Analytic Bethe-Salpeter description of the lightest pseudoscalar mesons

Wolfgang Lucha and Franz F. Schöberl

Phys. Rev. D 93, 056006 – Published 17 March 2016

Abstract

Within the Bethe-Salpeter formalism for instantaneous interactions, we describe, along a totally analytic route, the lightest pseudoscalar mesons by quark-antiquark bound states which show at least three indispensable general features—namely, the (almost) masslessness required for pions and kaons to be interpretable as (pseudo-)Goldstone bosons, the suitable asymptotic behavior in the limit of large spacelike relative momenta as determined by the relationship between quark mass function and Bethe-Salpeter amplitudes, and a pointwise behavior for finite spacelike relative momenta suited for guaranteeing color confinement.

Received 7 February 2016

DOI:https://doi.org/10.1103/PhysRevD.93.056006

Physics Subject Headings (PhySH)

Bound states Strong interaction

Mesons

Particles & Fields

Authors & Affiliations

Wolfgang Lucha^*

Institute for High Energy Physics, Austrian Academy of Sciences, Nikolsdorfergasse 18, A-1050 Vienna, Austria

Franz F. Schöberl^†

Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria

^*wolfgang.lucha@oeaw.ac.at
^†franz.schoeberl@univie.ac.at

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Vol. 93, Iss. 5 — 1 March 2016

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Figure 1
Independent Salpeter component fully defining, at least for any Salpeter equation (3) characterized by the Lorentz structure $2 Γ \otimes Γ = γ_{μ} \otimes γ^{μ} + γ_{5} \otimes γ_{5} - 1 \otimes 1$ of the interaction kernel, the Salpeter amplitude (5) for pseudoscalar mesons, shown in adequate units of $μ$ in its (a) momentum-space representation, $φ_{2} (p) \propto (p^{2} + 1)^{- 3 / 2} + η (p^{2} + \frac{1}{4}) (p^{2} + 1)^{- 5 / 2}$ , and its (b) configuration-space representation, $φ (r) \propto K_{0} (r) + η [K_{0} (r) - r K_{1} (r) / 4]$ , at the values $η = 0$ (black solid line, Sec. IV. C of Ref. [6]), $η = 0.5$ (red dotted line), $η = 1$ (magenta short-dashed line), $η = 1.5$ (blue long-dashed line) and $η = 2$ (violet dot-dashed line) of our mixing parameter.
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Figure 2
Configuration-space potential $V (r)$ extracted from the Salpeter equation (3) with interaction-kernel Lorentz structure $2 Γ \otimes Γ = γ_{μ} \otimes γ^{μ} + γ_{5} \otimes γ_{5} - 1 \otimes 1$ by assuming the ansatz $φ_{2} (p) \propto (p^{2} + 1)^{- 3 / 2} + η (p^{2} + \frac{1}{4}) (p^{2} + 1)^{- 5 / 2}$ for the nonvanishing component of the Salpeter amplitude (5) to describe massless pseudoscalar bound states of fermions with mass $m = 0$ : $V (r) = - N (r) / D (r)$ with the two abbreviations $D (r) \equiv 2 r [4 (1 + η) K_{0} (r) - η r K_{1} (r)]$ and $N (r) \equiv π [4 + η (4 + r^{2})] [I_{0} (r) - L_{0} (r)] + π (4 + 5 η) r [I_{1} (r) - L_{1} (r)] - 4 (2 + 3 η) r$ for notational ease introduced for the denominator and numerator, respectively, depicted for the values $η = 0$ (black solid line, Sec. V. A of Ref. [6]), $η = 0.5$ (red dotted line), $η = 1$ (magenta short-dashed line), $η = 1.5$ (blue long-dashed line) and $η = 2$ (violet dot-dashed line) of our mixing parameter.
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Figure 3
Configuration-space potential $V (r)$ extracted from the Salpeter equation (3) with interaction-kernel Lorentz structure $2 Γ \otimes Γ = γ_{μ} \otimes γ^{μ} + γ_{5} \otimes γ_{5} - 1 \otimes 1$ by assuming the ansatz $φ_{2} (p) \propto (p^{2} + 1)^{- 3 / 2} + η (p^{2} + \frac{1}{4}) (p^{2} + 1)^{- 5 / 2}$ for the nonvanishing component of the Salpeter amplitude (5) to describe massless pseudoscalar bound states of fermions with mass $m = 0$ , $V (r) = - N (r) / D (r)$ with the two abbreviations $D (r) \equiv 2 r [4 (1 + η) K_{0} (r) - η r K_{1} (r)]$ and $N (r) \equiv π [4 + η (4 + r^{2})] [I_{0} (r) - L_{0} (r)] + π (4 + 5 η) r [I_{1} (r) - L_{1} (r)] - 4 (2 + 3 η) r$ for the numerator and denominator, respectively, for a few $η$ -parameter choices from the interval $- 1 \leq η \leq 0$ : $η = 0$ (black solid line, again Sec. V. A of Ref. [6]), $η = - 0.25$ (red dotted line), $η = - 0.5$ (magenta short-dashed line), $η = - 0.75$ (blue long-dashed line), and $η = - 1$ (violet dot-dashed line).
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Figure 4
Configuration-space potential $V (r)$ extracted from the Salpeter equation (3) with interaction-kernel Lorentz structure $2 Γ \otimes Γ = γ_{μ} \otimes γ^{μ} + γ_{5} \otimes γ_{5} - 1 \otimes 1$ by assuming the ansatz $φ_{2} (p) \propto (p^{2} + 1)^{- 3 / 2} + η (p^{2} + \frac{1}{4}) (p^{2} + 1)^{- 5 / 2}$ for the nonvanishing component of the Salpeter amplitude (5) to describe massless pseudoscalar bound states of fermions with mass $m = 0$ , $V (r) = - N (r) / D (r)$ with the two abbreviations $D (r) \equiv 2 r [4 (1 + η) K_{0} (r) - η r K_{1} (r)]$ and $N (r) \equiv π [4 + η (4 + r^{2})] [I_{0} (r) - L_{0} (r)] + π (4 + 5 η) r [I_{1} (r) - L_{1} (r)] - 4 (2 + 3 η) r$ for the numerator and denominator, respectively, for a few $η$ -parameter values from the range $- \infty < η \leq - 1$ : $η = - 1$ (black solid line, as a benchmark), $η = - 1.25$ (red dotted line), $η = - 1.5$ (magenta short-dashed line), $η = - 1.75$ (blue long-dashed line), and $η = - 2$ (violet dot-dashed line).
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Figure 5
Configuration-space potential $V (r)$ extracted from the Salpeter equation (3) with interaction-kernel Lorentz structure $2 Γ \otimes Γ = γ_{μ} \otimes γ^{μ} + γ_{5} \otimes γ_{5} - 1 \otimes 1$ by assuming the ansatz $φ_{2} (p) \propto (p^{2} + 1)^{- 3 / 2} + η (p^{2} + \frac{1}{4}) (p^{2} + 1)^{- 5 / 2}$ for the nonvanishing component of the Salpeter amplitude (5) to describe massless pseudoscalar bound states of fermions with mass $m = 1$ : $V (r) = - {π [8 + η (8 - 3 r)] \exp (- r)} / {4 r [4 (1 + η) K_{0} (r) - η r K_{1} (r)]}$ , for the values $η = 0$ (black solid line, Sec. V. B of Ref. [6]), $η = 0.5$ (red dotted line), $η = 1$ (magenta short-dashed line), $η = 1.5$ (blue long-dashed line) and $η = 2$ (violet dot-dashed line) of our mixing parameter.
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Figure 6
Configuration-space potential $V (r)$ extracted from the Salpeter equation (3) with interaction-kernel Lorentz structure $2 Γ \otimes Γ = γ_{μ} \otimes γ^{μ} + γ_{5} \otimes γ_{5} - 1 \otimes 1$ by assuming the ansatz $φ_{2} (p) \propto (p^{2} + 1)^{- 3 / 2} + η (p^{2} + \frac{1}{4}) (p^{2} + 1)^{- 5 / 2}$ for the nonvanishing component of the Salpeter amplitude (5) to describe massless pseudoscalar bound states of fermions with mass $m = 1$ , $V (r) = - {π [8 + η (8 - 3 r)] \exp (- r)} / {4 r [4 (1 + η) K_{0} (r) - η r K_{1} (r)]}$ , for “confining,” yet negative mixing: $η = - 1$ (black solid line), $η = - 1.25$ (red dotted line), $η = - 1.5$ (magenta short-dashed line), $η = - 1.75$ (blue long-dashed line), and $η = - 2$ (violet dot-dashed line).
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