String Phenomenology: Past, Present and Future Perspectives †
Abstract
:1. Introduction
- The logarithmic running of the Standard Model parameters, which is compatible with observations in the gauge sectors [8,9] and the heavy generation Yukawa couplings [10]. Logarithmic running in the scalar sector is spoiled by radiative corrections from the Standard Model cut-off scale. Restoration of the logarithmic running mandates the existence of a new symmetry. Supersymmetry is a concrete example that fulfils the task. The observation of a scalar resonance at 125 GeV and the fact that no other particles have been observed up to the multi-TeV energy scale indicate that the resonance is a fundamental scalar rather than a composite state [11]. This outcome agrees with the Higgs states in heterotic-string vacua.
- Further evidence for the validity of the renormalisable Standard Model up to a very high energy scale stems from the suppression of proton decay mediating operators. The Standard Model should be regarded as providing a viable effective parametrisation, but not as a fundamental accounting of the observable phenomena; the reason being that it does not provide a complete description. Obviously, gravitational effects are not accounted for. Moreover, the Standard Model itself is not mathematically self-consistent. It gives rise to singularities in the ultraviolet limit. For these reasons, the Standard Model can only be regarded as an effective theory below some cut-off. A plausible cut-off is the Planck scale, at which the gravitational coupling is of comparable strength to the gauge couplings. The renormalisability of the Standard Model is not valid beyond its cut-off scale. Non-renormalisable operators are induced by whatever theory extends the Standard Model at and beyond the cut-off scale. We should therefore take into account all the non-renormalisable operators that are allowed by the Standard Model gauge symmetries and that are suppressed by powers of the cut-off scale. Such dimension six operators that are invariant under the Standard Model gauge symmetries lead to proton decay. They indicate that the cut-off scale must be above GeV, unless they are forbidden by some new symmetries. As global symmetries are, in general, expected to be violated by quantum gravity effects, the new symmetries should be either gauge symmetries or local discrete symmetries [12,13].
- Suppression of left-handed neutrino masses is compatible with the generation of heavy mass to the right-handed neutrinos by the seesaw mechanism.
2. Past
2.1. NAHE-Based Models
2.2. Phenomenology of String Unification
- The analysis of fermion masses entails the calculation of cubic and higher order terms in the superpotential that are reduced to dimension four terms in Equation (1). The Standard Model fermion mass terms arise from couplings to the electroweak Higgs with an assumed VEV of the order of the electroweak scale. Other fermion mass terms arise from coupling to other scalar fields, and their mass scales may therefore be higher than the electroweak scale. Analysis of Standard Model fermion masses yielded a viable prediction for top quark mass prior to its experimental observation [45]. The calculation proceeds as follows. First the top quark Yukawa coupling is calculated at the cubic level of the superpotential, giving , where g is the gauge coupling at the unification scale. Subsequently, the Yukawa couplings for the bottom quark and tau lepton are obtained from quartic order terms. The magnitude of the quartic order coefficients are calculated using standard CFT techniques, and the VEV of the Standard Model singlet field in the relevant terms is extracted from analysis of the F- and D-flat directions. This analysis yields effective Yukawa couplings for the bottom quark and tau lepton in terms of the unified gauge coupling given by [45]. This result for the top quark Yukawa coupling is common in a large class of free fermionic models, whereas those for the bottom quark and tau lepton differ between models. Similarly, the Yukawa coupling for the two lighter generations differ between models and depend on the flat direction VEVs. Subsequent to extracting the Yukawa couplings at the string scale, they are run to the electroweak scale using the Minimal Supersymmetric Standard Model (MSSM) Renormalisation Group Equations (RGEs). It is further assumed that the unified gauge coupling at the string scale is compatible with the value required by the gauge coupling data at the electroweak scale. The bottom Yukawa is further run to the bottom mass scale, which is used to extract a value for , with and being the VEVs of the two MSSMelectroweak Higgs doublets. The top quark mass is then given by:
- The analysis of the effective Yukawa couplings for the lighter two generations proceeds by analysing higher order terms in the superpotential and extracting the effective dimension four operators [27,28,29]. The analysis should be regarded as demonstrating in principle the potential of string models to explain the detailed features of the Standard Model flavour parameters. It is still marred by too many uncertainties and built in assumptions to be regarded as a predictive framework. Nevertheless, once an appealing model is constructed the methodology is in place to attempt a more predictive analysis. The explorations to date included, for example, the demonstration of the generation mass hierarchy [46], Cabibbo–Kobayashi–Maskawa (CKM) mixing [47,48], light generation masses [49] and neutrino masses [50,51].
- An important issue in heterotic-string models is compatibility with the experimental gauge coupling data at the electroweak scale. The perturbative heterotic-string predicts that the gauge couplings unify at the string scale, which is of the order of GeV. On the other hand extrapolation of the gauge couplings, assuming MSSM spectra, from the Z-boson mass scale to the GUT scale, shows that the couplings converge at a scale of the order of GeV. Thus, the two scales differ by a factor of about 20. This extrapolation should be taken with caution, as the the parameters are extrapolated over 14 orders of magnitude, with rather strong assumptions on the physics in the region of extrapolation. Indeed, in view of the more recent results from the LHC, the analysis needs to be revised, as the assumption of MSSM spectrum at the Z-boson scale has been invalidated. Nevertheless, the issue can be studied in detail in perturbative heterotic-string models, and a variety of possible effects have been examined, including heavy string threshold corrections, light supersymmetry (SUSY) thresholds, additional gauge structures and additional intermediate matter states [40]. Within the context of the free fermionic models, only the existence of additional matter states may resolve the discrepancy, and such states indeed exist in the spectrum of concrete string models [52]. This result may be relaxed in the non-perturbative heterotic-string [42] or if the moduli are away from the free fermionic point [53].
- Proton longevity is an important problem in quantum gravity, in general, and in string models in particular. The reason being that we expect only gauge symmetries, or local discrete symmetries that arise as remnants of broken gauge symmetries, to be respected in quantum gravity. Within the Standard Model itself baryon and lepton are accidental global symmetries at the renormalisable level. Thus, we expect, in general, all operators that are compatible with the local gauge and discrete symmetries in given string models to be generated from non-renormalisable terms. Such terms can then give rise to dimension four, five and six baryon and lepton number violating operators that may lead to rapid proton decay. Possible resolutions have been studied in specific free fermionic models and include the existence of an additional light symmetry [54,55,56,57,58,59] and local discrete symmetries [12,13].
- String models may, in general, lead to non-degenerate squark masses, depending on the specific SUSY breaking mechanism. For example, SUSY breaking mechanism which is dominated by the moduli F-term will lead to non-degenerate squark masses, because of the moduli dependence of the flavour parameters. Similarly, D-term SUSY breaking depends on the charges of the Standard Model fields under the gauge symmetry in the SUSY breaking sector, and those are in general family non-universal. Free fermionic models can give rise to a family universal anomalous [60]. If the SUSY breaking mechanism is dominated by the anomalous D-term it may produce family universal squark masses of order 1 TeV [61].
- Three generation semi-realistic string models produce, in general, additional massless vector-like states that are charged under the Standard Model gauge symmetries. Some of these additional vector-like states arise from the Wilson line breaking of the GUT symmetry and therefore carry fractional charge with respect to the remnant unbroken symmetries. In particular, they may carry fractional electric charge, which is highly constrained by observations. These fractionally charged states must therefore be sufficiently massive or diluted to evade the experimental limits. Mass terms for the vector-like states may arise from cubic and higher level terms in the superpotential. In the model of [34] its has been demonstrated in [62] that all the exotic fractionally charged states couple to a set of singlets. In [63,64] F- and D-flat solutions that incorporate this set of fields have been found. Additionally, all the extra standard-like fields in the model, beyond the MSSM, receive mass terms by the same set of VEVs. These solutions therefore give rise to the first known string solutions that produce in the low energy effective theory of the observable sector solely the states of the MSSM, and are dubbed Minimal Standard Heterotic-String Model (MSHSM). Three generation Pati–Salam free fermionic models in which fractionally charged exotic states arise only in the massive spectrum were found in [65]. Flat directions that lead to MSHSM with one leading Yukawa coupling were found in an exemplary model in this class [66].
- An important issue in string models is that of moduli stabilisation. The free fermionic models are formulated near the self-dual point in the moduli space. However, the geometrical moduli that allow deformation from that point exist in the spectrum and can be incorporated in the form of Thirring worldsheet interactions [24]. The correspondence of the free fermionic models with orbifold implies that the geometrical moduli correspond to three complex and three Kähler structure moduli. String theory as a theory of quantum geometry, rather than classical geometry, allows for assignment of asymmetric boundary conditions with respect to the worldsheet fermions that correspond to the internal dimensions. These correspond to the asymmetric bosonic identifications under . In the free fermionic models, and consequently in orbifolds, it is possible to assign asymmetric boundary conditions with respect to six circles of the six-dimensional compactified torus. In such a model all the complex and Kähler moduli of the untwisted moduli are projected out [67,68]. Additionally, the breaking of the worldsheet supersymmetry in the bosonic sector of the heterotic-string results in projection of the would-be twisted moduli [67,68]. Thus, all the fields that are naively identified as moduli in models with (2,2) worldsheet supersymmetry can be projected out in concrete models. However, the identification of the moduli in models with (2,0) worldsheet supersymmetry is not well understood and there may exist other fields in the spectrum of such models that may be identified as moduli fields. Furthermore, as long as supersymmetry remains unbroken in the vacuum there exist moduli fields associated with the supersymmetric flat directions. However, it has been proposed that there exit quasi-realistic free fermionic models which do not admit supersymmetric flat directions [69,70]. This is obtained when both symmetric and asymmetric twistings of the internal dimensions are implemented, resulting in reduction of the number of moduli fields. In the relevant models supersymmetry is broken due to the existence of a Fayet–Iliopoulos term, which is generated by an anomalous symmetry. It was argued in [69,70] that the relevant models do not admit exact flat directions and therefore supersymmetry is broken at some level. In such models all the moduli are fixed. It should be noted that this possibility arises only in very particular string models, rather than in a generic string vacua [71].
3. Present
3.1. Classification of Fermionic Orbifolds
3.1.1. Spinor-Vector Duality
4. Other Approaches
5. Future
5.1. Toward String Predictions
5.2. Cosmological Evolution
5.3. Dualities and Fundamental Principles
5.3.1. The Classical Limit
5.3.2. Where is the Connection with String Theory?
6. Conclusions
Acknowledgements
Conflicts of Interest
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Faraggi, A.E. String Phenomenology: Past, Present and Future Perspectives. Galaxies 2014, 2, 223-258. https://doi.org/10.3390/galaxies2020223
Faraggi AE. String Phenomenology: Past, Present and Future Perspectives. Galaxies. 2014; 2(2):223-258. https://doi.org/10.3390/galaxies2020223
Chicago/Turabian StyleFaraggi, Alon E. 2014. "String Phenomenology: Past, Present and Future Perspectives" Galaxies 2, no. 2: 223-258. https://doi.org/10.3390/galaxies2020223