%Paper: hep-th/9206083
%From: casas@vxcrna.cern.ch (J.A.CASAS)
%Date: Tue, 23 Jun 92 18:25:10 +0200



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\vspace{-2.0cm}
\rightline{{ CERN--TH.6507/92}}
\rightline{{ IEM--FT--57/92}}
\rightline{{ US--FT/5--92}}

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\begin{center}
{\bf FITTING THE QUARK AND LEPTON MASSES IN STRING THEORIES}
\vspace{1cm}

J.A. CASAS${}^{*,**}\;$, F. GOMEZ${}^{***}$ and C. MU\~NOZ${}^{*}$
\vspace{1.1cm}

%\baselineskip=16pt

${}^{*}$ CERN, CH--1211 Geneva 23, Switzerland
\vspace{0.5cm}

${}^{**}$ Instituto de Estructura de la Materia (CSIC),\\
Serrano 123, 28006--Madrid, Spain
\vspace{0.5cm}

${}^{***}$ Dept. de Fisica Te\'orica,\\
Universidad de Santiago, 15706--Santiago, Spain
\vspace{0.5cm}

\end{center}
%\baselineskip=18pt

\centerline{\bf Abstract}

\vspace{0.5cm}
%\textwidth=16cm
\noindent The capability of string theories to reproduce at low energy
the observed pattern of quark and lepton masses and mixing angles
is examined, focusing the attention on orbifold constructions, where
the magnitude of Yukawa couplings
depends on the values of the deformation parameters
which describe the size and shape of the compactified space.
A systematic exploration shows that for $Z_3$, $Z_4$,
$Z_6$--I and possibly $Z_7$ orbifolds a correct fit of the
physical fermion masses is feasible. In this way
the experimental masses, which are low--energy quantities, select
a particular size and shape of the compactified space, which turns
out to be very reasonable (in particular the modulus $T$ defining
the former is $T=O(1)$).
The rest of the $Z_N$ orbifolds are rather hopeless and should be
discarded on the assumption of a minimal $SU(3)\times SU(2)\times
U(1)_Y$ scenario. On the other hand,
due to stringy selection rules, there
is no possibility of fitting the Kobayashi--Maskawa parameters
at the renormalizable level, although it is remarked that
this job might well be done by non--renormalizable couplings.

\vspace{0.5cm}
\begin{flushleft}
{CERN--TH.6507/92} \\
{IEM--FT--57/92} \\
{US--FT/5--/92} \\
{May 1992}
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\section{Introduction}

One of the most intriguing facts of particle physics is the peculiar
experimental pattern of quark and lepton masses and mixing angles.
In the framework
of the Standard Model these are just initial parameters put by hand
without any possible hint about their origin. Grand Unification theories
(GUTs) impose certain relations between them. For instance, in the minimal
SU(5) model, $m_e=m_d$, $m_\mu=m_s$, $m_\tau=m_b$ at $M_{GUT}$. Only the
third equality is compatible with experiments. This is in fact a major
shortcoming of GUTs (also shared by supersymmetric GUTs) which may
only be bypassed by complicating the Higgs sector in an artificial way.
On the other hand, if Superstring Theories are the fundamental theory
from which the Standard Model is derived as a energy limit, they
should be able to give an answer to this fundamental question. This
is the main motivation of the present work. In this sense a crucial
ingredient to relate theory and observation is the knowledge of the
theoretical Yukawa couplings predicted by Superstrings. Actually, there
are several ways to construct four--dimensional strings, but perhaps
%and not for all of them
%the Yukawa couplings have been calculated.
the most complete study of the Yukawa couplings has been carried out
for orbifold compactifications
[1--6], which on the other hand have proved to possess very interesting
properties from the phenomenological point of view [7]. So, we will
focus in this letter on this kind of scenarios. Furthermore, orbifold
Yukawa couplings for twisted matter (see below) present a
very rich range, which is extremely attractive as the geometrical
origin of the observed variety of fermion masses [2,8,9].
We will assume throughout that the effective four--dimensional field
theory has $N=1$ supersymmetry, $SU(3)\times SU(2)\times U(1)_Y$ observable
gauge group and three generations of particles with the correct gauge
representations. (We do not consider a GUT theory to avoid the above
mentioned problems.) All these properties have been obtained in explicit
orbifold constructions [10].
%We denote $\{Q_u,Q_c,Q_t\}$,
%$\{L_e,L_\mu,L_\tau\}$ the $SU(2)$ quark and lepton doublets, and
%$\{u_L^c,d_L^c,c_L^c,s_L^c,t_L^c,b_L^c\}$,
%$\{e_L^c,\mu_L^c,\tau_L^c\}$ the $SU(2)$ quark and lepton singlets.
Moreover we will assume a unique generation
of Higgses $\{H_1,H_2\}$ (necessary to get a correct Weinberg
angle [11]) and that all the observable matter is of the twisted
type (as was argued in ref.[9], observable untwisted matter
is not phenomenologically viable).

We have made a study, as systematic as possible, for the complete set
of Abelian $Z_N$ orbifolds, i.e. $Z_3$, $Z_4$, $Z_6$--I, $Z_6$--II,
$Z_7$, $Z_8$--I, $Z_8$--II, $Z_{12}$--I, $Z_{12}$--II. We will not
enter here into the details about the construction of these schemes
(these can be found in refs.[1]). Let us recall, however, that a $Z_N$
orbifold is constructed by dividing $R^6$ by a six--dimensional lattice
$\Lambda$ modded by some $Z_N$ symmetry, called the point group $P$.
The space group $S$ is defined as $S=\Lambda\times P$, i.e. $S=\{
(\gamma,u);\ \gamma\in P,\ u\in\Lambda\}$. A twisted string satisfies
$x(\sigma=2\pi)=gx(\sigma=0)$ as the boundary condition, where $g$ is
an element (more precisely a conjugation class) of the space group
whose point group component is non--trivial. Owing to the boundary
condition, a twisted string is attached to a fixed point (sometimes
to a fixed torus) $f$ of $g$. Roughly speaking, the form of $g$ is
$g=(\theta^k, (1-\theta^k)(f+v))$, where $\theta$ is the generator
of $Z_N$ ($\theta^N=1$) and $v\in\Lambda$. It is said
that the string belongs to the $\theta^k$ sector. For a Yukawa
coupling to be allowed the product of the three relevant space group
elements, say $g_1 g_2 g_3$, must contain the identity. This implies
two important equalities:
%
\begin{equation}
k_1+k_2+k_3=0\;\;{\rm mod}\ N
\label{pgroup}
\end{equation}
%
\begin{equation}
(1-\theta^{k_1})(f_1+v_1)+\theta^{k_1}(1-\theta^{k_2})(f_2+v_2)-
(1-\theta^{k_1+k_2})(f_3+v_3)=0,\;\;v_i\in\Lambda
\label{sgroup}
\end{equation}
%
The first one is the so--called point group selection rule, which
implies that the coupling must be of the $\theta^{k_1}\theta^{k_2}
\theta^{-(k_1+k_2)}$ type. The second one is the so--called space
group selection rule, which can have different characteristics
depending on the orbifold under consideration. Some additional
complications appear when a fixed point $f$ under $\theta^k$ is
not fixed under $\theta$ [12,6]. The space group selection
rules for all the $Z_N$ orbifolds have been classified in refs.[5,6].
Likewise, the expressions for the different Yukawa couplings have been
calculated in refs.[2--6]. Their characteristics are summarized in Table 1.
They contain suppression factors that depend
on the relative positions of the fixed points to which the fields
involved in the coupling are attached (i.e. $f_1, f_2, f_3$), and on
the size and shape of the orbifold\footnote{The size and shape of the
orbifold are
given by the vacuum expectation values (VEVs) of certain fields (moduli)
and, consequently, they are dynamical parameters. It has been shown
[13,14] that supersymmetry breaking effects could determine their actual
values.}.
As mentioned above, this has been
suggested as the possible origin of the observed hierarchy of fermion
masses.
We have explored that possibility in this letter, finding that for
certain schemes it can be successfully realized. In section 2 we
discuss the possibility of getting the correct Kobayashi--Maskawa
parameters at the renormalizable level. This turns out to be out
of reach even for non--prime orbifolds, where the mass matrices
are allowed to be non--diagonal. It is remarked, however, that
this job might well be done by non--renormalizable couplings,
at the same time as they account for the masses of
the first generation (which should come from off--diagonal entries
in the mass matrices). However,
renormalizable couplings should still be responsible for
the masses of the second and third generations. Whether this
is possible or not is studied in section 3. As a first step,
a renormalization group analysis is performed, which presents
(slight) differences from the ordinary GUT one. Then it is shown that
for a reasonable size and shape of the compactified space,
the $Z_3$, $Z_4$, $Z_6$--I and possibly $Z_7$ orbifolds can fit the physical
quark and lepton masses adequately. The rest of the $Z_N$
orbifolds, however, should be discarded under the previous
minimal assumptions.
We present our conclusions in section 4.


\section{Mixing angles and geometrical selection rules}

In order to reproduce the mixing angles and the CP violating phase of
the experimental Kobayashi--Maskawa (KM) matrix the quark mass matrices
must have off--diagonal entries. Consequently, the first question is whether
it is possible or not to get non--diagonal mass matrices. The answer to
this question is intimately related to the space--group selection rule
(see eq.(\ref{sgroup})). For prime orbifolds ($Z_3$, $Z_7$), this is of the
so--called diagonal type. This means that given two fields associated
with two fixed points $f_1, f_2$, they can only couple to a unique
third fixed point $f_3$. Of course the coupling, to be allowed,
must satisfy other requirements, in particular gauge invariance.
On the other hand, the couplings must satisfy the point group selection
rule, eq.(\ref{pgroup}), which is also diagonal and, in addition,
in a $\theta^k$ sector the matter associated with a given fixed point
is not degenerate, i.e. all fields have different gauge quantum numbers.
Consequently, for $Z_3$ and $Z_7$ orbifolds the mass matrices are diagonal.
For instance, $Q_uH_2$ (where $Q_u$ denotes the $(u,d)_L$ doublet)
can only couple to a unique field with the gauge
quantum numbers of $u^c$, although this does not mean that such a
coupling must be present. If this were the whole story we should conclude
that mixing angles cannot be obtained within the $Z_3$ and $Z_7$
frameworks. Fortunately, things are quite different when the gauge group
is spontaneously broken after compactification and, in fact, this is
what happens in all the phenomenologically interesting models so far
constructed [10]\footnote{Several sources for this breakdown have been
explored, namely Fayet--Iliopoulos breaking [10], flat directions,
and gaugino condensation induced breaking [14,15].}. Then there
appear new effective trilinear couplings coming from higher order
operators in which some of the fields get non--vanishing VEVs\footnote{
Another (model--dependent) mechanism for mixings, after the breaking,
is explained in refs.[9,16]}. These couplings have a strong
exponential damping [17], but they are no longer subjected to the
trilinear selection rule (examples of this can be found in ref.[10]).
This leads to a natural ansatz for quark and
lepton mass matrices:
%
\begin{eqnarray}
M=\left( \begin {array}{ccc}
\epsilon & a & b\\
\tilde a & A & c\\
\tilde b & \tilde c & B
\end {array} \right)
\label{CM}
\end{eqnarray}
%
where $\epsilon,a,\tilde{a},b,\tilde{b},c,\tilde{c}<<A<<B$ in magnitude,
lower--case letters denoting entries generated by higher order operators.
Here we have assumed that the (1,1) entry is zero at the renormalizable
level. As is known, this is extremely convenient to obtain the Cabibbo
angle in a natural manner (more precisely, $\sin\theta_c\sim\sqrt{m_d/m_s}$).
Notice that in this way the masses of the first generation should also be
caused by higher order operators. It is not difficult to construct explicit
models with this property (see e.g. ref.[18]). On the other hand, the entries
$A,B$ should essentially
be generated by renormalizable couplings since non--renormalizable ones are
too small to fit the second and third generation masses properly. Of course,
one has to require $A,B$ to be the correct ones in order to reproduce
those masses. Whether this is possible or not will be studied in the
next section. The ansatz (\ref{CM}) was obtained in ref.[9] in the context
of the $Z_3$ orbifold. It was shown there that it gives correct KM
parameters and first generation masses
for reasonable values of the off--diagonal entries (in
particular it is highly desirable that $\epsilon=0$). Of
course, the precise values of these have to be calculated in each
particular case, but at least this shows that there is no
incompatibility {\em ab initio} between prime orbifolds and the observed
KM parameters. In some sense eq.(\ref{CM}) (with
$\epsilon=0$) is a "stringy" alternative to the Fritzsch ansatz [19]
%
\begin{eqnarray}
M=\left( \begin {array}{ccc}
0 & A & 0\\
A & 0 & B\\
0 & B & C
\end {array} \right)
\label{Frit}
\end{eqnarray}
%
(with $|A|<<|B|<<|C|$), which is the most extensively discussed
form for $u$ and $d$--type quark mass matrices.

Things go in a different way for even orbifolds. The reason is twofold.
First, it is clear from Table 1 that, for an even orbifold, Yukawa
couplings are not necessarily of a unique $\theta^{k_1}\theta^{k_2}
\theta^{k_3}$ type. Second, the space group selection rule for a
given $\theta^{k_1}\theta^{k_2}\theta^{k_3}$ coupling is not,
in general, of the diagonal type [6], i.e. for two given fixed points
$(f_1, f_2)$, $f_3$ is not uniquely selected. These two features
in principle open the possibility of having non--diagonal mass matrices
at the renormalizable level, and this is indeed what happens. However,
we will argue now that the structure of these matrices is still
strongly constrained by the selection rules, so that, as for prime
orbifolds, no realistic prediction for the KM parameters can emerge
at the renormalizable level.

Let us first show that the point group selection rule implies that
any viable form for the quark mass matrices should be built up
with Yukawa couplings of a unique $\theta^{k_1}\theta^{k_2}
\theta^{k_3}$ type. Consider for example the $d$--quark mass matrix and suppose
that $H_1, d^c,s^c,b^c$ correspond to the $\theta^l,\theta^{m_1},
\theta^{m_2},\theta^{m_3}$ sectors respectively. Notice now that
if one row of the mass matrix contains more than one entry different
from zero, say $M_{ij_1}, M_{ij_2}\neq 0$, then the point group
selection rule (\ref{pgroup}) requires $l+m_{j_1}=l+m_{j_2}\ \rightarrow
\ m_{j_1}=m_{j_2}$, otherwise the two $SU(2)$ singlet quarks involved
here could not be coupled to the same quark doublet. Now, it is easy
to apply this rule to check that a mass matrix of the Fritzsch type,
eq.(\ref{Frit}), or of the type of eq.(\ref{CM})
cannot be obtained unless all the Yukawa couplings
involved are of the same $\theta^{k_1}\theta^{k_2}\theta^{k_3}$
class. In fact, it is hardly conceivable a phenomenologically
viable mass matrix which does
not contain rows with more
than one non--vanishing entry involving the three generations, so
this rule is general.

Now, we will show that the space group selection rule induces
an important property in the mass matrix which we call
"box--closing" for short. This property means that if we have a $2\times 2$ box
in the mass matrix with three entries different from zero, then the
fourth entry must also be different from zero, e.g.
%
\begin{eqnarray}
\left[ \begin {array}{cc}
{} & \times \\
\times & \times
\end {array} \right]\;\rightarrow\;
\left[ \begin {array}{cc}
\times & \times \\
\times & \times
\end {array} \right]
\label{BC}
\end{eqnarray}
%
To see this, suppose that the three initial entries correspond to the
couplings
%
\begin{equation}
Q_aH_1q_1^c\;,\;\;\;Q_aH_1q_2^c\;,\;\;\;Q_bH_1q_2^c
\label{tresc}
\end{equation}
%
Calling $\theta^p$, $\theta^l$ the sectors to which $Q_{a,b}$\ ,\  $H_1$
belong, the space group selection rule (\ref{sgroup}) implies
%
\begin{equation}
(1-\theta^p)(Q_a+v_1)+\theta^{p}(1-\theta^{l})(H_1+v_2)-
(1-\theta^{l+p})(q^c_1+v_3)=0
\label{sgr1}
\end{equation}
%
\begin{equation}
(1-\theta^p)(Q_a+\tilde v_1)+\theta^{p}(1-\theta^{l})(H_1+\tilde v_2)-
(1-\theta^{l+p})(q^c_2+v_4)=0
\label{sgr2}
\end{equation}
%
%
\begin{equation}
(1-\theta^p)(Q_b+v_5)+\theta^{p}(1-\theta^{l})(H_1+\tilde{\tilde v}_2)-
(1-\theta^{l+p})(q^c_2+\tilde v_4)=0
\label{sgr3}
\end{equation}
%
where $v_i,\tilde v_i,\tilde{\tilde v}_i\in \Lambda$ and we have denoted,
for simplicity, a field and its corresponding fixed point by the same
symbol. Now, (\ref{sgr1})$+$(\ref{sgr3})$-$(\ref{sgr2}) reads
%
\begin{equation}
(1-\theta^p)(Q_b+v_1+v_5-\tilde v_1)+
\theta^{p}(1-\theta^{l})(H_1+v_2+\tilde{\tilde v}_2-\tilde v_2)-
(1-\theta^{l+p})(q_1^c+v_3+\tilde v_4-v_4)=0
\label{sgr4}
\end{equation}
%
which implies that the coupling $Q_bH_1q_1^c$ is also allowed\footnote{
The above analysis is more involved when some of the fixed points
are not invariant under $\theta$. However, after an exhaustive study,
it turns out that the box--closing property holds in all cases.}.
This excludes the possibility of obtaining the Fritzsch matrix
(\ref{Frit}) at the renormalizable level (starting with (\ref{Frit})
and applying the box--closing property four times we fill all the
entries). Also the matrix of eq.(\ref{CM}) with $\epsilon=0$
is not allowed. Again, it is hard to imagine any viable mass matrix
satisfying the box--closing property.

This seems to exclude any possibility of having a reasonable mass
matrix at the renormalizable level. One could still try to get something
similar to the Fritzsch matrix, for example, but with very suppressed
couplings instead of zeros. However, this is hopeless since
the selection rule not only imposes the "closing" of any $2\times 2$
box with three non--vanishing entries, see (\ref{BC}), but it usually
relates the value of the fourth entry to those of the three initial ones.
When this is not so, the corresponding suppression factors for the
would--be zero entries should be very
strong, thus requiring high values for the moduli (see next section)
and making virtually impossible to fit the fermion masses correctly .
In view of these results we have to give up fitting the KM parameters
at the renormalizable level. As was mentioned above, this job can
be realized by the (model--dependent) non--renormalizable operators
for the mass matrix of eq.(\ref{CM}), at the same time as they account
for the first generation masses\footnote{This mechanism has also been
considered in ref.[20] in the context of a flipped string model [21].}.
However, the renormalizable couplings should be able to fit the fermion
masses of the second and third generations, which is still extremely
restrictive. This is what we study in the next section.

\section{Fermion masses}
\subsection{Renormalization group analysis}

It is customary to give the experimental values of fermion masses [22]
(except $m_t$) at $1\ GeV$, see first row of Table 2. On the other
hand, the Yukawa couplings in orbifolds are calculated at the
string scale $M_{Str}=0.527\times g\times 10^{18}\ GeV$ [23], where
$g\simeq 1/\sqrt{2}$ is the corresponding value of the gauge coupling
constant. Thus, in order to compare theory and
experiment a renormalization group (RG) running between these
two scales is necessary. This RG analysis differs from the ordinary
GUT one since, in GUTs, the running of
Yukawa couplings is performed between $M_{GUT}$ and $1\ GeV$, where
$M_{GUT}$ is the scale at which gauge interactions are unified.
On the other hand, in string theories, there are "stringy" (no GUT)
threshold corrections on the value of the gauge coupling constants,
shifting the actual scale at which they are unified. More precisely,
%the value of the gauge couplings at $M_{Str}$ is
%%
%\begin{equation}
%g_a^{-2}=k_a S+...
%\label{thres}
%\end{equation}
%%
%where $S$ ....
the value of these threshold corrections depends on the VEVs of
some moduli\footnote{
Explicit expressions for the threshold corrections can be
found in ref.[23].} that, in general, are not the ones involved
in the Yukawa couplings (see next subsection) [6]. It has been
shown in ref.[24] that, for appropriate VEVs of these moduli
the gauge couplings still unify at an effective unification
scale $M_X\simeq 10^{16}\ GeV$, as is phenomenologically required [25].
However, the running of the Yukawa
couplings has still to be made from $M_{Str}$. This fact, for example,
modifies (slightly) the traditional relation $m_b/m_\tau$ at low energy
when one sets $m_b=m_\tau$ at tree level, as will be seen shortly.

Let us write, for the sake of definiteness, the Yukawa Lagrangian
for the second and third generations, at a scale $\mu$
%
\begin{eqnarray}
{\cal L}_{Yuk}\ & = & \ h_c(\mu)Q_cH_2c^c + \ h_s(\mu)Q_cH_1s^c
+\ h_\mu(\mu)L_\mu H_1\mu^c
\nonumber \\
& + & \ h_t(\mu)Q_tH_2t^c + \ h_b(\mu)Q_tH_1b^c
+\ h_\tau(\mu)L_\tau H_1\tau^c
\label{Lyuk}
\end{eqnarray}
%
where the capital letters denote $SU(2)$ doublets and the
$h$'s are the Yukawa couplings. The physical masses
at $1\ GeV$ are then given by
%
\begin{equation}
m_\alpha=h_\alpha (1\ GeV)\nu_1\ ,\; m_\beta=h_\beta (1\ GeV)\nu_2
\label{masfis}
\end{equation}
%
where $\alpha=s,b,\mu,\tau$\ ; $\ \beta=c,t$ and $\nu_{1,2}=
\langle H_{1,2}\rangle$
are subjected to the bound
%
\begin{equation}
\nu_1^2+\nu_2^2 = 2\left(\frac{M_W}{g_2}\right)^2=(175\ GeV)^2
\label{cotanus}
\end{equation}
%
Moreover, electroweak symmetry breaking in the context of the minimal
supersymmetric standard model suggests $\nu_2 >\nu_1$ [26].
In order to relate $h_{\alpha,\beta}(1\ GeV)$ to
$h_{\alpha,\beta}(M_{Str})$ we have to make use of the RG equations
for the Yukawa couplings between these two scales (see e.g. [27,28]).
This has to be done
in several steps since the matter content is not the same
at any intermediate scale. In particular, we assume as usual a unique
supersymmetric mass $M_{S}$ for all the supersymmetric partners
of the standard matter. (Allowing for a differentiation
of the various supersymmetric masses does not modify the results
substantially.) Besides this, there are, of course, the ordinary
quark thresholds. Following a standard RG analysis and working
in the usual limit $h_b,h_\tau<<h_t$,
we find, for the first two generations of quarks and the three
generations of leptons, the following expressions
%
\begin{eqnarray}
h_{u,c}(1\ GeV) & = & F_u\ h_{u,c}(M_{Str}) =
\left(\frac{\alpha_3(1\ GeV)}{\alpha_3(m_c)}\right)^{\frac{4}{9}}
\left(\frac{\alpha_3(m_c)}{\alpha_3(m_b)}\right)^{\frac{12}{25}}
\left(\frac{\alpha_3(m_b)}{\alpha_3(M_Z)}\right)^{\frac{12}{23}}
\nonumber \\
& \times &
\left(\frac{\alpha_3(M_Z)}{\alpha_3(M_{S})}\right)^{\frac{4}{7}}
\left(\frac{\alpha_3(M_{S})}{\alpha_3(M_{Str})}\right)^{\frac{8}{9}}
\left(\frac{\alpha_2(M_Z)}{\alpha_2(M_{S})}\right)^{\frac{3}{8}}
\left(\frac{\alpha_2(M_{S})}{\alpha_2(M_{Str})}\right)^{\frac{-3}{2}}
\nonumber \\
& \times &
\left(\frac{\alpha_1(M_Z)}{\alpha_1(M_{S})}\right)^{\frac{-17}{168}}
\left(\frac{\alpha_1(M_{S})}{\alpha_1(M_{Str})}\right)^{\frac{-13}{198}}
h_{u,c}(M_{Str})
\label{RGh1}
\end{eqnarray}
%
\begin{equation}
h_{d,s}(1\ GeV)= F_u
\left(\frac{\alpha_1(M_Z)}{\alpha_1(M_{S})}\right)^{\frac{12}{168}}
\left(\frac{\alpha_1(M_{S})}{\alpha_1(M_{Str})}\right)^{\frac{6}{198}}
h_{d,s}(M_{Str})
\label{RGh2}
\end{equation}
%
\begin{eqnarray}
h_{e,\mu,\tau}(1\ GeV) & = &
\left(\frac{\alpha_2(M_Z)}{\alpha_2(M_{S})}\right)^{\frac{3}{8}}
\left(\frac{\alpha_2(M_{S})}{\alpha_2(M_{Str})}\right)^{\frac{-3}{2}}
\nonumber \\
& \times &
\left(\frac{\alpha_1(M_Z)}{\alpha_1(M_{S})}\right)^{\frac{-45}{168}}
\left(\frac{\alpha_1(M_{S})}{\alpha_1(M_{Str})}\right)^{\frac{-3}{22}}
h_{e,\mu,\tau}(M_{Str})
\label{RGh3}
\end{eqnarray}
%
where $\alpha_1,\alpha_2,\alpha_3$ are the gauge couplings
of $U(1)_Y$, $SU(2)$ and $SU(3)$ respectively. For $h_t$, $h_b$ the
RG equations are more complicated since the effect of the
top Yukawa interactions are not negligible here (see e.g. [27]).
After some algebra one arrives at
%
\begin{eqnarray}
h_t(M_Z) & = &
%\left(\frac{\alpha_3(1\ GeV)}{\alpha_3(m_c)}\right)^{\frac{4}{9}}
%\left(\frac{\alpha_3(m_c)}{\alpha_3(m_b)}\right)^{\frac{12}{25}}
%\left(\frac{\alpha_3(m_b)}{\alpha_3(M_Z)}\right)^{\frac{12}{23}}
%\nonumber \\
\left(\frac{E_1'(M_Z)}{1+\frac{9}{16\pi^2}
h_t^2(M_{S})F_1'(M_Z)}\right)^{\frac{1}{2}}
\nonumber \\
& \times & \left(\frac{E_1(M_{S})}{1+\frac{6}{16\pi^2}
h_t^2(M_{Str})F_1(M_{S})}\right)^{\frac{1}{2}}
h_t(M_{Str})
\label{RGh4}
\end{eqnarray}
%
\begin{eqnarray}
h_b(1\ GeV) & = &
\left(\frac{\alpha_3(1\ GeV)}{\alpha_3(m_c)}\right)^{\frac{4}{9}}
\left(\frac{\alpha_3(m_c)}{\alpha_3(m_b)}\right)^{\frac{12}{25}}
\left(\frac{\alpha_3(m_b)}{\alpha_3(M_Z)}\right)^{\frac{12}{23}}
\nonumber \\
& \times & \left(\frac{E_2'(M_Z)}{[1+\frac{9}{16\pi^2}
h_t^2(M_{S})F_1'(M_Z)]^{1/9}}\right)^{\frac{1}{2}}
\nonumber \\
& \times & \left(\frac{E_2(M_{S})}{[1+\frac{6}{16\pi^2}
h_t^2(M_{Str})F_1(M_{S})]^{1/6}}\right)^{\frac{1}{2}}
h_b(M_{Str})
\label{RGh5}
\end{eqnarray}
%
where
%
\begin{eqnarray}
E_1(Q) & = & \left(1-3\frac{\alpha_3(M_{Str})}{4\pi}t\right)^{\frac{-16}{9}}
\left(1+\frac{\alpha_2(M_{Str})}{4\pi}t\right)^3
\left(1+\frac{33}{5}\frac{\alpha_1(M_{Str})}{4\pi}t\right)^{\frac{13}{99}}
\nonumber \\
E_1'(Q') & = & \left(1-7\frac{\alpha_3(M_{S})}{4\pi}t'\right)^{\frac{-8}{7}}
\left(1-3\frac{\alpha_2(M_{S})}{4\pi}t'\right)^{\frac{-3}{4}}
\left(1+\frac{42}{10}\frac{\alpha_1(M_{S})}{4\pi}t'\right)^{\frac{17}{84}}
\nonumber \\
F_1(M_{S}) & = & \int_{Q=M_{Str}}^{Q=M_{S}} E_1(Q)dt\;,\;\;\;\;\;\;
F_1'(M_{Z})=\int_{Q'=M_S}^{Q'=M_{Z}} E_1'(Q')dt'
\nonumber \\
E_2(Q) & = & E_1(Q)
\left(1+\frac{33}{5}\frac{\alpha_1(M_{Str})}{4\pi}t\right)^{\frac{-12}{198}}
\nonumber \\
E_2'(Q') & = & E_1'(Q')
\left(1+\frac{42}{10}\frac{\alpha_1(M_{S})}{4\pi}t'\right)^{\frac{-24}{168}}
\label{Es}
\end{eqnarray}
%
with
%
\begin{equation}
t=2\log \frac{M_{Str}}{Q}\;,\;\;\;t'=2\log \frac{M_{S}}{Q'}
\label{tes}
\end{equation}
%
The experimental values of $\alpha_i(M_Z)$ are (see e.g. ref.[25]):
%
\begin{equation}
\alpha_1(M_Z)=0.016930(80),\;\;
\alpha_2(M_Z)=0.03395(52),\;\;
\alpha_3(M_Z)=0.125(5)
\label{alfexp}
\end{equation}
%
from which $\alpha_i$ can be obtained at any scale following
a standard RG analysis. It has been shown in ref.[25] that a
correct perturbative unification demands\footnote{It can be easily checked
that with $M_{S}= 10^3\ GeV$ the gauge coupling constants
(\ref{alfexp}) are unified at $M_X\simeq2.3\times 10^{16}\ GeV$ with
$\alpha(M_X)\simeq 0.0393$.}
$M_{S}\sim10^3\ GeV$, which is the value we insert in
eqs.(\ref{RGh1}--\ref{tes}) (variations of $M_{S}$ within the
errors are negligible for our purposes). It is interesting to
calculate the values of $h_c,h_s,h_b,h_t,h_\mu,h_\tau$ at $M_{Str}$
which would give the measured values of the corresponding
fermion masses. We have represented them in Fig.1 as
functions of $\nu_2$. Of course, these values for the $h$'s should
emerge from the theory. Whether this happens or not is studied
in the next subsection.

Let us finally note that the relation $m_b(1\ GeV)/m_\tau(1\ GeV)$
is obtained from (\ref{RGh5}) and (\ref{RGh3}). If one imposes,
following the usual GUT ansatz, $h_b(M_{Str})=h_\tau(M_{Str})$,
an expression similar to the GUT one is obtained, but only after
substituting
$M_{GUT}$ by $M_{Str}$ and taking into account that $\alpha_3(M_{Str})
\neq \alpha_2(M_{Str}) \neq \alpha_1(M_{Str})$. Of course, the numerical
results are not substantially affected.

\subsection{The fits}

The theoretical Yukawa couplings $h(M_{Str})$ to be inserted
in eqs.(\ref{RGh1}--\ref{RGh5}) have been calculated in refs.[2--6].
In order to get a feeling of their main characteristics, let us take
the $Z_4$ orbifold based on an $[SO(4)]^3$ root lattice as a
useful example. The action of $\theta$ on the lattice basis
$(e_1,...,e_6)$ is simply $\theta e_l = e_{l+1}$, $\theta e_{l+1} =
-e_l$ with $l=1,3$ and $\theta e_5 = -e_5$, $\theta e_6 = -e_6$.
Let us call $R_i\equiv |e_i|$ and $\alpha_{ij}
=\cos \theta_{ij}$ with $e_i e_j= R_iR_j \cos \theta_{ij}$.
In the orbifold without deformations $\alpha_{ij}=0$ ($i\neq j$).
However the orbifold can consistently be deformed by a modification
of the values of the so--called deformation parameters\footnote{
The values of the deformation parameters correspond to the VEVs
of certain singlet fields with perturbative flat potential,
called moduli and usually denoted by $T_i$.} [9]. For the $Z_4$
orbifold these are $R_1$, $R_3$, $R_5$, $R_6$, $\alpha_{13}$,
$\alpha_{14}$, $\alpha_{56}$ with $(\alpha_{13}+\alpha_{14})^2\leq 1$.
The sizes of the Yukawa couplings
depend on the values of some of them called
effective deformation parameters [9]. For the $Z_4$ these are
%
\begin{equation}
R_1,\;R_3,\;\alpha_{13},\;\alpha_{14}
\label{EDP}
\end{equation}
%
On the other hand, for this orbifold all the twisted couplings are
of the $\theta\theta\theta^2$ type and the selection rule reads
%
\be
f_1 + f_2 - (1 + \theta) f_3 \in \Lambda ,
\label{Z4selrul}
\ee
%
where $f_3$ is the $\theta^2$ fixed point. The classification of
the fixed points in terms of the lattice basis can be found in ref.[6].
The value of an allowed Yukawa coupling at $M_{Str}$ turns out to be
%
\begin {eqnarray}
h_{\theta\theta\theta^2} = gN \sum_{v \in (f_2-f_3+\Lambda)_\bot}
\exp [-\frac {1}{4\pi} \vec{v}^{\top} M \vec{v}]
 = gN\;\;  \vartheta \left[
\begin {array}{c}
\vec{f_{23}} \\
0
\end{array}
\right]
[ 0 , \Omega ] ,
\label{ac41}
\end {eqnarray}
%
where the subscript $\bot$ denotes projection on the $(e_1,...,e_4)$
$D=4$ space, $f_{23}=f_2-f_3$,
the arrow means the corresponding 4--plet of components
% $g$ is the gauge coupling constant ($\sim 1/26$ )
and $\vartheta$ is the Jacobi theta function. Moreover
%
\begin{eqnarray}
\begin{array}{c}
N = \sqrt{V_{\perp}}\; \frac{1} {2\pi}\; \frac {\Gamma ^2
 (\frac{3}{4})}{\Gamma^2 (\frac
{1}{4})}
\\ \\
M= (-4\pi^2i) \Omega =
\left(
\begin{array}{cccc}
R_1^2 & 0 & R_1R_3\alpha_{13} & R_1R_3\alpha_{14} \\
0     & R_1^2 & -R_1R_3\alpha_{14} & R_1R_3\alpha_{13} \\
R_1R_3\alpha_{13} & -R_1R_3\alpha_{14} & R_3^2 & 0 \\
R_1R_3\alpha_{14} & R_1R_3\alpha_{13} & 0 & R_3^2
\end{array}
\right)
\end{array}
\label{mac41}
\end{eqnarray}
%
where $V_{\perp}=R_1^2R_3^2(1-\alpha_{13}^2-
\alpha_{14}^2)$ is the volume of the unit cell of the
$(e_1,...,e_4)$ lattice. If $f_3$ is not fixed by $\theta$ the
result for $h_{\theta\theta\theta^2}$ is exactly the same but multiplied
by $\sqrt{2}$. Notice that $h_{\theta\theta\theta^2}$ depends
on the relative positions in the lattice of the relevant fixed
points to which the physical fields are attached. This
information is condensed in $\vec{f_{23}}$. In addition,
$h_{\theta\theta\theta^2}$ depends on the size and shape of
the compactified space, which is reflected in the effective
deformation parameters $(R_1, R_3, \alpha_{13}, \alpha_{14})$
appearing in $\Omega$ and in $V_{\perp}$. Note
that both pieces of information appear in a completely distinguishable way
from each other in eq.(\ref{ac41}). It is also interesting
to say that the number of allowed couplings is 160. The number
of {\em different} Yukawa couplings is 10. These characteristics
are summarized in Table 1 for all the orbifolds.

In order to calculate the value of a specific Yukawa coupling,
say $h_s$ (see eq.(\ref{Lyuk})) we need to know what the
$\theta^k$ sectors are and fixed points to which $Q_c,H_1$ and $s^c$
are associated. Actually, it is an empirical fact that, because of the
huge proliferation of scenarios within a given compactification scheme
($Z_4$ in this case), the observable fields can correspond to
any choice of $\{\theta^k,f\}$ sectors, see e.g. ref.[18]. Consequently,
we will take the freedom to assign the physical fields to $\theta^k$
sectors and fixed points at convenience. Of course, a particular
assignment will only be realized in certain scenarios. It is
interesting to notice, however, that not for all the assignments
are the physical Yukawa couplings (see eq.(\ref{Lyuk})) allowed
from the point group and space group selection rules. In order
to illustrate this, suppose that $H_1$ and $H_2$ belong to the
$\theta$ and $\theta^2$ sectors respectively. Then, writting
the space group selection rule (\ref{Z4selrul}) for the quark couplings
of ${\cal L}_{Yuk}$ (see eq.(\ref{Lyuk})),
one finds after some algebra
%
\be
(H_2-t^c) - (1 + \theta) (b^c-H_1)=
(H_2-c^c) - (1 + \theta) (s^c-H_1) + \Lambda
\label{assrul}
\ee
%
where we have denoted the fields and their corresponding fixed points
by the same symbols. Eq.(\ref{assrul}) sets severe restrictions on the
possible assignments and, hence, on the possible correspondences of the
physical Yukawa couplings to the above mentioned 10 different Yukawa
couplings. Similar expressions appear if we initially assign $H_1,H_2$
to other $\theta^k$ sectors.

The final step is to let the effective deformation parameters
vary in order to see whether for some choices of them the theoretical
masses, calculated using
eqs.(\ref{masfis}, \ref{RGh1}--\ref{RGh5}, \ref{ac41}), coincide with
the experimental ones (see also Fig.1). In this fit $\nu_1$ has also
to be considered as a free parameter (within the limits mentioned
in the previous subsection) while $\nu_2$ is given by (\ref{cotanus}).
Of course, a different fit has to be made for each possible assignment.

If a satisfactory fit is found, this means that the corresponding
orbifold scheme (in this case $Z_4$) is compatible with the observed
spectrum of fermion masses, which is highly non--trivial as will
be seen shortly. On the other hand, if no such fit is found the
orbifold scheme should be discarded. Obviously, orbifolds with
a higher number of deformation parameters and different Yukawa
couplings (see Table 1) are in a better position to fit the
experimental masses, but this is not a guarantee. For the particular
case of the $Z_4$ orbifold, we have found that, for most of the
possible assignments, the fits are not satisfactory. However,
provided
%
\begin {eqnarray}
\vec f_{23}(c) & = & (00\frac{1}{2}\frac{1}{2})\ ,\
\vec f_{23}(s) = (0\frac{1}{2}\frac{1}{2}\frac{1}{2})\ ,\
\vec f_{23}(t) = (\frac{1}{2}000)
\nonumber \\
\vec f_{23}(b) & = & (\frac{1}{2}\frac{1}{2}00)\ ,\
\vec f_{23}(\mu) = (\frac{1}{2}\frac{1}{2}0\frac{1}{2})\ ,\
\vec f_{23}(\tau) = (0\frac{1}{2}0\frac{1}{2})
\label{Z4ass}
\end {eqnarray}
%
where $\vec{f_{23}}(\phi)$ is the corresponding $\vec{f_{23}}$ (see
eq.(\ref{ac41})) for the $h_\phi$ coupling, remarkably good fits
can be found. Eq.(\ref{Z4ass}) is satisfied (up to spurious lattice
vectors) by the following assignment of physical fields to
$\{\theta^k,f\}$ sectors
%
\begin {eqnarray}
Q_c & : & (\frac{1}{2}\frac{1}{2}00),\;\;Q_t:(0000),\;\;
c^c:(\frac{1}{2}\frac{1}{2}\frac{1}{2}\frac{1}{2}),\;\;
s^c:(0\frac{1}{2}\frac{1}{2}\frac{1}{2})
\nonumber \\
b^c & : & (\frac{1}{2}\frac{1}{2}00),\;\;t^c:(0\frac{1}{2}00),\;\;
L_\mu:(00\frac{1}{2}\frac{1}{2}),\;\;
L_\tau:(\frac{1}{2}\frac{1}{2}\frac{1}{2}\frac{1}{2})
\nonumber \\
\mu^c & : & (\frac{1}{2}\frac{1}{2}0\frac{1}{2}),\;\;\tau^c:(0\frac{1}{2}
0\frac{1}{2}),\;\;
H_1:(0000),\;\;
H_2:(\frac{1}{2}\frac{1}{2}00)
\label{Z4ass2}
\end {eqnarray}
%
where the $\phi^c$ fields are understood to belong to the $\theta^2$ sector
and the rest to the $\theta$ one\footnote{There are other possible
assignments consistent with (\ref{Z4ass}).}.
The values for the deformation parameters (in string units) and $\nu_1$
for an illustrative fit are
%
\begin {eqnarray}
R_1 &=& 13.280\;,\;\;R_3 = 15.077\;,\;\;\alpha_{13}=-0.2395
\nonumber \\
\alpha_{14} &=&0\;,\;\;\nu_1=71.8\ GeV
\label{Z4fit}
\end {eqnarray}
%
and the corresponding fermion masses are shown in the third row of Table 2.
(Oscillations around these values with the subsequent variations of the
fermions masses are of course possible.)
This result is rather remarkable, specially when one notices that
the number of free parameters is lower than the number of physical
masses fitted. In some sense the experimental masses, which are low--energy
quantities, are selecting a particular assignment of the physical fields
to fixed points and the values of the deformation parameters that
define the size and shape of the compactified space (e.g. the "preferred"
$\theta_{14}$ angle is the cartesian one). We find this
quite encouraging. Notice also that the numbers of eq.(\ref{Z4fit})
are quite sensible for a compactified space. The hierarchy of masses
which emerges from them has to do with the exponential dependence
of the Yukawa couplings (see eq.(\ref{ac41})).

On the technical side, let us comment that the fit has been performed
with the help of a MINUIT program, choosing for the minimization
function the total $\chi^2$. The major obstacle we have found was
to control the convergence of the Jacobi $\vartheta$ function of
eq.(\ref{ac41}), particularly when the $\alpha_{13}$, $\alpha_{14}$
parameters are close to the boundary of their definition range.
This requires to sum up to 10000 terms of the series: it is
by no means significant to keep only a few terms. Finally, we
have increased the usual experimental errors of $m_\mu$ and
$m_\tau$ up to $1\%$ to incorporate, to some extent, the errors
attributable to the calculation. The "experimental" error of $m_b$ was
conservatively set at $\Delta m_b=10\%$.

Let us now comment the results for the other $Z_N$ orbifolds.
It turns out that, besides $Z_4$, the only ones that can work
are the $Z_3$ and the $Z_6$--I. Our best fits for them are
shown in the second and fourth rows of Table 2.
For the $Z_6$--I one, all the couplings considered were of the
$\theta^2\theta^2 \theta^2$ type. Consequently, all the physical
fields are understood to belong to the $\theta^2$ sector in this case.
On the other hand, in the $Z_3$ orbifold there is a unique $\theta$ sector.
The corresponding
assignments, given in the respective lattice basis, are
%
\begin {eqnarray}
&f_{23}(c)&  \left\{\begin{array}{cc}
Z_3 & :  (00\frac{1}{3}\frac{2}{3}\frac{1}{3}\frac{2}{3}) \\
Z_6 & :  (0\frac{1}{3}0\frac{1}{3}00)
\end{array}
\right.
,\;\;
f_{23}(s)    \left\{\begin{array}{cc}
Z_3 & :  (\frac{1}{3}\frac{2}{3}\frac{2}{3}\frac{1}{3}\frac{1}{3}
\frac{2}{3}) \\
Z_6 & :  (0\frac{1}{3}00\frac{1}{3}\frac{2}{3})
\end{array}
\right.
,\;\;
f_{23}(t)  \left\{\begin{array}{cc}
Z_3 & :  (0000\frac{1}{3}\frac{2}{3}) \\
Z_6 & :  (0000\frac{1}{3}\frac{2}{3})
\end{array}
\right.
\nonumber \\
&f_{23}(b)& \left\{\begin{array}{cc}
Z_3 & :  (\frac{1}{3}\frac{2}{3}\frac{2}{3}\frac{1}{3}00) \\
Z_6 & :  (0\frac{1}{3}0000)
\end{array}
\right.
,\;\;
f_{23}(\mu)  \left\{\begin{array}{cc}
Z_3 & :  (\frac{1}{3}\frac{2}{3}\frac{1}{3}\frac{2}{3}\frac{1}{3}
\frac{2}{3}) \\
Z_6 & :  (000\frac{1}{3}\frac{1}{3}\frac{2}{3})
\end{array}
\right.
,\;\;
f_{23}(\tau)   \left\{\begin{array}{cc}
Z_3 & :  (\frac{1}{3}\frac{2}{3}\frac{1}{3}\frac{2}{3}00) \\
Z_6 & :  (000\frac{1}{3}00)
\end{array}
\right.
\label{Z36ass}
\end {eqnarray}
%
where, again, $f_{23}(\phi)$ is the difference between two of the fixed
points involved in the $h_\phi$ coupling. The corresponding values for
the nine deformation parameters that the $Z_3$ orbifold possesses (see
Table 1 and ref.[6]) and for $\nu_1$ are
%
\begin {eqnarray}
R_1 &=&13.039 \;,\;\;R_3 = 30.460\;,\;\;R_5=13.076
\nonumber \\
\alpha_{13} &=&-0.6055\;,\;\alpha_{14} = -0.2395\;,\;\;\nu_1=39.3\ GeV
\nonumber \\
\alpha_{15} &=& \;\;\alpha_{16} = \;\;\alpha_{35} = \;\;\alpha_{36}
= 0\; (fixed)
\label{Z3fit}
\end {eqnarray}
%
Notice that four of them have not been used in the fit. This has been done
to improve the convergence of the MINUIT program. Clearly, a better
fit could be obtained once these four parameters are also considered.
Similarly, the values for the five deformation parameters of the $Z_6$--I
orbifold and $\nu_1$ are
%
\begin {eqnarray}
R_1 &=& 18.349\;,\;\;R_3 = 17.588\;,\;\;R_5=13.073
\nonumber \\
\alpha_{13} &=& -0.3438\;,\;\alpha_{14} = 0.2978\;,\;\;\nu_1=70\ GeV
\label{Z6fit}
\end {eqnarray}
%

For the rest of the orbifolds, after an exhaustive exploration,
we have not found sensible fits. This should not be surprising
since they have a smaller number of effective deformation parameters,
see Table 1. This consideration makes the $Z_4$ case the most
remarkable one. Just for completeness we have given in Table 2
our best fits for these orbifolds. It is worth noticing that
the $Z_7$ orbifold (which has only four different couplings and three
deformation parameters) works very acceptably for all fermions masses,
except for the strange one, which on the other hand has a large experimental
uncertainty.

Let us finally give the values of the moduli $T_i$ corresponding
to eqs.(\ref{Z4fit},\ref{Z3fit},\ref{Z6fit}). As usual, we define
the normalization of $T_i$ in such a way that, under a duality
transformation, they transform as $T_i\rightarrow 1/T_i$ [29]. This
implies Re $T_{i}= \alpha R_{i}^2$ with $\alpha=\frac{\sqrt{3}}{16\pi^2}$
for $Z_3$ and $Z_6$, and $\alpha=\frac{\sqrt{2}}{8\pi^2}$ for
$Z_4$.\footnote{Notice that these values of $\alpha$ are
consistent with the definition of our lattice. This
is the usual one, coinciding with that of the first paper
of ref.[2], while in the second paper of the same reference
the lattice is redefined as $e_i\rightarrow 2\pi e_i$.
Of course the values of $T_i$ are independent on these
redefinitions.} Hence, $T_1=1.86$, $T_3=10.17$,
$T_5=1.87$ (for $Z_3$); $T_1=3.16$, $T_3=4.07$ (for $Z_4$) and
$T_1=3.69$, $T_3=3.39$, $T_5=1.87$ (for $Z_6$).

\section{Summary and Conclusions}

We have explored the capability of string theories to reproduce
the observed pattern of quark and lepton masses and mixing angles.
We have focused our attention on orbifold constructions since,
apart from their phenomenological merits, there is at present a
good knowledge of the theoretical Yukawa couplings in these scenarios.
A first conclusion is that, due to stringy selection rules, there
is no possibility of fitting the Kobayashi--Maskawa parameters
at the renormalizable level. This is so even for non--prime orbifolds,
where the mass matrices are allowed to have a non--diagonal structure.
It is, however, argued that (model--dependent) non--renormalizable couplings
might well do this job at the same time as they account for the masses of
the first generation (which should come from off--diagonal entries
in the mass matrices).
On the other hand, non--renormalizable couplings are
too suppressed to adequately fit the fermion masses of the second
and third generations ($m_\mu$, $m_\tau$, $m_c$, etc.), which, in consequence,
should be accounted for by renormalizable ones (this is still extremely
restrictive). We then examined this issue.

As a first step, a renormalization group running of the Yukawa couplings
between the string scale ($M_{Str}$) and the low--energy scale ($1\ GeV$)
has to be performed. This running is slightly different from the
ordinary GUT one since the gauge couplings are not unified at $M_{Str}$
due to string threshold corrections. This modifies, for example,
the traditional $m_b/m_\tau$ relation at low energy, although not
substantially. The magnitude of the orbifold Yukawa couplings
depends on the values of the so--called deformation parameters,
which describe the size and shape of the compactified space. A
systematic exploration allows us to check whether there is a choice
of these deformation parameters for which the physical fermion masses
are properly fitted. Not all the $Z_N$ orbifolds are here on a
same footing. It turns out that this fit is possible only for the
$Z_3$, $Z_4$ and $Z_6$--I orbifolds. Besides these, the $Z_7$ orbifold
is able to fit all the fermion masses except the strange one, which
on the other hand has a large experimental uncertainty.

The corresponding values of the
deformation parameters are quite reasonable (they correspond to
values for the moduli $T_i=O(1)$). The case of the $Z_4$ orbifold
is specially remarkable since the number of free parameters is
lower than the number of physical masses fitted. In some sense
the experimental masses, which are low--energy quantities, are selecting
a particular size and shape of the compactified space. We find this
quite encouraging. It should be stressed, however, that this only
shows the compatibility of certain string schemes with the
low--energy measurements, although this is certainly non--trivial.
The rest of $Z_N$ orbifolds, however, are rather hopeless
and should be discarded on these grounds. Finally, let us remark that
all these results have been obtained under the assumptions explained
in the Introduction, in particular within a minimal $SU(3)\times
SU(2)\times U(1)_Y$ scenario.


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\newpage

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\textheight=22cm
\oddsidemargin=-1cm
\evensidemargin=0cm
\topmargin=0cm
\parindent=1cm


\begin{table}
\underline{\bf TABLE 1}
$\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline
Orb. & Twist\;\theta  & Lattice & \#DP & Coupling  & \#AC
&\#EDP  &\#DCR  & \#DCD \\ \hline
Z_3  & (1,1,-2)/3    & SU(3)^3 & 9 & \theta\theta\theta    & 729 &
9 & 4 & 14 \\ \hline
Z_4  & (1,1,-2)/4    & SU(4)^2 & 7 & \theta\theta\theta^2    & 160 &
4 & 6 & 10 \\
  &     & SO(4)^3 & 7 & \theta\theta\theta^2    & 160 &
4 & 6 & 8 \\ \hline
Z_6-{\rm I} & (1,1,-2)/6 & G_2^2\times SU(3) & 5 & \theta\theta^2\theta^3 & 90
&
4 & 10 & 30 \\
      &  &  & & \theta^2\theta^2\theta^2 & 369 &
5 & 8 & 12 \\ \hline
Z_6-{\rm II} & (1,2,-3)/6 & SU(6)\times SU(2) & 5 & \theta\theta^2\theta^3
& 48 & 1 & 4 & 4 \\
      &  &  & & \theta\theta\theta^4 & 72 &
2 & 4 & 4 \\ \hline
Z_7 & (1,2,-3)/7 & SU(7) & 3 & \theta\theta^2\theta^4
& 49 & 3 & 2 & 4 \\ \hline
Z_8-{\rm I} & (1,2,-3)/8 & SO(5)\times SO(9) & 3 & \theta^2\theta^2\theta^4
& 80 & 2 & 8 & 8 \\
      &  &  & & \theta\theta^2\theta^5 & 40 &
3 & 8 & 9 \\ \hline
Z_8-{\rm II} & (1,3,-4)/8 & SO(4)\times SO(8) & 5 & \theta\theta\theta^6
& 24 & 2 & 3 & 3 \\
                                                &  &  & &
\theta^2\theta^3\theta^3 & " &
" & " & " \\
      &  &  & & \theta\theta^3\theta^4 & 48 &
2 & 6 & 6 \\ \hline
Z_{12}-{\rm I} & (1,4,-5)/12 & SU(3)\times F_4 & 3 & \theta\theta^2\theta^9
& 6 & 2 & 2 & 2 \\
      &  &  & & \theta^2\theta^3\theta^7 & " &
" & " & " \\
      &  &  & & \theta\theta^4\theta^7 & 27 &
3 & 4 & 6 \\
      &  &  & & \theta^2\theta^4\theta^6 & 36 &
2 & 7 & 12 \\
      &  &  & & \theta^4\theta^4\theta^4 & 135 &
3 & 8 & 12 \\ \hline
Z_{12}-{\rm II} & (1,5,-6)/12 & SO(4)\times F_4 & 5 & \theta\theta\theta^{10}
& 4 & 2 & 1 & 1 \\
      &  &  & & \theta^2\theta^5\theta^5 & " &
" & " & " \\
      &  &  & & \theta\theta^3\theta^8 & 24 &
2 & 6 & 6 \\
      &  &  & & \theta^3\theta^4\theta^5 & 24 &
2 & 6 & 6 \\
      &  &  & & \theta^3\theta^3\theta^6 & 40 &
2 & 6 & 8 \\
      &  &  & & \theta\theta^5\theta^6 & 16 &
2 & 3 & 4 \\ \hline
\end{array}$
\caption{Characteristics of twisted Yukawa couplings for $Z_n$
orbifolds. The twist $\theta$ is
specified by the three $c_i$ parameters (one for each complex
plane rotation) appearing in $\theta=\exp (\sum c_i J_i)$.
$\#DP\equiv$ No. of deformation parameters, $\#AC\equiv$ No. of
allowed couplings, $\#EDP\equiv$ No. of effective deformation
parameters, $\#DCR\equiv$ No. of different Yukawa couplings for
the non--deformed (rigid) orbifold, $\#DCD\equiv$ No. of different
Yukawa couplings when deformations are considered. Quotation marks
denote equivalent couplings.}

\end{table}

\newpage

\begin{table}
\underline{\bf TABLE 2}
$\begin{array}{|c|c|c|c|c|c|c|c|} \hline
{} & m_\mu  & m_\tau & m_s & m_c  & m_b
& m_t &\chi^2_{tot} \\ \hline
Exp.  & 0.1056 & 1.784 & 0.199 & 1.35 & 5 & 130 & -  \\ \hline
Z_3   & 0.1055 & 1.786 & 0.252 & 1.35 & 4.1 & 125 & 6.7  \\ \hline
Z_4   & 0.1062 & 1.774 & 0.173 & 1.35 & 4.34 & 104 & 3.86  \\ \hline
Z_6-{\rm I}   & 0.1056 & 1.785 & 0.252 & 1.35 & 4.04 & 122 & 6.8  \\ \hline
Z_6-{\rm II}  & 1.13 & 64 & 5.1 & 17 & 8.3 & 173 & 3\times 10^8  \\ \hline
Z_7   & 0.104 & 1.783 & 0.466 & 1.35 & 5.2 & 133 & 81  \\ \hline
Z_8-{\rm I}   & 0.087 & 2.00 & 0.280 & 2.17 & 6.3 & 36 & 705  \\ \hline
Z_8-{\rm II}  & 0.1058 & 1.82 & 0.009 & 0.89 & 2.47 & 172 & 158  \\ \hline
Z_{12}-{\rm I}   & 0.103 & 1.83 & 0.107 & 1.40 & 11 & 88 & 177  \\ \hline
Z_{12}-{\rm II}  & 0.036 & 2.15 & 0.045 & 1.0 & 30 & 24 & 7236  \\ \hline
\end{array}$
\caption{Fits for each $Z_N$ orbifold of the first and second
generation fermion masses and total $\chi^2$. The masses,
given in $GeV$, are to be understood at the $1\ GeV$ scale, except
the top mass, which is at the $M_Z$ scale.
The first row corresponds to the present central
experimental values (errors are not shown). For the top mass, the
recent estimations based on the size of the electroweak radiative
corrections was considered. Only the $Z_3$, $Z_4$ and $Z_6$--I
orbifolds are compatible with the experiment.}

\end{table}

\vspace{3cm}
\noindent {\bf FIGURE CAPTION}

\vspace{2cm}
\noindent {\bf FIG.1}:  Values of $h_c,h_s,h_b,h_t,h_\mu,h_\tau$
at $M_{Str}$ versus $\nu_2=<H_2>$ giving the measured values of
the corresponding fermion masses. Errors are not included.

\end{document}


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825
 98 l 818 91 l 828 91 l s 860 134 m 857 131 l 854 127 l 852 121 l 850 114 l 850
 108 l 852 101 l 854 95 l 857 91 l 860 88 l s 890 121 m 889 124 l 886 127 l 883
 128 l 877 128 l 874 127 l 872 124 l 870 121 l 869 117 l 869 110 l 870 105 l
872
 103 l 874 100 l 877 98 l 883 98 l 886 100 l 889 103 l 890 105 l 890 110 l s
883
 110 m 890 110 l s 899 110 m 916 110 l 916 113 l 915 115 l 913 117 l 910 118 l
 906 118 l 903 117 l 900 114 l 899 110 l 899 107 l 900 103 l 903 100 l 906 98 l
 910 98 l 913 100 l 916 103 l s 922 128 m 933 98 l s 945 128 m 933 98 l s 950
 134 m 953 131 l 956 127 l 959 121 l 960 114 l 960 108 l 959 101 l 956 95 l 953
 91 l 950 88 l s 596 413 m 596 381 l s 596 397 m 601 401 l 604 403 l 609 403 l
 612 401 l 613 397 l 613 381 l s 628 384 m 625 373 l s 621 382 m 622 383 l 625
 384 l 633 384 l s 752 430 1181 214 bl 1185 290 m 1192 291 l 1200 291 l 1207
291
 l 1215 292 l 1222 292 l 1230 292 l 1237 293 l 1245 293 l 1252 293 l 1260 294 l
 1267 294 l 1275 295 l 1282 295 l 1290 295 l 1297 296 l 1305 296 l 1312 296 l
 1320 297 l 1328 297 l 1335 298 l 1343 298 l 1350 299 l 1358 299 l 1365 300 l
 1373 300 l 1380 301 l 1388 301 l 1395 302 l 1403 302 l 1410 303 l 1418 303 l
 1425 304 l 1433 304 l 1440 305 l 1448 305 l 1455 306 l 1463 307 l 1470 307 l
 1478 308 l 1485 309 l 1493 309 l 1500 310 l 1508 311 l 1515 311 l 1523 312 l
 1530 313 l 1538 314 l 1546 315 l 1553 316 l s 1553 316 m 1561 316 l 1568 317 l
 1576 318 l 1583 319 l 1591 320 l 1598 321 l 1606 322 l 1613 323 l 1621 324 l
 1628 326 l 1636 327 l 1643 328 l 1651 329 l 1658 331 l 1666 332 l 1673 334 l
 1681 335 l 1688 337 l 1696 339 l 1703 340 l 1711 342 l 1718 344 l 1726 346 l
 1733 348 l 1741 350 l 1749 353 l 1756 355 l 1764 358 l 1771 361 l 1779 364 l
 1786 367 l 1794 370 l 1801 374 l 1809 378 l 1816 382 l 1824 387 l 1831 392 l
 1839 397 l 1846 403 l 1854 410 l 1861 418 l 1867 425 l 1869 426 l 1874 433 l
 1876 436 l 1881 443 l 1884 448 l 1888 455 l 1891 461 l 1896 469 l s 1896 469 m
 1899 477 l 1903 486 l 1905 494 l 1906 496 l 1910 506 l 1912 514 l 1914 521 l
 1916 531 l 1919 541 l 1921 551 l 1921 554 l s 1921 554 m 1929 601 l s 1181 214
 m 1181 644 l s 1213 214 m 1181 214 l s 1197 242 m 1181 242 l s 1197 271 m 1181
 271 l s 1197 299 m 1181 299 l s 1197 327 m 1181 327 l s 1213 355 m 1181 355 l
s
 1197 383 m 1181 383 l s 1197 412 m 1181 412 l s 1197 440 m 1181 440 l s 1197
 468 m 1181 468 l s 1213 496 m 1181 496 l s 1197 524 m 1181 524 l s 1197 553 m
 1181 553 l s 1197 581 m 1181 581 l s 1197 609 m 1181 609 l s 1213 637 m 1181
 637 l s 1213 637 m 1181 637 l s 1127 225 m 1124 224 l 1122 221 l 1121 216 l
 1121 213 l 1122 208 l 1124 205 l 1127 204 l 1129 204 l 1132 205 l 1134 208 l
 1135 213 l 1135 216 l 1134 221 l 1132 224 l 1129 225 l 1127 225 l cl s 1055
366
 m 1052 365 l 1050 362 l 1049 357 l 1049 354 l 1050 349 l 1052 346 l 1055 344 l
 1057 344 l 1060 346 l 1062 349 l 1063 354 l 1063 357 l 1062 362 l 1060 365 l
 1057 366 l 1055 366 l cl s 1071 347 m 1070 346 l 1071 344 l 1072 346 l 1071
347
 l cl s 1086 366 m 1083 365 l 1081 362 l 1080 357 l 1080 354 l 1081 349 l 1083
 346 l 1086 344 l 1088 344 l 1091 346 l 1093 349 l 1094 354 l 1094 357 l 1093
 362 l 1091 365 l 1088 366 l 1086 366 l cl s 1106 366 m 1103 365 l 1101 362 l
 1100 357 l 1100 354 l 1101 349 l 1103 346 l 1106 344 l 1108 344 l 1111 346 l
 1113 349 l 1114 354 l 1114 357 l 1113 362 l 1111 365 l 1108 366 l 1106 366 l
cl
 s 1124 362 m 1126 363 l 1129 366 l 1129 344 l s 1055 507 m 1052 506 l 1050 503
 l 1049 498 l 1049 495 l 1050 490 l 1052 486 l 1055 485 l 1057 485 l 1060 486 l
 1062 490 l 1063 495 l 1063 498 l 1062 503 l 1060 506 l 1057 507 l 1055 507 l
cl
 s 1071 487 m 1070 486 l 1071 485 l 1072 486 l 1071 487 l cl s 1086 507 m 1083
 506 l 1081 503 l 1080 498 l 1080 495 l 1081 490 l 1083 486 l 1086 485 l 1088
 485 l 1091 486 l 1093 490 l 1094 495 l 1094 498 l 1093 503 l 1091 506 l 1088
 507 l 1086 507 l cl s 1106 507 m 1103 506 l 1101 503 l 1100 498 l 1100 495 l
 1101 490 l 1103 486 l 1106 485 l 1108 485 l 1111 486 l 1113 490 l 1114 495 l
 1114 498 l 1113 503 l 1111 506 l 1108 507 l 1106 507 l cl s 1122 502 m 1122
503
 l 1123 505 l 1124 506 l 1126 507 l 1130 507 l 1132 506 l 1133 505 l 1134 503 l
 1134 501 l 1133 499 l 1131 496 l 1121 485 l 1135 485 l s 1055 648 m 1052 647 l
 1050 644 l 1049 639 l 1049 636 l 1050 630 l 1052 627 l 1055 626 l 1057 626 l
 1060 627 l 1062 630 l 1063 636 l 1063 639 l 1062 644 l 1060 647 l 1057 648 l
 1055 648 l cl s 1071 628 m 1070 627 l 1071 626 l 1072 627 l 1071 628 l cl s
 1086 648 m 1083 647 l 1081 644 l 1080 639 l 1080 636 l 1081 630 l 1083 627 l
 1086 626 l 1088 626 l 1091 627 l 1093 630 l 1094 636 l 1094 639 l 1093 644 l
 1091 647 l 1088 648 l 1086 648 l cl s 1106 648 m 1103 647 l 1101 644 l 1100
639
 l 1100 636 l 1101 630 l 1103 627 l 1106 626 l 1108 626 l 1111 627 l 1113 630 l
 1114 636 l 1114 639 l 1113 644 l 1111 647 l 1108 648 l 1106 648 l cl s 1123
648
 m 1134 648 l 1128 640 l 1131 640 l 1133 639 l 1134 638 l 1135 635 l 1135 632 l
 1134 629 l 1132 627 l 1129 626 l 1126 626 l 1123 627 l 1122 628 l 1121 630 l s
 1181 214 m 1933 214 l s 1456 247 m 1456 214 l s 1513 230 m 1513 214 l s 1571
 230 m 1571 214 l s 1629 230 m 1629 214 l s 1687 230 m 1687 214 l s 1745 247 m
 1745 214 l s 1456 247 m 1456 214 l s 1398 230 m 1398 214 l s 1340 230 m 1340
 214 l s 1282 230 m 1282 214 l s 1224 230 m 1224 214 l s 1745 247 m 1745 214 l
s
 1803 230 m 1803 214 l s 1860 230 m 1860 214 l s 1918 230 m 1918 214 l s 1431
 189 m 1433 190 l 1436 193 l 1436 171 l s 1459 193 m 1448 178 l 1464 178 l s
 1459 193 m 1459 171 l s 1475 193 m 1472 192 l 1470 189 l 1469 184 l 1469 181 l
 1470 175 l 1472 172 l 1475 171 l 1477 171 l 1480 172 l 1482 175 l 1483 181 l
 1483 184 l 1482 189 l 1480 192 l 1477 193 l 1475 193 l cl s 1720 189 m 1722
190
 l 1725 193 l 1725 171 l s 1751 190 m 1750 192 l 1747 193 l 1745 193 l 1742 192
 l 1740 189 l 1739 184 l 1739 178 l 1740 174 l 1742 172 l 1745 171 l 1746 171 l
 1749 172 l 1751 174 l 1752 177 l 1752 178 l 1751 182 l 1749 184 l 1746 185 l
 1745 185 l 1742 184 l 1740 182 l 1739 178 l s 1764 193 m 1761 192 l 1759 189 l
 1758 184 l 1758 181 l 1759 175 l 1761 172 l 1764 171 l 1766 171 l 1769 172 l
 1771 175 l 1772 181 l 1772 184 l 1771 189 l 1769 192 l 1766 193 l 1764 193 l
cl
 s 1761 118 m 1765 118 l 1764 110 l 1762 103 l 1761 98 l s 1779 118 m 1778 114
l
 1777 111 l 1774 107 l 1769 103 l 1765 100 l 1761 98 l s 1785 102 m 1785 103 l
 1786 104 l 1787 105 l 1788 106 l 1791 106 l 1792 105 l 1793 104 l 1794 103 l
 1794 102 l 1793 100 l 1792 98 l 1784 91 l 1795 91 l s 1827 134 m 1824 131 l
 1821 127 l 1818 121 l 1817 114 l 1817 108 l 1818 101 l 1821 95 l 1824 91 l
1827
 88 l s 1857 121 m 1855 124 l 1853 127 l 1850 128 l 1844 128 l 1841 127 l 1838
 124 l 1837 121 l 1835 117 l 1835 110 l 1837 105 l 1838 103 l 1841 100 l 1844
98
 l 1850 98 l 1853 100 l 1855 103 l 1857 105 l 1857 110 l s 1850 110 m 1857 110
l
 s 1865 110 m 1883 110 l 1883 113 l 1881 115 l 1880 117 l 1877 118 l 1873 118 l
 1870 117 l 1867 114 l 1865 110 l 1865 107 l 1867 103 l 1870 100 l 1873 98 l
 1877 98 l 1880 100 l 1883 103 l s 1888 128 m 1900 98 l s 1911 128 m 1900 98 l
s
 1917 134 m 1920 131 l 1923 127 l 1926 121 l 1927 114 l 1927 108 l 1926 101 l
 1923 95 l 1920 91 l 1917 88 l s 1563 430 m 1563 398 l s 1563 413 m 1568 417 l
 1571 419 l 1575 419 l 1578 417 l 1580 413 l 1580 398 l s 1591 400 m 1587 384 l
 s 1591 397 m 1590 393 l 1590 391 l 1591 389 l 1593 389 l 1594 390 l 1596 392 l
 1597 395 l s 1599 400 m 1597 395 l 1597 392 l 1597 390 l 1597 389 l 1599 389 l
 1601 391 l 1601 393 l s
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