  We present results on the nucleon tensor form factors and first moment of the
transversity distribution using maximally twisted mass fermions. We analyze two
Nf=2+1+1 ensembles having pion masses of 213 MeV and 373 MeV with lattice
spacing a=0.064 fm and a=0.082 fm, respectively. First results using an Nf=2
ensemble of twisted mass fermions with a clover term at a physical pion mass
and lattice spacing a=0.094 fm are also presented. The renormalization function
for the local tensor form factors is evaluated non-perturbatively with a
perturbative subtraction of O(a^2)-terms, while for the first moment of the
transversity we use a perturbative estimate. Results are given in the MS-bar
scheme at a scale of mu=2 GeV, and are compared with recent results obtained
using different discretization schemes.
