These are questions I have after reading the Rajaraman's book "Solitons and instantons". So I think you must have read the book if want to answer. And also know about quantum solitons. Rajaraman derives classically (page 82) an interaction potential energy $V(R)$ of the kink-antikink pair.

Then, he says some things that are not obvious for me, but he talk about they as obvious things:

"[The potential (3.109) is clearly reminiscent of a one-meson exchange potential extracted from quantum field theory. In fact, in chapter 5 where we quantise this theory, it will be seen to carry a meson of mass $ \sqrt 2 m$, and to yield a kink-kink-meson vertex of order $1/\sqrt\lambda$. If we remember that two such vertices are involved when a meson is exchanged between kinks, and put in sufficient factors of m to meet dimensional requirements, the one- meson-exchange Born amplitude must clearly yield a potential $V(R)$"

What is a kink-kink-meson vertex? And how we can show that it is of order $1/\sqrt\lambda$?

How are related two such vertices with the one-meson-exchange Born amplitude? and what is this one-meson-exchange Born amplitude he talks about?

and finally How is obvious that this amplitude yield a potential $V(R)$?

These are questions I have after reading the Rajaraman's book "Solitons and instantons". So I think you must have read the book if want to answer. And also know about quantum solitons. Rajaraman derives classically (page 82) an interaction potential energy $V(R)$ of the kink-antikink pair.

Then, he says some things that are not obvious for me, but he talk about they as obvious things:

"[The potential (3.109) is clearly reminiscent of a one-meson exchange potential extracted from quantum field theory. In fact, in chapter 5 where we quantise this theory, it will be seen to carry a meson of mass $ \sqrt 2 m$, and to yield a kink-kink-meson vertex of order $1/\sqrt\lambda$. If we remember that two such vertices are involved when a meson is exchanged between kinks, and put in sufficient factors of m to meet dimensional requirements, the one- meson-exchange Born amplitude must clearly yield a potential $V(R)$"

What is a kink-kink-meson vertex? And how we can show that it is of order $1/\sqrt\lambda$?

How are related two such vertices with the one-meson-exchange Born amplitude? and what is this one-meson-exchange Born amplitude he talks about?

and finally How is obvious that this amplitude yields a potential $V(R)$?