Question 26.3: Range of the Weak Force Use the masses of the weak-force med...
Range of the Weak Force
Use the masses of the weak-force mediators to estimate the range of the weak force.
ORGANIZE AND PLAN Equation 26.2 relates particle mass m and interaction range R:
m=\frac{h}{2 \pi R c} (Mass of force-mediating particle; SI unit: kg) (26.2).
m=\frac{h}{2 \pi R c}.
\text { Known: Rest energies } m_{ W ^{+}}{ }^{2}=m_{ W ^{-} c^{2}}=80.4 GeV ; m_{Z^{0}} c^{2}= 91.2 GeV.
We first convert rest energies into masses, so we can work in SI units. From Chapter 25, you know the conversion factor 1 u \cdot c^{2}=931.5 MeV =0.9315 GeV \text {. Converting the } W ^{+} \text {and } W ^{-} \text {rest } energy:
\frac{80.4 GeV }{c^{2}} \times \frac{1 u \cdot c^{2}}{0.9315 GeV } \times \frac{1.661 \times 10^{-27} kg }{1 u }=1.43 \times 10^{-25} kg.
Thus the range of the weak interaction mediated by a W particle is approximately
R=\frac{h}{2 \pi m c}=\frac{6.626 \times 10^{-34} J \cdot s }{2 \pi\left(1.43 \times 10^{-25} kg \right)\left(3.00 \times 10^{8} m / s \right)}.
=2.46 \times 10^{-18} m.
\text { A similar calculation for the more massive } Z^{0} particle gives a slightly smaller range:
R=2.16 \times 10^{-18} m.
REFLECT The weak force has an extremely short range. This makes sense physically, given that the beta decay takes place within a single nucleon.