Single-particle space-momentum angle distribution effect on two-pion HBT correlation with Coulomb interaction

  • We calculate the HBT radius Rs for π+ with Coulomb interaction using the string melting version of a multiphase transport (AMPT) model. We study the relationship between the single-particle space-momentum angle and the particle sources and discuss HBT radii without single-particle space-momentum correlation. Additionally, we study the Coulomb interaction effect on the numerical connection between the single-particle space-momentum angle distribution and the transverse momentum dependence of Rs.
  • The Hanbury-Brown Twiss (HBT) method is a useful tool in relativistic heavy ion collisions, and it can probe the dynamically generated geometry structure of the emitting system. It is also called the two-pion interferometry method, for it is often used with pions, which are the most investigated particles in high-energy collisions. The method of measuring the photons correlation to extract angular sizes of stars was invented by Hanbury Brown and Twiss in astronomy in the 1950s [1]. Several years later, this method was extended to¯p+p collisions by G. Goldhaber, S. Goldhaber, W. Lee and A. Pais [2]. After years of improvement, the HBT method has become a precision tool for measuring space-time and the dynamic properties of the emitting source [35], and it has been used in e++e, hadron+hadron, and heavy ion collisions [3, 6, 7].

    At the high energies of heavy-ion collisions, the normal matter transforms into the Quark-Gluon Plasma (QGP), which is a new state of matter consisting of deconfined quarks and gluons [8, 9]. There are two kinds of phase transition between the low-temperature hadronic phase and the high-temperature quark-gluon plasma phase: cross-over transition and first-order transition. The critical end point (CEP) is the point where the first-order phase transition terminates [10, 11]. Searching for the CEP at lower energies on the QCD phase diagram is one of the main goals of the Beam Energy Scan (BES) program [12, 13]. There will be critical behavior near the CEP [14], and the transport coefficients will change violently, which will lead to changes in HBT radii; then, the CEP can be estimated by the HBT analysis [15].

    The space-momentum correlation is important in HBT research, which is caused by the collective expanding behavior of the collision source [16]. Moreover, the space-momentum correlation can lead to changes in the HBT radii with the transverse momentum of pion pairs [17]. This phenomenon is called the transverse momentum dependence of HBT radii. Our research focuses on the connection between the space-momentum correlation and this phenomenon. As we need a tool to quantify this space-momentum correlation, the normalized single-particle space-momentum angle distribution was introduced in our previous work [18, 19]. This distribution consists of a series of angles belonging to freeze-out pions in the same energy sections and the same transverse momentum pion pair sections. We use the projection angle Δθ on the transverse plane in our study, as shown in Fig. 1. With this angle distribution, we can obtain more information about the source from the transverse momentum dependence of HBT radii.

    Figure 1

    Figure 1.  The diagram of the Δφ and Δθ. Δφ is the angle between r and p, and Δθ is the angle between rT and pT, at the freeze-out time. The origin is the center of the source.

    In our previous work, we treated three kinds of pions as one, considered them to have the same mass, and neglected the Coulomb interaction [18, 19]. Meanwhile, in experiments, the three kinds of pions can be distinguished and the charged ones are more easily detected. Therefore, in this study, we focus only on the π+ and discuss the influence of the Coulomb interaction on the numerical connection between the single-particle space-momentum angle Δθ distribution and the transverse momentum dependence of HBT radius Rs. We use a multiphase transport (AMPT) model to generate the freeze-out π+ at different collision energies. This model contains the physical processes of the relativistic heavy-ion collisions. Besides, it has already been widely used in HBT research [2022].

    The paper is organized as follows. In Sec. II, the string melting AMPT model and the HBT correlation are briefly introduced. In Sec. III, the HBT radius Rs for π+ is calculated for two conditions: with and without Coulomb interaction. In Sec. IV, we discuss the influence of the single-particle space-momentum angle Δθ distribution on the HBT radius Rs and build the numerical connections between them, with and without Coulomb interaction. In the final section, we give the summary.

    In this study, we use the string melting AMPT model, which can give a better description of the correlation function in HBT research [23, 24]. The string melting AMPT model contains four main parts. The first part uses the HIJING model to produce the partons and strings, where the strings will fragment into partons. The second part uses the Zhang’s parton-cascade (ZPC) model to describe the interactions among these partons. Then, in the third part, a quark coalescence model is used to combine these partons into hadrons. In the last part, the interaction of these hadrons is described by a relativistic transport (ART) model till the hadrons freeze out. With the models in these four main parts, the string melting AMPT model can be used to describe heavy-ion collisions.

    The HBT radii are important in HBT research, and they can be extracted by the HBT three-dimensional correlation function, whereas the normal form can be written as [7]

    C(q,K)=1+λeq2oR2o(K)q2sR2s(K)q2lR2l(K)2qoqlR2ol(K).

    (1)

    In addition, in this study, we use the form with Coulomb interaction as [25]

    C(q,K)=1λ+λKcoul[1+eq2oR2o(K)q2sR2s(K)q2lR2l(K)2qoqlR2ol(K)],

    (2)

    where q=p1p2, K=(p1+p2)/2, C is the two pion correlation function, Kcoul is the squared Coulomb wave function, and λ is the coherence parameter. R is for the HBT radius. It is usually used in the 'out-side-long' coordinate system, and this system is used for the pair particles, as shown in Fig. 2. l represents the longitudinal direction and beam direction. o and s are outward and sideward directions that are both defined on the transverse plane. The pair particles momentum direction is defined as the outward direction, and the side direction is the direction perpendicular to the outward direction. Rol is the cross term, which will vanish at mid-rapidity in a symmetric system. In this study, we set the biggest impact parameter as 1.4 fm and chose the mid-rapidity range 0.5<η<0.5. We use Eq. (1) to fit the correlation function without Coulomb interaction and Eq. (2) to fit the correlation function with Coulomb interaction.

    Figure 2

    Figure 2.  The diagram of 'out-side-long'(o-s-l) coordinate system.

    We use the Correlation After Burner (CRAB) code to generate the correlation function, which can read the phase space information from the string melting AMPT model [26]. It generates the correlation function by the formula

    C(q,K)=1+d4x1d4x2S1(x1,p2)S2(x2,p2)|ψrel|2d4x1d4x2S1(x1,p2)S2(x2,p2),

    (3)

    where ψrel is the relative two particles wave function, which includes the interaction between two particles. S(x,p) is the single particle emission function.

    We produced the particles of Au+Au collisions at SNN=14.5, 19.6, 27, 39, 62.4, 200 GeV by the string melting AMPT model. These are the BES energies, each one of them has more than sixty thousand collision events. We separate the pair transverse momentum 125–625 MeV/c into 9 bins. While the last bin has fewer particles, we set it for 525–625 MeV/c, and the width is twice that of the others. Then we use the CRAB code to calculate the correlation functions for the π+. The Coulomb interaction can change the correlation functions, which are shown in Fig. 3. If the Coulomb interaction is neglected, because of the collective flow created by the collisions, two π+ are easily frozen out along the same direction, and the correlation function will be a Gaussian form. While considering the Coulomb interaction, the two π+ have the Coulomb potential between them, and it will change the two π+ wave function ψrel. The closer are the two π+, the bigger the Coulomb potential between them will be. When two π+are close to each other, the probability density of the two π+ will decrease, and will cause the decrease of the Correlation function in the direction along the momentum difference direction. qs is related to the momentum difference on the transverse plane, so there is a little gap in the correlation function in the qs direction.

    Figure 3

    Figure 3.  (color online) Correlation functions of π+ in the qo and qs directions, generated by the string melting AMPT model for the Au+Au collisions at SNN= 200 GeV. The KT range is 325–375 MeV/c, and the ql range is -3–3 MeV/c. Figure (a) neglects the Coulomb interaction, and figure (b) is the situation with the Coulomb interaction.

    We only focus on the HBT radius Rs; it is least affected by other physical factors and related to the transverse size of the source [27]. In addition, the transverse momentum dependence of Rs with and without Coulomb interaction is shown in Fig. 4. The obvious difference are the values of Rs calculated with the Coulomb interaction, which are smaller than those without the Coulomb interaction at small pair momentum. Besides, the strength of KT dependence of Rs, especially for the lower collision energies, can barely be distinguished. And it is even worse for the situation with the Coulomb interaction. At low collision energies, there are similar collective flow velocities in each KT bin, which leads to similar single-particle momentum-space angle distributions [19]. Moreover, the single-particle momentum-space angle distributions can affect the HBT radii [18], so the values of Rs are similar. To distinguish the strength of KT dependence of Rs, we introduced a parameter b in the fit function

    Figure 4

    Figure 4.  (color online) Transverse momentum dependence of Rs in AMPT model.

    Rs=aKbT,

    (4)

    where parameter a is just a common constant. The parameter b can be used to describe the strength for the KT dependence of Rs, as shown in Fig. 5.

    Figure 5

    Figure 5.  (color online) Collision energy dependence of parameter b.

    In Fig. 5, All the |b| values increase with increasing collision energies. It indicates that, at high collision energies, the changes in Rs values are more intense with the transverse pair momenta. The black dots for all pions are taken from our last work, which treated all pions as one kind, generated with the 0 fm impact parameter. Comparing to the 0–1.4 fm for all pions, the |b| values decrease with a higher impact parameter. The latter can decrease the HBT radii [28], which indicates that the impact parameter has different influences on Rs in different KT sections. Comparing π+ without Coulomb interaction and all pions, shows that the kind of pions can also affect the strength of the KT dependence of Rs. The |b| values are smaller with the Coulomb interaction, which means the Coulomb interaction inhibits the strength of the transverse momentum dependence of HBT radii. Furthermore, with increasing the collision energies, the differences of the |b| values for π+ between the two situations are also increased, which indicates this inhibition is stronger at high collision energies for the Coulomb interaction.

    In HBT research, the correlation function is often approximated by the on-shell momenta [29, 30]. So the correlation can also be written as [31]

    C(q,K)1+|d4xS(x,K))eiqx|2|d4xS(x,K)|21+|<eiqx>|2,

    (5)

    where K0=EK=(m2+|K|2), q=p1p2, then we get

    eiqx=exp[i(p1xp2x)],

    (6)

    for the single-particle, we have already defined the angle Δφ, which is between the momentum direction and space direction, then we define two more angles,

    cos(Δα)p2x|p2||x|,

    (7)

    cos(Δβ)p1p2|p1||p2|,

    (8)

    where Δα is the angle between the momentum direction and radius direction for two particles, and Δβ is the angle between the momentum directions for two particles. The smoothness assumption is |p1||p2|12|K| [32], then that

    eiqxexp[i2(|K||x|cos(Δφ)|K||x|cos(Δα))],

    (9)

    the Δα angle is not independent, its values are related to the Δβ and the Δφ, Δα=f(Δφ,Δβ), so the correlation function can be written as

    C(K,Δφ,Δβ)1+|exp[i2|K||x|[cos(Δφ)cos(f(Δφ,Δβ))]]|2.

    (10)

    When we calculate the HBT correlation function, we need to limit the values of momentum difference, |q|<qmax. If we choose the pair particles in one transverse momentum section, and because we focus on the mid-rapidity range, the pair momenta are also screened, kmin<|K|<kmax. At the same time, the angle Δβ is also limited, cos(Δβ)>2k2mink2min+q2max1. So the Δβ values are related to the |K| values and the interaction betweenthe particles. Therefore, the single-particle space-momentum angle distribution can directly affect the correlation function. Our research is focused on the transverse plane, so we discuss the connection between the Δθ distribution and Rs.

    For a source, the collective flow is created by the expansion, and the transverse flow velocities are different at different layers of the source, which leads to different normalized Δθ distributions, as shown in Fig. 6.

    Figure 6

    Figure 6.  (color online) The normalized Δθ distributions in different transverse radii, π+ are generated by the string melting AMPT model for the Au+Au collisions at SNN= 62.4 GeV/c.

    In Fig. 6, the average transverse flow velocity can be calculated by

    <vT>=pTrTE|rT|.

    (11)

    N and NR are the particle numbers in each bin, whereas NR is obtained with the random p and r particles. The normalization process is using the cos(Δθ) distribution divided by the random cos(Δθ) distribution, and we let N=NR. The particles with lower transverse momenta are closer to N/NR=1. This phenomenon indicates that the source is approaching a random freeze-out source. While with higher flow velocities, the distributions are closer to cos(Δθ)=1, it means the particles tend to freeze out along the radius direction. For the particles located at 8–9 and 11–12 fm, their <νT> values are closer, so they have similar normalized Δθ distributions.

    Furthermore, for the particles with different transverse momenta pT, they correspond to the parts of the whole source. For the particles with higher pT, the part sources have bigger collective flow, as shown in Fig. 7. The <νT> increase with rT=x2+y2 inside of the source, and at the outside of the source, <νT> almost becomes a constant. So the sizes of part sources are changing with the pT.

    Figure 7

    Figure 7.  (color online) The transverse flow velocities for the transverse radii in different transverse momenta, π+ are generated by the string melting AMPT model for the Au+Au collisions at SNN= 62.4 GeV/c.

    Before particle freezing out from the source, the system goes through a series of processes, containing parton production and interaction, hadronization, and hadron cascade. The expansion of the source will be reflected in the collective flow of the freeze-out particles. The collective flow is different at different locations in the source, meanwhile, the Δθ is related to the flow. Thus, different Δθ have different freeze-out sources, as shown in Fig. 8.

    Figure 8

    Figure 8.  (color online) The freeze-out π+ sources in different cos(Δθ) sections. The particles are generated by the string melting AMPT model for the Au+Au collisions at SNN= 62.4 GeV, and the pT range is 250–450 MeV/c.

    For the π+ that have large Δθ angles, their freeze-out directions are reversed to the space directions. They have a small freeze-out source, which indicates they are almost freezing out from the center of the whole π+ source. Those π+ freeze-out directions are perpendicular to the space directions and have a bigger source, as shown in Fig. 8(b). While in Fig. 8(c), for the π+ whose momentum directions are along to the space directions, their freeze-out source has a ring shape, and has the biggest number density. So in this pT section, most of the π+ freeze out from the shell. The closer to the center of the source, the higher the probability of producing large Δθ angle particles. For we limited the differences of the pair particles qmax, the normalized cos(Δθ) distribution in KTmin<KT<KTmax section only contains KTminqmax/2<pT<KTmax+qmax/2 particles. Besides, different cos(Δθ) in the KT section has different partial sources, similar to Fig. 8. The normalized cos(Δθ) distribution corresponds to the superposition of these partial sources. Therefore, the distribution is related to the HBT radius Rs.

    If there is no correlation between the space and momentum, the relation between Δθ and the source will break, and the differences in the normalized cos(Δθ) distributions will disappear. Then the Δθ angle will be completely random, as shown in Fig. 9. The normalized "no x-p correlation" distribution becomes a line and is the same in other KT sections.

    Figure 9

    Figure 9.  (color online) The original distribution and the no x-p correlation distribution at SNN= 62.4 GeV, and the KT range is 325–375 MeV/c.

    In Fig. 9, we disrupt the space and momentum correlation in each collision event. For example, original data are x1p1 and x2p2, and after the disruption, they are x1p2 and x2p1. This is a rough but effective method, and it can change the correlation between the space and the momentum, which means the particles can freeze out from the source in any direction. Then we calculate the HBT radius Rs, as shown in Fig. 10. The results show that the phenomenon of transverse dependence of Rs for "no x-p correlation" has almost disappeared. The Rs values for "no x-p correlation" are closer to each other, because they have the same normalized cos(Δθ) distribution. This method destroys the physical process, there will be fluctuations in calculating the correlation function, which leads to the emergence of the big error bars.

    Figure 10

    Figure 10.  (color online) The original Rs and no x-p correlation Rs at SNN= 62.4 GeV.

    Moreover, we can build a numerical connection between the normalized cos(Δθ) distribution and HBT radius Rs. The AMPT model does not contain the Coulomb interaction at the stage of the hadron cascade [23], so we discuss the Coulomb interaction only for the HBT analysis. Moreover, the two conditions are using the same string melting AMPT data, leading to the same normalized cos(Δθ) distribution for different KT regions. In order to describe the normalized cos(Δθ) distributions, we introduce the fit function

    f=0.002exp{c1exp[c2cos(Δθ)]}.

    (12)

    c1 and c2 are the fit parameters. c1 is influenced by the proportion of particles whose cos(Δθ)=0, and c2 is influenced by the strength of the distribution approaching cos(Δθ)=1. The value of 0.002 is settled by us to get good fitting results, and this value is different from that in our last work because the normalized cos(Δθ) distributions have changed, as shown in Fig. 11. We can see in two KT sections, π+ and all pions with 0–1.4 fm impact parameter have similar distributions, while the impact parameter has a big influence on the distributions. Thus, the impact parameter plays an important role in the single-particle space-momentum angle correlation. For we change this settled value, it leads to the changing of the fitting results in the subsequent analysis. The parameters c1 and c2 are changing with the KT, so we can use the fitting functions to describe their changing pattern. The fit functions are

    Figure 11

    Figure 11.  (color online) The normalized Δθ distributions, generated by the string melting AMPT model for the Au+Au collisions at SNN= 39 GeV/c.

    c1=k1exp[6×(KT1000)2]+j1,

    (13)

    c2=k2exp[4.5×(KT1000)2]+j2,

    (14)

    where –6 and –4.5 were chosen by us to obtain good fitting results, k and j are fit parameters. k1 and j1 are related to c1. k1 is influenced by the changing range of the proportion of pions with cos(Δθ)=0, and j1 is influenced by this lowest proportion. Besides, k2 and j2 are related to c2, k2 is influenced by the changing range of the strength of the distribution approaching cos(Δθ)=1, and j2 is influenced by its highest strength.

    The collision energies not only have influences on the KT dependence of Rs, but also on the KT dependence of normalized cos(Δθ) distributions. Then we can plot the parameter b, as a function of the parameters k and j, as shown in Fig. 12. The changing patterns indicate parameter b has an extremum; we set b1=0.2 for the situation without the Coulomb interaction, and b2=0.3 for considering the Coulomb interaction. We use the following functions to fit these patterns,

    Figure 12

    Figure 12.  (color online) The numerical connections between the strength of the transverse momentum dependence of Rs and the parameters related to the normalized cos(Δθ) distribution, the lines are fit lines.

    b(km)=μm1|km|μm2+bn,

    (15)

    b(jm)=νm1|jm|νm2+bn,

    (16)

    where m=1 or 2, for distinguishing the parameters of c1 or c2. Also n=1 or 2, then 1 is for the situation without the Coulomb interaction, and 2 refers to the Coulomb interaction. μ and ν are fit parameters, and the fitting results are shown in Table 1. The results show that the Coulomb interaction has an influence on these fit parameters, it can decrease |μm1| and |νm1|, and increase |μm2| and |νm2|. It is caused by the changing of the differences of b values. The differences in b values between the two situations are increasing with increasing collision energies. Compared to our previous work, the changing patterns are different, so these numerical connections are also related to the kinds of particles.

    Table 1

    Table 1.  Fit results for parameters μ and ν.
    n=1 n=2
    m=1 μ11 0.0004±0.0003 0.00004±0.00003
    μ12 3.3±0.5 4.7±0.4
    ν11 0.086±0.004 0.084±0.003
    ν12 0.68±0.09 0.94±0.08
    m=2 μ21 0.024±0.003 0.014±0.003
    μ22 2.3±0.3 3.2±0.3
    ν21 0.021±0.004 0.012±0.002
    ν22 2.4±0.3 3.3±0.3
    DownLoad: CSV
    Show Table

    With these fit functions and parameters, the numerical connections between the normalized cos(Δθ) distribution and the strength of the transverse momentum dependence of Rs have been built. Comparing to our previous work, we found that the fit parameters are related to the impact parameter, the kinds of pions, and the Coulomb interaction. We improve the fit parameters, which makes our numerical correlation applied to the experiment closer. With further research, the experiment data can be directly used to study the single-particle space-momentum correlation.

    With the string melting AMPT model, we calculated the HBT radius Rs for π+ in two situations, with and without Coulomb interaction. The results indicate that the Coulomb interaction can decrease the Rs values and inhibits the KT dependence of R. We compared the strength of KT dependence of R in two impact parameters and found that the impact parameters can also inhibit KT dependence of R. Then, we showed that the transverse flow can affect the normalized cos(Δθ) distributions, and the flow changes with the location of the source, which leads to particles with bigger Δθ angles tending to freeze out from the inner of the source. We also demonstrated that the normalized cos(Δθ) distributions can influence the KT dependence of Rs by disrupting the space and the momentum correlation. The inhibition of the transverse momentum dependence of Rs caused by Coulomb interaction leads to changes in the fit results. Different impact parameters and particles have different numerical connections. Moreover, with these numerical connections, we can get more information about the final stage of the Au+Au collision at the freeze-out time by the HBT analysis. With more collision energies, more impact parameters, and more accurate fits, the numerical connection can be improved. There are anomalies near the CEP, which may change this connection, so the connection improvement may be a probe to detect the CEP. The Coulomb interaction can weaken the strength of the transverse momentum dependence of Rs, it may also weaken the sensitivity of this probe.

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  • [1] R. H Brown and R. Twiss, Nature 178, 1046-1048 (1956) doi: 10.1038/1781046a0
    [2] G. Goldhaber, S. Goldhaber, W. Lee et al., Phys. Rev. 120(1), 300-312 (1960) doi: 10.1103/PhysRev.120.300
    [3] M. A. Lisa, S. Pratt , R. Soltz et al., Annu. Rev. Nucl. Part. Sci. 55, 357-402 (2005) doi: 10.1146/annurev.nucl.55.090704.151533
    [4] U. Heinz and B. V. Jacak, Annu. Rev. Nucl. Part. Sci. 49, 529-579 (1999) doi: 10.1146/annurev.nucl.49.1.529
    [5] U. A. Wiedemann and U. Heinz 1999 Phys. Rep. 319 145 – 230 ISSN 0370-1573
    [6] W. Kittel, Acta. Phys. Pol. B 32, 3927 (2001), arXiv:hep-ph/0110088
    [7] G. Alexander, Rep. Prog. Phy. 66, 481-522 (2003) doi: 10.1088/0034-4885/66/4/202
    [8] J. Adams, M. Aggarwal, Z. Ahammed et al. (STAR Collaboration) , Nucl. Phys. A 757, 102 (2005)
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Hang Yang, Qichun Feng and Jingbo Zhang. Single-particle space-momentum angle distribution effect on two-pion HBT correlation with Coulomb interaction[J]. Chinese Physics C. doi: 10.1088/1674-1137/ac8c23
Hang Yang, Qichun Feng and Jingbo Zhang. Single-particle space-momentum angle distribution effect on two-pion HBT correlation with Coulomb interaction[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ac8c23 shu
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Single-particle space-momentum angle distribution effect on two-pion HBT correlation with Coulomb interaction

    Corresponding author: Jingbo Zhang, jinux@hit.edu.cn
  • School of Physics, Harbin Institute of Technology, Harbin 150001, China

Abstract: We calculate the HBT radius Rs for π+ with Coulomb interaction using the string melting version of a multiphase transport (AMPT) model. We study the relationship between the single-particle space-momentum angle and the particle sources and discuss HBT radii without single-particle space-momentum correlation. Additionally, we study the Coulomb interaction effect on the numerical connection between the single-particle space-momentum angle distribution and the transverse momentum dependence of Rs.

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    I.   INTRODUCTION
    • The Hanbury-Brown Twiss (HBT) method is a useful tool in relativistic heavy ion collisions, and it can probe the dynamically generated geometry structure of the emitting system. It is also called the two-pion interferometry method, for it is often used with pions, which are the most investigated particles in high-energy collisions. The method of measuring the photons correlation to extract angular sizes of stars was invented by Hanbury Brown and Twiss in astronomy in the 1950s [1]. Several years later, this method was extended to¯p+p collisions by G. Goldhaber, S. Goldhaber, W. Lee and A. Pais [2]. After years of improvement, the HBT method has become a precision tool for measuring space-time and the dynamic properties of the emitting source [35], and it has been used in e++e, hadron+hadron, and heavy ion collisions [3, 6, 7].

      At the high energies of heavy-ion collisions, the normal matter transforms into the Quark-Gluon Plasma (QGP), which is a new state of matter consisting of deconfined quarks and gluons [8, 9]. There are two kinds of phase transition between the low-temperature hadronic phase and the high-temperature quark-gluon plasma phase: cross-over transition and first-order transition. The critical end point (CEP) is the point where the first-order phase transition terminates [10, 11]. Searching for the CEP at lower energies on the QCD phase diagram is one of the main goals of the Beam Energy Scan (BES) program [12, 13]. There will be critical behavior near the CEP [14], and the transport coefficients will change violently, which will lead to changes in HBT radii; then, the CEP can be estimated by the HBT analysis [15].

      The space-momentum correlation is important in HBT research, which is caused by the collective expanding behavior of the collision source [16]. Moreover, the space-momentum correlation can lead to changes in the HBT radii with the transverse momentum of pion pairs [17]. This phenomenon is called the transverse momentum dependence of HBT radii. Our research focuses on the connection between the space-momentum correlation and this phenomenon. As we need a tool to quantify this space-momentum correlation, the normalized single-particle space-momentum angle distribution was introduced in our previous work [18, 19]. This distribution consists of a series of angles belonging to freeze-out pions in the same energy sections and the same transverse momentum pion pair sections. We use the projection angle Δθ on the transverse plane in our study, as shown in Fig. 1. With this angle distribution, we can obtain more information about the source from the transverse momentum dependence of HBT radii.

      Figure 1.  The diagram of the Δφ and Δθ. Δφ is the angle between r and p, and Δθ is the angle between rT and pT, at the freeze-out time. The origin is the center of the source.

      In our previous work, we treated three kinds of pions as one, considered them to have the same mass, and neglected the Coulomb interaction [18, 19]. Meanwhile, in experiments, the three kinds of pions can be distinguished and the charged ones are more easily detected. Therefore, in this study, we focus only on the π+ and discuss the influence of the Coulomb interaction on the numerical connection between the single-particle space-momentum angle Δθ distribution and the transverse momentum dependence of HBT radius Rs. We use a multiphase transport (AMPT) model to generate the freeze-out π+ at different collision energies. This model contains the physical processes of the relativistic heavy-ion collisions. Besides, it has already been widely used in HBT research [2022].

      The paper is organized as follows. In Sec. II, the string melting AMPT model and the HBT correlation are briefly introduced. In Sec. III, the HBT radius Rs for π+ is calculated for two conditions: with and without Coulomb interaction. In Sec. IV, we discuss the influence of the single-particle space-momentum angle Δθ distribution on the HBT radius Rs and build the numerical connections between them, with and without Coulomb interaction. In the final section, we give the summary.

    II.   THE STRING MELTING AMPT MODEL AND METHODOLOGY
    • In this study, we use the string melting AMPT model, which can give a better description of the correlation function in HBT research [23, 24]. The string melting AMPT model contains four main parts. The first part uses the HIJING model to produce the partons and strings, where the strings will fragment into partons. The second part uses the Zhang’s parton-cascade (ZPC) model to describe the interactions among these partons. Then, in the third part, a quark coalescence model is used to combine these partons into hadrons. In the last part, the interaction of these hadrons is described by a relativistic transport (ART) model till the hadrons freeze out. With the models in these four main parts, the string melting AMPT model can be used to describe heavy-ion collisions.

      The HBT radii are important in HBT research, and they can be extracted by the HBT three-dimensional correlation function, whereas the normal form can be written as [7]

      C(q,K)=1+λeq2oR2o(K)q2sR2s(K)q2lR2l(K)2qoqlR2ol(K).

      (1)

      In addition, in this study, we use the form with Coulomb interaction as [25]

      C(q,K)=1λ+λKcoul[1+eq2oR2o(K)q2sR2s(K)q2lR2l(K)2qoqlR2ol(K)],

      (2)

      where q=p1p2, K=(p1+p2)/2, C is the two pion correlation function, Kcoul is the squared Coulomb wave function, and λ is the coherence parameter. R is for the HBT radius. It is usually used in the 'out-side-long' coordinate system, and this system is used for the pair particles, as shown in Fig. 2. l represents the longitudinal direction and beam direction. o and s are outward and sideward directions that are both defined on the transverse plane. The pair particles momentum direction is defined as the outward direction, and the side direction is the direction perpendicular to the outward direction. Rol is the cross term, which will vanish at mid-rapidity in a symmetric system. In this study, we set the biggest impact parameter as 1.4 fm and chose the mid-rapidity range 0.5<η<0.5. We use Eq. (1) to fit the correlation function without Coulomb interaction and Eq. (2) to fit the correlation function with Coulomb interaction.

      Figure 2.  The diagram of 'out-side-long'(o-s-l) coordinate system.

      We use the Correlation After Burner (CRAB) code to generate the correlation function, which can read the phase space information from the string melting AMPT model [26]. It generates the correlation function by the formula

      C(q,K)=1+d4x1d4x2S1(x1,p2)S2(x2,p2)|ψrel|2d4x1d4x2S1(x1,p2)S2(x2,p2),

      (3)

      where ψrel is the relative two particles wave function, which includes the interaction between two particles. S(x,p) is the single particle emission function.

    III.   THE COULOMB EFFECT ON Rs
    • We produced the particles of Au+Au collisions at SNN=14.5, 19.6, 27, 39, 62.4, 200 GeV by the string melting AMPT model. These are the BES energies, each one of them has more than sixty thousand collision events. We separate the pair transverse momentum 125–625 MeV/c into 9 bins. While the last bin has fewer particles, we set it for 525–625 MeV/c, and the width is twice that of the others. Then we use the CRAB code to calculate the correlation functions for the π+. The Coulomb interaction can change the correlation functions, which are shown in Fig. 3. If the Coulomb interaction is neglected, because of the collective flow created by the collisions, two π+ are easily frozen out along the same direction, and the correlation function will be a Gaussian form. While considering the Coulomb interaction, the two π+ have the Coulomb potential between them, and it will change the two π+ wave function ψrel. The closer are the two π+, the bigger the Coulomb potential between them will be. When two π+are close to each other, the probability density of the two π+ will decrease, and will cause the decrease of the Correlation function in the direction along the momentum difference direction. qs is related to the momentum difference on the transverse plane, so there is a little gap in the correlation function in the qs direction.

      Figure 3.  (color online) Correlation functions of π+ in the qo and qs directions, generated by the string melting AMPT model for the Au+Au collisions at SNN= 200 GeV. The KT range is 325–375 MeV/c, and the ql range is -3–3 MeV/c. Figure (a) neglects the Coulomb interaction, and figure (b) is the situation with the Coulomb interaction.

      We only focus on the HBT radius Rs; it is least affected by other physical factors and related to the transverse size of the source [27]. In addition, the transverse momentum dependence of Rs with and without Coulomb interaction is shown in Fig. 4. The obvious difference are the values of Rs calculated with the Coulomb interaction, which are smaller than those without the Coulomb interaction at small pair momentum. Besides, the strength of KT dependence of Rs, especially for the lower collision energies, can barely be distinguished. And it is even worse for the situation with the Coulomb interaction. At low collision energies, there are similar collective flow velocities in each KT bin, which leads to similar single-particle momentum-space angle distributions [19]. Moreover, the single-particle momentum-space angle distributions can affect the HBT radii [18], so the values of Rs are similar. To distinguish the strength of KT dependence of Rs, we introduced a parameter b in the fit function

      Figure 4.  (color online) Transverse momentum dependence of Rs in AMPT model.

      Rs=aKbT,

      (4)

      where parameter a is just a common constant. The parameter b can be used to describe the strength for the KT dependence of Rs, as shown in Fig. 5.

      Figure 5.  (color online) Collision energy dependence of parameter b.

      In Fig. 5, All the |b| values increase with increasing collision energies. It indicates that, at high collision energies, the changes in Rs values are more intense with the transverse pair momenta. The black dots for all pions are taken from our last work, which treated all pions as one kind, generated with the 0 fm impact parameter. Comparing to the 0–1.4 fm for all pions, the |b| values decrease with a higher impact parameter. The latter can decrease the HBT radii [28], which indicates that the impact parameter has different influences on Rs in different KT sections. Comparing π+ without Coulomb interaction and all pions, shows that the kind of pions can also affect the strength of the KT dependence of Rs. The |b| values are smaller with the Coulomb interaction, which means the Coulomb interaction inhibits the strength of the transverse momentum dependence of HBT radii. Furthermore, with increasing the collision energies, the differences of the |b| values for π+ between the two situations are also increased, which indicates this inhibition is stronger at high collision energies for the Coulomb interaction.

    IV.   THE SINGLE-PARTICLE SPACE-MOMENTUM ANGLE DISTRIBUTION
    • In HBT research, the correlation function is often approximated by the on-shell momenta [29, 30]. So the correlation can also be written as [31]

      C(q,K)1+|d4xS(x,K))eiqx|2|d4xS(x,K)|21+|<eiqx>|2,

      (5)

      where K0=EK=(m2+|K|2), q=p1p2, then we get

      eiqx=exp[i(p1xp2x)],

      (6)

      for the single-particle, we have already defined the angle Δφ, which is between the momentum direction and space direction, then we define two more angles,

      cos(Δα)p2x|p2||x|,

      (7)

      cos(Δβ)p1p2|p1||p2|,

      (8)

      where Δα is the angle between the momentum direction and radius direction for two particles, and Δβ is the angle between the momentum directions for two particles. The smoothness assumption is |p1||p2|12|K| [32], then that

      eiqxexp[i2(|K||x|cos(Δφ)|K||x|cos(Δα))],

      (9)

      the Δα angle is not independent, its values are related to the Δβ and the Δφ, Δα=f(Δφ,Δβ), so the correlation function can be written as

      C(K,Δφ,Δβ)1+|exp[i2|K||x|[cos(Δφ)cos(f(Δφ,Δβ))]]|2.

      (10)

      When we calculate the HBT correlation function, we need to limit the values of momentum difference, |q|<qmax. If we choose the pair particles in one transverse momentum section, and because we focus on the mid-rapidity range, the pair momenta are also screened, kmin<|K|<kmax. At the same time, the angle Δβ is also limited, cos(Δβ)>2k2mink2min+q2max1. So the Δβ values are related to the |K| values and the interaction betweenthe particles. Therefore, the single-particle space-momentum angle distribution can directly affect the correlation function. Our research is focused on the transverse plane, so we discuss the connection between the Δθ distribution and Rs.

      For a source, the collective flow is created by the expansion, and the transverse flow velocities are different at different layers of the source, which leads to different normalized Δθ distributions, as shown in Fig. 6.

      Figure 6.  (color online) The normalized Δθ distributions in different transverse radii, π+ are generated by the string melting AMPT model for the Au+Au collisions at SNN= 62.4 GeV/c.

      In Fig. 6, the average transverse flow velocity can be calculated by

      <vT>=pTrTE|rT|.

      (11)

      N and NR are the particle numbers in each bin, whereas NR is obtained with the random p and r particles. The normalization process is using the cos(Δθ) distribution divided by the random cos(Δθ) distribution, and we let N=NR. The particles with lower transverse momenta are closer to N/NR=1. This phenomenon indicates that the source is approaching a random freeze-out source. While with higher flow velocities, the distributions are closer to cos(Δθ)=1, it means the particles tend to freeze out along the radius direction. For the particles located at 8–9 and 11–12 fm, their <νT> values are closer, so they have similar normalized Δθ distributions.

      Furthermore, for the particles with different transverse momenta pT, they correspond to the parts of the whole source. For the particles with higher pT, the part sources have bigger collective flow, as shown in Fig. 7. The <νT> increase with rT=x2+y2 inside of the source, and at the outside of the source, <νT> almost becomes a constant. So the sizes of part sources are changing with the pT.

      Figure 7.  (color online) The transverse flow velocities for the transverse radii in different transverse momenta, π+ are generated by the string melting AMPT model for the Au+Au collisions at SNN= 62.4 GeV/c.

      Before particle freezing out from the source, the system goes through a series of processes, containing parton production and interaction, hadronization, and hadron cascade. The expansion of the source will be reflected in the collective flow of the freeze-out particles. The collective flow is different at different locations in the source, meanwhile, the Δθ is related to the flow. Thus, different Δθ have different freeze-out sources, as shown in Fig. 8.

      Figure 8.  (color online) The freeze-out π+ sources in different cos(Δθ) sections. The particles are generated by the string melting AMPT model for the Au+Au collisions at SNN= 62.4 GeV, and the pT range is 250–450 MeV/c.

      For the π+ that have large Δθ angles, their freeze-out directions are reversed to the space directions. They have a small freeze-out source, which indicates they are almost freezing out from the center of the whole π+ source. Those π+ freeze-out directions are perpendicular to the space directions and have a bigger source, as shown in Fig. 8(b). While in Fig. 8(c), for the π+ whose momentum directions are along to the space directions, their freeze-out source has a ring shape, and has the biggest number density. So in this pT section, most of the π+ freeze out from the shell. The closer to the center of the source, the higher the probability of producing large Δθ angle particles. For we limited the differences of the pair particles qmax, the normalized cos(Δθ) distribution in KTmin<KT<KTmax section only contains KTminqmax/2<pT<KTmax+qmax/2 particles. Besides, different cos(Δθ) in the KT section has different partial sources, similar to Fig. 8. The normalized cos(Δθ) distribution corresponds to the superposition of these partial sources. Therefore, the distribution is related to the HBT radius Rs.

      If there is no correlation between the space and momentum, the relation between Δθ and the source will break, and the differences in the normalized cos(Δθ) distributions will disappear. Then the Δθ angle will be completely random, as shown in Fig. 9. The normalized "no x-p correlation" distribution becomes a line and is the same in other KT sections.

      Figure 9.  (color online) The original distribution and the no x-p correlation distribution at SNN= 62.4 GeV, and the KT range is 325–375 MeV/c.

      In Fig. 9, we disrupt the space and momentum correlation in each collision event. For example, original data are x1p1 and x2p2, and after the disruption, they are x1p2 and x2p1. This is a rough but effective method, and it can change the correlation between the space and the momentum, which means the particles can freeze out from the source in any direction. Then we calculate the HBT radius Rs, as shown in Fig. 10. The results show that the phenomenon of transverse dependence of Rs for "no x-p correlation" has almost disappeared. The Rs values for "no x-p correlation" are closer to each other, because they have the same normalized cos(Δθ) distribution. This method destroys the physical process, there will be fluctuations in calculating the correlation function, which leads to the emergence of the big error bars.

      Figure 10.  (color online) The original Rs and no x-p correlation Rs at SNN= 62.4 GeV.

      Moreover, we can build a numerical connection between the normalized cos(Δθ) distribution and HBT radius Rs. The AMPT model does not contain the Coulomb interaction at the stage of the hadron cascade [23], so we discuss the Coulomb interaction only for the HBT analysis. Moreover, the two conditions are using the same string melting AMPT data, leading to the same normalized cos(Δθ) distribution for different KT regions. In order to describe the normalized cos(Δθ) distributions, we introduce the fit function

      f=0.002exp{c1exp[c2cos(Δθ)]}.

      (12)

      c1 and c2 are the fit parameters. c1 is influenced by the proportion of particles whose cos(Δθ)=0, and c2 is influenced by the strength of the distribution approaching cos(Δθ)=1. The value of 0.002 is settled by us to get good fitting results, and this value is different from that in our last work because the normalized cos(Δθ) distributions have changed, as shown in Fig. 11. We can see in two KT sections, π+ and all pions with 0–1.4 fm impact parameter have similar distributions, while the impact parameter has a big influence on the distributions. Thus, the impact parameter plays an important role in the single-particle space-momentum angle correlation. For we change this settled value, it leads to the changing of the fitting results in the subsequent analysis. The parameters c1 and c2 are changing with the KT, so we can use the fitting functions to describe their changing pattern. The fit functions are

      Figure 11.  (color online) The normalized Δθ distributions, generated by the string melting AMPT model for the Au+Au collisions at SNN= 39 GeV/c.

      c1=k1exp[6×(KT1000)2]+j1,

      (13)

      c2=k2exp[4.5×(KT1000)2]+j2,

      (14)

      where –6 and –4.5 were chosen by us to obtain good fitting results, k and j are fit parameters. k1 and j1 are related to c1. k1 is influenced by the changing range of the proportion of pions with cos(Δθ)=0, and j1 is influenced by this lowest proportion. Besides, k2 and j2 are related to c2, k2 is influenced by the changing range of the strength of the distribution approaching cos(Δθ)=1, and j2 is influenced by its highest strength.

      The collision energies not only have influences on the KT dependence of Rs, but also on the KT dependence of normalized cos(Δθ) distributions. Then we can plot the parameter b, as a function of the parameters k and j, as shown in Fig. 12. The changing patterns indicate parameter b has an extremum; we set b1=0.2 for the situation without the Coulomb interaction, and b2=0.3 for considering the Coulomb interaction. We use the following functions to fit these patterns,

      Figure 12.  (color online) The numerical connections between the strength of the transverse momentum dependence of Rs and the parameters related to the normalized cos(Δθ) distribution, the lines are fit lines.

      b(km)=μm1|km|μm2+bn,

      (15)

      b(jm)=νm1|jm|νm2+bn,

      (16)

      where m=1 or 2, for distinguishing the parameters of c1 or c2. Also n=1 or 2, then 1 is for the situation without the Coulomb interaction, and 2 refers to the Coulomb interaction. μ and ν are fit parameters, and the fitting results are shown in Table 1. The results show that the Coulomb interaction has an influence on these fit parameters, it can decrease |μm1| and |νm1|, and increase |μm2| and |νm2|. It is caused by the changing of the differences of b values. The differences in b values between the two situations are increasing with increasing collision energies. Compared to our previous work, the changing patterns are different, so these numerical connections are also related to the kinds of particles.

      n=1 n=2
      m=1 μ11 0.0004±0.0003 0.00004±0.00003
      μ12 3.3±0.5 4.7±0.4
      ν11 0.086±0.004 0.084±0.003
      ν12 0.68±0.09 0.94±0.08
      m=2 μ21 0.024±0.003 0.014±0.003
      μ22 2.3±0.3 3.2±0.3
      ν21 0.021±0.004 0.012±0.002
      ν22 2.4±0.3 3.3±0.3

      Table 1.  Fit results for parameters μ and ν.

      With these fit functions and parameters, the numerical connections between the normalized cos(Δθ) distribution and the strength of the transverse momentum dependence of Rs have been built. Comparing to our previous work, we found that the fit parameters are related to the impact parameter, the kinds of pions, and the Coulomb interaction. We improve the fit parameters, which makes our numerical correlation applied to the experiment closer. With further research, the experiment data can be directly used to study the single-particle space-momentum correlation.

    V.   CONCLUSIONS
    • With the string melting AMPT model, we calculated the HBT radius Rs for π+ in two situations, with and without Coulomb interaction. The results indicate that the Coulomb interaction can decrease the Rs values and inhibits the KT dependence of R. We compared the strength of KT dependence of R in two impact parameters and found that the impact parameters can also inhibit KT dependence of R. Then, we showed that the transverse flow can affect the normalized cos(Δθ) distributions, and the flow changes with the location of the source, which leads to particles with bigger Δθ angles tending to freeze out from the inner of the source. We also demonstrated that the normalized cos(Δθ) distributions can influence the KT dependence of Rs by disrupting the space and the momentum correlation. The inhibition of the transverse momentum dependence of Rs caused by Coulomb interaction leads to changes in the fit results. Different impact parameters and particles have different numerical connections. Moreover, with these numerical connections, we can get more information about the final stage of the Au+Au collision at the freeze-out time by the HBT analysis. With more collision energies, more impact parameters, and more accurate fits, the numerical connection can be improved. There are anomalies near the CEP, which may change this connection, so the connection improvement may be a probe to detect the CEP. The Coulomb interaction can weaken the strength of the transverse momentum dependence of Rs, it may also weaken the sensitivity of this probe.

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