Grand unification through gravitational effects

Xavier Calmet, Stephen D. H. Hsu, and David Reeb

Phys. Rev. D 81, 035007 – Published 4 February 2010

Abstract

We systematically study the unification of gauge couplings in the presence of (one or more) effective dimension-5 operators $c H G_{μ ν} G^{μ ν} / 4 M_{pl}$ , induced into the grand unified theory by gravitational interactions at the Planck scale. These operators alter the usual condition for gauge-coupling unification, which can, depending on the Higgs content $H$ and vacuum expectation value, result in unification at scales $M_{X}$ significantly different than naively expected. We find nonsupersymmetric models of $S U (5)$ and $S O (10)$ unification, with natural Wilson coefficients $c$ , that easily satisfy the constraints from proton decay. Furthermore, gauge-coupling unification at scales as high as the Planck scale seems feasible, possibly hinting at simultaneous unification of gauge and gravitational interactions. In the Appendix we work out the group theoretical aspects of this scenario for $S U (5)$ and $S O (10)$ unified groups in detail; this material is also relevant in the analysis of nonuniversal gaugino masses obtained from supergravity.

Received 17 November 2009

DOI:https://doi.org/10.1103/PhysRevD.81.035007

Authors & Affiliations

Xavier Calmet^1,*, Stephen D. H. Hsu^2,†, and David Reeb^2,‡

¹Physics and Astronomy, University of Sussex, Falmer, Brighton, BN1 9QH, United Kingdom
²Institute of Theoretical Science, University of Oregon, Eugene, Oregon 97403, USA

^*x.calmet@sussex.ac.uk
^†hsu@uoregon.edu
^‡dreeb@uoregon.edu

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Vol. 81, Iss. 3 — 1 February 2010

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Images

Figure 1
Numerical gauge-coupling unification results for the $S U (5)$ model with a $24$ and a $75$ Higgs. Left panel: For different fixed ratios $v_{24} ∶ v_{75}$ of the Higgs VEVs, the curves characterize the size of the Wilson coefficients $c_{i}$ necessary to achieve unification at $M_{X}$ . The straight horizontal dashed and solid lines indicate two possible choices for the parameter $ξ^{run} = 1.6$ and $ξ_{red}^{run} = 8.0$ in (2) (cf. also Table ), and the vertical lines the corresponding suppression scales $M_{pl} = 1.2 \times 10^{19} GeV / ξ$ (it may not make sense to consider scales $M_{X}$ larger than $M_{pl}$ ). Right: The $ϵ_{s}$ in (6) necessary to achieve unification at a given $M_{X}$ (not dependent on the ratio $v_{24} ∶ v_{75}$ of Higgs VEVs).Reuse & Permissions
Figure 2
Similar to the left panel of Fig. 1, but for the models with $24$ , $200$ Higgses (left: $ξ^{run} = 2.42$ , $ξ_{red}^{run} = 12.1$ ) and $75$ , $200$ Higgses (right: $ξ^{run} = 2.68$ , $ξ_{red}^{run} = 13.6$ ). For the $24$ , $200$ model, the ratio of Higgs VEVs that achieves any chosen $M_{X}$ with the smallest Wilson coefficients is roughly $v_{24} ∶ v_{200} = 1 ∶ 3$ (solid bold curve), and for the $75$ , $200$ model this ratio depends on $M_{X}$ and lies between $1 ∶ 3$ and $3 ∶ 1$ , requiring no fine-tuning of the Higgs potential. In both models, $| ϵ_{s} | < 0.5$ at all unification scales $M_{X}$ shown here (cf. right panel of Fig. 1).Reuse & Permissions
Figure 3
For a model with the three ( $h = 3$ ) Higgs multiplets $24$ , $75$ , and $200$ , a random sample is taken of 30 000 points ${c_{i}, v_{i}}$ of the parameter space which yield gauge-coupling unification. The point plot on the left characterizes the required sizes of the Wilson coefficients $c_{i}$ to achieve unification, as in Figs. 1, 2; unification at smaller scales $M_{X}$ requires proportionally larger Wilson coefficients. To illustrate the validity of (13), the right panel shows, for each sample point, the $O (0.1)$ constant to achieve equality in (13); numerical gauge-coupling unification at any $10^{13} GeV < M_{X} < 10^{20} GeV$ can be achieved by choosing 0.5 as this constant, and a lower bound is obtained by the choice 0.1. However, this should not be mistaken: unification at small $M_{X} ≲ 10^{16} GeV$ requires large dimensionless Wilson coefficients $c_{i}$ (see left panel).Reuse & Permissions
Figure 4
For the $S O (10)$ model with a $210$ Higgs, a random sample is taken of 30 000 points ${c_{210}, v_{(210) j}}$ of the parameter space which yield gauge-coupling unification; cf. Figure 3. Estimate (13) is valid here as well (see right panel), and numerical gauge-coupling unification at any $10^{13} GeV < M_{X} < 10^{20} GeV$ can be achieved with the $O (0.1)$ constant in (13) being at most 0.5 (the caveat from Fig. 3 applies here as well).Reuse & Permissions
Figure 5
Similar to Fig. 3, but for the supersymmetric $S U (5)$ model with the three ( $h = 3$ ) Higgs multiplets $24$ , $75$ , and $200$ . Again, numerical gauge-coupling unification at any $10^{13} GeV < M_{X} < 10^{20} GeV$ can be achieved by choosing 0.5 as the constant in (13) (but cf. caveat from Fig. 3). Most of the randomly chosen unification points lead to unification scales $M_{X} \sim 2 \times 10^{16} GeV$ , where, in supersymmetric models, the standard model gauge couplings almost meet.Reuse & Permissions

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