R symmetries and a heterotic MSSM
aa r X i v : . [ h e p - ph ] N ov R symmetries and a heterotic MSSM Rolf Kappl, Hans Peter Nilles, Matthias Schmitz
Bethe Center for Theoretical PhysicsandPhysikalisches Institut der Universit¨at BonnNussallee 12, 53115 Bonn, Germany
Abstract
We employ powerful techniques based on Hilbert- and Gr¨obner bases to analyzeparticle physics models derived from string theory. Individual models are shown tohave a huge landscape of vacua that differ in their phenomenological properties. Weexplore the (discrete) symmetries of these vacua, the new R symmetry selection rulesand their consequences for moduli stabilization. Introduction
Recently new results on R symmetries in heterotic orbifolds have been obtained [1, 2]. Wediscuss the implications of these results on phenomenologically appealing models like theones obtained in [3–5]. Our purpose is twofold. On the one hand we show that with thenew R symmetries successful models remain. On the other hand we employ techniquesdeveloped in [6] to study the vacuum configuration of a given model. These techniquesenable us to determine the structure of the superpotential in form of building monomials.The advantage to previous attempts is, that one can immediately see which standardmodel singlets induce which couplings. We further show that this approach is also usefulfor general particle physics models with continuous or discrete symmetries. All of thesesymmetries lead to Diophantine equations whose solutions are given by so-called Hilbertbasis elements. After the determination of the superpotential we search for suitable SUSYpreserving minima of our model. We find that some of the standard model singlets will getstabilized at a non-trivial value. In contrast to [7] many singlets will have flat directions.This can be understood in terms of remnant symmetries.The paper is outlined as follows. In section 2 we show how symmetries result in Dio-phantine equations and how these can be solved to determine the structure of the under-lying physical model. In section 3 we briefly review new results for R symmetries in theheterotic orbifold context obtained in [1, 2] and comment on their consequences. We finallyapply the developed techniques to a phenomenologically appealing model in section 4 andsummarize our results. It has been outlined in [6] how to use Hilbert bases to compute a basis of all allowedmonomials in the superpotential. In this section we will use this technique to derive whichcouplings are allowed by the R symmetries and the other well known string selection rules.We will explain the approach by an example from flavor model building [8] and apply thetechnique to our concrete string model. Given some continuous or discrete symmetries one is usually interested in the questionwhich couplings are allowed by the given symmetries. Let us assume a theory with a U(1)symmetry and fields φ i with charges q i under the symmetry. An example for an allowedmonomial would be W ⊃ φ φ φ ⇔ q + q + 2 q = 0 . (1)We can generalize this to W ⊃ φ n . . . φ n M M ⇔ q T · n = 0 , (2)where q T = ( q , . . . , q M ) and n T = ( n , . . . , n M ) ∈ N M . Because of the restriction n i ∈ N this is called a Diophantine equation. That means every Abelian gauge symmetry leads2aturally to a Diophantine equation. It is not possible to give an analytical basis for allsolution vectors n because the natural numbers only form a monoid. Nevertheless it ispossible to algorithmically determine a basis. We can write n = X i α i x i , x i ∈ H , α i ∈ N . (3)Note that the total number of Hilbert basis elements x i cannot be determined analytically.The basis spanned by the elements x i is called Hilbert basis and several freely availableprograms [9, 10] exist to determine it. This was shown and outlined in more detail in [6].Let us consider an example from flavor model building, namely the model discussed in [8].Some of its fields and their respective charges under a large number of discrete symmetriesare given in table 1. We are interested in Yukawa couplings involving two fields T i and a Z Z Z Z Z Z Z Z H T T T θ θ θ θ H . We can compute the Hilbert basis H for these couplings with thehelp of normaliz [9, 11] and find the following elements in this model W = H (cid:0) T T ( θ θ + θ θ + θ θ + θ θ θ + . . . ) + T T ( θ θ + . . . ) + T T ( θ + . . . )+ T T ( θ + . . . ) + T T (1 + . . . )) . (4)This result is in agreement with the superpotential given in [8] and shows the applicationsof our approach. The dots denote monomials which consist of higher order couplings inthe flavon fields θ i . Note that while we considered only Abelian symmetries here, it wouldalso be interesting to consider Hilbert bases for non-Abelian discrete symmetries like theones discussed for example in [12]. R symmetries R symmetries result in inhomogeneous Diophantine equations. In this section we will al-ready have a concrete string model in mind (benchmark model 1 from [4]). Nonethe-less, the results are applicable to any model with R symmetries. Our approach also de-viates slightly from [6]. Let us assume that our theory gives rise to the R symmetry Z R × Z R × Z R . This is for example the case for models in the mini-landscape and has As outlined, for example, in [7] the charges under these symmetries are not integers, which meansthat a redefinition seems to be natural. Nevertheless we keep this form for easier comparison with theliterature. R T = ( R , R , R ). Thus the R symmetry of the modelwill result in the constraint X i R Ti = ( − , − , −
1) mod (6 , , , (5)where R i denotes the R charge of a field φ i and we assume that the superpotential trans-forms with R charge one. Let us illustrate our approach with an example. If we want toknow which singlets ˜ s i induce the Yukawa term W ⊃ ¯ φ q ¯ u f (˜ s i ) , (6)we have to solve the inhomogeneous system of Diophantine equationsˆ R · n = − − − mod , n i ∈ N ∀ i , (7)with ˆ R = (cid:0) R . . . R M +3 (cid:1) , R Ti = ( R i , R i , R i ) , (8)where M is the number of different standard model singlets in our model. During thisdiscussion we use the notation from [4] and call the up type Higgs ¯ φ and the down typeHiggs φ . Further, as in section 2.1, n i denotes the exponent of a given field W ⊃ ¯ φ n q n ¯ u n (˜ s ) n (˜ s ) n . . . (˜ s M ) n M +3 . (9)Only solutions with n = n = n = 1 are physical, therefore we can simplify this equationsystem by substituting ˜ R = R + R + R + (1 , , T . (10)This results in a homogeneous system of equations with less variables (cid:0) ˜ R R . . . R M +3 (cid:1) · n n ... n M +3 = mod , n i ∈ N ∀ i . (11)We have discussed how to find solutions to such homogeneous Diophantine equation sys-tems in section 2.1. After determining a Hilbert basis H one can further split it to obtainphysically viable solutions. Elements with n = 1 are assigned to H inhom ⊂ H and basiselements with n = 0 are assigned to H hom ⊂ H . All physical solutions to equation (11)are then given by the vectors x = x inhom (1 + x hom + x + . . . ) , (12)where . . . denote higher powers in x hom . One has to take all possible combinations ofelements x inhom ∈ H inhom and x hom ∈ H hom . For practical purposes it seems reasonable totruncate the solution at some finite order in the fields φ i .4 .3 The superpotential to all orders We can now determine all monomials which give W for the string model under considera-tion. We follow the approach of [6] and view all string selection rules as gauge and discrete Z N symmetries. Note that the space group is an infinite discrete non-Abelian group. How-ever for Z -II models, it is possible to rephrase it in terms of the finite discrete group Z × Z × Z ′ [13].Our model enjoys the symmetries Z R × Z R × Z R × Z × Z × Z × Z ′ × U(1) . The R symmetries are remnants of the internal Lorentz symmetries, the Z symmetry resultsfrom the so-called point group selection rule, whereas the space group selection rule resultsin Z × Z × Z ′ . In addition we have to take care of the gauge symmetries resulting fromthe E × E . The non-Abelian symmetries can be discussed along the lines of [6] and wefocus on the Cartan generators leading to the U(1) factors given here. With the discussionof the R symmetries from section 2.2 we get the Diophantine equation system ˜ R R . . . R M +3 ˜ R R . . . R M +3 ˜ R R . . . R M +3 q Z + q Z + q Z q Z . . . q Z M +3 q Z + q Z + q Z q Z . . . q Z M +3 q Z + q Z + q Z q Z . . . q Z M +3 q Z ′ + q Z ′ + q Z ′ q Z ′ . . . q Z ′ M +3 q + q + q q . . . q M +3 ... ... q + q + q q . . . q M +3 · n n ... n M +3 = mod . (13)We use normaliz [9, 11] to compute the Hilbert basis H of this Diophantine equationsystem. Compared to other systems like [10] it has the advantage that the algorithmused is suited better for our problem [14] and is also able to deal with congruences. We usethe so-called ”primal” algorithm [14] which is much faster in our case then the so-called”dual” algorithm which is implemented in .To make our results comparable with the results from the literature we reconsiderbenchmark model 1 from [4] in the following. We apply the rederived R symmetries whichhave been determined in [1, 2] and which we review in section 3. We consider a vacuumconfiguration in which we give the 14 standard model singlets˜ s = { h , h , h , h , s , s , s , s , s , s , s , s , s , s } (14)a vacuum expectation value (VEV). To get the superpotential W of these fields to allorders, we can consider which fields induce the µ -term. This is possible because φ ¯ φ formsa complete singlet under all symmetries in this model [4, 15, 16]. We find that up to order10 in standard model singlets ˜ s i the superpotential is given by W = ( M , inhom + M , inhom + M , inhom )(1 + M , hom + M , hom )= (( s s + s s ) s + h h s s )(1 + h h s s s s + ( s s ) ) , (15) There is an additional rule for some exceptional cases in the G orbifold plane [13]. We have not beenable to write it as a discrete symmetry and checked the invariance of the basis elements separately caseby case a posterior. M , inhom = s s s , M , hom = ( s s ) ,M , inhom = s s s , M , hom = h h s s s s ,M , inhom = h h s s . (16)The monomials M i, inhom are given by their exponent vectors x i, inhom which are elementsof the corresponding part of the Hilbert basis x i, inhom ∈ H inhom . The same is true forthe homogeneous part. In principle we know all Hilbert basis elements and therefore allmonomials which means we know the exact form of W to all orders. We also see themanifestation of the D symmetry [17, 18] present in this model which relates M , inhom and M , inhom . There are two doublets˜ D = ( s , s ) , ˜ D = ( s , s ) , (17)which result in two invariants˜ D · ˜ D T s = ( s s + s s ) s = M , inhom + M , inhom . (18) R symmetries in heterotic orbifolds In this section we briefly review the origin of discrete R symmetries in heterotic orbifolds[1, 2, 19]. These symmetries arise from automorphisms of the orbifold space group. For thecase of the Z -II orbifold the generators R α can be written in terms of the Cartan generatorsof the internal SO(6) by means of twist vectors v α according to R α = e π i( v α J + v α J + v α J ) .They are given by v = (cid:0) , , (cid:1) , v = (cid:0) , , (cid:1) , v = (cid:0) , , (cid:1) . (19)In order to determine the charges of the corresponding fields with respect to the R sym-metries, the transformation behavior of the string states under these generators needs tobe worked out.String states of heterotic orbifolds are characterized by their left- and right-movingmomenta p sh and q sh , their left-moving oscillator excitations ˜ α and their locus. As theproperties of the strings, described by these quantum numbers, are mainly independent ofeach other it is customary to write such states as | Ψ i = ˜ α N L ( ˜ α ∗ ) ¯ N L | p sh i ⊗ | q sh i ⊗ | locus i . (20)Here N L and ¯ N L count the number of holomorphic and anti-holomorphic oscillators of thestate. Recall that twisted strings have their center of mass momentum localized at fixedpoints of the space group S . The constructing element g ∈ S of a fixed point z f is definedby gz f = z f . Untwisted strings can propagate freely and hence the factor describing thelocalization in (20) is absent for such states.Note that we know the action of the generators (19) on the covering space C of theorbifold only. However, on C , there are infinitely many copies of each fixed point, whichare related to z f by the action of space group elements h that do not commute with g .The corresponding constructing elements form the conjugacy class [ g ] = { hgh − | h ∈ S } g . Therefore, if we denote the locus of a string state located at the fixed point z f withconstructing element g by | g i , a physical state corresponds to the linear combination | locus i = X g ′ ∈ [ g ] e π i˜ γ ( g ′ ) | Ψ g ′ i . (21)The phases ˜ γ are fixed by the requirement of invariance of the string states under elements h that do not commute with g , such that γ ( g, h ) = ˜ γ ( g ) − ˜ γ ( hgh − ) = − p sh · V h + v h · ( q sh − N L + ¯ N L ) mod 1 . (22)Here v h denotes the twist vector of the space group element h and V h denotes its gaugeembedding.The crucial observation is now, that the rotations (19) map each space group element g to a conjugate one, i.e. for each g there exists an h g such that R ( g ) = h g g h − g . Therefore,thetransformation behavior of the linear combination (21) is given by | locus i R e − π i γ ( g,h g ) | locus i . (23)For the other parts of the string states the transformation behavior follows from the trans-formation of the space time coordinates. A string state (20) transforms according to | Ψ i R exp (cid:2) π i v · (cid:0) q sh − N L + ¯ N L (cid:1) − π i γ ( g, h g ) (cid:3) | Ψ i (24)under an R symmetry generator R with shift vector v . For the case we are consideringhere, asking string correlators corresponding to superpotential terms to transform triviallyunder the R symmetries one arrives at the R charge selection rules [1, 2] L X i =1 R i = L X i =1 h q (boson) 1sh i − N i + ¯ N i − γ ( g i , h g i ) i = − , L X i =1 R i = L X i =1 h q (boson) 2sh i − N i + ¯ N i − γ ( g i , h g i ) i = − , L X i =1 R i = L X i =1 h q (boson) 3sh i − N i − ¯ N i − γ ( g i , h g i ) i = − . (25)Here we have made use of the fact that the internal right-moving momenta q sh of bosonsand fermions within the same chiral multiplet are related by a shift of . We denote themomentum of the respective bosons by q (boson)sh and define the R charge of a string state tobe that of the bosonic component of the chiral multiplet.We modified the program orbifolder [20] to incorporate the outlined R symmetries.The result for a concrete model can be found in appendix A. Differences due to the new R symmetries arise already at order 3 in the superpotential.Several new terms appear which were forbidden by the old rules. In the following we show7 , old R , old R , old R , new R , new R , new s
013 116 − −
12 116 − − s
014 116 − −
12 116 − − s − − − R charges for some singlet fields.a simple example. We look at the allowed couplings in the so-called benchmark model 1Aof [4]. The R charges of the considered fields are displayed in table 2. We immediately seethat the coupling s s s was forbidden by the old rules but is allowed under the new R symmetries.This has important phenomenological implications. If all fields s , s and s geta VEV in a chosen configuration, the superpotential W also gets a VEV. As alreadydiscussed, the superpotential in such models is linked to the µ -term. Therefore, in thisconfiguration, the µ -term will be generically too large. We will discuss this issue for ourVEV configuration later in more detail. There are also couplings forbidden by the new R symmetries which were allowed by the old symmetries. The first example occurs at order4 in the superpotential. In this section we want to briefly comment on the phenomenological properties of the con-sidered model. Instead of doing a complete scan over different vacua we stick to benchmarkmodel 1 (the field content of this model can be found in appendix A). A full scan overdifferent configurations is beyond the scope of this work. Using the vacuum configurationgiven in equation (14) we obtain the following features. F -flatness, Gr¨obner bases and D -flatness To analyze F -flatness we use techniques and methods known in computational algebraicgeometry and discussed in high energy physics for example in [21] and in string theoryin [22, 23]. We are looking for solutions to the equation system F i = ∂W∂ ˜ s i = 0 , ∀ i . (26)To find a solution we truncate the superpotential W at a given order. We take W up toorder 10 in standard model singlets which was given in equation (15) and set all couplingcoefficients to unity for simplicity. In principle these coefficients are calculable functionsof the geometric moduli of the compactification. This dependence can be used to stabilizethe geometric moduli [16].We use Singular [24] to compute the Gr¨obner basis of the ideal generated by the F -term equations. Afterwards we compute the primary decomposition and search for F -flat solutions F i = 0. We find only one non-trivial branch of solutions. Trivial solutions8 ˜ s i i = 0, would violate our assumption that all fields ˜ s i given in equation (14) obtain anon-vanishing VEV. The only non-trivial branch can be solved, for example, by (cid:10) s (cid:11) = − h s i h s i h h i h h i h s i h s i h s i , (cid:10) s (cid:11) = − h h i h h i h s i h s ih s i h s i + h s i h s i , (27)which results in F i = h W i = 0. That means we can solve all 14 F -term equations simulta-neously by fixing only two VEVs. This is nearly the opposite behavior to the one discussedin [7] where a remnant Z R symmetry [25] has been used to restrict the superpotential .There it seems to be more fertile to look for minima in which all singlets get fixed bythe F -term equations. It is interesting to study how the symmetries of the superpotentialdetermine the solution structure of the F -term equations. In our case many F -term equa-tions are degenerate because of the remnant D symmetry. The necessary breaking of thissymmetry at a lower scale (see section 4.2) can therefore be potentially used to stabilizeadditional moduli. The detailed study of this mechanism is beyond the scope of this work.At this stage we are satisfied with finding a consistent, non-trivial solution.We also would like to emphasize that the µ -term in the minima determined above,vanishes because h W i = 0. This is well known to be related to approximate R symmetries[15]. As we know the superpotential not only to order 10 in singlets, but to all orders, itseems to be possible to address the question of F -flatness to all orders. It is effective toattack this problem in terms of the Hilbert basis monomials. In this way, the given non-trivial solution can be extended to all orders. We can split the superpotential according tothe Hilbert basis monomials into two pieces W (˜ s i ) = W inhom (˜ s i )(1 + W hom (˜ s i )) . (28)Here W inhom is given by the linear combination of all inhomogeneous Hilbert basis monomi-als M i, inhom . Furthermore W hom denotes all possible combinations of homogeneous mono-mials M i, hom . The F -term equations in this picture are given by F i = ∂W inhom ∂ ˜ s i (1 + W hom ) + W inhom ∂W hom ∂ ˜ s i , ∀ i . (29)It is obvious that all F -term equations vanish if W inhom (˜ s i ) = 0 and 1 + W hom (˜ s i ) = 0simultaneously. Our concrete solution at order 10 is exactly of this kind. This behavior isnot limited to any order and in general such a solution will exist for the superpotential athigher order. The disadvantage that generically only two fields are fixed in this solutionbranch nevertheless remains. Different strategies to find minima like the ones consideredin [28, 29] may also help to address the computational difficulties.Let us also comment on D -flatness. It is possible to satisfy D -flatness along the linesof [30, 31]. We have explicitly checked that we can cancel the Fayet-Iliopoulos term bya monomial which carries negative charge under the anomalous U(1). D -flatness will alsohelp to fix some VEVs (see for example [7]) but in our concrete example some flat directionsremain. Further applications of this symmetry to flavor model building are discussed in [26], whereas therelation to R parity violation is studied in [27]. .2 Yukawa couplings For the Yukawa interactions we obtain W Yuk = Y u (˜ s i ) q ¯ u ¯ φ + Y d (˜ s i ) q ¯ dφ + Y e (˜ s i ) l ¯ eφ , (30)where the Yukawa matrices Y i (˜ s i ) depend on the singlet fields ˜ s i to which we have assignedVEVs (see equation (14)). We computed for each coupling the corresponding Hilbert basisand found that, at lowest order in singlets, the structure is Y u = M M M + M M M M + M M + M + M + M M + M + M + M , (31)with M = h h s s s s , M = h h s s s s ,M = s M , M = s M ,M = s M , M = s M ,M = s M , M = s s s s ( s ) s ,M = s ( s ) s ( s ) s , M = ( s ) s s ( s ) s ,M = s M , M = s s s s ( s ) s ,M = s ( s ) s ( s ) s , M = s ( s ) s ( s ) s . (32)Further, for the down quarks and leptons Y d = M M M + M M M M M M M + M , Y e = M M M + M M M M M M M + M , (33)with M = h h s s s , M = h h s s s ,M = s M , M = s M ,M = s M , M = s s M ,M = s s M , M = s s s M ,M = s s s M , M = ( s ) s s s ,M = ( s ) s s s , M = s s s s s ,M = s M , M = s M ,M = s M , M = s M . (34)Let us note that, as expected, the top quark coupling is of order one. The reason for thisbehavior is the connection of the coupling to the higher dimensional gauge coupling [32].Thus a realistic top quark mass in this model is guaranteed.As a consequence of the localization of the first two generations in the extra dimensionalspace, these fields form a doublet under the D symmetry [17, 18]. This manifests itselfin the appearance of the monomials M and M in the Yukawa matrices. However, thissymmetry needs to be broken at a lower scale to explain the different masses between thefirst and second generation [33]. 10 .3 Additional features Also with the new R symmetries all exotics can be made massive. We will discuss oneexample in detail and skip the details for the other exotics. As can be seen from table 3the R charges of y and y have changed under the new R symmetry. Nevertheless the R , old R , old R , old R , new R , new R , new y − − − − −
13 12 y − − − − −
13 12
Table 3: Differences between the R charges of y and y .mass matrix does not change, because both fields always appear in pairs and we have X i R i = − − R symmetries. Thus, neutrino masses can be generated with the see-sawmechanism for many right-handed neutrinos [34, 35].Our results for proton decay do not differ substantially from the ones described in [4].We find that qqql operators as well as couplings with massive exotic triplets like q l ¯ δ and q q δ are allowed. After integrating out the exotic triplets the trilinear operators mightbe a further source of proton decay. A full clarification of these questions needs a detailedexamination of the vacuum configurations of benchmark model 1 and is beyond the scopeof this paper. We have shown that with the recently rederived R symmetries the phenomenological at-tractive models of the mini-landscape remain viable. All appealing features survive andcan be understood also from the viewpoint of Hilbert basis monomials. We have shownhow symmetries lead to Diophantine equations and how one can find all solutions. It waspossible to use this approach for a model in the mini-landscape and its large number of R and non- R symmetries. With this method we have been able to determine the couplingstructure to all orders in singlet fields. As an example we have shown that this approachcan also be useful for flavor model building and not only for string model building. Thesymmetries of a given model directly constrain the superpotential and its Hilbert basisbuilding blocks. As has been shown, this can be useful in understanding the F -flatnessconditions to find supersymmetry preserving vacua. More work in this direction seems tobe interesting. Especially to find point-like minima where all fields and thus all moduli arestabilized. On the other hand the extension of the Hilbert basis approach to non-Abeliandiscrete symmetries for flavor model building seems to be desirable. An application of ourapproach to heterotic models based on different orbifold geometries like Z × Z is anotherpotential extension. 11 cknowledgments We want to thank Dami´an Mayorga Pe˜na for discussions. This work was supported by theSFB-Transregio TR33 ”The Dark Universe” (Deutsche Forschungsgemeinschaft).
A Details of model 1
Model 1 is based on the gauge embedding V = ( , − , − , , , , , , − , − , − , − , − , − , ) ,W = W = ( − , − , , , , , , )( , , , , , , − , ,W = ( 0 , − , − , − , , , , , − , − , − , − , − , − , ) . Here we list the complete spectrum of massless string states including their R charges.Those R charges that differ from the “old” ones are marked by red color. label k n n n ′ q γ R R R representation q Y q q q q q q q q q anom q B − L ¯ n − , , , ) 0 − −
12 12 52 − − e − , , , ) − −
12 12 −
12 12 − u − ¯3 , , , ) −
12 12 −
12 12
23 13 ¯ f − , , ¯4 , ) 0 0 0 0 0 −
12 12 53 f − , , , ) 0 0 0 0 0 − − −
12 23 − φ − , , , ) − φ − , , , ) − − − s − , , , ) 0 0 0 0 0 − − − s − , , , ) 0 0 0 0 0 − q − , , , ) −
16 12 − −
12 12 − n −
23 23 , , , ) 0 −
56 12 − − −
13 23 − f −
23 23 , , ¯4 , ) 0 − − −
56 16 −
13 12 −
16 89 δ
12 73 − , , , ) −
13 23 −
13 23 −
19 23 ¯ n
12 73 − , , , ) 0 − − −
56 23 23 − − − η
12 73 − , , , ) 0 − − − − − − − − d − − ¯3 , , , ) −
13 16 12 −
16 16 −
13 23
13 89 13 s
12 73 23 , , , ) 0
13 53 −
13 23
13 89 δ − − , , , ) − − −
79 23 h − − , , , ) 0 − − δ − − ¯3 , , , ) − − − − h − − , , , ) 0 − s − − , , , ) 0 − − − s − − , , , ) 0 − s
12 43 − , , , ) 0 s −
23 23 , , , ) 0 δ
12 73 23 ¯3 , , , ) − − − − − − − − − ¯ n − − , , , ) 0 −
12 16 56 −
13 43 −
13 59 − n − − , , , ) 0
16 12 −
56 56 − − −
13 149 − η −
23 23 , , , ) 0 −
12 16 56 23 13 − − − abel k n n n ′ q γ R R R representation q Y q q q q q q q q q anom q B − L ¯ f
12 73 − , , ¯4 , ) 0 −
12 16 56 16 13 −
12 16 − s
12 73 − , , , ) 0 − − − − −
13 29 l
12 73 − , , , ) −
12 16 12 16 − − − − − − n
12 43 − , , , ) 0 −
12 16 56 − − − − − n − − , , , ) 0 −
12 16 56 − − − − − n −
23 23 , , , ) 0 −
56 12 − − −
13 23 − f −
23 23 , , ¯4 , ) 0 − − −
56 16 −
13 12 −
16 89 d − − ¯3 , , , ) −
13 16 12 −
16 16 −
13 23
13 89 13 δ − − , , , ) − − −
79 23 h − − , , , ) 0 − − δ − − ¯3 , , , ) − − − − h − − , , , ) 0 − s − − , , , ) 0 − − − s − − , , , ) 0 − s −
23 23 , , , ) 0 n − − , , , ) 0 −
12 16 56 −
13 43 −
13 59 − n − − , , , ) 0
16 12 −
56 56 − − −
13 149 − η −
23 23 , , , ) 0 −
12 16 56 23 13 − − − n − − , , , ) 0 −
12 16 56 − − − − − s +14 − − ( , , , ) − − − − − s − − − ( , , , ) − − − − s +12 − − ( , , , ) − − s − − − ( , , , ) − f +2 −
13 32 ( , , ¯4 , ) − − − − − f − −
13 32 ( , , , )
12 12 s +11 − − ( , , , ) − − − − − s − − − ( , , , ) − − − − s +9 − − ( , , , ) − − s − − − ( , , , ) − f +1 −
13 32 ( , , ¯4 , ) − − − − − f − −
13 32 ( , , , )
12 12 h −
13 32 − ( , , , ) 0 0 −
12 12 − h −
13 32 − ( , , , ) 0 0 − − χ −
13 32 − ( , , , ) 0 0 −
12 12 − χ −
13 32 − ( , , , ) 0 0 − − − h −
13 32 − ( , , , ) 0 0 −
12 12 − h −
13 32 − ( , , , ) 0 0 − − χ −
13 32 − ( , , , ) 0 0 −
12 12 − χ −
13 32 − ( , , , ) 0 0 − − − s +13 − − ( , , , ) − − s − − − ( , , , ) − s +10 − − ( , , , ) − − s − − − ( , , , ) − s −
13 13 , , , ) 0 −
13 53 13 23 − f −
13 13 , , , ) 0 −
16 12 − − − −
13 12 −
16 79 abel k n n n ′ q γ R R R representation q Y q q q q q q q q q anom q B − L l − − , , , ) − − −
16 16 13 23
13 49 n
12 83 − , , , ) 0 − −
12 56 −
56 13 23 − n
12 83 − , , , ) 0 −
16 12 − −
56 13 − − η
12 83 43 , , , ) 0 −
16 12 − − − −
13 19 δ −
13 43 , , , )
13 13
13 23 13 23
13 19 23 n
23 13 , , , ) 0 −
16 12 − −
56 13 23
13 19 n
12 83 − , , , ) 0 −
16 12 − −
56 13 23
13 19 h
12 83 − , , , ) 0 − − δ
12 83 − ¯3 , , , ) −
13 13 − − h
12 83 − , , , ) 0 − − − δ
12 83 − , , , )
13 13 − −
19 23 s
12 83 − , , , ) 0 − − − s
12 83 − , , , ) 0 − s − , , , ) 0 − − − s
12 83 43 , , , ) 0 − − − s −
13 43 , , , ) 0 − − −
53 13 − − − d
12 83 13 , , , ) − −
12 16 −
16 13 − − − − η −
13 13 , , , ) 0 −
16 12 16 56 13 13
23 19 f
12 83 43 , , , ) 0 −
16 12 16 56 −
16 13 −
12 16 − n
12 83 43 , , , ) 0 −
12 16 56 13 − −
13 19 − n − − , , , ) 0 −
16 12 16 56 − −
23 19 δ − − ¯3 , , , ) −
13 13 − −
23 13 − −
13 19 − s −
13 13 , , , ) 0 −
13 53 13 23 − f −
13 13 , , , ) 0 −
16 12 − − − −
13 12 −
16 79 l − − , , , ) − − −
16 16 13 23
13 49 δ −
13 43 , , , )
13 13
13 23 13 23
13 19 23 n
23 13 , , , ) 0 −
16 12 − −
56 13 23
13 19 s − , , , ) 0 − − − s −
13 43 , , , ) 0 − − −
53 13 − − − η −
13 13 , , , ) 0 −
16 12 16 56 13 13
23 19 n − − , , , ) 0 −
16 12 16 56 − −
23 19 δ − − ¯3 , , , ) −
13 13 − −
23 13 − −
13 19 − s +4 −
16 23 12 ( , , , ) − − −
23 23 16 −
13 12 13 3718 ν −
16 23 − ( , , , ) − − −
13 16 −
13 12 13 118 23 s − − − − ( , , , )
12 16 − −
16 16 16 − − − − − m − −
13 12 ( , , , ) 0
16 12 − −
56 16 −
13 12 13 1918 s +3 − − − ( , , , ) −
12 23
13 23 16 −
13 12 13 1918 s −
56 23 12 ( , , , )
12 16 − −
16 16 16 −
13 12 13 2518 s +2 −
16 23 12 ( , , , ) − − −
23 23 16 −
13 12 13 3718 nu −
16 23 − ( , , , ) − − −
13 16 −
13 12 13 118 23 s − − − − ( , , , )
12 16 − −
16 16 16 − − − − − m − −
13 12 ( , , , ) 0
16 12 − −
56 16 −
13 12 13 1918 s +1 − − − ( , , , ) −
12 23
13 23 16 −
13 12 13 1918 s −
56 23 12 ( , , , )
12 16 − −
16 16 16 −
13 12 13 2518 η −
16 23 − ( , , , ) 0 − − − −
13 1918 abel k n n n ′ q γ R R R representation q Y q q q q q q q q q anom q B − L ¯ n −
16 23 − ( , , , ) 0 − − −
13 23 −
23 1918 − n − − − ( , , , ) 0 − −
56 23 23
13 1918 η −
16 23 − ( , , , ) 0 − − − −
13 1918 n −
16 23 − ( , , , ) 0 − − −
13 23 −
23 1918 − n − − − ( , , , ) 0 − −
56 23 23
13 1918 y − −
13 12 ( , , , ) 0 − m − −
13 12 ( , , , ) 0 − − − m − −
13 12 ( , , , ) 0 − − − − y − −
13 12 ( , , , ) 0 − m − −
13 12 ( , , , ) 0 − − − m − −
13 12 ( , , , ) 0 − − − − d − − − ( ¯3 , , , ) −
13 16 − − −
518 13 ¯ e − − − ( , , , ) − − − u − − − ( ¯3 , , , )
23 16 −
718 13 l − − − ( , , , )
12 16 − − − q − − − ( , , , ) −
16 16 − − ¯ n − − − ( , , , ) 0 − − s − − ( , , , ) 0 −
12 12 − − s − − ( , , , ) 0
23 12 − − s − − ( , , , ) 0 − − − − s − − ( , , , ) 0 −
13 12 12 − − s −
16 23 − ( , , , ) 0 − − − − s −
16 23 − ( , , , ) 0 −
13 12 12 − − d − − − ( ¯3 , , , ) −
13 16 − − −
518 13 ¯ e − − − ( , , , ) − − − u − − − ( ¯3 , , , )
23 16 −
718 13 l − − − ( , , , )
12 16 − − − q − − − ( , , , ) −
16 16 − − ¯ n − − − ( , , , ) 0 − − s − − ( , , , ) 0 −
12 12 − − s − − ( , , , ) 0
23 12 − − s − − ( , , , ) 0 − − − − s − − ( , , , ) 0 −
13 12 12 − − s −
16 23 − ( , , , ) 0 − − − − s −
16 23 − ( , , , ) 0 −
13 12 12 − − ν − − − ( ¯3 , , , ) − −
13 13 16 13 12 −
13 2518 − s − − −
43 12 ( , , , ) − −
23 16 13 12 − − s +7 −
16 23 12 ( , , , ) −
12 16 −
12 16 −
16 16 13 −
12 23 − m −
16 23 − ( , , , ) 0
16 12 16 56 16 13 12 −
13 1918 − s − − − − ( , , , )
12 23 − −
23 16 13 12 −
13 3118 s +8 − − ( , , , ) −
12 16 −
12 16 −
16 16 13 12 −
13 1318 − ν − − − ( ¯3 , , , ) − −
13 13 16 13 12 −
13 2518 − s − − −
43 12 ( , , , ) − −
23 16 13 12 − − s +5 −
16 23 12 ( , , , ) −
12 16 −
12 16 −
16 16 13 −
12 23 − m −
16 23 − ( , , , ) 0
16 12 16 56 16 13 12 −
13 1918 − abel k n n n ′ q γ R R R representation q Y q q q q q q q q q anom q B − L s − − − − ( , , , )
12 23 − −
23 16 13 12 −
13 3118 s +6 − − ( , , , ) −
12 16 −
12 16 −
16 16 13 12 −
13 1318 − f −
16 23 − ( , , , ) 0 −
13 56 16 13 −
12 16 118 f −
16 23 − ( , , ¯4 , ) 0 −
13 56 16 13 12 16 3718 η − − ( , , , ) 0 −
13 56 −
13 13 −
13 1918 − n − − ( , , , ) 0 −
13 56 − −
23 1918 n
56 23 − ( , , , ) 0 −
13 56 23 − −
13 718 − f −
16 23 − ( , , , ) 0 −
13 56 16 13 −
12 16 118 f −
16 23 − ( , , ¯4 , ) 0 −
13 56 16 13 12 16 3718 η − − ( , , , ) 0 −
13 56 −
13 13 −
13 1918 − n − − ( , , , ) 0 −
13 56 − −
23 1918 n
56 23 − ( , , , ) 0 −
13 56 23 − −
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