In this blog entry I summarize the data analysis results and phenomenological investigations related to two-pion interferometry with Levy-stable sources at low center-of-mass collision energies.

Preliminary figures in pdf:

Example correlation function fit at 3.9 GeV
lambda(mT) at 3.9 GeV
alpha(mT) at 3.9 GeV
R(mT) at 3.9 GeV
lambda(mT) at 3.2 GeV
alpha(mT) at 3.2 GeV
R(mT) at 3.2 GeV
alpha0 vs sqrt(sNN)

Previous PWG presentations:
May 2nd, 2024 (preliminary request)
April 25th, 2024
April 18th, 2024
April 11th, 2024

Below further details follow about various aspects.

Phenomenological studies with UrQMD

I conducted a detailed study of low energy heavy-ion collision events generated with UrQMD.
For the 3.2 GeV results, I used the files I found at "/gpfs01/star/pwg_tasks/simulations/UrQMD/3p2GeV/40M_cascade/".
For the 3.9 GeV results, I used the files at "/star/u/yghuang/urqmd_pwg_task/3p9GeV_cascade/10M_cascade/".
I am not sure though what were the initial settings of these simulations, as I could not find the scripts that were used to generate these files.
It seems that the simulation stopped around 50 fm/c, but it would be great if someone could confirm this (EDIT: this was confirmed.).

I followed the method described in Entropy 24 (2022) 3, 308 and Phys.Lett.B 847 (2023) 138295.
The steps of the analysis are the following:

1. Using the freeze-out coordinates, create the angle-averaged one-dimensional relative-distance distribution of pion pairs in the LCMS frame (also called pair source-function).
   This is done for the 0-10% centrality selection, using a midrapidity selection |eta| < 1, in different average transverse momentum kT bins, corresponding to the ones used in the experimental data analysis.
    
2. As the multiplicity is much lower than that of the high-energy collisions used in the referenced papers, the single-event fitting is not feasible.
   However, to still be able to say something about the event-by-event distribution of the source parameters, I applied averages of 20 events, thousands of times, for 20 different events each time. 
   Each of these 20 events averaged source functions are then fitted with a Levy-stable distribution, in the [2,40] fm range, which already captures nicely the power-law tail. -> see example fit
   
   Figure 1 - example fit to a reconstructed D(r) source function

   
    

3. These type of fits are repeated thousands of times, and the 2-dimensional distribution of the resulting fit parameters (alpha, R) are measured. -> see example (alpha,R) distribution

   Figure 2 - example (alpha, R) distribution from the source function fits in a given kT range
   
   
   The mean and standard deviation of the (alpha,R) distributions are extracted for each kT bin. -> see figure
   The mean is plotted with markers, and the standard deviation is plotted on top with a colored band:

   Figure 3 - mT dependence of the mean and std.deviaton of the source parameters 

   

Observations:

• R shows a similar decreasing trend with both kT and beam energy as the usual experimental results.
  The magnitude of R is also close to the experimental measurements.
• Alpha also shows very similar trends with kT and beam energy dependence as the experimental observations. 
  It depends very little on kT, and increases towards lower energies. 
  The magnitude of alpha is very close to the experimental measurements.
• Results for positive and negative charged pairs are compatible.
  For 3.2 GeV results, R is slightly lower for positive pairs.

This source function fitting method is different from the usual method of calculating correlation functions;
however, it is superior to it as when calculating correlation functions from a source the bin-by-bin (q-by-q) statistical uncertainties will be very correlated.
Hence fitting such a reconstructed correlation function via the usual chisqure or log-likelihood methods is not ideal.
Nevertheless, as a cross-check, I also analyzed correlation functions, calculated from the measured source distributions.

I used a numerical integral calculation method, as described in Phys.Rev.C 102 (2020) 6, 064912.
Instead of assuming an angle averaged Levy distribution in the integral calculation, I plugged in the measured D(r) histograms.
As the UrQMD simulation stopped at 50 fm/c, this will not be the 'true' correlation function, as the integral is 'cut short'.
However, one can still fit these calculated correlation functions with the formula used in the experimental fits.
An example fit is shown here:

Figure 4 - Example reconstructed correlation function, using the C(Q) = integral D(r)*|Psi|^2 dr equation, for the same kT class as Figure 1 & 2:

This analysis is much more time-consuming than the source-function analysis (hence for now, I only did it for 3.9 GeV).
Due to the limited available range in pair-separation, and the fully correlated nature of the uncertainties, it is not reliable.
However, the results are still very close to those from the source-function analysis, which is reassuring.

Figure 5 - Source parameters extracted from fitting reconstructed correlation functions are shown with blue x markers on top of Figure 3

Note finally that in neither of the above described methods can the lambda parameter reliably be measured:

• When fitting the pair-source distribution D(r), lambda is related to the integral of the source, and is hence by definition unity if one normalizes D(r) to have an integral of 1.
  The only way to measure lambda would be to determine the fraction of primordial pions (including pions from very short lived resonances such as rho, Delta or K*),
  but that is not necessarily the same as the experimental lambda - not mentioning the fact that in the investigated UrQMD simulation particles with large lifetimes (K0S, eta, Lambda, Sigma, etc) are not decayed.
  With the simulation stopping at 50 fm/c, there is essentially no halo contribution.
• When fitting the correlation function, lambda in principle could be extracted, but this also turns out not to be comparable to the experimental lambda, because of two issues.
  (1) The non-decayed nature of resonances with a lifetime above 50 fm/c makes the fraction of primordial pions unrealistically large (basically only the omega is decayed out of the halo contributors, and probably the phi).
  (2) Because of the perfect pair resolution in pure UrQMD, the correlation function would still go to two, i.e., C(q->0)->2, and one would have to determine lambda by not fitting the small q portion of C(q).
  However, the correlation functions visibly have no extra bump at small q, rendering the usual core-halo interpretation of the intercept parameter meaningless.

Experimental data analysis

Physics message:

The trend in the parameters as a function of beam energy continues towards the lower energies, no obvious deviation, no non-monotonic trend is observed yet. The change of the mean alpha with collision energy implies that the shape is important to consider as well when analyzing the change of source size with energy.
The UrQMD vs data comparison at the two FXT energies implies, that the alpha parameter is explained by rescattering processes, no additional effect is needed to reproduce the results.
This is in contrast with high energies (200 GeV), where there is an obvious disagreement between the measurement and the result of a similar source function analysis.
A comprehensive UrQMD energy scan would be interesting, to compare with the alpha0 vs sqrt(sNN) preliminary results; this is foreseeable in the near future.

Figure 6 - Example correlation function fit at 3.9 GeV to experimental data:


Figure 7 - Source parameters vs average transverse mass (mT):

Figure 8 - alpha0 vs center-of-mass collision energy:

Compatibility check with existing preliminary results:

As my 1D measurement is done in the LCMS frame, the R parameter is comparable with the corresponding average 3D radii. 
I compared my results when the Levy exponent alpha is fixed to 2 to 3D Gaussian preliminary results by Youquan Qi, and found that they are compatible: