Flat directions, doublet-triplet splitting, the monopole problem, and all that

We discuss a supersymmetric SU(6) grand unified theory with the GUT flat direction being lifted by soft supersymmetry breaking, and the doublet-triplet splitting being achieved with Higgs as a pseudo-Goldstone boson. The theory offers a simple solution to the false vacuum and monopole problems.


I. INTRODUCTION
The doublet-triplet splitting (D-T) problem and the origin of the unification scale are the outstanding problems of grand-unification. Both of them seem to cry for low energy supersymmetry which, miraculously enough, leads automatically to the unification of couplings [1][2][3][4] and the dynamical generation of electroweak scale [5].
Among various proposals to understand the lightness of the Higgs doublets, the mechanism that stands out is based on the beautiful idea of Higgs being pseudo-Goldstone boson of some accidental global symmetry [6][7][8][9]. A particular simple realization of this scenario is realized in an SU (6) GUT with an anomalous U(1) A symmetry [10].
On the other hand the most elegant mechanism for generating the GUT scale seems to be based on the idea of flat directions [11], often naturally present in supersymmetric gauge theories. These flat directions are lifted after supersymmetry breaking and their large vevs can be traced to the logarithmic running of coupling constants and masses.
In this paper we show how these appealing scenarios could be married in a realistic grand unified model. Our strategy is the following.
As in [10] we separate various sectors of the theory through U(1) A in order to maintain the lightness of the Higgs doublets. Next, in order to guarantee the existence of flat directions we employ (an) additional global symmetries(y). This is the easiest way to achieve the right pattern of symmetry breaking.
On the other hand it is not so appealing to believe in global symmetries free from 1/M P l suppressed corrections. Thus we make an attempt to avoid completely global symmetries, i.e. to use only the local anomalous U(1) A . While we cannot rigorously derive the correct symmetry breaking pattern in this case, we do believe that this is the most appealing possibility, worth pursuing further in future. If proven correct this would mean a realistic grand unified theory with a natural doublet-triplet splitting and the GUT scale determined dynamically.
All this sounds nice. However, this program from its beginning suffered from a lack of phenomenological predictions and thus it becomes almost a question of semantics and not physics. Fortunately there is a possibility, as we show in this paper, that magnetic monopoles produced in the early universe are detectable in future experiments such as MACRO. The number of monopoles cannot be precisely calculated at this stage, but it could be comparable with the dark matter density of the universe.
Last, but not least, we briefly comment on various other cosmological issues such as the false vacuum, gravitino and moduli problems.

II. A PROTOTYPE MODEL
Before presenting a realistic theory we wish to discuss the generic features of the lifting of flat directions. The simplest GUT example is based on SU(6) gauge symmetry with the adjoint representation Σ and the following superpotential: The absence of the mass term is simply a desire to determine masses dynamically and can be accounted by an appropriate R symmetry. It is clear that the direction is a flat direction since it disappears from the superpotential. It is also clear that this can only work in SU(2n) theories and thus not in SU (5). In this scenario one imagines the soft terms to originate at the Planck scale and to be positive as in the simplest models of supergravity. As in the MSSM the Higgs mass can change the sign [5] and due to the larger number of fields this can now happen close to the GUT scale M GU T of the order 10 16 GeV [12][13][14][15].
To complete the symmetry breaking down to the standard model the minimal set of Higgs scalars is a fundamental (H) and antifundamental (H) representation. This can be achieved by nonrenormalizable terms in the superpotential or through D-terms. The latter case is preferred if one wants to avoid the introduction of arbitrary mass terms. An appealing possibility is to have H =H as a flat direction, but the trouble is the absence of enough running to change the sign of the soft mass terms at high enough scale. The way out is to introduce an extra (anomalous) gauge U(1) A symmetry, under which H andH are charged. A nonzero Fayet-Iliopoulos D-term then forces nonvanishing (and equal) vevs for H andH: where from string theory What about the doublet-triplet splitting? Interestingly enough, it is achieved, but it ends up being a disaster: the SU(2) doublets are superheavy, while the colour triplets are light. Namely, if you do not couple H andH to Σ, the F part of the potential has enlarged global symmetry SU(6) Σ × SU(6) H,H . Let us imagine that Σ first gets a vev, breaking the local SU (6) The above example shows that it seems to be easier to find flat directions than to achieve natural doublet-triplet splitting. Therefore we now focus our attention on the model of D-T splitting which works and look for the implementation of flat directions.

III. A REALISTIC THEORY
What we learned in the previous example is that is not good to break SU(3) down to SU(2) with H andH, since the doublets get eaten and the SU(3) triplets remain light. We need SU(3) triplets to be eaten, and this can happen naturally when SU(4) is broken down to SU(3). In fact this is what Dvali and Pokorski do: they break SU(6) down to SU(4)×SU(2)×U(1) through the vev of Σ. At the next stage H andH break SU(4), which as we said, makes the SU(3) triplets eaten and allows for the doublets to be light. A simple counting of Goldstone bosons demonstrates that the doublets are really light.
Of course, the order of symmetry breaking is irrelevant for the above arguments; if anything in supersymmetry one prefers to go through the SU(5) stage, i.e. to have first H andH develop vevs (or simultaneously with Σ).
It is easy to achieve the desired symmetry breaking [10]; it is enough to choose the complete superpotential for Σ: One of the degenerate vacua is then The question of course is how to make it flat. The simplest possibility is to promote m into a dynamical variable, i.e. a singlet field S. The trouble is that F S = 0 will make σ vanish. Of course one can add a cubic self-interaction for S, but the equations F Σ = F S = 0 over determine the system, forcing again the vevs to vanish. Notice that we are not allowed to introduce quadratic terms with our philosophy of generating masses dynamically.
We see then that unfortunately the prize for achieving both the flatness and D-T splitting is to double the number of adjoints. Regarding the flat directions the situation here mimics the one encountered in SU(5) [14]. It is the D-T splitting problem that points to the elegant solution which requires SU(6) symmetry.

A. The model
A simple model that implements our program requires two adjoint superfields A, B and two singlet ones S, S ′ with the following renormalizable superpotential A physical minimum of the potential is given by with < S > and < S ′ > undetermined. The global symmetries of this superpotential are a U(1) R-symmetry and a U(1) global symmetry with charges (1, 1, 1, 1) and (1, −2, 1, 4), respectively, for (A, B, S, S ′ ), which forbids all other terms to all orders in 1/M P l .
One is clearly tempted to get rid of one of the singlet superfields, for example S ′ . This is readily achieved with λ S ′ = 0. This is a disaster for gauge coupling unification since both (4,2) and (4,2) multiplets under SU(4)×SU(2) subgroup of SU(6) from B would remain light. Of course you could add a term such as T rAB 2 , but then no symmetry could forbid the T rA 3 term, which spoils flatness.
Strictly speaking one can do without a U(1) R-symmetry. The reason is that only B field carries a negative U(1) global charge and thus the nonrenormalizable terms will involve at least two powers of B. The vev of B, as is readily seen, still remains zero and the flatness is not spoiled.
The Higgs as a pseudogoldstone boson program requires, as we mentioned before and as is well known, a separation in the superpotential of various sectors of the theory (for a systematic and careful study of this issue as a perturbation in powers of 1/M P l see for example [16]). In particular H andH must decouple from A, B, S and S ′ . This can be achieved simply by giving nonzero and not opposite U(1) A charges only to H andH as in [10].

B. Fixing the scales
A few words are in order regarding the determination of < A > which defines the GUT scale. Since the couplings λ A,S,S ′ are not known, the GUT scale cannot be determined from the first principles. However, since the number of fields in A and B is large compared to the situation in the MSSM, it is not surprising, that the running from M P l down may be speeded up enough in order to flip the sign of the soft mass of the flat direction around the GUT scale. Furthermore, A is also coupled to matter fields [17] and this can only help. For more details on similar models see [12][13][14][15].
In summary, it appears, at least in the high energy sector of the theory, that everything works. The important ingredient though was at least one continuous global symmetry. Strictly speaking this is OK since we do not know the fate of global symmetries in the presence of quantum gravity. It is often suspected that only gauge symmetries are protected from gravitatonal, i.e. 1/M P l -like effects. If one took this seriously it would be impossible to speak of Peccei-Quinn symmetry and the axion solution to the strong CP problem [18][19][20].

C. No global symmetries?
Still, it would be reassuring to be able to get us rid of continuous global symmetries. It would also be much more elegant and physical to do so. The simplest and most appealing possibility is to use only the gauge (anomalous) U(1) A . Actually, this could work in principle. Namely, in this case the U(1) A charges of (A, B, S, S ′ ) would be (1,-2,1,4) instead of zero, and the H,H charges should be large enough and positive. As before, the fact that only B has negative charge guarantees that the mixing between the two sectors involves more powers of B. For example, if the charges of (H,H) are (2,2), the lowest order mixing would be proportional to B 2 HH, which is not harmful, since the vev of B vanishes.
There is however a new potential problem. In the original version, since only H andH has nonvanishing U(1) A charges and since the SU(6) D-term has to be zero before supersymmetry breaking, both of these fields are forced to have nonvanishing and equal vevs (see (3-(4)). Now on the other hand the U(1) A D-term takes the form (10) and the issue who and when gets a vev becomes somewhat tricky. In order to answer this question the RG improved effective potential should be calculated using the running from M P l to M GU T . This is a difficult task beyond the scope of this paper.

D. The matter sector
The theory can be made realistic with the proper inclusion of light matter superfields. A realistic theory can be shown to require three families of 15 f ,6 f and6 ′ f (a minimal anomaly free set). Also, one needs a self-conjugate 20 of SU(6) in order to get a large top Yukawa coupling. This is discussed at length in [17,21]. A particular attention must be paid to neutrino mass as in general SU(6) models. Fortunately one has more than one option at disposal. Right-handed neutrinos can be the SU(5) singlet components of the6 and6 ′ matter fields as for example in [10] or additional SU(6) singlets as in [21]. In both cases one ends up with the usual mechanism for generation of small neutrino masses [22].

IV. COSMOLOGICAL ISSUES: THE MONOPOLE PROBLEM AND THE PROBLEM OF THE FALSE VACUUM
Besides the well known monopole problem, SUSY GUTs are also plagued by the problem of the false vacuum. Namely, normally one gets a set of degenerate vacua which includes the unbroken one. At sufficiently high temperature the unbroken vacuum becomes the global minimum and the large barrier between the vacua prevents the tunneling to our world [23].
The theories with flat directions offer a natural solution to both of these problems. First, the monopole problem. The point is remarkably simple [24][25][26][27]: the critical temperature of the GUT phase transition becomes very small and the usual Kibble [28] mechanism production gets suppressed. On top of that, the phase transition is of the first order and the number of monopoles can get suppressed. For a small flat direction σ (<< T ) the one-loop high temperature correction to the effective potential is where N is proportional to the degrees of freedom to which σ is coupled and α is positive.
In the opposite limit, when σ >> T , ∆V T ≈ exp(−c|σ|/T ) (c > 0), i.e. in this limit σ is coupled only to superheavy fields (>> T ) and is out of thermal equilibrium. Thus, for sufficiently high T the σ = 0 minimum wins and the symmetry is restored just as in the case with no flat directions [29][30][31]. Since the energy difference between the σ = 0 and σ = M GU T vacua is only of order m 2 3/2 M 2 GU T , it is clear that the transition can take place not before the temperature drops down to at least T c ≈ (m 3/2 M GU T ) 1/2 ≈ 10 9 − 10 10 GeV. If the phase transition was of the second order, the ratio between the energy of monopoles and baryons today would be approximately for the GUT monopoles with a mass of the order 10 17 GeV. Clearly, even if this was true, the number of monopoles would be small enough not to be in conflict with cosmology. At first glance, the usual curse of grandunification would be turned into the blessing: monopoles could be the dark matter of the universe. Even more important, this is not far from the MACRO limit [32] and the old dream of detecting magnetic monopoles could be relized in not so far future. In our case the phase transition is of first order and the monopole production could be suppressed, although not necessarily (see for example [33][34][35]).
Of course, all this is relevant if we do manage to tunnel into our world. In a sense, we are saying that the solution to the false vacuum problem automatically resolves the monopole problem. The quasi flat direction may imply no barrier at all and so no problem whatsoever. However, this is in principle model dependent. Also, it is conceivable that the production of monopoles happens only after the false vacuum stops being a local minimum, i.e. for T ≈ m 3/2 . Obviously, the number of monopoles could then be completely negligible, similar to the case of inflation 1 . A more careful study of these issues is on its way.

V. SUMMARY AND OUTLOOK
In short, the SU(6) × U(1) A theory discussed here achieves the determination of the GUT scale through the lifting of the flat direction after supersymmetry breaking. Also, it allows for a simple solution of the doublet-triplet splitting problem with the Higgs being a pseudo-Goldstone boson of an accidental global symmetry.
Furthermore, independently of when the inflation takes place and what the reheating temperature is, the theory is free from the monopole and false vacuum problems. Of course, we believe that inflation did take place at some point for the usual reasons of horizon and flatness problems. By this we mean the usual inflationary scenario of at least 60 e-folds, but now with a reheating temperature higher than M GU T .
For this reason one must face the gravitino problem. This is however easily solved by assuming a short inflation before the first order GUT phase transition discussed throughout this paper. It is enough to wash out the gravitinos thermally produced before it: the reaheating temperature will be smaller or at least equal to the critical temperature T C ≈ 10 9 GeV, which is safe.
The main physical implication of flat directions is the existence of moduli-like fields with masses of order m 3/2 and 1/M P l suppressed interactions. It is well known (for the original work see [36]) that it poses a serious cosmological problem. It can be solved with a short inflation (this time after the phase transition) as suggested in [37][38][39].
To be honest, both problems could be more severe through the non-thermal production of relics, as emphasized [40,41]. If so, one would need a low scale inflation at a later stage. The issue however is very subtle and recently an opposite point of view was raised [42], according to which the non-thermal production is suppressed in realistic models.
The reader may feel uneasy about this multi-inflation scenario. We do not believe one should worry about it, since inflation is a natural scenario and often it is more of a problem to get out of it [43] than to experience it. In our scenario there is an award of having a possibility of detecting magnetic monopoles. This is the major point of our work. Magnetic monopoles as much as proton decay, if not more, provide a test of the idea of grand unification. After all, these are the only generic properties of GUTs. Of course, the usual inflation with low reheating temperature solves the monopole problem, but at the tragic prize of implying no monopoles left in the whole universe.