# What is the speed of a neutral pion and how is it measured?

(As pointed out by responders the speed of a neutral pion in a lab setting depends on the nature of the reaction which produced it. So the remaining question is "how is the speed determined?".)

Alvager et al 1964 reported evidence against Ritz's emitter theory in an experiment that generated neutral pions ($$\pi_0$$'s) with a velocity of $$v \approx 0.99.c$$. How is the velocity of a neutral pion determined? Does the determination invoke special relativity?

Alvager et al (http://mysite.verizon.net/cephalobus_alienus/papers/Alvager_et_al_1964.pdf) directed a pulsing beam of high energy ($$19.2 GeV$$) protons at a beryllium crystal and measured arrival times of resulting emitted $$\gamma$$ ray photons (energy $$>= 6GeV$$) at downstream detectors. The photons were emitted from the decay of $$\pi_0$$'s produced from collisions of the protons with the beryllium nuclei. The $$\pi_0$$'s had speeds $$v = \beta.c$$. They state that $$1/(1-\beta^2) >45$$ and so $$\beta >= 0.989$$. Later they use the value $$\beta=0.99975$$.

Time intervals between the peaks of successive photon bursts (interval ~105 ms) detected at two separate detectors were interpreted as indicating that, if the photons were travelling at some speed other than $$c$$ and given by $$v=c+k.\beta.c$$, then the value of $$k$$ must be very small (-3+/- 13 * 10e-5). (I think that this formula for altered photon velocity harks back to Fizeau experiments investigating the effect of moving water on the velocity of light. Applying simple galilean relativity in the Alvager et al experiments would predict a range of photon velocities $$v$$ between $$c -\beta.c$$ and $$c + \beta.c$$).

In a contemporary paper Velocity of Gamma Rays from a Moving Source T. A. Filippas and J. G. Fox, Phys. Rev. 135, B1071, 1964 ) used $$\pi_0$$'s with velocity $$0.2c$$ which decayed to produce 68 MeV $$\gamma$$ ray photons. They declare that "independently of relativity, and indeed of nuclear theory, there can be no reasonable doubt about the velocity of the ($$\pi_0$$) source(s)" based on reported aberration angle and Doppler energy shift for $$\pi_0$$'s produced by the same reaction ($$\pi- + p \rightarrow \pi_0$$).

I pose the question because it seems that Alvager et al were trying to confirm special relativity (SR) by assuming pion velocities which were themselves determined assuming SR.

Regarding the interpretation of the Alvager experiment the following note is interesting: (http://worldnpa.org/pipermail/memberschat_worldnpa.org/attachments/20090115/db6f6bc5/attachment.pdf)(since deleted).

The answer by AnnaV is very helpful. I made a follow-up question in which I was led to obtain a formula for $$v$$ myself:- $$v=\sqrt(2KE/m_0)$$. The community declared this solution correct but then deemed the question a pointless homework question and deleted/hid it. As it happens the formula does not seem to be correct when I try to apply the Filippas & Fox 1964 data ($$\pi_0$$ rest mass $$m_0=135MeV/c^2$$, $$KE=68 MeV$$) it gives $$v_{calc} = 1.004c$$ instead of the authors' value $$v=0.2c$$. Applying the Alvager et al data to it ($$KE= 6000 MeV$$) it gives $$v_{calc}=9.428c$$. So I am still in the dark on the Special Relativity approach. (The aberration method mentioned by Fox & Filippas 1964 is relatively simple to understand for me.)

• Speed in what process? Relative to what? Sep 12, 2013 at 23:08

If you go to this link you will see that the lifetime of the pi0 is orders of magnitude shorter than of the charged pions.

8.4 ± 0.6 × 10^−17 seconds, a time characteristic of electromagnetic reactions.

It decays to two photons, which can be measured in the laboratory.

If it is produced with some energy in the laboratory system, its speed can be estimated by measuring the four momenta of the photons and equating the sum to the four momentum of the pi0. Its speed then can be found for that individual measurement. There is no general "speed" of the pi0, as there is no general speed of any elementary particle, their four momenta being dependent of the interaction that produced them and very variable.

To have a speed a fraction of the speed of light any pion or other elementary particle should have an energy given by the relativistic formulae. Have a look here where they calculate the energy necessary for a velocity 1% of the velocity of light for various particles.

• Alvager et al 1964 quote pion energy >6GeV and gamma>45. and pion velocity v of $> 0.99975c$. Sep 14, 2013 at 20:22
• This is for a specific experiment and a specific reactions studied in their publication. Not for any pion any interaction. One has to find the energy for the pi0 as I outlined above, (experimentally or by a thought experiment), and then see what its speed is with respect to c. Sep 15, 2013 at 2:57
• Thanks, I had realized that these were specific from your answer. I would love to see a detailed example of such a calculation. Could pi0's simply be a fiction invented to rescue SR from photons travelling at non-c velocities? Sep 15, 2013 at 15:17
• Here is a link dealing in detail with the pi0 decay hep.princeton.edu/~mcdonald/examples/piondecay.pdf . No, they are not fiction, they are measured over and over again and have the same mass as p+ and pi- and are in the same isotopic spin vector. en.wikipedia.org/wiki/Pion Sep 15, 2013 at 15:59
• In a pi0 decay are the two daughter photons produced simultaneously? In the rest frame of the pi0 do these two photons travel in diametrically opposite directions? Sep 16, 2013 at 19:50