Breakdown of Lorentz invariance for spin-$1/2$ particle motion in curved space-time with applications to muon decay

This paper explores the properties of the Pauli-Lubanski spin vector for the general motion of spin-$1/2$ particles in curved space-time. Building upon previously determined results in flat space-time, it is shown that the associated Casimir scalar for spin possesses both gravitational contributions and frame-dependent contributions due to noninertial motion, where the latter represents a possible quantum violation of Lorentz invariance that becomes significant at the Compton wavelength scale. When applied to muon decay near the event horizon of a microscopic Kerr black hole, it is shown that its differential cross section is strongly affected by curvature, with particular sensitivity to changes in the black hole’s spin angular momentum. In the absence of curvature, the noninertial contributions to the decay spectrum are also identified and explored in detail, where its potential for observation is highest for large electron opening angles. It is further shown how possible contributions to noncommutative geometry can emerge from within this formalism at some undetermined length scale. Surprisingly, while the potential exists to identify noncommutative effects in muon decay, the relevant terms make no contribution to the decay spectrum, for reasons which remain unknown.

Figure 1

Muon in circular orbit around a Kerr black hole, subject to electroweak decay. The classical orbit is defined by $(r_0,{\theta }_0,{\varphi }_0)$, where quantum fluctuations about it are defined by $(r,\theta ,\varphi )$. While ${\theta }_0=\pi /2$ for this paper, the inclined orbit here emphasizes the point that the decaying muon is subject to polar variations in the curvature that need to be accounted for properly.

Figure 2

The muon differential decay rate as a function of electron energy fraction for the muon near the event horizon of a Kerr black hole with mass $M=2\times {10}^{-11}\mathrm{cm}$ and orbital radius $r_0\sim M$. Figures 2a, 2b display comparative plots of the decay spectrum due to curvature and noninertial effects, for $\alpha =1$ and $\alpha =0.99$, respectively. It is clear that the plots are very sensitive to changes in $\alpha$ for fixed $r_0$.

Figure 3

Net differential decay rate due to the noninertial dipole operator for various decay angles, after subtracting the flat space-time background contribution. It is evident that the contribution of $\mathit{R}$ to the decay spectrum becomes identifiable for large emission angles.

Figure 4

Net differential decay rate due to various combinations of curvature and the noninertial dipole operator after subtracting the flat space-time background contribution.

Figure 5

Michel spectrum with contributions due to curvature and the noninertial dipole operator for $M=4\times {10}^{-11}\mathrm{cm}$ and $r_0\sim M$. While the combined curvature and noninertial effect makes a larger change to the profile compared to the noninertial effect alone, the fact that there exists a slight but discernible change in its curve compared to the flat background contribution is noteworthy.

Figure 6

Michel spectrum with contributions due to curvature and the noninertial dipole operator for $M=2\times {10}^{-11}\mathrm{cm}$ and $r_0\sim M$. Comparison between Figs. 6a, 6b shows that the profile is extremely sensitive to changes in $\alpha$, while the increase in $r_0$ leads to greater distinction between the noninertial curve compared to the flat background.

Figure 7

Three-dimensional plot of the Michel spectrum for varying $\alpha$ and $x$, with $M=2\times {10}^{-10}\mathrm{cm}$. Figure 7a with ${\Gamma }_1\equiv \frac{1}{{\Gamma }_0}\widehat{\Gamma }(x)$ starts to develop a minimum value due to curvature and noninertial contributions, while Fig. 7b with ${\Gamma }_2\equiv \frac{1}{{\Gamma }_0}\Delta \widehat{\Gamma }(x)$ shows the net contribution after subtracting off the exclusively curvature-dependent contribution.

Figure 8

Three-dimensional plot of the Michel spectrum for varying $\alpha$ and $x$, with $M=4\times {10}^{-11}\mathrm{cm}$. The minimum in Fig. 8a due to curvature and noninertial effects becomes more pronounced, with a correspondingly larger net effect found in Fig. 8b.

Figure 9

Three-dimensional plot of the Michel spectrum for varying $\alpha$ and $x$, with $M=2\times {10}^{-11}\mathrm{cm}$. As $M$ becomes progressively smaller, curvature effects on the plot strongly determines the plot’s structure and magnitude.

Figure 10

Relative total decay rate as a function of $M$ for a Schwarzschild black hole ($\alpha =0$). Figure 10a corresponds to $r_0=6M$ for $N_0=1/\sqrt{2}$, while Fig. 10b implies that $r_0\sim 3M$ for $N_0={10}^{-2}$. It is evident that curvature strongly enhances the lifetime of the muon as $r_0\to 2M$, the Schwarzschild radius, while the contribution due to $\mathit{R}$ indicates strong enhancement for $r_0\sim {10}^{-12}\mathrm{cm}$, the Compton wavelength scale for the muon.

Figure 11

Relative total decay rate as a function of $M$ for an extreme Kerr black hole. Figure 11a denotes $r_0\sim M$ for a corotating black hole ($\alpha =1$) with respect to the muon’s orbit, while Fig. 11b refers to $r_0\sim 4M$ for a counter-rotating black hole ($\alpha =-1/4$). The spikes on the main curve shown in Fig. 11a, along with the apparent upward translation of the curve due to $\mathit{R}$, are numerical artifacts that should be discounted.

Figure 12

Differential decay rate displaying contributions due to curvature and the noninertial dipole operator (forward emission).

Figure 13

Differential decay rate displaying contributions due to curvature and the noninertial dipole operator (backward emission).

Figure 14

Net differential decay rate due to curvature and the noninertial dipole operator after subtracting the flat space-time background contribution (forward emission).

Figure 15

Net differential decay rate due to curvature and the noninertial dipole operator after subtracting the flat space-time background contribution (backward emission).