Article 1 Section 8 specifically enumerates Congress with the power to establish the rules of naturalization. That includes:
- who can never be naturalized
- who can be naturalized
- the process of that naturalization
- who is a citizen at birth and does not need to be naturalized (i.e. naturally born citizen)
Congress has expressed its authority in Title 8 section 1401 - Citizen at birth. Under sub section A of that code, anyone born within the jurisdiction of the US (jus soli per the 14th Amendment) is naturally born a citizen. Further under subsection G, Sen Cruz is naturally born a citizen via his mother’s citizenship.