Extracting a less model dependent cosmic ray composition from X max distributions
EExtracting a less model dependent cosmic ray composition from X max distributions
Simon Blaess, Jose A. Bellido, and Bruce R. Dawson
Department of Physics, University of Adelaide, Adelaide, Australia
At higher energies the uncertainty in the estimated cosmic ray mass composition, extracted fromthe observed distributions of the depth of shower maximum X max , is dominated by uncertainties inthe hadronic interaction models. Thus, the estimated composition depends strongly on the particularmodel used for its interpretation. To reduce this model dependency in the interpretation of the masscomposition, we have developed a novel approach which allows the adjustment of the normalisationlevels of the proton (cid:104) X max (cid:105) and σ ( X max ) guided by real observations of X max distributions. Inthis paper we describe the details of this approach and present a study of its performance andits limitations. Using this approach we extracted cosmic ray mass composition information fromthe published Pierre Auger X max distributions. We have obtained a consistent mass compositioninterpretation for Epos-LHC, QGSJetII-04 and Sibyll2.3. Our fits suggest a composition consistingof predominantly iron. Below 10 . eV, the small proportions of proton, helium and nitrogen vary.Above 10 . eV, there is little proton or helium, and with increasing energy the nitrogen componentgradually gives way to the growing iron component, which dominates at the highest energies. Thefits suggest that the normalisation level for proton (cid:104) X max (cid:105) is much deeper than the initial predictionsof the hadronic interaction models. The fitted normalisation level for proton σ ( X max ) is also greaterthan the model predictions. When fixing the expected normalisation of σ ( X max ) to that suggestedby the QGSJetII-04 model, a slightly larger fraction of protons is obtained. These results remainsensitive to the other model parameters that we keep fixed, such as the elongation rate and the (cid:104) X max (cid:105) separation between p and Fe. I. INTRODUCTION
A common parameter used to extract mass compo-sition information is X max , the atmospheric depth ing / cm from the top of the atmosphere where the longitu-dinal development of an air shower reaches the maximumnumber of particles or the maximum of the energy de-posited in the atmosphere. Different cosmic ray primariespropagate through the atmosphere differently, resultingin different observed distributions of X max [1]. Due tostatistical variability in the interaction between cosmicrays of a specific primary mass and the atmosphere, acosmic ray’s primary mass cannot be determined on anevent by event basis by examining X max . Instead westudy the X max distribution of cosmic rays of similar en-ergy to infer the mass composition distribution of theevents. Differences in the mode, width and tail of the X max distribution provide information on the mass com-position distribution of the events and on the hadronicinteraction properties [2, 3].Fig. 1 shows the X max distribution resulting from theCONEX v4r37 simulation of 750 proton events accord-ing to the Epos-LHC model, and separately 750 protonevents according to the QGSJetII-04 model, of energy10 eV. The figure illustrates the differences in the X max distribution predicted by different hadronic interactionmodels. Most noticeable is the difference in the modes ofthe distributions, but there are also marginal differencesin the width and tails of the distributions. These dif-ferences between the hadronic interaction models changewith energy to some degree. Although the dissimilaritybetween these predicted distributions may appear minor,applying a parameterisation based on these different pre-dictions to data can have a considerable impact on the ] Xmax [ g/cm
600 800 1000050100
EPOS-LHCQGSJetII-04
FIG. 1: An X max distribution of 750 Epos-LHCsimulated proton events (red), and separately 750QGSJetII-04 simulated protons events (blue), of energy10 eV.mass composition inferred. Consequently, typical masscomposition studies of X max are strongly dependent onthe hadronic interaction model assumed.The algorithm CONEX v4r37 [4, 5], along with thehadronic interaction packages Epos-LHC [6], QGSJetII-04 [7] and Sibyll2.3 [8], were used to simulate air showersto obtain X max distributions according to each of thesemodels. We have developed a parameterisation for des-cribing these expected X max distributions for cosmic raysof some energy and mass. Our parameterisation of the a r X i v : . [ a s t r o - ph . H E ] M a r X max distributions can then be used to fit observed X max distributions, to extract primary mass information (com-position fractions) from each energy bin. By includingsome of the coefficients of our X max parameterisation inthe fit, mass composition results are obtained which aresomewhat independent of the hadronic interaction modelassumed.Assuming the Epos-LHC, QGSJetII-04 or Sibyll2.3hadronic models, the Auger X max distributions can bewell reproduced assuming a composition of at least fourcomponents consisting of proton, Helium, Nitrogen andIron [9–11]. Therefore, in this work we have used mockdata sets to evaluate the performance of our method forretrieving the true relative amounts of p, He, N, Fe (com-position fractions). The results of applying this methodto interpret the published Auger X max distributions in [9]in terms of the mass composition of cosmic rays are pre-sented. II. PARAMETERISATION OFX max
DISTRIBUTIONS An X max distribution of some primary energy andmass can be modelled as the convolution of a Gaus-sian with an exponential [12]. Three shape parameters( t , σ, λ ) define the X max distribution: dNd X max ( t ) = 12 λ exp (cid:18) t − tλ + σ λ (cid:19) Erf c (cid:32) t − t + σ λ σ √ (cid:33) (1)where t defines the mode of the Gaussian component, σ defines the width of the Gaussian component and λ defines the exponential tail of the X max distribution, and t is the X max bin. The mode and spread of the distri-bution defined in Equation (1) is sensitive to t and σ respectively.We fit Equation (1) to the X max distributions fromCONEX v4r37 simulations of cosmic rays of a particularprimary energy, mass (either proton, Helium, Nitrogenor Iron primaries) and hadronic interaction model, ob-taining the values of t , σ and λ for that distribution(see Appendix A). The fit results as a function of energyare displayed in Figs. 2, 3 and 4. The solid lines are fitsto the shape parameters ( t , σ and λ ) as a function ofenergy. The functions fitted are defined as follows: t ( E ) = t norm + B · log (cid:18) log E log E (cid:19) ,σ ( E ) = σ norm + C · log (cid:18) EE (cid:19) ,λ ( E ) = λ norm − K + K · (cid:18) log E log E (cid:19) L ln 10 , (2)where E is the energy in eV and E = 10 . eV, theenergy at which we choose to normalise the equations.This energy corresponds to the energy at which Auger ( E/eV ) log
17 17.5 18 18.5 19 19.5 20 ] [ g / c m t p He N Fe ( E/eV ) log
17 17.5 18 18.5 19 19.5 20 ] [ g / c m s ( E/eV ) log
17 17.5 18 18.5 19 19.5 20 ] [ g / c m l FIG. 2: Fits to the shape parameter as a function ofenergy according to the Epos-LHC model.has measured λ for a proton dominated composition [3].This means that λ norm for proton can be directly com-pared with Λ η , the exponential tail measured by Auger,which is shown in Equation (3). We even consideredadopting Λ η as the value for λ norm , but this could poten-tially break self consistency in the models.Λ η = [55 . ± . stat ) ± . sys )] g / cm (3)The coefficients in Equation (2) are specified in AppendixC for each mass component and hadronic model.The functions of Equation (2) consist of two parts, thefirst part defining the value of a shape parameter at thenormalisation energy, and the second part defining thechange in the shape parameter as a function of energy.For example, for protons t norm would be the value of t for protons at 10 . eV, and similarly σ norm would bethe value of σ at 10 . eV. A. Accounting for the detector resolution andacceptance
The expected X max distributions are affected by thedetector resolution and the detector acceptance. ThePierre Auger X max publication [9] provides parametrisa-tions for the average detector X max resolution as a func-tion of energy ( Res ( E )) and the detector acceptance asa function of X max for each energy bin, Acc ( E, t ), where t is the X max bin as in Equation (1).The detector X max resolution is accounted for byadding it in quadrature with the corresponding σ ( E ),to provide the total expected value of σ ( E ) tot for some ( E/eV ) log
17 17.5 18 18.5 19 19.5 20 ] [ g / c m t p He N Fe ( E/eV ) log
17 17.5 18 18.5 19 19.5 20 ] [ g / c m s ( E/eV ) log
17 17.5 18 18.5 19 19.5 20 ] [ g / c m l FIG. 3: Fits to the shape parameter as a function ofenergy according to the QGSJetII-04 model. ( E/eV ) log
17 17.5 18 18.5 19 19.5 20 ] [ g / c m t p He N Fe ( E/eV ) log
17 17.5 18 18.5 19 19.5 20 ] [ g / c m s ( E/eV ) log
17 17.5 18 18.5 19 19.5 20 ] [ g / c m l FIG. 4: Fits to the shape parameter as a function ofenergy according to the Sibyll2.3 model.primary: σ ( E ) tot = (cid:112) σ ( E ) + Res ( E ) (4)We can combine Equations (1), (2), (4) and the detec-tor acceptance Acc ( E, t ) to obtain the expected X max dis-tribution for cosmic rays of a mixture of primary massesin a particular energy bin according to a hadronic inter-action model: dNd X max ( E, t ) (cid:12)(cid:12)(cid:12)(cid:12) total = N ( E ) Acc ( E, t ) (cid:88) i = p,He,N,F e f i ( E ) dNd X max ( E, t ) (cid:12)(cid:12)(cid:12)(cid:12) i (5)where f p ( E ), f He ( E ), f N ( E ) and f F e ( E ) are the frac-tions of proton, Helium, Nitrogen and Iron events re-spectively, and N ( E ) is the total number of events. Thefractions f p , f He , f N and f F e are all correlated. Fur-thermore, the range of allowed values is not always [0 , f p were 0 .
9, the allowed rangefor any of the other fractions would be [0 , . f p , f He , f N and f F e interms of η , η and η as follows: f p ( E ) = η f He ( E ) = (1 − η ) η f N ( E ) = (1 − η )(1 − η ) η f F e ( E ) = 1 − f p ( E ) − f He ( E ) − f N ( E ) (6)Therefore, each energy bin has a set of η , η and η which defines the mass fractions of that energy bin. Theallowed range for η , η and η is always [0 , η , η and η to determine thecorresponding fractions ( f p , f He , f N , f F e ).Fig. 5 displays the (cid:104) X max (cid:105) and σ ( X max ) predictionsof the three parameterisations for each primary. Thepredicted (cid:104) X max (cid:105) separation of each adjacent mass com-ponent (eg. proton vs. helium, helium vs. nitrogen)within a parameterisation is approximately 30 g / cm to40 g / cm . The predicted σ ( X max ) of the primaries ismuch larger for the QGSJetII-04 and Sibyll2.3 param-eterisations than the Epos-LHC parameterisation. B. Validation of the parameterisation
Fig. 6 displays the mass composition results of fittingthe mass fractions using our Epos-LHC, QGSJetII-04or Sibyll2.3 X max parameterisations and the X max datameasured by the Pierre Auger Observatory fluorescencedetector (FD) [9]. The fits took into account the detec-tor resolution and acceptance. The mass composition ob-tained using our X max parameterisations are consistentwith the Auger analysis of the 2014 FD X max data set[10], where X max distribution templates from hadronicinteraction models were compared to the data. The com-patibility of our results with the 2014 Auger analysis val-idates the accuracy of our X max parameterisations. ( E/eV ) log ] [ g / c m æ X m ax Æ EPOS-LHCQGSJetII-04Sibyll2.3 ( E/eV ) log ] ( X m ax ) [ g / c m s FIG. 5: The (cid:104) X max (cid:105) and σ ( X max ) predictions of the Epos-LHC, QGSJetII-04 and Sibyll2.3 X max parameterisationsfor proton (black), helium (red), nitrogen (green) and iron (blue). III. METHOD
The parameters of Equation (5) are fitted to energybinned X max distributions. The coefficients of Equa-tion (2) shown in Appendix C were obtained with a globalfit which included all energy bins.When fitting (the X max distribution data) for the massfraction parameters using our Epos-LHC, QGSJetII-04or Sibyll2.3 parameterisation with the coefficients fixed(as in Fig. 6), the resulting mass composition reflectsthe characteristics of the corresponding hadronic model.Therefore, the estimated composition depends on whichhadronic model is used. Additionally, the mass composi-tion fitted to each energy bin is independent of the masscomposition fitted to other energy bins. However, by in-cluding some of the coefficients shown in Appendix C inthe fit, in addition to the mass composition fractions, themass composition obtained has a reduced dependence onthe hadronic interaction model assumed. In this alterna-tive case the mass composition fitted at each energy binhas some dependence with the fits at other energy bins.This is because the fitted coefficients (from the X max pa-rameterisation) are fitted using all energy bins, while inthe first case these coefficients were fixed.In principle, if we were able to use the Auger X max datato perform a global fit of the mass composition and allof the coefficients from Equation (2), the resulting com-position would be independent of the hadronic models,depending only on the assumed functional forms of theequations. However, the degeneracy between the fittedmass fractions and the coefficients makes it impossibleto unambiguously constrain all of these parameters (i.e. the solution would be degenerate). Therefore, we need toidentify which coefficients are most relevant for interpret-ing the mass composition, and evaluate whether we canunambiguously fit these coefficients and the mass compo-sition. One way to identify which coefficients to includein a global fit is to compare the values of t , σ and λ between different models. This comparison will identifythe parameters that are well or poorly constrained byour current knowledge of the high energy hadronic inter-action physics.Figs. 7, 8 and 9 illustrates the t , σ and λ difference be-tween the Epos-LHC, QGSJetII-04 and Sibyll2.3 parame-terisations at some energy and mass. The differences as afunction of energy are relatively small. For example, theslope of ∆ t as a function of energy is less than ∼ / cm /energy-decade, which is small compared with an elon-gation rate of 60 g / cm /energy-decade. We have alsoverified that the separation between different primariesin the t , σ and λ space is similar for the three testedmodels. The main differences between our Epos-LHC,QGSJetII-04 and Sibyll2.3 X max parameterisations arethe normalisation of t and σ . The difference in the nor-malization of λ is not negligible, but it has little impacton the mass composition interpretation. Therefore, whenincluding t norm and σ norm in the global fit, we should ob-tain a similar interpretation of the mass composition witheither the Epos-LHC, QGSJetII-04 or Sibyll2.3 X max dis-tribution parameterisation. We choose to fit t norm and σ norm in the following way: • t norm is fitted such that the absolute values of t norm for each primary change by the same amount.Therefore, the difference in t norm between pri- ( E/eV ) log
18 18.5 19 19.5 m ass f r ac t i on s pHeNFe This work: MarkersAuger fits: - - -
EPOS-LHC ( E/eV ) log
18 18.5 19 19.5 m ass f r ac t i on s QGSJetII-04 ( E/eV ) log
18 18.5 19 19.5 m ass f r ac t i on s Sibyll2.3
FIG. 6: Fitting only the mass fractions of ourparameterisations to FD X max data measured by thePierre Auger Observatory. The error bars represent thestatistical error of the fits. Included is the masscomposition results for each hadronic model from thePierre Auger Observatory analysis (labelled ‘Augerfits’). [10].maries is conserved. • σ norm is fitted such that the ratio of σ between pri-maries remains similar to the initial ratio over theenergy range (differences in C between primariesprevents the exact conservation of the initial ra-tio). Therefore, if σ norm for protons changes by ∆, σ norm for other primaries will change by ∆ multi-plied by the initial average ratio of σ between thatprimary and proton.Fitting t norm and σ norm in this way assumes thehadronic models are correctly predicting the separation ( E/eV ) log
17 17.5 18 18.5 19 19.5 t D -2002040 p He N Fe ( E/eV ) log
17 17.5 18 18.5 19 19.5 s D -10010 ( E/eV ) log
17 17.5 18 18.5 19 19.5 l D -100 FIG. 7: Epos-LHC shape parameter value minusQGSJetII-04 shape parameter value for some mass andenergy. ( E/eV ) log
17 17.5 18 18.5 19 19.5 t D -2002040 p He N Fe ( E/eV ) log
17 17.5 18 18.5 19 19.5 s D -10010 ( E/eV ) log
17 17.5 18 18.5 19 19.5 l D -100 FIG. 8: Epos-LHC shape parameter value minusSibyll2.3 shape parameter value for some mass andenergy.in t between different species, and the ratio of σ betweendifferent species, over the fitted energy range.In Equation (2), the values of the shape parametersfor Helium, Nitrogen and Iron can be expressed in termsof the corresponding values for protons, therefore fitting t norm and σ norm in the way described above can be im-plemented by simply fitting t norm and σ norm for protons.In order to avoid unphysical fit results, we constrainthe possible fitted values for t norm and σ norm . Theseconstraints are significantly wider than the separationbetween the Epos-LHC, QGSJetII-04 and Sibyll2.3 X max parameterisation predictions for these coefficients. Thepredicted value of t norm for protons according to Epos-LHC is ∼
703 g / cm , according to QGSJetII-04 is ∼
688 g / cm , and according to Sibyll2.3 is ∼
714 g / cm .The minimum and maximum limits of t norm for pro- ( E/eV ) log
17 17.5 18 18.5 19 19.5 t D -2002040 p He N Fe ( E/eV ) log
17 17.5 18 18.5 19 19.5 s D -10010 ( E/eV ) log
17 17.5 18 18.5 19 19.5 l D -100 FIG. 9: Sibyll2.3 shape parameter value minusQGSJetII-04 shape parameter value for some mass andenergy.tons are set to 670 g / cm and 765 g / cm respectively.The predicted value of σ norm for protons according toEpos-LHC, QGSJetII-04 and Sibyll2.3 is ∼
22 g / cm , ∼
25 g / cm and ∼
28 g / cm respectively. The minimumand maximum limits of σ norm for protons are set to5 g / cm and 55 g / cm respectively.With a suitable shift in t norm and σ norm , many primarymixtures which produce a fairly smooth total distributioncan be fitted well with a single dominant distribution,instead of a sum of distributions. On the other hand, adistribution dominated by a single primary can be wellfitted by a balanced mixture of distributions when t norm and σ norm are shifted appropriately. It is common that X max distributions can be fitted with a value of t norm for protons much larger than the true t norm of the distri-butions, which results in the primary mass of the eventsbeing overestimated (i.e. biased towards heavier masses).Therefore, it is important that appropriate shape coeffi-cient limits are chosen.We have evaluated the performance of fitting t norm and σ norm in addition to the mass fractions using sim-ulated X max distributions of a known composition (seedetails in Sec. IV). Provided there is enough dispersionof masses in the data, it is possible to fit with good ac-curacy, t norm , σ norm and the corresponding abundance(fractions) of p, He, N and Fe. An important achieve-ment from including t norm and σ norm in the fit is thatthe mass composition interpretation becomes consistentwhether using the predicted Epos-LHC, QGSJetII-04 orSibyll2.3 parameterisation.The requirement of a large dispersion of masses is eval-uated over the entire energy range. For example, a dataset consisting of a pure proton composition at higher en-ergies can be fitted, provided that at lower energies wehave populations consisting of other primaries. If thestatistics or mass dispersion were not large enough, therewould be some degeneracy in the fit between the mass fractions and t norm and σ norm . A greater change in themass composition with energy improves the accuracy ofthe fit.Apart from the dispersion of masses in the data, theperformance of the fit depends on the intrinsic valuesfor σ of the data. This is nature’s width for the X max distribution of the different primaries. The separationof the distribution modes between primaries remains un-changed in the fit, therefore primary X max distributionsof larger width will increase the X max distribution over-lap of adjacent primaries, resulting in the fit of t norm , σ norm and the mass composition becoming more uncer-tain.We have also evaluated the performance of fitting t norm , B , and σ norm in addition to the mass fractions,where B defined in Equation (2) describes the changein t with energy. As the predicted mass composition isparticularly sensitive to the predicted values of t , B isa powerful coefficient which can significantly affect thefitted mass composition. We fit B such that for each pri-mary the value of B changes by the same amount fromthe initial predicted value, thus the initial predicted dif-ferences among primaries in the rate of change of t withenergy are conserved (identical to how t norm is fitted).Our X max parameterisations have similar values for B ,therefore we do not expect fits of B to yield results sig-nificantly different from the initial prediction of B whenwe are fitting Epos-LHC, QGSJetII-04 or Sibyll2.3 sim-ulated X max data. However, if the values of B predictedby our parameterisations are significantly incorrect forthe data being fitted, considerable systematics would beintroduced to the reconstructed mass composition if B remains fixed.Data sets that can be fitted with t norm and σ norm may not be accurately fitted when B is included in thefit, as fitting extra coefficients increases the degeneracybetween the fitted variables. Fitting these three coeffi-cients accurately requires a greater spread of primariesand/or statistics than fitting just t norm and σ norm . Thepredicted value of B for protons according to Epos-LHC, QGSJetII-04 and Sibyll2.3 is ∼ / cm , ∼ / cm and ∼ / cm respectively. With t norm normalised at 10 . eV, a change in B of 350 g / cm cor-responds to a change in t at 10 . eV of ∼
10 g / cm .The fitting range limits of B for protons is 1000 g / cm to 4000 g / cm .We have also considered constraining t at 10 eV,where the hadronic models are more reliable, and fitting B and σ norm . Fitting B in this way can also provide aconsistent mass fraction result between the Epos-LHC,QGSJetII-04 and Sibyll2.3 parameterisation fits of sim-ulated X max data, as the t prediction of the fitted en-ergy range adjusts in a way that is similar to the t norm fit, with the added advantage that unlike the t norm fit,the resulting fitted parameterisation of t is consistentwith the hadronic model predictions at lower energies.We have found that over the energy range of interest(10 . eV to 10 eV), fitting t norm and σ norm results ina more accurate mass composition reconstruction com-pared to fitting B and σ norm . This is because there is lessdegeneracy between the fitted mass fractions and shapeparameters when fitting t norm and σ norm . Additionally,a t parameterisation constrained at 10 . eV describesthe energy range of interest better than a t parameteri-sation extrapolated from 10 eV. If a wider energy rangewas being fitted, then a t norm and σ norm fit would be lessaccurate, because the t and σ parameterisations of dif-ferent models do not adequately align over a wider energyrange by only adjusting their normalisations. It is alsoimportant to recognise that this fit of B is restricted, aswe are fixing how t changes with energy, and only fit-ting the rate of change of the log (cid:16) log E log E (cid:17) factor. Toproperly fit the slope of t with energy would require thefit of a third t parameter (for example, fitting B and x in B · log (cid:16) log E log E (cid:17) x , where x currently equals 1).We have evaluated the effect of different X max bin sizesand energy bin sizes on the performance of the fit. Whenfitting only the mass fractions, 1 g / cm X max binninggives marginally more accurate results than 20 g / cm X max binning (20 g / cm is the X max bin size of the Auger X max distributions published in [9]). The absolute im-provement in the fitted mass fractions is no greater than3% in an energy bin. However, when fitting t norm and σ norm in addition to the mass fractions, using a small X max binning is more important, otherwise the chosencenter of the X max bins may significantly affect the fit-ted results, especially if the statistics are not large. Thepredicted separation between different primaries in t norm and σ norm can be very small. For example, our Epos-LHC parameterisation predicts the difference in t norm between proton and helium is only ∼ / cm . There-fore, a 20 g / cm X max binning (as published in [9]) canbe too coarse, and can shift the apparent (cid:104) X max (cid:105) of thedistribution, which affects the fit of t norm .Due to similar reasons, the energy bin size is also im-portant. Energy binning that is too large can result indata from the same primary mass, but on opposite ex-tremes of the energy bin, being evaluated as data fromdifferent primaries. This is because the separation be-tween the predicted X max distributions of different pri-maries is small compared to the shift in these X max dis-tributions with energy. We find that an energy binningof 0 . ( E/ eV) is reasonable. IV. PERFORMANCE
Using CONEX v4r37, 100 X max data sets were gen-erated according to both the Epos-LHC and QGSJetII-04 hadronic interaction models for a number of differentmass compositions. The data consists of 17 energy bins,of which there are 13 energy bins of a width of 0 . ( E/ eV) between 10 eV and 10 . eV, and 4 fixed energy bins at 10 . eV, 10 . eV, 10 eV and 10 . eV.Each energy bin contains approximately 750 events. Thebinning of the simulated X max distributions is 1 g / cm .We have fitted only the mass fractions (all coefficientsfrom the X max parameterisation were kept fixed) to dataof a single primary generated with the same hadronic in-teraction model the parameterisation fitted is based on.Figs. 10 to 13 summarises the results (of these 100 fits) forthe Epos-LHC hadronic model and Figs. 14 to 17 for theQGSJetII-04 model. The markers represent the mediansof the fitted mass fractions, and the error bars representthe standard deviation. The results show that our X max parameterisations are an accurate description of the ex-pected X max distribution of a primary according to theEpos-LHC or QGSJetII-04 hadronic interaction models.Both our Epos-LHC and QGSJetII-04 X max parameteri-sation fits can accurately determine the mass compositionof data from the same hadronic model. ( E/eV ) log
17 17.5 18 18.5 19 19.5 R ec . m ass p He N Fe pure proton FIG. 10: Fitting only the mass fractions to mock datasets of X max distributions. The data sets have beengenerated using the Epos-LHC model and assuming a proton primary composition over the whole energyrange. The composition fits were performed using our X max parameterisations for the Epos-LHC modelpredictions. ‘Rec. mass’ refers to the mass fractionsfitted to the data. ( E/eV ) log
17 17.5 18 18.5 19 19.5 R ec . m ass p He N Fe pure helium FIG. 11: Same as Fig. 10, but assuming a
Helium primary composition over the whole energy range.Fig. 18 to Fig. 20 summarises the results of fits to 100 X max data sets with a true mass composition consisting ( E/eV ) log
17 17.5 18 18.5 19 19.5 R ec . m ass p He N Fe pure nitrogen FIG. 12: Same as Fig. 10 but assuming a
Nitrogen primary composition over the whole energy range. ( E/eV ) log
17 17.5 18 18.5 19 19.5 R ec . m ass p He N Fe pure iron FIG. 13: Same as Fig. 10, but assuming an
Iron primary composition over the whole energy range. ( E/eV ) log
17 17.5 18 18.5 19 19.5 R ec . m ass p He N Fe pure proton FIG. 14: Fitting only the mass fractions to mock datasets of X max distributions. The data sets have beengenerated using the QGSJetII-04 model and assuming a proton primary composition over the whole energyrange. The composition fits were performed using our X max parameterisations for the QGSJetII-04 modelpredictions.of 50% proton and helium in the first 8 energy bins, and50% helium and nitrogen in the remaining 9 energy bins.When fitting only the mass fractions (i.e. keeping fixedthe coefficients of the X max distribution parameterisa-tion) of our parameterisations to CONEX v4r37 X max data based on the same model, the fits are able to recon- ( E/eV ) log
17 17.5 18 18.5 19 19.5 R ec . m ass p He N Fe pure helium FIG. 15: Same as Fig. 14, but assuming a
Helium primary composition over the whole energy range. ( E/eV ) log
17 17.5 18 18.5 19 19.5 R ec . m ass p He N Fe pure nitrogen FIG. 16: Same as Fig. 14 but assuming a
Nitrogen primary composition over the whole energy range. ( E/eV ) log
17 17.5 18 18.5 19 19.5 R ec . m ass p He N Fe pure iron FIG. 17: Same as Fig. 14, but assuming an
Iron primary composition over the whole energy range.struct the mass composition to within an absolute offsetin the median of 10% from the true mass (as seen inFigs. 18 and 19).Fig. 20 shows the results of fitting t norm and σ norm , inaddition to the mass fractions, of the QGSJetII-04 pa-rameterisation to QGSJetII-04 data. These QGSJetII-04 X max distributions do not provide sufficient constraintson our fitted parameterisation, resulting in a mass com-position reconstruction that does not resemble the truemass composition. In order to successfully fit t norm and σ norm to data of a similar distribution, a wider range of ( E/eV ) log
17 17.5 18 18.5 19 19.5 R ec . m ass p He N Fe FIG. 18: Fitting only the mass fractions of ourEpos-LHC parameterisation to Epos-LHC X max data. ( E/eV ) log
17 17.5 18 18.5 19 19.5 R ec . m ass p He N Fe FIG. 19: Fitting only the mass fractions of ourQGSJetII-04 parameterisation to QGSJetII-04 X max data. ( E/eV ) log
17 17.5 18 18.5 19 19.5 R ec . m ass p He N Fe FIG. 20: Fitting t norm , σ norm and the mass fractions ofour QGSJetII-04 parameterisation to QGSJetII-04 X max data.primary masses over the energy range of the data is re-quired (wider than the one in the given example). Forexample, in Fig. 21 we have increased the range of pri-mary masses by replacing helium with iron in the lastenergy bin. The resulting fit of the mass fractions (with t norm and σ norm also fitted) have an absolute offset in themedian of less than ∼
15% from the true values, whichis comparable to a fit of only the mass fractions to dataof a similar composition. ( E/eV ) log
17 17.5 18 18.5 19 19.5 R ec . m ass p He N Fe FIG. 21: Fitting t norm , σ norm and the mass fractions ofour QGSJetII-04 parameterisation to QGSJetII-04 X max data. Helium has been replaced by Iron in thelast energy bin to increase the mass dispersion. ( E/eV ) log
17 17.5 18 18.5 19 19.5 R ec . m ass p He N Fe FIG. 22: Fitting t norm , σ norm and the mass fractions ofour Epos-LHC parameterisation to QGSJetII-04 X max data. Helium has been replaced by Iron in the lastenergy bin to increase the mass dispersion. ( E/eV ) log
17 17.5 18 18.5 19 19.5 R ec . m ass p He N Fe FIG. 23: Fitting only the mass fractions (i.e. t norm and σ norm are kept fixed) of our Epos-LHC parameterisationto QGSJetII-04 X max data. Compare this Fig. withFig. 22 where t norm and σ norm were included in the fit. A. Fitting data originating from a different model.
Compare Fig. 22 with Fig. 23, which shows the com-position fits when using the Epos-LHC parameterisationto fit QGSJetII-04 data, with t norm and σ norm fitted in0 Init.) - (Rec. t -20 -10 0 1001020 Init.) - (Rec. norm s -10 0 10050 Init.) - (Rec. t -20 -10 0 10 I n i t. ) - ( R e c . no r m s -10010 05101520 FIG. 24: Change in t norm and σ norm for protons fromthe fits in Fig. 21. Init.) - (Rec. t -20 -10 0 10020 Init.) - (Rec. norm s -10 0 10050 Init.) - (Rec. t -20 -10 0 10 I n i t. ) - ( R e c . no r m s -10010 05101520 FIG. 25: Change in t norm and σ norm for protons fromthe fits in Fig. 22.the former, and t norm and σ norm fixed in the latter. Fit-ting these two coefficients is enough to result in a recon-structed mass much closer to the true mass, despite thefitted data originating from a different model. By fit-ting t norm and σ norm , there is no longer a significant ironcomponent where there should only be 50% helium andnitrogen, and in the 50% proton and helium range thereis no longer a fitted nitrogen component larger than thehelium fraction.Figs. 24 and 25 show the difference between the fittedvalues and initial values of t norm and σ norm (and theircorrelation) when fitted to the data with iron added inthe last energy bin. Fig. 24 displays the results of fittingQGSJetII-04 data with our QGSJetII-04 parameterisa-tion, and as expected the difference between the recon-structed and initial values of our coefficients is minimal.Fig. 25 displays the results of fitting the same QGSJetII-04 data with our Epos-LHC parameterisation (the re-constructed mass is shown in Fig. 22), and we see that t norm and σ norm are shifted towards the QGSJetII-04 val-ues for these coefficients. The initial Epos-LHC proton t norm and σ norm values are ∼
703 g / cm and ∼
22 g / cm respectively, while the initial QGSJetII-04 proton t norm and σ norm values (and therefore the approximate val-ues of the QGSJetII-04 MC data) are ∼
688 g / cm and ∼
25 g / cm respectively.Notice that in Fig. 18 to Fig. 22 the bins containinga helium and nitrogen mix are reconstructed better thanthe bins containing a proton and helium mix. Protonand helium distributions are harder to reconstruct dueto their wider spread and their larger overlap. A widerspread means that for a given number of events, lessevents will populate individual X max bins. Therefore,proton and helium fits have larger statistical uncertain- ties. Additionally, the X max parameterisations for lightermasses do not describe the CONEX v4r37 Epos-LHCand QGSJetII-04 simulated data as accurately. Fig. 58in Appendix A illustrates that as the primary mass of thedistribution increases, the X max parameterisations repro-duce the true (cid:104) X max (cid:105) and σ ( X max ) of the distributionswith better accuracy. Appendix A shows that for protonand helium data especially, the fits of Equation (1) toMC data of either hadronic model tend to overestimatethe number of events at the mode of the distribution.When fitting mixes of protons and helium, our fits tendto have a reconstruction bias towards protons.As the absolute separation between σ for different pri-maries is similar in the Epos-LHC and QGSJetII-04 pa-rameterisations (like t ), marginally better results wouldbe obtained in Fig. 22 if instead of fitting σ norm suchthat the initial ratios of σ among primaries are conserved, σ norm was fitted such that the initial separation between σ norm among primaries was conserved (like t norm ). How-ever, conserving the initial ratios of σ is the more phys-ical approach, because if σ norm for protons changes by10 g / cm , we would not expect that σ norm for iron wouldalso change by 10 g / cm . Additionally, nature does notnecessarily conform to the Epos-LHC or QGSJetII-04predictions of the absolute separation of σ norm amongprimaries. V. t norm AND σ norm PARAMETER SPACE SCANOF THE AUGER FD X max DATA
Fig. 26 shows the minimised Poisson log likelihoodspace of the mass fraction fit of a parameterisation toAuger FD X max data, where t norm and σ norm have beenfixed to some particular value (indicated by the x and yaxes). The z-axis shows the difference between the min-imised probability for some value of t norm and σ norm ,and the absolute minimised probability obtained fromthe t norm and σ norm values which best fitted the data fora particular parameterisation. A difference of 1 in theminimised Poisson log likelihood corresponds to 1 σ . Theabsolute minima of the Epos-LHC and QGSJetII-04 fitsto the Auger FD data correspond to a similar value of t norm for protons, whereas the absolute minimum of theSibyll2.3 fit is located at a significantly larger value of t norm for protons. Between the three fitted parameteri-sations, when estimating the heavier nuclei t norm valuesthere is more similarity. This is because the separationbetween the proton t prediction and heavier nuclei islarger in the Sibyll2.3 parameterisation than Epos-LHCor QGSJetII-04 (see Figs. 7, 8 and 9). This is also truefor σ .These scans show that the fits of the Auger FD X max data performed in Section VII did not become stuck ina local minimum. The scans can also reveal secondarysolutions which are not as deep as the deepest minimum.1 t
700 750 no r m s EPOS-LHCQGSJetII-04Sibyll2.3
FIG. 26: The t norm and σ norm parameter space scanover the Auger FD X max data. For each modelparameterisation, at specific values of t norm and σ norm ,the mass fractions are fitted to the data, and the first5 σ contours of the minimised Poisson log likelihood areshown. The scanned shape coefficient values for protonare shown. The coefficient values of the heavier nucleichange (relative to protons) in the way the shapecoefficient would be fitted, outlined in Section III. VI. EVALUATING THE FIT PERFORMANCEFOR A MASS COMPOSITION CONSISTENTWITH THE AUGER RESULTS
The performance of fitting t norm , σ norm and the massfractions of our parameterisations to the Auger FD X max data is evaluated by fitting mock X max data sets thatresemble the Auger FD X max distributions. This wasachieved by fitting t norm , σ norm and the mass fractionsof a particular parameterisation to the Auger FD X max data, and then using this fitted parameterisation to gen-erate the mock data sets. Appendix C displays the t norm and σ norm values fitted to the Auger data, values whichcorrespond to the absolute minima found from the scansin Section V. These mock data sets have a true mass com-position which is defined by the parameterisation usedto generate them, therefore we can evaluate the abilityof our t norm , σ norm and mass fraction fit to accuratelyreconstruct the true mass fractions. The binning of themock Auger X max distributions is 20 g / cm .The measured FD X max distributions are broadened bythe X max resolution of the detector, and are affected bythe detector acceptance, therefore the mock X max datagenerated from the fitted parameterisation are convolvedwith the same detector effects. The X max resolution andacceptance of the Auger data is taken into account when fitting this mock Auger X max data. Our mock X max dis-tributions and the X max distributions measured by Augerare treated with exactly the same approach. A. Fitting t norm , σ norm and the mass fractions Figs. 27, 28 and 29 display the mass composition re-sults from fitting the mass fractions, t norm and σ norm of either the Epos-LHC, QGSJetII-04 or Sibyll2.3 pa-rameterisations respectively, to 100 data sets generatedfrom the parameterisation which resulted when the massfractions, t norm and σ norm of the Epos-LHC parame-terisation were fitted to Auger FD X max data (as willbe shown in Section VII). The true mass composition ofthe mock data is therefore the mass composition whichresulted from the Epos-LHC fit to the Auger FD X max data. Figs. 30, 31 and 32 display the fitted proton val-ues of t norm and σ norm relative to the original values ofthe model applied, compared to the change required tomatch the true proton values of the mock data. Thered lines indicate the mock data input values and theblue histograms are the reconstructed values. The cor-relations between the reconstructed t norm and σ norm arealso shown in Figs. 30, 31 and 32. There are no recon-struction systematics when using the Epos-LHC param-eterisation to fit Epos-LHC generated data (Fig. 30), butthere are some systematics when using the QGSJetII-04or Sibyll2.3 parameterisations to fit Epos-LHC generateddata (Figs. 31 and 32). These systematics in t norm and σ norm translate into relative small systematics of the re-constructed mass fractions (as seen in Figs. 28 and 29).Figs. 28 and 31 show that despite the differences be-tween the Epos-LHC and QGSJetII-04 parameterisations(which are not limited to different t norm and σ norm pre-dictions), by allowing t norm and σ norm of the QGSJetII-04 X max parameterisation to be fitted to mock data basedon the Epos-LHC parameterisation, the true mass frac-tions are reconstructed with an overall accuracy compa-rable to the Epos-LHC fits of Epos-LHC data. The ab-solute offsets in the median mass fractions from the truemass are less than 10% in most energy bins. This demon-strates that fitting t norm and σ norm significantly reducesthe differences between the Epos-LHC and QGSJetII-04 X max parameterisations. As we are fitting the QGSJetII-04 parameterisation to mock data based on the Epos-LHC parameterisation, we do not expect the average fit-ted values of t norm and σ norm to be centred on the redlines even if no systematic offset was present in the massfractions reconstruction. This is because the separationof these coefficients between masses differs between theEpos-LHC and QGSJetII-04 parameterisations, thus ifthe fitted QGSJetII-04 value of t norm for protons wasequal to the Epos-LHC value of t norm for protons, theaccordingly adjusted t norm values of other masses woulddiffer between these parameterisations.The mass composition reconstruction accuracy of theEpos-LHC fit to Epos-LHC based data changes less with2 ( E/eV ) log
18 18.5 19 19.5 R ec . m ass p He N Fe true: - - -fitted: markers FIG. 27: Epos-LHC fit of X max data generated from theEpos-LHC parameterisation fit of Auger data. ( E/eV ) log
18 18.5 19 19.5 R ec . m ass p He N Fe true: - - -fitted: markers FIG. 28: QGSJetII-04 fit of X max data generated fromthe Epos-LHC parameterisation fit of Auger data. ( E/eV ) log
18 18.5 19 19.5 R ec . m ass p He N Fe true: - - -fitted: markers FIG. 29: Sibyll2.3 fit of X max data generated from theEpos-LHC parameterisation fit of Auger data.energy than the accuracy of the QGSJetII-04 fit to theEpos-LHC data. This is because the Epos-LHC t pa-rameterisation fit to the Epos-LHC based data is offsetby a constant value at all energies from the true t ofthe mock data, whereas the difference between the fittedQGSJetII-04 t parameterisation and the true t of themock data (based on Epos-LHC) changes with energy.Fig. 29 shows the Sibyll2.3 fit to the Epos-LHC dataresults in a reconstructed mass that is very represen-tative of the true mass, but this mass reconstructionis not as accurate as the Epos-LHC and QGSJetII-04 Init.) - (Rec. t Init.) - (Rec. norm s -10 0 10 2001020 Init.) - (Rec. t I n i t. ) - ( R e c . no r m s -1001020 05101520 FIG. 30: Change in t norm and σ norm for protons fromthe fits in Fig. 27. Init.) - (Rec. t Init.) - (Rec. norm s -10 0 10 2002040 Init.) - (Rec. t I n i t. ) - ( R e c . no r m s -1001020 05101520 FIG. 31: Change in t norm and σ norm for protons fromthe fits in Fig. 28. Init.) - (Rec. t Init.) - (Rec. norm s -10 0 10 2001020 Init.) - (Rec. t I n i t. ) - ( R e c . no r m s -1001020 05101520 FIG. 32: Change in t norm and σ norm for protons fromthe fits in Fig. 29.fits to this data. This is because a t norm and σ norm shift of the Sibyll2.3 parameterisation does not align theSibyll2.3 t and σ parameterisations with the Epos-LHC(or QGSJetII-04) descriptions as adequately as the Epos-LHC or QGSJetII-04 descriptions can be aligned witheach other (compare Figs. 7, 8 and 9). Larger differ-ences in the λ Sibyll2.3 parameterisation relative to theother parameterisations further hinders an accurate massreconstruction of data based on these other parameteri-sations.Similar to the earlier figures presented, Figs. 33, 34and 35 display the mass composition results from fit-ting the mass fractions, t norm and σ norm of either theEpos-LHC, QGSJetII-04 or Sibyll2.3 parameterisationsrespectively, to 100 data sets generated from the pa-rameterisation which resulted when the mass fractions, t norm and σ norm of the QGSJetII-04 parameterisationwere fitted to Auger FD X max data. The true masscomposition of the mock data is the mass compositionfrom this QGSJetII-04 fit to the Auger FD X max data.The QGSJetII-04 based mock X max distributions willbe slightly different to the Epos-LHC based mock dis-tributions, because the X max parameterisations do not3 ( E/eV ) log
18 18.5 19 19.5 R ec . m ass p He N Fe true: - - -fitted: markers FIG. 33: Epos-LHC fit of X max data generated from theQGSJetII-04 parameterisation fit of Auger data. ( E/eV ) log
18 18.5 19 19.5 R ec . m ass p He N Fe true: - - -fitted: markers FIG. 34: QGSJetII-04 fit of X max data generated fromthe QGSJetII-04 parameterisation fit of Auger data. ( E/eV ) log
18 18.5 19 19.5 R ec . m ass p He N Fe true: - - -fitted: markers FIG. 35: Sibyll2.3 fit of X max data generated from theQGSJetII-04 parameterisation fit of Auger data.perfectly fit the Auger data, and the respective param-eterisations consist of differences which can not be com-pensated for by an appropriate t norm and σ norm shift.Figs. 36, 37 and 38 display the fitted values of t norm and σ norm for the Epos-LHC, QGSJetII-04 or Sibyll2.3 fitsrespectively to the QGSJetII-04 based data.The fits to QGSJetII-04 based mock data produce sim-ilar results to the fits of Epos-LHC based mock data.The mass fraction, t norm and σ norm fit of the Epos-LHCparameterisation to QGSJetII-04 based mock data re-constructs the mass composition above 10 . eV with an Init.) - (Rec. t Init.) - (Rec. norm s -10 0 10 2001020 Init.) - (Rec. t I n i t. ) - ( R e c . no r m s -1001020 05101520 FIG. 36: Change in t norm and σ norm for protons fromthe fits in Fig. 33. Init.) - (Rec. t Init.) - (Rec. norm s -10 0 10 20020 Init.) - (Rec. t I n i t. ) - ( R e c . no r m s -1001020 05101520 FIG. 37: Change in t norm and σ norm for protons fromthe fits in Fig. 34. Init.) - (Rec. t Init.) - (Rec. norm s -10 0 10 2001020 Init.) - (Rec. t I n i t. ) - ( R e c . no r m s -1001020 05101520 FIG. 38: Change in t norm and σ norm for protons fromthe fits in Fig. 35.accuracy almost as good as the QGSJetII-04 parameter-isation fit to the same data. For both the Epos-LHC andQGSJetII-04 fits, the absolute offsets in the median massfractions from the true mass are less than 10% in mostenergy bins. As noted before, due to the differences be-tween the Epos-LHC and QGSJetII-04 t descriptions asa function of energy, the mass reconstruction accuracyof the Epos-LHC fit varies more with energy than theQGSJetII-04 fit. Again the Sibyll2.3 fit, in this case toQGSJetII-04 based data, does not reconstruct the masscomposition as accurately as the Epos-LHC or QGSJetII-04 fits.Figs. 39, 40 and 41 display the mass composition re-sults from fitting the mass fractions, t norm and σ norm ofeither the Epos-LHC, QGSJetII-04 or Sibyll2.3 parame-terisations respectively, to 100 data sets generated fromthe parameterisation which resulted when the mass frac-tions, t norm and σ norm of the Sibyll2.3 parameterisa-tion were fitted to Auger FD X max data. Figs. 42, 43and 44 display the respective t norm and σ norm from thesefits. The Epos-LHC and QGSJetII-04 fits to the Sibyll2.3based data do not reconstruct the true mass compositionas accurately as the Sibyll2.3 fit, but they do accurately4 ( E/eV ) log
18 18.5 19 19.5 R ec . m ass p He N Fe true: - - -fitted: markers FIG. 39: Epos-LHC fit of X max data generated from theSibyll2.3 parameterisation fit of Auger data. ( E/eV ) log
18 18.5 19 19.5 R ec . m ass p He N Fe true: - - -fitted: markers FIG. 40: QGSJetII-04 fit of X max data generated fromthe Sibyll2.3 parameterisation fit of Auger data. ( E/eV ) log
18 18.5 19 19.5 R ec . m ass p He N Fe true: - - -fitted: markers FIG. 41: Sibyll2.3 fit of X max data generated from theSibyll2.3 parameterisation fit of Auger data.represent the general transition of the mass composition.The Sibyll2.3 fit to Sibyll2.3 based data (see Figs. 41 and44) results in absolute offsets in the median mass frac-tions from the true mass of less than 10%.The data fitted in this section sufficiently constrainsthe fitted values of t norm and σ norm , regardless of theparameterisation fitted. If different populations of t norm and σ norm were present in a histogram plot, it would in-dicate the data is unable to adequately constrain the fit,due to the degeneracy between the fitted shape coeffi-cients and the mass fractions. Init.) - (Rec. t Init.) - (Rec. norm s -10 0 10 2001020 Init.) - (Rec. t I n i t. ) - ( R e c . no r m s -1001020 05101520 FIG. 42: Change in t norm and σ norm for protons fromthe fits in Fig. 39. Init.) - (Rec. t Init.) - (Rec. norm s -10 0 10 20020 Init.) - (Rec. t I n i t. ) - ( R e c . no r m s -1001020 05101520 FIG. 43: Change in t norm and σ norm for protons fromthe fits in Fig. 40. Init.) - (Rec. t Init.) - (Rec. norm s -10 0 10 2001020 Init.) - (Rec. t I n i t. ) - ( R e c . no r m s -1001020 05101520 FIG. 44: Change in t norm and σ norm for protons fromthe fits in Fig. 41.Data consisting of predominantly iron, such as the datasets fitted in this section, are easier to fit than data con-sisting of predominately protons and helium.The ability of a t norm and σ norm fit of these parame-terisations to reconstruct the general mass compositiontrend of data based on any of these three parameterisa-tions, indicates that the normalisations of t and σ arethe most relevant differences between these parameterisa-tions in regards to reconstructing the mass composition.The results of the t norm , σ norm and mass fraction fits ofthe Auger FD X max data [9] are presented in Section VII. B. Fitting t norm , B , σ norm and the mass fractions The coefficient B (which defines the energy dependenceof t ) can also be fitted with t norm and σ norm providedthe data consists of an adequate dispersion of massesand statistics. This three-coefficient fit will generally beless precise than the two-coefficient fit of only t norm and σ norm . Fitting additional coefficients increases the de-generacy between the fitted variables, unless there is sig-nificant mass diversity and statistics. Our Epos-LHC,5QGSJetII-04 and Sibyll2.3 predictions of B are fairly sim-ilar among primaries, therefore we do not expect to seea significant improvement in the systematics of the re-constructed mass composition when adding B to our pa-rameterisation fits of data based on any of these threemodels. However, it is possible that nature has a dif-ferent energy dependence for t (different from the threemodels), so by including B in the fit we reduce consid-erably the model dependence of the mass compositioninterpretation of the X max distributions.Figs. 45 and 46 display the reconstructed mass com-position and fitted coefficient values from fitting t norm , B and σ norm of our Epos-LHC parameterisations to datagenerated from the Epos-LHC t norm and σ norm fit of theFD X max data set. Comparing this result to Fig. 27,the systematic offsets in the median reconstructed masscomposition from the true mass for the three-coefficientfit are similar to the two-coefficient fit. Fig. 46 showsthat the three fitted shape coefficients are accurately fit-ted and are well constrained.However, as mentioned previously, data consisting ofpredominantly iron are easier to fit than data consist-ing of predominately proton and helium. The t norm , B , σ norm and mass fraction fit of the latter data can resultin a reconstructed mass composition which is consider-ably less accurate than a fit where B is fixed to the truevalue of the data. This is because the degeneracy be-tween the fitted parameters can result in the fitted shapecoefficients shifting away from the true values. ( E/eV ) log
18 18.5 19 19.5 R ec . m ass p He N Fe true: - - -fitted: markers FIG. 45: Epos-LHC fit of X max data generated from theEpos-LHC parameterisation fit of Auger data. C. Effect of X max systematic uncertainties whenfitting t norm and σ norm Fitting t norm can compensate for systematic offsets in X max , while fitting σ norm can compensate for system-atic errors in the estimation of the detector resolutionof X max . Figs. 47 and 48 shows the results of fittingthe mass fractions, t norm and σ norm of our QGSJetII-04parameterisation to 100 data sets generated from the pa-rameterisation which resulted when the mass fractions, t norm and σ norm of the QGSJetII-04 parameterisation Init.) - (Rec. t Init.) - (Rec. norm s -10 0 10 2001020 Init.) - (Rec. /10 B -1 0 10102030 Init.) - (Rec. t I n i t. ) - ( R e c . no r m s -1001020 05101520 Init.) - (Rec. t I n i t. ) - ( R e c . / B -101 05101520 FIG. 46: Change in t norm , B and σ norm for protonsfrom the fits in Fig. 45.were fitted to Auger FD X max data. Across the wholeenergy range, the mock data was shifted by a system-atic offset of -10 g / cm , and also smeared by a Gaussiandistributed random variable of σ = 10 g / cm (this addi-tional smearing is not accounted for in the resolution ofthe applied X max parameterisation), to test if the fit of t norm and σ norm can compensate for these systematics.The red lines in Fig. 48 indicate the true t norm and σ norm values of the data (relative to the initial QGSJetII-04 pa-rameterisation being fitted) before the X max systematicswere applied.The mean shift in the fitted t norm values from the orig-inal t norm values of the data is ∼ -10 g / cm (Fig. 48),to compensate mainly for the -10 g / cm X max system-atic offset applied to the data. As t changes by thesame amount for each primary when t norm is fitted, andthe X max systematic was applied consistently to all data,the t norm fit is capable of completely accounting for the X max systematic offset. However, σ norm for each pri-mary is changed by different absolute amounts when fit-ting this coefficient, but all of the data is smeared (allmasses are consistently smeared), consequently the cor-rect σ norm cannot be fitted for each primary, which mayalso effect the fit of t norm . The shift in σ norm for protonsfrom the original σ norm is only ∼ +2 g / cm . Despite thefit of σ norm being unable to thoroughly account for the10 g / cm systematic in the resolution, the absolute off-sets in the median reconstructed mass fractions from thetrue mass are less than 10% in most energy bins, due toa combined shift of t norm and σ norm in the appropriatedirections.The accuracy of the reconstructed mass fractions fromthe fit of this shifted and smeared data is similar to thesame fit of the un-shifted and un-smeared data in Fig. 34.Reasonable detector resolution systematics and system-atic offsets in X max will not significantly effect the accu-6 ( E/eV ) log
18 18.5 19 19.5 R ec . m ass p He N Fe true: - - -fitted: markers FIG. 47: Fits of t norm and σ norm to X max dataconsisting of a −
10 g / cm systematic offset in X max .The X max data was also smeared by a Gaussiandistributed random variable of σ = 10 g / cm , which wasunaccounted for in the initial X max parameterisationfitted. Init.) - (Rec. t Init.) - (Rec. norm s -10 0 10 20020 Init.) - (Rec. t I n i t. ) - ( R e c . no r m s -1001020 05101520 FIG. 48: Change in t norm and σ norm for protons fromthe fits in Fig. 47.racy of the reconstructed mass composition.If the data was not smeared by a Guassian randomvariable, and only shifted by a constant X max offset, the t norm and σ norm fit of this shifted data would result in achange in the fitted t norm (compared to the t norm fittedto the un-shifted data) which is very close to the valueof the X max offset. Shifting the X max data by a constantvalue has essentially the same effect on the fit as shiftingthe parameterisation by a constant value, with a veryminuscule difference arising if the detector acceptance of X max is not shifted by the same offset to account for theapplied X max offset (this is not an issue when fitting themeasured Auger data). VII. RESULTS
We have applied our Epos-LHC, QGSJetII-04 andSibyll2.3 X max parameterisations separately to X max data measured by the Pierre Auger Observatory fluores-cence detector (FD) [9].Fig. 49 displays the results from fitting the mass frac-tions and the coefficients t norm and σ norm of our Epos-LHC, QGSJetII-04 and Sibyll2.3 X max distribution pa-rameterisations. The top three panels display the fit-ted mass fractions for each model, and the bottom panel shows the p-values for these fits. The fits of these pa-rameterisations to the X max distributions are shown inAppendix B.The p-value is defined as the probability of obtaininga worse fit (larger likelihood ratio L ) than that obtainedwith the data. The resulting parameterisation and frac-tions from the fit of the X max distributions were usedto generate sets of mock X max distributions to deter-mine the p-values, and to calculate the mass composi-tion statistical errors. Fitting t norm and σ norm improvesthe goodness of the fit of the X max distributions (bot-tom panel Fig. 49). This is evident by comparing theQGSJetII-04 p-values for the t norm and σ norm fit to theQGSJetII-04 p-values for the fit of only the mass frac-tions.We find that the Epos-LHC, QGSJetII-04 and Sibyll2.3parameterisation fits of the X max distributions give a con-sistent mass composition result. Fig. 50 shows the corre-sponding moments of the ln A distribution. The resultssuggest a composition consisting of predominantly iron.Below 10 . eV, the small proportions of proton, heliumand nitrogen vary. Above 10 . eV, there is little pro-ton or helium, and with increasing energy the nitrogencomponent gradually gives way to the growing iron com-ponent, which dominates at the highest energies. Theredoes not appear to be a distinct feature near the ankle( ∼ . eV), where it is assumed cosmic rays transi-tion from Galactic to extragalactic [14]. Considering theupper limits on the large scale anisotropy [15] indicateprotons below 10 . eV are most likely of extragalacticorigin, the fitted proton fractions below the ankle aresuitably small if cosmic rays below the ankle are Galac-tic. A significant modification of the hadronic models isrequired to accommodate a proton dominant compositionabove 10 eV [16].The first two moments of the Auger X max distribu-tions from [9] and their predictions (for proton andFe) as a function of energy are shown in Fig. 51. Itshows that the t norm and σ norm fits reduce the differ-ence between the predictions from the Epos-LHC andQGSJetII-04 hadronic models. For t and σ , the separa-tion between the proton prediction and heavier nuclei islarger in the Sibyll2.3 parameterisation than the Epos-LHC or QGSJetII-04 parameterisations, consequentlythe Sibyll2.3 proton predictions from the fit are in dis-agreement with the two other parameterisations. Thevalues of the coefficients in Equation (2) for proton, he-lium, nitrogen and iron primaries for the Epos-LHC,QGSJetII-04 and Sibyll2.3 models (assuming a normali-sation energy of E = 10 . eV) can be found in Table Iof Appendix C. The values fitted to the data for t norm and σ norm are also shown in Table I. The statistical errorsin the estimated value of (cid:104) X max (cid:105) for protons or iron overthe energy range are the same as the statistical error inthe fitted value of t norm , while for σ ( X max ) the statisticalerror is less than 1 g / cm for protons and iron.The fitted values of t norm are much larger than the ini-tial parameterisation predictions, consequently the pre-7 ( E/eV ) log
18 18.5 19 19.5 m ass f r ac t i on s pHeNFe EPOS-LHC ( E/eV ) log
18 18.5 19 19.5 m ass f r ac t i on s QGSJetII-04 ( E/eV ) log
18 18.5 19 19.5 m ass f r ac t i on s Sibyll2.3 ( E/eV ) log
18 18.5 19 19.5 p - va l u e -2 -1
10 1
EPOS-LHC QGSJetII-04 Sibyll2.3 QGSJetII-04 fixed
FIG. 49: Fitting t norm , σ norm and the mass fractions of our parameterisations to FD X max data measured by thePierre Auger Observatory. The fitted mass fractions and p-values for each fitted model are shown. The red solidsquares show the p-values for QGSJetII-04 when fitting only the mass fractions ( t norm and σ norm fixed).dicted (cid:104) X max (cid:105) from the fits are much larger than the ini-tial predictions. The fitted σ norm values are also largerthan the initial predictions, consequently the predicted σ ( X max ) from the fit is larger. After the fit of t norm and σ norm , our Epos-LHC, QGSJetII-04 and Sibyll2.3 param-eterisations still have different predictions for the X max distribution shape properties as a function of mass andenergy, but despite this there is reasonable agreement on the reconstructed mass composition from these fits.An observed shift in the fitted values of t norm and σ norm from the initial parameterisation prediction could be dueto the initial parameterisation inadequately describingnature, systematics in the measured X max values, or acombination of both factors. Degeneracy between thefitted parameters could also contribute to a shift in thefitted coefficients, however the performance analysis in8 ( E/eV ) log
18 19 æ l n A Æ EPOS-LHCQGSJetII-04Sibyll2.3 ( E/eV ) log
18 18.5 19 19.5 V ( l n A ) FIG. 50: First two moments of the ln A distributionestimated from the fitted fractions of the t norm , σ norm and mass fraction fit of the FD X max distributionsmeasured by the Pierre Auger Observatory.Section VI indicates that the results presented here areunlikely to be affected by degeneracy.The mass composition results are sensitive to the as-sumed values of the X max distribution properties whichare not affected by the fit of t norm and σ norm (such asthe elongation rate and the (cid:104) X max (cid:105) separation between pand Fe). The results are also sensitive to the fitting rangelimits. As our knowledge of the hadronic physics occur-ring at the highest energies progresses, the coefficientswhich are fitted and the fitting range limits applied maychange. For example, a reduced upper limit of t norm would result in the t norm , σ norm and mass fraction fit ofthe Auger data reconstructing a mass composition con-sisting of predominantly proton and helium. An increasein the statistics of the Auger X max data, and/or an in-creased energy range, can reveal additional informationregarding the shape coefficients.Using the fitted values of t norm and σ norm , the pa-rameters of the equations in [17], to convert the X max moments into ln A moments, have been determined andare shown in Tables II and III of Appendix D.Given the large t norm and σ norm values fitted to theAuger data when the mass fractions, t norm and σ norm are fitted, a second set of fits were performed where only t norm and the mass fractions were fitted to the Augerdata, using the same t norm fitting range. These fitsof the three parameterisations each used the standardQGSJetII-04 σ prediction. The resulting mass composi- ( E/eV ) log
18 18.5 19 19.5 ] [ g / c m æ X m ax Æ EPOS-LHCQGSJetII-04Sibyll2.3Auger ( E/eV ) log
18 18.5 19 19.5 ] ( X m ax ) [ g / c m s FIG. 51: The black lines show the (cid:104) X max (cid:105) and σ ( X max )initially predicted by the X max parameterisations forproton and iron. The red, blue and green lines show thenew predictions for the (cid:104) X max (cid:105) and σ ( X max ) after fits of t norm , σ norm and the mass fractions to FD X max distributions measured by the Pierre AugerObservatory.tion, ln A and X max moments are shown in Figs. 52, 53and 54 respectively. The fitted values of t norm are shownin Table I of Appendix C, and using these values the pa-rameters of the equations in [17] have been determinedand are shown in Tables IV and V of Appendix D.As the fitted values of t norm are not as large compared9 ( E/eV ) log
18 18.5 19 19.5 m ass f r ac t i on s pHeNFe EPOS-LHC ( E/eV ) log
18 18.5 19 19.5 m ass f r ac t i on s QGSJetII-04 ( E/eV ) log
18 18.5 19 19.5 m ass f r ac t i on s Sibyll2.3 ( E/eV ) log
18 18.5 19 19.5 p - va l u e -2 -1
10 1
EPOS-LHC QGSJetII-04 Sibyll2.3
FIG. 52: Fitting t norm and the mass fractions of our parameterisations to FD X max data measured by the PierreAuger Observatory. The fitted mass fractions and p-values for each fitted model are shown.to the two-coefficient fit, the predicted (cid:104) X max (cid:105) of the fitsare not as large, but still quite large compared to theinitial parameterisation predictions. The reconstructedmass composition from the fits of only t norm (Fig. 52)consists of a larger abundance of nitrogen and protons,at the expense of iron and helium, compared to that ofthe t norm and σ norm fit (Fig. 49). The general transitionof the mass composition for the three parameterisations isconsistent between the one-coefficient and two-coefficient fits.0 ( E/eV ) log
18 19 æ l n A Æ EPOS-LHCQGSJetII-04Sibyll2.3 ( E/eV ) log
18 18.5 19 19.5 V ( l n A ) FIG. 53: First two moments of the ln A distributionestimated from the fitted fractions of the t norm andmass fraction fit of the FD X max distributions measuredby the Pierre Auger Observatory. ( E/eV ) log
18 18.5 19 19.5 ] [ g / c m æ X m ax Æ EPOS-LHCQGSJetII-04Sibyll2.3Auger ( E/eV ) log
18 18.5 19 19.5 ] ( X m ax ) [ g / c m s FIG. 54: The black lines show the (cid:104) X max (cid:105) and σ ( X max )initially predicted by the X max parameterisations forproton and iron. The red, blue and green lines show thenew predictions for the (cid:104) X max (cid:105) and σ ( X max ) after fits ofthe mass fractions and t norm (applying the standardQGSJetII-04 σ prediction) to FD X max distributionsmeasured by the Pierre Auger Observatory.1 VIII. CONCLUSIONS
We have presented a novel method to estimate themass composition (from X max distributions) which is lessdependent on hadronic models. The method uses pa-rameterisations of X max distributions according to dif-ferent hadronic interaction models. Provided that themeasured X max distributions consist of different primarymasses and sufficient statistics over a large energy range(which seems to be the case for the Auger X max data),two shape coefficients, of the X max distribution parame-terisation, can be fitted together with the mass fractions,reducing the model dependency in the mass compositioninterpretation (we have tested the Epos-LHC, QGSJetII-04 and Sibyll2.3 models). The main differences betweenthe predicted X max distributions from different modelsare the normalisation values of the mode and spread foreach primary. So, by fitting two coefficients ( t norm and σ norm ) which adjust the normalisation of the mode andspread for each primary in an appropriate manner, theresulting mass composition is consistent for the threehadronic models tested here. A third coefficient, “ B ”, which adjust the energy dependence of the (cid:104) X max (cid:105) can befitted, further reducing the systematic model uncertaintyin the fitted mass composition. However, given the cur-rent statistics and limited energy range of the publishedAuger X max distributions and the possible distribution ofmasses, fitting this third parameter may introduce largesystematic uncertainties in the composition.The mass fraction, t norm and σ norm fits reconstruct amass composition trend with energy that is consistent be-tween the three models. There is a dominant abundanceof iron over the energy range, particularly at the high-est energies where there is almost pure iron. By fittingonly t norm and adopting the QGSJetII-04 σ predictionfor the three models, the relative abundance of protonsincreases.The results are sensitive to the other model parame-ters that we keep fixed, such as the elongation rate andthe (cid:104) X max (cid:105) separation between p and Fe. It is impor-tant to note that systematics in the measured X max val-ues are absorbed by the fits of t norm and σ norm . Thus,the composition fractions are not significantly affected bysystematics in X max . [1] T. K. Gaisser and A. M. Hillas, Proc. 15th ICRC , 353(1977).[2] K.-H. Kampert and M. Unger, Astropart. Phys. , 660(2012), arXiv:1201.0018 [astro-ph.HE].[3] P. Abreu et al. (Pierre Auger), Phys. Rev. Lett. ,062002 (2012), arXiv:1208.1520 [hep-ex].[4] T. Bergmann, R. Engel, D. Heck, N. N. Kalmykov,S. Ostapchenko, T. Pierog, T. Thouw, and K. Werner,Astropart. Phys. , 420 (2007), arXiv:astro-ph/0606564[astro-ph].[5] T. Pierog et al. , Nucl. Phys. Proc. Suppl. , 159 (2006),astro-ph/0411260.[6] T. Pierog, I. Karpenko, J. M. Katzy, E. Yatsenko,and K. Werner, Phys. Rev. C92 , 034906 (2015),arXiv:1306.0121 [hep-ph].[7] S. Ostapchenko, Phys. Rev.
D83 , 014018 (2011),arXiv:1010.1869 [hep-ph].[8] F. Riehn, R. Engel, A. Fedynitch, T. K. Gaisser, andT. Stanev, PoS(ICRC2015) (2016). [9] A. Aab et al. (Pierre Auger), Phys. Rev.
D90 , 122005(2014), arXiv:1409.4809 [astro-ph.HE].[10] A. Aab et al. (Pierre Auger), Phys. Rev.
D90 , 122006(2014), arXiv:1409.5083 [astro-ph.HE].[11] A. Aab et al. (Pierre Auger), JCAP , 038 (2017),arXiv:1612.07155 [astro-ph.HE].[12] C. J. Todero Peixoto, V. de Souza, and J. A. Bellido,Astropart. Phys. , 18 (2013), arXiv:1301.5555 [astro-ph.HE].[13] B. Peters, Nuovo Cimento , 800 (1961).[14] J. Linsley, Proc. 8th ICRC , 77 (1963).[15] Pierre Auger Collaboration, Astrophys. J Suppl. , 34(2012), arXiv:1210.3736 [astro-ph.HE].[16] V. S. Berezinsky and S. I. Grigor’eva, Astron. Astrophys. , 1 (1988).[17] P. Abreu et al. (Pierre Auger), JCAP , 026 (2013),arXiv:1301.6637 [astro-ph.HE]. Appendix A: Fits to X max distributions The fits of Equation (1) to energy binned X max data are shown in Figs. 55, 56 and 57. The differences in the (cid:104) X max (cid:105) and σ ( X max ) of the data versus the fitted equation are shown in Fig. 58. For the fitted equation, (cid:104) X max (cid:105) fit = t + λ and σ (X max ) fit = √ σ + λ . Although the fitted function (red line) does not always precisely overlap the data (blueline), we see (cid:104) X max (cid:105) fit is always within 0 . of (cid:104) X max (cid:105) data . The (cid:104) X max (cid:105) of the distribution is the main propertywe endeavour to accurately define. σ (X max ) fit is always within 3 g/cm of σ (X max ) data which is acceptable.
1. Epos-LHC X max distribution fits
600 800 1000 logE = 17.0 p
600 800 1000 logE = 17.5
600 800 1000 logE = 18.0
600 800 1000 logE = 18.5
600 800 1000 logE = 19.0
600 800 1000 logE = 19.5
600 800 1000 He
600 800 1000600 800 1000600 800 1000600 800 1000600 800 1000 600 800 1000 N
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600 800 1000600 800 1000600 800 1000600 800 1000600 800 1000 ] Xmax [ g/cm
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FIG. 55: Energy binned Epos-LHC X max distributions (blue line) fitted with Equation (1) (red line).3
2. QGSJetII-04 X max distribution fits
600 800 1000 logE = 17.0 p
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600 800 1000 logE = 18.0
600 800 1000 logE = 18.5
600 800 1000 logE = 19.0
600 800 1000 logE = 19.5
600 800 1000 He
600 800 1000600 800 1000600 800 1000600 800 1000600 800 1000 600 800 1000 N
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600 800 1000600 800 1000600 800 1000600 800 1000600 800 1000 ] Xmax [ g/cm
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FIG. 56: Energy binned QGSJetII-04 X max distributions (blue line) fitted with Equation (1) (red line).4
3. Sibyll2.3 X max distribution fits
600 800 1000 logE = 17.0 p
600 800 1000 logE = 17.5
600 800 1000 logE = 18.0
600 800 1000 logE = 18.5
600 800 1000 logE = 19.0
600 800 1000 logE = 19.5
600 800 1000 He
600 800 1000600 800 1000600 800 1000600 800 1000600 800 1000 600 800 1000 N
600 800 1000600 800 1000600 800 1000600 800 1000600 800 1000 600 800 1000 Fe
600 800 1000600 800 1000600 800 1000600 800 1000600 800 1000 ] Xmax [ g/cm
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FIG. 57: Energy binned Sibyll2.3 X max distributions (blue line) fitted with Equation (1) (red line).5 X max moment comparison between the fitted parameterisation and the data ( E/eV ) log
17 17.5 18 18.5 19 19.5 ( f i t - da t a ) æ X m a x > Æ p He N Fe EPOS-LHCQGSJETII-04Sibyll2.3 ( E/eV ) log
17 17.5 18 18.5 19 19.5 ( X m a x ) ( f i t - da t a ) s FIG. 58: Difference in the (cid:104) X max (cid:105) and σ (X max ) between the data and the fitted equation.6 Appendix B: Mass fraction, t norm and σ norm fits of the Auger FD X max data The t norm , σ norm and mass fraction fits of each parameterisation to the Auger FD X max distributions are shownin the following plots. The magenta lines illustrate the measured X max distributions, while the teal lines illustratethe fitted parameterisation. The black, red, green and blue lines are the fitted proton, helium, nitrogen and ironparameterisations respectively.
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FIG. 59: Fit of the Epos-LHC X max parameterisation.7
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FIG. 60: Fit of the QGSJetII-04 X max parameterisation.8
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FIG. 61: Fit of the Sibyll2.3 X max parameterisation.9 Appendix C: Table of coefficients for the X max distribution parameterisations TABLE I: Coefficients of Equation (2) for the Epos-LHC, QGSJetII-04 and Sibyll2.3 X max distribution predictions,assuming a normalisation energy of E = 10 . eV. Also in the table, we show the t norm and σ norm fitted to theAuger data from the t norm , σ norm and mass fraction fit of each of the three models, and the t norm fitted to theAuger data from the t norm and mass fraction fit of each model. Epos-LHC Proton Helium Nitrogen Iron t norm
703 697 680 650 B σ norm C -0.63 -1.81 -1.67 -1.36 λ norm K L -25.93 0.063 0.035 0.027fitted t norm
740 (stat.) +2 −
734 717 688fitted σ norm
37 (stat.) +2 −
40 32 22fitted t norm only 731 (stat.) +1 −
725 708 678QGSJetII-04 Proton Helium Nitrogen Iron t norm
688 679 660 635 B σ norm C -1.32 -1.24 -0.99 -0.91 λ norm K L -17.63 -6.08 0.041 0.040fitted t norm
738 (stat.) +1 −
730 711 685fitted σ norm
32 (stat.) +1 −
35 30 21fitted t norm only 729 (stat.) +1 −
721 702 676Sibyll2.3 Proton Helium Nitrogen Iron t norm
715 701 678 650 B σ norm C -1.08 -0.82 -1.20 -0.77 λ norm K L -27.47 -6.84 0.083 0.044fitted t norm
748 (stat.) +1 −
735 712 684fitted σ norm
42 (stat.) +1 −
36 29 21fitted t norm only 741 (stat.) +1 −
727 704 676 Appendix D: X max moments in terms of ln A moments. The first two X max moments can be parameterised in terms of ln A as follows [17]: (cid:104) X max (cid:105) = X + D log 10 (cid:18) EE A (cid:19) + ξ ln A + δ ln A log 10 (cid:18) EE (cid:19) , (D1)and σ ( X max ) = σ p [1 + a (cid:104) ln A (cid:105) + b (cid:104) (ln A ) (cid:105) ] , (D2)where σ p = p + p log 10 (cid:18) EE (cid:19) + p (cid:20) log 10 (cid:18) EE (cid:19)(cid:21) ,a = a + a log 10 (cid:18) EE (cid:19) . (D3)Using the t norm and σ norm fit results of the 2014 FD dataset (see Table I), the parameters of Equations (D1), (D2)and (D3) have been determined, and are displayed in Tables II and III. The mean and maximum (cid:104) X max (cid:105) residualsof the fit are ∼ / cm and ∼ . / cm respectively. The mean and maximum σ ( X max ) residuals of the fit are ∼ / cm and ∼ . / cm respectively. parameter Epos-LHC QGSJetII-04 Sibyll2.3 X ± ± ± D ± ± ± ξ -0.10 ± ± ± δ ± ± ± TABLE II: Parameters of Equation (D1), obtained by fitting the predicted (cid:104) X max (cid:105) from the t norm and σ norm fit ofthe 2014 FD data set. All values are in g / cm . parameter Epos-LHC QGSJetII-04 Sibyll2.3 p × g − cm ±
19 4402 ±
32 5222 ± p × g − cm -361 ±
20 -427 ±
33 -413 ± p × g − cm ±
33 71 ±
54 87 ± a -0.377 ± ± ± a -0.0038 ± ± ± b ± ± ± TABLE III: Parameters of Equation (D2) and Equation (D3), obtained by fitting the predicted σ ( X max ) from the t norm and σ norm fit of the 2014 FD data set.1Using the results from the fit of only t norm and the mass fractions to the 2014 FD dataset (see Table I), theparameters of Equations (D1), (D2) and (D3) are displayed in Tables IV and V. The (cid:104) X max (cid:105) and σ ( X max ) residualsof these results are similar to those from the t norm , σ norm and mass fraction fit results. parameter Epos-LHC QGSJetII-04 Sibyll2.3 X ± ± ± D ± ± ± ξ -0.10 ± ± ± δ ± ± ± TABLE IV: Parameters of Equation (D1), obtained by fitting the predicted (cid:104) X max (cid:105) from the t norm fit of the 2014FD data set parameter Epos-LHC QGSJetII-04 Sibyll2.3 p × g − cm ±
35 3990 ±
44 4049 ± p × g − cm -355 ±
36 -411 ±
45 -392 ± p × g − cm ±
61 74 ±
76 89 ± a -0.459 ± ± ± a -0.0022 ± ± ± b ± ± ± TABLE V: Parameters of Equation (D2) and Equation (D3), obtained by fitting the predicted σ ( X max ) from the t normnorm