Loop quantum gravity effects on electromagnetic properties of charged leptons

Abstract The efforts in this contribution consist in reassessing a modified Dirac equation that incorporates a $$\gamma ^0 \gamma _5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>γ</mml:mi> <mml:mn>0</mml:mn> </mml:msup> <mml:msub> <mml:mi>γ</mml:mi> <mml:mn>5</mml:mn> </mml:msub> </mml:mrow> </mml:math> -Lorentz-symmetry violating (LSV) term induced as a Loop Quantum Gravity (LQG) effect. Originally, this equation has been applied and considered as a good scenario for describing a number of investigations on the flight time of cosmic neutrinos, which suggests that the speed, in vacuum, in connection with the geometry that describes a granular space-time, takes the energy-dependent form, e.g., $$v(E) =1 \pm E/E_{\text {\tiny LSV}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>v</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>E</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>±</mml:mo> <mml:mi>E</mml:mi> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow> <mml:mspace/> <mml:mtext>LSV</mml:mtext> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , with $$E_{\text {\tiny LSV}} \approx 6,5 \times 10^{17}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow> <mml:mspace/> <mml:mtext>LSV</mml:mtext> </mml:mrow> </mml:msub> <mml:mo>≈</mml:mo> <mml:mn>6</mml:mn> <mml:mo>,</mml:mo> <mml:mn>5</mml:mn> <mml:mo>×</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>17</mml:mn> </mml:msup> </mml:mrow> </mml:math> GeV for neutrinos. Once LQG provides a viable way to understand this picture consistently, we pursue an analysis of this effective Dirac equation to inspect some of its properties. These include: the derivation of the modified fermionic propagator, attainment of the Gordon decomposition of the vector current with minimal electromagnetic coupling to obtain information on the form factors, examination of the non-relativistic limit of the equation, evaluation of the spin- and velocity-dependent corrections to the Coulomb potential due to LQG effects, and the modified Hamiltonian in the low-relativistic regime. The study of the form factors may open up paths to set up bounds on the LQG parameters from the precision measurements of electromagnetic attributes of the charged leptons, such as their respective electric and magnetic dipole moments.