5.2. MONTE CARLO SIMULATIONS
all other correlations contained in the model. The CF most closely resembles an experimental correlation function.
Moreover, since all the distributions are calculated in three dimensions in LCMS, following the approach of the experiments [193, 263], we employ a spherical harmonic decomposition of the measured correlation functions, as described in Sec. 4.5.
All the correlation functions have been calculated for seven ranges of pair transverse mo-mentum kT, which were exactly the same as in the analysis of experimental data. The kT inter-vals are introduced in Sec. 6.4.1. An example of all three correlation functions, calculated for two (low and high) kT ranges are shown in Fig. 5.1.
) c (GeV/
q
0 0.5 1 1.5 2
)q(0 0C
1 1.2 1.4
= 5.02 TeV sNN
EPOS 3.076, p-Pb
c < 0.3 GeV/
kT
, 0.2 <
π+
π+
) c (GeV/
q
0 0.5 1 1.5 2
)q(0 2C
-0.04 -0.02 0 0.02 0.04
CF
Full correlation CQS
Pure correlation CB
Background correlation
) c (GeV/
q
0 0.5 1 1.5 2
)q(2 2C
-0.04 -0.02 0 0.02 0.04
) c (GeV/
q
0 0.5 1 1.5 2
)q(0 0C
1 1.5
2 EPOS 3.076, p-Pb sNN = 5.02 TeV c < 1.0 GeV/
kT
, 0.8 <
π+
π+
) c (GeV/
q
0 0.5 1 1.5 2
)q(0 2C
-0.1 0 0.1
CF
Full correlation CQS
Pure correlation CB
Background correlation
) c (GeV/
q
0 0.5 1 1.5 2
)q(2 2C
-0.1 0 0.1
Figure 5.1: First three non-vanishing components of the SH representation of the π+π+ correlation functions from EPOS model for 0.2 < kT< 0.3 GeV/c (left plot) and 0.8 < kT< 1.0 GeV/c (right plot).
5.2.3 Extracting the femtoscopic information
With the three correlation functions calculated we proceed to treat them with an experimental-ist’s recipe. We use the Gaussian and EGE forms of correlation function derived in Sec. 4.4 and employ a fitting procedure to extract the femtoscopic radii. Equations (4.20) and (4.21) for the Gaussian and EGE forms, respectively, are fitted directly to the calculated correlation functions CQS to extract the "true" femtoscopic radii. We use both fitting functions in order to see if a particular shape of the femtoscopic effect influences the background estimation procedure.
In the presence of additional non-femtoscopic correlations, the forms given by Eq. (4.20) or Eq. (4.21) will produce unreliable results. Those effects must be taken into account with additional factors in the fitting equation. Following the discussion in Sec. 5.2.2 such factor should be multiplicative with the QS+FSI effect. A modified fitting function for the Gaussian and the EGE fit case is then
Cf(q)= NCqs(q)Ω(q), (5.1)
where N is the normalization factor andΩ term contains the "non-femtoscopic" effects. Obvi-ously, the exact form ofΩ is not known. Ω will also naturally introduce new fitting parameters.
The main purpose of this chapter is to propose a recipe to obtain a form for Ω. We will then apply this procedure to our model calculation and try to extract the "realistic" femtoscopic radii by fitting Eq. (5.1) to the calculated CF. By comparing these "realistic" radii with the "true" ones obtained from the fit of CQS we will be able to judge the correctness of the procedure to extract Ω as well as the correctness of the fitting process itself. We will also estimate the theoretical systematic uncertainty coming from the presence of the background.
5.2.4 Characterizing the background
In order to propose a function for theΩ term needed in Eq. (5.1) and accounting for the non-femtoscopic effects in the fitting procedure, we need to calculate CB that contains only non-femtoscopic correlations. Examples of the CBcalculated for selected pair kT ranges are shown in Fig. 5.1. The background at low kT is flat at low q, where the femtoscopic effect is most prominent. It shows a rise at q > 1.0 GeV/c due to the momentum conservation in mini-jet mechanism, however this behavior is not relevant for the femtoscopic analysis. At large kTthere is a significant correlation, wide in q, approximately Gaussian in shape, with prominent
5.2. MONTE CARLO SIMULATIONS
contribution to the low q, where the femtoscopic effect is located. Its three-dimensional shape is reflected in the (2,0) and (2,2) SH components. They differ from zero, but not strongly, indicating that the shape is approximately spherically symmetric in LCMS.
Fixing the background with the MC calculation introduces a model dependence in the anal-ysis. Therefore we propose several options for the parametrization of CB, with varying degree of such model dependence. We propose that the background has a Gaussian shape:
Ω00(q)= 1 + a00exp − q2 2(σ00)2
!
, (5.2)
where a00 is a free parameter describing the magnitude of the correlation and σ00 is another free parameter describing its width. In the first scenario, with minimal model dependence, we only fix σ00, separately for each kT range, from the fit to the CB. In the second scenario, we fix both the σ00, as well as a00 for each kT range. In the third scenario we also account for the full three-dimensional shape of the background, with the parametrization of the (2,0) and (2,2) components of the background:
Ω02(q)= a02exp − q2 2(σ02)2
!
+ β02, (5.3)
Ω22(q)= a22exp − q2 2(σ22)2
!
+ γ22q+ β22, (5.4) where a02, σ02, a22, σ22, γ22, β02, and β22are free parameters of the fit to CB. All of them but β02 and β22, which are kept free, are then fixed in the fitting of the full correlation function. The overall fitting formula is therefore of the following form:
Cf(q)= N · Cqs(q) ·hΩ00(q) · Y00(θ, ϕ)+ Ω02(q) · Y20(θ, ϕ)+ Ω22(q) · Y22(θ, ϕ)i , (5.5) where Y00(θ, ϕ), Y20(θ, ϕ), and Y22(θ, ϕ) are the corresponding spherical harmonics.
We stress that this particular functional form has been derived for this particular EPOS MC calculation and is by no means a universal one. Each time such analysis is performed, a new functional form should be proposed, corresponding to the particular background shape observed in data or MC calculation. Nevertheless, the three scenarios proposed represent three rather general cases of background characterization. Scenario 1 (also referred to as "Background 1") corresponds to only constraining the background shape in q, scenario 2 (also referred to as
"Background 2") corresponds to constraining also the background magnitude, while scenario 3 (also referred to as "Background 3") corresponds to fixing the full three-dimensional shape and magnitude of the background.