Superstatistics in high energy physics: Application to cosmic ray energy spectra and e+e- annihilation
aa r X i v : . [ h e p - ph ] F e b Superstatistics in high energy physics: Application tocosmic ray energy spectra and e + e − annihilation Christian Beck
School of Mathematical Sciences, Queen Mary University of London, MileEnd Road, London E1 4NS, UK.
Abstract
We work out a superstatistical description of high-energy scat-tering processes that takes into account temperature fluctuations insmall volume elements. For Γ-distributed fluctuations of the inversetemperature one effectively obtains formulas similar to those used innonextensive statistical mechanics, whereas for other temperature dis-tributions more general superstatistical models arise. We consider twomain examples: Scattering processes of cosmic ray particles and e + e − annihilation processes. In both cases one obtains excellent fits of ex-perimentally measured energy spectra and cross sections. Introduction
Superstatistical techniques have been recently successfully applied to a largevariety of complex systems, for example hydrodynamic turbulence [1, 2, 3, 4],defect turbulence [5], share price dynamics [6, 7], random matrix theory [8, 9],random networks [10], wind velocity fluctuations [11, 12], hydro-climatic fluc-tuations [13], the statistics of train departure delays [14] and models of themetastatic cascade in cancerous systems [15]. The basic idea underlyingthis approach is that there is an intensive parameter, for example the in-verse temperature β or the energy dissipation in turbulent systems, thatexhibits fluctuations on a large time scale (large as compared to internal re-laxation times of the system under consideration). As a consequence, onecan model these types of complex systems by a kind of superposition of ordi-nary statistical mechanics with varying temperature parameters, in short asuperstatistics [16, 17, 18, 19, 20, 21, 22, 23]. The stationary distributions ofsuperstatistical systems deviate from ordinary Boltzmann-Gibbs statisticalmechanics and can exhibit asymptotic power laws, stretched exponentials, orother functional forms in the energy E [18].In this paper we work out potential applications of this concept in highenergy physics. In scattering processes at high energies, the effective inter-action volume, as well as the number of particles involved, can be rathersmall, and hence temperature fluctuations can play a very important role[24, 25, 26]. These can either be temperature fluctuations from scatteringevent to scattering event, i.e. temporal fluctuations, or they may be relateda nonequilibrium situtations where different spatial regions have differenttemperature, i.e. spatial fluctuations.Our starting point for a suitable thermodynamic model will be Hagedorn’stheory [27, 28], which yields a statistical description of a selfsimilar ‘fireball’of particles produced in scattering events. Hagedorn’s theory models thehadronization cascade from a statistical mechanics point of view (of courseQCD was not known at the time when he wrote the seminal paper [27]). Histheory is regularly in use to describe heavy ion collisions [29, 30, 31]. Thebasic assumption is that in the scattering region the density of states growsso rapidly that the effective temperature cannot exceed a certain maximumtemperature, the Hagedorn temperature T H [32]. The value is approximately T H ≈
180 MeV and it describes the confinement phase transition. TheHagedorn phase transition is also of fundamental interest in string theories[33, 34, 35]. 1n this paper we will consider a superstatistical extension of the Hage-dorn theory, starting from a q -generalized version previously introduced in[36]. The predictions of this generalized statistical mechanics model are inexcellent agreement with measured experimental data. We will illustratethis for two main examples: i) observed energy spectra of cosmic rays [24] ii)experimentally measured cross sections in e + e − annihilation [36].The Hagedorn theory of scattering processes is known to give correctpredictions of cross sections and energy spectra for center of mass energies E CMS < GeV , whereas for larger energies there is experimental evidencefrom various collision experiments that power-law behaviour of differentialcross sections sets in. This power-law is not contained in the original Hage-dorn theory but can be formally obtained if one extends the original Hagedorntheory to a superstatistical one, along the lines sketched in this paper. Infact, if fluctuations are taken into account then power laws can arise in quitea natural way in various ways, and often lead to Tsallis type of generalizeddistribution functions[37, 38, 39, 1, 16]. While in this paper we concentrateon cosmic ray statistics and transverse momentum spectra in e + e − annihila-tion, it is quite clear that related superstatistical techniques can be appliedto other scattering data as well, for example heavy ion collisions [31, 40], pp collisions [41] and p ¯ p collisions [26]. In the superstatistics approach [16] one assumes that locally the system un-der consideration reaches local equilibrium but on a large spatio-temporalscale there are temperature fluctuations, described by a probability density f ( β ) which describes the probability to observe a certain inverse temperature β in a given spatial area. One thus gets a kind of mixing (or superposition) ofmany equilibrium distributions which effectively decribe the driven nonequi-librium system with a stationary state (see [42] for a recent review discussingvarious applications).For high energy scattering processes, it is clear that the larger the centerof mass energy E CMS of the collision process is, the smaller is the volume r probed, due to the uncertainty relation c E CMS · r ∼ O ( ~ ). This means,the effective interaction volume where a thermodynamic description of the2ollision process makes sense will become smaller and smaller with increasing E CMS . However, a smaller volume means larger temperature fluctuations.This is in particular relevant if we repeat our scattering experiment severaltimes or if we have different scattering events in different spatial regionswhich all contribute to our data. It thus makes sense to consider at largeenergies E CMS a superstatistical description of scattering events which takesinto account local temperature fluctuations.Assume that locally some value of the fluctuating inverse temperature β is given. We then expect the momentum of a randomly picked particle inthis region to be distributed according to the relativistic Maxwell-Boltzmanndistribution p ( E | β ) = 1 Z ( β ) E e − βE . (1)Here p ( E | β ) denotes the conditional probability to observe a particle withenergy E = p ~p c + m c , given some value of β . For highly relativisticparticles we can neglect the rest mass m so that E = c | ~p | , where ~p is themomentum. The normalization constant is given by Z ( β ) = Z ∞ E e − βE dE = 2 β . (2)Now let us take into account local temperature fluctuations in the smallinteraction volumes where scattered particles are produced. We have toconsider some suitable probability density f ( β ) of the inverse temperature inthe various interaction volumes. In a long series of experiments we will thenobserve the marginal distribution obtained by integrating over all βp ( E ) = Z ∞ p ( E | β ) f ( β ) dβ. (3)These types of distributions are generally studied in superstatistical models,and are known to exhibit fat tails whose asymptotic decay with E dependson how f ( β ) behaves for β → f ( β ) that are relevant for large classes of complex sys-tems: These are the χ -distribution, the inverse χ distribution, and the3ognormal distribution. Lognormal superstatistics is often observed in hy-drodynamic turbulence, due to the multiplicative nature of the Richardsoncascade [3, 4]. Inverse χ superstatistics plays an important role in randommatrix theory [8, 9], as well as in medical statistics [15]. For high energyphysics, χ superstatistics is most relevant, although other superstatisticsmay play a role as well (for example, one could think about a superstatisticsgenerating Kaniadakis statistics [43]).For χ superstatistics (or equivalently Γ superstatistics), the distribution f ( β ) is given by the χ -distribution of degree n , i.e. f ( β ) = 1Γ (cid:0) n (cid:1) (cid:26) n β (cid:27) n β n − exp (cid:26) − nβ β (cid:27) . (4)The χ -distribution is a typical distribution that naturally arises in many cir-cumstances, for example if n independent Gaussian random variables X i , i =1 , . . . , n with average 0 and the same variance are squared and added. If wewrite β := n X i =1 X i (5)then β has the probability density function (4). The average of the fluctuating β is given by h β i = n h X i i = Z ∞ βf ( β ) dβ = β (6)and the variance by h β i − β = 2 n β . (7)The integral (3) with f ( β ) given by (4) and p ( E | β ) given by (1) is easilyevaluated and one obtains p ( E ) ∼ E (1 + b ( q − E ) q − (8)where q = 1 + 2 n + 6 (9)and b = β − q . (10)4ote that the partition function Z ( β ) entering into eq. (3) is β -dependent.Different β -dependencies of the partition function Z ( β ) lead to different an-swers if the integration over β is performed. In particular, the precise relationbetween q and n depends on this. This was for the first time correctly workedout in [1].The distribution (8) is a q -generalized relativistic Maxwell-Boltzmann dis-tribution in the formalism of nonextensive statistical mechanics [37]. Thesekind of distributions can be directly obtained by maximizing the q -entropies[37] S q = k q − − X i p qi ) (11)and multiplying with the available phase space volume. The p i are the prob-abilities of the microstates i . The q -entropies contain the Shannon entropy S = − k P i p i ln p i underlying ordinary statistical mechanics as a specialcase for q = 1. Whereas q = 1 corresponds to the usual canonical en-semble with constant temperature, the Tsallis-canonical ensemble obtainedfor q > β is χ -distributed. We thus have a plausible physical mechanism why aTsallis-like statistical description makes sense if a suitable intensive param-eter fluctuates [1, 38]. Tsallis statistics with q = 3 (respectively q = − When experimental scattering data are collected, one often looks at momen-tum distributions of a particular particle that are produced by repeating thesame experiment many times. In each scattering event, the effective temper-ature (given by the heat bath of surrounding particles) will fluctuate fromevent to event. Since the scattering experiment is repeated many times inan independent way, and since we concentrate on the statistics of just oneparticle rather then many-particle states, it makes sense to directly integrateout the temperature fluctuations and to consider effective 1-particle super-statistical Boltzmann factors.Let us consider particles of different types and label the particle types byan index j . Each particle can be in a certain momentum state labelled by5he index i . The energy associated with this state is ǫ ij = q ~p i + m j , (12)using units where c = 1. We may now define an effective 1-particle Boltzmannfactor by x ij := Z ∞ dβ f ( β ) e − βǫ ij . (13)For example, for Tsallis statistics f ( β ) is a χ distribution and one has x ij = (1 + ( q − bǫ ij ) − q − , (14)where b − is proportional to the average temperature, and q − x ij approaches the ordinary Boltzmann factor e − bǫ ij for q → Z = X ( ν ) Y ij x ν ij ij (15)Here ν ij denotes the number of particles of type j in momentum state i . Thesum P ( ν ) stands for a summation over all possible particle numbers. Forbosons one has ν ij = 0 , , , . . . , ∞ , whereas for fermions one has ν ij = 0 , X ν ij x ν ij ij = 11 − x ij ( bosons ) (16)whereas for fermions X ν ij x ν ij ij = 1 + x ij ( f ermions ) (17)Hence the partition function can be written as Z = Y ij − x ij Y i ′ j ′ (1 + x i ′ j ′ ) , (18) To describe 2-particle states with the same temperature fluctuations surrounding bothparticles one has to proceed in a slightly different way, see [1]. Z = − X ij log(1 − x ij ) + X i ′ j ′ log(1 + x i ′ j ′ ) . (19)One may actually proceed to continuous variables by replacing X i [ ... ] → Z ∞ V πp h [ ... ] dp = V π Z ∞ p [ ... ] dp ( ~ = 1) (20)( V : volume of the interaction region) and X j [ ... ] → Z ∞ ρ ( m )[ ... ] dm, (21)where ρ ( m ) is the mass spectrum.Let us now formally calculate the average occupation number of a particleof species j in the momentum state i . We obtain¯ ν ij = x ij ∂∂x ij log Z = x ij ± x ij . (22)For example, for the case of Tsallis statistics one obtains¯ ν ij = 1(1 + ( q − bǫ ij ) q − ± − sign is for bosons and the + sign for fermions.In order to single out a particular particle of mass m , one can formallywork with the mass spectrum ρ ( m ) = δ ( m − m ). To obtain the probabilityto observe a particle of mass m in a certain momentum state, we have tomultiply the average occupation number with the available volume in mo-mentum space. An infinitesimal volume in momentum space can be writtenas dp x dp y dp z = dp L p T sin θdp T dθ (24)where p T = p p y + p z is the transverse monentum and p x = p L is the longi-tutinal one. By integrating over all θ and p L one finally arrives at a proba-bility density w ( p T ) of transverse momenta given by w ( p T ) = const · p T Z ∞ dp L q − b p p T + p L + m ) q − ± . (25)7ince the Hagedorn temperature is rather small (of the order of the π mass),and b − is of the order of the Hagedorn temperature, under normal circum-stances one has b p p T + p L + m >>
1, and hence the ± q is close to 1. One thus obtains for both fermions and bosons the statistics w ( p T ) ≈ const · p T Z ∞ dp L (cid:18) q − b q p T + p L + m (cid:19) − q − (26)which, if our model assumptions are satisfied, should determine the p T de-pendence of experimentally measured particle spectra. The differential crosssection σ − dσ/dp T is expected to be proportional to w ( p T ).In nonextensive statistical mechanics, it is sometimes of advantage toconsider normalized q -expectation values [45], thus implementing a formal-ism that is based on so-called escort distributions [44]. If the escort formalismis used then the power q − in the above formula is replaced by qq − , a simplereparametrization. For more general classes of superstatistics described bygeneral f ( β ) the q -exponential is replaced by R ∞ f ( β ) exp {− β p p T + p L + m } dβ . Let us now look at measured data. Fig. 1 shows the experimentally measuredenergy spectrum of primary cosmic ray particles as observed on the earth[50, 51, 52, 53, 54, 55]. The measured spectrum is very well fitted over manydecades of different energies by the distribution (8) [24] or other generalizednonextensive distributions [56]. The best fit is obtained if the entropic index q is chosen as q = 1 .
215 (27)and if the effective temperature parameter is given by kT = b − = 107 MeV . (28)Let us now predict a plausible value of q using the superstatistical ap-proach. The variables X i in eq. (5) describe the independent degrees offreedom contributing to the fluctuating inverse temperature β . At very largecenter of mass energies E CMS → ∞ , the interaction region is very small,and all relevant degrees of freedom are basically represented by the 3 spatial8 e-301e-251e-201e-151e-101e-0511000001e+06 1e+08 1e+10 1e+12 1e+14 1e+16 1e+18 1e+20 1e+22 F l u x E [eV] <-- kneeankle -->
Figure 1: Measured energy spectrum of primary cosmic rays (in units of m − s − sr − GeV − ) [24]. The solid line is the formula (8) with q = 1 . b − = kT = 107 MeV and C = 5 · − in the above units. The dashed lineis eq. (8) with q = 11 / kT = 107 MeV and C smaller by a factor 1 / X i as theheat loss in the spatial i -direction, i = x, y, z , during the collision processthat generates the primary cosmic ray particle. The more heat is lost, thesmaller is the local kT , i.e. the larger is the local β given by (5). The 3spatial degrees of freedom yield n = 3 as the smallest possible value of n or,according to (9), q = 119 = 1 . . (29)For cosmic rays E CMS is very large, hence we expect a q -value that is close tothis asymptotic value. The fit in Fig. 1 in fact uses q = 1 . q -values werealso obtained in [56]).For smaller center of mass energies, according to c E CMS · r ∼ O ( ~ ), theinteraction region will be bigger and more effective degrees of freedom withinthis bigger interaction region will contribute to the fluctuating temperature.Hence we expect that for smaller E CMS n will be larger than 3, or q will besmaller than 11 / E CMS → q → r where athermodynamic description makes sense becomes very large, and within thislarge region a large number of independent degrees of freedom n contributesto the fluctuating temperature, represented by many different particles. Ac-cording to eq. (9), the limit n → ∞ is equivalent to q → χ -distribution degenerates to a delta function δ ( β − β ).It is reasonable to assume [51] that the ‘knee’ at E ≈ eV is due to thefact that one has reached the maximum energy scale to which typical galacticaccelerators can accelerate. This then implies a rapid fall in the number ofobserved events with a higher energy, i.e. a steeper slope in Fig. 1 betweenabout 10 and 10 eV. The ‘ankle’ at E ≈ eV may then be due tothe fact that a higher energy population of cosmic ray particles takes overfrom a lower energy population. This higher energy population may havea different origin (for example, extragalactic origin). The new populationhas a significantly smaller flux rate but can reach much larger energies. Asa matter of fact the cosmic accelerators underlying the production processof this new species of cosmic rays must have a much larger center of massenergy E CMS than the ankle energy ∼ eV, so q should be given by itsasymptotic value 11 /
9, whereas the effective temperature T should be thesame as before. The dashed line in Fig. 1 corresponds to our formula with10igure 2: Measured cosmic ray energy spectrum E · dN/dE at largest en-ergies (data from [51, 52, 53, 54]). The straight line is a power law withexponent α = 5 / q = 11 / q = 11 / k ˜ T = 107 MeV and a flux rate that is smaller by a factor 1 /
50 ascompared to the high-flux generation of cosmic rays. This is consistent withthe data.Formula (8) predicts asymptotic power-law behavior of the measured en-ergy spectrum. For large E one has p ( E ) ∼ E − α where the index α is givenby α = 1 q − − . (30) q = 1 .
215 implies α = 2 .
65 (for moderately large energies), whereas forlargest energies the asymptotic value q = 11 / α = 5 /
2. As shownin Fig. 2, the largest-energy events are compatible with such an asymptoticpower law exponent. 11
Comparison with experimentally measureddifferential cross sections in e + e − annihila-tion Let us now proceed to our second example, differential cross sections fortransverse momenta in e + e − annihilation experiments [36, 57]. If one usesthe escort formalism [44, 45] then formula (26) implies1 σ dσdp T = Cu Z ∞ dx (cid:16) q − q x + u + m β (cid:17) − qq − (31)Here x = p L /T , u = p T /T and m β := m /T are the longitudinal momen-tum, transverse momentum and mass in units of a temperature parameter T that is of the same order of magnitude as the Hagedorn temperature. C is a suitable constant related to multiplicity.Generally, one knows that the interaction energy in a nonextensive systemincreases with increasing entropic index q . Since this energy must be takenfrom somewhere, it is most natural to assume that it is taken from the heatbath. So the average temperature parameter T should slighly decrease withincreasing q . In [36] a linear dependence of T on q was postulated, of thesimple form T = (cid:16) − q (cid:17) T H (32)where T H = 180 M eV is the Hagedorn temperature.We also have to clarify the dependence of q on E CMS . Clearly, for E CMS → ∞ our argument in section 2 and 4 suggests the value q = 11 / E CMS → q = 1, i.e the ordinary Hagedorn theorywithout fluctuations. In [36], a smooth interpolation between these valueswas suggested, of the form q ( E CMS ) = 11 − e − E CMS /E e − E CMS /E (33)where E ≈ . GeV is about half of the Z mass.Finally, one also needs to know the multiplicity (the average numberof produced charged particles) in order to estimate the cross section. Themultiplicity M as a function of E CMS has been independently measured inmany experiments[58]. A good fit of the experimental data in the relevant12 .010.1110 0.1 1 10’Theory’’14GeV’’22GeV’’35GeV’’44GeV’’91GeV’’133GeV’’161GeV’
Figure 3: Differential cross section as a function of the transverse momentum p T for various center of mass energies E . The data correspond to measure-ments of the TASSO ( E ≤
44 GeV) and DELPHI ( E ≥
91 GeV) collabora-tion. The solid lines are given by the analytic formula (35).13nergy region is the formula [36] M = (cid:18) E CMS T q =10 (cid:19) / (34)The final formula derived for the cross section in [36] involves a further ap-proximation step to perform the integration over x and is then finally givenby 1 σ dσdp T = 1 T M p ( u ) . (35)where p ( u ) is the normalized probability density p ( u ) = 1 Z q u / (1 + ( q − u ) − qq − + (36)with normalization constant Z q = ( q − − / B (cid:18) , qq − − (cid:19) . (37)Formula (35) with p ( u ) given by eq. (36), q ( E CMS ) given by eq. (33), T ( q ) given by eq.(32) and multiplicity M ( E CMS ) given by eq. (34) turnsout to very well reproduce the experimental results of measured transversalcross sections for all energies E CMS . This is illustrated in Fig. 3 which showsthe measured differential cross section versus p T . The solid lines are givenby formula (35) for the various center of mass energies E CMS . One obtainsexcellent agreement with the measured data of the TASSO and DELPHIcollaborations [59, 60]. Remarkably, for the largest energies the best fittingparameter q is again given by q ≈ /
9, in agreement with the data for thecosmic rays. Moreover, at the largest energies, according to eq. (32) T isgiven by 107 M eV , again in agreement with what was used for the cosmicray data in Fig. 1. Our superstatistical argument in section 2 and 4, relating q to temperature fluctuatuons and predicting from this q ≈ / E CMS , is indeed generally applicable.
In this paper we have dealt with a superstatistical generalization of the Hage-dorn theory which is generally applicable to describe the statistics of scattered14articles produced in high-energy collisions. Our approach is based on takinginto account temperature fluctuations in small interaction volumes. For χ -distributed inverse temperature one effectively ends up with formulas thatare similar to those used in non-extensive versions of statistical mechanics.At large energies the χ -superstatistical approach implies that energyspectra of particles and cross sections decay with a power law. This powerlaw is indeed observed for various experimental data. We obtained formulasthat are in very good agreement with experimentally measured data for cos-mic rays and e + e − annihilation. In particular, the superstatistical approachallowed us to give a concrete prediction for q at largest center of mass en-ergies, namely q ≈ /
9. Generally, it appears that high-energy scatteringdata do not only yield valuable information on elementary particle physics,but they may also be regarded as test grounds to further develop generalizedversions of statistical mechanics.
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