Why does neutral/EM meson decay scale as the cube of mass?

Question

For beta decay of mesons, the heavy quark approximation and surely other techniques can be used to show that the decay width scales as the quintic power of mass. This can be checked from pdg files very easily with this gnuplot script:

  set logscale xy
  set key noautotitle
  plot "<cat Downloads/mass_width_2020.mcd|grep -v ^* | cut -c 32-" using "%20lf%*8lf%*8lf%16lf"
  replot 2.4952*(x/91.1876)**3
  replot 2.99591E-19*(x/0.105658)**5

The plot includes all the particles having mass and decay width listed in the particle data group datafile: mesons and baryons, some of their excited states, and also the Z0 and the W and the two unstable leptons, muon and tau. And the top quark. Higgs boson does not appear because its decay width is not yet in the table, proton does not appear because it is stable AFAWK, and well, the neutron does not appear because I have zoomed it out)

Units in the table are GeV (mass in the horizontal, decay width in the vertical). And you can appreciate the aforementioned quintic scaling as the blue line here.

But surprise, there is also an alignment for neutral, purely electromagnetic, decays. It is a cubic scaling. Is it an artifact, or can it be shown analytically?

To put more detail: the particles on the green line are $\pi^0, \eta, \Sigma^0, J/\Psi, \Psi(2S)$ and, rather peculiarly, the gauge bosons.

Particle	Mass (Gev)	Total Width (GeV)	"Reduced Width" $MeV^{-2}$
$\pi^0$	0.135	7.73E-9	3.1
$\eta$	0.548	1.31E-6	8.0
$\Sigma^0$	1.19	8.90E-6	5.3
$D^*$	2.01	8.34E-5	10.3
$J/\Psi$	3.10	9.29E-5	3.1
$\Psi(2S)$	3.69	2.94E-4	5.9
$B(s2)^*$	5.84	1.49E-3	7.5
$Z^0$	91.2	2.50	3.3

The particles above the green line are all the mesons and baryons that have strong decay. For some of these, when the partial EM decay width is known, usually to $\gamma\gamma$, it also aligns on the green line. I have checked this for $\omega, \phi$ and $\eta'$.

It would seem that the hadronic scale, extracted via the "reduced width" $\Gamma/M^3$, has some general bound, the only exception being the three first states in the bottom quark sector: $\Upsilon(1S,2S,3S)$ happen to have some extra of stability allowing them to lie under the cubic line. But note $\Upsilon(4S)$ is already above it.

Answer 1

You might get better answers if you actually identified the particles involved and their decays.

I'll just remind you of the dimensional analysis aspect of all of them, which you drill in an introductory HEP course:

For weak decays involving one dominant scale, Γ with units of energy must go as the amplitude-squared, involving an exchange of a virtual W in the amplitude, so $1/M_W^2$, so $$ \Gamma\propto \frac {1}{M_W^4} ; $$
for dimensional consistency, for a single scale problem, the mass of the decaying particle, must involve the 5th power of its mass, as you first learned in muon decay, $$ \Gamma \sim m^5/M_W^4. $$ This might hold for mesons, dominated by heavy quarks, etc...

By contrast, for e.g. em decays of the pion, $$ \Gamma_{\pi^0} \sim \alpha^2 \frac{m_\pi^3}{f_\pi^2}, $$ by dint of the PCAC nature of the pion, which resolves to a triangle of quarks inversely as $f_\pi$ in the amplitude. This might be at the root of your observation, but I can't tell which meson decays you are talking about.

• Note added after clarification OK, for the pseudoscalar mesons, all decay constants are a low energy hadronic scale not too different from $f_\pi$. For the J/Ψ, the EM decays spring to prominence because of the OZI suppression of the strong modes, and, as in the case of the $\Sigma^0$ hadronic decay, the mass constant in the denominator of the amplitude is some low hadronic scale quantifying how a meson couples to a quark-antiquark pair current coupling to a photon, so not that dissimilar to neutral pion decay. You'd get the square of that scale in the denominator, and so the cube of the dominant mass, that of the decaying particle, in the numerator. There is such a plethora of circumstances, around, however, wavefucntions at the origin, etc... that I am not sure what general systematics could be seen to prevail.