# Coulomb Corrections for Bose–Einstein Correlations from One- and Three-Dimensional Lévy-Type Source Functions

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Two-Particle Correlation Functions

#### 1.2. Lévy Sources

## 2. Methodology

#### 2.1. Coulomb Interaction

- We could assume that the ${R}_{\sigma \nu}^{2}$ matrix, and thus the whole source function, is the same in the PCMS and the LCMS frames. This is essentially an approximation where $\overrightarrow{K}\approx 0$. However, this is a rather strong approximation, and one of the goals of HBT measurements is indeed to explore the average momentum (or transverse mass) dependence of the parameters that describe the source.
- There are two objects, one in the PCMS (the wave function) and the other in the LCMS (the source function). We could try to transform the wave function from the PCMS to the LCMS and then use the simple form of the source function and obtain the result in LCMS coordinates. However, the two-particle wave function of Equation (17) is not a relativistic expression; thus, we refrained from trying to come up with the right transformation of this object.
- The third option was to evaluate the integral in the PCMS, as the two-particle Coulomb wave function is only known in the PCMS. This meant that the Lévy source had to be transformed from the LCMS to the PCMS.

#### 2.2. Numerical Simulations

## 3. Results

#### 3.1. Three-Dimensional Calculations

#### 3.2. Spherical (One-Dimensional) HBT Measurements

- Simply use ${C}_{2}^{\left(C\right)}({q}_{\mathrm{LCMS}},{R}_{\mathrm{LCMS}})$, which means that one formally substitutes ${q}_{\mathrm{LCMS}}={q}_{\mathrm{inv}}$ and ${R}_{\mathrm{PCMS}}={R}_{\mathrm{LCMS}}$.
- Take into account the fact that ${q}_{inv}\ne {q}_{\mathrm{LCMS}}$ but neglect the same for the scale parameters, and use the weighting method of Equation (33); however, implement this not for the Coulomb correction, but for the correlation function instead. Thus, use ${C}_{2,\mathrm{weighted}}({q}_{\mathrm{LCMS}},{R}_{\mathrm{LCMS}})$ for the fitting:$${C}_{2,\mathrm{weighted}}({q}_{\mathrm{LCMS}},{R}_{\mathrm{LCMS}})=\frac{\int A({q}_{\mathrm{LCMS}},{q}_{\mathrm{inv}}){C}_{2}({q}_{\mathrm{inv}},{R}_{\mathrm{LCMS}})\mathrm{d}{q}_{\mathrm{inv}}}{\int A({q}_{\mathrm{LCMS}},{q}_{\mathrm{inv}})\mathrm{d}{q}_{\mathrm{inv}}}.$$
- Following the same approach as above, use ${R}_{\mathrm{LCMS}}$ for the Coulomb correction and use a weighted average, though for the Coulomb correction this time. This approach is more sensible if one considers Figure 1, where we saw that the correlation functions could look rather different even if in Figure 2 the Coulomb corrections looked very much the same. Now, one uses ${K}_{\mathrm{weighted}}({q}_{\mathrm{LCMS}},{R}_{\mathrm{LCMS}})\xb7{C}_{2}^{\left(0\right)}({q}_{\mathrm{LCMS}},{R}_{\mathrm{LCMS}})$ for fitting.
- One improvement to the methods mentioned above would be to consider the transformation of scale parameters; thus, use the average as in Equation (32). The simpler version is the same as no. 3 above, i.e., weighing the correlation function and using ${C}_{2,\mathrm{weighted}}({q}_{\mathrm{LCMS}},{R}_{\mathrm{PCMS}})$ for fitting. Here, however, one loses the explicit form of ${C}_{2}^{\left(0\right)}$ in the LCMS, which is known.
- The most sophisticated option would be to use ${R}_{\mathrm{PCMS}}$ only for the Coulomb correction and use the weighting of Equation (33). The function used for fitting is now ${K}_{\mathrm{weighted}}({q}_{\mathrm{LCMS}},{R}_{\mathrm{PCMS}})\xb7{C}_{2}^{\left(0\right)}({q}_{\mathrm{LCMS}},{R}_{\mathrm{LCMS}})$.
- Finally, an approach that is easier to implement than the previous methods making use of a distribution $A({q}_{\mathrm{LCMS}},{q}_{\mathrm{inv}})$ is to make an approximation for the ${q}_{\mathrm{LCMS}}$-${q}_{\mathrm{inv}}$ relationship that is appropriate for the Coulomb correction. One could be motivated by the left-hand plot of Figure 3, as the one-dimensional Coulomb correction with ${R}_{\mathrm{PCMS}}$ and the angle-averaged three-dimensional calculation were in relatively good agreement. The relationship ${q}_{\mathrm{inv}}=\sqrt{1-{\beta}_{T}^{2}/3}{q}_{\mathrm{LCMS}}$ could be used, as it would hold for the diagonal line ${q}_{\mathrm{out}}={q}_{\mathrm{side}}={q}_{\mathrm{long}}$. Therefore, the function that could be used for fitting would be $K(\sqrt{1-{\beta}_{T}^{2}/3}{q}_{\mathrm{LCMS}},{R}_{\mathrm{PCMS}})\xb7{C}_{2}^{\left(0\right)}({q}_{\mathrm{LCMS}},{R}_{\mathrm{LCMS}})$.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Csörgö, T. Particle Interferometry from 40 Mev to 40 TeV; NATO Science Series C: Mathematical and Physical Sciences; Springer: Dordrecht, The Netherlands, 2000; Volume 554, Chapter 8; pp. 203–257. [Google Scholar] [CrossRef] [Green Version]
- Bolz, J.; Ornik, U.; Plumer, M.; Schlei, B.; Weiner, R. Resonance decays and partial coherence in Bose-Einstein correlations. Phys. Rev. D
**1993**, D47, 3860–3870. [Google Scholar] [CrossRef] [PubMed] - Adamczyk, L.; Adkins, J.K.; Agakishiev, G.; Aggarwal, M.M.; Ahammed, Z.; Alekseev, I.; Alford, J.; Anson, C.D.; Aparin, A.; Arkhipkin, D.; et al. Beam-energy-dependent two-pion interferometry and the freeze-out eccentricity of pions measured in heavy ion collisions at the STAR detector. Phys. Rev. C
**2015**, C92, 014904. [Google Scholar] [CrossRef] [Green Version] - Adare, A.; Aidala, C.; Ajitanand, N.N.; Akiba, Y.; Akimoto, R.; Alexander, J.; Alfred, M.; Al-Ta’ani, H.; Angerami, A.; Aoki, K.; et al. Lévy-stable two-pion Bose-Einstein correlations in $\sqrt{{s}_{NN}}$ = 200 GeV Au+Au collisions. Phys. Rev. C
**2018**, C97, 064911. [Google Scholar] [CrossRef] [Green Version] - Adler, S.S.; Afanasiev, S.; Aidala, C.; Ajitanand, N.N.; Akiba, Y.; Alexander, J.; Amirikas, R.; Aphecetche, L.; Aronson, S.H.; Averbeck, R.; et al. Evidence for a long-range component in the pion emission source in Au + Au collisions at $\sqrt{{s}_{NN}}$ = 200 GeV. Phys. Rev. Lett.
**2007**, 98, 132301. [Google Scholar] [CrossRef] [Green Version] - Afanasiev, S.; Aidala, C.; Ajitanand, N.N.; Akiba, Y.; Alexander, J.; Al-Jamel, A.; Aoki, K.; Aphecetche, L.; Armendariz, R.; Aronson, S.H.; et al. Source breakup dynamics in Au+Au Collisions at $\sqrt{{s}_{NN}}$ = 200 GeV via three-dimensional two-pion source imaging. Phys. Rev. Lett.
**2008**, 100, 232301. [Google Scholar] [CrossRef] [Green Version] - Shapoval, V.M.; Sinyukov, Y.M.; Karpenko, I.A. Emission source functions in heavy ion collisions. Phys. Rev. C
**2013**, 88, 064904. [Google Scholar] [CrossRef] [Green Version] - Adhikary, H.; Adrich, P.; Allison, K.K.; Amin, N.; Andronov, E.V.; Antićić, T.; Arsene, I.-C.; Bajda, M.; Balkova, Y.; Baszczyk, M.; et al. Measurements of Two-pion HBT Correlations in Be+Be Collisions at 150A GeV/c Beam Momentum, at the NA61/SHINE Experiment at CERN. arXiv
**2023**, arXiv:2302.04593. [Google Scholar] - Tumasyan, A.; Wolfgang, A.; Andrejkovic, J.W.; Bergauer, T.; Chatterjee, S.; Damanakis, K.; Dragicevic, M.; Escalante Del Valle, A.; Hussain, P.S.; Manfred, J.; et al. Two-particle Bose-Einstein correlations and their Lévy parameters in PbPb collisions at $\sqrt{{s}_{NN}}$ = 5.02 TeV. arXiv
**2023**, arXiv:2306.11574. [Google Scholar] - Kurgyis, B. Three dimensional Lévy HBT results from PHENIX. Acta Phys. Polon. Supp.
**2019**, 12, 477. [Google Scholar] [CrossRef] [Green Version] - Sinyukov, Y.; Lednicky, R.; Akkelin, S.V.; Pluta, J.; Erazmus, B. Coulomb corrections for interferometry analysis of expanding hadron systems. Phys. Lett.
**1998**, B432, 248–257. [Google Scholar] [CrossRef] [Green Version] - Bowler, M.G. Coulomb corrections to Bose-Einstein correlations have been greatly exaggerated. Phys. Lett. B
**1991**, B270, 69–74. [Google Scholar] [CrossRef] - Csanád, M.; Lökös, S.; Nagy, M. Expanded empirical formula for Coulomb final state interaction in the presence of Lévy sources. Phys. Part. Nucl.
**2020**, 51, 238–242. [Google Scholar] [CrossRef] - Csörgo, T.; Hegyi, S.; Zajc, W.A. Bose-Einstein correlations for Levy stable source distributions. Eur. Phys. J.
**2004**, C36, 67–78. [Google Scholar] [CrossRef] - Pratt, S.; Csörgo, T.; Zimányi, J. Detailed predictions for two pion correlations in ultrarelativistic heavy ion collisions. Phys. Rev. C
**1990**, C42, 2646–2652. [Google Scholar] [CrossRef] - Bertsch, G.; Gong, M.; Tohyama, M. Pion Interferometry in Ultrarelativistic Heavy Ion Collisions. Phys. Rev. C
**1988**, C37, 1896–1900. [Google Scholar] [CrossRef] [Green Version] - Landau, L.D.; Lifshitz, L.M. Quantum Mechanics Non-Relativistic Theory, 3rd ed.Pergamon Press: Oxford, UK, 1977; Volume 3. [Google Scholar] [CrossRef]
- Metropolis, N.; Rosenbluth, A.W.; Rosenbluth, M.N.; Teller, A.H.; Teller, E. Equation of state calculations by fast computing machines. J. Chem. Phys.
**1953**, 21, 1087–1092. [Google Scholar] [CrossRef] [Green Version] - Hastings, W.K. Monte Carlo Sampling Methods Using Markov Chains and Their Applications. Biometrika
**1970**, 57, 97–109. [Google Scholar] [CrossRef] - Lökös, S. Probing the QCD Phase Diagram with HBT Femtoscopy. Acta Phys. Polon. Supp.
**2022**, 15, 30. [Google Scholar] [CrossRef] - Mukherjee, A. Kaon femtoscopy with Lévy-stable sources from $\sqrt{{s}_{NN}}$ = 200 GeV Au + Au collisions at RHIC. Universe
**2023**, 9, 300. [Google Scholar] [CrossRef] - Porfy, B. Femtoscopic correlation measurement with symmetric Lévy-type source at NA61/SHINE. Universe
**2023**, 9, 298. [Google Scholar] [CrossRef] - Kórodi, B. Centrality dependent Lévy HBT analysis in $\sqrt{{s}_{NN}}$ = 5.02 TeV PbPb collisions at CMS. Universe
**2023**, 9, 318. [Google Scholar] [CrossRef] - Kovács, L. Charged kaon femtoscopy with Lévy sources in $\sqrt{{s}_{NN}}$ = 200 GeV Au+Au collisions at PHENIX. Submitted to Universe. 2023. [Google Scholar]

**Figure 1.**On the left-hand side, the two-particle correlation functions are shown in a spherical case for the three-dimensional calculation in comparison with one-dimensional calculations in the presence of Coulomb interactions in final-state interactions. On the right-hand side, a non-spherical three-dimensional calculation is shown alongside one-dimensional calculations with quadratic and arithmetic mean scale parameters.

**Figure 2.**The Coulomb corrections and the Coulomb-corrected three-dimensional two-particle correlation function is shown in two non-spherical cases.

**Figure 3.**The Coulomb corrections and the Coulomb-corrected three-dimensional two-particle correlation function are shown in the LCMS when the source was spherical in the LCMS but not for the calculation. On the left-hand side, we took the three-dimensional Coulomb correction along a diagonal line, and on the right-hand side along the ${q}_{\mathrm{out}}$ axis.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kurgyis, B.; Kincses, D.; Nagy, M.; Csanád, M.
Coulomb Corrections for Bose–Einstein Correlations from One- and Three-Dimensional Lévy-Type Source Functions. *Universe* **2023**, *9*, 328.
https://doi.org/10.3390/universe9070328

**AMA Style**

Kurgyis B, Kincses D, Nagy M, Csanád M.
Coulomb Corrections for Bose–Einstein Correlations from One- and Three-Dimensional Lévy-Type Source Functions. *Universe*. 2023; 9(7):328.
https://doi.org/10.3390/universe9070328

**Chicago/Turabian Style**

Kurgyis, Bálint, Dániel Kincses, Márton Nagy, and Máté Csanád.
2023. "Coulomb Corrections for Bose–Einstein Correlations from One- and Three-Dimensional Lévy-Type Source Functions" *Universe* 9, no. 7: 328.
https://doi.org/10.3390/universe9070328