Treatment for Puffiness Under Right Eye?

In your case, this is likely swelling if it occured after dental work. Bags under the lower eyelids are usually caused by displaced fat, redundant muscle, redundant skin, or sometimes even a prominent vascular plexus giving the aesthetics of lower lid bags or dark circles.
Early volume loss in your upper cheek and tear trough area can exacerbate even a small amount of genetic predisposition to having eye bags. There are numerous, good options to address this issue ranging from nonsurgical to surgical. Which treatment option is best will depend on the underlying cause of the bags
It is important to realize that it is displacement of fat, NOT just excess fat that can cause lower lid bags. Old school blepharoplasty approach took skin and fat out of the lower lids by cutting right through the skin, which gave long term poor results with hollowed out lower lids and more potential for surgical complications. 
We know that we tend to lose fat as we age in our face, so taking more out as part of a surgical procedure may look good immediately, but it is a long term bad idea. Newer surgical approaches to lower lid blepharoplasty do not cut through the skin and leave no external scars, this is called the transconjunctival approach. More important yet, the existing fat is repositioned instead of removed, giving much better long term results with a more conservative surgical technique that also has lower potential for complications. 
Nonsurgical options include strategically placed volume to replace early loss, which can have a dramatic impact on the appearance of eye bags and the junction between your eyelids and cheek. Volume can come in the form of fillers or as micro fat grafting. 
Laser resurfacing can also address these areas when skin changes are the prominent player.To ensure you are receiving the highest level of care, seek out a modernly trained, new-school dermatologic surgeon, oculoplastic surgeon, facial plastic surgeon or plastic surgeon who is board certified and fellowship trained in one of these “core four” cosmetic specialties. Membership in organizations like the American Academy of Facial Plastic and Reconstructive Surgery help to identify a highly trained surgeon. Cameron Chesnut #realself500 Physician 

The face forms from two separate halves. This leaves us with two very different halves and is responsible for many of our facial asymmetries.

Looking at your picture, you have separation of the cheek fat pad on the right. As we age the fat pad begins to separate and fall creating a deeper fold between the cheek and the mouth. Notice the extra skin on the right cheek/mouth area compared to your left side. You appear to be aging more on the right side. Most of us will age differently from one side to the other. You may sleep on this side, which may be accelerating the aging process, that is the extra lines and puffiness.

 I would recommend a hyaluronic acid based filler to soften the separation and help smooth the cheek. The development of the crease most likely had nothing to do with the dental procedure.

You may get the best results from a combination of treatments. Fillers can be used to help improve symmetry and disguise contour irregularities. Other treatments such as Morpheus8 can be used to stimulate collagen production and improve skin tone and texture.

The under eye area is one of the best places to treat with fillers.  Much like you would apply makeup to conceal the shadows, I would place fillers under the skin to remove the indentation causing the 'u' shaped line. 

Restylane, Perlane, and Radiesse may be place both under the eyes and in the cheeks to treat puffiness and dark circles under the eyes.

Treatments are maintained at 6 month intervals.

Good luck in your search for information!

There are many effective surgical and non-surgical treatments for excess skin and fat of the lower eyelids depending upon your particular complaints. It is not uncommon for patients to begin to notice something that has possibly been there for years after undergoing a "facial" procedure - even one that just creates swelling such as routine dental work.
In looking at your attached photo, it appears that you have a dark area beneath both the lower lids and the right side does appear slightly more noticeable. However, you are unequally lit in the picture with your left side being more exposed to the sun than the right. This could easily produce differential shadowing.
The best way to get a definitive answer to your problem would be to make an appointment with a plastic surgeon in your area to have your eyes fully evaluated. You appear young in the photos, but
due to genetics or environmental factors, some women are appropriate candidates for eyelid surgery even in their 20s.Tightening these areas can result in less of a shadow along the lower lid "U" and soften your appearance.
Good luck.