Principles and prospects of Bose-Einstein correlation study at BESIII

The hadron production process in electron-positron collisions involves $e^{+}$ and $e^{-}$ annihilating into a virtual photon ($\gamma^{\ast}$), which splits into an initial quark-antiquark pair ($q\bar{q}$). The hadronization process consists of three phases: (1) perturbative Quantum Chromodynamics (pQCD) cascade evolutions of quarks and gluons, (2) hadron source formation and preliminary hadron emission, and (3) the decay of unstable hadrons.

In the energy region of the Beijing Electron Positron Collider II (BEPCII), which ranges from 2 to 5 GeV, the preliminary hadron states include continuous states and resonant states characterized by the quantum numbers $J^{PC}=1^{--}$. Because the hadronization process cannot be predicted comprehensively and quantitatively by pQCD, phenomenological models are used to describe this phase. One of the models commonly used to accomplish this task is the semi-classical Lund string fragmentation model lundmodel ; bohu . The Lund model describes the hadron source as a string, which fragments into hadrons.

The production of vector resonant states such as $\rho$, $\omega$, $\phi$, $J/\psi$, and $\psi(3686)$, can be summarized as follows: the $\gamma^{\ast}$ evolves into a $q\bar{q}$ pair, which constitutes the vector meson. The meson then decays into final states, a process described by the vector meson dominance model ioffe .

In quantum mechanics, the propagation of hadrons can be described using wave functions. The wave function of a system of identical bosons is symmetric under the interchange of the space-time coordinates of any two bosons. This symmetry results in a higher probability of finding identical bosons in a small phase-space element than finding different bosons, as illustrated in Fig.1. This phenomenon is known as Bose-Einstein correlation (BEC), which is independent of specific interactions. Measurements of the BEC effect provide valuable space-time information about the hadron source boal ; weiner .

Theoretical studies of the BEC effect have been conducted for many years, leading to various models of the space-time distribution of the hadron source, including Gaussian- and ellipsoid-type distributions boal ; weiner , as well as the more generalized L$\acute{e}$vy distribution phenix ; adare ; csorgo .

Numerous collaborations have conducted experimental measurements of the BEC effect involving $e^{+}e^{-}$, $p\bar{p}$ collisions, and $ep$ and $Kp$ scattering to select various signal samples at energies ranging from tens of GeV to TeV. A selection of these studies are listed in Table 1, and additional details on other experiments are documented in Ref.boutemeur . All these experiments demonstrate the universal characteristics of the BEC effect.

Table 1. Status of measurements of BEC effect.

However, in the BEPCII energy region, measurements of the BEC effect are still lacking. The BESIII detector at BEPCII has collected data samples with large-sample statistics at certain energies, providing the opportunity to measure the BEC effect. In this study, we first introduce the basic concepts of the BEC effect. Then we derive the emission amplitude of the hadron source based on the Lund model, as well as the wave functions of a two-pion system. Finally, we perform feasibility studies using the Monte Carlo (MC) method via the Lund model generator to explore the properties relevant for measuring the BEC effect.

In relativistic terms, a free and stable particle can be described by a plane wave function (strictly speaking, however, the motion of particles should be described by wave packets):

The lifetime of certain hadrons, such as $\pi^{\pm}$ and $K^{\pm}$, are sufficiently long to be considered stable in experiments. For an unstable particle with a short life-time (resonance), the wave function can be written as

where $m$ is the resonance peak position (the mass of the resonance), $s$ is the square of the center-of-mass energy, and $\Gamma$ is the decay width (which indicates the lifetime of the particle via $\tau=\hbar/\Gamma$. For simplicity, we neglect the phase factor $\exp(i\theta)$ as it does not affect the analysis of the BEC effect.

However, if a particle is influenced by Coulomb and strong interactions, the corresponding wave function becomes significantly more complex hepph9606365 . This study does not discuss the situation of final state interaction.

For the BESIII detector, the spatial coordinate origin $O$ was defined as the center of the detector, which also served as the collision point of $e^{+}e^{-}$, and the time origin was synchronized with the event trigger clock of the data acquisition system.

As shown in Fig.3, two pions ($\pi^{+}\pi^{-}$ and $\pi^{\pm}\pi^{\pm}$) are emitted from the hadron source points at $x_{i}=({\bf r_{i}},t_{i})$ and $x_{j}=({\bf r_{j}},t_{j})$, respectively, and are detected by detectors Da at $x_{a}=({\bf r_{a}},t_{a})$ and detector Db at $x_{b}=({\bf r_{b}},t_{b})$. Here, ${\it\bf r}_{a}$ and ${\it\bf r}_{b}$ are the central positions of Da and Db relative to $O$, respectively, and $t_{a}$ and $t_{b}$ represent the times required for particles to travel from $O$ to Da and Db, respectively.

For a real detector, Da is not a point but has a finite range, and time and momentum are measured with uncertainties. Thus, the detected interval $x_{ia}$ and $p_{a}$ can be expressed as

where $\delta x_{ia}$ and $\delta p_{a}$ denote the detected space-time and energy-momentum uncertainties. The wave function of $\pi^{+}$ arriving at $x_{a}$ is given by

where $\delta_{ia}=p_{a}\delta x_{ia}+\delta p_{a}x_{ia}$ is a random phase angle, possibly with observable effects on the measurement of the correlation functions.

Similarly, the wave function of $\pi^{-}$ emitted from $x_{j}$ with momentum $p_{j}$ and detected by Db at $x_{b}$ with momentum $p_{b}$ is given by

where $x_{jb}=x_{b}-x_{j}+\delta x_{jb}$, $p_{b}=p_{j}+\delta p_{b}$, and $\delta_{jb}=p_{b}\delta x_{jb}+\delta p_{b}x_{jb}$. The joint wave function of $\pi^{+}\pi^{-}$ arriving at Da and Db is expressed as

This expression for $\psi_{d}$ also applicable to $\pi^{\pm}\pi^{0}$.

As depicted in Fig.3(B), $\pi^{+}$ and $\pi^{+}$ are detected by Da and Db with momenta $p_{a}$ and $p_{b}$ respectively. The two identical pions are indistinguishable according to measurements, meaning Da and Db cannot determine whether a $\pi^{+}$ was emitted from $x_{i}$ or $x_{j}$. Consequently, the joint wave function of $\pi^{+}\pi^{+}$ has a symmetrized form for interchanging source points $i$ and $j$:

The expression for $\psi_{s}$ is also applicable to $\pi^{-}\pi^{-}$ and $\pi^{0}\pi^{0}$.

If the preliminary state contains unstable particles that decay into stable final states, owing to the fact that the lifetimes of most unstable particles are only a few femtometer (much smaller than the spatial resolution of the detector), the decay vertex is considered as part of the experimental uncertainty in the emitting position.

The inclusive detection probability amplitudes for $\pi^{+}$ and $\pi^{-}$, emitted from points $x_{i}$ and $x_{j}$ and detected by detectors Da and Db with momenta $p_{a}$ and $p_{b}$, respectively, are expressed as

In these calculations, we assume that $p_{a}=p_{i}$ and $p_{b}=p_{j}$ (incorporating measurement uncertainties into the random phase angles $\delta_{ia}$ and $\delta_{jb}$), and that they can be absorbed into the functions $f(x_{i};p_{a})$ and $f(x_{j};p_{b})$, respectively. Therefore, the detection probability amplitudes not only encode information about the hadron sources but are also influenced by measurement uncertainties.

The joint probability amplitude that $\pi^{+}$ and $\pi^{-}$ will be detected by Da and Db is given by

For the case in which two identical $\pi^{+}\pi^{+}$ are detected by Da and Db, four probability amplitudes exist, as shown in Fig.3(B). Besides the two already given by Eqs.(16) and (17), the additional amplitudes are

The joint probability amplitude for detecting $\pi^{+}\pi^{+}$ is then

which is symmetric with respect to the interchange of source points $i$ and $j$.

The expression for the correlation function for a two-pion system varies depending on whether the measurement is coherent or incoherent.

The BEC effect is a natural consequence of the Lund model lundmodel ; npb5131998627 . In this model, multiple string configurations can fragment into the same hadron state. For example, as shown in Fig.14.1 of Ref.lundmodel , there are two configurations with light-cone areas ${\cal A}_{12}$ and ${\cal A}_{21}$, both of which fragment into the same hadron final state but with different momenta. In these configurations, two identical bosons are produced at convex vertices 1 and 2. The difference between these two configurations is simply the exchange of the two identical bosons. The total matrix element for this hadron state can be expressed as

leading to

where the term ${\cal H}$ is defined as

where $\Delta{\cal A}$ is the difference between ${\cal A}_{12}$ and ${\cal A}_{21}$. This shows that the interference between the two configurations ${\cal A}_{12}$ and ${\cal A}_{21}$ results in an enhancement factor of $1+{\cal H}$. This factor increases as $\Delta{\cal A}$ decreases, which occurs when the momenta $p_{1}$ and $p_{2}$ of the two identical bosons become more similar in magnitude.

In quantum field theory, the distribution functions are related to the differential cross sections. For the inclusive process $a+b\to c+X$ (where $X$ represents all other particles except $c$), the differential cross section is defined as

where $d\phi$ is the phase-space element of particle $c$, $<n>$ is the average multiplicity, and $\rho_{1}(p)$ is the single-particle density distribution. Similarly, for the two-particle differential cross section in the process $a+b\to c_{1}+d_{2}+X$, we have

where $\rho_{2}(p_{1},p_{2})$ is the two-particle density distribution.

The relationships between the dimensionless probability distributions and the density distributions are given by

In theoretical and modeling studies, the correlation function often uses the ratio

which is a six-dimensional representation. However, it is more convenient to use the one-dimensional two-particle correlation function, which is expressed as

where $Q^{2}$ is defined as the invariant squared difference of the four-momenta of the two particles:

The correlation functions are obtained via

where the pseudo-correlation in $\rho_{2}(p_{1},p_{2})$ is deducted.

In experimental studies, BECFs are often parameterized as

where $\lambda$ is known as the incoherent or chaotic parameter, and $R$ is often interpreted as the source radius boal ; weiner . The line-shapes of Eqs.(53) and (54) are shown in Fig.5. In the figure, $\lambda=0.75$ and $R=0.4$ fm are used, but the actual values are determined by fitting the experimental data. According to Eqs. (53) and (54), the correlation function has two important features: it goes to $1$ at large $Q$ values, and it goes to $1+\lambda$ at $Q=0$.

Because $Q^{2}$ is defined as a four-dimensional form in Eq.(50), the parameter $R$ in Eq.(53) and (54) does not necessarily reflect the true spatial scale of the hadron source as it also depends on the lifetime of the hadron source. Various models attempt to provide a physical explanation for the parameter $\lambda$ weiner ; boal ; bolz ; csorgo . While $\lambda$ may consist of multiple physical factors, the term expressed in Eq.(39) is certainly a component of them.

To the shape of the BECF, as measured by the experimental data, can be fit using the correlation density, which is expressed as

where $\Delta n(Q_{i})$ is the number of particle pairs falling within the interval $(Q_{i},Q_{i}+\Delta Q_{i})$, and $K$ is a normalization factor.

The observed density correlation functions are obtained by selecting signal and reference particle pairs from the experimental data. Some preliminary hadrons are unstable and decay into final-state hadrons before reaching the detector. The correlation functions obtained from the decay final-state hadrons are given by

The final state BECF is defined as

Ideally, the BECF should be measured using preliminary-state hadrons. If the decayed final states can be traced back to the preliminary states, the preliminary-state density correlation functions are given by

and the observed preliminary BECF is

Both the BECFs expressed by Eqs.(58) and (61) depend on the data analysis method, and the physical results account for the efficiency correction.

To derive the physical BECF, $C^{\rm phy}(Q)$, from the observed $C_{\rm pre}^{\rm datobs}(Q)$ and $C_{\rm fin}^{\rm datobs}(Q)$, two sets of corrections are required: one that accounts for the efficiency in the analysis-dependent quantities, and another that converts the final-state $C_{\rm fin}(Q)$ into a preliminary-state one $C_{\rm pre}(Q)$.

If MC simulations align well with the experimental data, then the relationship

holds, where

are the BECFs obtained by MC simulations at the generator and detector levels, respectively.

For both preliminary- and final-states hadrons simulated at the generator and detector levels, we have:

The experimental physics results of the BECFs can be derived from the observed values by applying the efficiency corrections

where the efficiencies $\epsilon_{\rm pre}$ and $\epsilon_{\rm fin}$ are obtained via MC simulations:

The relationship of BECFs between the experimental raw data and physics result is

where the recover factor is

The MC generators for the Lund model are LUARLW (for the low-to medium-energy region) bohu and JETSET (for the high-energy region) jetset .

The LUBOEI subroutine in in LUARLW and JETSET employs algorithms based on mean-field potential attraction between identical bosons to simulate the BEC effect. This is achieved by shifting the final-state momenta of identical bosons closer together while maintaining energy-momentum conservation within the event EurPhysJC21998165 ; plb3511995293 ; BECinMP . LUBOEI offers two options for simulating the BEC effect: the exponential form (given in Eq.(53)) and the Gaussian form (given in Eq.(54)). An analysis of the BESIII data indicated that the Gaussian form agreed well with the experimental data; thus, it was used in the MC simulations.

In this section, we present various BECF line shapes simulated at the generator level. The signal samples typically include $\pi^{+}\pi^{+}$, $\pi^{-}\pi^{-}$, and $\pi^{0}\pi^{0}$, while the reference samples include $\pi^{+}\pi^{-}$ and $\pi^{\pm}\pi^{0}$. We provide comparisons between the preliminary and final states for different signal and reference samples at the generator level.

In the MC simulations, we could generate either preliminary hadron states only or allow unstable hadrons to decay into final states. The inclusive line shapes of the normalized density distributions $\rho_{\rm presig}^{\rm simgen}(Q)$ and $\rho_{\rm preref}^{\rm simgen}(Q)$ in Eq.(65) and those of $\rho_{\rm finsig}^{\rm simgen}(Q)$ and $\rho_{\rm finref}^{\rm simgen}(Q)$ in Eq.(66) are shown in Fig.6. The line shapes of $\rho_{\rm presig}^{\rm simgen}(Q)$ and $\rho_{\rm finsig}^{\rm simgen}(Q)$ tend towards the low $Q$ end, which is a characteristic feature of the BEC effect.

In the reference sample, some $\pi^{+}\pi^{-}$ pairs are the decay final states of resonances such as $\rho$, $\eta$, $\eta^{\prime}$, $\omega$ and $\phi$. In Fig.6(B), the convexity of $\rho_{\rm finref}^{\rm simgen}(Q)$ is caused by resonant effects. We can exclude these $\pi^{+}\pi^{-}$ pairs by requiring their squared invariant mass $q^{2}=(p_{\pi^{+}}+p_{\pi^{-}})^{2}$ to lie outside the resonance peak $[(M_{R}-\Gamma)^{2},(M_{R}+\Gamma)^{2}]$. However, this requirement inevitably removes some $\pi^{+}\pi^{-}$ pairs from non-resonant states.

Various schemes for selecting reference samples have been investigated by different collaborations eurphysjc822022608 . However, these schemes can violate energy-momentum conservation. As noted in Ref.boal , the correlations arising from energy-momentum conservation are stronger than those arising from the BEC effect. It is expected that the biases introduced by these schemes can be corrected using high-quality MC simulations and the correction factor given in Eq.(70).

Figure 7 shows the density correlation functions $\rho(Q)$ of the preliminary and final states for $\pi^{+}\pi^{-}$, $\pi^{\pm}\pi^{0}$ and $K^{+}K^{-}$. If $\rho(Q)$ consists of only $\pi^{+}\pi^{-}$ pairs or all of the $\pi^{+}\pi^{-}$, $\pi^{\pm}\pi^{0}$ and $K^{+}K^{-}$ pairs, the line shape of $C(Q)$ differs.

Figure 8 illustrates the BECF of the inclusive preliminary and final states, along with the corresponding ratio as presented in Eq.(70) at 2 GeV and 3 GeV. The figures indicate that the line shapes of $C(Q)$ are energy-dependent within the BEPCII energy range. Figure 12 shows the dependence of the BECF preliminary and final states for the semi-exclusive continuous states at 3 GeV.

The $J/\psi$ particle is produced via $e^{+}e^{-}\to\gamma^{\ast}\to c\bar{c}\to J/\psi$, and it decays to preliminary states and/or stable final states. It is interesting to examine whether the BEC effect manifests in the preliminary and secondary final states of $J/\psi$ decay. MC simulations indicate that both preliminary and secondary decay final states exhibit the BEC effect, as shown in Fig.9, which shows that the line shapes of $C(Q)$ and $J/\psi$ similar to those of the continuous states.