Intersoliton forces in weak-coupling quantum field theories

We offer a procedure for evaluating the forces exerted by solitons of weak-coupling field theories on one another. We illustrate the procedure for the kink and the antikink of the two-dimensional ${\phi }^4$ theory. To do this, we construct analytically a static solution of the theory which can be interpreted as a kink and an antikink held a distance $R$ apart. This leads to a definition of the potential energy $U\left(R\right)\mathrm{}$ for the pair, which is seen to have all the expected features. A corresponding evaluation is also done for $U\left(R\right)\mathrm{}$ between a soliton and an antisoliton of the sine-Gordon theory. When this $U\left(R\right)\mathrm{}$ is inserted into a nonrelativistic two-body problem for the pair, it yields a set of bound states and phase shifts. These are found to agree with exact results known for the sine-Gordon field theory in those regions where $U\left(R\right)\mathrm{}$ is expected to be significant, i.e., when $R$ is large compared to the soliton size. We take this agreement as support that our procedure for defining $U\left(R\right)\mathrm{}$ yields the correct description of the dynamics of well-separated soliton pairs. An important feature of $U\left(R\right)\mathrm{}$ is that it seems to give strong intersoliton forces when the coupling constant is small, as distinct from the forces between the ordinary quanta of the theory. We suggest that this is a general feature of a class of theories, and emphasize the possible relevance of this feature to real strongly interacting hadrons.