{'fraction_non_alphanumeric': 0.029614616734224695, 'fraction_numerical': 0.010920200859096134, 'mean_word_length': 4.508915180803199, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Neutrinos, their Partners and Unification

Efforts to unify group-theoretically the standard-model gauge interactions with the generation structure of fermions and their mirror partners should be accompanied with the unification of the corresponding gauge couplings. In this paper the possibility of such a unification is studied and conclusions on possible symmetry-breaking channels and scales, as well as on the fermion content of the theory, are drawn. The breaking of some of the symmetries allows various Majorana masses for neutrinos and their mirror partners, so these are studied next. Implications to neutrino mixings and mass hierarchies in connection with recent experimental results, as well as to electroweak precision tests, are then discussed.

The quantum numbers of the known fermions under the standard-model gauge structure allow their partial classification under the fundamental representations of the corresponding symmetry groups. This motivates efforts to complete this classification by studying unifying symmetry groups large enough to accommodate all the fundamental-particle generations which have been observed so far. Apart from the purely theoretical interest in such a possibility, quests for such a unification usually lead to predictions on the existence of new particles, like extra fermions and gauge bosons [1]- [3]. In particular, the new fermions are usually referred to as the "mirror partners" of the standard-model fermions.
Since there are currently several theoretical and possibly also some experimental indications hinting for the existence of physics beyond the standard model at scales on the order of 1 TeV [1], it is worthwhile investigating whether extensions of particle theories in a direction compatible with generation unification could be related to these indirect indications at TeV-energy scales. Furthermore, in view of the fact that accelerators designed to operate in the next decade plan to cover such high energies, it is quite important to investigate their discovery potential by producing directly the particles predicted by the aforementioned extensions.
Before embarking in such a detailed production-and-decay study however, one should first check the internal consistency of the proposed theories and their compatibility with current experimental constrains. A first effort to reproduce the observed charged-fermion mass hierarchy, the quark mixing matrix elements and the weak scale while staying in agreement with the electro-weak precision data within such a framework was recently presented [1]. The purpose of the present work is to tackle some related, equally important open issues.
One of these issues is to calculate the evolution of the gauge couplings to very high energies, in order to see if there is a sequence of symmetry breakings consistent with the unification picture which motivated the proposed extension in the first place. In all cases discussed, the symmetries in question are taken to break spontaneously, and getting into the details of the breaking mechanism, like it being of dynamical or fundamental nature and the transformation properties of its non-zero vacuum-expectation-value, is avoided because this usually involves a high degree of arbitrariness and speculation in an area of no phenomenological input.
Since the energy scales of these breakings could be associated with the lightness of the standard-model neutrinos via the see-saw mechanism, the question of neutrino masses and mixings left open in [1] has to be studied next. This also allows the calculation of novel "oblique" contributions to the electro-weak parameters due to the possible Majorana nature of the mirror neutrinos. An effort to address these different but closely related issues follows next.

Coupling unification 2.1 Preliminary considerations
The starting point of the discussion could be either of the unification gauge groups E 8 1 × E 8 2 or SO(16) 1 × SO(16) 2 without change in the final results, with gauge couplings g 1 and g 2 corresponding to the groups with subscripts 1 and 2 respectively. These symmetries are taken to break at the unification scale Λ GU T down to SO(10) 1 × SU (4) 1G × SO(10) 2 × SU (4) 2G . The fermions and mirror fermions of interest transform under the above groups like (16,4,1,1) and (1,1,16,4) respectively.
It is imagined next that around the same unification scale, SO(10) 1 ×SO(10) 2 breaks to its diagonal subgroup SO(10) D , which has accordingly a gauge coupling at that scale equal to g = g 1 g 2 (1) This in its turn is taken to break again at Λ GU T to its maximal subgroup SU (4) P S × SU (2) L × SU (2) R . In addition, SU (4) 1G × SU (4) 2G could be taken to break down at the same scale to either of the three groups SU (3) 2G , SU (3) 2G × U (1) G , or SU (4) 2G . Under the resulting group SU (4) P S × SU (2) L × SU (2) R × SU (3) 2G for instance, the standard-model-type fermions transform like four copies ("generations") of (4, 2, 1, 1) and (4, 1, 2, 1), and the mirror fermions like (4, 1, 2, 3) + (4, 1, 2, 1) and (4, 2, 1, 3) + (4, 2, 1, 1). The two other possibilities will be discussed in subsection 2.4. In all these cases it is assumed that the fourth generation standard-model-type fermions pair-up with their mirror partners, acquiring gaugeinvariant masses on the order of the SU (4) 2G breaking scale. (This is Λ GU T for the first two cases and about 1 TeV for the third case, as will be seen in subsection 2.4.) One first notes that the only way to unify the generation-group coupling with the other gauge couplings is to satisfy the relation g 1 ≫ g 2 at Λ GU T , because then the common unification coupling is g ≈ g 2 according to Eq. 1. Therefore, the generation group SU (4) 1G , which is taken to break completely at Λ GU T , is strongly coupled at that scale. The situation with generation group SU (3) 2G with coupling g 2 is first investigated. The basis of the analysis of the gauge-coupling renormalization that follows is more of a qualitative nature and limited to the oneloop β-function, due to the many uncertainties of the dynamics influencing the running of these couplings. These are mainly due not only to our ignorance of the exact masses of the mirror fermions and of the type of new physics needed to break the gauge groups involved in this picture, which could be Higgs particles in various presently unpredictable representations, but also to the possible existence of supersymmetric partners of the standard-model fermions and to threshold effects near the unification scale.
These uncertainties lead one to take all the mirror fermions to have the same mass Λ M at around 1 TeV for simplicity, since the coupling unification is found to be anyway quite insensitive to this scale. Below Λ M the couplings evolve like in the standard model. Above that scale, one has to take the mirror fermions into account.
It is then assumed that there exists a "desert" between Λ M and the Pati-Salam scale Λ P S where SU (4) P S is broken, with no new dynamics or particles able to influence the evolution of the gauge couplings with energy.
The β-function describing the evolution of the gauge coupling g of an SU (N ) group with N f fermion N -plets with respect to momentum p is given by where r stands for higher-than-one-loop corrections and p 0 is some reference scale. If the same fermions transform also under the fundamental representation of another unitary gauge group SU (N ′ ) with coupling g ′ much larger than g, the quantity r at two loops is approximately given by [4] Therefore, when the SU (3) 2G interactions become strong at around 2 TeV and break SU (2) L × U (1) Y dynamically by an effective Higgs mechanism induced by fermion condensates [1], the corresponding fine-structure-constant is α G ≈ 1.
One therefore gets r ≈ 0.3, which is still much smaller than the one-loop contribution to the other couplings, even for the smaller groups considered like SU (2) L or U (1) Y (the influence of the other couplings to each other is of course even more negligible due to their smallness). In addition, since SU (3) 2G is taken to break just after it becomes strong [1], it has a rather limited energy region where it can influence substantially the β functions of interest, so large deviations from the one-loop renormalization of the rest of the gauge couplings are not expected. This issue is investigated further by presenting a particular example in subsection 2.3.
Moreover, a fundamental Higgs mechanism for breaking SU (3) 2G is avoided by evoking the mechanism conjectured in the Appendix of [1]. In any case, a minimal fundamental Higgs mechanism breaking the generation symmetry, apart from all the naturalness problems it carries with it, would make the corresponding gauge coupling run slightly slower. The generation-coupling unification with the rest of the gauge couplings at Λ GU T would then still be achievable by slightly lowering the maximal value this coupling reaches before SU (3) 2G breaks and/or lowering the mirror-mass scale Λ M . An effort to estimate the energy scales entering this problem without fundamental scalars is presented in the next subsection.
However simple, the approach adopted allows to draw general conclusions, not depending on particular details, on the way the unification groups break down to the standard-model gauge structure. It has to be stressed nevertheless that the class of symmetry-breaking channels of interest here has an additional degree of freedom compared to the usual or to the supersymmetric unification, and that is the Pati-Salam symmetry breaking scale Λ P S . This can be in most cases slightly adjusted to allow unification of couplings even after the correct inclusion of these corrections, unless one introduces unnaturally large Higgs sectors to break the gauge symmetries. The results that follow should therefore be seen not as exact predictions but rather as order-of-magnitude estimates.

Calculation of Λ GU T and proton life-time
The analysis presented here is based on different alternative breakings of the gauge symmetry SU (4) P S , since a priori there is no obvious reason to expect a specific breaking channel. The subsequent analysis will show that only one alternative seems to be viable if one takes proton-lifetime bounds and the order of magnitude of the weak scale into consideration. It is particularly interesting therefore to note that, under certain assumptions, current phenomenological input is able to constrain the number of different group-breaking channels, even when these appear at scales much higher than the ones directly accessible at present.
In particular, the Pati-Salam group is taken to break at the scale Λ P S either along the channel or along the channel if the breaking SU (2) R −→ U (1) R has already occurred at Λ GU T . A third possibility is also examined, namely one of a Pati-Salam symmetry breaking like at Λ GU T , which is followed by the breaking of SU In all these alternative scenarios, the Pati-Salam 4-plets are each broken to a QCD triplet and a lepton, while simultaneously giving rise to a "predecessor" of the electromagnetic charge.
It is also noted that, as a first approximation, below the mirror-mass thresh- It is more convenient to work in the following with the inverse structure constants α −1 = 4π/g 2 since their evolution is linear with ln (p/p 0 ). The value of the hypercharge coupling α Y (Λ P S ) in the first two cases is computed via the relation which is evaluated at the Pati-Salam scale Λ P S , where α R is the coupling corresponding to SU (2) R or U (1) R respectively. In the third case the hypercharge coupling is given by the relation Furthermore, the first and third cases are based on the working assumption of unbroken discrete left-right symmetry above the scale where SU (2) R is broken, i.e.
α R = α L . As was said in the introduction, a discussion on the possible breaking mechanisms of these symmetries is here avoided, since the purpose of the analysis is to allow general qualitative conclusions to be drawn.
The starting point of the calculation is based on the approximate experimental values for these quantities listed below [5] Moreover, at scale Λ M the SU (3) 2G coupling is taken to be equal to α 2G (Λ M ) = 1.
This coupling is not plotted for α 2G > 1, because then higher-order corrections to the renormalization of this coupling become important. This is expected to have a limited effect to the other couplings however, since the generation group breaks at around the same scale. This issue is examined again later. for which unification is possible, along with the value of the unification coupling are also given in Table 1.
To begin with, we discuss the first two cases. The energy scales of interest are found to be Λ M = 10 2.75 GeV and Λ P S = 10 13.65 GeV in the first case, and Λ M = of the hypercharge-coupling running at that scale, since at scales higher than Λ P S , The inclusion of a minimal Higgs field able to break these symmetries spon- The third alternative breaking sequence is as already said to break the Pati-

Symmetry breaking sequence
Energy scales (GeV)  Table 1: The energy scales required to achieve unification assuming three different symmetry-breaking channels, and the corresponding value of the unification coupling. The second channel is disfavored because of the low unification scale, and the third channel is also disfavored because of the large mirror-fermion masses implied by Λ M .
Salam group at the unification group like SU (4 This possibility is drawn in Figure 3. The relevant scales are found to be Λ M = 10 4 GeV, Λ R = 10 10.05 GeV and Λ GU T = 10 17.3 GeV. The scale Λ R with the present fermion content is quite large (compare for instance solutions with alternative fermion contents [7]), supporting a see-saw mechanism for the neutrino masses. The unification scale is in this case quite large, a result reminiscent of [8], and the common coupling at that scale is α GU T = 0.034. The main reason for the largeness of Λ M is the effort to unify the generation-group gauge coupling with the other couplings.
If this unification condition is relaxed, in the same way it is relaxed in connection with a SU (4) 2G generation group that is discussed in the next subsection, the rest of the couplings can be unified with a smaller Λ M and this channel is still viable. In the present case nonetheless, the largeness of Λ M used would correspond to an unacceptably large weak scale. The fact that SU (2) R breaks far away from Λ GU T would also render effects coming from the mechanism to which its breaking is due and which are here neglected, like the existence of scalar particles, more important. Such effects however are not expected to alleviate the problem of the large scale Λ M . We can therefore conclude here that the third breaking channel is improbable, unless generation-coupling unification with the rest of the couplings is abandoned. Another way to keep Λ M small would be of course to add a large Higgs sector transforming non-trivially under SU (3) 2G , but this alternative is not investigated since it is foreign to the present conceptual framework and would raise naturalness problems.
Gauge-coupling unification in connection with bounds on the proton life-time is discussed next, since the breaking of SO(10) D at Λ GU T can induce proton decay via effective four-fermion operators. This issue could actually help us decide between the first two breaking channels proposed. From the proton life-time experimental constraint [5] τ (p → e + π 0 ) > 5.5 × 10 32 yr (10) and the theoretical order-of-magnitude estimate one gets the inequality This proton life-time bound makes clear that the second breaking channel possibility is disfavored due to a small unification scale. Nevertheless, it cannot be at this point definitely excluded due to the limited level of accuracy of the current rather qualitative analysis. Note that this result is reminiscent of the result of [8] in an analysis with the same breaking sequence in a left-right symmetric context but without mirror fermions. One is consequently left with the first alternative as the one corresponding to the most probable symmetry breaking sequence.

The effect of strong dynamics
Strong dynamics can alter the results quoted above, since higher-order corrections to the various β-functions due to the strong SU (3) 2G interactions could become important if the quantity r introduced before is not negligible. However, as also noted in [9], the fermion content of the theory implies that this effect, however large, would be uniform for all standard-model couplings, as is shown in Figure 4 for an exaggerated effect corresponding to r = 40. From Eq.3, this would correspond to a highly-non-perturbative generation coupling α G ≈ 126 (this number is of course purely indicative, since the perturbative β-function has no meaning in this regime), something which has the same influence on each of the other relatively weak SU (N ) couplings as the introduction of 20 new fermion N -plets.
Such strong dynamics can shift the unification coupling α GU T to larger values, but cannot shift the unification scale. In reality the coupling-evolution curves shown should be smooth, without angles, but r is here taken to become suddenly important for illustration purposes, in a -perhaps overambitious -effort to simulate the relevant effect. There is no guarantee of course that the non-perturbative effects of SU (3) 2G can be limited even by such large r-values, but it is assumed that they are in order to maintain the conclusions presented in this work. of the mirror fermions.

Mirror-generation groups other than SU(3) 2G
The issue of the generation groups comes next. Even if there is an abelian generation group U (1) G surviving down to TeV-scales along with SU (3) 2G , unification requires that its coupling is negligibly small at low energies, so its running is neglected and its evolution with energy not plotted. Note however that if one wants the see-saw mechanism to work for the standard-model neutrinos, as will be seen in the next section, this symmetry would have to be broken at very large energy scales to allow for ultra-light neutrinos. In the first case considered for instance, one could think of a breaking channel involving These produce after diagonalization the following lepton and mirror-lepton mass For the neutrinos the situation is more complicated, since they could be of a Majorana nature. The neutrino 12 × 12 mass matrix M N is introduced next, and it is taken to have the following form: Since the present model does not predict the existence of a light sterile neutrino, contrary to other "mirror" models [11], attention is restricted to current experimental data regarding solar and atmospheric neutrino anomalies which imply differences of masses-squared ∆m 2 ij ≡ m 2 i −m 2 j , with i, j = 1, 2, 3, and mixing angles θ between only three mass eigenstates m 1,2,3 of standard-model left-handed neutrinos, if one accepts the view that they are due to quantum-mechanically coherent oscillations between different neutrino-flavour eigenstates. Assuming for instance that the small-angle MSW solution to the solar-neutrino deficit involves the lefthanded electron and muon standard-model neutrinos ν e and ν µ , one gets experimental bounds which according to [10] are given by with a mixing angle sin θ sun ≈ 0.03 − 0.05.
Further information on neutrino masses and mixing coming from the atmospheric neutrino anomaly assuming it involves the left-handed standard-model neutrinos ν τ and ν µ , gives the bounds with a mixing angle sin θ atm ≈ 0.49 − 0.71 associated with it [10]. It has to be noted that this ν µ − ν τ mixing is unusually large compared with charged fermion mixings observed so far. This experimental input constitutes the basis which determines the form of the neutrino submatrices given below.

Two numerical examples of mass matrices
The working assumption is made next that the SU ( equal to 0.04 and 0.53 respectively, which is compatible with experiment [10].
Whereas the form of m B is consistent with the corresponding mirror chargedlepton matrix M L , the matrix m I is slightly problematic, since it has a gaugeinvariant mass termν µ L ν M µ R for the second generations which is larger than the correspondingc L c M R quark-mass term in [1]. Qualitatively one would generally expect such terms involving quarks to be larger than the corresponding lepton ones. Although this is generally expected in analogy with the standard-model case and based to possibly QCD-related contributions to particle masses, the lack of a definite calculational scheme for these gauge-invariant masses pose limits to such arguments. There are nevertheless several solutions to this naturalness issue, and these are presented below.
One is to consider lighter mirror leptons, which would then allow for smaller entries in m I . This by itself would however not be enough to remove this discrepancy without exceeding the lower mass bounds on direct production of new weak-doublet fermions set by the LEP experiments. Another solution is to consider heavier mirror quarks and charged leptons, which would then require larger entries in the corresponding gauge-invariant quark-mass submatrices [1]. This solution would also help indirectly to reduce the problems with the electroweak precision tests, as is shown in the next section.
A third alternative solution to this naturalness question is here investigated, i.e. smaller Majorana masses for the fields ν R . This is not such a severe assumption, It is worth noting here that, in order to get the observed fermion mass hierarchies and mixings [1], one is lead to consider gauge-invariant fermion mass terms ψ M L ψ R , denoted by m I ψ for ψ = t, c, b, s, τ, µ, ν τ , ν µ , exhibiting the hierarchy It therefore seems that the more gauge interactions a fermion flavour has, the larger the gauge-invariant-mass splitting between its third-and second-generation representatives, even when these gauge interactions are relatively weak compared to the SU (3) 2G one. This could be an indication of near-critical four-fermion interactions in the sense explained in [12] and which are contained in the scenario presented in [1]. The large neutrino mixing suggested by the SuperKamiokande data is therefore consistent with the existence of such type of interactions, as made clear by the generic pattern noted in Eq. 21.

The mixing parameters
The symmetric charged-and neutral-lepton mass matrices used above are diagonalized by the 6 × 6 and 12 × 12 matrices which we denote by K i , i = L, N respectively, via the relations where the J i denote diagonal matrices. The lepton mixing information is therefore contained in a 6 × 12 matrix U defined by the relation where m-summation is implied, with l, m = e, µ, τ, e M , µ M , τ M , with ν m running only over the SU (2) L -doublet neutrino flavours, and j = 1, ..., 12. Note that, following the convention of [10] and contrary to the neutrino case, we keep flavour indices for the charged-lepton mass eigenstates due to the assumed small mixing between them.
It is therefore clear that the three ordinary-neutrino and three mirror-neutrino flavor eigenstates ν l which are weak doublets are given in terms of the twelve neutrino mass eigenstates ν j via the relation ν l = U lj ν j , with j-summation implied.
The first three neutrino mass eigenstates are light enough to allow their superposition to be considered as coherent. Furthermore, since the matrix M L is almost diagonal, K L is close to the unit matrix and the form of U is mostly affected by Experimentally one has presently information only on some of the elements Its form is quasi-symmetric as expected by the form of the mass matrices assumed.
This is consistent with the matrix given in [10] for the small-angle MSW solution to the solar-neutrino deficit, even though in the present case U SM is not rigorously unitary because of the existence of mirror leptons which slightly mix with the ordinary ones. Moreover, it is observed that the smallness of the element U SM e3 justifies in the current example the assumption that the two oscillations are practically decoupled [10].
Larger non-diagonal entries in the matrix m I can further increase the entries (2,3) and (3,2) and the corresponding ν µ − ν τ mixing. A similar analysis therefore could also be easily performed for the large mixing-angle MSW and the vacuum- having a charged lepton of mass m l is given by [13] S 0 where This result is identical to the one given in [14] in this mass-limit only if the quantities c θ and s θ are correctly defined as In the above, corrections due to the fact that one mirror neutrino is not much heavier than the Z boson are neglected, since the purpose of this example is to just illustrate an effect which depends only on mass ratios and not on independent masses, and since one has poor knowledge of the overall mirror fermion mass normalization anyway. Note moreover that, contrary to [13]- [14], the mirror neutrino masses m a,b do not correspond to pure weak eigenstates due to mixing with ordinary neutrinos. This mixing is however small due to the relative smallness of the elements of m I compared to the m B entries, and its effects are therefore also neglected.
As regards the ∆ρ parameter which measures the isospin breaking in the new sector, it is shown in [1] that there exists no problem rendering it small enough to fit experiment, even though some fine tuning might be needed. Since whatever leptonic contributions due to Majorana mirror neutrinos as described in [13]- [14] can be compensated by a corresponding shift to the up-down mass splitting of the mirror fermions, there is no use of further discussing it in the present context when the precise mirror-fermion mass spectrum remains experimentally unknown.
The total oblique correction S 0 to S in this model assuming QCD-like dynamics is the sum of the contributions S 0 q and S 0 l coming from mirror quarks and leptons respectively, i.e. [1] S 0 = S 0 q + S 0 l = 0.9 + 0.3. After analyzing the above formula for S, it is concluded that contributions to S l are not very sensitive to m b , but depend drastically on m a /m l . Heavier mirror leptons would produce an even smaller S parameter, but assuming that the mirror quarks are at least as heavy as them would render difficult the correct reproduction of the weak scale after a certain point. It is nevertheless clear that a larger m a /m l hierarchy could facilitate the reproduction of a small or even negative S parameter in accordance with experiment. This could be achieved now with the assistance of vertex corrections and non-QCD-like dynamics in these models as described in [1] without having to introduce very large top-quark anomalous couplings. Such a situation would also reduce the fine-tuning needed to keep the ∆ρ parameter small.

Conclusions
Mirror fermions near the weak scale offer rich possibilities for the study of new physics. However, the absence of direct experimental evidence on the existence of mirror partners to the standard-model fermions lead to the present qualitative study of various unification possibilities and related neutrino physics without a prior knowledge of the exact mirror mass hierarchies. This did not prevent however very useful general conclusions pertinent to such types of models to be drawn.
There are two basic results to be kept in mind. One is that unification of all the gauge couplings, including the generation group coupling, is possible within this group-theoretical context and consistent not only with the weak scale but also with current bounds on the proton life-time. The other is that neutrino masses and mixings consistent with the observed solar and atmospheric neutrino anomalies are naturally achieved.
In particular, it is made clear that with the proposed fermion content extension, not only SU (5) unification is disfavored, but also unification with an SU (3) 2G generation group is possible. Within the group-theoretic framework chosen, this is possible by taking the gauge coupling (g 1 ) of one sector to be much larger than the other (g 2 ) at the unification scale. This unification is a priori not at all obvious, and constitutes a highly non-trivial result within the context of dynamical symmetry breaking theories. There exist nevertheless no direct indications that the scales Λ M and Λ P S assume indeed the exact values needed for this to happen, and no guarantee that this coupling crossing is not just a coincidence with no particular importance for the embedding of the standard-model gauge structure.
Moreover, the existence of "mirror fermions" which are weak singlets is also not favored. This is a clear manifestation of the "sinθ W " problem in [6], [16] which does not appear when weak-doublet mirrors of the type introduced in The unification investigation conducted makes also apparent a problem in this theory having to do with the generation of lepton masses. In particular, the Pati-Salam scale Λ P S is found to be too large to allow quark masses to be fed down to leptons via effective four-fermion operators associated to the SU (4) P S breaking.
If one does not want to use a fundamental Higgs mechanism to break the generation group at the TeV-scale, a solution to this problem would be a strong U (1) G at the TeV scale [1]. To avoid a Landau pole to the corresponding gauge coupling, the group U (1) G has to be soon embedded into a larger non-abelian group, like SU (4) 2G .
However, if one insists on unifying the generation coupling with the rest of the gauge couplings the solution above is unfortunately not viable, since as was noted in subsection 2.4 the SU (4) 2G coupling runs to fast to unify with the other gauge couplings. The group SU (4) 2G has therefore to be broken at the unification scale and the U (1) G coupling at low energies is consequently very weak. A way out for lepton-mass generation could in principle be the existence of gauge-invariant operators generated beyond tree-level which feed down quark masses to leptons.
As already stressed, in the unification analysis presented several effects are neglected. These are related to unification threshold effects, to the Higgs content needed to break the SU (4) P S and SU (2) R symmetries, to 2-and higher-loop contributions to the β functions, to the fact that the mirror fermions are taken to be all degenerate in mass, and to the fact that SU (3) 2G becomes strongly coupled at around 1 TeV, something that could influence the rest of the couplings. It is not expected however that these effects can spoil the qualitative results of the analysis above. Unification could still be achieved if these effects were correctly taken into account, since one has the freedom to adjust the scales Λ M and Λ P S without influencing considerably the unification scale Λ GU T . This is particularly true for the favored possibility presented in Figure 1, since the proximity of the scales Λ P S and Λ GU T does not leave room for large adjustments.
One could of course claim that the freedom to adjust Λ P S in order to achieve unification makes this exercise easier to complete and reduces the predictability of the theory by adding an extra free parameter. On the other hand, the most favored scenario described connects this scale with the breaking not only of SU (4) P S , but also of the SU (2) R symmetry. The examples involving Majorana neutrinos which are presented above indicate however that this scale is expected to be several orders of magnitude smaller than the unification scale. This not only speaks against a "desert" reaching up to Λ GU T , but is also consistent with the scenario analyzed here.
As already noted, the unification considerations above indicate a favored SU (2) R breaking scale usually associated with the mass of heavy Majorana neutri-nos in the context of the see-saw mechanism. This leads to the study of neutrino masses and mixings in this framework and in connection with recent experimental results. It is found that, in order to have the neutrino masses and mixings compatible with experiment and unification, heavier charged mirror fermions than the ones quoted in [1] might be needed, unless the heavy standard-model Majorana neutrinos are quite lighter than the scale where SU (2) R breaks. Heavier mirror fermions imply furthermore not only a more difficult detection of their indirect effects, since their mixing with the standard-model fermions is smaller, but also smaller need of fine-tuning of their masses [1]. It is further interesting to show that the above observation is perfectly consistent with a small S parameter which is currently favored by electroweak precision tests. A small S-parameter could furthermore be an indication that the lightest mirror neutrino, i.e. the field denoted as ν M 1R , is so light that it could lie just beyond the reach of present high-energy collider experiments.