SO(10) grand unified theory  is probably the best motivated candidate for the unification of the strong and electro-weak interactions. It unifies the family of fermions; it includes the SU(4) C quark-lepton symmetry  and the left-right (LR) symmetry  in the form of charge conjugation as a finite gauge symmetry; it predicts the existence of right-handed neutrinos and through the see-saw mechanism  offers an appealing explanation for the smallness of neutrino masses. Due to the success of supersymmetric unification, and the use of supersymmetry in controlling the gauge hierarchy, most of the attention in recent years has focused on the supersymmetric version of SO(10). However, supersymmetry may not be there. After all, it controls the Higgs gauge hierarchy, but not the cosmological constant. The long standing failure of understanding the smallness of the cosmological constant suggests that the unwelcome fine-tuning may be necessary..
What about grand unification without supersymmetry? At first glance, one may worry about the unification of gauge couplings in this case. Certainly, in the minimal SU(5) theory, the gauge couplings do not unify without low-energy supersymmetry. What happens is the following: the colour and weak gauge couplings meet at around 1016GeV, an ideal scale from the point of view of the proton stability and perturbativity (i.e., sufficiently below MPlanck). The problem is the U(1) coupling. Without supersymmetry it meets the SU(2) L coupling at around 1013 GeV ; with low-energy supersymmetry the onestep unification works as is well known .
On the other hand, the fact that neutrinos are massive indicates strongly that SU(5) is not the right grand unified theory: it simply requires too many disconnected parameters in the Yukawa sector [8, 9]. The SO(10) theory is favored by the neutrino oscillation data. Most interestingly, SO(10) needs no supersymmetry for a successful unification of gauge couplings. On the contrary, the failure of ordinary SU(5) tells us that in the absence of supersymmetry there is necessarily an intermediate scale such as the left-right symmetry breaking scale MR. Namely, in this case the SU(2)L and SU(3)C couplings run as in the standard model or with a tiny change depending whether or not there are additional Higgs multiplets at MR (recall that the Higgs contribution to the running is small). However, the U(1) coupling is strongly affected by the embedding in SU(2)R above MR. The large contributions of the right-handed gauge bosons makes the U(1) coupling increase much slower and helps it meet the other two couplings at the same point. The scale MR typically lies between 1010 GeV and 1014 GeV (see for example [10, 11] and references therein), which fits very nicely with the see-saw mechanism. Now, having no supersymmetry implies the loss of a dark matter candidate. One may be even willing to introduce an additional symmetry in order to achieve this. In this case it should be stressed that SO(10) provides a framework for the axionic dark matter: all one needs is a Peccei-Quinn  symmetry U(1)PQ which simultaneously solves the strong CP problem
This seems to us more than sufficient motivation to carefully study ordinary non-supersymmetric SO(10). What is missing in this program is the construction of a well defined predictive theory with the realistic fermionic spectrum. This paper is devoted precisely to this task.
(Submitted on 11 Oct 2005 (v1), last revised 1 Nov 2005 (this version, v2))
Asking burning questions waiting for realistic answers
In particular, the search of the minimal realistic Yukawa sector is a burning question. In the absence of higher dimensional operators at least two Higgs multiplets with the corresponding Yukawa matrices are needed, otherwise there would be no mixings. The Yukawa Higgs sector can contain 10 H , 120 H and 126 H representations, since
16×16 = 10+120+126 . (1)
One version of the theory with only 10H and 126H was studied in great detail in the case of low-energy supersymmetry [13, 14, 15]. In spite of having a small number of parameters it seems to be consistent with all the data [16, 17, 18, 19, 20]. For the type II seesaw it predicts furthermore the 1−3 leptonic mixing angle not far from its experimental limit: |Ue3| > 0.1 [18, 20] and it offers an interesting connection between b−τ unification and the large atmospheric mixing angle [21, 22]...
Thus, a first obvious possibility in ordinary SO(10) is to address the model with 10H + 126H, and to see whether or not it can continue to be realistic...
In this work, we stick to the renormalizable version of the see-saw mechanism (for alternatives using a radiatively-induced see-saw, see), which makes the representation 126H indispensable, since it breaks the SU(2)R group and gives a see-saw neutrino mass. By itself it gives no fermionic mixing, so it does not suffice. The realistic fermionic spectrum requires adding either 10H or 120H...
Before starting out, it is convenient to decompose the Higgs fields under the SU(2)L× SU(2)R× SU(4)C Pati-Salam (PS) group:10 = (2, 2, 1) + (1, 1, 6)
126 = (1, 3, 10) + (3, 1, 10) + (2, 2, 15) + (1, 1, 6)
120 = (1, 3, 6) + (3, 1, 6) + (2, 2, 15) + (2, 2, 1) + (1, 1, 20)
As is well known, the 126H provides mass terms for right-handed and left-handed neutrinos:
MνR = <1, 3, 10>Y126, MνL = <3, 1, 10> Y126
which means that one has both type I and type II seesaw:
MN = −MνD MνR-1MνD + MνL
In the type I case it is the large vev of (1, 3, 10) that provides the masses of right-handed neutrino whereas in the type II case, the left-handed triplet provides directly light neutrino masses through a small vev [26, 27]. The disentangling of the two contributions is in general hard.
With the minimal fine-tuning the light Higgs is in general a mixture of, among others, (2,2,1) of 10H and (2,2,15) of 126H. This happens at least due to the large (1,3,10) vev in the term (126H)2126H†10H. In any case, their mixings require the breaking of SU(4)C symmetry at a scale MP S, and it is thus controlled by the ratio MP S/M, where M corresponds to the mass of the heavy doublets. Thus, if M≃MGUT, and MPS≪ MGUT , this would not work; we come to the conclusion that one needs to tune-down somewhat M...
If the model with real 10H does fail eventually, one could simply complexify it. This of course introduces new Yukawa couplings which makes the theory less predictive. Certain predictions may remain, though, such as the automatic connection between b−τ unification and large atmospheric mixing angle in the type II seesaw. This is true independently of the number of 10 dimensional Higgs representations, since 10H cannot distinguish down quarks from charged leptons... It is a simple exercise to establish the above mentioned connection between |mb|≈|mτ| and large θatm... In the non-supersymmetric theory, b−τ unification fails badly, mτ∼2mb . The realistic theory will require a Type I seesaw, or an admixture of both possibilities.
A complex 10H means, as we said, an extra set of Yukawa couplings. At the same time this non-supersymmetric theory cannot account for the dark matter of the universe, since there are no cosmologically stable neutral particles and, as is well known, light neutrinos cannot too. It is then rather suggestive to profit from the complex 10H and impose the U(1)PQ Peccei-Quinn symmetry:
16 → e iα16 , 10 → e-2iα10 , 126 → e-2iα126 , (10)
with all other fields neutral. The Yukawa structure has the form (5) with 10H now complex. This resolves the inconsistency in fermion masses and mixings discussed above, and gives the axion as a dark matter candidate as a bonus . The neutrality of the other Higgs fields under U(1)PQ emerges from the requirement of minimality of the Higgs sector that we wish to stick to. Namely, 126H is a complex representation and 10H had to be complexified in order to achieve realistic fermion mass matrices and to have U(1)PQ. It is desirable that the U(1)PQ be broken by a nonzero <126H>, i.e. the scale of SU(2)R breaking and right-handed neutrino masses , otherwise 10H would do it an give MPQ≈MW, which is ruled out by experiment. Actually, astrophysical and cosmological limits prefer MPQ in the window 1010 − 1013 GeV . Now, a single 126H just trades the original Peccei-Quinn charge for a linear combination of U(1)PQ, T3R and B−L [31, 33]. Thus in order to break this combination and provide the Goldstone boson an additional Higgs multiplet is needed. One choice is to add another 126H and decouple it from fermions, since it must necessarily have a different PQ charge . An alternative is to use a (complex) GUT scale Higgs as considered for SU(5) by , with MPQ≃MGUT, which however implies too much dark matter or some amount of fine-tuning. Of course, the Peccei-Quinn symmetry does more than just providing the dark matter candidate: it solves the strong CP problem and predicts the vanishing θ. The reader may object to worrying about the strong CP and not the Higgs mass hierarchy problem; after all, they are both problems of fine-tuning. Actually, the strong CP problem is not even a problem in the standard model, at least not in the technical sense . Namely, although divergent, in the standard model θ is much smaller than the experimental limit: θ ≪ 10-10 for any reasonable value of the cutoff Λ, e.g. θ≈10-19 for Λ=MPlanck.
The physical question is really the value of θ. PecceiQuinn symmetry fixes this arbitrary parameter of the SM. The situation with supersymmetry and the Higgs mass is opposite. Low energy SUSY helps keep Higgs mass small in perturbation theory, but fails completely in predicting it. If we do not worry about the naturalness we can do without supersymmetry. On the other hand, if we wish to predict the electron dipole moment of the neutron, U(1)PQ is a must, unless we employ the spontaneous breaking of P or CP in order to control θ [36, 37, 38].
Scrutinizing the patterns of symmetry breaking and neutrino mass
In the over-constrained models discussed in this paper, the Dirac neutrino Yukawa couplings are not arbitrary. Thus one must make sure that the pattern of intermediate mass scale is consistent with a see-saw mechanism for neutrino masses. More precisely, the B−L-breaking scale responsible for right-handed neutrino masses cannot be too low. On the other hand, this scale, strictly speaking, cannot be predicted by the renormalization group study of the unification constraints. The problem is that the right-handed neutrinos and the Higgs scalar responsible for B−L breaking are Standard Model singlets, and thus have almost no impact (zero impact at one-loop) on the running. Fortunately, we know that the B−L breaking scale must be below SU(5) breaking, since the couplings do not unify in the Standard Model. Better to say,
MB-L≤ MR, the scale of SU(2)R breaking, and hence one must make sure that MR is large enough. This, together with proton decay constraints, will allow us to select between a large number of possible patterns of symmetry breaking. Our task is simplified by the exhaustive study of symmetry breaking in the literature, in particular the careful two-loop level calculations of Ref. . Recall, though, that the (2, 2, 15) field must lie below the GUT scale ...and although its impact on the running is very tiny, it must be included. The lower limit on MR stems from the heaviest neutrino mass mν ≥ mt2 / MR , (26) which gives MR≥ 1013 GeV or so. One can now turn to the useful table of Ref. , where the most general patterns of SO(10) symmetry breaking with two intermediate scales consistent with proton decay limits are presented...
The above limit on MR immediately rules out a number of the remaining possibilities; the most promising candidates are those with an intermediate SU(2)L×SU(2)R×SU(4)C×P symmetry breaking scale (that is, PS group with unbroken parity). This is the case in which the breaking at the large scale is achieved by a Pati-Salam parity even singlet, for example contained in 54H. In the searching for a realistic symmetry breaking pattern one does not need to stick to the global minimum of the potential as in , but a local metastable minimum with a long enough lifetime will do the job as well. It has to be stressed however, that a big uncertainty is implicit in all models with complicated or unspecified Higgs sector, due to possibly large and uncontrolled threshold corrections . In any case, the nature of the GUT Higgs and the pattern of symmetry breaking will also enter into the fitting of fermion masses, since they determine the decomposition of the light (fine-tuned) Higgs doublet...This point is often overlooked but it is essential in the final test of the theory. At this point, for us it is reassuring that both the pattern of symmetry breaking and the nature of Yukawa interactions allow for a possibly realistic, predictive minimal model of non-supersymmetric SO(10).
Since we know nothing about the existence of supersymmetry or the nature of its breaking, it is mandatory to study the non-supersymmetric version, as a part of the search for the SO(10) GUT. This was the scope of our paper. We have identified two potentially realistic, predictive Yukawa structures for the case of the renormalizable see-saw mechanism, based on a 126H. This choice is motivated by the fact that the alternative radiative seesaw seems to favor split supersymmetry . We have focused on the renormalizable version simply in order to be predictive, without invoking unknown physics.
The models require adding 10H or 120H fields. The latter is particularly interesting, due to the small number of Yukawa couplings. Both models seem to require adding U(1)PQ. While this may be appealing since it provides the axion as a dark matter candidate, it is against the spirit of sticking to pure grand unification...
***Talking about dark matter it is worth reminding the reader that other solutions requiring no new particle but modification of gravitation have been proposed. It just so happens that one of them called mimetic dark matter (mentioned recently for the first time in a post at Quantum diaries ;-) could be provided naturally by non-commutative geometry.