### How far could be the intermediate partial unification scale?

A few words about partial unification

Two recent developments have inspired the investigations reported in this paper. First, it was noted that unlike the SU(2)L*U(1) mode, the U(1) vector generator in the left-right-symmetric model can be identified with B-L symmetry. One implication of this observation is that the mass scale associated with spontaneous breakdown of parity could be associated with spontaneous breakdown of local B-L electroweak symmetry…

Furthemore, the left-right-symmetric model enables us to study the phenomenological implications of local B-L breaking and the possible existence of intermediate mass scales without reference to grand unification models. The second development is the observation by Senjanovic and one of the authors [Mohapatra] that relating the breaking of B-L symmetry to that of the discrete parity symmetry implies a Majorana neutrino and, furthermore, the smallness of neutrino mass is then related to the dominant V-A nature of the weak interaction at low energy…
In the present paper we extend the above considerations by generalizing the model to include the full quark-lepton correspondence. The most elegant formulation appears to be in terms of the “partial unification” group SU(2)L×SU(2)R×SU(4') [The SU(4') “color” group was introduced by J.C Pati and A. Salam with L as the fourth color; in our scheme B-L is the fourth “color”]…
We note that the breakdown of SU(4’) to U(1)B-L×SU(3)could be achieved either dynamically or via a Higgs multiplet Σ that belongs to the representation  (1,1,15) under the gauge group with <Σ>=diag(1,1,1,-3)mX/f [in the base with SU(4’) indices, mX mass scale of the breakdown of SU(4’), f the  SU(4’) coupling]. It is in the remaining Higgs multiplets that the full implications of (B-L)-symmetry breakdown surfaces... We consider the “minimal” model, namely only those kinds of Higgs multiplets that can rise as bound states of existing fermion multiplets. Our results therefore do not depend on the existence of physical Higgs mesons …
We stress that in our “minimal” model (without any additional Higgs beyond those already introduced) the proton is stable [The reason … is the existence of a hidden discrete symmetry in the model under which the Lagrangian is invariant even after spontaneous symmetry breakdown. This hidden discrete symmetry is broken by adding extra Higgs bosons which are antisymmetric in the interchange of SU(4’) indices thereby allowing for proton decay (with B-L conservation). It should be emphasized that the “symmetric“ Higgs bosons are essential in our model to explain the parity nonconservation of the weak interaction at low energy whereas the “antisymmetric” Higgs bosons play no such role].
R. N. Mohapatra, R. E. Marshak,

Which phenomenologist would give her kingdom for an estimated 1012 GeV scale?

Among the possible GUT groups, SO(10)[2] is particularly interesting for several reasons: it is the smallest group for which the fifteen fermions of one Standard Model (SM) generation can fit  within a single irreducible representation (the spinorial 16), it predicts the existence of one right-handed (RH) neutrino per family which in turn, via the seesaw mechanism, can account for massive light neutrinos, it can allow for non-supersymmetric gauge coupling unification and for a sufficiently long nucleon lifetime ... and, being the group rank five, it can allow for an intermediate scale a few order of magnitude below the GUT scale where the gauge group reduces to rank 4. Neutrino masses, the mechanism generating the baryon asymmetry of the Universe (BAU), and possibly also dark matter, might all be related with this scale.
Several connections between SO(10) high energy parameters and observables can be pinned down by studying the Yukawa sector. Vacuum expectation values (vevs) giving rise to fermion masses must belong to conjugate representations of 16⊗16 = 10⊕120⊕126. In a renormalizable model, the 126 is in any case unavoidable since it is the only representation containing a SM singlet, which is needed to implement the seesaw mechanism (otherwise neutrino would have Dirac masses of the order of the charged fermion masses). However, the minimal choice of just one Yukawa coupling is not viable, because it is always possible to rotate the fermionic 16 to a basis in which the Yukawa matrix is diagonal, with the result that the up- and down-quark masses would be diagonal in the same basis and all the Cabibbo-Kobayashi-Maskawa (CKM) mixings would thus vanish. The possibility of 120⊕126 was suggested in [9] but later found, by dedicated numerical analyzes, to be not viable [1,10]. The option 10⊕126 has been instead found to allow fitting consistently all the low energy data [1810]. This can be achieved under the assumption that the neutrino masses are dominated by type I seesaw contributions, and after promoting the fields in the 10 to complex fields [We refer to [4, 9] for details and implications of complexifying the 10 while forbidding an additional 161610∗ Yukawa coupling]. Moreover, as it was recently found in [11], in this model both the requirements of gauge coupling unification and of a proton lifetime above the experimental limits can be satisfied.
... In this work we will confront the model with one more test, namely we will study if the 10⊕126 SO(10) model is able to account for the observed amount of the BAU via the standard mechanism of CP violating decays of the heavy Majorana neutrinos (N) and leptogenesis [1213a13b]. Our main results are that the model is indeed compatible with BAU observations. As a byproduct, we perform a complete disentanglement between the Yukawa coupling matrices and the values of the vevs related to fermion masses (something that cannot be achieved with low energy fits alone) and we also obtain some information about the structure of the intermediate scale particle spectrum. Various studies related to leptogenesis in SO(10) that rely on different sets of assumptions and/or on variations of the minimal model have appeared in the literature [81415]. Here we stick to the minimal SO(10) model constrained only by the condition that the numerical values of the model parameters are such that all the low energy observables are ﬁtted correctly...
The heavy RH neutrinos N acquire an intermediate scale mass via the vev of the SM singlet scalar S that sits in the 126. We deﬁne: σ=<S†S> ... Under [the Pati-Salam groupGPS the 126 branches to (1,1,6)⊕(3,1,10)⊕(1,3,10)⊕(2,2,15). S is the neutral component and SU(3)c singlet of (1,3,10) so that σ also breaks GPS →GSM...
The scalar ﬁelds Σu,d belong to (2,2,15) of GPS and have a mass unrelated to the vev σ. Naturalness considerations then suggest that MΣ>>MN in which case the decays N → lΣu are kinematically forbidden. Note however, that the neutral components of these bi-doublets will acquire induced vevs proportional to the EW vevs residing in the 10, via the coupling (126126)(12610) [4]. These induced vevs are of fundamental importance to achieve the correct fermion mass relations...
To estimate the baryon asymmetry yield of the minimal SO(10) model we need the nu- merical values of the partial decay widths and CP asymmetries, and to compute these quantities we need to know the values of the Yukawa coupling matrices and of the vev σ that ﬁxes the scale of the N’s masses. As we will now argue, the values of these parameters can be ﬁxed almost univocally in terms of measured low energy observables, with only one single high energy parameter left free...
Although the number of parameters exceeds by one the number of constraints from the data, SO(10) numerical ﬁts are able to determine all 19 parameters (and thus to yield also predictions for yet unmeasured quantities like the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) Dirac phase and the 0ν2β effective neutrino mass [1,10]). This is due in part to the nonlinearity of the relations that connect [the set of Yukawa coupling matrices and coefficients] {H,F,r,rR,s} [H=Y10 and F=Y126] to the data, and also to the fact that the number of real observables (moduli) is sizably larger than that of real free parameters... While low energy data are only sensitive to the product of the coupling matrices times the vev <Σd>, leptogenesis is directly sensitive to the values of f alone, and then it can allow to disentangle the Yukawa couplings from the vevs...  As regards σ, it is in general complex, but its overall phase does not affect low energy data or leptogenesis, and therefore for simplicity we take σ to be real...
... the RH neutrino spectrum is univocally ﬁxed solely by the low energy data. As we have already mentioned, the RH neutrino masses obtained from the numerical ﬁts generally fall in the range 109−1012 GeV, which is a favourable one for leptogenesis, but it should be remarked that this type of results always implies a certain degree of tuning. The mass of the lightest Majorana neutrino N1 would in fact more naturally lie in a mass range well below 109GeV...
If from one hand it is somewhat unpleasant that the numerical values that we will use result from a certain amount of tuning in ﬁtting SO(10) parameters, on the other hand we ﬁnd intriguing that, without any knowledge of what is required for leptogenesis to be successful, low energy data alone force all the Mj’s in the correct ballpark.
We take the {H,F,r,rR,s} data points from the ﬁts of Dueck and Rodejohann (DR) [1] to non-supersymmetric SO(10) with scalars in 10126, that are labeled as MN (minimal non-supersymmetric). Starting from the low energy data set (SM fermion masses and mixings) at the scale µ = MZ, DR perform two different types of ﬁts to the MN model. In the ﬁrst approach, which they denote as MN-RGE, the observable are evolved from the high energy scale down to MZ, integrating out the heavy neutrinos Nj one by one at the appropriate scale. The outcome of the running is then compared with the experimental data. This is the most sophisticated approach, and in particular is expected to yield a more reliable ﬁt to the heavy neutrino masses. This, besides having sizable effects on the neutrino parameters [32], is a quite crucial ingredient in leptogenesis. The [heavy Right handed Neutrinos] spectrum obtained with this procedure is:
{MN1, MN2, MN3} = {1.2×1011, 2.0×1011, 3.6×1012} GeV.
As regards the numerical values of the set {H,F,r,rR,s}, they can be found in appendix A of [1] labeled as MN-RGE and [recopied below].
The main approximation in the DR analysis is that of neglecting effects of the intermediate scale states on gauge coupling uniﬁcation and on the running of the Yukawa matrices, and it is quite hard to estimate the related uncertainty on the ﬁtted parameters. [A preliminary tentative in this direction, although in a slightly different setup, has been done in [33], where it has been shown that threshold effects at the intermediate scale can produce effects on the fermion observables at the electroweak scale as large as 30%.]
... In summary, we ﬁnd that the DR RGE ﬁt to the minimal non-supersymmetric SO(10) model is fully consistent with the requirement that the observed value of the BAU is produced via leptogenesis, if the two conditions (i) M > MNN2 = 2.0×1011 GeV and (ii) 0.3≤√[<Σ†dΣd>/≤H†dHd>] 1 are satisﬁed...

In this work we have considered leptogenesis in a non-supersymmetric SO(10) GUT with fermion masses from the 10⊕126 Higgs representations, which can (i) ﬁt well all the low energy data, (ii) successfully account for uniﬁcation of the gauge couplings, and (iii) allow for a sufficiently long lifetime of the proton. We have shown that, once the model parameters are ﬁxed in terms of measured low energy observables, the requirement of successful leptogenesis can ﬁx the only one remaining high energy parameter... We have shown that the values of the model parameters obtained from the ﬁts to low energy observables given in ref. [1] yield a baryon asymmetry in agreement with observations.
(Submitted on 15 Dec 2014 (v1), last revised 29 Jan 2015 (this version, v2))

The article by Fong et al. does not show any numerical value for the vev of σ since it is the last remaining high energy parameter mentioned in the conclusion. Nevertheless as it is stressed in the article but not quoted in this post, any value of the parameter td  in the interval [0.3;1] gives a B-L asymmetry prediction in agreement with experimental values. That gives a σ vev in the range 1.6×1012 GeV to 5.3×1012 GeV if I am correct...