Hi Your Ground loop is really not a good . and not recommended .
Not to Say It won't work
You would be better making the transform into a Center tap arrangement and use just 2 diodes . from a centre tap with a common
ground direct to the centre tap will give a full wave rectification . Also it will increase affiance .
using LM317 LM337 . The large cap replace with 2 x 480uf . The 1uf at the input of the LMs 0.1uf
also this arrangement with a 15volt transform will work better on full load centre taped
As someone else mentioned there are dozens of designs on line that are proven .
As your circuit stands It will suffer . Theory and Practice don't always work well in real life.
Have fun
![Popcorn :popcorn:](https://www.eevblog.com/forum/Smileys/default/EatingPopcorn.jpg.pagespeed.ce.yJ6TmC5fSa.jpg)