Influence of Collective Nuclear Vibrations on Initial State Eccentricities in Pb + Pb Collisions

Abstract

We study within the Monte Carlo Glauber model the influence of collective quantum effects in the Pb nucleus on the azimuthal anisotropy coefficients ${{\epsilon }_{{2,3}}}$ in Pb + Pb collisions at the LHC energies. To account for the quantum effects, we modify the sampling of the nucleon positions by applying suitable filters that guarantee that the colliding nuclei have the mean squared quadrupole and octupole moments consistent with the ones extracted from the experimental quadrupole and octupole strength functions for the Pb nucleus with the help of the energy weighted sum rule. Our Monte Carlo Glauber model with the modified sampling of the nucleon positions leads to ${{\epsilon }_{2}}${2}/${{\epsilon }_{3}}${2} ≈ 0.8 at centrality $\lesssim$1%, which allows to resolve the ${{{v}}_{2}}$-to-${{{v}}_{3}}$ puzzle.

Appendices

APPENDIX

Calculation of the Mean Squared Multipole Moments of the ²⁰⁸Pb Nucleus

For completeness, in this Appendix we briefly review the method of [31] for calculation of the mean squared multipole moments of the ²⁰⁸Pb nucleus with the help of the EWSR [27, 33], and give the ratios between the mean squared multipole moments obtained using the ordinary MC WS sampling of the nucleon positions and those calculated using the EWSR.

We assume that in the ground state the ²⁰⁸Pb nucleus is spherical. We write the nuclear density in the WS form

$${{\rho }_{A}}(r) = \frac{{{{\rho }_{0}}}}{{1 + \exp [(r + {{R}_{A}}){\text{/}}d]}}$$

(A.1)

with R_A = 6.62 fm, and d = 0.546 fm [55, 56]. We define the quadrupole and octupole moments in terms of the spherical harmonics, Y_Lm, with L = 2 and 3. The needed isosinglet L-multipole operator reads (see, e.g. [27, 28, 30])

$${{F}_{L}} = \sum\limits_{i = 1}^A {r_{i}^{L}{{Y}_{{Lm}}}({{{\mathbf{n}}}_{i}})} $$

(A.2)

with n_i = r_i/|r_i|. The mean squared L-multipole moment, 〈$Q_{L}^{2}$〉, of a nucleus in the ground state can be defined quantum mechanically as

$$\langle Q_{L}^{2}\rangle = \langle 0{\text{|}}F_{L}^{ + }{{F}_{L}}{\text{|}}0\rangle .$$

(A.3)

Classical calculation of 〈$Q_{L}^{2}$〉 for the uncorrelated WS nuclear density gives³

$${{\langle Q_{L}^{2}\rangle }_{{WS}}} = {{\langle F_{L}^{ + }{{F}_{L}}\rangle }_{{WS}}} = \frac{{A(2L + 1)\langle {{r}^{{2L}}}\rangle }}{{4\pi }}.$$

(A.4)

Of course, this formula becomes invalid if one includes the effect of the short range hard core NN correlations. But their effect is not very strong (see below). To perform quantum calculation of 〈$Q_{L}^{2}$〉 of the ²⁰⁸Pb nucleus we use the EWSR (for a review, see [33]) for strength function, S(ω) of the operator F_L. It is defined as

$$S(\omega ) = \sum\limits_n^{} {{\text{|}}\langle n{\text{|}}{{F}_{L}}{\text{|}}0\rangle {{{\text{|}}}^{2}}\delta (\omega - {{\omega }_{n}}),} $$

(A.5)

where ω_n = E_n – E₀ and E_n are the nucleus state energies. In terms of moments of the strength function, given by

$${{m}_{k}} = \int\limits_0^\infty {d\omega {{\omega }^{k}}S(\omega ),} $$

(A.6)

we can write 〈0|$F_{L}^{ + }$F_L|0〉 = m₀. It is convenient to rewrite it as

$$\langle 0{\text{|}}F_{L}^{ + }{{F}_{L}}{\text{|}}0\rangle = \frac{{{{m}_{1}}}}{{{{E}_{c}}}},$$

(A.7)

where

$${{E}_{c}} = {{m}_{1}}{\text{/}}{{m}_{0}}$$

(A.8)

is the so called the centroid energy E_c, which can be viewed as the typical excitation energy for the operator F_L acting on the ground state. The representation (A.7) is more convenient than the one via m₀, because the experimental errors in the normalization of the strength function are not important for the ratio m₁/m₀, and the moment m₁ can be exactly calculated with the help of the EWSR, which for L ≥ 2 [27, 30, 33] gives

$${{m}_{1}} = \frac{{AL{{{(2L + 1)}}^{2}}\langle {{r}^{{2L - 2}}}\rangle }}{{8\pi {{m}_{N}}}},$$

(A.9)

where m_N is the nucleon mass. Thus, we have

$${{\langle Q_{L}^{2}\rangle }_{{EWSR}}} = \frac{{AL{{{(2L + 1)}}^{2}}\langle {{r}^{{2L - 2}}}\rangle }}{{8\pi {{m}_{N}}{{E}_{c}}}}.$$

(A.10)

Comparing (A.10) with (A.4), we see that the ratio between the mean squared multipole moments for the ordinary MC sampling of the nucleon positions and that for quantum calculation with the help of the EWSR reads

$${{r}_{L}} = \frac{{{{{\langle Q_{L}^{2}\rangle }}_{{WS}}}}}{{{{{\langle Q_{L}^{2}\rangle }}_{{EWSR}}}}} = \frac{{2{{m}_{N}}{{E}_{c}}\langle {{r}^{{2L}}}\rangle }}{{L(2L + 1)\langle {{r}^{{2L - 2}}}\rangle }}.$$

(A.11)

We calculate the centroid energy using the Breit-Wigner parametrization of the strength function. Since the strength function is proportional to the imaginary part of the polarisability (susceptibility) α (which, as usual, should satisfy the relation α(–ω*) = α*(ω) [57]) for the operator F_L, for each resonance a double Breit-Wigner parametrization with the poles at ±ω_R – iΓ_R/2 (with the same residues) should be used (see Eq. (20) of [32]). For N resonances this gives

$${{E}_{c}} = {{\left[ {\sum\limits_{i = 1}^N {\frac{{2{{f}_{i}}}}{{\pi {{\omega }_{i}}}}} \arctan 2{{\omega }_{i}}{\text{/}}{{\Gamma }_{i}}} \right]}^{{ - 1}}}$$

(A.12)

with f_i the fraction of the ith resonance contribution to the EWSR.

For the isoscalar F₂ operator, the EWSR for the ²⁰⁸Pb nucleus is practically exhausted by the isoscalar giant quadrupole resonance with ω ≈ 10.89 MeV and Γ ≈ 3 MeV [58]. Formula (A.12) with these parameters gives E_c ≈ 11.9 MeV, then from (A.11) one can obtain r₂ ≈ 2.25. Thus, we see that probabilistic treatment of the ²⁰⁸Pb nucleus with the WS nuclear density overestimates the 3D quadrupole fluctuations. It is clear that this can lead to incorrect predictions for the 2D fluctuations of the initial QGP fireball in AA collisions as well. As in [31], our strategy to cure this problem is to modify the MC sampling of the nucleon positions by applying a suitable filter that generates the nuclear configurations with the mean squared quadrupole moment consistent with the EWSR.

To calculate r₃ we need the strength function for F₃. For the ²⁰⁸Pb nucleus, the function S(ω) for the operator F₃ is distributed in a broad range of ω. There are several very narrow peaks in the low-energy region ω $\lesssim$ 7 MeV [59–61], in which the low lying 3^– state with ω ≈ 2.615 MeV exhausts ~20–25% of the EWSR [59–61] and several more states in the region 4.7 MeV $\lesssim$ ω $\lesssim$ 7 MeV (so called the low-energy octupole resonance (LEOR) region) that exhaust about 8–13% of the EWSR [59, 60]. In the high-energy region there is a broad resonance at ω ~ 16–20 MeV with Γ ~ 5–8 MeV [58, 61–65]. The measured EWSR fraction of the high-energy octupole resonance (HEOR) varies from ~20–50% [63, 64] to ~60– 90% [58, 61, 62, 65]. Using the data from [60], that give 21% for the EWSR fraction of the 2.615 MeV 3^– state, and 8.3% for the EWSR fraction of the LEOR region, together with parameters of the HEOR from [58] (ω ≈ 19.6 ± 0.5 MeV, ω ≈ 7.4 ± 0.6 MeV with the EWSR fraction 70 ± 14%) we obtain r₃ ≈ 0.84. However, if we take 25% for the EWSR fraction of the 2.615 MeV state as obtained in [61], and the parameters of the HEOR obtained in [63] (ω = 16 MeV, Γ = 6 MeV), then we obtain r₃ ≈ 0.7. Thus, we see that the experimental data on the octupole strength function of the ²⁰⁸Pb nucleus support that r₃ $\lesssim$ 1. But due to the experimental uncertainties for the octupole strength function, we have uncertainties in the value of r₃ about 15–20%. In the present analysis we perform calculations for two values r₃ = 0.84 and 0.7.

The above values of the coefficients r₂ and r₃ correspond to the MC sampling of the nucleon positions with the uncorrelated WS nuclear density. Calculations using the WS distribution with restrictions on the minimum nucleon–nucleon distances, to mimic the NN hard core, give somewhat di.erent values of r_2,3. However, the effect of the NN hard core on r_2,3 is relatively small: we obtained the reduction of r₂ by the factor 0.78 (0.926), and the reduction of r₃ by the factor 0.81 (0.928) for the core radius r_c = 0.9 (0.6) fm.

In the present analysis we modify the MC sampling of the nucleon positions only with the filters for the isoscalar L = 2 and 3 moments, which correspond to the nuclear shape fluctuations. We do not use a filter for the L = 0 mode, that corresponds to the pure radial fluctuations. The radial fluctuations may be characterized by the squared moment for the monopole isoscalar operator F₀ = $\sum\nolimits_{i = 1}^A {(r_{i}^{2}}$ – 〈r²〉). The EWSR for this operator gives m₁ = 2〈r²〉/m_N [66]. Using this formula, for the uncorrelated WS nuclear density, we obtain for the analogue of (A.11) in the case of the L = 0 mode

$${{r}_{0}} = \frac{{{{m}_{N}}{{E}_{c}}}}{2}\left[ {\frac{{\langle {{r}^{4}}\rangle }}{{\langle {{r}^{2}}\rangle }} - \langle {{r}^{2}}\rangle } \right].$$

(A.13)

For the isoscalar L = 0 mode the EWSR is practically exhausted by the isoscalar giant monopole resonance with ω ≈ 13.6–13.9 MeV and Γ ≈ 3 MeV [58, 67]. These parameters give E_c ≈ 15 MeV, and calculation using (A.13) for the WS distribution (A.1) gives r₀ ~ 1.6. This means that for the MC sampling of the nuclear configurations with the uncorrelated WS nuclear density the magnitude of the pure radial fluctuations is somewhat overpredicted as compared to that extracted from the experimental monopole strength function. However, we have found that adding the filtering for the L = 0 mode, that decreases the mean squared L = 0 moment to its EWSR value, practically does not affect the azimuthal coefficients ${{\epsilon }_{{2,3}}}$. Therefore we do not use filtering for the L = 0 fluctuations.