The π^∘ decays electromagnetically with τ(π^∘)= 8.7 ×10^-17 s, but the π^+ decays only weakly w

step 1 The time required for quirk to move inside the volume of the pi zero meson is proportional to 1

The π^∘ decays electromagnetically with τ(π^∘)= 8.7 ×10^-17...

The $\pi^{\circ}$ decays electromagnetically with $\tau\left(\pi^{\circ}\right)=$ $8.7 \times 10^{-17} \mathrm{~s}$, but the $\pi^{+}$ decays only weakly with $\tau\left(\pi^{+}\right)=2.6 \times 10^{-8} \mathrm{~s}$. You can explain the large difference between these lifetimes in terms of the different ranges of the electromagnetic and weak interactions: First, recall that any meson is composed of two quarks. Next, note that, according to the electroweak theory, the intrinsic strengths of the electromagnetic and weak interactions are the same; this means that the probabilities for either kind of interaction are about the same, provided that two quarks are within the relevant range. Now the two quarks in a meson are always within the (infinite) range of the electromagnetic force; on the other hand, they are very seldom within the range of the weak force. Given that the two quarks move more or less randomly inside the volume of the meson (radius of order $1 \mathrm{fm}$ ) and that the range of the weak force is of order $10^{-3} \mathrm{fm}$, estimate what fraction of the time the two quarks are within the range required for weak interaction. Use these considerations to estimate the ratio $\tau\left(\pi^{\circ}\right) / \tau\left(\pi^{+}\right)$, and show that your answer is of the same order as the observed ratio.

Key Concepts

Meson Structure

Mesons are hadronic particles composed of a quark and an antiquark bound together by the strong force. This underlying two-body structure is key to understanding how different interactions (like electromagnetic and weak) affect their decay processes, since the decay rates depend on the probability of these constituents interacting within certain spatial ranges.

Range of Forces

The electromagnetic and weak interactions have different effective ranges. While the electromagnetic force acts over an infinite range, the weak force is very short-ranged (typically about 10?³ fm). This disparity in interaction range is fundamental in determining how frequently the constituents within a meson are close enough to engage in a weak decay process.

Geometric Probability Based on Interaction Volume

Due to the random motion of the quarks inside a meson (with a typical radius of about 1 fm), the likelihood that both quarks are within the short range required for the weak interaction can be estimated by comparing the volume corresponding to the weak force range to the total meson volume. This ratio, essentially a geometric probability, quantifies how rarely the quarks find themselves close enough for weak interactions to occur.

Lifetime Estimation via Interaction Probability

Given that the intrinsic strengths of the electromagnetic and weak interactions are similar, the vastly different observed lifetimes of mesons such as the ?? and ?? can be attributed largely to the spatial probability factor. The electromagnetic decay of the ?? is allowed throughout its entire volume, while the ?? decay via the weak interaction is suppressed by the small probability that its quarks come within the tiny weak-interaction range. This geometric suppression factor leads to an estimated ratio of lifetimes that is consistent with the observed difference.

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