Changes in extensive air showers from isotropic Lorentz violation in the photon sector

We consider a theory with isotropic nonbirefringent Lorentz violation in the photon sector and explore the effects on the development of the electromagnetic component of extensive air showers in the Earth atmosphere. Specifically, we consider the case of a “fast” photon with a phase velocity larger than the maximum attainable velocity of a massive Dirac fermion (this case corresponds to a negative Lorentz-violating parameter $\kappa$ in the action). Shower photons with above-threshold energies decay promptly into electron-positron pairs, instead of decaying by the conventional production of electron-positron pairs in the background fields of atomic nuclei. This rapid production of charged leptons accelerates the shower development, decreasing the atmospheric depth of the shower maximum ($X_{\max }$) by an amount which could be measured by cosmic-ray observatories. Precise measurements of $X_{\max }$ could then improve existing limits on the negative Lorentz-violating parameter $\kappa$ by several orders of magnitude.

Figure 1

Photon decay length $l_{\mathrm{PhD}}\equiv 1/{\mathrm{\Gamma }}_{\mathrm{PhD}}$ as a function of the photon energy $\omega$ for the process $\tilde{\gamma }\to e^{-}+e^{+}$. Two values of the Lorentz-violating parameter $\kappa$ are considered, $\kappa =-9\times {10}^{-16}$ and $\kappa =-9\times {10}^{-20}$, and the vertical lines indicate the respective threshold energies, defined by (2.5) in terms of $\kappa$ and the electron mass $m$. For $\kappa \uparrow 0$, the photon-decay threshold energy moves towards infinity.

Figure 2

Shower-maximum depth (3.7) of the electromagnetic shower for different values of $\kappa$ as a function of the initial photon energy ${\omega }_0$. The blue-dashed curve for $\kappa =-9\times {10}^{-20}$ has a larger photon-decay threshold energy than the red-solid curve for $\kappa =-9\times {10}^{-16}$.

Figure 3

Relative modification (3.10) of the shower-maximum depth (3.9) for different values of the Lorentz-violating parameter $\kappa$. If an electromagnetic shower is initiated by a $10^7\text{−}\mathrm{GeV}$ photon, a reduction of approximately 15% on $X_{\max }$ is expected from the current limit value $\kappa =-9\times {10}^{-16}$, whereas a 10% reduction is expected from $\kappa =-9\times {10}^{-18}$ and a reduction by 5% from $\kappa =-9\times {10}^{-20}$. The three different curves start at different threshold energies defined by (2.5) in terms of $\kappa$ and the electron mass $m$.

Figure 4

Lorentz-violating parameter $\kappa$ as a function of the initial photon energy ${\omega }_0$ producing different relative modifications $\delta$ of the shower-maximum depth. Also shown is the line ${\omega }_0={\omega }_{\mathrm{th}}(\kappa )$ with ${\omega }_{\mathrm{th}}(\kappa )$ given by (2.5) and only photons to the right of this line are considered. If an electromagnetic shower is initiated by a $10^8\text{−}\mathrm{GeV}$ photon, a reduction of 15% on $X_{\max }$ would be produced by $\kappa \sim -{10}^{-16}$, a reduction of 10% by $\kappa \sim -5\times {10}^{-19}$, and a reduction of 5% by $\kappa \sim -2\times {10}^{-21}$.