Multiparticle azimuthal cumulants from transverse momentum conservation and collective flow

We calculate the $n$th order of $2k$-particle azimuthal cumulants $c_n\{2k\}$ based on transverse momentum conservation (TMC) and collective flow $v_n$(n=2,3). We demonstrate that the TMC effect only leads to a nonzero $c_n\{2k\}$ with the sign of $(-1)^{nk}$ and the magnitude inversely proportional to $(N-2k)^{nk}$. The interplay between TMC and collective flow can change the signs of $c_2\{4\}$, $c_3\{2\}$ and $c_3\{4\}$ at some values of multiplicity $N$, which could provide a good probe to study the onset of collectivity and search for the substructure of proton in small colliding systems.