energy is defined as and is our Lagrange multiplier. The time dependence of the Q-ball solution can be obtained easily if one rewrites the functional as where . Since the first term in the functional is now positive, minimization of this terms implies We therefore interpret the Lagrange multiplier as the frequency of oscillation of the field within the Q-ball. The theory contains Q-ball solutions if there are any values of at which the potential is less than . In this case, a volume of space with the field at that value can have an energy per unit charge that is less than , meaning that it cannot decay into a gas of individual particles. Such a region is a Q-ball. If it is large enough, its interior is uniform and is called "Q-matter". (For a review see Lee et al. (1992). Thin-wall Q-balls The thin-wall Q-ball was the first to be studied, and this pioneering work was carried out by Sidney Coleman in 1986. For this reason, Q-balls of the thin-wall variety are sometimes called "Coleman Q-balls". We can think of this type of Q-ball a spherical ball of nonzero vacuum expectation value. In the thin-wall approximation we take the spatial profile of the field to be simply In this regime the charge carried by the Q-ball is simply . Using this fact, we can eliminate from the energy, such that we have Minimization with respect to gives Plugging this back into the energy yields Now all that remains is to minimize the energy with respect to . We can therefore state that a Q-ball solution of the thin-wall type exists if and only if for . When the above criterion is satisfied the Q-ball exists and by construction is stable against decays into scalar quanta. The mass of the thin-wall Q-ball is simply the energy Although this kind of Q-ball is stable against decay into scalars, it is not stable against decay into fermions if the scalar field has nonzero Yukawa couplings to some fermions. This decay rate was calculated in 1986 by Andrew Cohen, Sidney Coleman, Howard Georgi, and Aneesh Manohar. History Configurations of a charged scalar field that are classically stable (stable against small perturbations) were constructed by Rosen in 1968. Stable configurations of multiple scalar fields were studied by Friedberg, Lee, and Sirlin in 1976. The name "Q-ball" and the proof of quantum-mechanical stability (stability