1. Dalian University of Technology, Dalian, China
Multi-pion intensity correlations for the pion-emitting sources with Bose-Einstein condensation
Ghulam Bary
Collaborators: Peng Ru, Wei-Ning Zhang
Dalian University of Technology, Dalian, China

2. Outline
Introduction
B-E condensation of pion gas in harmonic oscillator potentials
Density distribution and multi-pion HBT correlations
The effects of condensation on correlation functions and normalized correlators
Summary and outlook

3. Introduction
HBT interferometry is a tool to study the space & time of particle-emitting sources. Because HBT correlation occurs only for chaotic sources. So, it can be used to probe the source chaotic degree.
Pions are the most copiously produced particles in high-energy collisions. In the heavy-ion collisions at the RHIC and LHC, The detected identical pions are about hundreds and thousands.
PRL106(2011)032301,
arXiv:

4. The high pion event multiplicity may possibly lead to occurrence of the pion condensate in ultra-relativistic heavy-ion collisions.
Recently, the three-pion HBT measurements for the =2.76 TeV Pb-Pb at the LHC (PRC89,2014,024911) indicate that the pion sources are not chaotic completely and with considerable degree of coherence as shown in Fig.
r3 < 2 indicates partially coherent sources

5. Recently, the three and four-pion HBT measurements for the 𝑠 𝑁𝑁 = 2
Recently, the three and four-pion HBT measurements for the 𝑠 𝑁𝑁 = 2.76 TeV Pb-Pb at the LHC (PRC93,054908,2016)also indicate that the pion sources are not chaotic completely and with considerable degree of coherence.
Experimental values of three- and four-pion correlation functions are smaller than the expected values
Why ?
It is our motivation to study the possible pion Bose-Einstein condensation in ultra-relativistic heavy-ion collisions and investigate the effects of the condensation on pion HBT measurements.

6. B-E condensation of pion gas in harmonic oscillator potentials
We consider an expanding pion source, the time-dependent harmonic oscillator potential is given by
V r,t = 1 2 ℏ𝜔(𝑡) 𝑟 𝑎(𝑡) 2
Assuming the system relaxation time is smaller than source evolution time, we may deal the pion gas as a quasi-static adiabatic expansion, and we have
ℏ𝜔 𝑡 = ℏ 2 𝑚 (𝑎(𝑡)) 2
𝑎(𝑡)= ℏ 𝑚𝜔(𝑡)

7. In a canonical ensemble, the total number of bosons
fugacity, which is related to and can be solved as a function of
Introduce the condensation and chaotic fractions
𝑓 0 = 𝑁 0 𝑁 = Ζ 1−Ζ N
𝑓 𝑇 = 𝑁 𝑇 𝑁 =1− 𝑓 0
Condensation fraction f0 decreases with increasing temperature.
At high temperature, f0=0, corresponding to completely uncondensed.
For N=2000, the system has large condensation fraction, which may reach 0.7 at the temperatures MeV.
However, for N=500, the system has only small condensation fraction at low temperature and for smaller source C1=0.35.

8. Density distribution and pion HBT correlations
Using one- and two-body density matrices, we can calculate the pion momentum density distributions
One-body density matrices in momentum space
Density distributions in momentum space
Two-body density matrix in momentum space
In the limit of

9. Two-particle momentum correlation function
In BE correlation measurement, we normalize the probability relative to the probability of detecting particle 𝑝 1 and 𝑝 2 , and define the momentum correlation function C
C 𝑝 1 , 𝑝 2 = 𝐺 𝑝 1 , 𝑝 2 ; 𝑝 1 , 𝑝 2 𝐺 𝑝 1 , 𝑝 1 𝐺 𝑝 2 , 𝑝 2
C 𝑝,𝑞 =C 𝑝 1 , 𝑝 2 =1+ 𝐺 𝑝 1 , 𝑝 −𝑁 𝑢 0 𝑝 𝑢 0 𝑝 𝐺 𝑝 1 , 𝑝 1 𝐺 𝑝 2 , 𝑝 2
This is the general Bose-Einstein correlation for all situations
1: Coherent
2: Chaotic
3: The case between them
Further JPG . 41 (2014)

10. Three-pion correlation function
=1+ 𝐺 𝑝 1 , 𝑝 −𝑁 𝑢 0 𝑝 𝑢 0 𝑝 𝐺 𝑝 1 , 𝑝 1 𝐺 𝑝 2 , 𝑝 2
𝐺 𝑝 2 , 𝑝 −𝑁 𝑢 0 𝑝 𝑢 0 𝑝 𝐺 𝑝 3 , 𝑝 3 𝐺 𝑝 2 , 𝑝 𝐺 𝑝 1 , 𝑝 −𝑁 𝑢 0 𝑝 𝑢 0 𝑝 𝐺 𝑝 3 , 𝑝 3 𝐺 𝑝 1 , 𝑝 𝐺 𝑝 1 , 𝑝 2 𝐺 𝑝 2 , 𝑝 3 𝐺 𝑝 1 , 𝑝 3 −𝑁 𝑢 0 𝑝 𝑢 0 𝑝 𝑢 0 𝑝 𝐺 𝑝 1 , 𝑝 1 𝐺 𝑝 3 , 𝑝 3 𝐺 𝑝 2 , 𝑝 2
TWO-pion correlators

11. Two and three- pion correlators
𝑅 12 = 𝐺 1 𝑝 1 , 𝑝 2 2 −𝑁 0 2 𝑢 0 𝑝 1 2 𝑢 0 𝑝 2 2 𝐺 1 𝑝 1 , 𝑝 1 𝐺 1 𝑝 2 , 𝑝 2 𝑅 23 = 𝐺 1 𝑝 2 , 𝑝 3 2 −𝑁 0 2 𝑢 0 𝑝 2 2 𝑢 0 𝑝 3 2 𝐺 1 𝑝 3 , 𝑝 3 𝐺 1 𝑝 2 , 𝑝 2 𝑅 13 = 𝐺 1 𝑝 1 , 𝑝 3 2 −𝑁 0 2 𝑢 0 𝑝 1 2 𝑢 0 𝑝 3 2 𝐺 1 𝑝 3 , 𝑝 3 𝐺 1 𝑝 1 , 𝑝 1 𝑅 33 =2 𝐺 1 𝑝 1 , 𝑝 2 𝐺 1 𝑝 2 , 𝑝 3 𝐺 1 𝑝 1 , 𝑝 3 −𝑁 0 3 𝑢 0 𝑝 1 2 𝑢 0 𝑝 2 2 𝑢 0 𝑝 3 2 𝐺 1 𝑝 1 , 𝑝 1 𝐺 1 𝑝 3 , 𝑝 3 𝐺 1 𝑝 2 , 𝑝 2 𝐶 3 𝑝 1 , 𝑝 2 , 𝑝 3 =1+ 𝑅 12 + 𝑅 13 + 𝑅 23 + 𝑅 33
Three-pion correlator
Three-pion correlation function

12. Principle of symmetrization
Correlation exist
Coherent
Chaotic
Correlation exist
Chaotic
No correlation
Coherent
Symmetrization
Symmetrization absent
Symmetrization
Principle of symmetrization

13. Chaotic pool
Coherent pool
𝐶 3 =1+ 1 1 1 2 2 2 1 1 1 2 2 2 3 3 3 3 3 3 1 1 1 1 1 1 1 1 2 3 2 3 2 3 2 3 3 2 3 2 3 2 3 2

14. Results
The influence of the pion sources Bose -Einstein condensation in the HBT measurements at various pion number and temperature are investigated
The results indicate that a source with thousands of identical pions may appear to possess as substantial degree of condensation
Large number of particles and low temperature welcome to the coherence in case of small source
The effect of coherence manifests itself more directly in the suppression of Bose-Einstein correlations

15. 𝐶 1 =0.35
𝐶 1 =0.40
This finite condensation may decrease the correlation functions in three-pion interferometry measurements at low pion transverse momenta, but influence only slightly at high pion transverse momentum

16. Comparison with experimental data
𝐶 3 𝑄 3 =1+ 𝑅 12 + 𝑅 13 + 𝑅 23 + 𝑅 33
𝑐 3 𝑄 3 =1+ 𝑅 33
Cumulant 3-pion can be obtained by removing all 2-pion symmetrization terms

17. Normalized three-pion correlator
One may isolate the 3-pion phase factor by normalizing 𝒄 𝟑 with the appropriate 2-pion correlation factors given below
𝑟 3 𝑄 3 = 𝑅 𝑅 12 × 𝑅 23 ×𝑅 31
𝑟 3 𝑝 1 , 𝑝 2 , 𝑝 3 = 𝑐 −1 [𝐶 12 −1]× [𝐶 23 −1] ×[𝐶 13 −1]
𝑟 3 𝑝, 𝑄 3 =0 = 2 𝜖(𝑝) 3−2𝜖(𝑝) 2−𝜖(𝑝) 3/2
The intercept of three-pion correlator is expected be 2 in the case of fully chaotic particle emitting sources and less than 2 in the case of partially coherent sources
ϵ p = 𝜌 𝐶ℎ (𝑝) 𝜌(𝑝)

18. Four-pion correlation functions
Higher order QS correlations contain additional information about the source which cannot be learned from 2- and 3- particle correlations alone
𝐶 4 𝑄 4 =1+ 𝑅 12 + 𝑅 23 +𝑅 34 + 𝑅 41 + 𝑅 13 + 𝑅 24 + 𝑅 12 𝑅 34 + 𝑅 23 𝑅 14 +𝑅 13 𝑅 24 + 𝑅 33 + 𝑅 44
The four-pion correlation function contains four sequences of symmetrization
1: Single-pair
2: Double-pair
3: Triplet
4 :Quadruplet
Multi-pion Bose-Einstein correlations could provide an increased sensitivity to coherent as the expected suppression increase with the order of correlation function

19. 𝐶 1 =0.35
𝐶 1 =0.40

20. Comparison with experimental data
Cumulant 4-pion can be obtained by removing all 2- and 3- pion symmetrization terms

21. Summary and outlook
Recently, the suppression of multi-pion HBT correlation has been observed by ALICE which may indicates the coherence in the pion emitting sources
In this work, we investigated the 3- and 4-pion HBT correlations in the evolving pion gas (EPG) model , in which the pion source is regarded as a gas within harmonic oscillator mean-field and evolving hydrodynamically
we found that with the source coherence, the strengths of multi-pion correlations increase with increasing average momentum of the particles and decrease with increasing particle number of system
The dependence of coherent fraction on pion average momentum is consistent with the experimental multi-pion HBT measurements 
Further investigations based on more realistic source models in ultra-relativistic heavy-ion collisions will be of great interest.

22. Thanks for your attention!

23. Backup

24. The RMSR of the sources for N=2000
The RMSR of the sources for N=2000. Considering that the RMSR in Pb-Pb collisions at LHC to be on the order 10 fm, the value of parameter C1 being between 0.35 and 0.40 appear reasonable.

25. Possible origins of the suppression
CGC
Multi-pion phases
Coulomb repulsion
Multi-pion distortions