标题： 阈值处的精确多粒子振幅在$φ^4$理论中软破缺$O(\infty)$对称性 显示英文标题

标题： Exact Multiparticle Amplitudes at Threshold in $φ^4$ Theories with Softly Broken $O(\infty)$ Symmetry

摘要： 我们考虑在具有$\phi^4$对称性的$O(N_1$$+$$N_2)$ 理论中，当粒子产生处于阈值时的多粒子产生问题，该对称性被不同质量软地破缺为$O(N_1)\times O(N_2)$。我们推导出多粒子振幅之间的递归关系，这些关系在大$N$极限下，以固定产生的粒子数求和所有相关图的任意环数。我们将它转化为一个量子力学问题，并展示如何通过在大$N$处应用因子化直接从运动方程的算符中得到它。我们通过使用薛定谔算子的对角化预解式的 Gelfand--Dikiĭ 表示法，找到了该问题的精确解。结果与树振幅一致，而环效应则是耦合常数和质量的重整化。 解的形式是由于当对入射粒子的$O(N_{1,2})$-指标进行平均时，过程$2$$\ra$$n$ 在质量壳上的$n$$>$$2$ 处精确振幅消失。我们讨论这一性质背后的动力学对称性。 我们还给出了在大-$N$极限下，具有反射对称性自发破缺的$O(N)$$+$$singlet$ 标量粒子模型的精确解。 显示英文摘要

摘要： We consider the problem of multiparticle production at threshold in a $\phi^4$-theory with an $O(N_1$$+$$N_2)$ symmetry softly broken down to $O(N_1)\times O(N_2)$ by nonequal masses. We derive the set of recurrence relations between the multiparticle amplitudes which sums all relevant diagrams with arbitrary number of loops in the large-$N$ limit with fixed number of produced particles. We transform it into a quantum mechanical problem and show how it can be obtained directly from the operator equations of motion by applying the factorization at large $N$. We find the exact solutions to the problem by using the Gelfand--Diki\u{\i} representation of the diagonal resolvent of the Schr\"{o}dinger operator. The result coincides with the tree amplitudes while the effect of loops is the renormalization of the coupling constant and masses. The form of the solution is due to the fact that the exact amplitude of the process $2$$\ra$$n$ vanishes at $n$$>$$2$ on mass shell when averaged over the $O(N_{1,2})$-indices of incoming particles. We discuss what dynamical symmetry is behind this property. We also give an exact solution in the large-$N$ limit for the model of the $O(N)$$+$$singlet$ scalar particle with the spontaneous breaking of a reflection symmetry.