On the relation between the proton-air cross section and fluctuations of the shower longitudinal profile
aa r X i v : . [ a s t r o - ph ] J un TH I NTERNATIONAL C OSMIC R AY C ONFERENCE
On the relation between the proton-air cross section and fluctuations of the showerlongitudinal profile
R. U
LRICH , J. B L ¨ UMER , , R. E NGEL , F. S CH ¨ USSLER AND
M. U
NGER Institut f¨ur Kernphysik, Forschungszentrum Karlsruhe, Postfach 3640, 76021 Karlsruhe, Germany Universit¨at Karlsruhe, Institut f¨ur Experimentelle Kernphysik, 76128 Karlsruhe, Germany
Abstract:
The current status and prospects of deducing the proton-air cross section from fluorescencetelescope measurements of extensive air showers are discussed. As it is not possible to observe the pointof first interaction, X , directly, other observables closely linked to X must be inferred from the mea-sured longitudinal profiles. This introduces a dependence on the models used to describe the shower de-velopment. Systematic uncertainties arising from this model dependence, from the reconstruction methoditself and from a possible non-proton contamination of the selected shower sample are discussed. Introduction
Indirect cosmic ray measurements by means of ex-tensive air shower (EAS) observations are difficultto interpret. Models needed for a deeper under-standing of the data have to be extrapolated overmany decades in energy. This is the case for highenergy (HE) interaction models, but also appliesto the primary composition of cosmic rays. Un-fortunately a changing primary composition andchanges in the HE interaction characteristics canhave similar effects on EAS development and aredifficult to separate.One of the key parameters for EAS development isthe cross section σ p − air of a primary proton in theatmosphere. Of course, only the part of the crosssection leading to secondary particle production isrelevant for EAS development, which we call forsimplicity here σ p − air . But also the productioncross section contains contributions which cannotbe observed in EAS. As diffractive interactions ofprimary particles with air nuclei do not (target dis-sociation) or weakly (projectile dissociation) influ-ence the resulting EAS, any measurement based onEAS is insensitive to these interactions. Therefore,we define an effective cross section to require aninelasticity k inel = 1 − E max E tot of at least 0.05 σ ∗ p − air = σ p − air ( k inel ≥ . . (1) In the following the amount of traversed matter be-fore an interaction with k inel ≥ . is called X .Taking this into account the reconstructed valueof σ ∗ p − air needs to be altered by a model depen-dent correction σ modelp − air ( k inel < . . This cor-rection amounts to 2.4 % for SIBYLL [1], 3.9 %for QGSJETII.3 [2] and 5.5 % for QGSJET01 [3],resulting in a model uncertainty of ∼ X max is folded with an Gaussian function having20 gcm − width, which corresponds roughly to theresolution of the Pierre Auger Observatory [5]. X max -distribution ansatz The most prominent source of shower fluctuationsis the interaction path length of the primary particlein the atmosphere. However the EAS developmentitself adds a comparable amount of fluctuations toobservables like X max . This is mainly due to theshower startup phase, where the EAS cascade isdominated by just a few particles. Our approachto fit the full distribution of X max does thereforehandle the primary interaction point explicitly and ROTON - AIR CROSS SECTION FROM LOGITUDINAL PROFILES (Energy/eV) log
10 11 12 13 14 15 16 17 18 19 C r o ss sec t i on ( p r o t on - a i r ) [ m b ] · p-air s QGSJETII, 1.1 · p-air s QGSJETII, 0.9 · p-air s QGSJETII,
Energy [GeV]
10 [GeV] pp sEquivalent c.m. energy 10 Mielke et al. 1994Baltrusaitis et al. 1984Nam et al. 1975Siohan et al. 1978Aglietta et al. 1999Honda et al. 1999Knurenko et al. 1999Belov et al.
Tevatron (p-p) LHC (p-p) LHC (C-C)
Figure 1: Impact of a 10 % change of σ p − air inQGSJETII at 10 EeV. Data from [6, 8, 9, 10, 11,12, 13].the EAS development in a parametric way dPdX expmax = Z dX max Z dX e − X /λ ∗ p − air λ ∗ p − air × P ∆ X (∆ X + X shift , λ ∗ p − air ) × P X max ( X expmax − X max ) , (2)where ∆ X was introduced as X max − X . Thusthe X max -distribution is written as a double con-volution, with the first convolution taking care ofthe EAS development and the second convolutionhandling the detector resolution. In this model wehave two free parameters λ ∗ p − air , which is directlyrelated to σ ∗ p − air , and X shift , needed to reduce themodel dependence. Note that Eq. (2) differs fromthe HiRes approach [6] and that used in the simula-tion studies in [7] by explicitly including the crosssection dependence in P ∆ X .The simulated P ∆ X -distributions can beparametrized efficiently with the Moyal func-tion P ∆ X (∆ X ) = e − ( t + e − t ) β √ π and t = ∆ X − αβ (3)using the two free parameters α and β . Impact of σ p − air on EAS development To include the cross section dependence of P ∆ X in a cross section analysis at 10 EeV, we modified ] -2 X [gcm D
600 700 800 900 1000 1100 1200 1300 X D dn / d -5 -4 -3 -2 /ndf=1.57) c · p-air s QGSJETII, /ndf=4.62) c · p-air s QGSJETII, /ndf=15.7) c · p-air s QGSJETII, Moyal function
Figure 2: Example fits of Eq. (3) to simulated P ∆ X -distributions at 10 EeV.CONEX for several HE models such that the crosssection used in the simulation is replaced by σ modifiedp − air ( E ) = σ p − air ( E ) · (1 + f ( E )) , (4)with the energy dependent factor f ( E ) , which isequal to for E ≤ f ( E ) = ( f − · log ( E/ PeV )log (1 EeV / PeV ) (5)for E > PeV, reaching f at E = 10 EeV.This modification accounts for the increasing un-certainty of σ p − air for large energies (see Fig. 1).Below 1 PeV (Tevatron energy), σ p − air is pre-dicted within a given HE model by fits to the mea-sured p ¯ p cross section.The cross section dependence of P ∆ X and the cor-responding parametrizations are shown in Fig. 2. ] -2 [gcm a
700 720 740 760 780 800 820 ] - [ g c m b i n c r eas i ng c r o ss sec t i on QGSJETIIQGSJETSIBYLL(QGSJET) p-air s (QGSJETII) p-air s (SIBYLL) p-air s Figure 3: Resulting σ p − air -dependence of theparametrized P ∆ X -distribution. The markers de-note the location of the original HE model crosssections. TH I NTERNATIONAL C OSMIC R AY C ONFERENCE [mb] model s - modified s -300 -200 -100 0 100 200 300 [ m b ] m od i f i e d s - r ec s -150-100-50050100150 (QGSJET) max X(QGSJET), X D (QGSJETII) max X(QGSJETII), X D (SIBYLL) max X(SIBYLL), X D other model permutations Figure 4: Sensitivity and HE model dependence ofthe σ p − air reconstruction for a pure proton compo-sition at 10 EeV.At large ∆ X , the simulated distributions are notperfectly reproduced by the parametrizations. Thiseffect worsens for large cross sections, as can beobserved from the increasing χ /ndf (see Fig. 2).Also the deviation of the Moyal function from the P ∆ X -distribution depends on the HE model. It isbiggest for QGSJETII and smallest for SIBYLL.Unfortunately this disagreement produces a sys-tematic overestimation of ∼
30 mb for the recon-structed σ p − air . This is visible in all the follow-ing results and will be addressed in future work bymaking the parametrization more flexible.The dependence of α and β on σ p − air can be in-terpolated with a polynomial of 2nd degree. Fig. 3gives an overview of this interpolation in the α - β plane. Obviously the P ∆ X predicted by differentHE model are not only a consequence of the dif-ferent model cross sections. Results
Pure proton composition
In Fig. 4 we show the reconstructed σ recp − air for sim-ulated showers with modified high energy modelcross section , σ modifiedp − air . The original HE crosssection σ modifiedp − air − σ modelp − air = 0 can be recon-structed with a statistical uncertainty of ∼
10 mb,whereas the uncertainty caused by the HE mod-els is about ±
50 mb. At smaller cross sectionsthe reconstruction results in a slight overestima-tion ( < mb). But for larger cross sections thereoccurs a significant underestimation of the inputcross section. This is mainly due to the worse de- Fraction of gamma [%] [ m b ] t r u e s - r ec s -300-250-200-150-100-50050 QGSJETQGSJETIISIBYLL / nd f c Figure 5: Systematic caused by photon primariesat 10 EeV.
Fraction of helium [%]
10 20 30 40 50 60 70 80 90 [ m b ] t r u e s - r ec s QGSJETQGSJETIISIBYLL
10 20 30 40 50 60 70 80 90 / nd f c Figure 6: Systematic caused by helium primariesat 10 EeV.scription of P ∆ X by the used Moyal function forlarge values of σ p − air (see last section). Photon primaries
Primary photons generate deeply penetratingshowers. Even a small fraction of photon show-ers has a noticeable effect on the tail of the X max -distribution [7]. Fig. 5 demonstrates how much afew percent of photons could influence the recon-structed σ p − air . The current limit on the photonflux is 2 % at 10 EeV [14]. Note that there is aclear trend of an increasing χ /ndf with increas-ing photon fraction, meaning the photon signal isnot compatible with the proton model. Helium primaries
On the contrary, helium induced EAS are very sim-ilar to proton showers. Therefore their impact on σ p − air is significant and very difficult to suppress,see Fig. 6. Interestingly, even for large helium con-tributions there is no degradation of the quality ofthe pure proton model fit ( χ /ndf is flat). Thus it ROTON - AIR CROSS SECTION FROM LOGITUDINAL PROFILES ] -2 [gcm max X600 700 800 900 1000 1100 1200 1300 m ax dn / d X -5 -4 -3 -2 pHeFe QGSJET01QGSJETII.3SIBYLL2.1
Figure 7: Composition impact on X max at 10 EeV.is not possible in a simple way to distinguish be-tween a 25% proton / 75% helium mixture or justa pure proton composition with a cross section in-creased by about 150 mb. Outlook: Mixed primary composition
Fluctuations and the mean value of the X max -distribution are frequently utilized to infer the com-position of primary cosmic rays [15]. It is well un-derstood how nuclei of different mass A produceshower maxima at different depth X max ( A ) andhow shower-to-shower fluctuations decrease with A (semi-superposition model).The relative change of the X max -distribution froma pure proton to a pure mass A primary composi-tion can be evaluated using CONEX. To fit X max -distributions we use the formula [16] dPdX max ( A ) = N · e − (cid:16) √ X max − X peak) γ · ( X max − X peak+3 · δ ) (cid:17) (6)with four parameters N , X peak , γ and δ . The nor-malization constant N was not fitted, but set to re-produce the known number of events. Fig. 7 showshow the X max -distributions for proton, helium andiron primaries are positioned relative to each otherfor several HE models. This relative alignment canbe utilized during σ p − air -fits to reduce the com-position dependence.The total mixed composition X max -distribution is then the weighted sum of theindividual primaries dPdX mixmax ( X max ) = X i ω i dPdX max ( A i , X max ) (7) where the weights ω i are additional free parame-ters to be fitted together with X shift and λ ∗ p − air .The shape of dPdX max ( A ) for A > is always as-sumed to change relative to the proton distribution.First studies indicate that the correlation betweenthe reconstructed composition and the correspond-ing σ p − air does not allow a measurement of thecross section. The situation is expected to be morepromising if the parameter X shift is fixed, however,the model dependence of the analysis will then belarger than shown here. References [1] R. Engel et al. volume 1, page 415.26 th ICRC Utah, 1999.[2] S. Ostapchenko.
Nucl. Phys. (Proc.Suppl.) ,151:143, 2006.[3] N.N. Kalmykov et al.
Nucl. Phys. B (Proc.Suppl.) , 52B:17, 1997.[4] T. Bergman et al.
Astropart. Phys. 26 , 26:420,2007.[5] B.R. Dawson [Pierre Auger Collaboration].30 th ICRC, these proceedings,
Nucl. Phys. (Proc. Suppl.) ,151:197, 2006.[7] R. Ulrich et al. 14 th ISVHECRI Weihai,astro-ph/0612205, 2006.[8] H.H. Mielke et al.
J. Phys. G , 20:637, 1994.[9] M. Aglietta et al.
Nucl. Phys. A (Proc.Suppl.) ,75A:222, 1999.[10] R.M. Baltrusaitis et al.
Phys. Rev. Lett. ,15:1380, 1984.[11] S.P. Knurenko et al. volume 1, page 372.26 th ICRC Utah, 1999.[12] M. Honda et al.
Phys. Rev. Lett. , 70:1993,1993.[13] T.K. Gaisser et al.