'Running' under tight constraints in pionless effective field theory

The contents of renormalization group invariance and equations under tight constraints are explored and demonstrated with closed-form on-shell $T$ matrices of pionless effective field theory for nuclear forces right within the effective field theory philosophy. The 'running' couplings under such tight constraints are presented in $^1S_0$ and uncoupled $P$ channels up to truncation order $\mathcal{O}(Q^4)$. Some linear relations are exposed and in turn employed in the pursuit of nonperturbative 'running' solutions which serves as an alternative choice without resorting to special prescription and additional operations or treatments. The utility of such 'running' behaviors inherent in the closed-form $T$-matrices of pionless effective field theory is remarked and a number of important issues related to effective field theory constructed with various truncations are interpreted or discussed from the underlying theory perspective.