4. Results 119
4.2. Construction of pp references
4.2.1. pp reference for p
s= 2.76 TeV
With the analyzed data sample, the cross section measurement of ppp
s= 2.76 TeV is limited by statistics topT <32GeV/c, with considerable statistical uncertainties of about 30% at the largest pT. For the calculation of the nuclear modification factorRAA this would limit the pT reach and introduce large statistical uncertainties at high pT. Recall, that in Pb–Pb collisions the pT reach is up to 50 GeV/c. In order to show RAA also up to 50 GeV/c, a pp reference spectrum is needed also forpT >32GeV/c.
To extend the pp reference up to 50GeV/c, the measured spectrum has been parameterized by the so-called modified Hagedorn function [190]
d2σ
dηd pT =A· pT mT
1+ pT
p0 −n
(4.2)
forpT>5 GeV/c. Here, themTis the transverse massmT=p
m2+pT2andm=140 MeV/c, corresponding to the pion mass, is used.
At low pT the modified Hagedorn function behaves like an exponential in pT, while at high pT it has a power law behavior. For the pT range used in the fit (>5 GeV/c), the spectrum exhibits an approximate power law behavior. In this case the exponential part of the modified Hagedorn function acts as a correction that improves the fit at the lower end of thepTrange.
In the range0.15 < pT <5 GeV/c the measured differential cross section is used as the pp reference while for 5 < pT < 50 GeV/c the parameterization is used. Above 32 GeV/c in pT this parameterization is actually an extrapolation. The fit range of the parameterization has been varied to pT > 3 GeV/c and pT > 7 GeV/c to estimate the additional systematic uncertainty related to this procedure. Systematic and statistical uncertainties from the spectra are propagated to the parameterization/extrapolation.
Figure 4.7 shows the measured cross section for η
< 0.8at p
s = 2.76 TeV together with the modified Hagedorn parameterization of the high pT part and the extrapolation up to 50
128 4. Results
GeV/cin pT. The bottom panel of Figure 4.7 shows the ratio fit/data that demonstrates the quality of the fit. Systematic uncertainties of both, the measurement and the parameteriza-tion/extrapolation are also shown.
The totalpT-dependent systematic uncertainty of the pp reference spectrum ranges from 6.4%
at low pT up to 19% at pT = 50 GeV/c. In addition, the normalization to inelastic events comes with a relative uncertainty of 1.9%, fully correlated between thepTbins.
Considering that the reference is not measured in the fullpT range and based partially on an parameterization/extrapolation it is desirable to check the consistency with other approaches.
Figure 4.8 shows the ratio of alternative references to the reference described above. The alter-native references shown are: a measurement by CMS [196], the results atp
s= 7 TeV scaled to 2.76 TeV using the ratio obtained from NLO calculations, simulations with PYTHIA8 [179]
and the interpolation between yields as used for the first publication ofRAA [172]. System-atic uncertainties of the reference are shown without the overall normalization uncertainty of 1.9%.
The CMS collaboration has measured thepT distribution of charged particles in pp collisions atp
s= 2.76 TeV [196], though in a slightly wider pseudorapidity range of η
<1. It agrees within the overall uncertainty (additional systematic uncertainty of the CMS reference is 6%).
For pT <6 GeV/cthe two measurements are in excellent agreement, while for larger pT the CMS data is below the ALICE reference.
As discussed in the previous section, the NLO calculations are able to describe the evolution of the differential cross section with p
s. This allows to scale the differential cross section measured atp
s = 7 TeV, by the the ratio from NLO calculations to obtain the cross section at ps= 2.76 TeV:
d2σ dηd pT
ps=2.76TeV
=
d2σ dηd pT
ps=7TeV
| {z }
measurement
· d2σ
dηd pT
NLO,p
s=2.76TeV
d2σ dηd pT
NLO,p s=7TeV
| {z }
ratio from NLO calculation
(4.3)
The pT interval of3< pT <50 GeV/cfor the scaled reference is the maximal overlap of the NLO calculation (starting from 3GeV/cinpT) and the 7 TeV measurement (up to 50GeV/cin pT). The reference obtained with this approach agrees well with the actual pp reference over the fullpT range.
As described in the previous section PYTHIA8 is able to describe the shape of thepTspectrum atp
s= 2.76 TeV very well forpT>1 GeV/c, but with a normalization about 10% above the data. A reference obtained from PYTHIA8 simulations agrees with the nominal reference for pT>15 GeV/c. The shape is well described also at lowerpT.
The power law interpolation between the differential yields atp
s = 0.9 TeV and 7 TeV used in [172] differs from the current reference.
The final pp reference atp
s= 2.76 TeV, including the extrapolation is shown in Figure 4.9.
4.2. Construction of pp references 129
pp Reference Paper The ALICE Collaboration
)2c -2 ) (mb GeV Tp dη)/(dchσ2 ) (d Tp π1/(2
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
1 10 102
103
104
| < 0.8 η ALICE, pp, INEL, |
= 2.76 TeV s
) c > 5 GeV/
pT
mod. Hagedorn fit (
) c > 32 GeV/
pT
mod. Hagedorn extrapol. (
) c (GeV/
pT
10-1 1 10
data / fit
0.6 0.8 1 1.2
1.4 syst. uncert. spectrum syst. uncert. reference
Fig. 3: (color online) Top: Differential cross section of charged particles in INEL pp collisions at√
s=2.76 TeV as a function of pTtogether with the parametrization (pT>5 GeV/c) described in the text. Bottom: Ratio of data to parametrization. The grey band indicates the total pTdependent systematic uncertainty of the data, open circles show the data points not used in the parametrization.
d
2σ
chpp/d η dp
Thas been parametrized by a so-called modified Hagedorn function [14]
1 2 π p
Td
2σ
chppd η dp
T= A p
Tm
T1 + p
Tp
T,0 −n(2) where m
Tdenotes the transverse mass m
T= q
m
20+ p
2T, with m
0= 140 MeV/c assumed for all tracks.
For small p
T, the term
1 +
ppTT,0
−nbehaves like an exponential function with an inverse slope parameter of p
T,0/n while for large p
Tthe Hagedorn function behaves like a power-law function.
To determine the extrapolation to high p
T, d
2σ
chpp/d η dp
Tis parametrized for p
T> 5 GeV/c. For 5 GeV/c < p
T< 10 GeV/c the exponential part of the Hagedorn function acts as a correction term to the power-law part in the function.
Figure 3 shows the differential cross section in INEL pp collisions as a function of p
Tfor √
s = 2.76 TeV together with the parametrization for p
T> 5 GeV/c. The ratio between data and parametrization in the lower panel demonstrates the good agreement of the parametrization with the data. The grey band indicates the total p
Tdependent systematic uncertainty of the measured spectrum as presented in Table 1.
To estimate the systematic uncertainty of the parametrization and extrapolation, the lower boundary of the fit range of the Hagedorn parametrization is varied between p
T= 3 GeV/c and p
T= 7 GeV/c, while
7
Figure 4.7.: Top: Differential cross section of charged particles measured in inelastic pp colli-sions atp
s= 2.76 TeV. The line shows the parameterization by a modified Hage-dorn function fitted to the data for pT > 5 GeV/c. Above pT = 32 GeV/c the parameterization is extrapolated to pT = 50 GeV/c, indicated as a dashed line.
Bottom: Ratio of the data points to the parameterization. The filled band around unity shows the systematic uncertainties of the data (excluding the normalization uncertainty). The lines show the systematic uncertainty of the reference, includ-ing the uncertainty of the parameterization and extrapolation. Data points which are use for the fit are shown as full circles. To estimate the uncertainty of the parameterization the fit range was changed to pT > 3 GeV/c, these data points are shown as open circles.
Figure published in [178].
130 4. Results
pp Reference Paper The ALICE Collaboration
) c (GeV/
pT
0 10 20 30 40 50
alt. ref. / ref.
0.6 0.8 1 1.2
1.4 Pythia 8 NLO scaled s = 7 TeV CMS Interp. ref. ALICE [171]
Fig. 4: (color online) Ratio of alternative references to the new constructed pp reference at √
s = 2.76 TeV as discussed in the text. The grey band indicates the total p
Tdependent systematic uncertainty as discussed in the text. The overall normalization systematic uncertainties ± 1.9% ( ± 6%) for ALICE (CMS) are not shown.
the upper boundary is fixed to the highest data point measured at p
T= 32 GeV/c. Together with the systematic uncertainties on the measured differential cross section as shown in Table 1 this results in a total systematic uncertainty on the reference at √
s = 2.76 TeV of 6.4% for low p
Tup to 19% at p
T= 50 GeV/c.
The final pp reference for the determination of R
AAat √
s = 2.76 TeV is constructed from the measured data points up to p
T= 5 GeV/c and the parametrization for p
T> 5 GeV/c. Statistical uncertainties in the extrapolated part of the reference are obtained from the covariance matrix of the parametrization.
The systematic uncertainties on the spectrum are propagated to the reference by application of the full extrapolation procedure using the measured data points shifted up and down by the total systematic uncertainty.
This reference is compared to alternative measurements and approaches. Figure 4 shows the ratio be-tween alternative pp references and the reference at √
s = 2.76 TeV presented in this paper. Above p
T= 20 GeV/c, all references agree within the systematic uncertainties. Simulations with the PYTHIA8 generator [15] agree with the new reference for p
T> 15 GeV/c. Below p
T= 20 GeV/c, the shape of the PYTHIA8 spectrum is similar to the measured reference. A pp reference presented by the CMS collab-oration [16] agrees best for p
T< 6 GeV/c. The overall normalization systematic uncertainties ± 1.9%
( ± 6%) for ALICE (CMS) are not included in the comparison. A reference based on an interpolation be-tween measured yields at √
s = 0.9 and 7 TeV as discussed in [4] does not agree with the new reference for p
T> 6 GeV/c. Finally a scaling of the measured differential cross section in INEL pp collisions at
√ s = 7 TeV with the ratio of pQCD calculations (as shown in Figure 2)
d
2σ
chpp/d η dp
T|
2.76TeV= d
2σ
chpp/d η dp
T|
NLO,2.76TeVd
2σ
chpp/d η dp
T|
NLO,7TeV× d
2σ
chpp/d η dp
T|
7TeV(3) agrees well in shape and normalization with the measured data over a wide range in p
T. The systematic uncertainty of the new reference is indicated in Figure 4 as a grey band for comparison.
5 Construction of a pp reference for √
s = 5.02 TeV
Similar to R
AA, a nuclear modification factor R
pAin proton-lead collisions has been studied [17] at
√ s = 5.02 TeV. No measured pp reference is available at this collision energy. Due to the asymmetric p-Pb collision system, the η coverage of the detector is shifted with respect to the symmetric pp or Pb–
Pb collisions. To obtain a maximum overlap between the pp and p-Pb systems, a pp reference is needed for | η | < 0.3. To construct the pp reference at this energy, different methods for three p
T-ranges are combined.
0.15 < p
T< 5 GeV/c: As NLO-pQCD becomes unreliable for small p
T, the measured differential cross 8
Figure 4.8.: Comparison of alternative pp references atp
s= 2.76 TeV discussed in the text to the chosen one in term of the ratio. The gray band around unity shows thepT de-pendent uncertainty of the reference. Additional normalization uncertainties are not shown. Error-bars shown for the CMS reference are statistical and systematic uncertainties added in quadrature.
Figure published in [178].