quantum field theory - Why helicity for massless particles is Lorentz invariant?

By definition helicity is projection of spin onto the 3 momentum.

$$h={\bf J} \cdot {\mathbf{P }}$$ where ${\mathbf{P }}=(P_1,P_2,P_3)$ is the momentum operator and ${\mathbf{J }}=(J_1,J_2,J_3)$ the angular operator.

Now under a Lorentz transformation massless particles transform like this:

$$U(\Lambda)|p,\sigma\rangle=e^{i\theta\sigma}| \Lambda p,\sigma\rangle.$$

As we can see the momentum is changing but the spin not.

Suppose that state $|p,\sigma\rangle$ is a state of helicity $\sigma$ such that we have

$$h|p,\sigma\rangle=J_3P_3|p,\sigma\rangle=\sigma p_3|p,\sigma\rangle$$

But for the state $U(\Lambda)|p,\sigma\rangle=e^{i\theta\sigma}| \Lambda p,\sigma\rangle$, we would have

$$h|\Lambda p,\sigma\rangle=\sigma p'_3e^{i\theta\sigma}| \Lambda p,\sigma\rangle|$$ So for conservation of helicity we would require $p_3=p'_3$ which is not always the case.

So why do people say that helicity is Lorentz invariant?