{'fraction_non_alphanumeric': 0.03973253472401508, 'fraction_numerical': 0.0266174420405845, 'mean_word_length': 4.284447476125512, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 9, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Higgs sector of the next-to-minimal renormalizable SUSY SO(10)

We study the Higgs potential of the next-to-minimal renormalizable SUSY SO(10) GUT with 120 Higgs representation on top of the"standard"minimal model Higgs sector spanning over 10, 126bar+126 and 210. All the GUT-scale Higgs sector mass matrices for the 592 Higgs states of the model are written down in detail with all the conventions fully specified. The consistency of the results is checked by the decoupling of 120 and independently by the analysis of the relevant Goldstone modes. The matching of the Yukawa sector sum-rules driving the matter fermion masses and mixing at the level of the effective theory is described thoroughly.


I. INTRODUCTION
The idea of grand unification [1] is one of the simplest and very powerful strategies of extending the Standard Model (SM) of the elementary particle interactions and the grand-unified theories (GUTs), though most of them formulated already in 1970's, receive still a lot of attention of the high energy physics community. Apart from accounting for all the Standard Model phenomena within a unified picture (accommodating the plethora of the SM multiplets into just a few irreducible representation of the unified gauge group G ⊃ SU (3) c ⊗ SU (2) L ⊗ U (1) Y ) the GUTs typically provide answers to the questions the Standard theory is incapable to answer, like e.g. the hypercharge quantization or the hierarchy of the gauge (and some of the Yukawa) couplings at the electroweak scale. On top of that, GUTs generally predict distinctive new phenomena like proton decay, nucleon-antinucleon oscillations, monopole production in the early stage of the universe evolution etc., which makes these frameworks potentially testable and thus physical [2].
One of the most popular grand-unified frameworks is based on the SO(10) gauge group [3] which is assumed to govern physics at very high scales (typically above 10 16 GeV) providing for the the Standard Model dynamics once the GUT symmetries get spontaneously broken. Perhaps the main virtue of the SO(10) models is that all the SM matter fermions of each generation resides in a 16-dimensional spinorial multiplet 16 F , which, however, contains the SM singlet playing the role of the right-handed neutrino, it is then very natural that small neutrino masses [4] are generated by means of some variant of the seesaw mechanism [5]. Furthermore, as a rank-5 group, SO(10) admits several different symmetry breaking patterns (see e.g. [6,7,8,9]), which is a feature that renders these theories slightly more flexible than the other popular scenarios based on the SU (5) gauge symmetry [10], alleviating to some degree the problems with proton decay (see e.g. [11,12]) generically emerging in the supersymmetric (SUSY) SU (5) models.
in three copies of 16 F of SO(10) is indeed the minimal potentially realistic supersymmetric GUT (MSGUT) with only 26 (real) parameters driving the MSSM physics (up to the soft SUSY-breaking sector) [14] and an automate R-parity as a direct consequence of the B − L symmetry breakdown by means of 126 H ⊕ 126 H [19]. Moreover, it was observed that in case of a dominant triplet contribution in the seesaw formula for the neutrino masses, there is a natural case for the maximality of the atmospheric mixing in the lepton sector due to the GUT-scale b-and τ -Yukawa coupling convergence [20]. On top of that, a generic tendency for a somewhat larger 13 mixing in the lepton sector (|U e3 | 0.1) has been observed, providing a clear experimental signal within the reach of the near-future facilities [21]. This, of course, triggered an enormous boost to the field with tens of papers concerning various aspects of the model, like e.g. proton decay [11], effective Yukawa sector [15], gauge coupling unification [8] etc.
However, the real breakthrough came in 2004 with the study by Bajc, Melfo, Senjanović and Vissani [22] (preceded by [23] accounting for some partial results) providing a complete analysis of the Higgs potential of the minimal model, deriving in particular the mass matrices of all the SM components of the 472-dimensional Higgs sector in terms of the parameters of the underlying SO (10) lagrangian. With the extra information at hand it was possible to improve the existing analyses to such a degree a tension between the minimal theory and observation has been revealed [24,25,26]. It was argued that the need for a relatively low singlet Majorana neutrino mass scale (of the order of 10 13−14 GeV) in order to satisfy the absolute neutrino mass constraints is indeed in clash with the fits of the effective Yukawa sector sum-rules. Basically, the trouble stems from the dual role of the 126 H multiplet in the Yukawa sector: the need for the B − L breakdown close to the GUT-scale (M G ) which is mandatory [22] in order to prevent the Higgs sector thresholds from spoiling gauge-coupling unification requires a relatively small Yukawa coupling of 126 H which is typically at odds with the second generation SM matter hierarchy.
Perhaps the most conservative extension of the minimal model 1 that could alleviate this tension, yet retaining a significant amount of its predictive power, consists in invoking an extra 120-dimensional three-index antisymmetric tensor in the Higgs sector [29,30,31]. Such a scenario, sometimes called 'the next-to minimal SUSY SO(10)' (or 'new minimal SUSY SO(10)', NMSGUT [31]) has the appealing feature of accounting for all three possible types of renormalizable couplings the matter bilinear 16 F .16 F can potentially develop (recall that 16⊗16 = 10⊕126⊕120) and the price to be paid for the extra multiplet can be acceptable -the Yukawa coupling of 120 H is antisymmetric and thus the minimal model Yukawa sector predictivity is challenged in a minimal way. With an extra set of couplings in the game, the effective SM Yukawa sum-rules are altered, which provides an option to account for the second generation hierarchies by means of the Yukawa of 120 H while keeping f 126 relatively free to deal with the MSGUT neutrino mass scale problem [31]. Since the full description of the SM quark and lepton masses based on the interplay of the 10 H and 120 H only seems troublesome [32], the subleading effects from 126 H deferred almost entirely for the sake of the neutrino sector can be just the right cure to reconcile the first and second generation structures with the observation, which makes this extension of the minimal model particularly interesting and worth further scrutiny.
However, due to the further relaxation of the effective sum-rules via an extra Yukawa in the model the oldfashioned effective analyses of the Yukawa sum-rules became indecisive in the NMSGUT case (though some interesting results in this direction have been obtained under various extra assumptions like e.g. perturbatively small 120 H -effects [30], in a setting with decoupled 126 H [32] or imposing furhter constraints on the Yukawa couplings, see e.g. [33]). Any extra information about the GUT-scale Higgs spectra and mixings (as almost the only means of constraining the parametric space of the model) became a crucial ingredient of any quantitative analysis of the model.
The main scope of this work is to study the Higgs sector of the NMSGUT case (with the total of 10 H ⊕ 126 H ⊕ 126 H ⊕ 210 H ⊕ 120 H in the Higgs sector) as thoroughly as possible, i.e. calculate all the Higgs sector mass matrices (encoding the information about the GUT-scale Higgs spectra and mixings which is the key to a reliable analysis of the effective Yukawa sector). We shall pass through all the SM submultiplets of the 592-dimensional Higgs sector of NMSGUT providing the relevant mass matrices for each sector in the form that emerges right after the breakdown of the GUT symmetry and comment on some of the features of the different multiplets. Last, but not least, in the limit when 120 H decouples from the GUT-scale physics, our results provide an independent cross-check of the MSGUT formulae given in [22].
The study is organized as follows: after recapitulating in brief some of the salient features of the Higgs sector of the minimal model we write down an upgraded form of the Higgs superpotential relevant for the NMSGUT case (including the extra 120 H ) in Section II and argue (Section III) that despite from the extra 5 terms therein the vacuum structure of the extended model is the same like that of the MSGUT. This allows us to inherit the minimal model parametrization of the vacuum manifold from [22] and write down all the mass matrices in Section IV in the 'familiar' notation. Each of the mass matrices is augmented with a complete information on the states it is spanned over and on the phase convention employed upon its derivation so that the study is maximally self-contained and verifiable. Section V is then devoted to the identification of the Goldstone modes in the relevant sectors, which provides a further consistency check of our results. Focusing on the Yukawa sector of the MSSM in Section VI, we provide a detailed description of the matching between the effective theory and the underlying SO(10) model. Most of the technicalities (namely notation issues & comments on conventions) are deferred to an Appendix.

II. THE NMSGUT HIGGS SECTOR SUPERPOTENTIAL
Using the "group theory notation" the Higgs part of the superpotential of the minimal SUSY SO(10) model reads: which is the piece that has been studied in detail in [22]. In terms of the SO(10) tensorial components: W min H can be transcribed as: where the notation of [22] has been fully inherited and summation over all repeating indices is understood. There is in total 7 parameters in W min H : {m H , m Σ , m Φ , α, α, η, λ}. However, 1 of them (m Σ ) can be traded for a dimensionless parameter called x from the requirements on the GUT symmetry breaking chain and another (conventionally m H ) can be fixed by the need for a zero determinant of the SU (2) L doublet mass matrix to arrange the pair of light MSSM Higgs doublets.
In NMSGUT, the extra 120-dimensional three index fully antisymmetric tensor is added to the Higgs sector giving rise to a new set of vertices with 120 H : This piece of the total NMSGUT superpotential W H = W min H +W 120 H brings in 5 extra parameters: {m Ψ , β, β, γ, ρ}. All together, there is 11 relevant parameters in the next-to-minimal setting, namely {x, m Φ , m Ψ , α, α, η, λ, β, β, γ, ρ} that we shall further refer to as "microscopic".
In what follows, we shall mostly use the notation and conventions defined in the Appendix A.1 of [22]. However, since the embedding of the SM states into the SO(10) multiplets is defined only up to global phases, there will be some differences in the choice of phase convention for the basis states the mass matrices are spanned over. For sake of completeness, we will comment on this issue wherever appropriate. Further information on accounting for some of the convention differences is given in Appendix A. provided is being used as a convention for the Gell-Mann Nishijima formula defining the weak hypercharge normalization.
As it was shown in [22] there are 7 distinct SUSY preserving minima of the minimal model Higgs potential corresponding to SO(10), SU respectively and there is a single dimensionless parameter called x in [22] governing all the relevant breaking chains. The Standard Model vacuum manifold then corresponds to those values of x which generate "generic" patterns amongst the VEVs of the Standard Model singlet VEVs and in which case SO(10) is broken down to SU (3) c ⊗ SU (2) L ⊗ U (1) Y essentially in one step while the intermediate symmetries can be obtained if x is tuned to receive one of the particular values identified in [22]. Let us remind the reader that the GUT-scale D-flatness implies σ =σ up to an overall phase. In view of the complicated structure of the minimal model vacuum it is rather welcome that the extra 120 H decomposing under the Pati-Salam subgroup as 120 = does not bring in any extra SM singlets because the only colour singlet of 10 of Pati-Salam has a non-zero B − L charge, c.f. formulae (7). This means that the vacuum structure of the minimal model remains unaffected even with an extra 120 H in the game, and in particular the parametrization (9) can be entirely inherited for sake of the discussion of the extended model. This feature saves us a lot of tedium indeed.

IV. HIGGS SECTOR MASS MATRICES
In this section we shall provide the mass matrices for the (fermionic components of the) Higgs multiplets after the SO(10) breakdown to the Standard Model SU (3) c ⊗ SU (2) L ⊗ U (1) Y . In order to simplify the situation considerably we shall, as usual, focus on the fermionic (i.e. higgsino) components of the superfields and invoke SUSY (which remains unbroken at the GUT scale) to transfer the results of interest (i.e. the eigenvalues and mixings) into the bosonic sector. The main advantage of this approach is that the higgsino mass matrices can be obtained directly from the VEVs of the double superpotential derivatives and there is no need to pass through the lengthy computation of all the F -terms and D-terms providing the building blocks of the SUSY scalar potential.
For sake of illustration let us remark that there is in total 13,321,010 terms in the sums in (3) and (5) (out of which 2,111 thousand terms come from the new piece W 120 H ), but fortunately 'only ' 1,190,170 of them are nonzero by antisymmetry of the tensors under consideration (W 120 H then accounts for 338,400 out of this number). Thus, perhaps the only reasonable strategy of handling all these contributions is to work with the antisymmetrized combinations rather than with the very components of the antisymmetric tensors (this is why all the results in [22] are given just in terms of such structures), c.f. Appendix A.  In every sector (i.e. for every combination of C c , C L , Y ), we shall first specify the choice of the representative (i.e. the values of T c 3 , T c 8 , T L 3 ) and comment on the dimensionality and origin of the relevant contributions. Every subsequent mass matrix shall be equipped with a table of states corresponding to its rows (denoted generically by R i ) and columns (C i ), unless the representation is real, in which case a single symbol S i shall be used. For each C i and R i (or S i ) we shall also display a chunk of the map of the SM components of 10 H , 126 H , 126 H , 210 H and 120 H (i.e. the submultiplets with definite SM quantum numbers) onto the defining basis states H i , Σ ijklm , Σ ijklm , Φ ijkl and Ψ ijk (typically we present only the "lowest" relevant permutation of indices and defer an interested reader to Appendix A or to [22] for further details) in order to provide an information about the phase convention used in derivation of the mass matrix under consideration. (Note that for sake of simplicity we always choose our phase convention in such a way there are no pending imaginary units in the mass matrices.) For sake of a simple bookkeeping the top-left box of each table shall indicate the full dimensionality of the sector under consideration.

B. Mass matrices -colour singlets
There are in total 40 colour singlet states in the Higgs sector of the minimal model. With an extra 120 H this number is increased up to 50. In what follows we shall pass through the three categories corresponding to SU (2) L singlets, doublets and triplets respectively.
As in the minimal model case, the Y = 0 sector of the NMSGUT consists of 5 entirely SM-neutral Higgses (and thus the representation is real and the mass matrix symmetric) because there is no full SM singlet in 120 H . Note also that these are the only fields that can receive GUT-scale VEVs triggering the SO(10) breakdown. The Y = ±1 sector of the NMSGUT differs from the minimal model one due to an extra state in 120 H and the relevant 3 × 3 mass matrix is hermitean accounting for 6 degrees of freedom in total. Notice that these are exactly the quantum numbers of the SU (2) R gauge bosons (W R ) that should be made superheavy via the standard Higgs mechanism once the SU (2) R symmetry gets broken. In order to be able to achieve that, there must be a Goldstone boson in this sector corresponding to a zero in the spectrum of the relevant mass matrix. For further details the reader is deferred to Section V. Finally, the 2 components with Y = ±2 are identical to the minimal model ones (coming from 126 H ⊕126 H ) as well as the corresponding mass matrix.
The relevant mass matrices and the corresponding physical basis vectors (and the phase convention) these matrices are spanned on follow: 2 One can easily check that in all three cases our results agree with the relevant matrices in [22] up to row/column reshuffling and the phase conventions.
There are in total 14 different colourless doublets accommodating 28 independent components. This cathegory breaks down into 2 subspaces with hypercharges Y = ± 1 2 and ± 3 2 . The Y = ± 1 2 sector corresponds to the MSSM-like Higgs doublets driving the Dirac masses of the matter sector. Due to the two extra Pati-Salam-bidoublets residing in 120 H , the original MSGUT 4 × 4 mass matrix for each charge is in NMSGUT upgraded to 6 × 6, accounting for 12 doublets (i.e. 24 states in total). We give the relevant mass matrix for the Q = 0 sector only; the same for Q = ±1 states is readily obtained from the SU (2) L symmetry considerations. (Since this is the key to any Yukawa sector analysis, a detailed study of this sector is subject of a dedicated Section VI.) The remaining 4 states (i.e. 2 isodoublets) with Y = ± 3 2 are the same like in the minimal model. [1,5,7,9] + . . . [1,5,7,9] + . . .
and the results in the decoupling limit (i.e. m Φ → ∞) are again the same as in [22], up to phase conventions.
In the NMSGUT Higgs sector there are 3 colourless triplets belonging to this cathegory (accounting in total for 9 degrees of freedom) breaking into 2 subspaces with hypercharges 0 and ±1. This sector is identical to the case of the minimal model.
As before, it can be checked our results correspond to those of [22] wherever appropriate.
60 Pati-Salam origin (C c , T c 3 , T c 8 , B − L, C L , T L 3 , C R , T 24 The coloured triplet-antitriplet pairs that transform like triplets under SU (2) L account in total for 54 states with hypercharges Y = ∓ 2 3 and ± 1 3 . The only difference between the minimal model and the extended scenario is one extra set of states in the Y ± 1 3 sector though. 18 3 states: : As before, in the m Ψ → ∞ decoupling limit we do reconstruct the result of the minimal model study [22].

D. Mass matrices -colour octets
There is in total 152 states in 19 different colour octets belonging into this cathegory, to be compared with 15 such octets in the minimal model. The new states from 120 H enter only the (8, 2, − 1 2 ) ⊕ (8, 2, + 1 2 ) sector while the rest is identical to the MSGUT situation. 16

E. Mass matrices -colour sextets & antisextets
There are 132 degrees of freedom corresponding to the colour sextet-antisextet pairs in the next to minimal SUSY SO(10) (to be compared to 120 in the MSGUT [22]). Since all sextets descend from the Pati-Salam   This subspace is identical to the relevant part of the MSGUT, see e.g. [22] and references therein.
a. Sector (6, 2, − 5 6 ) ⊕ (6, 2,  To conclude, in this section we have written down the mass matrices for all the 592 bosonic degrees of freedom (up to gauge transformations) of the Higgs sector of the next-to-minimal supersymmetric SUSY SO(10) model. We have checked that in the m Ψ → ∞ limit (corresponding to the decoupling of the extra 120 H multiplet) the minimal model effective particle content is recovered (i.e. exactly 120 states decouple) and the resulting mass matrices are reduced to the relevant MSGUT formulae. This provides a non-trivial consistency cross-check of the results given previously in [22] as well as ours.  [22] are easily recovered. In what follows we shall focus on the former case and provide an evidence that the zeros are still in the spectra of the extended mass matrices, as required by consistency.

A. Relevant mass matrices in the Standard model vacuum
Adopting the SM vacuum notation along the lines of [22] (recall that the vacuum structure of the model with 120 H is identical to the one of the MSGUT because there is no extra full SM singlet in there; c.f. also formulae (9) and (10) in Section III) the relevant mass matrices can be recast in terms of the microscopic parameters as follows: It is not hard to see that with (10) all these mass matrices have indeed a zero in the spectra irrespective of the values of r ≡ m Ψ /m Φ , β and β.

VI. THE NMSGUT YUKAWA SECTOR
The Yukawa structure of the theory provides an important and (at least in principle) testable imprint of the GUT-scale physics on the electroweak scale observables. In this section, we shall discuss some of the salient features of this part of the NMSGUT model. With an extra 120 H at hand, the MSGUT [14] Yukawa piece of the superpotential is extended into where f 10 , f 126 are general complex 3 × 3 matrices, the former two being symmetric in the family space while f 120 is antisymmetric. After the SO (10)   If, on the other hand, the Higgs sector of the model is fully specified, the weights above become computable in terms of the parameters entering the Higgs scalar potential (1), (5) and the VEVs driving the breakdown of the GUT symmetry (8). In case of the minimal scenario this has been done in full generality by Bajc, Melfo, Senjanović and Vissani in their 2004 paper [22] and further extended in [18]. Subsequently, it has been pointed out [24,25,26] that the minimal setting is indeed incompatible with the low-energy data.
B. Microscopic structure of the effective MSSM mass sum-rules in NMSGUT In this section we consider the extended Yukawa sector (42) and compute the effective projections of the electroweak VEVs entering (43) by means of the microscopic parameters of the theory.

Bidoublet mass matrix in NMSGUT
In order to do that we should look at the shape of the SU (2) L doublet mixing arising from the generic 6×6 bidoublet mass matrix (15) emerging after spontaneous breakdown of the GUT symmetries. Notice that the upperleft 4×4 sub-block of (15) corresponding to the doublets in 10 H , 126 H ⊕ 126 H , 210 H of the original MSGUT is indeed identical (up to an irrelevant global rephasing) to the bidoublet mass matrix given in [22], which provides a non-trivial consistency check of both ours and Bajc & co.'s analysis. The last two rows/columns in (15) are due to the pair of bidoublets in 120 H and we have chosen our convention in such a way there are no pending i factors. Adopting the SM vacuum notation along the lines of [22] the VEVs in the mass matrix under consideration can be recast in terms of the microscopic parameters (9) as: where now also σ and σ are functions of the basic parameters and obey (c.f. [22]), c.f. (10).

Arranging the light MSSM Higgs doublets
As in the case of the minimal model, m H can be fixed from the zero-determinant condition which is necessary to arrange the pair of light MSSM-like Higgs doublets h u and h d . In the mass basis, these are also the only zero modes of this mass matrix while the orthogonal states correspond to the five heavy (typically GUT-scale) Higgs doublets H (1) u,d , . . . , H (5) u,d . Up to an overall normalization, the light Higgs doublets of the MSSM correspond to the following combinations of the defining basis states (using H u,d for the doublets within (1, 2, 2) 10 and so on, i.e. Σ u,d ∈ (15, 2, 2) 126 , Σ u,d ∈ (15, 2, 2) 126 , Φ u ∈ (10, 2, 2) 210 , Φ d ∈ (10, 2, 2) 210 , Ψ u,d (1) ∈ (1, 2, 2) 120 and Ψ u,d (2) ∈ (15, 2, 2) 120 ): where the generic w u,d R factors stand for the numerical weights (or projections) of the various components of h u,d in the relevant 6-dimensional SM-doublet space. Recall that in order for the change of basis (44) to be unitary, the weights above should obey the normalization condition R |w u,d R | 2 = 1 for each of the hypercharges.

Projecting the MSSM Higgs doublets onto the defining MSGUT basis
Defining a "decoupling parameter" r ≡ m Φ /m Ψ which is a quantity that can trace the "strength of interaction" between the minimal model bidoublets residing in 10 H , 126 H ⊕ 126 H and 210 H , and the extra bidoublets within 120 H . (indeed, for m Ψ → ∞ we get r → 0 and the minimal model situation should be recovered) the relevant weight factors can be shown to obey the following formulae (up to an overall normalization): and similarly in the down-sector: ρ ηλ P f 12 + + r 2 ᾱβ 2β2 η 3 λ 2 P g 12 + αββ 3 η 3 λ 2 P h 12 + ββ 2 γ η 2 λ 2 P d 13 + αβ 2 ρ η 2 λ 2 P e 13 +ᾱ ββρ η 2 λ 2 P f 13 +β γρ ηλ 2 P g 13 +ᾱ where the P x n factors denote polynomials in x of order n that are given (in terms of their -irreducible components) in Tables I and II. Consistency requires that in the limit r → 0 the minimal model situation should be recovered. Indeed, in such a case one obtains: which is (up to an overall factor −Q b 3 (x − 1) 4 and signs we shall comment on below) identical to the minimal scenario relations, c.f. formulae (C18), (C19) in [22] in view of the comments given e.g. in [26]. We can also see that in such a case the 120 H Higgs multiplet decouples as desired.
Note on convention and phase factors: Notice that the phase convention used to derive formulae (15), (45) and (46) has been chosen in such a way that the mass matrix is "optically simple", i.e. with positive multiples of the superpotential mass terms on the diagonal and without explicit i-factors in the 5th and 6th rows and columns.  II: Tables of -irreducible polynomials Q x n used in the definitions of polynomials P x n in Table I . This is also the reason why we have got some of the signs in (47) different from those in the corresponding formulae in [22] (notice in particular the extra minus sign at the 44 position of the relevant mass matrix in there). As far as only the mass matrices are concerned, this has, of course, no physical significance and the physical spectra remain intact. One must, however, pay attention to these effects upon getting to the interaction vertices, which are of course sensitive to the particular choice of phases in (15) and (44). This, in particular, is relevant for the matching of the "microscopic theory" (i.e. the SO(10) model) onto the effective Yukawa sector sum-rules, that shall be studied thoroughly in the next section.
C. Matching NMSGUT to the Yukawa sector of the MSSM

Note on phases & conventions
Let us now focus on the form of the effective Yukawa sum-rules in the specific convention adopted in this study, which leads in particular to an "optically simple" form of (not only) the doublet mass matrix (15). It is important to notice that the would-be simple-minded identification 2 v u,d 10 = w u,d * and v u,d(2) 120 = w u,d(2) * 120 v u,d is contrived because of its convention-dependence. Indeed, changing for instance the convention in such a way that the symbol Ψ u (1) would be instead used for what we would call −Ψ u (1) in the original convention (which, being a phase transformation, would be as good basis vector as the original one), the functional dependence of the corresponding w u(1) * 120 weight factor on the "microscopic" parameters (x, r, α . . .) in the new convention would get an extra minus sign with respect to the form given in formula (45). Such a change, however, should lead to a change of the relative sign of terms in the square bracket in the first two equations in (43), which can not be absorbed into a redefinition of the Yukawa matrix 3 . Apart from the phases, there could also be Clebsch-Gordon coefficients popping-up in the matching. Note also that this is a general issue of any matching between an effective and a "microscopic" theory.
Therefore, in order for the information provided in this analysis to be self-contained and verifiable, the matching conditions should be carefully inspected and all the utilized conventions have to be fully specify. That is also why we have devoted a significant portion of Section (IV) to the detailed specification of the relevant SM eigenvectors.
The main virtue of this definition is that these factors are indeed simple functions of the decomposition weights in and thus, given the relevant MSSM doublet weight factors in formulae (45) and (46), can be readily computed from the underlying theory. In order to be fully specific, let us also note that in our convention the electroweak doublet VEVs (47) are spread over the following components of the defining SO(10) tensors (2): This information shall be used later on in Section VI C 3 when it comes to the derivation of the matching conditions for the effective sum-rule VEV factors v u,d R in (43) in terms of the u u,d R symbols fixing our convention for the neutral components of the defining basis SU (2) L doublets (47).
b. Induced SU (2) L -triplet VEV: Apart from the SU (2) L doublet VEVs, there is a pair of VEVs relevant for the Majorana sector of the model, namely those of the colourless SU (2) L singlet ∆ 0 R corresponding to the neutral component of the SU (2) R triplet ∆ R = (10, 1, 3) 126 (this VEV is actually responsible for the SU (2) R ⊗ U (1) B−L → U (1) Y breakdown) and the colourless SU (2) L triplet ∆ 0 L residing in ∆ L = (10, 3, 1) 126 . Denoting the former by (which is just a more "physical" notation for σ; indeed, as one can see from (8) these symbols are equivalent and we use the latter only for sake of clarity 4 ). Vanishing of the GUT-scale F -terms then requires a non-zero VEV to be induced on 5 ∆ 0 L : In particular, the relevant formula reads (c.f. for example [18]) : where, as in the case of the other doublets (48), v u 210 = w u * 210 v u and the relevant weight factor is given in formula (45). Note also that if 120 H decouples (i.e. for β → 0) this formula is reduced (up to the minus sign in the second term corresponding to a different convention being used) to the minimal model result given in [18]. It is remarkable that both factors, i.e. the minimal model contribution ∝ αu u 10 − √ 6ηu u 126 as well as the extra piece due to 120 H proportional to v u(1) 120 share a common feature that the Clebsch-Gordon coefficients therein exactly cancel the overall constants in the relevant weight factors and the resulting polynomials admit for a great further simplification, which can be viewed as an indication of consistency of our results.
As before, to be fully explicit, let us remark that the B − L breaking VEVs U R and u L from (50) and (51) are spread over the following components of the relevant SO(10) tensors: 3. Matching NMSGUT to the effective Yukawa sum-rules The rest of this section will be devoted to a derivation of the effective sum-rules of the form (43) in terms of these VEVs and the identification of the "effective" Yukawa couplings Y 10 , Y 126 and Y 120 in terms of the superpotential couplings f 10 , f 126 and f 120 in W Y , c.f. formula (42).
In what follows we shall use the method [34] to work out the three relevant structures. In particular, we shall be using the embedding of the SO(10) spinorial matter multiplets into the ψ + sector of [34]. Note that our convention is such that the "colourless" indices {1, 2, 3, 4} used in this study correspond to the {7, 8, 9, 10} sector of Mohapatra and Sakita and similarly our {5, 6, 7, 8, 9, 10} indices can be identified with their {2, 1, 4, 3, 6, 5} sector. With this at hand one can write the effective MSSM matter mass terms in a generic form (suppressing the family indices): where the B and Γ i matrices as well as the bra's ψ * + | and ket's |ψ + are defined in [34]. a. SU (2) L -doublet VEVs & Dirac mass sum-rules: Working out the structure (54) together with (49) one first obtains the effective Dirac mass matrices in the form: .
Denoting Y 10 ≡ √ 2if 10 , Y 126 ≡ 2if 126 / √ 3 and Y 120 ≡ √ 2if 120 , the effective sum-rules in the notation (43) can be then recovered provided: which, together with formula (48) yields the desired matching between the weight factors given by formulae (45) and (46) Note that due to the properties of f 126 the Majorana masses are indeed symmetric in the family space, as desired. Restoring the effective Yukawa couplings, the matching conditions for the effective VEV factors in the Majorana sector of (43) read: Note that these results correspond to those obtained previously in [18] combined with [17,24] and thus provide a further nontrivial consistency check of our calculation.

D. Final remarks
With all these results at hand, all the ingredients necessary for fitting the effective sum-rules (43) in the framework of the full-featured NMSGUT have been discussed. Indeed, matching conditions (56) and (58), when supplemented with the prescriptions (50) and (52) together with the translation tables (9) and (48) and the explicit formulae for the weight factors (45) and (46) can be used to rewrite all the weight factors in (43) in terms of the "microscopic" parameters {x, m Φ , r, α, α, η, λ, β, β, γ, ρ}.

VII. CONCLUSIONS
It has been shown several years ago that the renormalizable SUSY SO(10) grand-unified model with the simplest potentially realistic Higgs sector spanned over the representations 10 H , 126 H ⊕ 126 H and 210 H experiences severe difficulties in accommodating the low-energy Yukawa sector constraints stemming from the observed patterns of the quark and lepton masses and mixing parameters. It was argued that the troubles emerge due to the antagonism between the need for a B − L breaking scale to be slightly suppressed relative to the GUT scale (which is vital in order to bring the type-I seesaw contribution to the neutrino masses into play) on one side and the significant difference between the second generation quark and lepton masses calling for essentially the opposite on the other.
One of the most popular simple extensions of the minimal framework that could in principle resolve this issue consists in employing an extra 120-dimensional three-index antisymmetric tensor representation in the Higgs sector providing for a new contribution to the effective Yukawa sector sum-rules, thus relaxing the tight link between the seesaw scale and the second generation hierarchy.
Due to the rather complicated structure of the Higgs sector, the key ingredient of any quantitative analysis of such kind of models is a thorough understanding of the relevant Higgs spectra and the corresponding Higgs mixing patterns. It is remarkable that even in the case of the minimal model (which was formulated at the beginning of 1980's) the mathematical complexity of the 472-dimensional Higgs sector did not admit for drawing reliable statements about the viability of the theory until a couple of years ago when the complete analysis of the 10 H , 126 H ⊕ 126 H and 210 H Higgs sector was first published [22].
In the current paper we provided a very detailed and maximally self-contained analysis of the 592-dimensional Higgs potential of the next-to minimal SUSY SO(10) model (consisting of 10 H , 126 H ⊕ 126 H , 210 H and 120 H multiplets) focusing on the effects of the extra 120 H in the game. Since there are no extra Standard Model singlets in 120 H the symmetry breaking pattern of the minimal model remains intact which, in turn, simplifies the analysis considerably and admits for adopting the useful notation of [22] for the case under consideration.
In particular, all the GUT-scale Higgs sector mass matrices have been written in detail together with a thorough description of the basis states and the relevant conventions in each of the different SM sectors. Focusing subsequently on the sum-rules for the effective Yukawa sector emerging under the GUT-scale, and in particular to the masses of the matter fermions, the matching of all the relevant effective building blocks to the microscopic structure of the model was investigated. A set of nontrivial consistency checks was also provided: 1) in the limit of a decoupling 120 H all the mass matrices given in this study reduce to the minimal models structures of [22] (up to unphysical rearrangements) and 2) the 33 Goldstone modes were shown to be present in the Higgs sector spectra so that the proper Standard Model gauge structure emerges at low energies.

Note added
A day before finishing ver.1 of this manuscript the author's attention was drawn to the preprint [35] where the relevant part of the next-to minimal SUSY SO(10) model has been previously studied from a similar perspective. As far as one can see through the jungle of different notation, normalization and conventions the results therein agree with those given in this study. Moreover, since the method we employed is different I believe that the current analysis is worth and does indeed provide a valuable and an independent survey of many of the crucial and technically rather demanding prerequisites of any numerical analysis of the NMSGUT scenario. Despite from that, there could still be a good case for even a further check of ours as well as Aulakh & Garg's results, in particular when it comes to the phases and matching(s).
Apart from this, the current study is rather detailed on various aspects and we aimed onto making it maximally self-contained so that a careful and patient reader could potentially reproduce all the results with just the ingredients given here and in the 'canonical' MSGUT reference [22]. On top of that, some of the extra information provided in Sections IV and VI and in particular in Sections V and in Appendix A does not have any counterpart in [35], due to a different method used therein. which is the correct result 7 conforming our phase and notation conventions. In a similar manner one can construct all the remaining maps 8 .
2. SU (3)c ⊗ SU (2)L ⊗ U (1)Y content of 120 of SO (10) The SM decomposition and the mapping of bases for 120 of SO(10) has not been provided in [22] though. Thus, in this section we shall give explicit decompositions of all the submultiplets of 120 H that have been used in the text as a basis for the various mass matrices. We shall give just a representative of each of the sectors -an interested reader can obtain all the other weights within the multiplet under consideration from the simple roots of the relevant Lie-algebras.
As before, we shall use the Ψ[i, j, k] symbols to represent the properly normalized totally antisymmetric combination of the defining components of 120, i.e.
The results are given in Table III. The phase convention used therein is the same as the one used to derive the mass matrices in section IV.