Natural Gauge and Gravitational Coupling Unification and the Superpartner Masses

The possibility to achieve unification at the string scale in the context of the simplest supersymmetric grand unified theory is investigated. We find conservative upper bounds on the superpartner masses consistent with the unification of gauge and gravitational couplings, M_{\tilde G}<5 TeV and M_{\tilde f}<3 \times 10^7 GeV, for the superparticles with spin one-half and zero, respectively. These bounds hint towards the possibility that this supersymmetric scenario could be tested at future colliders, and in particular, at the forthcoming LHC.

However, it is interesting to look for alternative ways to test the idea of the unification of all fundamental forces in nature. In this letter we investigate if the unification of the gauge and gravitational couplings at the string scale can give us some new insight in our quest for unification. We study the possibility to achieve unification of gauge couplings and gravity in the context of the simplest supersymmetric grand unified theory. We show that such a unification leads at one-loop level to a unique relation between the superpartner masses. Using the electroweak precision data and the current limits on the SUSY partner masses, we find upper bounds on the sfermion and fermionic superpartner masses. We conclude that in this minimal framework the fermionic superpartner masses are naturally at or below the TeV scale.

II. UPPER BOUND ON THE SUPERPARTNER MASSES
In this section we will explain the possibility to find upper bounds on the superpartner masses once the unification of all forces is assumed in the context of heterotic string scenarios. In a weakly-coupled heterotic string theory, gauge and gravitational couplings unify at tree level [2], where str = g 2 str =4 is the string-scale unification coupling constant, G N is the Newton constant, 0 is the Regge slope, i = g 2 i =4 (i= 1;2;3)are the gauge couplings and k i are the so-called affine or Kač-Moody levels at which the group factors U (1) Y , SU (2) L and SU (3) C are realized in the four-dimensional string [4]. Including one-loop string effects, the unification scale M str is predicted as [3] M str = p 4 str s ; where s 5: 27 10 17 GeV.
Our main goal is to investigate the possibility to achieve unification of all interactions in the context of the minimal supersymmetric SU (5)theory. The relevant one-loop renormalization group equations are given by where iZ i (M Z ) D R are the couplings defined in the D R renormalization scheme. The masses M R are the different thresholds included in the running. We recall that in the Standard Model b SM i = (41=6; 19=6; 7). The coefficients R i are the additional contributions associated to each mass threshold M R . In Table I we list their values for the minimal SUSY SU (5) theory considered here. In the above  equations we have used M str as the most natural value for the superheavy gauge boson masses as well as for the mass of the colored triplets in5 H and^ 5 H , relevant for proton decay [7]. Notice that since the contribution of the colored triplets to b 1 b 2 (b 2 b 3 )is positive (negative) the upper bounds presented below are the most conservative bounds. In other words, the lower the colored triplet mass scale is, the lighter the superpartner masses have to be to achieve unification.
The affine levels k i are those corresponding to the standard SU (5) theory, i.e. the canonical values k 1 = 5=3, k 2 = 1 and k 3 = 1. We remark that considering a higher Kač-Moody level k (as required, for instance, in string models having a G G structure [8]) simply corresponds to the redefinition s ! p k s .
This pushes the string scale M str up and would require slightly lower values of the adjoint scalar masses and somewhat heavier sfermions to achieve unification.
Assuming a common mass M e G for gauginos and Higgsinos, as well as a common mass Mf for sfermions and the extra Higgs doublet, and using M 3 = M 8 M as predicted by the minimal supersymmetric SU (5)model, the system of Eqs. (3) has the solution with the unification scale M str given by where W 0 (x)is the principal branch of the Lambert function [9,10].
Notice that from Eqs. (4) and (6) we can find a unique relation between the gaugino and sfermion masses in this minimal framework. Our main result reads then as In the case when MG = Mf M S U S Y , from Eq. (4) we find that the common superpartner mass is  Fig. 2 for the case of a split-SUSY scenario [12] with a common gaugino mass MG = 200 GeV, which corresponds to the presently available experimental lower bound [11]. In the latter case, we obtain from Eqs. approximation to a realistic spectrum that is produced in several scenarios of supersymmetry breaking as, for instance, in models based on minimal supergravity (mSUGRA) [13]. Our approximation represents averages of the mass spectra in these models. A more realistic analysis of the sparticle masses will not change the main conclusions of our work. We may ask ourselves how a mass splitting between the superpartners could modify the unification picture. In particular, one could expect different masses for the gluino (g), the weak-gauginos (W ) and Higgsinos (h). To illustrate the dependence of our results on the gaugino spectrum, and without committing ourselves to any specific SUSY breaking scenario, we present in Fig. 3 If we take into account the presently allowed s uncertainty, then there is a sizable shift of the curves (see dotted lines). Clearly, these bounds could also be subject to modifications if gaugino masses are nondegenerate, as can be seen from the figure. The result of Eq. (10) is consistent with the upper bound on the scalar masses of p m 3=2 M P l 10 10 GeV [14] in SUGRA and string models coming from the cancellation of vacuum energy. We recall that m 3=2 is the gravitino mass. We also notice that the upper bound on the superpartner masses is in agreement with the cosmological constraints on the gluino lifetime [15].
In a similar way, one can consider the case when the adjoint scalars 3 and 8 have different masses [16].
In Fig. 4 we present the solutions for MG and Mf consistent with unification. We notice that when the mass splitting is small the fermionic superpartner masses in agreement with unification are in the interesting region for LHC. However, if we restrict ourselves to the minimal supersymmetric SU (5), where these adjoint fields have to be degenerate, the upper bounds given in Eqs. (9) and (10) hold.
Let us also comment on some other relevant effects. As explained before, when the colored triplets in 5 H and^ 5 H are below the unification scale the masses of the superpartners have to be smaller. Therefore, the upper bounds on the superpartner masses are indeed those coming from the case when the colored triplets are at the unification scale. String threshold effects as well as two loop effects have been neglected in our analysis. These effects could be important and we will be studied elsewhere. However, as we have pointed out, there are other relevant effects at one-loop level, such as the mass splitting between the fermionic superpartners, which already indicate that only in the simplest scenario conservative upper bounds on the superpartner masses can be found.

III. SUMMARY
We have investigated the possibility to achieve unification of the gauge and gravitational couplings at the perturbative string scale in the context of the simplest supersymmetric grand unified theory. We have pointed out a unique one-loop relation between the superpartner masses consistent with the unification of all interactions. Conservative upper bounds on the superpartner masses were found, namely, MG . 5 TeV and Mf . 3 10 7 GeV, for the spin-1/2 and spin-0 superpartners, respectively. These bounds hint towards the possibility that this supersymmetric scenario could be tested at future colliders, and in particular, at the forthcoming LHC.