# Why is the typical lifetime so different depending on the type of interaction?

Why is $\tau$ so different when it comes to Quantum Electrodynamics ($10^{-16}$s), Quantum Flavordynamics ($>10^{-13}$s) and Quantum Chromodynamics ($10^{-23}$s)?

Does this have something to do with the coupling constants as suggested in the comments below? They are $\alpha=1/137\approx 0.0073$ (for QED), $\alpha_W\approx 1/29\approx 0.034$ (QFD) and $\alpha_S\approx 0.12$ (QCD). QFD has the longest typical lifetime although it has the middle coupling constant so I am not sure to see a direct link.

• Hint: have you also written down the relative couplings of each interaction? – Cosmas Zachos Feb 13 '18 at 20:40
• @CosmasZachos By the relative couplings you mean to calculate for example $\alpha_W / \alpha_S$? – Jxx Feb 13 '18 at 21:47
• @CosmasZachos I guess it's better for me if I try to compare QED and QFD... So in my course notes the propagator for weak interactions is $\propto\frac{g_W^2}{q^2-M_W^2}$ right? And that the QED propagator is $\propto\frac{1}{q^2-M_X^2}$ where $M_X$ can be either $0$ if the virtual particle is a photon or $m_e$ for an electron. For simplicity I would compare $\frac{g_W^2}{q^2-M_W^2}$ and $\frac{1}{q^2}$, but the result is then $\frac{g_W^2}{1-M_W^2/q^2}$ and I don't manage to eliminate $q^2$ in my calculations. Did I understand you correctly, and if yes am I mistaken with the propagators? – Jxx Feb 13 '18 at 22:12
• Yes, broadly... if you want a reductionist summary, which is hard... Ideally, you'd take decays that are comparable in the effective q involved, and square the amplitude.... most of the strong decays aren't thusly suppressed, so they might serve as the yardstick--the 1 to compare to... – Cosmas Zachos Feb 13 '18 at 22:27
• What is QFD? Hint: In know what the "Q" and "D" are already. – JEB Feb 13 '18 at 22:42

The short answer is the above relative strength of the couplings; except, ordinarily, a weak decay will involve a W propagator in the amplitude, and so the square of that in the rate. So, for momentum transfers q smaller than the mere hundreds of MeVs of available energy transferred, involved in light hadron decays, this could easily give you 8 orders of magnitude of suppression in decay rates of weak versus EM. So the W propagator, the fermi constant, is what really makes weak decays weak (long lived). Recall we compare through dimensionless quantities.

For light meson decays, we are so far from perturbative QCD, so we might as well take the low energy effective strong coupling to be 1. The QCD running coupling is yet to become relevant here. Often, the momentum available, also displayed in the PDG listings, controls the phase space and the rates.

But strong lifetimes are all over the map, if you inspect the PDG tables. I prefer widths, as $MeV \approx 1/(6 \cdot 10^{-22} s)$. The strong ππ decay of the ρ is 150 MeV wide. By contrast, the EM sibling of this is ππγ, suppressed by 2 orders of magnitude in rate, so just an α, alright.

Now, look at the strong decay of the φ, 40 times slower than that of the ρ. There is a lot of slop in a type of decay--see below.

Look at the weak decay of the charged K, to ππ, suppressed by 14 orders of magnitude w.r.t. the strong ρ decay rate. So, can you get 7 orders of magnitude suppression in the amplitude, playing wildly with numbers? Well, yeah: 2 orders from $\alpha_W$, and 5 from the $(q/M_W)^2$ in the propagator... The W is 160 times more massive than the maximum available momentum of 200 MeV in the decay.

Nothing beats the exact formulas, of course, but you should be able to infer the PDG numbers within a couple of orders of magnitude, especially by comparison of dimensionless ratios, "all other factors being equal".

Of course, unexpectedly suppressed or enhanced rates may be a key to something. (The relative narrowness of the φ seen above gave Zweig the heroic happy idea of quarks over half a century ago.) The extremely slow weak decay of the neutron, 15 minutes, is a feature of the phase-space suppression by the absurdly small available momentum transfer q ≪1MeV.