Effect of partonic cascade on two-pion HBT parameters of the core-halo source in relativistic heavy-ion collisions

Experiments at the Relativistic Heavy-Ion Collider provide the opportunity to investigate the properties of quark–gluon plasma (QGP) and have demonstrated that a new form of QCD matter, the strongly coupled quark–gluon plasma (sQGP), exists [1, 2]. Two-pion Hanbury–Brown–Twiss (HBT) interferometry is a useful tool in the quest to understand the spacetime structure and dynamical information of the emission source in relativistic heavy-ion collisions [3, 4]. The common HBT analysis fits the measured correlation function by the parametrized formula of a Gaussian form to extract the HBT radii and intercept parameter [5]. These obtained HBT results are model dependent, which require an approximately Gaussian source, but many studies show that the particle emission source is far from Gaussian distributed [6–8]. The conventional Gaussian parametrization of the HBT correlation function is inappropriate for the non-Gaussian source. Accumulating evidence indicates that the pion emission source in relativistic heavy-ion collisions may be a core–halo structure [7, 9]. For the core–halo source, the pions are emitted from a central core, and a halo of long-lived resonance decays [10]. The long-lived resonances produced in the hot and dense medium decay far outside the core, e.g., ω, η, $\eta ^{\prime}$, K_S⁰, etc. The pions emitted from the decay of η, $\eta ^{\prime}$ and K_S⁰ effect the HBT correlation function in a vary narrow region with momentum difference $q\lt 10$ MeV c⁻¹. The main contribution to the halo is from ω decays, which give rise to the production of pions with low momentum. Such pions from ω decays deflect the emission source from Gaussian distributed. It can be assumed that the emitting points of pions in the core and halo are all the Gaussian distribution with different characteristic length-scales. The spacetime and dynamical information of the core, where the time evolution of the matter can be thought of as a violent cascading of binary collisions or hydrodynamical behavior of strongly interacting matter, are directly affected by the processes of partonic interaction and the existence of QGP [11–13].

In this paper, we apply the double-Gaussian parametrization with a two-scale approach to fit the HBT correlation function of charged pions and study the effect of partonic cascade on the evolution time scale of the pion emission source extracted by the HBT parameters. The string melting version of the multi-phase transport (AMPT) model is used to simulate Au+Au collisions at $\sqrt{{{s}_{NN}}}=200$ GeV, which contain the partonic transport processes, including only two-body scatterings, and provides a convenient way to study the effect of a partonic system with various cross sections [14]. In section 2 we compare HBT parameters by double-Gaussian fit with results by Gaussian fit and obtain the spherical harmonic analysis result. The evolution time as a function of the partonic cross section is presented in section 3. We conclude in section 4.

In common HBT analysis, the correlation functions are often fitted by a Gaussian formula to obtain HBT parameters, but the Gaussian parametrization is inappropriate to the non-Gaussian source. For the core–halo source, the HBT correlation function, which is calculated by the Correlation After Burner program without the final state interactions [15], can be fitted by the double-Gaussian parametrization in the 'out-side-long' coordinate system [10],

$$\begin{eqnarray}\begin{array}{rcl} {{C}_{2}}({\bf q}) & = & 1+{{\lambda }_{{\rm C}}}{{{\rm e}}^{-q_{{\rm o}}^{2}R_{{\rm oC}}^{2}-q_{{\rm s}}^{2}R_{{\rm sC}}^{2}-q_{{\rm l}}^{2}R_{{\rm lC}}^{2}}} \\ {} & {} & +{{\lambda }_{{\rm H}}}{{{\rm e}}^{-q_{{\rm o}}^{2}R_{{\rm oH}}^{2}-q_{{\rm s}}^{2}R_{{\rm sH}}^{2}-q_{{\rm l}}^{2}R_{{\rm lH}}^{2}}} \\ {} & {} & +2\sqrt{{{\lambda }_{{\rm C}}}}\sqrt{{{\lambda }_{{\rm H}}}}{{{\rm e}}^{[-q_{{\rm o}}^{2}(R_{{\rm oC}}^{2}+R_{{\rm oH}}^{2})-q_{{\rm s}}^{2}(R_{{\rm sC}}^{2}+R_{{\rm sH}}^{2})-q_{{\rm l}}^{2}(R_{{\rm lC}}^{2}+R_{{\rm lH}}^{2})]/2}}, \\ \end{array}\end{eqnarray} \tag{ 1 }$$

where ${{q}_{{\rm o}}}$, ${{q}_{{\rm s}}}$, ${{q}_{{\rm l}}}$ are three components of the relative momentum difference ${\bf q}={{{\bf p}}_{{\bf 1}}}-{{{\bf p}}_{{\bf 2}}}$ of a pair of pions, which are defined in the 'out-side-long' system. The size parameters ${{R}_{{\rm iC}}}$ and ${{R}_{{\rm iH}}}$ (${\rm i}={\rm o},{\rm s},{\rm l}$) are the HBT radii of the core and the halo. The second, third, and fourth terms of the HBT correlation function in equation (1) correspond to the contributions of two pions from the core, the halo, or one pion from the core and another from the halo, respectively. There are too many parameters to extract a unique set of best-fit values in the double-Gaussian formula. We use a procedure of the two-scale approach which assumes that equation (1) can be separated into two scales to fit the correlation data by a cut of the relative momentum difference ${{q}_{{\rm cut}}}$ [16]. First, we fit the correlation data, of which the relative momentum differences of the bins are larger than a cutoff (${{q}_{{\rm o}}}\gt {{q}_{{\rm cut}}}$, ${{q}_{{\rm s}}}\gt {{q}_{{\rm cut}}}$, and ${{q}_{{\rm l}}}\gt {{q}_{{\rm cut}}}$), by the front half of equation (1), which ignores the cross term and only consists of the first and second terms to extract a set of the fitting parameters, ${{\lambda }_{{\rm C}}}$, ${{R}_{{\rm oC}}}$, ${{R}_{{\rm sC}}}$, and ${{R}_{{\rm lC}}}$. Then we fit all the data by the full function equation (1), including the cross term with the obtained fitting parameters of the core Gaussian, to extract the halo parameters, ${{\lambda }_{{\rm H}}}$, ${{R}_{{\rm oH}}}$, ${{R}_{{\rm sH}}}$, and ${{R}_{{\rm lH}}}$. Fitting with different ${{q}_{{\rm cut}}}$, we can obtain sets of the parameters by the two-scale approach. One set of the parameters corresponding to the minimum of ${{\chi }^{2}}/{\rm NDF}$ is the best-fit value of the double-Gaussian parametrization and the obtained size parameters are the HBT radii of the core and halo. The intercept parameter of the core ${{\lambda }_{{\rm C}}}$ is related to the core fraction [10],

$$\begin{eqnarray}&&{{\lambda }_{{\rm C}}}({\bf K})={{[{{N}_{{\rm C}}}({\bf K})/N({\bf K})]}^{2}},\end{eqnarray} \tag{ 2 }$$

where the average momentum ${\bf K}=({{{\bf p}}_{{\bf 1}}}+{{{\bf p}}_{{\bf 2}}})/2$, $N({\bf K})={{N}_{{\rm C}}}({\bf K})+{{N}_{{\rm H}}}({\bf K})$, and N is the number of pions from source. The pion emission source radii can be calculated from the emission function by the curvature of the CF at ${\bf q}=0$ [17],

$$\begin{eqnarray}&&R_{ij}^{2}({\bf K})=-{{\left. \frac{1}{2}\frac{{{\partial }^{2}}C({\bf q},{\bf K})}{\partial {{q}_{i}}\partial {{q}_{j}}} \right|}_{{\bf q}=0}}={{D}_{{{x}_{i}},{{x}_{j}}}}({\bf K})-{{D}_{{{x}_{i}},{{\beta }_{j}}t}}({\bf K})-{{D}_{{{\beta }_{i}}t,{{x}_{j}}}}({\bf K})+{{D}_{{{\beta }_{i}}t,{{\beta }_{j}}t}}({\bf K}),\end{eqnarray} \tag{ 3 }$$

where the variance ${{D}_{x,y}}=\langle xy\rangle -\langle x\rangle \langle y\rangle$, $\langle x\rangle$ denotes the average value of x, and $\beta ={\bf K}/{{K}_{0}}$, with K₀ being the average energy of the two pions.

Figure 1 shows the projections of the three-dimensional correlation function and corresponding fits for ${{m}_{{\rm T}}}-{{m}_{0}}=[0,0.1]{\rm GeV}{{{\rm C}}^{-2}}$. The deviations between the correlation function (CF) and the Gaussian fit (GF) reflect the fact that the pion emission source is not Gaussian distributed and produces a non-Gaussian correlation function. The double-Gaussian fit (DGF) is closer to the correlation function, which suggests the source of the core–halo structure.

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Figure 2 shows the extracted HBT parameters for charged pions from the Gaussian and double-Gaussian fit to the correlation function, respectively, as well as the pion emission source radii at midrapidity as a function of the transverse mass ${{m}_{{\rm T}}}$ in central Au+Au collisions at $\sqrt{{{s}_{NN}}}=200$ GeV, of parton cross section ${{\sigma }_{{\rm p}}}$ = 3 mb and impact parameter b = 0 fm from the AMPT model with string melting. It can be seen in figure 2(a) that the HBT radii of the halo (${{R}_{{\rm oH}}}$, ${{R}_{{\rm sH}}}$, and ${{R}_{{\rm lH}}}$) are a factor of 3–5 larger than those of the core (${{R}_{{\rm oC}}}$, ${{R}_{{\rm sC}}}$, and ${{R}_{{\rm lC}}}$). The errors of the halo radii become much larger with the increasing ${{m}_{{\rm T}}}$ as the result of the pions from resonance decays produced mainly with the low transverse momentum. In figure 2(b), ${{\chi }^{2}}/{\rm NDF}$ of the Gaussian parametrization is much larger than that of the double-Gaussian parametrization in the low transverse mass range, which indicates the double-Gaussian fit is more appropriate for the pion emission source, and the core–halo structure is closer to the source distribution obtained by long-lived resonance decays in the AMPT model. The intercept parameters ${{\lambda }_{{\rm C}}}$ and ${{\lambda }_{{\rm H}}}$ also suggest that the scale of the pions produced from the halo is larger with the decreasing transverse mass. In figures 2(c) and (d), we note that the HBT radii of the core and the halo are closer in magnitude to the Gaussian fitted radii and the pion emission source radii, respectively. This is due to the characteristic length-scales of the core and halo, which give rise to the correlation function in different regions of momentum difference $q\lt 80$ MeV c⁻¹ and $q\lt 20$ MeV c⁻¹. The Gaussian parametrization of the correlation function works in the same momentum difference region with that of the core. The pion emission source radii are calculated by the curvature of the correlation function at ${\bf q}=0$, where the influence degree of the halo on the correlation function is according to the halo fraction of the source. The halo length-scale gives rise to a sharp peak of the correlation function.

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In order to check the fitted results of the extracted HBT parameters from the double-Gaussian parametrization, we apply the spherical harmonic analysis method to decompose the HBT correlation function. We obtained the spherical harmonic decomposition coefficients ${{A}_{l,m}}$, which provide an efficient way to visualize the full 3D structure of ${{C}_{2}}({\bf q})$ [18–20]. The spherical coordinates $Q=|{\bf q}|$, θ and ϕ relate to the Cartesian ones as

$$\begin{eqnarray}&&{{q}_{{\rm o}}}=Q{\rm sin} \theta {\rm cos} \phi ,\;\;\;{{q}_{{\rm s}}}=Q{\rm sin} \theta {\rm sin} \phi ,\;\;\;{{q}_{{\rm l}}}=Q{\rm cos} \theta ,\end{eqnarray} \tag{ 4 }$$

and the coefficients ${{A}_{l,m}}$ are defined as [19]

$$\begin{eqnarray}&&{{A}_{l,m}}(Q)=\frac{1}{\sqrt{4\pi }}\int {\rm d}\phi {\rm d}({\rm cos} \theta ){{C}_{2}}(Q,\theta ,\phi ){{Y}_{l,m}}(\theta ,\phi ).\end{eqnarray} \tag{ 5 }$$

The coefficient ${{A}_{0,0}}(Q)$ represents the overall angle-integrated strength of ${{C}_{2}}({\bf q})$. ${{A}_{2,0}}(Q)$ reflects the relative magnitude of the longitudinal radii ${{R}_{{\rm l}}}$ to the transverse radii ${{R}_{{\rm T}}}$, and ${{A}_{2,2}}(Q)$ reflects the nonidentical values of the ${{R}_{{\rm o}}}$ and ${{R}_{{\rm s}}}$.

Figure 3 shows the spherical harmonic decomposition coefficient of the correlation function for ${{m}_{{\rm T}}}-{{m}_{0}}=[0,0.1]{\rm GeV}{{{\rm C}}^{-2}}$ from the string melting AMPT model in central Au+Au collisions at $\sqrt{{{s}_{NN}}}=200$ GeV, with ${{\sigma }_{{\rm p}}}$ = 3 mb, b = 0 fm, and different curves represent the three parametrizations of the HBT correlations used in the fit. We can see that only the double-Gaussian parameterized results can be coincident with the coefficient ${{A}_{l,m}}$ in the total $|{\bf Q}|$ range, and the other two parameterized results deflect the ${{A}_{l,m}}$ of the correlation function, especially at the low momentum difference region of the second-order coefficients. Obviously, the double-Gaussian parametrization can extract more detailed femtoscopic information from the correlation function calculated from the AMPT model.

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Based on the relativistic kinematic theory, the source function of pions from resonance decays can be expressed in terms of the source function of resonances [21]. Theoretically, the momentum distribution and the corresponding Herrmann–Bertsch formula are related to the source function of resonances [22, 23]. Considering the complicacy of the source function of pions from resonance decays, the expression cannot be used to obtain a formula related to the momentum distribution or the correlation function. In reference [24], an effective suppression factor, which measures the fraction of pion pairs containing no pions from resonance decays, could suppress the measured correlation functions. Applying the double-Gaussian parametrization, the correlation functions are not required to make corrections by a suppression factor and can be extracted the HBT parameters of the core and halo, contributed from direct pions or resonance decays. The momentum spectra of pions can be separated into two spectra of direct pions and pions from resonance decays by the extracted intercept parameter ${{\lambda }_{{\rm C}}}$.

For the transverse mass ${{m}_{{\rm T}}}$ spectra with an exponential form, we can parametrize it as

$$\begin{eqnarray}&&\frac{{{{\rm d}}^{2}}N}{2\pi {{m}_{{\rm T}}}{\rm d}{{m}_{{\rm T}}}{\rm d}y}=A{\rm exp} (-\frac{{{m}_{{\rm T}}}}{T}),\end{eqnarray} \tag{ 6 }$$

where T is the inverse slope parameter and A is a normalization parameter. The freeze-out temperature T₀ is related to the parameter T [25, 26],

$$\begin{eqnarray}&&T={{T}_{0}}+m{{\langle {{\beta }_{{\rm T}}}\rangle }^{2}},\end{eqnarray} \tag{ 7 }$$

where $\langle {{\beta }_{{\rm T}}}\rangle$ is the average transverse flow velocity.

Figure 4 shows the transverse mass spectra of the pions from the core, the halo, and the total charged pions, as well as the pions and ω mesons without ω decays from the AMPT model for Au+Au central collisions at $\sqrt{{{s}_{NN}}}=200$ GeV. In order to separate the ${{m}_{{\rm T}}}$ spectra of pions which are produced in the core or the halo, we can use equation (2) to quantify the core fraction from the number of the total pions produced in the source. In figure 4, the transverse mass spectra of the pions from the core is close to that of the total pions without ω decays, and the result indicates that pions from the halo are indeed decayed from the resonance, and ω decays are the main contribution to the halo fraction, which is in accordance with reference [27]. Through the extracted inverse slope parameters T and the average transverse flow velocity $\langle {{\beta }_{{\rm T}}}\rangle$, we can calculate the freeze-out temperature T₀ by equation (7). It is noteworthy that although pions from the halo can be separated from the total spectra, the spectra of pions from resonance decays are not thermal under an assumption of sharp kinetic freeze-out. They are produced in effective post freeze-out four-volume, which depends also on resonance widths [24]. It is approximate that the freeze-out temperature of ω mesons, produced by stopping the ω decays in the AMPT model, is considered as the temperature of pions from the halo by assuming a fast freeze-out of pions after resonance decays. ${{\langle {{\beta }_{{\rm T}}}\rangle }_{{\rm C}}}$ and ${{\langle {{\beta }_{{\rm T}}}\rangle }_{\omega }}$ are the average transverse flow velocity of the core and the halo, which are calculated on the average of pions and ω mesons from the AMPT model, excluding ω decays. Corresponding to the value of T in figure 4, the calculated $\langle {{\beta }_{{\rm T}}}\rangle$ and T₀ are ${{\langle {{\beta }_{{\rm T}}}\rangle }_{{\rm tot}}}=0.638\pm 0.001$, ${{\langle {{\beta }_{{\rm T}}}\rangle }_{{\rm C}}}={{\langle {{\beta }_{{\rm T}}}\rangle }_{{\rm ex}}}=0.692\pm 0.002$, ${{\langle {{\beta }_{{\rm T}}}\rangle }_{\omega }}=0.403\pm 0.003$, and ${{T}_{0,{\rm tot}}}=83.5\pm 0.1\;{\rm MeV}$, ${{T}_{0,{\rm C}}}=94.6\pm 0.2\;{\rm MeV}$, ${{T}_{0,\omega }}=65.1\pm 0.1\;{\rm MeV}$, and ${{T}_{0,{\rm ex}}}=87.9\pm 0.2\;{\rm MeV}$.

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It is considered that ${{R}_{{\rm l}}}$ is sensitive to the details of expansion, which helps to discriminate between various cases [27]. The evolution time scale of pions from core and halo can be calculated by the longitudinal HBT radius ${{R}_{{\rm l}}}$ and the corresponding freeze-out temperature T₀, respectively. From the dependence of the longitudinal HBT radius ${{R}_{{\rm l}}}$ on ${{m}_{{\rm T}}}$ and the corresponding freeze-out temperature T₀, we can extract the evolution time scale τ of the source by the Herrmann–Bertsch formula [22, 28, 29],

$$\begin{eqnarray}&&{{R}_{{\rm l}}}=\tau \sqrt{\frac{{{T}_{0}}}{{{m}_{{\rm T}}}}\frac{{{K}_{2}}({{m}_{{\rm T}}}/{{T}_{0}})}{{{K}_{1}}({{m}_{{\rm T}}}/{{T}_{0}})}}.\end{eqnarray} \tag{ 8 }$$

The Herrmann–Bertsch formula is derived by assuming the thermal spectrum of particles satisfied with the Cooper–Frye formula and the equilibrium thermal distribution as required by the Boltzmann approximation [23, 30]. Equation (8) can be used to calculate the evolution time of cole pions that were produced in the hot medium and emitted out of local thermal equilibrium. Unfortunately, the formula fails to calculate the evolution time of the halo pions directly. Halo pions are from resonance decays and do not have a well-defined temperature, because halo pions produced in effective post freeze-out four-volume, which depends also on resonance widths, are not thermal under an assumption of sharp kinetic freeze-out. In theory, the source function of halo pions can be expressed in terms of that of ω resonances, and the Herrmann–Bertsch formula for halo pions can be derived by the source function of ω resonances. It is too complicated to derive the formula for halo pions that are almost produced from three-body decays of ω resonances. We assume a crude approximation to obtain a qualitative trend of the evolution time of halo pions for comparing with the reliable result of core pions. We suppose that halo pions freeze out immediately after ω decays, and their freeze-out positions are approximate, as those of ω resonances. Using the temperature of ω mesons from figure 4 and the longitudinal HBT radii of halo pions, we can apply equation (8) to obtain an approximate evolution time of ω mesons, which have a similar trend with halo pions.

Figure 5 shows the longitudinal HBT radii that are fitted by equation (8) and the extracted time scales. ${{\tau }_{{\rm H}}}$ is much larger than ${{\tau }_{{\rm tot}}}$, and the long-lived results correspond to the lifetime of the resonance where ${{\tau }_{\omega }}$ = 23.4 fm c⁻¹. Excluding ω decays, ${{\tau }_{{\rm ex}}}$ is smaller than ${{\tau }_{{\rm tot}}}$ and corresponds to ${{\tau }_{{\rm C}}}$, which is also smaller than ${{\tau }_{{\rm tot}}}$. The evolution time indicates that the lifetime of the core is smaller than that of the whole pion emission source extracted by the fitted longitudinal radius ${{R}_{{\rm l}}}$, using the Gaussian parametrization.

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Figure 6 shows the freeze-out temperature T₀ as a function of the partonic cross section ${{\sigma }_{{\rm p}}}$ for Au+Au central collisions at $\sqrt{{{s}_{NN}}}=200$ GeV. For each ${{\sigma }_{{\rm p}}}$, ${{T}_{0,\omega }}$ is smaller than ${{T}_{0,{\rm tot}}}$, and ${{T}_{0,{\rm C}}}$ is larger than ${{T}_{0,{\rm tot}}}$. With increasing ${{\sigma }_{{\rm p}}}$, ${{T}_{0,{\rm C}}}$ and ${{T}_{0,\omega }}$ deflect the value of ${{T}_{0,{\rm tot}}}$, and their differences are larger. It suggests that pions from the core freeze out at a higher temperature than those from the halo with larger ${{\sigma }_{{\rm p}}}$.

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Figure 7 shows the evolution time τ as a function of the partonic cross section ${{\sigma }_{{\rm p}}}$ for Au+Au and p+p central collisions at $\sqrt{{{s}_{NN}}}=200$ GeV. We note that ${{\tau }_{{\rm tot}}}$ for Au+Au central collisions decreases with increasing ${{\sigma }_{{\rm p}}}$. However, we cannot see the drop of τ for p+p collisions, which is unchanged with increasing ${{\sigma }_{{\rm p}}}$. The result indicates that the strength of the partonic interaction processes via two-body scattering actually affects the evolution time of the pion emission source for Au+Au collisions. The vanished drop of τ for p+p collisions suggests that the decreased evolution time for Au+Au collisions may be related to the existence of sQGP, which does not exist in p+p collisions. It is noteworthy that the larger ${{\sigma }_{{\rm p}}}$ which corresponds to the longer evolution time of the partonic system, leads to a shorter lifetime of the pion emission source for Au+Au collisions. The behavior of drop only appears in the evolution time result ${{\tau }_{{\rm C}}}$ of the core. Compared with the drop of ${{\tau }_{{\rm C}}}$, ${{\tau }_{{\rm H}}}$ increases with increasing ${{\sigma }_{{\rm p}}}$. The different behavior of the halo shows that the drop of ${{\tau }_{{\rm tot}}}$ is due to the evolution time of the core, and the partonic interaction processes directly affect ${{\tau }_{{\rm C}}}$. We think that the larger ${{\sigma }_{{\rm p}}}$, leads to a longer evolution time of the partonic system, and a delayed peak value of the production time for pions produced in the core. But the delayed produced pions freeze out more quickly, because that the larger ${{\sigma }_{{\rm p}}}$ leads to a larger mean free path of pions produced in the core. The cascade of the partonic system has a larger number density to raise the number of parton scatterings and attains a larger mean free path with a shorter evolution time than that of the hadron system. The effect of the larger mean free path is weak on the evolution time of resonances, which have a long enough lifetime before decays; hence the freeze-time of pions from the resonance decays is hardly affected. The shorter evolution time of source is due to the earlier average freeze-out time of pions from the core with the more intense parton scattering processes.

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Based on the string melting AMPT model, we apply the double-Gaussian parametrization with a two-scale approach to fit the HBT correlation function of charged pions and extract the HBT parameters of the core–halo source. The spherical harmonic analysis result indicates that the double-Gaussian fit is more appropriate for the pion emission source, and the core–halo structure is closer to the source distribution that obtained long-lived resonance decays in the AMPT model. Through the dependence of ${{R}_{{\rm l}}}$ on ${{m}_{{\rm T}}}$ and the corresponding freeze-out temperature T₀, we extract the evolution time scale τ of the source. The evolution time as a function of ${{\sigma }_{{\rm p}}}$ for Au+Au and p+p central collisions at $\sqrt{{{s}_{NN}}}=200$ GeV shows that the drop of the source evolution time with increasing ${{\sigma }_{{\rm p}}}$ is due to the lifetime of the core. The partonic cascade with a larger ${{\sigma }_{{\rm p}}}$ leads to a shorter lifetime of the pion emission source.

This work was supported by the National Natural Science Foundation of China (NSFC U1332125) and the Program for Innovation Research of Science in Harbin Institute of Technology (PIRS OF HIT B201408).