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The first variable we will investigate is the angle θA between the two outgoing leptons in the laboratory frame. Some basic algebraic manipulations lead us from Eq. (5.28)

5.4 Dilepton polarization analysis in V → `<sup>+</sup>`⁻ decays where θ is the angle between p and the beam axis (not to be confused with θ_Apreviously defined) and φ is the azimuthal angle of p⁰ with respect to p. The lepton energies are given by E_p² − p² = E_p²0 − p⁰² = m²<sub>`</sub> and p<sup>0</sup> = |~p<sup>0</sup>|(θ<sub>A</sub>, p, M, m`) can be found from the decay kinematics that lead to the following equation:

E_pE_p⁰ − pp⁰cos θ_A= M²

2 − m²_`. (5.35)

In the left panels of Fig. 5.5 we present the results of the dilepton production in the ρ channel as a function of its invariant mass M integrating small bins of

∆ cos θA = 0.2. The different curves displayed in the plot correspond to cos θA ∈ [−0.2, 0], [0, 0.2], [0.2, 0.4], [0.4, 0.6] and [0.6, 0.8]. The upper panel corresponds to the vacuum case with µ₅ = 0 while the lower one represents a parity-breaking medium with µ5 = 300 MeV. The LPB effect produces a secondary peak corresponding to a transverse polarization in the low invariant mass region in addition to the vacuum resonance. This graphic does not imply a larger abundance of transversely polarized mesons with respect to the longitudinal ones but the fact that transverse mesons can be selected (and almost isolated) with a suitable angular coverage. If the effective tem-perature T increased, we would see an enhancement of the upper mass tail making the secondary peak smaller in comparison to the vacuum one. As cos θ_Aincreases, the out-going leptons are more collimated, meaning a higher meson 3-momentum and from the dispersion relation in Eq. (5.27), a larger separation between the two peaks together with a Boltzmann suppression. Note that the highest peaks in the figures with µ₅ 6= 0 have a value ' 0.1, this is a 10% of the total production at the vacuum peak when the total phase space is considered (except for the detector cuts). We took a representative value µ₅ = 300 MeV so as to show two different visible peaks. If the axial chemical potential acquires smaller values, the secondary peak approaches the vacuum one. The smaller µ₅ is taken, the more collinear the leptons have to be in order to obtain a large meson momentum and thus, a visible and significant secondary peak (recall Eq.

(5.27)).

Dealing with quantities smaller than 10% may be tricky. As the main information related to LPB is focused in cos θ_A≈ 1, in the right panels of Fig. 5.5 we show a some-what more experimentally oriented set of plots where we integrated cos θ_Adown to -0.2, 0, 0.2 and 0.4. The upper panel corresponds to vacuum and the lower one represents a parity-breaking medium with µ₅ = 300 MeV. The number of events grows when more

Figure 5.5: The ρ spectral function is presented depending on the invariant mass M in vacuum (µ5 = 0) and in a parity-breaking medium with µ5 = 300 MeV (up-per and lower panels, respectively) for different ranges of the angle between the two outgoing leptons in the laboratory frame θ_A. We display the curves corresponding to cos θ_A ∈ [−0.2, 0], [0, 0.2], [0.2, 0.4], [0.4, 0.6] and [0.6, 0.8] in the left panels, and cos θ_A ≥

−0.2, 0, 0.2, 0.4 in the right ones. The total production at the vacuum peak is normalized to 1 when the entire phase space is considered. Results are presented for the experimental cuts quoted by PHENIX [47].

and more separated leptons are considered but the secondary peak smears and becomes much less significant. Therefore, an optimization process should be performed in every different experiment so as to find the most suitable angular coverage.

At this point, we recover the discussion about introducing a realistic cut in single-electron rapidity. In principle, all accelerator experiments have to account for this cut (independently on the particle type) in order not to accumulate a large number of tracks and energy amounts in the forward direction that would be rather tricky to isolate. However, for the PHENIX experiment there is no clear information about this cut [47] and therefore we decided to omit it. In Fig. 5.6 we show one of the curves already presented in Fig. 5.5 including a single-electron cut |ye| < 0.35 and without it. One may observe that the qualitative behaviour already discussed holds if this cut

5.4 Dilepton polarization analysis in V → `<sup>+</sup>`⁻ decays

is added. In fact, the vacuum peak slightly decreases whereas the secondary peak due to LPB does not substantially change. Indeed, this small distortion does not affect the results already explained regarding the ρ spectral function measured by NA60 or the improvement of the PHENIX cocktail.

Figure 5.6: One of the curves presented in Fig. 5.5 is displayed (solid line) together with its modification when an additional single-electron cut |y_e| < 0.35 is included (dashed line).

No significant changes are observed concerning the significance of the secondary peak that appears due to a non-trivial µ₅.

Another issue that could be experimentally addressed is the analysis of µ₅ for a particular (and fixed) coverage of θA. If the secondary peaks due to the transversally polarized mesons were found, their positions would be an unambiguous measurement of µ₅ (more precisely, |µ₅|). In Fig. 5.7 we integrate the forward direction of the two outgoing leptons, i.e. cos θA ≥ 0, and examine how the transverse polarized peaks move with respect to the vacuum one when we change the values of the axial chemical potential. The ρ and ω spectral functions are displayed in the upper panels for µ₅ = 100, 200 and 300 MeV. The same tendency is found in both graphics except for the fact that both transverse peaks are observed in the ω channel (one below and one above the vacuum resonance). For small values like µ₅ ' 100 MeV the vacuum peak hides the transverse one due to the ρ width, being impossible to discern if one or two resonances are present. In the ω channel all the peaks are visible even for such small µ5. The latter channel could be the most appropriate one in order to search for polarization asymmetry. For completeness, we also present their combination in the lower panel normalized to PHENIX data. The normalized number of events are defined in this case as

N_θ^PH(M ) = dN^PH

dM (M = m_V) R

∆θA

dN

dM d cos θ(M, cos θ) d cos θ R+1

−1 dN

dM d cos θ(M = mV, cos θ) d cos θ, (5.36) where ^dN_dM^PH(M ) is the theoretical spectral function used in the PHENIX hadronic

cocktail. As it was discussed in Section 5.3 an enhancement factor 1.8 is included for the ratio ρ/ω due to the ρ meson regeneration into pions within the HIC fireball [151, 155]

and the in-medium ω suppression [156]. As already mentioned, our conclusions do not depend substantially on the precise value of this ratio as the experimental signal that we propose is amply dominated by the ω when the two resonances are considered together (see. Fig. 5.7).

Figure 5.7: The ρ (upper-left panel) and ω (upper-right panel) spectral functions and their combination (lower panel) are showed depending on the invariant mass M and integrating the forward direction cos θA ≥ 0 for µ5= 100, 200 and 300 MeV. In the upper panels the total production at the vacuum peak is normalized to 1 when the entire phase space is considered whereas the lower panel is normalized to PHENIX data. Results are presented for the experimental cuts quoted by PHENIX [47].