source
∝exp
−x2+y2+z2 2R2G
(6.33) results in the distribution for the distance r∗ of the emitters
∝exp − r~∗2 4RG2
!
. (6.34)
The Gaussian shape for the source yields an analytical solution for the theoretical two-particle correlation function
C2th.(k∗) = 1 +λ·X
S
ρS 1
2
f0S(k∗)2
1− dS0 2√
πRG
+2RfS(k∗)
√π F1(2k∗RG)−IfS(k∗)
RG F2(2k∗RG)
,
(6.35)
where F1(ζ) = Rζ
0 dξexp(ξ2 −ζ2)/ζ, F2(ζ) = (1−exp(−ζ2)/ζ), and ρS is the spin density.
Following [294,314,339,340] it is assumed that pairs are produced unpolarized, i. e.ρ0= 1/4 and ρ1 = 3/4 for the singlet S= 0 and triplet S= 1 respectively; the λparameter is introduced to accomodate for a possible non-Gaussian shape of the correlation function by strongly decaying resonances (see Section 6.8.2) and allow for a cross-check of the purity correction procedure performed via Eq. 6.25.
As the pΛ interaction parameters are sufficiently well known (see the systematic evaluation in Section 6.9), Eq. 6.35 directly links the correlation function to the source radius. To facilitate the comparison with other experiments [294, 314, 339, 340], we stick to the commonly used parametrization given in [163].
6.8 Momentum Resolution and Resonances 125
for Monte-Carlo generators, any after-burner [301, 341, 342] operates with weights to account for Coulomb and strong final-state interaction and quantum statistical effects. The standard way [287, 343–346] of studying detector resolution effects on the correlation function thus is to assign each pair a weight according to its generated relative momentum and compare it to the recreated weights as a function of the reconstructed relative momentum. The problem with this method is, that a source radius has to be plugged into the analysis of the Monte-Carlo sample.
The experimental radius from Pb-Pb data can then only be determined in an iterative procedure, since the correction of the data depends on the outcome of the analysis of the corrected data.
An additional, more fundamental problem is that the uncertainty evaluated by the fit procedure can conceptually only be wrong. To retrieve the uncertainty, the fit parameter, i. e. the radius, is varied and compared to the data. However, the momentum-resolution correction on the data is no longer valid for the varied fit parameter. Within this thesis, an improved method was developed that avoids both the iterative procedure and the inaccurate uncertainties by means of a response matrix. It has the additional advantage of a possible parametrization yielding practically infinite statistics for the response matrix whereas the traditional approach severely suffers from the limited statistics available in Monte-Carlo simulations.3
Fig. 6.12 shows the momentum resolution in form of a Monte-Carlo response matrix. The axes represent the diagonal and the perpendicular to it of aqrec vs. qsim graphic. The slightly less intuitive variables (qrec−qgen)/√
2 vs. (qrec+qgen)/2 were decided on since they illustrate more clearly a feature of the matrix, namely the independence of the momentum resolution on q in the studied regionq .0.3 GeV/c. Since bothqrec andqgen can only be bigger than zero, the small region indicated by the black lines in Fig. 6.12 can not be populated. Taking this into account, no variation of the momentum resolution, i. e. the spread in (qrec−qgen)/√
2, can be observed. This allows us to determine aq-integrated momentum resolution correction.
Fig. 6.13 shows the momentum resolution (qrec−qgen)/√
2 integrated over 0.0 ≤(qrec+ qgen)/2 (GeV/c)<0.1 for the three centrality bins investigated within this work. Also shown is a Gaussian fit to each of the datasets, which quantifies the momentum resolution to be about 7 MeV/c in (qrec−qgen)/√
2 for all centrality classes. The centrality dependence shows only a very mild degradation of less than 2% towards more central events.
The pair statistics as a function of (qrec+qgen)/2 were described by a polynomial fit. Taking into account the increase of the number of pairs with increasing q is equally important since
— with a finite momentum resolution — the less correlated pairs at high q will dilute the femtoscopic effect at low q even more when they are more abundant. Fig. 6.14 shows the momentum resolution matricesMqqrecgen generated from the aforementioned parametrization for 0–10, 10–30, and 30–50% most central events. Generating pairs from a flat distribution in (qrec+qgen)/2 and using the pairs statistics as a weight when obtaining the matrices, allows
3The traditional approach is impossible for this pΛ study. Only a centrality andmTintegrated correction factor could have been obtained. The situation is of course better for highly abundant particles like pions.
smearing probability
3
10− 2
10− 1
10−
) )/2 (GeV/c qgen rec+ (q
0 0.05 0.1 0.15 0.2 0.25 0.3
)c (GeV/2)/genq-recq(
−0.3
−0.2
−0.1 0 0.1 0.2 0.3
≥ 0 qgen
≥ 0
rec
q
Figure 6.12: Momentum resolution as obtained from a Hijing Monte-Carlo simulation.
The histogram was normalized such that the sum over one column inydirection equals one. The axes were chosen due to the symmetry in a qrec vs. qsim representation, they are the diagonal and the perpendicular to the diagonal in theqrec vs. qsim histogram.
2 gen)/
-q qrec ( 0.3
− −0.2 −0.1 0 0.1 0.2 0.3
0 10000 20000 30000 40000 50000 60000 centrality
0-10%
gen)/2 +q qrec ( 0.0-0.1 GeV/c
Gauss (MeV) 0.02 -0.32± x
0.02 7.26± σ
2 gen)/
-q qrec ( 0.3
− −0.2 −0.1 0 0.1 0.2 0.3
0 2000 4000 6000 8000 10000
centrality 10-30%
gen)/2 +q qrec ( 0.0-0.1 GeV/c
Gauss (MeV) 0.04 -0.32± x
0.04 7.24± σ
2 gen)/
-q qrec ( 0.3
− −0.2 −0.1 0 0.1 0.2 0.3
0 500 1000 1500 2000 2500
centrality 30-50%
gen)/2 +q qrec ( 0.0-0.1 GeV/c
Gauss (MeV) 0.08 -0.38± x
0.07 7.13± σ
Figure 6.13: Momentum difference distribution integrated over (qrec+qgen)/2 ¡ 0.1 GeV/cfor the three centrality ranges investigated with in this thesis.
to significantly reduce the statistical fluctuations in the low q region and obtain the high granularity shown in Fig. 6.14. The matrices Mqqrecgen allow to properly translate the theoretical C2th.(qgen) of Eq. 6.35 into a correlation as a function of the reconstructed momentumC2th.(qrec) via a simple sum
C2th.(qrec) =X
qgen
MqqrecgenC2th.(qgen) (6.36)
6.8 Momentum Resolution and Resonances 127
and thus allows to directly compare to the measured correlation function. By performing the sum of Eq. 6.36 during the fitting procedure, appropriate uncertainties are obtained. The standard approach of correcting the correlation function [287, 343–346] needs an analysis of the full Monte-Carlo sample to alter the source radius used in the momentum resolution correction;
this has a typical timescale of 100 days of running time. The procedure developed here only needs∼1 second, corresponding to a speed-up factor of about 109.
# pairs
1 10 102 103 104 105 106
) c (GeV/
qgen 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 )c (GeV/recq
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
This Thesis Pb-Pb, 2.76 TeV
Monte Carlo 0-10% central
# pairs
1 10 102 103 104 105
) c (GeV/
qgen 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 )c (GeV/recq
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
This Thesis Pb-Pb, 2.76 TeV
Monte Carlo 10-30% central
# pairs
1 10 102 103 104
) c (GeV/
qgen 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 )c (GeV/recq
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
This Thesis Pb-Pb, 2.76 TeV
Monte Carlo 30-50% central
Figure 6.14: Momentum resolution matrix generated from a parametrization of a Monte-Carlo study for 0–10, 10–30, and 30–50%.
6.8.2 Strongly Decaying Resonances
Aλparameter (see Eq. 6.35) below unity is expected from the contribution of strongly decaying resonances [347, 348]. The source size at the LHC for primary particles, excluding strong decay products, is about 12 fm, see [349] for a set of√sNN-dependent emission functions. The ∆(1232) has a width of about 117 MeV [32] and thus acτ of 1.69 fm. This decay length is roughly 10%
of the just mentioned source size. The protons coming from decays of ∆(1232) resonances will therefore be offset relative to the truly primary protons, which leads to non-Gaussian shape of the source; specifically long non-Gaussian tails develop [350]. Fig. 6.15 shows correlation functions obtained from Fourier transformations of a source with the shape of a Gaussian (left), a Lorentzian (center), and a mix of the two (right). Pairs were sampled according to a q2 functional for the phase-space dependence and the Fourier transformations were used as weights for these pairs. An equivalent for mixed events was formed by generating 15 times more pairs following theq2 dependence but without the weights. In order to avoid a bias, the same random pair distributions were used for the Gaussian, Lorentzian, and mixed source. The correlation functions resulted from the division of the weighted set by the unweighted one; therefore, the correlation functions appropriately display the feature of larger statistical fluctuations for small
relative momentumq. Also shown in the panels of the figure is a Gaussian fit to the correlation functions, where the height and width were left as free parameters. The Gaussian fit nicely describes the data of the Gaussian source with a λ parameter, i. e. the height of the fitted Gaussian, of one. Deviations between the fit and the correlation function of the Lorentzian source are apparent. The fit can not describe the shape of the correlation function; importantly for this study, the λ parameter is lowered to about 0.6. In the case of the admixture of a Lorentzian component to the Gaussian source, the Gaussian fit describes the data; however the height of the Gaussian fit is reduced toλ≈0.9.
(a.u.) q
0 5 10 15 20 25 30 35 40 45 50
2C
1 1.2 1.4 1.6 1.8 2 2.2
(a.u.) q
0 5 10 15 20 25 30 35 40 45 50
2C
1 1.2 1.4 1.6 1.8 2 2.2
(a.u.) q
0 5 10 15 20 25 30 35 40 45 50
2C
1 1.2 1.4 1.6 1.8 2 2.2
Figure 6.15: Correlation function according to a Fourier transformation of a source with the profile of a Gaussian (left), a Lorentzian (center), and a mix of the two (right) sampled with finite statistics mimicking the q2 phase-space dependence of pair
abundances together with a Gaussian fit.
A realistic model which incorporates the effect of resonances is Therminator [301]. It inherited from SHARE [351] its database of more than 300 particles and their decay tables.
Therminator events are generated by performing a statistical hadronization at the freeze-out hypersurface given by a hydrodynamic calculation and tracking the subsequent evolution of the decay cascades. An analysis of these events showed [352] that the pΛ source (including the feed-down from strong decays) is very Gaussian compared to the pion source [287]. Furthermore