▁, ▁so ▁that ▁a ▁plot ▁of ▁the ▁production ▁rate ▁of ▁the ▁un stable ▁particle ▁as ▁a ▁function ▁of ▁energy ▁traces ▁out ▁the ▁shape ▁of ▁the ▁relativ istic ▁Bre it – W igner ▁distribution . ▁ ▁Note ▁that ▁for ▁values ▁of ▁ ▁off ▁the ▁maximum ▁at ▁ ▁such ▁that ▁, ▁( h ence ▁ ▁for ▁), ▁the ▁distribution ▁ ▁has ▁at ten u ated ▁to ▁half ▁its ▁maximum ▁value , ▁which ▁just ifies ▁the ▁name ▁for ▁ Γ , ▁width ▁at ▁half - maxim um . ▁ ▁In ▁the ▁limit ▁of ▁van ishing ▁width , ▁ Γ   →   0 , ▁the ▁particle ▁becomes ▁stable ▁as ▁the ▁Lor ent z ian ▁distribution ▁ ▁sharp ens ▁infinitely ▁to ▁. ▁ ▁In ▁general , ▁ Γ ▁can ▁also ▁be ▁a ▁function ▁of ▁; ▁this ▁dependence ▁is ▁typically ▁only ▁important ▁when ▁ Γ ▁is ▁not ▁small ▁compared ▁to ▁ ▁and ▁the ▁phase ▁space - dep end ence ▁of ▁the ▁width ▁needs ▁to ▁be ▁taken ▁into ▁account . ▁( For ▁example , ▁in ▁the ▁decay ▁of ▁the ▁r ho ▁mes on ▁into ▁a ▁pair ▁of ▁p ions .) ▁The ▁factor ▁of ▁ 2 ▁that ▁multip lies ▁ Γ 2 ▁should ▁also ▁be ▁replaced ▁with ▁ 2 ▁ ▁( or ▁▁ 4 / 2 , ▁etc .) ▁when ▁the ▁reson ance ▁is ▁wide . ▁ ▁The ▁form ▁of ▁the ▁relativ istic ▁Bre it – W igner ▁distribution ▁ar ises ▁from ▁the ▁propag ator ▁of ▁an ▁un stable ▁particle , ▁which ▁has ▁a ▁denomin ator ▁of ▁the ▁form ▁.