First look at mass constraint [How simple can it be?] Two decays: i) pi 0 -> g g ii) K 0 -> pi+ pi- - Mother particle momenta: P = 1, 2, 4, 8, 16 GeV - Daughter momentum cut: p1, p2 > 0.5 GeV - Simple ATLAS resolution smearing - Very simple refitting of mass: change p1 and p2 by the same scale (angular resolution not considered) Note that a complete treatment involves relatively complex and slow chi-square fitting!
Plots. 1. Mother particle momentum resolution (r) for truth, reconstructed, and refitted daughters r = ( p meas – p true ) / p true This will hopefully improve after refitting 2. Invariant mass for truth, reconstructed, and refitted daughters. After refitting M should equal Mtruth
i) pi 0 -> g g M 2 = (E 1 +E 2 ) 2 – P 2 =>M 2 = 2 E 1 E 2 ( 1 – cos A ) for massless photons =>M 2 = p 1 p 2 k p=E, and we wont change A “Refit”: simply scaling p1 and p2 by M pi0 /M will constrain the invariant mass to M pi0 Here we assume p 1 p 2 (true-ish given that we have a 0.5 GeV momentum cut) and so we don’t care too much about weighting the scale for different resolutions.
P = 1 GeV Truth Recon Refit
P = 2 GeV Truth Recon Refit
P = 4 GeV Truth Recon Refit
P = 8 GeV Truth Recon Refit
P = 16 GeV Truth Recon Refit
ii) K 0 -> pi+ pi- M 2 = (E 1 +E 2 ) 2 – P 2 =>M 2 = 2 E 1 E 2 ( 1 – cos A ) for massless pions!!! =>M 2 = p 1 p 2 k p=E!!, and we wont change A “Refit”: simply scaling p1 and p2 by M K0 /M will constrain the invariant mass to M K0 Here we assume p 1 p 2 (true-ish given that we have a 0.5 GeV momentum cut) and so we don’t care too much about weighting the scale for different resolutions. BUT we assume massless pions! So this will not work so well.
P = 1 GeV Truth Recon Refit
P = 2 GeV Truth Recon Refit
P = 4 GeV Truth Recon Refit
P = 8 GeV Truth Recon Refit
P = 16 GeV Truth Recon Refit
Conclusion Simple scaling works well for the pi0 because E 1 and E 2 are similar (after energy cuts). Weighting the scale seems to be not very important. - But I didnt compare yet with the standard mass constraint method (Ahmet?) - Try a simple weighting scheme? When we have massive daughters this method is not so good (the invariant mass doesn’t constrain properly). - Try adding the mass effects in a simple way? - Ahmet does the scaling in the c.o.m frame, again without weights This works well!