The Flipped SU(5) × U(1) Model from Four-Dimensional Strings

Abstract

We review the construction of a flipped $SU\left(5\right)\times U\left(1\right)$ model in the context of the Free-Fermionic Formulation of four-dimensional strings and present its main phenomenological consequences in particle physics and cosmology.

1. Introduction

During the so-called first string revolution in the mid-1980s, string theory was shown to be a consistent theory of quantum gravity [1,2], at least in the sense of being ultraviolet finite in perturbation theory. Moreover, with the discovery of the heterotic string, it was realised that string theory also provides a framework of unification of all fundamental interactions that could eventually lead to a ‘theory of Nature’. Such a fundamental theory should be able to describe all physical phenomena from very short distances in the microcosmos to very large distances in our universe. Although a fully satisfactory explicit example is still lacking, several important results and milestones have been accomplished over the years, and the present review describes a promising avenue towards this goal.

Initially, string model building began with the aim of unveiling a unified theory of particle physics, describing the fundamental constituents of matter and their binding forces, extending the Standard Model of electroweak and strong interactions to high energies. More precisely, there were two main approaches to string model building: one was based on compactifying the ten-dimensional theory on a Calabi–Yau three-fold, leaving one unbroken supersymmetry [3] and another involved in directly constructing four-dimensional (4D) superstrings [4]. Using the second approach, the compactification space is described by a two-dimensional (super)conformal field theory with the appropriate central charge; 22 in the bosonic string case and 9 in the superstring case. A particularly attractive and calculable framework consists of the Free-Fermionic Formulation (FFF), where the internal degrees of freedom are described by two-dimensional fermions. The arbitrariness of the compactification is then encoded in the choice of the boundary conditions and corresponding coefficient-phases associated to projections in the Hilbert space. They both correspond to boundary conditions along the two cycles of the world-sheet torus defining the (one-loop) partition function. They are subject to consistency constraints stemming from one-loop modular invariance and two-loop factorization, or, alternatively, particle interpretation. Their solution leads to a systematic set of rules for constructing consistent four-dimensional (super)string vacua [5,6,7,8].

Another interesting property of the Free-Fermionic Formulation is that all string compactification moduli, except the dilaton, are fixed at the self-dual point of duality transformations, where extra symmetries (either gauge or discrete) emerge. These points are, in general, extrema of the scalar potential induced by non-perturbative or other effects.

The FFF of 4D heterotic superstring was shown to be an ideal framework for implementing (supersymmetric) Grand Unification (GUT) [9]. Indeed, all non-abelian gauge couplings are equal at the string tree level, while the maximum rank of the gauge group is 22, which is sufficiently large to accommodate the Standard Model or its GUT extensions. Moreover, massless gauge representations are limited to the fundamental or the two-index antisymmetric, in agreement with the observed matter content of the Standard Model. On the other hand, there are no massless adjoints, which are usually required in GUTs to break the unified group down to the Standard Model. It turns out that the flipped $SU\left(5\right)\times U\left(1\right)$ is a minimal variation of the standard $SU\left(5\right)$ to address this problem [10,11,12]. This involves exchanging the up and down anti-quarks between the $\mathbf{10}$ and $\overline{\mathbf{5}}$ representations of $SU\left(5\right)$, as well as swapping the positron with the right-handed neutrino in $\mathbf{10}$ and leaving the positron as an outlying singlet. The gauge group is extended by an extra $U\left(1\right)$ to ensure the correct embedding of the hypercharge. Actually, the flipped $SU\left(5\right)\times U\left(1\right)$ is a maximal subgroup of $SO\left(10\right)$ and does not require adjoint Higgs fields to achieve symmetry breaking, but it relies on a pair of ${\mathbf{10}}_H+{\overline{\mathbf{10}}}_{\overline{H}}$ instead. The required representations are thus compatible with the massless spectrum restrictions of 4D strings.

Besides determining the spectrum and ensuring automatic gauge coupling unification, 4D heterotic string models provide a mechanism for obtaining the observed pattern of fermion masses and Yukawa couplings realizing the Froggatt–Nielsen proposal [13]: one fermion generation (the heaviest) obtains masses via tree-level Yukawa couplings using two Higgs doublets, while the masses of the other two generations and their mixings are generated by vacuum expectation values (VEVs) of gauge singlets whose couplings are constrained by appropriate gauge and discrete symmetries. This mechanism relies on the presence of a universal anomalous $U\left(1\right)$ symmetry, whose anomaly cancellation dictates, via the D-term flatness conditions, non-vanishing VEVs for a set of singlets [14,15]. The scale of these VEVs is determined by the one-loop anomaly and is therefore suppressed by a loop factor compared to the string scale. Note also that the tree-level top quark Yukawa coupling is equal to the gauge coupling (both being determined by the 4D string coupling), leading to a prediction of the top quark mass being near the infrared fixed point of $m_t/m_W$, in agreement with experimental data [16].

Indeed, a supersymmetric flipped $SU\left(5\right)\times U\left(1\right)$ model sharing all the properties mentioned above was constructed in the early days of string phenomenology using the Free-Fermionic Formulation of 4D heterotic superstrings [16,17,18]. A few decades later, it was shown that this model also features an inflaton sector successfully describing early cosmology before the gauge symmetry breaking that occurs during a first-order phase transition [19]. The inflaton sector contains two singlet superfields, the inflaton and the goldstino, yielding a cosmology very similar to a supersymmetric extension of the Starobinsky $R^2$ model [20,21,22].

This review is dedicated to our friend and collaborator Dimitri Nanopoulos and is submitted to the Themed Issue in his honour of the journal Universe. The outline is the following. Section 2 presents a brief overview of the FFF of four-dimensional heterotic strings. Section 3 describes the main properties of the supersymmetric flipped $SU\left(5\right)\times U\left(1\right)$ field theory model. Section 4 presents the string construction, massless spectrum, and tree-level superpotential. In Section 5, we discuss the inflaton sector and its properties, while in Section 6, we describe briefly the particle physics phenomenology and some concluding remarks.

2. The Free-Fermionic Formulation of the 4-Dimensional Heterotic Strings

The Free-Fermionic Formulation was developed in the late 1980s as a method for string theory compactification, which allows for the construction of string models in any number of dimensions $d\le 10$ [6,7,8]. In this formalism, all bosonic degrees of freedom associated with the internal space are replaced by fermionic fields, utilizing the fermion–boson equivalence in two dimensions. The FFF is particularly efficient for constructing heterotic string vacua in four dimensions and analyzing their phenomenological features [23].

Within the context of FFF, a string model is specified by a set of basis vectors $B=\{v_1,\dots ,v_N\}$ and a set of phases $c\left[\begin{array}{c}{v}_i\\ v_j\end{array}\right]$, $i,j=1,\dots N$, where N is an integer. The basis vectors describe the phases acquired by the fermions as they wrap around the two non-contractible cycles of the world-sheet torus, which are referred to as the spin structures. The phases $c\left[\begin{array}{c}{v}_i\\ v_j\end{array}\right]$, known also as spin structure coefficients, encompass information related to generalised Gliozzi–Scherk–Olive (GSO) projections [24].

For the purposes of our analysis, we will focus on the heterotic string vacua in four space–time dimensions. Following the commonly used conventions, we assume that the left-moving sector bears $N=1$ world-sheet supersymmetry. The left-moving Ramond–Neveu–Schwarz (RNS) fermions encompass 20 real fields. These are denoted as ${\psi }^{\mu }$, $x^I,y^I,z^I$, $I=1,\dots ,6$, where the first stands for the two space–time fermions in the light cone gauge, while the others refer to the six fermionic internal coordinates $x^I$ and their fermionized partners $y^I,z^I$. The right-moving (RNS) fermions comprise 12 real fermions ${\overline{y}}^I,{\overline{z}}^I$, $I=1,\dots ,6$ pertaining to the fermionized internal bosonic coordinates, and 16 complex fermions, expressed as ${\overline{\psi }}^1,\dots ,{\overline{\psi }}^5$, ${\overline{\eta }}^1,\dots ,{\overline{\eta }}^3$, ${\overline{\varphi }}^1,\dots ,{\overline{\varphi }}^8$. In this parametrization, every basis vector $v\in B$ has 44 components:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill v=\left\{a\left({\psi }^{\mu }\right),a\left(x^{12}\right),a\left(x^{34}\right),a\left(x^{56}\right),a\left(y^1\right),\dots ,a\left(y^6\right),a\left(z^1\right),\dots ,a\left(z^6\right)|a\left({\overline{y}}^1\right),\dots ,a\left({\overline{y}}^6\right),\right.\\ \hfill \left.a\left({\overline{z}}^1\right),\dots ,a\left({\overline{z}}^6\right),a\left({\overline{\psi }}^1\right),\dots ,a\left({\overline{\psi }}^5\right),a\left({\overline{\eta }}^1\right),\dots ,a\left({\overline{\eta }}^3\right),a\left({\overline{\varphi }}^1\right),\dots ,a\left({\overline{\varphi }}^8\right)\right\}=\left\{v_L|v_R\right\},\end{array}\end{array}$$

(1)

where we treat ${\psi }^{\mu }$ and $x^{12},x^{34},x^{56}$ pairs as complex fermions and the vertical bar is used to separate the subset of left-moving fermions $v_L$ from the subset of right-moving fermions $v_R$. The phase $a\left(f\right)\in \left(-1,1\right]$ stands for a fractional number, describing the parallel transportation properties of the fermion f:

$$\begin{array}{c}\hfill f\to -e^{i\pi a\left(f\right)}f,\end{array}$$

(2)

where $a=1$ corresponds to periodic fermions (R) and $a=0$ to antiperiodic fermions (NS). The world-sheet supersymmetry in the left-moving sector is non-linearly realized among the fermions $x^I,y^I,x^I,I=1,\dots ,6$ [5]. The most general implementation of this scenario involves a world-sheet supercharge that is trilinear in these fermions, with coefficients determined by the structure constants of a semi-simple Lie group G of dimension 18. The discussion here is restricted to the simplest possibility, $G={SU\left(2\right)}^6$, which has been extensively explored in the literature. This corresponds to a supercurrent of the form

$$\begin{array}{c}\hfill T_F={\psi }^{\mu }\partial X_{\mu }+\sum _{I=1}^6{x}^I{y}^I{z}^I.\end{array}$$

(3)

The basis vectors in $B=\{v_1,\dots ,v_N\}$ generate the spin structure finite group:

$$\begin{array}{c}\hfill \Xi =\left\{m_1{v}_1+\dots +m_N{v}_N\right\},m_i=0,1,\dots ,N_i\end{array}$$

(4)

with $N_i$ being the smallest integer, such as $N_i{v}_i=0$. The partition function of a string model defined by a basis B and projection coefficients $c\left[\begin{array}{c}{v}_i\\ v_j\end{array}\right]$ can be written as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill Z={\displaystyle \frac{1}{2^N}}{\int }_{\mathcal{F}}{\displaystyle \frac{d^2\tau }{{\tau }_2^3}}{\displaystyle \frac{1}{{\eta }^{12}{\overline{\eta }}^{24}}}\sum _{\alpha ,\beta \in \Xi }c\left[\begin{array}{c}\alpha \\ \beta \end{array}\right]& \prod _{f\in f_L\mathrm{real}}{\Theta }^{{\displaystyle \frac{1}{2}}}\left[\begin{array}{c}\alpha \left(f\right)\\ \beta \left(f\right)\end{array}\right]\prod _{f\in f_L\mathrm{complex}}\Theta \left[{\displaystyle \genfrac{}{}{0pt}{}{\alpha \left(f\right)}{\beta \left(f\right)}}\right]\hfill \\ \hfill & \prod _{f\in f_R\mathrm{real}}{\overline{\Theta }}^{{\displaystyle \frac{1}{2}}}\left[\begin{array}{c}\alpha \left(f\right)\\ \beta \left(f\right)\end{array}\right]\prod _{f\in f_R\mathrm{complex}}\overline{\Theta }\left[{\displaystyle \genfrac{}{}{0pt}{}{\alpha \left(f\right)}{\beta \left(f\right)}}\right],\hfill \end{array}\end{array}$$

(5)

where $\mathcal{F}$ is the fundamental domain, $\Theta \left[\begin{array}{c}\alpha \left(f\right)\\ \beta \left(f\right)\end{array}\right]$ denotes the Jacobi theta function with characteristics, and $\eta$ stands for the Dedekind eta function. The symbols $f_L$ and $f_R$ refer to the sets of left- and right-moving fermion fields, respectively.

To ensure the consistency of string theory, it is necessary to impose invariance under modular transformations and ensure the factorization of string amplitudes. These requirements translate into a set of constraints on possible basis vectors and GSO phases. The constraints on the basis vectors can be summarized as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill N_{ij}{v}_i·v_j& =0\mathrm{mod}4,\mathrm{with}{N}_{ij}=\mathrm{lcm}(N_i,N_j),\hfill \\ \hfill N_i{v}_i·v_i& =0\mathrm{mod}8,\mathrm{if}{N}_i\mathrm{is}\mathrm{even},\hfill \end{array}\end{array}$$

(6)

where we have introduced the Lorentzian inner product:

$$\begin{array}{c}\hfill \alpha ·\beta =\left\{{\displaystyle \frac{1}{2}}\sum _{\begin{array}{c}f\in f_L\mathrm{real}\end{array}}+\sum _{\begin{array}{c}f\in f_L\mathrm{complex}\end{array}}-{\displaystyle \frac{1}{2}}\sum _{\begin{array}{c}f\in f_R\mathrm{real}\end{array}}-\sum _{\begin{array}{c}f\in f_R\mathrm{complex}\end{array}}\right\}\alpha \left(f\right)\beta \left(f\right).\end{array}$$

(7)

Additionally, the overlap condition must be imposed: The number of real fermions that are simultaneously periodic under any combination of four basis vectors is required to be even. Furthermore, to ensure unambiguous periodicity of the supercharge (4), the following condition must hold:

$$\begin{array}{c}\hfill v\left(x^I\right)+v\left(y^I\right)+v\left(z^I\right)=v\left({\psi }^{\mu }\right)\mathrm{mod}2,\forall I=1,\dots ,6,\forall v\in B.\end{array}$$

(8)

The constraints on the projection coefficients $c\left[\begin{array}{c}{v}_i\\ v_j\end{array}\right]$ dictate that

$$\begin{array}{c}\hfill \begin{array}{c}\hfill c\left[\begin{array}{c}\alpha \\ \beta +\gamma \end{array}\right]={\delta }_{\alpha }c\left[\begin{array}{c}\alpha \\ \beta \end{array}\right]c\left[\begin{array}{c}\alpha \\ \gamma \end{array}\right],\\ \hfill c\left[\begin{array}{c}\alpha \\ \beta \end{array}\right]=e^{\mathrm{i}\pi (\alpha ·\beta )/2}c{\left[\begin{array}{c}\beta \\ \alpha \end{array}\right]}^{*},\\ \hfill c\left[\begin{array}{c}\alpha \\ \alpha \end{array}\right]=e^{\mathrm{i}\pi (\alpha ·\alpha +11·11)/4}c\left[\begin{array}{c}\alpha \\ 11\end{array}\right],\end{array}\end{array}$$

(9)

with ${\delta }_{\alpha }={(-1)}^{\alpha \left({\psi }^{\mu }\right)}$, and $\alpha ,\beta ,\gamma \in \Xi$. In the last equation, $11$ denotes the vector in which all fermions are periodic. These conditions allow all projection coefficients to be expressed in terms of $1+N(N-1)/2$-independent phases, involving the basis vectors, specifically, $c\left[\begin{array}{c}11\\ 11\end{array}\right]$, assuming $11\in B$, and $c\left[\begin{array}{c}{v}_i\\ v_j\end{array}\right]$, $i>j=1,\dots ,N$. The possible choices of the independent coefficients are determined from the relation

$$\begin{array}{c}\hfill c\left[\begin{array}{c}{v}_i\\ v_j\end{array}\right]={\delta }_{v_i}{e}^{2\mathrm{i}\pi n/N_j}={\delta }_{v_j}{e}^{\mathrm{i}\pi v_i·v_j}{e}^{2\mathrm{i}\pi m/N_i},n=0,1,\dots ,N_j,m=0,1,\dots ,N_i,i\ne j.\end{array}$$

(10)

This leads to $g_{ij}=\gcd (N_i,N_j)$ possible values for $(n,m)$ pairs.

The Hilbert space of the string states contributing to (6) can be expressed as

$$\begin{array}{c}\hfill \mathcal{H}=\underset{\alpha \in \Xi }{⨁}\prod _{i=1}^N\left\{e^{\mathrm{i}\pi v_i{F}_{\alpha }}={\delta }_{\alpha }c{\left[\begin{array}{c}\alpha \\ v_i\end{array}\right]}^{*}\right\}{\mathcal{H}}_{\alpha },\end{array}$$

(11)

where

$$\begin{array}{c}\hfill v_i{F}_{\alpha }=\left\{\sum _{f\in {\alpha }_L}-\sum _{f\in {\alpha }_R}\right\}{v}_i\left(f\right)F_{\alpha }\left(f\right),\end{array}$$

(12)

and ${\mathcal{H}}_{\alpha }$ represents the Hilbert space associated with the vector $\alpha$. The projector in the curly brackets retains only states that satisfy $e^{\mathrm{i}\pi v_i{F}_{\alpha }}={\delta }_{\alpha }c{\left[\begin{array}{c}\alpha \\ {\beta }_i\end{array}\right]}^{*}$. In the conventions used here, the fermion number operator $F_{\alpha }\left(f\right)$ yields $+1$ or $-1$ on f and $f^{*}$, respectively, and vanishes on the vacuum when it is non-degenerate. Otherwise, it takes the values 0 or $-1$ on vacuum states annihilated by $f_0$ or $f_0^{*}$, respectively.

The masses of string states in the sector ${\mathcal{H}}_{\alpha }$ are given by

$$\begin{array}{cc}\hfill M_{\alpha }^2& =-{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{8}}{\alpha }_{\mathrm{L}}·{\alpha }_{\mathrm{L}}+N_{\mathrm{L}}=-1+{\displaystyle \frac{1}{8}}{\alpha }_{\mathrm{R}}·{\alpha }_{\mathrm{R}}+N_{\mathrm{R}},\hfill \end{array}$$

where $\alpha =\left\{{\alpha }_L|{\alpha }_R\right\}$ and $N_L,N_R$ represent the sums of the left and right oscillator frequencies, respectively. The contribution to $N_L$ or $N_R$ of a fermion with parallel transportation properties, as in (2), is $[\left(1+\alpha \left(f\right)\right)/2+\mathrm{integer}]$ for f and $[\left(1-\alpha \left(f\right)\right)/2+\mathrm{integer}]$ for $f^{*}$. It can be demonstrated that the sector with all fermions being antiperiodic, ${\mathcal{H}}_0$, gives rise to a state of the form ${\psi }_{1/2}^{\mu }{\left(\overline{\partial X}\right)}_1^{\mu }\left|0\right.〉$, which survives all GSO projections. This state encompasses the graviton, the dilaton, and the two-index antisymmetric tensor.

By imposing that S belongs to the basis B, where S is the vector with only $\left\{{\psi }^{\mu },x^1,\dots ,x^6\right\}$ being periodic, together with an appropriate choice of $c\left[\begin{array}{c}S\\ v_i\end{array}\right]$ such that the gravitino multiplet in ${\mathcal{H}}_S$ survives, can ensure the presence of space–time supersymmetry and the absence of tachyons. Under some additional mild assumptions on the projection coefficients associated with S the space–time supersymmetry can be specified as $\mathcal{N}=1$. In this case, the effective low-energy theory is an $\mathcal{N}=1$ no-scale supergravity model [25,26,27,28]. The associated Kähler potential can be calculated exactly to all orders in ${\alpha }^{\prime }$ at the string tree level [29,30,31,32], while the superpotential receives only contributions at the string tree level that can be computed order by order in the ${\alpha }^{\prime }$ expansion [33,34]. The computation of superpotential couplings reduces to evaluating correlation functions of the primary fields participating in the associated vertex operators. In general, the coupling of a superpotential term given by

$$\begin{array}{c}\hfill \int d^2\theta {\mathcal{S}}_1\dots {\mathcal{S}}_n,n\ge 3\end{array}$$

(13)

is proportional to the correlation function

$$\begin{array}{c}\hfill \langle {\mathsf{\Psi }}_1{\mathsf{\Psi }}_2{\mathsf{\Phi }}_3\dots {\mathsf{\Phi }}_n\rangle ,\end{array}$$

(14)

where ${\mathsf{\Psi }}_1,{\mathsf{\Psi }}_2$ are the fermionic components of the superfields ${\mathcal{S}}_1,{\mathcal{S}}_2$, respectively, and ${\mathsf{\Phi }}_3,\dots ,{\mathsf{\Phi }}_n$ denote the bosonic components of the superfields ${\mathcal{S}}_3,\dots ,{\mathcal{S}}_n$, respectively. Here, the order n corresponds to $n-3$ in the ${\alpha }^{\prime }$ expansion, that is, $n=0$ describes the tree-level superpotential terms [35,36]. When unpaired real fermions are present, these correlators involve Ising fields [37], leading to the nontrivial elimination of superpotential couplings that would otherwise be permitted by gauge symmetries. As an example, focusing on the order operator, ${\sigma }_{+}$, the disorder operator, ${\sigma }_{-}$, and fermion operators, f and $\overline{f}$, the only non-vanishing correlation functions involving two or three Ising fields are

$$\begin{array}{c}\hfill \langle {\sigma }_{+}{\sigma }_{+}\rangle =\langle {\sigma }_{-}{\sigma }_{-}\rangle =\langle ff\rangle =\langle \overline{f}\overline{f}\rangle =1,\end{array}$$

(15)

$$\begin{array}{c}\hfill \langle {\sigma }_{+}{\sigma }_{-}f\rangle =\langle {\sigma }_{+}{\sigma }_{-}\overline{f}\rangle ={\displaystyle \frac{1}{\sqrt{2}}}.\end{array}$$

(16)

Additional restrictive selection rules arise from correlators associated with the internal fermionic coordinates $\left\{x^1,\dots ,x^6\right\}$, related to a conserved $U\left(1\right)$ current of the $\mathcal{N}=2$ world-sheet supersymmetry algebra [38,39].

3. The Supersymmetric Flipped $\mathit{SU}\left(\mathbf{5}\right)\times \mathit{U}\left(\mathbf{1}\right)$ Model

The flipped $SU\left(5\right)\times U\left(1\right)$ model was initially introduced as an alternative route for breaking the $SO\left(10\right)$ grand unified symmetry down to the Standard Model symmetry $SU\left(3\right)\times SU\left(2\right)\times U\left(1\right)$, where the electric charge generator does not lie within the $SU\left(5\right)$ subgroup of $SO\left(10\right)$ [10]. The flipped $SU\left(5\right)\times U\left(1\right)$ symmetry was afterwards explored in [11] as a standalone GUT group, independently of $SO\left(10\right)$ unification. In [12], a supersymmetric version of the model was proposed that employs a minimal gauge symmetry breaking mechanism using only a pair of Higgs fields in the antisymmetric representation of the $SU\left(5\right)$, ${\mathbf{10}}_{+1}+{\overline{\mathbf{10}}}_{-1}$.

Following [12], the three fermion matter generations are accommodated in $\mathbf{10}+\overline{\mathbf{5}}+\mathbf{1}$ of $SU\left(5\right)$. More specifically, the $SU\left(5\right)\times U\left(1\right)$ assignments are as follows:

$$\begin{array}{cc}\hfill F_i\left(\mathbf{10},+1\right)& ={\left\{Q,d^c,{\nu }^c\right\}}_i,{\overline{f}}_i{\left(\overline{\mathbf{5}},-3\right)}_i={\left\{u^c,L\right\}}_i,{\ell }_i^c\left(\mathbf{1},+5\right)={\left\{e^c\right\}}_i,i=1,2,3.\hfill \end{array}$$

(17)

The matter field assignments are “flipped" with respect to the standard $SU\left(5\right)$ model in the sense that $e_i^c\leftrightarrow {\nu }_i^c$ and $d_i^c\leftrightarrow u_i^c$ have been interchanged. The Standard Model (SM)-breaking Higgs doublets, $H_d,H_u$, are assigned to the $\mathbf{5}+\overline{\mathbf{5}}$ representations of $SU\left(5\right)$:

$$\begin{array}{c}\hfill h\left(\mathbf{5},-2\right)=\left\{{\overline{d}}_h^c,H_d\right\},\overline{h}\left(\overline{\mathbf{5}},+2\right)=\left\{d_h^c,H_u\right\},\end{array}$$

(18)

accompanied by an extra pair of triplets, $d_h^c,{\overline{d}}_h^c$. Moreover, the $SU\left(5\right)\times U\left(1\right)$-breaking Higgs fields content is

$$\begin{array}{c}\hfill H\left(\mathbf{10},+1\right)=\left\{Q_H,d_H^c,{\nu }_H^c\right\},\overline{H}\left(\overline{\mathbf{10}},-1\right)=\left\{{\overline{Q}}_H,{\overline{d}}_H^c,{\overline{\nu }}_H^c\right\},\end{array}$$

(19)

where we have added the subscript H to distinguish from SM fields. The model also comprises a number of singlet fields ${\varphi }_a\left(\mathbf{1},0\right),a=1,\dots ,n,$ where $n\ge 3$.

The superpotential of the model is given by

$$\begin{array}{cc}\hfill W=& {\lambda }_1^{ij}{F}_i{F}_{j}h+{\lambda }_2^{ij}{F}_i{\overline{f}}_j\overline{h}+{\lambda }_3^{ij}{\overline{f}}_i{\ell }_i^{c}h+{\lambda }_{4}HHh\hfill \\ \hfill & +{\lambda }_5\overline{H}\overline{H}\overline{h}+{\lambda }_6^{ia}{F}_i\overline{H}{\varphi }_a+{\lambda }_7^{a}h\overline{h}{\varphi }_a+{\lambda }_8^{abc}{\varphi }_a{\varphi }_b{\varphi }_c,\hfill \end{array}$$

(20)

where only terms invariant under the ${\mathbb{Z}}_2$ symmetry $H\to -H$ have been retained in order to avoid the mixing of the fields accommodated into the Higgs multiplets (19) with the SM particles. The GUT gauge symmetry breaking to the SM gauge group is achieved through a high-scale VEV, to the neutral components of the associated Higgs scalars, $\langle {\nu }_H^c\rangle =\langle {\overline{\nu }}_H^c\rangle =M_{\mathrm{GUT}}$, along an F- and D-flat direction of the superpotential (20):

$$\begin{array}{c}\hfill SU\left(5\right)\times U\left(1\right)\stackrel{\langle H\rangle =\langle \overline{H}\rangle }{\to }SU\left(3\right)\times SU\left(2\right)\times U{\left(1\right)}_{\mathrm{Y}}.\end{array}$$

(21)

In the conventions of (17) and (18), the hypercharge generator is given by

$$\begin{array}{c}\hfill Y={\displaystyle \frac{1}{15}}{T}_{24}+{\displaystyle \frac{1}{5}}{Q}_X,\end{array}$$

(22)

where $T_{24}$ stands for the $SU\left(5\right)$ Cartan generator $T_{24}=\left(2,2,2,-3,-3\right)/\sqrt{60}$ and $Q_X$ is the $U\left(1\right)$ charge. Subsequently, the $SU\left(3\right)\times SU\left(2\right)\times {U\left(1\right)}_{\mathrm{Y}}$ symmetry is broken to $SU\left(3\right)\times {U\left(1\right)}_{\mathrm{em}}$ via a low-scale VEV to the neutral component of the doublets in $h,\overline{h}$. In turn, the first term in (20), associated with the ${\lambda }_1$ coupling, provides masses for down quarks, while the second, associated with the ${\lambda }_2$ coupling, gives rise to up quark masses. Charged leptons obtain their masses (independently of the quarks) through the ${\lambda }_3$ coupling. Left-handed neutrinos stay naturally light by mixing with the total singlets ${\varphi }_a$ through an elegant see-saw mechanism of the form

$$\begin{array}{c}\hfill \left({\nu }_i{\nu }_i^c{\varphi }_a\right)\left(\begin{array}{ccc}0& <\overline{h}>& 0\\ <h>& 0& \langle H\rangle \\ 0& \langle H\rangle & 0\end{array}\right)\left(\begin{array}{c}{\nu }_i\\ {\nu }_i^c\\ {\varphi }_a\end{array}\right),\end{array}$$

(23)

resulting from the couplings ${\lambda }_2$ and ${\lambda }_6$, where ${\nu }_i$ stands for the left-handed neutrino in $L_i$. Finally, the couplings ${\lambda }_4,{\lambda }_5$ account for a natural solution to the doublet-triplet splitting problem as they generate heavy masses for the extra triplets accommodated in $H/\overline{H}$ and $h,\overline{h}$ leaving the SM Higgs doublets in $h,\overline{h}$ massless. This is achieved through a minimal and elegant realization of the so-called missing partner mechanism in the specific symmetry-breaking pattern in the flipped $SU\left(5\right)\times U\left(1\right)$ model (21), which employs Higgs multiplets lacking states carrying Standard Model Higgs doublet quantum numbers.

4. The Flipped $\mathit{SU}\left(\mathbf{5}\right)\times \mathit{U}\left(\mathbf{1}\right)$ String Model

Besides its appealing phenomenological characteristics, the flipped $SU\left(5\right)\times U\left(1\right)$ variant discussed in Section 3 also possesses the notable feature of being compatible with embedding within string constructions. In fact, the flipped $SU\left(5\right)\times U\left(1\right)$ was amongst the first models built in the context of the Free-Fermionic Formulation of the heterotic string [16,17,18], a framework summarized in Section 2, and has since become one of the most studied models. In this section, we will focus on the model derived in [16], which is derived using the basis $B=\{{\beta }_1,\dots ,{\beta }_8\}$, where

$$\begin{array}{cc}\hfill {\beta }_1=\zeta & =\left\{{\overline{\varphi }}^{1\dots 8}\right\},\hfill \\ \hfill {\beta }_2=S& =\left\{{\psi }^{\mu },x^{1\dots 6}\right\},\hfill \\ \hfill {\beta }_3=b_1& =\left\{{\psi }^{\mu },x^{12},y^{3\dots 6};{\overline{y}}^{3\dots 6},{\overline{\psi }}^{1\dots 5},{\overline{\eta }}^1\right\},\hfill \\ \hfill {\beta }_4=b_2& =\left\{{\psi }^{\mu },x^{34},y^{12},{\omega }^{56};{\overline{y}}^{12},{\overline{w}}^{56},{\overline{\psi }}^{1\dots 5}{\overline{\eta }}^2\right\},\hfill \\ \hfill {\beta }_5=b_3& =\left\{{\psi }^{\mu },x^{56},{\omega }^{1\dots 4};{\overline{w}}^{1\dots 4},{\overline{\psi }}^{1\dots 5},{\overline{\eta }}^3\right\},\hfill \\ \hfill {\beta }_6=b_4& =\left\{{\psi }^{\mu },x^{12},y^{36},{\omega }^{45};{\overline{y}}^{36},{\overline{w}}^{45},{\overline{\psi }}^{1\dots 5},{\overline{\eta }}^1\right\},\hfill \\ \hfill {\beta }_7=b_5& =\left\{{\psi }^{\mu },x^{34},y^{26},{\omega }^{15};{\overline{y}}^{26},{\overline{w}}^{15},{\overline{\psi }}^{1\dots 5},{\overline{\eta }}^2\right\},\hfill \\ \hfill {\beta }_8=\alpha & =\left\{y^{46},{\omega }^{46};{\overline{y}}^{46},{\overline{w}}^{2346},\underset{{\displaystyle \frac{1}{2}},\dots ,{\displaystyle \frac{1}{2}}}{\underbrace{{\overline{\psi }}^{1\dots 5},{\overline{\eta }}^{123},{\overline{\varphi }}^{1\dots 4}}},{\overline{\varphi }}^{56}\right\}.\hfill \end{array}$$

(24)

In the notation employed here, the included fermions are periodic while all others are antiperiodic, with the exception of the under-braced fermions in vector ${\beta }_8$, where ${\displaystyle \frac{1}{2}}$ represents twists of $-i$. The associated model defining GSO phases are given by

$$\begin{array}{c}\hfill C_{ij}=c\left[{\displaystyle \genfrac{}{}{0pt}{}{{\beta }_i}{{\beta }_j}}\right]=e^{i\pi v_{ij}},\end{array}$$

(25)

with

$$v=\begin{array}{cc}& \begin{array}{cccccccc}\zeta & S& b_1& b_2& b_3& b_4& b_5& \alpha \end{array}\\ \begin{array}{c}\zeta \\ S\\ b_1\\ b_2\\ b_3\\ b_4\\ b_5\\ \alpha \end{array}& \left(\begin{array}{cccccccc}1& 1& 1& 1& 1& 1& 1& 1\\ 1& 0& 0& 0& 0& 0& 0& 1\\ 1& 1& 1& 1& 1& 1& 1& -1/2\\ 1& 1& 1& 1& 1& 1& 1& -1/2\\ 1& 1& 1& 1& 1& 1& 1& 1\\ 1& 1& 1& 1& 1& 1& 1& -1/2\\ 1& 1& 1& 1& 1& 1& 1& +1/2\\ 1& 1& 1& 1& 1& 1& 0& -1/2\end{array}\right)\end{array}.$$

(26)

The basis vectors $11=\zeta +b_1+b_2+b_3$, S and $\zeta$ yield an $\mathcal{N}=4$ supersymmetric model possessing $SO\left(28\right)\times E_8$ gauge symmetry, with the vector $\zeta$ associated with the group enhancement in the second factor from $SO\left(16\right)$ to $E_8$. A ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ orbifold twist induced by the vectors $b_1$ and $b_2$ decreases the space–time supersymmetry to $\mathcal{N}=1$, produces chiral fermions, and reduces the gauge symmetry to $SO\left(10\right)\times SO{\left(6\right)}^3\times E_8$. The gauge symmetry is further reduced by the vectors $b_4$ and $b_5$, which break the $SO{\left(6\right)}^3$ group factor down to $SO{\left(4\right)}^2\times U\left(1\right)$ and by the vector $\alpha$, which breaks the $SO\left(10\right)$ factor, leading to the final gauge group

$$\begin{array}{c}\hfill G=SU\left(5\right)\times U{\left(1\right)}^{\prime }\times {U\left(1\right)}^4\times SU\left(4\right)\times SO\left(10\right).\end{array}$$

(27)

In the sequel, we will refer to the group factor $SU\left(5\right)\times U{\left(1\right)}^{\prime }$, associated with ${\overline{\psi }}^{1\dots 5}$, as the “observable” gauge group and to $SU\left(4\right)\times SO\left(10\right)$, pertaining to ${\overline{\varphi }}^{1\dots 8}$, as the “hidden” gauge group.

The massless matter spectrum comprises four $SO\left(10\right)\supset SU\left(5\right)\times U{\left(1\right)}^{\prime }$ spinorial multiplets $\mathbf{16}=(\mathbf{10},+1)+(\overline{\mathbf{5}},-3)+(\mathbf{1},+5)$ and one anti-spinorial $\overline{\mathbf{16}}=(\overline{\mathbf{10}},-1)+(\mathbf{5},+3)+(\mathbf{1},-5)$ coming from the sectors $(S+)b_i,i=1,\dots ,5$, which can accommodate three fermion generations and one pair of $SU\left(5\right)\times U{\left(1\right)}^{\prime }$-breaking Higgs fields. Moreover, four pairs of $SU\left(5\right)\times U{\left(1\right)}^{\prime }$ vector representations $\left(\mathbf{5},-2\right)+\left(\overline{\mathbf{5}},+2\right)$, suitable for accommodating the SM-breaking Higgs fields, arise from the sectors $(S+)0$,$(S+)b_4+b_5$. The same sectors also produce several non-abelian gauge singlets. The full “observable” massless matter spectrum is listed in

Table A1. Quantum numbers under $SU\left(5\right)\times U{\left(1\right)}^{\prime }\times {U\left(1\right)}^4\times SU\left(4\right)\times SO\left(10\right)$ of “observable” sector massless matter states of the string model. Here, $I=1,\dots ,5$ and $i=1,\dots ,4$ and $b_{45}=b_4+b_5$.

Sector		$\mathit{SU}\left(5\right)$	$\mathit{U}{\left(1\right)}^{\prime }$	${\mathit{U}\left(1\right)}_1$	${\mathit{U}\left(1\right)}_2$	${\mathit{U}\left(1\right)}_3$	${\mathit{U}\left(1\right)}_4$	$\mathit{SU}\left(4\right)$	$\mathit{SO}\left(10\right)$
*S	$h_1$	$\mathbf{5}$	$-1$	$+1$	0	0	0	$\mathbf{1}$	$\mathbf{1}$
	${\overline{h}}_1$	$\overline{\mathbf{5}}$	$+1$	$-1$	0	0	0	$\mathbf{1}$	$\mathbf{1}$
	$h_2$	$\mathbf{5}$	$-1$	0	$+1$	0	0	$\mathbf{1}$	$\mathbf{1}$
	${\overline{h}}_2$	$\overline{\mathbf{5}}$	$+1$	0	$-1$	0	0	$\mathbf{1}$	$\mathbf{1}$
	$h_3$	$\mathbf{5}$	$-1$	0	0	$+1$	0	$\mathbf{1}$	$\mathbf{1}$
	${\overline{h}}_3$	$\overline{\mathbf{5}}$	$+1$	0	0	$-1$	0	$\mathbf{1}$	$\mathbf{1}$
	${\mathsf{\Phi }}_{12}$	$\mathbf{1}$	0	$-1$	$+1$	0	0	$\mathbf{1}$	$\mathbf{1}$
	${\overline{\mathsf{\Phi }}}_{12}$	$\mathbf{1}$	0	$+1$	$-1$	0	0	$\mathbf{1}$	$\mathbf{1}$
	${\mathsf{\Phi }}_{31}$	$\mathbf{1}$	0	$+1$	0	$-1$	0	$\mathbf{1}$	$\mathbf{1}$
	${\overline{\mathsf{\Phi }}}_{31}$	$\mathbf{1}$	0	$-1$	0	$+1$	0	$\mathbf{1}$	$\mathbf{1}$
	${\mathsf{\Phi }}_{23}$	$\mathbf{1}$	0	0	$-1$	$+1$	0	$\mathbf{1}$	$\mathbf{1}$
	${\overline{\mathsf{\Phi }}}_{23}$	$\mathbf{1}$	0	0	$+1$	$-1$	0	$\mathbf{1}$	$\mathbf{1}$
	${\mathsf{\Phi }}_I$	$\mathbf{1}$	0	0	0	0	0	$\mathbf{1}$	$\mathbf{1}$
$b_1$	$F_1$	$\mathbf{10}$	$+\frac{1}{2}$	$-\frac{1}{2}$	0	0	0	$\mathbf{1}$	$\mathbf{1}$
	${\overline{f}}_1$	$\overline{\mathbf{5}}$	$-\frac{3}{2}$	$-\frac{1}{2}$	0	0	0	$\mathbf{1}$	$\mathbf{1}$
	${\ell }_1^c$	$\mathbf{1}$	$+\frac{5}{2}$	$-\frac{1}{2}$	0	0	0	$\mathbf{1}$	$\mathbf{1}$
$b_2$	$F_2$	$\mathbf{10}$	$+\frac{1}{2}$	0	$-\frac{1}{2}$	0	0	$\mathbf{1}$	$\mathbf{1}$
	${\overline{f}}_2$	$\overline{\mathbf{5}}$	$-\frac{3}{2}$	0	$-\frac{1}{2}$	0	0	$\mathbf{1}$	$\mathbf{1}$
	${\ell }_2^c$	$\mathbf{1}$	$+\frac{5}{2}$	0	$-\frac{1}{2}$	0	0	$\mathbf{1}$	$\mathbf{1}$
$b_3$	$F_3$	$\mathbf{10}$	$+\frac{1}{2}$	0	0	$+\frac{1}{2}$	$-\frac{1}{2}$	$\mathbf{1}$	$\mathbf{1}$
	${\overline{f}}_3$	$\overline{\mathbf{5}}$	$-\frac{3}{2}$	0	0	$+\frac{1}{2}$	$+\frac{1}{2}$	$\mathbf{1}$	$\mathbf{1}$
	${\ell }_3^c$	$\mathbf{1}$	$+\frac{5}{2}$	0	0	$+\frac{1}{2}$	$+\frac{1}{2}$	$\mathbf{1}$	$\mathbf{1}$
$b_4$	$F_4$	$\mathbf{10}$	$+\frac{1}{2}$	$-\frac{1}{2}$	0	0	0	$\mathbf{1}$	$\mathbf{1}$
	$f_4$	$\mathbf{5}$	$+\frac{3}{2}$	$+\frac{1}{2}$	0	0	0	$\mathbf{1}$	$\mathbf{1}$
	${\overline{\ell }}_4^c$	$\mathbf{1}$	$-\frac{5}{2}$	$+\frac{1}{2}$	0	0	0	$\mathbf{1}$	$\mathbf{1}$
$b_5$	${\overline{F}}_5$	$\overline{\mathbf{10}}$	$-\frac{1}{2}$	0	$+\frac{1}{2}$	0	0	$\mathbf{1}$	$\mathbf{1}$
	${\overline{f}}_5$	$\mathbf{5}$	$-\frac{3}{2}$	0	$-\frac{1}{2}$	0	0	$\mathbf{1}$	$\mathbf{1}$
	${\ell }_5^c$	$\mathbf{1}$	$+\frac{5}{2}$	0	$-\frac{1}{2}$	0	0	$\mathbf{1}$	$\mathbf{1}$
$S+b_{45}$	$h_{45}$	$\mathbf{5}$	$-1$	$-\frac{1}{2}$	$-\frac{1}{2}$	0	0	$\mathbf{1}$	$\mathbf{1}$
	${\overline{h}}_{45}$	$\overline{\mathbf{5}}$	$+1$	$+\frac{1}{2}$	$+\frac{1}{2}$	0	0	$\mathbf{1}$	$\mathbf{1}$
	${\varphi }_{45}$	$\mathbf{1}$	0	$+\frac{1}{2}$	$+\frac{1}{2}$	$+1$	0	$\mathbf{1}$	$\mathbf{1}$
	${\overline{\varphi }}_{45}$	$\mathbf{1}$	0	$-\frac{1}{2}$	$-\frac{1}{2}$	$-1$	0	$\mathbf{1}$	$\mathbf{1}$
	${\varphi }_{+}$	$\mathbf{1}$	0	$+\frac{1}{2}$	$-\frac{1}{2}$	0	$+1$	$\mathbf{1}$	$\mathbf{1}$
	${\overline{\varphi }}_{+}$	$\mathbf{1}$	0	$-\frac{1}{2}$	$+\frac{1}{2}$	0	$-1$	$\mathbf{1}$	$\mathbf{1}$
	${\varphi }_{-}$	$\mathbf{1}$	0	$+\frac{1}{2}$	$-\frac{1}{2}$	0	$-1$	$\mathbf{1}$	$\mathbf{1}$
	${\overline{\varphi }}_{-}$	$\mathbf{1}$	0	$-\frac{1}{2}$	$+\frac{1}{2}$	0	$+1$	$\mathbf{1}$	$\mathbf{1}$
	${\varphi }_i$	$\mathbf{1}$	0	$+\frac{1}{2}$	$-\frac{1}{2}$	0	0	$\mathbf{1}$	$\mathbf{1}$
	${\overline{\varphi }}_i$	$\mathbf{1}$	0	$-\frac{1}{2}$	$+\frac{1}{2}$	0	0	$\mathbf{1}$	$\mathbf{1}$

. The “hidden” massless matter spectrum encompasses five multiplets in the representations $\left(\mathbf{6},\mathbf{1}\right)+\left(\mathbf{1},\mathbf{10}\right)$ of $SU\left(4\right)\times SO\left(10\right)$, arising from the sectors $(S+)b_i+2\alpha (+\zeta ),i=i,\dots ,5$. These are detailed in

Table A2. Quantum numbers under $SU\left(5\right)\times U{\left(1\right)}^{\prime }\times {U\left(1\right)}^4\times SU\left(4\right)\times SO\left(10\right)$ of “hidden” sector massless matter states of the string model.

Sector		$\mathit{SU}\left(5\right)$	$\mathit{U}{\left(1\right)}^{\prime }$	${\mathit{U}\left(1\right)}_1$	${\mathit{U}\left(1\right)}_2$	${\mathit{U}\left(1\right)}_3$	${\mathit{U}\left(1\right)}_4$	$\mathit{SU}\left(4\right)$	$\mathit{SO}\left(10\right)$
$b_1+2\alpha \left(+\zeta \right)$	$D_1$	$\mathbf{1}$	0	0	$-\frac{1}{2}$	$+\frac{1}{2}$	0	$\mathbf{6}$	$\mathbf{1}$
	$T_1$	$\mathbf{1}$	0	0	$-\frac{1}{2}$	$+\frac{1}{2}$	0	$\mathbf{1}$	$\mathbf{10}$
$b_2+2\alpha \left(+\zeta \right)$	$D_2$	$\mathbf{1}$	0	$-\frac{1}{2}$	0	$+\frac{1}{2}$	0	$\mathbf{6}$	$\mathbf{1}$
	$T_2$	$\mathbf{1}$	0	$-\frac{1}{2}$	0	$+\frac{1}{2}$	0	$\mathbf{1}$	$\mathbf{10}$
$b_3+2\alpha \left(+\zeta \right)$	$D_3$	$\mathbf{1}$	0	$-\frac{1}{2}$	$-\frac{1}{2}$	0	$+\frac{1}{2}$	$\mathbf{6}$	$\mathbf{1}$
	$T_3$	$\mathbf{1}$	0	$-\frac{1}{2}$	$-\frac{1}{2}$	0	$-\frac{1}{2}$	$\mathbf{1}$	$\mathbf{10}$
$b_4+2\alpha \left(+\zeta \right)$	$D_4$	$\mathbf{1}$	0	0	$-\frac{1}{2}$	$+\frac{1}{2}$	0	$\mathbf{6}$	$\mathbf{1}$
	$T_4$	$\mathbf{1}$	0	0	$+\frac{1}{2}$	$-\frac{1}{2}$	0	$\mathbf{1}$	$\mathbf{10}$
$b_5+2\alpha \left(+\zeta \right)$	$D_5$	$\mathbf{1}$	0	$+\frac{1}{2}$	0	$-\frac{1}{2}$	0	$\mathbf{6}$	$\mathbf{1}$
	$T_5$	$\mathbf{1}$	0	$-\frac{1}{2}$	0	$+\frac{1}{2}$	0	$\mathbf{1}$	$\mathbf{10}$

. Finally, we have six pairs of exotic fractional charge multiples transforming as $\left(\mathbf{1},\pm {\displaystyle \frac{5}{4}},\mathbf{4}\right)+\left(\mathbf{1},\pm {\displaystyle \frac{5}{4}},\overline{\mathbf{4}}\right)$ under $SU\left(5\right)\times U{\left(1\right)}^{\prime }\times SU\left(4\right)$, originating from the sectors $\left(S\right)+b_1\pm \alpha (+\zeta )$, $\left(S\right)+b_1+b_4+b_5\pm \alpha (+\zeta )$, $\left(S\right)+b_2+b_3+b_5\pm \alpha (+\zeta )$, $\left(S\right)+b_1+b_2+b_4\pm \alpha (+\zeta )$, $\left(S\right)+b_2+b_4\pm \alpha (+\zeta )$, and $\left(S\right)+b_4\pm \alpha (+\zeta )$. These are presented in

Table A3. Quantum numbers under $SU\left(5\right)\times U{\left(1\right)}^{\prime }\times {U\left(1\right)}^4\times SU\left(4\right)\times SO\left(10\right)$, the exotic fractionally charged massless matter states of the string model. Here, $b_{145}=b_1+b_4+b_5$, $b_{235}=b_2+b_3+b_5$, $b_{124}=b_1+b_2+b_4$, and $b_{24}=b_2+b+4$.

Sector		$\mathit{SU}\left(5\right)$	$\mathit{U}{\left(1\right)}^{\prime }$	${\mathit{U}\left(1\right)}_1$	${\mathit{U}\left(1\right)}_2$	${\mathit{U}\left(1\right)}_3$	${\mathit{U}\left(1\right)}_4$	$\mathit{SU}\left(4\right)$	$\mathit{SO}\left(10\right)$
$b_1\pm \alpha \left(+\zeta \right)$	${\overline{X}}_1$	$\mathbf{1}$	$-\frac{5}{4}$	$-\frac{1}{4}$	$+\frac{1}{4}$	$+\frac{1}{4}$	$+\frac{1}{2}$	$\overline{\mathbf{4}}$	$\mathbf{1}$
	${\overline{X}}_2$	$\mathbf{1}$	$-\frac{5}{4}$	$-\frac{1}{4}$	$+\frac{1}{4}$	$+\frac{1}{4}$	$-\frac{1}{2}$	$\overline{\mathbf{4}}$	$\mathbf{1}$
$b_{145}\pm \alpha \left(+\zeta \right)$	$Y_1$	$\mathbf{1}$	$+\frac{5}{4}$	$-\frac{1}{4}$	$+\frac{1}{4}$	$-\frac{1}{4}$	$+\frac{1}{2}$	$\mathbf{4}$	$\mathbf{1}$
	$Y_2$	$\mathbf{1}$	$+\frac{5}{4}$	$-\frac{1}{4}$	$+\frac{1}{4}$	$-\frac{1}{4}$	$-\frac{1}{2}$	$\mathbf{4}$	$\mathbf{1}$
$b_{235}\pm \alpha \left(+\zeta \right)$	$Z_2$	$\mathbf{1}$	$+\frac{5}{4}$	$-\frac{1}{4}$	$+\frac{3}{4}$	$+\frac{1}{4}$	0	$\mathbf{4}$	$\mathbf{1}$
	${\overline{Z}}_2$	$\mathbf{1}$	$-\frac{5}{4}$	$-\frac{3}{4}$	$+\frac{1}{4}$	$-\frac{1}{4}$	0	$\overline{\mathbf{4}}$	$\mathbf{1}$
$b_{124}\pm \alpha \left(+\zeta \right)$	$Y_2^{\prime }$	$\mathbf{1}$	$+\frac{5}{4}$	$-\frac{1}{4}$	$+\frac{1}{4}$	$-\frac{1}{4}$	$-\frac{1}{2}$	$\mathbf{4}$	$\mathbf{1}$
	${\overline{Y}}_1$	$\mathbf{1}$	$-\frac{5}{4}$	$+\frac{1}{4}$	$-\frac{1}{4}$	$+\frac{1}{4}$	$-\frac{1}{2}$	$\overline{\mathbf{4}}$	$\mathbf{1}$
$S+b_{24}\pm \alpha \left(+\zeta \right)$	$Z_1$	$\mathbf{1}$	$-\frac{5}{4}$	$+\frac{1}{4}$	$+\frac{1}{4}$	$-\frac{1}{4}$	$+\frac{1}{2}$	$\mathbf{4}$	$\mathbf{1}$
	${\overline{Z}}_1$	$\mathbf{1}$	$+\frac{5}{4}$	$-\frac{1}{4}$	$-\frac{1}{4}$	$+\frac{1}{4}$	$-\frac{1}{2}$	$\overline{\mathbf{4}}$	$\mathbf{1}$
$b_4\pm \alpha \left(+\zeta \right)$	$X_1$	$\mathbf{1}$	$+\frac{5}{4}$	$+\frac{1}{4}$	$-\frac{1}{4}$	$-\frac{1}{4}$	$-\frac{1}{2}$	$\mathbf{4}$	$\mathbf{1}$
	${\overline{X}}_2^{\prime }$	$\mathbf{1}$	$-\frac{5}{4}$	$-\frac{1}{4}$	$+\frac{1}{4}$	$+\frac{1}{4}$	$-\frac{1}{2}$	$\overline{\mathbf{4}}$	$\mathbf{1}$

.

The full tree-level superpotential is

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill W_3=& g_s\sqrt{2}[F_1{F}_1{h}_1+F_2{F}_2{h}_2+F_4{F}_4{h}_1+{\overline{F}}_5{\overline{F}}_5{\overline{h}}_2+F_4{\overline{f}}_5{\overline{h}}_{45}+F_3{\overline{f}}_3{\overline{h}}_3\hfill \\ \hfill & +{\overline{f}}_1{\ell }_1^c{h}_1+{\overline{f}}_2{\ell }_2^c{h}_2+{\overline{f}}_5{\ell }_5^c{h}_2+f_4{\overline{\ell }}_4^c{\overline{h}}_1+{\displaystyle \frac{1}{\sqrt{2}}}\left(F_4{\overline{F}}_5{\varphi }_3+f_4{\overline{f}}_5{\overline{\varphi }}_2+{\overline{\ell }}_4^c{\ell }_5^c{\overline{\varphi }}_2\right)\hfill \\ \hfill & +h_1{\overline{h}}_2{\mathsf{\Phi }}_{12}+{\overline{h}}_1{h}_2{\overline{\mathsf{\Phi }}}_{12}+h_2{\overline{h}}_3{\mathsf{\Phi }}_{23}+{\overline{h}}_2{h}_3{\overline{\mathsf{\Phi }}}_{23}+h_3{\overline{h}}_1{\mathsf{\Phi }}_{31}\hfill \\ \hfill & +{\overline{h}}_3{h}_1{\overline{\mathsf{\Phi }}}_{31}+h_3{\overline{h}}_{45}{\overline{\varphi }}_{45}+{\overline{h}}_3{h}_{45}{\varphi }_{45}+{\displaystyle \frac{1}{2}}{h}_{45}{\overline{h}}_{45}{\mathsf{\Phi }}_3\hfill \\ \hfill & +\left({\varphi }_1{\overline{\varphi }}_2+{\overline{\varphi }}_1{\varphi }_2\right){\mathsf{\Phi }}_4+\left({\varphi }_3{\overline{\varphi }}_4+{\overline{\varphi }}_3{\varphi }_4\right){\mathsf{\Phi }}_5+{\displaystyle \frac{1}{2}}{\varphi }_{45}{\overline{\varphi }}_{45}{\mathsf{\Phi }}_3+{\displaystyle \frac{1}{2}}\left({\varphi }_{+}{\overline{\varphi }}_{+}+{\varphi }_{-}{\overline{\varphi }}_{-}\right){\mathsf{\Phi }}_3\hfill \\ \hfill & +{\mathsf{\Phi }}_{12}{\mathsf{\Phi }}_{23}{\mathsf{\Phi }}_{31}+{\overline{\mathsf{\Phi }}}_{12}{\overline{\mathsf{\Phi }}}_{23}{\overline{\mathsf{\Phi }}}_{31}+{\mathsf{\Phi }}_{12}{\varphi }_{+}{\varphi }_{-}+{\overline{\mathsf{\Phi }}}_{12}{\overline{\varphi }}_{+}{\overline{\varphi }}_{-}\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\mathsf{\Phi }}_3\sum _{i=1}^4{\varphi }_i{\overline{\varphi }}_i+{\mathsf{\Phi }}_{12}\sum _{i=1}^4{\varphi }_i^2+{\overline{\mathsf{\Phi }}}_{12}\sum _{i=1}^4{\overline{\varphi }}_i^2\hfill \\ \hfill & +D_1^2{\overline{\mathsf{\Phi }}}_{23}+D_2^2{\mathsf{\Phi }}_{31}+D_4^2{\overline{\mathsf{\Phi }}}_{23}+D_5^2{\overline{\mathsf{\Phi }}}_{31}+{\displaystyle \frac{1}{\sqrt{2}}}{D}_4{D}_5{\overline{\varphi }}_3\hfill \\ \hfill & +T_1^2{\overline{\mathsf{\Phi }}}_{23}+T_2^2{\mathsf{\Phi }}_{31}+T_4^2{\mathsf{\Phi }}_{23}+T_5^2{\mathsf{\Phi }}_{31}+{\displaystyle \frac{1}{\sqrt{2}}}{T}_4{T}_5{\varphi }_2\hfill \\ \hfill & +{\displaystyle \frac{1}{\sqrt{2}}}\left(Y_1{\overline{X}}_2{\varphi }_4+Y_2{\overline{X}}_1{\varphi }_1\right)+Y_2{\overline{X}}_2{\varphi }_{+}+{\displaystyle \frac{1}{2}}{Z}_1{\overline{Z}}_1{\mathsf{\Phi }}_3+Z_2{\overline{Z}}_2{\overline{\mathsf{\Phi }}}_{12}+Z_1{\overline{X}}_2^{\prime }{\ell }_2^c+Y_2^{\prime }{Z}_1{D}_1].\hfill \end{array}\end{array}$$

(28)

As concluded from an inspection of the massless spectrum abelian charges, the $U{\left(1\right)}^4={\prod }_{j=1}^4{U\left(1\right)}_j$ gauge group factor can be redefined as to obtain tree orthogonal $U\left(1\right)$ linear combinations:

$$\begin{array}{cc}\hfill U{\left(1\right)}_1^{\prime }& ={U\left(1\right)}_3+2{U\left(1\right)}_4,\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill U{\left(1\right)}_2^{\prime }& ={U\left(1\right)}_1-3{U\left(1\right)}_2,\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill U{\left(1\right)}_3^{\prime }& =3{U\left(1\right)}_1+{U\left(1\right)}_2+4{U\left(1\right)}_3-2{U\left(1\right)}_4,\hfill \end{array}$$

(31)

which are free from both gauge and gravitational anomalies, and one anomalous linear combination

$$\begin{array}{cc}\hfill U{\left(1\right)}_A& =-3{U\left(1\right)}_1-{U\left(1\right)}_2+2{U\left(1\right)}_3-{U\left(1\right)}_4,\mathrm{Tr}\left(U{\left(1\right)}_A\right)=180.\hfill \end{array}$$

(32)

This anomaly induces a large Fayet–Iliopoulos D-term, leading to supersymmetry breaking and destabilization of the vacuum. Nevertheless, such a situation can be avoided via the Dine–Seiberg–Witten (SDW) mechanism [14,15]. This entails an F- and D-flat direction in the scalar potential, along which a set of charged fields ${\phi }_j$ acquire VEVs that lead to $U{\left(1\right)}_A$ gauge symmetry breaking and the re-stabilization of the vacuum at one loop. In this case, the $U{\left(1\right)}_A$ D-flatness condition is recast in the form

$$\begin{array}{c}\hfill D_A=\sum _j{q}_A^j{\left|{\phi }_j\right|}^2+{\xi }^2=0,\mathrm{with}{\xi }^2={\displaystyle \frac{1}{192{\pi }^2}}{\displaystyle \frac{2}{{\alpha }^{\prime }}}\mathrm{Tr}\left(U{\left(1\right)}_A\right)={\displaystyle \frac{15}{16{\pi }^2}}{\displaystyle \frac{2}{{\alpha }^{\prime }}},\end{array}$$

(33)

where $q_A^j$ stands for the $U{\left(1\right)}_A$ charge of the singlet field ${\phi }_j$. As a result of this constraint, the DSW mechanism leads to the emergence of an additional scale, $\xi$, which is approximately one order of magnitude lower than the string scale. This scale plays a central role in low-energy phenomenology, as it is related to the appearance of hierarchical Yukawa couplings.

A phenomenological analysis of the model begins with deriving a nontrivial solution of the F- and D-flatness constraints. As mentioned earlier, the anomalous D-flatness Equation (33) imposes that some singlet fields acquire non-vanishing VEVs. Next, the candidate Standard Model doublet mass matrix is analyzed to determine which doublet pair(s) survive in low energies. As explained before, doublets are accommodated in $h_i,{\overline{h}}_i,i=1,2,3,45$. Following this, fermion mass matrices are computed. A detailed analysis requires the computation of higher-order non-renormalizable contributions to the superpotential. The phenomenology of the flipped $\mathrm{SU}\left(5\right)$ string model has been extensively studied in the literature [16,19,40,41,42,43,44,45,46,47,48,49,50].

5. Cosmological Background of the Model

String theory, as a candidate for a unified theory of all fundamental interactions, must account for both particle physics and cosmology within a single coherent framework. Over the past few decades, significant progress in string phenomenology has advanced the development of realistic particle physics models and cosmological scenarios. However, constructing models that simultaneously address both realms remains a significant hurdle. A recent paper has tackled this challenge in the context of the string-derived flipped $SU\left(5\right)\times U\left(1\right)$ model discussed in Section 4 [19]. It was shown that it can accommodate Starobinsky-type inflation [20] implemented in the context of $\mathcal{N}=1$ no-scale supergravity [51,52,53,54,55], which arises as an effective theory in the low-energy limit of string theory in four dimensions [29]. In general, the Starobinsky inflationary model is derived by extending the Einstein action with a term that is quadratic in the scalar curvature. After linearization and supersymmetrization, the model results in a no-scale-type supergravity theory with Kähler potential

$$\begin{array}{c}\hfill K=-3ln\left(T+\overline{T}-C\overline{C}\right),\end{array}$$

(34)

and superpotential

$$\begin{array}{c}\hfill W=\mu C\left(T-k\right),\end{array}$$

(35)

where $\mu$ denotes the inflation scale and k is a constant. The field T contains the inflaton $\phi$, which is expressed as $\mathrm{Re}T=e^{\sqrt{2/3}\phi }$, and C is the goldstino superfield associated with supersymmetry breaking.

In the context of the free-fermionic construction, an $\mathcal{N}=1$ supersymmetric string model corresponds to $Z_2\times Z_2$ asymmetric orbifold on an $\mathcal{N}=4$ theory. It comprises three $Z_2$ elements, $h_1,h_2$, and $h_3$, each of which results in an $\mathcal{N}=2$ theory. The untwisted field Kähler potential can be cast in the form

$$\begin{array}{c}\hfill K_{\mathrm{UT}}=K_0+K_1+K_2+K_3,\mathrm{with}{K}_0=-ln(S+\overline{S}),\end{array}$$

(36)

where S stands for the dilaton superfield and

$$\begin{array}{c}\hfill K_i=-ln\left(1-\sum _{A=1}^{n_i}{\left|{\mathsf{\Phi }}_{i,A}\right|}^2+{\displaystyle \frac{1}{4}}{\left|\sum _{A=1}^{n_i}{\mathsf{\Phi }}_{i,A}\right|}^2\right)\end{array}$$

(37)

parametrize an $SO(2,n_i)/SO\left(2\right)\times SO\left(n_i\right)$ manifold. Here, ${\mathsf{\Phi }}_{i,A},i=1,\dots ,n_i$ refers to the chiral superfields of the i-$\mathcal{N}=2$ sector that survive the other two $h_j,j\ne i$ orbifold projections.The twisted field contribution to the Kähler potential, in the weak field limit, can be written as

$$\begin{array}{c}\hfill K_{\mathrm{T}}=K_{T_1}+K_{T_2}+K_{T_3},\mathrm{with}{K}_{T_i}\simeq \sum _{\left\{{\mathsf{\Phi }}_{T_i}\right\}}{\left|{\mathsf{\Phi }}_{T_i}\right|}^2{e}^{K_j+K_{\ell }},i,j,\ell \mathrm{pairwise}\mathrm{dinstict},\end{array}$$

(38)

where $\left\{{\mathsf{\Phi }}_{T_i}\right\}$ denotes the set of the chiral superfields of the i-th twisted sector. Restricting to the case of a single superfield z arising from the third untwisted sector and a single field ${\mathsf{\Phi }}_{T_3}$ coming from the third twisted plane and assuming all other fields are set to zero, which corresponds to the fermionic point, the full Kähler potential and superpotential can be reduced to

$$\begin{array}{cc}\hfill K& =-ln\left(S+\overline{S}\right)-2ln\left(1-{\displaystyle \frac{1}{2}}{\left|z\right|}^2\right)-2ln\left(1-{\left|y\right|}^2\right),\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill W& =\mu zy(1-y),\hfill \end{array}$$

(40)

where $y={\mathsf{\Phi }}_{T_3}/\sqrt{2}$. Introducing $y=\rho e^{i\theta },\rho \in [0,1)$, the associated scalar potential reads [19]

$$\begin{array}{cc}\hfill V& =e^K\left(\sum _{\mathsf{\Phi }\in U}\left|D_{\mathsf{\Phi }}{W|}^2-3{\left|W\right|}^2\right|\right),U=\{S,y,z\}\hfill \\ \hfill & ={\mu }^2{g}_s^2\left[{\displaystyle \frac{{\rho }^4\left({\left|z\right|}^4-2{\left|z\right|}^2+4\right)-2\rho cos\left(\theta \right)\left({\rho }^2\left({\left|z\right|}^4+4\right)+4{\left|z\right|}^2\right)+{\rho }^2\left({\left|z\right|}^4+8{\left|z\right|}^2+4\right)+2{\left|z\right|}^2}{{\left({\rho }^2-1\right)}^2{\left({\left|z\right|}^2-2\right)}^2}}\right],\hfill \end{array}$$

(41)

where $D_{\mathsf{\Phi }}$ is the Kähler covariant derivative, given by $D_{\mathsf{\Phi }}={\partial }_{\mathsf{\Phi }}+\left({\partial }_{\mathsf{\Phi }}K\right)$, and $g_s$ is the string coupling, assumed to be constant due to an appropriate dilaton stabilization mechanism. Minimizing the potential with respect to $\theta$ yields $\theta =0$, rendering the potential in the following form:

$$\begin{array}{c}\hfill {\left.V\right|}_{\theta =0}={\displaystyle \frac{{\mu }^2{g}_s^2}{{(1+\rho )}^2}}\left[{\rho }^2+{\displaystyle \frac{{(1-\rho )}^2{\left|z\right|}^2}{2{\left(1-\frac{{\left|z\right|}^2}{2}\right)}^2}}\right]\end{array}$$

(42)

Further investigation of the potential shows that the phase fluctuation decouples, acquiring a mass of the order of the inflation scale. Moreover, the goldstino direction z is stabilized as a result of the contribution of the dilaton field in (39), and the mass of z is proportional to the inflaton slope. The goldstino direction z is stabilized at $z=\theta =0$, where the potential reduces to

$$\begin{array}{c}\hfill {\left.V\right|}_{z=\theta =0}={\mu }^2{g}_s^2{\displaystyle \frac{{\rho }^2}{1+{\rho }^2}}.\end{array}$$

(43)

Introducing the canonically normalised inflaton field $\phi =ln{\displaystyle \frac{1+\rho }{1-\rho }}$ that leads to $\rho =tanh{\displaystyle \frac{\phi }{2}}$ we recover a Starobinsky-type potential

$$\begin{array}{c}\hfill V={\displaystyle \frac{{\mu }^2{g}_s^2}{4}}{\left(1-e^{-\phi }\right)}^2.\end{array}$$

(44)

According to [54], a potential of this form results in a spectral index of primordial density perturbations of $n_s=0.965$ and a ratio of tensor-to-scalar modes of $r=2.4\times {10}^{-23}$ for $N_{*}=55$ e-folds of inflation. The potential minimum is attained at $\phi =0$, around which the inflaton acquires a mass $m_{\phi }=\mu g_s/\sqrt{2}$.

A concrete realization of the inflationary scenario described above was presented in [19] within the context of the flipped $SU\left(5\right)\times U\left(1\right)$ string model discussed in Section 4. The field ${\mathsf{\Phi }}_{T_3}$, linked to the inflaton field, is identified with a linear combination of fields including ${\varphi }_3$, which enters the superpotential coupling $F_4{\overline{F}}_5{\varphi }_3$ (see (28)), also referred to as ${\lambda }_6$ coupling [56]. This coupling is also linked to right-handed neutrino masses. As required, the singlet ${\varphi }_3$ arises from the third twisted sector $T_3$, while the superfield z corresponds to a singlet of the third untwisted sector. The GUT symmetry breaking takes place after the end of inflation, proceeding through a first-order phase transition.

As explained in Section 4, the presence of an anomalous $U\left(1\right)$ requires some fields to develop VEVs approximately one order of magnitude below the Planck scale. These VEVs are subject to nontrivial F- and D-flatness conditions. These are

$$\begin{array}{cc}\hfill {\mathcal{F}}_k& ={\displaystyle \frac{\partial W}{\partial {\phi }_k}}=0,\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill {\mathcal{D}}_I& =\sum _k{q}_I^k{\left|{\phi }_k\right|}^2=0,I=1,2,3,\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill {\mathcal{D}}_A& =\sum _k{q}_A^k{\left|{\phi }_k\right|}^2+{\xi }^2=0,\hfill \end{array}$$

(47)

where ${\phi }_k$ stands for a field in the massless spectrum, ${\xi }^2$ is defined in (33), and W is the superpotential. The charges of the field ${\phi }_k$ under $U{\left(1\right)}_I^{\prime }$ and $U{\left(1\right)}_A$ are denoted by $q_I^k$ and $q_A^k$, respectively. Assuming

$$\begin{array}{c}\hfill {\phi }_k\in \left\{{\mathsf{\Phi }}_{12},{\mathsf{\Phi }}_{23},{\mathsf{\Phi }}_{31},{\overline{\mathsf{\Phi }}}_{12},{\overline{\mathsf{\Phi }}}_{23},{\overline{\mathsf{\Phi }}}_{31},{\varphi }_1,\dots ,{\varphi }_4,{\varphi }_{+},{\varphi }_{-},{\varphi }_{45},{\overline{\varphi }}_1,\dots ,{\overline{\varphi }}_4,{\overline{\varphi }}_{+},{\overline{\varphi }}_{-},{\overline{\varphi }}_{45},\right.\\ \hfill \left.{\mathsf{\Phi }}_1,\dots ,{\mathsf{\Phi }}_5,D_1,\dots ,D_5,T_1,\dots ,T_5\right\},\end{array}$$

(48)

and restricting to tree-level superpotential, the F-flatness conditions read

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\mathsf{\Phi }}_{12}}}=0\Rightarrow & \sum _{i=1}^4{\varphi }_i^2+{\mathsf{\Phi }}_{23}{\mathsf{\Phi }}_{31}+{\varphi }_{+}{\varphi }_{-}=0,\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\overline{\mathsf{\Phi }}}_{12}}}=0\Rightarrow & \sum _{i=1}^4{\overline{\varphi }}_i^2{\overline{\mathsf{\Phi }}}_{23}{\overline{\mathsf{\Phi }}}_{31}+{\overline{\varphi }}_{+}{\overline{\varphi }}_{-}=0,\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\mathsf{\Phi }}_{31}}}=0\Rightarrow & {\mathsf{\Phi }}_{12}{\mathsf{\Phi }}_{23}+D_2^2+T_2^2+T_5^2=0,\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\overline{\mathsf{\Phi }}}_{31}}}=0\Rightarrow & {\overline{\mathsf{\Phi }}}_{12}{\overline{\mathsf{\Phi }}}_{23}+D_5^2=0,\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\mathsf{\Phi }}_{23}}}=0\Rightarrow & {\mathsf{\Phi }}_{12}{\mathsf{\Phi }}_{31}+T_4^2=0,\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\overline{\mathsf{\Phi }}}_{23}}}=0\Rightarrow & {\overline{\mathsf{\Phi }}}_{12}{\overline{\mathsf{\Phi }}}_{31}+D_1^2+D_4^2+T_1^2=0,\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\mathsf{\Phi }}_3}}=0\Rightarrow & \sum _{i=1}^4{\varphi }_i{\overline{\varphi }}_i+{\varphi }_{45}{\overline{\varphi }}_{45}+{\varphi }_{-}{\overline{\varphi }}_{-}+{\varphi }_{+}{\overline{\varphi }}_{+}=0,\hfill \end{array}$$

(55)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\mathsf{\Phi }}_4}}=0\Rightarrow & {\varphi }_2{\overline{\varphi }}_1+{\varphi }_1{\overline{\varphi }}_2=0,\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\mathsf{\Phi }}_5}}=0\Rightarrow & {\varphi }_4{\overline{\varphi }}_3+{\varphi }_3{\overline{\varphi }}_4=0,\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\varphi }_1}}=0\Rightarrow & 2{\varphi }_1{\mathsf{\Phi }}_{12}+{\mathsf{\Phi }}_3{\overline{\varphi }}_1+{\mathsf{\Phi }}_4{\overline{\varphi }}_2=0,\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\varphi }_2}}=0\Rightarrow & 2{\varphi }_2{\mathsf{\Phi }}_{12}+{\mathsf{\Phi }}_4{\overline{\varphi }}_1+{\mathsf{\Phi }}_3{\overline{\varphi }}_2+T_4·T_5=0,\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\varphi }_3}}=0\Rightarrow & 2{\varphi }_3{\mathsf{\Phi }}_{12}+{\mathsf{\Phi }}_3{\overline{\varphi }}_3+{\mathsf{\Phi }}_5{\overline{\varphi }}_4=0,\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\varphi }_4}}=0\Rightarrow & 2{\varphi }_4{\mathsf{\Phi }}_{12}+{\mathsf{\Phi }}_5{\overline{\varphi }}_3+{\mathsf{\Phi }}_3{\overline{\varphi }}_4=0,\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\overline{\varphi }}_1}}=0\Rightarrow & 2{\overline{\varphi }}_1{\overline{\mathsf{\Phi }}}_{12}+{\mathsf{\Phi }}_3{\varphi }_1+{\mathsf{\Phi }}_4{\varphi }_2=0,\hfill \end{array}$$

(62)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\overline{\varphi }}_2}}=0\Rightarrow & 2{\overline{\varphi }}_2{\overline{\mathsf{\Phi }}}_{12}+{\mathsf{\Phi }}_4{\varphi }_1+{\mathsf{\Phi }}_3{\varphi }_2=0,\hfill \end{array}$$

(63)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\overline{\varphi }}_3}}=0\Rightarrow & 2{\overline{\varphi }}_3{\overline{\mathsf{\Phi }}}_{12}+{\mathsf{\Phi }}_3{\varphi }_3+{\mathsf{\Phi }}_5{\varphi }_4+D_4·D_5=0,\hfill \end{array}$$

(64)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\overline{\varphi }}_4}}=0\Rightarrow & 2{\overline{\varphi }}_4{\overline{\mathsf{\Phi }}}_{12}+{\mathsf{\Phi }}_5{\varphi }_3+{\mathsf{\Phi }}_3{\varphi }_4=0,\hfill \end{array}$$

(65)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\varphi }_{+}}}=0\Rightarrow & {\mathsf{\Phi }}_3{\overline{\varphi }}_{+}+{\mathsf{\Phi }}_{12}{\varphi }_{-}=0,\hfill \end{array}$$

(66)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\overline{\varphi }}_{+}}}=0\Rightarrow & {\mathsf{\Phi }}_3{\varphi }_{+}+{\overline{\mathsf{\Phi }}}_{12}{\overline{\varphi }}_{-}=0,\hfill \end{array}$$

(67)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\varphi }_{-}}}=0\Rightarrow & {\mathsf{\Phi }}_3{\overline{\varphi }}_{-}+{\mathsf{\Phi }}_{12}{\varphi }_{+}=0,\hfill \end{array}$$

(68)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\overline{\varphi }}_{-}}}=0\Rightarrow & {\mathsf{\Phi }}_3{\varphi }_{-}+{\overline{\mathsf{\Phi }}}_{12}{\overline{\varphi }}_{+}=0,\hfill \end{array}$$

(69)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\varphi }_{45}}}=0\Rightarrow & {\mathsf{\Phi }}_3{\overline{\varphi }}_{45}=0,\hfill \end{array}$$

(70)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\overline{\varphi }}_{45}}}=0\Rightarrow & {\mathsf{\Phi }}_3{\varphi }_{45}=0,\hfill \end{array}$$

(71)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial T_1}}=0\Rightarrow & 2{\overline{\mathsf{\Phi }}}_{23}{T}_1=0,\hfill \end{array}$$

(72)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial T_2}}=0\Rightarrow & 2{\mathsf{\Phi }}_{31}{T}_2=0,\hfill \end{array}$$

(73)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial T_4}}=0\Rightarrow & 2{\mathsf{\Phi }}_{23}{T}_4+{\varphi }_2{T}_5=0,\hfill \end{array}$$

(74)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial T_5}}=0\Rightarrow & {\varphi }_2{T}_4+2{\mathsf{\Phi }}_{31}{T}_5=0,\hfill \end{array}$$

(75)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial D_1}}=0\Rightarrow & 2D_1{\overline{\mathsf{\Phi }}}_{23}=0,\hfill \end{array}$$

(76)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial D_2}}=0\Rightarrow & 2D_2{\mathsf{\Phi }}_{31}=0,\hfill \end{array}$$

(77)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial D_4}}=0\Rightarrow & 2D_4{\overline{\mathsf{\Phi }}}_{23}+D_5{\overline{\varphi }}_3=0,\hfill \end{array}$$

(78)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial D_5}}=0\Rightarrow & 2D_5{\overline{\mathsf{\Phi }}}_{31}+D_4{\overline{\varphi }}_3=0.\hfill \end{array}$$

(79)

The D-flatness conditions can be rewritten in the form

$$\begin{array}{cc}\hfill {\mathcal{D}}_1^{\prime }=& {\left|{\varphi }_{45}\right|}^2-{\left|{\overline{\varphi }}_{45}\right|}^2-{\displaystyle \frac{1}{2}}\left({\left|D_3\right|}^2+{\left|T_3\right|}^2\right)-{\xi }^2=0,\hfill \end{array}$$

(80)

$$\begin{array}{cc}\hfill {\mathcal{D}}_2^{\prime }=& {\left|{\varphi }_{+}\right|}^2-{\left|{\varphi }_{-}\right|}^2-{\left|{\overline{\varphi }}_{+}\right|}^2+{\left|{\overline{\varphi }}_{-}\right|}^2+{\displaystyle \frac{1}{2}}\left({\left|D_3\right|}^2-{\left|T_3\right|}^2\right)-{\xi }^2=0,\hfill \end{array}$$

(81)

$$\begin{array}{cc}\hfill {\mathcal{D}}_3^{\prime }=& {\left|{\mathsf{\Phi }}_{31}\right|}^2-{\left|{\overline{\mathsf{\Phi }}}_{31}\right|}^2-{\left|{\mathsf{\Phi }}_{23}\right|}^2+{\left|{\overline{\mathsf{\Phi }}}_{23}\right|}^2-{\displaystyle \frac{1}{2}}\left({\left|D_1\right|}^2+{\left|D_2\right|}^2+{\left|D_3\right|}^2+{\left|D_4\right|}^2-{\left|D_5\right|}^2\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\left({\left|T_1\right|}^2+{\left|T_2\right|}^2+{\left|T_3\right|}^2-{\left|T_4\right|}^2+{\left|T_5\right|}^2\right)-3{\xi }^2=0,\hfill \end{array}$$

(82)

$$\begin{array}{cc}\hfill {\mathcal{D}}_4^{\prime }=& {\left|{\mathsf{\Phi }}_{23}\right|}^2-{\left|{\overline{\mathsf{\Phi }}}_{23}\right|}^2-{\left|{\mathsf{\Phi }}_{12}\right|}^2+{\left|{\overline{\mathsf{\Phi }}}_{12}\right|}^2+{\displaystyle \frac{1}{2}}\sum _{i=1}^4\left({\left|{\varphi }_i\right|}^2-{\left|{\overline{\varphi }}_i\right|}^2\right)+{\left|{\varphi }_{+}\right|}^2-{\left|{\overline{\varphi }}_{+}\right|}^2\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\left({\left|D_1\right|}^2+{\left|D_3\right|}^2+{\left|D_4\right|}^2\right)+{\displaystyle \frac{1}{2}}\left({\left|T_1\right|}^2-{\left|T_4\right|}^2\right)=0.\hfill \end{array}$$

(83)

As shown in [19], the following choice of VEVs is compatible with all but four of the F-flatness constraints and retains a sufficient number of free parameters to satisfy both the remaining F-flatness constraints and the D-flatness requirements:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathsf{\Phi }}_{12}={\overline{\mathsf{\Phi }}}_{12}={\mathsf{\Phi }}_{23}={\overline{\mathsf{\Phi }}}_{23}={\overline{\mathsf{\Phi }}}_{31}=0,\\ \hfill {\mathsf{\Phi }}_3={\mathsf{\Phi }}_4={\mathsf{\Phi }}_5=0,\\ \hfill {\varphi }_1={\varphi }_2={\varphi }_3={\overline{\varphi }}_1={\overline{\varphi }}_2={\overline{\varphi }}_3=0,\\ \hfill {\overline{F}}_5=F_1=F_2=F_3=F_4=0,\\ \hfill D_2=T_2=T_5=0,\\ \hfill T_4^2=D_5^2=D_4·D_5=0.\end{array}\end{array}$$

(84)

Focusing on the singlet masses, a detailed analysis shows that the above solution, in general, provides masses for all ${\mathsf{\Phi }}_{12},{\overline{\mathsf{\Phi }}}_{12},{\mathsf{\Phi }}_3,{\mathsf{\Phi }}_5$ via tree-level superpotential terms (28). Similarly, four linear combinations of ${\varphi }_3,{\varphi }_4,{\overline{\varphi }}_3,{\overline{\varphi }}_4,{\varphi }_{+},{\varphi }_{-},{\overline{\varphi }}_{+},{\overline{\varphi }}_{-},{\varphi }_{45},{\overline{\varphi }}_{45}$, and ${\mathsf{\Phi }}_{23}$ become superheavy. The field ${\varphi }_3$ mixes with ${\mathsf{\Phi }}_5$ through the superpotential terms ${\mathsf{\Phi }}_5({\varphi }_3{\overline{\varphi }}_4+{\varphi }_4{\overline{\varphi }}_3)$, leading to one massless linear combination, ${\varphi }_0$, and one massive linear combination, ${\varphi }_m$, as follows:

$$\begin{array}{cc}\hfill {\varphi }_0& =sin\omega {\varphi }_3-cos\omega {\overline{\varphi }}_3,\hfill \end{array}$$

(85)

$$\begin{array}{cc}\hfill {\varphi }_m& =cos\omega {\varphi }_3+sin\omega {\overline{\varphi }}_3,\hfill \end{array}$$

(86)

with $tan\omega =\langle {\varphi }_4\rangle /\langle {\overline{\varphi }}_4\rangle$. As a result, the inflaton field y is identified with the massless combination ${\varphi }_0$.

Non-renormalizable (NR) contributions to the superpotential (28) play an important role in the realization of the proposed cosmological scenario. In general, a non-renormalizable term of the order $N>3$ is of the following form:

$$\begin{array}{c}\hfill g_s^{N-2}{C}_N{\phi }_1{\phi }_2{\phi }_3\dots {\phi }_N\end{array}$$

(87)

where $C_N$ is a numerical constant determined by a correlator of the form (14). In [19], non-renormalizable superpotential terms have been fully analyzed up to and including the $N=6$ order using computer-assisted scan. It turns out that the solution (84) persists if supplemented with the conditions

$$\begin{array}{c}\hfill {\mathsf{\Phi }}_2=0,T_1^2=T_3^2=T_3·T_4=0,D_3=D_1^2=D_4^2=0.\end{array}$$

(88)

After applying (84) and (88), the sixth-order F-flatness constraints reduce to

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\mathsf{\Phi }}_{12}}}=0\Rightarrow & {\varphi }_4^2+{\varphi }_{+}{\varphi }_{-}+\left\{{\left(D_1·D_5\right)}^2+{\left(D_4·D_5\right)}^2\right\}=0,\hfill \end{array}$$

(89)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\overline{\mathsf{\Phi }}}_{12}}}=0\Rightarrow & {\overline{\varphi }}_4^2+{\overline{\varphi }}_{+}{\overline{\varphi }}_{-}=0,\hfill \end{array}$$

(90)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\mathsf{\Phi }}_3}}=0\Rightarrow & {\varphi }_4{\overline{\varphi }}_4+{\varphi }_{45}{\overline{\varphi }}_{45}+{\varphi }_{+}{\overline{\varphi }}_{+}+{\varphi }_{-}{\overline{\varphi }}_{-}=0,\hfill \end{array}$$

(91)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\overline{\varphi }}_3}}=0\Rightarrow & D_4·D_5+\left\{\left(D_1·D_5\right)\left(T_1·T_4\right)\right\}=0,\hfill \end{array}$$

(92)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial T_1}}=0\Rightarrow & \left\{\left({\overline{\varphi }}_4^2+{\overline{\varphi }}_{+}{\overline{\varphi }}_{-}\right){\mathsf{\Phi }}_{31}{T}_1\right\}=0,\hfill \end{array}$$

(93)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial D_1}}=0\Rightarrow & \left\{\left({\overline{\varphi }}_4^2+{\overline{\varphi }}_{+}{\overline{\varphi }}_{-}\right){\mathsf{\Phi }}_{31}{D}_1\right\}=0,\hfill \end{array}$$

(94)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial D_4}}=0\Rightarrow & \left\{\left({\overline{\varphi }}_4^2+{\overline{\varphi }}_{+}{\overline{\varphi }}_{-}\right){\mathsf{\Phi }}_{31}{D}_4\right\}=0,\hfill \end{array}$$

(95)

where we have omitted order-one numerical coefficients for the terms included in the curly brackets. Equations (89), (90) and (92)–(95) can be solved perturbatively assuming

$$\begin{array}{cc}\hfill D_4·D_5& =-\left\{(D_1·D_5)(T_1·T_4)\right\}\sim {\xi }^4,\hfill \end{array}$$

(96)

$$\begin{array}{cc}\hfill {\varphi }_4^2+{\varphi }_{+}{\varphi }_{-}& ={(D_1·D_5)}^2\sim {\xi }^4,\hfill \end{array}$$

(97)

$$\begin{array}{cc}\hfill {\overline{\varphi }}_4^2& =-{\overline{\varphi }}_{+}{\overline{\varphi }}_{-}\sim {\xi }^4,\hfill \end{array}$$

(98)

leaving only Equation (91) as a constraint among the non-vanishing VEVs.

Next, we consider higher-order contributions to the inflaton mass and mixings. To this end, we focus on non-renormalizable superpotential couplings comprising ${\varphi }_3,{\overline{\varphi }}_3$, that is, superpotential terms of the form $\left({\varphi }_3\phi \right)$${\phi }_1\dots {\phi }_{N-2}$ and $\left({\overline{\varphi }}_3\phi \right)$${\phi }_1\dots {\phi }_{N-2}$, where $\phi \in \left\{{\mathsf{\Phi }}_1,{\mathsf{\Phi }}_2,{\mathsf{\Phi }}_4,{\mathsf{\Phi }}_{31},{\mathsf{\Phi }}_{23},{\overline{\mathsf{\Phi }}}_{31},{\overline{\mathsf{\Phi }}}_{23},{\varphi }_i,{\overline{\varphi }}_i\right\},i=1,2,3,45$, and ${\phi }_a$ is another singlet or some hidden sector field VEV. It turns out that, up to and including sixth-order non-renormalizable superpotential terms, only one coupling of this type exists

$$\begin{array}{c}\hfill \left[{\overline{\varphi }}_3{\mathsf{\Phi }}_4\right](D_1·D_5)(T_1·T_4).\end{array}$$

(99)

Moreover, to this order, there is a single mixing term involving the singlet ${\mathsf{\Phi }}_4$. This term is $\left[{\mathsf{\Phi }}_4{\mathsf{\Phi }}_{12}\right]{(D_1·D_5)}^2$, which corresponds to mixing with ${\mathsf{\Phi }}_{12}$. However, this mixing can be ignored as ${\mathsf{\Phi }}_{12}$ is superheavy due to a tree-level mass term. Following the discussion above, the term (99) identifies ${\mathsf{\Phi }}_4$ as the goldstino superfield. An additional scan for inflaton–goldstino superfield non-renormalizable interactions reveals a single relevant term at the eighth order:

$$\begin{array}{c}\hfill \left[{\overline{\varphi }}_3^2{\mathsf{\Phi }}_4\right](D_1·D_4)(T_1·T_4){\mathsf{\Phi }}_{31}.\end{array}$$

(100)

Putting the inflaton-related non-renormalizable interactions together, we end up with the effective superpotential

$$\begin{array}{cc}\hfill W_I=& g_s{C}_6{\left(g_s\sqrt{2{\alpha }^{\prime }}\right)}^3{\overline{\varphi }}_3{\mathsf{\Phi }}_4\langle D_1\rangle ·\langle D_5\rangle \langle T_1\rangle ·\langle T_4\rangle +\hfill \\ \hfill & g_s{C}_8{\left(g_s\sqrt{2{\alpha }^{\prime }}\right)}^5{\overline{\varphi }}_3^2{\mathsf{\Phi }}_4\langle D_1\rangle ·\langle D_4\rangle \langle T_1\rangle ·\langle T_4\rangle \langle {\mathsf{\Phi }}_{31}\rangle ,\hfill \end{array}$$

(101)

where we denote by $C_6$ and $C_8$ the numerical values of the associated $N=6$ and $N=8$ correlators, respectively, and we have restored the mass units. This superpotential can be recast in the form (40)

$$\begin{array}{c}\hfill W_I=M_{I}z(y-\lambda y^2),\end{array}$$

(102)

where $y={\varphi }_0,z={\mathsf{\Phi }}_4$, and

$$\begin{array}{c}\hfill M_I={\zeta }^4{\displaystyle \frac{\gamma }{g_s}}{\displaystyle \frac{1}{\sqrt{2{\alpha }^{\prime }}}},\lambda =-g_s\zeta {\displaystyle \frac{\delta }{\gamma }}\sqrt{2{\alpha }^{\prime }}{M}_P,\end{array}$$

(103)

with

$$\begin{array}{cc}\hfill \gamma & =-g_s{C}_6{\displaystyle \frac{\langle D_1\rangle ·\langle D_5\rangle \langle T_1\rangle ·\langle T_4\rangle }{\langle {\varphi }^4\rangle }}cos\omega ,\hfill \end{array}$$

(104)

$$\begin{array}{cc}\hfill \delta & =g_s{C}_8{\displaystyle \frac{\langle D_1\rangle ·\langle D_4\rangle \langle T_1\rangle ·\langle T_4\rangle }{\langle {\varphi }^5\rangle }}\langle {\mathsf{\Phi }}_{31}\rangle {cos}^2\omega ,\hfill \end{array}$$

(105)

$$\begin{array}{cc}\hfill \zeta & =\langle \varphi \rangle g_s\sqrt{2{\alpha }^{\prime }}.\hfill \end{array}$$

(106)

Here, $\langle \varphi \rangle$ denotes a typical VEV satisfying all D- and F-flatness conditions (80)–(84) and (89)–(95). Substituting the constants’ numerical values and assuming typical VEVs compatible with flatness conditions, we obtain $M_I\sim {10}^{13}$ GeV and $\lambda \sim g_s{\displaystyle \frac{M_P}{M_s}}$, a tuneable parameter of the order of one. It is curious to note that, by tuning $\lambda$ to unity, one can produce, in the string spectrum, a scalar that has exactly the properties of an $R^2$ term in the effective action. Altogether, we conclude that the Starobinsky-like scenario of inflation can be successfully realized in the context of the string-derived flipped $SU\left(5\right)\times U\left(1\right)$ model.

6. Breaking the Flipped $\mathit{SU}\left(\mathbf{5}\right)\times \mathit{U}\left(\mathbf{1}\right)$ Symmetry and Particle Phenomenology

As explained in Section 5, our cosmological scenario assumes that the flipped $SU\left(5\right)\times U\left(1\right)$ symmetry remains unbroken during the inflation era. GUT symmetry breaking takes place through a first-order phase transition at a temperature below the inflation scale [51]. A detailed analysis of the vacuum structure of the theory after the end of the inflation is presented in Ref. [49]. This analysis is based on solving the modified F- and D-flatness conditions while incorporating additional constraints arising from phenomenological requirements, such as doublet–triplet splitting. It also accounts for the modifications due to the VEV development of a pair of $\mathbf{10}+\overline{\mathbf{10}}$ fields associated with flipped $SU\left(5\right)\times U\left(1\right)$ symmetry breaking.

As explained previously, in the presence of several $\mathbf{5}+\overline{\mathbf{5}}$ $SU\left(5\right)$ pairs, an extension of the doublet–triplet flipped $SU\left(5\right)\times U\left(1\right)$ mechanism is necessary for successful phenomenology. Assuming that ${\overline{F}}_5$ and $F_1,F_3$ develop VEVs, with $F={\alpha }_1{F}_1+{\alpha }_3{F}_3=\overline{F}$, $|a_1{|}^2+{\left|a_3\right|}^2=1$, the tree-level Higgs doublet and extra triplet mass matrices are

$$M_2^{\left(3\right)}=\begin{array}{cc}& \begin{array}{cccc}{H}_{d1}& H_{d2}& H_{d3}& H_{d45}\end{array}\\ \begin{array}{c}{H}_{u1}\\ H_{u2}\\ H_{u3}\\ H_{u45}\end{array}& \left(\begin{array}{cccc}0& {\overline{\mathsf{\Phi }}}_{12}& {\mathsf{\Phi }}_{31}& 0\\ {\mathsf{\Phi }}_{12}& 0& {\overline{\mathsf{\Phi }}}_{23}& 0\\ {\overline{\mathsf{\Phi }}}_{31}& {\mathsf{\Phi }}_{23}& 0& {\varphi }_{45}\\ 0& 0& {\overline{\varphi }}_{45}& {\mathsf{\Phi }}_3\end{array}\right)\end{array},$$

(107)

and

$$M_3^{\left(3\right)}=\begin{array}{cc}& \begin{array}{ccccc}{\overline{d}}_{h1}^c& {\overline{d}}_{h2}^c& {\overline{d}}_{h3}^c& {\overline{d}}_{h45}^c& {\overline{d}}_H^c\end{array}\\ \begin{array}{c}{d}_{h1}^c\\ d_{h2}^c\\ d_{h3}^c\\ d_{h45}^c\\ d_H^c\end{array}& \left(\begin{array}{ccccc}0& {\overline{\mathsf{\Phi }}}_{12}& {\mathsf{\Phi }}_{31}& 0& 0\\ {\mathsf{\Phi }}_{12}& 0& {\overline{\mathsf{\Phi }}}_{23}& 0& 2{\overline{F}}_5\\ {\overline{\mathsf{\Phi }}}_{31}& {\mathsf{\Phi }}_{23}& 0& {\varphi }_{45}& 0\\ 0& 0& {\overline{\varphi }}_{45}& {\mathsf{\Phi }}_3& 0\\ 2F_1& 0& 0& 0& 0\end{array}\right)\end{array},$$

(108)

respectively. Here, ${\mathsf{\Phi }}_3,{\mathsf{\Phi }}_{12},{\mathsf{\Phi }}_{31},{\mathsf{\Phi }}_{23},{\varphi }_{45},{\overline{\mathsf{\Phi }}}_{12},{\overline{\mathsf{\Phi }}}_{31},{\overline{\mathsf{\Phi }}}_{23},{\overline{\varphi }}_{45},F_1,F_3,{\overline{F}}_5$ stand for the VEVs of the associated fields, and $d_H^c,{\overline{d}}_H^c,$ refer to the extra triplets residing in $F_1,{\overline{F}}_5$, respectively. In this notation, the doublet–triplet spitting condition can be written as

$$\begin{array}{c}\hfill det{M}_2^{\left(3\right)}=0,det{M}_3^{\left(3\right)}\ne 0.\end{array}$$

(109)

An additional phenomenological constraint arises from the requirement that the top quark acquires its mass from the tree-level superpotential (28), and more specifically from the term $F_{4}5{\overline{h}}_{45}$, which, in turn, dictates that $H_{u45}$ must remain massless. A detailed analysis performed in Ref. [49] shows that, when restricting to cases with, at most, two vanishing VEVs (out of the eight entering the doublet mass matrix in (108)), these constraints are satisfied by only two configurations:

(a)

${\mathsf{\Phi }}_{12}={\mathsf{\Phi }}_3=0$ that yields $det{M}_3^{\left(3\right)}=-4F_1{\overline{F}}_5{\varphi }_{45}{\overline{\varphi }}_{45}{\overline{\mathsf{\Phi }}}_{12}$ and a pair of massless doublets $H_d,H_u$:

$$\begin{array}{cc}\hfill H_d& ={\varphi }_{45}{H}_{d1}-{\overline{\mathsf{\Phi }}}_{31}{H}_{d45},\hfill \end{array}$$

(110)

$$\begin{array}{cc}\hfill H_u& ={\overline{\varphi }}_{45}{H}_{u2}-{\overline{\mathsf{\Phi }}}_{23}{H}_{u45}.\hfill \end{array}$$

(111)

(b)

${\mathsf{\Phi }}_{12}={\overline{\mathsf{\Phi }}}_{31}=0$ that has $det{M}_3^{\left(3\right)}=-4F_1{\overline{F}}_5\left({\mathsf{\Phi }}_{23}{\mathsf{\Phi }}_{31}{\mathsf{\Phi }}_3+{\varphi }_{45}{\overline{\varphi }}_{45}{\overline{\mathsf{\Phi }}}_{12}\right)$, and

$$\begin{array}{cc}\hfill H_d& =H_{d1},\hfill \end{array}$$

(112)

$$\begin{array}{cc}\hfill H_u& ={\mathsf{\Phi }}_{23}{\overline{\mathsf{\Phi }}}_{23}{\mathsf{\Phi }}_3{H}_{u1}-\left({\mathsf{\Phi }}_{31}{\mathsf{\Phi }}_{23}{\mathsf{\Phi }}_3+{\overline{\mathsf{\Phi }}}_{12}{\varphi }_{45}{\overline{\varphi }}_{45}\right)H_{u2}-{\overline{\mathsf{\Phi }}}_{12}{\overline{\mathsf{\Phi }}}_{23}{\mathsf{\Phi }}_3{H}_{u3}+{\overline{\mathsf{\Phi }}}_{12}{\overline{\mathsf{\Phi }}}_{23}{\varphi }_{45}{H}_{u45}.\hfill \end{array}$$

(113)

In the sequel, focus will be placed on the solution (a), as the requirement ${\mathsf{\Phi }}_3\ne 0$ in solution (b) leads, among other issues, to an excessive number of additional non-renormalizable contributions to the superpotential, owing to the fact that ${\mathsf{\Phi }}_3$ is a total singlet.

The D-flatness conditions (80)–(83) are slightly modified due to the VEVs of $F_1,F_3,{\overline{F}}_5$, as follows:

$$\begin{array}{cc}\hfill {\mathcal{D}}_1^{\prime }& \to {\mathcal{D}}_1^{\prime }+{\displaystyle \frac{1}{2}}{\left|F_3\right|}^2=0,\hfill \end{array}$$

(114)

$$\begin{array}{cc}\hfill {\mathcal{D}}_2^{\prime }& \to {\mathcal{D}}_2^{\prime }-{\displaystyle \frac{1}{2}}{\left|F_3\right|}^2=0,\hfill \end{array}$$

(115)

$$\begin{array}{cc}\hfill {\mathcal{D}}_3^{\prime }& \to {\mathcal{D}}_3^{\prime }=0,\hfill \end{array}$$

(116)

$$\begin{array}{cc}\hfill {\mathcal{D}}_4^{\prime }& \to {\mathcal{D}}_4^{\prime }-{\displaystyle \frac{1}{2}}{\left|{\overline{F}}_5\right|}^2=0,\hfill \end{array}$$

(117)

where (82) remains invariant. These are supplemented by the non-abelian D-flatness constraints associated with the breaking of $SU\left(5\right),SU\left(4\right)$, and $SO\left(10\right)$:

$$\begin{array}{c}\hfill |{\alpha }_1{F}_1{|}^2+|{\alpha }_3{F}_3{|}^2={\left|{\overline{F}}_5\right|}^2,\end{array}$$

(118)

$$\begin{array}{c}\hfill \sum _{i=1}^6{D}_i^{*}{\tau }^a{D}_i=0,a=1,\dots ,15,\end{array}$$

(119)

$$\begin{array}{c}\hfill \sum _{i=1}^6{T}_i^{*}{\lambda }^A{T}_i=0,A=1,\dots ,45,\end{array}$$

(120)

where ${\tau }^a$ and ${\lambda }^A$ are the generators of $SU\left(4\right)\sim SO\left(6\right)$ and $SO\left(10\right)$, respectively.

Higher-order $N>3$ non-renormalizable superpotential contributions, defined in (87), play an important role in phenomenology, although they are suppressed by inverse powers of the string scale. They are tasked with the role of providing heavy masses for the pairs of surplus states $\overline{f_{\mathrm{i}}}\left(\overline{\mathbf{5}},-{\displaystyle \frac{3}{2}}\right),f_4\left(\mathbf{5},+{\displaystyle \frac{3}{2}}\right)$ and ${\ell }_j^c\left(\mathbf{1},+{\displaystyle \frac{5}{2}}\right),{\overline{\ell }}_4^c\left(\mathbf{1},-{\displaystyle \frac{3}{2}}\right)$, where i and j each take exactly one value from $1,2,3,5$, which appear in the string model massless spectrum (see Table A1), as well as provide mass terms for the lighter fermion generations. Moreover, these are expected to preserve the structure of the tree-level Higgs doublet mass matrix, $M_2^{\left(3\right)}$, and, most importantly, to preserve $det{M}_2^{\left(3\right)}=0$.

A comprehensive computer-assisted scan for candidate NR superpotential terms uncovers a total of 15 couplings for $N=4$ and 256 couplings for $N=5$. Among the $N=4$ NR couplings, we identify the following suitable surplus state mass term:

$$\begin{array}{c}\hfill \left(f_4{\overline{f}}_3+{\overline{\ell }}_4^c{\ell }_3^c\right)T_3·T_4,\end{array}$$

(121)

which, in turn, requires assigning VEVs to hidden sector states $T_3,T_4$ such that $T_3·T_4\ne 0$.

The aforementioned phenomenological constraints, combined with a detailed analysis of superpotential terms [49], lead to

$$\begin{array}{c}\hfill {\mathsf{\Phi }}_{12}={\mathsf{\Phi }}_3={\mathsf{\Phi }}_4={\mathsf{\Phi }}_5=0,\end{array}$$

(122)

$$\begin{array}{c}\hfill {\varphi }_1={\varphi }_2={\varphi }_3={\overline{\varphi }}_2={\overline{\varphi }}_3=0,\end{array}$$

(123)

$$\begin{array}{c}\hfill T_2=T_5=T_4^2=T_1·T_4=0,\end{array}$$

(124)

$$\begin{array}{c}\hfill D_2=D_5=0,F_2=F_4=0,\end{array}$$

(125)

accompanied by

$$\begin{array}{c}\hfill \left\{{\mathsf{\Phi }}_{31},{\mathsf{\Phi }}_{23},{\varphi }_{45},{\overline{\mathsf{\Phi }}}_{12},{\overline{\mathsf{\Phi }}}_{31},{\overline{\mathsf{\Phi }}}_{23},T_3·T_4\right\}\ne 0.\end{array}$$

(126)

Under these assignments, the F-flatness conditions including up to $N=5$ NR terms, reduce to

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\mathsf{\Phi }}_{12}}}=0\Rightarrow & {\varphi }_1^2+{\varphi }_4^2+{\mathsf{\Phi }}_{23}{\mathsf{\Phi }}_{31}=0\hfill \end{array}$$

(127)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\overline{\mathsf{\Phi }}}_{12}}}=0\Rightarrow & {\overline{\varphi }}_1^2+{\overline{\varphi }}_4^2+{\overline{\varphi }}_{+}{\overline{\varphi }}_{-}+{\overline{\mathsf{\Phi }}}_{23}{\overline{\mathsf{\Phi }}}_{31}+\left\{F_1^2{\overline{F}}_5^2\right\}=0\hfill \end{array}$$

(128)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\mathsf{\Phi }}_{31}}}=0\Rightarrow & \left\{\left({\overline{\varphi }}_1^2+{\overline{\varphi }}_4^2+{\overline{\varphi }}_{+}{\overline{\varphi }}_{-}\right)\left(D_1^2+D_4^2+T_1^2\right)\right\}=0\hfill \end{array}$$

(129)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\overline{\mathsf{\Phi }}}_{31}}}=0\Rightarrow & {\overline{\mathsf{\Phi }}}_{12}{\overline{\mathsf{\Phi }}}_{23}=0\hfill \end{array}$$

(130)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\overline{\mathsf{\Phi }}}_{23}}}=0\Rightarrow & {\overline{\mathsf{\Phi }}}_{12}{\overline{\mathsf{\Phi }}}_{31}+D_1^2+D_4^2+T_1^2=0\hfill \end{array}$$

(131)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\mathsf{\Phi }}_3}}=0\Rightarrow & {\varphi }_1{\overline{\varphi }}_1+{\varphi }_4{\overline{\varphi }}_4+{\varphi }_{-}{\overline{\varphi }}_{-}+{\varphi }_{+}{\overline{\varphi }}_{+}+{\varphi }_{45}{\overline{\varphi }}_{45}=0\hfill \end{array}$$

(132)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial F_1}}=0\Rightarrow & \left\{F_1{\overline{F}}_5^2{\overline{\mathsf{\Phi }}}_{12}\right\}=0\hfill \end{array}$$

(133)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\overline{F}}_5}}=0\Rightarrow & \left\{{\overline{F}}_5{\overline{F}}_1^2{\overline{\mathsf{\Phi }}}_{12}\right\}=0\hfill \end{array}$$

(134)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\varphi }_3}}=0\Rightarrow & \left\{\left(F_1{\overline{F}}_5\right)(T_1·T_4)\right\}=0\hfill \end{array}$$

(135)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\overline{\varphi }}_1}}=0\Rightarrow & 2{\overline{\varphi }}_1\left[{\overline{\mathsf{\Phi }}}_{12}+\left\{{\mathsf{\Phi }}_{31}(D_1^2+D_4^2+T_1^2)\right\}\right]=0\hfill \end{array}$$

(136)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\overline{\varphi }}_4}}=0\Rightarrow & 2{\overline{\varphi }}_4\left[{\overline{\mathsf{\Phi }}}_{12}+\left\{{\mathsf{\Phi }}_{31}(D_1^2+D_4^2+T_1^2)\right\}\right]=0\hfill \end{array}$$

(137)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\overline{\varphi }}_{+}}}=0\Rightarrow & {\overline{\varphi }}_{-}\left[{\overline{\mathsf{\Phi }}}_{12}+\left\{{\mathsf{\Phi }}_{31}(D_1^2+D_4^2+T_1^2)\right\}\right]=0\hfill \end{array}$$

(138)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial {\overline{\varphi }}_{-}}}=0\Rightarrow & {\overline{\varphi }}_{+}\left[{\overline{\mathsf{\Phi }}}_{12}+\left\{{\mathsf{\Phi }}_{31}(D_1^2+D_4^2+T_1^2)\right\}\right]=0\hfill \end{array}$$

(139)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial T_1}}=0\Rightarrow & 2T_1\left[{\overline{\mathsf{\Phi }}}_{23}+\left\{{\mathsf{\Phi }}_{31}({\overline{\varphi }}_1^2+{\overline{\varphi }}_4^2+{\overline{\varphi }}_{+}{\overline{\varphi }}_{-})\right\}\right]=0\hfill \end{array}$$

(140)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial T_4}}=0\Rightarrow & 2T_4\left[{\mathsf{\Phi }}_{23}+\left\{{\overline{\mathsf{\Phi }}}_{31}({\varphi }_1^2+{\varphi }_4^2+{\varphi }_{+}{\varphi }_{-})\right\}\right]=0\hfill \end{array}$$

(141)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial D_1}}=0\Rightarrow & 2D_1\left[{\overline{\mathsf{\Phi }}}_{23}+\left\{{\mathsf{\Phi }}_{31}({\overline{\varphi }}_1^2+{\overline{\varphi }}_4^2+{\overline{\varphi }}_{+}{\overline{\varphi }}_{-})\right\}\right]=0\hfill \end{array}$$

(142)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial D_1}}=0\Rightarrow & 2D_4\left[{\overline{\mathsf{\Phi }}}_{23}+\left\{{\mathsf{\Phi }}_{31}({\overline{\varphi }}_1^2+{\overline{\varphi }}_4^2+{\overline{\varphi }}_{+}{\overline{\varphi }}_{-})\right\}\right]=0\hfill \end{array}$$

(143)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial W}{\partial D_5}}=0\Rightarrow & \left\{D_3\left(F_3{\overline{F}}_5\right)\right\}=0\hfill \end{array}$$

(144)

These equations admit a perturbative solution in terms of $\xi$, which is of the order of ${10}^{-1}$ in string-scale units. Assuming, to leading order,

$$\begin{array}{c}\hfill {\mathsf{\Phi }}_{31},{\overline{\mathsf{\Phi }}}_{31},{\varphi }_{45},{\varphi }_4,{\overline{\varphi }}_1,{\varphi }_{+},{\varphi }_{-},{\overline{\varphi }}_{+},{\overline{\varphi }}_{-}\sim \xi ,\end{array}$$

(145)

$$\begin{array}{c}\hfill D_1,D_4,T_1\sim \xi ,F_1{\overline{F}}_5\sim {\xi }^3,\end{array}$$

(146)

and utilizing specific superpotential coupling relations established in Ref. [49], the F-flatness conditions to order ${\xi }^4$ reduce to

$$\begin{array}{cc}\hfill {\overline{\mathsf{\Phi }}}_{12}& =-c_4{D}_4^2{\mathsf{\Phi }}_{31},\hfill \end{array}$$

(147)

$$\begin{array}{cc}\hfill T_1^2& =-{\mathsf{\Phi }}_{31}{\mathsf{\Phi }}_{12}-D_1^2-D_4^2,\hfill \end{array}$$

(148)

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_{23}& =-\left\{{\mathsf{\Phi }}_{31}({\varphi }_{+}{\varphi }_{-}+{\varphi }_4^2\right\},\hfill \end{array}$$

(149)

$$\begin{array}{cc}\hfill {\overline{\mathsf{\Phi }}}_{23}& =-2c_4{\mathsf{\Phi }}_{31}({\overline{\varphi }}_{+}{\overline{\varphi }}_{-}-{\overline{\varphi }}_1^2-{\overline{\varphi }}_4^2),\hfill \end{array}$$

(150)

$$\begin{array}{cc}\hfill {\overline{\varphi }}_1^2& =-{\overline{\varphi }}_{+}{\overline{\varphi }}_{-}-{\overline{\varphi }}_4^2-{\overline{\varphi }}_{31}{\overline{\varphi }}_{23},\hfill \end{array}$$

(151)

and

$$\begin{array}{c}\hfill {\varphi }_4^2+{\varphi }_{-}{\varphi }_{+}=0,\end{array}$$

(152)

$$\begin{array}{c}\hfill {\varphi }_4{\overline{\varphi }}_4+{\varphi }_{+}{\overline{\varphi }}_{+}+{\varphi }_{-}{\overline{\varphi }}_{-}+{\varphi }_{45}{\overline{\varphi }}_{45}=0,\end{array}$$

(153)

$$\begin{array}{c}\hfill \left\{D_3\left(F_3{\overline{F}}_5\right)\right\}=0.\end{array}$$

(154)

Here, $c_4$ is the coupling of the term $D_1^2{\mathsf{\Phi }}_{31}{\overline{\varphi }}_{+}{\overline{\varphi }}_{-}$, given by $c_4={\displaystyle \frac{g_s^3\sqrt{2}}{{\left(2\pi \right)}^2}}{I}_{c_4}$, where $I_{c_4}\approx 189.07$ is a numerical constant. The D-flatness conditions related to the abelian symmetries (114)–(117) are found to be compatible with this solution, while the non-abelian D-flatness conditions (118) and (119) can be explicitly solved by choosing real VEVs for $D_1,D_3,D_4$ and $T_1,T_3,T_4$ using the following ansatz:

$$\begin{array}{cc}\hfill T_1& =\left(r{e}^{-i\theta },ir{e}^{-i\theta },b,0,\dots ,0\right),\hfill \end{array}$$

(155)

$$\begin{array}{cc}\hfill T_3& =\left(ic,c,c+ie,0,\dots ,0\right),\hfill \end{array}$$

(156)

$$\begin{array}{cc}\hfill T_4& =\left(a,ia,0,\dots ,0\right),\hfill \end{array}$$

(157)

where $r,a,b,c,e,\theta$ are free real parameters with $r^2=c^2-a^2,e^2=b^2{r}^2/c^2-c^2,tan\theta =c/e$.

As expected, the doublet mass matrix (108) receives contributions from NR superpotential terms of the form $h_i{\overline{h}}_j{\phi }^{N-3}$, where $N>3$, $i,j=1,2,3,45$, and ${\phi }^{N-3}$ stands for a combination of $N-3$ field VEVs. An exhaustive computer-aided investigation up to $N=7$ yields, under the additional assumptions of

$$\begin{array}{c}\hfill {\mathsf{\Phi }}_1={\mathsf{\Phi }}_2=0,{\overline{\varphi }}_{45}\lesssim {\xi }^2,\end{array}$$

(158)

that, to the fifth order in the $\xi$ expansion,

$$M_2^{\left(7\right)}=\begin{array}{cc}& \begin{array}{cccc}{H}_{d1}& H_{d2}& H_{d3}& H_{d45}\end{array}\\ \begin{array}{c}{H}_{u1}\\ H_{u2}\\ H_{u3}\\ H_{u45}\end{array}& \left(\begin{array}{cccc}0& {\overline{\mathsf{\Phi }}}_{12}& {\mathsf{\Phi }}_{31}& 0\\ 0& a^{\left(7\right)}& {\overline{\mathsf{\Phi }}}_{23}& 0\\ {\overline{\mathsf{\Phi }}}_{31}& {\mathsf{\Phi }}_{23}& 0& {\varphi }_{45}\\ 0& {\overline{b}}^{\left(5\right)}& {\overline{\varphi }}_{45}& 0\end{array}\right)\end{array},$$

(159)

with ${\overline{b}}^{\left(5\right)}=\left\{{\overline{\varphi }}_{45}(D_1^2+D_4^2+T_1^2)\right\}\sim {\xi }^2{\overline{\varphi }}_{45}$, and $a^{\left(7\right)}=\left\{({\overline{\varphi }}_1^2+{\overline{\varphi }}_4^2+{\overline{\varphi }}_{-}{\overline{\varphi }}_{+})(D_1^2+D_4^2\right.$$\left.+T_1^2){\mathsf{\Phi }}_{31}\right\}\sim {\xi }^5$, where the first term comes from $N=5$ and the second from $N=7$ superpotential couplings. The mass matrix $M_2^{\left(7\right)}$ results in a pair of massless eigenstates:

$$\begin{array}{cc}\hfill H_d& =cos\theta H_{d1}-sin\theta H_{d45},\hfill \end{array}$$

(160)

$$\begin{array}{cc}\hfill H_u& =sin\overline{\theta }sin\overline{\vartheta }{\overline{H}}_{u1}+sin\overline{\theta }cos\overline{\vartheta }{\overline{H}}_{u2}+cos\overline{\theta }{H}_{u45},\hfill \end{array}$$

(161)

where $\theta ,\overline{\theta },\overline{\vartheta }$ are defined using the following relations:

$$\begin{array}{c}\hfill tan\theta ={\displaystyle \frac{{\overline{\mathsf{\Phi }}}_{31}}{{\varphi }_{45}}},tan\overline{\vartheta }={\displaystyle \frac{{\overline{b}}^{\left(5\right)}{\overline{\mathsf{\Phi }}}_{23}-a^{\left(7\right)}{\overline{\varphi }}_{45}}{{\overline{\mathsf{\Phi }}}_{12}{\overline{\varphi }}_{45}-{\overline{b}}^{\left(5\right)}{\mathsf{\Phi }}_{31}}},tan\overline{\theta }cos\overline{\vartheta }={\displaystyle \frac{{\overline{\mathsf{\Phi }}}_{12}{\overline{\varphi }}_{45}-{\overline{b}}^{\left(5\right)}{\mathsf{\Phi }}_{31}}{a^{\left(7\right)}{\mathsf{\Phi }}_{31}-{\overline{\mathsf{\Phi }}}_{12}{\overline{\mathsf{\Phi }}}_{23}}}.\end{array}$$

(162)

In terms of the massless doublets $H_d,H_u$ we have

$$\begin{array}{cc}\hfill H_{d1}& =cos\theta H_d+\dots ,H_{d45}=sin\theta H_d+\dots ,\hfill \end{array}$$

(163)

$$\begin{array}{cc}\hfill H_{u1}& =sin\overline{\theta }sin\overline{\vartheta }{\overline{H}}_u+\dots ,H_{u2}=sin\overline{\theta }cos\overline{\vartheta }{\overline{H}}_u+\dots ,H_{u45}=cos\overline{\theta }{\overline{H}}_u+\dots ,\hfill \end{array}$$

(164)

where the dots represent superheavy massive doublets. The subsequent analysis focuses on the case $cos\overline{\theta }\sim cos\overline{\vartheta }\sim 1$, for simplicity.

Given the surviving doublets, it is possible to identify the superpotential terms providing masses to charged fermions. An extensive and thorough computer search, including relevant 7th-order NR terms, yields the following candidate couplings:

$$\begin{array}{cc}\mathrm{Up}\mathrm{quarks}:\hfill & F_4{\overline{f}}_5{\overline{h}}_{45}+F_2{\overline{f}}_2{\overline{h}}_{45}{\overline{\varphi }}_4+F_1{\overline{f}}_1{\overline{h}}_2{\mathsf{\Phi }}_{31}{\varphi }_{45}{\overline{\varphi }}_1+\hfill \\ & F_3{\overline{f}}_{45}{\overline{h}}_{45}(D_3·D_4){\overline{\mathsf{\Phi }}}_{23}+F_3{\overline{f}}_1{\overline{h}}_{45}\left[(T_1·T_3){\mathsf{\Phi }}_{31}{\overline{\varphi }}_{-}+(D_1·D_3){\mathsf{\Phi }}_{31}{\overline{\varphi }}_1\right],\hfill \end{array}$$

(165)

$$\begin{array}{c}\mathrm{Down}\mathrm{quarks}:F_4{F}_4{h}_1+F_2{F}_2{h}_1\left({\overline{\varphi }}_1^2-{\overline{\varphi }}_{-}{\overline{\varphi }}_{+}+\lambda {\overline{\varphi }}_4^2\right)+\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{F}_1{F}_3{h}_1(D_1·D_3){\overline{\mathsf{\Phi }}}_{23}+F_3{F}_4{h}_1(D_3·D_4){\overline{\mathsf{\Phi }}}_{23},\end{array}$$

(166)

$$\begin{array}{c}\mathrm{Charged}\mathrm{leptons}:{\overline{f}}_1{\ell }_1^c{h}_1+{\overline{f}}_2{\ell }_2^c{h}_1\left({\overline{\varphi }}_1^2-{\overline{\varphi }}_{-}{\overline{\varphi }}_{+}+\lambda {\overline{\varphi }}_4^2\right)+{\overline{f}}_5{\ell }_5^c{h}_1\left({\overline{\varphi }}_1^2+{\overline{\varphi }}_{-}{\overline{\varphi }}_{+}-{\overline{\varphi }}_4^2\right)\hfill \\ \hfill +{\overline{f}}_5{\ell }_5^c\left(h_1{\overline{\mathsf{\Phi }}}_{31}+h_{45}{\varphi }_{45}\right)\left({\overline{\varphi }}_1^2+{\overline{\varphi }}_4^2+{\overline{\varphi }}_{-}{\overline{\varphi }}_{+}\right).\end{array}$$

(167)

Here, coupling coefficients have been omitted except for a cocycle factor, $\lambda$, which could distinguish between couplings.

These couplings lead to quite successful predictions for fermion masses provided the following fermion identification:

$$\begin{array}{c}\hfill F_4={\left\{(t,b),b^c,{\nu }_{\tau }^c\right\}}_L,F_2={\left\{(u,d),d^c,{\nu }_e^c\right\}}_L,F_3={\left\{(u,d),d^c,{\nu }_e^c\right\}}_L,\end{array}$$

(168)

$$\begin{array}{c}\hfill {\overline{f}}_5={\left\{t^c,\left({\nu }_e,e\right)\right\}}_L,{\overline{f}}_2={\left\{c^c,({\nu }_{\mu },\mu )\right\}}_L,{\overline{f}}_1={\left\{u^c,({\nu }_{\tau },e)\right\}}_L,\end{array}$$

(169)

$$\begin{array}{c}\hfill {\ell }_1^c={\tau }_L^c,{\ell }_2^c={\mu }_L^c,{\ell }_5^c=e_L^c.\end{array}$$

(170)

Using the F/D-flatness relations, the detailed massless doublet expression, and the above assignments together with the $\xi$-expansion, we obtain the following expressions for the fermion masses at the string scale:

$$\begin{array}{cccc}& m_t=g{v}_{45},\hfill & \hfill m_b=g{v}_1,& m_{\tau }=m_b,\hfill \\ & m_c\sim {\zeta }^2{m}_t,\hfill & \hfill m_s\sim {\zeta }^2{m}_b,& m_{\mu }=m_s,\hfill \\ & m_u\sim {\zeta }^5{m}_t,\hfill & \hfill m_d\sim {\zeta }^3{m}_b,& m_e\sim {\zeta }^4{m}_{\tau },\hfill \end{array}$$

where $v_{45}$ and $v_1$ denote the VEVs of doublets a$H_u$ and $H_d$, respectively; $g=g_s\sqrt{2}$ is the GUT gauge coupling; and, typically, $\zeta \sim 1/10$. Predictions include a heavy mass for the top quark as a result of the tree-level superpotential mass term, the successful GUT relation $m_b=m_{\tau }$, and the suppression of second-generation masses relative to the third (heaviest) generation, as well as the suppression of first-generation masses relative to the second. With the exception of the relation $m_{\mu }=m_s$, which could be the subject of a separate analysis along with neutrino masses, these fermion mass predictions can be considered highly successful.

The breaking of the GUT symmetry brings to light an additional challenge that plagues GUT models: proton decay. While the aforementioned analysis ensures that all extra triplets acquire masses at the tree level, it must be verified whether dimension-5 baryon- and lepton-number-violating operators, endemic to SUSY GUTs, are adequately suppressed. In the string-derived flipped $SU\left(5\right)\times U\left(1\right)$ model under consideration, there are two sources of such operators. The first arises from effective field theory couplings that induce dimension-5 operators via triplet exchange, similar to those in conventional supersymmetric GUTs. The second source is attributed to effective operators generated by non-renormalizable (NR) couplings associated with the exchange of massive string modes.

The triplet-mass matrix arising from the NR interactions up to $N=7$ can be recast in the form

$$M_3^{\left(7\right)}=\begin{array}{cc}& \begin{array}{ccccc}{\overline{d}}_{h1}^c& {\overline{d}}_{h2}^c& {\overline{d}}_{h3}^c& {\overline{d}}_{h45}^c& {\overline{d}}_H^c\end{array}\\ \begin{array}{c}{d}_{h1}^c\\ d_{h2}^c\\ d_{h3}^c\\ d_{h45}^c\\ d_H^c\end{array}& \left(\begin{array}{ccccc}0& {\overline{\mathsf{\Phi }}}_{12}& {\mathsf{\Phi }}_{31}& 0& {\overline{s}}_1^{\left(5\right)}\\ {\mathsf{\Phi }}_{12}& 0& {\overline{\mathsf{\Phi }}}_{23}& 0& 2{\overline{F}}_5\\ {\overline{\mathsf{\Phi }}}_{31}& {\mathsf{\Phi }}_{23}& 0& {\varphi }_{45}& {\overline{s}}_3^{\left(5\right)}\\ 0& 0& {\overline{\varphi }}_{45}& {\mathsf{\Phi }}_3& 0\\ 2F_1& s_2^{\left(5\right)}& 0& s_4^{\left(5\right)}& s\end{array}\right)\end{array}+\mathcal{O}{\left(\xi \right)}^6,$$

(171)

in which ${\overline{s}}_1^{\left(5\right)}=\left\{{\overline{F}}_5({\varphi }_4^2+{\varphi }_{-}{\varphi }_{+}\right\}$, ${\overline{s}}_3^{\left(5\right)}=\left\{{\overline{F}}_5(D_1^2+D_4^2+T_1^2)\right\}$, $s_2^{\left(5\right)}=\left\{F_1({\varphi }_4^2+{\varphi }_{-}{\varphi }_{+})\right\}$, $s_4^{\left(5\right)}=\left\{F_1{\mathsf{\Phi }}_{31}{\varphi }_{45}\right\}$, and $s=\left\{F{\overline{F}}_5{\overline{\mathsf{\Phi }}}_{12}\right\}$. A laborious calculation of eigenvalues leads, at the lowest order in $\xi$, to eigenstate masses $\xi ,\xi ,{\xi }^{3/2},{\xi }^{3/2},{\xi }^5$, ensuring that all extra triplets are sufficiently massive. However, these triplets are associated with the first source of dimension-5 operators. As discussed in [57], in the case of the general flipped $SU\left(5\right)\times U\left(1\right)$ model, including one pair of $F,\overline{F}$ Higgs fields and four pairs of $\mathbf{5}+\overline{\mathbf{5}}$ and superpotential

$$\begin{array}{c}\hfill W=f_{ij}^a{F}_i{F}_j{h}_a+y_{ij}^a{F}_i\overline{f_j}{\overline{h}}_a+{\overline{h}}_a{\left(M_3\right)}_{ab}{\overline{h}}_b,\end{array}$$

(172)

where $f_{ij}^a,y_{ij}^a$ are coupling constants and $M_3$ is the triplet mass matrix, we have triplet-exchange dimension-5 operators of the form $QQQL$ that are proportional to

$$\begin{array}{c}\hfill O_{ijkl}^{QQQL}\sim {\displaystyle \frac{1}{det{M}_3}}\sum _{a,b=1}^5{y}_{kl}^a\mathrm{cofactor}{\left(M_3\right)}_{ab}{f}_{ij}^b.\end{array}$$

(173)

A detailed investigation of these operators, incorporating the $\xi$ expansion, reveals that the dominant contributions arise from the tree-level superpotential terms $F_2{F}_2{h}_2$ and $F_4{\overline{f}}_5{\overline{h}}_{45}$. These terms give rise to an effective operator $F_2{F}_2{F}_4{\overline{f}}_5$, which is proportional to

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{cofactor}{\left(M_3^{\left(7\right)}\right)}_{2,45}}{det{M}_3^{\left(7\right)}}}\sim {\displaystyle \frac{1}{{\xi }^5}}.\end{array}$$

(174)

However, considering the additional suppression due to the assignment of $F_2,F_4$ to second- and third-generation quarks, respectively, and assuming $m_{\mathrm{susy}}\gtrsim {10}^2\mathrm{TeV}$ [58], these operators turn out to lead to a proton lifetime that exceeds current experimental limits. The second type of dimension-5 operators corresponds to NR terms of the form $F^3\bar{f}{\langle \phi \rangle }^{N-4}$, where $\langle \phi \rangle$ represents singlet field VEVs. An explicit search shows that such operators first appear at $N=5$, yielding

$$\begin{array}{c}\hfill F_2^2{F}_3{\overline{f}}_3{\overline{\mathsf{\Phi }}}_{23}+F_4^2{F}_3{\overline{f}}_3{\mathsf{\Phi }}_{31}.\end{array}$$

(175)

The contribution of these operators is negligible, as they involve the superheavy state ${\overline{f}}_3$ (see (121)). Higher-order operators also exist, but they have been explicitly shown to vanish in the flipped $SU\left(5\right)\times U\left(1\right)$ model due to permutation symmetries [44]. Summarizing in the string flipped $SU\left(5\right)\times U\left(1\right)$ model under consideration, proton decay from all sources can remain consistent with experimental bounds, provided that the SUSY-breaking scale is $m_{\mathrm{susy}}\gtrsim {10}^2\mathrm{TeV}$.

7. Conclusions

We have reviewed the cosmology of the flipped $SU\left(5\right)\times U\left(1\right)$ model derived within the framework of the free-fermionic formulation of 4D heterotic superstrings. It has been demonstrated that the model can support successful Starobinsky-type inflation, with the inflaton identified as one of the gauge singlet massless states. The inflaton’s superpartner mixes with the right-handed neutrino. The breaking of an anomalous $U\left(1\right)$ symmetry, a characteristic feature of string-derived heterotic models, induces effective superpotential interactions involving the inflaton and the goldstino field arising at the 6th and 8th orders in the ${\alpha }^{\prime }$-expansion. The VEVs of the fields involved are consistent with F- and D-flatness conditions. The GUT symmetry $SU\left(5\right)\times U\left(1\right)$ remains unbroken during inflation with gauge symmetry breaking occuring at the end of the inflation via a second-order phase transition.

We have analyzed the post-inflationary phenomenology of the model through a detailed examination of higher-order superpotential terms. The breaking of an anomalous $U\left(1\right)$ symmetry necessitates the introduction of field VEVs, which are typically of the order $\xi =1/10$ in string units. Using a pertubative expansion in the small parameter $\xi$, we solve the F- and D- flatness conditions of the model. Subsequently, we identify the massless Higgs doublets responsible for generating fermion masses via electroweak symmetry breaking and explore the implementation of an extended doublet–triplet splitting mechanism. The analysis demonstrates that the model successfully reproduces the observed fermion mass hierarchy while addressing the proton decay problem.

In conclusion, the $SU\left(5\right)\times U\left(1\right)$ model is a very promising high-energy extension of the Standard Model, providing an explicit working example of string unification of all fundamental interactions and testable experimental predictions in both particle physics and cosmology.