Quantum-gravity induced Lorentz violation and dynamical mass generation

In the eprint by Jean Alexandre [arXiv:1009.5834], a minimal extension of ($3+1$)-dimensional quantum electrodynamics has been proposed, which includes Lorentz violation (LV) in the form of higher-(spatial)-derivative isotropic terms in the gauge sector, suppressed by a mass scale $M$. The model can lead to dynamical mass generation for charged fermions. In this article, I elaborate further on this idea and I attempt to connect it to specific quantum-gravity models, inspired from string/brane theory. Specifically, in the first part of the article, I comment briefly on the gauge dependence of the dynamical mass generation in the approximations of J. Alexandre [arXiv:1009.5834.], and I propose a possible avenue for obtaining the true gauge-parameter-independent value of the mass by means of pinch technique argumentations. In the second part of the work, I embed the LV QED model into multibrane world scenarios with a view to provide a geometrical way of enhancing the dynamical mass to phenomenologically realistic values by means of bulk warp metric factors, in an (inverse) Randall-Sundrum hierarchy. Finally, in the third part of this paper, I demonstrate that such Lorentz-violating QED models may represent parts of a low-energy effective action (of Finsler-Born-Infeld type) of open strings propagating in quantum D0-particle stochastic space-time foam backgrounds, which are viewed as consistent quantum-gravity configurations. To capture correctly the quantum-fluctuating nature of the foam background, I replace the D0-recoil-velocity parts of this action by appropriate gradient operators in three-space, keeping the photon field part intact. This is consistent with the summation over world-sheet genera in the first-quantized string approach. I identify a class of quantum orderings which leads to the LV QED action of J. Alexandre, arXiv:1009.5834. In this way I argue, following the logic in that work, that the D-foam can lead to dynamically generated masses for charged-matter (fermionic) excitations interacting with it.

Figure 1

Left: A multibrane scenario, in which our world is represented as a positive tension brane (at $z=z_2$), having from the left branes with alternating-sign tensions, which shield a bulk naked singularity [which may be thought of as a limiting (singular) case of a negative tension brane]. To the right of the brane world, on the other hand, the bulk dimension extends to infinity. Right: A multibrane scenario, in which our world is represented as a positive tension brane (at $z=0$), surrounded by branes with alternating-sign tensions that shield two symmetrically positioned bulk naked singularities.

Figure 2

Schematic representation of a generic D-particle space-time foam model. The model of Ref. 8, which acts as a prototype of a D-foam, involves two stacks of D8-branes, each stack being attached to an orientifold plane. Owing to their special reflective properties, the latter provide a natural compactification of the bulk dimension. The bulk is punctured by D0-branes (D-particles), which are allowed in the type IA string theory of 8. The presence of a D-brane is essential due to gauge flux conservation, since an isolated D-particle cannot exist. Open strings live on the brane world, representing standard model matter and they can interact in a topologically nontrivial way with the D-particle defects in the foam.

Figure 3

Graphic representation (Feynman rules) for gauge boson (photon) propagator in the Lorentz-violating (LV) QED model. The curly line denotes the full propagator in the covariant gauge $\xi$, while the wavy line denotes the photon propagator in the Feynman gauge $\xi =1$.

Figure 4

Auxiliary propagatorlike structures, needed for diagrammatic proof of certain identities in the LV QED model.

Figure 5

Simple diagrammatic relation, which can be proven using the structures of Figs. 3, 4.

Figure 6

The elementary Ward identity in Lorentz-violating QED.

Figure 7

Combination of graphs used in the diagrammatic representation of Eqs. (a4, a5, a6).

Figure 8

One-loop diagram contributing to the Lorentz-violating QED fermion self-energy. The diagram is formally the same as in the Lorentz-invariant case due to the specific form of the action (2.1).

Figure 9

A typical diagrammatic relation in (LV) QED, used to demonstrate the appearance of unphysical vertices in off-shell processes.

Figure 10

Feynman rule for the unphysical effective vertex describing an interaction of the form $\gamma \gamma \overline{\psi }\psi$, where $\gamma$ denotes a photon field and $\mu$ is the index of the external current.

Figure 11

Diagrammatic representation of the Ward identity used to demonstrate the cancellation of unphysical contributions inside the two-point fermion-current correlation function $I_{\mu \nu }$, Eq. (a9).