\displaystyle A_{\mu}|_{H_{q}}$
$\displaystyle=\begin{pmatrix}\Lambda_{\mu}&0&0\\\ 0&-\Lambda_{\mu}&0\\\
0&0&Q_{\mu}\end{pmatrix}\otimes 1_{3},$ $\displaystyle
A_{\mu}|_{H_{\overline{q}}}$
$\displaystyle=-\left(\overline{V}_{\mu}+\frac{1}{3}\Lambda_{\mu}\right)\otimes
1_{4}\,,$

where we have defined

$\displaystyle\Lambda_{\mu}\equiv$
$\displaystyle-i\overline{\lambda}\partial_{\mu}\overline{\lambda}^{\prime}\,,$
$\displaystyle Q_{\mu}\equiv$ $\displaystyle-iq\partial_{\mu}q^{\prime}\,,$
$\displaystyle V_{\mu}\equiv$ $\displaystyle-im\partial_{\mu}m^{\prime}\,.$

They can be compassed into the total gauge field $B_{\mu}\equiv
A_{\mu}-J_{F}A_{\mu}J_{F}^{\dagger}$, whose action on the particle side is
given by

$\displaystyle B_{\mu}|_{H_{l}\oplus
H_{q}}=\operatorname{diag}\left[-2\Lambda_{\mu},Q_{\mu}-\Lambda_{\mu}1_{2},V_{\mu}+\frac{4}{3}\Lambda_{\mu}\otimes
1_{3},V_{\mu}-\frac{2}{3}\Lambda_{\mu}\otimes 1_{3},V_{\mu}\otimes
1_{2}+\left(Q_{\mu}+\frac{1}{3}\Lambda_{\mu}1_{2}\right)\otimes 1_{3}\right].$

Note that the coefficients of $\Lambda_{\mu}$ are the hypercharges of the
corresponding SM fermions.

On the other hand, with the hermiticity of $\phi$, we have that the action of
the Higgs field,

$\displaystyle{\Phi}\equiv D_{F}+\phi+J_{F}\phi J_{F}^{\dagger}\,,$

on the particles, is given by a general 2HDM Yukawa structure with four
complex degrees of freedoms

$\displaystyle\Phi|_{H_{l}}$
$\displaystyle=\begin{pmatrix}0&\overline{\varphi}_{1}Y_{l,1}^{\dagger}+\overline{\varphi}_{2}Y_{l,2}^{\dagger}&\overline{\varphi}_{3}Y_{l,1}^{\dagger}+\overline{\varphi}_{4}Y_{l,2}^{\dagger}\\\
\varphi_{1}Y_{l,1}+\varphi_{2}Y_{l,2}&0&0\\\
\varphi_{3}Y_{l,1}+\varphi_{4}Y_{l,2}&0&0\end{pmatrix},$
$\displaystyle\Phi|_{H_{q}}$
$\displaystyle=\begin{pmatrix}0&0&{\varphi}_{4}Y_{q,11}^{\dagger}-{\varphi}_{3}Y_{q,21}^{\dagger}&-{\varphi}_{2}Y_{q,11}^{\dagger}+{\varphi}_{1}Y_{q,21}^{\dagger}\\\
0&0&\overline{\varphi}_{1}Y_{q,12}^{\dagger}+\overline{\varphi}_{2}Y_{q,22}^{\dagger}&\overline{\varphi}_{3}Y_{q,12}^{\dagger}+\overline{\varphi}_{4}Y_{q,22}^{\dagger}\\\
\overline{\varphi}_{4}Y_{q,11}-\overline{\varphi}_{3}Y_{q,21}&\varphi_{1}Y_{q,12}+\varphi_{2}Y_{q,22}&0&0\\\
-\overline{\varphi}_{2}Y_{q,11}+\overline{\varphi}_{1}Y_{q,21}&\varphi_{3}Y_{q,12}+\varphi_{4}Y_{q,22}&0&0\end{pmatrix},$
(10)

where, using Eqs. (5), we define

$\displaystyle\varphi_{1}$
$\displaystyle\equiv\alpha(\overline{\lambda}^{\prime}-{\alpha^{\prime}})+\beta\overline{\beta}^{\prime}+1\,,$
$\displaystyle\varphi_{2}$
$\displaystyle\equiv\beta(\overline{\lambda}^{\prime}-\overline{\alpha}^{\prime})-\alpha\beta^{\prime}\,,$
$\displaystyle\varphi_{3}$
$\displaystyle\equiv\overline{\beta}(\alpha^{\prime}-\overline{\lambda}^{\prime})+\overline{\alpha}\overline{\beta}^{\prime}\,,$
$\displaystyle\varphi_{4}$
$\displaystyle\equiv\overline{\alpha}(\overline{\lambda}^{\prime}-\overline{\alpha}^{\prime})+\overline{\beta}\beta^{\prime}+1\,.$

We observe that with the identification

$\displaystyle\Phi_{1}=$ $\displaystyle\begin{pmatrix}\varphi_{1}\\\
\varphi_{3}\end{pmatrix},$ $\displaystyle\Phi_{2}=$
$\displaystyle\begin{pmatrix}\varphi_{2}\\\ \varphi_{4}\end{pmatrix},$ (11)

the Higgs sector in NCG is in general composed of two complex Higgs doublets,
as mentioned in [26]. To avoid the appearance of flavor changing neutral
currents (FCNC) at tree-level, we use the usual approach of impose a (softly
broken [29, 31]) $\mathbb{Z}_{2}$ type-II 2HDM symmetry [3] enforced with
$\Phi_{1}\rightarrow-\Phi_{1}$, $d_{R}\to-d_{R}$, and $e_{R}\to-e_{R}$. This
implies that $Y_{l,2}=Y_{q,21}=Y_{q,22}=0$, so that by defining $Y_{u}\equiv
Y_{q,11}$, $Y_{d}\equiv Y_{q,12}$ and $Y_{e}\equiv Y_{l,1}$, we can express
the Yukawa interaction as777This is a piece of the total fermionic action
which is given by the bilinear form $\langle
J\Psi,\mathcal{D}_{A}\Psi\rangle$, where
$\mathcal{D}_{A}=\mathcal{D}+a[\mathcal{D},b]+\mathcal{J}a[\mathcal{D},b]\mathcal{J}^{\dagger}$
is the fluctuation of the total Dirac operator [43].

$\displaystyle\frac{1}{2}\langle
J\Psi,\left(\gamma_{5}\otimes{\Phi}\right)\Psi\rangle$ $\displaystyle=\langle
J_{M}\overline{u_{R}},Y_{u}^{\dagger}\left({\varphi}_{4}{u}_{L}-{\varphi}_{2}{d}_{L}\right)\rangle+\langle
J_{M}\overline{d_{R}},Y_{d}^{\dagger}\left(\overline{\varphi}_{1}{u}_{L}+\overline{\varphi}_{3}{d}_{L}\right)\rangle$
$\displaystyle+\langle
J_{M}{\overline{e_{R}}},Y_{e}^{\dagger}\left(\overline{\varphi}_{1}{\nu}_{L}+\overline{\varphi}_{3}{e}_{L}\right)\rangle-\text{h.c}\,,$
(12)

where the minus sign preceding the ‘h.c.’ acronym means that the Yukawa
matrices are anti-Hermitian. Note that the Lagrangian in Eq. (2) corresponds
to the type-II 2HDM Yukawa interaction [45, 46].

Now, we examine the spectral action principle, from which we will deduce the
constraints on the parameter space of the model at the unification scale.

##  3  Spectral action and constrains

The action functional describing the dynamics of the SM bosons is given by the
spectral action $\operatorname{Tr}\left(f(\mathcal{D}_{A}/\Lambda)\right)$,
where $\Lambda$ is a cut-off scale, and $f$ is a real, even, and positive
function such that $f\rightarrow 0$ when $\Lambda\rightarrow\infty$. By
writing $f$ in terms of its Laplace transform, we can use the heat kernel
expansion [47] to expand the spectral action, so that we concentrate only in
the following piece [48, 49]

$\displaystyle\operatorname{Tr}\left[f\left(\frac{D_{A}}{\Lambda}\right)\right]\supset\int_{M}\frac{f(0)}{8\pi^{2}}\left\\{\frac{1}{3}\operatorname{Tr}\left(F^{\mu\nu}F_{\mu\nu}\right)+\operatorname{Tr}\left[\left(\widetilde{D}_{\mu}\Phi\right)\left(\widetilde{D}^{\mu}\Phi\right)\right]+\operatorname{Tr}\left({\Phi}^{4}\right)\right\\}\sqrt{|g|}\operatorname{d}^{4}x\,,$
(13)

where $F^{\mu\nu}$ is the curvature of $B_{\mu}$, and the second term is given
by $\widetilde{D}_{\mu}{\Phi}=\partial_{\mu}\Phi+i[B_{\mu},\Phi]$.

Having into account the three SM generations of fermions and anti-fermions,
the first term in Eq. (13) is given by

$\displaystyle\operatorname{Tr}\left(F^{\mu\nu}F_{\mu\nu}\right)$
$\displaystyle=24\left[\frac{10}{3}\operatorname{Tr}\left(\Lambda_{\mu\nu}\Lambda^{\mu\nu}\right)+\operatorname{Tr}\left(Q_{\mu\nu}Q^{\mu\nu}\right)+\operatorname{Tr}\left(V_{\mu\nu}V^{\mu\nu}\right)\right],$

where $\Lambda_{\mu\nu}$, $Q_{\mu\nu}$ and $V_{\mu\nu}$ are the curvature of
$\Lambda_{\mu}$, $Q_{\mu}$ and $V_{\mu}$, respectively. We introduce the gauge
couplings $g_{1}$, $g_{2}$ and $g_{3}$ by rescaling the gauge fields in terms
of the SM vector fields $Y_{\mu}$, $W_{\mu}$ and $G_{\mu}$ as

$\displaystyle\Lambda_{\mu}\equiv$ $\displaystyle\frac{g_{1}}{2}Y_{\mu}\,,$
$\displaystyle Q^{i}_{\mu}\equiv$ $\displaystyle\frac{g_{2}}{2}W^{i}_{\mu}\,,$
$\displaystyle V^{i}_{\mu}\equiv$ $\displaystyle\frac{g_{3}}{2}G^{a}_{\mu}\,,$

with $i=1,2,3$ and $a=1,2,\ldots,8\,$. The gauge couplings can be properly
normalized so that [40]

$\displaystyle\frac{g_{2}^{2}f(0)}{2\pi^{2}}$ $\displaystyle=\frac{1}{4}\,,$
$\displaystyle g\equiv\frac{5}{3}g_{1}^{2}$
$\displaystyle=g_{2}^{2}=g_{3}^{2}\,,$ (14)

where we have introduced $g$ as an unified gauge coupling constant. Since the
constrain on the gauge couplings, established by Eq. (14), is the same that
holds for grand unified theories, we assume that the cut-off scale, $\Lambda$,
of the spectral action is the grand unification scale.

The last term in Eq. (13) is given by

$\displaystyle\operatorname{Tr}\left({\Phi}^{4}\right)=$ $\displaystyle
4\left[\operatorname{Tr}\left(Y_{e}^{\dagger}Y_{e}\right)^{2}+3\operatorname{Tr}\left(Y_{d}^{\dagger}Y_{d}\right)^{2}\right]\left(\Phi^{\dagger}_{1}\Phi_{1}\right)^{2}+12\operatorname{Tr}\left(Y_{u}^{\dagger}Y_{u}\right)^{2}\left(\Phi^{\dagger}_{2}\Phi_{2}\right)^{2}$
$\displaystyle+12\left[\operatorname{Tr}\left(Y_{u}^{\dagger}Y_{u}Y_{d}^{\dagger}Y_{d}\right)+\operatorname{Tr}\left(Y_{u}Y_{u}^{\dagger}Y_{d}Y_{d}^{\dagger}\right)\right]\left[\left(\Phi^{\dagger}_{1}\Phi_{1}\right)\left(\Phi^{\dagger}_{2}\Phi_{2}\right)-\left(\Phi^{\dagger}_{1}\Phi_{2}\right)\left(\Phi^{\dagger}_{2}\Phi_{1}\right)\right].$
(15)

To get the kinetic terms for the scalar fields, we first calculate the
commutator $[B_{\mu},\Phi]$, from which we obtain

$\displaystyle\frac{f(0)}{8\pi^{2}}\operatorname{Tr}\left[\left(\widetilde{D}_{\mu}\Phi\right)\left(\widetilde{D}^{\mu}\Phi\right)\right]$
$\displaystyle=\frac{\operatorname{Tr}\left(Y_{e}^{\dagger}Y_{e}\right)+3\operatorname{Tr}\left(Y_{d}^{\dagger}Y_{d}\right)}{4g^{2}}\left(D_{\mu}\Phi_{1}\right)^{\dagger}\left(D_{\mu}\Phi_{1}\right)+\frac{3\operatorname{Tr}\left(Y_{u}^{\dagger}Y_{u}\right)}{4g^{2}}\left(D_{\mu}\Phi_{2}\right)^{\dagger}\left(D_{\mu}\Phi_{2}\right),$
(16)

where
$D_{\mu}\Phi_{j}=\partial_{\mu}\Phi_{j}+iQ_{\mu}^{i}\sigma_{i}\Phi_{j}+i\Lambda_{\mu}\Phi_{j}$,
with $j=1,2$, stands for the covariant derivative acting on the Higgs
doublets. To have a proper normalization for this equation, we redefine the
scalar doublets as

$\displaystyle\Phi_{1}$
$\displaystyle\to\frac{2g}{\sqrt{\operatorname{Tr}\left(Y_{e}^{\dagger}Y_{e}\right)+3\operatorname{Tr}\left(Y_{d}^{\dagger}Y_{d}\right)}}\Phi_{1},$
$\displaystyle{\Phi}_{2}$
$\displaystyle\to\frac{2g}{\sqrt{3\operatorname{Tr}\left(Y_{u}^{\dagger}Y_{u}\right)}}{\Phi}_{2}\,,$
(17)

and from now on

$\displaystyle\Phi_{2}=$ $\displaystyle\begin{pmatrix}\phi_{2}^{+}\\\
\phi_{2}^{0}\\\ \end{pmatrix},$ $\displaystyle\Phi_{1}=$
$\displaystyle\begin{pmatrix}\phi_{1}^{+}\\\ \phi_{1}^{0}\end{pmatrix}.$ (18)

In terms of these normalized fields, we can re-write Eq. (3) to get the
quartic scalar terms as follows

$\displaystyle\frac{f(0)}{8\pi^{2}}\operatorname{Tr}({\Phi}^{4})$
$\displaystyle=4g^{2}\frac{\operatorname{Tr}\left(Y_{e}^{\dagger}Y_{e}\right)^{2}+3\operatorname{Tr}\left(Y_{d}^{\dagger}Y_{d}\right)^{2}}{\left[\operatorname{Tr}\left(Y_{e}^{\dagger}Y_{e}\right)+3\operatorname{Tr}\left(Y_{d}^{\dagger}Y_{d}\right)\right]^{2}}\left(\Phi_{1}^{\dagger}\Phi_{1}\right)^{2}+\frac{4}{3}g^{2}\left(\Phi_{2}^{\dagger}\Phi_{2}\right)^{2}$
$\displaystyle+4g^{2}\frac{\operatorname{Tr}\left(Y_{u}^{\dagger}Y_{u}Y_{d}^{\dagger}Y_{d}\right)+\operatorname{Tr}\left(Y_{u}Y_{u}^{\dagger}Y_{d}Y_{d}^{\dagger}\right)}{\operatorname{Tr}\left(Y_{u}^{\dagger}Y_{u}\right)\left[\operatorname{Tr}\left(Y_{e}^{\dagger}Y_{e}\right)+3\operatorname{Tr}\left(Y_{d}^{\dagger}Y_{d}\right)\right]}\left[\left(\Phi_{1}^{\dagger}\Phi_{1}\right)\right(\Phi_{2}^{\dagger}\Phi_{2}\left)-\left(\Phi_{1}^{\dagger}\Phi_{2}\right)\left(\Phi_{1}^{\dagger}\Phi_{2}\right)\right].$

By comparing this expression with the quartic scalar potential terms in Eq
(1), we make the following identification

$\displaystyle{\lambda_{2}(\Lambda)}$ $\displaystyle=\frac{8}{3}g^{2}\,,$
(19a) $\displaystyle\lambda_{1}(\Lambda)$
$\displaystyle=8g^{2}\frac{\operatorname{Tr}\left(Y_{e}^{\dagger}Y_{e}\right)^{2}+3\operatorname{Tr}\left(Y_{d}^{\dagger}Y_{d}\right)^{2}}{\left[\operatorname{Tr}\left(Y_{e}^{\dagger}Y_{e}\right)+3\operatorname{Tr}\left(Y_{d}^{\dagger}Y_{d}\right)\right]^{2}}\,,$
(19b) $\displaystyle\lambda_{3}(\Lambda)$
$\displaystyle=-\lambda_{4}(\Lambda)=4g^{2}\frac{\operatorname{Tr}\left(Y_{u}^{\dagger}Y_{u}Y_{d}^{\dagger}Y_{d}\right)+\operatorname{Tr}\left(Y_{u}Y_{u}^{\dagger}Y_{d}Y_{d}^{\dagger}\right)}{\operatorname{Tr}\left(Y_{u}^{\dagger}Y_{u}\right)\left[\operatorname{Tr}\left(Y_{e}^{\dagger}Y_{e}\right)+3\operatorname{Tr}\left(Y_{d}^{\dagger}Y_{d}\right)\right]}\,,$
(19c)

which, in turn, constrain the values of the quartic scalar couplings at the
unification scale $\Lambda$.

From our normalization in Eq. (17), we find that the (anti-Hermitian) Yukawa
matrix for the quark-up in Eq. (2) is given by

$\displaystyle Y_{u}$
$\displaystyle\equiv-i\frac{\sqrt{3\operatorname{Tr}(Y_{u}^{\dagger}Y_{u})}}{\sqrt{2}gv_{2}}m_{u}\,,$
(20)

where $m_{u}$, is a Hermitian mass matrix.

From now on, we concentrate only on the third SM generation, so that the
indices $u$, $d$, $e$ correspond to top ($t$), bottom ($b$), and tau ($\tau$),
respectively. Therefore, we can rewrite Eqs. (19) as

$\displaystyle\lambda_{1}(\Lambda)$
$\displaystyle=\frac{8(1+3\rho^{2})}{(1+3\rho)^{2}}g^{2},$
$\displaystyle\lambda_{2}(\Lambda)$ $\displaystyle=\frac{8}{3}g^{2},$
$\displaystyle\lambda_{3}(\Lambda)$
$\displaystyle=\frac{8\rho}{1+3\rho}g^{2},$
$\displaystyle\lambda_{4}(\Lambda)$
$\displaystyle=-\frac{8\rho}{1+3\rho}g^{2}\,,$ (21)

where we have introduced the parameter

$\displaystyle\rho=\frac{|y_{b}|^{2}}{|y_{\tau}|^{2}}\,.$ (22)

Analogously, by multiplying Eq. (20) times its own Hermitian conjugate, we
obtain

$\displaystyle y_{t}(\Lambda)=\frac{2g}{\sqrt{3}}\,.$ (23)

Eqs. (21) and (23) are interpreted as the boundary conditions imposed by the
spectral action on the quartic and the Yukawa couplings at the grand
unification scale, respectively. However, the coefficients $\mu_{12}^{2}$ and
$\lambda_{5}(\Lambda)$ are still unconstrained by the spectral action, and we
consider them as the free parameters of the model. Although the terms
$\mu_{1}^{2}\Phi_{1}^{\dagger}\Phi_{1}+\mu_{2}^{2}\Phi_{2}^{\dagger}\Phi_{2}$
are in general present in the spectral action expansion (which could have
cosmological implications [28, 50]), their values are determined after
symmetry breaking by the tadpole equations described in the next section.

##  4 Scalar mass eigenstates

Before going through the calculation of the mass terms of the scalar fields of
the model, we present the additional constraints on the parameters
$\lambda_{i}$ which guarantee the boundedness from below [51, 52] of the
potential in Eq. (1)

$\displaystyle\lambda_{1}$ $\displaystyle>0\,,$ $\displaystyle\lambda_{3}$
$\displaystyle>-\sqrt{\lambda_{1}\lambda_{2}}\,,$ (24a)
$\displaystyle\lambda_{2}$ $\displaystyle>0\,,$
$\displaystyle\lambda_{3}+\lambda_{4}-|\lambda_{5}|$
$\displaystyle>-\sqrt{\lambda_{1}\lambda_{2}}\,.$ (24b)

In the two-Higgs-doublet model, after electroweak symmetry breaking, a total
of eight scalar degrees of freedom appear. Three of them are the massless
Goldstone modes $G^{\pm}$ and $G^{0}$, which are absorbed to get the masses
for SM gauge bosons $W^{\pm}$ and $Z$. The remaining fields are two $CP$-even
(real-uncharged) scalars $h$, and $H$, one $CP$-odd pseudoscalar $A$, and two
charged Higgs bosons $H^{\pm}\,$.

By expanding the two Higgs doublets in Eq. (18) around the VEVs we get

$\displaystyle\Phi_{j}=\begin{pmatrix}\phi_{j}^{+}\\\
\frac{1}{\sqrt{2}}\left(h_{j}+v_{j}+i\eta_{j}\right)\end{pmatrix},\hskip
9.95863pt\text{for}\hskip 7.11317ptj=1,2\,,$

where $\Re(\phi^{0}_{j})=h_{j}+v_{j}$ and $\Im(\phi^{0}_{j})=\eta_{j}$. The
ratio between the two VEVs, $\tan\beta$ in Eq. (2), is defined for
$0\leq\beta<{\pi}/{2}$, with $v=\sqrt{v_{1}^{2}+v_{2}^{2}}=246.2$ GeV. The
minimum conditions ${\partial
V/\partial\Phi_{j}}|_{\langle\Phi_{j}\rangle_{0}}=0$ for the potential give
rise to

$\displaystyle\mu_{1}^{2}$
$\displaystyle=\mu_{12}^{2}\tan{\beta}-\frac{\lambda_{1}}{2}v_{1}^{2}-\frac{\lambda_{345}}{2}v_{2}^{2},$
$\displaystyle\mu_{2}^{2}$
$\displaystyle=\mu_{12}^{2}\cot{\beta}-\frac{\lambda_{2}}{2}v_{2}^{2}-\frac{\lambda_{345}}{2}v_{1}^{2}\,,$

where $\lambda_{345}\equiv\lambda_{3}+\lambda_{4}+\lambda_{5}$.

The physical scalar states are obtained by means of the following rotation
matrices

$\displaystyle\begin{pmatrix}H\\\ h\end{pmatrix}$
$\displaystyle={R_{\alpha}}\begin{pmatrix}h_{1}\\\ h_{2}\end{pmatrix},$
$\displaystyle\begin{pmatrix}G^{+}\\\ H^{+}\end{pmatrix}$
$\displaystyle={R_{\beta}}\begin{pmatrix}\phi_{1}^{+}\\\
\phi_{2}^{+}\end{pmatrix},$ $\displaystyle\begin{pmatrix}G^{0}\\\
A\end{pmatrix}$ $\displaystyle={R_{\beta}}\begin{pmatrix}\eta_{1}\\\
\eta_{2}\end{pmatrix},$

which are given in terms of the mixing angles $\alpha$ and $\beta$

$\displaystyle{R_{\alpha}}$
$\displaystyle=\begin{pmatrix}\cos\alpha&\sin\alpha\\\
-\sin\alpha&\cos\alpha\end{pmatrix},$ $\displaystyle{R_{\beta}}$
$\displaystyle=\begin{pmatrix}\cos\beta&\sin\beta\\\
-\sin\beta&\cos\beta\end{pmatrix}.$

The angle $\alpha$ is given by

$\displaystyle\tan
2\alpha=\frac{2\mathcal{M}^{2}_{12}}{\mathcal{M}^{2}_{11}-\mathcal{M}^{2}_{22}}\,,$
(25)

where $-\pi/2<\alpha<\pi/2$ and $\mathcal{M}_{ij}^{2}$ are the components of
the real uncharged square-mass matrix

$\displaystyle\mathcal{M}^{2}=\begin{pmatrix}\phantom{-}\mu_{12}^{2}\tan{\beta}+\lambda_{1}v_{1}^{2}&-\mu_{12}^{2}+\lambda_{345}v_{1}v_{2}\\\
-\mu_{12}^{2}+\lambda_{345}v_{1}v_{2}&\phantom{-}\mu_{12}^{2}\cot{\beta}+\lambda_{2}v_{2}^{2}\end{pmatrix}.$

The mass expressions for the scalar fields are

$\displaystyle m^{2}_{H,h}$
$\displaystyle=\frac{1}{2}\left[\mathcal{M}_{11}^{2}+\mathcal{M}_{22}^{2}\pm\sqrt{\left(\mathcal{M}_{11}^{2}-\mathcal{M}_{22}^{2}\right)^{2}+4\left(\mathcal{M}_{12}^{2}\right)^{2}}\right],$
(26a) $\displaystyle m^{2}_{H^{\pm}}$
$\displaystyle=\left(\frac{\mu_{12}^{2}}{v_{1}v_{2}}-\frac{\lambda_{4}+\lambda_{5}}{2}\right)v^{2},$
(26b) $\displaystyle m^{2}_{A}$
$\displaystyle=\left(\frac{\mu_{12}^{2}}{v_{1}v_{2}}-\lambda_{5}\right)v^{2}\,.$
(26c)

We identify the $CP$-even state $h$ with the SM-like Higgs boson, whereas the
other is the heavy $CP$-odd real scalar not yet discovered at the LHC. The
main bounds in the scalar mass spectrum in Eqs. (26) arises from direct Higgs
searches and the coupling of the SM-like Higgs boson $h$ to the gauge bosons
$W^{\pm}$ and $Z$. In the 2HDM, the coupling associated with this vertex is
affected by the factor $\sin(\beta-\alpha)$. The SM couplings are recovered in
the alignment limit [29] $\sin(\beta-\alpha)\approx 1$, so that the vector
bosons acquire their masses mainly by interacting with the observed 125 GeV
$CP$-even state. We will make use of these restrictions in the next section,
where we perform a RGEs analysis in order to derive the 2HDM mass spectrum in
NCG.

##  5 Phenomenology

We fix the boundary conditions at $\Lambda=10^{16}\ \text{GeV}$ for the RGEs
of the 2HDM couplings in Eq. (21) and the top Yukawa in Eq. (23), and use the
typical unification gauge coupling value $g=0.53$.

We start by evolving the RGEs for [53]

$\displaystyle 16\pi^{2}\beta_{y_{b}}$
$\displaystyle=y_{b}\left(-8g_{3}^{2}-\frac{9}{4}g_{2}^{2}-\frac{5}{12}g_{1}^{2}+\frac{9}{2}y_{b}^{2}+\frac{1}{2}y_{t}^{2}+y_{\tau}^{2}\right),$
$\displaystyle 16\pi^{2}\beta_{y_{\tau}}$
$\displaystyle=y_{\tau}\left(-\frac{9}{4}g_{2}^{2}-\frac{15}{4}g_{1}^{2}+\frac{5}{2}y_{\tau}^{2}+3y_{b}^{2}\right),$

from the electroweak scale, $m_{Z}$

$\displaystyle y_{b}(m_{Z})$
$\displaystyle=\frac{2.85\sqrt{2}}{v\cos\beta}\,,$ $\displaystyle
y_{\tau}(m_{Z})$ $\displaystyle=\frac{1.78\sqrt{2}}{v\cos\beta}\,,$ (27)

to $\Lambda$. Then, we fix $y_{t}(\Lambda)$ according to Eq. (23) and we
obtain $\tan\beta$ after get the proper value of $y_{t}(m_{Z})$ by using [53]

$\displaystyle 16\pi^{2}\beta_{y_{t}}$
$\displaystyle=y_{t}\left(-8g_{3}^{2}-\frac{9}{4}g_{2}^{2}-\frac{17}{12}g_{1}^{2}+\frac{9}{2}y_{t}^{2}+\frac{1}{2}y_{b}^{2}\right).$
(28)

The direct measurements on top quark mass $172.9\pm 0.4$ GeV [54] at $3\sigma$
restrict $2.0\lesssim\tan\beta\lesssim 2.2$.

In the same way, we vary the free parameter $\lambda_{5}(\Lambda)$ and use the
boundary conditions, $\lambda_{i}(\Lambda)$, in Eq. (21) to obtain the quartic
scalar couplings of the scalar potential at $m_{Z}$ from (see appendix A of
both refs. [53, 52])

$\displaystyle 16\pi^{2}\beta_{\lambda_{1}}$
$\displaystyle=12\lambda_{1}^{2}+4\lambda_{3}^{2}+4\lambda_{3}\lambda_{4}+2\lambda_{4}^{2}+2\lambda_{5}^{2}$
$\displaystyle+\frac{3}{4}g_{1}^{4}+\frac{3}{2}g_{1}^{2}g_{2}^{2}+\frac{9}{4}g_{2}^{4}-3\left(g_{1}^{2}+3g_{2}^{2}\right)\lambda_{1}\,,$
(29a) $\displaystyle 16\pi^{2}\beta_{\lambda_{2}}$
$\displaystyle=12\lambda_{2}^{2}+4\lambda_{3}^{2}+4\lambda_{3}\lambda_{4}+2\lambda_{4}^{2}+2\lambda_{5}^{2}$
$\displaystyle+\frac{3}{4}g_{1}^{4}+\frac{3}{2}g_{1}^{2}g_{2}^{2}+\frac{9}{4}g_{2}^{4}-3\left(g_{1}^{2}+3g_{2}^{2}-4y_{t}^{2}\right)\lambda_{2}-12y_{t}^{4}\,,$
(29b) $\displaystyle 16\pi^{2}\beta_{\lambda_{3}}$
$\displaystyle=\left(6\lambda_{3}+2\lambda_{4}\right)\left(\lambda_{1}+\lambda_{2}\right)+4\lambda_{3}^{2}+2\lambda_{4}^{2}+2\lambda_{5}^{2}$
$\displaystyle+\frac{3}{4}g_{1}^{4}-\frac{3}{2}g_{1}^{2}g_{2}^{2}+\frac{9}{4}g_{2}^{4}-3\left(g_{1}^{2}+3g_{2}^{2}-2y_{t}^{2}\right)\lambda_{3}\,,$
(29c) $\displaystyle 16\pi^{2}\beta_{\lambda_{4}}$
$\displaystyle=2\lambda_{1}\lambda_{4}+2\lambda_{2}\lambda_{4}+8\lambda_{3}\lambda_{4}+4\lambda_{4}^{2}+8\lambda_{5}^{2}$
$\displaystyle+3g_{1}^{2}g_{2}^{2}-3\left(g_{1}^{2}+3g_{2}^{2}-2y_{t}^{2}\right)\lambda_{4}\,,$
(29d) $\displaystyle 16\pi^{2}\beta_{\lambda_{5}}$
$\displaystyle=\left(2\lambda_{1}+2\lambda_{2}+8\lambda_{3}+12\lambda_{4}\right)\lambda_{5}-3\left(g_{1}^{2}+3g_{2}^{2}-2y_{t}^{2}\right)\lambda_{5}\,.$
(29e)

At the electroweak scale, we fix the other free parameter $\mu_{12}^{2}$.

To determine the scalar mass spectrum of this model, we scan the parameter
space according to the ranges shown in Table 2. We have chosen
$|\lambda_{5}(\Lambda)|<0.5$ to preserve the perturbativity of the potential,
and $\mu_{12}^{2}>10^{4}$ GeV2 to get rid of taquionic masses and to have a
sufficiently charged Higgs mass compatible with LEP constraints [55, 56, 57].

Parameter | Range
---|---
$\operatorname{sgn}(\lambda_{5}(\Lambda))$ | $\pm 1$
$|\lambda_{5}(\Lambda)|$ | $(10^{-3},0.5)$
$\mu_{12}^{2}$ | $(10^{4}\ \text{GeV}^{2},10^{6}\ \text{GeV}^{2})$
$\tan\beta$ | $(2.0,2.2)$
Table 2: Scan range of the free parameters. We fix $\Lambda=10^{16}$ GeV and
$g=0.53$ .

The explored parameter space for $\lambda_{5}(\Lambda)$ and $\mu_{12}^{2}$ is
displayed as the cyan region in Fig. 1. After impose the theoretical
constraints in Eqs. (24) in order to keep the boundedness from below of the
potential, we get the green band in Fig. 1. We use the implemented type-II
2HDM in SARAH [58] to obtain the input file and the customized code SPheno
[59, 60], from which we can obtain the masses and mixing for each set of input
parameters. The points compatible with a SM-like Higgs boson mass
($m_{h}=125.10\pm 0.14$ GeV [54]) at $3\sigma$, are shown as the yellow and
dark-blue region in Fig. 1. For each parameter point in those regions, we use
the SPheno interface to HiggsBounds-5 [61], to test the neutral and charged
Higgs sectors against the current exclusion bounds from the Higgs searches at
the LEP, Tevatron and LHC experiments. With this tool, we determine that the
parameter points in the dark-blue region are excluded at 95% C.L. In this way,
the remaining yellow region satisfy all the constraints, including the proper
mass for the SM-like Higgs.

Figure 1:  Scan of input parameters $\lambda_{5}(\Lambda)$ versus
$\mu_{12}^{2}$. The green area corresponds to a potential which is bounded
from below. The points within the yellow and dark-blue region satisfy the SM-
like Higgs mass at $3\sigma$, whereas the dark blue region is excluded at 95%
C.L.

The parameter space in Fig. 1 where the $m_{h}$ eigenvalue is less than $180\
\text{GeV}$ is shown in Fig. 2, where the dark-blue and yellow region
corresponds to the very same points as in Fig. 1, such that the yellow region
can be identified with the SM-like Higgs compatible with all the current
collider constraints888Our general scan includes another region with
$m_{h}>185\ \text{GeV}$ and $\sin(\beta-\alpha)<0.6$ which is excluded by the
LHC.. In our approach, it is no possible to recover exactly the SM Higgs
scalar potential since in our scan at least one of the couplings of the 2HDM
at $m_{Z}$ is different from zero. However, we can see that the larger $m_{h}$
masses around $160\ \text{GeV}$ require $\sin(\beta-\alpha)=1$, corresponding
to the alignment well inside the decoupling limit [32, 62], and hence are in
the limit of the noncommutative SM Higgs scalar potential with a too heavy
mass. In this way, we recover the rule-out prediction for $m_{h}$ of the SM in
terms of NCG [19]. Conversely, we need to stay well inside the region of 2HDM
to have a consistent SM-like Higgs boson mass. In particular, we can achieve
the alignment limit without decoupling [32, 62, 29, 63, 64] as illustrated
below.

Figure 2: Scan of $m_{h}$ versus $\sin(\beta-\alpha)$. The 3$\sigma$ region
for the SM-like Higgs mass corresponds to the yellow and dark blue region,
while the yellow region satisfy all the current constraints as in Fig. 1.

In Fig. 3, we observe the behavior of the charged Higgs boson mass as a
function of the coupling $\sin(\beta-\alpha)$, with $\mu_{12}^{2}$ in the
color code. In this case, we only consider the parameter space that is totally
consistent with the HiggsBounds constrains corresponding to the yellow region
in Figs. 1 and 2. We see also that the non-decoupling region, $350$ GeV
$\lesssim m_{H^{\pm}}\lesssim 600$ GeV, is achieved for $\mu_{12}^{2}\lesssim
10^{5}$ GeV2.

Figure 3: $m_{H}^{\pm}$ vs $\sin(\beta-\alpha)$ with the color code indicating
the value of $\log(\mu_{12}^{2}/\text{GeV}^{2})$. We only consider the
parameter space that is fully consistent with collider searches.

We can see that the noncommutative 2HDM is compatible with the scalar
discovered at the LHC with a value of $\sin(\beta-\alpha)\gtrsim 0.99$, which
is in accordance with the alignment limit including a spectrum of non-
decoupled extra scalars.

##  6 Conclusions

The noncommutative geometry approach to the standard model display a 2HDM
structure when the second-order axiom is used rather than the massless photon
condition. By working with the type-II 2HDM as a simple realization, we are
able to recover the SM in the decoupling limit with the rule-out too heavy SM
Higgs mass. By varying the only remaining two free parameters of the 2HDM
scalar potential with $CP$ conservation: $\lambda_{5}$ at the grand
unification scale $\Lambda$, and the softly broken type-II $\mathbb{Z}_{2}$
term, $\mu_{12}^{2}$, we are able to obtain regions with a SM-like Higgs in
the alignment limit which is not excluded by collider constraints. There, the
extra scalar masses can be sufficiently low to be within reach of future LHC
runs. Other 2HDM realizations without tree-level FCNC, could also be
implemented in this framework.

## Acknowledgments

The work of DR is supported by Sostenibilidad UdeA, the UdeA/CODI Grants
2017-16286 and 2020-33177, and FAPESP funding Grant 2021/11383-4. FJ is
supported by COLCIENCIAS (Doctorado Nacional-757) and would like to thank
Shane Farnsworth for useful discussions. AR is supported by COLCIENCIAS
through the ESTANCIAS POSTDOCTORALES program.

## References

  * [1] Georges Aad _et al._ (ATLAS), “Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC,” Phys. Lett. B 716, 1–29 (2012), arXiv:1207.7214 [hep-ex]
  * [2] Serguei Chatrchyan _et al._ (CMS), “Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC,” Phys. Lett. B 716, 30–61 (2012), arXiv:1207.7235 [hep-ex]
  * [3] G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, Marc Sher, and Joao P. Silva, “Theory and phenomenology of two-Higgs-doublet models,” Phys. Rept. 516, 1–102 (2012), arXiv:1106.0034 [hep-ph]
  * [4] David Eriksson, Johan Rathsman, and Oscar Stal, “2HDMC: Two-Higgs-Doublet Model Calculator Physics and Manual,” Comput. Phys. Commun. 181, 189–205 (2010), arXiv:0902.0851 [hep-ph]
  * [5] Mark Trodden, “Electroweak baryogenesis: A Brief review,” in  _33rd Rencontres de Moriond: Electroweak Interactions and Unified Theories_ (1998) pp. 471–480, arXiv:hep-ph/9805252
  * [6] N. Aghanim _et al._ (Planck), “Planck 2018 results. VI. Cosmological parameters,” Astron. Astrophys. 641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO]
  * [7] Daniel A. Camargo, Miguel D. Campos, Tessio B. de Melo, and Farinaldo S. Queiroz, “A Two Higgs Doublet Model for Dark Matter and Neutrino Masses,” Phys. Lett. B 795, 319–326 (2019), arXiv:1901.05476 [hep-ph]
  * [8] Ernest Ma, “Linkage of Dirac Neutrinos to Dark U(1) Gauge Symmetry,” Phys. Lett. B 817, 136290 (2021), arXiv:2101.12138 [hep-ph]
  * [9] Georges Aad _et al._ (ATLAS), “Higgs boson production cross-section measurements and their EFT interpretation in the $4\ell$ decay channel at $\sqrt{s}=$13 TeV with the ATLAS detector,” Eur. Phys. J. C 80, 957 (2020), [Erratum: Eur.Phys.J.C 81, 29 (2021), Erratum: Eur.Phys.J.C 81, 398 (2021)], arXiv:2004.03447 [hep-ex]
  * [10] Albert M Sirunyan _et al._ (CMS), “Measurements of production cross sections of the Higgs boson in the four-lepton final state in proton–proton collisions at $\sqrt{s}=13\,\text{Te}\text{V}$,” Eur. Phys. J. C 81, 488 (2021), arXiv:2103.04956 [hep-ex]
  * [11] Johannes Haller, Andreas Hoecker, Roman Kogler, Klaus Mönig, Thomas Peiffer, and Jörg Stelzer, “Update of the global electroweak fit and constraints on two-Higgs-doublet models,” Eur. Phys. J. C 78, 675 (2018), arXiv:1803.01853 [hep-ph]
  * [12] Alain Connes and John Lott, “Particle Models and Noncommutative Geometry (Expanded Version),” Nucl. Phys. B Proc. Suppl. 18, 29–47 (1991)
  * [13] Alain Connes,  _Noncommutative geometry_ (1994) ISBN 978-0-12-185860-5
  * [14] A. Connes, “Noncommutative geometry and reality,” J. Math. Phys. 36, 6194–6231 (1995)
  * [15] Thomas Krajewski and Igor Pris, “Towards a Z-prime gauge boson in noncommutative geometry,” Lett. Math. Phys. 39, 187–202 (1997), arXiv:hep-th/9607005
  * [16] Christoph A. Stephan, “Almost-commutative geometries beyond the standard model. II. New Colours,” J. Phys. A 40, 9941 (2007), arXiv:0706.0595 [hep-th]
  * [17] Romain Squellari and Christoph A. Stephan, “Almost-Commutative Geometries Beyond the Standard Model. III. Vector Doublets,” J. Phys. A 40, 10685–10698 (2007), arXiv:0706.3112 [hep-th]
  * [18] Ali H. Chamseddine and Alain Connes, “The Spectral action principle,” Commun. Math. Phys. 186, 731–750 (1997), arXiv:hep-th/9606001
  * [19] Ali H. Chamseddine and Alain Connes, “Noncommutative Geometry as a Framework for Unification of all Fundamental Interactions including Gravity. Part I,” Fortsch. Phys. 58, 553–600 (2010), arXiv:1004.0464 [hep-th]
  * [20] Ali H. Chamseddine and Alain Connes, “Resilience of the Spectral Standard Model,” JHEP 09, 104 (2012), arXiv:1208.1030 [hep-ph]
  * [21] Hosein Karimi, “Implications of the complex Singlet field for Noncommutative Geometry model,” JHEP 12, 040 (2017), arXiv:1705.01605 [hep-ph]
  * [22] Agostino Devastato, Fedele Lizzi, and Pierre Martinetti, “Grand Symmetry, Spectral Action, and the Higgs mass,” JHEP 01, 042 (2014), arXiv:1304.0415 [hep-th]
  * [23] Agostino Devastato, Fedele Lizzi, and Pierre Martinetti, “Higgs mass in Noncommutative Geometry,” Fortsch. Phys. 62, 863–868 (2014), arXiv:1403.7567 [hep-th]
  * [24] Ali H. Chamseddine, Alain Connes, and Walter D. van Suijlekom, “Beyond the Spectral Standard Model: Emergence of Pati-Salam Unification,” JHEP 11, 132 (2013), arXiv:1304.8050 [hep-th]
  * [25] Ali H. Chamseddine, Alain Connes, and Walter D. van Suijlekom, “Inner Fluctuations in Noncommutative Geometry without the first order condition,” J. Geom. Phys. 73, 222–234 (2013), arXiv:1304.7583 [math-ph]
  * [26] Shane Farnsworth and Latham Boyle, “Rethinking Connes’ approach to the standard model of particle physics via non-commutative geometry,” New J. Phys. 17, 023021 (2015), arXiv:1408.5367 [hep-th]
  * [27] Latham Boyle and Shane Farnsworth, “Non-Commutative Geometry, Non-Associative Geometry and the Standard Model of Particle Physics,” New J. Phys. 16, 123027 (2014), arXiv:1401.5083 [hep-th]
  * [28] Ali H. Chamseddine, Alain Connes, and Matilde Marcolli, “Gravity and the standard model with neutrino mixing,” Adv. Theor. Math. Phys. 11, 991–1089 (2007), arXiv:hep-th/0610241
  * [29] Jérémy Bernon, John F. Gunion, Howard E. Haber, Yun Jiang, and Sabine Kraml, “Scrutinizing the alignment limit in two-Higgs-doublet models: mh=125 GeV,” Phys. Rev. D 92, 075004 (2015), arXiv:1507.00933 [hep-ph]
  * [30] I. F. Ginzburg and I. P. Ivanov, “Tree level unitarity constraints in the 2HDM with CP violation,” (12 2003), arXiv:hep-ph/0312374
  * [31] Ilya F. Ginzburg and Maria Krawczyk, “Symmetries of two Higgs doublet model and CP violation,” Phys. Rev. D 72, 115013 (2005), arXiv:hep-ph/0408011
  * [32] John F. Gunion and Howard E. Haber, “The CP conserving two Higgs doublet model: The Approach to the decoupling limit,” Phys. Rev. D 67, 075019 (2003), arXiv:hep-ph/0207010
  * [33] Giovanni Landi,  _An Introduction to noncommutative spaces and their geometry_, Vol. 51 (1997) arXiv:hep-th/9701078
  * [34] Jose M. Gracia-Bondia, J. C. Varilly, and H. Figueroa,  _Elements of noncommutative geometry_ (2001)
  * [35] Koen van den Dungen and Walter D. van Suijlekom, “Particle Physics from Almost Commutative Spacetimes,” Rev. Math. Phys. 24, 1230004 (2012), arXiv:1204.0328 [hep-th]
  * [36] Shane Farnsworth,  _Standard Model Physics and Beyond from Non-Commutative Geometry_ , Ph.D. thesis, Waterloo U. (2015)
  * [37] Bruno Iochum, Thomas Krajewski, and Pierre Martinetti, “Distances in finite spaces from noncommutative geometry,” J. Geom. Phys. 37, 100–125 (2001), arXiv:hep-th/9912217
  * [38] Alain Connes, “Gravity coupled with matter and foundation of noncommutative geometry,” Commun. Math. Phys. 182, 155–176 (1996), arXiv:hep-th/9603053
  * [39] Shane Farnsworth, “The geometry of physical observables,” J. Math. Phys. 61, 101702 (2020), arXiv:2003.01708 [hep-th]
  * [40] Walter D. van Suijlekom,  _Noncommutative geometry and particle physics_, Mathematical Physics Studies (Springer, Dordrecht, 2015) ISBN 978-94-017-9161-8, 978-94-017-9162-5
  * [41] Michael Atiyah, “K-theory past and present,” (2000), arXiv:math/0012213 [math.KT]
  * [42] F. Lizzi, G. Mangano, G. Miele, and G. Sparano, “Fermion Hilbert space and fermion doubling in the noncommutative geometry approach to gauge theories,” Phys. Rev. D 55, 6357–6366 (1997), arXiv:hep-th/9610035
  * [43] Alain Connes, “Noncommutative geometry and the standard model with neutrino mixing,” JHEP 11, 081 (2006), arXiv:hep-th/0608226
  * [44] John W. Barrett, “A Lorentzian version of the non-commutative geometry of the standard model of particle physics,” J. Math. Phys. 48, 012303 (2007), arXiv:hep-th/0608221
  * [45] Lawrence J. Hall and Mark B. Wise, “FLAVOR CHANGING HIGGS - BOSON COUPLINGS,” Nucl. Phys. B 187, 397–408 (1981)
  * [46] Oliver Atkinson, Matthew Black, Alexander Lenz, Aleksey Rusov, and James Wynne, “Cornering the Two Higgs Doublet Model Type II,” (7 2021), arXiv:2107.05650 [hep-ph]
  * [47] D. V. Vassilevich, “Heat kernel expansion: User’s manual,” Phys. Rept. 388, 279–360 (2003), arXiv:hep-th/0306138
  * [48] Bruno Iochum, Cyril Levy, and Dmitri Vassilevich, “Global and local aspects of spectral actions,” J. Phys. A 45, 374020 (2012), arXiv:1201.6637 [math-ph]
  * [49] Wentao Fan, Farzad Fathizadeh, and Matilde Marcolli, “Spectral Action for Bianchi Type-IX Cosmological Models,” JHEP 10, 085 (2015), arXiv:1506.06779 [hep-th]
  * [50] Christoph A. Stephan, “New Scalar Fields in Noncommutative Geometry,” Phys. Rev. D 79, 065013 (2009), arXiv:0901.4676 [hep-th]
  * [51] I. P. Ivanov, “Minkowski space structure of the Higgs potential in 2HDM,” Phys. Rev. D 75, 035001 (2007), [Erratum: Phys.Rev.D 76, 039902 (2007)], arXiv:hep-ph/0609018
  * [52] P. M. Ferreira and D. R. T. Jones, “Bounds on scalar masses in two Higgs doublet models,” JHEP 08, 069 (2009), arXiv:0903.2856 [hep-ph]
  * [53] Phillipp Basler, Pedro M. Ferreira, Margarete Mühlleitner, and Rui Santos, “High scale impact in alignment and decoupling in two-Higgs doublet models,” Phys. Rev. D 97, 095024 (2018), arXiv:1710.10410 [hep-ph]
  * [54] M. Tanabashi _et al._ (Particle Data Group), “Review of Particle Physics,” Phys. Rev. D 98, 030001 (2018)
  * [55] Felix Kling, Honglei Li, Adarsh Pyarelal, Huayang Song, and Shufang Su, “Exotic Higgs Decays in Type-II 2HDMs at the LHC and Future 100 TeV Hadron Colliders,” JHEP 06, 031 (2019), arXiv:1812.01633 [hep-ph]
  * [56] Shuailong Li, Huayang Song, and Shufang Su, “Probing Exotic Charged Higgs Decays in the Type-II 2HDM through Top Rich Signal at a Future 100 TeV pp Collider,” JHEP 11, 105 (2020), arXiv:2005.00576 [hep-ph]
  * [57] Tao Han, Tong Li, Shufang Su, and Lian-Tao Wang, “Non-Decoupling MSSM Higgs Sector and Light Superpartners,” JHEP 11, 053 (2013), arXiv:1306.3229 [hep-ph]
  * [58] F. Staub, “SARAH,” (6 2008), arXiv:0806.0538 [hep-ph]
  * [59] Werner Porod, “SPheno, a program for calculating supersymmetric spectra, SUSY particle decays and SUSY particle production at e+ e- colliders,” Comput. Phys. Commun. 153, 275–315 (2003), arXiv:hep-ph/0301101
  * [60] W. Porod and F. Staub, “SPheno 3.1: Extensions including flavour, CP-phases and models beyond the MSSM,” Comput. Phys. Commun. 183, 2458–2469 (2012), arXiv:1104.1573 [hep-ph]
  * [61] Philip Bechtle, Daniel Dercks, Sven Heinemeyer, Tobias Klingl, Tim Stefaniak, Georg Weiglein, and Jonas Wittbrodt, “HiggsBounds-5: Testing Higgs Sectors in the LHC 13 TeV Era,” Eur. Phys. J. C 80, 1211 (2020), arXiv:2006.06007 [hep-ph]
  * [62] B. Grzadkowski, O. M. Ogreid, and P. Osland, “Measuring CP violation in Two-Higgs-Doublet models in light of the LHC Higgs data,” JHEP 11, 084 (2014), arXiv:1409.7265 [hep-ph]
  * [63] Siddhartha Karmakar and Subhendu Rakshit, “Alignment Limit in 2HDM: Robustness put to test,” JHEP 09, 142 (2018), arXiv:1802.03366 [hep-ph]
  * [64] Bohdan Grzadkowski, Howard E. Haber, Odd Magne Ogreid, and Per Osland, “Heavy Higgs boson decays in the alignment limit of the 2HDM,” JHEP 12, 056 (2018), arXiv:1808.01472 [hep-ph]