Correlation femtoscopy of small systems

# Correlation femtoscopy of small systems

Yu.M. Sinyukov    V.M. Shapoval Bogolyubov Institute for Theoretical Physics, Metrolohichna str. 14b, 03680 Kiev, Ukraine
###### Abstract

The basic principles of the correlation femtoscopy, including its correspondence to the Hanbury Brown and Twiss intensity interferometry, are re-examined. The main subject of the paper is an analysis of the correlation femtoscopy when the source size is as small as the order of the uncertainty limit. It is about 1 fm for the current high energy experiments. Then the standard femtoscopy model of random sources is inapplicable. The uncertainty principle leads to the partial indistinguishability and coherence of closely located emitters that affect the observed femtoscopy scales. In thermal systems the role of corresponding coherent length is taken by the thermal de Broglie wavelength that also defines the size of a single emitter. The formalism of partially coherent phases in the amplitudes of closely located individual emitters is used for the quantitative analysis. The general approach is illustrated analytically for the case of the Gaussian approximation for emitting sources. A reduction of the interferometry radii and a suppression of the Bose-Einstein correlation functions for small sources due to the uncertainty principle are found. There is a positive correlation between the source size and the intercept of the correlation function. The peculiarities of the non-femtoscopic correlations caused by minijets and fluctuations of the initial states of the systems formed in and collisions are also analyzed. The factorization property for the contributions of femtoscopic and non-femtoscopic correlations into complete correlation function is observed in numerical calculations in a wide range of the model parameters.

###### pacs:
03.65.Ta, 25.75.Gz

Keywords: correlation femtoscopy, Hanbury Brown - Twiss intensity interferometry, proton-proton collisions, uncertainty principle, coherence, HBT correlations.

Corresponding author: Yu.M. Sinyukov, Bogolyubov Institute for Theoretical Physics, Kiev 03680, Metrolohichna 14b, Ukraine. E-mail: sinyukov@bitp.kiev.ua

## I Introduction

The correlation femtoscopy, or intensity interferometry method, is the direct tool to measure the spatial and temporal scales of extremely small and short-lived systems created in particle and nuclear collisions with accuracy of m and sec, respectively. The method Gold (); Kopylov (); Coc (), is grounded on the Bose-Einstein (BE) or Fermi-Dirac (FD) symmetric properties of the quantum states. It has a deep analogy with the intensity interferometry telescope that was proposed by Hanbury Brown and Twiss for measurements of angular sizes of remote stars HBT (). In distinction on standard telescopic and microscopic techniques based on the registration of intensities of light or particles, e.g., electrons, coming from (or through) the object, this method deals with the correlation between intensities of the source radiation registered by two (many) spatially separated parts of devices, such as telescopes, reflectors, particle detectors, etc. In fact, it measures the correlations between numbers of emitted identical particles detected in separated parts of the detector.

The femtoscopic space-time structure of the systems is typically represented in terms of the interferometry radii. They are result of the Gaussian fit of the correlation function defined as a ratio of the two- (identical) particle spectra to the product of the single-particle ones. In the pioneer papers Kopylov (); Coc () the measured interferometry radii were interpreted as the geometrical sizes of the systems. Later on it was found Pratt (); MakSin (); Hama () that for typical systems formed in experiments with heavy ions, the above geometrical interpretation needs to be generalized. The treatment of the interferometry radii as the homogeneity lengths Sin (); AkkSin () in the systems and the crucial suggestion for femtoscopy scanning of the source radiation in different momentum bins bring the possibility to analyze different parts of the source and explain the behavior of the interferometry radii. In addition, the practical method how to use the final state interactions (FSI) and effects of long-lived resonances to extract the BE correlations in relatively large systems created in heavy ion collisions has been proposed FSI ().

The other challenge, which is still actual, concerns the femtoscopy analysis of relatively small systems created in particle interactions such as and , where the observed femtoscopic scales are approximately 1 fm or smaller Kittel (). Typically, the suppression of the correlation function is fairly large in these processes. Here we will analyze the femtoscopy of such small systems accounting for the uncertainty principle, coherence of the radiation from spatially very closely set emitters and non-femtoscopic (non BE, FD and FSI) correlations. The latter appear due to the energy-momentum conservation law and incoherent contributions to the two- and single-particle spectra induced by particle clusterization in momentum space and fluctuations of initial conditions of the collision processes. The detailed analysis of these theoretical problems can help to provide the correct femtoscopy study of the small systems.

## Ii The basic ideas of the Intensity Interferometry Telescope

The intensity interferometry method for the measurement of the stars’ angular sizes was proposed and realized first by Hanbury Brown and Twiss HBT () at the end of the 1950s. The electromagnetic radiation from the star is the mixture of different, almost monochromatic wave trains, which are mutually incoherent at the moment of radiation. To see the principal aspects, let us consider the emission from the different sites of a radiating object. If the stellar object is close to the observer, like our Sun, then one can easily select, say, the opposite sites (edges) of it. If the two telescopes are directed to those different sites and one measures the correlations between photon numbers coming to each of the two telescopes, then the waves from different sites of the source do not mix in the telescopes, and the correlations between them are absent — the signals coming to the two telescope reflectors are mutually incoherent. If, however, the stellar object — let us consider now the double star system — is very remote, so that it is impossible to select only one of the two stars by a telescope, the light from both stars will come to each of the two reflectors and become mutually (partially) coherent. For simplicity let us imagine that at some moment in time the telescopes register only one wave train from each star, and both these trains are equally polarized and follow each other continuously. Ipso facto we ignore the real problems of the intensity interferometry method — how to extract the signal from the noise — but preserve the principal point of this method.

### ii.1 The basic formalism

One can decompose the electric field strength into positive and negative frequency parts, , . The ideal photon counter reacts just to the product Glauber (). Far from the stellar object the light is described well by the plane waves, so that

 E(+)(x,t)=Aei(k1⋅x−ω1t)+Bei(k2⋅x−ω2t). (1)

The complex amplitudes and have stochastic independent phases: , , which are roughly constants during coherence time of the wave train, e.g., sec, so that being averaged over a period of time , the amplitudes and their product become zero: , . The intensity of light that is proportional to the number of photons registered by the telescope/reflector and photo-multipliers at point is

 ⟨Ik1,k2(x)⟩ = ⟨E2⟩=⟨E(−)(x,t)E(+)(x,t)⟩=⟨|E(+)(x,t)|2⟩ = ⟨|A|2+|B|2+AB∗ei((k1−k2)⋅x−(ω1−ω2)t)+A∗Be−i((k1−k2)⋅x−(ω1−ω2)t)⟩ = |A|2+|B|2,

where we supposed that are almost independent on .

The statistically averaged intensity registered by one of the telescopes contains the information only about the (averaged) squared modulus of the amplitudes. However, the correlation of intensities , defined as the ratio of the averaged product of intensities registered at the two space-time points and to the product of averaged intensities registered at these points, depends already on the differences of momenta and energies of the light quanta. To simplify notation, we put and get

 C = = 1+12cos[(k1−k2)(x1−x2)−(ω1−ω2)(t1−t2)] ≈ 1+12cos[θ|k|d+((xL1−xL2)/c−t1+t2)(ω1−ω2)],

where one took into account that , etc. Here the is the angular size of the double star system with “transverse” distance between stars in the plane perpendicular to the direction to the system, is the distance to the system, is the “transverse” distance between two telescopes, is the mean detected wave number, is the “longitudinal”, directed to the system, coordinate of the i-telescope and is the signal delay between points and . It is worth noting that the time resolution in the method has to be smaller than the coherence time, , in order to provide correlation measurement of photon numbers during the mutual coherence time of the electromagnetic waves in points and . For a stationary process the averaging over a large period of time plays the role of the averaging over the ensemble of events with duration time . Also note that the condition of the validity of formula (II.1) is

 x2L−x1Lc−(t2−t1)<τcoh. (4)

Otherwise, , and the correlations disappear, , because of the mutual incoherence of waves coming to the two telescopes.

### ii.2 The nature of the HBT effect

The formula (II.1) demonstrates the principle of measurement the differences in momenta (and energies) of photons radiated by remote stars by measuring the correlation function depending on distances between telescopes/reflectors and time delay. In fact, the momentum difference in the transverse plane is connected with the angular size of a stellar object. As for the difference in the energy of the photons, it is possible, in principle, to measure it with the restriction given by Eq. (4), and this difference would be associated with different temperatures of the two stars, if such a situation could take place. However, there is no direct connection of the difference in the energy of radiated photons with the time and space scales of the stellar system, and, therefore, only the angular size of this object can be extracted in this way. The latter is the basic application of the intensity interferometry telescope, and the method was used to measure the angular sizes of single remote stars 111In this case one deals not with two emitters, but with many emitters at the whole stellar disk, and formula (II.1) is modified at as follows: . Determining the distance corresponding to the first zero of the correlation function in this representation Hanbury Brown and Twiss have measured the angular size of Sirius and some other stars..

Despite the classical description of the Hanbury Brown and Twiss (HBT) method, the detailed analysis of the measurements accounting for the principle of photon registration pointed to the quantum nature of the effect Glauber (). The relationship between classical and quantum descriptions of the electromagnetic waves has been established through the formalism of coherent states Glauber (). Appealing to the quantum nature of the electromagnetic fields, one can say that the method of the measurement of the star angular size is based on the positive correlations between numbers of photons registered in two close space-time points because of Bose-Einstein statistics for these quanta.

If a double star system is so far from the detectors that the latter can register only a few photons per , then one has to use the amplitudes of registration at points and of the two photons emitted with mutually random phases from the two stars and : and . Then taking the ratio of the averaged modulus squared of symmetric (over photons permutation) two-photon amplitude to the product of the averaged single-photon amplitudes one can get again the correlation structure (II.1). Note that describing a registration of the two neutrinos from the star, one should use the antisymmetrized amplitude .

## Iii The basic ideas of the correlation interferometry

In particle physics the positive correlations between numbers of identical pions with close momenta emitted from an interaction region in proton-antiproton annihilations were found in 1960 by Goldhaber et al. (GGLP effect) Gold (). It was understood that the nature of the effect lies in quantum statistics for identical particles demanding the symmetrization of bosonic wave function. Later on, based on this fundamental principle, Kopylov and Podgoretsky Kopylov () developed the method of the pion interferometry microscope, or correlation femtoscopy/interferometry. They found an analogy between the correlation interferometry and Hanbury Brown and Twiss (HBT) stellar intensity interferometry HBT (). As one can see below, the basic mathematical structure of the two methods can be presented in identical form. However, at the HBT measurements, the interference of intensities/photon numbers happen near the detector (telescope pair), where the correlations form, while in the femtoscopy the quantum statistical (QS) correlations arise in the emitting object. The HBT is based on the analysis of particle correlations as they depend on the space-time separation of the telescopes/reflectors, while the correlation interferometry deals with dependence of the correlations on the particles’ momenta differences. Correspondingly, the measurands are different. The HBT method measures differences of momenta and energies of photons radiated by stellar objects, and the only angular sizes associated with observed momenta difference, but not spatiotemporal scales can be extracted 222If the distance to the double star system is known, the only transverse (to the direction of the stellar system) projection of the distance between the stars is possible to restore from the angular size.. In contrast, the correlation femtoscopy measures the space and time separation of the emission points, and so extracts all the sizes and three-dimensional geometrical shape of the source, as well as a duration of the emission. In this sense the term “HBT radii” that one often uses to present the femtoscopy measurements is not quite adequate, as is stressed in Ref. Ledn1 ().

### iii.1 Standard approach

The basic ideas of the correlation femtoscopy are described in many publications, e.g., Ref. Gyulassy (). We reproduce them here with some important remarks. Let us suppose that two identical bosons (e.g., pions) are emitted from the two space-time points, and , and then propagate freely. The wave function of a single particle at the initial time in the configuration representation is333We use notation () for current Minkowski coordinates to escape confusion with emission points . . At some time in the momentum representation with , it is , where . Here and below we use dimensionless units: .

In the momentum representation () the two-boson wave function is symmetrized and has the form

 ψx1,x2(p1,p2;τ)=1√2(2π)3[eip1x1eip2x2+eip2x1eip1x2]e−i(E1+E2)τ. (5)

Then the probability to find the two pions with momenta , is expressed through the scalar product of 4-momentum and 4-coordinate differences of the two-pion emission:

 Wx1,x2(p1,p2)=∣∣ψx1,x2(p1,p2;τ)∣∣2∝1+cos[(p1−p2)⋅(x1−x2)]. (6)

Comparing Eq. (6) with result (II.1), related to the interferometry telescope, one can see that they differ from each other by a factor of before the cosine. The attentive reader can notice the difference between the two cases. In the case of emission from a double star system the correlation of intensities accounts for all the possibilities: this is the correlation between intensities of the wave trains coming from different stars as well as from the same star. In contrast, when formula (6) is derived, it is supposed that if one boson (with momentum ) is emitted from the point , then another boson (with momentum ) is emitted from different point and vice versa, but the possibility when the two bosons are emitted from the same point, or , is excluded. In the case of independent particles’ radiation all the possibilities have to be taken into account. For such a case one usually demonstrates the idea of the correlation femtoscopy method by means of factorization of the two-particle normalized emission function, and integrates the two-point probability (6) over the space-time region. Then for the correlation function one has (we ignore here possible correlations between coordinates and momenta of the emitted particles)

 C(p1,p2) = W(p1,p2)W(p1)W(p2)=1W(p1)W(p2)∫d4x1d4x2ρ(x1)ρ(x2)Wx1,x2(p1,p2) (7) = 1+1W(p1)W(p2)∣∣∣∫d4xρ(x)eiq⋅x∣∣∣2,

where . So, the probability to find the particles with momenta , at very large times is expressed through the Fourier image of the emission function. This is the typical basis of the correlation interferometry method, allowing one to analyze the size and shape of small systems. If is the Gaussian-like emission probability that in some reference frame has the form , then (7) reads as

 C(p1,p2)=1+exp[−3∑i=1q2iR2i]. (8)

For example, the typical resolving width of the correlation function MeV corresponds to the size m Fm (Fermi) fm (femtometer). The origination of the method’s name — the correlation femtoscopy — is obvious from such estimates.

### iii.2 Correlation femtoscopy formalism under microscope

The problem, however, appears when we apply the basic formula (7) to the system with small number of emitters. For example, if there are only two different emitting points: , then from (7) follows ()

 Wx1,x2(p1,p2) ∝ (9) = 1+cos2[12(p1−p2)(x1−x2)].

The result is incorrect as we shall show.

To analyze the situation in detail, let us consider single-particle radiation from the two points. If the emission from the point is associated with the quantum state that is distinguishable and independent from the state corresponding to emission from the point , then the two orthogonal states  are allowed to be realized with the following known probabilities :

 ρ1:\ \ \ A1(p)=eipx1e−iEτ\ \ \ and \ \ \ ρ2:\ \ \ A2(p)=eipx2e−iEτ; \ % \ \ ρ1+ρ2=1. (10)

Here and below we omit multiplier since these factors cancel in the correlation function (see below). Note that . Since two identical bosons are emitted independently from the two points, there are the three different final states (amplitudes) that are distinguishable and realized with the probabilities :

 ρ11: \ A11(p1,p2)=eip1x1eip2x1e−i(E1+E2)τ; \ ρ22: \ A22(p1,p2)=eip1x2eip2x2e−i(E1+E2)τ; \ (11)

If , then and . As a result, the probability of finding the two particles with momenta that are emitted independently from the two points and is

 C(p1,p2)=Wx1,x2(p1,p2)Wx1,x2(p1)Wx1,x2(p2)=∑i1⩽i2=1,2ρi1i2∣∣Ai1i2∣∣2=1+12cos[(p1−p2)(x1−x2)]. (12)

The last result coincides with Eq.(II.1) for stellar intensity interferometry telescope. The result (12) takes into account that the emission of both quanta from the same point (say, ) brings no interference, while the wrong account for the interference effect leads to the results like (9). Only in the case of a very large number of emitters, when contributions to the correlation function from one-point two-particle radiations are relatively small, can one use the idealized continuous limit  (7). Therefore the simplest ”derivation” of the intensity interferometry method (7) has to be corrected to exclude the double accounting in the correlation function for the contribution associated with the particle pairs emitted from very close points. This correction is significant if the number of independent emitters is not large. For small systems with not flat momentum spectrum, there is essentially finite number of independent incoherent emitters because of the uncertainty principle. We shall consider these effects in detail in the next subsections.

If the emission from points and is not independent at all (full coherence), then one has to use not the probabilities , but the pure state only to account for the interference between the different states: . The amplitude of the two identical bosons symmetrized over in this case is: and

 C(p1,p2)=Wx1,x2(p1,p2)Wx1,x2(p1)Wx1,x2(p2)=1. (13)

### iii.3 Correlation femtoscopy in random phase representation

Both these cases of completely independent and fully coherent radiation from points and , which correspond to mixed and pure quantum states, can be reproduced in the formalism of partially coherent phases Tolst (). To demonstrate it, let us consider the two-point source with coordinates and and some undetermined phases () and express amplitude of single boson emission with momentum ,

 Ax1,x2(p)=e−iEτ2∑i=1eipxieiϕ(xi). (14)

The symmetrized amplitude of the two-boson radiation is

 Ax1,x2(p1,p2)=Ax1,x2(p1)Ax1,x2(p2). (15)

The probability of registering the two identical particles with momenta and is

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯W(p1,p2)=⟨Ax1,x2(p1)Ax1,x2(p2)A∗x1,x2(p1)A∗x1,x2(p2)⟩= 142∑i1,i2,i′1,i′2=1ei(p1xi1+p2xi2−p1xi′1−p2xi′2)⟨ei(ϕ(xi1)+ϕ(xi2)−ϕ(xi′1)−ϕ(xi′2)⟩, (16)

where brackets mean the averaging over all combinations of the two-boson radiation from the two points with probabilities ( as in previous subsection) and over all events with mutually different phases in separate points. The random emission corresponds to the two-point phase average:

 ⟨ei(ϕ(x)−ϕ(x′))⟩=δ4K(x−x′), (17)

where is the Kronecker delta, , , and we omit here the discrete index : , , etc., aiming to apply formalism to continuously distributed emitters. The other two-point phase averages are zero. The four-point phase average in chaotic sources is expressed through the sum of the products of the two-point ones (17):

 ⟨ei(φ(x1)+φ(x2)−φ(x′1)−φ(x′2))⟩=δ4K(x1−x′1)δ4K(x2−x′2)+ +δ4K(x1−x′2)δ4K(x2−x′1)−δ4K(x1−x′1)δ4K(x1−x′2)δ4K(x2−x′1). (18)

The last subtracted term eliminates double counting in the four-point phase average in (16) and (18), when and is usually omitted at the large number of emitters, but for essentially small number it is important. It corresponds to considered earlier elimination of the double counting in the model of independent emitters (cf. (9) vs (12)) when the two bosons are emitted from the same point. With accounting for the subtracted term in (18), the correct results (12) in the formalism of random phases follow directly from (16).

If the source emission is not independent and fully coherent, it means that , and Eq. (16) leads to the result (13).

In the case of independent emitters the phase average (17) can be presented in the form Tolst ():

 Gx′x=⟨ei(ϕ(x)−ϕ(x′))⟩=δ4(x−x′)=Ix′xδ(t−t′), (19)

where is the overlap integral,

 Ix′x=∣∣∣∫d3rψx(τ,r)ψ∗x′(τ,r)∣∣∣/N, (20)

where is the normalization of the wave function. As is demonstrated, the two-particle emission with fully random phases describes the mixed state corresponding to different combinations of the two-particle emission from points and with probabilities . Such a description is possible only if both bosons are emitted independently from different points in distinguishable/orthogonal quantum states 444For indistinguishable quantum states one cannot define/measure the classical probability of the separate state in the system. Note also that the orthogonality is the necessary but not sufficient condition for the independence of emission. . The latter requirement is satisfied in the above case of the flat momentum spectrum for each emitter, , since the initial quantum states taken in different points are orthogonal.

### iii.4 Uncertainty principle and formalism of partially coherent phases

If the momentum spectrum is essentially not flat, this corresponds to the wave packets characterized by their centers and some finite width. In this realistic case one cannot consider quantum states with very close distances between emitter centers as distinguishable and independent because similar to the situation when two identical bosons are emitted from the same point, such a system gives no contribution to the Bose-Einstein correlation function Lyub (), like a fully coherent state (13). One can discriminate between the different states with emission centers and only if they are approximately orthogonal: the overlap integral (20) is small, . In other words, the distance between the centers of the emitters has to be larger than the width of the emitted wave packets. Since the latter is the inverse of the variance of the momentum spectrum, so . The latter expresses the uncertainty principle: one can discriminate the wave packet with center without noticeable violation of the particle spectrum if the measurement that localizes the particle’s position somewhere inside the sphere with the center and the diameter not less than , unambiguously points to the quantum state with center , but not . So, the distance between the emitter centers should satisfy the uncertainty principle in the form . Then the states are almost distinguishable and orthogonal, so the radiation from both emitters can be considered as independent and well approximated within the random phase approach with probabilities for separate quantum states.

In relativistic physics there is another uncertainty principle: the measurement of the particle’s momentum has accuracy depending on the duration of the measurement , Lifshits ()555In fact, Ref. Lifshits () presents one of the forms of the uncertainty principle for energy-time measurement.. One can measure the time of particle emission without noticeable violation of the momentum spectrum with accuracy not better than and, then, the -function in Eq. (19) has to be smeared when one deals with wave packets. So, for normalized wave packets, one cannot use the random phase approximation if the distance in space and time between emitters is less than the width of the wave packet (in units ). For example, if , then at , and . The last term expresses the uncertainty principle for energy-momentum & time measurements.

Therefore, the fully random phases, corresponding to the standard femtoscopy approach Kopylov (); Coc (), are possible only under some conditions which we have discussed: the phase average for quantum states emitted from the points and , written in the form of Eqs. (19), (20),

 Gx′x=⟨ei(ϕ(x)−ϕ(x′))⟩=Ix′xe−Δp2(t−t′)22, (21)

has to be close to zero. Here is the overlap integral (20), and is a variance of the momentum spectrum of emitters.

In the opposite case, when , the states are indistinguishable and overlap integral (21) at . Then the result looks like the one for fully coherent radiation (13), which takes place for very close emitters Lyub (). So, in both limiting cases of chaotic and fully coherent emission Eq. (21) leads to physically obvious results, and therefore the quantity (21), , is the natural measure of distinguishability/indistinguishability and mutual coherence of the two emitted states caused by the uncertainty principle at any distance between the emission centers . In this way we solve the problem of distinguishability of the quantum states associated with different emission points : we always sum -amplitudes, not -probabilities, and average the resulting distributions/spectra over partially coherent phases in correspondence with the requirement of maximal possible distinguishability and the independence of different -states compatible with the uncertainty principle for momentum & position and energy-momentum & time measurements.

### iii.5 Femtoscopic correlations in multiparticle systems

We discussed above the simplest situation with only two emitted bosons (pions). Typically, the basic formalism of correlation interferometry in multipion systems for inclusive momentum spectra is similar Heinz (): one can use the same basic formula (7) with the following substitutions: , , . The correlation between the space-time position of the emission point and momentum of emitted particle, reflected in -dependence of the emission function , leads to dependence of the interferometry radii in formula (8) on the mean momentum of pion pair, MakSin (). The correlation appears due to fast expansion of the multiparticle systems created in high energy (or even ) collisions. The radii are associated then with homogeneity lengths in -direction in inhomogeneous expanding systems Sin (). So, one can carry the two-pion correlation results from the two-pion system to multiparticle systems if one applies the above substitutions. In fact, our quantum-mechanical consideration is related to the rest frame of the homogeneous emitting subsystem forming the spectra in a vicinity of some momentum .

Another aspect of multiparticle emission is that the two-pion wave function may not always be factorized out as for independent subsystems: , where state is related to the residual part of the total system. The symmetrization/antisymmetrization procedure has to be applied to the total system wave function . However, it can be difficult to provide it if one supposes the distinguishability of the radiation points as in the standard correlation interferometry method of independent sources Gold (); Kopylov (). For example, let us suppose that one pion with momentum , is the primary particle emitted from the fireball near the point , and another pion with the momentum is emitted together with particle at the resonance decay, (say, ) near the point . Such a system may not have symmetry because

(i) if at the exchange the momentum of the particle is preserved, then the momentum of emitted resonance is changed and gets the other value . This new resonance state has typically another probability to form as compared to the previous resonance state. (In extreme cases it can even happen that the value instead of excludes the resonance decay into the pair , because of the kinematics). In general, all that suppresses the interference effect Ledn2 ();

(ii) if the exchange of momenta is accompanied by the change of by the value , , then the two configurations and become distinguishable and so cannot interfere;

(iii) if at the particles together with are emitted by the resonance from the fireball near point and primary pion radiates from , the requirement of distinguishability of the emission points may lead to principal distinguishability of the emission points and of the particle that excludes the pion interference effect.

So, the exact symmetry of the total system at exchange can be lost in similar situations, and the two configurations and of the two-pion subsystem can interfere only partially. This has to be taken into account in the transport models used for high energy collisions.

The attempt to estimate the limits of applicability of the standard correlation femtoscopy is done in Ref. Gol (). It is argued that when then the classical phase-space position of the residual part of the system is changed by the value along each direction . If this change exceeds the size of the elementary cell of the phase-space per one degree of freedom, then the residual system moves to another quantum state, and it can be, in principle, measured. Therefore the pion interference should disappear when (), where is the effective system size/homogeneity length in -direction. However, this argumentation fails already in the simplest case when the system contains only identical particles. Such a system can be symmetrized with distinguishable and independent radiation points, and there is no principal possibility to measure the exchange of the momenta between two identical pions at any value of using for this aim the residual system containing the same sort of pions. We think that the real picture of the pion interference can be restored on the basis of symmetrized/antisymmetrized amplitude of the total system with partially coherent phases in the way described in subsection III.4.

## Iv The correlation femtoscopy in Gaussian approximation

Here we apply the basic ideas discussed in the previous section to construct the simple analytical model accounting for uncertainty principle in the correlation femtoscopy. We use the non-relativistic approximation in the rest frame of the source which has Gaussian sizes corresponding to the homogeneity lengths in the corresponding part of the total expanding system. The transformation of the results to a global reference frame, where the source moves with 4-velocity , depends on the concrete model used.

### iv.1 Analytical model for fixed emitters

Let us introduce the quantum state corresponding to a boson with mass emitted at the time from the effectively finite space region with the center as a wave packet with momentum dispersion , and then propagating freely:

 ψxi(p,τ)=eipxi−iEτeiφ(xi)~f(p), (22)

where is some phase and defines the primary momentum spectrum that we take in the Gaussian form with variance :

 f(p)=~f2(p)=1(2πk2)3/2e−p22k2. (23)

The amplitude of the single-particle radiation from some 4-volume at very large times can be written as a superposition of the wave functions with some coefficients/distribution for emission 4-points . In the continuous limit of a very large number of emitters we can omit discrete index , and in the momentum representation the superposition looks like:

 A(p,τ)=c∫d4xψx(p,τ)ρ(x), (24)

where is the normalization constant. Let us select the two directions: parallel to -axis (for systems that are created in collision processes it is the beam axis), which is marked by the index “L” and orthogonal to it, the transverse axis “T”. The distribution of the emission centers is supposed to be the Gaussian one, so that the quantity , being the probability distribution in the case of random phases , is normalized to unity:

 ρ(x)=12πRT√RLTe−x2T4R2T−x2L4R2L−t24T2. (25)

The single-particle momentum distribution averaged over events with different phase distributions is

 ¯¯¯¯¯¯¯¯¯¯¯¯¯W(p)=c2∫d4xd4x′eip(x−x′)ρ(x)ρ(x′)⟨ei(φ(x)−φ(x′)⟩f(p). (26)

To calculate the phase average one needs first to calculate the overlap integral (20). In non-relativistic approach the wave function of the particle emitted at the moment from the point x in coordinate representation is at some time

 ψx(τ,r)=1(2π)3/2∫˜f(p)eip(r−x)e−ip22m(τ−t)d3p. (27)

Then the modulus of the overlap integral (20) is

 Ix′x=∣∣∣∫d3rψx(τ,r)ψ∗x′(τ,r)∣∣∣=e−k2(x−x′)22(1+k4(t−t′)2/m2)(1+k4(t−t′)2/m2)3/4. (28)

To provide calculations in analytical form, we substitute the squared time difference by the constant proportional to its mean value over emission region

 (t−t′)2→a⟨(t−t′)2⟩=a∫d4xd4x′ρ(x)ρ(x′)(t−t′)2=4aT2≡αT2. (29)

As we will demonstrate later, the results obtained within such a prescription reproduce with good accuracy the exact numerical calculations for momentum spectra and correlation functions with the values of depending on the basic parameters of emission. Then the overlap integral (21) that accounts for the uncertainty principle takes the form

 Gxx′=⟨ei(φ(x)−φ(x′))⟩=e−k2(x−x′)22(1+αk4T2/m2)(1+αk4T2/m2)3/4e−k2(t−t′)2/2. (30)

Now one can obtain the one-particle spectrum (26). It is presented below for the case when homogeneity lengths are equal, :

 ¯¯¯¯¯¯¯¯¯¯¯¯¯W(p)=Ne−p22k2−2p2R21+4k20R2−p4T22m2(1+4k2T2), (31)

where

 k20=k2/(1+αk4T2/m2). (32)

The two-particle spectrum averaged over events with partially coherent phases is

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯W(p1,p2) = c4∫d4x1d4x2d4x′1d4x′2ei(p1x1+p2x2−p1x′1−p2x′2)f(p1)f(p2)ρ(x1)ρ(x2)ρ(x′1)ρ(x′2)⋅ (33) ⋅⟨ei(φ(x1)+φ(x2)−φ(x′1)−φ(x′2))⟩.

The 4-point phase correlator supposing emitters to be chaotic and independent in a maximum possible way permitted by uncertainty principle is decomposed into the sum of products of the two-point correlators (21) and contains the three terms

 ⟨ei(φ(x1)+φ(x2)−φ(x′1)−φ(x′2))⟩=Gx1x′1Gx2x′2+Gx1x′2Gx2x′1−Gx1x2Gx1x′2Gx2x′1, (34)

where the third term removes the double accounting which appears in the sum of the first two terms when . We emphasize that just the second term — factor at crossing interference term (with cosine) — has to be compensated for when all four points are very close to each other. The detailed discussion about the subtracted term is presented earlier in the subsection III.3. Note that we use the product of the minimal number (three) of the two-point correlators in the subtracted term. The structure of (34) coincides with one in Eq. (18) and is symmetric under simultaneous interchange & as it is required by the -symmetry of Eq. (33).

Let us split transverse direction “T” into two: out which is directed along the total transverse momentum of the pair and side, which is orthogonal to the out-direction as well as to the longitudinal “L” axis. Then the calculations of the one- and two-particle spectra lead to the following correlation function in the variables of the half of the bosonic pair momenta sum and the momenta difference

 C(p,q)=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯W(p1,p2)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯W(p1)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯W(p2)=1+e−q2TR2T4k20R2T1+4k20R2T−q2LR2L4k20R2L1+4k20R2L−(q⋅p)2T2m24k2T21+4k2T2−Cd(p,q), (35)

where is defined by (26), is (32), and the subtracted function eliminates the double accounting at the averaging of the 4-points phase correlator, it is associated with the third term in Eq. (34). This term reduces the intercept of the correlation function as one can see below.

The coefficient in the expression for should be chosen from the requirement of a good agreement between the correlation function calculated using the approximation (30) and the one calculated numerically using the exact expression for the space part of the phase correlator (28). For the typical freeze-out temperature the value of is GeV. Having fixed the value of this parameter, we present in Fig.1 the comparison of the side-projections of the correlation functions calculated numerically using (28) without subtraction of the double accounting and corresponding analytical expressions (the first two terms in (35)). The values of at different system sizes are presented. For fairly small systems, fm, they are about unity, , and are decreasing, , when the system size grows up.

As follows from Eq. (35), the observed Gaussian interferometry radii of the system are reduced as compared to the standard results (we mark it by st index) for the interferometry radii for the Gaussian source. The reductions for side- (), out- () and long- () interferometry radii in the LCMS system (