Bounds on the Slope and Curvature of Isgur-Wise Function in a QCD-Inspired Quark Model

Abstract

The quantum chromodynamics-inspired potential model pursued by us earlier has been recently modified to incorporate an additional factor ‘c’ in the linear cum Coulomb potential. While it felicitates the inclusion of standard confinement parameter b = 0.183 GeV² unlike in previous work, it still falls short of explaining the Isgur-Wise function for the B mesons without ad hoc adjustment of the strong coupling constant. In this work, we determine the factor ‘c’ from the experimental values of decay constants and masses and show that the reality constraint on ‘c’ yields bounds on the strong coupling constant as well as on slope and curvature of Isgur-Wise function allowing more flexibility to the model.

Appendix

X ₁, X ₂ and X ₃ are evaluated as

$$ \begin{array}{rll}\label{eq26} X_{1}&=&64\pi c^{2}A_{0}^{2}a_{0}^{3}+64+\mu^{2}b^{2}a_{0}^{6}\\ &&\times\left(8-2\epsilon\right)\left(7-2\epsilon\right)\left(6-2\epsilon\right)\left(5-2\epsilon\right)\\ &&+128cA_{0}\sqrt{\pi a_{0}^{3}} \! - \! 16cA_{0}\mu b\sqrt{\pi a_{0}^{9}}\left(6 \! - \! 2\epsilon\right)\left(5 \! - \! 2\epsilon\right)\\ &&-16\mu ba_{0}^{3}\left(6-2\epsilon\right)\left(5-2\epsilon\right), \end{array} $$

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$$ \begin{array}{rll} \label{eq27} X_{2}&=&64\pi c^{2}A_{0}^{2}a_{0}^{3}+64+\mu^{2}b^{2}a_{0}^{6}\\ &&\times\left(6-2\epsilon\right)\left(5-2\epsilon\right)\left(4-2\epsilon\right)\left(3-2\epsilon\right)\\ &&+128cA_{0}\sqrt{\pi a_{0}^{3}} \! - \! 16cA_{0}\mu b\sqrt{\pi a_{0}^{9}}\left(4 \! - \! 2\epsilon\right)\left(3 \! - \! 2\epsilon\right)\\ &&-16\mu ba_{0}^{3}\left(4-2\epsilon\right)\left(3-2\epsilon\right), \end{array} $$

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and

$$ \begin{array}{rll}\label{eq28} X_{3}&=&64\pi c^{2}A_{0}^{2}a_{0}^{3}+64+\mu^{2}b^{2}a_{0}^{6}\\ &&\times\left(10-2\epsilon\right)\left(9-2\epsilon\right)\left(8-2\epsilon\right)\left(7-2\epsilon\right)\\ &&+128cA_{0}\sqrt{\pi a_{0}^{3}} \! - \! 16cA_{0}\mu b\sqrt{\pi a_{0}^{9}}\left(8 \! - \! 2\epsilon\right)\left(7 \! - \! 2\epsilon\right)\\ &&-16\mu ba_{0}^{3}\left(8-2\epsilon\right)\left(7-2\epsilon\right). \end{array} $$

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Not only the above expressions but also all the integrals in the analysis are evaluated with the help of Gamma function, given by:

$$\label{eq29} \frac{\Gamma\left(n+1\right)}{\alpha^{n+1}}=\int_{0}^{+\infty} r^{n}e^{-\alpha r} dr. $$

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