#### Abstract

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.

DOI: http://dx.doi.org/10.1103/PhysRevD.15.2866

- Received 25 August 1976
- Published in the issue dated 15 May 1977

© 1977 The American Physical Society