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  • Open Access

Bethe-Salpeter amplitudes of Upsilons

Rasmus Larsen, Stefan Meinel, Swagato Mukherjee, and Peter Petreczky
Phys. Rev. D 102, 114508 – Published 11 December 2020
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  • INTRODUCTION
  • BETHE-SALPETER AMPLITUDES AT T=0
  • BETHE-SALPETER AMPLITUDES AT T>0
  • COMPARISONS BETWEEN T>0 AND T=0…
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
    • References

    Abstract

    Based on lattice nonrelativistic QCD (NRQCD) studies, we present results for Bethe-Salpeter amplitudes for ϒ(1S), ϒ(2S), and ϒ(3S) in vacuum as well as in quark-gluon plasma. Our study is based on 2+1 flavor 483×12 lattices generated using the Highly Improved Staggered Quark action and with a pion mass of 161 MeV. At zero temperature, the Bethe-Salpeter amplitudes follow the expectations based on nonrelativistic potential models. At nonzero temperatures, the interpretation of Bethe-Salpeter amplitudes turns out to be more nuanced but consistent with our previous lattice QCD study of excited Upsilons in quark-gluon plasma.

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    • Received 7 August 2020
    • Accepted 18 November 2020

    DOI:https://doi.org/10.1103/PhysRevD.102.114508

    Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

    Published by the American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Lattice field theoryQuantum chromodynamicsQuark modelQuark-gluon plasma
    Particles & Fields

    Authors & Affiliations

    Rasmus Larsen1,*, Stefan Meinel2,3, Swagato Mukherjee1, and Peter Petreczky1

    • 1Physics Department, Brookhaven National Laboratory, Upton, New York 11973, USA
    • 2Department of Physics, University of Arizona, Tucson, Arizona 85721, USA
    • 3RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973, USA

    • *rlarsen@bnl.gov

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    Vol. 102, Iss. 11 — 1 December 2020

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    • Figure 1
      Figure 1

      The BS amplitudes for ϒ(1S), ϒ(2S), and ϒ(3S) states at T=0 as function of r (filled symbols) compared with the corresponding trial wave functions (open symbols).

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    • Figure 2
      Figure 2

      The effective bottom quark mass, mb, in the potential approach determined for different quark-antiquark separations r (see text). The horizontal solid line is the fitted value of mb, while the dashed lines indicate the corresponding uncertainty.

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    • Figure 3
      Figure 3

      The potential, V(r), obtained from the BS amplitude of ϒ(1S), ϒ(2S), and ϒ(3S) states compared to the phenomenological Cornell potential [41] shown as a solid line as well as to the energy of the static quark-antiquark pair obtained from Wilson loops using a=0.06  fm lattice [38]. All the lattice results were normalized to coincide with the Cornell potential at r=0.4  fm.

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    • Figure 4
      Figure 4

      The BS amplitude for ϒ(nS) states as function of r (filled symbols) compared with the nonrelativistic wave functions obtained from potential model with mb=6  GeV (open symbols).

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    • Figure 5
      Figure 5

      The effective masses Meffr(τ,T) in GeV of the ϒ(1S) correlator at T=151  MeV (left) and T=334  MeV (right) as a function of τ and r.

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    • Figure 6
      Figure 6

      The effective masses Meffr(τ,T) in GeV of the ϒ(3S) correlator for r≃0.25  fm (left) and r≃0.65  fm (right) at different temperatures as function of τ.

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    • Figure 7
      Figure 7

      Norm of the squared BS wave function at different temperatures for the ϒ(1S) (top), ϒ(2S) (middle), and ϒ(3S) (bottom) states.

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    • Figure 8
      Figure 8

      Effective mass MeffNα in GeV at different temperatures for ϒ(1S) (top), ϒ(2S) (middle), and ϒ(3S) (bottom) correlators.

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    • Figure 9
      Figure 9

      The BS amplitudes times r for the ϒ(1S),ϒ(2S), and ϒ(3S) at T=0  MeV (filled symbols) and T=151  MeV (open symbols) for τ=0.653  fm.

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    • Figure 10
      Figure 10

      BS amplitude times r for the ϒ(1S),ϒ(2S), and ϒ(3S) at T=334  MeV (filled symbols) and T=151  MeV (open symbols) at τ∼0.4  fm (see text).

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    • Figure 11
      Figure 11

      The BS amplitude times r for the ϒ(2S) (filled symbols) and ϒ(3S) (open symbols) at T=251  MeV for τ=0.197, 0.393, and 0.653 fm. Also shown as crosses is the result for the ϒ(1S).

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