This question, as old and insane as my science teacher was, will never be answered correctly, because those who want it aren't usually happy and want it while those who have it hide their distrust in those around them that don't in materialistic crap that makes them even more miserable. Take every, or just about, lottery winner. They won all that money and what the hell happened. They became overwhelmed with something that they've never had and destroyed their chances at being comfortable. Now, comfortable you could actually be.. but this is about happiness.. something that only comes in doses. Some so small you don't even realize they're there sometime.
Science might be able to make a test or check brain patterns or the psychological aspects of money after the fact, but it'll never come out the same and you'd have to wear the testing devise for quite some time due to the above mentioned. Too many variables. Also, are you talking about short term or long term happiness. I know if I were given a ton of cash I'd be happy, for a while. Then I'd probably get bored with buying things and having people use me to get what I have and having to keep an eye out for everyone who might steal my cash or give me some sob story to get a few thousand so they wouldn't have to do anything to improve their own situation. In the long run I'd probably do something really good with the money, help out family and friends, and vanish. Maybe making people happy with what I'd have would make me happy in the long run. Maybe people could see through my example that money is worthless, except for who holds the money.
Arthur Hoppe said, "We ought to change the legend on our money from "In God We Trust" to "In Money We Trust." Because, as a nation, we've got far more faith in money these days than we do in God," and this was in 1963! The only thing about the dollar that's changed since then is the value. If only people would stop going to war and fighting over it and actually get together and do away with what has caused so much suffering, envy, and greed in this world, cause it seems most of the problems come from those who have it and don't know what to do with it.