A recent example of such a model is Chih-Hsun Lin, Jurgen Ulbricht, Jian Wu, Jiawei Zhao, "Experimental and Theoretical Evidence for Extended Particle Models" (2010). The abstract, in part, for the 147 page paper with 41 figures reads as follows:

We review the experimental searches on those interactions where the fundamental particles could exhibit a non point-like behavior. In particular we focus on the QED reaction measuring the differential cross sections for the process $ \EEGG $ at energies from sqrt{s} =55 GeV to 207 GeV using the data collected with the VENUS, TOPAZ, ALEPH, DELPHI L3 and OPAL from 1989 to 2003.

The global fit to the data is 5 standard deviations away from the standard model expectation for the hypothesis of an excited state of the electron, corresponding to the cut-off scale Lambda =12.5 TeV.

Assuming that this cut-off scale restricts the characteristic size of QED interaction to 15.7x10^{-18} cm, we perform an effort to assign in a semi-mechanical way all available properties of fundamental particles to a hypothetical classical object. Such object can be modeled as a classical gyroscope consisted of a non rotating inner massive kernel surrounded by an outer rotating massive layer equipped with charged sorted in a way to match the charge contents for different interactions. The model size of an electron agrees with 1.86x10^{-17} cm with the experiment. The introduction of a particle like structure related to gravity allows to estimate the inner mass kernel of an electron to 1.7x10^{-19} cm and the mass of a scaler to 154 GeV. The extension of the model to electrical charged particle-like structure in nonlinear electrodynamics coupled to General Relativity confirms the model in the global geometrical structure of mass and field distribution.

Some of the same authors have explored the same ideas in papers here (2009), here (2003), here (2001), and here (1999).

The 2009 paper is much shorter and has a nice survey of the research in the field, with this particular approach related closely to non-linear electrodynamics coupled to gravity (NED-GR) theories and the Born-Infeld Lagrangian, discussed, for example, here (2010) and here (2009). Most of the literature related to this focuses on modeling atypical hypothetical types of black holes.

Ultimately, the matter goes to fundamental issues of quantum gravity discused, for example, in this paper (2000) that systematically explores different possible couplings of the electromagnetic field and gravity.

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