Unarmed strikes normally do damage equal to (1 + Str mod).
If I punch someone and have a -2 Strength modifier, will I technically heal the target by 1 HP?
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The general rules for unarmed strikes (unless overridden by a feat or racial/class/subclass feature) state:
Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons). On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength modifier. You are proficient with your unarmed strikes.
The attack roll for an unarmed strike is (d20 + Str + proficiency bonus).
The damage is (1 + Str mod).
If you have a negative Str mod, as in your case, the result of that calculation may even be negative. However, the rules on damage rolls clarify:
With a penalty, it is possible to deal 0 damage, but never negative damage.
Thus, the minimum possible damage on an attack is 0; you can never deal "negative damage". Any damage calculation that would result in a negative value (after applying all relevant modifiers) actually results in 0 damage.