Grand Pleromal Transmutation : condensates via Konsishi anomaly, dimensional transmutation and ultraminimal GUTs

We show that dimensional transmutation via gaugino condensation {\emph{in the ultraviolet}} drives gauge symmetry breaking in a large class of {\emph{asymptotically strong}} Supersymmetric gauge theories. For Adjoint multiplet type chiral superfields $\Phi$ (transforming as $r \times \bar r$ representations of a non Abelian gauge group G), solution of the Generalized Konishi Anomaly(GKA) equations allows calculation of quantum corrected vevs in terms of the dimensional transmutation scale $\Lambda \simeq M_X \, e^{\frac{8\pi^2}{ g^2(M_X) b_0}} $ which determines the gaugino condensate. Thus the gauge coupling at the perturbative unification scale $M_X$ generates GUT symmetry breaking vevs by non-perturbative dimensional transmutation. This obviates the need for large(or any) input mass scales in the superpotential. Rank reduction can be achieved by including pairs of chiral superfields transforming as either $({\bf Q}(r),{ \bf\bar Q}(\bar r))$ or $ (\Sigma((r\otimes r)_{symm})), {\overline{\Sigma}}(({\bar r \otimes\bar r})_{symm})$, that form trilinear matrix gauge invariants $\bar Q\cdot \Phi\cdot Q, {\overline{\Sigma}} \cdot \Phi\cdot \Sigma $ with $\Phi$. Novel, robust and {\emph{ultraminimal}} Grand unification algorithms emerge from the analysis. We sketch the structure of a realistic Spin(10) model, with the $16$-plet of Spin(10) as the base representation $r$, which mimics the realistic Minimal Supersymmetric GUT but contains even fewer free parameters. We argue that our results point to a large extension of the dominant and normative paradigms of Asymptotic Freedom$/$IR colour confinement and potential driven spontaneous symmetry breaking that have long ruled gauge theories.


Introduction
It has been a longstanding dream [1] to provide a mechanism for dynamical generation of the Grand Unification scale from the low energy (i.e. Electro-weak) data. Asymptotic freedom(AF) of the Grand Unified gauge coupling(s) has been a generally unquestioned requirement for acceptable unification models. Some years ago, motivated by the glaring Asymptotic strength(AS) of the couplings of the phenomenologically satisfactory Minimal Supersymmetric Spin(10) model (MSGUT) [2,3], we proposed [4] that this 'defect' is actually a signal from the model that it generates its own UV cutoff in the form of a Landau polar scale associated with dynamical symmetry breaking of the Spin(10) gauge symmetry driven by gaugino condensates in the Ultraviolet. Inspired by their defining role in our proposal for robust parameter counting ultra-minimal AS Grand Unification we called such condensates pleromal. We were particularly enthused by the observation that -in contrast to, say, Susy QCD-the nearly exact Supersymmetry at the GUT scale implies that the UV dynamics of the MSGUT, and other AS Susy GUTs, are physically the best justified and most realistic context for the use of the powerful methods [5] for analysing strongly coupled Susy theories.
A toy AS GUT model with gauge group SU (2) and a single symmetric chiral 5-plet was used to explore the derivation of symmetry breaking vevs from the gaugino condensate assumed to exist in Susy Yang mills theories at strong coupling in the UV. The use of the Konishi Anomaly(KA) allowed us to argue that a vev driven by the gaugino condensate might well develop. Shortly thereafter, a sophisticated and powerful method based on the Generalized Konishi anomaly (GKA) was invented [6] which allows the calculation of the condensates of the "Chiral Ring" generators tr((W α W α ) n Φ m )|n = 0, 1, m ∈ Z ≥0 (where W α is the gaugino-field strength multiplet, Φ the N × N adjoint multiplet of the Unitary gauge group and the trace is in the N-dimensional fundamental since the adjoint has been written as an N × N matrix). Thereafter this method enjoyed a great vogue and was also extended [7,8] by the addition of pairs of fundamental ("quark" ) chiral multiplets leading to either Higgs or "pseudo-confining" vacua, or [9] by other sets of chiral supermultiplets, such as (anti)symmetric representations : which provide more general rank breaking scenarios with several novel and non-trivial features in their strong coupling dynamics. In this letter we argue that GKA techniques allow calculation of vevs in our [4] asymptotically strong(AS) dynamical symmetry breaking scenario in an vast class of Susy gauge theories with chiral multiplet Φ transforming as r ⊗r for any representation r of any gauge group G, further supplementable by representation pairs {Q,Q}; {Σ, Σ} that can form singlets Q · Φ n · Q, Σ · Φ n · Σ. Crucially, Φ, Σ, Σ can be written as matrices (with rows and columns labelled by indices running over the dimension d(r) of a general representation r larger than the fundamental : which we call the base representation of the model). The basic idea of using a tensor product to define a matrix type representation can be extended to gauge groups other than Unitary groups. In particular it applies to the product of spinorial representations of Spin(N ). By imposing trace constraints on the matrices Φ, Σ, Σ that set some, or all but one, irreps contained in the direct products r ×r, r × r etc to zero one can project out smaller reducible or irreducible contents. However due to the increase in calculational load, we defer such refinements to sequels.
The focus in the literature has been on the restricted class of asymptotically free models and the derivation of effective Wilsonian superpotentials to describe the (strongly coupled) low energy theory. In the GUT application case one rather wishes to know the Higgs vacuum expectation values(vevs) that must be substituted in the Lagrangian to derive the perturbative effective MSSM. Thus rather than an effective superpotential for IR condensation one needs a definition for the quantum corrected vevs and associated effective superpotential(whose equations of motion the quantum corrected vevs solve). On the other hand, since the gaugino condensation occurs at huge scales in in the UV, the issue of calculating supersymmetric condensates for the low energy gauge group is vacuous in scenarios where supersymmetry is broken at scales above TeV : as is the case with all realistic Susy GUT models. Thus our focus here is to explore the generation of the GUT scale vevs by dimensional transmutation, especially when the mass parameters in the superpotential are absent or negligible.
In the semi-classical case the vev of Φ is diagonal (to zero the D-terms) and carries the critical points a i of the superpotential W (z) placed on the diagonal with multiplicity N i ( i N i = N = d(r)). We propose that the classical (quantum) vev is given by a contour integral of z T (z) around the critical points(branch cuts) of the (quantum corrected) superpotential. In the classical case this obviously yields the vevs. In the quantum corrected case the analysis of [6] shows that the poles are resolved into branch cuts that form the A-cycles of a Riemann surface associated with the factorisation problem of a certain quantum corrected polynomial given by the solution of the GKA equations for certain resolvents. Correspondingly the effective superpotential, at least for purposes of calculation of the spectrum of the perturbative effective theory obtained at low energies after the dynamically driven spontaneous symmetry breaking, should be obtained by modifying the coefficients of the original superpotential so that the new vevs are solutions of the F-term conditions of the effective superpotential. We assume that asymptotic strength results in gaugino condensation for the gauge group G as a whole, though, as we will see, the condensates of gaugino submultiplets lying in the decomposition of the adjoint of G w.r.t. the little group H are distinguishable from each other due precisely to the gauge symmetry breaking. In this work we shall outline the generic features of our proposal applicable to a large class of models with arbitrary gauge group, noting in conclusion only the principal features of its application to a realistic SO(10) model. In short, we suggest a quite novel approach to the hoary problem of GUT symmetry breaking which realises the old dream of symmetry breaking scale determination by dimensional transmutation, not in a engineered perturbative model [1,11], but generically and robustly in an infinite class of models of a type hitherto largely neglected.
In Section 2. we discuss how the techniques of [6] for Adjoint Multiplets extend to an infinite class of "Adjoint multiplet type"(AM) models based on a generalized notion of baser tensors transforming as direct products r ×r of the gauge group G. In Section 3. we give a simple example of this extension with gauge group SU(3) and base representation r = 6 and illustrate calculations with some numerical results using the formulas developed in Section 2. In Section 4. we discuss how rank breaking may be implemented by introducing additional (r,r) or (r × r,r ×r) pairs of Chiral supermultiplets whose vevs reduce the rank of the little group H. In Section 5. we outline a realistic Spin(10) GUT model with base representation r = 16. In Section 6. we discuss our results and the outlook for further work from a general view point.

Generalized Adjoint models
In a supersymmetry preserving vacuum of a super-Yang Mills theory with gauge group G coupled to a Chiral multiplet Φ in an arbitrary representation R of the gauge group, and with superpotential W (Φ), the GKA implies [6,9,13] the following relation for condensates of chiral gauge invariants formed from the Gaugino-Field strength Weyl spinor chiral multiplet W A α , A = 1...dim(G), α = 1, 2 and the chiral fields Φ I , I = 1....dim(R) : Here f (W α , Φ) I is an arbitrary chiral variation of the field Φ in the representation R with generators M A . Repeated indices I, J, K are summed over dim(R) values. The important constraint equation whereby W α on the "vector form" of the general representation Φ ∼ R is equivalent to zero in the "Chiral Ring" [6] is frequently used in simplifying expressions for chiral expectation values. Henceforth, we drop angular brackets to indicate expectation values of operators since that is all we ever consider. For the case where Φ transforms as the traceful adjoint of U (N ) the method of [6] allowed a complete solution for the generators of the Chiral ring of gauge invariants t n,m ∼ tr((W α W α ) n Φ m )(n = 0, 1; m ∈ Z ≥0 ). The solution proceeds by solving for the generating functions of the Chiral Ring generators defined as the resolvents tr(W α W α ) n (z − Φ) −1 . These have obvious expansions as power series, valid for large z, whose coefficients are the Chiral Ring Generators t n,m . In [7,8] the extension to the case with additional ("quark") superfield pairsQ, Q which can form a singlet with Φ asQ·Φ n ·Q and in [9] the similar case of the Adjoint with conjugate pairs of (anti)symmetric representations (Σ, Σ) are resolved. These additional models allow consideration of rank breaking supersymmetric Higgs vacua not available with just an adjoint. Thus tr is taken so that the matrix indices run over 1...d(r) = N . The semi classical vacuum of the model is defined by distributing the critical values a i (i = 1..n|W ′ (a i ) = 0) of the superpotential function W (z) (degree n + 1) over the diagonal slots of Φ and setting the off-diagonal elements to zero (this ensures minimisation of the D-term contributions to the potential via D A (Φ, Φ * ) ∼ [Φ, Φ † ] = 0). One can extract various interesting quantities, such as the number N i of times a critical point a i is repeated or the value of gauge invariant chiral ring generators, via integrals of z m {R(z), T (z)} etc. around suitable contours C i . Our basic observation is that the equations for resolvents derived via the GKA in [6] also hold for the Adjoint Multiplet type(AM ) field transforming as r ×r for any representation r of any simple/semisimple gauge group G with the sole replacement of the trace (tr) in the fundamental of SU (N ) by the trace (T r) in the representation r of G. Note that d(r) = N for the model of [6] but their results generalise easily using the expanded notion of an "base-r Adjoint", written as a d(r) × d(r) matrix, that transforms as r ×r rather than as N ×N. In other words, under a gauge transformation where U d(r) are d(r) × d(r) dimensional Unitary matrices in the representation r of G. In general, r ×r contains one or more irreps of G, besides the singlet and the adjoint always present, and such representations generally carry large S 2 (R) >> 3C 2 (G). By constraining the matrix Φ so as to single out one or more irreps ( which are still AS) one can work in terms of irreps of G. This complicates the calculations and hence we shall defer such procedures to sequels.
Where S is the gaugino condensate and S 2 (r) the index of the representation r : T r(T A (r) T B (r) ) = δ AB S 2 (r). It is obvious that v cl i = a i = I[C i , z T (z)]/N i extracts the vev v i = Φ ii of an N i -fold replicated diagonal Φ component from the semiclassical T (z). This motivates the vev definition in the quantum corrected case when C i encircles the pairs of branch points into which the semi-clasiscal critical points split under the influence of quantum corrections [6].
In the presence of quantum corrections the GKA method of [6] solves for the generating functions R(z) using the position independence and complete factorizability [6] of Chiral ring operator correlators to reduce the GKA (for the case where δΦ ∼ (W α W α (z − Φ) −1 ) to a quadratic equation for R(z) : where f (z) is a degree n − 1 polynomial which can be determined and W (z) is just the superpotential (of degree n + 1) as a function of z. Since our entire focus is on the relevance to renormalizable GUTs, we shall only consider cubic superpotentials (W (z) = λz 3 /3 + m z 2 /2+µ 2 z). Then f (z) is a linear function f (z) = f 0 +f 1 z = −4λ(R 1 +zR 0 )−4mR 0 . The coefficients R 0 , R 1 are not determined by the GKA and should be regarded as dynamical moduli of the vacuum manifold of the theory which are to be determined by an appropriate numerical investigation of gaugino condensation at strong coupling. Some information about the main contribution to R 1 /R 4/3 0 can however be gleaned by surveying the GKA constraints numerically.
In the familiar case of asymptotically free Susy YM with Adjoint Chiral Higgs the gauge coupling runs to a Landau pole in the infrared at a (RG invariant) scale Λ approximated by the one loop value (exact for the Wilsonian gauge coupling) where b 0 = −2N for the adjoint-SYM. There are good arguments [12] to support the conjecture that strong coupling causes the development of a gaugino condensate at this scale Where the constant b is scheme dependent and may be chosen to be 1 [6]. Our core assumption is that gaugino condensation also occurs when the gauge coupling runs to a Landau pole in the ultraviolet i.e. for the case b 0 (R) = S 2 (R) − 3C 2 (G) > 0. Note that S 2 (r ×r) = 2 d(r) S 2 (r), which grows fast with d(r), so that b 0 (r ×r) > 0 for most base representations r. We employ it to deduce, via the GKA relations and consistency conditions, the symmetry breaking quantum vevs of Φ that define the effective low energy theory as functions of the basic gaugino condensates and the superpotential parameters. The analysis is performed at a scale where all the degrees of freedom of the Super YMH theory are retained with the RG invariant gaugino condensates as a given background. Scale dependent quantities such as superpotential parameters and vevs should be regarded as defined at such an intermediate scale where the gauge and superpotential couplings are still perturbative. Note that the beta function for the cubic superpotential coupling is Thus the strong divergence of g 2 in the ultraviolet drives dλ/dt negative. We will not try to explore the λ-dependence near the Landau pole but confine ourselves to the influence of the gaugino condensate in the perturbative region.
By assumption, a gaugino condensate S for the group as a whole develops in the ultraviolet. This gaugino condensate then requires (via the GKA and its solution) that the fields develop vevs (calculated below) which break the gauge symmetry in a manner dictated by the placement of these quantum corrected vevs v (q) i on the diagonal of the Φ vev in any way we choose. This placement of vevs determines the little group H in practice. The different gaugino bilinears condense in a pattern determined by the vev placement.The assumption that one may choose the N i (subject to d(r) = i N i ) to be fixed integers is an important constraint on dynamical symmetry breaking.
The solution of the GKA equation for R(z) is For a cubic superpotential (n = 2) the expansion of T (z) for large z and the above solution The square root of the polynomial y 2 (z) = W ′ (z) 2 + f (z) encodes the 'quantum' superpotential derivative, which is distinct from the classical one if R 0,1 = 0. The higher coefficients R n , T n which are not present in the solutions y(z), T (z) are determined in terms of R 0,1....n−1 , T 0,1...n−1 by the GKA relations. For the cubic case (n = 2) the unknowns are thus R 0,1 , T 1 . The gaugino condensate R 0 = 2S 2 (R)S = 2S 2 (R)Λ 3 sets the scale of all other condensates and is estimated directly from the running of the perturbative gauge coupling in the full theory. Thus, for numerical work, we can conveniently rescale all our expectation values and integration variables to dimensionless forms using units of R 1/3 0 . For a cubic superpotential R(z), T (z) the first few dependent R n , T n are : The superconformal case where m = µ = 0 is particularly simple and interesting : If, restoring angular brackets for a moment, we separate Φ(x) =< Φ > +Φ(x) wherẽ Φ has zero vev, we see that This sort of interplay between non-gauge invariant quantum correlators and non-gauge invariant vevs can yield gauge invariant resolvent coefficients R n , T n . Thus although we shall use vacuum expectation values deducible from T (z) to calculate masses and define a effective theory with spontaneously broken gauge group at low energies, we must keep in mind that due to the strongly non-perturbative physics at high energies the GKA relations imply a host of strong constraints on higher order chiral correlators whose implications for the effective theory remain to be explained. The phenomenological implications of these constraints for correlators involving quantum fields which are known to describe light particles, like SM fields, are not clear to us. We regard this hybrid of a perturbative effective theory with strong correlations due to underlying microscopic strong coupling as one of the most puzzling and intriguing implications of our results : which may provide fresh insight on how to think about vacua arising non-perturbatively in strongly correlated systems.
We argue below that consistency of the GKA relations with the choice of N i also determines T 1 in terms of contour integrals involving y(z) and the integer repetition numbers .n on the diagonal of Φ which, following [6], we assume to be free inputs stable against quantum deformation. This only leaves R 0 and R 1 = κ T r ΦW α W α undetermined. We shall see below that the GKA equations offer insight and an estimate even for this dynamical modulus : at least in cases where the effects of the condensation are primarily encoded in the quantum vevs and gaugino condensates.

Determination of Quantum vevs
The essence of the analysis of [6] in the AF case is that the influence of gaugino condensation modifies the polynomial y 2 (z) such that its zeros are bifurcates a which are the branch points of y(z) (i.e. they merge when the quantum condensates R 0 , R 1 are sent to 0 and y(z) reverts to its classical value W ′ (z)). The polynomial y 2 (x) defines a two sheeted Riemann surface(of genus 1 when n = 2 since y 2 is then quartic). The contours C i enclosing the semi-classical critical points a i now become contours A i enclosing branch cuts running between the corresponding pairs of branch points a [6] a (once redundant) basis for the A-cycles of the Riemann surface defined by y 2 (z). The definitions of the sub-condensates in the quantum case involve integrals around the A-cycles : (2.14) It is clear that as f (z) → 0 the vevs approach their classical values v (q) i → a i . As emphasized in [6], the integers N i should not change even when evaluated as In contrast to the r = N, Φ ∼ N × N case studied in the AF case in [6], the subcondensates R 0i correspond not to unbroken subgroup factors' gaugino condensates but instead to certain combinations of the gauginos of the unbroken gauge sub-group H and the G/H coset gaugino condensates. The combination relevant for R 0i can be identified by evaluating only the unit diagonal elements in the sector corresponding to the cycle A i are retained and the others are set to zero. This is illustrated explicitly with an example in the next section. The consistency of these contour integrals with the Laurent expansion even in the quantum corrected case is ensured by the fact that since the region enclosed by the curves ∪ j A j ∪ (−C ∞ ) is free of singularities or branch cuts. For n = 2, T (z) = (d(r)(zλ + m) + λ T 1 )y −1 , and we can solve the definition of N 1 to obtain a consistency condition of the assumption [6] that the value of the integrals giving N i around the critical points a i do not change under quantum corrections. From eqn(2.15)we get The equation for N 2 gives nothing fresh because of the complementarity of the integrals around C ∞ and the union of the A cycles. When n > 2 we can similarly obtain equations for T 1 ....T n−1 by using the definitions of N 1 ...N n−1 and solving the n − 1 linear equations for the T i . The dynamically determined condensate R 1 (at or near the perturbative unification scale where the whole symmetry breaking evaluation of the low energy effective Lagrangian is performed) is required to proceed further. R 1 can be rigorously determined in terms of Λ and the superpotential parameters only by an numerical calculation of condensates in the full Super YMH theory. However in favourable cases (see the section on Rank Breaking models below) even R 1 can emerge determined in terms of R 0 .
In the pure r ×r case, analysis of the GKA equations also yields insight and constraints upon the form of R 1 . Closed form evaluation of the elliptic or hyper-elliptic type integrals involved is difficult since λ, R 1 are, in general, complex. Numerical evaluation of expressions for for which the dimensionless semi-classicality parameter is small : While we cannot calculate the dimensionless condensate R 1 /R 4/3 0 we do find that there are regions of the dimensionless parameter space (λ, where δ sc is very small. Such regions in the parameter space are thus candidate vacua where the gaugino condensate and the quantum corrected vevs encode most of the dynamical information relevant for defining the effective perturbative theory below the scale of dynamical gauge symmetry breaking. We propose that the vevs v (q) i be used to define a consistent effective Lagrangian for calculating mass spectra in the spontaneously broken theory by modifying the semiclassical superpotential W (z) to a quantum modified or effective superpotential W (q) (z)by changing the coefficients in W (z) so that the vevs obtained are zeros of W (q) ′ (z) i.e. W (q) ′ (v (q) i ) = 0. Thus, for example for a cubic superpotential, we take : and one recovers the original superpotential. This proposal has the virtue that the cubic coupling describing the interactions has not been modified and the changes made are only in the soft super-renormalizable couplings. Use of such a superpotential will ensure that the super-Higgs effect and spectrum calculations using v (q) i are consistent.
For evaluating the contour integrals around the A i cycles we should first define the square root branched function y(z) to lie unambiguously on the first sheet : where θ(z) ∈ (−π, π] is the quadrant wise correct polar angle of the complex number z. Then the integral over the A i -cycle which encloses the branch cut running from z 2i to z 2i−1 is achieved by (P.V. denotes principal value and the function g(z) should be such such that the contribution from the end circles around the branch points is zero) If the dynamical behaviour supports the emergence of an effective spontaneously broken perturbative theory then we expect that j R 0j v j ≃ R 1 upto small quantum corrections due to irreducible three point correlation functions of two gaugino superfields and Φ. We can scan the parameter space of superpotential couplings together with R 1 (in units of R We show instances of such parameter regions in an explicit example below (see Table 1.).

A Simple Example
The diagonal 3 × 3, 2 × 2, 1 × 1 blocks of Φ are occupied by representations that are Y singlets and contain SU(2) singlets so that where V I , I = 1.
2 the mass term will vanish. In case A this can happen for 5 index pairs(1/2/3/4/5 paired with 6), Case B for 8 base rep index pairs (4/5 paired with 1/2/3/6) and in Case C for 9 (1/2/3 with 4/5/6). Thus these are the numbers of off-diagonal index pairs (out of the total of 15 ) which remain massless. Since we expect Dirac partners only for the 4 gauginos of the coset SU (3)/(SU (2) × U (1) Y ), pseudo Goldstone(PG) multiplets arise. The 4 coset gauginos transform as two doublets of SU (2) with Y = ±3 and pair up with 2 doublet pairs from the above enumerated massless pairs of conjugate fields. The 6 diagonal pairs are of course always massive. For case A ψ 33 α3 (ψᾱ 3 33 ) teams up with λᾱ 3 (λ 3 α ) respectively while ψ 33 αβ ( and ψᾱβ 33 ) remain massless. However we shall see that introduction of rank breaking fields gives these putative PG (PPG) multiplets a mass.

Gaugino sub-condensates
The gaugino sub-condensate patterns are described by the values of the SU(2) triplet con-  with the total of the coefficients of each gaugino squared equal to S 2 (6 SU (3) ) = 5/2 as expected. The computation of the trace can be easily carried out by setting up the 6×6 generators of SU (3) in the 6-plet representation using the symmetric 6-plet generators obtained from the symmetrized tensor product:

Numerical investigation of semi-classicality
In Table 1 we give instances of the calculation of the vacuum expectation values for this model taking m = µ = 0, for simplicity, for which δ sc defined in eqn(2.18) is indeed small and the semiclassical approximation R 1 ≃ j R 0j v j is good. Note that the nonzero values of v

Rank Reduction
Since the elements of Φ ∼ r ⊗r with vevs are neutral w.r.t. all the Cartan subalgebra generators the gauge group rank cannot decrease due to symmetry breaking in any purely AM type model. However GUT models (such as those based on SO(10) ) with rank ≥ 5 require rank reduction to break the gauge symmetry to the SM gauge group which has rank 4. In the Minimal SO(10) GUT [2,3], just such a rank reduction is achieved by including a pair of conjugate representations (126, 126) whose role is precisely to break SO(10) → SU (5). The authors of [6] have also provided [7,8] an analysis for Adjoint-SYM with Flavours, i.e. super SU (N c ) YM with an adjoint as well as N f pairs of Quark fundamental -anti-fundamental N c -plets Q f ,Q f , f = 1...N f . Such models possess "Higgs vacua" with < Q f >, <Q f > = 0, f = 1.., n, which imply rank reduction N c −1 → N c −n−1 which is unachievable with AM type fields alone.
Consider N f = 1 i.e. with a pair of complex representations Q,Q transforming as r,r added to AM Φ ∼ r ×r. We modify the superpotential while maintaining renormalizability by adding gauge invariant terms (∆W = W Q = −ηQ · Φ · Q). We have omitted a Quark mass term by shifting Φ. The model then admits semi-classical "Higgs Vacua" in which parallel components of the complex multiplets Q,Q, say Q 1 ,Q 1 , obtain vevs (of equal magnitude to cancel the D term contributions) leading to a lowering of the rank of the little group by one : in the Chiral ring, the loop equation containing R 2 (z) is unchanged from the pure AM case(eqns(2.5,2.8)) but the equation for T (z) is modified to : The GKA for δQ ≡ 1 z−Φ · Q or δQ ≡Q · 1 z−Φ give Then if we impose we findQ Which reverts to its classical value W ′ (0)/η as y(z) → W ′ (z) i.e. in the absence of quantum effects a.k.a gluino condensate. The vevs of Q,Q lead to useful mass matrix contributions. For example for a SU(3) model based on the 6-plet (i.e φ ∼ 6 ⊗6) discussed above we take Q 6 = Q (33) = σ,Q 6 =Q (33) =σ. These vevs Dirac-pair Q (33) , Q (α3) with gauginos λ 3 3 , λ α respectively. Moreover they modify the masses of the putative pseudo-Goldstone (PPG) multiplets. For example in case A where V (ᾱβ) = Vᾱ 3 = V (33) , the PPGs Φ Dirac-pair withQ (ᾱβ) , Q (ᾱβ) and become massive. We can define an effective cubic superpotential that works to reproduce the quantum vevs along the same lines as in the pure adjoint case by modifying the parameters of the cubic superpotential so as to support a Higgs vacuum solution with Φ 11 = 0 even in the quantum case but with:

r × r based rank reduction
Besides rank reduction based on complex representations ∼ r,r, one can also consider more complicated scenarios based upon pairs of representations ∼ (r × r) s , (r ×r) s which prove useful in realistic scenarios(see the next section). Following and extending [9,10], we survey the solution of the resolvent system for this case. Introduce a pair of chiral supermultiplets Σ ij = Σ ji , Σ ij = Σ ji ; i, j = 1...d(r) transforming as (r × r) symm , (r ×r) symm and an additional superpotential As in the QQΦ case there is a semiclassical Higgs vacuum where one conjugate component pair from Σ, Σ (say (Σ M M , Σ M M )) gets a vev : For example in the case of the SU(3) model based on the 6-plet S ij = S ji , Σ, Σ are 6 × 6 symmetric matrices and we can break SU (3) → SU (2) by Since S (33) is an SU(2) singlet but has Y = −4 it is clear that the vevs of σ,σ = 0 reduce the rank by 1. As in the case with d(r)-plet rank-breaker pairsQ, Q the PPG spectrum becomes massive due to the rank breaking vevs. By considering a combination of the loop equations for GKA variations δΦ = κW α W α (z± Φ) −1 and δΣ = 2κW α (z − Φ) −1 · Σ · (W α (z + Φ) −1 ) T and using the Chiral ring constraints W α j (i Σ k)j = 0 (and similarly for Σ) one derives [9] the loop equation (notation For a cubic superpotential r 1 = f 0 /2 = −2κT r(λ Φ + m)W α W α . Due to branch cuts in the z plane (that emerge further on) the resolvent function R(z) is not even in z. Introducing a new resolvent (the analogue was automatically zero in the previous case due to the Chiral ring constraint W α · Q ≃ 0) : we obtain a modified equation for R 2 : and a similar equation forR. One can then show [9,10] that by substituting where u a , a = 1, 2, 3 are solutions of a cubic equation of a special form Here s(z) = s 0 (z) + s 1 (z) and s 0 , s 1 are polynomials of degree 3n, 2n − 2 which can be explicitly calculated [9] given W (z) : Thus one finds In spite of the cube root, the Riemann surface branching structure for ω(z),ω(z) is still two sheeted provided ω − (z) = ω + (−z) = r(z) 3ω + . The third root ω(z) +ω(z) occupies an isolated 'singleton sheet'. Since s(z), r(z) are even polynomials the condition on ω ± can be satisfied provided √ ∆ ≡ s 2 4 − r 3 27 is an odd function. This can be ensured by imposing a constraint fixing a higher R n coefficient in terms of a lower R n coefficient. Writing ∆(z) = z 2 Q(z 2 ) = z 2 P (z), one finds that for non-zero m, µ 2 the polynomial P (z) defining the branch cuts and Riemann surface is quartic in z 2 i.e. even and of degree 8 in z and has 4 square-root branch cuts defining a Riemann surface of genus 3 . Contour integrals around these branch cuts play the same role as in the pure AM case. Thus we expect 4 possible quantum vevs when the tree level superpotential is cubic, even though at tree level there are just two semi-classical vevs v ± = 0 besides the vanishing vev of Φ 11 . This indicates that the effective superpotential in the general case with m, µ 2 = 0 will need to be quintic in Φ.
Since the analysis becomes quite involved for the general case we here present the explicit solution of the resolvent system only for m = µ 2 = 0. In some sense this solution is more interesting since it eliminates all explicit mass scales completely so that all masses arise purely by dimensional transmutation and the classical theory will be superconformal. Moreover, since ∆(z) is sextic and P (z) is quartic, there are only two branch cuts and thus two quantum vevs. A cubic quantum effective super-potential can still be defined as in the earlier cases studied.
We now trace the determination of resolvent coefficients {U, R, T, S} n using the available GKA equations. Firstly the expansion of eqn(4.11) and then eqn(4.10) for large z determines U n , R 2n+3 , n = 0, 1, 2.... in terms of R 2n , R 1 : Next imposing ∆(0) = 0 fixes R 1 and with ∆(z) = z 2 P 4 (z) we have (s 1,2 = ±1) Thus the two branch cuts in this (degenerate) case run between z ++ , z +− and z −+ , z −− . We emphasize that the determination of R 1 in terms of R 0 is a novel consequence of adding Σ, Σ. This simplifies the numerical analysis significantly since-after rescaling to dimensionless form-only the dimensionless coupling λ remains free. Contrast this with the pure AM case where R 1 was to be dynamically determined. Now we can also obtain all the even coefficients R 2n , n = 1, 2, ... by expanding the cubic equation for ω(z) = R(z) − ω r (z) for large z. This gives ; .... Finally we define the pure rank-breaker resolvent S(z) ≡ Σ · (z − Φ) −1 · Σ. As in the ΦQQ case one derives the system of GKA resolvents where f (z), c(z) are as before. Thus given T (z) one can derive S(z). To find T (z) we use the equation [9] which is the analogue of eqn(4.10) derived using the same Φ variations but with κW α W α factors omitted : Motivated by the solution of theQQΦ case where the corresponding equation differs only by the absence, on the r.h.s., of the mixing (third) term and the factor of 2 in the fourth term and has solution T (z) = (2R − W ′ (z)) −1 ((R − ηQQ)/z + c(z)/4) we propose where ζ(z) is to begin with an arbitrary odd function of z. However the behaviour of T (z) as z → ∞ allows only ζ(z) = −λT 2 /z. Here c(z) = −4λ(T 1 + T 0 z). By expanding the solution eqn(4.21) for large z we get T n>3 . If, following the pattern of the Higgs vacuum solution in theQQΦ case we demand that the residue of T (z) at z = 0 be unity, corresponding to the rank breaking Higgs vacuum, we determine T 2 : Just as in theQQΦ case eqn(4.21) gives the correct T (z) in the semiclassical limit where R,R → 0. Of course the semi-classical limit is trivial in this massless case in the sense that all vevs are then zero. The quantum superpotential derivative is now Note that the square root branching structure of y is now hidden inside the expressions for ω ± which contain P (z). Where P (z) is the quartic(m = µ = 0)/octic (m, µ non-zero) polynomial which defines the branch cuts. The resolvent for S(z) is also determined via eqn(4.19) once R(z), T (z) are known so that the coefficients S n of z −n−1 in the large z expansion can be read off. Thus we get and so on. The residue of ηS(z) at z = 0 is (−4λR 2 0 ) 1/3 . Since T 0 = d(r) we have only R 0 , T 1 left undetermined. The former is set by the gaugino condensation S = Λ 3 . The same argument as for the case with the Adjoint gives where y(z) was given explicitly above. Thus with input parameters λ, N 1 (N 2 ≡ d(r) − N 1 −1) and using units of R 1 3 0 ∼ Λ for dimensionful quantities we can evaluate the quantum vevs by performing the contour integrals numerically. Details will be given in a sequel.
Although we also defer detailed consideration of the case with m, µ = 0 to the sequel it is important to underline that it presents new features not observed in the degenerate case described above. One finds √ ∆ = z Q 4 (z 2 ) = z P 8 (z) so that one has 4 rather than 2 branch cuts. This opens the possibility of cases where the quantum spontaneous symmetry breaking includes vev patterns with no semi-classical antecedent. With cubic W (z) and thus two critical points, the contour integrals around the 4 branch cuts define 4 different vevs : which may be placed at will on the diagonal of the quantum corrected vev of Φ giving a symmetry breaking pattern without a semi-classical analog. Thus the corresponding "perturbative effective quantum superpotential " will need to be quintic rather than cubic.

Realistic MSGUT type model
We next come full circle and consider our motivating problem: the gauge UV Landau pole in the successful MSGUT [3]. To illustrate how the AS dynamics permits novel realistic GUT scenarios with dimensional transmutation and dynamical symmetry breaking, we propose a Spin(10) gauge model with 3 matter 16-plets and a Higgs structure generated by base representation r = 16 . We take Φ ∼ 16 × 16 = 1 + 45 + 210, Σ ∼ 16 × 16 = 10+120+126, Σ = 16×16 = 10+120+126. As before we note that one may choose to work with just the irreps of the MSGUT, or some extended set thereof, by applying projectors to select only the irreps one wishes to keep at the cost of an increase in calculational overhead. The required SO(10) decompositions w.r.t. G P S are explicitly available in [14]. But we will not work with projections here. The superpotential for the complete model as where Ψ A , A = 1, 2, 3 are the three matter 16-plets and it is understood that in the last term only the real 10-plet and 120-plet parts of the tensor product will be present since there is no invariant between two matter 16 plets (Ψ A )and a 126-plet. Notice the remarkable economy of AM type couplings. We recall our proposal [15] to further reduce the number of matter Yukawa couplings by making Σ, Σ carry the generation indices. We do not discuss the matter couplings further except to note that as per our arguments the Konsishi anomaly will also force the development of large trilinear condensates involving Ψ A ·Σ·Ψ B , Ψ A ·Σ·Ψ B , even provided the solution has no 16-plet vevs. The theoretical and phenomenological implications of such condensates are not clear to us. We follow the conventions and use the results of [14] to explicitly calculate the decomposition of Spin(10) invariants. Here µ =μ, 4;μ = 1, 2, 3 refer to the SU(4) indices of the Pati-Salam maximal subgroup G 422 = SU (4) × SU (2) L × SU (2) R ⊂ SO (10).
3) with Φ 11 ≡ Φ ν c ν c * = V 1 = 0 as per the 16-plet labels introduced above. Then the rank breaking vevs will be Σ = Diag(σ, / 0 15 ) i.e. Σ 4 * 2 ,4 * 2 = σ, Σ = Diag(σ, / 0 15 ), i.e.Σ 42 * ,42 * = σ, all other component vevs zero. If we insist on a cubic tree level superpotential for Φ and set also m = µ = 0 then in addition to the vanishing singleton vev in the rank breaking(Φ ν c ν c = Φ ν c ν c * ) sector we will have only two possible vevs emerging from the pair of branch cuts that develop. This case will the easiest to analyse. Moreover the effective potential can then also be chosen to be just cubic.
The symmetry breaking patterns corresponding to the various vev distribution possibilities can be easily worked out. The easiest way of identifying the unbroken symmetry is to look at the gaugino masses that arise for a given distribution of v ± over the last 5 (Φ 11 = Φ ν c ν c * = 0 for the Higgs vacuum solution we are focussed on here) diagonal blocks of Φ that violates these equalities will break SO(10) → G 123 . For instance Φ = Diag(0, v (q) Thus both break to the same SM little group even though the mass patterns are dissimilar.
The mass spectrum is straightforward to evaluate given the contractions (A µναβ is the (6,2,2) of G 422 and A µν αβ is its SU(4) dual) [14] 16 · 16 * = 16 µα (16 * ) µ * α * + 16 µ * α (16  The analysis, while tedious due to the reducible reps used, is straightforward and similar to the case of the Adjoint of SU(N) with a symmetric representation except for the crucial fact that the 16-plet is not the fundamental of Spin (10). The evaluation of the resolvents and quantum vevs proceeds along the lines discussed above leading again to spontaneous breaking SO(10) → G 321 via dimensional transmutation. Details will be given in the sequel. We note that the general features of the model are similar to the MSGUT except for the extreme economy with regard to superpotential parameters since the cubic potential for Φ has just 3 (if m = 0) complex parameters and just one in the massless case. Since the matter Yukawa couplings h AB , h ′ AB include couplings of two 16-plets to both 10, 126plets (symmetric Yukawas) and to 120-plets(antisymmetric Yukawas) they are general 3×3 complex matrices and thus have ample scope for fitting the observed matter fermion mass parameters and mixings. Note again the new feature, not present in the MSGUT, that Σ ∼ 16 × 16 = 10 + 126 + 120 and it is possible to couple the 10, 120 plets generated in this way to the matter 16 bilinear although 16 · 16 · 126 ≡ 0 as before.

Discussion
Using Generalized Konishi Anomaly relations obeyed by gluino and scalar condensates in the supersymmetric vacua we have shown that asymptotically strong Supersymmetric Yang Mills Higgs theories with matrix type Higgs multiplets transforming as general matrix type base r tensors of G provide a calculable implementation of spontaneous symmetry breaking of the fundamental gauge group, including rank reduction, via dimensional transmutation. As such they provide a robust and novel method of making sense of AS Susy GUT models and justify the surmise that the AS exhibited by phenomenologically successful and minimal models such as the MSGUT is a signal of nontrivial UV behaviour that makes the theory consistent and yields a sensible low energy limit. This demonstration calls for deep revision of our notions of the relation between strong coupling behaviour in the microscopic theory and a phenomenologically acceptable low energy effective theory.
For any given YM gauge group G, the number of asymptotically free models is strictly limited, whereas we have shown that the number of asymptotically strong models with sensible low energy limits is essentially unlimited. Thus it should be clear that our approach points the way to to a vast expansion of admissible microscopic theories beyond the narrow set of currently canonical AF type models. The signal successes of QCD and the amiable ease of analysis of AF models have led, over the half century since their discovery and dominance, to the hardening of a Dogma that sees AF as the necessary condition for a field theory to be physically sensible and relevant as a fundamental microscopic theory. On the other hand we continue, especially in Condensed matter Physics, to be challenged by the need to tackle quantum systems that are strongly correlated or massively entangled at the microscopic level. The success of the AdS-CFT [17] conjecture, and the Seiberg-Witten analysis [16] of monopole condensation leading to confinement in N = 2 supersymmetric YMH theories, in providing fruitful working paradigms in manifold non-supersymmetric strongly coupled field contexts, even though the original contexts which allowed calculation were specifically supersymmetric, suggests that even our analysis which is rooted in supersymmetry may, in the long run, motivate a more broad minded view of the way in which microscopic condensation due to strong coupling can generate sensible low energy behaviour, such as the effective perturbative gauge model we found after GUT dynamical ssb, even when Supersymmetry is absent. After all, the strong coupling dynamics underlying satisfaction of the "kinematic" GKA constraints must enforce the development of vevs driven by the gaugino condensate. This phenomenon may well persist even when one moves off the supersymmetric point in coupling space , and even, perhaps, for "small" structural differences w.r.t the fields present. These matters require the development of Lattice methods applicable to AS theories for their definitive resolution. The recent development of Lattice methods applicable to supersymmetric gauge theories encourages us to hope that such methods will be developed. Workers on the lattice will then have a plethora of AS toy models to choose from. For example even the behaviour of our original SU (2) model which is a projection of a symmetric 3× 3 matrix Φ corresponding to a triplet of SU(2) as the base representation r awaits investigation. We also note that discovery of weakly coupled models dual to the AS models of the type we have suggested would open up fruitful avenues towards a deeper understanding of the possibilities we have almost blindly raised on the basis of the Konsihi Anomaly alone.
We have emphasised that in Susy GUTs, in sharp contrast to say SQCD, the smallness of the ratio of the Susy breaking scale M S to the GUT scale M X implies that Supersymmetry may be assumed to be essentially exact at the scales where the theory becomes strongly coupled. Nevertheless it is clear that the issue of supersymmetry breaking must be tackled for such AS GUT models to make contact with reality. The soft supersymmetry breaking terms typically invoked in the MSSM and Susy GUTs, can be introduced by spurion Chiral supermultiplets that take fixed values thus breaking supersymmetry (S F = θ 2 A, S M = θ 2 Mg, S D = θ 2θ2 m 2 f and so on ). Since the Konishi anomaly relations involve the lowest components of Chiral multiplets the new terms should not affect the GKA relations directly. Thus after carrying out the GUT ssb using the robust dimensional transmutation studied here and defining a consistent superpotential for the effective SSM generated therefrom one may simply add in the small susy breaking terms and proceed as usual to study electroweak breaking and low energy phenomenology.
One novel and mysterious implication of the GKA relations is that even the superpotential terms containing matter chiral multiplets(along with Hoggs multiplets) must participate at least in trilinear condensates with superheavy values, even though vevs for the smatter fields are phenomenologically unacceptable. The phenomenological implications of such three point correlators are not clear to us. Perhaps such novel quantum background contaminations of the perturbative theory will eventually yield novel signals of the dynamical symmetry breaking origin of GUT spontaneous symmetry breaking.
for encouragement. The support of the ICTP, Trieste Senior Associates program through an award during 2013-2019 and the hospitality of the ICTP High Energy Group, and in particular G. Senjanovic, both when this idea was first conceived in 2002 and during the summer of 2019, is gratefully acknowledged.