Effects of finite coverage on global polarization observables in heavy ion collisions

In non-central relativistic heavy ion collisions, the created matter possesses a large initial orbital angular momentum. Particles produced in the collisions could be polarized globally in the direction of the orbital angular momentum due to spin-orbit coupling. Recently, the STAR experiment has presented polarization signals for $\Lambda$ hyperons and possible spin alignment signals for $\phi$ mesons. Here we discuss the effects of finite coverage on these observables. The results from a multi-phase transport and a toy model both indicate that a pseudorapidity coverage narrower than $|\eta|<\sim 1$ will generate a larger value for the extracted $\phi$-meson $\rho_{00}$ parameter; thus a finite coverage can lead to an artificial deviation of $\rho_{00}$ from 1/3. We also show that a finite $\eta$ and $p_T$ coverage affect the extracted $p_H$ parameter for $\Lambda$ hyperons when the real $p_H$ value is non-zero. Therefore proper corrections are necessary to reliably quantify the global polarization with experimental observables.

Introduction. The dense matter created in non-central relativistic heavy ion collisions processes a large orbital angular momentum. It has been predicted that the orbital angular moment may result in a net-polarization of produced particles along the direction of the initial angular moment due to the spin-orbit coupling [1,2]. Recently, the RHIC-STAR collaboration claimed the discovery of a global Λ polarization in heavy ion collisions [3]. The estimated vorticity is much larger than all other known fluids, which suggests that the quark-gluon plasma is the most vortical fluid produced in the laboratory. Meanwhile the measured spin alignment parameter ρ 00 of φ mesons is systematically larger than 1/3 [4,5]. A deviation of ρ 00 from 1/3 indicates a spin alignment of the vector mesons, and whether ρ 00 is greater or smaller than 1/3 depends on the hadronization mechanism [2]. More detailed measurements of polarization observables, e.g., as a function of rapidity, pseudorapidity (η), or transverse momentum (p T ), will be important to better understand the novel polarization phenomena.
The angular distribution of the Λ decay products with respect to the orbital angular momentum of system L can be written as [6] dN d cos where θ ⋆ is the angle between L and the momentum of the daughter proton in the rest frame of the parent Λ hyperon, and α H is the decay parameter α Λ = -α Λ = 0.642 ± 0.013 [7]. Therefore the global polarization parameter p H for Λ can be calculated by The angle brackets above represent the averaging over all Λ decays. Assuming a perfect detector acceptance, one can also write [8] where φ ⋆ p is the azimuth of the daughter proton momentum vector in the Λ rest frame and φ L is the azimuth of the system angular momentum.
For vector mesons such as the φ meson, the polarization information including the spin alignment is described by a spin density matrix ρ. A deviation of the diagonal element ρ m,m (m = −1, 0, 1) from 1/3 is the signal of spin alignment. For a diagonal matrix ρ, since the elements ρ −1,−1 and ρ 1,1 are degenerate, ρ 0,0 is the only independent element [9]. It can be extracted from the following angular distribution of the decay products of vector mesons with respect to L [10]: So far lots of theoretical progresses have been made on the evolution of vorticity and polarization [11][12][13][14]. Since the experimental coverage of the phase space is always finite, in this paper we study the effects of a finite η and p T coverage on the measured global polarization parameter p H of Λ hyperons and spin alignment parameter ρ 00 of φ mesons.
The simulation method. We have modified a multiphase transport (AMPT) model [15] for this study. The string melting version of the AMPT model [16] used here consists of a fluctuating initial condition, an elastic parton cascade, a quark coalescence model for hadronization, and a hadron cascade. We use the same model parameters as those used for the top RHIC energy in an earlier study [17]. In the hadron cascade of AMPT, φ mesons and Λ hyperons are assumed to be uniformly polarized according to Eqs.(1) and (4) when they decay. The included channels of polarized decays are φ → K +K (branching ratio ∼ 83%) and Λ → p + π − and n + π 0 (branching ratio ∼ 100%). While experimentally the direction of the angular momentum is often estimated by the normal of the reconstructed event plane [8,9], we directly calculate the initial angular momentum vector of the participant nucleons event-by-event in the AMPT calculations. All results in this study are for minimum bias (impact parameter from 0 to 15.6 fm) Au+Au collisions.
We follow the experimental procedure to extract the ρ 00 parameter of φ mesons: pairs of KK from each event are used to reconstruct the φ-meson candidates, while pairs from different events are constructed to estimate the background; these same event pairs and mixed event pairs are both divided into multiple cos θ ⋆ bins. We then extract the ρ 00 parameter by using Eq. (4) to fit the | cos θ ⋆ | distribution of the φ-meson signal, i.e., the difference between the same event and mixed event pairs. For Λ hyperons, we found that the systematic uncertainty of background subtraction (up to a fraction of a percent) is too large when the input p H is small or zero; it is also not straightforward to apply topological cuts to decays from the AMPT model as done in the experimental analysis. Therefore we use the Λ decay information from the AMPT model directly to extract the p H parameter. We have found that hadronic rescatterings have negligible effects on the value of the extracted p H parameter for Λ hyperons or the ρ 00 parameter for φ mesons; this is expected due to their long lifetimes before decay. We have also used the φ decay information directly to extract the ρ 00 parameter from the AMPT model and obtained consistent values as those extracted from the background subtraction method.
When there are no phase space cuts on particles (i.e. when we have full coverage), we have found that the ex-tracted ρ 00 or p H value is consistent with the input value as expected. However, the experimental acceptance is always limited. The detected η range for finial state particles is typically from ±0.5 to ±2, e.g., the STAR experiment covers |η| < 1; and the p T of detected particle tracks is typically larger than 0.1 -0.2 GeV/c. Not only are the candidate decay daughters for φ and Λ reconstructions within a certain η and p T coverage, but additional y (or η) and p T cuts are also often applied to the reconstructed parent particles (φ and Λ). Therefore we need to investigate whether and how the phasespace cuts may influence the extracted value of the global polarization/spin-alignment parameters. Note that the vorticity in a nuclear collision depends on the transverse position and pseudorapidity in principle [14], thus the real polarization may depend on the phase space variables such as η and p T ; but that is outside the scope of this study.
Results. Figure 1 shows the AMPT model results for the extracted ρ 00 of φ mesons as a function of the upper limit of the |η| coverage, η max , for Au+Au collisions at √ s N N = 200 GeV. Results are shown for two input ρ 00 values, 0.2 and 1/3, where we study the η-cut effects by applying the cut |η| < η max to both kaons and φ mesons (open circles), only kaons (triangles), or only φ mesons (stars). We see that, when the |η| cut is only applied to the parent φ mesons, the extracted ρ 00 has almost no deviation from the input value for any η cut. On the other hand, the extracted ρ 00 strongly depends on the η cut applied to the decay daughter candidates, where an η coverage narrower than ∼ 1 gives a significantly larger extracted ρ 00 than the input value. This is because, with the angular momentum L in the transverse plane, a narrow η cut on kaons tends to exclude some kaons along the beam directions and thus excludes those φ-meson decays with daughter kaons around θ ⋆ ∼ 90 • . Note that such loss of decay daughters close to the beam direction due to finite acceptance and its effect on the hyperon global polarization parameter p H have been pointed out earlier for Lambda decays [8]. Also shown in Fig. 1 are the results where both the η cuts and the STAR p T cuts (p K T > 0.1 GeV/c and 0.4 < p φ T < 3 GeV/c) are applied to the kaons and φ mesons (filled circles), where we see that the p T cuts lead to a small reduction of the extracted ρ 00 values. We also see in Fig.1 that the extracted ρ 00 value converges to the input value at large η max for all the considered cuts.
To further illustrate the η-cut effect on ρ 00 , we also use a toy model, where we sample the p T , η and azimuth distributions of φ mesons according to the AMPT results and decay the hadrons with PYTHIA [18]. As shown in Fig.2a, the effects of η cuts from the toy model are essentially the same as those from the AMPT model. Figure 2b shows the two-dimensional distribution (in cos θ ⋆ and p T ) of kaons within |η| < 0.3 from φ decays. We can clearly observe that the η cut excludes more kaons from φ decays around θ ⋆ ∼ 90 • , i.e., around cos θ ⋆ ∼ 0. The figure also demonstrates that the η-cut mostly affects the low p T region. Furthermore, a different η-cut effect is observed for η max roughly between 1.3 and 4 from both AMPT and the toy model, where the extracted ρ 00 can be smaller than the input value. This could be the result of the finite p T and finite anisotropic flow of the parent φ mesons, which complicate the relation between the η cut (in the center-of-mass frame) and the θ ⋆ distribution of the decay daughters (in the φ rest frame). As a test, we show in Fig.3 the effect of η cuts on φ mesons at rest from the toy model (circles), where the extracted ρ 00 monotonically decreases with η max and finally approaches the input value (1/3 here). The solid line in Fig.3 represents the analytical result on the η-cut effect for φ mesons at rest, as shown in the appendix, which agrees with the toy model result.
For Λ hyperons, Fig.4 shows the η-cut effect on the p H parameter when Eq.(3) is used to extract p H , as often done in the experimental measurements [3,8]. Figures  4a and 4b show the toy model results for the input value p H = 0 and p H = 0.01, respectively; while Fig.4c shows the AMPT results for the input value p H = 0.2. In the toy model, we sampled Λ hyperons using p T , η and azimuth distributions that have been fitted to the AMPT results, where the azimuth distribution includes the elliptic flow. Note that the case p H = 0 represents no real global polarization, while the case p H =0.01 roughly corresponds to the STAR data [3]. When the input p H is zero, we see that the extracted p H values are consistent with zero, with or without the phase space cuts. This is in line with the expectation from the up-down symmetry with respect to the direction of L. However, when the input p H is finite (and positive), the effects of η cuts on p H for Λ hyperons are similar as those on ρ 00 for φ mesons: a finite |η| cut applied to the decay-daughter candidates generates an extracted p H value larger than the input value, but a finite |η| cut applied to parent Λ hyperons has little effect on the extracted p H . Also, the extracted p H value approaches the input value for η max greater than about 2 (when there are no p T cuts).
We also see from Figs.4b and 4c that p T cuts can lead to deviations of the extracted p H from the input value, even when there is no η cut. This is quite different from the φ-meson case, where the extracted ρ 00 value is not affected by the p T cuts when there is no η cut. Note that the p T cuts in Fig.4 are the STAR cuts: p daughter T > 0. 15 GeV/c and 0.4 < p Λ T < 3 GeV/c. Furthermore, we see that the effects of the phase-space cuts from the AMPT model for input p H = 0.2 in Fig.4c are qualitatively the same as those from the toy model for input p H = 0.01 in Fig.4b.
We have analytically derived the η-cut effect on the extracted p H value for Λ decays at rest (see the appendix), which is shown as lines in Fig.5 for the input value p H = 0.01 when the p H value is calculated with Eq.(2) (solid) or Eq.(3) (dashed). Symbols represent the corresponding results from the toy model, which agree with the analytical results. We also see that the η-cut effect is smaller when using Eq.(3) instead of Eq.(2).
Discussions and summary.
While the global spin alignment and polarization signals can be respectively described by Eq.(1) and Eq.(4), phase-space cuts such as the η and p T cuts discussed here modify the functional forms. We show in Fig.6 examples after the cut |η| < 0.4 is applied, where the cos θ ⋆ distribution for φ mesons (when plotted versus cos 2 θ ⋆ ) and the cos θ ⋆ distribution for Λ hyperons deviate from a straight line with a large χ 2 per degree of freedom. This method can be used in the experimental analysis to probe the effect of phasespace cuts, e.g., by selecting a narrow η range within the experimental coverage and observing a deviation from the expected straight line.
A natural question is whether the experimentally observed signals of spin alignment and polarization could be mostly due to the phase-space cuts. Figure 7 shows the results on the extracted ρ 00 for φ mesons from AMPT and the toy model for the case of no spin alignment (i.e. input ρ 00 = 1/3) after the STAR η and p T cuts are applied. We see that the extracted ρ 00 parameters are systematically higher than 1/3 and have a weak energy dependence, similar to the STAR preliminary measurements [4,5]. This indicates that the effect of the phase-space cuts may be a dominant contribution to the deviation of the current experimental data of ρ 00 from 1/3.
For Λ hyperons, we show in Fig.8     The STAR data [3] on the p H values of Λ andΛ are shown for comparisons, where the effects of phase-space cuts have already been corrected [3,8].
In summary, we have used a modified AMPT model that includes the decays of polarized φ mesons and Λ hyperons as well as a toy model to study the effects of phase-space cuts on the extracted polarization parame-ters ρ 00 and p H . We find that a finite η coverage narrower than η <∼ 1 leads to a larger extracted value of ρ 00 than the input value, and a similar behavior is observed for the extracted p H when the input p H is positive. The narrower the η coverage, the larger the extracted values. A finite p T coverage also affects the extracted values of ρ 00 and p H . When we assume no global polarization by setting the input ρ 00 to 1/3, the model results after the STAR η and p T cuts give extracted ρ 00 values of φ mesons that significantly deviate from the value 1/3. These extracted ρ 00 values are similar to the preliminary experimental data, suggesting that the phase-space cuts may be a dominant contribution to the observed higherthan-1/3 values. For Λ hyperons, we find that, when assuming no polarization, a finite coverage in η and/or p T does not lead to a non-zero extracted p H ; however η and p T cuts affect the magnitudes of the Λ polarization parameter p H when the input p H is non-zero. Our study thus shows that measured polarization observables need to be corrected for the effects of finite acceptance before quantitative conclusions on the global polarization can be reliably made.