Shower center of gravity and interaction characteristics

Transcription

1 EPJ Web of Conferences, (25 DOI:.5/epjconf/25 c Owned by the authors, published by EDP Sciences, 25 Shower center of ravity and interaction characteristics Lev Kheyn a Institute of Nuclear Physics, Moscow State University, Moscow Abstract. The shower center of ravity is used for studyin the interconnection between shower lonitudinal profile and hadronic interaction characteristics. The equations for the shower oriinated by hih enery proton in the atmosphere are written and, within certain simplifications, solved for the case of loarithmically decreasin interaction lenth of hadrons in the air. The obtained expression explicitely splits into center of ravity of the purely electromanetic cascade at the primary proton enery and modification of that by hadronic cascadin and provides transparent view of the way in which hadronic interaction characteristics determine the lonitudinal shower development.. Introduction Since lon ao, numerous attempts have been undertaken to explicitely connect the air shower lonitudinal profile, in particular the shower maximum depth (X max, with hadronic interaction characteristics. Very approximate approaches were tried, from toy models to extensions of the Heitler model for the electromanetic shower [] to the hadronic shower [2 4] and all these have proven to be of not bi quantitative help. A direct way to establish the connection is the use of the cascade theory. A problem is that the shower maximum is an inconvenient quantity for cascade equations. Rather, a convenient quantity is the shower center of ravity (CG. Althouh the shift between X max and CG is enery and enerator dependent, havin been successfully solved equations for CG would allow to look in detail into the dynamics of shower lonitudinal development, which overns both X max and CG, and to see explicitely how interaction characteristics enter that dynamics. 2. Expression for the center of ravity We consider dependence on enery of the shower center of ravity: X(E = XN(X dx/ N(X dx = XN(X dx/ E, ( where N(X is the number of chared particles in a shower. The proton primary is considered. In simplifyin assumptions of i Feinman scalin, ii cascadin only two types of hadrons, baryons (nucleons and pions, iii nelectin pion decay the system of equations for the nominator of ( = XN(X dx looks a kheyn@mail.cern.ch (arxiv:22.4: N (E = dn N N (x N (Ex dn N π (x π (Ex π (E = dn N (x (Ex E λ N (E (E = 2 dn π π (x π (Ex dn π (x (Ex E λ π (E x γ (Ex. (2 On the assumptions of loarithmically decreasin proton and pion interaction lenths an exact solution of that system of equations is possible. The obtained expression for the proton shower center of ravity looks: X N (E = X ( ln E E c δ 2 { λ N (E ef f N X µ N NN Nπ ππ [ λπ (E ef f π X µ π ] } (3 E is the enery of the proton, E c is the critical enery and X is the radiation depth in air, δ =.7 and N A is Avoadro number. This is an Open Access article distributed under the terms of the Creative Commons Attribution License 4., which permits unrestricted use, distribution, and reproduction in any medium, provided the oriinal work is properly cited. Article available at or

2 EPJ Web of Conferences -x*lo(x*dn/ Fraction(>X TeV 433 TeV TeV 433 TeV X X Fiure. Inclusive distributions for all particles production in proton-air interactions at two eneries from LHC enerator: x lox dn/ (left and the interal distibution, i.e. the fraction of enery contained above some x (riht. Interaction lenths have proven to be taken not at the primary enery but at some lower effective eneries: ( E ef f γnn N = E/ exp (4 NN and E ef f π ( = E ef f N / exp γnπ Nπ γ ππ ππ (5 As one can see, the expression for the proton shower center of ravity explicitly splits into two terms: the center of ravity of the purely electromanetic cascade at the proton primary enery and a modification of this by the hadronic cascade, X N (E = X e (E X had.the latter is determined by two competin oppositely directed processes: i carryin throuh enery by hadrons, that elonates the shower, ii enery dissipation in the hadronic interactions due to which electromanetic subshowers start at smaller eneries, and because of the loarithmic enery dependency of their center of ravity that results in shortenin of the total shower. The first proccess is represented by λ terms in the final expression, the second one is represented by µ terms. Characteristics of multiple production enter two kinds of expressions. The first kind reflects the enery transition between different sorts of hadrons, i.e. the from the baryon to the chared or neutral pions or from the chared pion to the neutral pions (i,j below denote sort. The obtained expressions are simply mean relative eneries contained in the produced particles of some sort (like the inelasticity: ij = x dn i j (x. (6 Enery transition overns the pace of the shower elonation over hadron cascadin. The second kind reflects the rate of enery dissipation: γ ij = x ln x dn i j (x, (7 µ i = x ln x dn i X (x. ( The meanin of weiht x lnx is clear: each particle contributes to the total center of ravity proportionally to its enery and the lonitudinal width of the produced at that enery electromanetic shower which is proportional to the loarithm of enery. These factors are neative: the dissipation of enery in hadronic cascade results in electromanetic subcascades startin at smaller and smaller eneries and thus in reducin the total center of ravity relative to the absense of dissipation. The total mutiplicity is usually considered as a shower property definin enery dissipation. Instead, enery dissipation here is represented by the above interals which are in fact the forward multiplicities. It should be noted that althouh equations are written for the case of Feynman scalin, interals enterin final formula would be obtained for each primary proton enery from MC simulations for that particular enery and accordinly with accountin for whatever physics these MC manaes. includin scalin violation. Thus scalin violation proves to be partially accounted for (may be considered as to first order. 3. Results The shower center of ravity is calculated for five enerators of hadronic interactions: Sibyll[5], QGSJETII- 3[6, 7], QGSJETII-4[, ],.[, ] and LHC[2]. Fiure illustrates the statement that the interals definin enery dissipation could be considered as the forward multiplicities. In the left, the distribution x lo x dn/ is shown for all produced particles in interactions of protons with air at two eneries. From the riht, where the distribution of the fraction of the enery contained above a certain value of x is shown, it can be extracted that the main enery is contained in the forward reion, about 75% at x >.. Fiure 2 compares the dependence on enery of the total and forward multiplicities for two enerators, -p.2

3 ISVHECRI 24 Multiplicity QGSJETII-4 Forward multiplicity QGSJETII Fiure 2. Dependence on enery of the total (left and forward (riht multiplicities for QGSJETII-4 and LHC. Ni.6 πi.7 ππ.5 NN N Nπ QGSJETII QGSJETII π QGSJETII QGSJETII-4 Fiure 3. Dependence on the enery of the interals definin the enery transition for proton-air (left and pion-air (riht interactions. µ N -2.4 π µ QGSJETII QGSJETII QGSJETII QGSJETII-4 Fiure 4. Dependence on the enery of the interals definin the enery dissipation for proton-air (left and pion-air (riht interactions. -p.3

4 EPJ Web of Conferences Fiure 5. Enery dependence of the shower maximum (left and center of ravity (riht of the the proton shower. 2 Contributions (/cm 3 2 N π N QGSJETII π QGSJETII N QGSJETII-4 π QGSJETII-4 N π.. N π Fiure 6. Enery dependence of two types of contributions. Full lines are for nucleon and dotted lines for pion contributions. Eiht upper lines are X λ contributions, eiht bottom lines are X µ contributions, eiht middle lines are the sums X µ X λ. -p.4

5 ISVHECRI 24 QGSJETII-4 and LHC. The forward multiplicity is less dependent on enery, the spread between enerators is also smaller for the forward multiplicity. Dependence on enery of the interals of the two above kinds for five enerators is presented in the next two plots. Fiure 3 compares the dependence on enery of the interals definin enery transition ij = x dn i j (x for proton and pion interactions with air. One can see a bi spread between different enerators which is present even at LHC eneries. Fiure 4 compares the dependence on enery of the interals definin enery dissipation. µ i = ln x dn i X (x. The behavior for pion and proton (nucleon is very similar. The spread between enerators sinificantly increases with enery. Fiure 5 compares the dependence on enery of the shower maximum and the center of ravity. The disposition of enerators relative to one another is the same for X max and CG. However the spread for CG is sinificantly larer. The reason for that is that CG represents the whole shower while X max represents the developin stae of the shower. Accordinly, the number of contributin enerations of interactions is larer for CG. Since the effect of variation of interaction characteristics multiplicates with enerations the spread in CG exceeds the spread in X max. Fiure 6 shows contributions of four different terms to the hadronic part of CG: Xλ N = λ N (E ef f N, Xµ N = X µ N NN NN Xλ π = λ π(eπ ef f Nπ NN, Xµ π = X µ π ππ NN x Nπ ππ, such that Xλ N X µ N X λ π X µ π = X had. The spread of the pion contributions is reater than the spread of the nucleon contributions for both λ and µ terms. The elonation rate of the summed pion contribution is reater than the elonation rate of the summed proton contributions. The spread of the summed contributions sinificantly increases with enery, i.e. the elonation rate of CG is different for different enerators. 4. Summary Cascade equations for the proton shower center of ravity were solved under the assumptions of Feinman scalin, cascadin only two types of hadrons, baryons (nucleons and pions, nelectin pion decay and loarithmically decreasin proton and pion interaction lenths. The obtained expression for the proton shower center of ravity explicitely splits into the center of ravity of the purely electromanetic cascade at the primary proton enery and a modification of that by hadronic cascadin. The characteristics of interaction enter the expression throuh interaction lenths (cross-sections of protons and pions taken at some effective, lower than primary, eneries and interals over inclusive distributions of two types: interals definin enery transition between different sorts of hadrons which overn the cascade elonation and interals definin enery dissipation which define cascade shortenin. The latter are the forward multiplicities rather than the total multiplicities which are usually assummed to be responsible for enery dissipation. Althouh the relations between interaction characteristics (interals and shower properties for X max should be different from those for CG, presumably, more relevant for accountin for enery dissipation should be the forward multiplicities than the total multiplicities for X max as well. The center of ravity collects contributions from the whole shower, in particular essential contribution provides lon tail, whereas for X max matters the developin stae of shower. Accordinly, the number of contributin enerations of particle production is larer for CG. Since the effect of the variation of interaction characteristics multiplicates with enerations the difference between predictions of CG while usin enerators with different hadronic interaction models proves be larer than the difference between predictions of X max. Since the main contribution to the tail of the shower comes from pions CG exaerates the effect of the difference in characteristics of interactions of pions relative to X max. On the other hand contribution of pions to the CG elonation rate is larer than that of nucleons. It could be that for X max as well as the importance of the correct description of the pion interactions is not much less than of the proton interactions and tunin of the pion interactions deserves more attention than at present when the main attention in tunin is paid to the proton interactions. References [] W. Heitler. The Quantum theory of Radiation, Oxford Univ. Press (54 [2] J. Matthews, Astrop. Phys. 22 (25 37 [3] J.A.J. Matthews et al., J. Phys. G: Nucl. Part. Phys. 37, 2522 (2 [4] Stanev, T., 2, Hih Enery Cosmic Rays: Spriner Praxis Books, Volume. ISBN Spriner-Verla Berlin Heidelber, 2 (Spriner Praxis [5] E.-J. Ahn, R. Enel, T.K. Gaisser, P. Lipari, and T. Stanev, Phys. Rev. D (2 43 [6] S. Ostapchenko, Phys. Rev. D 74, 426 (26, hep-ph/5525 [7] S. Ostapchenko, Phys. Lett. B 636, 4 (26, hep-ph/623 [] S. Ostapchenko, Phys.Rev. D 3, 4 (2,.6 [] S. Ostapchenko, Phys. Rev. D 3 (2 4 [] K. Werner, F.M. Liu, T. Piero, Phys. Rev. C 74, 442 (26, hep-ph/56232 [] T. Piero, K. Werner, Nucl. Phys. Proc. Suppl. 6, 2 (2, 5. [2] T. Piero et al., arxiv:36.2 -p.5