Abstract
As discussed in the opening chapter, one of the best ways to understand the realm of validity for an effective theory is to calculate the energy scale where perturbative unitarity breaks down. In the first section of this chapter we do exactly this for the effective theory of gravity coupled to matter as given by the action (1.1.10). In the second section we apply the bound to various grand unified theories. In the third section we incorporate renormalisation group (RG) effects into the bounds and are then able to compare the scale at which unitarity breaks down with the scale of strong coupling. We discuss the consequences of the RG improved bounds for various models of particle physics and introduce two models which can lower the scale of quantum gravity in four dimensions. The unitarity bound derived here will also provide an important basis for later chapters.
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Notes
- 1.
Note that as in the case for \(WW\) scattering in the standard model (see Sect. 1.2.1) we might think that including longitudinally polarised vector bosons in the external states may lead to the largest high energy behaviour of the scattering amplitudes. However, as can be seen by considering the Goldstone boson equivalence theorem, there should be cancellations that happen in the calculation of such amplitudes so that the high energy behaviour is no stronger than for transversely polarised vector bosons. Indeed this is the case and we have verified it for the scattering amplitudes presented here.
- 2.
We remark here that despite the rigorous heat kernel derivation of Eq. (2.3.1), a recent publication [11] has criticised attempts to define a running Planck mass. The main argument is that a precise definition of the running is not independent of the process from which it was derived. This therefore leads to difficulty in defining a universally applicable running. If true, these criticisms could cast doubt on the validity of our arguments here. However, we only consider a single process, \(s\)-channel scattering via graviton exchange, and so we need not worry about universality of the definition of the running. The running we employ is defined from exactly the process we wish to consider and should therefore be applicable everywhere we have used it, even if it were not applicable for other processes.
- 3.
- 4.
We remark again that the concerns raised in Ref. [9] about defining a universal running Planck mass are not relevant here since the argument given in this section turns out to be independent of the specific running employed.
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Atkins, M. (2014). Unitarity of Gravity Coupled to Models of Particle Physics. In: Bounds on the Effective Theory of Gravity in Models of Particle Physics and Cosmology. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-06367-6_2
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DOI: https://doi.org/10.1007/978-3-319-06367-6_2
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