The Bose-Einstein distribution functions and the multiparticle production at high energies

The evolution properties of propagating particles produced at high energies in a randomly distributed environment are studied. The finite size of the phase space of the multiparticle production region as well as the chaoticity can be derived.


Introduction
Interest in (charged) particles "moving" in an environment of quantum fields taking into account the relations between quantum fluctuations and chaoticity is still attracted by the particle physics society. The events of high multiplicity at high energies is an interesting subject because of increasing energies of current and future colliders (e + e − ,pp, pp) of particles. One of the most important tasks of superhigh energy particle studies is to analyze fluctuations and correlations such as the Bose-Einstein (BE) correlation [1] of produced particles. This is a rather instructive tool to study high multiplicity hadron processes in detail. We understand the multiparticle (MP) production as the process of colliding particles where the kinetic energy is dissipated into the mass of produced particles [2]. We consider the incident energy √ s ≫ Λ, where Λ means the quantum chromodynamics (QCD) scale. Phenomenological models [3,4] describing the crucial properties of multiparticle correlations are very useful for systematic investigations of the properties caused by fluctuations and correlations. By considering them, one can obtain the characteristic properties of the internal structure of disordering of produced particles in order to extract the information on the space-time size of the multiparticle production region (MPPR), to estimate the lifetime of the particle emitter, etc. There were used the analysis of correlation functions (CF) and distriburion functions (DF) in [5,6] to understand the possible view on the quark-gluon plasma formation. In this paper, we present the model to describe the very high multiplicity effects at high energies. The most characteristic point of our model is that the BE both DF's and CF's are taken into account on the quantum level (the operators of production and annihilations are used) with the random source contributions coming from the environment. We define the average mean multiplicity N via MP CF w( k) as N = d 3 k w( k), where k is the spatial momentum of a particle. Following a natural way we suppose N ≪ N, while N ≪ N 0 = √ s/m, where m ∼ O(0.1 GeV ). The main object in this investigation is the MP thermal DFW (k µ , k ′ µ ) related to N as where N is defined as the scale of the produced-particle multiplicity N at four-momentum k µ (µ is the Lorenz index), the normalized DF f (k µ ) is finite, i.e. d 4 k f (k µ ) < ∞ and the label β in (1.1)) means the temperature T (of the phase space occupied by operators b + (k µ ) and b(k µ )) inverse. The nature of operators b + (k µ ) and b(k µ ) will be clarified in Sec.3.
In this paper, we claim that the observation of the size effect in a MP production is derived via MP CF's and DF's as well as the so-called chaoticity which will be introduced in Sec.4. The MP CF formalism concerns the statistical physics based on the Langevin-type of equations. The Langevin equation to be introduced in Sec.3 is considered as a basis for studying the approach to equilibrium of the particle(s). It is assumed that the heat bath being in essence infinite in size remains for all times in equilibrium as well. We use the method applied to the model where a relativistic particle moving in the Fock space is described by the number representation underlying the second quantization formulation of the canonical field theory. We deal with the microscopic look at the problem with the elements of quantum field theory (QFT) at the stochastic level with the semiphenomenological noise embedded into the evolution dissipative equation of motion.

The model of dual representation within the dissipative dynamics
Let us suppose that the evolution of particles produced at high energies is described by the solutions of the model Hamiltonian where any physical system of particles is described by the doublet of field operators. We introduce the dual states in the eigenbasis {|k , |k(ǫ) } of the Hamiltonian H, where 0 ≤ ǫ < ∞ is identified as the frequency representing the energy of the object and |k is a discrete eigenstate. We consider the simple dual model where H is given within the damped harmonic oscillator where Here, we identify |k , |k(ǫ) and k|, k(ǫ)| with the special mode operators of annihilation a, b(ǫ) and creation a + , b + (ǫ), respectively, e.g., for "quarks" and "gluons" or their combinations. In the interaction Hamiltonian (2.3), ρ is the coupling constant, while g(ǫ) provides the transition between discrete and continuous states. The equations of motion obeying (2.1) with (2.2) and (2.3) are where the label k means | k| as the momentum. We demand that a k (t) and b k (ǫ, t) satisfy the following natural conditions: . The next step is to give the relations between the variables and couplings of the model (2.1) and the phenomenological constants, e.g., the frequency E and the damping constant κ involved into the dissipative dynamics given by the Boltzmann-type equations in the relaxation time approximation: Here, as τ → −∞. One can conclude that the phenomenological constants E and κ in (2.4) (2.5) are nothing else but ǫ 0 and 2 π ρ 2 g 2 (ǫ 0 ), respectively, i.e. the microscopic parameters coming from the model Hamiltonian (2.1).

Stochastic model. Langevin equation
As it was pointed out in the Introduction, to derive the characteristic features of the MP production physics at high energies, one should specify the model on the quantal level. Let us assume that only the particles p i of the same kind of statistics labeled by index i are produced just after the high energy collision process occured, e.g., pp,pp → p i . In order to extend the method of stochastically distributed particles in the environment, we propose that rather complicated real physical processes happened in the MP formation region should be replaced by a single-constituent propagation of particles provided by a special kernel operator (in the stochastically evolution equation) considered as an input of the model and disturbed by the random force F [5,6]. We assume that F can be the external source being both a c-number function and an operator. In such a hypothetical system of excited (thermal) local phase we deal with the canonical operator a( k, t) and its Hermitian conjugate a + ( k, t). We formulate DF of produced particles in terms of a point-to-point equal-time temperature-dependent thermal CF of two operators Here, ... β means the procedure of thermal statistical averaging; k and t are, respectively, momentum and time variables, e −Hβ /T r(e −Hβ ) stands for the standard density operator in equilibrium, and the Hamiltonian H is given by the squared form of the annihilation a p and creation a + p operators for Boseand Fermi-particles, H = p ǫ p a + p a p (the energy ǫ p and operators a p , a + p carry some index p, where p α = 2π n α /L, n α = 0, ±1, ±2, ...; V = L 3 is the volume of the system considered). Here, we use the canonical formalism in a stationary state in the thermal equilibrium (SSTE), and a closed structural resemblance between the SSTE and the standard QFT is revealed. We define the thermal boson field as and ∆( k) is an element of the invariant phase volume. The standard canonical commutation relation (CCR) at every time t is used as usual for Bose (-) and Fermi (+)-operators. The probability to find the particles in MPPR with momenta k and k ′ in the same event at the time t is: where the MP DF W ( k, t) in the simple version fluctuates only its normalization, e.g., the mean multiplicity N : Here, the one-particle thermal DF is defined as defining the single spectrum, while for i-and j-types of particles. Here, δ ij = 1 if i = j and 0 otherwise. DF's f ( k, t) and f ( k, k ′ , t) look like (hereafter the label β will be omitted in the sense of (1.1) and (3.1)) under the assumption of the random source-function R( k, t) being an operator, in general. One can rewrite (3.4) in the following form For simplicity, we deal with operators a and b as if they are the single boson or fermion operators. Considering the "propagation" of a particle with the momentum k in the quantum equilibrium phase space under the infuence of a random force coming from surrounding particles, the dissipative dynamics of the relevant system is described by the equation containing only the first order time derivatives of the dynamic degrees of freedom, the operators b( k, t) and b + ( k, t) [5]: Here, the interaction of particles under consideration with surroundings as well as providing the propagation is given by the operator A( k, t) defined as the one closely related to the dissipation force: The particle transitions are provided by the random source operator F ( k, t) while P stands for a stationary external force. An interplay of particles with surroundings is embedded into the interaction complex kernel K( k, t), while the real physical transitions are provided by the random source operator F ( k, t) with the zeroth value of the statistical average , F = 0. The random evolution field operator K( k, t) in (3.7) stands for the random noise and it is assumed to vary stochastically with a δ-like equal time correlation function where both the strength of the noise κ and the positive constant ρ → ∞ define the effect of the Gaussian noise on the evolution of particles in the thermalized environment. The formal solutions of (3.5) and (3.6) in the operator form in S( respectively, where the operatorã(k µ ) is expressed via the Fourier transformed operatorF (k µ ) and the Fourier transformed kernel functionK(k µ ) (coming from (3.7)) asã In our model, we suppose that a heat bath (an environment) is an assembly of damped oscillators coupled to the produced particles which in turn are distributed by the random forcẽ F (k µ ). In addition, there is the assumption that the heat bath is statistically distributed. The random force operator F ( k, t) can be expanded by using the Fourier integral where the form ψ(k µ )·ĉ(k µ ) is just the Fourier operatorF (k µ ) = ψ(k µ )·ĉ(k µ ), and the canonical operatorĉ(k µ ) obeys the commutation relation

DF ratio enhancement.
The enhanced probability for emission of identical particles is given by the ratio R of DF's in S(ℜ 4 ) as follows: . Using Fourier solutions of equations (3.5) and (3.6) in S(ℜ 4 ), one can get the R-ratio for DF where and the Bose-Einstein CF Ξ(k µ , k ′ µ ) looks like Inserting CF (4.4) into (4.3) and taking into account the trick with δ 4 (k µ − k ′ µ )-function to be replaced by the δ-like consequence like Ω(r) · exp[−(k − k ′ ) 2 r 2 ] [7], one can get the following expression for D-function instead of (4.3) where while q 2 ≡ Q 2 r 2 and the function Ω(r)·n(ω; T )·exp(−q 2 /2) in (4.5) describes the space-time size of the multiparticle production region. Choosing the zaxis along the pp orpp collision axis one can put , Ω(r) = 1 π 2 r 0 · r z · r 2 t , where r 0 , r z and r t are time-like, longitudinal, and transverse "size" components of MPPR. To derive (4.5), the Kubo-Martin-Schwinger condition ( µ is the chemical potential) has been used, and the thermal statistical averages for theĉ(k µ )-operator should be represented in the following form: for Bose (+)-and Fermi (-)-statistics, where n(ω, T ) = {exp[(ω −µ)β]±1} −1 . Formula (4.5) indicates that the chaotic multiparticle source emanating from the thermalized MPPR exists. Taking into account (4.6) and (4.7), it is easy to see that the correlation functions containing the random force functions F ( k, t) (3.8) carry the quantum features in the termalized stationary equilibrium, namely The quantitative information (longitudinal r z and transverse r t components of MPPR, the temperature T of the environment) could be extracted by fitting the theoretical formula (4.5) to the measured D-function and estimating the errors of the fit parameters. Hence, the measurement of the space-time evolution of the multiparticle source would provide information of the multiparticle process and the general reaction mechanism. The temperature of the environment enters into formula (4.5) through the two-particle CF Ξ(k µ , k ′ µ ; T ). If T is unstable, the R-functions (4.1) will change due to a change of DFf which, in fact, can be considered as an effective density of the multiparticle source. Formula (4.1) looks like the fitting R-ratio using a source parametrization: , where r t (r z ) is the transverse (longitudinal) radius parameter of the source with respect to the beam axis, λ F stands for the effective intercept parameter (chaoticity parameter) which has a general dependence of the mean momentum of the observed particle pair. Here, the dependence on the source lifetime is omitted. The chaocity parameter λ F is the temperature-dependent and the positive one defined by .

Comparing (4.3) and (4.4) one can identify
Hence, CF Ξ(k µ , k ′ µ ) defines uniquely the size r of MPPR. There is no satisfactory tool to derive the precise analytic form of the random source functioñ R(k µ ) in (4.3), but one can put (see (4.4) and taking into accountR(k µ ) where α is of the order O P 2 /n(ω, T ) · |ψ(k µ )| 2 . Thus, It is easy to see that, in the vicinity of q 2 ≈ 0, one can get the full correlation if α = α ′ = 0 andλ(ω; T )=1. Putting α = α ′ in (4.8), we find the formal lower bound on the space-time dimensionless size of MPPR of the bosons In fact, the function D(k µ , k ′ µ ; T ) in (4.5) could not be observed because of some model uncertainties. In the real world, D-function has to contain background contributions which have not been included in the calculation. To be close to the experimental data, one has to expand the D-function as projected on some well-defined function (in S(ℜ 4 )) of the relative momentum of bosons produced D(k µ , k ′ µ ; T ) → D(Q 2 µ ; T ). Thus, it will be very instructive to use the polynomial expansion which is suitable to avoid any uncertainties as well as to characterize the degree of deviation from the Gaussian distribution, for example. In (−∞, +∞), a complete orthogonal set of functions can be obtained with the help of the Hermite polynomials in the Hilbert space of the square integrable functions with the measure dµ(z) = exp(−z 2 /2)dz. The function D corresponds to this class if +∞ −∞ dq exp(−q 2 /2) |D(q)| n < ∞ , n = 0, 1, 2, ... .
The expansion in terms of the Hermite polynomials H n (q) is well suited for the study of possible deviations both from the experimental shape and from the exact theoretical form of the function D (4.5). The coefficients c n in (4.9) are defined via the integrals over the expanded functions D because of the orthogonality condition Thus, the observation of the multiparticle correlation enables one to extract the properties of the structure of q 2 , i.e. the space-time size of MMPR. The other possibility is related with the replacement of R-function (4.2) with respect to the cylindrical symmetry angles θ and φ which are non-observable ones at fixed Q t : It remains to sketch how one goes about calculating the thermodynamical quantities in a local thermalized system of produced particles. Taking into account the positive-and negative-frequency parts of the boson field operator (3.2) to be applied to the energy-momentum tensor T µν (x) =: we can calculate the energy density E(β), the pressure V (β) and the entropy density S(β) in the local system of the volume v for bosons in the equilibrium thermalized phase space. The simple straightforward calculations give (see also [9]) Here, we suppose that the thermalized MPPR is isotropic, and one can use the space-averaged operators normalized to the volume v, taking ensemble averages (4.6) and (4.7). It is easy to see that both E(β) and V (β) tend to their maximum values with rising T , while the entropy S(β) changes not so much essentially even if T → ∞.
From a widely accepted point of view, at high energies, there are two channels, at least, for multiparticle production where produced particles occupy the MMPR consisting of i elementary cells. These main channels are a) a direct channel assuming that all particles p j are produced directly within the quark (q)-antiquark (q) annihilation or the gauge-boson fusion, e.g., qq → p j p j ...; b) an indirect channel which means that the particles are produced via the decays of intermediate vector bosons χ * in both heavy and light sectors in the kinematically allowed region, e.g., qq → χ * χ * ... → p j p j ... All the particles produced are classified by the like-sign constituents that are labeled as p + , p − , p 0 -subsystems, where p : µ , π , K... The mean multiplicity N and the mean energy E of the p j subsystem are defined as [3] where ǫ j is the energy of a p-particle in the jth elementary cell and ζ (m j ) j stands for the probability of finding m j p particles in the jth cell and is normalized as In the direct channel, for charged produced mesons N is defined uniquely for a given β as Going into y-rapidity space in the longitudinal phase space with a lot of cells of equal size δy, the energy ǫ j should be expressed in terms of the transverse mass m t = k t 2 + m 2 p ( k t and m p are the transverse average momentum and the mass of a p-particle): Here, the four-momentum of p-particle is given as k µ = ( k t 2 + m 2 p cosh y, k t cos ϕ, k t sin ϕ, k t 2 + m 2 p sinh y) , where the azimuthal angle of k t is in the range 0 < ϕ < 2 π. Our model produces an enhancement of R(Q, β) in the small enough region of Q where R is defined only by the model parameter α and the mean multiplicity N(s) at fixed value of β, namely: cannot exceed 2 because of α = 0 and ξ(N(s)) < 1 even at large multiplicity. The Boltzmann behaviour should be realized in the case when α → ∞, i.e. the main contribution to the fluctuating behaviour of the R(Q, β)-function should come from the random source contribution (see (4.5) and (4.8)). We found that the enhancement of the R(Q, β)-function, mainly, the shape of this function, strongly depends on the transverse size r t of the phase space and has a very weak dependence of the δy size of a separate elementary cell. The increasing of r t makes that the shape of the R(Q, β)-function becomes more crucial.
Obviously, ξ( N(s) ) is the normalization constant in (4.2), where N(s) should be derived at the origion of Q 2 precisely from R(Q = 0, β) ≡ R 0 (s) as can be extracted from the experiment at some chosen value of α (α = 0 should be taken into account as well). On the other hand, the R(Q, β)function allows one to measure α = α ′ which parametrizes the random source contribution as well as the splitting between α and α ′ . Neglecting the random source contribution (i.e., putting α = α ′ = 0) we can estimate the chaoticitỹ λ(ω, β) by measuring R(Q, β) as Q 2 → 0. In fact, the theoretical prediction that D(Q, β) > 1 means that in MPPR one should select the single boson "dressing" of some quantum numbers, and the particles suited near it in the phase space are "dressed" with the same set of quantum numbers. The amount of such neighbour particles has to be as many as possible. This allows to form a cell in the space-time occupied by equal-statistics particles only. Such a procedure can be repeated while all the particles will occur in the MPPR. This leads to the space-time BE distribution of produced particles in the phase-space cells formed only for bosons. In fact, there is no restriction of the number of bosons occupying the chosen elementary cells. It means that D(Q, β)-function are defined for all orders.

Summary and discussion
We investigated the finite temperature BE correlations of identical particles in the multiparticle production using the solutions of the operator field Langevin-type equation in S(ℜ 4 ), the quantum version of the Nyquist theorem and the quantum statistical methods. The model considered states that all the particles are produced directly from the a high-energy collision process. There was presented the crucial role of the model in describing the BE correlations via calculations of DF's as functions of the mean multiplicity and chaoticity at each four-momentum Q 2 µ . Based on this model, one can compare the effects on single particle spectra and multiparticle distribution caused by multiparticle correlations. There are several parameters in the model: β, δy, α(α ′ ). One can focus on the statement that the deviation of the D(Q, β)-function from zero at finite values of the physical variables q 2 and the model parameter α indicates that the MPPR should be considered as the phase-space consisting of the elementary cells (with the size δy of each) which are occupied by the particles of identical statistics. The Boltzmann behaviour of R-function is available only at large enough values of α which means the leading role of the random source contribution to the distribution function. An important feature of the model is getting the information on the space-time structure of the multiparticle production region. We are able to predict the source size and the intercept parameter -the chaoticity λ as well. We have found that DF R(Q, β) depends on the number of elementary defined by the size δy in the rapidity y-space.
Of course, the best check of any model could be done if various kinds of high energy experimental data on the multiparticle correlations would be well reproduced by the model in consideration. Our model can be applied to the data which are available from the ALEPH experiment [10] at √ s=91.2 GeV.