  In this paper, we calculate the QCD nonperturbative contributions of the
neutrino-quark tensor operators to the neutrino magnetic moments by matching
onto the chiral perturbation theory at low energies. These nonperturbative
contributions can be compared to the perturbative ones, which are induced from
one-loop mixing when performing the renormalization group evolutions from
$\mu=m_W$ down to $\mu=2~\mathrm{GeV}$. We then constrain the dipole and tensor
Wilson coefficients of the low-energy neutrino effective field theory (LNEFT)
separately from the neutrino-electron scattering with Borexino data and
coherent elastic neutrino-nucleus scattering (CE$\nu$NS) with COHERENT data to
show the competition between these two contributions, at the renormalization
scales $\mu=2~\mathrm{GeV}$ and $\mu=m_W$ in the $\bar{\mathrm{MS}}$ scheme. In
the neutrino-electron scattering, it is found that the nonperturbative
contributions dominate for the coefficients involving up and down quarks, while
they are expected to be of the same order of magnitude as the perturbative
contributions for the coefficients involving strange quark. As for constraints
in the CE$\nu$NS, the tensor operators can contribute to the process through
either direct or indirect way. As a result, the indirect contributions
including nonperturbative and perturbative parts for all couplings become
negligible in comparison with the direct ones. As the nonperturbative
contributions crucially depend on the value of $c_T$, its inputs will affect
the extraction of limits on the tensor LNEFT Wilson coefficients. We compute
the upper bounds on these coefficients with $c_T$ quoting from the model and
lattice estimates.
