Recently I stumbled upon this question my self and since then it is making me restless to understand/answer happiness or any other emotion in terms of its physicality for example- There are forms to measure Speed, light, time, distance and they also have a standard quantifiable unit that defines them, however we can measure happiness or any other emotion through the means of questionnaire but the major problem that stands out there is - there is no standard set of questions which can answer or measure emotion in its truest glory. For example GRAVITY = 9.8M/S ( standard unit atleast for \U0001f30d Earth)
Also thinking about the question from a different angle makes me ponder into a deeper thought of confusion - for example - Every living creature projects happiness or other basic emotions (anger) in a quite similar way - with happiness they smile, feel good, display positive mood and everything nice then, I belive there has to be a measurement or a standard unit for simple things called emotion.
Also, I'm keen to know what could be the direction to start looking into merging physics, psychology and mathematics to get a easier and sustainable answer for such question.
At this point- I do not have any definitive answer for your question except for mentioning happiness could be measured through different forms of questionnaires also it be measured through MRI scanners - it finds different areas of brain light up when we are happy or display other emotions having said this there is 100% possibility to discover a standard set of quantifiable unit of happiness or any other emotions in that case. Logically it does make sense to have a standard quantifiable unit for emotions.
Anyways - thanks for your question. I will try to update my answer as soon as I have the answer.