Phenomenological Quantum Gravity

Planck scale physics represents a future challenge, located between particle physics and general relativity. The Planck scale marks a threshold beyond which the old description of spacetime breaks down and conceptually new phenomena must appear. In the last years, increased efforts have been made to examine the phenomenology of quantum gravity, even if the full theory is still unknown. Quantum gravity is probably the most challenging and fascinating problem of physics in the 21st century. The most impressive indicator is the number of people working on it, even though so far there is no experimental evidence that might guide us from mathematics to physical reality. During the last years, the priority in the field has undergone a shift towards the investigation of possible observable effects1. The phenomenology of physics beyond the Standard Model (SM) has been studied using effective models. Such top-down inspired bottom-up approaches, which use the little knowledge that we have about quantum gravity, might soon become experimentally accessible. These models have in common that they are not a fundamentally and fully consistent description of nature, but able to capture essential features that allow to make testable predictions. Models with extra dimensions are one such approach that has been extensively studied in the last decade. In some scenarios, the Planck scale can be lowered to values soon accessible at the LHC. These models predict a vast number of quantum gravitational effects at the lowered Planck scale, among them the production of TeV-mass black holes and gravitons. Another approach which is currently under study is to extend the SM by one of the most important and general features of quantum gravity that we know: a minimal invariant length scale that acts as a regulator in the ultraviolet [3, 4, 5]. Such a minimal length scale leads to a generalized uncertainty relation and it requires a deformation of Lorentz-invariance which becomes important at high boost parameters. Within a scenario with large compatified extra dimensional, also the minimal length comes into the reach of experiment and sets a fundamental limit to short distance physics [6]. A particle with an energy close to the Planck mass, mp, is expected to significantly disturb space-time on a distance scale comparable to its own Compton wavelength and thereby make effects of quantum gravity become important. A concentration of energy high enough to cause strong curvature will result in significant quantum effects of gravity. Such a concentration of energy might most intuitively be seen as an interaction 1 For an extensive list of references see [1, 2] and references therein. process. In that process, the relevant energy is that in the center of mass (com) system, which we will denote with √ s. Note that this is a meaningful concept only for a theory with more than one particle. However, it can also be used for a particle propagating in a background field consisting of many particles (like e.g. the CMB). The scale at which effects of quantum gravity become important is when √ s/mp ∼ 1, for small impact parameters √ t ∼ 1/b ∼ mp. The construction of a quantum field theory (QFT) that self-consistently allows such a minimal length makes it necessary to carefully retrace all steps of the standard scheme. Let us consider the propagation of a particle with wave-vector kμ , when it comes into a spacetime region in which its presence will lead to a com energy close to the Planck scale. The concrete picture we want to draw is that of inand outgoing point particles separated far enough and without noticeable gravitational interaction that undergo a strong interaction in an intermediate region which we want to describe in an effective way2. We denote the asymptotically free in(out)-going states with Ψ+ (Ψ−), primes are used for the momenta of the outgoing particles. This is schematically shown in Fig. 1. In the central collision region, the curvature of spacetime is non-negligible and the scattering process as described in the SM is accompanied by gravitational interaction. An effective description of this gravitational interaction will modify the propagator, not of the asymptotically free states, but of the particles that transmit the interaction. The exchange particle has to propagate through a region with strong curvature, and the particle’s propagation will dominantly be modified by the energy the particle carries. The aim is then to examine the additional gravitational interaction by means of an concrete model3. As laid out in [2, 7] this can be done by either using an energy dependent metric in the intermediate region, or by assuming a non-linear relation between the wave-vector k and the momentum p = f (k), both of which result in a modified dispersion relation (MDR), and make it necessary to adopt a deformation of Lorentztransformations [1, 8, 9]. The functional form of the unknown relation f (k) is where knowledge from an underlying theory has to enter. The essential property of this relation is that in the high energy limit the wave-vector becomes asymptotically constant at a finite value, which is the inverse of the minimal length. The relation between momentum and wave-vector can be expanded in a power series where mp sets the scale for the higher order terms. The wave-vectors coincide with the momenta of the inor outgoing particles far away from the interaction region, where space-time is approximately flat g ∼ η . We denote these asymptotic momenta by pi. Putting the interaction in a box and forgetting about it, ∑i pi is a conserved quantity. The unitary operators of the Poincaré group act as usual on the asymptotically free states. In particular, the whole box is invariant under translations aν and the translation operator has the form exp(−iaν pν) when applied to Ψ±. In contrast to the asymptotic momenta p, the wave-vector k of the particle in the interaction region will behave non-trivially because strong gravitational effects disturb the propagation of the wave. In particular, it will not transform as a standard (flat space) Lorentz-vector, and obey the MDR. 2 An effective description as opposed to going beyond the theory of a point-like particle. 3 Explicit production of real or virtual gravitons, or black hole formation are not considered. FIGURE 1. In addition to the SM-interaction under investigation, strong gravitational effects accompany the processes in the collision region. These effects are described in the effective QFT model with a MDR for the exchange particle. Shown is the example of fermion scattering f + f− → f + f− (s-channel). Under quantization, the local quantity k will be translated into a partial derivate. Consequently, under quantization the momentum p results in an higher order operator δ ν = i f ν(−i∂ ). From this one can further define the operator g(∂α)∂μ∂ν = δ ν ∂ν , which plays the role of the propagator in the quantized theory. It captures the distortion of the exchange-particles in the strongly disturbed background. It is convenient to use the higher order operator δ ν in the setup of a field theory, instead if having to deal with an explicit infinite sum. Note, that this sum actually has to be infinite when the relation pν = f ν(k) has an asymptotic limit as one would expect for an UV-regulator. Such an asymptotic behavior could never be achieved with a finite power-series. As an example, the action for a scalar field4 takes the form