Quick question about about a Pion decay

[venomxx]

Reading on pions the book glanced over this idea:

Aloud:
Neutral pion --> photon and photon

Not aloud:
Neutral pion --> photon

the book says its due to energy momentum conservation? but if photons are massless how does this work?

Cheers

 

[ansgar]

You can try this by just putting up the equations of momentum- and energy conservation.. it is really simple.

 

[venomxx]

cheers for the reply, but if the photon is massless how does it have any momentum to carry away?

 

[Bob S]

If you apply the relativistic rest mass, momentum, total energy equation to photons:

E² = (βγ m₀c²)² + (m₀c²)² to photons, we get

E = pc, or p = E/c

So massless photons of energy E have momentum E/c.

Bob S

 

[arivero]

Still, to understand the complete problem we need other question: can a Z0 meson decay to two photons?

 

[venomxx]

So, using p=E/c for the photon and using the energy and momentum of neutral pion, you can determine that it needs two photons to carry away the momentum as one would not be sufficient. Would i be correct in saying that?

arivero: i believe a Z0 is able decay into two photons with no problems...at least from what iv just looked up!

 

[Bob S]

So, using p=E/c for the photon and using the energy and momentum of neutral pion, you can determine that it needs two photons to carry away the momentum as one would not be sufficient. Would i be correct in saying that?

Consider a pi zero from the proton capture of a pi minus at rest going to a neutron plus pi zero. The pi zero momentum is miniscule. How could it decay to a single 135 MeV photon?

arivero: i believe a Z0 is able decay into two photons with no problems...at least from what iv just looked up!

The Z₀ branching ratio to two photons is less than ~5 x 10^-5.
Bob S

 

[ansgar]

So, using p=E/c for the photon and using the energy and momentum of neutral pion, you can determine that it needs two photons to carry away the momentum as one would not be sufficient. Would i be correct in saying that?

arivero: i believe a Z0 is able decay into two photons with no problems...at least from what iv just looked up!

no Z0 BOSON does not have couplings to the photon (at tree level), they are orthogonal states..

 

[ansgar]

still, to understand the complete problem we need other question: Can a z0 meson decay to two photons?

z0 meson?

 

[ansgar]

So, using p=E/c for the photon and using the energy and momentum of neutral pion, you can determine that it needs two photons to carry away the momentum as one would not be sufficient. Would i be correct in saying that?

No, the correct answer is that energy and momentum conservation can not be simoultatneously fulfilled in the case of pi0 decaying into one photon.

 

[arivero]

z0 meson?

 

[arivero]

Following the reasoning in Chapter IV.2 and using the erroneus (10) show that the decay amplitude for the decay $\pi^0 \to \gamma + \gamma$ would vanish in the ideal world in which the $\pi^0$ is massless. ... since our world is close to the ideal world, this provided the first indication historically that (10) cannot posibly be valid

Some textbooks propose, to deep in the understanding of the "triangle", to compare the decay where the spin 0 pion goes to quark antiquark and then to two photons with other two triangles: the one with an spin 1 meson (pho, sigma, J/Psi, etc) decaying to the same mechanism, and the one with the Z0 decaying to any of their particle/antiparticle couplings and then to two photons.

The point I want to make is that, while the kinematic answer is enough for the Original Post goal, the OP has unfortunately chosen the most famously complicated example of electromagnetic decay.

 

[Holychikenz]

Consider this senario. The pion is initially at rest in its frame. The pion has a certain amount of rest energy mc^2 and when it decays into photons all of this energy is converted into the momentum of the photon E=pc. In order to uphold conservation of momentum, if the pion with initial momentum of zero were to decay into one photon, then that photon would necessarily be at rest. Well we know that this can't be the case because the photon must have velocity c and some energy corresponding to its energy (from the pion rest mass).
From that simple argument its straight forward to see that the number of photons must be greater than one (and using the quantum charge operator, must be even number of photons).

Hope this helps.

 

[venomxx]

That helps a lot, cheers for the great replys! its so simple when you think about it...:)