The neutral vector mesons can decay leptonically through a virtual photon, for example by \(V(q \bar{q}) \rightarrow \gamma \rightarrow\) \(\mathrm{e}^{+} \mathrm{e}^{-}\). The matrix element for this decay is proportional to \(\left\langle\psi\left|\hat{Q}_{q}\right| \psi\right\rangle\), where \(\psi\) is the meson flavour wavefunction and \(\hat{Q}_{q}\) is an operator that is proportional to the quark charge. Neglecting the relatively small differences in phase space, show that $$ \Gamma\left(\rho^{0} \rightarrow \mathrm{e}^{+} \mathrm{e}^{-}\right): \Gamma\left(\omega \rightarrow \mathrm{e}^{+} \mathrm{e}^{-}\right): \Gamma\left(\phi \rightarrow \mathrm{e}^{+} \mathrm{e}^{-}\right) \approx 9: 1: 2 $$

Neutral vector mesons are short-lived subatomic particles made up of a quark and an antiquark (q-antiq). They are called 'neutral' because their net electric charge is zero.
Examples include the \rho^{0}, \text{ω}, and \text{φ} mesons. A \rho^{0} meson, for instance, consists of an up quark (u) and an anti-up quark (\bar{u}) paired with a down quark (d) and an anti-down quark (\bar{d}). Similarly, an ω meson is composed of a symmetric combination of the same quarks while the \(\text{φ}\) meson consists solely of a strange quark (s) and an anti-strange quark (\bar{s}).
These neutral vector mesons are subject to decay through various channels, one of which is leptonic decay involving a virtual photon.

Virtual photon decay is a mechanism where a particle decays into a pair of leptons (like an electron e^{-} and a positron e^{+}) through an intermediary particle called a 'virtual photon.'
In the context of neutral vector mesons, the decay process looks like this: \( V(q\bar{q}) \rightarrow \gamma \rightarrow e^{+}e^{-} \).
The 'virtual' aspect means the photon exists temporarily during the decay process. This intermediate state allows mesons to decay into leptons, although the photon does not appear as a final product.
The probability of this decay can be calculated and is proportional to matrix elements involving the quark charge operator acting on the meson's flavor wavefunction.

The quark charge operator (\( \hat{Q}_q \)) is crucial in calculating the matrix elements \( \langle \psi | \hat{Q}_q | \psi \rangle \) for meson decays.
This operator represents the electric charge of quarks. Different quarks have different charges:

• Up quark (u): \( Q_u = +\frac{2}{3} \)
• Down quark (d): \( Q_d = -\frac{1}{3} \)
• Strange quark (s): \( Q_s = -\frac{1}{3} \)

The matrix element for each meson flavor wavefunction can be calculated using these charge values. For instance, for \rho^{0} meson: \( \rho^{0} = \frac{1}{\sqrt{2}}(u\bar{u} - d\bar{d}) \), the matrix element \( \langle \rho^{0} | \hat{Q}_q | \rho^{0} \rangle \) involves these specific charges and their combinations.
The resulting values influence the decay rates and allow us to compare different meson decays.

Decay rate ratios are used to compare the probabilities of different decay processes. For neutral vector mesons, the decay rate into leptons is proportional to the square of the matrix elements.
Using the matrix elements found earlier, we have: \begin{aligned} &\Gamma( \rho^{0} \rightarrow e^{+}e^{-} ) \propto \left( \frac{\sqrt{2}}{3} \right)^{2} = \frac{2}{9} \ &\Gamma( \omega \rightarrow e^{+}e^{-} ) \propto \left( \frac{2}{3\sqrt{2}} \right)^{2} = \frac{1}{9} \ &\Gamma( \phi \rightarrow e^{+}e^{-} ) \propto \left( -\frac{2}{3} \right)^{2} = \frac{4}{9} \ \text{This gives us a ratio:} \ &\Gamma( \rho^{0} \rightarrow e^{+}e^{-} ) : \Gamma( \omega \rightarrow e^{+}e^{-} ) : \Gamma( \phi \rightarrow e^{+}e^{-} ) \approx 9 : 1 : 2 \ \text{This ratio helps us understand relative decay probabilities for different paths.} \ \text{For example, a \rho^{0} meson is nine times as likely to decay into an \( e^{+}e^{-} \) pair compared to an ω meson. This is important for predicting particle behavior in experiments.} \ }