For the heartbroken I've found a report that indicates your "psychological" pain is just as real as if were physical: Pain of Rejection (transcript).

So what we're looking at. This activation this red block here is in the anterior cingulate cortex. And this is basically the same part of the brain that's involved in physical pain. So when we feel distressed or when we feel bothered by some kind of physical pain, that's the same activation that you'll see.

So you've got a broken heart and clearly you already know it's no different than physical pain, but what you may not know is what to do about it.

According to the transcript the brain also has a kind of safety valve that protects the brain against feeling too rejected:

We also found in out study that this area, the right ventral prefrontal cortex which is sort of right in over my eyebrow, we also found activation here. .../... and this area has typically been shown to be associated with regulating people's distress .../... And so people who had more activity in this area actually reported feeling less badly after they were rejected."

So perhaps this safety valve that is the key to mending a broken heart.

And I think this is part of the reason why people tend to write poetry or write in diaries more when they're sad than when they're happy, because there's something about turning on this region of the brain that helps turn off the parts that produce the distress itself.

If you're not a poet/artist/blogger, (or if you've been forbidden from writing on the subject) it also mentions that a drop of alcohol can help ease the pain.