(warning: long and preachy)
I don't have any studies on headphones to offer, but I went and consulted my copy of
Slack by (fairly well-known software/technical management consultant) Tom DeMarco. It is essentially a book about the difference between being
busy and being
productive, in a knowledge/technology workplace. There's nothing directly relevant to music in it, but this section seemed applicable nonetheless to the headphones-at-work and productivity issue:
My own experience consulting inside some highly successful companies (Microsoft, Apple, Hewlett-Packagd, IBM, Dupont, to name a few) cannot corroborate a relationship between busyness and success. Very successful companies have never struck me as particularly busy; in fact, they are, as a group, rather laid-back. Energy is evident in the workplace, but it's not the energy tinged with fear that comes from being slightly behind on everything.
A later chapter notes,
It's the stressed-out organization that elevates the use of pressure to its prominent status. it makes its managers apply far too much. The long-term effect of too much pressure is demotivation, burnout, and loss of key people. The best managers use pressure only rarely and never over extended periods.
Whether this fits your situation or not, I think the underlying problem is that your managers have decided to impose an unpleasant rule on you based on an unproven belief that it will help them meet organizational goals. I'm not necessarily condemning that
per se; managers are pretty much always having to make decisions based on unproven models of the world, it just goes with the territory. But a good manager will be open to adjusting his or her model based on results. If your managers have wrong ideas about what contributes to and detracts from productivity
and are dogmatic about them, then you've got poor managers on your hands.
If you have poor managers, you've got some decisions to make. Does the organization seem to actually encourage the manager's dysfunctions? If so, it's a sign that you may have a dysfunctional organization rather than just a dysfunctional manager. It's worth making the distinction, because a dysfunctional manager has a chance of being replaced by a good organization, but can thrive in a dysfunctional one. I don't know what you can do about a dysfunctional organization except make preparations for departure. (That's not to say it's necessarily best to hurry out the door. I've usually stuck around and focused long enough to do a show-off job finishing some project, and treated that success as my resume for the next job.)
For what it's worth, I've managed technology teams to success over a fair number of years, at Apple, at Sun, and now for the government. If you ask me, once you've selected good talent, and have a decently worthwhile project for them to work on, the thing that has just about the highest correlation to productivity -- way higher than any conjecture about the influence of headphones, for crying out loud -- is confidence in the supportiveness of the organization and its management. People who can do their work without worrying about what their org or their boss is doing while their back is turned get a lot more done. People who feel good rip through work, while people who don't, don't. The corollary is that caring about people's happiness is good business (as well as suiting my personal values anyway). If wearing headphones makes you happy at work, I'm all for it; here, audition some of mine.
But maybe they're not seriously bad managers, just scared or in over their heads or overreacting. Maybe they're open to adjusting their beliefs, or cutting some kind of deal. If I were in your shoes, I'd ask what the real issue in the manager's mind was (presumably productivity, but better to find out than to guess). I'd see if I could get the manager to agree to some kind of deal based on some metric relevant to his/her concerns -- if it's productivity, then, for example, let me guarantee to meet project milestone X with headphones on. If I do, I keep them. If I don't, we try it his way. If all the boss really wants are the milestones and the guarantees, this is a good offer, so maybe they'll take it.
