Gauged Q-ball decay rates into fermions

We derive the decay rate of a gauged Q-ball into fermions, applying the leading semiclassical approximation. We find that more particles come out from the surface of a gauged Q-ball, compared to the case of a global Q-ball, due to the electric repulsion. We show, however, that the decay rate of a gauged Q-ball is bounded from above due to the Pauli blocking at the surface of the Q-ball, just as in the case of a global Q-ball. We also find that there is a further suppression due to the Coulomb potential outside the Q-ball, which we find to play the role of a potential barrier for the fermions coming from the inside the Q-ball.

Figure 1

Examples of profile of the gauged Q-ball of gauge mediation type. The dimensionful parameters are in units of $m_{\varphi }$. The dashed line denotes the global Q-ball with the same charge. We see that the profile of $\varphi$ is pushed outward due to the electric repulsion. We can also see that ${\varphi }^{\prime \prime }(r=R)$ becomes singular for a large gauged Q-ball, just as for a global Q-ball, even when the Coulomb potential has a non-negligible effect on the profile.

Figure 2

The plots of $\omega =dE/dQ$ and $R$ as functions of $Q$. We plot those for global Q-balls by dashed lines, for comparison. We see that $\omega$ and $R$ become large as the charge grows, due to the electric repulsion. We also present the Coulomb energy at the surface, $e^{2}Q/4\pi R$, which is denoted by a dotted line in the left figure.

Figure 3

The production rates of fermions from gauged Q-balls. We can see the enhancement due to the electric repulsion. The dashed line indicates the saturated rates for global Q-balls. We can see that for global Q-balls, the production rates saturate as the charge grows, while for gauged Q-balls the saturation is unclear from the figure.

Figure 4

The production rate from a gauged Q-ball as a function of $g{\varphi }_0/\omega$. we see that the production rate saturates for $g{\varphi }_0/\omega \gg 1$. We took a gauged Q-ball with large charge, in order to identify the size of the Q-ball as clearly as possible, so that we can compare the production rate to the classically defined saturated rate, which is illustrated by a dotted line. We note that the actual saturated rate is larger than the classical formula, since the fermions with classically forbidden momenta are produced at infinity by quantum tunneling, where Coulomb potential outside effectively becomes potential barrier for fermions coming from inside. We can see that the production rate is suppressed compared to the saturated rate when the Coulomb barrier outside does not exist.

Figure 5

The production rate as a function of momentum. We can see that the fermions with classically forbidden momenta are produced by quantum effect at infinity.

Figure 6

The production rates as a function of $g{\varphi }_0/\omega$, normalized by the classical saturated rates. We find the enhancement of the production rates for $g{\varphi }_0/\omega \ll 1$ when $e^2$ or $Q$ becomes large, which can be explained by the analogy to the case of the global Q-ball, where the production rate enhances if $\varphi$ has a large value near the surface, since the electric repulsion deforms the profile of the gauged Q-ball so that the charge is pushed outward.