Scalar or vector tetraquark state candidate: $Z_c(4100)$

In this article, we separate the vector and axialvector components of the tensor diquark operators explicitly, construct the axialvector-axialvector type and vector-vector type scalar tetraquark currents and scalar-tensor type tensor tetraquark current to study the scalar, vector and axialvector tetraquark states with the QCD sum rules in a consistent way. The present calculations do not favor assigning the $Z_c(4100)$ to be a scalar or vector tetraquark state. If the $Z_c(4100)$ is a scalar tetraquark state without mixing effects, it should have a mass about $3.9\,\rm{GeV}$ or $4.0\,\rm{GeV}$ rather than $4.1\,\rm{GeV}$; on the other hand, if the $Z_c(4100)$ is a vector tetraquark state, it should have a mass about $4.2\,\rm{GeV}$ rather than $4.1\,\rm{GeV}$. However, if we introduce mixing, a mixing scalar tetraquark state can have a mass about $4.1\,\rm{GeV}$. As a byproduct, we obtain an axialvector tetraquark candidate for the $Z_c(4020)$.