Self-similarity in jet events following from p-p collisions at LHC
aa r X i v : . [ h e p - ph ] O c t Self-similarity in jet events following from p-p collisions at LHC
Grzegorz Wilk a , Zbigniew Włodarczyk b a National Centre for Nuclear Research, Department of Fundamental Research, Ho˙za 69, 00-681 Warsaw, Poland b Institute of Physics, Jan Kochanowski University, ´Swi¸etokrzyska 15, 25-406 Kielce, Poland
Abstract
Using a Tsallis nonextensive approach, we simultaneously analyze recent data obtained by the LHC ATLAS ex-periment on distributions of transverse momenta of jets, p jetT , together with distributions of transverse momenta ofparticles produced within these jets (defined relative to the jet’s axis), p relT , and their multiplicity distributions, P ( N ).The respective nonextensivity parameters for distributions of jets, q jet , for distributions of particles in jets, q rel andthe global nonextensivity parameter obtained from P ( N ), q N , were then compared with nonextensivity parameters q obtained from minimum bias pp collisions at energies corresponding to the energies of these jets. The values ofthe corresponding nonextensivity parameters were found to be similar, strongly indicating the existence of a commonmechanism behind all these processes. We tentatively identify this as a self-similarity property known to be presentthere and resulting in Tsallis type distributions. If confirmed, this would considerably strengthen the nonextensiveTsallis approach. Keywords: p − p collisions, jets, nonextensivity, self-similarity
1. Introduction
For some time now it is known that transverse momen-tum spectra of di ff erent kinds measured in multiparticleproduction processes, which change character from expo-nential at small values of p T to power-like at large p T , canbe described by a simple two-parameter formula, h ( p T ) = C (cid:18) + p T nT (cid:19) − n . (1)This was first proposed in [1] as the simplest formula ex-trapolating the large p T power behavior expected fromparton collisions to exponential behavior observed for p T →
0. At present it is known as the QCD-based
Hage-dorn formula [2] and was used in many fits to recent data.However, in many branches of physics Eq. (1), with n replaced by n = / (1 − q ), is more widely known as the Email addresses: [email protected] (Grzegorz Wilk), [email protected] (Zbigniew Włodarczyk)
Tsallis formula [3]. In this case, q is known as a nonex-tensivity parameter. In this form, Eq. (1) is usually sup-posed to represent a nonextensive generalization of theBoltzmann-Gibbs exponential distribution, exp( − p T / T ),used in a statistical description of multiparticle produc-tion processes, with q being a new parameter, in addi-tion to previous ”temperature” T . Such an approach isknown as nonextensive statistics [3] in which the parame-ter q summarily describes all features causing a departurefrom the usual Boltzmann-Gibbs statistics (in particularit can be shown that it is directly related to the possibleintrinsic, nonstatistical fluctuations of the temperature T [4, 6]). However, the Tsallis distribution also emergesfrom a number of other more dynamical mechanisms, forexample see [5] for more details and references. In allpossible scenarios leading to Eq. (1), the ”temperature”,or, in general, scale parameter T , is given by the meanvalue of the transverse momentum, h p T i = nT / ( n − Preprint submitted to Elsevier August 12, 2018 f transverse momenta, p T >> nT , Eq. (1) becomes scalefree (independent of T ) distribution. The Tsallis distribu-tion was successfully used for a description of all kinds ofmultiparticle production processes in a wide range of inci-dent energy (from few GeV up to few TeV) and in a broadrange of transverse momenta (see, for example, reviews[5, 6]. In particular, it turned out that it also successfullydescribes transverse momenta of charged particles mea-sured by LHC experiments, the flux of which changes byover 15 orders of magnitude [7] .The Tsallis distribution was recently used in an analy-sis of the distribution of the longitudinal component ofmomenta of particles within jets produced in pp colli-sions [10] which, from this point of view, is similar towhat was found in e + e − collisions [11]. Recent ATLASdata [12, 13] allow us to extend such an analysis to trans-verse characteristics of jets and charged particles withinthem. This is because they provide both the distributionsof transverse momenta of jets produced at LHC energies, p jetT , and distributions of transverse momenta of particlesproduced within these jets (defined relative to the jets), p relT . One can then retrieve and discuss the respectivenonextensivity parameters of jets, q jet , and particles pro-duced within them, q rel . In addition, because [12] at thesame time also provides multiplicity distributions withinjets, P ( N ), it is possible to confront both nonextensivitieswith that obtained from an analysis of P ( N ), q N . This isthe subject of the present work .
2. Transverse momentum distributions of jets andparticles within jets
In what follows we shall concentrate on ATLAS data[12]. They were taken at energy 7 TeV and in rapiditywindow | y | < . R = p ∆ η + ∆ φ (where ∆ φ In [8] these results were derived from QCD considerations. It turnsout that, although one gets a Tsallis-like formula, there is a p T dependentprefactor, the presence of which a ff ects the value of the q parameter.Also, in the low p T domain, Tsallis distribution with p T seems to dobetter than the one with p T . Both choices are possible, depending on thecircumstances, cf. [9] for details. In our case both would result in thesame conclusions. The other two LHC experiments, ALICE and CMS, do not providesuch results for the same experimental conditions and using the samecriteria for data selection. and ∆ η are, respectively, the azimuthal angle and thepseudorapidity of the hadrons relative to that of the jet, η = − ln tan θ , with θ being the polar angle), namely R = .
6. Distributions of transverse momenta, p jestT , ofjets of charged particles were observed, f (cid:16) p jetT (cid:17) = N jet dN jet d p jetT (2)and also distributions of transverse momenta p relT = (cid:12)(cid:12)(cid:12) ~ p × ~ p jet (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ~ p jet (cid:12)(cid:12)(cid:12) (3)of all N particles (only charged) in the jet, f (cid:16) p relT (cid:17) = N dNd p relT . (4)In addition, [12] also provides multiplicity distributionsof particles produced within observed jets, P ( N ).It should be stressed that the pure power law distribu-tion, f ( p T ) ∼ p − γ T , is not experimentally observed forjets. The observed slope parameter γ depends on p T , γ = γ ( p T ). However, a Tsallis distribution (1) emergesif one accounts for this dependence and assumes it in thefollowing two parameter ( n and T ) form, γ ( p T ) = n ln ( nT + p T )ln ( p T ) + ( n −
1) ln( nT ) + ln( n − p T ) . (5)In this case, the transverse momentum distribution for jetscan be fitted by a Tsallis formula (1) with n ≃ T = .
45 GeV, cf. Fig. 1.Data on distributions of transverse momenta p relT of par-ticles produced within the jet are presented in two papers.In [12] are data for p jetT ≤
40 GeV and in [13] for p jetT > T and n from fits because bothvariables are correlated. One also has to remember thatdata from [13] presented in Fig. 3 di ff er from those from[12] and presented in Fig. 2. Namely, they were collectedfor | η | < . p trackT > . | η | < . p trackT > .
10 10010 -6 -4 -2 f( p T j e t ) / p T j e t p Tjet [GeV] data ATLASTsallis fit
Figure 1: (Color online) Distribution of p jetT for jets at √ s = T = .
45 GeV and n = q = . q = + / n . p jetT [GeV] T [GeV] n q T .Notice the negative values of the parameter n (or, cor-respondingly, q < p jetT , i.e., for small values ofthe energy of such jets seen in Fig. 2. This fact is con-nected with the limitation of the available phase space inthis case. Actually, maximal values for the ratios p relT / p jetT for data in Fig. 2 are in the range 0 . − .
15 and in Fig. 3in the range 0 . − .
09. The nonextensivity parameterdrops below unity for distributions with p relT / p jetT > .
3. Multiplicity distributions within jets
From our experience with applications of Tsallis statis-tics to multiparticle production processes, we know [14]that multiplicity distribution of particles energy spectra of -2 f( p T r e l ) / p T r e l p Trel [GeV] ]10-15 GeV [x10 ]15-24 GeV [x10 ]24-40 GeV [x10 ] Figure 2: (Color online) Distributions of p relT particles inside the jetswith di ff erent values of p jetT obtained in [12], fitted using Tsallis dis-tribution (1). To make distributions readable, the consecutive curves i = , , , . . . were multiplied by 10 i . For all curves T = .
18 GeV. Thecorresponding values of the parameter n (and q = + / n ) are listed inTable1. Data are taken from [12]. which follow Tsallis distribution has Negative Binomialform (NBD) , P ( N ) = Γ ( N + k ) Γ ( N + Γ ( k ) (cid:16) h N i k (cid:17) N (cid:16) + h N i k (cid:17) k + N , (6)with 1 k = Var ( N ) h N i − h N i = q N − . (7)Whereas for NBD q > k in (7) is positive,for the q < k becomes negative ( k → − κ ) and NBDbecomes a binomial distribution (BD), P ( N ) = Γ ( κ + Γ ( N + Γ ( κ − N + (cid:16) h N i κ (cid:17) N (cid:16) − h N i κ (cid:17) κ − N , (8)and 1 κ = − q N . (9) Cf., also [15] where similar results were obtained from apparentlydi ff erent point of views. In fact there is a parameter equivalent to q anda resulting distribution can be written in Tsallis form. .1 110 -3 -1 Tjet f( p T r e l ) / p T r e l p Trel [GeV]
Figure 3: (Color online) The same as in Fig. 2 but now for jests withlarger p jetT for which all curves have T = .
25 GeV. The correspondingvalues of the parameter n (and q = + / n ) are given in Table 2. Dataare taken from [13]. For both the NBD and BD we expect the following tohold: ( N + P ( N + P ( N ) = a + bN (10)with a = h N i kk + h N i , b = ak for NBD , (11) a = h N i , b = , (12) a = h N i κκ − h N i , b = a κ for BD , (13)From data on multiplicity distributions, P ( N ), mea-sured in jets [12] (for p jetT ≤
40 GeV only) one can checkthe behavior of Eq. (10). As can be seen from in Fig. 4this relation is linear, i.e., the corresponding P ( N ) are in-deed of NBD or BD type (the deviation from linearity oc-curs only for N =
1, for which one encounters experimen-tal di ffi culties and which, in fact, can be omitted from ouranalysis). From parameters a and b obtained this way wecan deduce, using Eqs. (11) - (13), values of h N i , Var ( N )and k or κ (i.e., values of the corresponding nonextensivityparameter q N ) which are presented in Table 3. Notice thattheir values correspond closely to those obtained from thedistributions of p T in jets presented in Table 1. Table 2: Fit parameters for Fig. 3; q = + / n . p jetT [GeV] T [GeV] n q
25 - 40 0.25 70 1.01440 - 60 0.25 25 1.04060 - 80 0.25 18 1.05680 - 110 0.25 15 1.067110 - 160 0.25 12 1.083160 - 210 0.25 10 1.100210 - 260 0.25 9 1.111260 - 310 0.25 9 1.111310 - 400 0.25 9 1.111400 - 500 0.25 7.5 1.133
Table 3: P ( N ) characteristics for jets with di ff erent p jetT . p jetT [GeV] h N i Var ( N ) q N −
14 - 6 4.41 2.31 -0.116 - 10 5.72 3.83 -0.05810 - 15 7.11 6.61 -0.009815 - 24 7.56 11.2 0.06324 - 40 7.80 18.1 0.097
4. Self-similarity property of the multiparticle pro-duction processes
The values of nonextensivity parameters obtained froman analysis of multiplicity distributions and distributionsof p T of jets and in jets can now be compared with the re-spective nonextensivity parameters obtained in measure-ments of p T distributions in other experiments on mini-mum bias pp collisions in which the range of p T and mul-tiplicities were similar and energies of which were simi-lar to energies of the jets investigated. The correspondingresults for the dependence of the resulting nonextensiveparameters q as a function of the measured mean mul-tiplicity h N i are presented in Fig. 5. The approximatesimilarity of these results is clearly visible . A word of comment on Fig. 5 is in order here. So far we wereestimating the parameter q from distributions of p T or N and discussingits energy dependence, q ( s ), as obtained from di ff erent experiments [5, ( N + ) P ( N + ) / P ( N ) N Figure 4: (Color online) ( N + P ( N + P ( N ) in function of multiplicity N in jetswith di ff erent values of p jetT as measured in [12] and presented in Fig. 2.No such information on P ( N ) is available for jets analyzed in Fig. 3. The results presented here can be summarized in thefollowing way: ( i ) A Tsallis distribution successfully de-scribes inclusive p T distributions in a wide range of trans-verse momenta for all energies measured so far [5, 6, 7].( ii ) This is also true for the distribution of transverse mo-menta of jets as shown in Fig. 1. The nonextensivity pa-rameter in this case, q = .
14, is comparable to q = . iii ) The Tsallisdistribution also describes transverse momenta distribu-tions of particles in jets. The values of q obtained in thiscase are roughly the same as those obtained from an an-alyzes of multiplicity distributions in these jets. It shouldbe noted that, as seen in Fig. 5, values of the nonexten-sivity parameter q for particles in jets correspond ratherclosely to values of q obtained from the inclusive distri-butions measured in pp collisions (for the correspondingenergies available for production) in the similar ranges oftransverse momenta . p + p collisions to those in jets, for which, unfortunately, we do not know thecorresponding energy √ s . On the other hand, we know h N i both for p + p collisions and for particles produced in jets, so it is reasonableinstead to show q as a function of h N i . Results discussed here could be regarded as related to the phe- q -
Trel ) in jet
Figure 5: (Color online) Compilation of values of q as obtained from p relT distributions (triangles) and from multiplicity distributions (circles).Triangles at small h N i are obtained from data [12], those for larger h N i from [13]. Full squares and circles are from data on multiparticle pro-duction in p + p collisions and, correspondingly, squares (inelastic data)are from compilation for LAB energy 3 . −
303 GeV presented in [16],whereas circles (non-single di ff ractive data) are from compilation pre-sented in [17]. To conclude, one observes a kind of similarity (in whatconcerns values of the corresponding nonextensivity pa-rameters) of multiplicity distributions P ( N ) and transversemomentum distributions f ( p T ) of particles produced inminimum bias collisions pp and particles in jets of com-parable energies. This can be interpreted as a demonstra-tion that the mechanisms of particle production in bothcases are the same or, at least, are similar and containsome common part [19]. This common part, in turn, canbe identified with the self-similarity character of the pro-duction process in both cases, resulting in a kind of cas-cade process, which always results in a Tsallis distribution[20]. Actually, this is a very old idea, introduced alreadyby Hagedorn in [21]. He assumed that the production ofhadrons proceeds through formation of fireballs which are nomenon of geometrical scaling for p T distributions discussed recently(cf. [24] and references therein), apparently being a consequence ofgluon saturation at some scale Q s . It turns out that scaled distributionscan be described by a Tsallis formula [25] with the saturation scale be-ing hidden in the parameter T (not q ); in fact to get scaling one has toallow for T being dependent on p T . One should, however, be aware ofthe fact that in the energy domain discussed here scaling seems to beviolated [26]. statistical equilibrium of an undetermined number of allkinds of fireballs, each of which in turn is considered to bea fireball . In fact this was used as a justification in the firstproposed generalization of the Hagedorn model, consid-ered as a statistical model, to q -statistics, cf., [22]. In thepure dynamical QCD approach to hadronization, one en-counters the same idea, as, for example, that presentedin [23]. In it partons fragment into final state hadronsthrough multiple sub-jet production. As a result one has a self-similar behavior of cascade of jets to sub-jets to sub-sub-jets . . . to final state hadrons .
5. Summary
Using the Tsallis nonextensive approach, we have anal-ysed recent data found by the LHC ATLAS experiment[12, 13] on transverse momentum distributions of jets,particles within jets and their multiplicity distributions.The values of the respective nonextensivity parametersobtained this way, when compared with the correspondingvalues obtained from the inclusive distributions measuredin pp collisions for the corresponding energies availablefor production and in similar ranges of transverse mo-menta, were found to be similar. This can be consid-ered as strong evidence of the existence of some com-mon mechanism behind all these processes which we ten-tatively identify with a self-similarity property and cas-cade type processes based on multiplicative noise [20].They are known to lead to a Tsallis distribution (with n − = h η i / Var ( η ) given by fluctuations of multiplicativenoise η [20]) of the same type as those describing statisti-cal or thermodynamical systems (with q − = Var ( T ) / h T i given by fluctuations of temperature T [4, 5]) . A word of caution should be added. The observed self-similar be-havior of distributions of particles inside the jets and particles producedin inelastic pp collisions can be indications of self-similarity in multi-particle production processes. Jets being a part of all produced particlesare approximately similar to inelastic collisions. However, in reality weare not able to observe the whole process of hadronization or to analyzeall its subprocesses to really speak of self-similarity in multiparticle pro-duction processes. We only have information on one such subprocess,i.e., on the production in jets. We observe similarities between themand multiparticle production in innelastic collisions in total. This ob-servation is the basis of our claim that we are dealing with a processwhich shows the same statistical properties at many scales. This is ourself-similarity.
It is worth reminding at this point that both Tsallis dis-tribution and the Negative Binomial Distribution can beregarded as a consequence of using a gamma distribu-tion for clusters formed before fragmentation. Whereasthe former arises from the fluctuations of temperature in aBoltzman-Gibs distribution, the latter arises from the fluc-tuations of mean multiplicity in a Poissonian distribution.The common feature is that in both cases fluctuations aregiven by a gamma distribution which is stable under thesize distribution, i.e., exhibits self-similarity and scalingbehavior (actually, NBD is also a self-similar distribution[27]). This indicates once more that self-similarity en-countered in processes under consideration is the physicalground of the observed similarities discusses here. Re-sults presented here could possibly open discussion aboutthe validity of thermal models [28].
Acknowledgments