After recent discussion with Jacques van de Wiele and Tingting (who discovered in fact some inconsistencies in the boosting procedure) we have realized that general double Lorentz boost is not trivial because it does not result in the Lorentz tranformation only, but also involves Wigner rotation (i.e. see the attached document at the bottom). Therefore all procedures of helicity calculation are affected if boosting is done more than once (i.e. also pseudohelicity analysis if there are boosts, first to the CM system, then to γ*).
The distribution 1 + cos2θ is expected in γ* reference system. In our procedure we have included more boosts (Δ not at rest) distorting the proper angular distribution. Please, read also nice and short explanation of helicity from BaBar experiment, found by Beatrice.
This brings the following questions:
We measure in LAB: proton, e+, e-. We create Δ (= proton + electron + positron or beam + target - proton) in the LAB. We create γ* (= electron + positron) in the LAB.
We boost in the following way:
Sometimes is addition all objects (Δ, γ*, e+, e-) are boosted to CM first, what biases the anisotropy even more!
We boost in the following way:
In this procedure there is only one Lorentz boost (of leptons, and later, of γ*). This is exactly in line with BaBar definition: in the reaction Δ -> γ* -> e+ + e-, the helicity angle of particle e+ / e- is the angle measured in the rest frame of the decaying parent particle, γ*, between the direction of the decay daughter e+ / e-and the direction of the grandparent particle Δ (boosted also to γ *).
In the pictures below "one boost" (usually green) means the correct boosting procedure while "sequence boost" or "two boosts" (usually red) means the old, incorrect procedure.
First, I draw the Δ helicity in the 4pi (PLUTO ver.5.30c simulation). It should indicate what kind of boosting / production is implemented inside the PLUTO generator. Please note, that I do not select the proton coming from the Δ, I assume ambiguity in proton selection (I take the mean of both) - but it changes (the proton coming from Δ selection or not) the anisotropy distribution only very little. Red distribution seems to indicate, because the defined 1 + cos2θ distribution has been reconstructed, that in PLUTO the decaying objects are boosted to the parent rest frame and within the parent reference frame a given distribution is applied. The result is such that if we apply correct boosting procedure we get significantly less anisotropic helicity distribution (compare anisotropy factor 0.97 to 0.48).
To quantify the importance of the various boosting procedures here you can see the opening angle (the second picture) between the electron (positron) when it was boosted once (properly) to γ* and the same electron (positron) when it was boosted (incorrectly) via Δ to γ*. The opening angle distribution spans up to around 30 degrees, therefore it is rather not negligent effect. In addition (the first picture) one can see that the difference becomes larger at more anisotropic range of cos θ.
* TwoLorentzBoosts: Two Lorentz boosts and Wigner rotation
-- WitoldPrzygoda - 16 Dec 2009
I | Attachment | Action | Size | Date | Who | Comment |
---|---|---|---|---|---|---|
![]() | TwoLorentzBoosts | manage | 133.9 K | 15 Dec 2009 - 14:39 | WitoldPrzygoda | Two Lorentz boosts and Wigner rotation |
![]() | helicity_all_boost30_exp-sim.gif | manage | 11.6 K | 16 Dec 2009 - 13:09 | WitoldPrzygoda | Delta helicity (corrected event by event with 30-bin matrix) with a sequence of boosts - experiment |
![]() | helicity_all_boost_exp-sim.gif | manage | 12.3 K | 15 Dec 2009 - 14:05 | WitoldPrzygoda | Delta helicity (corrected) with a sequence of boosts - experiment |
![]() | helicity_all_ok_exp.gif | manage | 10.4 K | 15 Dec 2009 - 14:04 | WitoldPrzygoda | Delta helicity (uncorrected) - experiment |
![]() | helicity_all_ok_sim.gif | manage | 9.2 K | 15 Dec 2009 - 14:02 | WitoldPrzygoda | simulation (uncorrected) - Delta helicity |
![]() | helicity_all_ok_sim_4pi.gif | manage | 10.6 K | 15 Dec 2009 - 14:01 | WitoldPrzygoda | simulation in 4pi |
![]() | helicity_components_all_boost.gif | manage | 16.4 K | 15 Dec 2009 - 17:38 | WitoldPrzygoda | Δ helicity components: sequence boost |
![]() | helicity_components_ok_boost.gif | manage | 10.4 K | 15 Dec 2009 - 17:38 | WitoldPrzygoda | Δ helicity components: one boost |
![]() | helicity_correction.gif | manage | 8.7 K | 15 Dec 2009 - 14:03 | WitoldPrzygoda | acc&eff correction (1-dim) matrix for Delta helicity |
![]() | helicity_correction30.gif | manage | 13.9 K | 16 Dec 2009 - 13:08 | WitoldPrzygoda | Delta helicity correction acc+eff with more (30) bins |
![]() | helicity_ok_boost30_exp-sim.gif | manage | 10.8 K | 16 Dec 2009 - 13:10 | WitoldPrzygoda | Delta helicity (corrected event by event with 30-bin matrix) with one boost - experiment |
![]() | helicity_ok_boost_exp-sim.gif | manage | 11.8 K | 15 Dec 2009 - 14:06 | WitoldPrzygoda | Delta helicity (corrected) with one boost - experiment |
![]() | oa_2dim.gif | manage | 16.3 K | 15 Dec 2009 - 19:29 | WitoldPrzygoda | opening angle (Δpee and Δmiss): lepton ("one boost") and lepton ("two boosts") versus cos θ (helicity in the one boost case) |
![]() | oa_electrons.gif | manage | 8.0 K | 15 Dec 2009 - 17:33 | WitoldPrzygoda | opening angle (Δpee): electron ("one boost") and electron ("two boosts") |
![]() | oa_electrons_miss.gif | manage | 8.8 K | 15 Dec 2009 - 17:34 | WitoldPrzygoda | opening angle (Δmiss): electron ("one boost") and electron ("two boosts") |
![]() | oa_leptons.gif | manage | 9.0 K | 15 Dec 2009 - 19:30 | WitoldPrzygoda | opening angle (Δpee and Δmiss): lepton ("one boost") and lepton ("two boosts") |
![]() | oa_positrons.gif | manage | 8.0 K | 15 Dec 2009 - 17:35 | WitoldPrzygoda | opening angle (Δpee): positron ("one boost") and positron ("two boosts") |
![]() | oa_positrons_miss.gif | manage | 8.4 K | 15 Dec 2009 - 17:36 | WitoldPrzygoda | opening angle (Δmiss): positron ("one boost") and positron ("two boosts") |