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Elementary Charge and Neutrino’s Mass from Planck Length

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Abstract

It is shown that the postulation of a minimum length for the horizons of a black hole leads to lower bounds for the electric charges and magnetic moments of elementary particles. If the minimum length has the order of the Planck scale, these bounds are given, respectively, by the electronic charge and by \(\mu \sim 10^{-21} \mu _B\). The latter implies that the masses of fundamental particles are bounded above by the Planck mass, and that the smallest non-zero neutrino mass is \(m_{\nu } \sim 10^{-2}\)eV. A precise estimation in agreement to the area quantisation of Loop Quantum Gravity predicts a mass for the lightest massive state in concordance with flavor oscillation measurements, and a Barbero–Immirzi parameter in accordance to horizon entropy estimations.

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Notes

  1. This is valid for Dirac neutrinos in the minimally extended Standard Model with right-handed singlets. Majorana neutrinos do not have magnetic moments.

  2. The last digit in this figure is affected by higher order corrections to (5) that depend on the neutrinos mixing angles and Dirac phase [20]. Using the current best-fits for these quantities [21], we find \(m_{\nu } \approx 8.662\,(8) \times 10^{-3}\, \text {eV}\). Note, however, that higher order loops lead to corrections of the same order.

  3. We are using natural units \(\hbar = c = G = 1\), in which \(l_P = 1\).

  4. A general correspondence between angular momenta and spin network lines is conjectured in [24, 25].

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Acknowledgements

I am thankful to G. A. Mena Marugán for a critical reading and helpful suggestions. My thanks also to J.C. Fabris, R. Gambini, P.C. de Holanda, J. Olmedo, O.L.G. Peres, C. Pigozzo, A. Saa, R. Woodard and J. Zanelli for useful discussions. Work partially supported by CNPq.

Funding

Funding was provided by Conselho Nacional de Desenvolvimento Científico e Tecnológico (Grant No. 307467/2017-1).

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  1. Instituto de Física Gleb Wataghin, UNICAMP, Campinas, SP, 13083-970, Brazil

    Saulo Carneiro

  2. Instituto de Física, Universidade Federal da Bahia, Salvador, Bahia, 40210-340, Brazil

    Saulo Carneiro

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Correspondence to Saulo Carneiro.

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Carneiro, S. Elementary Charge and Neutrino’s Mass from Planck Length. Found Phys 50, 1376–1381 (2020). https://doi.org/10.1007/s10701-020-00383-z

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