Robustness and Predictivity of 4 TeV Unification

The stability of the predictions of two of the standard model parameters, $\alpha_3(M_Z)$ and $\sin^2 \theta(M_Z)$, in a $M_U \sim 4$ TeV unification model is examined. It is concluded that varying the unification scale between $M_U \simeq 2.5$ TeV and $M_U \simeq 5$ TeV leaves robust all predictions within reasonable bounds. Choosing $M_U = 3.8 \pm 0.4$ TeV gives, at lowest order, accurate predictions at $M_Z$. The impact of threshold effects on unification depends on the spectrum of states beyond the standard model.


Introduction
One of the principal motivations for extending the standard model is the GUT gauge hierarchy between the weak scale and the grand unification or GUT scale. A related concern, not addressed here, is the Planck hierarchy between the weak scale and the Planck scale; the model we consider has flat spacetime, vanishing Newton's constant and infinite Planck scale.
In a recently-proposed model [5], grand unification occurs differently. The three couplings run from µ = M Z up to a lower unification scale M U ∼ 4 TeV, at which scale the theory is embedded in a larger gauge group G ≡ SU(3) 12  This low-scale unification model also has a top-down inspiration from string theory through the AdS/CFT correspondence [6][7][8] arising from consideration of a Type IIB superstring in d = 10 dimensional spacetime compactified on AdS 5 × S 5 . Using a finite group Γ = Z 12 in an abelian orbifold AdS 5 × S 5 /Γ gives a quiver gauge theory [9] with gauge group SU(N) 12 either with no supersymmetry N = 0 [5] or with N = 1 supersymmetry [10] Several issues were left open in [5]: robustness of the predictions under variations of the scale M U (conversely, the accuracy of the predictions at µ = M Z ); the size of flavor-changing effects, and the consistency of the additional states around M ∼ M U with constraints imposed by precision low-energy data. In this article we shall address all of these issues.

Robustness of Predictions to Variation in M U
The calculations of [5] were done in the one-loop approximation to the renormalization group equations without threshold effects. Because the couplings remain weak this can be self-consistent provided the masses of the new states in the model are sufficiently close to M U . Other corrections due to non-perturbative effects, and the effects of large extra dimensions, are outside of the scope of this paper. In one sense the robustness of this TeVscale unification is almost self-evident, in that it follows from the weakness of the coupling constants in the evolution from M Z to M U . That is, in order to define the theory at M U , one must combine the effects of threshold corrections ( due to O(α(M U )) mass splittings ) and potential corrections from redefinitions of the coupling constants and the unification scale.
We can then impose the coupling constant relations at M U as renormalization conditions and this is valid to the extent that higher order corrections do not destabilize the vacuum state.
We shall approach the comparison with data in two different but almost equivalent ways. The first is "bottom-up", where we use as input the requirement that the values of α 3 (µ)/α 2 (µ) and sin 2 θ(µ) are expected to be 5/2 and 1/4, respectively, at µ = M U . Using the experimental ranges allowed for sin 2 θ(M Z ) = 0.23113 ± 0.00015, α 3 (M Z ) = 0.1172 ± 0.0020 and α −1 em (M Z ) = 127.934±0.027 from [11] we have plotted in Figure   For contribution of new gauge bosons, we refer to the analysis in [13]. In the limit where the bilepton gauge bosons are degenerate M ++ = M + the contribution to S vanishes except for the subtlety of the pinch contribution. From the formula presented in [13] we find ( S| P is the pinch contribution): The first term in Eq.(2) is explicitly: in whichF 0,3 are given by: The second term in Eq.(2) is: From these equations, we find that the contributions of gauge bosons to S are suppressed by (M Z /M U ) 2 ∼ 10 −4 and so even for many such new gauge bosons the contribution to S is acceptably small provided the SU(2) doublets are adequately degenerate.

Threshold Effects
In  Table 1 are  Table 2.  All of the scalar representations are real under 3 C 2 L 1 Y , indeed under SU(3) 12 , so all will naturally acquire a mass ∼ M U . One SU(2) L doublet from the WH row of Table 2 must, however, remain light as the standard Higgs doublet; this is the hierarchy problem.
Threshold effects are generally larger for fermions than for scalars, as seen from Table 1 and 2. Let us therefore illustrate how fermion masses below M U can effect the unification of α 3C , α 2L and α Y .
Without any threshold corrections, the consistent unification of teh three couplings, Fig. 4.
trate this by Fig. 5-7. Fig.5 shows all the vector like CH fermions at 2 TeV; Fig. 6 shows all the vector-like WH fermions at 2 TeV. In both cases, unification fails. Fig. 7 shows all the vector-like CW fermions at 2 TeV; here, the unification is consistent at a higher scale  Table 2.

Discussion
The plots we have presented clarify the accuracy of the predictions of this TeV unification scheme for the precision values accurately measured at the Z-pole. The predictivity is as accurate for sin 2 θ as it is for supersymmetric GUT models [1][2][3][4]. There is, in addition, an accurate prediction for α 3 which is used merely as input in SusyGUT models.
At the same time, the accuracy of the predictions remains robust if we allow the unification scale to vary from about 2.5 TeV to 5 TeV.
Threshold effects are large in some cases and may spoil unification, which depend on the spectrum of new states.
In conclusion, since this model ameliorates the GUT hierarchy problem and naturally accommodates three families, it provides a viable alternative to the widely-studied GUT models which unify by logarithmic evolution of couplings up to much higher scales.