标题： 关于中微子力的短程行为：从$1/r^{5}$到$1/r^{4}$、$1/r^2$和$1/r$ 显示英文标题

标题： On the short-range behavior of neutrino forces: from $1/r^{5}$ to $1/r^{4}$, $1/r^2$, and $1/r$

摘要： 一对中微子在相距$r$的两个物体之间交换，会导致一个与$1/r_{}^5$成正比的长程有效势，假设中微子质量为零且存在四费米子接触相互作用。 在本文中，我们研究如果距离$r$足够短，对应足够大的动量转移，可能会使接触相互作用失效，这种已知的中微子介导势会如何改变。 我们考虑两种可能的情景，通过引入$t$通道或$s$通道的媒介粒子来打开接触相互作用。 我们推导出一个通用公式，只要外部粒子保持非相对论性，该公式就可以描述所有区域的势。 在两种情景中，势在长程极限下都按照$1/r_{}^5$减少，正如预期的那样。 在短程极限下，$t$通道的势表现出库仑类似的行为（即） 与$1/r$成正比，而$s$通道电势表现出$1/r_{}^4$和$1/r_{}^2$的行为。 显示英文摘要

摘要： The exchange of a pair of neutrinos between two objects, seperated by a distance $r$, leads to a long-range effective potential proportional to $1/r_{}^5$, assuming massless neutrinos and four-fermion contact interactions. In this paper, we investigate how this known form of neutrino-mediated potentials might be altered if the distance $r$ is sufficiently short, corresponding to a sufficiently large momentum transfer which could invalidate the contact interactions. We consider two possible scenarios to open up the contact interactions by introducing a $t$-channel or an $s$-channel mediator. We derive a general formula that is valid to describe the potential in all regimes as long as the external particles remain non-relativistic. In both scenarios, the potential decreases as $1/r_{}^5$ in the long-range limit as expected. In the short-range limit, the $t$-channel potential exhibits the Coulomb-like behavior (i.e. proportional to $1/r$), while the $s$-channel potential exhibits $1/r_{}^4$ and $1/r_{}^2$ behaviors.