M Chemtob
email: marc.chemtob@cea.fr

P Hosteins
email: pierre.hosteins@ph.tum.de

Searching singlet extensions of the supersymmetric standard model in Z 6-II orbifold compactification of heterotic string *

Keywords: numbers: 12, 60, -i,12, 60, Jv

We search for supersymmetric standard model realizations with extra singlets and extra U (1) using the heterotic string compactification on the Z6-II orbifold with two Wilson lines. The effective superpotential produced through the vacuum restabilization mechanism is examined for three representative Pati-Salam string models obtained in the literature. An automated selection of semirealistic vacua along flat directions in the non-Abelian singlet modes field space is performed by requiring the presence of massless pairs of electroweak Higgs bosons having trilinear superpotential couplings with massless singlet modes and the decoupling of color triplet exotic modes needed to suppress B and L number violating processes.

I. INTRODUCTION

Model building with the heterotic string [1][2][3][4][5][6][7][8][9] has received a vigorous stimulus thanks to the recent focus [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] on anisotropic compactifications on the orbifold Z 6-II . Recent reviews are given in [27,28]. Satisfactory representative vacuum solutions are also reported for the Z 12-I orbifold [29][30][31][32][33]. The exploration of the vast moduli space of the Z 6-II orbifold with two authorized Wilson lines [17,20] was found to give access to a fertile mini-landscape of vacua for minimal supersymmetric standard models descending from 5-d or 6-d grand unified theories. Sampling the still wider space of solutions with three Wilson lines [34] leads to equally hopeful conclusions.

One characteristic feature of these top-down constructions is the profusion of U (1) gauge factors which remain after the first gauge symmetry breaking from the orbifold gauge twist and the presence of Wilson lines. One of them is anomalous, creating a one-loop Fayet-Iliopoulos D-term a bit below the string scale that needs to be cancelled. This step usually leaves a lot of freedom due to the large vacuum degeneracy. In the vacuum restabilisation process, the U (1) gauge symmetries will get broken and most of the extra modes beyond the MSSM (minimal supersymmetric standard model) matter content, but hopefully not all of them, will decouple with large masses. Except for searches of extensions including right-handed neutrino supermultiplets [35,36], most studies have restricted to solutions in which the whole set of Standard Model singlet modes acquire large masses and decouple from the low energy field theory. It is natural, however, to ask whether the large moduli space of these compactifications does not include regions where extra singlets and extra U (1) symmetries might realize the NMSSM (next-to-minimal supersymmetric standard model) or one of the related versions proposed in the literature [37][38][39][40][41][42] and analyzed over the years in phenomenological [43][44][45][46][47][48] and formal [49][50][51][52][53] studies. (After completion of the present work, there appeared a study of NMSSM realizations complying with the severe selection criteria of [17,20] which reports that solutions exist only in the case with three Wilson lines [54].)

Filling this gap is the purpose of the present work. Rather than pursuing a statistical exploration of the moduli space of heterotic string compactifications on the Z 6-II orbifold, we restrict consideration in this preliminary study to the three representative 5-d Pati-Salam models, designated in [10] as models A1, A2, B, and apply techniques developed in past works. The early studies along these lines focused on flat directions for the non-Abelian singlet modes realizing the minimal supersymmetric standard model [55][56][57][58] or the Pati-Salam model [59], while subsequent studies have explored field directions along scalar modes charged under the non-Abelian gauge group factors [60][61][62][63].

Having in hand the string massless spectra for a set of representative solutions, we use a Fortran computer program to implement an automated search of the flat directions which exhibit a satisfactory supersymmetric effective action. Our study is somewhat close in spirit to that of [59]. Concentrating on string theory realizations [64][65][66][67] of the Pati-Salam model [68] comes with a number of well-known advantages. We shall not detail in the present work the Pati-Salam to Standard Model (P S → SM ) gauge symmetry breaking, but note that the use of Pati-Salam group multiplets merely serves a convenient book-keeping purpose, since the representation content with respect to the Standard Model gauge group is then uniquely determined.

The contents of the present work are organized into four sections. Section II discusses general features of the low energy theory with Pati-Salam gauge symmetry. A brief review of the supersymmetry preserving flat directions in the presence of an anomalous U (1) A gauge symmetry is provided in Appendix A. In order to clarify certain fine points that have caused us much confusion, we found useful to include in Appendix B a brief review of the string theory construction for the Z 6-II orbifold complementing the detailed discussions in [10,13,20]. The core part of the present work figures in Section III, where we present the results of our searches of singlet extended supersymmetric standard models for the Pati-Salam string models A1, A2, B of [10]. A general discussion of results and a summary of our conclusions figures in Section IV. with the corresponding Abelian factors, U (1) Na , [N a = 5, 6]. The gauge group representations consist of three chiral generations of bifundamental matter supermultiplets, f i ∼ (4, 2, 1), f c i ∼ ( 4, 1, 2), a number of electroweak Higgs bidoublet supermultiplets, h i ∼ (1, 2, 2), and a number of vector bifundamental supermultiplets, f ′ i , f ′ i and f ′ c i , f ′ c i , which accomplish the P S → SM Higgs mechanism for the gauge symmetry breaking, SU (4) × SU (2) R → SU (3) c × U (1) Y , along with a set of non-Abelian singlet modes, φ i ∼ (1, 1, 1)(1 ′ , 1 ′ ). There also occur modes with exotic quantum numbers: C i ∼ (6, 1, 1), d r i ∼ (1, 1, 2), d l i ∼ (1, 2, 1), q i ∼ (4, 1, 1), qi ∼ ( 4, 1, 1); (II.3) hidden sector modes:

Σ ′ i ∼ (16 ′ , 1 ′ ), Σ′ i ∼ (16 ′ , 1 ′ ), T ′ i ∼ (10 ′ , 1 ′ ), d ′ i ∼ (1 ′ , 2 ′ ); (II.4)
along with modes charged under the observable and hidden groups:

h ′ l i ∼ (1, 2, 1)(1 ′ , 2 ′ ), h ′ r i ∼ (1, 1, 2)(1 ′ , 2 ′ ).
(II.5)

Among the set of non-Abelian singlets, φ i , we distinguish the modes without and with oscillator excitations, S i and Y i . We group the singlets into two subsets, the first including the fields φi excited along the flat directions with large vacuum expectation values (VEVs) of the order of the string or Planck mass scales, m s or M ⋆ , and the second including the dynamical fields, φi and φ i , with undetermined or vanishing VEVs and masses. The P S → SM phase transition is assumed to occur at a mass scale m P S near the string scale. This hypothesis is in harmony with the renormalization group analysis of the gauge coupling constants [10] and with related phenomenological studies [65][66][67]. At the P S → SM gauge symmetry breaking, the above listed multiplets decompose into Standard Model group

G 321 = SU (3) × SU (2) L × U (1) Y multiplets as, f ∼ (4, 2, 1) → q + l = (u d) + (ν e) ∼ (3, 2) 1 6 + (1, 2) -1 2 ,
• One or several non-Abelian gauge group singlets, φ k , coupled to one or several pairs of electroweak Higgs boson bidoublets by trilinear effective superpotential terms, h i h j φ k ;

• Extra U (1) ′ symmetries [48] possibly broken at mass scales much lower than O(M ⋆ );

• A secluded sector with additional singlets [39,42] belonging to an approximate flat direction and charged under the extra U (1) ′ only. The simplest version includes in addition to s three singlets [39,42] s 1 , s 2 , s 3 belonging to a flat direction approximately lifted by a small coupling constant λ ′ in the effective renormalizable superpotential,

W EF F = λH u H d s + λ ′ s 1 s 2 s 3 .
The superpotential is decomposed into three main components, W s , W 2 , W 3 , associated to the couplings of the non-Abelian singlet fields, φ i , alone or multiplied by two and three fields charged under the Pati-Salam group. Each component is built as a power expansion in the φ i of the form

W s (φ k ) = n W (n) s (φ k ) = n,m s (n) m i,j S p (m) i i Y q (m) j j , W 2 (φ k ) = r ij (φ k )f c i f c j + p ij (φ k )f i fj + µ ij (φ k )h i h j + τ ij (φ k )C i C j + σ ij q i qj , W 3 (φ k ) = C k [c ij k (φ l )f c i f c j + c ′ ij k (φ l ) f c i f c j + l ij k (φ l )f i f j + e ij k (φ l )q i q j + e ′ ij k (φ l )q i qj ] +λ ij k (φ l )h k f i f c j + λ ′ ij k (φ l )h k fi f c j , (II.8) where p (m) i , q (m) j ∈ Z + , i,j (p (m) i + q (m) j
) ≥ n and the summations over f i , f c i (and their complex conjugates) include both the matter and Higgs multiplets. The coefficient functions µ ij (φ k ), σ ij (φ k ), τ ij (φ k ), • • • are given by infinite power expansions in the singlet fields of same form as that displayed above for W

W EF F = (l k (τ -1 ) kl ) l l f f f f + (l k (τ -1 ) kl c l ) f f f c f c + (c k (τ -1 ) kl c l ) f c f c f c f c → 1 M ⋆ [l 2 (φ)qqql + l(φ)c(φ)qqd c u c + c 2 (φ)u c d c d c ν c ] + m P S M ⋆ c 2 (φ)u c d c d c , (II.9)
may compete with similar couplings already present at the compactification scale.

Our search strategy consists of four stages.

• Firstly, we construct (up to some fixed order) the superbasis of holomorphic invariant monomials

P α (φ) = i φ r α i i = i S n α i i Y m α i i , [r α i , n α i , m α i ∈ Z + ] (II.10) which solve the D flatness conditions, Q a (P α ) = 0, Q A (P α ) × sign (ξ A ) < 0, where Q a , [a = 1, • • • , N a -1]
denote the anomaly free Abelian charges and Q A the single anomalous Abelian charge with Fayet-Iliopoulos parameter ξ A defined in Eqs. (A.1) and (B.9).

• Secondly, we construct the superpotential W s in the non-Abelian singlet modes (up to some fixed order) satisfying the gauge symmetries and the string selection rules in Eqs. (B.10).

• Thirdly, we scan the D flat monomials P α (φ i ) and compare each of these in turn with the allowed couplings of singlets in W s (φ i ), in order to determine which of the D flat monomials obey the type A or B F flatness, such that there are no allowed couplings in the superpotential W s with all field factors, or all but a single field factor, included in P α (φ i ). (Said otherwise, type A or B lifting of a flat direction P α (φ i ) occurs whenever some superpotential monomial with only 1 or no field having zero expectation value is allowed.) Our terminology slightly deviates from the original one in [55,56,62] which referred to the subset of monomials in W s allowed by the gauge invariance rules alone.

The conservative approach of exploring the space of supersymmetric vacua consists in restricting to the type B directions, on the grounds that these preserve local supersymmetry, thanks to the vanishing scalar potential or cosmological constant, < V >= 0, and can be made flat to all orders by imposing a finite number of conditions. The existing applications [56,57,59,62] typically stop at the trilinear or fourth orders terms of W s (φ). An alternative procedure is pursued in [15,17,31] which consists in selecting a (typically sizeable) set of singlet field directions providing for a satisfactory effective action and solving next the D and F flatness equations out to high orders for these singlets. In the present work, we follow an intermediate approach in which we restrict to flat directions and superpotential monomials of low orders only but include both type A and B flat directions. Since supersymmetry must get broken anyway, we argue that it makes sense to consider the approximate flat directions which are lifted by non-renormalizable F terms of reasonably high orders. To motivate our choice we note that the condition < W s >≈ 0 is automatically satisfied for models preserving an approximate R symmetry [70]. We also observe that an initially non F flat direction can be repaired into a flat one by using the cancellation with contributions from higher order and/or moduli dependent couplings [71]. One may tie the order of the F term lifting operator to the supersymmetry breaking scale F by considering the case of a single flat direction φ lifted by the superpotential, W s (φ) = φ p+3 /M p ⋆ . Assuming that φ picks up from radiative corrections a tachyonic soft supersymmetry breaking mass term, V (φ) = -m 2 W |φ| 2 , then the scalar potential minimization,

V (φ) = -m 2 W |φ| 2 + (p + 3) 2 M 2p ⋆ |φ p+2 | 2 =⇒ |F | 1 2 = V 1/4 (φ min ) ≃ (m p+2 W M p ⋆ ) 1 2(p+1) , (II.11)
yields the following acceptable range for the breaking scale, |F | 1 2 ≤ (10 6 -10 8 ) GeV, for p = (1 -3). Larger p yield higher scales. It should be noted that the F term lifting by non-renormalizable operators also plays a useful rôle in string theory models realizing gauge symmetry breaking [72,73] or supersymmetry breaking [74] at intermediate scales.

• Fourthly, we select some supersymmetry preserving flat directions, identified by sets of excited singlet fields φi , and evaluate the effective superpotential between the non-singlet modes, while treating the fields φi VEVs as free parameters. Our main focus will be on the bilinear and trilinear couplings of the electroweak Higgs modes, h i h j (µ ij ( φ) + λ k ij ( φ)φ k ). However, for completeness, we shall also examine the bilinear couplings of the modes carrying the bifundamental, sextet and quartet representations, in order to test the decoupling of mirror generations and exotic modes and the absence of baryon number violation.

We pause briefly at this point to clarify some technical issues. The non-Abelian singlet modes VEVs are 0dimensional if they are completely fixed by the gauge constraints, in which case they are roughly set at, φ = O((-ξ A /Q A ( φ)) ). Otherwise, they may be 1-d or higher if they depend on a single or several free complex parameters.

A given holomorphic monomial of the non-Abelian singlets,

P α = n i=1 φ r α i i , is D-flat if it carries vanishing anomaly free charges, Q a (P α ) = n i=1 Q a (φ i )r α i = 0, and a finite anomalous charge, Q A (P α ) = n i=1 Q A (φ i )r α i = 0 of opposite sign to the charge anomaly, T race(Q A ) ∝ ξ A . A superpotential monomial, W x = n i=1 φ s x i i , is allowed if it carries
vanishing anomaly free and anomalous charges,

n i=1 Q a (φ i )s x i = 0, n i=1 Q A (φ i )s x i = 0
, and satisfies the string selection rules listed in Eq. (B.10).

We define the order n of a holomorphic D flat or superpotential monomial, P α ( φi ) or W x ( φk , φ l ), by the number of field factors this contains, independently of the values of the fields positive integer exponents, r α i , s x i . The number of inequivalent monomial solutions rises fast with the order n considered, reaching orders of hundreds already at the relatively low order, n = 4. The solutions will be ordered according to increasing values of the number of field factors, n = 1, 2, • • •, while allowing the power exponents to range over the discrete set of values, r α i , s x i ∈ [0, 1,2]. Note that with this definition, the order n of a monomial is always bounded by its effective order, n i=1 r α i or n i=1 s x i . To avoid dealing with an overwhelming number of solutions, we generally restrict to orders n ≤ 4.

The unbroken (anomaly free) Abelian symmetries U (1) x along a flat direction are identified by the kernel of the charge matrix, A ai = Q a ( φi ), [a = 1, N a -1] where φi ranges over the modes excited along the flat direction and the index label a ranges over the anomaly free Abelian symmetries. In practice, we proceed by evaluating the left eigenvectors with zero eigenvalues of the matrix, A ai , [a, i = 1, • • • , N a -1] namely, x T A = 0, for some selected subset of the φi , and next retain among the Abelian charges, Q x = a x a Q a , those yielding zero charges, Q x ( φi ) = 0, for the full set of φi .

The effective trilinear interactions descending from non-renormalizable operators of order n + 3 are assigned string tree level coupling constants given by the approximate formula [57]

W (n) 3 = c(n) ijk (φ)Φ i Φ j Φ k , c(n) ijk (φ) = c (n) ijk < n i=1 φ i >, c (n) ijk ≃ g s C n I n (πM ⋆ ) n (II.12)
where C n are constant coefficients of O(1) and

I n = n i=1 d 2 z i f (z i , zi
) are integrals over the location of vertex operators on the world sheet sphere surface. The integrals for the four and five point string amplitudes evaluate numerically to [75][76][77]: I 1 ≃ 70, I 2 ≃ 400. The string coupling constant is commonly set at, g s = g X / √ 2 ≃ 1/2. For the case of a 0-d flat direction, assigning the singlet fields the common VEV φ determined by the Fayet-Iliopoulos term, one infers the following formula for the coefficients

| φ| ≃ - ξ A Q A (φ) 1 2 ≃ g s M ⋆ π - T race(Q A ) 192Q A ( φ) √ k A 1 2 =⇒ c(n) ijk ≃ g n+1 s C n I n (π 2 √ 192) n - T race(Q A ) Q A ( φ) √ k A n/2 . (II.13)
With increasing order of the non-renormalizable operators, we may thus anticipate a strong or mild suppression depending on the unknown extrapolation of I n for large n and the model dependent size of the anomalous charges. Using, say, I n = 10

) ij = µ (n) /M ⋆ .
Bounding the coupling constant by the requisite ratio between Planck and effective supersymmetry breaking mass scales, µ (n) /M ⋆ ≤ TeV/M ⋆ ≃ 10 -15 , yields, n ≥ 8, which shows that non-renormalizable effective operators starting from O( φ(8) )h i h j can be tolerated. Considering instead the tentative extrapolation, I n ≃ 10 n 2 , [C n = 1], and the numerical estimates for the anomalous charge appropriate to model A1 (to be detailed below)

-T race(Q A ) √ k A Q A ( φ) 1 2 ≃ 10, | φ| g s M ⋆ ≃ 1 π √ 192 -T race(Q A ) √ k A Q A ( φ) 1 2
≃ 0.23, (II. 14) then the corresponding condition on the µ-term coupling, µ (n) /M ⋆ ≃ 1 2 10 -n ≤ 10 -15 , gives the stronger bound on the order of non-renormalizable operators, n ≥ 14. Note that lowering φi should weaken the bounds on n.

III. SINGLET EXTENDED SUPERSYMMETRIC STANDARD MODELS

The Pati-Salam models A1, A2, B of [10] for the orbifold Z 6-II of T 6 with two Wilson lines only (since W ′ 2 = 0), are described by the 16-d shift vectors for the gauge twist and Wilson lines of the E 8 × E 8 group lattice:

Model A1 : V = 1 6 (2, 2, 2, 0 5 )(1, 1, 0 6 ), W 2 = 1 6 (3, 0 4 , 3 3 )(0 8 ), W 3 = 1 6
(2, -2, 0 6 )(0 2 , 4, 0 5 ).

Model A2 : V = 1 6 (2 3 , 0 5 )(1 2 , 0 6 ), W 2 = 1 6 (3, 0 4 , 3 3 )(0 8 ), W 3 = 1 6 (4, 2, -2, 0 5 )(0, 4, 2 2 , 0 4 ). Model B : V = 1 6 (4, 1 2 , 0 5 )(2 2 , 0 6 ), W 2 = 1 6 (0 3 , 3 2 , 0 3 )(3, 0, 3, 0 5 ), W 3 = 1 6 (0, 2, -2, 0 5 )(0 2 , 4, 0 5 ). (III.1)
The 4-d gauge groups for the above three models are:

Model A 1 : SU (4) × SU (2) 1 × SU (2) 2 × U (1) 5 × (SO(10) ′ × SU (2) ′ ). (III.2) Model A 2 : SU (4) × SU (2) 1 × SU (2) 2 × U (1) 5 × (SO(10) ′ × SU (2) ′ ). (III.3) Model B : SU (4) × SU (2) 1 × SU (2) 2 × U (1) 6 × SO(10) ′ .
(III.4)

Our results for the massless string spectra of models A1, A2 and B match to those of [10], except for certain assignments of oscillator numbers, N L I, Ī , hence of R I charges. The numerical data for the linear and cubic traces of the anomalous U (1) A gauge charge and the Fayet-Iliopoulos term,

ξ A /(g s M ⋆ ) 2 ≡ T race(Q A )/(192π 2 √
k A ) are displayed in the following table. This also includes results for the total number of massless string states Φ i and the degeneracies of the various modes. The massless string spectrum of model A1 is displayed in Table I. The set of holomorphic invariant monomials of order n ≤ 2 consist of eleven solutions,

Model T r(Q A ) T r(Q 3 A ) T r(Q A Q 2 a )/2 k A ξ A /(g s M ⋆ ) 2 Φ i (S i , Y i ) h i C i f i fi f c i f c i (q i , qi ) A1 -2 -0.
P α = [Y 11 S 8 , S 3 S 6 , Y 14 Y 2 18 , Y 14 Y 2 19 , Y 15 Y 2 19 , Y 15 Y 2 18 , Y 2 18 , Y 18 Y 19 , Y 2 18 Y 19 , Y 18 Y 2 19 , Y 2 19 ]
. None of these D flat directions is found to be lifted by the superpotential couplings in W s at orders n ≤ 4. Rather than analyzing each monomial P α individually, we combine these multiplicatively into what we term as the 'composite' D-flat direction, P (2) = P 1 • • • P 11 , which then includes the eight modes, Y 11 , S 8 , S 3 , S 6 , Y 14 , Y 18 , Y 19 , Y 15 . The set of excluded couplings along the composite direction is clearly the union of all the sets excluded by the individual couplings of order n. All these directions are 0-d ones, hence with a common VEV given by, | φ|/(g s M ⋆ ) ≃ 0.23. The effective superpotential components obtained by assigning all the fields along P (2) a common value φi = φ read at order n ≤ 4

W (2) s = φS 1 S 10 + φ 3 Y 3 Y 9 + φ 2 S 11 S 13 + φ 2 S 10 S 15 . W h = h 1 h 1 [φ 5 Y 17 + φ 5 S 4 ] + h 2 h 2 [φ 4 S 10 ] + h 3 h 3 [S 1 ] + h 1 h 2 [φ 2 S 10 ] + h 1 h 3 [φ 4 + S 9 ] + h 2 h 3 [φ 6 + φ 4 S 9 ]. W C = C 1 C 1 [φ 3 Y 10 ] + C 2 C 2 [φ 4 S 10 ] + C 3 C 3 [S 1 ] + C 4 C 4 [φ 4 Y 7 ] + C 1 C 2 [0] + C 1 C 3 [φ 2 Y 3 ] + C 1 C 4 [φY 3 Y 9 ] +C 2 C 3 [φ 6 + φ 4 S 9 ] + C 2 C 4 [φ 5 Y 9 ] + C 3 C 4 [Y 5 S 11 ]. W q = q 1 q1 [0] + q 1 q2 [φ 5 ] + q 2 q1 [0] + q 2 q2 [S 14 ]. W f = f c 1 f c 1 [0] + f c 1 f c 2 [S 2 ] + f c 2 f c 1 [0] + f c 2 f c 2 [φ 2 Y 3 ] + f c 3 f c 1 [S 12 ] + f c 3 f c 2 [0]. W hf f c = h 1 f 1 f c 1 [1] + h 1 f 1 f c 2 [φY 9 ] + h 1 f 1 f c 3 [φ 2 Y 9 ] + h 1 f 2 f c 1 [0] + h 1 f 2 f c 2 [0] + h 1 f 2 f c 3 [0] +h 2 f 1 f c 1 [φ 4 ] + h 2 f 1 f c 2 [φ 3 Y 9 ] + h 2 f 1 f c 3 [φ 4 Y 9 ] + h 2 f 2 f c 1 [0] + h 2 f 2 f c 2 [0] + h 2 f 2 f c 3 [0] +h 3 f 1 f c 1 [0] + h 3 f 1 f c 2 [0] + h 3 f 1 f c 3 [Y 7 ] + h 3 f 2 f c 2 [Y 1 ] + h 3 f 2 f c 1 [Y 1 + Y 2 + Y 4 ] + h 3 f 2 f c 3 [φ 2 ]. (III.5)
The results for W h through W hf f c are illustrative ones in the sense that they include only a subset of the couplings at most linear or quadratic in the singlets which are not part of the flat direction. For convenience, we have listed the results for the bilinear couplings along P (2) in the second column of Table II. The flavor bases for the massless fermions consist of the 3 modes f 1,2 and the 3 linear combinations of the 5 modes f c 1,2,3 which are left unpaired. The fermion mass matrices are sums of 3 × 5 matrices

< h k > f 1,2 f c 1,2,3 , [k = 1, 2, 3
] involving the combinations of 7 light bidoublets which acquire finite VEVs at the electroweak transition. The presence of an unsuppressed coupling h 1 f 1 f c 1 suggests that h 1 should be the dominant component of the light electroweak doublets and f 1 , f c 1 should be associated

d l i ∼ (1, 2, 1)(1, 1) ′ , d r i ∼ (1, 1, 2)(1, 1) ′ , d ′ i ∼ (1, 1, 1)(1 ′ , 2 ′ ); the bidoublets, hi ∼ (1, 2, 2)(1, 1) ′ ; the quartets, qi ∼ (4, 1, 1)(1, 1) ′ , qi ∼ ( 4, 1, 1)(1, 1) ′ ; the sextets, Ci ∼ (6, 1, 1)(1, 1) ′ ; the bifundamentals, f c i ∼ ( 4, 1, 2)(1, 1) ′ , f c i ∼ (4, 1, 2)(1, 1) ′ , fi ∼ (4, 2, 1)(1, 1) ′ , fi ∼ ( 4, 2, 1)(1, 1) ′ , ; the decuplets, T ′ i ∼ (1, 1, 1)(10 ′ , 1 ′ ); and the sextuplets, Σ ′ i ∼ (1, 1, 1)(16 ′ , 1 ′ ), Σ′ i ∼ (1, 1, 1)(16 ′ , 1 ′
). The first column lists the twisted sectors label g; the second column lists the modes name; the third column consists of three subcolumns which list the twisted subsector specified by the excited Wilson lines, with , 4]; and N32 = n2 + 1, [g = 3], the complex phase γ(g) describing the orbifold twist action on the fixed points in the T 2 1 plane and the modes multiplicity (degeneracy) D; the third column consists of three subcolumns which list the shifted H-momenta specified by the charges, 6R I , [I = 1, 2, 3]; and the fourth column consists of five subcolumns which list the Abelian gauge charges in the rotated basis starting with the anomaly free charges Qa and ending with the single anomalous charge QA. In order to quote integer charges, the Abelian charges have been rescaled as, Q1/0.1667, Q2/0.02083 , Q3/0.0104, 4 × Q4/0.006944, QA/0.008333. (Note that the first column entry for g is omitted in the right-hand part of the table .)

N32 = 1, [g = 0]; N32 = 3n2 + n3 + 1, [g = 1]; N32 = n3 + 1, [g = 2
g M N32 γ D 6(R1 R2 R3) Q1 Q2 Q3 Q4 QA M N32 γ D 6(R1 R2 R3) Q1 Q2 Q3 Q4 QA 0 h1 1 1 1 0 6 0 0 3 -3 -6 3 S1 1 1 1 0 6 0 0 0 -6 24 -12 T ′ 1 1 1 1 6 0 0 0 0 -3 12 -6 f c 1 1 1 1 0 0 6 3 3 0 0 0 f1 1 1 1 6 0 0 3 0 3 6 -3 f c 1 1 1 1 0 0 6 -3 -3 0 0 0 1 Y1 1 1 2 1 2 9 2 2 -1 4 -2 Y2 1 1 2 1 2 -3 2 2 -1 4 -2 Y3 1 1 2 -5 -4 3 2 2 -1 4 -2 Y4 1 1 2 -17 2 3 2 2 -1 4 -2 Y5 1 1 2 -5 2 3 2 -4 -1 4 -2 Y6 1 1 2 -5 2 3 -4 -1 2 10 -5 Y7 1 1 2 -5 2 3 -4 -1 -4 -2 1 f c 2 1 1 2 1 2 3 -1 -1 -1 4 -2 f2 1 1 2 1 2 3 -1 -1 -1 4 -2 Y8 2 1 2 -5 2 3 4 1 -2 -10 -5 d ′ 1 2 1 2 1 2 3 4 1 1 -4 7 Y9 2 1 2 -5 2 3 -2 -2 1 -4 -8 d ′ 2 2 1 2 1 2 3 -2 -2 4 2 4 Y10 3 1 2 -5 2 3 0 3 0 18 1 d ′ 3 3 1 2 1 2 3 0 3 3 -12 1 Y11 3 1 2 -5 2 3 0 -3 0 18 1 d ′ 4 3 1 2 1 2 3 0 -3 3 -12 1 q1 4 1 2 -5 2 3 -1 2 -1 4 -2 q1 4 1 2 1 2 3 -1 -1 2 10 -5 q2 4 1 2 1 2 3 -1 -1 -4 -2 1 d l 1 4 1 2 1 2 3 2 2 2 10 -5 d l 2 4 1 2 1 2 3 2 2 -4 -2 1 d r 1 4 1 2 1 -4 3 2 -1 -1 4 -2 d r 2 4 1 2 -11 2 3 2 -1 -1 4 -2 d l 3 4 1 2 -5 2 3 -4 -1 -1 4 -2 q2 5 1 2 1 2 3 1 1 -2 -10 -5 d r 3 5 1 2 1 2 3 -2 1 1 -4 -8 d l 4 5 1 2 1 2 3 -2 -2 -2 -10 -5 d r 4 6 1 2 -5 2 3 0 0 0 18 1 h ′ r 1 6 1 2 1 2 3 0 0 3 -12 1 2 S2 1 -1 1 2 4 0 4 4 -2 8 -4 Y12 1 1 2 -4 4 0 4 -2 -2 8 -4 Y13 1 -1 1 2 10 0 4 -2 -2 8 -4 S3 1 -1 1 2 4 0 4 -2 4 -16 8 C1 1 -1 1 2 4 0 -2 -2 -2 8 -4 f c 3 1 1 2 2 4 0 1 1 -2 8 -4 Y14 2 1 2 -4 4 0 0 0 0 0 -10 Y15 2 -1 1 2 10 0 0 0 0 0 -10 T ′ 1 2 -1 1 2 4 0 0 0 0 18 -4 d ′ 5 2 1 2 2 10 0 0 0 3 6 2 d ′ 6 2 -1 1 -4 4 0 0 0 3 6 2 S4 2 -1 1 2 4 0 0 0 6 -24 2 Σ ′ 1 2 1 2 2 4 0 0 0 3 -3 -1 S5 2 -1 1 2 4 0 0 0 -6 -12 -4 d ′ 7 2 1 2 2 4 0 0 0 -3 -6 8 S6 3 -1 1 2 4 0 -4 2 -4 16 2 d ′ 8 3 1 2 2 4 0 -4 2 -1 -14 2 3 S7 1 ω 2 3 0 3 0 3 -6 6 -3 S8 1 ω 2 2 3 0 3 0 3 0 -18 9 h2 1 ω 2 2 3 0 3 0 0 -3 12 -6 h3 1 1 4 3 0 3 0 0 3 -12 6 C2 1 ω 2 2 3 0 3 0 0 -3 12 -6 C3 1 1 4 3 0 3 0 0 3 -12 6 S9 1 1 4 3 0 3 0 -3 0 18 -9 S10 1 ω 2 3 0 3 0 -3 6 -6 3 4 C4 1 1 2 4 2 0 2 2 2 -8 4 S11 1 1 2 4 2 0 -4 2 -4 16 -8 Y16 1 1 2 4 -4 0 -4 2 2 -8 4 Y17 1 -1 1 10 2 0 -4 2 2 -8 4 S12 1 1 2 4 2 0 -4 -4 2 -8 4 f c 2 1 -1 1 4 2 0 -1 -1 2 -8 4 d ′ 9 2 -1 1 4 2 0 4 -2 1 14 -2 S13 2 1 2 4 2 0 4 -2 4 -16 -2 d ′ 10 3 -1 1 4 2 0 0 0 3 6 -8 S14 3 1 2 4 2 0 0 0 6 12 4 S15 3 1 2 4 2 0 0 0 -6 24 -2 d ′ 11 3 1 2 10 2 0 0 0 -3 -6 -2 d ′ 12 3 -1 1 4 -4 0 0 0 -3 -6 -2 T ′ 2 3 1 2 4 2 0 0 0 0 -18 4 Y18 3 1 2 4 -4 0 0 0 0 0 10 Y19 3 -1 1 10 2 0 0 0 0 0 10 Σ′ 1 3 -1 1 4 2 0 0 0 -3 3 1
to the third generation fermions. The degenerate pairs of modes f 2 and f c 2 must then be associated to the first two generations. As long as the Dihedral D 4 discrete symmetry of [10,86] is unbroken, the fermions mass matrices f 2 f c i would include pairs of identical rows, which means that their rank is ≤ 2. This seemingly good starting point motivates one to search for mechanisms breaking the D 4 symmetry. As discussed in [10,86], the non-renormalizable operators,

h k f i f c j k,l,m,m ′ S k Y l O c mm ′ , involving the composite singlet fields, O c ij = f c i f c j
, produce contributions at the P S → SM transition which can improve predictions for the fermions mass matrices. We note that an alternative way to break the D 4 symmetry is by switching on the third Wilson line, n ′ 2 W ′ 2 . That no constant or linear terms appears in W indicates that all the n = 2 flat directions are indeed unlifted. The decoupled massive singlets are, S 1,10,11,13,15 and Y 3,9 . The µ-term bilinear coupling in W h is a rank 2 matrix with the massless mode, (h 2φ 2 h 1 ). However, the suppressed component is along h 1 , which is the preferred mode whose VEV supposedly dominates the fermion masses. The trilinear couplings W = (φ 4 h 2 h 2 + φ 2 h 1 h 2 )S 10 are seen to select the massive singlet S 10 , which figures among the modes which acquire large masses along P (2) . However, it should be noted that S 10 remains massless for a subset of the n = 2 directions P α . The above result for W f shows that no mass pairings among the matter modes are allowed. The mass matrices M f ij f i f c j have the single non-vanishing entry, M f 11 =< h 1 >, if we omit the massive h 3 mode. Examination of the exotic modes mass couplings, shows that q 1 , q2 and C 2 , C 3 pair up, while the pairs q 2 , q1 and C 1 , C 4 remain massless. We also find that except for the coupling,

C 1 f c 2 f c 2 [φ 2 ], all other trilinear couplings, C 1 f c i f c j , C 1 f c i f c j in W Cf f
are suppressed. The 4 × 8 matrix of the Abelian charges for modes along P (2) is found to have column rank 2. The existence of a secluded sector requires finding three singlets, φ I , φ J , φ K , charged under the unbroken U (1) x with a mildly suppressed trilinear coupling, c( φk )φ I φ J φ K . A wide range of choices is clearly possible given the sizeable number of singlets orthogonal to the moduli space of P (2) , and the abundant number of monomial solutions in W s . An inspection of the allowed couplings lets us select the following effective cubic or quartic order couplings Y ]. These resuls indicate that a cubic self-coupling, S 3 10 φ 4 < S 2 7 >, cannot arise without an unacceptably large mass term, S 2 10 φ 3 < S 7 >. Since this correlation between the quadratic and cubic couplings of S 10 also holds for the individual flat directions, we conclude that the occurrence of a trilinear coupling, h i h j S 10 , with a finite cubic self-coupling is not the favored option.

We also found useful to study a class of solutions on an individual basis. We present in Table II a subset of the bilinear couplings for four randomly selected F flat monomial solutions of order n = 4. The monomials P (4)

II , P (4)
III , P (4) IV are seen to allow a pair of massless bidoublets, h 1 , h 2 , with preferred trilinear coupling, W h = h 1 h 2 S 10 . Along P (4) II , the diagonal trilinear coupling, W h = φ 5 h 2 h 2 S 13 , is also present. The direction P

(3) I has all three bidoublets massless. However, some subsets of the exotic modes always remain massless.

To complete the present study, we have scanned the 70 and 48 flat monomial solutions of orders n = 3 and n = 4 in search of the bidoublets couplings of form,

W h = h i h j (µ ij + λ k ij φ k ).
The global results combining the contributions from the various individual flat directions read:

• n = 3 : W h = h 1 h 1 [0] + h 2 h 2 [φ 4 S 10 ] + h 3 h 3 [φS 11 ] + h 1 h 2 [φ 2 S 13 ] + h 1 h 3 [φ 2 Y 15 ] + h 2 h 3 [φ 6 + φ 4 Y 15 ]. • n = 4 : W h = h 1 h 1 [0] + h 2 h 2 [φ 5 S 12,13 ] + h 3 h 3 [φ + φ 2 Y 12,14 ] + h 1 h 2 [S 10 + φ 2 Y 6,11 ] +h 1 h 3 [S 9 + φ 5 Y 6 ] + h 2 h 3 [φ 4 Y 12,14 ]. (III.6)
Happily, it appears that the cases with no trilinear couplings, µ ij = 0, λ k ij = 0, are outnumbered by those with no bilinear couplings, µ ij = 0, λ k ij = 0. The results for the directions n = 3 favor a scenario with a single light bidoublet h 1 having trilinear couplings, φ 2 h 1 h 2 S 13 or φ 2 h 1 h 3 Y 15 . The results for the directions n = 4 favor three light bidoublets with canonical and non-canonical type trilinear couplings.

B. Model A2

The mass spectrum of model A2 is displayed in Table III.

The D flatness conditions are solved at order n = 2 by the single monomial, Y 9 S 8 , and at order n = 3 by the 7 II: Superpotential couplings for model A1 of bilinear order in hi, Ci, qi, qi and orders n ≤ 4 in the singlets. The column entries refer to the composite flat direction P (2) and the four randomly selected individual flat directions:

P (3) I = Y12Y16Y 2 19 , P (4)
II = S1S3Y16Y 2 19 , P (4)
III = S1S 2 3 S4S 2 6 , P (4)
IV = S1S4Y 2 18 . Empty entries correspond to cases where no coupling is present up to order n = 4.

W P (2) P (3) I P (4) II P (4) III P (4) IV h1h1 φ 5 Y17 + φ 5 S4 h2h2 φ 5 S13 h3h3 S1 + φS15 + φ 5 S7 φS11 + S1 φ + φ 2 Y12 φ φ + φ 2 Y14 h1h2 S10 + φ 2 S10 S10 S10 S10 S10 h1h3 φ 4 + S9 + φ 3 S11 S9 S9 + φ 5 Y6 S9 S9 + φ 5 Y11 h2h3 φ 6 + φ 4 S9 φ 4 Y12 + φ 4 Y14 C1C1 φ 3 Y10 φ 5 Y3 φ 5 Y3 C2C2 φ 4 S10 φ 5 S13 C3C3 S1 + φS15 + φ 5 S7 φS11 + S1 φ + φ 2 Y12 φ φ + φ 3 Y14 C4C4 φ 4 Y7 φ 3 Y5 φ 3 Y9 C1C2 C1C3 φ 2 Y3 + φ 4 Y4 φ 3 Y3 φ 3 Y3 C1C4 C2C3 φ 6 + φ 4 S9 φ 4 Y12 + φ 4 Y14 C2C4 φ 5 Y9 C3C4 φ 2 Y5 φ 2 Y9 q1 q1
q1 q2 φ 3 S9 + φ 5 φ 4 Y14 + φ 4 Y12 q2 q1 q2 q2 S14 φ 2 S14 S14 S14 S14
VEVs. The effective superpotential component W s and those of bilinear and trilinear couplings obtained by assigning a common VEV φ to the modes φi ∈ P (2) × P (3) are given by

W s = [φ 3 S 9 ] + [φS 1 S 9 + φ 2 (Y 2 1 + Y 2 2 + Y 2 4 ) + φ 2 Y 6 S 17 + φ 2 S 13 S 17 + φ 2 Y 10 S 13 ]
.

W h = h 1 h 1 [φ 3 + φ 3 Y 5 ] + h 2 h 2 [0] + h 1 h 2 [φ 2 Y 3 ]. W C = C 1 C 1 [φ 2 Y 3 ] + C 2 C 2 [φY 6 ] + C 1 C 2 [φ 2 Y 3 ]. W q = q 1 q1 [φ 3 Y 1 + φ 3 Y 2 ] + q 2 q2 [φ 3 Y 8 ] + q 3 q3 [φY 1 Y 7 + φY 2 Y 7 ] +q 1 q2 [φ + φ 2 S 9 + φ 2 Y 12 ] + q 1 q3 [φ 4 Y 4 ] + q 2 q3 [0] + q1 q 2 [0] + q1 q 3 [φ 3 S 9 S 17 ] + q2 q 3 [φ 2 Y 1 Y 7 ]. W f = f c 1 f c 1 [φY 3 (Y 12 + Y 13 + S 13 )] + f c 2 f c 1 [0]. W hf f c = h 1 f 1 f c 1 [Y 3 S 4 ] + h 1 f 2 f c 2 [0] + h 1 f 1 f c 2 [0] + h 1 f 2 f c 1 [0] + h 2 f 1 f c 1 [0] + h 2 f 2 f c 2 [φY 10 ] + h 2 f 1 f c 2 [0] + h 2 f 2 f c 1 [0]. W Cf f = C 1 f c 1 f c 1 [0] + C 1 f c 2 f c 2 [φ 3 Y 14 ] + C 2 f c 1 f c 1 [φY 12 ] + C 2 f c 2 f c 2 [φ 2 Y 5 ]. (III.7)
We see that the singlet superpotential W s has no constant term but includes a harmless tadpole for the massive field, S 9 . The massive singlets are S 1,9,13,17 and Y 1,2,4,6,10 . The Higgs bidoublet couplings in W h show that h 1 decouples, while h 2 is left massless but lacks trilinear couplings to singlets. Indeed, the quartic coupling, φY 10 S 13 h 2 h 2 , cannot generate an effective trilinear coupling at lower scales, since both of the singlets Y 10 , S 13 are massive. No large mass pairing takes place between the conjugate bifundamental modes. The suppressed couplings in W hf f c preclude the possibility of discriminating between the light and heavy flavors. Some zero entries may be possibly lifted by assuming a finite VEV for the singlet Y 10 . The sextet and quartet modes, except the pair, q 1 q2 , remain massless. The couplings in W Cf f give no contributions to baryon number violating operators.

We have also scanned the individual D flat directions of orders n = 2, 3 and n = 4 in search of the bidoublet modes couplings. The full set of 15 monomials at orders n = 2, 3 are F flat. Of the 100 monomials at order n = 4, about 5 , with the pair of group factors, SU (2)1,2, identified with SU (2)L,R. Same notational conventions as in Table I are used. The Abelian charges (given by integers except for Q4) are defined in the rotated charge basis by the rescalings, Q1/0.1667, Q2/0.1667 , 4 Q3/0.00260, Q4/0.0007667, 18 QA/0.1244. 80 are lifted by F-terms. The results obtained by combining the contributions from the various flat directions are

) ′ × SU (2) ′ ) × U (1)
g M N32 γ D 6(R1 R2 R3) Q1 Q2 Q3 Q4 QA M N32 γ D 6(R1 R2 R3) Q1 Q2 Q3 Q4 QA 0 Σ ′ 1 1 1 1 6 0 0 0 0 -18
• n = 2, 3 : W h = h 1 h 1 [Y 18 ] + h 2 h 2 [0] + h 1 h 2 [0]. • n = 4 : W h = h 1 h 1 [φ 4 Y 18,19 ] + h 2 h 2 [φ 4 Y 14,15 + φ 3 S 13 ] + h 1 h 2 [φ 3 S 1 + φ 2 Y 5 + φ 4 S 2 ]. (III.8)
We see again a clear trend in favor of several light bidoublets coupled by trilinear terms to massless singlets.

C. Model B

The massless spectrum for model B is displayed in Table IV. We find no holomorphic invariant monomials at order n = 2. At order n = 3, there appears 16 solutions of which a few representative monomials are: No Abelian charges remain unbroken along P (3) . Assigning a common VEV φ to these modes yields the reduced superpotential

P α = [S 11
W s = φY 3 S 8 + φY 4 S 5 + φS 3 Y 7 + φS 5 Y 5 + φS 6 Y 10 + φS 8 Y 8 . W h = h 1 h 1 [S 16 ] + h 2 h 2 [0] + h 3 h 3 [0] + h 1 h 2 [0] + h 1 h 3 [S 2 ] + h 2 h 3 [0]. W C = C 1 C 1 [0] + C 2 C 2 [S 16 ] + C 1 C 2 [0]. W q = q 1 q1 [φ] + q 2 q2 [0] + q 1 q2 [φ] + q 2 q1 [0]. W f = f 1 f1 [0] + f 1 f2 [0] + f 2 f1 [0] + f 2 f2 [φY 3 + φY 8 ] + f c 1 f c 1 [S 1 (Y 19 + Y 20 )] + f c 1 f c 2 [0] +f c 2 f c 1 [S 1 ] + f c 2 f c 2 [S 1 Y 21 ] + f c 3 f c 1 [S 1 Y 19 ] + f c 3 f c 2 [φ 2 Y 1 Y 7 ]. W hf f c = h 1 f 1 f c 1 [Y 19 ] + h 1 f 1 f c 2 [Y 15 + Y 15 Y 19 ] + h 1 f 1 f c 3 [φY 12 + φY 14 ] +h 1 f 2 f c 1 [0] + h 1 f 2 f c 2 [0] + h 1 f 2 f c 3 [0] + h 2 f 1 f c 1 [0] + h 2 f 1 f c 2 [0] + h 2 f 1 f c 3 [0] +h 2 f 2 f c 1 [0] + h 2 f 2 f c 2 [0] + h 2 f 2 f c 3 [0] + h 3 f 1 f c 1 [φY 12 + φY 14 ] + h 3 f 1 f c 2 [Y 15 ] + h 3 f 1 f c 3 [Y 15 ] +h 3 f 2 f c 1 [0] + h 3 f 2 f c 2 [0] + h 3 f 2 f c 3 [0]. W Cf f = C 1 f 1 f 1 [0] + C 1 f 2 f 2 [0] + C 1 f c 1 f c 1 [Y 19 + Y 20 ] + C 1 f c 2 f c 2 [Y 15 ] + C 1 f c 3 f c 3 [φY 12 + φY 14 ] +C 2 f 1 f 1 [0] + C 2 f 2 f 2 [0] + C 2 f c 1 f c 1 [0] + C 2 f c 2 f c 2 [0] + C 2 f c 3 f c 3 [0]. (III.9)
The identical couplings for h 1 and C 2 is a consequence of the fact that these modes have same quantum numbers. The singlets superpotential W s contains no constant or linear terms, while bilinear terms appear for the singlets S 3,5,6,8 and Y 3,4,5,7,8,10 , which thus pick up large masses. The bidoublets couplings in W h show that the three sets of h i modes remain massless, and have trilinear couplings (hhS 2 + hhS 10 ) involving the massless singlets S 2 and S 10 . No mass pairings arise between the conjugate bifundamental modes that would remove some of the f i , f c i modes or decouple some of the mirror fermions, f i , fi and f c i , f c i . The bidoublet-fermions couplings in W hf f c give vanishing mass matrices for the matter fermions. The flavor structure is problematic since it gives no hint on how to discriminate between the heavy and light flavors or between the matter and Higgs modes, independently of the dominant linear combinations of the bidoublets. The results for W C and W q show that all the sextet modes, C i , and the single pair of quartet modes, q 2 , q1 , remain unpaired.

The special role of the singlet S 16 motivates us in studying its self-couplings at orders n ≤ 4. The results for P (3)

: W s (S 16 ) = S 16 [0] + S 2 16 [0] + S 3 16 [0]
, indicate that the mode S 16 is light but has a suppressed cubic coupling. The results for P (4)

, W s (S 16 ) = S 16 [0] + S 2 16 [φ 5 Y 13 + φ 5 Y 11 ] + S 3 16 [φ 4 Y 3 + φ 4 Y 8 ]
would also allow for a light mode S 16 provided that the massive singlets Y 3,8,11 acquire small VEVs.

Results for seven randomly selected individual flat directions are displayed in Table V. For all monomials other than P IV , we see that all the h i remain light and have trilinear couplings to singlets, but that the sextets and a subset of the quartets fail to decouple. We have also performed global type scans of the D flat monomial solutions restricted to the bidoublets couplings. Of the 16 and 100 solutions at orders n = 3 and 4, we find 0 and 10 monomials lifted by F-terms. The bidoublets couplings combining the contributions from the various flat directions are given by [10] with gauge group, G422 × (SO(10) ′ ) × U (1) 6 . We use same notational conventions as in Table I, except that the pair of group factor SU (2)1,2 identify here with SU (2)R,L.

• n = 3 : W h = h 1 h 1 [S 16 ] + h 2 h 2 [0] + h 3 h 3 [0] + h 1 h 2 [0] + h 1 h 3 [S 2 ] + h 2 h 3 [0]. • n = 4 : W h = h 1 h 1 [S 16 + φY 19 + φY 20 ] + h 2 h 2 [0] + h 3 h 3 [φY 15 + φS 2 ] + h 1 h 2 [0] +h 1 h 3 [φ + φS 2 + φS 16 ] + h 2 h 3 [φ 3 Y 13 + φ 2 S 5 + φ 2 Y 1 ]. (III.10)
The quoted integer rescaled charges are related to the initial ones by the rescalings, 4 Q1/0.005208, 25 Q2/0.002042 , 12 Q3/0.009615, 9 Q4/0.00347, Q5/0.000672, QA/0.017796.

g M N32 γ D 6(R1 R2 R3) Q1 Q2 Q3 Q4 Q5 QA M N32 γ D 6(R1 R2 R3) Q1 Q2 Q3 Q4 Q5 QA 0 S1 1
+ φ 2 S16 φS2 + φ 2 S16 h1h2 h1h3 S2 S2 S2 φ S2 + φS16 S2 + φS16 S2 h2h3 φ 3 Y13 φ 2 Y1 + φ 2 S5 C1C1 C2C2 S16 S16 S16 S16 + φY19,20 S16 S16 S16 C1C2 q1 q1 φ Y25 Y25 + φ 2 Y11,14 + φ 2 S19 Y25 q2 q2 q1 q2 Y23 φ φ + φ 2 Y12 Y25 + φ 2 Y11 + φ 2 S10 Y25 + φ 2 Y11 + φ 2 S19 Y25 + φ 3 Y22 + φ 4 Y19,20 q2 q1 Y23 Y23 Y23 φ + φ 2 Y21
These results again favor the scenario in which light bidoublets interact by trilinear couplings to singlets. A scan of the flat directions reveals the existence of a large majority of individual monomial directions along which the mode S 16 is light but with a suppressed cubic self-coupling.

IV. DISCUSSION AND CONCLUSIONS

Let us first state some general features of our results. The top-down construction for the Z 6-II orbifold has a large number of flat directions and a rich structure of superpotential couplings. This contrasts with the situation prevailing for the Z 3 orbifold models [1,7,35], but is in harmony with that in non-prime orbifolds or free fermions. Comparing with the string spectra obtained in intersecting brane models [69,78] would not be very teaching because the existing type I string models are more akin to bottom-up type constructions.

We have carried the F flatness test indiscriminately for both types A and B flat directions, restricting to the superpotential monomials with at most four distinct singlet field factors,

W s (φ i ) = n i=1 φ si i , [n ≤ 4, s i ≤ 2]
. These contain a subset of the monomials of absolute order ≤ 2n = 8. It is intuitively clear that the F flatness condition is more severe for the lower order superpotential monomials, since these are more likely to contain all (or all but one) of the fields excited along the flat direction. Upon pushing the F flatness test in model B to the order n = 5 of W s , we find that out of 100 order n = 4 flat monomials, 34 are lifted by the 3 monomials of order n = 3 in

W s = S 1 Y 2 19 , Y 4 S 5 Y 25 , S 5 Y 5 Y 23 .
Including all the monomials in W s up to order 8, selects, 0, 6, 40 flat monomials of orders n = 2, 3, 4.

To determine which of the singlets decouple and which remain massless, we have derived the mass matrices for the S i and Y i by scanning over each individual flat direction up to absolute order 8. We found that one cannot decouple all the singlets at the string scale by these means in any of the models. However, taking the combinations of all the singlets of a given order, we found that all singlets of models A1 and A2 can be decoupled if the order is large enough, n ≥ 5 and 7 respectively, and the number of fields involved is itself large enough. One might prefer vacua defined by smaller sets of fields, and indeed our aim here is not to decouple all the singlets but leave a few of them massless in order to obtain an NMSSM like model. We thus prove that there is room for decoupling all but a few relevant modes, should one perform a thorough scan over the flat directions. As far as model B is concerned, the number of singlet fields is too high to decouple all of them at the order considered, and making this model viable on a phenomenological basis would require giving a lot of singlet modes a mass at the level of the Pati-Salam breaking down to the Standard Model gauge group.

Our study of the fermion mass generation has been rather sketchy. The preferred candidate for the electroweak Higgs bidoublet is obvious only in model A1. To discriminate between the heavy and light flavors in models A2 and B requires a closer analysis of the couplings of bidoublets and bifundamentals. Such a task would be warranted once one has really in hand a benchmark type model. All three models satisfy at the string mass scale a D 4 family symmetry with a number of bifundamental modes f or f c from the twisted sectors T 1 or T 3 transforming as doublets.

An acceptable description of the fermions flavor structure can be achieved only if the D 4 symmetry is broken. On side of the promising mechanism using the condensation at the Pati-Salam breaking scale of the composite singlet fields,

O c ij = f c i f c j
, stringy mechanisms can be envisionned to lift the degeneracy of the modes in the T 1,3 sectors. One could use blow up submanifolds of different sizes at the two fixed points, or consider the so far lightly explored models with three Wilson lines [34].

The existence of a light pair of Higgs bosons seems to be correlated with the presence of massless exotic color triplets descending from sextet and bifundamental modes, C i , f ′ c i . This is unavoidable in models A1 and B because of certain h i and C i having identical quantum numbers. A single pair of q i , qi modes always fails to decouple. Since the exotic modes are vector like, they could pair off or couple to lighter modes into which they decay, through lower scale physics, including the P S → SM transition. We have not examined here the decoupling of the weak doublet and singlet exotic modes, d l = (N l , E l ), d r = (E r , N r ), since these represent perhaps a lesser threat on the low energy theory.

Our main purpose in this study was to establish an existence proof, based on three representative string models, of supersymmetric models where suppressed bilinear couplings of the electroweak Higgs doublets coexist with unsuppressed trilinear couplings to singlets, µh k h l +λh k h l φ, [µ ≤ O(φ 8 ), ]. Rather than scanning over the flat directions, we have followed an approximate procedure making use of what we term as composite flat directions. This is admittedly open to criticism, especially at high orders, because of the large number of excited singlets. However, the qualitative orientation this approach gives does not seem invalidated by selecting randomly, or scanning in a global way, a subset of individual flat directions. We frequently encounter canonical and non-canonical type trilinear couplings, λ k ij h i h j φ k , however, without any obvious correlations between these bidoublets and those dominating the fermion-Higgs Yukawa couplings.

Additional U (1) ′ gauge symmetries are generally present in the individual flat directions of order n ≤ 4, but they are absent in most composite directions. It is difficult to decide if a gaugino supersymmetry breaking mediation or a secluded sector scenario is favored since our searches give no clue on the size of the Z ′ boson mass scale and the Z -Z ′ mixing angle.

In summary, the search of models with extra singlets is considerably facilitated in the Z 6-II orbifold by the rich mini-landscape of vacua. We find a clear preference towards an active role for extra singlets, but the presence of sizeable trilinear couplings hhφ along with strongly suppressed bilinear couplings hh is not systematic. The study of solutions was made tractable thanks to the restriction to low orders, n ≤ 4. Beyond this order one must consider representative samples of the flat directions. However, the composite flat directions of low orders, n ≤ 4, seem to capture general features not invalidated by global scans of the individual directions, as confirmed by the similar conclusions for models A1 and B. The strong O(φ 8 ) suppression required for the µ couplings suggests, however, that definitive conclusions could only be made until the searches include higher order couplings. Weak and strong points are present in all three models, with no model faring best on all issues.

V. ACKNOWLEDGEMENTS

We would like to thank Dr. M. Ratz for advice and useful discussions. One of the authors (P.H.) thanks also R. Kappl for useful discussions and is grateful to the Alexander Von Humboldt Foundation for the financial support granted to him during the elaboration of the present paper.

Appendix A: Flat directions in heterotic string compactification Given a gauge theory with the Lie group G a and superpotential W , which includes the set of massless chiral supermultiplets, φ i , [i = 1, • • • , N ] then the D flatness conditions are given by: 0 = D a = (φ ⋆ i (T a ) ij φ j ), where T a are the Lie algebra generators, and the F flatness conditions are given by: W i ≡ ∂W ∂φi = 0. In the scalar fields φ i vector space, the D flat directions are parameterized by the gauge invariant monomials, P α (φ i ), satisfying the equations, P α,i ≡ ∂Pα ∂φi = c α φ ⋆ i , for suitably chosen complex constants, c α , with the φ i substituted by their (constant) VEV [1,80]. (This statement is easily checked by writing the G a gauge invariance condition as, 0 = δ a P α = i ∂Pα ∂φi δ a φ i = c α i φ ⋆ i (T a φ) i .) For an anomalous gauge symmetry [81][82][83] U (1) A with the φ i charges, Q i A = Q A (φ i ), and a finite charge anomaly implying the presence of a Fayet-Iliopoulos term with coupling constant ξ A , the D-flatness condition is modified to

0 = D A = i φ ⋆ i Q i A φ i + ξ A , ξ A = e 2φ M 2 ⋆ δ A GS = g 2 s M 2 ⋆ 192π 2 1 √ k A T r(Q A ), k A = 2 I (Q I A ) 2 (A.1)
where g s =< e φ > is the string coupling constant, with φ the 10-d dilaton field, and 16] are the orthogonal basis components of the anomalous charge generator in the E 8 × E 8 group weight lattice. For the set of gauge group factors, G = ( a G a ) × U (1) A , the D flatness conditions is solved by considering holomorphic invariant monomials, P α (φ i ), uncharged with respect to the anomaly free gauge factors G a , but with finite anomalous U (1) A charge, Q A (P α ) = 0, of opposite sign to ξ A ∝ T race(Q A ). In the supergravity context, the F flatness equations are, W = 0, F i = ∂W/∂φ i = 0, where W is built from holomorphic monomials invariant under the complete set of non-Abelian and Abelian gauge symmetries, including the global type string theory symmetries.

Q I A , [I = 1, • • • ,
Most of our applications deal with the field space of non-Abelian singlets charged under the Abelian gauge groups, G = ( Na a=1 U (1) a ) × U (1) A . The D-flat directions are parameterized by the monomials, P α = i φ r α i i , solving the simultaneous equations: Q a (P α ) = 0, Q A (P α ) = 0 of sign opposite to ξ A . By contrast, the superpotential couplings are constructed from the gauge invariant monomials with respect to both non-Abelian and Abelian (anomaly free and anomalous) gauge groups which obey the string selection rules, listed for the orbifold Z 6-II in Eqs. (B.10). The submanifold of vacua for non-Abelian singlets [55] is parameterized by the holomorphic invariant monomials,

P α = i φ r α i i , [r α i ≥ 0] described by the column vectors, r α = (r α 1 , • • • , r α N ) ∈ Z N + , subject to the equations, Q a (P α ) = i ∂P α ∂φ i (Q a φ) i = P α (φ) × i r α i Q i a = 0, Q A (P α ) = P α (φ) × i r α i Q i A = -c A ξ A , [c A > 0]. (A.2)
These equations are solved by the simple solution, |φ i |/ r α i = |φ| = 1/c A . The flat directions may have 0, 1 or more dimensions, where the 0-d case involves a fixed common VEV, φ = (-ξ

A / i Q i A r α i ) 1 2
, and the 1-d or higher dimension cases involve a single or more free complex parameters. When the system of linear equations is underdetermined, the solutions depend on free continuous parameters whose number identifies with the moduli space dimension, D ≤ Nrank(A). The vector space of column vectors, r, is generated by the basis of column vectors,

r A = (r A 1 , • • • , r A N ), associated to the holomorphic invariant monomials, M A = i φ r A i i , [A = 1, • • • , D]
such that any solution can be written as, P n = α M nA A , for n ≥ 1 and n A ∈ Z (positive or negative signs). More conveniently, one can also consider the superbasis of one-dimensional invariant monomials, P α , satisfying the condition that they cannot be factorized into products of two or more invariant monomials.

The D flat direction described by the monomial solution, P α (φ) = i φ

r i α i , can be lifted by F-terms of type A, W A s = ( ′ i∈Pα φ r i α i ) n or type B, W B s = Φ( ′ i∈Pα φ i ) n , [Φ ∈ P α ]
consisting of gauge invariant superpotential monomials with all, or all but one, field factors included in P α (φ). In the terminology of [56,57,62], these refer to the gauge invariant non-renormalizable couplings in the field theory action, prior to applying the string selection rules. For the F flatness to remain valid to arbitrary orders of the full superpotential, an infinite number of conditions must be imposed for a type A direction but a finite number for a type B direction. The F flatness conditions are, of course, more restrictive when set on the string theory action. 4)). The line entries refer to the twisted sectors, Tg, [g = 1, • • • , 4] of Z6-II with T1 ∼ T5, and the associated Tg 1 ,g 2 sectors of the orbifold, Z2 × Z3. The second colum displays the number of fixed points N I f p in the three complex planes T 2 I . The third column consists of three subcolumns which display for the lattice Λ(G2), the fixed points and shift vectors, fa, ua and f ab , u ab , for the elements Θ 2,4 and Θ 3 , the linear combination eigenvectors, |γ(g) >, and the eigenvalues γ(g), with ω = e 2iπ/3 . The fourth column consists of two subcolumns which display for the lattice Λ(SU (3)), the fixed points and shift vectors, fn 3 , un 3 , for the elements Θ 1 and Θ 2,4 , and the range of n3. The fifth column consists of three subcolumns which display for the lattice Λ(SO(4)), the fixed points and shift vectors, f n 2 ,n ′ 2 , u n 2 ,n ′ 2 , for the elements Θ 1 and Θ 3 , and the range of n2, n ′ 2 . The last column displays the modes multiplicities D assigned in the various twisted sectors Tg, in the case W ′ 2 = 0 with unresolved quantum number n ′ 2 = 0, 1 for the orbifold group action on T 2 1 . The entries for D correspond to the entries for the eigenvectors, |γ(g) >, of eigenvalues γ(g).

Z 6-II Z 2 ×Z 3 N I f p T 2 1 = Λ(G2) |γ(g) > γ(g) T 2 2 = Λ(SU(3)) n3 T 2 3 = Λ(SO(4)) n2 n ′ 2 D T 1 +T 5 T 1,2 +T 1,1 1,3,4 fα =0 uα=0 1 1 fn 3 = [0, 2 3
f n 2 ,n ′ 2 = n 2 2 e 5 + n ′ 2 2 e 6 u n 2 ,n ′ 2 =-n 2 e 5 -n ′ 2 e 6 0, 1 0, 1 2
T 2 +T 4 T 0,1 +T 0,2 3, 3, 1 fa = a 3 e 1 , ua=-ae 2 [a=0,1,2] f 0 , 1 √ 2 (f 1 +f 2 ) 1 √ 2 (f 1 -f 2 ) 1 -1 fn 3 , un 3 0,1,2 fα =0 uα=0 0 0 2 1 T 3 T 1,0 4,1,4 f ab = 1 2 (ae 1 +be 2 ) u ab =ae 1 +be 2 [a,b=0,1]   f00, f 01 +f 10 +f 11 √ 3 f 01 +ωf 01 +ω 2 f 11 √ 3 f 10 +ω 2 f 01 +ωf 11 √ 3   1 ω 2 ω fα=0 uα=0 0 f n 2 ,n ′ 2 u n 2 ,n ′ 2 0, 1 0, 1 4 2 2
vectors u f and sublattices Λ I Θ g = (1 -Θ g )Λ(T 2 I ) of the twisted sectors g are defined as follows for the three 2-d tori,

T 2 I , with G 2 , SU (3)
, SO(4) group weight lattices:

Λ(G 2 ) : (1 -Θ 2,4 )f a = u a = -ae 2 , f a = a 3 e 1 , [a = 0, 1, 2]; (1 -Θ 3 )f a,b = u a,b = a 2 e 1 + b 2 e 2 , [a, b = (0, 1)]; Λ 1 Θ ∼ (e 1 , e 2 ), Λ 1 Θ 2,4 ∼ (e 1 , 3e 2 ), Λ 1 Θ 3 ∼ (2e 1 , 2e
(1 -Θ g )f n2,n ′ 2 = u n2,n ′ 2 = -n 2 e 5 -n ′ 2 e 6 , f n2,n ′ 2 = 1 2 (n 2 e 5 + n ′ 2 e 6 ), [n 2 , n ′ 2 = 0, 1]; Λ 3 Θ g ∼ (2e 5 , 2e 6 ), [g = 1, 3]. (B.2)
The orbifold fundamental group allows for three discrete Wilson lines: W 2 , W ′ 2 of order N 2 = N ′ 2 = 2, around two dual one-cycles of T 2 3 , and W 3 of order N 3 = 3 around the e 3 one-cycle of torus T 2 2 . Upon turning on the Wilson lines, the T g twisted sectors split into twisted subsectors, (g, γ, n 3 , n 2 , n ′ 2 ), labelled by the discrete parameters, n 3 = 0, 1, 2 and n 2 = 0, 1, n ′ 2 = 0, 1, for the lattices Λ(T 2 2 ) ∼ Λ(SU (3)) and Λ(T 2 3 ) ∼ Λ(SO(4)). The complex phase parameters, γ(g), describing the action of Θ g on the fixed points of the lattice Λ(T

R i , N L i ∈ Z, [i = I, Ī; I, Ī = 1, 2 ,
3] for the 3 rightand left-moving complex coordinate fields X I of T 6 . The string squared mass spectrum is evaluated in terms of these quantum numbers by means of the formula

M 2 R 8m 2 s = I (N R I ω (g) I + N R Ī ω (g) Ī ) + 1 2 4 a=1 (r a + gv a ) 2 -E (g) 0 - 1 2 , M 2 L 8m 2 s = I (N L I ω (g) I + N L Ī ω (g) Ī ) + 1 2 16
I=1 (P I + X I g,n f ) 2 -E (g) 0 -1, [E (g) 0 = 1 2 I | ĝ v I |(1 -| ĝ v I |), ĝ v I = gv I mod 1 = gv I -[gv I ],
TABLE VII: The SO(8) group weight vectors, r l,r v and r l,r s , assigned to the massless right moving modes, M 2 R = 0, with left and right chiralities (l, r). The vector and spinor representations, (v, s) differ by the weight vectors ρ l,r assigned to the supercharge generators, r l,r s

= r l,r v + ρ l,r , [ρ l,r = (± 1 2 ± 1 2 ± 1 2 , ± 1
2 )]. The line entries refer to the untwisted and twisted sectors, g = 0, 1, • • • , 4. No massless mode solutions arise for right movers carrying oscillator exitations, N R I, Ī .

•

g = 3, h = 2 : ∆ h = (γφ) h e 2iπh[(P + 1 2 (gV +n2W2+n ′ 2 W ′ 2 ))•(V +n2W2+n ′ 2 W ′ 2 )-(r+ 1 2 gv)•v] = 1 h P • W 3 ∈ Z, [ P = P + X g,0,n2,n ′ 2 ] . (B.7)
This is the prescription for the GSO orbifold phase that we have used to obtain the string spectra of models A1, A1, B.

Use of the world sheet modular invariance (level matching conditions) allows replacing the conditions on the scalar products in Eqs. (B.7) by the simpler ones [10]:

P • W 2 ∈ Z, P • W ′ 2 ∈ Z for g = (2, 4
) and P • W 3 ∈ Z for g = 3. The following alternative projections are also proposed in [15] • g = (2, 4) :

3[(N L i -N R i )v i + q γ (g) -(r + gv) • v + (P + gV + n 3 W 3 ) • V ] ∈ Z (P + gV + n 3 W 3 ) • W2 W ′ 2 ∈ Z , • g = 3 : 2[(N L i -N R i )v i + q γ (g) -(r + gv) • v + (P + gV + n 2 W 2 + n ′ 2 W ′ 2 ) • V ] ∈ Z (P + gV + n 2 W 2 + n ′ 2 W ′ 2 ) • W 3 ∈ Z , (B.8)
where γ(g) = e 2iπqγ (g) . It is important to remark at this point that the orbifold projection for sectors with fixed T 2 subtori boils down to the invariance under the world sheet modular group. The modular invariance is vital if the mixed gauge and gravitational anomalies in the various Abelian gauge factors U (1) a are to satisfy the universality relations [81, 84]

1 24k 1/2 a T r(Q a ) = 1 3k 3/2 a T r(Q 3 a ) = 1 2k 1/2 a T r(Q a c b (R)) = 1 k 1/2 a k b T r(Q a Q 2 b ) = 8π 2 δ a GS , (B.9)
which are necessary for the Green-Schwarz type anomaly cancellation. mechanism to work. When the orbifold space group action by h = (Θ h , u h,f ) along the string time direction does not commute with that of the constructing orbifold element along the string space direction, g = (Θ g , u g,f ), the physical states must be constructed by summing over elements of the orbit, h n gh -n , and including suitable complex phase factors to ensure the SL(2, Z) modular group invariance [17,20,85]. This situation indeed occurs for the sectors, g = 2, 4 and g = 3, where the fixed points in T VI.

For non-commuting twists, [h, g] = 0, the complex phase γ(g, h) is not determined in a unique way by the GSO projection. Since γ(g, h) fixes the degeneracies of string modes, its choice has clearly an incidence on the U (1) a groups anomalies and hence on the modular invariance. The choice of γ(g, h) can be set uniquely by requiring the mixed gauge and gravitational anomalies to satisfy the anomalies universality relations in Eq. (B.9). Given a massless string state, specified by the quantum numbers, P I , R I , N L i , then the freedom on γ(g, h) is taken into account by redefining, γ(g, h) → χ(h)γ(g, h), in terms of the GSO modified condition, ∆ h χ h = 1, so that the mode multiplicities determined by γ(g, h) give anomaly coefficients satisfying the requisite universality relations. Although the natural choice, χ = 1, applies for the large majority of modes, there do occur cases where a non-trivial χ in ∆ h is needed to satisfy the universality relations. The factors χ(h) depend on the prescription used for the modified gamma phases. With our GSO projection prescrition in Eqs.(B.7), one exception occurs for three modes of sector T 3 in model A2 and another one for two modes of sectors T 2 and T 4 in model B.

Finally, we quote the string selection rules on the n-point world sheet correlators of vertex operators, < n l=1 V l (z l , zl ) >. Applied to the superpotential couplings, the conditions set by the gauge symmetries, the Hmomentum conservation, and the orbifold point and space symmetry groups read: (n 2 (l), n ′ 2 (l)) = (0, 0) mod 2. (B.10) Some of the above selection rules can be formulated as ordinary [11,86] or R type [87] discrete symmetries. We have omitted from the list in Eq.(B.10) the constraints on couplings from the space orbifold group action on the T 2 1 torus with lattice Λ(G 2 ). These consist of the following two selection rules, stated around Eq. (C.9) of [10]:

• Rule I on the gamma phases of physical modes: n l=1 γ l = 1. Using the definition, γ l ≡ γ(g l ) = e 2iπqγ l , this can also be written as, n l=1 q γ l = 0 mod 1. • Rule II for couplings involving the twisted sector modes T 2,4 only, or twisted sector modes T 3 only, times untwisted sector states. The version of this rule derived in [13] states that the column vector of gamma phases for fields in n-point couplings T p1 2,4 T p2 0 or T p1 3 T p2 0 , with p 1 = 0, n > p 2 > 0 must obey the condition, q γ l ∈ [(p, 0, • • • , 0) + perms], [p = 0]. Thus, the allowed couplings are those containing at least two physical modes with non-trivial gamma phases, γ l = 1. The version of this rule derived in [10] uses instead the conjugacy classes of the lattice cosets, Λ/Λ Θ g , associated to the fixed points f l a l = a l 3 e 1 , [a l = 0, 1, 2] in sectors g = 2, 4 and f l a l ,b l = a l 2 e 1 + b l 2 e 2 , [a l = 0, 1; b l = 0, 1] in sector g = 3, in the notational conventions of Table VI and Eq. (B.2). In terms of the fixed point content of the physical states (with fixed γ l eigenvalues), the orbifold space group constraints are described for the couplings T p1 2,4 T p2 0 by the Z 3 group selection rule, n l=1 a l = 0 mod 3, and for the couplings T p1 3 T p2 0 by the Z 2 × Z 2 group selection rule, n l=1 a l = 0 mod 2, n l=1 b l = 0 mod 2. Convincing arguments are given in [13,20,85] that Rule I is not a genuine selection rule, since it is found to be automatically satisfied by physical states obeying the GSO orbifold projection, P (g, n f , γ, φ) = 1, once the selection rules on Θ g l I , P I (l), R I (l) are imposed. In a consistent string model complying with the orbifold GSO projection, and hence with the world sheet modular group invariance, Rule I is redundant. However, the redundancy was proved with the specific prescription for the orbifold projection quoted in Eq.(B.8). Since our prescription in Eq.(B.7) is quite close but not strictly identical to that used in [13,20], there is no guarantee that Rule I is automatically implied by other rules in our case too. That a selection rule is superfluous in one projection prescription and not in another appears odd at first sight but cannot be excluded. It certainly is a logical possibility if several consistent string models arise from the same data set of shift vectors, V, W 2 , W 3 . To settle this issue on practical grounds, we have applied our search procedure for models A1, B with the rule n l=1 γ l = 1 included, and found that this did forbid certain superpotential couplings allowed by the rules in Eq.(B.10). However, the excluded couplings represent a very tiny fraction of the large set of allowed couplings, which cause insignificant changes on the effective couplings. We have also tested the Rule II above and found that this had a negligible incidence on the allowed couplings.

  II. LOW ENERGY THEORYA. General features of Pati-Salam string modelsOur interest is on the 4-d orbifold compactifications of the E 8 ×E ′ 8 heterotic string satisfying N = 1 supersymmetry, with gauge symmetry group, G × G ′ , including a Pati-Salam observable sector,G = G 422 × Na a=1 U (1) a , [G 422 = SU (4) × SU (2) L × SU (2) R ] (II.1) and a hidden sector, G ′ = SO(10) ′ × SU (2) ′ or G ′ = SO(10) ′ , (II.2)

  out the massive modes C i , by means of the classical fields equations, can produce baryon and/or lepton number violating couplings represented by local operators of dimension D = 4: lqd c , u c d c d c , lle c , or of dimension D = 5: qqql, u c d c d c ν c . The resulting dangerous local operators,

  S 18 Y 25 , S 11 S 18 Y 26 , S 11 Y 23 S 20 , S 11 Y 24 S 20 ]. The composite direction, P (3) , consists of the eight modes, S 11 , S 18 , Y 25 , Y 26 , Y 23 , S 20 , Y 24 , S 15 .

  2 ). Λ(SU (3)) : (1 -Θ g )f n3 = u n3 = [e 3 , e 3 + e 4 ] = n 3 e 3 , f n3 = [0g ∼ (e 3e 4 , 3e 3 ), [g = 1, 2, 4]. Λ(SO(4)) :

2 1 )

 1 ∼ Λ(G 2 ) are needed to specify the degeneracy of string modes in the sectors g = 2, 4 and g = 3. b The string states are described by the SO(8) group weight vectors, r a , [a = 1, • • • , 4] for the right-moving fermion fields; the E 8 × E 8 group weight vectors, P I , [I = 1, • • • , 16] for the left-moving 16 coordinate fields; and the oscillator numbers, N

2 1

 2 with lattice Λ(G 2 ) are reshuffled by the action of h. The physical string modes decompose then into eigenvectors of Θ g , [g = 2, 3, 4] with eigenvalues γ(g) = ±1 in the (g = 2, 4) sectors T 2 , T 4 and γ(g) = 1, ω, ω 2 , [ω = e 2iπ/3 ] in the (g = 3) sector T 3 . The degeneracy of eigenstates is D = 2 in T 1 ; D = (1, 2) for γ = (1, -1) in T 2 , T 4 , and D = (4, 2, 2) for γ = (1, ω 2 , ω) in T 3 , as displayed in Table

( 1 -

 1 (l) + X I g,n f (l)] ∈ Λ(E 8 × E 8 ), [X g,f = gV + n a g,f W a ]. mod (6, 3, 2), [R I (l) = r I (l) + g l v I (l) -N L I (l) + N L Ī (l)]. Θ g l I )(f l -Λ I ) = 0 mod N =⇒n l=1 n 3 (l) = 0 mod 3, n l=1

  3 , C n = 1, and dropping the factor depending on Q A by setting, -T race(QA)

	√ C n I n ≈ 10 -2n+3 /2 n+1 . A similar formula holds for the µ-term coefficient of the electroweak Higgs kAQA( φ) → 1, gives, bosons bilinear coupling, c(n) ijk ≃ 10 -2n g 1+n s c(n) ij h i h j , by substituting, c(n

TABLE I :

 I Spectrum of string massless modes for the Pati-Salam model A1 of[10] with gauge group, G422 ×(SO(10) ′ ×SU (2) ′ )× U (1)5 . The pair of group factors SU (2)1,2 identify with SU (2)L,R. The non-Abelian singlet modes φi ∼ (1, 1, 1)(1, 1) ′ without and with oscillator excitations are denoted by Si and Yi. The non-singlet modes include the doublets,

  1 Y 2 S 12 , Y 8 S 7 S 12 < S 14 >, < Y 2 4 > Y 12 Y 16 S 12 , Y 10 S 7 S 12 < S 13 >, that can give rise to a secluded sector with three massless singlets coupled by mildly suppressed trilinear interactions. A systematic search does not appear warranted at this stage.

The special role of S 10 motivates us to examine its low order self-couplings, which are given for n ≤ 4 by W s (S 10 ) = S 10 [0] + S 2 10 [φ 3 S 7 ] + S 3 10 [φ 4 S 2 7

  monomials, S 1 S 2 8 S 15 , Y 9 S 2 8 S 11 , S 3 Y 16 Y 18 , S 3 Y 17 Y 18 , S 3 Y 17 Y 19 , S 3 Y 16 Y 19 , S 8 S 10 S 15 .The number of solutions at order n = 4 exceeds O(100). The combined composite flat direction P(2) × P(3) for n ≤ 3 includes the eleven modes, Y 9 , S 8 , S 1 , S 15 , S 11 , S 3 , Y 16 , Y 18 , Y 19 , Y 17 , S 10 . This remains unlifted by all the singlet couplings in W s of order n ≤ 4. However, no unbroken U (1) charges survive if the above eleven modes simultaneously acquire finite

TABLE

TABLE III :

 III String spectrum of massless modes for model A2 of[10] of gauge group, G422 × (SO(10

  15.05 -432 d ′

	S1 1 Y1 Y3 Y5 Y7 f2 Σ ′ 2 d ′ 3 d ′ 4 d ′ 5 q1 q2 d l 2 d r 2 h ′ r 1 q2 d r 3 2 S2 Y13 1 -1 1 1 1 1 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 3 1 2 3 1 2 4 1 2 4 1 2 4 1 2 4 1 2 5 1 2 6 1 2 6 1 2 1 -1 1 C1 1 -1 1 d ′ 6 2 -1 2 S5 3 -1 1 S6 3 -1 1 T ′ 1 3 -1 1 Y15 3 -1 1 3 S7 1 ω 2 S9 1 ω 2 S11 1 1 4 q3 2 ω 2 2 d r 4 2 ω 4 4 C2 1 1 2 Y16 1 1 2 S14 1 1 2 d ′ 9 2 1 1 Y18 2 -1 2 d ′ 10 2 1 1 h2 2 -1 2 S17 3 1 2	0 1 -5 -4 3 6 0 2 9 -5 2 3 -5 2 3 1 2 3 1 2 3 -11 2 3 1 2 3 1 2 3 -5 2 3 1 2 3 1 2 3 -11 2 3 1 2 3 1 2 3 1 2 3 2 4 0 2 10 0 2 4 0 2 4 0 2 4 0 2 4 0 2 4 0 2 10 0 3 0 3 3 0 3 3 0 3 3 0 3 3 0 3 4 2 0 4 -4 0 4 2 0 4 2 0 4 -4 0 4 2 0 4 2 0 4 2 0	1 -18 f1 2 2 -20 -10.45 -96 Y2 0 -6 48 0.63 2 2 -20 -10.45 -96 Y4 2 -4 28 -9.82 -114 Y6 -4 2 28 -9.82 -114 f c 1 -1 -1 4 -10.14 -105 Y8 0 0 18 -15.05 36 d ′ 2 0 0 12 30.73 -90 Y9 4 -2 -4 -51.00 -120 Y10 3 1 1 1 1 1 1 2 2 2 -2 4 -4 -51.00 -120 Y11 3 -1 2 -20 -10.45 -96 q1 4 -1 2 28 -9.82 -114 d l 1 4 2 5 4 -10.14 -105 d r 1 4 2 -1 4 -10.14 -105 d l 3 4 0 3 -12 30.41 -81 h ′ l 1 5 1 -2 -4 71.28 -66 d l 4 6 -2 1 20 71.59 -75 4 4 -40 -20.90 -192 Y12 1 4 -2 8 -20.28 -210 S3 1 -1 1 1 1 1 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 -2 -2 8 -20.28 -210 f c 2 1 1 2 -4 -4 -8 20.28 -186 S4 2 1 1 0 6 -24 60.83 -162 h1 3 -1 1 0 -6 -24 60.83 -162 d ′ 7 3 1 2 0 0 36 -30.10 72 Y14 3 1 2 8 3 1 2 0 0 24 61.45 -180 d ′ 6 0 -12 -30.73 -306 S8 1 ω 2 2 0 6 -12 -30.73 -306 S10 1 ω 2 0 0 36 -30.10 -324 S12 1 ω 2 3 0 12 30.73 306 q3 2 ω 2 2 0 -3 -12 30.41 315 d r 5 2 1 2 2 2 -8 20.28 210 S13 1 1 2 -4 2 -8 20.28 210 Y17 1 -1 1 -4 -4 40 20.90 192 f c 1 1 -1 1 0 0 -48 -0.63 -378 T ′ 2 2 -1 2 0 0 -24 -61.45 180 Y19 2 1 1 0 0 -24 60.83 234 S15 2 -1 2 0 0 24 -60.83 162 S16 2 -1 2 4 4 -16 40.55 -372 d ′ 11 3 -1 1	6 6 1 -17 2 0 0 2 -3 0 0 3 -5 2 3 1 2 3 -5 2 3 1 -4 3 -5 2 3 -5 2 3 -5 2 3 1 2 3 1 2 3 1 -4 3 -5 2 3 1 2 3 1 2 3 -4 4 0 2 4 0 2 4 0 2 4 0 2 4 0 2 4 0 -4 4 0 2 4 0 3 0 3 3 0 3 3 0 3 3 0 3 3 0 3 4 2 0 10 2 0 4 2 0 4 2 0 10 2 0 4 2 0 4 2 0 4 2 0	0 0 3 3 24 0 -122.28 -54 0.31 -9 2 2 -20 -10.45 -96 2 2 -20 -10.45 -96 -4 -4 -20 -10.45 -96 -1 -1 4 -10.14 -105 0 0 12 -91.55 -144 0 0 12 30.73 -90 0 0 36 -30.10 468 4 -2 -4 71.28 -66 -2 4 -4 71.28 -66 -1 -4 -20 -10.45 -96 2 -1 -44 -10.76 -87 2 -1 4 -10.14 -105 -4 -1 4 -10.14 -105 0 -3 -12 30.41 -81 -2 1 -28 70.97 -57 4 -2 8 -20.28 -210 41.18 402 4 -2 32 1 1 -16 -20.59 -201 -4 -4 16 -40.55 372 0 0 -24 60.83 -162 0 0 24 -60.83 -234 0 0 24 61.45 -180 0 0 48 0.63 378 324 0 0 -36 30.10 30.73 306 0 -6 12 30.73 306 -6 0 12 -3 0 -12 -30.73 -306 0 3 12 -30.41 -315 -4 2 -32 -41.18 -402 20.28 210 -4 2 -8 20.59 201 -1 -1 16 0 0 -36 30.10 -72 0 0 -24 -61.45 180 0 6 24 -60.83 162 0 -6 24 -60.83 162 4 4 8 -20.28 186

TABLE IV :

 IV Massless string modes for model B of

TABLE V :

 V Bilinear superpotential couplings for model B in the modes hi, Ci, qi, qi of order, n ≤ 4 in the singlets. The column entries refer to the three flat directions of order n = 3 : PI = S11S18Y25, PII = S11Y23S20, PIII = S15Y23Y26, and the four randomly selected flat directions of order n = 4 : PIV = S2Y6S 2 11 S20, PV = Y 2 4 Y 2 15 S17Y26, PV I = Y6S15Y15Y26, PV II = Y9S15Y18Y23. Empty entries correspond to cases where no coupling is present up to order n = 4.

	W	PI PII	PIII	PIV	PV	PV I	PV II
	h1h1 S16 S16	S16	S16 + φY19,20	S16	S16	S16
	h2h2
	h3h3			φY15	φS2

TABLE VI :

 VI Geometric data for the orbifold Z6-II on the torus T 6 = T 2

	1 × T 2 2 × T 2 3 of lattice, Λ(G2) + Λ(SU (3)) + Λ(SO(

Appendix B: Review of string construction for Z6-II orbifold

The orbifold Z 6-II is described by the rotation vector, Θ I = e 2iπv I 6 , or the twist vector, v I 6 = 1 6 (1, 2, -3), acting on the complexified coordinate and fermion field components X I , ψ I associated to the complex planes of T 2 I . This orbifold is equivalent to the direct product of suborbifolds, Z 2 × Z 3 , with twist vectors,

. The anisotropic compactifications [10,12] are realized with 6-d tori which factorize on the three maximal tori T 2 I of the rank 2 semi-simple groups, G 2 , SU (3), SO(4), with lattice basis vectors e 1,2 , e 3,4 , e 5,6 , and orbifold point group action represented by the 2 × 2 matrices for the Coxeter operators

The input data for the orbifold fixed points and shift vectors in the various twisted sectors is displayed in Table VI. The shift vectors u g,f ∈ Λ, associated to the fixed points f in sectors T g ∼ Θ g , are defined by, (1-Θ g )f = u g,f , modulo elements of the sublattice, Λ Θ g = (1 -Θ g )Λ, of the torus lattice Λ. The shift vectors are in 1-to-1 correspondence with the orbifold space group elements, (Θ g , u g,f ), which act on the orthogonal frame coordinates as, X → Θ g X + u g,f . Each shift vector u g,f represents a conjugacy class element of the torus lattice coset, Λ/Λ Θ g . The fixed points f , shift (B.3) (in units of m s ) where X g,n f denote the shift vectors in the E 8 ×E 8 group lattice depending on the orbifold and Wilson lines gauge embeddings, V, W 3 , W 2 , W ′ 2 . The oscillator energies are defined by ω (g)

Ī ≤ 1. In the Z 6-II orbifold, the string states generally occur in CPT conjugate pairs of opposite 4-d chirality for the sectors, g and g ′ = 6g, except for the twisted sectors T 2 and T 4 with g = 2, 4, which admit both left and right chirality modes each (because of the supersymmetry N = 2), and the (self-conjugate) twisted sector T 3 with g = 3. The vector and spinor weight vectors r a v,s of the SO(8) symmetry group for right moving fermions are associated to boson and fermion superpartners of the massless chiral supermultiplets as displayed in Table VII.

The GSO projection on the orbifold singlet states is determined via the one-loop partition function by the condition, P (g, n f , γ, φ) = 1, where

The twists along the world sheet spatial and temporal directions are denoted by g and h and the terms ∆ h include the complex phases, γ(g, h) = γ h (g), eigenvalues of the orbifold twist action on the fixed points in the 2-d torus lattice Λ(T 2 1 ) = Λ(G 2 ), and the complex phases from oscillator excitations, φ(g, h) = φ h (g),

The level matching for right and left movers entails the conditions, N (X 2 g,n f -(gv) 2 )) = 0 mod 2, with additional conditions involving the scalar products of Wilson lines, W 3 , W 2 , W ′ 2 . For the twisted sectors, T 2,4 and T 3 , the projections for the Z 3 and Z 2 suborbifolds can be implemented by summing over the time twists in the subsectors (h = 3, m 2 , m ′ 2 ) and (h = 2, m 3 ) as follows

Except for the singly twisted sector, T 1 , where the conditions in Eq. (B.4) hold with fixed values of n 3 , n 2 , the projections in the other sectors can be concisely stated in terms of simple sets of conditions on the weight vectors P I . For the untwisted sector, these conditions are: P • V ∈ Z, P • W 2 ∈ Z, P • W ′ 2 ∈ Z, P • W 3 ∈ Z. For the twisted sectors, T 2,4 and T 3 , the projections are more easily implemented by imposing the conditions .