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Study on the Resonant Parameters of Y (4220) and Y (4390).

In the last decade, charmonium physics has gained renewed strong interest from both the theoretical and the experimental side, due to the observation of a series of charmonium-like states, such as the X(3872) [1], the Y(4260) [2], and the Y(4360) [3]. These states do not fit in the conventional level system of charmonium states and are good candidates for exotic states not encompassed by the naive quark model [4]. Moreover, many charged charmonium-like states or their neutral partners [5] were observed, which might indicate the presence of new dynamics in this energy region.

Y(4260) is the first charmonium-like state, which was observed in the process [e.sup.+][e.sup.-] [right arrow] [[pi].sup.+][[pi].sup.-]J/[psi] by the BABAR experiment using an initial-state-radiation (ISR) technique [2]. This observation was immediately confirmed by the CLEO [6] and Belle experiments [7] in the same process. Being produced in [e.sup.+][e.sup.-] annihilation, the Y state has quantum numbers [J.sup.PC] = [1.sup.--]. Y(4360) is the second Y state, which was observed in the [e.sup.+][e.sup.-] [right arrow] [[gamma].sub.ISR]Y(4360) [right arrow] [[gamma].sub.ISR][[pi].sup.+][[pi].sup.-][psi](3686) by BABAR [3] and subsequently confirmed by Belle experiment [8]. Belle also observed another structure, Y(4660), in the [[pi].sup.+][[pi].sup.-][psi](3686) [8]. The observation of these Y states has stimulated substantial theoretical discussions on their nature [4].

Recently, with higher statistic data, the [e.sup.+][e.sup.-] [right arrow] [[pi].sup.+][[pi].sup.-]J/[psi] cross section was measured by BESIII experiment more precisely [9]. The fine structure was observed for Y(4260) in [e.sup.+][e.sup.-] [right arrow] [[pi].sup.+][[pi].sup.-]J/[psi]. The Y(4260) structure is a combination of two resonances: the lower one is Y(4220) and the higher is Y(4320). Using the results for [e.sup.+][e.sup.-] [right arrow] [[pi].sup.+][[pi].sup.-][psi](3686) from Belle [10], BABAR [11], and BESIII experiments [12], the authors of [13] also observed the fine structure for Y(4360) in [e.sup.+][e.sup.-] [right arrow] [[pi].sup.+][[pi].sup.-][psi](3686), inferring that the Y(4360) structure is also a combination of two resonances: the lower one is Y(4220) and the higher is Y(4360). The Y(4220) state also is observed in the processes [e.sup.+][e.sup.-] [right arrow] [omega][[chi].sub.c0] [14, 15], [[pi].sup.+][[pi].sup.-][h.sub.c] [16], and [[pi].sup.+][D.sup.0][D.sup.*-] + c.c. [17] by BESIII experiment. In the [e.sup.+][e.sup.-] [right arrow] [[pi].sup.+][[pi].sup.-][h.sub.c] and [[pi].sup.+][D.sup.0][D.sup.*-] + c.c., besides the Y(4220), another Y state Y(4390) is observed [16, 17]. The parameters for Y(4220), Y(4320), Y(4360), and Y(4390) states in different processes are listed in Table 1. In addition, the authors of [18] have performed a combine fit to the cross sections of [e.sup.+][e.sup.-] [right arrow] [omega][[chi].sub.c0], [[pi].sup.+][[pi].sup.-][h.sub.c], [[pi].sup.+][[pi].sup.-]J/[psi], and [[pi].sup.+][D.sup.0][D.sup.*-] + c.c. to obtain the resonant parameters for Y(4220), Y(4320), and Y(4390) states.

These states challenge the understanding of charmonium spectroscopy as well as QCD calculations [4, 19, 20]. According to potential models, there are five vector charmonium states between the 1D state [psi](3770) and 4.7 GeV/[c.sup.2], namely, the 3S, 2D, 4S, 3D, and 5S states [4]. Besides the three well-established structures observed in the inclusive hadronic cross section [21], i.e., [psi](4040), [psi](4160), and [psi](4415), five Y states, i.e., Y(4220), Y(4320), Y(4360), Y(4390), and Y(4660), have been observed. These newly observed Y states exceed the number of vector charmonium states predicted by potential models in this energy region. They are thus good candidates for exotic states, such as hybrid states, tetraquark states, and molecule states [5].

Figure 1 shows the cross sections of [e.sup.+][e.sup.-] [right arrow] [omega][[chi].sub.c0] [14, 15], [[pi].sup.+][[pi].sup.-][h.sub.c] [16, 22], [[pi].sup.+][[pi].sup.-]J/[psi] [9, 23, 24], [[pi].sup.+][[pi].sup.- ][psi](3686) [10-12], and [[pi].sup.+][D.sup.0][D.sup.*-] + c.c. [17] measured by Belle, BABAR, CLEO, and BESIII experiments. For data from BESIII, "XYZ" data sample refers to the energy points with integrated luminosity larger than 40 [pb.sup.-1] and "scan" data sample refers to the energy points with integrated luminosity smaller than 20 [pb.sup.-1]. In this paper, we perform a combined fit to these cross sections.

These vector charmonium-like states in the fit are assumed to be resonances. We parameterize the cross section with the coherent sum of a few amplitudes, either resonance represented by a Breit-Wigner (BW) function or nonresonant production term parameterized with a phase space function or an exponential function. The BW function used in this article is [18]

[mathematical expression not reproducible], (1)

where M and [GAMMA] are the mass and total width of the resonance, respectively; [mathematical expression not reproducible] is the partial width to [e.sup.+][e.sup.-], [B.sub.f] is the branching fraction of the resonance decays into final state f, and PS([square root of (s)]) is the phase space factor that increases smoothly from the mass threshold with the [square root of (s)] [21]. In the fit, [mathematical expression not reproducible] and [B.sub.f] can not be obtained separately; we can only extract the product [mathematical expression not reproducible].

Ref. [18] has performed a combine fit to the cross sections of [e.sup.+][e.sup.-] [right arrow] [omega][[chi].sub.c0], [[pi].sup.+][[pi].sup.-][h.sub.c], [[pi].sup.+][[pi].sup.-]J/[psi], and [[pi].sup.+][D.sup.0][D.sup.*-] + c.c., while the cross section of [e.sup.+][e.sup.-] [right arrow] [[pi].sup.+][[pi].sup.-][psi](3686) is not included. In Ref. [18], the resonances Y(4320) and Y(4390) are regarded as different states in the fit, while, from Table 1, we notice that the parameters for Y(4320), Y(4360), and Y(4390) are relatively close. Although there are some differences in the obtained mass and width in different channels, there are only a few data points with small errors around 4.4 GeV. It is not reasonable that there are three states in such a close position. In addition, the analysis in [25] also indicates that the charmonium-like states Y(4360) in the [[pi].sup.+][[pi].sup.- ][psi](3686) and Y(4320) in the [[pi].sup.+][[pi].sup.-]J/[psi] should be the same state. Therefore, we consider Y(4320), Y(4360), and Y(4390) as the same state, which has been suggested in [26]. The same state is marked as "Y(4390)" in this paper. A least [chi square] fit method is used to perform a combined fit to the five cross sections using three resonances Y(4220), Y(4390), and Y(4660), assuming the two resonances Y(4220) and Y(4390) are the same two states in these processes. The fit functions are

[mathematical expression not reproducible], (2)

[mathematical expression not reproducible], (3)

[mathematical expression not reproducible], (4)

[mathematical expression not reproducible], (5)

[mathematical expression not reproducible], (6)

where [BW.sub.1], [BW.sub.2], and [BW.sub.3] denote the resonances Y(4220), Y(4390), and Y(4660), respectively; PS([square root of (s)]) is the phase space factor; [mathematical expression not reproducible] is an exponential function, where [p.sub.0] is free parameter, [M.sub.th] = 2[m.sub.[pi]] + [m.sub.J/[psi]] is the mass threshold of the [[pi].sup.+][[pi].sup.-]J/[psi] system; [[phi].sub.1], [[phi].sub.2], [[phi].sub.3], [[phi].sub.4], [[phi].sub.5], [[phi].sub.6], and [[phi].sub.7] are relative phases; [c.sub.1] and [c.sub.2] are amplitudes of exponential function term and phase space term.

We fit to the cross sections of [e.sup.+][e.sup.-] [right arrow] [omega][[chi].sub.c0], [[pi].sup.+][[pi].sup.-][h.sub.c], [[pi].sup.+][[pi].sup.-]J/[psi], [[pi].sup.+][[pi].sup.-][psi](3686), and [[pi].sup.+][D.sup.0][D.sup.*-] + c.c. simultaneously. The fits for [e.sup.+][e.sup.-] [right arrow] [omega][[chi].sub.c0], [[pi].sup.+][[pi].sup.-][h.sub.c], [[pi].sup.+][[pi].sup.-]J/[psi], [[pi].sup.+][[pi].sup.-][psi](3686), and [[pi].sup.+][D.sup.0][D.sup.*-] + c.c. are found to have one solution, two solutions, four solutions, four solutions, and four solutions with the same minimum values of [chi square], respectively. The masses and widths of the resonances are identical, but the [mathematical expression not reproducible] vary with the different solutions for each process.

Figure 2 shows the fit results with a goodness of the fit being [chi square]/ndf = 460/474 = 0.97, corresponding to a confidence level of 67%. Thegood fit indicates that the assumption that the two resonances Y(4220) and Y(4390) are same two states in these processes is reasonable. From fit results, we can get [M.sub.Y(4220)] = (4216.5 [+ or -] 1.4) MeV/[c.sup.2], [[GAMMA].sub.Y(4220)] = (61.1 [+ or -] 2.3) MeV; [M.sub.Y(4390)] = (4383.5 [+ or -] 1.9) MeV/[c.sup.2], [[GAMMA].sub.Y(4390)] = (114.5 [+ or -] 5.4) MeV; [M.sub.Y(4660)] = (4623.4 [+ or -] 10.5) MeV/[c.sup.2], [[GAMMA].sub.Y(4660)] = (106.1 [+ or -] 16.2) MeV. The all obtained resonant parameters from fit are listed in Table 2.

From the fit results, the obtained parameters of Y(4660) are quite different from Belle's results [10]. There are two main reasons. One is that the interference between Y(4390) and Y(4660) has large influence on Y(4660)'s parameters. We can see the obtained combined Y(4390)'s parameters are very different from the Y(4360)'s parameters from Belle's results; it will lead to the fact that Y(4660)'s parameters are also different. Another is that the data point at 4.6 GeV from BESIII has very small error, so the fitted Y(4660)'s BW curve is influenced greatly by this data point. From Figure 2, we can see that, in order to cover the data point, the Y(4660)'s BW curve has to have some deviations from Belle data points around 4.66 GeV.

The systematic uncertainties on the resonant parameters in the combined fit to the cross sections of [e.sup.+][e.sup.-] [right arrow] [omega][[chi].sub.c0], [[pi].sup.+][[pi].sup.-][h.sub.c], [[pi].sup.+][[pi].sup.-]J/[psi], [[pi].sup.+][[pi].sup.-][psi](3686), and [[pi].sup.+][D.sup.0][D.sup.*-] + c.c. are mainly from the uncertainties of the center-of-mass energy determination, parametrization of the BW function, background shape, and the cross section measurements.

Since the uncertainty of the beam energy is about 0.8 MeV at BESIII, so the uncertainty of the resonant parameters caused by the beam energy is estimated by varying [square root of s] within 0.8 MeV for BESIII data. Instead of using a constant total width, we assume an energy dependent width to estimate the uncertainty due to parametrization of BW function. To model the [e.sup.+][e.sup.-] [right arrow] [[pi].sup.+][[pi].sup.-]J/[psi] cross section near 4 GeV, a BW function is used to replace the exponential function, and the differences of the fit results in the two methods are taken as the uncertainty from background shape. The uncertainty of the cross section measurements will affect the resonant parameters in fit, we vary the cross sections within the systematic uncertainty, and the differences in the final results are taken as the uncertainty. By assuming all these sources of systematic uncertainties are independent, we add them in quadrature. The systematic uncertainty from the parametrization of the BWfunction for the parameters mass and width is dominant, while the systematic uncertainty from the cross section measurements for the parameter [mathematical expression not reproducible] is dominant.

Theleptonic decay width for a vector state is an important quantity for discriminating various theoretical models [27-29]. By considering the isospin symmetric modes of the measured channels, we can estimate the lower limits on the leptonic partial width of the Y(4220) and Y(4390) decays. For an isospin-zero charmonium-like state, we expect

[mathematical expression not reproducible], (7)

so we have

[mathematical expression not reproducible] (8)

and

[mathematical expression not reproducible] (9)

By inserting the numbers from Table 2, considering the solutions with the smallest [mathematical expression not reproducible], we obtain

[mathematical expression not reproducible], (10)

and

[mathematical expression not reproducible], (11)

where the first uncertainties are statistical, and the second systematic.

On the other hand, if we take the results with the largest [mathematical expression not reproducible] and in Table 2, we obtain [mathematical expression not reproducible]. This means that the leptonic partial widths of Y(4220) and Y(4390) can be as large as 200 and 1000 eV or even higher based on current information, because maybe there are some other decay channels for Y(4220) and Y(4390) that we have not observed.

In summary, a combined fit is performed to the cross sections of [e.sup.+][e.sup.-] [right arrow] [omega][[chi].sub.c0], [[pi].sup.+][[pi].sup.-][h.sub.c], [[pi].sup.+][[pi].sup.-]J/[psi], [[pi].sup.+][[pi].sup.-][psi](3686), and [[pi].sup.+][D.sup.0][D.sup.*-] + c.c. by using three resonances Y(4220), Y(4390), and Y(4660). The parameters are determined to be [M.sub.Y(4220)] = (4216.5 [+ or -] 1.4 [+ or -] 3.2) MeV/[c.sup.2], [[GAMMA].sub.Y(4220)] = (61.1 [+ or -] 2.3 [+ or -] 3.1) MeV; [M.sub.Y(4390)] = (4383.5 [+ or -] 1.9 [+ or -] 6.0) MeV/[c.sup.2], [[GAMMA].sub.Y(4390)] = (114.5 [+ or -] 5.4 [+ or -] 9.9) MeV; [M.sub.Y(4660)] = (4623.4 [+ or -] 10.5 [+ or -] 16.1) MeV/[c.sup.2], [[GAMMA].sub.Y(4660)] = (106.1 [+ or -] 16.2 [+ or -] 17.5) MeV, where the first uncertainties are statistical and the second systematic. We emphasize that two resonances Y(4220) and Y(4390) are sufficient to explain these cross sections below 4.6 GeV. The resonances Y(4320), Y(4360), and Y(4390) should be one state. The lower limits of Y(4220) and Y(4390)'s leptonic decay widths are also determined to be (36.4 [+ or -] 2.0 [+ or -] 4.2) and (123.8 [+ or -] 6.5 [+ or -] 9.0) eV. These results will be useful in understanding the nature of charmonium-like states in this energy region. Higher precision measurements around this energy region are desired, and this can be achieved in BESIII and BelleII experiments in the further.

https://doi.org/10.1155/2018/5428734

Data Availability

All the data used in this work are from Ref. [9-12, 14-17, 22-24].

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the Foundation of Henan Educational Committee (No. 19A140015), Nanhu Scholars Program for Young Scholars of Xinyang Normal University, and Open Research Program of Large Research Infrastructures (2017), Chinese Academy of Sciences.

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Jielei Zhang (iD), Limin Yuan, and Rumin Wang

College of Physics and Electronic Engineering, Xinyang Normal University, Xinyang 464000, China

Correspondence should be addressed to Jielei Zhang; zhangjielei@ihep.ac.cn

Received 31 July 2018; Revised 7 October 2018; Accepted 30 October 2018; Published 18 November 2018

Guest Editor: Tao Luo

Caption: Figure 1: Cross sections of [e.sup.+][e.sup.-] [right arrow] [omega][[chi].sub.c0], [[pi].sup.+][[pi].sup.-][h.sub.c], [[pi].sup.+][[pi].sup.-]J/[psi], [[pi].sup.+][[pi].sup.-][psi](3686), and [[pi].sup.+][D.sup.0][D.sup.*-] + c.c. measured by Belle, BABAR, CLEO, and BESIII experiments.

Caption: Figure 2: The results of the combined fit to the cross sections of [e.sup.+][e.sup.-] [right arrow] [omega][[chi].sub.c0], [[pi].sup.+][[pi].sup.-][h.sub.c], [[pi].sup.+][[pi].sup.-]J/[psi], [[pi].sup.+][[pi].sup.-][psi](3686), and [[pi].sup.+][D.sup.0][D.sup.*-] + c.c. (from the top to the bottom row). The solid red curves show the best fits, and the dashed green ones are individual components.
Table 1: The parameters for Y(4220) ([omega][[chi].sub.c0],
[[pi].sup.+][[pi].sup.-][h.sub.c], [[pi].sup.+][[pi].sup.-]J/[psi],
[[pi].sup.+][[pi].sup.-][psi](3686) and [[pi].sup.+][D.sup.0][D.sup.*-]
+ c.c.), Y(4320) ([[pi].sup.+][[pi].sup.-]J/[psi]), Y(4360)
([[pi].sup.+][[pi].sup.-][psi] (3686)), and Y(4390)
([[pi].sup.+][[pi].sup.-][h.sub.c] and [[pi].sup.+][D.sup.0][D.sup.*-]
+ c.c..) states in different processes. The first uncertainties are
statistical, and the second systematic.

                                                     Y(4220)

                                                M (MeV/[c.sup.2])

[omega][[chi].sub.c0] [15]                       4226 [+ or -] 8
                                                   [+ or -] 6
[[pi].sup.+][[pi].sup.-][h.sub.c] [16]     [4218.4.sup.+5.5.sub.-4.5]
                                                  [+ or -] 0.9
[[pi].sup.+][[pi].sup.-]J/[psi] [9]            4222.0 [+ or -] 3.1
                                                  [+ or -] 1.4
[[pi].sup.+][[pi].sup.-][psi](3686) [13]       4209.1 [+ or -] 6.8
                                                  [+ or -] 7.0
[[pi].sup.+][D.sup.0][D.sup.*-] +              4224.8 [+ or -] 5.6
c.c. [17]                                         [+ or -] 4.0

                                                     Y(4220)

                                                  [GAMMA] (MeV)

[omega][[chi].sub.c0] [15]                  39 [+ or -] 12 [+ or -] 2

[[pi].sup.+][[pi].sup.-][h.sub.c] [16]      [66.0.sup.+12.3.sub.-8.3]
                                                  [+ or -] 0.4
[[pi].sup.+][[pi].sup.-]J/[psi] [9]             44.1 [+ or -] 4.3
                                                  [+ or -] 2.0
[[pi].sup.+][[pi].sup.-][psi](3686) [13]       76.6 [+ or -] 14.2
                                                  [+ or -] 2.4
[[pi].sup.+][D.sup.0][D.sup.*-] +               72.3 [+ or -] 9.1
c.c. [17]                                         [+ or -] 0.9

                                             Y(4320)/Y(4360)/Y(4390)

                                               M (MeV/[c.sup.2])

[omega][[chi].sub.c0] [15]

[[pi].sup.+][[pi].sup.-][h.sub.c] [16]     [4391.5.sup.+6.3.sub.-6.8]
                                                  [+ or -] 1.0
[[pi].sup.+][[pi].sup.-]J/[psi] [9]           4320.0 [+ or -] 10.4
                                                  [+ or -] 7.0
[[pi].sup.+][[pi].sup.-][psi](3686) [13]       4383.7 [+ or -] 2.9
                                                  [+ or -] 6.2
[[pi].sup.+][D.sup.0][D.sup.*-] +              4400.1 [+ or -] 9.3
c.c. [17]                                         [+ or -] 2.1

                                             Y(4320)/Y(4360)/Y(4390)

                                                  [GAMMA] (MeV)

[omega][[chi].sub.c0] [15]

[[pi].sup.+][[pi].sup.-][h.sub.c] [16]     [139.5.sup.+16.2.sub.-20.6]
                                                  [+ or -] 0-6
[[pi].sup.+][[pi].sup.-]J/[psi] [9]        [101.4.sup.+25.3.sub.19.7]
                                                  [+ or -] 10.2
[[pi].sup.+][[pi].sup.-][psi](3686) [13]        94.2 [+ or -] 7.3
                                                  [+ or -] 2.0
[[pi].sup.+][D.sup.0][D.sup.*-] +              181.7 [+ or -] 16.9
c.c. [17]                                         [+ or -] 7.4

Table 2: The fitted parameters from the combined fit to the cross
sections of [e.sup.+][e.sup.-] [right arrow] [omega][[chi].sub.c0],
[[pi].sup.+][[pi].sup.-][h.sub.c], [[pi].sup.+][[pi].sup.-]J/[psi],
[[pi].sup.+][[pi].sup.-][psi] (3686), and
[[pi].sup.+][D.sup.0][D.sup.*-] + c.c. The first uncertainties are
statistical, and the second systematic.

(a)

Parameter                 Y(4220)               Y(4390)

M (MeV/[c.sup.2])   4216.5 [+ or -] 1.4   4383.5 [+ or -] 1.9
                       [+ or -] 3.2          [+ or -] 6.0
[GAMMA] (MeV)        61.1 [+ or -] 2.3    114.5 [+ or -] 5.4
                       [+ or -] 3.1          [+ or -] 9.9

Parameter                 Y(4660)

M (MeV/[c.sup.2])   4623.4 [+ or -] 10.5
                       [+ or -] 16.1
[GAMMA] (MeV)        106.1 [+ or -] 16.2
                       [+ or -] 17.5

(b)

Parameter                SolutionI            SolutionII

[mathematical        3.5 [+ or -] 0.4
expression not         [+ or -] 0.5
reproducible]
[mathematical        6.5 [+ or -] 0.5      3.1 [+ or -] 0.2
expression not         [+ or -] 1.1          [+ or -] 0.8
reproducible]
[mathematical        15.1 [+ or -] 1.0     7.5 [+ or -] 0.6
expression not         [+ or -] 2.8          [+ or -] 1.8
reproducible]
[mathematical        10.5 [+ or -] 0.5     12.3 [+ or -] 0.7
expression not         [+ or -] 1.7          [+ or -] 2.1
reproducible]
[mathematical        0.3 [+ or -] 0.1      12.1 [+ or -] 0.7
expression not         [+ or -] 0.1          [+ or -] 3.2
reproducible]
[mathematical        1.6 [+ or -] 0.3      1.5 [+ or -] 0.3
expression not         [+ or -] 0.3          [+ or -] 0.3
reproducible]
[mathematical        13.4 [+ or -] 1.1     9.9 [+ or -] 1.0
expression not         [+ or -] 1.4          [+ or -] 1.2
reproducible]
[mathematical        8.8 [+ or -] 1.2      3.0 [+ or -] 0.5
expression not         [+ or -] 1.4          [+ or -] 0.6
reproducible]
[mathematical        39.0 [+ or -] 2.5     7.1 [+ or -] 0.6
expression not         [+ or -] 3.1          [+ or -] 1.3
reproducible]
[mathematical        55.4 [+ or -] 5.7     32.4 [+ or -] 2.1
expression not         [+ or -] 7.8          [+ or -] 2.8
reproducible]

Parameter                SolutionIII            SolutionIV

[mathematical
expression not
reproducible]
[mathematical
expression not
reproducible]
[mathematical
expression not
reproducible]
[mathematical         3.7 [+ or -] 0.3       3.1 [+ or -] 0.3
expression not          [+ or -] 0.6           [+ or -] 0.6
reproducible]
[mathematical         10.4 [+ or -] 0.6      0.3 [+ or -] 0.1
expression not          [+ or -] 2.3           [+ or -] 0.1
reproducible]
[mathematical         1.6 [+ or -] 0.3       1.5 [+ or -] 0.3
expression not          [+ or -] 0.3           [+ or -] 0.3
reproducible]
[mathematical         13.4 [+ or -] 1.1      9.9 [+ or -] 1.0
expression not          [+ or -] 1.4           [+ or -] 1.2
reproducible]
[mathematical         8.8 [+ or -] 1.2       3.0 [+ or -] 0.5
expression not          [+ or -] 1.4           [+ or -] 0.6
reproducible]
[mathematical         57.5 [+ or -] 3.0     10.5 [+ or -] 1.1
expression not          [+ or -] 6.1           [+ or -] 2.7
reproducible]
[mathematical        313.6 [+ or -] 13.9   183.1 [+ or -] 11.2
expression not          [+ or -] 26.4         [+ or -] 19.3
reproducible]
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Article Details
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Title Annotation:Research Article
Author:Zhang, Jielei; Yuan, Limin; Wang, Rumin
Publication:Advances in High Energy Physics
Article Type:Report
Geographic Code:1USA
Date:Jan 1, 2018
Words:4606
Previous Article:A Probability Distribution for Quantum Tunneling Times.
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