Gauge Coupling Unification in the Exceptional Supersymmetric Standard Model

We consider the renormalisation group flow of gauge couplings within the so-called exceptional supersymmetric standard model (E$_6$SSM) based on the low energy matter content of 27 dimensional representations of the gauge group $E_6$, together with two additional non-Higgs doublets. The two--loop beta functions are computed, and the threshold corrections are studied in the E$_6$SSM. Our results show that gauge coupling unification in the E$_6$SSM can be achieved for phenomenologically acceptable values of $\alpha_3(M_Z)$, consistent with the central measured low energy value, unlike in the minimal supersymmetric standard model (MSSM) which, ignoring the effects of high energy threshold corrections, requires significantly higher values of $\alpha_3(M_Z)$, well above the experimentally measured central value.


Introduction
Unification of gauge couplings is probably one of the most appealing features of supersymmetric (SUSY) extensions of the standard model (SM). More than fifteen years ago it was found that the electroweak (EW) and strong gauge couplings extracted from LEP data (hence at the EW scale) and extrapolated to high energies using the renormalisation group equation (RGE) evolution do not meet within the SM but converge to a common value at some high energy scale (within α 3 (M Z ) uncertainties) after the inclusion of supersymmetry, e.g. in the framework of the minimal SUSY standard model (MSSM) [1]. This allows one to embed SUSY extensions of the SM into Grand Unified Theories (GUTs) (and superstring ones) that make possible partial unification of gauge interactions with gravity. Simultaneously, the incorporation of weak and strong gauge interactions within GUTs permits to explain the peculiar assignment of U(1) Y charges postulated in the SM and to address the observed mass hierarchy of quarks and leptons.
Due to the lack of direct evidence verifying or falsifying the presence of superparticles at low energies, gauge coupling unification remains the main motivation for low-energy supersymmetry based on experimental data. But since 1990 the uncertainty in the determination of α 3 (M Z ) has reduced significantly and the analysis of the two-loop RG flow of gauge couplings performed in [2]- [4] revealed that it is rather problematic to achieve exact unification of gauge couplings within the MSSM. This is also demonstrated in Fig. 1a, where we plot the running of the gauge couplings from the EW (M Z ) scale to the GUT (M X ) scale. Fig. 1b shows a blow-up of the crucial region in the vicinity of M X = 3 · 10 16 GeV. To ensure the correct breakdown of the EW symmetry requires an effective SUSY threshold scale around 250 GeV, which corresponds to a SUSY Higgs mass parameter µ ≃ 1.5 TeV. Dotted lines show the interval of variations of gauge couplings caused by 1 σ deviations of α 3 (M Z ) around its average value, i.e. α 3 (M Z ) ≃ 0.118 ± 0.002 [5]. From Fig. 1b it is clear that exact gauge coupling unification in the MSSM cannot be attained even within 2 σ deviations from the current average value of α 3 (M Z ). Recently, it was argued that it is possible to get the unification of gauge couplings in the minimal SUSY model for α 3 (M Z ) = 0.123 [6].
The above observation is in fact true for a whole class of GUTs that break to the SM gauge group in one step and which predict a so-called "grand desert" between the EW and GUT scales. This conclusion must be qualified, however, by the fact that in general there are non-negligible high energy GUT/string threshold corrections to the running of the couplings associated with heavy particle thresholds and higher dimension operator effects which we shall not consider here. Furthermore, in this paper, we restrict our considerations to the minimal scenario for GUT symmetry group breakdown -the aforementioned one-step GUTs -as this allows one to get a stringent prediction for α s (M Z ). In particular, we examine gauge coupling unification within an E 6 inspired extension of the MSSM, the exceptional supersymmetric standard model (E 6 SSM) of Refs. [7]- [8] in which the E 6 symmetry breaking proceeds uniquely at a single step through the SU(5) breaking direction. This results in a low energy SM gauge group augmented by a unique U(1) N gauge group under which right-handed neutrinos have zero charge, allowing them to be superheavy, shedding light on the origin of the mass hierarchy in the lepton sector and providing a mechanism for the generation of the lepton and baryon asymmetry of the Universe. The µ problem of the MSSM is solved within the E 6 SSM in a similar way to the NMSSM, but without the accompanying problems of singlet tadpoles or domain walls.
Thus the E 6 SSM is a low energy alternative to the MSSM or NMSSM.
In this paper we calculate the two-loop beta functions of the gauge couplings in the E 6 SSM, and then apply them to the question of gauge coupling unification, including the important effects of low energy threshold corrections. The structure of the twoloop contributions to the corresponding beta functions is such that the EW and strong couplings meet at some high energy scale for an α 3 (M Z ) value which is just slightly higher than the experimentally measured central value, with the low energy threshold effects pushing it further towards the central measured value. As the results in Fig. 1c,d will show, the unification of gauge couplings in the E 6 SSM is achieved for values of α 3 (M Z ) consistent with the measured central value, unlike in the MSSM which, ignoring the effects of high energy threshold corrections, requires significantly higher values of α 3 (M Z ), well above the experimentally measured central value.
The layout of the remainder of the paper is as follows. In section 2 we present an analytical approach to the solution of the RGEs for the gauge couplings that allows one to examine the unification of forces in SUSY models and we specialise to the MSSM case in section 3. In section 4 we briefly review the E 6 SSM and in section 5 we discuss the two-loop RG flow of the gauge couplings within this model, including the low energy threshold corrections, leading to the stated results. Section 6 concludes the paper.

RG flow of gauge couplings in SUSY models
In SUSY models the running of the SM gauge couplings is described by a system of RGEs which can be written in the following form: where b i andb i are one-loop and two-loop contributions to the beta functions [9]- [10], t = ln (µ/M Z ), µ is a renormalisation scale, with the index i running from 1 to 3 corre-sponding to U(1) Y , SU(2) W and SU(3) C interactions, respectively. One can obtain an approximate solution of the RGEs in Eq. (1) that at high energies can be written as [3] 1 where the third term in the right-hand side of Eq. (2) is the MS → DR conversion factor with C 1 = 0, C 2 = 2, C 3 = 3 [11], which exact gauge coupling unification takes place [12]: where ∆ s are combined threshold corrections whose precise form depends on the model under consideration.

MSSM
In this section we apply the results of the previous section to the MSSM. In the MSSM ∆ s takes the form [2]- [3], [12]- [13]: For example, assuming for simplicity that superpartners of all quarks are degenerate, i.e. their masses are equal to mq, and all sleptons have a common mass ml, we find: In Eq. (6) M 3 and M 2 are masses of gluinos and winos (superpartners of SU(2) W gauge bosons), whereas µ and m A are µ-term and masses of heavy Higgs states respectively. In general T 1 , T 2 and T 3 , obtained from Eq. (3), can be quite different.
We now perform a simplified numerical discussion of the previous results in order to illustrate the effect of threshold corrections on gauge unification in the MSSM. In our simplified discussion we shall assume the effective threshold scales T i be equal to each However the correct pattern of EW symmetry breaking (EWSB) requires µ to lie within the 1 − 2 TeV range, while from Eq. (6) it follows that M S ≃ µ/6, which implies that the effective threshold scale should be M S < 200 − 300 GeV [2]- [3], [12]- [14].

E 6 SSM -a brief review
In this section, in order to make the paper self-contained, we give a brief review of the E 6 SSM which was proposed recently in [7,8]. The E 6 SSM involves an additional low energy gauged U(1) N not present in the MSSM, and in order to ensure anomaly cancellation the particle content of the E 6 SSM is also extended to include three complete fundamental 27 representations of E 6 at low energies. These multiplets decompose under the SU(5) × U(1) N subgroup of E 6 as follows [15]: The first and second quantities in the brackets are the SU (5) representation and extra U(1) N charge while i is a family index that runs from 1 to 3. An ordinary SM family which contains the doublets of left-handed quarks Q i and leptons L i , right-handed upand down-quarks (u c i and d c i ) as well as right-handed charged leptons, is assigned to 10, 1 . Right-handed neutrinos N c i should be associated with the last term in Eq. (7), (1,0) i . The next-to-last term in Eq. (7) However these exotic quark states carry a B − L charge ± 2 3 twice larger than that of ordinary ones. Therefore in phenomenologically viable E 6 inspired models they can be either diquarks or leptoquarks. In addition to the complete 27 i multiplets the low energy particle spectrum of the E 6 SSM is supplemented by SU (2) (5) singlets with U(1) N charges. The presence of a Z ′ boson and exotic quarks predicted by the E 6 SSM provides spectacular new physics signals at the LHC which were discussed in [7]- [8], [16].
The superpotential in E  are odd. The Z H 2 symmetry reduces the structure of the Yukawa interactions to: where α, β = 1, 2 and i = 1, 2, 3 . The SU (2)  However the Z H 2 symmetry can only be an approximate one because it forbids all Yukawa interactions that would allow the exotic quarks to decay. Since models with stable charged exotic particles are ruled out by different experiments [17] the Z H 2 symmetry has to be broken. At the same time the breakdown of Z H 2 should not give rise to operators leading to rapid proton decay. There are two ways to overcome this problem. The resulting Lagrangian has to be invariant with respect to either a Z L 2 symmetry, under which all superfields except lepton ones are even, or a Z B 2 discrete symmetry, which implies that exotic quark and lepton superfields are odd whereas the others remain even.
Because Z H 2 symmetry violating operators may also give an appreciable contribution to the amplitude of K 0 − K 0 oscillations and give rise to new muon decay channels like µ → e − e + e − the corresponding Yukawa couplings are expected to be small. Therefore It is worth to emphasize that all the discrete symmetries Z H 2 , Z L 2 and Z B 2 that we use here to prevent rapid proton decay break E 6 because different components of the fundamental 27 representation transform differently under these symmetries. Another manifestation of the breakdown of the E 6 symmetry is the presence of the SU(2) W doublet H ′ and anti-doublet H ′ in the low energy particle spectrum of the E 6 SSM that comes from the splitting of extra 27 ′ and 27 ′ . Because the splitting of 27-plets is a necessary ingredient of the considered model, as it is required in order to attain gauge coupling unification, it seems to be very attractive to reduce all origins of the E 6 symmetry breakdown (including postulated discrete symmetries) to the splitting of different E 6 multiplets. The splitting of GUT multiplets can be naturally achieved in the framework of orbifold GUTs [18].
The E 6 GUT model whose incomplete multiplets form the particle content of the In the superpotential (9) we omit higher order terms that are suppressed as 1/M 2 P l or even stronger. If µ X << M P l the singlet field Σ may acquire vacuum expectation value which is many orders of magnitude smaller than the Planck scale. Non-zero vacuum expectation value of Σ breaks the Z H 2 symmetry spontaneously. Then the first three terms in Eq. (9) result in the Z H 2 symmetric part of the superpotential of the E 6 SSM at low energies while the next three terms give rise to couplings that violate the Z H 2 symmetry explicitly. In this case the effective Yukawa couplings which are induced after the breakdown of the Z H 2 symmetry are naturally suppressed by the small ratio < Σ > M P l leading to the desirable hierarchical structure of Yukawa interactions postulated in the E 6 SSM.

Gauge Coupling Unification in the E 6 SSM
We now turn to the central issue of this paper, that of gauge coupling unification in the E 6 SSM. We first present our results for the two-loop beta functions in this model, before going on to consider the question of gauge coupling unification in the presence of low energy threshold effects. The running of gauge couplings in the E 6 SSM is affected by a kinetic term mixing [7], [19]. As a result the RGEs can be written as follows: where B and G are 2 × 2 matrices As always the two-loop diagonal β i and off-diagonal β 11 beta functions may be presented as a sum of one-loop and two-loop contributions (see Eq. (1)). In the one-loop approximation the beta functions are given by The parameter N g appearing in Eq. (12)  functions in a general softly broken N = 1 SUSY model [10] we find the following twoloop beta functions for the E 6 SSM: . Because our previous analysis performed in [7] revealed that an off-diagonal gauge coupling g 11 being set to zero at the scale M X remains very small at any other scale below M X we neglect two-loop corrections to the off-diagonal beta function β 11 .
The results of our numerical studies of gauge coupling unification in this model are summarised in Fig. 1c-d where the two-loop RG flow of gauge couplings in the E 6 SSM is shown. As before we fix the effective SUSY threshold scale to be equal to 250 GeV, that on the one hand results in appreciable threshold corrections to the RG running of the gauge couplings but on the other hand does not spoil the breakdown of the EW symmetry. We also assume that the masses of the Z ′ and all exotic fermions and bosons predicted by the E 6 SSM are degenerate around 1.5 TeV. Thus we use the SM beta functions to describe the running of gauge couplings between M Z and M S , then we apply the two-loop RGEs of the MSSM to compute the flow of g i (t) from M S to M Z ′ and the two-loop RGEs of the E 6 SSM to calculate the evolution of g i (t) between M Z ′ and M X which is equal to 3.5 · 10 16 GeV in the case of the E 6 SSM. Again dotted lines in Fig. 1c-d where µ D i and mD i are the masses of exotic quarks and their superpartners, m Hα and µH α are the masses of non-Higgs and non-Higgsino fields of the first and second generation, while m H ′ and µH′ are the masses of the scalar and fermion components of H ′ and H ′ .
The value of strong gauge coupling at the EW scale that results in the exact gauge coupling unification can be predicted anew. It is given by Eq. (4) where the E 6 SSM beta functions and new threshold scalesT i should be substituted. Such replacement does not change the form of Eq. (4) because extra matter in the E 6 SSM form complete SU (5) representations which contribute equally to the one-loop beta functions of the SU (3) Here we also assume that non-Higgs fields of the first two generations have the same mass m Hα and the masses of non-Higgsinos of the first and second generation are equal to µH α while the masses of scalar non-Higgs fields and their superpartners from H ′ and H ′ are degenerate around µ ′ . In Fig. 1c- to be heavy 10 TeV. As a consequence, although the effective threshold scaleM S may be considerably less than µ ′ , the corresponding mass parameter can be always chosen so thatM S lies in a few hundred GeV range that allows to get the exact unification of gauge couplings for any value of α 3 (M Z ) which is in agreement with current data.

Conclusions
In this paper we have presented the two-loop RGEs of the E 6 SSM and examined gauge coupling unification in this model using both analytical and numerical techniques. We     in the scenario E 6 SSM III we ignore all Yukawa and U (1) N gauge couplings. Note that in all versions of the E 6 SSM the large individual contributions Θ i conspire to partially cancel when forming the quantity Θ s which describes the effect of the two-loop corrections to determining the low energy value of α(M Z ).