Summary
Adopting a Ũ12 four-quark interaction as the potential input to the Bethe-Salpeter equation for the quark-antiquark system we can obtain nonets of bound 0− and 1− meson states. Owing to the presence of a spin-splitting term in the kernel, the masses and coupling constants of the composites deviate from the common values one uses in a phenomenological Ũ12 field theory of quarks and mesons; the magnitudes of these changes are calculated in detail.
Riassunto
Inserendo un'interazione di quattro quark invariante rispetto a Ũ12 come valore del potenziale nell'equazione di Bethe-Salpeter per il sistema quark-antiquark si possono ottenere nonetti di stati mesonici 0− e 1− legati. A causa della presenza nel nocciolo di un termine di separazione dello spin, le masse e le costanti di accoppiamento delle particelle composte deviano dai comuni valori che si usano in una teoria fenomenologica di campo dell'U 12 dei quark e dei mesoni; si calcolano dettagliatamente le grandezze di queste differenze.
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References
In this context we are also including the current-algebras approach, although users of this technique now employ the commutation rules themselves and abandon the bilinear quark representation for which they are derived.
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One may even hope that the renormalized masses and couplings follow from degenerate bare values due to intermediate-state corrections which break down theŨ 4 symmetries. In this view we are estimating departures from thebare ≈U 12prescriptions.
If we identify ensuing masses with physics, ξ≈1/3 is not so small—the curvature ofK(S) becomes significant. Nevertheless the qualitative results expressed in (40) should be quite reliable.
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This is quite in keeping with the ideas expressed in ref. (4) The idea that mesons may be regarded as spin—1/2 composites is old..
In deriving the following results we make use of the asymptotic (α≫1) behaviours of the integrals,\(\int\limits_1^\infty {x^{ - \rho } \left( {x - 1} \right)^\lambda e^{ - \alpha x} dx = \Gamma \left( {\lambda + 1} \right)\rho ^{ - \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \lambda - 1} e^{ - \alpha } \alpha ^{\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \left( {1 - \lambda - \rho } \right)} + 0} \left( {e^{ - x} \alpha ^{ - \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \left( {\lambda + \rho + 1} \right)} } \right)\).
We thank Dr.H. Rubinstein for this observation.
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Daboul, J., Delbourgo, R. Departures from Ũ12 in a relativistic quark model. Nuovo Cimento A (1965-1970) 44, 1031–1046 (1966). https://doi.org/10.1007/BF02719347
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DOI: https://doi.org/10.1007/BF02719347