New minimal supersymmetric GUT emergence and sub-Planckian renormalization group flow

Consistency of trans-unification RG evolution is used to discuss the domain of definition of the New Minimal Supersymmetric SO(10) GUT (NMSGUT). We compute the 1-loop RGE β functions, simplifying generic formulae using constraints of gauge invariance and superpotential structure. We also calculate the 2 loop contributions to the gauge coupling and gaugino mass and indicate how to get full 2 loop results for all couplings. Our method overcomes combinatorial barriers that frustrate computer algebra based attempts to calculate SO(10) β functions involving large irreps. Use of the RGEs identifies a perturbative domain Q < ME , where ME < MPlanck is the scale of emergence where the NMSGUT, with GUT compatible soft supersymmetry breaking terms emerges from the strong UV dynamics associated with the Landau poles in gauge and Yukawa couplings. Due to the strength of the RG flows the Landau poles for gauge and Yukawa couplings lie near a cutoff scale ΛE for the perturbative dynamics of the NMSGUT which just above ME . SO(10) RG flows into the IR are shown to facilitate small gaugino masses and generation of negative Non Universal Higgs masses squared needed by realistic NMSGUT fits of low energy data. Running the simple canonical theory emergent at ME through MX down to the electroweak scale enables tests of candidate scenarios such as supergravity based NMSGUT with canonical kinetic terms and NMSGUT based dynamical Yukawa unification. ∗ aulakh@iisermohali.ac.in, aulakh@pu.ac.in 1 ar X iv :1 50 9. 00 42 2v 2 [ he pph ] 2 2 O ct 2 01 8

I.

INTRODUCTION
Renormalization group equations (RGE) are an important mathematical tool to study the evolution of the parameters (couplings and masses) of a quantum field theory with energy scale. For example the three gauge couplings of the Standard Model (SM) evolve with energy and tend to meet roughly around energy 10 15 GeV : this was the first dynamical hint supporting the "Grand Unification" vision [1][2][3]. However the SM has a problem in the sensitivity of the Higgs mass to quantum effects of superheavy particles which give rise to large loop corrections due to their circulation within loops correcting the Higgs propagator. This implies a mass correction : ∆m 2 H ∼ αM 2 X . Supersymmetry (Susy) is the best known tool to cure this problem. The two loop RGEs of gauge couplings, superpotential parameters [4] and soft terms [5,6] of a generic softly broken supersymmetric theory have long been available.
The exact relation between the beta functions for dimensionless and dimensionful couplings is also known [7]. In particular these results give the explicit formulas for the MSSM β functions which are routinely used to study the evolution of MSSM parameters from UV scales into physically meaningful quantities that describe physics near the electroweak scale.
The combination of supersymmetry and RG flows leads to nearly exact convergence of the three gauge couplings of the MSSM at M 0 X = 10 16.3 GeV. This striking and robust result has remained the most convincing hint of physics beyond the standard model for nearly 30 years since it was predicted to be possible by Marciano and Senjanovic [8] if the top quark mass was found to be near to 200 GeV and sin 2 θ W was larger than 0.23 : as was found to be the case after more than a two decades of searches and measurements [9]. Apart from the hints from neutrino oscillations this amazing convergence has for long stood as the unique guide post to the nature of extreme ultraviolet physics.
The closeness of the MSSM Unification scale to the Planck scale where gravity becomes strong has long tantalized theorists. We have advocated that induced gravity is a natural partner for Asymptotically strong GUTs [10,11] since their scale of Asymptotic Strength and UV condensation should function both as a UV cutoff for the perturbative GUT and set the scale for its contributions to the strength of gravity. Recent theoretical arguments [12] renew the old speculation [13] that the observed Planck mass may receive if there are a large number(N X ) of heavy particle degrees of freedom of mass M GU T . To our mind the most appealing scenario [10,11] is the interpretation of the strong coupling scale of the NMSGUT as a physical UV momentum cutoff. Simultaneously gravitational variables(metric,vierbein,gravitino) are demoted to the role of a background even as they are supplied with Kinetic terms by the effect of matter quantum fluctuations. Their effective action and strength are determined by GUT scale wave function renormalization of the dummy variables introduced firstly to implement general covariance. Situational boundary conditions relevant to large scale astrophysical and cosmological contexts which are, very plausibly, the only ones where gravity is actually relevant will then specify the stress densities that source the classical gravitational fields and waves. Such an acceptance of the secondary and induced nature of the gravitational field which needs no quantization might finally lay quantum gravity to rest as an irrelevant incubus, at least to the satisfaction of those concerned with testable hypotheses, provided it were anchored in a interpretation of the Planck cutoff as a physical cutoff arising from the breakdown of GUT perturbativity.
Induced gravitational kinetic terms have long been postulated [13] on grounds of perturbative wave function renormalization of the graviton due to heavy particles. If we take the Planck mass as its experimental value (10 18.4 GeV) and Λ = M X = 10 16.3 GeV this seems to indicate N ∼ 10 4 . Thus it is interesting to note that in the NMSGUT there are 640 chiral superfields and 45 Vector superfields. This large number of SO(10) coupled fields are precisely what make the couplings diverge strongly in the UV. If we count each chiral and vector superfield as 4 degrees of freedom we see that number of heavy particle degrees of freedom is N ∼ 10 3. 4 . This is in the right ball park to justify the claim that the NMSGUT corrections to the graviton propagator actually reduce gravity to the weakly coupled theory we observe. By this line of reasoning the Planck scale is determined by the unification scale of the NMSGUT or its flavour unifying generalization(the so called 'YUMGUT' [14] which has even more superfields). In this picture the Landau Pole(s) of the NMSGUT signal a physical cutoff for the perturbative GUT at a scale Λ X ∼ 10 17.0 − 10 17.5 GeV, are the scale of UV condensation driven by SO(10) gauge forces and moreover set the observed strength of gravity. Conversely the observed strength of gravity actually dictates the precise value of the UV momentum cutoff to be used when computing GUT quantum effects in any renormalization scheme. Thus the relation between different cutoff schemes is presumably deducible. However, it must be admitted that there are technical obstacles [15] in the way of these largely intuitive arguments which may not only render the Newton constant uncalculable but necessitate the introduction of the Planck Length as a fundamental parameter and require independent quantization of gravity.
In the Landau Polar region, the gauge coupling is strong and the theory has entered some sort of condensed phase [10,11]. Thus the range of scales where the gauge symmetry of the unified gauge group has unsuppressed play seems confined to a narrow range of scales ∼ 10 15.5 < Q < 10 17.5 GeV. The UV flows of asymptotically free GUTs (of which, in our opinion, no fully realistic example as successful as the realistic Asymptotically Strong Susy SO(10) models [16,17] really exists) cannot further constrain these scales and only seem to offer the picture of a weakly coupled gauge theory crushed as an irrelevance by the strength of gravity above M ef f P lanck . In contrast we argue that asymptotically strong GUTs (ASGUTs) [10,11] point to simple yet phenomenologically and calculationally viable linkage between gravity and Grand Unification of non gravitational forces and matter.
The very asymptotic strength of NMSGUT RG flows also hints how the weakly coupled gauge theory and a weakly coupled gravitational theory can emerge supernatant at large length scales upon the condensate of strongly coupled physics at the smallest length scales.
The IR flows of these theories very rapidly drive the coupling from arbitrarily strong coupling to the typical values found via RG analyses near the Supersymmetric Unification scale g 10 ∼ g 5 / √ 2 ∼ 0.5 (subscripts 5 and 10 refer to SU (5) and SO(10) normalizations for the running gauge coupling constant). From this point of view the trans-unification flows of the GUT gauge and Yukawa couplings that presumably underwrite the convergence of MSSM couplings(and third generation Yukawa couplings [18]) at or near M 0 X ∼ 10 16.3 GeV require the existence of a regime Q < M E where a perturbative unified theory actually operates as the proper renormalizable effective theory describing all particle phenomena except gravity.
The nature of the RG flows in the trans-unification or sub-Planckian regime has a vital bearing on many interesting physical questions such as flavour violation in Susy theories [19] and the freedom to choose soft Susy breaking parameters required by realistic fits beginning from simple and universal Susy breaking scenarios such as canonical Supergravity(cSUGRY) type parameters at the upper limit M E where the GUT emerges from the strongly coupled UV regime proper.
The so called New Minimal Supersymmetric SO(10) GUT(NMSGUT) based on SO (10) gauge group and the 210 ⊕ 126 ⊕ 126 ⊕ 10 ⊕ 120 Higgs system [16,17,[20][21][22] is the sim-plest and most phenomenologically successful ASGUT in existence. It has repaid thirty years of detailed investigation by exhibiting a remarkable flexibility to accommodate emergent phenomena and their associated data in one overarching calculable theoretical framework and resolve long outstanding problems of unification in terms of the quantum effects implied by its spontaneous symmetry breaking and associated mass spectra. This has resulted [16,17] in a realistic unification model which is compatible with the known data and with distinctive predictions for the Susy spectra one hopes to observe at the LHC and/or its successors.
Thus it is now topical to examine the RG flows of this theory in the sub-Planckian/transunification regime to see whether they allow consistent definition of a perturbative GUT over an appreciable energy range.
The NMSGUT requires [16,17] small gaugino masses, large squark masses and negative non universal Higgs mass squared (NUHM) soft parameters to accomplish EW symmetry breaking and fit fermion masses. Such parameters require justification, in particular for simple cSUGRY scenarios (gravity mediation with canonical gauge and scalar kinetic terms).
Soft Susy breaking parameters in minimal Supergravity (mSUGRY) are typically assumed to be generated well above the GUT scale i.e. near the Planck scale M P ≡ (8πG N ) −1/2 = 10 18.4 GeV. To consider the effect of renormalization from Planckian scales to GUT scale, when the GUT symmetry is unbroken, one needs the explicit form of GUT RGEs. As is well known the NMSGUT exhibits a Landau pole in the generic UV running of the gauge coupling [10,11] quite close to the perturbative scale of grand unification. In fact the large coefficients in the β functions of its other couplings imply the Landau Polar regime involves all couplings.
Thus there can only be a small energy interval M X < E < M E during which the NMSGUT RGEs are usable. Due to the strength of the running it can still have important effects even over the short energy range available in ASGUTs as compared to the evolution over three decades of energy in the flavour violation study of SU(5) SUGRY-GUTs [19]. If the unification program is carried out by running down simple and perturbative data initially defined at M E using first the ASGUT RGEs and then the effective MSSM RGEs(with added neutrino Seesaw and other exotic effective operators) then the rapid weakening of ASGUT couplings towards the IR ensures that a the trans-unification flow remains perturbative and the calculation well defined. On the other hand the UV flow of such theories enters the Landau Polar region just above M E implying that we must assume a physical UV cutoff Λ E M E for the whole Grand unification scenario. Beyond this energy lies the true cielo incognito where all couplings are no longer weak : "Whereof one cannot speak, one must be silent." In spite of their relevance the RGEs for the NMSGUT had so far never been presented.
In principle the application of the generic formulas of Martin and Vaughn is algorithmic and straightforward. However Computer Algebra programs [23] that aim to calculate the RG functions automatically given the Lagrangian cannot, in practice, handle the combinatorial complexity in theories with as many fields as the MSGUT or NMSGUT. Using the vertex structure of the superpotential and SO(10) gauge invariance as constraints makes the sums over the components of the large irreps (210, 126 ,126 and 120) required by the formulas of [6] tractable. The form of the RGEs for supersymmetric theories is governed by the supersymmetric non-renormalization theorem [24] whereby holomorphic(superpotential) couplings are free of renormalization except that arising from wave-function renormalization. A similar simplification is observable in the formulas for the soft couplings and masses. Once the tricks for computing the one loop anomalous dimensions are mastered the two loop anomalous dimensions and thus β functions also follow with some additional combinatorics.
In this paper we present the NMSGUT one loop β functions. However for the case of the gauge coupling and gaugino mass we also give the two loop results. We have also calculated the two loop results for the rest of the hard couplings and soft Susy breaking parameters [25] and we indicate how the methods used for the one loop calculation suffice to yield also the two loop results. The other explicit two loop formulas and their effect on running will be discussed in a sequel.
With strict assumptions such as canonical kinetic terms and canonical supergravity type soft breaking terms the gravitino mass parameter (m 3/2 ) and the universal trilinear scalar parameter A 0 (∼ m 3/2 ) are the only free parameters since then there are not even any gaugino masses, the common scalar soft hermitian mass is m 3/2 and the soft bilinear ("B" type) parameters are determined by A 0 , m 3/2 [26]. Then the soft Susy breaking parameters at GUT scale M X are determined by running down soft parameters of NMSGUT from M E with just these two soft parameters as input. Of course in general one may also consider introducing more general soft terms, but our idea here is to show the power of the SO(10) RG flow to generate suitable soft terms at M X even when placed under such strong constraints. The NMSGUT SSB and effective theory are explicitly calculable in terms of the fundamental parameters. In practice the extreme non linearity of the connection between these parameters and the low energy data implies that only a random search procedure (for parameters defined at M E ) combined with RG flows past intervening thresholds down to M Z can find acceptable fits of the SM data. The degree of confidence in the completeness of the search diminishes exponentially with the increase in number of fundamental parameters. Thus every reduction in the number of free parameters represents significant progress towards defining a falsifiable model. The present work may thus be seen as an attempt not only to improve the UV consistency but also to enforce a simplification of the fitting problem by using constraints from the consistency of trans-unification RG flows.
In fact we shall see that the SO(10) RG flow will identify an additional constraint or tuning that must be imposed to keep the soft holomorphic scalar bilinear ('B') terms for the MSSM Higgs pair in the TeV 2 region mandated by NMSGUT fits [16,17] as well as a RG flow based scenario whereby the values of the B parameters may naturally be left in this region. Various seemingly peculiar aspects of the NMSGUT parameter choices may find an explanation in terms of the RG flows at high scales. For instance the negative non universal Higgs mass squared parameters m 2 H,H which are found in NMSGUT are also justifiable by the RG flows between M E and M X . Minimal SUGRY predicts that all soft scalar masses squared are positive and equal to m 2 3/2 at the scale where they are generated. Soft gaugino masses will be generated at two loops from the other soft terms but do not arise at one loop if set to zero to begin with. This justifies the typical hierarchy we observed in NMSGUT fits whereby sfermions are in the 5-50 TeV range and are much heavier than the gauginos of the effective MSSM (which lie in 0.2-3 TeV range: depending on the lower limits imposed by hand in the search). Also the NUHM with negative masses are preferred to have controlled lepton flavor violation in Susy-GUTs [27]. Similarly choice of the SUGRY emergence scale below the Planck scale may also allow adjustment of the gaugino mass and other low energy parameters. The existence of (quasi) fixed points [28][29][30] of the RG flow is an important question with a bearing on the physical interpretation of the theory. We have analysed this question for the NMGUT RG equations but find that neither fixed nor quasi-fixed points exist.
In Section II we introduce the formulae of [6] and evaluate them in terms of the parameters in the superpotential of the NMSGUT. In Section III we present examples of running in the sub-Planckian domain. We discuss the possibility of fixed and quasi-fixed points of the NMSGUT RG flow in Section IV. A summary and discussion of our results is given in Section V. In the Appendix we collect the explicit form of the 1-loop RG β functions of the NMSGUT for soft and hard parameters.

II. APPLICATION OF MARTIN-VAUGHN FORMULAE TO THE NMSGUT
The generic renormalizable Superpotential without singlets is [ Here Φ i are chiral superfields which contain a complex scalars φ i and Weyl fermions ψ i . The generic collective indices i, j, k run over both the different SO(10) irreps of the NMSGUT and dimension of those irreps. The generic Lagrangian corresponding to Soft Susy breaking terms is given by h ijk are the soft supersymmetry breaking trilinear couplings, b ij the soft breaking bilinear masses,(m 2 ) i j the Hermitian scalar masses and M is the SO(10) gaugino mass parameter. The arrays Y ijk , h ijk , µ ij , b ij are all symmetric and we have allowed for SO(10) invariant universal gaugino masses only corresponding to canonical diagonal gauge kinetic term functions and SO(10) invariant 2-loop generation of gaugino masses.
The theory we now call the New Minimal Supersymmetric SO(10) GUT was proposed [21] by Mohapatra and one of us (CSA) in the early days of supersymmetric GUTs and was essentially the first complete and consistent supersymmetric SO(10) GUT. Its natural and minimal structure led another group [22] to independently propose it around the same time.
Following neutrino mass discovery and formulation [31] of high scale left-right and B-L breaking Minimal Left Right supersymmetric models in the last years of the last millenium, it was realized [32] in 2003 that it -and not another R-parity preserving supersymmetric SO (10) GUT based on 45 ⊕ 54 [33], nor any other competing model such as supersymmetric SU (5) with right handed neutrinos added -was the parameter counting minimal realistic Susy SO(10) model. In the same paper it was shown that the GUT SSB can be reduced to the solution of a simple cubic equation for one of the vevs. Thereafter it was the subject of intense study which calculated its spectra [34][35][36][37] and specified the roles(coupling magnitudes) required of the different Higgs representations for complete fermion fits [38][39][40] taking proper account of the role of threshold effects at M S and M X (on gauge unification). Recently we established [17,20] that if proper account was taken of the threshold effects at M X on the relation between effective MSSM and GUT Yukawa couplings then the latter -which determine fermion masses and proton decay-can emerge so small as to suppress the long standing problematic fast proton decay due to dimension 5 operators completely. The 210, 126 and 126 Higgs break Susy SO(10) to MSSM. The 10 and 120 Higgs are mainly responsible for the larger charged fermion masses while the small Yukawa couplings of 126 produce adequately large left handed neutrino masses via the Type I seesaw mechanism, instead of failing to do so due to too large right handed neutrino Majorana masses : as feared from the early days of this model [22]. Moreover these quantum corrected effective Yukawas restore a welcome freedom from the onerous constraints (such as y b − y τ y s − y µ , y b /y τ y s /y µ [41]) on fermion Yukawas imposed by the NMSGUT proposal [16,39] to use mainly the 10, 120 irreps for charged fermion masses. Thus the model as it stands [16,17] is fully realistic and invites further scrutiny. The current study is a part of an effort to simplify and unify the fitting procedure by matching the MSSM parameters implied by randomly chosen GUT parameters at M E to the electroweak and fermion mass data at M Z after RG evolution through, and threshold corrections at, the intervening scales(M X , M S ).
The Superpotential of the NMSGUT is : Here As familiar from the MSSM the chiral gauge invariants in the superpotential are the templates for the SO(10) invariant soft supersymmetry breaking terms. So corresponding to each term in the superpotential we have a soft term in L Sof tSusy . For example we havẽ λ corresponding to λ, b Φ corresponding to µ Φ and a Hermitian mass squared parameter for each Higgs representation. In all we have Our successful fits [16,17] show that fermion data and EW symmetry breaking requires negative Higgs soft masses m 2 H,H and soft parameter b H both negative with magnitudes ∼ 10 10 GeV 2 in (b H runs positive at low scales) along with gaugino masses in the TeV(gluino) and sub-TeV(Bino,Wino) range. In the following sections we will see that such initial values of the soft parameters can be generated by running of the SO(10) theory specified above even over the short range from M E to M X = M GU T and even when beginning from very restricted scenarios for the initial parameter values : such as those implied by cSUGRY.
We define the β functions at n-loop order for any parameter x after extracting n powers of 1/(16π 2 ) for convenience in presentation : -The one-loop β-functions for the SO(10) gauge coupling and gaugino mass parameter M have the generic form: here S(R) and C(G) are Dynkin index (including contribution of all superfields) and Casimir invariant respectively. Table I gives the Dynkin index and Casimir invariant for different representations of NMSGUT. We get a total index S(R)=1+(3× 2)+28+35+35+56=161.
So one-loop β functions for the SO(10) gauge coupling and gaugino mass parameter are  The general form of 1-loop beta function for Yukawa couplings is [6] : where γ (1) is the one loop anomalous dimension matrix. Thus we need to calculate the anomalous dimensions for each superfield. SO(10) gauge invariance implies that γ i j must be field-wise and (irrep) componentwise diagonal. This simplifies their computation enormously. The generic one-loop anomalous dimension parameters are given by To see what is involved in calculatingγ • For any given coupling vertex, calculate the number of ways the (conserved) chosen (1234) line gets wave function corrections from the fields it couples to in the considered vertex. Since it must emerge with the same SO(10) quantum numbers as it entered with and the counting will apply equally to every such field component, a little practice suffices to get all 1-loop anomalous dimensions.
Here M runs over remaining 6 values (M = 5, 6..10 since the 120 plet is totally antisymmetric). In this example we can have 18 possible combinations that couple to Φ 1234 . Therefore Similarly The six allowed index values for H (i.e. 5-10) give-in an obvious shorthand with SO (10) indices suppressed- The invariant kH I Θ JKL Φ IJKL will contribute to γ Thus the anomalous dimension matrix reduces to a common anomalous dimension for each independent component of each field and only for the triplicated matter 16-plets need one consider mixing.
In this way one finds that the one loop anomalous dimension for the 210-plet Φ is Using the anomalous dimensions one can compute the beta functions for all the superpotential parameters. For example the one loop β function for λ is : The formulas for the soft terms are closely analogous to those for the Superpotential couplings on which they are modelled. Indeed the exact prescription for obtaining the soft from hard beta functions is known [7] in terms of a differential operator in the couplings operating on the anomalous dimensions. This yields the generic formulae given in [6,7] [β whereγ The index patterns of the soft and hard couplings being identical one can calculate the oneloop β function for the soft parameterλ using the same counting rules used above to sum over independent loops. For example the β function for the soft trilinear analog of the 210 cubic superpotential coupling λ(calledλ) is given by : whereγ (1) (1) Φ = 1 2 Y Φmn h mnΦ are anomalous dimensions. The first was given above in eqn(15) while its soft (tilde) counterpart is γ (1) Φ = 4κκ * + 180λλ * + 2ρρ * + 240ηη * + 6(γγ * +γγ * ) + 60(ζζ * +ζζ * ) These are calculated in the way described earlier with substitution of a soft coupling (h) for a hard coupling (Y)(on which h is modelled) and thus the numerical coefficients followγ Φ closely.
The generic form of the β functions for the soft bilinear "B" terms is also known in terms of an exact relation given by the action of a differential operator in the couplings acting on the anomalous dimensions [7] and can be found in [6,7] [β Which can again be written in terms ofγ andγ . Then arguments similar to those given above yield : Similarly the Hermitian soft masses have generic β functions Again the previous results and a similar one for the doubly soft contribution (i.e. from h jpq h ipq ) yields for example for the 210 soft Hermitian mass : Thus for examplê γ (1) Φ = 240|η| 2 + 4|κ| 2 + 180|λ| 2 + 2|ρ| 2 + 6(|γ| 2 + |γ| 2 ) + 60(|ζ| 2 + |ζ| 2 ) As a final example of one loop functions consider matter field (Ψ A ) wave function renormalization due to the matter Higgs superpotential couplings where the SO(10) conjugation matrix C and Gamma matrices Γ I may be found in [37], α, β are Spin(10) spinor indices and A, B.. are the generation indices. To calculate the contribution to wavefunction renormalization we need to contract this vertex and its conjugate so as to leave Ψ Aα ,Ψ * A α as external fields. The remaining numerical factors are : Then [37] either C = C Similarly the 120 plet contributes 120(g * g T ) while the 126 − 126 pair give 252(f * f T ) (since there is a double counting of the 126-plet components due to duality within the 252 independent antisymmetric orderings of 5 vector indices).
Finally since h, f, g are either symmetric or anti-symmetric h * h T ≡ h † h, g * g T ≡ g † g etc.
The complete 1-loop anomalous dimensions and β functions are given in Appendix.

A. Two loop anomalous dimensions
In this paper we study RG flows at one loop level with two important exceptions. Firstly the gauge coupling is strongly driven to a Landau pole and it is natural to first ask what is the two loop correction to the huge positive coefficient in the one loop term. The generic two loop formula is where the factor in the last term simplifies as C(k)/d(G) = S(k)/d(k). Since for any given field type k ij Y ijk Y ijk is diagonal in field type it follows that the sum over k will just cancel the dimension of the representation (d(k)) leaving the index S(k) as an overall factor weighting the contribution of that field type in the last term in eqn. (29). This yields β (2) g 10 = 9709g 5 10 − 2g 3 10 (γ H + 28γ (1) Θ + 35γ (1) Σ + 35γ (1) The general formula for the two loop gaugino mass β function is very similar to the gauge beta function and this readily evaluates to   This concludes the β equations we need in this paper. However we have also computed the complete two loop results [25]. Here we indicate how they are computed. The two loop anomalous dimensions γ (2) are the building blocks of two loop β functions and have generic form : Thus the total contribution can be written with the help of one loop anomalous dimension parameters. For example : (1) Σ )) + g 2 10 (6240|η| 2 + 24|k| 2 + 4320|λ| 2 + 36|ρ| 2 +60|γ| 2 + 60|γ| 2 + 1320|ζ| 2 + 1320|ζ| 2 ) + 3864g  GUT compatible with the gauge and fermion data. These strong flows can explain and justify certain features of the parameter values assumed in the extensive studies we have performed elsewhere [16,17] to find fits of the standard model parameters by matching to the effective MSSM obtained from the GUT after RG flow from M X to M Z .
In our example we take SO(10) gauge and Yukawa couplings similar to those found in earlier NMSGUT fits [16,17]. Examples of these features from the two explicit fits found in [17] are quoted in Table II. We see that these parameter choices exhibit the following features : • Searches for parameters using combined trans and cis unification RG flows are not attempted here. We restrain our numerical investigations to showing that soft breaking parameters like those seen in Table II can  (for illustration) A 0 (M E )=2m 3/2 . We also require that the soft bilinears obey the strictest form of the gravity mediated scenario [26]: from M E to M X is shown in Fig. 1 and we can see that some of them become negative.
Moreover some of the B parameters also turn negative. In our realistic fits [16,17] we in fact find that the values of soft hermitian masses squared and B parameter relevant to the light MSSM Higgs at the GUT scales need to be negative (Table II)  H,H are constrained to be very light compared to the GUT scale by imposing detH = 0 on their mass matrix (H) which is calculated using the MSGUT vevs [16,32,[34][35][36][37]. The left and right null eigenvectors of H furnish the "Higgs Fractions" [16,32,34] whereby the composition of the light doublets in terms of 6 pairs of GUT doublets is specified and the rule for passing to the effective theory : h i → α i H,h i →ᾱ iH defined. Then the soft hermitian scalar mass terms will give Since m 2 i can turn negative when running from M E to M X we see that negative m 2 H,H can be achieved. However note that one also has the b ij terms for each of the GUT Higgs multiplets so that one will in fact also induce the B term for the light Higgs as An additional constraint(analogous to that imposed on µ) to maintain the B term at magnitudes less than 10 10 GeV 2 (rather than the RG evolved values which tend to have magnitude      parameters may provide a mechanism whereby the short flow lands these parameters closer to the TeV scale values required.

IV. CONCERNING INVARIANT PARAMETER SUBMANIFOLDS UNDER THE MSGUT RG FLOW
The seminal work of Pendleton and Ross [28] on quasi fixed points of the SM RG flow successfully estimated the approximate top quark mass before its discovery on the basis of the intuition of an approximate "quasi infra red fixed point" in the RG flow governing the ratio h 2 t /α s . Since then the same basic idea has been applied to the dimensionless and even dimensionful(i.e. soft) parameters of (Susy) GUTs to study [29] whether the quasi-fixed point structure of the GUT Yukawa and gauge couplings may be significant in fixing the couplings at the Unification scale. It was found that such structures are particularly relevant in the case where there are many fields so that the GUT model is strongly coupled in the ultraviolet. This is precisely the case for the MSGUT and NMSGUT. If such invariant structures could be identified they would obviously be an important criterion for comparing different unified models. In the present instance we have calculated the full set of RG equations for both hard and soft couplings of the MSGUT. Thus the optimistic view might be that these complex flow equations somehow support novel invariant structures when considered in their entirety. The generic form of the 1 loop beta functions for the dimensionless (gauge and Yukawa ) couplings of a supersymmetric model is where γ i is given by eqn(9) after using diagonality ( γ i j ≡ γ i δ i j ) of the anomalous dimension matrices , and b 0 is a large integer or rational number (b 0 = 137 for the MSGUT ). It is clear that combining these two equations and the 1-loop formula for γ i we can derive fixed point conditions in terms of Z ijk = |Y ijk | 2 /g 2 for the squared magnitudes of various couplings , while their phases remain free. These conditions are readily seen to be generically of the form where we have separated out the gauge (b ijk )and Yukawa(γ i,j,k ) components of the anomalous dimensions for fields i,j,k. Writingγ i = a I i |Y I | 2 where I runs over the different Yukawa couplings in the theory we get the fixed point conditions in the form where The question as to whether any quasi fixed points of the full set of RG equations can possibly exist then involves solving these equations subject to the constraints that all Z I are positive semi-definite. Unfortunately the huge value of the coefficient b 0 which is common to all the conditions makes a solution impossible to achieve.
We illustrate the difficulty for a simplified MSGUT model with negligible first generation matter Yukawas (h, f, g) 1A 0,diagonal h, f couplings h 2,3 , f 2,3 and g 32 = −g 23 . The relevant anomalous dimensions arē γ Φ g 2 = 4Z k + 180Z λ + 2Z ρ + 240Z η + 6(Z γ + Zγ) + 60(Z ζ + Zζ) γΣ Consider first the case where the couplings to the 16-plets have been set to zero(Z h 2,3 ,f 2,3 ,g 23 ≡ 0). The fixed point conditions for the other couplings are then Solving these fixes Z η,k,ζ,ζ,ρ in terms of Z λ,γ,γ and for them to be semipositive gives 5 inequalities which can be easily reduced by eliminating Z γ between them. However this yields the condition Z λ ≤ −227/2700 which is inconsistent with the semipositive values allowed for Z λ . So there is no fixed point.
One might hope that introducing the 16-plet couplings might help. Then we restore Z h 2,3 ,f 2,3 ,g 23 and obtain the additional conditions for the beta functions of these ratios : Solving these conditions one finds that Z ζ,ζ,h 3 ,f 2 ,f 3 ,g 23 ,ρ are determined in terms of Z η,γ,γ,k,λ,h 2 ≥ 0 which are themselves undetermined. The question is whether there are any semipositive values of these free parameters for which the dependent variables remain semipositive. Solution of the fixed point conditions yields the following solution vector An elementary reduction of this system of inequalities [42] leads to contradictory condition showing that again there is no fixed point for the system even when measuring in units of the (exploding in the UV) value of g 2 . Although we have not obtained a general proof it seems likely that no fixed point can be found. Support for this can be found in recent investigations [30], based on the so called non-perturbative a-theorem and the exact NSVZ beta function(see [30] for a concise introduction and a fairly complete list of references for these topics), of the possibility of non-trivial superconformal UV fixed points in the SO (10) MSGUT. They conclude that no such fixed points exist without rather artificial requirements being placed upon the couplings and R-charges of some of the SO(10) multiplets or by introducing very large numbers of additional multiplets and trivializing the superpotentials allowed. used in NMSGUT fits [16,17]. Note that the distinctive normal s-hierarchy at low scale is strongly correlated with the large negative M 2 H,H we use in the fits. Gaugino masses(M λ ) will be generated by two loop RG evolution between M X and M Z , even if M λ =0 at the scale M X . On the other hand the same applies to the evolution between M E and M X . Thus even canonical gauge kinetic terms in the GUT can still generate adequate gaugino masses. This is pleasing since we have always resisted invoking non canonical Kähler potential and gauge kinetic terms on grounds of minimality/predictivity and to preserve renormalizability of the gauge sector. The running of trilinear soft coupling and s-fermion mass squared parameters (m 2 Ψ ) will give distinct values at the GUT scale for the three generations (considered the same in earlier studies of NMSGUT [16,17]). In sequels we will integrate these RG flows with our previous code that incorporates the MSSM flows between M X and M Z . Then one will throw the core soft parameters m 3/2 , A 0 at M E and run down over thresholds to M Z with one additional fine tuning constraint. Thus the total number of soft parameters will be significantly reduced. Improvements would include the 2-loop RG coefficients we have already computed [25]. Finally the straightforward (since the superpotential vertex connectivity is preserved) generalization of these results to the case of YUMGUTs [14] will allow us also to perform the RG flows from the Planck scale for dynamical flavour generation models based on the MSGUT. These theories have around 6 times as many fields as the NMSGUT and are thus even more capable of separating M X and M P l . We note that the techniques we have used to actually evaluate the 2-loop RGEs have overcome the combinatorial complexity that prevented their calculation by automated means. They can be used for any Susy GUT.  One-loop beta functions for the SO(10) superpotential parameters and Yukawa couplings are: β (1) γ = γ(γ (1) (1) (1) (1) H + 4µ Hγ (1) (1) Φ m 2 Φ + 720m 2 Φ |λ| 2 + m 2 H (12|γ| 2 + 12|γ| 2 + 8|κ| 2 ) +m 2 Θ (8|ρ| 2 + 120(|ζ| 2 + |ζ| 2 ) + 8|κ| 2 ) + m 2 Σ (480|η| 2 + 12|γ| 2 + 120|ζ| 2 ) +m 2 Σ (480|η| 2 + 12|γ| 2 + 120|ζ| 2 ) + 2γ (1) H m 2 H + m 2 Φ (252(|γ| 2 + |γ| 2 ) + 168|κ| 2 ) + 168m 2 Θ |κ| 2 + 252m 2 Σ |γ| 2 +252m 2 Σ |γ| 2 + 2γ (1) (1) Θ m 2 Θ + m 2 Φ (14(|κ| 2 + |ρ| 2 ) + 210(|ζ| 2 + |ζ| 2 )) + 14m 2 Θ |ρ| 2 + 14m 2 H |κ| 2 +210m 2 Σ |ζ| 2 + 210m 2 Σ |ζ| 2 + 2γ (1)