You could use HoldForm
expr = (2 m^2)^(-2 ϵ) (s/(4 m^2))^
z Pi^(1/2) (Gamma[-z] Gamma[1 + z] Gamma[1 - ϵ] Gamma[
z + ϵ] Gamma[z + 2 ϵ])/(Gamma[
2 + z - ϵ] Gamma[1/2 + z + ϵ])
Original

Traditional Form
TraditionalForm[expr]

TeXForm with No HoldForm
TeXForm[expr]
$$
\frac{\sqrt{\pi } \left(m^2\right)^{-2 \epsilon } 2^{-2 z-2
\epsilon } \Gamma (-z) \Gamma (z+1) \Gamma (1-\epsilon )
\left(\frac{s}{m^2}\right)^z \Gamma (z+\epsilon ) \Gamma
(z+2 \epsilon )}{\Gamma (z-\epsilon +2) \Gamma
\left(z+\epsilon +\frac{1}{2}\right)}
$$
TeXForm with HoldForm
TeXForm[HoldForm[(2 m^2)^(-2 ϵ) (s/(4 m^2))^
z Pi^(1/2) (Gamma[-z] Gamma[1 + z] Gamma[1 - ϵ] Gamma[
z + ϵ] Gamma[z + 2 ϵ])/(Gamma[
2 + z - ϵ] Gamma[1/2 + z + ϵ])]]
$$
\frac{\left(2 m^2\right)^{-2 \epsilon } \left(\frac{s}{4 m^2}\right)^z \sqrt{\pi } (\Gamma (-z) \Gamma (1+z) \Gamma (1-\epsilon ) \Gamma (z+\epsilon )
\Gamma (z+2 \epsilon ))}{\Gamma (2+z-\epsilon ) \Gamma \left(\frac{1}{2}+z+\epsilon \right)}
$$
To use HoldForm
you'd have to apply the actual expression itself, not the variable expr
in there.