ACT-05-21

MI-HET-768

Particle physics and cosmology of the string derived no-scale flipped $SU(5)$

I. Antoniadis<EMAIL_ADDRESS>D.V. Nanopoulos
<EMAIL_ADDRESS>J. Rizos<EMAIL_ADDRESS>
a Laboratoire de Physique Théorique et Hautes Energies - LPTHE
Sorbonne Université, CNRS, 4 Place Jussieu, 75005 Paris, France;
Department of Mathematical Sciences, University of Liverpool,
Liverpool L69 7ZL, United Kingdom
b George P. and Cynthia W. Mitchell Institute for Fundamental Physics and
Astronomy, Texas A&M University, College Station, TX 77843, USA;
Astroparticle Physics Group, Houston Advanced Research Center (HARC)
Mitchell Campus, Woodlands, TX 77381, USA;
Academy of Athens, Division of Natural Sciences, Athens 10679, Greece
c Physics Department, University of Ioannina, 45110, Ioannina, Greece;
School of Science and Technology, Hellenic Open University
Tsamadou 13-15, GR-26222 Patras, Greece

In a recent paper, we identified a cosmological sector of a flipped $SU(5)$
model derived in the free fermionic formulation of the heterotic superstring,
containing the inflaton and the goldstino superfields with a superpotential
leading to Starobinsky type inflation, while $SU(5){\times}U(1)$ is still
unbroken. Here, we study the properties and phenomenology of the vacuum after
the end of inflation, where the gauge group is broken to the Standard Model.
We identify a set of vacuum expectation values, triggered by the breaking of
an anomalous $U(1)_{A}$ gauge symmetry at roughly an order of magnitude below
the string scale, that solve the F and D-flatness supersymmetric conditions up
to 6th order in the superpotential which is explicitly computed, leading to a
successful particle phenomenology. In particular, all extra colour triplets
become superheavy guaranteeing observable proton stability, while the Higgs
doublet mass matrix has a massless pair eigenstate with realistic hierarchical
Yukawa couplings to quarks and leptons. The supersymmetry breaking scale is
constrained to be high, consistent with the non observation of supersymmetric
signals at the LHC.

###### Contents

  1. 1 Introduction
  2. 2 The revamped flipped model and its cosmology sector
  3. 3 Triplet-doublet splitting in $SU(5)\times U(1)$ string model
  4. 4 Solution of the F/D-flatness conditions
  5. 5 Computation of higher order superpotential terms
  6. 6 Fermion masses and proton decay
    1. 6.1 Fermion masses
    2. 6.2 Proton decay
  7. 7 Concluding remarks
  8. A String construction and spectrum of the flipped $SU(5)\times{U(1)}$ model
  9. B F-flatness condition analysis
  10. C  Operator product expansions, correlators etc

##  1 Introduction

In this work, we make an important step further on the avenue we started
recently [1], towards a string derived microscopic model that provides a
simultaneous description of fundamental particle physics and cosmology. The
model was constructed in 1989 [2] within the framework of free fermionic
formulation of four-dimensional (4d) heterotic superstring [3] and has an
observable sector based on the flipped $SU(5)\times U(1)$ gauge group with
three chiral families of quarks and leptons [4, 5]. The basis vectors of
boundary conditions defining the model, as well as its full massless spectrum
are given again for self consistency in Appendix A.

In [1], we identified the inflaton among the gauge singlet massless states of
the model with the superparner of a fermion mixed with the Right-handed
neutrinos [6]. It acquires a superpotential together with the goldstino at 6th
and 8th order in the string slope $\alpha^{\prime}$-expansion, via vacuum
expectation values (VEVs) of fields generated by the breaking of an anomalous
$U(1)_{A}$ gauge symmetry (characteristic in heterotic models [7]) at a
calculable scale an order of magnitude below the string mass, related to the
anomaly. This scale in string units introduces therefore a small parameter
allowing perturbative computations around the free-fermionic point where all
fields and moduli are fixed at zero VEVs. As a result, the inflation scale
turns out to be about five orders of magnitude lower, in the range of
$10^{13}$ GeV, while the superpotential leads to a Starobinsky-type inflation
[8] due to the no-scale structure of the low energy effective supergravity [9]
which is calculable in our model to all orders in $\alpha^{\prime}$ [1, 10,
11]. Note that during inflation, $SU(5)\times U(1)$ remains unbroken because
its breaking occurs via a first order phase transition at a critical
temperature which is lower that the scale of inflation [12, 13, 14].

In this work, we extend the previous analysis to the study of the vacuum of
the theory, after the end of inflation, where $SU(5)\times U(1)$ is broken to
the Standard Model. More precisely, we find a consistent set of VEVs that
solve the F and D-flatness equations up to 6th order in the
$\alpha^{\prime}$-expansion of the superpotential. This vacuum obviously
preserves supersymmetry, whose breaking we don’t discuss here. An important
result to emphasise is that the requirement of gauge symmetry breaking by a
pair of ${\bf 10}+{\bf\overline{10}}$ leads to a slight reorganisation of the
choice of VEVs for the $SU(5)\times U(1)$ gauge singlet states, compared to
the set we had during the inflationary phase, in order to satisfy the flatness
conditions.

Our analysis for finding the choice of VEVs requires in some cases the
knowledge of the exact coefficients of higher dimensional operators, or
precise relations among them. Thus, besides applying selection rules to find
which of those are non-vanishing, we need to perform some explicit
computations of superpotential terms to 5th or 6th order which is highly non-
trivial. The challenging part involves correlation functions of several
primary operators in the Ising model for which we concentrate a dedicated
section of this paper.

Obviously, one of our goals is to identify the quarks and leptons among the
three chiral generations and one vector-like pair, as well as a pair of Higgs
doublets with the required Yukawa couplings. On the other hand, one should
also ensure that all colour triplet states acquire masses at high scale so
that there is no dangerous proton decay. For these reasons, we impose in our
choice of VEVs that the colour triplet mass matrix has a non-vanishing
determinant, while the weak doublet matrix has exactly one massless eigenstate
with components along the doublets that provide a realistic hierarchical
Yukawa matrix, taking into account the successful phenomenological analysis of
the model that has been done in the past [2, 15, 16, 17, 18, 19]. Moreover,
the constraints from proton decay impose that the supersymmetry breaking scale
$m_{\rm susy}$ should be at least of order of tens of TeV, compatible with an
independent analysis of reheating and nucleosynthesis requiring $m_{\rm susy}$
to lye in this energy region [13], possibly within the reach of the next
generation of high energy hadron colliders.

The outline of the paper is the following. In Section 2, we review briefly our
previous results [1], such as the identification of the inflaton and the
goldstino, as well as the choice of VEVs giving rise to the inflationary
superpotential. In Section 3, we impose the phenomenological constraints
realising the triplet-double splitting at the string level, making all colour
triplets superheavy while leaving massless one pair of massless doublets. In
Section 4, we compute and solve the D and F-flatness conditions up the 6th
order, taking into account the above constraints. Section 5 is devoted to the
explicit computation of some superpotential coefficients involving higher
point functions of primary operators in the Ising model. In Section 6, we
perform the phenomenological analysis by identifying the electroweak Higgs
doublets and the quarks and leptons, and by computing in particular the
structure of fermion masses. We also study proton decay by computing the
relevant dimension-five operators induced by the Higgs triplets exchange, as
well as those emerging directly at the string level. Finally, Section 7
contains some concluding remarks. The paper has also three Appendices.
Appendix A contains a brief summary of the ‘revamped’ flipped $SU(5)$ string
model, Appendix B contains the list and details on the flatness conditions up
to 5th order in the superpotential which we use in our analysis in Section 4,
while Appendix C contains operator product expansions and various correlators
of the Ising model that we use in Section 5.

##  2 The revamped flipped model and its cosmology sector

For convenience of the reader, we recall briefly the massless spectrum of the
‘revamped’ flipped $SU(5)$ model (with the original notation of [2]), which
consist of:

  1. 1.

A gauge group $[SU(5)\times U(1)]\times U(1)^{4}\times[SO(10)\times SO(6)]$,
where besides the observable and hidden sectors in the first and last bracket,
there are four $U(1)$s, one combination of which is anomalous and becomes
massive.

  2. 2.

Four generations and one anti-generation of chiral matter in the
representations ${\bf 10}+{\bf\bar{5}}+{\bf 1}$ (and ${\bf\overline{10}}+{\bf
5}+{\bf\bar{1}}$) of $SU(5)\times U(1)$, denoted as
$F_{i},\bar{f}_{i},l_{i}^{c}$ (and
$\overline{F}_{5},f_{4},\bar{\ell}^{c}_{4}$) where $i$ labels the
corresponding vector of boundary conditions.

  3. 3.

Four pairs of ${\bf 5}+{\bf\bar{5}}$ containing the electroweak Higgs
doublets, denoted by $(h_{i},\bar{h}_{i})_{i=1,2,3}$ for those coming from the
Neveu-Schwarz (NS) sector and $(h_{45}+\bar{h}_{45})$ for the pair coming from
the $b_{4}+b_{5}$ sector.

  4. 4.

Five vectors $({\bf 10},{\bf 1})+({\bf 1},{\bf 6})$ of the hidden gauge group,
denoted as $T_{i}$ and $D_{i}$, respectively.

  5. 5.

Six pairs $({\bf 4}+{\bf\bar{4}})$ of $SO(6)$ with fractional electric charges
$\pm 1/2$, denoted as $(Z_{1}+\bar{Z}_{1})$, $(Z_{2}+\bar{Z}_{2})$,
$(Y_{1}+\bar{Y}_{1})$, $(X_{1}+\bar{X}_{1})$, $Y_{2},Y_{2}^{\prime}$,
$\bar{X}_{2},\bar{X}_{2}^{\prime}$.

  6. 6.

Ten pairs of non-abelian gauge singlets but charged under $U(1)^{4}$, denoted
by $\Phi_{12},\Phi_{23},\Phi_{31}$ (with their conjugates) for those coming
from the NS sector, and $\phi_{1},\dots,\phi_{4},\phi_{\pm},\phi_{45}$ (with
their conjugates) for those coming from the $b_{4}+b_{5}$ sector.

  7. 7.

Five gauge singlets from the NS sector, denoted as $\Phi_{1},\dots,\Phi_{5}$.

The full tree-level (trilinear) superpotential reads

$\displaystyle W_{3}=\,$ $\displaystyle g_{s}\sqrt{2}\left[\vrule
height=12.0pt,depth=1.0pt,width=0.0ptF_{1}F_{1}h_{1}+F_{2}F_{2}h_{2}+F_{4}F_{4}h_{1}+\overline{F}_{5}\overline{F}_{5}\overline{h}_{2}+F_{4}\overline{f}_{5}\overline{h}_{45}+F_{3}\overline{f}_{3}\overline{h}_{3}\right.$
$\displaystyle+\overline{f}_{1}\ell^{c}_{1}h_{1}+\overline{f}_{2}\ell^{c}_{2}h_{2}+\overline{f}_{5}\ell^{c}_{5}h_{2}+f_{4}\overline{\ell}^{c}_{4}\overline{h}_{1}$
$\displaystyle+\frac{1}{\sqrt{2}}F_{4}\overline{F}_{5}\phi_{3}+\frac{1}{\sqrt{2}}f_{4}\overline{f}_{5}\overline{\phi}_{2}+\frac{1}{\sqrt{2}}\overline{\ell}^{c}_{4}\ell^{c}_{5}\overline{\phi}_{2}$
$\displaystyle+h_{1}\overline{h}_{2}\Phi_{12}+\overline{h}_{1}h_{2}\overline{\Phi}_{12}+h_{2}\overline{h}_{3}\Phi_{23}+\overline{h}_{2}h_{3}\overline{\Phi}_{23}+h_{3}\overline{h}_{1}\Phi_{31}$
$\displaystyle+\overline{h}_{3}h_{1}\overline{\Phi}_{31}+h_{3}\overline{h}_{45}\overline{\phi}_{45}+\overline{h}_{3}h_{45}\phi_{45}+\frac{1}{2}h_{45}\overline{h}_{45}\Phi_{3}$
$\displaystyle+\phi_{1}\overline{\phi}_{2}\Phi_{4}+\overline{\phi}_{1}\phi_{2}\Phi_{4}+\phi_{3}\overline{\phi}_{4}\Phi_{5}+\overline{\phi}_{3}\phi_{4}\Phi_{5}+\frac{1}{2}\phi_{45}\overline{\phi}_{45}\Phi_{3}$
$\displaystyle+\frac{1}{2}\phi_{+}\overline{\phi}_{+}\Phi_{3}+\frac{1}{2}\phi_{-}\overline{\phi}_{-}\Phi_{3}$
(2.1)
$\displaystyle+\Phi_{12}\Phi_{23}\Phi_{31}+\overline{\Phi}_{12}\overline{\Phi}_{23}\overline{\Phi}_{31}+\Phi_{12}\phi_{+}\phi_{-}+\overline{\Phi}_{12}\overline{\phi}_{+}\overline{\phi}_{-}$
$\displaystyle+\frac{1}{2}\Phi_{3}\sum_{i=1}^{4}\phi_{i}\overline{\phi}_{i}+\Phi_{12}\sum_{i=1}^{4}\phi_{i}^{2}+\overline{\Phi}_{12}\sum_{i=1}^{4}\overline{\phi}_{i}^{2}$
$\displaystyle+D_{1}^{2}\overline{\Phi}_{23}+D_{2}^{2}\Phi_{31}+D_{4}^{2}\overline{\Phi}_{23}+D_{5}^{2}\overline{\Phi}_{31}+\frac{1}{\sqrt{2}}D_{4}D_{5}\overline{\phi}_{3}$
$\displaystyle+T_{1}^{2}\overline{\Phi}_{23}+T_{2}^{2}\Phi_{31}+T_{4}^{2}\Phi_{23}+T_{5}^{2}\Phi_{31}+\frac{1}{\sqrt{2}}T_{4}T_{5}\phi_{2}$
$\displaystyle+\frac{1}{\sqrt{2}}Y_{1}\overline{X}_{2}\phi_{4}+\frac{1}{\sqrt{2}}Y_{2}\overline{X}_{1}\phi_{1}+Y_{2}\overline{X}_{2}\phi_{+}+\frac{1}{2}Z_{1}\overline{Z}_{1}\Phi_{3}$
$\displaystyle\left.+Z_{2}\overline{Z}_{2}\overline{\Phi}_{12}+Z_{1}\overline{X}_{2}^{\prime}\ell^{c}_{2}+Y_{2}^{\prime}Z_{1}D_{1}\vrule
height=12.0pt,depth=1.0pt,width=0.0pt\right]\,,$

where $g_{s}$ is the string coupling.

We now summarise the main results we obtained on the cosmology sector of the
model [1]. The identification of the inflaton comes from the single
superpotential coupling of the form $\bf{10}$ $\bf{\overline{10}}$ $\bf{1}$
with the singlet being a R-handed neutrino, which is
$F_{4}\bar{F}_{5}\phi_{3}$, implying that $\phi_{3}$ should be component of a
linear combination of fields defining the inflaton $y$. Since $\phi_{3}$
corresponds to a state from the third twisted sector, inspection of the
derived no-scale Kähler potential that leads to the required Starobinsky type
inflation implies that the goldstino superfield $z$ should come from the third
untwisted NS sector. Moreover, the inflationary superpotential should be (in
supergravity units)

$W_{I}=M_{I}z(y-\lambda y^{2})\,,$ (2.2)

with $M_{I}$ the scale of inflation and $\lambda$ a parameter close to 1111The
value $\lambda=1$ corresponds to the scalaron with the properties of an
$R^{2}$ term in the effective action.. Following a detailed analysis of the
various possibilities and a study of the relevant non-renormalisable (NR)
superpotential interactions up to 10th order in the
$\alpha^{\prime}$-expansion, we identified the goldstino with the gauge
singlet superfield $\Phi_{4}$ from the NS sector, while the inflaton is a
linear combination of $\phi_{3}$ and $\bar{\phi}_{3}$:

$\displaystyle y$
$\displaystyle=\sin\omega\,\phi_{3}-\cos\omega\,\bar{\phi}_{3}\quad;\quad\tan\omega=\langle\phi_{4}\rangle/\langle\bar{\phi}_{4}\rangle\,,$
(2.3) $\displaystyle z$ $\displaystyle=\Phi_{4}\,.$ (2.4)

The orthogonal linear combination to $y$ is massive, while phenomenological
constraints on appropriate reheating through the inflaton decay into
neutrinos, baryogenesis and light neutrino masses require
$\langle\phi_{4}\rangle/\langle\bar{\phi}_{4}\rangle\sim 10^{-3}$ [12, 13,
14].

The superpotential (2.2) is generated at the 6th and 8th order and reads:

$\displaystyle W_{I}$
$\displaystyle=g_{s}C_{6}\left(g_{s}\sqrt{2\alpha^{\prime}}\right)^{3}\,\overline{\phi}_{3}\Phi_{4}\langle{D_{1}}\rangle{\cdot}\langle{D_{5}}\rangle\,\langle{T_{1}}\rangle{\cdot}\langle{T_{4}}\rangle$
(2.5) $\displaystyle\hskip
22.76228pt+g_{s}C_{8}\left(g_{s}\sqrt{2\alpha^{\prime}}\right)^{5}\,\overline{\phi}_{3}^{2}\Phi_{4}\langle{D_{1}}\rangle{\cdot}\langle{D_{4}}\rangle\,\langle{T_{1}}\rangle{\cdot}\langle{T_{4}}\rangle\langle{\Phi_{31}}\rangle\,,$

where $C_{6}$ and $C_{8}$ stand for the numerical values of the correlators
associated to the $N=6$ and $N=8$ NR couplings, respectively. Note the
presence of several hidden sector fields. The reason is that the trilinear
superpotential involving only gauge singlets under the non-abelian gauge group
is exact and does not receive $\alpha^{\prime}$ corrections [20]. It turns out
that a typical VEV satisfying the D-term conditions is of order $\xi$:

$\xi\sim{M_{s}\over 2\pi}\quad;\quad
M_{s}\equiv{1\over\sqrt{2\alpha^{\prime}}}\,.$ (2.6)

Thus, defining $\alpha_{s}\equiv g_{s}/2\pi$, one gets the inflation scale
$M_{I}\simeq C_{6}\alpha_{s}^{5}M_{s}\sim 10^{13}$ GeV and $\lambda$ an order
one tuneable parameter. The choice of VEVs that solve the F and D-flatness
conditions, giving rise to the above superpotential are given in the left
panel of Table 1.

Field | Assignments in Ref. [1] | Assignments in Section 4
---|---|---
$\Phi_{12}$ | 0 | 0
$\overline{\Phi}_{12}$ | 0 | $\xi^{3}$
$\Phi_{23}$ | 0 | $\xi^{3}$
$\overline{\Phi}_{23}$ | 0 | $\xi^{3}$
$\Phi_{31}$ | $\xi$ | $\xi$
$\overline{\Phi}_{31}$ | 0 | $\xi$
$\Phi_{I}$ | $0$ | $0$
$\phi_{45}$ | $\xi$ | $\xi$
$\overline{\phi}_{45}$ | $\xi^{3}$ | $\xi^{3}$
$\phi_{1}$ | $0$ | $0$
$\phi_{2}$ | $0$ | $0$
$\phi_{3}$ | $0$ | $0$
$\phi_{4}$ | $\xi^{4}$ | $\xi$
$\overline{\phi}_{1}$ | $0$ | $\xi$
$\overline{\phi}_{2}$ | $0$ | $0$
$\overline{\phi}_{3}$ | $0$ | $0$
$\overline{\phi}_{4}$ | $\xi^{2}$ | $\xi^{2}$
$\phi_{+}$ | $\xi$ | $\xi$
$\phi_{-}$ | $\xi^{3}$ | $\xi$
$\overline{\phi}_{+}$ | $\xi^{3}$ | $\xi$
$\overline{\phi}_{-}$ | $\xi$ | $\xi$
$F_{1}$ | $0$ | $\xi^{3/2}$
$\overline{F}_{5}$ | $0$ | $\xi^{3/2}$
$F_{3}$ | $0$ | $\xi^{3}$
$D_{1}$ | $\xi$ | $\xi$
$D_{2}$ | $0$ | $0$
$D_{3}$ | 0 | $\xi^{2}$
$D_{4}$ | $\xi$ | $\xi$
$D_{5}$ | $\xi$ | $0$
$T_{1}$ | $\xi$ | $\xi$
$T_{2}$ | $0$ | $0$
$T_{3}$ | $\xi$ | $\xi^{2}$
$T_{4}$ | $\xi$ | $\xi^{2}$
$T_{5}$ | $0$ | $0$
| $D_{1}^{2}=D_{4}^{2}=D_{5}^{2}=0\ D_{4}{\cdot}D_{5}\sim\xi^{4}$ | $D_{1}^{2}+D_{4}^{2}+T_{1}^{2}=0$
| $T_{1}^{2}=T_{3}^{2}=T_{4}^{2}=T_{3}{\cdot}T_{4}=0$ | $T_{4}^{2}=T_{1}{\cdot}T_{4}=0$
Table 1:  Field VEV assignments in the F/D-flatness solution of Ref. [1]
(column 2) and the solution derived here, in Section 4 (column 3).

##  3 Triplet-doublet splitting in $SU(5)\times U(1)$ string model

In many grand unified theories (GUTs), including $SU(5)$ and $SO(10)$,
Standard Model (SM) Higgs doublets reside in the lowest dimensional gauge
group representation(s) together with colour triplets. Nevertheless, a
successful phenomenological model has to differentiate between Higgs doublets
and additional triplets. The former have to stay light down to the electroweak
scale in order to realise the electroweak symmetry breaking and provide
fermion masses. The latter mediate nucleon decay and, unless sufficiently
heavy, they could lead to proton lifetimes incompatible with existing data.
This is often referred to as the doublet-triplet splitting problem. Among
possible solutions one singles out the so-called missing partner mechanism
that involves the breaking of the GUT gauge symmetry via Higgs fields that
include unequal numbers of SM colour triplets and electroweak doublets. The
missing partner scenario is elegantly realised in the flipped $SU(5)$ model
[5]. In the minimal case, the GUT breaking Higgs fields entail a pair of
$\mathbf{10},\overline{\mathbf{10}}$ representations that include one pair of
d-type colour triplets $\mathbf{3},\overline{\mathbf{3}}$ but no SM Higgs
doublets. Interaction terms of these Higgs fields with a single pair of
$\mathbf{5},\overline{\mathbf{5}}$ representations,
$\mathbf{10}\times\mathbf{10}\times\mathbf{5}+\overline{\mathbf{10}}\times\overline{\mathbf{10}}\times\overline{\mathbf{5}}$,
provide masses for the triplet components leaving the associated doublets in
the $\mathbf{5},\overline{\mathbf{5}}$ unaffected, see e.g. [21].

A more complicated situation arises in the string implementation of the
flipped $SU(5)$ model [2]. Here, we have four pairs of
$\mathbf{5},\overline{\mathbf{5}}$ fields, namely
$h_{i},\overline{h}_{i},i=1,2,3,45$ and one pair of GUT breaking
$\mathbf{10},\overline{\mathbf{10}}$ Higgs fields, referred as
$F,\overline{F}$, where $F$ stands for a linear combination of
$F_{\alpha},\alpha=1,2,3,4$ and $\bar{F}\equiv\bar{F}_{5}$. SM Higgs doublets
and additional triplets are assigned to $SU(5)\times{U(1)}$ representations as
follows

$\displaystyle h_{i}\left(\mathbf{5},-1\right)$
$\displaystyle=H_{i}\left(\mathbf{1},\mathbf{2},-\frac{1}{2}\right)+D_{i}\left(\mathbf{3},\mathbf{1},-\frac{1}{3}\right)\,,$
(3.1) $\displaystyle F\left(\mathbf{10},+\frac{1}{2}\right)$
$\displaystyle=Q_{H}\left(\mathbf{3},\mathbf{2},+\frac{1}{6}\right)+d^{c}_{H}\left(\overline{\mathbf{3}},\mathbf{1},+\frac{1}{3}\right)+\nu^{c}_{H}\left(\mathbf{1},\mathbf{1},0\right)\,,$
(3.2)

and similarly for the conjugate fields $\overline{h}_{i}$, $\overline{F}$.
Taking into account the tree-level superpotential (2.1), the Higgs doublet
mass matrix reads

$\displaystyle
M_{H}^{(3)}=\bordermatrix{~{}&H_{1}&H_{2}&H_{3}&H_{45}\cr\bar{H}_{1}&0&\overline{\Phi}_{12}&\Phi_{31}&0\cr\bar{H}_{2}&\Phi_{12}&0&\overline{\Phi}_{23}&0\cr\bar{H}_{3}&\overline{\Phi}_{31}&\Phi_{23}&0&\phi_{45}\cr\bar{H}_{45}&0&0&\overline{\phi}_{45}&\Phi_{3}}\,,$
(3.3)

where $\Phi_{3}$,
$\Phi_{12},\overline{\Phi}_{12},\Phi_{23},\overline{\Phi}_{23},\Phi_{31},\overline{\Phi}_{31}$
stand for flipped $SU(5)$ singlets that can acquire VEVs.

The associated triplet mass matrix depends on the choice of $F$, which is the
linear combination of the $F_{\alpha}$ that acquires VEV. Notice, that there
are two candidate up-quark couplings at the tree-level superpotential (2.1),
namely $F_{4}\overline{f}_{5}\overline{h}_{45}$ and
$F_{3}\overline{f}_{3}\overline{h}_{3}$. Taking into account the absence of
associated down-quark couplings of the form $F_{3}F_{3}h_{i}$, $i=1,2,3,45$,
both at tree-level and at higher order $N=4,5$ superpotential terms [15, 17,
22], we are led to identify the heaviest generation of quarks and leptons with
$F_{4},\overline{f}_{5}$. Moreover, in (2.1) there exist three terms providing
mass to the down quarks namely $F_{1}F_{1}h_{1}$, $F_{4}F_{4}h_{1}$ and
$F_{2}F_{2}h_{2}$. Bearing in mind that the first two involve the same doublet
$H_{1}$ (residing in $h_{1}$) and $F_{4}$ is assigned to the third generation
we deduce that $F_{1}$ has to be associated with the GUT breaking Higgs field
$F$. In this case the additional triplet mass matrix can be recast in the form

$\displaystyle
M_{D}^{(3)}=\bordermatrix{~{}&D_{1}&D_{2}&D_{3}&D_{45}&\overline{d}^{c}_{H}\cr\bar{D}_{1}&0&\overline{\Phi}_{12}&\Phi_{31}&0&0\cr\bar{D}_{2}&\Phi_{12}&0&\overline{\Phi}_{23}&0&2\overline{F}_{5}\cr\bar{D}_{3}&\overline{\Phi}_{31}&\Phi_{23}&0&\phi_{45}&0\cr\bar{D}_{45}&0&0&\overline{\phi}_{45}&\Phi_{3}&0\cr
d^{c}_{H}&2F_{1}&0&0&0&0\cr}\,,$ (3.4)

where $d^{c}_{H},\overline{d}^{c}_{H}$ are the triplets residing in
$F_{1},\overline{F}_{5}$ respectively.

The doublet-triplet splitting conditions can now be expressed as

$\displaystyle\det{M_{H}^{(3)}}=0\ ,\ \det{M_{D}^{(3)}}\neq 0\,,$ (3.5)

where

$\displaystyle\det{M_{H}^{(3)}}$
$\displaystyle=\left(\Phi_{12}\Phi_{23}\Phi_{31}+\overline{\Phi}_{12}\overline{\Phi}_{23}\overline{\Phi}_{31}\right)\Phi_{3}+\Phi_{12}\overline{\Phi}_{12}\phi_{45}\overline{\phi}_{45}\,,$
(3.6) $\displaystyle\det{M_{D}^{(3)}}$
$\displaystyle=-4F_{1}\overline{F}_{5}\left(\Phi_{23}\Phi_{31}\Phi_{3}+\phi_{45}\overline{\phi}_{45}\overline{\Phi}_{12}\right)\,,$
(3.7)

are the determinants of relevant mass matrices. Additional constraints arise
from low energy phenomenology and more particularly from the requirement to
have a top quark mass coupling at the tree-level superpotential and in
particular from the aforementioned coupling
$F_{4}\overline{f}_{5}\overline{h}_{45}$. Expressing the massless Higgs
doublet pair as

$\displaystyle H$
$\displaystyle=c_{1}H_{1}+c_{2}H_{2}+c_{3}H_{3}+c_{45}H_{45}\,,$ (3.8)
$\displaystyle\bar{H}$
$\displaystyle=\bar{c}_{1}\bar{H}_{1}+\bar{c}_{2}\bar{H}_{2}+\bar{c}_{3}\bar{H}_{3}+\bar{c}_{45}\bar{H}_{45}\,,$
(3.9)

where $c_{i},\overline{c}_{i}$ depend on the $SU(5)\times{U(1)}$ singlet VEVs,
this requirement is translated to

$\displaystyle\bar{c}_{45}\neq 0\,.$ (3.10)

The constraints (3.5), (3.10) depend on $F_{1},\overline{F}_{5}$ and nine
additional parameters, namely
$\Phi_{12},\overline{\Phi}_{12},\Phi_{23},\overline{\Phi}_{23},\Phi_{31},\overline{\Phi}_{31},\phi_{45},\overline{\phi}_{45},\Phi_{3}\,$.
As these VEVs are subject to non-trivial F/D flatness conditions we seek for
solutions of (3.5), (3.10) with the minimum number of vanishing VEVs and no
fine-tunings. After some algebra we find no solution with a single vanishing
VEV and only two solutions with two vanishing VEVs. These are

  1. (a)

$\\!\Phi_{12}=\Phi_{3}=0$, with
$\det{M_{D}^{(3)}}=-4F_{1}\overline{F}_{5}\phi_{45}\overline{\phi}_{45}\overline{\Phi}_{12}$
and massless doublets

$\displaystyle H$ $\displaystyle=\phi_{45}H_{1}-\overline{\Phi}_{31}H_{45}\,,$
(3.11) $\displaystyle\bar{H}$
$\displaystyle=\overline{\phi}_{45}\overline{H}_{2}-\overline{\Phi}_{23}\overline{H}_{45}\,.$
(3.12)

  2. (b)

$\\!\Phi_{12}=\overline{\Phi}_{31}=0$, with
$\det{M_{D}^{(3)}}=-4F_{1}\overline{F}_{5}\left(\Phi_{23}\Phi_{31}\Phi_{3}+\phi_{45}\overline{\phi}_{45}\overline{\Phi}_{12}\right)$
and massless doublets

$\displaystyle H$ $\displaystyle\\!=\\!H_{1}\,,$ (3.13)
$\displaystyle\overline{H}$
$\displaystyle\\!=\\!\Phi_{23}\overline{\Phi}_{23}\Phi_{3}\overline{H}_{1}\\!-\\!\left(\Phi_{31}\Phi_{23}\Phi_{3}\\!+\\!\overline{\Phi}_{12}\phi_{45}\overline{\phi}_{45}\right)\overline{H}_{2}\\!-\\!\overline{\Phi}_{12}\overline{\Phi}_{23}\Phi_{3}\overline{H}_{3}\\!+\\!\overline{\Phi}_{12}\overline{\Phi}_{23}\phi_{45}\overline{H}_{45}$
(3.14)

##  4 Solution of the F/D-flatness conditions

In supersymmetric string models build in the framework of the free fermionic
formulation, many features of low energy phenomenology are specified by the
VEVs of Standard Model singlet fields. These in turn, are subject to non-
trivial F/D flatness constraints dictated by $N=1$ supersymmetry. For the
flipped string model [2] the solution and phenomenological consequences of F/D
flatness conditions have been thoroughly studied in the past [2, 15, 20, 16,
17, 19]. In this section we re-examine solutions of flatness equations taking
into account: (i) compatibility with the recent results concerning string
cosmology [1] as explained in Section 2 and (ii) the results of Section 3
regarding doublet-triplet splitting and in particular the requirement to
generate triplet masses at tree-level in order to efficiently suppress proton
decay.

For the flipped string model under consideration the full gauge group is
$SU(5)\times{U(1)}\times{U(1)^{\prime}}^{4}\times{SU(4)}\times{SO(10)}$ The
D-flatness conditions associated with the ${U(1)^{\prime}}^{4}$ gauge group
factor are of the form

$\displaystyle{\cal D}_{I}$
$\displaystyle=\sum_{i}q^{i}_{I}\left|\varphi_{i}\right|^{2}+\frac{1}{192\pi^{2}}{2\over\alpha^{\prime}}\,{\rm
Tr}Q_{I}=0\,,I=1,2,3,A\,,$ (4.1)

where $\varphi_{i}$ denotes a field with charges $q^{i}_{I}$ under
${U(1)^{\prime}}_{I}$ and ${\rm Tr}{Q_{1}}={\rm Tr}{Q_{2}}={\rm Tr}{Q_{3}}=0$,
${\rm Tr}{Q_{A}}=180$ are the traces of the associated abelian group
generators [2, 23]. After some algebra Eqs. (4.1) can be unravelled by taking
appropriate linear combinations [17]

$\displaystyle\left|\phi_{45}\right|^{2}-\left|\overline{\phi}_{45}\right|^{2}-\frac{1}{2}\left(\left|D_{3}\right|^{2}+\left|T_{3}\right|^{2}\right)+\frac{1}{2}\left|F_{3}\right|^{2}=\xi^{2}\,,$
(4.2)
$\displaystyle\left|\phi_{+}\right|^{2}-\left|\phi_{-}\right|^{2}-\left|\overline{\phi}_{+}\right|^{2}+\left|\overline{\phi}_{-}\right|^{2}+\frac{1}{2}\left(\left|D_{3}\right|^{2}-\left|T_{3}\right|^{2}\right)-\frac{1}{2}\left|F_{3}\right|^{2}=\xi^{2}\,,$
(4.3)
$\displaystyle\left|\Phi_{31}\right|^{2}-\left|\overline{\Phi}_{31}\right|^{2}-\left|\Phi_{23}\right|^{2}+\left|\overline{\Phi}_{23}\right|^{2}-\frac{1}{2}\left(\left|D_{1}\right|^{2}+\left|D_{2}\right|^{2}+\left|D_{3}\right|^{2}+\left|D_{4}\right|^{2}-\left|D_{5}\right|^{2}\right)$
$\displaystyle\
-\frac{1}{2}\left(\left|T_{1}\right|^{2}+\left|T_{2}\right|^{2}+\left|T_{3}\right|^{2}-\left|T_{4}\right|^{2}+\left|T_{5}\right|^{2}\right)=3\xi^{2}\,,$
(4.4)
$\displaystyle\left|\Phi_{23}\right|^{2}-\left|\overline{\Phi}_{23}\right|^{2}-\left|\Phi_{12}\right|^{2}+\left|\overline{\Phi}_{12}\right|^{2}+\frac{1}{2}\sum_{i=1}^{4}\left(\left|\phi_{i}\right|^{2}-\left|\overline{\phi}_{i}\right|^{2}\right)+\left|\phi_{+}\right|^{2}-\left|\overline{\phi}_{+}\right|^{2}$
$\displaystyle\
+\frac{1}{2}\left(\left|D_{1}\right|^{2}+\left|D_{3}\right|^{2}+\left|D_{4}\right|^{2}\right)+\frac{1}{2}\left(\left|T_{1}\right|^{2}-\left|T_{4}\right|^{2}\right)-\frac{1}{2}\left(\left|F_{2}\right|^{2}-\left|\overline{F}_{5}\right|^{2}\right)=0\,,$
(4.5)

where

$\xi^{2}=\frac{1}{16\pi^{2}}{2\over\alpha^{\prime}}\,.$ (4.6)

The F-flatness equations are derived from the superpotential $W$

$\displaystyle{\cal F}_{i}=\frac{\partial W}{\partial\varphi_{i}}=0\,.$ (4.7)

At tree-level $W=W_{3}$ given in Eq. (2.1). However, the superpotential
receives additional contributions from non-renormalisable (NR) terms at higher
orders. These come from terms of the form
$\varphi_{1}\varphi_{2}\varphi_{3}\varphi^{N-3},N>3$, where $\varphi^{N-3}$
denotes a product of $N-3$ field VEVs. Apart from gauge invariance these terms
are subject to intricate string selection rules ascribed to world-sheet
superalgebra [20, 24]. Using a computer program that successively applies all
selection criteria we find 15 candidate NR superpotential couplings for $N=4$
and 256 couplings for $N=5$. However, for the sake of simplicity we start our
analysis from the tree-level superpotential (2.1) and we will take into
account higher order non-renormalisable contributions at a later stage. The
tree-level F-flatness equations give

$\displaystyle\frac{\partial W_{3}}{\partial\Phi_{12}}$
$\displaystyle=\sum_{i=1}^{4}\phi_{i}^{2}+\Phi_{23}\Phi_{31}+\phi_{+}\phi_{-}=0\,,$
(4.8) $\displaystyle\frac{\partial W_{3}}{\partial\overline{\Phi}_{12}}$
$\displaystyle=\sum_{i=1}^{4}\overline{\phi}_{i}^{2}+\overline{\Phi}_{23}\overline{\Phi}_{31}+\overline{\phi}_{+}\overline{\phi}_{-}=0\,,$
(4.9) $\displaystyle\frac{\partial W_{3}}{\partial\Phi_{31}}$
$\displaystyle=\Phi_{12}\Phi_{23}+D_{2}^{2}+T_{2}^{2}+T_{5}^{2}=0\,,$ (4.10)
$\displaystyle\frac{\partial W_{3}}{\partial\overline{\Phi}_{31}}$
$\displaystyle=\overline{\Phi}_{12}\overline{\Phi}_{23}+D_{5}^{2}=0\,,$ (4.11)
$\displaystyle\frac{\partial W_{3}}{\partial\Phi_{23}}$
$\displaystyle={\Phi}_{12}{\Phi}_{31}+T_{4}^{2}=0\,,$ (4.12)
$\displaystyle\frac{\partial W_{3}}{\partial\overline{\Phi}_{23}}$
$\displaystyle=\overline{\Phi}_{12}\overline{\Phi}_{31}+D_{1}^{2}+D_{4}^{2}+T_{1}^{2}=0\,,$
(4.13) $\displaystyle\frac{\partial W_{3}}{\partial\Phi_{3}}$
$\displaystyle=\sum_{i=1}^{4}\phi_{i}\overline{\phi}_{i}+\phi_{45}\overline{\phi}_{45}+\phi_{-}\overline{\phi}_{-}+\phi_{+}\overline{\phi}_{+}=0\,,$
(4.14) $\displaystyle\frac{\partial W_{3}}{\partial\Phi_{4}}$
$\displaystyle=\phi_{2}\overline{\phi}_{1}+\phi_{1}\overline{\phi}_{2}=0\,,$
(4.15) $\displaystyle\frac{\partial W_{3}}{\partial\Phi_{5}}$
$\displaystyle=\phi_{4}\overline{\phi}_{3}+\phi_{3}\overline{\phi}_{4}=0\,,$
(4.16) $\displaystyle\frac{\partial W_{3}}{\partial\phi_{1}}$
$\displaystyle=2\phi_{1}\Phi_{12}+\Phi_{3}\overline{\phi}_{1}+\Phi_{4}\overline{\phi}_{2}=0\,,$
(4.17) $\displaystyle\frac{\partial W_{3}}{\partial\phi_{2}}$
$\displaystyle=2\phi_{2}\Phi_{12}+\Phi_{4}\overline{\phi}_{1}+\Phi_{3}\overline{\phi}_{2}+T_{4}{\cdot}T_{5}=0\,,$
(4.18) $\displaystyle\frac{\partial W_{3}}{\partial\phi_{3}}$
$\displaystyle=F_{4}\overline{F}_{5}+2\phi_{3}\Phi_{12}+\Phi_{3}\overline{\phi}_{3}+\Phi_{5}\overline{\phi}_{4}=0\,,$
(4.19) $\displaystyle\frac{\partial W_{3}}{\partial\phi_{4}}$
$\displaystyle=2\phi_{4}\Phi_{12}+\Phi_{5}\overline{\phi}_{3}+\Phi_{3}\overline{\phi}_{4}=0\,,$
(4.20) $\displaystyle\frac{\partial W_{3}}{\partial\overline{\phi}_{1}}$
$\displaystyle=2\overline{\phi}_{1}\overline{\Phi}_{12}+\Phi_{3}{\phi}_{1}+\Phi_{4}{\phi}_{2}=0\,,$
(4.21) $\displaystyle\frac{\partial W_{3}}{\partial\overline{\phi}_{2}}$
$\displaystyle=2\overline{\phi}_{2}\overline{\Phi}_{12}+\Phi_{4}{\phi}_{1}+\Phi_{3}{\phi}_{2}=0\,,$
(4.22) $\displaystyle\frac{\partial W_{3}}{\partial\overline{\phi}_{3}}$
$\displaystyle=2\overline{\phi}_{3}\overline{\Phi}_{12}+\Phi_{3}{\phi}_{3}+\Phi_{5}{\phi}_{4}+D_{4}{\cdot}D_{5}=0\,,$
(4.23) $\displaystyle\frac{\partial W_{3}}{\partial\overline{\phi}_{4}}$
$\displaystyle=2\overline{\phi}_{4}\overline{\Phi}_{12}+\Phi_{5}{\phi}_{3}+\Phi_{3}{\phi}_{4}=0\,,$
(4.24) $\displaystyle\frac{\partial W_{3}}{\partial\phi_{+}}$
$\displaystyle=\Phi_{3}\overline{\phi}_{+}+\Phi_{12}\phi_{-}=0\,,$ (4.25)
$\displaystyle\frac{\partial W_{3}}{\partial\overline{\phi}_{+}}$
$\displaystyle=\Phi_{3}\phi_{+}+\overline{\Phi}_{12}\overline{\phi}_{-}=0\,,$
(4.26) $\displaystyle\frac{\partial W_{3}}{\partial\phi_{-}}$
$\displaystyle=\Phi_{3}\overline{\phi}_{-}+\Phi_{12}\phi_{+}=0\,,$ (4.27)
$\displaystyle\frac{\partial W_{3}}{\partial\overline{\phi}_{-}}$
$\displaystyle=\Phi_{3}\phi_{-}+\overline{\Phi}_{12}\overline{\phi}_{+}=0\,,$
(4.28) $\displaystyle\frac{\partial W_{3}}{\partial\phi_{45}}$
$\displaystyle=\Phi_{3}\overline{\phi}_{45}=0\,,$ (4.29)
$\displaystyle\frac{\partial W_{3}}{\partial\overline{\phi}_{45}}$
$\displaystyle=\Phi_{3}\phi_{45}=0\,,$ (4.30) $\displaystyle\frac{\partial
W_{3}}{\partial F_{4}}$ $\displaystyle=\overline{F}_{5}\phi_{3}=0\,,$ (4.31)
$\displaystyle\frac{\partial W_{3}}{\partial\overline{F}_{5}}$
$\displaystyle=F_{4}\phi_{3}=0\,,$ (4.32) $\displaystyle\frac{\partial
W_{3}}{\partial T_{1}}$ $\displaystyle=2\overline{\Phi}_{23}T_{1}=0\,,$ (4.33)
$\displaystyle\frac{\partial W_{3}}{\partial T_{2}}$
$\displaystyle=2\Phi_{31}T_{2}=0\,,$ (4.34) $\displaystyle\frac{\partial
W_{3}}{\partial T_{4}}$ $\displaystyle=2\Phi_{23}T_{4}+\phi_{2}T_{5}=0\,,$
(4.35) $\displaystyle\frac{\partial W_{3}}{\partial T_{5}}$
$\displaystyle=\phi_{2}T_{4}+2\Phi_{31}T_{5}=0\,,$ (4.36)
$\displaystyle\frac{\partial W_{3}}{\partial D_{1}}$
$\displaystyle=2D_{1}\overline{\Phi}_{23}=0\,,$ (4.37)
$\displaystyle\frac{\partial W_{3}}{\partial D_{2}}$
$\displaystyle=2D_{2}{\Phi}_{31}=0\,,$ (4.38) $\displaystyle\frac{\partial
W_{3}}{\partial D_{4}}$
$\displaystyle=2D_{4}\overline{\Phi}_{23}+D_{5}\overline{\phi}_{3}/\sqrt{2}=0\,,$
(4.39) $\displaystyle\frac{\partial W_{3}}{\partial D_{5}}$
$\displaystyle=2D_{5}\overline{\Phi}_{31}+D_{4}\overline{\phi}_{3}/\sqrt{2}=0\,.\hskip
85.35826pt$ (4.40)

The massless spectrum of the string derived flipped $SU(5)$ model comprises
four fermion generations
$F_{\alpha},\overline{f}_{\beta},\ell^{c}_{\beta},\alpha=1,2,3,4,\beta=1,2,3,5$
and one anti-generation $\overline{F}_{5},f_{4},\overline{\ell}^{c}_{4}$. The
solution of the F/D-flatness equations depends on the choice of the flipped
$SU(5)$ breaking Higgs fields $F,\overline{F}$ and the assignment of the
fermion generations. Following the discussion in Section 3 we choose
$F=A_{1}F_{1}+A_{3}F_{3},\overline{F}=\overline{F}_{5}$ with
$A_{1}^{2}+A_{4}^{2}=1$ and set

$\displaystyle F_{2}=F_{4}=0\,.$ (4.41)

Combined with (4.2), this implies $\overline{\phi}_{45}\neq 0$, which, in
conjunction with (4.29), leads to $\Phi_{3}=0\,$. This rejects the second
solution of Section 3 leaving us with

$\displaystyle\Phi_{12}=\Phi_{3}=0$ (4.42)

and
$\overline{\Phi}_{12},\Phi_{31},\overline{\Phi}_{31},\Phi_{23},\overline{\Phi}_{23},\phi_{45},\overline{\phi}_{45}\neq
0$ in order solve the doublet-triplet problem at tree-level. Eq. (4.12) then
implies

$\displaystyle T_{4}^{2}=0\,,$ (4.43)

and (4.31) yields

$\displaystyle\phi_{3}=0\,$ (4.44)

as $\overline{F}_{5}\neq 0\,$. To protect $\overline{f}_{5}$, associated in
Section 3 with the 3rd fermion generation, from receiving tree-level mass via
the superpotential term $f_{4}\overline{f}_{5}\overline{\phi}_{2}$ we have to
impose also

$\displaystyle\overline{\phi}_{2}=0\,.$ (4.45)

Next we solve Eqs. (4.15), (4.16) by choosing

$\displaystyle\phi_{2}=\overline{\phi}_{3}=0\,$ (4.46)

and Eqs. (4.19), (4.22) using

$\displaystyle\Phi_{4}=\Phi_{5}=0\,.$ (4.47)

Keeping in mind that $\Phi_{31},\overline{\Phi}_{31}\neq 0$ Eqs. (4.34),
(4.36), (4.38), (4.40) lead to

$\displaystyle T_{2}=T_{5}=D_{2}=D_{5}=0\,.$ (4.48)

The remaining equations can be solved after considering the aforementioned
non-renormalisable corrections to the superpotential (2.1). Taking into
account the full superpotential $W_{5}$ incorporating NR contributions up to
order $N=5$ and utilising the tree-level partial solution (4.41)-(4.48) the
F-flatness equations yield

$\displaystyle\frac{\partial W_{5}}{\partial\Phi_{12}}$
$\displaystyle=\phi_{1}^{2}+\phi_{4}^{2}+\phi_{+}\phi_{-}+\Phi_{23}\Phi_{31}=0\,,$
(4.49) $\displaystyle\frac{\partial W_{5}}{\partial\overline{\Phi}_{12}}$
$\displaystyle=\overline{\phi}_{1}^{2}+\overline{\phi}_{4}^{2}+\overline{\phi}_{+}\overline{\phi}_{-}+\overline{\Phi}_{23}\overline{\Phi}_{31}+\left\\{F_{1}^{2}\overline{F}_{5}^{2}\right\\}=0\,,$
(4.50) $\displaystyle\frac{\partial W_{5}}{\partial\Phi_{31}}$
$\displaystyle=\left\\{\left(\overline{\phi}_{1}^{2}+\overline{\phi}_{4}^{2}+\overline{\phi}_{+}\overline{\phi}_{-}\right)\left(D_{1}^{2}+D_{4}^{2}+T_{1}^{2}\right)\right\\}=0\,,$
(4.51) $\displaystyle\frac{\partial W_{5}}{\partial\overline{\Phi}_{31}}$
$\displaystyle=\overline{\Phi}_{12}\overline{\Phi}_{23}=0\,,$ (4.52)
$\displaystyle\frac{\partial W_{5}}{\partial\overline{\Phi}_{23}}$
$\displaystyle=\overline{\Phi}_{12}\overline{\Phi}_{31}+D_{1}^{2}+D_{4}^{2}+T_{1}^{2}=0\,,$
(4.53) $\displaystyle\frac{\partial W_{5}}{\partial\Phi_{3}}$
$\displaystyle=\phi_{1}\overline{\phi}_{1}+\phi_{4}\overline{\phi}_{4}+\phi_{-}\overline{\phi}_{-}+\phi_{+}\overline{\phi}_{+}+\phi_{45}\overline{\phi}_{45}=0\,,$
(4.54) $\displaystyle\frac{\partial W_{5}}{\partial F_{1}}$
$\displaystyle=\left\\{F_{1}\overline{F}_{5}^{2}\overline{\Phi}_{12}\right\\}=0\,,$
(4.55) $\displaystyle\frac{\partial W_{5}}{\partial\overline{F}_{5}}$
$\displaystyle=\left\\{\overline{F}_{5}F_{1}^{2}\overline{\Phi}_{12}\right\\}=0\,,$
(4.56) $\displaystyle\frac{\partial W_{5}}{\partial\phi_{3}}$
$\displaystyle=\left\\{\left(F_{1}\overline{F}_{5}\right)\left(T_{1}{\cdot}T_{4}\right)\right\\}=0\,,$
(4.57) $\displaystyle\frac{\partial W_{5}}{\partial\overline{\phi}_{1}}$
$\displaystyle=2\overline{\phi}_{1}\left[\overline{\Phi}_{12}+\left\\{\Phi_{31}\left(D_{1}^{2}+D_{4}^{2}+T_{1}^{2}\right)\right\\}\right]=0\,,$
(4.58) $\displaystyle\frac{\partial W_{5}}{\partial\overline{\phi}_{4}}$
$\displaystyle=2\overline{\phi}_{4}\left[\overline{\Phi}_{12}+\left\\{\Phi_{31}\left(D_{1}^{2}+D_{4}^{2}+T_{1}^{2}\right)\right\\}\right]=0\,,$
(4.59) $\displaystyle\frac{\partial W_{5}}{\partial\overline{\phi}_{+}}$
$\displaystyle=\overline{\phi}_{-}\left[\overline{\Phi}_{12}+\left\\{\Phi_{31}\left(D_{1}^{2}+D_{4}^{2}+T_{1}^{2}\right)\right\\}\right]=0\,,$
(4.60) $\displaystyle\frac{\partial W_{5}}{\partial\overline{\phi}_{-}}$
$\displaystyle=\overline{\phi}_{+}\left[\overline{\Phi}_{12}+\left\\{\Phi_{31}\left(D_{1}^{2}+D_{4}^{2}+T_{1}^{2}\right)\right\\}\right]=0\,,$
(4.61) $\displaystyle\frac{\partial W_{5}}{\partial T_{1}}$
$\displaystyle=2T_{1}\left[\overline{\Phi}_{23}+\left\\{\Phi_{31}\left(\overline{\phi}_{1}^{2}+\overline{\phi}_{4}^{2}+\overline{\phi}_{+}\overline{\phi}_{-}\right)\right\\}\right]=0\,,$
(4.62) $\displaystyle\frac{\partial W_{5}}{\partial T_{4}}$
$\displaystyle=2T_{4}\left[\Phi_{23}+\left\\{\overline{\Phi}_{31}\left(\phi_{1}^{2}+\phi_{4}^{2}+\phi_{+}\phi_{-}\right)\right\\}\right]=0\,,$
(4.63) $\displaystyle\frac{\partial W_{5}}{\partial D_{1}}$
$\displaystyle=2D_{1}\left[\overline{\Phi}_{23}+\left\\{\Phi_{31}\left(\overline{\phi}_{1}^{2}+\overline{\phi}_{4}^{2}+\overline{\phi}_{+}\overline{\phi}_{-}\right)\right\\}\right]=0\,,$
(4.64) $\displaystyle\frac{\partial W_{5}}{\partial D_{4}}$
$\displaystyle=2D_{4}\left[\overline{\Phi}_{23}+\left\\{\Phi_{31}\left(\overline{\phi}_{1}^{2}+\overline{\phi}_{4}^{2}+\overline{\phi}_{+}\overline{\phi}_{-}\right)\right\\}\right]=0\,,$
(4.65) $\displaystyle\frac{\partial W_{5}}{\partial D_{5}}$
$\displaystyle=\left\\{D_{3}\left(F_{3}\overline{F}_{5}\right)\right\\}=0\,,$
(4.66)

where we have omitted order one numerical coefficients of the NR terms (terms
in curly brackets).

Higher order superpotential terms lead to additional constraints on the SM
singlet VEVs in order to assure compatibility with low energy phenomenology.
First, superpotential terms of the form
$F_{1}\overline{f}_{j}\overline{h}_{k}\varphi^{n},j=1,2,3,5,k=1,2,3,45$ could
induce mixings of leptons with Higgs doublets. The effective superpotential
$W_{5}$ comprises three such terms

$\displaystyle
F_{1}\overline{f}_{1}\overline{h}_{45}\phi_{1}(1+\Phi_{2})+F_{1}\overline{f}_{5}\overline{h}_{45}\left(T_{1}{\cdot}T_{4}\right)\,,$
(4.67)

where we dropped curly brackets (denoting omission of numerical coefficients),
for simplicity. To eliminate them we set

$\displaystyle\phi_{1}=0\ ,\ T_{1}{\cdot}T_{4}=0\,.$ (4.68)

Second, the fifth order superpotential $W_{5}$ includes two terms that provide
mass for the surplus pairs of fermions $f,\overline{f}$ and
$\ell^{c},\overline{\ell}^{c}$

$\displaystyle\left(f_{4}\overline{f}_{3}+\overline{\ell}^{c}_{4}\ell^{c}_{3}\right)\left(T_{3}\cdot{T_{4}}\right)\,,$
(4.69)

as long as

$\displaystyle T_{3}\cdot{T_{4}}\neq 0\,.$ (4.70)

Summarizing the VEV assignments above, we have

$\displaystyle F_{2}=F_{4}=0\,,$
$\displaystyle\Phi_{12}=\Phi_{3}=\Phi_{4}=\Phi_{5}=\phi_{1}=\phi_{2}=\phi_{3}=\overline{\phi}_{2}=\overline{\phi}_{3}=0\,,$
(4.71) $\displaystyle
T_{2}=T_{5}=D_{2}=D_{5}=T_{4}^{2}=T_{1}{\cdot}T_{4}=0\,,$

together with

$\displaystyle\overline{\Phi}_{12},\Phi_{31},\overline{\Phi}_{31},\Phi_{23},\overline{\Phi}_{23},\phi_{45},T_{3}\cdot{T_{4}}\neq
0\,.$ (4.72)

Let us now proceed with the solution of the remaining equations. Notice that
the system of D/F-flatness constraints contains a single fixed parameter $\xi$
(4.6), which is about 0.1 in string units. We can thus attempt to find a
perturbative solution expressing all remaining VEVs as power series in $\xi$.
Assuming that (to leading order)
$\Phi_{31},\overline{\Phi}_{31},\phi_{45},\phi_{4},\phi_{+},\phi_{-},\overline{\phi}_{1},\overline{\phi}_{\pm}\sim\xi$,
$F_{1}\overline{F}_{5}\sim\xi^{3}$, Eq. (4.63) can be solved with respect to
$\Phi_{23}$

$\displaystyle\Phi_{23}=-\left\\{\overline{\Phi}_{31}\left(\phi_{4}^{2}+\phi_{+}\phi_{-}\right)\right\\}\sim\xi^{3}\,,$
(4.73)

Eqs. (4.62), (4.64), (4.65) with respect to $\overline{\Phi}_{23}$

$\displaystyle\overline{\Phi}_{23}=-\left\\{\Phi_{31}\left(\overline{\phi}_{1}^{2}+\overline{\phi}_{4}^{2}+\overline{\phi}_{+}\overline{\phi}_{-}\right)\right\\}\sim\xi^{3}\,,$
(4.74)

and Eqs. (4.58)-(4.61) with respect to $\overline{\Phi}_{12}$

$\displaystyle\overline{\Phi}_{12}=-\left\\{\Phi_{31}\left(D_{1}^{2}+D_{4}^{2}+T_{1}^{2}\right)\right\\}\sim\xi^{3}\,.$
(4.75)

The solutions (4.74), (4.75) need further clarification, since both involve
three different equations that should be compatible ((4.60) and (4.61) come
from the same superpotential term). Actually, their validity depends on the
numerical factors of the fifth order terms that we have omitted (terms in
curly brackets). A detailed analysis of Eqs. (4.50)-(4.53), (4.58)-(4.62) and
(4.64),(4.65) taking into account all numerical coefficients is presented in
the next section and Appendix B where we compute the F-flatness solutions of a
generic superpotential of the form

$\displaystyle w$
$\displaystyle=\overline{\Phi}_{23}\overline{\Phi}_{31}\overline{\Phi}_{12}+\overline{\Phi}_{12}\left(\overline{\phi}_{1}^{2}+\overline{\phi}_{2}^{2}+\overline{\phi}_{4}^{2}+\overline{\phi}_{+}\overline{\phi}_{-}\right)$
$\displaystyle+\left(D_{1}^{2}+D_{4}^{2}+T_{1}^{2}\right)\overline{\Phi}_{23}+\left(\alpha_{1}\overline{\phi}_{1}^{2}+\alpha_{2}\overline{\phi}_{2}^{2}+\alpha_{3}\overline{\phi}_{4}^{2}+\alpha_{4}\overline{\phi}_{+}\overline{\phi}_{-}\right)D_{1}^{2}\Phi_{31}$
(4.76)
$\displaystyle+\left(\\!\beta_{1}\overline{\phi}_{1}^{2}\\!+\\!\beta_{2}\overline{\phi}_{2}^{2}\\!+\\!\beta_{3}\overline{\phi}_{4}^{2}\\!+\\!\beta_{4}\overline{\phi}_{+}\overline{\phi}_{-}\\!\right)\\!D_{4}^{2}\Phi_{31}\\!+\\!\left(\gamma_{1}\overline{\phi}_{1}^{2}+\gamma_{2}\overline{\phi}_{2}^{2}+\gamma_{3}\overline{\phi}_{4}^{2}+\gamma_{4}\overline{\phi}_{+}\overline{\phi}_{-}\right)\\!T_{1}^{2}\Phi_{31}.$

As shown in Section 5 an explicit calculation of the related fifth order
superpotential couplings supports the following coupling relations

$\displaystyle-\frac{\alpha_{1}}{3}=-\frac{\gamma_{1}}{3}=-\frac{\beta_{1}}{2}=\frac{\beta_{3}}{2}=\frac{\beta_{4}}{2}=\alpha_{4}\,,$
(4.77)
$\displaystyle\alpha_{2}=\alpha_{3}=\gamma_{2}=\gamma_{3}=\gamma_{4}=\alpha_{4}\
,\ \beta_{2}=0\,.$ (4.78)

These in turn, following the analysis of Appendix B, lead to the subsequent
solution of the associated F-flatness equations to order $\xi^{4}$

$\displaystyle\overline{\Phi}_{12}$
$\displaystyle=-\alpha_{4}D_{4}^{2}\Phi_{31}$ (4.79) $\displaystyle T_{1}^{2}$
$\displaystyle=-D_{1}^{2}-D_{4}^{2}-\overline{\Phi}_{31}\overline{\Phi}_{12}\,,$
(4.80) $\displaystyle\Phi_{23}$
$\displaystyle=-\left\\{\overline{\Phi}_{31}\left(\phi_{4}^{2}+\phi_{+}\phi_{-}\right)\right\\}$
(4.81) $\displaystyle\overline{\Phi}_{23}$
$\displaystyle=-2\Phi_{31}\alpha_{4}\left(-\overline{\phi}_{1}^{2}+\overline{\phi}_{4}^{2}+\overline{\phi}_{+}\overline{\phi}_{-}\right)$
(4.82) $\displaystyle\overline{\phi}_{1}^{2}$
$\displaystyle=-\overline{\phi}_{4}^{2}-\overline{\phi}_{+}\overline{\phi}_{-}-\overline{\Phi}_{23}\overline{\Phi}_{31}\,,$
(4.83)

where in (4.82) we used the exact coefficients from the computation.

The remaining F and D-flatness equations to order $\xi^{4}$ give

$\displaystyle\phi_{4}^{2}+\phi_{+}\phi_{-}$ $\displaystyle=0\,,$ (4.84)
$\displaystyle\phi_{4}\overline{\phi}_{4}+\phi_{-}\overline{\phi}_{-}+\phi_{+}\overline{\phi}_{+}+\phi_{45}\overline{\phi}_{45}$
$\displaystyle=0\,,$ (4.85)
$\displaystyle\left\\{D_{3}\left(F_{3}\overline{F}_{5}\right)\right\\}$
$\displaystyle=0\,,$ (4.86)

and

$\displaystyle\left|\phi_{45}\right|^{2}-\left|\overline{\phi}_{45}\right|^{2}+\frac{1}{2}\left(\left|F_{3}\right|^{2}-\left|D_{3}\right|^{2}-\left|T_{3}\right|^{2}\right)=\xi^{2}\,,$
(4.87)
$\displaystyle\left|\phi_{+}\right|^{2}-\left|\phi_{-}\right|^{2}-\left|\overline{\phi}_{+}\right|^{2}+\left|\overline{\phi}_{-}\right|^{2}+\frac{1}{2}\left(\left|D_{3}\right|^{2}-\left|T_{3}\right|^{2}-\left|F_{3}\right|^{2}\right)=\xi^{2}\,,$
(4.88)
$\displaystyle\left|\Phi_{31}\right|^{2}\\!-\\!\left|\overline{\Phi}_{31}\right|^{2}\\!-\\!\frac{1}{2}\left(\left|D_{1}\right|^{2}\\!+\\!\left|D_{3}\right|^{2}\\!+\\!\left|D_{4}\right|^{2}\right)\\!-\\!\frac{1}{2}\left(\left|T_{1}\right|^{2}\\!+\\!\left|T_{3}\right|^{2}\\!-\\!\left|T_{4}\right|^{2}\right)=3\xi^{2}\,,$
(4.89)
$\displaystyle\frac{1}{2}\left(\left|\phi_{4}\right|^{2}-\left|\overline{\phi}_{1}\right|^{2}-\left|\overline{\phi}_{4}\right|^{2}\right)+\left|\phi_{+}\right|^{2}-\left|\overline{\phi}_{+}\right|^{2}+\frac{1}{2}\left(\left|D_{1}\right|^{2}+\left|D_{3}\right|^{2}+\left|D_{4}\right|^{2}\right)$
$\displaystyle\ \
+\frac{1}{2}\left(\left|T_{1}\right|^{2}-\left|T_{4}\right|^{2}-\left|\overline{F}_{5}\right|^{2}\right)=0\,,$
(4.90)
$\displaystyle\left|A_{1}F_{1}\right|^{2}+\left|A_{3}F_{3}\right|^{2}=\left|\overline{F}_{5}\right|^{2}\,.$
(4.91)

In addition, we have the $SU(4)\simeq SO(6)$ and $SO(10)$ D-flatness which can
be cast in the form

$\displaystyle\sum_{i=1,4}D_{i}^{\ast}\tau^{a}D_{i}=0\ ,\ a=1,\dots,15\ ,$
(4.92) $\displaystyle\sum_{i=1,3,4}T_{i}^{\ast}\lambda^{A}T_{i}=0\ ,\
A=1,\dots,45\ ,$ (4.93)

where $\tau^{a},\lambda^{A}$ are the $SO(6),SO(10)$ generators respectively.
These can be solved by utilising an antisymmetric representation of the
$SO(2n)$ group generators

$\displaystyle\left(M_{ab}\right)_{ij}=-i\left(\delta_{ai}\delta_{bj}-\delta_{aj}\delta_{bi}\right)\
;\ b>a=1,\dots,2n\ ;\ i,j=1,\dots,2n\,.$ (4.94)

In this way, Eqs. (4.92) can be easily solved by choosing the VEVs
$D_{1},D_{3},D_{4}$ real. Although this is not applicable for Eqs. (4.93)
because of the constraints $T_{4}^{2}=T_{1}{\cdot}T_{4}=0$, we can find other
explicit solutions. For example, using the ansatz

$\displaystyle T_{1}$
$\displaystyle=\left(re^{-i\theta},ire^{-i\theta},b,0,\dots,0\right)\,,$
(4.95) $\displaystyle T_{4}$ $\displaystyle=\left(a,ia,0,0,\dots,0\right)\,,$
(4.96) $\displaystyle T_{3}$
$\displaystyle=\left(ic,c,c+ie,0,\dots,0\right)\,,$ (4.97)

Eqs. (4.93) reduce to three independent constraints solved by

$\displaystyle r^{2}=c^{2}-a^{2}\ ,\ e^{2}=b^{2}r^{2}/c^{2}-c^{2}\ ,\
\tan\theta=\frac{c}{e}\,,$ (4.98)

with $a,b,c$ free real parameters. Moreover, this solution guarantees
$T_{4}^{2}=T_{1}{\cdot}T_{4}=0$ and gives $T_{1}^{2}=b^{2}$ and
$T_{3}{\cdot}T_{4}=2iac$, depending on the free parameters.

A natural question to ask at this point is whether higher order terms in the
superpotential could destabilise our flatness solution. A definite answer to
this would require the calculation of the full superpotential to a rather high
order, e.g. $N=10$, and the solution of the associated flatness equations,
which is a very difficult task from the technical point of view. Here, we have
shown that our perturbative solution is valid to order $\xi^{4}$ when taking
into account superpotential terms up to and including $N=5$. Moreover, we have
checked the $N=6$ contributions and it turns out that the aforementioned
flatness solution holds to order $\xi^{5}$ provided $\Phi_{2}=0$,
$D_{3}(F_{3}\overline{F}_{5})\lesssim\xi^{6}$, $T_{3}T_{4}\lesssim\xi^{3}$ and
an additional condition relating the VEVs of $D_{1},T_{1}$. This supports the
perturbative validity of our solution at higher orders. The above VEV
assignments that solve the F/D-flatness equations are summarised in the right
panel of Table 1.

##  5 Computation of higher order superpotential terms

In the context of the free fermionic formulation of the heterotic superstring
the effective $N=1$ superpotential is fully calculable. Actually, the
trilinear coupling constants are fixed by conformal symmetry alone [25]; the
non-vanishing ones are of the form $kg_{s}$, where $g_{s}$ is the string
coupling and $k\in\left\\{1/{\sqrt{2}},1,\sqrt{2}\right\\}$. The computation
of the coupling constants of higher order NR terms, $N>3$, is in general more
intricate as it entails the calculation of $N$-point correlation functions
[26]. Though suppressed by inverse powers of the string scale, these terms
turn out to play an important role in low energy phenomenology. For example,
they can account for fermion mass hierarchies and provide intermediate scale
masses for exotic states. In our analysis they are also important in ensuring
the existence of a particular solution of the F-flatness constraints as
explained in Section 4 and Appendix B. For this purpose, we compute in this
section, the coupling constants $\alpha_{i},\beta_{i},\gamma_{i},i=1,4$ of the
fifth order NR terms appearing in Eq. (4.76)

$\displaystyle\\!\\!\\!\left(\alpha_{1}\overline{\phi}_{1}^{2}\\!+\alpha_{2}\overline{\phi}_{2}^{2}+\\!\alpha_{3}\overline{\phi}_{4}^{2}\\!+\\!\alpha_{4}\overline{\phi}_{+}\overline{\phi}_{-}\right)\\!D_{1}^{2}\Phi_{31}\\!+\\!\left(\beta_{1}\overline{\phi}_{1}^{2}\\!+\beta_{2}\overline{\phi}_{2}^{2}\\!+\beta_{3}\overline{\phi}_{3}^{2}\\!+\\!\beta_{4}\overline{\phi}_{+}\overline{\phi}_{-}\right)\\!D_{4}^{2}\Phi_{31}\\!$
$\displaystyle+\\!\left(\gamma_{1}\overline{\phi}_{1}^{2}\\!+\gamma_{2}\overline{\phi}_{2}^{2}\\!+\gamma_{3}\overline{\phi}_{4}^{2}\\!+\\!\gamma_{4}\overline{\phi}_{+}\overline{\phi}_{-}\right)\\!T_{1}^{2}\Phi_{31}\,.$
(5.1)

Following [26] the coupling constant of a NR superpotential term of the form

$\displaystyle\int d^{2}\theta\,{\mathds{X}}_{1}\dots\mathds{X}_{N}\ ,\ N\geq
3\,$ (5.2)

is proportional to the correlation function

$\displaystyle\langle{\Psi_{1}\Psi_{2}\Phi_{3}\dots\Phi_{N}}\rangle\,,$ (5.3)

where $\Psi_{1},\Psi_{2}$ are the fermionic components of the superfields
${\mathds{X}}_{1},{\mathds{X}}_{2}$ respectively and $\Phi_{i}$ stand for the
bosonic components of ${\mathds{X}}_{i},i=3,\dots,N$. Evaluation of the
correlator requires knowing the vertex operators of the associated fields
which (in the 0-ghost picture) are world-sheet operators of conformal
dimensions $(1,1)$. The vertex operators are expressed in terms of the world-
sheet degrees of freedom. These include 22 real left-moving fermions
$\big{\\{}\psi^{\mu}$, $\chi^{I},y^{I},\omega^{I},I=1,\dots,6\\}$ together
with 12 real $\big{\\{}\overline{y}^{I},\overline{\omega}^{I},I=1,\dots,6\\}$
and 16 complex
$\big{\\{}\bar{\psi}^{1,\dots,5},\bar{\eta}^{1,2,3},\bar{\phi}^{1,\dots,8}\big{\\}}$
right-moving fermions. In the case at hand, the world-sheet supercurrent takes
the form

$\displaystyle T_{F}=\psi^{\mu}\partial
X_{\mu}+i\sum_{I=1}^{6}\chi^{I}y^{I}\omega^{I}\,.$ (5.4)

Furthermore, the particular choice of basis vectors allows for the
bosonisation of the fermionic fields $\chi^{1},\dots,\chi^{6}$ as follows

$\displaystyle e^{\pm iS_{k,k+1}}=\frac{1}{\sqrt{2}}\left(\chi^{k}\pm
i\chi^{k+1}\right)\ ;\ k=1,3,5\,.$ (5.5)

In terms of the bosonised fields, the supercurrent takes the form

$\displaystyle T_{F}=T^{0}_{F}+T^{-}_{F}+T^{+}_{F}\,,$ (5.6)

where

$\displaystyle T^{0}_{F}$ $\displaystyle=\psi^{\mu}\partial X_{\mu}\,,$ (5.7)
$\displaystyle T^{-}_{F}$
$\displaystyle=\sum_{k=1,3,5}e^{-iS_{k,k+1}}\tau_{k,k+1}\,,$ (5.8)
$\displaystyle T^{+}_{F}$ $\displaystyle=-\left(T^{-}_{F}\right)^{\ast}\,,$
(5.9)

and

$\displaystyle\tau_{k,\ell}=\frac{i}{\sqrt{2}}\left(y^{k}\omega^{k}+iy^{\ell}\omega^{\ell}\right)\,.$
(5.10)

From the remaining real fermions only two pairs can be complexified, namely
$\omega^{2},\omega^{3}$ and $\overline{\omega}^{2},\overline{\omega}^{3}$. The
rest of them can be grouped as nine left-right moving fermion pairs
$\\{y^{I},\overline{y}^{I}\\},I=1,\dots,6$ and
$\\{\omega^{I},\overline{\omega}^{I}\\},I=1,4,6$, ascribed to critical Ising
models.

In this framework, a general vertex operator of the bosonic component of a
chiral superfield in the canonical picture (ghost charge -1) and vanishing
momentum is of the form

$\displaystyle V^{B}_{-1}(z)$
$\displaystyle=e^{-c}(z)e^{i\alpha_{1}S_{12}}(z)e^{i\alpha_{2}S_{34}}(z)e^{i\alpha_{3}S_{56}}(z)G(z,\overline{z})\,,$
(5.11)

where $c$ stands for the ghost field. Here,
$\alpha_{i}\in\left\\{0,\pm\frac{1}{2},\pm 1\right\\}$ and $G(z,\bar{z})$
stands for a conformal field with dimensions
$\left(\frac{1}{2}-\sum_{i=1}^{3}\frac{\alpha_{i}^{2}}{2},1\right)$ comprised
of exponentials of the remaining bosonised fields and of primary Ising fields
$\sigma_{+}(z,\overline{z}),\sigma_{-}(z,\overline{z}),f(z),\bar{f}(\overline{z})$,
corresponding to the order, disorder and the left/right fermion operators
respectively. The fermionic partner of (5.11) in the canonical -1/2 picture
takes the form

$\displaystyle V^{F}_{-1/2}(z)$
$\displaystyle=e^{-c/2}(z)S_{a}e^{i(\alpha_{1}-1/2)S_{12}}(z)e^{i(\alpha_{2}-1/2)S_{34}}(z)e^{i(\alpha_{3}-1/2)S_{56}}(z)G(z,\overline{z})\,,$
(5.12)

where $S_{a}$ represents a space-time spinor field. In the computation of the
correlator (5.3) the bosonic fields $\Phi_{4},\dots,\Phi_{N}$ have to be
picture changed to the 0-ghost picture. This procedure can be carried out
using the standard picture-changing formula

$\displaystyle V_{0}^{B}(z,\overline{z})=\lim_{w\to
z}e^{c}(w)T_{F}(w)V_{-1}^{B}(z,\overline{z})$ (5.13)

and the relevant operator product expansions.

Let us now proceed with the computation of the correlator

$\displaystyle\langle{{\psi^{F}_{1(-1/2)}}{\psi^{F}_{2(-1/2)}}\Phi_{31(-1)}^{B}\varphi^{B}_{1(0)}\varphi^{B}_{2(0)}}\rangle\,,$
(5.14)

where
$\psi^{F}_{1}\psi^{F}_{2}\in\left\\{\overline{\phi}_{1}^{2},\overline{\phi}_{2}^{2},\overline{\phi}_{4}^{2},\overline{\phi}_{+}\overline{\phi}_{-}\right\\}$
and
$\varphi^{B}_{1}\varphi^{B}_{2}\in\left\\{D_{1}^{2},D_{4}^{2},T_{1}^{2}\right\\}$.
The vertex operators of the fields involved are (in the canonical
fermionic/bosonic picture)

$\displaystyle\overline{\phi}^{F}_{1(-1/2)}$
$\displaystyle=e^{-c/2}S_{a}e^{-\frac{i}{2}S_{56}}\sigma_{+}^{y_{2}}\sigma_{+}^{y_{3}}\overline{y}^{5}\sigma_{+}^{\omega_{1}}\sigma_{-}^{\omega_{4}}e^{-\frac{i}{2}\overline{H}_{1}}e^{+\frac{i}{2}\overline{H}_{2}}\,,$
(5.15) $\displaystyle\overline{\phi}^{F}_{2(-1/2)}$
$\displaystyle=e^{-c/2}S_{a}e^{-\frac{i}{2}S_{56}}\sigma_{+}^{y_{2}}\sigma_{+}^{y_{3}}{\overline{\omega}}^{5}\sigma_{+}^{\omega_{1}}\sigma_{-}^{\omega_{4}}e^{-\frac{i}{2}\overline{H}_{1}}e^{+\frac{i}{2}\overline{H}_{2}}\,,$
(5.16) $\displaystyle\overline{\phi}^{F}_{4(-1/2)}$
$\displaystyle=e^{-c/2}S_{a}e^{-\frac{i}{2}S_{56}}\sigma_{-}^{y_{2}}\sigma_{-}^{y_{3}}{\overline{\omega}}^{6}\sigma_{-}^{\omega_{1}}\sigma_{+}^{\omega_{4}}e^{-\frac{i}{2}\overline{H}_{1}}e^{+\frac{i}{2}\overline{H}_{2}}\,,$
(5.17) $\displaystyle\overline{\phi}^{F}_{+(-1/2)}$
$\displaystyle=e^{-c/2}S_{a}e^{-\frac{i}{2}S_{56}}\sigma_{+}^{y_{2}}\sigma_{-}^{y_{3}}\sigma_{+}^{\omega_{1}}\sigma_{+}^{\omega_{4}}e^{-\frac{i}{2}\overline{H}_{1}}e^{+\frac{i}{2}\overline{H}_{2}}e^{-i\overline{H}_{4}}\,,$
(5.18) $\displaystyle\overline{\phi}^{F}_{-(-1/2)}$
$\displaystyle=e^{-c/2}S_{a}e^{-\frac{i}{2}S_{56}}\sigma_{+}^{y_{2}}\sigma_{-}^{y_{3}}\sigma_{+}^{\omega_{1}}\sigma_{+}^{\omega_{4}}e^{-\frac{i}{2}\overline{H}_{1}}e^{\frac{i}{2}\overline{H}_{2}}e^{+i\overline{H}_{4}}\,,$
(5.19) $\displaystyle\Phi_{31(-1)}^{B}$
$\displaystyle=e^{-c}e^{+iS_{34}}e^{+i\overline{H}_{1}}e^{-i\overline{H}_{3}}$
(5.20) $\displaystyle D_{1(-1)}^{B}$
$\displaystyle=e^{-c}e^{+\frac{i}{2}S_{34}}e^{+\frac{i}{2}S_{56}}\sigma_{+}^{y_{3}}\sigma_{+}^{y_{4}}\sigma_{+}^{y_{5}}\sigma_{+}^{y_{6}}e^{-\frac{i}{2}\overline{H}_{2}}e^{+\frac{i}{2}\overline{H}_{3}}e^{iJ_{\mathbf{6}}{\cdot}W_{\mathbf{6}}}\,,$
(5.21) $\displaystyle D_{4(-1)}^{B}$
$\displaystyle=e^{-c}e^{+\frac{i}{2}S_{34}}e^{+\frac{i}{2}S_{56}}\sigma_{+}^{y_{3}}\sigma_{+}^{y_{6}}\sigma_{+}^{\omega
4}\sigma_{+}^{\omega_{5}}e^{-\frac{i}{2}\overline{H}_{2}}e^{+\frac{i}{2}\overline{H}_{3}}e^{iJ_{\mathbf{6}}{\cdot}W_{\mathbf{6}}}\,,$
(5.22) $\displaystyle T_{1(-1)}^{B}$
$\displaystyle=e^{-c}e^{+\frac{i}{2}S_{34}}e^{+\frac{i}{2}S_{56}}\sigma_{+}^{y_{3}}\sigma_{-}^{y_{4}}\sigma_{-}^{y_{5}}\sigma_{+}^{y_{6}}e^{-\frac{i}{2}\overline{H}_{2}}e^{+\frac{i}{2}\overline{H}_{3}}e^{iJ_{\mathbf{10}}{\cdot}W_{\mathbf{10}}}\,,$
(5.23)

where $\sigma_{\pm}^{f}$ denotes the order/disorder operators of the Ising
pair $\left\\{f,\overline{f}\right\\}$, $e^{i\overline{H}_{i}},i=1,2,3$ and
$e^{i\overline{H}_{4}}$ stand for the bosonised fermions
$\overline{\eta}^{i},i=1,2,3$ and
$\overline{\omega}^{2},\overline{\omega}^{3}$ respectively;
$e^{iJ_{\mathbf{6}}{\cdot}W_{\mathbf{6}}}$ and
$e^{iJ_{\mathbf{10}}{\cdot}W_{\mathbf{10}}}$ represent the bosonised fermions
ascribed to the vectorial representations of the hidden gauge group
$\mathbf{6}$ of $SO(6)\simeq SU(4)$ and $\mathbf{10}$ of $SO(10)$,
respectively, where $J_{\mathbf{6}},J_{\mathbf{10}}$ the associated bosonic
fields and $W_{\mathbf{6}},W_{\mathbf{10}}$ the corresponding charges. Using
(5.13), we can then derive the expressions for the picture-changed bosonic
fields that contribute to the correlator (5.14)

$\displaystyle\\!\\!\\!\\!D_{1(0)}^{B}$
$\displaystyle\\!=\\!\frac{i}{2}e^{-\frac{i}{2}S_{34}}e^{+\frac{i}{2}S_{56}}\bigl{[}\sigma_{-}^{y_{3}}\sigma_{+}^{y_{4}}\sigma_{+}^{y_{5}}\sigma_{+}^{y_{6}}\omega^{3}\\!+\\!i\sigma_{+}^{y_{3}}\sigma_{-}^{y_{4}}\sigma_{+}^{y_{5}}\sigma_{+}^{y_{6}}\omega^{4}\bigl{]}e^{-\frac{i}{2}\overline{H}_{2}}e^{+\frac{i}{2}\overline{H}_{3}}e^{iJ_{\mathbf{6}}{\cdot}W_{\mathbf{6}}}$
(5.24) $\displaystyle\\!\\!\\!\\!D_{4(0)}^{B}$
$\displaystyle\\!=\\!\frac{i}{2}e^{-\frac{i}{2}S_{34}}e^{+\frac{i}{2}S_{56}}\bigl{[}\sigma_{-}^{y_{3}}\sigma_{+}^{y_{6}}\sigma_{+}^{\omega_{4}}\sigma_{+}^{\omega_{5}}\omega^{3}\\!+\\!i\sigma_{+}^{y_{3}}\sigma_{+}^{y_{6}}\sigma_{-}^{\omega_{4}}\sigma_{+}^{\omega_{5}}y^{4}\bigl{]}e^{-\frac{i}{2}\overline{H}_{2}}e^{+\frac{i}{2}\overline{H}_{3}}e^{iJ_{\mathbf{6}}{\cdot}W_{\mathbf{6}}}$
(5.25) $\displaystyle\\!\\!\\!\\!T_{1(0)}^{B}$
$\displaystyle\\!=\\!\frac{i}{2}e^{-\frac{i}{2}S_{34}}e^{+\frac{i}{2}S_{56}}\bigl{[}\sigma_{-}^{y_{3}}\sigma_{-}^{y_{4}}\sigma_{-}^{y_{5}}\sigma_{+}^{y_{6}}\omega^{3}\\!+\\!i\sigma_{+}^{y_{3}}\sigma_{+}^{y_{4}}\sigma_{-}^{y_{5}}\sigma_{+}^{y_{6}}\omega^{4}\bigl{]}e^{-\frac{i}{2}\overline{H}_{2}}e^{+\frac{i}{2}\overline{H}_{3}}e^{iJ_{\mathbf{10}}{\cdot}W_{\mathbf{10}}}\,.$
(5.26)

Putting everything together, we get

$\displaystyle\alpha_{1}$
$\displaystyle=\langle{{\overline{\phi}^{F}_{1(-1/2)}}{\overline{\phi}^{F}_{1(-1/2)}}\Phi_{31(-1)}^{B}D_{1(0)}D_{1(0)}}\rangle$
$\displaystyle=\frac{g_{s}^{3}\sqrt{2}}{\left(2\pi\right)^{2}}\prod_{i=1}^{5}\int
d^{2}z_{i}\,\,\langle{e^{-c/2}(1)}{e^{-c/2}(2)}{e^{-c}(3)}\rangle\langle{S_{a}(1)S_{b}(2)}\rangle$
$\displaystyle\quad\times\langle{e^{iS_{34}}(3)e^{-\frac{i}{2}S_{34}}(4)e^{-\frac{i}{2}S_{34}}(z_{5})}\rangle\langle{e^{-\frac{i}{2}S_{56}}(1)e^{-\frac{i}{2}S_{56}}(2)e^{+\frac{i}{2}S_{56}}(4)e^{+\frac{i}{2}S_{56}}(5)}\rangle$
$\displaystyle\quad\times\langle{e^{-\frac{i}{2}\overline{H}_{1}}(1)e^{-\frac{i}{2}\overline{H}_{1}}(2)e^{+i\overline{H}_{1}}(3)}\rangle\langle{e^{+\frac{i}{2}\overline{H}_{2}}(1)e^{+\frac{i}{2}\overline{H}_{2}}(2)e^{-\frac{i}{2}\overline{H}_{2}}(4)e^{-\frac{i}{2}\overline{H}_{2}}(5)}\rangle$
$\displaystyle\quad\times\langle{e^{-\i\overline{H}_{3}}(1)e^{+\frac{i}{2}\overline{H}_{3}}(4)e^{+\frac{i}{2}\overline{H}_{3}}(5)}\rangle\times\Sigma(1,2,5,6)\times\langle{e^{iJ_{\mathbf{6}}{\cdot}W_{\mathbf{6}}}(5)e^{iJ_{\mathbf{10}}{\cdot}W_{\mathbf{10}}}(6)}\rangle$
$\displaystyle\qquad\quad\langle{e^{iJ_{\mathbf{10}}{\cdot}W_{\mathbf{10}}}(5)e^{iJ_{\mathbf{10}}{\cdot}W_{\mathbf{10}}}(6)}\rangle\,,$
(5.27)

where

$\displaystyle\Sigma(1,2,5,6)=-\frac{1}{4}\left(\Sigma_{1}-\Sigma_{2}\right)\,,$
(5.28)

with

$\displaystyle\Sigma_{1}$
$\displaystyle=\langle{\sigma_{+}^{y_{2}}(1)\sigma_{+}^{y_{2}}(2)}\rangle\langle{\sigma_{+}^{\omega_{1}}(1)\sigma_{+}^{\omega_{1}}(2)}\rangle\langle{\sigma_{+}^{y_{4}}(5)\sigma_{+}^{y_{4}}(6)}\rangle\langle{\sigma_{+}^{y_{6}}(5)\sigma_{+}^{y_{6}}(6)}\rangle\,,$
$\displaystyle\quad\times\langle{\sigma_{+}^{y_{3}}(1)\sigma_{+}^{y_{3}}(2)\sigma_{-}^{y_{3}}(5)\sigma_{-}^{y_{3}}(6)}\rangle\langle{\overline{y}^{5}(1)\overline{y}^{5}(2)\sigma_{+}^{y_{5}}(5)\sigma_{+}^{y_{5}}(6)}\rangle\langle{\sigma_{-}^{\omega_{4}}(1)\sigma_{-}^{\omega_{4}}(2)}\rangle\langle{\omega^{3}(5)\omega^{3}(6)}\rangle$
(5.29) $\displaystyle\Sigma_{2}$
$\displaystyle=\langle{\sigma_{+}^{y_{2}}(1)\sigma_{+}^{y_{2}}(2)}\rangle\langle{\sigma_{+}^{\omega_{1}}(1)\sigma_{+}^{\omega_{1}}(2)}\rangle\langle{\sigma_{-}^{y_{4}}(5)\sigma_{-}^{y_{4}}(6)}\rangle\langle{\sigma_{+}^{y_{6}}(5)\sigma_{+}^{y_{6}}(6)}\rangle$
$\displaystyle\quad\times\langle{\sigma_{+}^{y_{3}}(1)\sigma_{+}^{y_{3}}(2)\sigma_{+}^{y_{3}}(5)\sigma_{+}^{y_{3}}(6)}\rangle\langle{\overline{y}^{5}(1)\overline{y}^{5}(2)\sigma_{+}^{y_{5}}(5)\sigma_{+}^{y_{5}}(6)}\rangle\langle{\sigma_{-}^{\omega_{4}}(1)\sigma_{-}^{\omega_{4}}(2)\omega^{4}(5)\omega^{4}(6)}\rangle\,.$
(5.30)

Substituting the conformal correlators from Appendix C, we get

$\displaystyle\alpha_{1}=-\frac{g_{s}^{3}\sqrt{2}}{4\left(2\pi\right)^{2}}\,I_{\alpha_{1}}\,,$
(5.31)

with

$\displaystyle I_{\alpha_{1}}=\int d^{2}w\int
d^{2}z\left[K_{1}^{\alpha_{1}}(w,z)-K_{2}^{\alpha_{1}}(w,z)\right]\,.$ (5.32)

Here

$\displaystyle K_{1}^{\alpha_{1}}(w,z)$
$\displaystyle=\frac{1}{2\sqrt{2}}\left|w\right|^{-1}\left|1-w\right|^{-1}\left|w-z\right|^{-1}|z|^{-2}|1-z|^{-3/4}$
$\displaystyle\quad\times\left(1-|z|+|1-z|\right)^{1/2}\frac{2-\bar{z}}{\sqrt{1-\bar{z}}}\,,$
(5.33) $\displaystyle K_{2}^{\alpha_{1}}(w,z)$
$\displaystyle=\frac{1}{4\sqrt{2}}\left|w\right|^{-1}\left|1-w\right|^{-1}\left|w-z\right|^{-1}|z|^{-2}|1-z|^{-7/4}$
$\displaystyle\quad\times\left(1+|z|+|1-z|\right)^{1/2}|2-z|^{2}\,,$ (5.34)

where we have utilised conformal invariance to set
$z_{1}=\infty\,,z_{2}=1\,,z_{3}=w$, $z_{4}=z\,,z_{5}=0$. It should be
emphasised that both $K_{1}^{\alpha_{1}}$ and $K_{2}^{\alpha_{1}}$ exhibit a
pole at $z=0$, however, the poles cancel exactly in the expression of Eq.
(5.32) ensuring the convergence of the integral. Next, we change variables
$z\to 1-z,w\to 1-w$ in $K_{1}^{\alpha_{1}}(w,z)$

$\displaystyle K_{1}^{\alpha_{1}}(w,z)$
$\displaystyle=\frac{1}{2\sqrt{2}}\left|w\right|^{-1}\left|1-w\right|^{-1}\left|w-z\right|^{-1}|z|^{-3/4}|1-z|^{-2}\left(1-|z|+|1-z|\right)^{1/2}$
$\displaystyle\times\left(\sqrt{z}+\sqrt{\bar{z}}|z|\right)$ (5.35)

and then replace $K_{1}^{\alpha_{1}}(w,z)$ with
$(K_{1}^{\alpha_{1}}(w,z)+K_{1}^{\alpha_{1}}(\bar{w},\bar{z}))/2$ in (5.32)
that enables us to recast $K_{1}^{\alpha_{1}}$ in a purely real form:

$\displaystyle K_{1}^{\alpha_{1}}(w,z)$
$\displaystyle=\frac{1}{2\sqrt{2}}\left|w\right|^{-1}\left|1-w\right|^{-1}\left|w-z\right|^{-1}|z|^{-3/4}|1-z|^{-2}\left(1-|z|+|1-z|\right)^{1/2}$
$\displaystyle\times\left(1+|z|\right)\text{Re}\sqrt{z}\,.$ (5.36)

In a similar way, we find

$\displaystyle\alpha_{4}$
$\displaystyle=\langle{{\overline{\phi}^{F}_{+(-1/2)}}{\overline{\phi}^{F}_{-(-1/2)}}\Phi_{31(-1)}^{B}D_{1(0)}D_{1(0)}}\rangle=\frac{g_{s}^{3}\sqrt{2}}{\left(2\pi\right)^{2}}I_{\alpha_{4}}\,,$
(5.37)

with

$\displaystyle I_{\alpha_{4}}=\int d^{2}w\int
d^{2}z\left[K_{1}^{\alpha_{4}}(w,z)-K_{2}^{\alpha_{4}}(w,z)\right]\,,$ (5.38)

and

$\displaystyle K_{1}^{\alpha_{4}}(w,z)$
$\displaystyle=\frac{1}{\sqrt{2}}\left|w\right|^{-1}\left|1-w\right|^{-1}\left|w-z\right|^{-1}|z|^{-2}\left|1-z\right|^{-3/4}\left(1+|z|+|1-z|\right)^{1/2}\,,$
(5.39) $\displaystyle K_{2}^{\alpha_{4}}(w,z)$
$\displaystyle=\frac{1}{2\sqrt{2}}\left|w\right|^{-1}\left|1-w\right|^{-1}\left|w-z\right|^{-1}\left|z\right|^{-2}\left|1-z\right|^{-3/4}\left(1-|z|+|1-z|\right)^{1/2}$
$\displaystyle\times\left(1+|z|\right)\text{Re}\sqrt{z}\,.$ (5.40)

Repeating the calculation for

$\displaystyle\alpha_{2}$
$\displaystyle=\langle{{\overline{\phi}^{F}_{2(-1/2)}}{\overline{\phi}^{F}_{2(-1/2)}}\Phi_{31(-1)}^{B}D_{1(0)}D_{1(0)}}\rangle\,,$
(5.41) $\displaystyle\alpha_{3}$
$\displaystyle=\langle{{\overline{\phi}^{F}_{4(-1/2)}}{\overline{\phi}^{F}_{4(-1/2)}}\Phi_{31(-1)}^{B}D_{1(0)}D_{1(0)}}\rangle\,,$
(5.42)

we find

$\displaystyle\alpha_{2}=\alpha_{3}=\alpha_{4}\,.$ (5.43)

The correlator functions $\beta_{1}$, $\beta_{2}$ and $\beta_{4}$ can be
computed similarly

$\displaystyle\beta_{1}$
$\displaystyle=\langle{{\overline{\phi}^{F}_{1(-1/2)}}{\overline{\phi}^{F}_{1(-1/2)}}\Phi_{31(-1)}^{B}D_{4(0)}D_{4(0)}}\rangle=\frac{g_{s}^{3}\sqrt{2}}{\left(2\pi\right)^{2}}I_{\beta_{1}}\,,$
(5.44) $\displaystyle\beta_{4}$
$\displaystyle=\langle{{\overline{\phi}^{F}_{+(-1/2)}}{\overline{\phi}^{F}_{-(-1/2)}}\Phi_{31(-1)}^{B}D_{4(0)}D_{4(0)}}\rangle=\frac{g_{s}^{3}\sqrt{2}}{\left(2\pi\right)^{2}}I_{\beta_{4}}\,,$
(5.45)

where

$\displaystyle I_{\beta_{1}}=-I_{\beta_{4}}=\int d^{2}w\int
d^{2}z\left[K_{1}^{\beta_{1}}(w,z)-K_{2}^{\beta_{1}}(w,z)\right]\,,$ (5.46)

and

$\displaystyle K_{1}^{\beta_{1}}(w,z)$
$\displaystyle=\frac{1}{2}\left|w\right|^{-1}\left|1-w\right|^{-1}\left|w-z\right|^{-1}\left|z\right|^{-2}\left|1-z\right|^{-1}\left(|1-z|-|z|+1\right)\,,$
(5.47) $\displaystyle K_{2}^{\beta_{1}}(w,z)$
$\displaystyle=\frac{1}{2}\left|w\right|^{-1}\left|w-z\right|^{-1}\left|1-w\right|^{-1}\left|z\right|^{-2}\left|1-z\right|^{-1}\left(|1-z|+|z|+1\right)\,.$
(5.48)

The two terms in the integrand are here nicely combined, cancelling explicitly
the pole at $z=0$ and yielding

$\displaystyle I_{\beta_{1}}=-I_{\beta_{4}}=-\int d^{2}w\int
d^{2}z\left|w\right|^{-1}\left|1-w\right|^{-1}\left|w-z\right|^{-1}\left|z\right|^{-1}\left|1-z\right|^{-1}\,.$
(5.49)

Repeating for

$\displaystyle\beta_{2}=\langle{{\overline{\phi}^{F}_{2(-1/2)}}{\overline{\phi}^{F}_{2(-1/2)}}\Phi_{31(-1)}^{B}D_{4(0)}D_{4(0)}}\rangle\,,$
(5.50)
$\displaystyle\beta_{3}=\langle{{\overline{\phi}^{F}_{4(-1/2)}}{\overline{\phi}^{F}_{4(-1/2)}}\Phi_{31(-1)}^{B}D_{4(0)}D_{4(0)}}\rangle\,,$
(5.51)

we obtain

$\displaystyle\beta_{2}=0\ ,\ \beta_{3}=\beta_{4}.$ (5.52)

Finally, an analysis of the $\gamma_{1},\gamma_{4}$ correlators gives

$\displaystyle\gamma_{1}$
$\displaystyle=\langle{{\overline{\phi}^{F}_{1(-1/2)}}{\overline{\phi}^{F}_{1(-1/2)}}\Phi_{31(-1)}^{B}T_{1(0)}D_{4(0)}}\rangle=\frac{g_{s}^{3}\sqrt{2}}{\left(2\pi\right)^{2}}I_{\beta_{1}}=\frac{g_{s}^{3}\sqrt{2}}{\left(2\pi\right)^{2}}I_{\alpha_{1}}=\alpha_{1}\,,$
(5.53) $\displaystyle\gamma_{4}$
$\displaystyle=\langle{{\overline{\phi}^{F}_{1(-1/2)}}{\overline{\phi}^{F}_{1(-1/2)}}\Phi_{31(-1)}^{B}T_{1(0)}D_{4(0)}}\rangle=\frac{g_{s}^{3}\sqrt{2}}{\left(2\pi\right)^{2}}I_{\beta_{4}}=\frac{g_{s}^{3}\sqrt{2}}{\left(2\pi\right)^{2}}I_{\alpha_{4}}=\alpha_{4}\,.$
(5.54)

In the same way we get

$\displaystyle\gamma_{2}$
$\displaystyle=\langle{{\overline{\phi}^{F}_{2(-1/2)}}{\overline{\phi}^{F}_{2(-1/2)}}\Phi_{31(-1)}^{B}T_{1(0)}D_{4(0)}}\rangle=\gamma_{4}=\alpha_{4}\,,$
(5.55) $\displaystyle\gamma_{3}$
$\displaystyle=\langle{{\overline{\phi}^{F}_{4(-1/2)}}{\overline{\phi}^{F}_{4(-1/2)}}\Phi_{31(-1)}^{B}T_{1(0)}D_{4(0)}}\rangle=\gamma_{4}=\alpha_{4}\,.$
(5.56)

Summarising, the couplings under consideration involve three independent
integrals $I_{\alpha_{1}},I_{\alpha_{4}}$ and $I_{\beta_{1}}$ defined in
(5.32), (5.38) and (5.46), respectively. We can calculate them numerically
after changing to polar coordinates
$w=\left|w\right|e^{iu},z=\left|z\right|e^{iv}$. Using Mathematica we get

$\displaystyle I_{\alpha_{1}}=-567.20\ ,\ I_{\alpha_{4}}=189.07\ ,\
I_{\beta_{1}}=-378.14$ (5.57)

with an error of the order of 0.1%. This supports the relations

$\displaystyle I_{\alpha_{1}}=-3I_{\alpha_{4}}\ ,\
I_{\beta_{1}}=-2I_{\alpha_{4}}\,,$ (5.58)

yielding

$\displaystyle\alpha_{1}=-3\alpha_{4}\ ,\ \beta_{1}=3\beta_{4}\,.$ (5.59)

When combined with Eqs. (5.43), (5.52),(5.53), (5.54), (5.56) the last
equation gives a set of coupling relations

$\displaystyle-\frac{\alpha_{1}}{3}=-\frac{\gamma_{1}}{3}=-\frac{\beta_{1}}{2}=\frac{\beta_{3}}{2}=\frac{\beta_{4}}{2}=\gamma_{3}=\gamma_{4}=\alpha_{3}=\alpha_{4}\,,$
(5.60) $\displaystyle\alpha_{2}=\gamma_{2}=\alpha_{4}\ ,\ \beta_{2}=0\,.$
(5.61)

##  6 Fermion masses and proton decay

In this section we study the phenomenological consequences of the flatness
solution of Section 4. Of key importance to low energy phenomenology are the
surviving SM Higgs doublets. At tree-level these are given in Eqs. (3.11),
(3.12). However, both the doublet and the triplet mass matrices receive
additional contributions from NR terms of the form
$h_{i}\overline{h}_{j}\varphi^{N-3}\,,i,j=1,2,3,45$, where $\varphi^{N-3}$ is
a combination of VEVs, arising from a NR term of order $N>3$. A comprehensive
computer search up to and including $N=7$ order yields numerous NR terms of
this form. However, after applying the flatness solution of Section 4 with the
additional simplifying assumptions

$\displaystyle\Phi_{1}=\Phi_{2}=0\,,\quad\overline{\phi}_{45}\mathrel{\lower
2.5pt\vbox{\hbox{$<$}\hbox{$\sim$}}}\xi^{2}\,,$ (6.1)

the doublet mass matrix reduces to

$\displaystyle
M_{H}^{(7)}=\bordermatrix{~{}&H_{1}&H_{2}&H_{3}&H_{45}\cr{\overline{H}}_{1}&0&\overline{\Phi}_{12}&\Phi_{31}&0\cr{\overline{H}}_{2}&0&{A^{(7)}}&\overline{\Phi}_{23}&0\cr{\overline{H}}_{3}&\overline{\Phi}_{31}&\Phi_{23}&0&\phi_{45}\cr{\overline{H}}_{45}&0&\overline{B}^{(5)}&\overline{\phi}_{45}&0\cr}+{\cal
O}\left(\xi^{6}\right)\,,$ (6.2)

where

$\displaystyle\overline{B}^{(5)}$
$\displaystyle=\left\\{\overline{\phi}_{45}\left(D_{1}^{2}+D_{4}^{2}+T_{1}^{2}\right)\right\\}\sim\xi^{2}\overline{\phi}_{45}\,,$
(6.3) $\displaystyle A^{(7)}$
$\displaystyle=\left\\{\Phi_{31}\left(\overline{\phi}_{1}^{2}+\overline{\phi}_{4}^{2}+\overline{\phi}_{+}\overline{\phi}_{-}\right)\left(D_{1}^{2}+D_{4}^{2}+T_{1}^{2}\right)\right\\}\sim\xi^{5}\,.$
(6.4)

Diagonalising we get a massless doublet pair

$\displaystyle H$
$\displaystyle\propto\phi_{45}H_{1}-\overline{\Phi}_{31}H_{45}\,,$ (6.5)
$\displaystyle\overline{H}$
$\displaystyle\propto\left(\overline{B}^{(5)}\overline{\Phi}_{23}\\!-\\!A^{(7)}\overline{\phi}_{45}\right)\\!\overline{H}_{1}\\!+\\!{\left(\overline{\Phi}_{12}\overline{\phi}_{45}\\!-\\!\overline{B}^{(5)}\Phi_{31}\right)\\!\overline{H}_{2}}\\!+\\!\left(A^{(7)}\Phi_{31}\\!-\\!\overline{\Phi}_{12}\overline{\Phi}_{23}\right)\\!\overline{H}_{45}\,$
(6.6)

In the last equation the coefficient of $\overline{H}_{1}$ is of order ${\cal
O}(\xi^{5}\overline{\phi}_{45})$ while those of the components
$\overline{H}_{2}$ and $\overline{H}_{45}$ are of orders ${\cal
O}(\xi^{3}\overline{\phi}_{45})$ and ${\cal O}(\xi^{6})$ respectively. We
therefore impose

$\displaystyle\overline{\Phi}_{12}\overline{\phi}_{45}\\!-\\!\overline{B}^{(5)}\Phi_{31}\mathrel{\lower
2.5pt\vbox{\hbox{$<$}\hbox{$\sim$}}}\xi^{6}\,,$ (6.7)

so that $\overline{H}_{45}$ is the dominant component in the $\overline{H}$
Higgs doublet. This can be done for instance by choosing
$\overline{\phi}_{45}\sim\xi^{3}$ and tuning slightly the combination
$\left(\overline{\Phi}_{12}-\\{D_{1}^{2}+D_{4}^{2}+T_{1}^{2}\\}\right)$. As a
result, we obtain:

$\displaystyle H$
$\displaystyle=\cos\theta\,H_{1}-\sin\theta\,H_{45}\quad;\quad\tan\theta=\langle\overline{\Phi}_{31}\rangle/\langle\phi_{45}\rangle$
(6.8) $\displaystyle\overline{H}$
$\displaystyle=\cos\bar{\theta}\,\overline{H}_{45}+\sin\bar{\theta}\cos\bar{\vartheta}\,\overline{H}_{2}+\sin\bar{\theta}\sin\bar{\vartheta}\,\overline{H}_{1}\quad;$
(6.9)
$\displaystyle\tan\bar{\theta}\cos\bar{\vartheta}=\frac{\langle\overline{\Phi}_{12}\rangle\langle\overline{\phi}_{45}\rangle\\!-\\!\overline{B}^{(5)}\langle\Phi_{31}\rangle}{A^{(7)}\langle\Phi_{31}\rangle\\!-\\!\langle\overline{\Phi}_{12}\rangle\langle\overline{\Phi}_{23}\rangle}\quad;\quad\tan\bar{\vartheta}=\frac{\overline{B}^{(5)}\langle\overline{\Phi}_{23}\rangle\\!-\\!A^{(7)}\langle\overline{\phi}_{45}\rangle}{\langle\overline{\Phi}_{12}\rangle\langle\overline{\phi}_{45}\rangle\\!-\\!\overline{B}^{(5)}\langle\Phi_{31}\rangle}\,.$

It follows that

$\displaystyle H_{1}$ $\displaystyle=\cos\theta
H+\dots\,\,;\,\,H_{45}=\sin\theta H+\dots$ $\displaystyle\overline{H}_{45}$
$\displaystyle=\cos\bar{\theta}\,\overline{H}+\dots\,\,;\,\,\overline{H}_{2}=\sin\bar{\theta}\cos\bar{\vartheta}\,\overline{H}+\dots\,\,;\,\,\overline{H}_{1}=\sin\bar{\theta}\sin\bar{\vartheta}\,\overline{H}+\dots$
(6.10)

where the dots stand for (superheavy) massive doublets. To simplify the
analysis, in the following we will take $\cos\theta\sim\cos\bar{\theta}\sim
1$.

###  6.1 Fermion masses

We notice that the physical Higgs which provides masses to the charged $Q=2/3$
up quarks is a mixture of $\overline{H}_{45}$ (as a necessary leading
component), $\overline{H}_{2}$ and $\overline{H}_{1}$, while for $Q=-1/3$ down
quarks and charged leptons, the physical Higgs is a mixture of $H_{1}$ (as a
necessary leading component) and $H_{45}$. Indeed, the leading electroweak
Higgs components are exactly what the trilinear superpotential (2.1) suggests
in order to provide masses to the heaviest third generation: $t$, $b$ and
$\tau$:

$W_{3}\supset
g_{s}\sqrt{2}\left\\{F_{4}\overline{f}_{5}\overline{H}_{45}+(F_{4}F_{4}+\overline{f}_{1}\ell^{c}_{1})H_{1}\right\\}\,.$
(6.11)

Actually, more than thirty years ago, this model predicted that the mass of
the top quark is around $\sim$ 170-180 GeV [2], as was observed in 1995 in
Fermilab. This is a consequence of the fact that the top Yukawa coupling
evolves at low energies towards a fixed point for its ratio to the QCD gauge
coupling. Furthermore, we get the highly successful relation at the GUT scale
$m_{b}=m_{\tau}$, following the equality of the corresponding Yukawa
couplings, which is apparent from the above expression of the trilinear
superpotential. Hence, we have the particle identification for the third
generation:

$F_{4}=\left\\{(t,b),b^{c},\nu^{c}_{\tau}\right\\}_{L}\quad;\quad\overline{f}_{5}\supset
t^{c}_{L}\quad;\quad\overline{f}_{1}\supset(\nu_{\tau},\tau)_{L}\quad;\quad
l^{c}_{1}=\tau^{c}_{L}$ (6.12)

We next look for possible fermion mass terms at higher NR orders in the
superpotential involving the massless Higgs doublets (6.5) and (6.6). Omitting
corrections to the Yukawa couplings of the 3rd generation, we find the
following list up to 5th order:

  * •

Up quarks:

$\displaystyle F_{2}\overline{f}_{2}\overline{h}_{45}\overline{\phi}_{4}$
(6.13)

  * •

Down quarks:

$\displaystyle
F_{2}F_{2}h_{1}\left(\overline{\phi}_{1}^{2}-\overline{\phi}_{+}\overline{\phi}_{-}+\lambda\overline{\phi}_{4}^{2}\right)$
(6.14)

  * •

Charged leptons:

$\displaystyle\overline{f}_{2}\ell^{c}_{2}h_{1}\left(\overline{\phi}_{1}^{2}-\overline{\phi}_{+}\overline{\phi}_{-}+\lambda\overline{\phi}_{4}^{2}\right)+\overline{f}_{5}\ell^{c}_{5}h_{1}\left(\overline{\phi}_{1}^{2}+\overline{\phi}_{+}\overline{\phi}_{-}-\overline{\phi}_{4}^{2}\right)$
(6.15)

where we displayed only the non-vanishing dominant contributions for the
choice of VEVs that solved the flatness conditions in Table 1, and we made a
convenient choice for the co-cycle factors ambiguity in the 5-point amplitudes
fixing the relative signs, while $\lambda$ is an (irrelevant) order one
constant (see below).

Using the flatness condition (4.50):

$\overline{\phi}_{1}^{2}+\overline{\phi}_{+}\overline{\phi}_{-}+\overline{\phi}_{4}^{2}=-\overline{\Phi}_{31}\overline{\Phi}_{23}={\cal
O}(\xi^{4})\,,$ (6.16)

one can identify all members of the 2nd generation:

$F_{2}=\left\\{(c,s),s^{c},\nu^{c}_{\mu}\right\\}_{L}\quad;\quad\overline{f}_{2}=\left\\{c^{c},(\nu_{\mu},\mu)\right\\}_{L}\quad;\quad
l^{c}_{2}=\mu^{c}_{L}\,,$ (6.17)

with Yukawa couplings suppressed by two orders of magnitude compared to those
of the 3rd generation. Indeed, (6.13) provides a successful mass to the charm
quark when $\overline{\phi}_{4}\sim{\cal O}(\xi^{2})$, while (6.14) and (6.15)
provide successful masses to the strange quark and muon. Moreover, a direct
computation of the corresponding coefficients of the above 5th order operators
shows that those of $F_{2}F_{2}h_{1}$ are equal to those of
$\overline{f}_{2}l^{c}_{2}h_{1}$ [16]. One thus obtains the successful mass
relation $m_{s}=m_{\mu}$.

It remains the identification of the first generation. Taking into account the
mass term (4.69), the identification (6.12), (6.17) and the leftover operator
for the charged leptons Yukawa couplings, one gets:

$F_{3}=\left\\{(u,d),d^{c},\nu^{c}_{e}\right\\}_{L}\quad;\quad\overline{f}_{5}\supset(\nu_{e},e)_{L}\quad;\quad\overline{f}_{1}\supset
u^{c}_{L}\quad;\quad l_{5}=e^{c}_{L}\,.$ (6.18)

Combining (6.12) and (6.18), we thus obtain:

$\overline{f}_{5}=\left\\{t^{c},(\nu_{e},e)\right\\}_{L}\quad;\quad\overline{f}_{1}=\left\\{u^{c},(\nu_{\tau},\tau)\right\\}_{L}\,.$
(6.19)

It is amazing how in this string model the three generations are distributed
with rather interesting consequences for inter-generation mixing, proton decay
modes and possibly flavour changing lepton number. Up to this order, there are
no Yukawa coupling for the up and down quarks, which is compatible with the
order of our approximation. On the other hand, the second operator in (6.15)
leads to a Yukawa coupling for the electron. Using the flatness relation
(6.16) and the identification (6.19), one finds that the electron mass is two
orders of magnitude lower than the muon mass, which is a remarkable successful
relation [16].

Focussing now to possible mass terms for the quarks of the first generation,
we extracted all relevant NR terms up to 7th order (included) for our choice
of VEVs, that we display below, omitting higher order corrections to existing
Yukawa couplings:

  * •

Up quarks:

$\displaystyle
F_{1}\overline{f}_{1}\overline{h}_{2}\Phi_{31}\phi_{45}\overline{\phi}_{1}+{F_{3}\overline{f}_{5}\overline{h}_{45}(D_{3}D_{4})\overline{\Phi}_{23}}$
$\displaystyle+F_{3}\overline{f}_{1}\overline{h}_{45}\left[\,\overline{\phi}_{-}\left(T_{1}T_{3}\right)\Phi_{31}+\overline{\phi}_{1}\left(D_{1}D_{3}\right)\Phi_{31}\right]$
(6.20)

  * •

Down quarks:

$\displaystyle
F_{1}F_{3}h_{1}\left(D_{1}D_{3}\right)\overline{\Phi}_{23}+F_{3}F_{4}h_{1}\left(D_{3}D_{4}\right)\overline{\Phi}_{23}$
(6.21)

  * •

Charged leptons:

$\displaystyle\overline{f}_{5}\ell^{c}_{5}\left(h_{1}\overline{\Phi}_{31}+h_{45}\phi_{45}\right)\left(\overline{\phi}_{1}^{2}+\overline{\phi}_{4}^{2}+\overline{\phi}_{+}\overline{\phi}_{-}\right)\Phi_{31}$
(6.22)

Eq. (6.22) gives a correction to the electron Yukawa coupling of order ${\cal
O}(\xi^{6})$. The term in the second line of (6.20) leads to an up quark
Yukawa coupling of the right order of magnitude ${\cal O}(\xi^{5})$, provided
$T_{3}$ and $D_{3}$ are of order ${\cal O}(\xi^{2})$, consistently with the
6th order flatness conditions discussed at the end of section 4. Finally, the
second term in the first line of (6.20), as well as the second term of (6.21)
lead to quark mixing between the first and third generation of order ${\cal
O}(\xi^{6})$.

A down quark Yukawa coupling can appear when $F_{3}$ gets a small VEV which is
consistent with the flatness conditions when $\langle F_{3}\rangle/\langle
F_{1}\rangle<\xi$, as explained in the end of section 4. In this case, the GUT
Higgs $F$ and the first generation $F_{3}^{\prime}$ are given by the linear
combinations:

$\displaystyle F$ $\displaystyle=$ $\displaystyle A_{1}F_{1}+A_{3}F_{3}\simeq
F_{1}+\varepsilon F_{3}$ $\displaystyle F_{3}^{\prime}$ $\displaystyle=$
$\displaystyle-A_{3}F_{1}+A_{1}F_{3}\simeq F_{3}-\varepsilon F_{1}$ (6.23)

where the constants $A_{i}$ satisfy $\sum_{i}|A_{i}|^{2}=1$ and enter in the
flatness condition (4.91), while $\varepsilon\sim{\cal O}(\xi^{3/2})$. Thus,
the tree-level superpotential term $F_{1}F_{1}h_{1}$ in (2.1) generate a
Yukawa coupling for the down quark of the right order of magnitude ${\cal
O}(\xi^{3})$.

Our results on the masses of quarks and leptons and their relations (at the
string scale) are summarised below:

$\begin{matrix}m_{t}=gv_{45}&m_{b}=gv_{1}&m_{\tau}=m_{b}\\\
m_{c}\sim\xi^{2}m_{t}&m_{s}\sim\xi^{2}m_{b}&m_{\mu}=m_{s}\\\
m_{u}\sim\xi^{5}m_{t}&m_{d}\sim\xi^{3}m_{b}&m_{e}\sim\xi^{4}m_{\tau}\\\
\end{matrix}$ (6.24)

where $g=g_{s}\sqrt{2}$ is the GUT gauge coupling and $v_{45},v_{1}$ denote
the VEVs of $\overline{H}_{45},H_{1}$ Higgs doublets, respectively (see Eqs.
(6), (6.8)). It follows that $\tan\beta\sim m_{t}/m_{b}\simeq 40$.

We consider that the above equation belongs to the highlights of this work,
and as such, we need to pause and reflect on its importance. We have achieved,
for the first time ever to our knowledge, to compute explicitly the mass
spectrum of quarks and (charged) leptons of the Standard Model in String
Theory. By calculating the superpotential at the $N=3$ (tree) level, we
identified the content of the third generation, i.e. the particles that get
Yukawa couplings proportional to the string coupling constant $g_{s}$ at this
level. We discussed above about the top quark Yukawa coupling and its
consequences of predicting in 1989 the top quark mass in the 170-180 GeV
range, as observed in 1995 at FNAL to be around 173 GeV. This particular top-
quark Yukawa coupling triggers the radiative electroweak breaking of
$SU(2)\times U(1)$ at low energies, thus explaining the gauge hierarchy
$M_{W}/M_{\rm GUT}\sim{\cal O}(10^{-16})$ a natural way. Actually, because our
string model is of no-scale type, it leads to a determination of the SUSY
breaking scale in the ${\cal O}({\rm TeV})$ region. Concerning the masses of
the bottom quark and $\tau$-lepton, we get the very successful relation
$m_{b}=m_{\tau}$ at the string scale, as well as the equality of the top and
bottom Yukawa couplings that leads to the determination of $\tan\beta\sim
m_{t}/m_{b}\simeq 40$, which eventually would be determined dynamically
through the no-scale mechanism.

For the next two generations, we need to calculate non renormalisable
corrections in $\alpha^{\prime}$ in the superpotential, corresponding to
$N=4,5,6,\dots$, that we have done using the general method of Ref.[26]. The
$N$-th order NR terms contain $(N-3)$ fields that will need to get VEVs. The
way that all these fields get their VEVs is through the endemic, in the string
models we are considering, existence of an ‘anomalous’ abelian gauge symmetry
$U(1)_{A}$ that enforces non-trivial VEVs for some ‘charged’ fields.
Eventually, in order to satisfy the F- and D-type flatness conditions, a set
of fields get dynamically VEVs of the order $\xi\approx 1/10M_{s}$, and thus
we have a perturbative expansion parameter! Thus, all the masses of the second
and first generations are found to be determined, involving powers of
$\xi^{n}$, $n=2,3,4,5$, multiplying, for normalisation, the corresponding
masses of the third generation. In other words, the masses of the second and
first generations are dynamically determined as $m_{t},m_{b},m_{\tau}$ and
$\xi$ and are dynamically fixed!

Now, we can really appreciate the structure of Eq. (6.24), as it provides a
very successful mass estimation for all quarks and charged leptons.

###  6.2 Proton decay

Let us now turn to the problem of proton decay. Proton decay has been for more
than forty years a real headache for theorists. Basically, it is a main
prediction of GUTs that has not been vindicated experimentally. The present
lower limits on proton decay are of the order of $10^{34}-10^{35}$ years
depending on the particular decay mode. Unlike the Standard Model where one
can show that, because of its particle content, contains no baryon (B) and
lepton (L) number violating interactions [27] in Grand Unified Theories, these
interactions are endemic. They led to the demise of non-supersymmetric GUTs,
while supersymmetric (SUSY) GUTs need enforcements to avoid rapid proton
decay, including matter-parity. Furthermore, SUSY GUTs contain dangerous $d=5$
B and L violating interactions that may lead to very rapid proton decay. With
the advent of superstring theory, generally, the proton decay problem became
more acute. The reason being that the low energy spectrum contains a plethora
of particles that may provide B, L violating interactions leading to a rather
rapid proton decay.

In the case of the string derived flipped $SU(5)$ model under consideration
there are two sources of $d=5$ baryon number violating operators. The first
consists of the usual dimension five operator $QQQL$ ascribed to the exchange
of additional triplets in the massless string spectrum. The second comprises
effective $QQQL$ operators generated from non-renormalisable string couplings
arising from the exchange of massive string modes.

Let us start with the triplet exchange induced dimension-five operators. As
explained in Section 3 we have five pairs of additional triplets accommodated
in the fields $h_{i},\overline{h}_{i},i=1,2,3,45$ and the flipped $SU(5)$
breaking Higgs multiplets $F,\overline{F}$ defined in this section. At tree-
level the triplets mass matrix is given by (3.4) where $d^{c}_{H}$ now stands
for the additional triplet combination $d^{c}_{1}+\varepsilon d^{c}_{3}$ and
$F_{1}$ is replaced by $F$. For our flatness solution of Table 1 and taking
into account non-renormalisable interactions up to $N=7$, the extra triplet
mass matrix to order $\xi^{5}$ reads

$\displaystyle
M_{D}^{(7)}=\bordermatrix{~{}&D_{1}&D_{2}&D_{3}&D_{45}&\overline{d}^{c}_{H}\cr\bar{D}_{1}&0&\overline{\Phi}_{12}&\Phi_{31}&0&\overline{s}_{1}^{(5)}\cr\bar{D}_{2}&0&A^{(7)}&\overline{\Phi}_{23}&0&2\overline{F}_{5}\cr\bar{D}_{3}&\overline{\Phi}_{31}&\Phi_{23}&0&\phi_{45}&\overline{s}_{3}^{(5)}\cr\bar{D}_{45}&0&\overline{B}^{(5)}&\overline{\phi}_{45}&0&0\cr
d^{c}_{H}&2F&s_{2}^{(5)}&0&s_{4}^{(5)}&s\cr}+{\cal O}\left(\xi\right)^{6}\,,$
(6.25)

where

$\displaystyle\overline{s}_{1}^{(5)}$
$\displaystyle=\left\\{\overline{F}_{5}\left(\phi_{4}^{2}+\phi_{+}\phi_{-}\right)\right\\}\,,$
(6.26) $\displaystyle\overline{s}_{3}^{(5)}$
$\displaystyle=\left\\{\overline{F}_{5}\left(D_{1}^{2}+D_{4}^{2}+T_{1}^{2}\right)\right\\}\,,$
(6.27) $\displaystyle{s}_{2}^{(5)}$
$\displaystyle=\left\\{F\left(\phi_{4}^{2}+\phi_{+}\phi_{-}\right)\right\\}\,,$
(6.28) $\displaystyle{s}_{4}^{(5)}$
$\displaystyle=\left\\{F\Phi_{31}\phi_{45}\right\\}\,,$ (6.29) $\displaystyle
s$ $\displaystyle=\left\\{F\overline{F}_{5}\overline{\Phi}_{12}\right\\}\,.$
(6.30)

Assuming $F\overline{F}_{5}\sim\xi^{3}$ the determinant is given by

$\displaystyle\det\left(M_{d}^{(7)}\right)$
$\displaystyle\sim\left(F\phi_{45}-\overline{\Phi}_{31}{s}_{4}^{(5)}\right)\left(\overline{B}^{(5)}\overline{F}_{5}\Phi_{31}-\overline{F}_{5}\overline{\Phi}_{12}\overline{\phi}_{45}-\overline{B}^{(5)}\overline{s}_{1}^{(5)}\overline{\Phi}_{23}+A_{2}^{(7)}{\overline{s}}_{1}^{(5)}\overline{\phi}_{45}\right)$
$\displaystyle\sim
F\overline{F}_{5}\phi_{45}\left(\overline{B}^{(5)}\Phi_{31}-\overline{\Phi}_{12}\overline{\phi}_{45}\right)+\dots\sim\xi^{10}$
(6.31)

ensuring that all triplets are massive. A detailed calculation shows that the
orders of magnitude of the triplets mass eigenstates are:
$\xi,\xi,\xi^{\frac{3}{2}},\xi^{\frac{3}{2}},\xi^{5}$. This is consistent with
our approximation utilised in (6.25), as the lightest eigenvalue is of order
${\cal O}(\xi^{5})$ rendering higher order contributions in (6.25) irrelevant.

Following the analysis of [28], triplet exchange $QQQL$ type dimension-five
operators for a general superpotential of the form

$\displaystyle
f_{ij}^{a}F_{i}F_{j}h_{a}+y_{ij}^{a}F_{i}\bar{f}_{j}\bar{h}_{a}+\overline{h}_{a}\left(M_{D}\right)_{ab}h_{b}$
(6.32)

involving $n$ additional fiveplets are proportional to

$\displaystyle{\cal
O}^{QQQL}_{ijkl}\sim\frac{1}{\det\left(M_{D}^{(7)}\right)}\sum_{a,b=1}^{n}y_{kl}^{a}\,{\rm
cof}\left(M_{D}^{(7)}\right)_{ab}f_{ij}^{b}\,.$ (6.33)

In our case, a direct computation yields

$\displaystyle{\rm cof}\left(M_{D}^{(7)}\right)_{1,1}$
$\displaystyle=\overline{F}_{5}\phi_{45}\overline{\phi}_{45}s_{2}^{(5)}+\dots\sim\xi^{9}\,,$
(6.34) $\displaystyle{\rm cof}\left(M_{D}^{(7)}\right)_{1,2}$
$\displaystyle=-\phi_{45}\overline{\phi}_{45}s_{2}^{(5)}\overline{s}_{1}^{(5)}+\dots\sim\xi^{11}\,,$
(6.35) $\displaystyle{\rm cof}\left(M_{D}^{(7)}\right)_{1,3}$
$\displaystyle=\overline{F}_{5}s_{4}^{(5)}\left(\Phi_{12}\phi_{45}-\overline{B}^{(5)}\Phi_{31}\right)+\dots\sim\xi^{11}\,,$
(6.36) $\displaystyle{\rm cof}\left(M_{D}^{(7)}\right)_{1,45}$
$\displaystyle=-\overline{F}_{5}\Phi_{31}\phi_{45}s_{2}^{(5)}+\dots\sim\xi^{7}\,,$
(6.37) $\displaystyle{\rm cof}\left(M_{D}^{(7)}\right)_{2,1}$
$\displaystyle=-F_{1}\overline{F}^{5}\phi_{45}\overline{\phi}_{45}s_{2}^{(5)}+\dots\sim\xi^{7}\,,$
(6.38) $\displaystyle{\rm cof}\left(M_{D}^{(7)}\right)_{2,2}$
$\displaystyle=F_{1}\phi_{45}\overline{\phi}_{45}\overline{s}_{1}^{(5)}+\dots\sim\xi^{9}\,,$
(6.39) $\displaystyle{\rm cof}\left(M_{D}^{(7)}\right)_{2,3}$
$\displaystyle=0\,,$ (6.40) $\displaystyle{\rm
cof}\left(M_{D}^{(7)}\right)_{2,45}$
$\displaystyle=F_{1}\overline{F}_{5}\Phi_{31}\phi_{45}+\dots\sim\xi^{5}\,,$
(6.41) $\displaystyle{\rm cof}\left(M_{D}^{(7)}\right)_{3,1}$
$\displaystyle=F_{1}\overline{F}_{5}\phi_{45}\overline{B}^{(5)}+\dots\sim\xi^{9}\,,$
(6.42) $\displaystyle{\rm cof}\left(M_{D}^{(7)}\right)_{3,2}$
$\displaystyle=-F_{1}\phi_{45}\overline{B}^{(5)}\overline{s}_{1}^{(5)}+\dots\sim\xi^{11}\,,$
(6.43) $\displaystyle{\rm cof}\left(M_{D}^{(7)}\right)_{3,3}$
$\displaystyle=0\,,$ (6.44) $\displaystyle{\rm
cof}\left(M_{D}^{(7)}\right)_{3,45}$
$\displaystyle=-F_{1}\overline{F}_{5}\phi_{45}\overline{\Phi}_{12}+\dots\sim\xi^{7}$
(6.45) $\displaystyle{\rm cof}\left(M_{D}^{(7)}\right)_{45,1}$
$\displaystyle=-\overline{F}_{5}\overline{\phi}_{45}\left(\Phi_{31}s_{2}^{(5)}-F_{1}\Phi_{23}\right)+\dots\sim\xi^{9}\,,$
(6.46) $\displaystyle{\rm cof}\left(M_{D}^{(7)}\right)_{45,2}$
$\displaystyle=\left(\overline{\Phi}_{31}s_{2}^{(5)}-F_{1}\Phi_{23}\right)\overline{\phi}_{45}\overline{s}_{1}^{(5)}\\!+\\!\left(\overline{\Phi}_{12}\overline{\phi}_{45}-\Phi_{31}\overline{B}^{(5)}\right)F_{1}\overline{s}_{3}^{(5)}\\!+\\!\dots\sim\xi^{11}\,,$
(6.47) $\displaystyle{\rm cof}\left(M_{D}^{(7)}\right)_{45,3}$
$\displaystyle=-F_{1}\overline{F}_{5}\left(\overline{\Phi}_{12}\overline{\phi}_{45}-\overline{B}^{(5)}\Phi_{31}\right)+\dots\sim\xi^{9}\,,$
(6.48) $\displaystyle{\rm cof}\left(M_{D}^{(7)}\right)_{45,45}$
$\displaystyle=\overline{F}_{5}\Phi_{31}\left(\overline{\Phi}_{31}s_{2}^{(5)}-F_{1}\Phi_{23}\right)+\dots\sim\xi^{7}\,.$
(6.49)

Triplet-exchange dimension-five operators can be generated via couplings of
the type $FFh_{2}\varphi^{N-3}$ and
$\overline{F}\,\overline{f}\,\overline{h}_{45}{\varphi^{\prime}}^{M-3}$,
arising at orders $N$ and $M$ respectively, where $\varphi^{N-3}$,
${\varphi^{\prime}}^{M-3}$ stand for combinations of field VEVs. At tree-level
($N=M=3$) we have a single pair of couplings of this type, namely
$F_{2}F_{2}h_{2},F_{4}\overline{f}_{5}\overline{h}_{45}$, that can give rise
to an effective $F_{2}F_{2}F_{4}\overline{f}_{5}$ operator. Actually, this
operator corresponds to the dominant contribution in the $\xi$ expansion,
among (6.34)-(6.49), arising from ${\rm cof}\left(M_{D}^{(7)}\right)_{2,45}$
which is proportional to

$\displaystyle\frac{{\rm
cof}\left(M_{D}^{(7)}\right)_{2,45}}{\det\left(M_{d}^{(7)}\right)}\sim\frac{1}{\xi^{5}}\,.$
(6.50)

However, the associated $d=5$ operator is further suppressed by at least a
factor $\xi^{3}$, since $F_{4},F_{2}$ accommodate third and second generation
quarks respectively. In fact the associated $QQQL$ operator is suppressed by
an effective triplet mass of order $\xi^{2}M_{P}\sim 10^{16}$ GeV which leads
to proton lifetime exceeding current experimental limits for a SUSY breaking
scale of the order of $m_{\rm susy}\gtrsim 10^{2}$ TeV [29]. Additional tree-
level operators could be generated via the $F_{1}-F_{3}$ mixing introduced in
Section 6.1 to generate down quark mass (see (6.23)). The term
$F_{1}F_{1}h_{1}$ induces an effective coupling of the form $F_{3}F_{3}h_{1}$
that combined with the $\overline{F}\,\overline{f}\,\overline{h}_{45}$ leads
to an effective $d=5$ operator of the form $F_{3}F_{3}F_{4}\overline{f}_{5}$.
However, this operator gets an extra suppression $\xi^{3}$ due to the mixing
and $\xi^{7}$ due to Eq. (6.45), leading to an effective triplet scale of the
order of $M_{P}$ and thus becomes subdominant.

At higher order ($N=3,M=4$) one could use the terms $F_{2}F_{2}h_{2}$ and
$F_{2}\overline{f}_{2}\overline{h}_{45}\overline{\phi}_{4}$ to form an
effective dimension five operator of the type
$F_{2}F_{2}F_{2}\overline{f}_{2}$. In this case we have smaller family mixing,
of the order of $\xi^{2}$, however, we get an additional suppression of order
$\xi^{2}$ from the additional VEV $\overline{\phi}_{4}$. Furthermore, higher
order contributions ($N,M>3$) are relatively suppressed by a factor of
$\xi^{2}$ in the worst case scenario. The same is true for all other operators
in Eqs. (6.34)-(6.50).

Let us now examine the string induced effective dimension-five operators [18].
These are of the form
${\mathbf{10}\\!\times\\!\mathbf{10}\\!\times\\!\mathbf{10}}\times{\overline{\mathbf{5}}}$.
An explicit search gives no candidate couplings of this type at the level of
$N=4$ non-renormalisable superpotential, while at $N=5$ we have the following
two terms

$\displaystyle
F_{2}^{2}F_{3}\overline{f}_{3}\overline{\Phi}_{23}+F_{4}^{2}F_{3}\overline{f}_{3}\Phi_{31}\,.$
(6.51)

However, both terms involve $\overline{f}_{3}$ which becomes superheavy in our
F/D-flatness solution (see (4.69)). As a result, we have no contributions to
proton decay at this level. At $N=6$ we find two non-vanishing terms

$\displaystyle
F_{3}^{2}F_{4}\overline{f}_{5}\left(\Phi_{31}\overline{\phi}_{-}+\overline{\Phi}_{23}\phi_{+}\right)\,.$
(6.52)

Following Table 1, these yield an affective operator
$F_{3}^{2}F_{4}\overline{f}_{5}$ with a coupling of order $\xi^{2}$ which is
translated to a dimension-five $QQQL$ effective operator with triplet scale of
the order of $\xi^{-2}M_{P}\sim 10^{20}$ GeV, which as explained above is safe
for proton decay. Moreover, all operators of the form (6.51) have been shown
to vanish explicitly in the case of the flipped $SU(5)$ model as a result of
permutation symmetries [18].

Summarising, the leading contribution to dimension-five proton decay operators
comes from the $F_{2}F_{2}F_{4}\overline{f}_{5}$ operator arising from
$h_{2},\overline{h}_{45}$ triplet pair mediation which is compatible with the
experimental bounds for a SUSY breaking scale $m_{\rm susy}\gtrsim{\cal
O}(10^{2})$ TeV. It is interesting to point out that a SUSY breaking scale in
the energy region of tens of TeV is also required for cosmological reasons,
following an analysis of reheating and nucleosynthesis in the flipped
$SU(5)\times U(1)$ model [13].

##  7 Concluding remarks

The quest for a Unified Theory of all interactions, including gravity, has
been for the last hundred years the ‘holly grail’ of High Energy physics. In
our times, it has been named the Theory of Everything (TOE), and as such it
should explain not only all of particle physics but also inflationary
cosmology in terms of some fundamental principles. Superstring theory has been
heralded as the fundamental framework that has the capacity to provide such a
Theory of Everything. There are different formulations of (compactified)
superstring theory in four dimensions, and for more than thirty years now, the
Free Fermionic Formulation (FFF) has been a very useful tool to perform
explicit calculations and construct models that may serve as a TOE.

Recently, we derived from the FFF of superstring theory, a Starobinsky like
inflationary model that fits all known cosmological data and connects the
inflation scale, calculated dynamically, to the Right-handed neutrino mass, as
the inflaton field is a mixture of the heavy sneutrino and some GUT singlet
fields provided by superstring theory. The framework is superstring derived
no-scale flipped $SU(5)$, that has some unique features, as we discussed in
previous works.

Here, we worked out in great detail all the possible physics issues that
needed to be resolved. We proved that the F- and D-flatness conditions are
satisfied at least to sixth order in the $\alpha^{\prime}$-expansion of the
superpotential, taking into account the fact that in our framework there is
always an ‘anomalous’ abelian gauge symmetry. The breaking mechanism of this
anomalous $U(1)_{A}$ entails several fields, mostly singlets, to get VEVs of
order $\xi\approx 1/10$ in string units. We use then $\xi$ as an expansion
parameter in perturbation theory, and thus we solve the F- and D-flatness
conditions and get a specific set of VEVs dynamically. Then, we are using this
‘vacuum’ to determine the triplet-doublet Higgs splitting and getting a pair
of ‘massless’ Higgs doublets that provides the radiative electroweak symmetry
breaking and the Yukawa couplings for the third generation at the tree level
of the superpotential. Actually, about 32 years ago, we predicted the mass of
the top-quark in the range of 170-180 GeV [2].

Furthermore, non-renormalisable terms, calculable in our framework, provide a
realistic hierarchical fermion mass spectrum with all quark and lepton masses
derived dynamically, consistent with the experimental hierarchies. As an
example, we mention that for the first time ever, the mass of the electron has
been calculated explicitly and in full agreement with its observed value,
which is rather remarkable. taking into account the fact of its tiny value,
vis a vis the top-quark mass. In addition, we derived some new relations
involving quarks and leptons that are experimentally satisfied. Furthermore,
the triplet Higgs masses are heavy enough as to provide a possible observable
proton decay in very specific modes.

We believe that, all in all, we have for the first time a framework that
provides not only a unified picture of particle physics and cosmology, but
also a dynamically derived hierarchical mass spectrum, probably observable
proton decay and inflationary cosmology in agreement with all cosmological
data. We cannot avoid but close with the same final statement used 32 years
ago in the first of reference [18]: “We leave it to the reader to decide how
many more miracles she wants to see before abandoning her doubts about flipped
$SU(5)$”.

## Acknowledgements

Work partially performed by I.A. as International Professor of the Francqui
Foundation, Belgium. The work of DVN was supported in part by the DOE grant
DE-FG02-13ER42020 at Texas A&M University and in part by the Alexander S.
Onassis Public Benefit Foundation. DVN would like to thank his beloved wife
Olga and the adorable young Odysseas for their continuous support,
encouragement and inspiration during the long period that this work was in
progress.

##  Appendix A String construction and spectrum of the flipped
$SU(5)\times{U(1)}$ model

In this appendix we briefly review the construction and the spectrum of the
revamped flipped $SU(5)\times{U(1)}$ string model [2]. The model is defined in
the free fermionic formulation of the heterotic superstring [3] using a set of
eight basis vectors $B=\left\\{\beta_{1},\beta_{2},\dots,\beta_{8}\right\\}$
and a set of phases
$c_{ij}=c\left[\beta_{i}\atop\beta_{j}\right]\,,i,j=1,\dots,8$ where

$\displaystyle\beta_{1}=S$
$\displaystyle=\left\\{\psi^{\mu},\chi^{1,\dots,6}\right\\}\,,$
$\displaystyle\beta_{2}=b_{1}$
$\displaystyle=\left\\{\psi^{\mu},\chi^{12},y^{3,\dots,6};\overline{y}^{3,\dots,6},\overline{\psi}^{1,\dots,5},\overline{\eta}^{1}\right\\}\,,$
$\displaystyle\beta_{3}=b_{2}$
$\displaystyle=\left\\{\psi^{\mu},\chi^{34},y^{12},\omega^{56};\overline{y}^{12},\overline{\omega}^{56},\overline{\psi}^{1,\dots,5}\overline{\eta}^{2}\right\\}\,,$
$\displaystyle\beta_{4}=b_{3}$
$\displaystyle=\left\\{\psi^{\mu},\chi^{56},\omega^{1,\dots,4};\overline{\omega}^{1,\dots,4},\overline{\psi}^{1,\dots,5},\overline{\eta}^{3}\right\\}\,,$
(A.1) $\displaystyle\beta_{5}=b_{4}$
$\displaystyle=\left\\{\psi^{\mu},\chi^{12},y^{36},\omega^{45};\overline{y}^{36},\overline{\omega}^{45},\overline{\psi}^{1,\dots,5},\overline{\eta}^{1}\right\\}\,,$
$\displaystyle\beta_{6}=b_{5}$
$\displaystyle=\left\\{\psi^{\mu},\chi^{34},y^{26},\omega^{15};\overline{y}^{26},\overline{\omega}^{15},\overline{\psi}^{1,\dots,5},\overline{\eta}^{2}\right\\}\,,$
$\displaystyle\beta_{7}=\zeta$
$\displaystyle=\left\\{\overline{\phi}^{1,\dots,8}\right\\}\,,$
$\displaystyle\beta_{8}=\alpha$
$\displaystyle=\Bigl{\\{}y^{46},\omega^{46};\overline{y}^{46},\overline{\omega}^{2346},\underbrace{\overline{\psi}^{1,\dots,5},\overline{\eta}^{1,2,3},\overline{\phi}^{1,\dots,4}}_{\frac{1}{2},\dots,\frac{1}{2}},\overline{\phi}^{56}\Bigr{\\}}\,,$

and $c_{ij}=c\left[\beta_{i}\atop\beta_{j}\right]=e^{i\pi\tilde{c}_{ij}}$,
with

$\displaystyle\tilde{c}=\bordermatrix{~{}&S&b_{1}&b_{2}&b_{3}&b_{4}&b_{5}&\zeta&\alpha\cr
S&0&0&0&0&0&0&1&1\cr b_{1}&1&1&1&1&1&1&1&-1/2\cr b_{2}&1&1&1&1&1&1&1&-1/2\cr
b_{3}&1&1&1&1&1&1&1&1\cr b_{4}&1&1&1&1&1&1&1&-1/2\cr
b_{5}&1&1&1&1&1&1&1&+1/2\cr\zeta&1&1&1&1&1&1&1&1\cr\alpha&1&1&1&1&1&0&1&-1/2\cr}\,.$
(A.2)

In the notation employed in (A.1) included fermions are periodic unless
indicated otherwise (for example, $\frac{1}{2}$ denotes fermions twisted by
$-i$). The string model under consideration possesses $N=1$ space-time
supersymmetry and
$SU(5)\times{U(1)}\times{U(1)}^{4}\times{SU(4)}\times{SO(10)}$ gauge symmetry.
We use the terms “observable” and “hidden” gauge symmetry to refer to the
$SU(5)\times{U(1)}$ and ${SU(4)}\times{SO(10)}$ group factors respectively.
The four extra abelian group factors ${U(1)}^{4}=\prod_{i=1}^{4}{U(1)}_{i}$
associated with the world-sheet currents
$\overline{\eta}^{1}\overline{\eta}^{1\ast},\overline{\eta}^{2}\overline{\eta}^{1\ast},\overline{\eta}^{3}\overline{\eta}^{3\ast},\overline{\omega}^{2}\overline{\omega}^{3}$
exhibit anomalies. As a matter of fact ${\rm Tr}{U(1)}_{1}=-36,{\rm
Tr}{U(1)}_{2}=-12,{\rm Tr}{U(1)}_{3}=-24,{\rm Tr}{U(1)}_{4}=-12$. However,
redefining appropriately we obtain three abelian combinations free of gauge
and mixed gravitational anomalies

$\displaystyle{U(1)}^{\prime}_{1}$ $\displaystyle={U(1)}_{3}+2U(1)_{4}\,,$
(A.3) $\displaystyle{U(1)}^{\prime}_{2}$
$\displaystyle={U(1)}_{1}-3U(1)_{2}\,,$ (A.4)
$\displaystyle{U(1)}^{\prime}_{3}$
$\displaystyle=3{U(1)}_{1}+{U(1)}_{2}+4{U(1)}_{3}-2{U(1)}_{4}\,,$ (A.5)

and one anomalous $U(1)$ group factor

$\displaystyle{U(1)}_{A}=-3{U(1)}_{1}-{U(1)}_{2}+2{U(1)}_{3}-{U(1)}_{4}\ \,,\
{\rm Tr}{U(1)}_{A}=180\,.$ (A.6)

The massless matter spectrum consists of: (i) “Observable” sector matter
particles listed in Table 2. These are states charged exclusively under the
$SU(5)\times{U(1)}$ and ${U(1)}^{4}$ gauge group factor. These comprise three
chiral fermion families as well as a family/anti-family pair, residing in
$SO(10)$ spinorials/anti-spinorials, coming from the sectors
$b_{1},b_{2},b_{3},b_{4},b_{5}$, and three fiveplet/anti-fiveplet pairs,
accommodated in $SO(10)$ vectorials, from the sectors $S,S+b_{4}+b_{5}$.
Bosonic partners come from the sectors $S+b_{i},i=1,\dots,4$ and
$0,b_{4}+b_{5}$ respectively. (ii) “Hidden” sector matter particles, that is,
states exclusively charged under ${SU(4)}\times{SO(10)}\times{U(1)}^{4}$
listed in Table 3. These arise from the sectors 222Here we employ a compact
notation to denote several sectors contributing to the same field multiplet,
for example $(S)+b_{i}+2\alpha(+\zeta)$ stands for four sectors:
$b_{i}+2\alpha,b_{i}+2\alpha+\zeta,S+b_{i}+2\alpha,S+b_{i}+2\alpha+\zeta$.
$(S)+b_{i}+2\alpha(+\zeta),i=1,\dots,4$. (iii) Exotic fractionally charged
states coming from the sectors $(S)+b_{1}\pm\alpha(+\zeta)$,
$(S)+b_{1}+b_{4}+b_{5}\pm\alpha(+\zeta)$,
$(S)+b_{2}+b_{3}+b_{5}\pm\alpha(+\zeta)$,
$(S)+b_{1}+b_{2}+b_{4}\pm\alpha(+\zeta)$. $(S)+b_{2}+b_{4}\pm\alpha(+\zeta)$,
$(S)+b_{4}\pm\alpha(+\zeta)$. These are listed in Table 4.

| $SU(5)$ | $U(1)$ | ${U(1)}_{1}$ | ${U(1)}_{2}$ | ${U(1)}_{3}$ | ${U(1)}_{4}$ | $SU(4)$ | $SO(10)$
---|---|---|---|---|---|---|---|---
$S$
$h_{1}$ | ${\mathbf{5}}$ | $-1$ | $+1$ | $0$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\bar{h}_{1}$ | ${\overline{\mathbf{5}}}$ | $+1$ | $-1$ | $0$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$h_{2}$ | ${\mathbf{5}}$ | $-1$ | $0$ | $+1$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\bar{h}_{2}$ | ${\overline{\mathbf{5}}}$ | $+1$ | $0$ | $-1$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$h_{3}$ | ${\mathbf{5}}$ | $-1$ | $0$ | $0$ | $+1$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\bar{h}_{3}$ | ${\overline{\mathbf{5}}}$ | $+1$ | $0$ | $0$ | $-1$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\Phi_{12}$ | ${\mathbf{1}}$ | $0$ | $-1$ | $+1$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{\Phi}_{12}$ | ${\mathbf{1}}$ | $0$ | $+1$ | $-1$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
${\Phi}_{31}$ | ${\mathbf{1}}$ | $0$ | $+1$ | $0$ | $-1$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{\Phi}_{31}$ | ${\mathbf{1}}$ | $0$ | $-1$ | $0$ | $+1$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\Phi_{23}$ | ${\mathbf{1}}$ | $0$ | $0$ | $-1$ | $+1$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{\Phi}_{23}$ | ${\mathbf{1}}$ | $0$ | $0$ | $+1$ | $-1$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\Phi_{I}\,,I=1,\dots,5$ | ${\mathbf{1}}$ | $0$ | $0$ | $0$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$b_{1}$
$F_{1}$ | ${\mathbf{10}}$ | $+{1/2}$ | $-{1/2}$ | $0$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{f}_{1}$ | ${\overline{\mathbf{5}}}$ | $-{3/2}$ | $-{1/2}$ | $0$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
${\ell}^{c}_{1}$ | ${\mathbf{1}}$ | $+{5/2}$ | $-{1/2}$ | $0$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$b_{2}$
$F_{2}$ | ${\mathbf{10}}$ | $+{1/2}$ | $0$ | $-{1/2}$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{f}_{2}$ | ${\overline{\mathbf{5}}}$ | $-{3/2}$ | $0$ | $-{1/2}$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
${\ell}^{c}_{2}$ | ${\mathbf{1}}$ | $+{5/2}$ | $0$ | $-{1/2}$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$b_{3}$
$F_{3}$ | ${\mathbf{10}}$ | $+{1/2}$ | $0$ | $0$ | $+{1/2}$ | $-{1/2}$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{f}_{3}$ | ${\overline{\mathbf{5}}}$ | $-{3/2}$ | $0$ | $0$ | $+{1/2}$ | $+{1/2}$ | ${\mathbf{1}}$ | $\mathbf{1}$
${\ell}^{c}_{3}$ | ${\mathbf{1}}$ | $+{5/2}$ | $0$ | $0$ | $+{1/2}$ | $+{1/2}$ | ${\mathbf{1}}$ | $\mathbf{1}$
$b_{4}$
$F_{4}$ | ${\mathbf{10}}$ | $+{1/2}$ | $-{1/2}$ | $0$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
${f}_{4}$ | ${{\mathbf{5}}}$ | $+{3/2}$ | $+{1/2}$ | $0$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{\ell}^{c}_{4}$ | ${\mathbf{1}}$ | $-{5/2}$ | $+{1/2}$ | $0$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$b_{5}$
$\overline{F}_{5}$ | $\overline{\mathbf{10}}$ | $-{1/2}$ | $0$ | $+{1/2}$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{f}_{5}$ | ${{\mathbf{5}}}$ | $-{3/2}$ | $0$ | $-{1/2}$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
${\ell}^{c}_{5}$ | ${\mathbf{1}}$ | $+{5/2}$ | $0$ | $-{1/2}$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$S+b_{4}+b_{5}$
$h_{45}$ | ${\mathbf{5}}$ | $-1$ | $-{1/2}$ | $-{1/2}$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{h}_{45}$ | $\overline{\mathbf{5}}$ | $+1$ | $+{1/2}$ | $+{1/2}$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
${\phi}_{45}$ | ${{\mathbf{1}}}$ | $0$ | $+{1/2}$ | $+{1/2}$ | $+1$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{\phi}_{45}$ | ${{\mathbf{1}}}$ | $0$ | $-{1/2}$ | $-{1/2}$ | $-1$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
${\phi}_{+}$ | ${{\mathbf{1}}}$ | $0$ | $+{1/2}$ | $-{1/2}$ | $0$ | $+1$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{\phi}_{+}$ | ${{\mathbf{1}}}$ | $0$ | $-{1/2}$ | $+{1/2}$ | $0$ | $-1$ | ${\mathbf{1}}$ | $\mathbf{1}$
${\phi}_{-}$ | ${{\mathbf{1}}}$ | $0$ | $+{1/2}$ | $-{1/2}$ | $0$ | $-1$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{\phi}_{-}$ | ${{\mathbf{1}}}$ | $0$ | $-{1/2}$ | $+{1/2}$ | $0$ | $+1$ | ${\mathbf{1}}$ | $\mathbf{1}$
${\phi}_{i}\,,i=1,\dots,4$ | ${{\mathbf{1}}}$ | $0$ | $+{1/2}$ | $-{1/2}$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
$\overline{\phi}_{i}\,,i=1,\dots,4$ | ${{\mathbf{1}}}$ | $0$ | $-{1/2}$ | $+{1/2}$ | $0$ | $0$ | ${\mathbf{1}}$ | $\mathbf{1}$
Table 2: “Observable” sector massless matter states and their $SU(5)\times{U(1)}\times{U(1)}^{4}\times{SU(4)}\times{SO(10)}$ quantum numbers. | $SU(5)$ | $U(1)$ | ${U(1)}_{1}$ | ${U(1)}_{2}$ | ${U(1)}_{3}$ | ${U(1)}_{4}$ | $SU(4)$ | $SO(10)$
---|---|---|---|---|---|---|---|---
$b_{1}+2\alpha\,\left(+\zeta\right)$
$D_{1}$ | ${\mathbf{1}}$ | $0$ | $0$ | $-{1/2}$ | $+{1/2}$ | $0$ | ${\mathbf{6}}$ | $\mathbf{1}$
$T_{1}$ | ${\mathbf{1}}$ | $0$ | $0$ | $-{1/2}$ | $+{1/2}$ | $0$ | ${\mathbf{1}}$ | $\mathbf{10}$
$b_{2}+2\alpha\,\left(+\zeta\right)$
$D_{2}$ | ${\mathbf{1}}$ | $0$ | $-{1/2}$ | $0$ | $+{1/2}$ | $0$ | ${\mathbf{6}}$ | $\mathbf{1}$
$T_{2}$ | ${\mathbf{1}}$ | $0$ | $-{1/2}$ | $0$ | $+{1/2}$ | $0$ | ${\mathbf{1}}$ | $\mathbf{10}$
$b_{3}+2\alpha\,\left(+\zeta\right)$
$D_{3}$ | ${\mathbf{1}}$ | $0$ | $-{1/2}$ | $-{1/2}$ | $0$ | $+{1/2}$ | ${\mathbf{6}}$ | $\mathbf{1}$
$T_{3}$ | ${\mathbf{1}}$ | $0$ | $-{1/2}$ | $-{1/2}$ | $0$ | $-{1/2}$ | ${\mathbf{1}}$ | $\mathbf{10}$
$b_{4}+2\alpha\,\left(+\zeta\right)$
$D_{4}$ | ${\mathbf{1}}$ | $0$ | $0$ | $-{1/2}$ | $+{1/2}$ | $0$ | ${\mathbf{6}}$ | $\mathbf{1}$
$T_{4}$ | ${\mathbf{1}}$ | $0$ | $0$ | $+{1/2}$ | $-{1/2}$ | $0$ | ${\mathbf{1}}$ | $\mathbf{10}$
$b_{5}+2\alpha\,\left(+\zeta\right)$
$D_{5}$ | ${\mathbf{1}}$ | $0$ | $+{1/2}$ | $0$ | $-{1/2}$ | $0$ | ${\mathbf{6}}$ | $\mathbf{1}$
$T_{5}$ | ${\mathbf{1}}$ | $0$ | $-{1/2}$ | $0$ | $+{1/2}$ | $0$ | ${\mathbf{1}}$ | $\mathbf{10}$
Table 3: “Hidden sector” massless matter states and their $SU(5)\times{U(1)}\times{U(1)}^{4}\times{SU(4)}\times{SO(10)}$ quantum numbers. | $SU(5)$ | $U(1)$ | ${U(1)}_{1}$ | ${U(1)}_{2}$ | ${U(1)}_{3}$ | ${U(1)}_{4}$ | $SU(4)$ | $SO(10)$
---|---|---|---|---|---|---|---|---
$b_{1}\pm\alpha\,\left(+\zeta\right)$
$\overline{X}_{1}$ | ${\mathbf{1}}$ | $-{5/4}$ | $-{1/4}$ | $+{1/4}$ | $+{1/4}$ | $+{1/2}$ | $\overline{\mathbf{4}}$ | $\mathbf{1}$
$\overline{X}_{2}$ | ${\mathbf{1}}$ | $-{5/4}$ | $-{1/4}$ | $+{1/4}$ | $+{1/4}$ | $-{1/2}$ | $\overline{\mathbf{4}}$ | $\mathbf{1}$
$b_{1}+b_{4}+b_{5}\pm\alpha\,\left(+\zeta\right)$
${Y}_{1}$ | ${\mathbf{1}}$ | $+{5/4}$ | $-{1/4}$ | $+{1/4}$ | $-{1/4}$ | $+{1/2}$ | ${\mathbf{4}}$ | $\mathbf{1}$
${Y}_{2}$ | ${\mathbf{1}}$ | $+{5/4}$ | $-{1/4}$ | $+{1/4}$ | $-{1/4}$ | $-{1/2}$ | ${\mathbf{4}}$ | $\mathbf{1}$
$b_{2}+b_{3}+b_{5}\pm\alpha\,\left(+\zeta\right)$
${Z}_{2}$ | ${\mathbf{1}}$ | $+{5/4}$ | $-{1/4}$ | $+{3/4}$ | $+{1/4}$ | $0$ | ${\mathbf{4}}$ | $\mathbf{1}$
$\overline{Z}_{2}$ | ${\mathbf{1}}$ | $-{5/4}$ | $-{3/4}$ | $+{1/4}$ | $-{1/4}$ | $0$ | $\overline{\mathbf{4}}$ | $\mathbf{1}$
$b_{1}+b_{2}+b_{4}\pm\alpha\,\left(+\zeta\right)$
${Y}_{2}^{\prime}$ | ${\mathbf{1}}$ | $+{5/4}$ | $-{1/4}$ | $+{1/4}$ | $-{1/4}$ | $-{1/2}$ | ${\mathbf{4}}$ | $\mathbf{1}$
$\overline{Y}_{1}$ | ${\mathbf{1}}$ | $-{5/4}$ | $+{1/4}$ | $-{1/4}$ | $+{1/4}$ | $-{1/2}$ | $\overline{\mathbf{4}}$ | $\mathbf{1}$
$S+b_{2}+b_{4}\pm\alpha\,\left(+\zeta\right)$
${Z}_{1}$ | ${\mathbf{1}}$ | $-{5/4}$ | $+{1/4}$ | $+{1/4}$ | $-{1/4}$ | $+{1/2}$ | ${\mathbf{4}}$ | $\mathbf{1}$
$\overline{Z}_{1}$ | ${\mathbf{1}}$ | $+{5/4}$ | $-{1/4}$ | $-{1/4}$ | $+{1/4}$ | $-{1/2}$ | $\overline{\mathbf{4}}$ | $\mathbf{1}$
$b_{4}\pm\alpha\,\left(+\zeta\right)$
${X}_{1}$ | ${\mathbf{1}}$ | $+{5/4}$ | $+{1/4}$ | $-{1/4}$ | $-{1/4}$ | $-{1/2}$ | ${\mathbf{4}}$ | $\mathbf{1}$
$\overline{X}_{2}^{\prime}$ | ${\mathbf{1}}$ | $-{5/4}$ | $-{1/4}$ | $+{1/4}$ | $+{1/4}$ | $-{1/2}$ | $\overline{\mathbf{4}}$ | $\mathbf{1}$
Table 4: Exotic fractionally charged massless matter states and their
$SU(5)\times{U(1)}\times{U(1)}^{4}\times{SU(4)}\times{SO(10)}$ quantum
numbers.

##  Appendix B F-flatness condition analysis

In this appendix we consider solutions of the F-flatness equations of a
superpotential of the form

$\displaystyle w$
$\displaystyle=\overline{\Phi}_{23}\overline{\Phi}_{31}\overline{\Phi}_{12}+\overline{\Phi}_{12}\left(\overline{\phi}_{1}^{2}+\overline{\phi}_{2}^{2}+\overline{\phi}_{4}^{2}+\overline{\phi}_{+}\overline{\phi}_{-}\right)$
$\displaystyle+\left(D_{1}^{2}+D_{4}^{2}+T_{1}^{2}\right)\overline{\Phi}_{23}+\left(\alpha_{1}\overline{\phi}_{1}^{2}+\alpha_{2}\overline{\phi}_{2}^{2}+\alpha_{3}\overline{\phi}_{4}^{2}+\alpha_{4}\overline{\phi}_{+}\overline{\phi}_{-}\right)D_{1}^{2}\Phi_{31}$
(B.1)
$\displaystyle+\left(\\!\beta_{1}\overline{\phi}_{1}^{2}\\!+\\!\beta_{2}\overline{\phi}_{2}^{2}\\!+\\!\beta_{3}\overline{\phi}_{4}^{2}\\!+\\!\beta_{4}\overline{\phi}_{+}\overline{\phi}_{-}\\!\right)\\!D_{4}^{2}\Phi_{31}\\!+\\!\left(\gamma_{1}\overline{\phi}_{1}^{2}+\gamma_{2}\overline{\phi}_{2}^{2}+\gamma_{3}\overline{\phi}_{4}^{2}+\gamma_{4}\overline{\phi}_{+}\overline{\phi}_{-}\right)\\!T_{1}^{2}\Phi_{31},$

where $\alpha_{i},\beta_{i},\gamma_{i},i=1,2,3,4$ are the coupling constants
of the associated fifth order terms. We focus on perturbative solutions
assuming $\Phi_{31},\overline{\Phi}_{31}\sim\xi$, where $\xi$ is a small
parameter in appropriate mass units. The F-flatness conditions read

$\displaystyle\frac{\partial w}{\partial\overline{\Phi}_{23}}$
$\displaystyle=D_{1}^{2}+D_{4}^{2}+T_{1}^{2}+\overline{\Phi}_{31}\overline{\Phi}_{12}=0\,,$
(B.2) $\displaystyle\frac{\partial w}{\partial\overline{\Phi}_{31}}$
$\displaystyle=\overline{\Phi}_{23}\overline{\Phi}_{12}=0\,,$ (B.3)
$\displaystyle\frac{\partial w}{\partial\overline{\Phi}_{12}}$
$\displaystyle=\overline{\phi}_{1}^{2}+\overline{\phi}_{2}^{2}+\overline{\phi}_{4}^{2}+\overline{\phi}_{+}\overline{\phi}_{-}+\overline{\Phi}_{23}\overline{\Phi}_{31}=0\,,$
(B.4) $\displaystyle\frac{\partial w}{\partial\Phi_{31}}$
$\displaystyle=\left(\alpha_{1}\overline{\phi}_{1}^{2}\\!+\\!\alpha_{2}\overline{\phi}_{2}^{2}\\!+\\!\alpha_{3}\overline{\phi}_{4}^{2}\\!+\\!\alpha_{4}\overline{\phi}_{+}\overline{\phi}_{-}\right)\\!D_{1}^{2}\\!+\\!\left(\beta_{1}\overline{\phi}_{1}^{2}\\!+\\!\beta_{2}\overline{\phi}_{2}^{2}\\!+\\!\beta_{3}\overline{\phi}_{4}^{2}\\!+\\!\beta_{4}\overline{\phi}_{+}\overline{\phi}_{-}\right)\\!D_{4}^{2}$
$\displaystyle+\left(\gamma_{1}\overline{\phi}_{1}^{2}\\!+\\!\gamma_{2}\overline{\phi}_{2}^{2}\\!+\\!\gamma_{3}\overline{\phi}_{4}^{2}\\!+\\!\gamma_{4}\overline{\phi}_{+}\overline{\phi}_{-}\right)T_{1}^{2}=0\,,$
(B.5) $\displaystyle\frac{\partial w}{\partial\overline{\phi}_{1}}$
$\displaystyle=2\overline{\phi}_{1}\left[\overline{\Phi}_{12}+\Phi_{31}\left(\alpha_{1}D_{1}^{2}+\beta_{1}D_{4}^{2}+\gamma_{1}T_{1}^{2}\right)\right]=0\,,$
(B.6) $\displaystyle\frac{\partial w}{\partial\overline{\phi}_{2}}$
$\displaystyle=2\overline{\phi}_{2}\left[\overline{\Phi}_{12}+\Phi_{31}\left(\alpha_{2}D_{1}^{2}+\beta_{2}D_{4}^{2}+\gamma_{2}T_{1}^{2}\right)\right]=0\,,$
(B.7) $\displaystyle\frac{\partial w}{\partial\overline{\phi}_{4}}$
$\displaystyle=2\overline{\phi}_{4}\left[\overline{\Phi}_{12}+\Phi_{31}\left(\alpha_{3}D_{1}^{2}+\beta_{3}D_{4}^{2}+\gamma_{3}T_{1}^{2}\right)\right]=0\,,$
(B.8) $\displaystyle\frac{\partial w}{\partial\overline{\phi}_{+}}$
$\displaystyle=\overline{\phi}_{-}\left[\overline{\Phi}_{12}+\Phi_{31}\left(\alpha_{4}D_{1}^{2}+\beta_{4}D_{4}^{2}+\gamma_{4}T_{1}^{2}\right)\right]=0\,,$
(B.9) $\displaystyle\frac{\partial w}{\partial\overline{\phi}_{-}}$
$\displaystyle=\overline{\phi}_{+}\left[\overline{\Phi}_{12}+\Phi_{31}\left(\alpha_{4}D_{1}^{2}+\beta_{4}D_{4}^{2}+\gamma_{4}T_{1}^{2}\right)\right]=0\,,$
(B.10) $\displaystyle\frac{\partial w}{\partial D_{1}}$
$\displaystyle=2D_{1}\left[\overline{\Phi}_{23}+\Phi_{31}\left(\alpha_{1}\overline{\phi}_{1}^{2}+\alpha_{2}\overline{\phi}_{2}^{2}+\alpha_{3}\overline{\phi}_{4}^{2}+\alpha_{4}\overline{\phi}_{+}\overline{\phi}_{-}\right)\right]=0\,,$
(B.11) $\displaystyle\frac{\partial w}{\partial D_{4}}$
$\displaystyle=2D_{4}\left[\overline{\Phi}_{23}+\Phi_{31}\left(\beta_{1}\overline{\phi}_{1}^{2}+\beta_{2}\overline{\phi}_{2}^{2}+\beta_{3}\overline{\phi}_{4}^{2}+\beta_{4}\overline{\phi}_{+}\overline{\phi}_{-}\right)\right]=0\,,$
(B.12) $\displaystyle\frac{\partial w}{\partial T_{1}}$
$\displaystyle=2T_{1}\left[\overline{\Phi}_{23}+\Phi_{31}\left(\gamma_{1}\overline{\phi}_{1}^{2}+\gamma_{2}\overline{\phi}_{2}^{2}+\gamma_{3}\overline{\phi}_{4}^{2}+\gamma_{4}\overline{\phi}_{+}\overline{\phi}_{-}\right)\right]=0\,.$
(B.13)

Using the coupling relations

$\displaystyle-\frac{\alpha_{1}}{3}=-\frac{\gamma_{1}}{3}=-\frac{\beta_{1}}{2}=\frac{\beta_{3}}{2}=\frac{\beta_{4}}{2}=\alpha_{4}\,,$
(B.14)
$\displaystyle\alpha_{2}=\alpha_{3}=\gamma_{2}=\gamma_{3}=\gamma_{4}=\alpha_{4}\
,\ \beta_{2}=0\,,$ (B.15)

Eqs. (B.4), (B.11), (B.12) and (B.13) take the form

$\displaystyle\overline{\phi}_{1}^{2}+\overline{\phi}_{2}^{2}+\overline{\phi}_{4}^{2}+\overline{\phi}_{+}\overline{\phi}_{-}+\overline{\Phi}_{23}\overline{\Phi}_{31}$
$\displaystyle=0\,,$ (B.16) $\displaystyle
D_{1}\left[\overline{\Phi}_{23}+\Phi_{31}\alpha_{4}\left(-3\overline{\phi}_{1}^{2}+\overline{\phi}_{2}^{2}+\overline{\phi}_{4}^{2}+\overline{\phi}_{+}\overline{\phi}_{-}\right)\right]$
$\displaystyle=0\,,$ (B.17) $\displaystyle
D_{4}\left[\overline{\Phi}_{23}+\Phi_{31}\alpha_{4}\left(-2\overline{\phi}_{1}^{2}+2\overline{\phi}_{4}^{2}+2\overline{\phi}_{+}\overline{\phi}_{-}\right)\right]$
$\displaystyle=0\,,$ (B.18) $\displaystyle
T_{1}\left[\overline{\Phi}_{23}+\Phi_{31}\alpha_{4}\left(-3\overline{\phi}_{1}^{2}+\overline{\phi}_{2}^{2}+\overline{\phi}_{4}^{2}+\overline{\phi}_{+}\overline{\phi}_{-}\right)\right]$
$\displaystyle=0\,.$ (B.19)

Solving (B.16) with respect to $\overline{\phi}_{1}^{2}$ and substituting into
(B.17) and (B.18), (B.19) we get

$\displaystyle
D_{1}\left[\overline{\Phi}_{23}+\Phi_{31}\alpha_{4}\left(4\overline{\phi}_{2}^{2}+4\overline{\phi}_{4}^{2}+4\overline{\phi}_{+}\overline{\phi}_{-}\right)\right]$
$\displaystyle=0\,,$ (B.20) $\displaystyle
D_{4}\left[\overline{\Phi}_{23}+\Phi_{31}\alpha_{4}\left(2\overline{\phi}_{2}^{2}+4\overline{\phi}_{4}^{2}+4\overline{\phi}_{+}4\overline{\phi}_{-}\right)\right]$
$\displaystyle=0\,,$ (B.21) $\displaystyle
T_{1}\left[\overline{\Phi}_{23}+\Phi_{31}\alpha_{4}\left(4\overline{\phi}_{2}^{2}+4\overline{\phi}_{4}^{2}+4\overline{\phi}_{+}4\overline{\phi}_{-}\right)\right]$
$\displaystyle=0\,,$ (B.22)

where we have neglected terms of order $\xi^{5}$.

Moreover, Eqs. (B.2) and (B.6), (B.7), (B.8), (B.9) yield

$\displaystyle
D_{1}^{2}+D_{4}^{2}+T_{1}^{2}+\overline{\Phi}_{31}\overline{\Phi}_{12}$
$\displaystyle=0\,,$ (B.23)
$\displaystyle\overline{\phi}_{1}\left[\overline{\Phi}_{12}-\alpha_{4}\Phi_{31}\left(3D_{1}^{2}+2D_{4}^{2}+3T_{1}^{2}\right)\right]$
$\displaystyle=0\,,$ (B.24)
$\displaystyle\overline{\phi}_{2}\left[\overline{\Phi}_{12}+\alpha_{4}\Phi_{31}\left(\overline{\Phi}_{12}+D_{1}^{2}+T_{1}^{2}\right)\right]$
$\displaystyle=0\,,$ (B.25)
$\displaystyle\overline{\phi}_{4}\left[\overline{\Phi}_{12}+\alpha_{4}\Phi_{31}\left(\overline{\Phi}_{12}+D_{1}^{2}+2D_{4}^{2}+T_{1}^{2}\right)\right]$
$\displaystyle=0\,,$ (B.26)
$\displaystyle\overline{\phi}_{+}\left[\overline{\Phi}_{12}+\alpha_{4}\Phi_{31}\left(\overline{\Phi}_{12}+D_{1}^{2}+2D_{4}^{2}+T_{1}^{2}\right)\right]$
$\displaystyle=0\,.$ (B.27)

Solving (B.23) with respect to $T_{1}^{2}$ and substituting into (B.24)-(B.27)
we get

$\displaystyle\overline{\phi}_{1}\left[\overline{\Phi}_{12}+\alpha_{4}\Phi_{31}D_{4}^{2}\right]$
$\displaystyle=0\,,$ (B.28)
$\displaystyle\overline{\phi}_{2}\left[\overline{\Phi}_{12}-\alpha_{4}\Phi_{31}D_{4}^{2}\right]$
$\displaystyle=0\,,$ (B.29)
$\displaystyle\overline{\phi}_{4}\left[\overline{\Phi}_{12}+\alpha_{4}\Phi_{31}D_{4}^{2}\right]$
$\displaystyle=0\,,$ (B.30)
$\displaystyle\overline{\phi}_{+}\left[\overline{\Phi}_{12}+\alpha_{4}\Phi_{31}D_{4}^{2}\right]$
$\displaystyle=0\,.$ (B.31)

These are not compatible unless $\overline{\phi}_{2}\lesssim\xi^{2}$. In this
case

$\displaystyle\overline{\Phi}_{12}$
$\displaystyle=-\alpha_{4}D_{4}^{2}\Phi_{31}\,,$ (B.32) $\displaystyle
T_{1}^{2}$
$\displaystyle=-D_{1}^{2}-D_{4}^{2}-\overline{\Phi}_{31}\overline{\Phi}_{12}\,,$
(B.33)

and the system (B.20)-(B.22) also yields

$\displaystyle\overline{\Phi}_{23}$
$\displaystyle=-2\Phi_{31}\alpha_{4}\left(-\overline{\phi}_{1}^{2}+\overline{\phi}_{4}^{2}+\overline{\phi}_{+}\overline{\phi}_{-}\right)\,,$
(B.34) $\displaystyle\overline{\phi}_{1}^{2}$
$\displaystyle=-\overline{\phi}_{4}^{2}-\overline{\phi}_{+}\overline{\phi}_{-}-\overline{\phi}_{2}^{2}-\overline{\Phi}_{23}\overline{\Phi}_{31}\,.$
(B.35)

Finally, we consider (B.5). Using, (B.11), (B.12), (B.13) all three factors in
parentheses equal to $-\frac{\overline{\Phi}_{23}}{\Phi_{31}}$. That is

$\displaystyle-\frac{\overline{\Phi}_{23}}{\Phi_{31}}\left(D_{1}^{2}+D_{4}^{2}+T_{1}^{2}\right)=0\Rightarrow\frac{\overline{\Phi}_{23}\overline{\Phi}_{12}\overline{\Phi}_{31}}{\Phi_{31}}=0$
(B.36)

which is satisfied at order $\xi^{6}$ by the use of (B.2). The same holds for
(B.3).

##  Appendix C  Operator product expansions, correlators etc

The conformal correlator functions appearing in Section 5 can be calculated
using techniques explained in [26],[30]. These include correlators of
exponentials which can be easily calculated using

$\displaystyle\langle{\prod_{i}e^{i\alpha_{i}\phi(z_{i})}}\rangle=\prod_{i<j}(z_{ij})^{\alpha_{i}\alpha_{j}}\,,$
(C.1)

the ghost correlator given by

$\displaystyle\langle{e^{-c/2}(z_{1})}{e^{-c/2}(z_{2})}{e^{-c}(z_{3})}\rangle=z_{12}^{-1/4}z_{13}^{-1/2}z_{23}^{-1}\,,$
(C.2)

and the space-time spinor correlator given by

$\displaystyle\langle{S_{a}(z_{1})S_{b}(z_{2})}\rangle=C_{ab}z_{12}^{-1/2}\,.$
(C.3)

In addition, we use the following Ising type correlators

$\displaystyle\langle{\sigma_{+}(1)\sigma_{+}(2)}\rangle$
$\displaystyle=\langle{\sigma_{-}(1)\sigma_{-}(2)}\rangle=\left|z_{12}\right|^{-1/4}\,,$
(C.4) $\displaystyle\langle{f(1)f(2)}\rangle$
$\displaystyle=\langle{\bar{f}(1)\bar{f}(2)}\rangle=\frac{1}{z_{12}}\,,$ (C.5)
$\displaystyle\langle{\sigma_{+}(1)\sigma_{-}(2)f(3)}\rangle$
$\displaystyle=\frac{e^{i\pi/4}}{\sqrt{2}}z_{12}^{+3/8}z_{13}^{-1/2}z_{23}^{-1/2}\bar{z}_{12}^{-1/8}$
(C.6) $\displaystyle\langle{\sigma_{+}(1)\sigma_{-}(2)\bar{f}(3)}\rangle$
$\displaystyle=\frac{e^{-i\pi/4}}{\sqrt{2}}\bar{z}_{12}^{+3/8}\bar{z}_{13}^{-1/2}\bar{z}_{23}^{-1/2}{z}_{12}^{-1/8}\,,$
(C.7)
$\displaystyle\langle{\sigma_{+}(1)\sigma_{+}(2)\sigma_{+}(3)\sigma_{+}(4)}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\frac{\left|z_{13}z_{24}\right|^{1/4}}{\left|z_{14}z_{23}z_{12}z_{34}\right|^{1/4}}\left(|x|+|1-x|+1\right)^{1/2}$
(C.8)
$\displaystyle=\frac{1}{\sqrt{2}}\left|z_{13}z_{24}\right|^{-1/4}\left|x(1-x)\right|^{-1/4}\left(|x|+|1-x|+1\right)^{1/2}\,,$
(C.9)
$\displaystyle\langle{\sigma_{+}(1)\sigma_{+}(2)\sigma_{-}(3)\sigma_{-}(4)}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left|z_{12}z_{34}\right|^{-1/4}\left|1-x\right|^{-1/4}\left(|1-x|-|x|+1\right)^{1/2}\,,$
(C.10) $\displaystyle\langle{\sigma_{+}(1)\sigma_{+}(2){f}(3){f}(4)}\rangle$
$\displaystyle=\frac{1}{2}\frac{\left|z_{12}\right|^{-1/4}}{{z}_{34}}\frac{2-{x}}{\sqrt{1-{x}}}\,,$
(C.11)
$\displaystyle\langle{\sigma_{+}(1)\sigma_{+}(2)\bar{f}(3)\bar{f}(4)}\rangle$
$\displaystyle=\frac{1}{2}\frac{\left|z_{12}\right|^{-1/4}}{\bar{z}_{34}}\frac{2-\bar{x}}{\sqrt{1-\bar{x}}}\,,$
(C.12)

where

$\displaystyle x=\frac{z_{12}z_{34}}{z_{13}z_{24}}\,.$ (C.13)

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