Are work lunches productive?

I like playing word games online that are timed. I noticed that I can come up with a bunch of words before time is up. But if I try doing it while I eat, or even a few minutes afterward, I have zero score. My brain just goes on vacation while I eat and digest.
So it had me wondering. How productive are people when they have work lunch?

Is it the quantity you eat that makes a difference. Or a boring meal vs. a good meal? Or starches vs. salads? Is it merely a distraction, or is because your body is pumping less blood to your brain and more towards digestion? The last one is my guess since I tend to get sleepy after eating. But especially starches makes me sleepy.