I have a 2008 copy of the ingredients for all of the Papa John's products and I would say that of the majors, PJ has the "cleanest" products. Their flour is a high protein flour that is milled from a Kansas wheat strain that was developed exclusively for them to achieve the characteristic flavor of the PJ crust. No one else has that flour. The flour is unbleached and unbromated (as are the flours used by most of the big chains) although PJ does use ascorbic acid in their flours. I estimate that the protein content is somewhere between 13-14%. Their pizza sauces are based on Stanislaus fresh pack tomatoes, which are generally considered to be among the best in the pizza industry. Their vegetables, such as peppers, onions, mushrooms and tomatoes, are fresh and free of preservatives or other chemicals. Their meats come from companies like Tyson so they will contain the same chemicals and preservatives as their retail level counterparts. Most of the rest of their toppings ingredients are processed foods and, as a result, will contain the usual suspects as additives and preservatives.
I believe the weak link of the PJ pizzas is their cheese. It comes from an industry cheese giant, Leprino's, which also has major accounts with other pizza chains. The basic pizza cheese used by PJ is a low-fat part-skim mozzarella cheese but it includes things like food starch, powdered cellulose and whey protein. All of the major pizza chains, especially those that use commissaries, use cheeses that are not pure cheeses. Shredding quality cheeses at the store level would be a big improvement in my opinion but none of the majors do that.
What I have also discovered is that it is best to buy pizzas from PJs when they are not being slammed. At times like that, they will form skins in advance and put them in racks pending orders. The skins might sit there for quite a long time and become puffy. Also, the workers have a tendency to dock the skins aggressively as they are being formed. This is especially true when they form skins out of dough balls that are cold right out of the cooler. Pizzas made from those skins will not be the best representation of their product. And they won't be their prettiest. The rims will be riddled with holes from the dough dockers.
At least PJs does not use frozen dough, as do Domino's (according to Tom Lehmann) and Pizza Hut. Frozen dough means a lot of dough conditioners and other chemical additives. Their doughs are nowhere as "clean" as the PJ doughs, if that is a matter of concern.