Bose-Einstein correlations and the stochastic scale of light hadrons emitter source
aa r X i v : . [ h e p - ph ] J a n Bose-Einstein correlations and thestochastic scale of light hadronsemitter source
G.A. Kozlov
Bogolyubov Laboratory of Theoretical PhysicsJoint Institute for Nuclear ResearchJoliot Curie st.6, Dubna, 141980 Russia
Abstract
Based on quantum field theory at finite temperature we carried out new results fortwo-particle Bose-Einstein correlation (BEC) function C ( Q ) in case of light hadrons.The important parameters of BEC function related to the size of the emitting source,mean multiplicity, stochastic forces range with the particle energy and mass depen-dence, and the temperature of the source are obtained for the first time. Not onlythe correlation between identical hadrons are explored but even the off-correlationbetween non-identical particles are proposed. The correlations of two bosons in 4-momentum space presented in this paper offer useful and instructive complimentaryviewpoints to theoretical and experimental works in multiparticle femtoscopy and in-terferometry measurements at hadron colliders. This paper is the first one to thenext opening series of works concerning the searching of BEC with experimental datawhere the parameters above mentioned will be retrieved. Introduction
For the aim to explore the correlations of Bose-Einstein type (BEC) one needs to usethe properties of a particle detector, e.g., its tracking system to study the hadronprocesses at some energy region. Such a study will be done soon in the next papers.This paper describes an attempt to address the problems of BEC within the the-oretical aspects prior the real data will be analyzed.Over the past few decades, a considerable number of studies have been done on thephenomena of multi-particle correlations observed in high energy particle collisions(see the review in [1]). It is well understood that the studies of correlations betweenproduced particles, the effects of coherence and chaoticity, an estimation of particleemitting source size and the temperature play an important role in this branch ofhigh energy physics.By studying the Bose-Einstein correlations of identical particles (e.g., like-signcharge particles of the same sort) or even off-correlations with respect to different-charge bosons, it is possible to predict and even experimentally determine the timeand spatial region over which particles do not have the interactions. Such a surfaceis called as decoupling one. In fact, for an evolving system such as, e.g., p ¯ p collisions,it is not really a surface, since at each time there is a spread out surface due tofluctuations in the final interactions, and the shape of this surface evolve even intime. The particle source is not approximately constant because of energy-momentumconservation constraint.More than half a century ago Hanbury-Brown and Twiss [2] used BEC betweenphotons to measure the size of distant stars. In the papers in [3] and [4] , the masterequations for evolution of thermodynamic system created at the final state of the(very) high multiplicity process were established. The equations have the form ofthe field operator evolution equation (Langevin-like [5]) that allows one to gain thebasic features of the emitting source space-time structure. In particular, it has beenconjectured and further confirmed that the BEC is strongly affected by non-classicaloff-shell effect.The shapes of BEC function were experimentally established in the LEP experi-ments ALEPH [6], DELPHI [7] and OPAL [8], and ZEUS Collaboration at HERA [9],which also indicated a dependence of the measured so-called correlation radius on thehadron ( π, K ) mass. The results for π ± π ± and π ± π ∓ correlations with p ¯ p collisionsat √ s = 1 . QF T β ) approach which tobe applied later to real experimental data on two-particle BEC. It is known that theeffective temperature of the vacuum or the ground state or even the thermalized stateof particles distorted by external forces is occurring in models quantized in externalfields. One of the main parameters of the model considered here is the temperatureof the particle source under the random source operator influence.Among the results obtained in this paper we mention a theoretical estimate ac-cessible to experimental measurements of two-particle BEC and proof that quantum-statistical evolution of particle-antiparticle correlations are not an artifact of the stan-2ard formalism but a quite general properties of particle physics. The effect (calledas surprized one) for non-identical particles correlations was predicted already in [11]. A pair of bosons with the mass m produced incoherently (in ideal nondisturbed,noninteracting cases) from an extended source will have an enhanced probability C ( p , p ) = N ( p , p ) / [ N ( p ) · N ( p )] to be measured (in terms of differentialcross section σ ), where N ( p , p ) = 1 σ d σd Ω d Ω (1)to be found close in 4-momentum space ℜ when detected simultaneously, as comparedto if they are detected separately with N i ( p i ) = 1 σ dσd Ω i , d Ω i = d ~p i (2 π ) E p i , E p i = q ~p i + m , i = 1 , . (2)The following relation can be used to retrieve the BEC function C ( Q ): C ij ( Q ) = N ij ( Q ) N ref ( Q ) , i, j = + , − , , (3)where N ij ( Q ) in general case refer to the numbers N ±± ( Q ) for like-sign chargeparticles (eg., π ± π ± , K ± K ± , . . . ); N ±∓ ( Q ) — for different charge bosons (eg., π ± π ∓ , K ± K ∓ , . . . ) or even for neutral charge particles N ( Q ) (eg., π π , K K , . . . )with Q = q − ( p − p ) µ · ( p − p ) µ = p M − m . (4)In formula (3) and (4) N ref is the number of pairs without BEC and p µ i ( i = 1 , M = q ( p + p ) µ is the invariant mass ofthe pair of bosons. For reference sample, N ref ( Q ), the like-sign pairs from differentevents can be used. It is commonly assumed that the maximum of two-particle BECfunction C ii ( Q ) is 2 for ~p = ~p if no any distortion and final state interactions aretaking into account.In general, the shape of BEC C ( Q ) function is model dependent. The most simpleform of Goldhaber-like parameterization for C ( Q ) [12] has been used for data fitting: C ( Q ) = C · (1 + λe − Q R ) · (1 + εQ ) , (5)where C is the normalization factor, λ is so-called the chaoticity strength factor,meaning λ = 1 for fully chaotic and λ = 0 for fully coherent sources; the parameter R is interpreted as a radius of the particle source, often called as the ”correlationradius”, and assumed to be spherical in this parameterization. The linear term in(5) is often supposed to be account for long-range correlations outside the region ofBEC. However, the origin of these long-range correlations as well as the value of ǫ are unknown yet. Note that distribution of, e.g., pions and kaons can be far from3sotropic, usually concentrated in narrow jets, and further complicated by the factthat the light particles with masses less than 1 GeV often come from decays of long-lived heavier resonances and also are under the random chaotic interactions causedby other fields in the thermal bath. In the parameterization (5) all of these problemsare embedded in the random chaoticity parameter λ .We obtained the C ( Q ) function within QF T β approach [3] in the form: C ( Q ) = ξ ( N ) · h α (1 + α ) q ˜Ω( Q ) + 1(1 + α ) ˜Ω( Q ) i · F ( Q, ∆ x ) , (6)where ξ ( N ) depends on the multiplicity N as ξ ( N ) = h N ( N − ih N i . (7)The consequence of the Bogolyubov’s principle of weakening of correlations at largedistances [13] is given by the function F ( Q, ∆ x ) of weakening of correlations at largespread of relative position ∆ xF ( Q, ∆ x ) = f ( Q, ∆ x ) f ( p ) · f ( p ) = 1 + r f Q + . . . (8)normalized as F ( Q, ∆ x = ∞ ) = 1. Here, f ( Q, ∆ x ) is the two-particle distributionfunction with ∆ x , while f ( p i ) are one-particle probability functions with i = 1 , r f is a measure of weakening of correlations with ∆ x : r f → x → ∞ .The important parameter α in (6) summarizes our knowledge of other than space-time characteristics of the particle emitting source.The ˜Ω( Q ) in (6) has the following structure in momentum space˜Ω( Q ) = Ω( Q ) · γ ( n ) , (9)where Ω( Q ) = exp( − ∆ p ℜ ) = exp [ − ( p − p ) µ ℜ µν ( p − p ) ν ] (10)is the smearing smooth dimensionless generalized function, ℜ µν is the (nonlocal)structure tensor of the space-time size (BEC formation domain), and it defines thespherically-like domain of emitted (produced) particles.The function γ ( n ) in (9) reflects the quantum features of BEC pattern and isdefined as γ ( n ) = n (¯ ω ) n ( ω ) n ( ω ′ ) , n ( ω ) ≡ n ( ω, β ) = 1 e ( ω − µ ) β − , ¯ ω = ω + ω ′ , (11)where n ( ω, β ) is the mean value of quantum numbers for BE statistics particles withthe energy ω and the chemical potential µ in the thermal bath with statistical equi-librium at the temperature T = 1 /β . The following condition P f n f ( ω, β ) = N isevident, where the discrete index f reflects the one-particle state f .Note that it is commonly assumed for a long time that there are no correlationeffects among nonidentical particles (e.g., among different charged particles). This4ssumption is often used in normalizing the experimental data on C ii with respectto C ij . In the absence of interference or correlation effects between, e.g., π + and π − mesons it is supposed that C + − = 1.In terms of time-like R , longitudinal R L and transverse R T components of thespace-time size R µ the distribution ∆ ijp ℜ looks like ( i, j = + , − , ijp ℜ → ∆ ijpR = (∆ p ) R + (∆ p L ) R L + (∆ p T ) R T . (12)Seeking for simplicity one has ( R L = R T = R )∆ iipR = ( p − p ) R + ( ~p − ~p ) ~R (13)for like-sign charge bosons, while∆ ijpR = ( p + p ) R + ( ~p + ~p ) ~R (14)for different charge particles.Obviously, the BEC effect with Ω ij = exp( − ∆ ijpR ) is smaller than that definedby Ω ii = exp( − ∆ iipR ). The distribution Ω ij gives rise to an off-correlation pat-tern between different charge particles. The evidence of C ij correlation representsa quantum-statistical correlation between a particle and an antiparticle. Since we didnot follow special assumptions on the quantum operator level for C from the initialstage, it may correspond to a physically real and observable effect. This pattern maylead to a new squeezing state of correlation region. We obtain that within the QF T β the BEC is more generally sensitive to particle-antiparticle correlations than it wouldbe expected from the two-particle (symmetrized) wave function which never leads tosuch the correlations. In this paper, we would like to focus on the role of the particle mass, which influencesthe correlations between particles. To explore this problem, one must derive thememory history of evolution of particles produced in high energy collisions using thegeneral properties of QFT at finite temperature.We consider the thermal scalar complex fields Φ( x ) that correspond to π ± mesonswith the standard definition of the Fourier transformed propagator F [ ˜ G ( p )] F [ ˜ G ( p )] = G ( x − y ) = T r { T [Φ( x )Φ( y )] ρ β } , (15)with ρ β = e − βH /T re − βH being the density matrix of a local system in equilibriumat temperature T = β − under the Hamiltonian H .We consider the interaction of Φ( x ) with the external scalar field given by thepotential U . In contrast to an electromagnetic field, this potential is a scalar one, butit is not a component of the four-vector. The Lagrangian density can be written L ( x ) = ∂ µ Φ ⋆ ( x ) ∂ µ Φ( x ) − ( m + U )Φ ⋆ ( x )Φ( x )5nd the equation of motion is( ∇ + m )Φ( x ) = − J ( x ) , (16)where J ( x ) = U Φ( x ) is the source density operator. A simple model like this allowsone to investigate the origin of the unstable state of the thermalized equilibriumin a nonhomogeneous external field under the influence of source density operator J ( x ). For example, the source can be considered as δ -like generalized function J ( x ) =˜ µ ρ ( x, ǫ )Φ( x ) in which ρ ( x, ǫ ) is a δ -like succession giving the δ -function as ǫ → µ is some massive parameter). This model is useful because the δ -like potential U ( x ) provides the model conditions for restricting the particle emission domain (orthe deconfinement region). We suggest the following form: J ( x ) = − Σ( i∂ µ ) Φ( x ) + J R ( x ) , where the source J ( x ) decomposes into a regular systematic motion part Σ( i∂ µ ) Φ( x )and the random source J R ( x ). Thus, the equation of motion (16) becomes[ ∇ + m − Σ( i∂ µ )]Φ( x ) = − J R ( x ) , and the propagator satisfies the following equation:[ − p µ + m − ˜Σ( p µ )] ˜ G ( p µ ) = 1 . (17)The random noise is introduced with a random operator η ( x ) = − m − Σ( i∂ µ ), forthat the equation of motion looks like: {∇ + m [1 + η ( x )] } Φ( x ) = − J R ( x ) . (18)We assume that η ( x ) varies stochastically with the certain correlation function(CF), e.g., the Gaussian CF h η ( x ) η ( y ) i = C exp( − z µ ch ) , z = x − y, where C is the strength of the noise described by the distribution function exp( − z /L ch )with L ch being the noise characteristic scale. Both C and µ ch define the influence ofthe (Gaussian) noise on the correlations between particles that ”feel” an action of anenvironment. The solution of Eq. (18) isΦ( x ) = − Z dy G ( x, y ) J R ( y ) , (19)where the Green’s function obeys the Eq. {∇ + m [1 + η ( x )] } G ( x, y ) = δ ( x − y ) . The final aim might having been to find the solution of Eq. (19), and then averageit over random operator η ( x ). Note that the operator M ( x ) = ∇ + m [1 + η ( x )] inthe causal Green’s function G ( x, y ) = 1 M ( x ) + i o δ ( x − y )6s not definitely positive. However, we shall formulate another approach, where therandom force influence is introduced on the particle operator level.We introduce the general non-Fock representation in the form of the operatorgeneralized functions b ( x ) = a ( x ) + r ( x ) , (20) b + ( x ) = a + ( x ) + r + ( x ) , (21)where the operators a ( x ) and a + ( x ) obey the canonical commutation relations (CCR):[ a ( x ) , a ( x ′ )] = [ a + ( x ) , a + ( x ′ )] = 0 , [ a ( x ) , a + ( x ′ )] = δ ( x − x ′ ) . The operator-generalized functions r ( x ) and r + ( x ) in (20) and (21), respectively,include random features describing the action of the external forces.Both b + and b obviously define the CCR representation. For each function f fromthe space S ( ℜ ∞ ) of smooth decreasing functions, one can establish new operators b ( f ) and b + ( f ) b ( f ) = Z f ( x ) b ( x ) dx = a ( f ) + Z f ( x ) r ( x ) dx,b + ( f ) = Z ¯ f ( x ) b + ( x ) dx = a + ( f ) + Z ¯ f ( x ) r + ( x ) dx. The transition from the operators a ( x ) and a + ( x ) to b ( x ) and b + ( x ), obeying thosecommutation relations as a ( x ) and a + ( x ), leads to linear canonical representations. Referring to [3] for details, let us recapitulate here the main points of our approach inthe quantum case: the collision process produces a number of particles, out of whichwe select only one (we assume for simplicity that we are dealing only with identicalbosons) and describe it by stochastic operators b ( ~p, t ) and b + ( ~p, t ), carrying the fea-tures of annihilation and creation operators, respectively. The rest of the particles arethen assumed to form a kind of heat bath, which remains in an equilibrium charac-terized by a temperature T (one of our parameters). We also allow for some external(relative to the above heat bath) influence on our system. The time evolution of sucha system is then assumed to be given by a Langevin-type equation [3] for stochasticoperator b ( ~p, t ) i∂ t b ( ~p, t ) = A ( ~p, t ) + F ( ~p, t ) + P (22)(and a similar conjugate equation for b + ( ~p, t )). We assume an asymptotic free undis-torted operator a ( ~p, t ), and that the deviation from the asymptotic free state is pro-vided by the random operator r ( ~p, t ): a ( ~p, t ) → b ( ~p, t ) = a ( ~p, t ) + r ( ~p, t ). This means,e.g., that the particle density number (a physical number) h n ( ~p, t ) i ph = h n ( ~p ) i + O ( ǫ ),where h n ( ~p, t ) i ph means the expectation value of a physical state, while h n ( ~p ) i denotesthat of an asymptotic state. If we ignore the deviation from the asymptotic state in7quilibrium, we obtain an ideal fluid. One otherwise has to consider the dissipationterm; this is why we use the Langevin scheme to derive the evolution equation, butonly on the quantum level. We derive the evolution equation in an integral form thatreveals the effects of thermalization.Equation (22) is supposed to model all aspects of the hadronization processes (oreven deconfinement). The combination A ( ~p, t ) + F ( ~p, t ) in the r.h.s of (22) representsthe so-called Langevin force and is therefore responsible for the internal dynamics ofparticle emission, as the memory term A causes dissipation and is related to stochasticdissipative forces [3] A ( ~p, t ) = Z + ∞−∞ dτ K ( ~p, t − τ ) b ( ~p, τ )with K ( ~p, t ) being the kernel operator describing the virtual transitions from one(particle) mode to another. At any dependence of the field operator K on the time,the function A ( ~p, t ) is defined by the behavior of the system at the precedent moments.The operator F ( ~p, t ) in (22) is responsible for the action of a heat bath of absolutetemperature T on a particle in the heat bath, and under the appropriate circumstancesis given by F ( ~p, t ) = Z + ∞−∞ dω π ψ ( p µ )ˆ c ( p µ ) e − iωt . The heat bath is represented by an ensemble of coupled oscillators, each describedby the operator ˆ c ( p µ ) such that (cid:2) ˆ c ( p µ ) , ˆ c + ( p ′ µ ) (cid:3) = δ ( p µ − p ′ µ ), and is characterizedby the noise spectral function ψ ( p µ ) [3]. Here, the only statistical assumption is thatthe heat bath is canonically distributed. The oscillators are coupled to a particle,which is in turn acted upon by an outside force. Finally, the constant term P in (22)(representing an external source term in the Langevin equation) denotes a possibleinfluence of some external force. This force would result, e.g., in a strong ordering ofphases leading therefore to a coherence effect.The solution of equation (22) is given in S ( ℜ ) by˜ b ( p µ ) = 1 ω − ˜ K ( p µ ) [ ˜ F ( p µ ) + ρ ( ω P , ǫ )] , (23)where ω in ρ ( ω, ǫ ) was replaced by new scale ω P = ω/P . It should be stressed that theterm containing ρ ( ω P , ǫ ) as ǫ → ρ ( ω P , ǫ ) indicates the continuous character of the spectrum, whilethe arbitrary small quantity ǫ can be defined by the special physical conditions or thephysical spectra. On the other hand, this ρ ( ω P , ǫ ) can be understood as temperature-dependent succession ρ ( ω, ǫ ) = R dx exp ( iω − ǫ ) x → δ ( ω ), in which ǫ → β − . Such asuccession yields the restriction on the β -dependent second term in the solution (23),where at small enough T there is a narrow peak at ω = 0.From the scattering matrix point of view, the solution (23) has the followingphysical meaning: at a sufficiently outgoing past and future, the fields described bythe operators ˜ a ( p µ ) are free and the initial and the final states of the dynamic systemare thus characterized by constant amplitudes. Both states, ϕ ( −∞ ) and ϕ (+ ∞ ), arerelated to one another by an operator S (˜ r ) that transforms state ϕ ( −∞ ) to state8 (+ ∞ ) while depending on the behaviour of ˜ r ( p µ ): ϕ (+ ∞ ) = S (˜ r ) ϕ ( −∞ ) . In accordance with this definition, it is natural to identify S (˜ r ) as the scatteringmatrix in the case of arbitrary sources that give rise to the intensity of ˜ r .Based on QFT point of view, relation (20) indicates the appearance of the termscontaining nonquantum fields that are characterized by the operators ˜ r ( p µ ). Hence,there are terms with ˜ r in the matrix elements, and these ˜ r cannot be realized viareal particles. The operator function ˜ r ( p µ ) could be considered as the limit on anaverage value of some quantum operator (or even a set of operators) with an intensitythat increases to infinity. The later statement can be visualized in the followingmathematical representation:˜ r ( p µ ) = q α Ξ( p µ , p µ ) , Ξ( p µ , p µ ) = h ˜ a + ( p µ ) ˜ a ( p µ ) i β , where α is the coherence (chaotic) function that gives the strength of the averageΞ( p µ , p µ ).In principal, interaction with the fields described by ˜ r is provided by the virtualparticles, the propagation process of which is given by the potentials defined by the˜ r operator function.The condition M ch → ( R ) ∼ M ch → ∞ ) in the representationlim p µ → p ′ µ Ξ( p µ , p ′ µ ) = lim Q → Ω ( R ) n (¯ ω, β ) exp( − q / → M ch n ( ω, β ) , with Ω ( R ) = 1 π R R L R T means that the role of the arbitrary source characterized by the operator function˜ r ( p µ ) in ˜ b ( p µ ) = ˜ a ( p µ ) + ˜ r ( p µ ) disappears. Let us go to the thermal field operator Φ( x ) by means of the linear combination ofthe frequency parts φ + ( x ) and φ − ( x )Φ( x ) = 1 √ (cid:2) φ + ( x ) + φ − ( x ) (cid:3) (24)composed of the operators ˜ b ( p µ ) and ˜ b + ( p µ ) as the solutions of equation (22) andconjugate to it, respectively: φ − ( x ) = Z d ~p (2 π ) ~p + m ) / ˜ b + ( p µ ) e ipx ,φ + ( x ) = Z d ~p (2 π ) ~p + m ) / ˜ b ( p µ ) e − ipx . x ) obeys the commutation relation[Φ( x ) , Φ( y )] − = − iD ( x )with [14] D ( x ) = 12 π ǫ ( x ) δ ( x ) − m q x µ Θ( x ) J (cid:16) m q x µ (cid:17) , where ǫ ( x ) and Θ( x ) are the standard unit and the step functions, respectively,while J ( x ) is the Bessel function. On the mass-shell, D ( x ) becomes [14] D ( x ) ≃ π ǫ ( x ) (cid:20) δ ( x ) − m x ) (cid:21) . One can easily find two equations of motion for the Fourier transformed operators˜ b ( p µ ) and ˜ b + ( p µ ) in S ( ℜ )[ ω − ˜ K ( p µ )]˜ b ( p µ ) = ˜ F ( p µ ) + ρ ( ω P , ǫ ) , (25)[ ω − ˜ K + ( p µ )]˜ b + ( p µ ) = ˜ F + ( p µ ) + ρ ⋆ ( ω P , ǫ ) , (26)which are transformed into new equations for the frequency parts φ + ( x ) and φ − ( x )of the field operator Φ( x ) (24) i∂ φ + ( x ) + Z ℜ K ( x − y ) φ + ( y ) dy = f ( x ) (27) − i∂ φ − ( x ) + Z ℜ K + ( x − y ) φ − ( y ) dy = f + ( x ) , (28)where f ( x ) = Z d ~p (2 π ) ( ~p + m ) / [ ˜ F ( p ) + ρ ( ω P , ǫ )] e − ipx , and f + ( x ) = Z d ~p (2 π ) ( ~p + m ) / [ ˜ F + ( p ) + ρ ⋆ ( ω P , ǫ )] e ipx . Here, the field components φ + ( x ) and φ − ( x ) are under the effect of the nonlocalformfactors K ( x − y ) and K + ( x − y ), respectively. In general, these formfactors canadmit the description of locality for nonlocal interactions.At this stage, it must be stressed that we have new generalized evolution Eqs. (27)and (28), which retain the general features of the propagating and interacting of thequantum fields with mass m that are in the heat bath (reservoir) and are chaoticallydistorted by other fields. For further analysis, let us rewrite the Eqs. (27) and (28)in the following form: i∂ φ + ( x ) + K ( x ) ⋆ φ + ( x ) = f ( x ) , (29) − i∂ φ − ( x ) + K + ( x ) ⋆ φ − ( x ) = f + ( x ) , (30)10here A ( x ) ⋆ B ( x ) is the convoluted function of the generalized functions A ( x ) and B ( x ). Applying the direct Fourier transformation to both sides of Eqs. (29) and (30)with the following properties of the Fourier transformation F [ K ( x ) ⋆ φ + ( x )] = F [ K ( x )] F [ φ + ( x )] , we get two equations [ p + ˜ K ( p µ )] ˜ φ + ( p µ ) = F [ f ( x )] , (31)[ − p + ˜ K + ( p µ )] ˜ φ − ( p µ ) = F [ f + ( x )] . (32)Multiplying Eqs. (31) and (43) by − p + ˜ K + ( p µ ) and p + ˜ K ( p µ ), respectively, wefind [ − p + ˜ K + ( p µ )][ p + ˜ K ( p µ )] ˜Φ( p µ ) = T ( p µ ) , (33)where T ( p µ ) = [ − p + ˜ K + ( p µ )] F [ f ( x )] + [ p + ˜ K ( p µ )] F [ f + ( x )] . We are now at the stage of the main strategy: we have to identify the field Φ( x )introduced in Eq. (15) and the field Φ( x ) (24) built up of the fields φ + and φ − as thesolutions of generalized Eqs. (27) and (28). The next step is our requirement thatGreen’s function ˜ G ( p µ ) in Eq. (17) and the function Γ( p µ ), that satisfies Eq. (44)[ − p + ˜ K + ( p µ )][ p + ˜ K ( p µ )]˜Γ( p µ ) = 1 , (34)must be equal to each other, where the full Green’s function ˜ G ( p , g , m )˜ G ( p µ ) → ˜ G ( p , g , m ) ≃ − g ξ ( p , m ) m − p − iǫ (35)has the same pole structure at p = m as the free Green’s function [14] with g beingthe scalar coupling constant and ξ is the one-loop correction of the scalar field. Thedimensioneless function 1 − g ξ ( p , m ) is finite at p = m .We define the operator kernel ˜ K ( p µ ) in (25) from the condition of the nonlocal co-incidence of the Green’s function ˜ G ( p µ ) in Eq. (17), and the thermodynamic function˜Γ( p µ ) from (34) in S ( ℜ ) F [ ˜ G ( p µ ) − ˜Γ( p µ )] = 0 . We can easily derive the kernel operator ˜ K ( p µ ) in the form˜ K ( p ) = m + ~p − g ξ ( p , m ) p − g ξ ( p , m ) (36)where [14] ξ ( m ) = 196 π m (cid:18) π √ − (cid:19) , p ≃ m , and ξ ( p , m ) = 196 π m i s − m p + π √ ! , p ≃ m . The ultraviolet behaviour at | p | >> m leads to ξ ( p , m ) ≃ − π p (cid:20) ln | p | m − π √ − i π Θ( p ) (cid:21) . Stochastic forces scale
In paper [4] it has been emphasized that two different scale parameters are in themodel which we consider here. One of them is the so-called ”correlation radius” R introduced in (5) and (6) with (9) and (12), (13), (14). In fact, this R -parametergives the pure size of the particle emission source without the external distortion andinteraction coming from other fields. The other (scale) parameter is the stochasticscale L st which carries the dependence of the particle mass, the α -coherence degreeand what is very important — the temperature T -dependence: L st = " α ( N ) | p − ˜ K ( p ) | ¯ n ( m, β ) . (37)It turns out that this scale L st defines the range of stochastic forces acting the particlesin the emission source. This effect is given by α ( N )-coherence degree which can beestimated from the experiment within the two-particle BE correlation function C ( Q )as Q close to zero, C (0), at fixed value of mean multiplicity h N i : α ( N ) ≃ − ¯ C (0) + p − ¯ C (0)¯ C (0) − , ¯ C (0) = C (0) /ξ ( N ) . (38)In formula (37) ¯ n ( m, β ) is the thermal relativistic particle number density¯ n ( m, β ) = 3 Z d ~p (2 π ) n ( ω, β ) = 3 µ + m π T ∞ X l =1 l K (cid:18) lT p µ + m (cid:19) , (39)where K is the modified Bessel function. For definite calculations we consider cor-relations between charge pions. The result can be extended to heavy particles case,e.g., for charge and neutral gauge bosons that is essential program for the LHC.The stochastic scale L st tends to infinity in case of particles are on mass-shell, i.e., | p − ˜ K ( p ) | → L ′ st s denominator (37). However, L st will be boundeddue to stochastic forces acting the particles where | p − ˜ K ( p ) | ≃ ∆ ǫ p = ǫ p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ǫ p − − g ξ ( p , m )2 − p ǫ p !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , ǫ p = p m + ~p as g ξ ( p , m ) < L st has the form L st ≃ e √ µ + m /T α ( N ) ∆ ǫ p ( µ + m ) / (cid:0) T π (cid:1) / (cid:18) T √ µ + m (cid:19) , (40)where the condition l β p m + µ > l in (39) was taken into account.The only lower temperatures will drive L st within the formula (40) even if µ = 0 and12 = 1 with the condition T < m . Note that the condition µ < m is a generalrestriction in the relativistic ”Bose-like gas”, and µ = m corresponds to the Bose-Einstein condensation.For large enough T no the dependence of the chemical potential µ is found for L st : L st ≃ (cid:20) π ζ (3) α ( N ) ∆ ǫ p T (cid:21) , (41)where the condition T > l p µ + m , l = 1 , , ... is taken into account. The origin offormula (41) comes from ¯ n ( m, β ) → ¯ n ( β ) = 3 T π ζ (3) (42)where neither a pion mass m - nor µ - dependence occurred; ζ (3) = P ∞ l =1 l − = 1 . p ≃ m ) theactual mass-dependence occurred for L st : L st ≃ e √ µ + m /T α ( N ) m ( µ + m ) / (cid:0) T π (cid:1) / (cid:18) T √ µ + m (cid:19) , (43)at low T, and L st ≃ (cid:20) π ζ (3) α ( N ) m T (cid:21) , (44)at high temperatures, if g ξ ( m ) << ξ ( m ) ∼ O (0 . /m ) and ( ~p / m ) << ∼ T behaviorwhich is the same as the thermal distribution (in terms of density) for a gas of freerelativistic massless particles. Such a behavior is expected anyway in high temperaturelimit if the particles can be considered as asymptotically free in that regime.Actually, the increasing of T leads to squeezing of L st , and L st ( T = T ) = R atsome effective temperature T . The higher temperatures, T > T , satisfy to moresqueezing effect and at the critical temperature T c the scale L st ( T = T c ) takes itsminimal value. Obviously T c defines the phase transition where the deconfinementwill occur. Since all the masses tend to zero (chiral symmetry restoration) and α → T > T c one should expect the sharp expansion of the region with L st ( T > T c ) → ∞ .The following condition ˜ n ( m, β ) · v π = 1 provides the phase transition (transition fromhadronizing phase to deconfinement one) with the volume v π = (4 π r π / r π is the pion charge radius. Actually, the temperature of phase transition essentiallydepends on the charge (vector) radius of the pion which is a fundamental quantity inhadron physics. A recent review on r π values is presented in [15].What we know about the source size estimation from experiments? DELPHIand L3 collaborations at LEP established that the correlation radius R decreaseswith transverse pion mass m t as R ≃ a + b/ √ m t for all directions in the Longi-tudinal Center of Mass System (LCMS). ZEUS collaboration at HERA did not ob-serve the essential difference between the values of R - parameter in π ± π ± , K s K s K ± K ± pairs, namely R ππ = 0 . ± . stat ) + 0 . − . syst. ) f m , R K s K s = 0 . ± . stat ) + 0 . − . syst. ) f m and R K ± K ± = 0 . ± . stat ) +0 . − . syst. ) f m , respectively. The ZEUS data are in good agreement with theLEP for radius R . However, no evidence for √ s dependence of R is found. It isevidently that more experimental data are appreciated. However, the comparisonbetween experiments is difficult mainly due to reference samples used and the MonteCarlo corrections.Finally, our theoretical results first predict the L st in (40) and (41), and bothmass- and temperature - dependence are obtained clearly. This can serve as a goodapproximation to explain the LEP, Tevatron and ZEUS (HERA) experimental data.We need that the pion energies at the colliders are sufficient to carry these studies out(since the ∆ ǫ p dependence). Careful simulation of their (pions) signal and backgroundare needed. The more precisely measured pion momentum may be of some help. Also,determination of the final state interactions may clarify what is happening. To summarize: we find the time dependence of the correlation function C ( Q ) calcu-lated in time-dependent external field provided by the operator r ( ~p, t ) and the chaoticcoherence function α ( m, β ). Based on this approach we emphasize the explanationof the dynamic origin of the coherence in BEC, the origin of the specific shape ofthe correlation C ( Q ) functions, and finding the dependence on the particle energy(and the mass) due to coherence function α , as seen from the QF T β . Actually, thestochastic scale L st decreases with the particle energy (the mass m ). It is alreadyconfirmed by the data of LEP, Tevatron and HERA (ZEUS) with respect to the sizeof particle source.In the framework of QF T β the numerical analysis of experimental data can becarried out with a result where important parameters of C −− ( Q ) and C ( Q )functions are retrieved (e.g., C , R, λ, ǫ, N, α, L st , T ).The correlations of non-identical particles pairs can be observed and the corre-sponding C parameters is retrieved. The off-correlation effect is given by the space-time distribution (14) containing the sum ~p + ~p , and this effect is sufficient if thefactor containing the sum p + p in (10) is not too small. The off-correlation effectis possible if the particle energies p i ( i = 1 ,
2) are small enough.Besides the fact that like, e.g., π ± π ± BEC the correlations π ± π ∓ can serve astools in the determination of parameters of the particle source. And besides the factthat these correlations play a particularly important role in the detection of randomchaotic correction to BEC.The stochastic scale L st decreases with increasing temperatures slowly at lowtemperatures, and it decreases rather abruptly when the critical temperature is ap-proached.We claim that the experimental measuring of R (in f m ) can provide the preciseestimation of the effective temperature T which is the main thermal character in theparticle’s pair emitter source (given by the effective dimension R ) with the particlemass and its energy at given α fixed by C ( Q = 0) and h N i . Actually, T is the14rue temperature in the region of multiparticle production with dimension R = L st ,because at this temperature it is exactly the creation of two particles occurred, andthese particles obey the criterion of BEC.We have found the squeezing of the particle source due to decreasing of the corre-lation radius R in the case of opposite charge particles. The off-correlated system ofnon-identical particles is less sensitive to the random force influence ( α - dependence).The results obtained in this paper can be compared with the static correlationfunction (see, e.g., [16] and the references therein relevant to heavy ion collisions).Finally, we should stress a new features of particle-antiparticle BEC which canemerge from the data. It is a highly rewarding task to experimental measurement ofnon-identical particles.There is much to be done for C ( Q ) investigation at hadron colliders. The timeis ripe for dedicated searches for new effects in C ( Q ) function at hadron colliders todiscover, or rule out, in particular, the α ( N ) dependence.In conclusion, the correlations of two bosons in 4-momentum space presented inthis paper offer useful and instructive complimentary viewpoints to theoretical andexperimental works in multiparticle femtoscopy and interferometry measurements athadron colliders. References [1] R.M.Weiner, Phys. Rep. (2000) 249.[2] R.Hanbury-Brown and R.Q.Twiss, Philos. Mag. (1954) 633; Nature (1956) 27; (1956) 1046; (1956) 1447.[3] G.A.Kozlov, O.V.Utyuzh and G.Wilk, Phys. Rev. C68 (2003) 024901;G.A.Kozlov, Phys. Rev.
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