Question about Neutral Pi-meson and Parity Conservation

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Discussion Overview

The discussion revolves around the decay of the neutral pi-meson (\(\pi^0\)) into two photons (\(\gamma + \gamma\)) and the implications for parity conservation in this process. Participants explore the theoretical aspects of this decay mode, including its electromagnetic nature and the associated quantum mechanical principles.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants argue that the decay process violates parity conservation, citing the left-hand side (L.H.S.) parity of -1 and the right-hand side (R.H.S.) parity of +1.
  • Others propose that parity is conserved due to the nature of photon spin and their relative orientations, suggesting that the decay can be understood through superpositions of photon states.
  • A participant mentions the requirement for the two-photon state to have total angular momentum \(J = 0\) and discusses the implications of photon spins being anti-parallel in the context of superposition.
  • One participant requests a Feynman diagram to illustrate the process, indicating a belief that it is typical of the Standard Model involving electroweak interactions.
  • Another participant references Sakurai's work on the intrinsic parity of photons and the conditions for the decay to conserve parity, discussing the necessary transformations of the final-state wave function.
  • Further contributions elaborate on the degrees of freedom of the photon and the implications of gauge conditions in quantum field theory, as well as the challenges posed by longitudinal modes.
  • One participant mentions chiral anomalies related to the decay, suggesting a connection to decay rate computations in the literature.

Areas of Agreement / Disagreement

Participants express differing views on whether parity is conserved in the decay of the neutral pi-meson to two photons. There is no consensus on the interpretation of the parity conservation in this context, and multiple competing explanations are presented.

Contextual Notes

The discussion includes references to specific theoretical frameworks and literature, indicating a reliance on advanced concepts in quantum mechanics and quantum field theory. Some assumptions about photon states and their interactions remain unaddressed, and the mathematical steps involved in the arguments are not fully resolved.

kakarukeys
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I have a question about the neutral Pi-meson.

One of the decay modes of the neutral pi-meson is to 2 photons.

\pi^0 \rightarrow \gamma + \gamma

it seems to me that this electromagnetic process violates parity conservation.

L.H.S. parity is -1.
R.H.S. parity is (-1)(-1) = +1

Is there anything wrong?
 
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kakarukeys said:
I have a question about the neutral Pi-meson.

One of the decay modes of the neutral pi-meson is to 2 photons.

\pi^0 \rightarrow \gamma + \gamma

it seems to me that this electromagnetic process violates parity conservation.

L.H.S. parity is -1.
R.H.S. parity is (-1)(-1) = +1

Is there anything wrong?


No,
The answer is situated in the photon spin and their direction with respect to the photon momentum : check out the bottom of : http://csm.jmu.edu/physics/giovanetti/particlePhysics/parity.htm

Parity IS conserved

regards
marlon
 
Last edited by a moderator:
Clue of this story is the fact that in order for the two-photon state to have J = 0 (conservation of J and J=0 for the neutral pi meson because it is a superposition of two quark anti-quark states) we must write it as a superposition of two states A and B. Each state has the two photons with anti-parallel spins and in state A the photonspin is aligned with the photon-momentum, in B photon spin is opposite wrt photon momentum. You can derive these states by using the Clebsch-Gordan coefficients, as is denoted in the text...

A photon has indeed parity -1 but since we are working with a superposition of TWO photonstates, the parity is relative. If you had just a single two-photon state then parity would be -1 * -1 but because of the superposition, it is the parity from state A with respect to state B that determins the actual parity of the entire wavefunction...

regards
marlon
 
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Could one of you please come up with the Feynman diagram...?I have a hunch this process typical SM-type (that is electroweak+quarks).

Daniel.
 
dextercioby said:
Could one of you please come up with the Feynman diagram...?I have a hunch this process typical SM-type (that is electroweak+quarks).

Daniel.

The Feynman diagram is one page four

http://www.oup.co.uk/pdf/0-19-852913-9.pdf

marlon
 
Last edited by a moderator:
Thanks,Marlon.I could have sworn it was a weak process,but it isn't...

Daniel.
 
dextercioby said:
I could have sworn it was a weak process

Daniel.

So did I...

As a matter of fact the OP asked a very good question and it took me a while to figure it out and find some good resources...

regards
marlon
 
It definitely took you less than it took me to come up with all sorts of Z,W+-,quarks & photons diagrams...


Daniel.
 
It is a good question. I took a look at Sakurai's Invariance Principles and Elementary Particles, p. 44-45. The intrinsic parity of the photon is based on the behavior of the polarization vector under parity, and is negative. For the pi0 decay, he says:

"We consider the decay of a spin zero pion at rest into two photons. The final-state wave function must transform like a J=0 system, and it must be linear in each of the two photon polarization vectors. We have two possibilities (even and odd parity):
<br /> \vec{\epsilon_1} \cdot \vec{\epsilon_2} <br />

or

<br /> (\vec{\epsilon_1} \times \vec{\epsilon_2} )\cdot k<br />

(k is the unit relative momentum vector of the two-photon system). The first expression is seen to be invariant under P while the second changes sign. In general the final state is expected to be a linear combination of these two forms. If parity is conserved in the decay, however, only one of the two forms is allowed, and we could determine, by measuring the relative orientations of the polarization vectors whether the pi0 is described by a scalar or pseudoscalar field."
 
  • #10
Yes, juvenal, and the two formula's that you gave are the only possibilties because they are invariant under the transformations generated by J : ie the rotations

We could also take the tensorproduct of k with one of the polarzation tensors but this option is useless because the longitudinal mode of photons vanishes... only the transverse (two) photon-polarizations remain. In QFT, you can kick out one degree of freedom by imposing a gauge-condition for the A field \partial^{\mu} A_{\mu} = 0

The A field (which is a Lorentz-object with four degrees of freedom) now has three remaining degrees of freedom.

This is very good because spin 1 means that this object will transform as a vector under the 3-D rotation group. The fundamental representations of this group are (3*1)-column matrices with indeed three variables : ie the components.

The reason why there are 'initially', three degrees of freedom for the photon (ie the three polrizations) has to do with the fact that in QFT a photon is described as the excitations of a MASSIVE spin one-meson (denoted by a vectorfield) because this field needs to couple to the vectorcurrent J (this field is disturbed by the J-current which expresses nothing more then the charged particles that will make the spin 1 field vibrate).

The J-current is a Lorentz vector with 4 degrees of freedom (ie the components) but one degree is knocked out because of this equation : \partial_{\mu} J^{\mu} = 0 : ie conservation of charge.


This J-current now has three degrees of freedom. Then we let m evolve towards zero and 'hope' things will not get awkward...Well, they do...damn it...In order to solve these problems (the longitudinal polarization will yield a negative expectance value) and Gupta-Bleuler-theory teaches us how to get rid of this third variable : the longitudinal polarization : it will decople from the physics at hand.



marlon
 
Last edited:
  • #11
Chapter 13,section 2 of Pokorski's book deals with chiral anomalies linked to \pi^{0}\rightarrow \gamma+\gamma [/tex] and a computation of the decay rate.<br /> <br /> Daniel.
 

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