E6 GUT through Effects of Dimension-5 Operators

In the effective field theory framework, quantum gravity can induce effective dimension-5 operators, which have important impacts on grand unified theories. Interestingly, one of main effects is the modification of the usual gauge coupling unification condition. We investigate the gauge coupling unification in $E_{6}$ under modified gauge coupling unification condition at scales $M_X$ in the presence of one or more dimension-5 operators. It is shown that nonsupersymmetric models of $E_6$ unification can be obtained and can easily satisfy the constraints from the proton lifetime. For constructing these models, we consider several maximal subgroups $H=SO(10)\times U(1), H=SU(3)\times SU(3)\times SU(3)$, and $H=SU(2)\times SU(6)$ of $E_{6}$ and the usual breaking chains for a specific maximal subgroup, and derive all of the Clebsch-Gordan coefficients $\Phi^{(r)}_{s,z}$ associated with $E_6$ breaking to the Standard Model, which are given in Appendix A.


Introduction
It is well-known that the problem of quantum gravity has not been solved yet, although superstring theory presents a beautiful promise. Can we examine the effects of quantum gravity nowadays? The answer is positive. The reason is that the scale of unification M 2 10 G 16´G eV in the minimal supersymmetric (SUSY) standard model (SM), as implicated by the experiment measurements [1][2][3][4], is smaller than the Planck scale M Pl (M G 8 2 . 4 1 0 N Pl 1 2 18 p =~-( ) GeV), where quantum gravity should come in, by about two orders of magnitudes so that one can build a field theoretical description of the unification of particle interactions without a full solution to the problem of quantum gravity. To describe the effects of quantum gravity, we could use the effective field theory approach in which non-renormalizable higher dimension operators are introduced. The d 5  operators induced by gravity should enter the Lagrangian, which are suppressed by factors of M d Pl 4 --( ) ( ) with coefficients at the order of 1  ( ) . They are only subject to the constraints of the symmetries (gauge invariance, supersymmetry in SUSY models, etc) of the low energy theory.
As it has been shown in  that the presence of higher dimension operators may have substantial impacts on grand unified theory (GUT) and its phenomenology for gauge groups SU (5) and SO (10). These operators modify the usual gauge coupling unification condition [5-10, 20, 27]. It is estimated that the effects of dimension-5 operators can be more important than two-loop corrections in the renormalization group (RG) analysis of gauge coupling unification [7]. With one or more dimension-5 operators, it is possible to achieve unification at scales M X much different than usually expected [27]. Higher dimension operators can lead to an acceptable value of sin w 2 q [5,11,12,20], affect SUSY particle spectrum in SUSY GUT and supergravity [14][15][16][17][18][19][20][21][22][23][24] and the analysis of proton decay [25,26]. Therefore, we should include them in the researches of GUT and SUSY GUT. In particular, in some cases, such studies are dramatically important. For example, the gauge coupling unification in the minimal SU(5) without supersymmetry cannot be realized and the minimal SUSY SU(5) has been already excluded by the limit (larger than y 10 34 ) of the proton lifetime from Super-Kamiokande [64] 4 , but they can be realized if effects of d 5  operators are included (see, for example, [20,27]).
The exceptional group E 6 is an attractive unification group among well-known unification groups. The main reasons are as follows. Firstly, from the viewpoint of superstring theory, the gauge and gravitational anomaly cancellation occurs only for the gauge groups SO (32) or E E 8 8 [31 -33] and compactification on a Calabi-Yau manifold with an SU (3) holonomy results in the breaking E SU E 3 8 6 ( ) [34]. Secondly, in terms of the phenomenology of low energy effective theories originated from E 6 GUT, there are several attractive features [35][36][37][38][39][40][41][42]. Moreover, if we assume the dynamical symmetry breaking scenario, we would have several constraints on the possible GUT models. It has been pointed out [43][44][45] that E 6 is uniquely selected among many GUT groups, if one demands that (1) the theory is automatically anomaly free, (2) every generation of quark/lepton fields belongs to a single irreducible representation (irrep) of the GUT group, and (3) the Higgs fields, which are necessary for inducing the symmetry breaking down to SU U 3 1 C eḿ ( ) ( ), fall in the representations that can be provided by the fermion bilinears. Recently some models and their low energy phenomenology originated from E 6 GUT generated further interests [46][47][48][49][50][51][52][53][54].
Effects of higher dimension operators to the unification of gauge couplings have also been investigated in the grand unified gauge group E 6 [20,55]. In the [20], E SU SU SU 6 3 3 3 ´( ) ( ) ( ) has been studied and the corresponding effective contributions to gauge kinetic terms have been given. However, their contributions to gauge kinetic terms in the SM have been not given, although it is not difficult to derive them from the results shown in table 4 of the paper. Moreover, the numerical analysis on the unification of gauge couplings and corresponding physical discussions have not been carried out in the paper. To construct E 6 unification models with effects of dimension-5 operators in [55], the author considers only the maximal subgroup H SO U 10 1 =( ) ( )and gives the Clebsch-Gordan coefficients s z r , F ( ) associated with E 6 breaking to the SM for that case. Nevertheless, same as the [20], the numerical analysis on the unification of gauge couplings and corresponding physical discussions have not been carried out.
In this paper, we investigate the unification of gauge couplings for the grand unified gauge group E 6 through effects of dimension-5 operators. Surveying the branching rules for the GUT group E 6 [56], we see that there are several maximal subgroups containing G 321 (e.g., H SO and H F 4 = ) and for a specific maximal subgroup there are usually several breaking chains. We consider all common maximal classical subgroups with SU U 3 1 C eḿ ( ) ( ) and the usual breaking chains for a specific maximal subgroup. We show that the gauge coupling unification condition is modified due to the effects of dimension-5 operators, which are the lowest higher dimension operators in E 6 GUT, and the gauge coupling unification at a higher scale can be realized without SUSY. Furthermore, we find that the constraints from the proton lifetime can be safely satisfied even at the higher unification scale M G near the Planck scale M Pl .
In section 2, we set up the notation to study effects of dimension-5 operators and point out how dimension-5 operators modify the usual gauge coupling unification condition. Section 3 is devoted to the numerical analysis of RG evolution of the gauge couplings and the corresponding physical results for these cases. In section 4, we discuss the constraints from the proton lifetime. Summary and conclusions are given in section 5. All of the Clebsch-Gordan coefficients s z r , F ( ) associated with E 6 breaking to the SM, in different bases s z , { }, up to a uniform normalization constant for different representations r, are derived, and results are given in appendix A. In appendix B, we present the structure constants of E 6 explicitly, which are mostly used by physicists and in the study of GUT.  (3), we denote the Higgs multiplet by a d-dimensional symmetric matrix r F ( ) , with d=d(G), the dimension of the adjoint representation (rep) G (we use the same letter G to denote the group and its adj. rep. for simplicity), and d=78 for G E 6 = . For our purpose, we find that all possible r F ( ) are invariant under the SM gauge group G SU

Dimension-5 operators and modified gauge coupling unification condition
That is, each of them is a SM singlet and then the matrix is largely simplified: it contains only a few independent entries.
There are different ways to define the hypercharge Y, which are consistent with the SM (see, e.g., the [59,60]). We consider two cases: Hereafter, we shall call the case (1) as 'normal embedding', and the case (2) as 'flipped embedding'. For example, for H SU SU SU    where R is a irrep of G, and X i ʼs are the generators of G, which satisfy with f ijk as the totally antisymmetric structure constants. The operator acts on the tensor product R×R so that is the eigenvalue of F in irrep r with C(r) and C(G) being the eigenvalues of the Casimir operator in irrep r and G respectively. C(r) depends on the normalization of the Casimir operator and consequently the choice of structure constants 6 . The structure constants of E 6 have been given in a Chevalley base [61]. For convenience of the study in GUT, they are transformed into the usual form and listed in appendix B.
The eigenvalues C(r) for several irreps r in E 6 , as well as SO(10) and SU (6) where v s z k , , v k are real. Equations (2), (11) lead to an alteration of the gauge coupling unification condition,  ¢ are also listed in tables in appendix A. Equation (12) is the boundary condition at the scale M GUT of the RG evolution of gauge couplings, and will be used in the numerical analysis in the next section. Table 1. Quadratic casimir invariants C(r) for some irreps r of E 6 , SO (10) and SU (6), in the conventions explained in the section 2 of the text.
for the singlet 1 of any group.

Grand unification in E 6
In this section, we study the unification of gauge couplings with one, two, or more dimension-5 operators numerically from different 650 and 2430 breaking chains.
As to the case of one-dimension-5 operator from 650 or 2430, because there are several maximal subgroups and for a specific maximal subgroup there are several breaking chains, the SM singlets H k ab á ñ, the non-zero vacuum expectation values of H k ab , are not determined for fixed k=2 or 3. Consequently it is clear from equation (13) and tables 3-10 that the ratio : :    cannot be determined fully. So one has much freedom to choose ratios among v s z k , . It is a boring and not necessary task to exhaust all possibilities. Instead, we just take some breaking chains as examples. Then, the unification scale M X (set M M X GUT = for simplicity) and Wilson coefficient c k are computable by means of the gauge coupling unification condition equation (12), when the running gauge couplings in the SM are given. We limit ourselves to one-loop case for running in the paper for simplicity and it is straightforward to generalize to two-loop case. Moreover, dimension-5 operators also affect analysis of proton decay [25,26]. As analyzed in the [27] and in the next section, the absolute value of Wilson coefficient should be less than 10, i.e., max c 10 k  | | , in order to satisfy the proton decay constraint. Hereafter, we name the unification with max c 10 k  | | as the successful unification. In , M X is as large as the Planck scale and c k | |is as small as −0.015. That means, we may have a perturbative theory up to the onset of quantum gravity. With only one-dimension-5 operator, we can also use two or more different breaking chains to achieve successful gauge coupling unification with continuously varied unification scale M X , as given in figures 1, 2 for illustrations. In figure 1, we choose two breaking chains 210 24  and 210 As it is easy to see, from equations (12), (13) and tables in appendix I and in the [55], that for a given specific breaking chain, when two of three numbers one cannot get successful unification. In the following, we study the unification of gauge couplings in the case of two-dimension-5 operators with two different Higgs multiplets from 650 and 2430 of E 6 respectively. In order to achieve continuously varied unification scale M X , we have four variables, two VEVs of Higgs multiplets, v 650 and v 2430 , and two Wilson coefficients. Without fine-tuning for the VEVs, we fix the ratio of 650 and 2430 in three cases, 1:5, 1 :1 and 5:1, in our figures for an illustration. Also, for the VEVs of the Higgs multiplets, we assume they account for half of the average gauge boson squared mass [27]. Then, we are left with two Wilson coefficients for the unification. The maximal absolute values of Wilson coefficients as a function of unification scale M X are given in figures 3-6 for different embedding of subgroup into E 6 .
The numerical results of the subgroup embedding SU > GeV. The needed òs for the unification are also given in figure 3, which are independent of the ratios of Higgs VEVs. The 2  is varied smoothly with the scale M X , but the needed 1  is negative and larger absolute value is required. Similar results for the other embeddings, , we have eight different ratios. As a result, the unification are much easier for the latter. Most of the points in both cases have satisfed the successful unification with max c 10 k  | | . Table 2. The breaking chain, unification scale M X , Higgs VEV v, Wilson coefficient c k , gauge coupling G a at the unification scale, and three 1,2,3  for unification with only one-dimension-5 operator.

1
, As a summary of this section, we conclude that non-SUSY models of E 6 grand unification can be obtained through effects of dimension-5 operators. Comparing with the other groups like SU(5) and SO (10), it is much easier to achieve the successful unification with natural Wilson coefficient c k and continuously varied unification scale M X . Thus including effects of quantum gravity provides a greater probability for building a realistic non-SUSY model in E 6 GUT.

The constraint from the proton decay
As it is well-known, the key predictions in GUT, despite of the details of model building, are the gauge coupling unification and the proton decay. It has been pointed out [25,26] that contributions to the proton decay mediated by superheavy Higgs, which are mainly due to color triplets and very model-dependent, are less important than those mediated by superheavy gauge bosons. Therefore, we limit ourselves to discuss only the contributions from the gauge dimension-6 operators. The grand unification breaking will bring in heavy thresholds. Assuming one-step breaking of the grand unified gauge group E 6 to the SM G 321 at the unification scale M X for simplicity, the average squared mass of the non-G 321 singlet gauge bosons (usually called 'superheavy' gauge bosons) is given by, for , 13, , 78, 14 where the sum runs over all Higgs multiplets i that acquire non-zero vacuum expectation value v i at M X , i.e., in addition to the non-G 321 singlet Higgs contained in equation (1), all other Higgs multiplets which are necessary to realize the gauge symmetry breaking chain in a specific model. In order to guarantee correct use of the running gauge couplings in the SM (see the previous section), it is necessary to require M HG close to the grand unified symmetry breaking scale, i.e., the unification scale M X , in the case of one-step breaking. For the purpose of definiteness and omitting heavy threshold effects, we take M M X HG = . Then, the proton lifetime due to superheavy gauge boson exchange can be written as [25,26,63], >´.    , which is very close to the bound equation (17) and can be easily adjusted (say, to increase the value of v) to satisfy the bound. With only one-dimension-5 operator, using two (or more) different breaking chains, one can achieve successful gauge coupling unification with continuously varied unification scale M X , as long as the ratio of two vevs varies with M X . Thus, the bound equation (17) is satisfied. Looking at the results shown in the last section, the same remains in the case of two-dimension-5 operators with two different Higgs multiplets from 650 and 2430 of E 6 respectively.

Summary and discussion
It has been pointed out that gauge coupling unification condition is modified due to the effects of dimension-5 operators. We have investigated the gauge coupling unification in E 6 without SUSY under modified gauge coupling unification condition. For this purpose, considering several maximal subgroups H SO ) of E 6 and the usual breaking chains for a specific maximal subgroup, we have derived and given all of the Clebsch-Gordan coefficients s z r , F ( ) associated with E 6 breaking to the SM. We have also presented the structure constants of E 6 in the usual form, which are mostly used by physicists and in the study of GUT. Results on the gauge coupling unification show that, for a single dimension-5 operator, realizing the gauge coupling unification under modified gauge coupling unification condition in E 6 GUT is easier than that in SO(10) GUT, since there are more maximal subgroups, and, for a specific maximal subgroup, there are more breaking chains, so that one has much freedom to choose ratios among the non-zero vacuum expectation values v s z k , of the Higgs multiplets H k . We have also analyzed the constraint on the unification scale M X from the newest data of the proton decay. It is shown that most of cases studied in the section IV satisfy the constraint easily.
In the effective field theory spirit, operators of dimension higher than 5 are also present, e.g., a dimension-6 operator generalization of equation (1)     where the corrections (see equation (13) = ( ), effects of dimension-6 operators might be the same order of those of dimension-5 operators. Therefore, the expansion of equation (19) might not or might be controlled perturbatively. If it is not, one cannot claim perturbative gauge-gravity unification at the Planck scale. However, the fact that the modification of the gauge coupling unification condition of equation (12) allows us in principle to adjust the unification scale to a higher scale M Pl could at least be taken as a hint that gauge-gravity unification is a possible scenario, even if the necessary parameter values or the last piece of the evolution cannot be computed perturbatively.
There is an interesting subject in the effective theory framework. That is, the following operators of dimension-5 are also probably present,          It is straightforward to generalize the study of the gauge coupling unification in E 6 including the effects of dimension-5 operators to SUSY GUT. In this case, gaugino mass ratios can be read from the tables shown in appendix A, since gauginos belong to the same multiplets as gauge bosons in SUSY. That is, for the flipped embedding,  where M a are the gaugino masses and a 3, 2, 1 = corresponds the SU(3), SU(2), U(1) of the SM, i.e., the gluino, wino and bino masses. The results agree with those in corresponding tables given by Martin [16].
For model building of E 6 GUT with effects of dimension-5 operators, there are several important problems, such as the doublet-triplet splitting, neutrino mass hierarchy, etc, which need to be answered. However, this is beyond the scope of this paper. One should study them in the future.