collision modelling, assumes that any baryon-antibaryon interaction has the same parameters as the pp system, expressed either as a function of relative momentum or pair energy at the center of mass.
4.7.2 Measuring two-baryon interactions
The idea of measuring unknown baryon-baryon interaction potentials is the following. At first, we perform the analysis of those systems where the interaction is known, such as proton-proton, and extract the system size. In the second step, assuming no significant changes of the femto-scopic radii between different baryon systems with similar mT, we move to other baryons and fix the system size. This means that in Eq. (4.6) the source function S is fixed and the parame-ters ofΨ are the ones which are kept free. If the described procedure succeeds, the parameters of the strong interaction in a considered baryon system will be obtained.
4.7.3 Lednicky & Lyuboshitz analytical model
The procedure described in Sec. 4.7.2 requires assumptions for the two-baryon interaction.
From the general form ofΨ given by Eq. (4.18), in the case when particles interact only with the strong FSI, we obtain:
Ψ−ks(+)∗(r∗, k∗)= eik∗r∗ + fs(k∗)eik∗r∗
r∗ , (4.24)
where fs(k∗) =
1 fs
0 + 12d0sk∗2− ik∗
, f0s is the scattering length, d0s is the effective radius, and s iterates over different spin states (singlet, triplet). The complex f0sand d0sare the main variables of interest, since they can used to calculate the interaction cross section:
σ = 4π| f (k∗)|2. (4.25)
We focus here on the baryon-antibaryon case. For these systems the parametrization of Ψ, called the Lednicky & Lyuboshitz analytical model, is provided in Ref. [281]. In the model the following form of the correlation function is derived:
C(k∗)= 1+X
s
ρs
" 1 2
fs(k∗) R
2
1 − d0s 2√
πR
!
+ 2< fs(k∗)
√πR F1(2k∗R) − = fs(k∗)
R F2(2k∗R)
#
, (4.26)
where F1(z) = R0zdxe(x2− z2)/z, F2(z) = (1 − ez)/z, and ρs corresponds to pair spin fractions.
Summation over spin orientations is neglected7, fsignlet(k∗)= ftriplet(k∗)= f (k∗), and the effective
7It is not possible to measure the spin dependence in the current experiments.
4.7. BEYOND THE SYSTEM SIZE – BARYON FEMTOSCOPY
radius is usually set to, d0 = 0. Parameter R corresponds to the size of the spherically symmetric source in PRF:
S(r∗)= exp −r∗2 4R2
!
. (4.27)
The imaginary part of the scattering length in Eq. (4.26) accounts for the baryon-antibaryon annihilation process. The correlation functions calculated from the Lednicky & Lyuboshitz model with non-zero and zero imaginary components of the scattering lengths are shown in Fig. 4.7 and Fig. 4.8, resepectively. When the imaginary part of the scattering length is set to zero, the anticorrelation width is limited to about 100 MeV. On the other hand, when we take into account the annihilation by introducing a non-zero imaginary component, a wide (few hundreds of MeV in k∗) anticorrelation appears. The observation of a similar structure in the experimental data would be an evidence for the presence of significant annihilation process between baryons in nature.
The current status and the preliminary results of the analysis of femtoscopic correlations of protons with antilambdas in ALICE are presented in Chapter 9.
Figure 4.7: The correlation function derived from the Lednicky & Lyuboshitz analytical model with the non-zero imaginary component of the scattering length, which implies that annihilation is taken into account. A clear wide anticorrela-tion extending up to hundreds of MeV in k∗is ob-served.
Figure 4.8: The correlation function derived from the Lednicky & Lyuboshitz analytical model with the imaginary component of the scattering length set to 0, which implies that annihilation is switched off. The anticorrelation effect is limited in k∗.
Non-femtoscopic correlations and fitting pro-cedure
This chapter introduces additional correlation sources (other than quantum statistics and final-state interactions) that are manifested in the femtoscopic correlation functions measured by experiments, especially in small systems like pp or p–A. These correlations are not part of the "standard" femtoscopic formalism but must be accounted for in order to extract reliable experimental results. Therefore, a robust fitting procedure must include in some way also these type of correlations
These "non-femtoscopic" correlations have been extensively studied using EPOS 3.076 Monte Carlo model [146, 147], which includes such additional correlations as well as allows to extract the femtoscopic information. In this chapter we propose a robust method which allows to take them into account in the procedure of extracting the femtoscopic radii.
We stress that all the studies, figures, and results presented in this chapter have been pub-lished in Ref. [1]. This publication have been prepared primarily by the author of this thesis, in collaboration with the supervisor and other co-authors. We also note that the text of this chapter, with slight modifications, is taken from the above reference.
5.1 Non-femtoscopic correlations
In the ideal case, as introduced in Chapter 4, the femtoscopic correlation rests on a flat base-line, reflecting the lack of other two-particle correlations. Such scenario is indeed realized for example for heavy-ion collisions, where all other correlations are either small or have a q scale vastly different than the femtoscopic effect. This is not the case for collisions in small systems, where a relatively small number of particles is produced. Measurements done by various
exper-5.1. NON-FEMTOSCOPIC CORRELATIONS
iments [193, 203, 229–231, 235–237, 263] show that significant additional correlation sources are contributing to the two particle correlation function. We will later collectively refer to such effects as non-femtoscopic background, to differentiate it from the femtoscopic signal, which we are primarily interested in extracting. These correlations have a magnitude and width in q comparable to the femtoscopic signal, and therefore the two cannot be easily disentangled.
The sources of such correlations are, among others, the energy-momentum conservation and the "mini-jet" phenomena [193].
Two main approaches have been taken by experiments to deal with the non-femtoscopic cor-relations, both relying on the modelling of the background by Monte Carlo (MC) models. The first is to construct a "double-ratio", where the experimental correlation function is divided by a corresponding one from the MC calculation. This technique relies on the fact that the particle production process in MC does not take into account the Bose-Einstein enhancement, but it does include other sources of correlation. The application of the "double-ratio" technique should be equivalent to "dividing out" the non-femtoscopic effects and leave a pure Bose-Einstein signal.
The second technique is to parametrize the background in MC calculation and then use it as an additional term in the fitting function applied to the experimental correlation function. We note that both approaches are, in perfect conditions (small size of q bins in the correlation function, large statistics both for data and MC, etc.), mathematically equivalent. However, the method with the additional term in the fitting function offers greater flexibility, which is needed for this work.
In this chapter we perform a methodological verification of the procedures used to account for the non-femtoscopic background. Using the EPOS 3.076 model [146, 147], we calculate the three-dimensional correlation functions in the LCMS frame [282, 283], where the pair mo-mentum along the beam vanishes. The correlation functions are calculated with (1) pure Bose-Einstein signal, (2) with the background effects only, and (3) with both correlation sources combined. We extract the source size from the "pure" correlations functions, and compare them with the ones extracted from the "full" calculation, where the background is constrained using the "only background" calculation. We propose several methods to parametrize the background.
We estimate the systematic uncertainty coming from their application and discuss their validity and stability. We conclude by selecting the method which is most reliable and introduces the smallest uncertainty in the procedure. This procedure is employed in the analysis of collision data in Chapter 7.