# EA HBT correlations

The measurement of Bose-Einstein Correlations (BEC) at the EIC could enable us to extract the source size of an excited color string and possibly gain information on the tension in the string. One way to study BECs is to look at correlations in the relative momentum between two identical particles. The correlation coefficient is defined as the ratio of the relative momentum spectrum between, say, two pi0's divided by the same spectrum with any correlations removed. Experimentally, this 'no-correlations' spectrum has to be generated with some care, but here we will be using simulation where we can easily turn the correlations on and off. These correlations have been observed at ZEUS (and other places) and we use some of that work to guide our studies here. For this study we looked at two-pion momentum correlations using the Pythia event generator. It turns out that Pythia version 6.4 comes with the capability to turn on these correlations.

To start, the first figure below shows the correlation function measured at ZEUS from ep events. The data sample used consisted mostly (about 80%) of pion tracks.

Fig 1. Plot from Zeus, arXiv:hep-ex/0311030v2 and Phys.Lett.B583:231-246,2004.
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To test the BEC part of Pythia we used the ZEUS kinematics (see Figure 2) and the average value of the BEC parameters from the ZEUS paper. We extracted the relative momentum from all pairs of positive pions in each event and generated histograms with BEC on and off (left-hand side of Figure 2). We used the Gaussian BEC simulation and the parameters shown in Figure 2. To get the correlation function we simply divided the BEC-on histogram by the BEC-off one. We did not run the simulation for very long, but we see a clear correlation at low relative momentum with a shape qualitatively similar to the ZEUS results. The pythia simulation has a similar, maximum correlation at zero relative momentum to the ZEUS results. The simulation is broader (crosses zero at Δ p = 1 GeV/c instead of 0.4 GeV/c for the ZEUS data), but this is a very simple analysis at this stage.

 . . Fig 2. Comparison of spectra from pythia BEC simulation at ZEUS energies (left panel) and correlation function for ZEUS energies (right panel).

Next, we took the same parameters and used possible EIC momenta for the electron and ion beams and did the same simulation. The results are shown in Figure 3. The correlation at small relative momentum has largely disappeared. This case is at much lower energies than the ZEUS so perhaps there is simply not enough energy to produce the large number of mesons needed to see this effect. This is a bit surprising since Hayk has observed a significant correlation among pion pairs in the EG2 data set at even lower energy transfers (see Figure 4).

 . . Fig 3. Comparison of spectra from pythia BEC simulation at EIC energies (left panel) and correlation function for EIC energies (right panel). Pythia parameters are the same as in Figure 2 (besides the incoming momenta).

Correlations from pi+-pi+ were extracted from the CLAS6 EG2 data and show a significant positive correlation at low relative momentum, but are hampered by the close-track inefficiencies in CLAS6. Figure 4 is from Will.

 . Fig 4. Extraction of correlation function from CLAS6 EG2 data.

Run the same calculations shown in Figs 2-3 for more (20000) events.

 . . Fig 5. Comparison of spectra from pythia BEC simulation at EIC energies (left panel) and correlation function for EIC energies (right panel). Pythia parameters are the same as in Figure 3.
 . . Fig 6. Comparison of spectra from pythia BEC simulation at EIC energies (left panel) and correlation function for EIC energies (right panel). Pythia parameters are the same as in Figure 2.

It's a bit disappointing the correlation in Figures 3 and 5 is so small using pe = 3 GeV/c so crank up the electron momentum to the maximum that will be available at the EIC and get the results in Figure 7. There is clearly a correlation, but only at the 8-10% level. At CLAS6 energies, there is no correlation at all which contradicts Hayk's observations in Fig 4.

 . . Fig 7. Comparison of spectra from pythia BEC simulation at EIC energies with pe= 11 GeV/c (left panel) and correlation function (right panel). Pythia parameters are the same as in Figure 2 except for the electron beam energy.
 . . Fig 8. Comparison of spectra from pythia BEC simulation at EG2 energies with pe= 5 GeV/c (left panel) and correlation function (right panel). Pythia parameters are the same as in Figure 2 except for the electron beam energy.

We next studied the effect of using EIC kinematics and looking at the effect of Q2 and &nu cuts that would limit the kinematic region so it overlaps with EG2. The results are shown in the next two figure. The lower limit on Q2 dramatically reduces the number of counts, but within the statistical uncertainty does not drastically change the correlation function. Most of the events are concentrated at low Q2.

 . Fig 9. Comparison of correlation functions from pythia BEC simulation at EIC energies with no Q2 cut (left panel) and with Q2 > 0.2 GeV2 (right panel).

Using the same kinematics as in the previous figure (EIC kinematics) study the effect of turning up the Failed to parse (unknown error): \lambda

coefficient in the parameterization


of the BEC in pythia. This increases the number of correlated particles in the data sample. In the right-hand panel of Figure 10, there is a spectacular increase in the size of the correlation peak at Q12=0.0 GeV. This peak persists even for low Q2 (Q2<0.2 GeV2) so tuning Failed to parse (unknown error): \lambda

should enable us to reproduce the results from Hayk for the EG2 data set.

 . Fig 10. Study of effect of increasing Failed to parse (unknown error): \lambda on the correlation functions from pythia BEC simulation at EIC energies with no Q2 cut (right panel). The left-hand panel shows the Q12 distributions with the correlations on and off. Same kinematics as Fig 9 with different Failed to parse (unknown error): \lambda  s in the correlation parameterization.

The plan here was originally to tune the BEC parameters (PARJ(91)-PARJ(93) in pythia) to reproduce the shape of the correlation function R from Hayk in Figure 4. At this point, we have not been able to reproduce the width of the distribution in Figure 4, but we can reproduce the value of R at Q12=0. The results are shown in Figure 11 below. The value of Failed to parse (unknown error): \lambda

required is much greater than the value at ZEUS kinematics.

 . Fig 11. Study of effect of increasing Failed to parse (unknown error): \lambda on the correlation functions from pythia BEC simulation at CLAS6 energies with no Q2 cut (right panel). The left-hand panel shows the Q12 distributions with the correlations on and off. Same kinematics as Fig 9 with different Failed to parse (unknown error): \lambda  s in the correlation parameterization.

Next, we change to EIC kinematics to see how things change while keeping the BEC parameters fixed. The result is shown in Figure 12. There is a spectacular increase in the correlation at Q12=0 which is probably not meaningful since this is much larger than any other comparable distribution (see Figure 13). What this does seem to show is that a small correlation at low energies (i.e. CLAS6 and Fig 4) gets amplified at the higher energies (EIC).

 . Fig 12. Study of effect of increasing Failed to parse (unknown error): \lambda on the correlation functions from pythia BEC simulation at EIC energies with no Q2 cut (right panel). The left-hand panel shows the Q12 distributions with the correlations on and off. 

Recent results from the LHC on BEC from PRL 105, 032001 (2010).

 . Fig 13. Recent LHC results on BEC showing the maximum value of the correlation at Q12=0 is only about 1.5 here.

Use the parameters from the LHC results above to see what pythia gives us at EIC energies. We see a smaller, but still significant correlation below.

 . Fig 14. Correlation at EIC kinematics using recent LHC results on BEC.

One of the interesting effects may be to study the size of the QCD string tension using BE correlations. The shape of the source would depend on direction so there would a longitudinal-transverse dependence on the source size. It turns out pythia has options to have asymmteric (transverse vs. longitudinal) momentum distributions for the fragmentation of a gluon. We tested this, but there is little effect on the correlation function. See next figure.

 . Fig 15. Correlation at EIC kinematics using different ratio of transverse/longitudinal momentum distributions.

Did a check on the pythia calculation at EG2 kinematics by setting the FRAME to FIXT (fixed target) in the optoins for PYINIT. In the calculations above I had set the FRAME to 3MOM which is for colliders and set the target momentum to zero for the EG2 kinematics. This happily makes sense.

 . Fig 16. Correlation at EG2 kinematics calculated with pythia and FIXT option in PYINIT.