Theory of Hadronic Interactions and Its Application to Modeling of Cosmic Ray Hadronic Showers Ralph Engel & Heinigerd Rebel Forschungszentrum Karlsruhe, Germany
Outline cosmic ray flux and energies physics of air showers physics of air showers and their characteristics basic methods for measuring energy and composition hadronic interaction models pomeron interpretation, minijets and parton densities unitarization and cross sections leading particle production generalization to nuclear projectiles and targets predictions for LHC (central and forward) constraints from cosmic ray data
Local interstellar cosmic ray flux all energies refer to total particle energy measured flux extends to E lab 3 10 20 ev highest energy particles extremely rare still unexplained: sources, knee, ankle... existence of GZK cutoff at E 5... 7 10 19 ev? hypothetical sources: below ankle: galactic above ankle: extra-gal. particle flux dn/dln(e) (cm -2 sr -1 s -1 ) 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-10 10-11 10-12 10-13 10-14 10-15 10-16 10-17 10-18 10-19 10-20 10-21 direct measurements 1 (m 2 sr hr) -1 1 (km 2 sr min) -1 1 (km 2 sr day) -1 1 (km 2 sr century) -1 10-22 10 11 10 12 10 13 10 14 10 15 10 16 10 17 10 18 10 19 10 20 10 21 particle energy E JACEE AGASA Akeno CASA-BLANCA CASA-MIA DICE Fly s Eye (mono) Fly s Eye (stereo) Grigorov Haverah Park HEGRA KASCADE MSU Tibet Tien Shan Yakutsk air shower measurements (ev / particle)
Hypothetical sources of high-energy cosmic rays Crab Nebula (SNR) and M87 (AGN)
Cosmic ray flux: energy comparison power-law spectra low-energy index: -2.7 high-energy index: -3.1 particle type low energy: mainly H... Fe ions high energy: unknown measured flux extends to spp 400 TeV knee coincides with Tevatron energy ankle above LHC energy scaled particle flux E 2.7 dn/de (cm -2 sr -1 s -1 GeV 1.7 ) 1000 100 10 1 0.1 fixed target s pp (GeV) 10 2 10 3 10 4 10 5 10 6 HERA RHIC TEVATRON AGASA Akeno CASA-BLANCA CASA-MIA DICE Fly s Eye (mono) Fly s Eye (stereo) Grigorov Haverah Park HEGRA JACEE KASCADE MSU Tibet Tien Shan Yakutsk LHC 0.01 10 11 10 12 10 13 10 14 10 15 10 16 10 17 10 18 10 19 10 20 10 21 E (ev / particle)
Introduction to extensive air showers
Simulation of extensive air showers particle production (hadronic, electroweak) propagation, decay energy loss and deflection in geomag. field public MC codes: CORSIKA (Heck et al.) AIRES (Sciutto et al.) SENECA (Drescher et al.) h (km) 35 30 25 20 15 10 γ p Fe problematic: simulation and extrapolation of hadronic multiparticle production 5 0-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x (km)
Characterization of extensive air showers shower size N e X 0 12 km 200 700 X max α ln(e 0 /A) N e(max) α E 0 secondaries mainly e ±, γ π 0 s transfer energy to em. component 6 km slant depth (g/cm 2 ) 1030 1200 sea level vertical shower sea level zenith angle of 30 deg. mean particle energy at maximum 0.6 GeV 70... 80% of primary energy converted to ionization energy
Model-dependence: energy/composition via shower profile simulation of shower profile: mean depth of shower maximum: particle multiplicity N e 3.5 x 10 11 3.0 x 10 11 2.5 x 10 11 2.0 x 10 11 1.5 x 10 11 1.0 x 10 11 5.0 x 10 10 E = 3.2 x 10 20 ev SIBYLL 2.1 Fly s Eye event proton iron gamma (mag.) <X max > (g/cm 2 ) 1000 900 800 700 600 500 CACTI CASA-BLANCA DICE Fly s Eye Haverah Park HEGRA HiRes-MIA SPASE-VULCAN Yakutsk SIBYLL 2.1 QGSjet01 p Fe 0.0 x 10 0 0 200 400 600 800 1000 1200 1400 atmospheric depth X (g/cm 2 ) 400 strong dependence on hadronic interaction model 300 10 14 10 15 10 16 10 17 10 18 10 19 10 20 primary energy E (ev)
Model-dependence: energy/composition via N e -N µ superposition model: proton of energy E muon number 10 6 10 5 DPMJET 2.55 nexus 2 QGSJET 01 SIBYLL 2.1 Fe 10 15 ev Fe 10 16 ev p Fe 10 17 ev p N µ E 0.89 N e E ln E 10 4 Fe 10 14 ev p nucleus of energy E and mass number A N µ A(E/A) 0.89 N e E ln(e/a) 10 3 p 10 3 10 4 10 5 10 6 10 7 electron number (D. Heck, ICRC 2001)
300 VENUS 200 Models: 100 charged particle multiplicity in p-n collisions 1997 N ch 500 400 DPMJET 2.5 nexus 2 QGSJET 01 SIBYLL 2.1 300 200 100 0 10 11 10 12 10 13 10 14 10 15 10 16 10 17 10 18 10 19 10 20 model predictions by more than factor 2 at high energy Average charged particle multiplicity as function of energy for p- N collisions. similar differences for π-n collisions mean muon multiplicities of showers differ only by 20% E lab (ev) (D.Heck, ICRC 2001) D. Heck heck@ik3.fzk.de 27 ICRC, Hamburg, 07-15 August 2001 13
Muon production: Heitler s model E only charged secondaries initiate new cascades E/n tot n tot = n ch+ n neut decay if critical energy E c is reached E c = E/(n tot ) N 2 E/(n ) tot E/(n ) N tot ( n ch ) 2 (n ch ) N charged particles decay producing one muon N µ = (n ch ) N eliminating N gives ( ) α E N µ = E c α = ln n ch ln n tot 0.85... 0.92 showers have reduced sensitivity to high energy interactions
Requirements on hadronic interaction models physics requirements: simulation of p, π, K,... collisions with air nuclei coverage of full energy range from production threshold to s 400 TeV minimum bias event simulation tuned to give optimal description of energetic secondary particles technical requirements: variable projectile/target combinations variable energy simulation fast simulation code
Cosmic ray hadronic interaction models high-energy models: DPMJET II.5 and III (Ranft / Roesler, RE & Ranft) nexus 2.0 and 3.0 (Drescher, Hladik, Ostapchenko, Pierog & Werner) QGSjet 98 and 01 (Kalmykov & Ostapchenko) SIBYLL 1.7 and 2.1 (Engel / RE, Fletcher, Gaisser, Lipari & Stanev) all QCD-inspired models (minijets) low-energy models: GHEISHA/GEANT (Fesefeldt) Hillas splitting algorithm (Hillas) FLUKA (Fasso, Ferrari, Ranft, Sala) UrQMD (Bass, Bleicher et al.) TARGET (RE, Gaisser, Protheroe & Stanev) HADRIN/NUCRIN (Hänßgen & Ranft) mostly parametrizations of data
What about PYTHIA, HIJING,...? most HEP models not designed for leading particle production other models lack completeness example: HIJING predictions and bubble chamber data (Pop, Gyulassy & Rebel: Astropart. Phys. 10 (1999) 211)
Partonic view: structure of QCD inspired models model for nucleon-nucleon interaction central particle production (soft interactions, minijet production) projectile leading particle production (projectile remnant, diffraction dissociation) generalization to hadron-nucleus and nucleus-nucleus interactions fragmentation and hadronization target intranuclear cascade, etc.
Nucleon-nucleon scattering: pomeron description color dipole scattering, exchange of vacuum quantum numbers: pomeron q q qq q q > q qq q two-gluon scattering in QCD-improved parton model: ~ > unitarity cut
Perturbative QCD: hard part of the pomeron conventional phenomenology: two-gluon scattering and collinear factorization projectile f(x,q ) 1 2 inclusive jet-pair cross section: σ jet 1 = f d 2 A (x 1, Q 2 ) f B (x 2, Q 2 ) dˆσ i,j k,l p 1 + δ kl d 2 p i,j,k,l parton density evolution (DGLAP): df(x, Q 2 ) dlogq 2 = α s(q 2 ) 2π P (x/y) f(y, Q2 ) factorization: parton densities are universal jet pair target nucleus (air) f(x,q 2 2 )
Minijet cross section: energy and cutoff dependence QCD expectation: f(x, Q 2 ) 1/x 1+ 12 13 14 15 16 17 18 19 20 100000 10000 total cross section: DL fit GRV 98 SIBYLL 2.1 EHLQ set 1, p t > 2 GeV log 10 ( E lab / ev ) σ jet s ln s old extrapolations: 0... 0.2 σ hard (mb) 1000 p t cutoff = 2 GeV 100 HERA measurements: 0.3... 0.4 10 100 1000 10000 100000 (GeV) E cm p t cutoff = 4 GeV inclusive cross section exceeds total cross section: many minijet pairs produced per p-p collision σ jet = n jet σ inel
Reliability of minijet phenomenology kinematics: x 1 x 2 4p 2 /s parton densities measured in limited x, Q 2 range x values 10 0 12 13 14 15 16 17 18 19 20 cutoff p T cutoff = 2 GeV p T cutoff = 4 GeV p T = 8 GeV 10-1 10-2 10-3 10-4 10-5 log 10 ( E lab / ev ) (x 1 = x 2 ) higher order corrections (k-factor) parton density saturation effects 10-6 10-7 10-8 10-9 (x 1 = 1) 100 1000 10000 100000 E cm (GeV)
Model implementations of hard scattering QGSjet: pre-hera parton densities, no saturation DPMJET III: energy-dependent transverse momentum cutoff (summation of lowest-order enhanced diagrams, string fusion) p cutoff ( ( s s) = p 0 + 0.12GeV log 10 50GeV ) 3 SIBYLL: energy-dependent transverse momentum cutoff p cutoff ( s) = p 0 {0.9 + 0.065GeV exp } ln(s) numerically similar to Golec-Biernat & Wüsthoff model ( ) λ ( ) λ/2 x s Q 2 s (x) = Q2 0 Q 2 0 λ 0.25... 0.37 x 0 s 0
Unitarization: eikonal approximation standard in literature: two-component eikonal models σ inel = d 2 b (1 exp { 2χs 2χ h }) χ s/h (s, b) = σ s/h A(s, b) with d 2 b A(s, b) = 1 profile function A(s, b) ( p t) ( B) ~ 1 R o R ~ log(s) low energy high energy
Cross section fits: model parameters 12 13 14 15 16 17 18 200 150 p t cutoff = 2.5 GeV 3.5 GeV log 10 ( E lab / ev ) R 0 2 = 3.5 GeV -2 10-1 10-2 E735 data p t cutoff = 2.5 GeV R 0 2 = 1.5 GeV -2 σ (mb) 100 50 σ tot R 0 2 = 1.5 GeV -2 dp nch /dn ch 10-3 10-4 p t cutoff = 3.5 GeV R 0 2 = 3.5 GeV -2 0 σ ela 10 2 10 3 10 4 s (GeV) 10-5 0 50 100 150 200 n ch reasonable cross section fits possible for many parameter combinations correlation between eikonal function and multiplicity distribution
Resummation according to unitarity cuts (AGK) analytical resummation possible for eikonal-type models cross section for n cut pomerons (i.e. n partonic interactions) σ (n) inel = d 2 (2χ) n b e 2χ n! P nch 0.1 0.01 UA5 data DPMJET III 1 cut Pomeron 2 cut Pomerons 3 cut Pomerons 4 cut Pomerons 5 cut Pomerons inelastic cross section σ inel = n=1 σ (n) ine 0.001 0.0001 0 10 20 30 40 50 60 70 80 n ch
Interpretation of the eikonal approximation unitarized amplitude a(s, b) = i 2 ( ) 1 e χ(s, b) χ(s, b) 1 2! [ χ(s, b) ] 2 + 1 3! [ χ(s, b) ] 3... problem: energy-momentum sharing not considered at amplitude level nexus: explicit calculation with energy-momentum conservation
Partonic view: structure of QCD inspired models model for nucleon-nucleon interaction central particle production (soft interactions, minijet production) projectile leading particle production (projectile remnant, diffraction dissociation) generalization to hadron-nucleus and nucleus-nucleus interactions fragmentation and hadronization target intranuclear cascade, etc.
Leading particle production: models DPMJET, QGSjet: leading particle distributions determined by pomeron/reggeon parameters (Mueller diagrams) f DPM nuc (x) x 1/2 q (1 x q ) 3/2, f DPM mes (x) x 1/2 q (1 x q ) 1/2 nexus, SIBYLL: fits to data f SIB nuc (x) (x2 q + µ2 /s) 1/4 (1 x q ) 3 distributions assumed to be energy-independent energy-momentum conservation influences distributions
Leading particle production: data fixed target experiments: dσ/dx lab (mb) E lab 5 10 11 ev 1000 100 10 Brenner et al. 100 GeV Brenner et al. 175 GeV Aguilar-Benitez et al. 400 GeV SIBYLL 2.1 QGSjet DPMJET II.5 HERA collider: sγp 200 GeV, E lab 2 10 13 ev dn/dx lab 10 1 ZEUS, high Q 2 ZEUS, low Q 2 SIBYLL 2.1 QGSjet DPMJET II.5 1 0.4 0.5 0.6 0.7 0.8 0.9 1 x lab = E p /E beam 0.1 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 x lab different parametrizations work well at low energy, however substantial differences in extrapolation
Gribov-Glauber approximation geometric picture: projectile b target (and its fluctuations) interact coherently with several nucleons σ inel = σ prod d 2 b [1 A k=1 d 2 b [1 exp ( 1 σ NN tot T N( b s k )) ] { σine NN T A( }] b) d 2 b [1 exp { σtot NN T A( }] b) known not to be a good approximation for central collisions first investigations of string fusion by Santiago group and Ranft show small impact on shower predictions
Current status of predictions: proton-air cross sections (mb) p-air σ inel 600 550 500 450 Mielke et al. Yodh et al. Baltrusaitis et al. Honda et al. Aglietta et al. rescaled by Block et al. 400 350 300 250 DPMJET II.5 nexus 2 QGSJET01 SIBYLL 2.1 Block et al. (D. Heck) 10 11 10 12 10 13 10 14 10 15 10 16 10 17 10 18 10 19 E lab (ev) difference QGSjet - SIBYLL: profile function A(s, b) difference nexus - QGSjet: pre- and post-hera parton densities
Current status of predictions: multiplicity distributions charged particle multiplicity distribution p- p collisions: s = 1800 GeV, E lab 1.7 10 15 ev dp nch /dn ch 10 0 10-1 10-2 10-3 E735 data SIBYLL 2.1 QGSjet 99 DPMJET III data became available only recently (no model tuning) crucial test of consistency of model parameters and cross sections 10-4 10-5 0 50 100 150 200 n ch
Current status of predictions: leading particle distributions 10 E lab = 10 12 ev s = 43 GeV SIBYLL 2.1 QGSjet 98 DPMJET 2.5 dn/dx lab 1 leading hadron distribution in p-air collisions model predictions similar at low energy significant divergence at high energy extremely inelastic collisions related to high multiplicity events popcorn-type effect dn/dx lab dn/dx lab 0.1 10 1 0.1 10 1 E lab = 10 18 ev s = 43 TeV E lab = 10 20 ev s = 433 TeV SIBYLL 2.1 QGSjet 98 DPMJET 2.5 SIBYLL 2.1 QGSjet 98 DPMJET 2.5 0.1 0 0.2 0.4 0.6 0.8 1 x lab
Model tuning: important phase space regions de T /dη (GeV) de/dη (GeV) 10 1 0.1 1000 100 10 CMS/TOTEM/Castor CDF, D0 s = 200 GeV s = 1800 GeV s = 14 TeV HEP: high-p jets and secondaries transverse energy leptonic secondaries air shower physics: total/inelastic cross section fraction of diff. dissociation energy flow particle multiplicity distribution hadronic secondaries 1-15 -10-5 0 5 10 15 pseudorapidity η
Model predictions: proton-proton at LHC transverse energy distribution de t /dη (GeV) 8 7 6 5 4 3 2 1 0-15 -10-5 0 5 10 15 transverse energy pseudorapidity η QGSjet01 SIBYLL 2.1 DPMJET 3 nexus 3 energy distribution de/dη (GeV) 2500 2000 1500 1000 500 total energy QGSjet01 SIBYLL 2.1 DPMJET 3 nexus 3 0-15 -10-5 0 5 10 15 pseudorapidity η
Model predictions: central vs. forward region particle distribution dn all /dη 14 12 10 8 6 4 QGSjet01 SIBYLL 2.1 DPMJET 3 nexus 3 particle distribution dn/dx F 10 1 QGSjet01 SIBYLL 2.1 DPMJET 3 nexus 3 2 0-15 -10-5 0 5 10 15 pseudorapidity η particle multiplicity 0.1 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Feynman-x particle multiplicity x F
Constraints from cosmic ray data difficulties with cosmic ray beams: no direct measurement of interactions possible selection of showers of fixed energy selection of showers of desired primary possible methods to constrain models: comparison of shower measurements to simulated showers assuming an energy spectrum and primary composition consistency checks within limits given by possible composition multiparameter measurements: check of parameter correlation model-independent limits on interaction characteristics impossible
Example: upper cross section bound? σ (mb) 12 13 14 15 16 17 18 19 20 250 200 150 100 pp/ pp data low p t cut high p t cut log 10 ( E lab / ev ) σ tot <X max > (g/cm 2 ) 1000 900 800 700 600 500 CACTI CASA-BLANCA DICE Fly s Eye Haverah Park HEGRA HiRes-MIA SPASE-VULCAN Yakutsk low p t cutoff high p t cutoff p Fe 50 σ ela 400 0 10 2 10 3 10 4 10 5 s pp (GeV) 300 10 14 10 15 10 16 10 17 10 18 10 19 10 20 primary energy E (ev) realistic model with steep cross section extrapolation (RE, ICHEP 2002) comparison of prediction with measurement (assumption: no photon primaries) shown example (based on CDF-like extrapolation) consistent with data exclusion of extreme cross section rise possible (but caveats)
Summary & conclusions cosmic rays and their interactions: interesting field of physics, many open questions measurements rely on detailed simulations hadron-nucleon interactions: models with high degree of self-consistency considerable uncertainty in extrapolation and reliability of minijet cross section no good theory of leading particle production hadron- and nucleus-nucleus interaction: need to understand RHIC data current phenomenology relies on forward hadron production data predictions for LHC central and forward particle production not directly correlated need measurements of central and forward region model-independent limits difficult to derive from cosmic ray data
Acknowledgments The authors thank all colleagues and friends who have contributed to this talk in various ways (discussions, plots, etc.): D. Heck, J. Alvarez-Muñiz, T.K. Gaisser, A. Haungs, S. Ostapchenko, J. Ranft, S. Roesler, and T. Stanev
Unitarity and optical theorem total cross section: σ tot = 1 Φ dp X M pp X 2 2 = = 1 dp X M pp X M + pp X Φ phase space integration d 3 k 2E = δ(k2 m 2 )d 4 k ( ) 1 Im d 4 k k 2 m 2 + iɛ results in particle propagators = Im = unitarity cut optical theorem: 1 s Im A pp pp(s, t = 0) = 1 Φ dp X M pp X 2 = σ tot
Topological expansion of QCD limit N c, N c /n f = const., g 2 N 2 c 1 example: ( t Hooft, Veneziano, Witten) planar graph (reggeon): q q q q > q qq unitarity cut q qq
Enhanced pomeron diagrams method of calculating gluon-gluon fusion different unitarity cuts possible: hard diffraction example: triple-pomeron graph 2 > resummation to all orders very difficult due to cancellations