Measurement of the e^{+}e^{-}\to D^{0}D^{*-}\pi^{+} cross section using initial-state radiation


Measurement of the  cross section using initial-state radiation

G. Pakhlova Institute for Theoretical and Experimental Physics, Moscow    H. Aihara Department of Physics, University of Tokyo, Tokyo    K. Arinstein Budker Institute of Nuclear Physics, Novosibirsk Novosibirsk State University, Novosibirsk    T. Aushev École Polytechnique Fédérale de Lausanne (EPFL), Lausanne Institute for Theoretical and Experimental Physics, Moscow    A. M. Bakich School of Physics, University of Sydney, NSW 2006    V. Balagura Institute for Theoretical and Experimental Physics, Moscow    E. Barberio University of Melbourne, School of Physics, Victoria 3010    A. Bay École Polytechnique Fédérale de Lausanne (EPFL), Lausanne    K. Belous Institute of High Energy Physics, Protvino    V. Bhardwaj Panjab University, Chandigarh    M. Bischofberger Nara Women’s University, Nara    A. Bondar Budker Institute of Nuclear Physics, Novosibirsk Novosibirsk State University, Novosibirsk    A. Bozek H. Niewodniczanski Institute of Nuclear Physics, Krakow    M. Bračko University of Maribor, Maribor J. Stefan Institute, Ljubljana    T. E. Browder University of Hawaii, Honolulu, Hawaii 96822    P. Chang Department of Physics, National Taiwan University, Taipei    A. Chen National Central University, Chung-li    B. G. Cheon Hanyang University, Seoul    R. Chistov Institute for Theoretical and Experimental Physics, Moscow    I.-S. Cho Yonsei University, Seoul    S.-K. Choi Gyeongsang National University, Chinju    Y. Choi Sungkyunkwan University, Suwon    J. Dalseno Max-Planck-Institut für Physik, München Excellence Cluster Universe, Technische Universität München, Garching    M. Danilov Institute for Theoretical and Experimental Physics, Moscow    M. Dash IPNAS, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061    A. Drutskoy University of Cincinnati, Cincinnati, Ohio 45221    W. Dungel Institute of High Energy Physics, Vienna    S. Eidelman Budker Institute of Nuclear Physics, Novosibirsk Novosibirsk State University, Novosibirsk    D. Epifanov Budker Institute of Nuclear Physics, Novosibirsk Novosibirsk State University, Novosibirsk    M. Feindt Institut für Experimentelle Kernphysik, Universität Karlsruhe, Karlsruhe    N. Gabyshev Budker Institute of Nuclear Physics, Novosibirsk Novosibirsk State University, Novosibirsk    A. Garmash Budker Institute of Nuclear Physics, Novosibirsk Novosibirsk State University, Novosibirsk    P. Goldenzweig University of Cincinnati, Cincinnati, Ohio 45221    B. Golob Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana J. Stefan Institute, Ljubljana    H. Ha Korea University, Seoul    J. Haba High Energy Accelerator Research Organization (KEK), Tsukuba    Y. Hasegawa Shinshu University, Nagano    K. Hayasaka Nagoya University, Nagoya    H. Hayashii Nara Women’s University, Nara    Y. Horii Tohoku University, Sendai    Y. Hoshi Tohoku Gakuin University, Tagajo    W.-S. Hou Department of Physics, National Taiwan University, Taipei    H. J. Hyun Kyungpook National University, Taegu    T. Iijima Nagoya University, Nagoya    K. Inami Nagoya University, Nagoya    R. Itoh High Energy Accelerator Research Organization (KEK), Tsukuba    M. Iwasaki Department of Physics, University of Tokyo, Tokyo    Y. Iwasaki High Energy Accelerator Research Organization (KEK), Tsukuba    T. Julius University of Melbourne, School of Physics, Victoria 3010    D. H. Kah Kyungpook National University, Taegu    J. H. Kang Yonsei University, Seoul    H. Kawai Chiba University, Chiba    T. Kawasaki Niigata University, Niigata    H. Kichimi High Energy Accelerator Research Organization (KEK), Tsukuba    C. Kiesling Max-Planck-Institut für Physik, München    H. O. Kim Kyungpook National University, Taegu    J. H. Kim Sungkyunkwan University, Suwon    S. K. Kim Seoul National University, Seoul    Y. I. Kim Kyungpook National University, Taegu    Y. J. Kim The Graduate University for Advanced Studies, Hayama    K. Kinoshita University of Cincinnati, Cincinnati, Ohio 45221    B. R. Ko Korea University, Seoul    S. Korpar University of Maribor, Maribor J. Stefan Institute, Ljubljana    P. Križan Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana J. Stefan Institute, Ljubljana    P. Krokovny High Energy Accelerator Research Organization (KEK), Tsukuba    T. Kuhr Institut für Experimentelle Kernphysik, Universität Karlsruhe, Karlsruhe    R. Kumar Panjab University, Chandigarh    T. Kumita Tokyo Metropolitan University, Tokyo    A. Kuzmin Budker Institute of Nuclear Physics, Novosibirsk Novosibirsk State University, Novosibirsk    Y.-J. Kwon Yonsei University, Seoul    S.-H. Kyeong Yonsei University, Seoul    S.-H. Lee Korea University, Seoul    T. Lesiak H. Niewodniczanski Institute of Nuclear Physics, Krakow T. Kościuszko Cracow University of Technology, Krakow    J. Li University of Hawaii, Honolulu, Hawaii 96822    C. Liu University of Science and Technology of China, Hefei    D. Liventsev Institute for Theoretical and Experimental Physics, Moscow    R. Louvot École Polytechnique Fédérale de Lausanne (EPFL), Lausanne    A. Matyja H. Niewodniczanski Institute of Nuclear Physics, Krakow    S. McOnie School of Physics, University of Sydney, NSW 2006    T. Medvedeva Institute for Theoretical and Experimental Physics, Moscow    K. Miyabayashi Nara Women’s University, Nara    H. Miyata Niigata University, Niigata    Y. Miyazaki Nagoya University, Nagoya    R. Mizuk Institute for Theoretical and Experimental Physics, Moscow    T. Müller Institut für Experimentelle Kernphysik, Universität Karlsruhe, Karlsruhe    Y. Nagasaka Hiroshima Institute of Technology, Hiroshima    E. Nakano Osaka City University, Osaka    M. Nakao High Energy Accelerator Research Organization (KEK), Tsukuba    S. Nishida High Energy Accelerator Research Organization (KEK), Tsukuba    K. Nishimura University of Hawaii, Honolulu, Hawaii 96822    O. Nitoh Tokyo University of Agriculture and Technology, Tokyo    T. Ohshima Nagoya University, Nagoya    S. Okuno Kanagawa University, Yokohama    S. L. Olsen Seoul National University, Seoul    P. Pakhlov Institute for Theoretical and Experimental Physics, Moscow    C. W. Park Sungkyunkwan University, Suwon    H. Park Kyungpook National University, Taegu    H. K. Park Kyungpook National University, Taegu    R. Pestotnik J. Stefan Institute, Ljubljana    L. E. Piilonen IPNAS, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061    A. Poluektov Budker Institute of Nuclear Physics, Novosibirsk Novosibirsk State University, Novosibirsk    Y. Sakai High Energy Accelerator Research Organization (KEK), Tsukuba    O. Schneider École Polytechnique Fédérale de Lausanne (EPFL), Lausanne    C. Schwanda Institute of High Energy Physics, Vienna    K. Senyo Nagoya University, Nagoya    M. Shapkin Institute of High Energy Physics, Protvino    V. Shebalin Budker Institute of Nuclear Physics, Novosibirsk Novosibirsk State University, Novosibirsk    C. P. Shen University of Hawaii, Honolulu, Hawaii 96822    J.-G. Shiu Department of Physics, National Taiwan University, Taipei    B. Shwartz Budker Institute of Nuclear Physics, Novosibirsk Novosibirsk State University, Novosibirsk    J. B. Singh Panjab University, Chandigarh    A. Sokolov Institute of High Energy Physics, Protvino    E. Solovieva Institute for Theoretical and Experimental Physics, Moscow    S. Stanič University of Nova Gorica, Nova Gorica    M. Starič J. Stefan Institute, Ljubljana    T. Sumiyoshi Tokyo Metropolitan University, Tokyo    G. N. Taylor University of Melbourne, School of Physics, Victoria 3010    Y. Teramoto Osaka City University, Osaka    I. Tikhomirov Institute for Theoretical and Experimental Physics, Moscow    K. Trabelsi High Energy Accelerator Research Organization (KEK), Tsukuba    S. Uehara High Energy Accelerator Research Organization (KEK), Tsukuba    T. Uglov Institute for Theoretical and Experimental Physics, Moscow    Y. Unno Hanyang University, Seoul    S. Uno High Energy Accelerator Research Organization (KEK), Tsukuba    P. Urquijo University of Melbourne, School of Physics, Victoria 3010    Y. Usov Budker Institute of Nuclear Physics, Novosibirsk Novosibirsk State University, Novosibirsk    G. Varner University of Hawaii, Honolulu, Hawaii 96822    K. E. Varvell School of Physics, University of Sydney, NSW 2006    K. Vervink École Polytechnique Fédérale de Lausanne (EPFL), Lausanne    A. Vinokurova Budker Institute of Nuclear Physics, Novosibirsk Novosibirsk State University, Novosibirsk    C. H. Wang National United University, Miao Li    P. Wang Institute of High Energy Physics, Chinese Academy of Sciences, Beijing    X. L. Wang Institute of High Energy Physics, Chinese Academy of Sciences, Beijing    Y. Watanabe Kanagawa University, Yokohama    R. Wedd University of Melbourne, School of Physics, Victoria 3010    E. Won Korea University, Seoul    B. D. Yabsley School of Physics, University of Sydney, NSW 2006    Y. Yamashita Nippon Dental University, Niigata    C. Z. Yuan Institute of High Energy Physics, Chinese Academy of Sciences, Beijing    C. C. Zhang Institute of High Energy Physics, Chinese Academy of Sciences, Beijing    Z. P. Zhang University of Science and Technology of China, Hefei    V. Zhilich Budker Institute of Nuclear Physics, Novosibirsk Novosibirsk State University, Novosibirsk    V. Zhulanov Budker Institute of Nuclear Physics, Novosibirsk Novosibirsk State University, Novosibirsk    T. Zivko J. Stefan Institute, Ljubljana    A. Zupanc J. Stefan Institute, Ljubljana    O. Zyukova Budker Institute of Nuclear Physics, Novosibirsk Novosibirsk State University, Novosibirsk
Abstract

We report measurements of the exclusive cross section for  as a function of center-of-mass energy from the  threshold to 5.2 with initial-state radiation. No evidence is found for decays. The analysis is based on a data sample collected with the Belle detector at or near a center-of-mass energy of 10.58 with an integrated luminosity of at the KEKB asymmetric-energy  collider.

pacs:
13.66.Bc,13.87.Fh,14.40.Lb

The Belle Collaboration

Studies of exclusive open charm production near threshold in  annihilation provide important information on the dynamics of charm quarks and on the properties of the states. During the past three years numerous measurements of exclusive  cross sections for charmed hadron pairs have been reported. Most of these measurements were performed at -factories using initial-state radiation (ISR). Belle presented the first results on the  cross sections to the ,  chcon (), ,  (including the first observation of  decays) belle:dd (); belle:dst (); belle:4415 () and  final states belle:x4630 (). BaBar measured  cross sections to  and recently to the ,  final states babar:dd (); babar:dd_new (). CLEO-c performed a scan over the energy range from 3.97 to 4.26 and measured exclusive cross sections for the ,  and  final states at thirteen points with high accuracy cleo:cs (). The measured open charm final states nearly saturate the total cross section for charm hadron production in  annihilation in the  region up to . In the energy range above some room for contributions to the  state from unmeasured channels still remains. The exclusive cross sections for charm strange meson pairs have been measured to be an order of magnitude smaller than charm meson production cleo:cs (). Charm baryon-antibaryon pair production occurs at energies above .

Another motivation for studying exclusive open charm production is the existence of a mysterious family of charmonium-like states with masses above open-charm threshold and quantum numbers . Although these have been known for over four years, the nature of these states, found in processes, remains unclear. Among them are the  state observed by BaBar babar:y4260 (); babar:y4260_08 (), confirmed by CLEO cleo:y4260_isr (); cleo:y4260_scan () and Belle belle:y4260 (); the discovered by BaBar babar:y4350 () and confirmed by Belle belle:y4350 (); and two structures, the and the seen by Belle belle:y4260 (); belle:y4350 ().

No clear evidence for open charm production associated with any of these states has been observed. In fact the  peak position appears to be close to a local minimum of both the total hadronic cross section bes:cs () and of the exclusive cross section for  belle:dst (); babar:dd_new (). The , recently found in the  cross section as a near-threshold enhancement belle:x4630 (), has a mass and width (assuming the to be a resonance) consistent within errors with those of the , supporting explanation that the is  x4630:bugg () or bound state x4630:molecule (). However, this coincidence does not exclude other interpretations of the , for example, as a conventional charmonium state x4630:charm () or as baryon-antibaryon threshold effect x4630:thresh (), point-like baryons x4630:point (), or as a tetraquark state x4630:tetra ().

The absence of open charm decay channels for states, large partial widths for decay channels to charmonium plus light hadrons and the lack of available charmonium levels are inconsistent with the interpretation of the states as conventional charmonia. To explain the observed peaks, some models assign the , with shifted masses y:ding (), other explore coupled-channel effects and rescattering of charm mesons voloshin:rescattering (). More exotic suggestions include hadro-charmonium hadroch (); multiquark states, such as a tetraquark y4260:tetra () and or  molecules y4260:molecule (). One of the most popular exotic options for the states are the hybrids expected by LQCD in the mass range from  y4260:hybryds (). In this context, some authors expect the dominant decay channels of the Y(4260) to be .

In this paper we report a measurement of the exclusive  cross section as a function of center-of-mass energy from the  threshold to 5.2, as part of our studies of the exclusive open-charm production in this mass range. The analysis is based on a data sample collected with the Belle detector det () at the resonance and nearby continuum with an integrated luminosity of at the KEKB asymmetric-energy  collider kekb ().

We employ the reconstruction method that was used for  and  exclusive cross section measurements belle:dd (); belle:4415 (). We select  signal events by reconstructing the , and mesons. In general the  is not required to be detected: instead, its presence in the event is inferred from a peak at zero in the spectrum of recoil mass squared against the  system. The square of the recoil mass is defined as:

(1)

Here  is the initial  center-of-mass () energy, and are the energy and momentum of the  combination, respectively. To suppress backgrounds two cases are considered: (1) the  is outside of the detector acceptance and the polar angle for the  combination in the c.m. frame is required to be ; (2) the fast  is within the detector acceptance (), in which case it is required to be detected and the mass of the combination must be greater than (). To suppress background from processes we exclude events that contain additional charged tracks that are not used in , or reconstruction.

All charged tracks are required to originate from the vicinity of the interaction point (IP); we impose the requirements and , where and are the impact parameters perpendicular to and along the beam direction with respect to the IP. Charged kaons are required to have a ratio of particle identification likelihood, , larger than 0.6 nim (). Charged tracks not identified as kaons are assumed to be pions.

candidates are reconstructed from pairs with an invariant mass within of the mass. The distance between the two pion tracks at the vertex must be less than , the transverse flight distance from the IP is required to be greater than , and the angle between the momentum direction and the flight direction in the plane should be smaller than .

Photons are reconstructed from showers in the electromagnetic calorimeter with energies greater than that are not associated with charged tracks. ISR photon candidates are required to have energies greater than . Pairs of photons are combined to form candidates. If the mass of a pair lies within of the mass, the pair is fitted with a mass constraint and considered as a candidate.

candidates are reconstructed using five decay modes: , , , and . A mass window is used for all modes except for where a requirement is applied ( in each case). candidates are reconstructed using and decay modes bckg (); a mass window is used for both modes. To improve the momentum resolution of meson candidates, final tracks are fitted to a common vertex with a mass constraint on the or mass. candidates are selected via the and (for background study) decay modes with a mass-difference window (). A mass- and vertex-constrained fit is also applied to candidates.

To remove contributions from the process, we exclude  combinations with invariant mass within of the nominal mass.

, , and mass sidebands are selected for the background study; these are four times as large as the signal region and are subdivided into windows of the same width as the signal. To avoid signal over-subtraction, the selected sidebands are shifted by ( for the mode) from the signal region. The candidates from these sidebands are refitted to the central mass value of each window. sidebands are shifted by to the higher mass side of the signal region.

The distribution of for the signal region in the data for is shown in Fig. 1 a). A clear peak corresponding to the  process is evident around zero. The shoulder at positive values is due to events. We define the signal region for by a tight requirement around zero to suppress the tail from such events. The invariant-mass distribution of combinations in the data after the requirement on and the polar angle distribution of  combinations shown in Fig. 1 b), c) are typical of ISR production and are in agreement with the MC simulation.

Figure 1: a) The distribution of . b) The mass spectrum of combinations. c) The polar angle distribution of  combinations. Histograms show the normalized  and  sideband contributions. The selected signal windows are illustrated by vertical dotted lines.

The  spectrum obtained after all the requirements is shown in Fig. 2.

Figure 2: The  spectrum. The histogram shows the normalized  and  sideband contributions.

The contribution of multiple entries after all the requirements is found to be less than . In such case the single  combination with the minimum value of is chosen, where and correspond to the mass fits for and candidates.

The following sources of background are considered:

  • combinatorial background under the () peak combined with a real () coming from the signal or other processes;

  • both and are combinatorial;

  • the reflection from the processes , where the is not reconstructed, including decays;

  • the reflection from the process , followed by , where the low-momentum is not reconstructed;

  •  where the energetic is misidentified as a single .

The contribution of background (1) is extracted using the and sidebands. Background (2) is present in both the  and  sidebands and is, thus, subtracted twice. To take into account this over-subtraction we use a two-dimensional sideband region, when events are selected from both the  and the  sidebands. The total contribution from the combinatorial backgrounds (1–2) is shown in Figs. 1, 2 as a hatched histogram.

Most of the background (3–4) events are suppressed by the tight requirement on . The remainder of background (3) is estimated directly from the data by applying a similar reconstruction method to the isospin-conjugate process . Since there is a charge imbalance in the  final state, only events with a missing extra can contribute to the signal window. To extract the level of background (3), the  mass spectrum is rescaled according to the ratio of and reconstruction efficiencies and an isospin factor of 1/2. A negligibly small contribution of background (3) is found: only one event with . Uncertainties in this estimate are included in the systematic error. The remainder of background (4) is estimated from the data assuming isospin symmetry. We measure the process () by applying a similar reconstruction method. Only three events with are found in the data. Thus the contribution of background (4) is also found to be negligibly small; uncertainties in this estimate are included in the systematic error.

The contribution of background (5) is determined from the data using fully reconstructed  events including the reconstruction of an energetic . Only one event with and is found in the data. Assuming a uniform polar angle distribution, this background contribution to the signal sub-sample (case 1) is 1 event/9 events in the entire  mass range, where is the reconstruction efficiency. The probability of misidentification due to asymmetric decays is also estimated to be . Thus the contribution of background (5) is found to be negligibly small; uncertainties in this estimate are included in the systematic error.

The  cross section is extracted from the background subtracted  mass distribution

(2)

where , is the mass spectrum obtained without corrections for resolution and higher-order radiation, is the total efficiency, and the factor is the differential ISR luminosity cs (). The total efficiency determined by MC simulation grows quadratically with energy from 0.007% near threshold to 0.036% at 5.2 . The resulting  exclusive cross section averaged over the bin width is shown in Fig. 3 with statistical uncertainties only. Since the bin width is much larger than the  resolution, which varies from around threshold to at , no correction for resolution is applied.

The systematic errors for the measurement are summarized in Table  1.

    Source Error,[]
    Background subtraction
    Cross section calculation
    
    Reconstruction
    Kaon identification
    Total
Table 1: Contributions to the systematic error on the  cross section.

The systematic errors associated with the background (1–2) subtraction are estimated to be 2% due to the uncertainty in the scaling factors for the sideband subtractions. This is estimated from fits to the  and  distributions in the data that use different signal and background parameterizations. Uncertainties in backgrounds (3–5) are estimated conservatively to be each smaller than 1% of the signal. The systematic error ascribed to the cross section calculation includes a 1.5% error on the differential luminosity and a 6% error in the total efficiency function. Another source of systematic errors is the uncertainties in track and photon reconstruction efficiencies (1% per track, 1.5% per photon and 5% per ). Other contributions come from the uncertainty in the identification efficiency and the absolute  and  branching fractions pdg (). The total systematic uncertainty is 10%.

We perform a likelihood fit to the  distribution where we parameterize a possible  signal contribution by an -wave relativistic Breit-Wigner (RBW) function with a free normalization. We use PDG values pdg () to fix its mass and total width. To take a non-resonant  contribution into account we use a threshold function with a free normalization. Finally, the sum of the signal and non-resonant functions is multiplied by a mass-dependent second-order polynomial efficiency function and differential ISR luminosity.

The fit yields signal events for the  state. The statistical significance for the  signal is determined to be from the quantity , where is the maximum likelihood returned by the fit, and is the likelihood with the amplitude of the RBW function set to zero. The goodness of the fit is . The systematic errors of the fit yield are obtained by varying the mass and total width within their uncertainties, histogram bin size and the parameterization of the background function and efficiency.

We calculate the peak cross section for the process at from the amplitude of the RBW function in the fit to be nb at the 90% C.L. Using and PDG values of the  mass, full width and electron width pdg () we found at the 90% C.L and at the 90% C.L. All presented upper limit values include systematic uncertainties. For illustration we include the corresponding fit function on the cross section distribution plot shown in Fig. 3.

Figure 3: The exclusive cross section for  averaged over the bin width with statistical uncertainties only. The fit function corresponds to the upper limit on  taking into account systematic uncertainties. The solid line represents the sum of the signal and threshold contributions. The threshold function is shown by the dashed line.

To obtain limits on the decays , where denotes , , or states, we perform four likelihood fits to the  spectrum each with one of the states, the  state and a non-resonant contribution. For fit functions we use the sum of two -wave relativistic RBW functions with a free normalization and a threshold function with a free normalization. The sum of the signal and non-resonant functions is multiplied by the mass-dependent second-order polynomial efficiency function and differential ISR luminosity. For masses and total widths of the  and  states we use PDG values pdg (). The corresponding parameters of the , and states are fixed from Ref. bellebabar:y (); belle:x4630 (), respectively.

The significances for the , , and signal are found to be , , and , respectively. The calculated upper limits (at the 90% C.L.) on the peak cross sections for processes at are presented in Table 2. Using fixed values of masses and full widths we obtain upper limits on the at the 90% C.L. Finally, for the  state we estimate the upper limit on at the 90% C.L. using  pdg (). For the and states we calculate at the 90% C.L. taking into account  bellebabar:y (). All upper limits presented in Table 2 are determined by choosing the maximum signal amplitudes that emerge from: varying masses and widths of the states within their uncertainties; varying the histogram bin size; and changing the parameterizations of the background & efficiency functions.

, [nb] 0.36 0.55 0.25 0.45
, [] 0.42 0.72 0.37 0.66
9
8 10
Table 2: The upper limits on the peak cross section for the processes at , and at the 90% C.L., where , , , .

To estimate the effects of possible interference between final states we also performed a fit to the  spectrum that includes complete interference between the  RBW amplitude and a non-resonant  contribution. We found two solutions both with ; the interference is constructive for one solution and destructive for the other. From the fit with destructive interference we find an upper limit on the peak cross section for process to be nb at the 90% C.L.

In addition we performed four likelihood fits to the  spectrum with complete interference between the and  states’ RBW amplitudes and a non-resonant  contribution. We found four solutions for each fit with similar goodness-of-fit (=1.39, 1.23, 1.39 & 1.21) and obtained the upper limits on the peak cross sections for process to be less than 1.44, 1.92, 1.38 and 0.98 nb at the 90% C.L. for , , and , respectively.

In summary, we report the first measurement of the  exclusive cross section over the center-of-mass energy range from 4.0 to 5.2. We calculate an upper limit on the peak cross section for the process at to be 0.76 nb at the 90% C.L. The values of the amplitude of the , , and signal function obtained in the fit to the  spectrum are found to be consistent with zero within errors. We see no evidence for decays as predicted by hybrid models and obtain the upper limit at the 90% C.L.

We thank the KEKB group for excellent operation of the accelerator, the KEK cryogenics group for efficient solenoid operations, and the KEK computer group and the NII for valuable computing and SINET3 network support. We acknowledge support from MEXT, JSPS and Nagoya’s TLPRC (Japan); ARC and DIISR (Australia); NSFC (China); DST (India); MEST, KOSEF, KRF (Korea); MNiSW (Poland); MES and RFAAE (Russia); ARRS (Slovenia); SNSF (Switzerland); NSC and MOE (Taiwan); and DOE (USA).

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