The Quark Model — Question 7

Question 7

Confining string energy predicts a pair-production length scale

Use a toy confining potential $V(r)=\kappa r$ with $\kappa=0.90$ GeV/fm. Producing an effective light quark-antiquark pair requires 0.60 GeV. Neglect kinetic and other field energies.

1. Find the separation at which the stored energy reaches this threshold.

2. Calculate the energy at 1.2 fm and the excess above threshold; explain why this favors fragmentation over isolating a quark.

Question 7 - Solution

Confining string energy predicts a pair-production length scale

(a) Method and calculation

$$\kappa r_{\mathrm{break}}=2m_{\mathrm{eff}}c^2=.60\ {\mathrm{GeV}}.$$$$\boxed{r_{\mathrm{break}}=.60/.90=.666667\ {\mathrm{fm}}}.$$

(b) Interpretation and further calculation

$$\boxed{V(1.2\ {\mathrm{fm}})=1.08\ {\mathrm{GeV}},\quad E_{\mathrm{excess}}=.48\ {\mathrm{GeV}}}.$$$$q\text{---}\bar q\ \longrightarrow\ (q\bar q_{\mathrm{new}})+(q_{\mathrm{new}}\bar q).$$

Physical check

The effective pair mass and linear potential are a model. Energy accounting suggests new color-singlet hadrons, not a sharp universal string-breaking distance.