Ajax调用中的cookie

I have this setting:

• html page at https://foo.bar.example.com
• with a cookie for the domain .example.com
• with a javascript doing an ajax get to retrieve a json from https://baz.example.com

specifically, I'm using jquery and doing a $.ajax("https://baz.example.com/.../resource.json")

I expected the ajax call to send the cookie to the server, but nothing is sent. If I add the option xhrFields: { withCredentials: true } to the jquery method I succesfully see the cookie sent, but I was expecting that this was not necessary.

Why is foo.bar.example.com a different domain from baz.example.com? Why isn't the cookie shared? if I do a "manual" get opening the JSON url with the browser from a different tab I see the cookie is sent.