Helicity 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 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:

• what is the impact of the wrong procedure on the result, mainly on the correction matrix but also on the uncorrected data
• how the angular distributions are treated in PLUTO (in which reference frame it is 1 + cos²θ) - 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

Please note one more difference in the procedure and I am checking right now the influence. It is not the same (if I boost as it is written in red below)

We boost in the following way:

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

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^*).

All what is below is wrong, please read e-mails I have written.

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 (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:
  

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 (Δ_pee and Δ_miss): lepton ("one boost") and lepton ("two boosts") versus cos θ (helicity in the one boost case):
  

• opening angle (Δ_pee and Δ_miss): lepton ("one boost") and lepton ("two boosts"):
  

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:
  

• acc&eff correction (1-dim) matrix for Delta helicity:
  

• Delta helicity (uncorrected) - experiment:
  

• Delta helicity (corrected) with a sequence of boosts - 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 (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:
  

Helicity angle separately for electron and postrion, the case of Δ_pee and Δ_miss:

• Δ helicity components: sequence boost:
  

• Δ helicity components: one boost:
  

The document about the double Lorentz boosting consequences

* TwoLorentzBoosts: Two Lorentz boosts and Wigner rotation

-- WitoldPrzygoda - 16 Dec 2009