Difference: DeltaHelicity (r7 vs. r6)

Helicity puzzle definition

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. 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 fact, the helicity calculations are done in the Δ our procedure we have included more boosts (Δ not at rest (therefore only one boost of leptons to γ rest) distorting the proper angular distribution. Please, read also *). 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:

  • how what the angular distributions are treated in PLUTO (in which reference frame it is 1 + cos the impact of the wrong procedure on the result, mainly on the correction matrix but also on the uncorrected data 2θ)
  • what how is the impact of the (possibly) wrong procedure on the result, mainly on the correction matrix but angular distributions are treated in PLUTO (in which reference frame it is 1 + cos also on the uncorrected data 2θ) - possible code fix necessary

The measured objects

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.

Helicity "old" (incorrect) procedure

We boost in the following way:

  • Δ stays in the LAB
  • γ* boosted to the Δ rest frame
  • e+ / e- boosted to the Δ first, then to the γ*
  • helicity: scalar product (cosine of angle) of e+ / e- and γ* is calculated

Sometimes is addition all objects (Δ, γ*, e+, e-) are boosted to CM first, what biases the anisotropy even more!

Helicity "new" (correct) procedure

We boost in the following way:

  • Δ stays in the LAB
  • e+ / e- boosted to the γ* (at the moment γ* in the LAB)
  • γ* boosted to the Δ rest frame
  • helicity: scalar product (cosine of angle) of e+ / e- and γ* is calculated

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.

Investigation and nomenclature

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 (the ( mean of both). Red distribution seems to indicate, because the defined 1 + cos 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).

  • simulation in 4pi:
    simulation in 4pi

Opening angle between electron (positron) after "one boost" and "two boosts" procedure

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.

  • opening angle (Δpee and Δmiss): lepton ("one boost") and lepton ("two boosts") versus cos θ (helicity in the one boost case):
    opening angle (Δ<sub>pee</sub> and Δ<sub>miss</sub>): lepton (
  • opening angle (Δpee and Δmiss): lepton ("one boost") and lepton ("two boosts"):
    opening angle (Δ<sub>pee</sub> and Δ<sub>miss</sub>): lepton (

Below - the influence of the boosting procedure on the simulation analysis, correction matrix, experimental data analysis, and finally, on data correction. Please bear in mind that the correction is consistent only with respect to the method (the same way of boosting) while the generator (PLUTO) input stays the same!

  • simulation (uncorrected) - Delta helicity:
    simulation (uncorrected) - Delta helicity
  • acc&eff correction (1-dim) matrix for Delta helicity:
    acc&eff correction (1-dim) matrix for Delta helicity
  • Delta helicity (uncorrected) - experiment:
    Delta helicity (uncorrected) - experiment
  • Delta helicity (corrected) with a sequence of boosts - experiment:
    Delta helicity (corrected) with a sequence of boosts - experiment
  • Delta helicity (corrected) with one boost - experiment:
    Delta helicity (corrected) with one boost - experiment

The same as above but to limit possible systematic error in correction procedure, matrices with more bins (30 bins) have been produced and the experimental data (in both procedures) corrected event by event, see pictures below:

  • Delta helicity correction acc+eff with more (30) bins:
    Delta helicity correction acc+eff with more (30) bins
  • Delta helicity (corrected event by event with 30-bin matrix) with a sequence of boosts - experiment:
    Delta helicity (corrected event by event with 30-bin matrix) with a sequence of boosts - experiment
  • Delta helicity (corrected event by event with 30-bin matrix) with one boost - experiment:
    Delta helicity (corrected event by event with 30-bin matrix) with one boost - experiment

Helicity angle separately for electron and postrion, the case of Δpee and Δmiss:

  • Δ helicity components: sequence boost:
    Δ helicity components: sequence boost
  • Δ helicity components: one boost:
    Δ helicity components: one boost
  • opening angle (Δpee and Δmiss): lepton ("one boost") and lepton ("two boosts") versus cos θ (helicity in the one boost case):
    opening angle (Δ<sub>pee</sub> and Δ<sub>miss</sub>): lepton (

The document about the double Lorentz boosting consequences

  • opening angle (Δpee and Δmiss): lepton ("one boost") and lepton ("two boosts"):
    opening angle (Δ<sub>pee</sub> and Δ<sub>miss</sub>): lepton (

* TwoLorentzBoosts: Two Lorentz boosts and Wigner rotation

-- WitoldPrzygoda- 15 16 Dec 2009

IAttachmentActionSizeDateWhoComment
TwoLorentzBoostsEXTTwoLorentzBoostsmanage 133.9 K 15 Dec 2009 - 14:39WitoldPrzygoda Two Lorentz boosts and Wigner rotation
helicity_all_boost30_exp-sim.gifgifhelicity_all_boost30_exp-sim.gifmanage 11.6 K 16 Dec 2009 - 13:09WitoldPrzygoda Delta helicity (corrected event by event with 30-bin matrix) with a sequence of boosts - experiment
helicity_all_boost_exp-sim.gifgifhelicity_all_boost_exp-sim.gifmanage 12.3 K 15 Dec 2009 - 14:05WitoldPrzygoda Delta helicity (corrected) with a sequence of boosts - experiment
helicity_all_ok_exp.gifgifhelicity_all_ok_exp.gifmanage 10.4 K 15 Dec 2009 - 14:04WitoldPrzygoda Delta helicity (uncorrected) - experiment
helicity_all_ok_sim.gifgifhelicity_all_ok_sim.gifmanage 9.2 K 15 Dec 2009 - 14:02WitoldPrzygoda simulation (uncorrected) - Delta helicity
helicity_all_ok_sim_4pi.gifgifhelicity_all_ok_sim_4pi.gifmanage 10.6 K 15 Dec 2009 - 14:01WitoldPrzygoda simulation in 4pi
helicity_components_all_boost.gifgifhelicity_components_all_boost.gifmanage 16.4 K 15 Dec 2009 - 17:38WitoldPrzygoda Δ helicity components: sequence boost
helicity_components_ok_boost.gifgifhelicity_components_ok_boost.gifmanage 10.4 K 15 Dec 2009 - 17:38WitoldPrzygoda Δ helicity components: one boost
helicity_correction.gifgifhelicity_correction.gifmanage 8.7 K 15 Dec 2009 - 14:03WitoldPrzygoda acc&eff correction (1-dim) matrix for Delta helicity
helicity_correction30.gifgifhelicity_correction30.gifmanage 13.9 K 16 Dec 2009 - 13:08WitoldPrzygoda Delta helicity correction acc+eff with more (30) bins
helicity_ok_boost30_exp-sim.gifgifhelicity_ok_boost30_exp-sim.gifmanage 10.8 K 16 Dec 2009 - 13:10WitoldPrzygoda Delta helicity (corrected event by event with 30-bin matrix) with one boost - experiment
helicity_ok_boost_exp-sim.gifgifhelicity_ok_boost_exp-sim.gifmanage 11.8 K 15 Dec 2009 - 14:06WitoldPrzygoda Delta helicity (corrected) with one boost - experiment
oa_2dim.gifgifoa_2dim.gifmanage 16.3 K 15 Dec 2009 - 19:29WitoldPrzygoda opening angle (Δpee and Δmiss): lepton ("one boost") and lepton ("two boosts") versus cos θ (helicity in the one boost case)
oa_electrons.gifgifoa_electrons.gifmanage 8.0 K 15 Dec 2009 - 17:33WitoldPrzygoda opening angle (Δpee): electron ("one boost") and electron ("two boosts")
oa_electrons_miss.gifgifoa_electrons_miss.gifmanage 8.8 K 15 Dec 2009 - 17:34WitoldPrzygoda opening angle (Δmiss): electron ("one boost") and electron ("two boosts")
oa_leptons.gifgifoa_leptons.gifmanage 9.0 K 15 Dec 2009 - 19:30WitoldPrzygoda opening angle (Δpee and Δmiss): lepton ("one boost") and lepton ("two boosts")
oa_positrons.gifgifoa_positrons.gifmanage 8.0 K 15 Dec 2009 - 17:35WitoldPrzygoda opening angle (Δpee): positron ("one boost") and positron ("two boosts")
oa_positrons_miss.gifgifoa_positrons_miss.gifmanage 8.4 K 15 Dec 2009 - 17:36WitoldPrzygoda opening angle (Δmiss): positron ("one boost") and positron ("two boosts")

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